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Preprints
16th Nordic Process Control
Workshop
Lund, Sweden
25–27 August 2010
25 August Workshop Tutorial
Department of Automatic Control
Lund University
26–27 August Nordic Process Control Workshop
AF-Borgen
Editor: Tore Hägglund
Department of Automatic Control
Lund University
Box 118
SE-221 00 Lund
Sweden
ISSN 0280–5316
ISRN LUTFD2/TFRT--3248--SE
Printed in Sweden,
Lund University, Lund 2010
The 16th Nordic Process Control
Workshop
Welcome the 16th Nordic Process Control Workshop (NPCW), and to Lund. The
aim of the NPCW is to bring the Nordic process control community together, and
to provide a rather informal forum for presenting recent and ongoing work in
the process control area. The workshops are arranged with a period of one and
a half year, circulating between Denmark, Finland, Norway, and Sweden.
The workshop is organized by the Nordic Working Group on Process Control,
where the current members are: Hans Aalto, Neste Jacobs, Finland; Jan Peter
Axelsson, Pfizer, Sweden; John Bagterp Jørgensen, DTU, Denmark; Claes Breitholtz, CTH, Sweden; Bjarne Foss, NTNU, Norway; Bjørn Glemmestad, Borealis,
Norway; Kurt Erik Häggblom, Åbo Akademi, Finland; Tore Hägglund, LU, Sweden; Alf Isaksson, ABB, Sweden; Elling W. Jacobsen, KTH, Sweden; Sirkka-Liisa
Jämsä-Jounela, Aalto Univ., Finland; Kaj Juslin, VTT, Finland; Annika Leonard,
Vattenfall, Sweden; Bernt Lie, Telemark Univ. College, Norway; Tommy Mølback,
Dong Energy, Denmark; Gürkan Sin, DTU, Denmark; and Sigurd Skogestad,
NTNU, Norway.
Another responsibility of the Nordic Working Group on Process Control is to
appoint recipients of the Nordic Process Control Award, one at each workshop.
The award should be given to someone who has made ”lasting and significant
contributions to the field of process control”. This year, professor Graham C.
Goodwin has received the award, and we are happy that he will participate in
the workshop and give a plenary talk with the title Architectural Issues in Control
System Design. The award has been given to the following recipients during the
last fifteen years.
1995 Howard H. Rosenbrock
1997 Karl Johan Åström
1998 Greg Shinskey
2000 Jens G. Balchen
2001 Charles R. Cutler
2003 Roger W. H. Sargent
2004 Ernst D. Gilles
2006 Manfred Morari
2007 Jasques Richalet
2009 John F. MacGregor
2010 Graham C. Goodwin
I hope this meeting in Lund will follow the tradition from the previous workshops and bring the Nordic process control community together and stimulate
further research in our field and contacts between the participants.
Lund, August 2010
Tore Hägglund
Program
Thursday, August 26 2010
8.00 Registration
9.00 Opening
9.10 Award ceremony
Award presented to Professor Graham C Goodwin
Award lecture:
Architectural Issues in Control System Design
Professor Graham C Goodwin
Abstract: Recent control literature places heavy emphasis on optimal
design. For example, MPC uses on-line optimization to achieve optimal
performance in the face of constraints, disturbances and modeling errors.
This talk will make the point that optimization is the final step in a
multi-stage design sequence. The first, and arguably the most important,
stage is deciding on the control architecture. This stage is usually the
most important and can make the difference between success and failure.
The ideas will be illustrated by several practical case studies including:
centre line thickness control in reversing mills, networked control over
communication channel, inner loop power control in WCDMA
telecommunication systems, and reference tracking in model predictive
control.
10.00 Break
10.30 Session 1: Modeling and Simulation
Accurate Dynamic Models for Type 1 Diabetes Identified from Novel
Clinical Data
9
Daniel A. Finan, Signe Schmidt, John Bagterp Jørgensen
Niels Kjølstad Poulsen, Kirsten Nørgaard, and Henrik Madsen
Commissioning a Distillation Column Simulator 11
Ramkrishna Ghosh and Kurt-Erik Häggblom
Dynamic Modeling of Combustion in a BioGrate Furnace: a
Sensitivity Analysis on the Fuel Quality and Combustion Air Supply
12
Alexandre Boriouchkine, Alexey Zakharov, and
Sirkka-Liisa Jämsä-Jounela
Data-Based Uncertainty Modeling of MIMO Systems 18
Hamed Jafarian and Kurt-Erik Häggblom
Modeling for Control of Beer Quality 20
D.R. Warnasooriya, P.G. Rathnasiri, and Bernt Lie
12.10 Lunch
5
13.10 Session 2: Model Predictive Control
A Two Phase MPC and its Application to a Grinding Process 21
Alexey Zakharov, Alexandre Boriouchkine, and
Sirkka-Liisa Jämsä-Jounela
ARX MPC for People with Type 1 Diabetes 27
Dimitri Boiroux, Daniel A. Finan, John Bagterp Jørgensen,
Niels Kjølstad Poulsen, and Henrik Madsen
Tuning of ARX-based Model Predictive Control for Offset-free
Tracking
29
Jakob Kjøbsted Huusom, Niels Kjølstad Poulsen,
Sten Bay Jørgensen, and John Bagterp Jørgensen
Comparison of Decentralized Controller and MPC in Control
Structure of a CO2 Capturing Process
31
Mehdi Panahi and Sigurd Skogestad
Potential of Economic Model Predictive Control for Management of
Multiple Power Producers and Consumers
37
Tobias Gybel Hovgaard and John Bagterp Jørgensen
14.50 Break
15.10 Session 3: Fault Detection and Diagnosis
Fault Detection for the Benfield Process using a Parametric
Identification Approach
39
Johannes P. Maree and Fernando R. Camisani-Calzolari
Diagnosis of Oscillation Due to Multiples Sources Using Wavelet
Transforms
41
Selvanathan Sivalingam and Morten Hovd
Availability Estimations for Utilities in the Process Industry 47
Anna Lindholm, Hampus Carlsson, and Charlotta Johnsson
Detection and Isolation of Oscillations Using the Dynamic Causal
Digraph Method
48
Vesa-Matti Tikkala, Alexey Zakharov, and
Sirkka-Liisa Jämsä-Jounela
16.10 End of sessions
19.00 Workshop dinner
6
Friday, August 27 2010
8.30 Session 4: Optimal and PID Control
Optimal Control of the Oil Reservoir Water-flooding Process 54
Eka Suwartadi, Stein Krogstad, and Bjarne Foss
State-Constrained Control Based on Linearization of the
Hamilton-Jacobi-Bellman Equation
55
Torsten Wik, Per Rutqvist, and Claes Breitholtz
Application of Optimal Control Theory to a Batch Crystallizer using
Orbital Flatness
61
Steffen Hofmann and Jörg Raisch
The Setpoint Overshoot Method: A Super-fast Approach to PI
Tuning
68
Mohammad Shamsuzzoha, Sigurd Skogestad, and Ivar J. Halvorsen
Comparing PI Tuning Methods in a Real Benchmark Temperature
Control System
74
Finn Haugen
10.10 Break
10.30 Session 5: Poster session
Tentative Dependence Analysis of Process Variables in a Circulating
Fluidized Bed Boiler
89
Laura Lohiniva and Kimmo Leppäkoski
Automated Controller Design using Linear Quantitative Feedback
Theory for Nonlinear systems
91
Roozbeh Kianfar and Torsten Wik
Optimal Controlled Variable Selection for Individual Process Units
in Self Optimizing Control with MIQP Formulations
98
Ramprasad Yelchuru and Sigurd Skogestad
Dynamic Characteristics of Counter-Current Flow Processes 106
Jennifer Puschke and Heinz A Preisig
Observer Design for the Activated Sludge Process 111
Marcus Hedegärd and Torsten Wik
Greenhouse Illumination Control 114
Anna-Maria Carstensen and Torsten Wik
Model Predictive Control for Plant-wide Control of a
Reactor-Separator-Recycle System
116
Dawid Jan Bialas, Jakob Kjøbsted Huusom,
John Bagterp Jørgensen, and Gürkan Sin
Fuel Quality Soft-Sensor for Control Strategy Improvement of the
Biopower 5 CHP Plant
117
Jukka Kortela and Sirkka-Liisa Jämsä-Jounela
Convex Approximation of the Static Output Feedback Problem with
Application to MIMO-PID
123
Henrik Manum and Sigurd Skogestad
11.30 Lunch 7
12.30 Session 6: Optimization
Modeling and Optimization of Grade Changes for Multistage
Polyethylene Reactors
125
Per-Ola Larsson, Johan Åkesson, Staffan Haugwitz, and
Niklas Andersson
Challenges in Optimization of Operation of LNG Plants 127
Magnus Glosli Jacobsen
Production Optimization for Two-Phase Flow in an Oil Reservoir 129
Carsten Völcker, John Bagterp Jørgensen, and Per Grove Thomsen
Convex Optimization for the Crystal Shape Manipulation 130
Naim Bajcinca, Ricardo Perl, Jörg Raisch, Christian Borchert, and
Kai Sundmacher
Comparison of Two Main Approaches for Operating Kaibel
Distillation Columns
132
Maryam Ghadrdan, Ivar J. Halvorsen, and Sigurd Skogestad
14.10 Break
14.30 Session 7: Control Strategies
Production of District Heating at Södra Cell Mörrum 133
Karin Axelsson and Veronica Olesen
Control of an HMR Pre-Combustion Gas Power Cycle 135
Lei Zhao, Finn A. Michelsen, and Bjarne Foss
Control of Industrial Chromatography Steps 136
Jan Peter Axelsson
Basic Control of Complex Distillation Columns 137
Deeptanshu Dwivedi, Ivar J. Halvorsen, Maryam Ghadrdan,
Mohammad Shamsuzzoha, and Sigurd Skogestad
15.50 Closing
16.00 End of Workshop
8
Accurate Dynamic Models for Type 1 Diabetes Identified from Novel Clinical Data
Daniel A. Finan,1 Signe Schmidt,2 John Bagterp Jørgensen,1 Niels Kjølstad Poulsen,1
Kirsten Nørgaard,2 and Henrik Madsen1
1Department of Informatics and Mathematical Modeling, Technical University of
Denmark, Kongens Lyngby, Denmark
2Department of Endocrinology, Hvidovre University Hospital, Hvidovre, Denmark
The practical, day-to-day treatment regimen for people with type 1 diabetes (T1DM)
entails self-administration of exogenous insulin in order to regulate blood glucose
concentrations as close to normal levels as possible. The most efficacious dosing is
achieved by measuring, as often as possible, blood glucose concentration through fingersticks, and delivering insulin accordingly. To manage T1DM properly, therefore, is
painful and requires constant decision making. By contrast, an artificial pancreas is a
biomedical device in development that will automatically regulate blood glucose
concentration while freeing patients of the daily burden of self-management.
A critical component of T1DM treatment is dosing insulin so as to offset the
carbohydrate content of meals. By current standards, this often involves administration of
insulin injections (called boluses) coincident with the meals. Moreover, these factors are
commonly taken in a prescribed ratio, known as the insulin-to-carbohydrate ratio.
The control algorithm for an artificial pancreas may well be based on a mathematical
model of the patient’s glucose-insulin dynamics, as in a model predictive control
framework. In such a model-based algorithm, it is advantageous to use an accurate, but
simple, model. Linear dynamic models may provide sufficiently accurate predictions, and
have other inherent advantages like straightforward, computationally tractable
identification, and potential to be re-estimated online, thereby adapting to the evolving
dynamics of the patient.
Unfortunately, the simultaneity and uniform proportionality of the meals and the insulin
boluses confounds accurate estimation of the model parameters. To avoid this pitfall, we
have devised a new in-clinic protocol based on design-of-experiment considerations that
yields information-rich data for model identification. The protocol involves separating
key factors that influence glucose concentration: meals, insulin boluses, and bouts of
exercise.
The novel data give rise to more accurately identified models. A variety of empirical
models was identified from the data: difference-equation models like autoregressive
exogenous-input (ARX) and autoregressive moving-average exogenous-input (ARMAX)
models, transfer-function (TF) models, and state-space models. In addition, “gray” forms
of these models were identified which incorporate simple physiological elements such as
estimates of subcutaneous-to-intravenous insulin absorption and appearance rate of
glucose in the blood from a carbohydrate meal.
9
The quality of the identified models was based on several measures including:
� Accuracy with which the model was fit to the training data
� Values of parameters and/or combinations of parameters (e.g., steady-state
insulin-to-glucose gain)
� Accuracy of predictions for independent (test) data
Identification results for one experiment are shown in Fig. 1. The experiment included
three inputs: a meal, an episode of exercise, and a correction bolus, as depicted in the
figure. The particular model shown was a transfer function model that was identified
from the experimental data. With the exception of the initial, minor glycemic excursion
between trial time -60 min and trial time 0 min, the model prediction is very accurate.
Fig. 1. Identification results for one subject. The TF model prediction is infinite-step. The
experimental inputs included an unbolused meal at trial time 0 min, a bout of moderate
exercise at trial time 150 min, and a correction bolus at trial time 300 min.
Financial support from the Danish Strategic Research Council is gratefully
acknowledged.
10
Commissioning a Distillation Column Simulator
Ramkrishna Ghosh and Kurt-Erik Häggblom
Process Control Laboratory, Department of Chemical Engineering,
Åbo Akademi University, Åbo, Finland
Abstract
Energy saving has become an extremely important issue in the chemical process industries.
Distillation columns, in particular, consume huge amounts of energy. One way of minimizing the
energy consumption is improved control, which enables operation closer to certain constraints.
It has been estimated that 75% of the cost associated with an advanced control project typically goes
into model development (Gevers, 2005). Hence, efficient modeling and system identification
techniques suited for industrial use and tailored for control design applications are needed. This task
is especially difficult for “ill-conditioned” MIMO systems such as distillation columns. Even for a linear
system, this ill-conditioning makes the system behavior resemble that of a (strongly) nonlinear
system. Because of this, identification, modeling and control of ill-conditioned systems are
demanding tasks.
In an industrial environment, the identification usually has to be carried out while the plant is in
normal operation. It is then essential to keep the variation of inputs and outputs within specified
limits and to limit the duration of the identification experiments. However, this also limits the
information available for system identification. Thus, there is a trade-off between how much one is
prepared to “pay” for the information and the information needed for system identification.
These kinds of identification issues can be investigated by means of a pilot-scale distillation column
at Åbo Akademi. However, in order to enable more effective identification studies, a distillation
column simulator has been constructed using MathWork’s Simulink as programming environment.
Because the simulator is to be used in conjunction with the real distillation column, it is desired that
the behavior of the simulator is close to that of the real column. A significant number of previously
performed identification experiments with the distillation column are available for the tuning of the
simulator.
In order facilitate the simulator tuning, we study the effects of column and mixture property
parameters appearing in the simulator model on observable static and dynamic properties. In
particular, we want to find out which parameters most strongly affect these properties in various
input-output relationships. Besides the tuning issue, this information also has more general interest.
11
Dynamic modeling of combustion in a BioGrate furnace: a sensitivity analysis on the fuel quality and
combustion air supply
Alexandre Boriouchkine, Alexey Zakharov, Sirkka-Liisa Jämsä-Jounela
Aalto University, School of Science and Technology, Department of Biotechnology and Chemical Technology, Process Automation Research
Group, 00076 Aalto, Finland. e-mail: aboriouc@cc.hut.fi
Abstract: This paper considers dynamic modeling of the bed combustion in a furnace of the
BioGrate boiler. The developed dynamic model is heterogeneous, including solid and gas
phases. Furthermore, the model considers chemical reactions in both, gas and solid phases. In
addition, fuel movement on the grate is included in to the model. The energy required by the
process is employed through a radiation function validated by industrial data from a BioGrate
boiler at Trolhättan, Sweden. The model is implemented in MATLAB environment and tested
with the industrial data. The results are presented and discussed.
1. Introduction
Increasing utilization of renewable energy has created new
energy efficiency challenges for industry. Biomass is one
of the most important raw material for renewable energy.
All the available biomass sources have to be considered
for energy production. Fuel properties of biomass vary a
lot depending on its origin, on processing and handling for
fuel. Variable properties cause fluctuations in combustion
and set challenges to develop new combustion control
strategies.
One of the latest successful processes developed,
which use wood waste as a fuel, is a BioGrate-boiler
technology developed by MWBiopower. The combustion
of wood waste is, however, a very complex process
involving several highly coupled chemical reactions.
Furthermore, operational conditions of the furnace greatly
affect the yields of chemicals produced during the
combustion process, i.e., fractions of tars, gases and char.
Moreover, not only the yields of chemicals differ under
different combustion conditions, but also their reactivity in
succeeding reactions. As a result of such complexity,
optimization of a boiler control strategy requires a detailed
process model [3].
The most recent publications, considering
modeling the combustion of solid fuel on a grate,
concentrate on the combustion of either straw or municipal
waste. Shin and Choi [21] have developed a 1-D model of
waste incineration to understand better phenomena
occurring inside a municipal solid waste (MSW)
incinerator. Van der Lans et al. [18] have developed a twodimensional, homogeneous model for design and operation
parameter optimization of a straw combustion process.
Goh et al. [19] have developed the model of grate
combustion of municipal solid waste in order to study the
process, since efficient incinerator design requires
extensive knowledge of the combustion process. Later,
Yang et al. [20] developed a 2-D model of a MSW
incinerator which was then verified using experimental
data obtained from a pot reactor. Kær [22] developed a
one-dimensional model to describe a fixed bed combustion
of straw and using walking grate concept extended it to
cover the moving bed of a straw boiler. Similarly, Zhou et
al. [23] assumed insignificant horizontal temperature
gradients in order to simplify the thermal conversion
model on the grate of a straw boiler to one dimension.
This paper describes the developed model of a
BioGrate furnace. The purpose for the modeling work was
to construct a dynamic model providing an insight into the
chemical and physical phenomena occurring inside the
process. This paper is organized as follows: Section 2
describes the structure of a BioGrate boiler process,
Section 3 presents the model and its aspects, Section 4
discusses the implementation details of the model, Section
5 presents the simulation results, Section 6 summarizes the
results of simulations.
2. Process description of a BioGrate boiler
A BioGrate consists of following parts: a water
filled ash space below the grate, while the grate itself is
located above the reservoir. The BioGrate is covered with
a heat insulating brick wall, which reflects the heat
radiation back to the grate [3].
The grate consists of several ring zones. These
zones are further divided into two types of rings: rotating
and fixed. A half of the grate rings are rotating and the rest
are fixed. Every second rotating ring rotates clockwise and
the others rotate counterclockwise. This structure helps
spreading fuel evenly upon the surface of the conical grate
[3].
Fuel is fed into the center of the grate from
below. In the middle of the cone the fuel dries as a result
of heat radiation, which is emitted by the combusting flue
gas and reflected back to the grate by the grate walls. The
dry fuel then proceeds to the outer shell of the grate, where
pyrolysis and char combustion occur. The ash and carbon
residues fall off the edge of the grate into the water-filled
ash pit [3].
The air required in combustion is fed into the
grate from the bottom of the grate (primary air) and from
the grate walls (secondary air). In addition, in order to
ensure clean combustion, additional air can be fed from
the top of the grate (tertiary air). Burning produces heat
that is absorbed in several steps. First, the evaporator
absorbs the energy in the flue gases. Next, part of the
energy of the flue gases is transferred to superheaters. In
the third phase, the heat is transferred to the convective
evaporator. Finally, economizers remove the remaining
flue-gas energy [3].
The operation principle of a power plant is based
on the steam generation. As any other bio power plant, a
BioGrate power plant comprises several parts, including a
boiler, a turbine generator, a feed-water tank, a water
treatment plant and a flue gas-cleaning system. Solid fuel
is fed into the furnace of the boiler, where it is combusted,
generating heat and flue gases. Flue gases contain fly ash
12
which comprises several harmful components. Therefore,
flue gases are purified of fly ash before the release into the
atmosphere. Therefore, the flue gases are subjected to
several steps of the cleaning procedure and then emitted
into the atmosphere. Instead, the heat acquired from the
fuel is used for the steam production.
The steam produced in the boiler is led to a
generator turbine, which converts its mechanical energy
into electricity. As steam performs mechanical work, its
pressure decreases, steam with decreased pressure is then
used for heating utility streams, such as water [2]. After
steam has released enough energy it condenses. The
condensed steam is called condensate which along with
pretreated feed water is fed into a feed-water tank. Inside
the tank, liquid is heated with a bled steam from the
turbine. This procedure increases energetic efficiency of
the process [1].
3. Dynamic modeling
The current model of a BioGrate uses walking grate
concept modified for a BioGrate furnace. In addition, the
chemical reaction kinetics were selected, especially, to fit
the operational conditions of the BioGrate. Furthermore,
an experimental model was used to model the radiation
distribution inside the furnace.
Biomass bed reacts in a series of four different chemical
reactions: drying, pyrolysis, char gasification and char
combustion [5]. Active drying starts when temperature of a
particle reaches the boiling point of water. Then, the high
temperature of a furnace initiates a pyrolysis reaction. The
pyrolysis reaction produces three products: gases, char and
tar. Gases are mainly composed of CO, CO2, H2, and C1C3 hydrocarbons. Tar contains many organic components,
such as levoglucosan, furfural, furan derivatives and
phenolic compounds [6]. Next, each reaction will be
discussed in details.
3.1. Continuity equation models, their parameters and
assumptions
3.1.1. Assumptions
Several assumptions are made in order to simplify the
modeling work. The assumptions are listed in descending
order of importance:
1. The system is one dimensional, because the length
of the grate is significantly longer than the height.
Therefore, the temperature gradient in the
horizontal direction is insignificant compared that in
the vertical direction.
2. Plug-flow gas assumption [9]. The gas phase is
assumed to be ideal [9], [10].
3. The solid is assumed to be a porous material [11].
4. Diffusion in the gas phase is neglected, since the
effect of convection on transportation of the gas is
significantly greater [5].
5. Pressure dynamics are ignored, because the release
of gaseous species is negligible compared to the
primary air flow and, as a result, pressure evolution
can be neglected [9]
6. Heat produced in char combustion is assumed to be
retained in the solid phase [9].
7. No volume reduction (shrinkage) occurs during
drying, pyrolysis and combustion [9], [7]
8. The temperature of the gas released from solids is
the same as that of the solids [9]
9. The temperature of solids in a discretized block is
uniform [9]
10. The heat capacity of the wood is assumed to be
constant [9]
11. No heat loss
Next, the simplified continuity equations are presented.
3.1.2. Solid Phase continuity equation
Solid phase reacts through drying, pyrolysis and char
combustion reactions:
f s R
t −=

∂ρ
(1)
where ρs is the density of the solid phase, and Rf the
overall reaction rate of the solids.
3.1.3. Energy continuity equation of the solid phase
Energy equation for the solid phase considers heat
conduction, heat exchange between phases, energy lost in
drying and pyrolysis reactions, and energy gained in char
combustion:
OHgasiOHgasiCOgasiCOgasi
CcombCcombpyrpyrevapevap
sfpconv
s condss
s HRHR
HRHRHR
TTvk
x T
k x C
t T
2,2,2,2,
,, ...
...)(
∆−∆
−∆+∆−∆
−−+








∂=

∂ ρ
(2)
where Ts is the temperature of solid phase, Cs the heat
capacity of the solid phase, ρs the density of the solid
phase, x the vertical coordinate, kcond the heat conduction
coefficient of the solid phase, kconv the heat convection
coefficient between the gas and solid phases, vp is the
density number, Tf the temperature of the gas phase, and
Revap and Rpyr the reaction rates of drying, pyrolysis.
Reaction rates Rcomb,C, Rgasi,CO2 and Rgasi,H2O correspond to
reaction rates of char combustion, gasification with carbon
dioxide and gasification with water steam, respectively.
∆Hevap and ∆Hpyr are the reaction enthalpies of drying,
pyrolysis. Reaction enthalpies ∆Hcomb,C, ∆Hgasi,CO2 and
∆Hgasi,H2O correspond to reaction enthalpies of char
combustion, gasification with carbon dioxide and
gasification with water steam, respectively.
The radiation reflected from the grate walls to the
fuel bed is described through boundary conditions. The
boundary conditions are defined as follows:
At the surface of the fuel bed, x = a
4
sin
ax s cond TeIx
T
k σ−=


=
(3)
where Iin is the energy flux into the system, and eσTs
4 the
energy flux out of the system.
To describe the energy flux, Iin, an experimental model
was used. The model was defined from the experimental
data of a BioGrate boiler located in Trolhättan, Sweden.
13
4
0
s x s cond Tex
T
k σ=


=
(4)
where eσTs
4 is the energy flux out of the system.
According to Yagi and Kunii [15], flowing fluid improves
the heat conduction of the bed due to effect of axial
dispersion; therefore. the overall heat conduction
coefficient will become:
PrRe0, ⋅+= αβcondcond kk (5)
where kcond, 0 is the heat conduction of the bed with a
stagnant fluid, and the right hand term represents the heat
conduction due to axial dispersion. Re is the Reynolds
number, Pr the Prandatl number, and αβ is a geometrical
constant for cylinders and spheres. The geometrical
constant is reported typically to take values of between 0.1
and 0.13 [15]. In the model, the average αβ = 0.115 was
used.
Heat conduction coefficient from the study of
Yagi and Kunii [15] was used to describe heat conduction
in the bed while heat conduction coefficient for wood
particles was based on the study [14].
3.1.4. Gas phase continuity equation
Reacted solid components of wood are transferred to gas
phase, in addition, gas phase continuity equation considers
gas flow:
iibffibf RYvx
Y
t =

∂−


)()( ερερ (6)
where ρf is the density of gas phase, εb the bed porosity, Yi
the mass fraction of the gaseous component i, vf the gas
flow velocity and Ri the rate of formation of gaseous
component i.
3.1.5. Energy continuity equation of the gas phase
Assuming no heat loss will occur, the energy continuity
equation can be denoted as follows:
( )
2,2,,,
...)(
HcombHcombCOcombCOcomb
sfpconvffbf
f HRHR
TTvkhv
xt h ∆+∆
+−−

∂−=


ερ
(7)
where hf is an enthalpy of the gas phase, ρf the density of
the gas phase, εb the bed porosity, vf the gas flow velocity,
and Ri the rate of formation of gaseous component i, kconv
is the heat convection coefficient between the gas and
solid phases, vp is the density number, Tf the temperature
of the gas phase and Ts is the temperature of the solid
phase.
3.2 Chemical reactions of the model
The thermal decomposition of wood comprises three main
chemical reactions: drying, pyrolysis and char gasification
with char combustion. In general, the chemical reactions
can be depicted using experimental or semi-experimental
models. However, since Arrhenius dependence equations
are simple to use, and also accurate; therefore, they have
been used in this work.
3.2.1 Moisture evaporation
Usually, fuels used in combustion processes contain
moisture. Depending on the type of fuel, a fuel particle can
contain various amounts of moisture. According to
Thunman et al. [4], fuel particles can contain up to 60 wt%
of moisture while char residue being as low as 10 wt% of
the wet wood. Water can be bound to the structure of a
wood particle or reside in its pores.
Di Blasi et al. [12] presented a simple, yet
accurate model to describe drying kinetics in the updraft
gasifiers, which use countercurrent combustion conditions:
[ ]( ) Watersevap TmolkJR ρ///88exp106.5 8 ℜ−⋅= (8)
where ρWater is the density of water, Ts the temperature of
the solid phase, and ℜ the gas constant.
3.2.2 Pyrolysis
After a particle has dried, the next reaction occurring is
pyrolysis. In the pyrolysis reaction, a dry wood particle is
decomposed into tar, volatile organic components and
char. However, fractions of tar, gas and char in the product
yield are strongly dependent on the reaction conditions of
a combustion process.
Alves and Figueiredo [13] presented a
mathematical model of wet wood pyrolysis. The
simulation results of this model were experimentally
validated in temperature range of 298 – 780 °C with a wet
pine cylinder having the radius of 18.5 mm, with water
content being 45-49 wt-%. Experimental and simulated
results agreed.
Cellulose is reported to react with the following reaction
kinetics [13]:
celsceldevol TmolkJR ρ⋅ℜ−⋅⋅= )//]/[146exp(102
9
, (9)
where ρcel is the density of cellulose, Ts the temperature of
the solid phase, and ℜ the gas constant.
Hemicellulose is reported to react with the following
rection kinetics [13]:
HemisHemidevol TmolkJR ρ⋅ℜ−⋅⋅= )//]/[83exp(107
4
, (10)
where ρHemi is the density of hemicellulose, Ts the the
temperature of the solid phase, and ℜ the gas constant.
3.2.3 Combustion of pyrolysis gases
The yield of pyrolytic gases is around 85 wt. % under the
operation conditions of a BioGrate boiler, since under
these conditions the gasifying pyrolysis is the dominant
pyrolysis mode. Therefore, significant amount of energy,
used by the boiler, comes from the combustion of gases;
this fact poses the combustion of pyrolytic gases as the
most important energy source. However, the composition
of the gaseous products of pyrolysis reported in the study
14
of Dupont et al. [25], suggests that carbon monoxide has
the highest concentration in the pyrolytic gas, while the
fraction of other combustible gases remains under 10 wt.
%. Therefore, in order to ensure the acceptable accuracy of
the model, while keeping the model simple, only the
oxidation of carbon monoxide to carbon dioxide is
considered.
In addition to the oxidation of carbon monoxide,
also the combustion hydrogen is included in the model
because of three facts. First, char is known to react with
water steam producing carbon monoxide and hydrogen.
Second, significant amount of water steam is released
during the drying reaction. Consequently, also the amount
of produced hydrogen in char gasification reaction can
become significant. Finally, hydrogen reacts rapidly in the
presence of oxygen, producing significant amounts of
energy.
In the presence of water steam and oxygen,
carbon monoxide is known to follow the following
kinetics [26]:
[ ]
5.0
2
5.0
2
14
, ...)///30exp(103.1
OHOCO
gCOcomb TmolekcalR
ρρρ ⋅⋅
⋅ℜ−⋅=
(11)
where ρCO is the density of carbon monoxide, ρO2 is the
density of oxygen, ρH2O is the density of water steam, Tg is
the temperature of the solid phase, and ℜ the gas
constant.
Hydrogen is reported to react through the following
kinetics [26]:
[ ]
92.0
22
11
2, ...)/6900exp(1096.2
OH
gOHcomb TKR
ρρ ⋅
⋅−⋅=
(12)
where ρH2 is the density of hydrogen, ρO2 is the density of
oxygen, Tg is the temperature of the solid phase, and ℜ
the gas constant.
3.2.4 Char conversion reactions
Char combustion in the model is accounted for with the
model presented in Janse et al. [24]. This model is chosen
because it is valid over the temperature range 573-773K,
which corresponds to the temperature of char combustion
in BioGrate. In addition, the pyrolysis conditions of char
particles, which were used to obtain the model parameters,
are similar to the pyrolysis conditions inside a BioGrate.
[ ]
CO
scomb
XP
TmolekJR
ρ49.053.02
5
)1(
...))/(/125exp(103.5

⋅ℜ−⋅=
(13)
where ρc is the density of the char, Ts is the temperature, of
the solid phase, ℜ is the gas constant, and PO2 the
pressure of oxygen, and X is the degree of the conversion
of char.
Matsumoto et al. [27] have reported, in their
study, that the random pore rate equations is the best
option to describe the gasification reactions of char with
carbon dioxide and water steam. Therefore, the rate
equations presented in the study of Matsumoto et al. [27]
for char gasification with carbon dioxide and water steam
are used in the model.
Gasification reaction of char with carbon dioxide [27]:
[ ]( )
( ) C
sCOCOgasi
XX
TmolekJPR
ρ⋅−−−⋅
ℜ−⋅⋅=
)1ln(1011
///9.93exp1024.2 322.0 22,
(14)
where ρc is the density of the char, Ts the temperature, of
the solid phase, ℜ is the gas constant, and PCO2 the
pressure of carbon dioxide, and X is the degree of the
conversion of char.
Gasification reaction of char with water steam [27]:
[ ]( )
( ) C
s OHOHgasi
XX
TmolekJ
PR
ρ⋅−−−⋅
⋅ℜ−
⋅⋅⋅=
)1ln(1011
...///136exp
...1099.9 422.022,
(15)
where ρc is the density of the char, Ts the temperature of
the solid phase, ℜ is the gas constant, and PH2O the
pressure of carbon dioxide, and X is the degree of the
conversion of char.
4 Implementation of the models and the
description of the testing environment
The model was implemented in the MATLAB
environment, in which a set of finite difference methods
was used to solve the continuity equations. The overall
solving algorithm is presented in Figure 1.
Figure 1. Model solving scheme.
5 Simulation Results
This section presents simulation results obtained with
different fuel parameter values.
5.1 Simulation case I, studying the effect of fuel quality
on the combustion of wood chips
Wood chips possess several quality properties, including
the fuel bed porosity, the bed density and the moisture
content. However, the moisture content is the most
important property, since it varies significantly between
different batches of the fuel. In contrast to the moisture
content, the bed porosity and bed density, in case of wood
chips, vary only insignificantly; therefore, they are of no
interest in the current study. The simulation results with
moisture contents of 40, 50 and 60 wt. % are shown in
Figure 2.
Mass Continuity
of Solids
Mass Continuity of
Gases
Energy Continuity
of Gas Phase
Energy Continuity
of Solid Phase
Runge-Kutta
2th order
Implicit
Finite
Difference
Method
CrankNicolson
+Second
Order
Boundary
Conditions
Next Time Step
Computation of
reaction rates
15
Figure 2. Temperature profiles of the fuel bed with fuel with a) 40 wt.%
b) 50 wt. % and c) 60 wt. % moisture content
The results presented in Figure 2 suggest that, although the
temperature of the fuel beds with a different moisture
content remains within the same range, there are
considerable differences in the combustion process. Not
only the combustion time increases with increasing
moisture content, but also the reaction front becomes
narrower. The increase in combustion time can be
explained by the fact that the higher the moisture content
the longer time it takes for the water to evaporate.
Furthermore, high moisture content prevents the heat flux
from penetrating deeper into the fuel bed, because a larger
amount of heat is required to dry the same amount of fuel.
Similar behavior was also observed in the study of Yang et
al. [28]. In the study [28], it was found that burning rate is
inversely proportional to the moisture content of a fuel
bed.
5.2 Simulation case II, studying the effect of the air
flow on the combustion of wood chips
The air flow also has a significant effect on the
combustion process, since it provides the oxygen required
for chemical reactions, such as char oxidation. Simulations
were conducted with the air flows of 0.5, 0.75, 1, 2 and 5
m3/s, while particle sizes were 20, 35 and 50 mm. Table 1
presents the result obtained from the simulation, while
Figure 3 visualizes the results presented in Table 1.
Combustion time (s)
Air flow m3/s
20 mm 35 mm 50 mm
0.5 1621 1590 1558
0.75 1398 1357 1336
1 1379 1320 1286
2 1373 1279 1230
5 1435 1269 1202
Figure 3 shows that, as the volume of the air flow
increases from 0.5 to 0.75 m3/s, the combustion times for
all particle sizes decrease significantly from around 1600 s
at 0.5 m3/s, to 1350 – 1400 second at 1 m3/s.
Figure 3. Combustion times as a function of the volumetric air flow.
This phenomenon is the result of oxygen deficiency at low
air supplies in the combustion process, i.e., not enough
oxygen is supplied to the process to burn the char at its
maximum rate.
Figure 3 indicates that, indeed, with low air flows char
burns slowly. However, as shown in Figure 3 also high air
flows can slow down the combustion of the fuel. This
phenomenon can be explained by the cooling property of
the air flow. The air supplied to the process has a
significantly lower temperature, especially, at the
combustion front where the temperature of the solid phase
is high. In addition to the significant temperature
difference, air is supplied in an opposite direction to the
reaction front, thus making heat conduction less efficient
and narrowing the reaction front. This finding is also
supported by the study of Thunman and Leckner [8].
Figure 4 shows the difference in reaction front thicknesses
between airflows of 0.5 and 5 m3/s.
Figure 4. Temperature profiles of beds with 0.5 and 5 m3/s airflows
Nevertheless, the cooling effect is not significant with
large fuel particle diameters, compared to that of 20 mm
particles. Figure 3 indicates that in case of 20 mm large
particles the combustion time decreases while air flow is
increased from 0.5 to 0.75 m3/s. However, when the air
flow is further increased above 2 m3/s, the combustion
Table 1. Combustion time of fuels with different air flows.
16
time starts decreasing. In contrast to 20 mm large particles,
larger particles seem to be less affected by the cooling of
the air flow. Furthermore, the combustion time continues
decreasing as the air flow is increased above 2 m3/s. These
findings can be explained by the fact that for smaller
particle sizes the density number, the ratio of particle area
to unit volume, is larger than that of large particles. In
addition, for small particle sizes, also the heat convection
coefficient, which is responsible for heat exchange
between the gas and solid phases, has a larger value than
the coefficient of large particles. Therefore, in case of
small particles the heat exchange between phases is more
efficient, compared to large particles, thus the cooling
property of the air flow affects small particles more than
the larger ones. Horttanainen et al. [29] have concluded in
their study that combustion air flow rate can be increased
as particle size is increased. This finding is in agreement
with the result obtained from simulation case II.
6 Conclusions
The investigation on the boiler furnace model was started
by studying phenomena occurring in the BioGrate furnace.
The furnace model was decomposed in to reaction rates
and governing equations for mass and energy
conservation.
The developed BioGrate model was than used to
study the process phenomena occurring inside the
BioGrate furnace with varying process conditions. In
addition, a sensitivity analysis was made for different
parameters, which affect the combustion process.
The sensitivity analysis showed that the
combustion time increased linearly with the increase of
moisture content. A study on the air flow effect indicated
that oxygen deficiency slowed down the combustion
process, however, excess air, on the other hand, increased
the combustion time by cooling the solid phase. The
results obtained from the simulator were found to be in
agreement with the results found in literature.
7 References
1 Kiameh, P., Power Generation Handbook, McGraw-Hill, 2002, 557 p.
2 Joronen, T., Kovács, J., Majanne, Y., Voimalaitosautomaatio,
Suomenautomaatioseura Oy, Helsinki 2007, 1st edition, 276 p.
3 WWW, Anon, Wärtsilä Oyj, Brochure,
http://service.wartsila.com/Wartsila/global/docs/en/power/media_publicat
ions/brochures/bioenergy_fi.pdf, 03.08.2009
4 Thunman, H., Davidsson, K., Leckner, B., Separation of drying and
devolatilization during conversion of solid fuels, Combustion and Flame
137 (2004), pp. 242 - 250
5 Peters, B., Bruch, C., A flexible and stable numerical method for
simulating the thermal decomposition of wood particles, Chemosphere 42
(2001), pp. 481 – 490
6 Di Blasi, C., Heat, Momentum and Mass Transport Through a
Shrinking Biomass Particle Exposed to Thermal Radiation, Chemical
Engineering Science 5 (1996), pp. 1121-1132
7 Di Blasi, C., Combustion and gasification rates of lignocellulosic
chars, Progress in Energy and Combustion Science 35 (2009), pp. 121 –
140
8 Thunman, H., Leckner, B., Ignition and propagation of a reaction front
in cross-current bed combustion of wet biofuels, Fuel 80 (2001), pp. 473481
9 Zhou H, Jensen, A. D., Glaborg, P., Jensen, P., A., Kavaliauskas, A.,
Numerical modeling of straw combustion in a fixed bed, Fuel 84 (2005),
pp. 389-403
10 Kær S., K., Straw combustion on slow moving grates-a comparison of
model predictions with experimental data, Biomass and Bioenergy 28
(2005), pp. 307-320
11 Yang, Y., B., Yamauchi, H., Nasserzadeh, V., Swithenbank, J., Effects
of fuel devolatilization on the combustion of wood chips and incineration
of simulation municipal solid wastes in a packed bed, Fuel 82 (2003), pp.
2205-2221
12 Di Blasi, C., Branca, C., Sparano, La Mantia, B., Drying
Characteristics of wood cylinders for conditions pertinent to fixed-bed
countercurrent gasification, Biomass and Bioenergy 25 (2003), pp. 45-58
13 Alves, S., S., Figueiredo, J., L., A model for pyrolysis of wet wood,
Chemical Engineering Science 44 (1989), pp. 2861-2869
14 Janssens, M., Douglas, B., Wood and Wood Products, Handbook of
Building Materials for Fire Protection, Edited by Harper, C., A.,
McGraw-Hill 2004, 542 p.
15 Yagi, S., Kunii, D., Studies on Effective Thermal Conductivities in
Packed Beds, A.I.Ch.E. Journal 3 (1957), pp. 373-381
16 Fjellerup, F., Henriksen, U., Heat Transfer in a Fixed Bed of Straw
Char, Energy & Fuels 17 (2003), pp. 1251-1258
17 Horttanainen M., Saastamoinen, J., Sarkomaa, P., Operational Limits
of Ignition Front Propagation against Airflow in Packed Beds of
Different Wood Fuels, Energy & Fuels 16 (2002), pp. 676-686
18 R.P. van der Lans, L. T. Pedersen, A. Jensen, P. Glarborg and K.
Dam-Johansen, Modelling and expirements of straw combustion in a
grate furnace, Biomass and Bioenergy 19 (2000) , pp. 199-208
19 Y. R. Goh, Y. B. Yang, R. Zakaria, R. G. Siddall, V. Nasserzadeh, J.
Swithenbank, Development of an Incinerator Bed Model for Municipal
Solid Waste Incineration, Combustion Science and Technology 162
(2001), pp. 37-58
20 Yang, Y., B., Goh,, Y.,R., Zakaria, R., Nasserzadeh, V., Swithenbank,
J., Mathematical modelling of MSW incineration on a traveling bed,
Waste Management 22 (2002), pp. 369-380
21 Shin, D, Choi, S, The Combustion of Simulated Waste Particles in a
Fixed Bed, Combustion and Flame 121 (2000), pp. 167-180
22 Kær S., K., Straw combustion on slow moving grates-a comparison of
model predictions with experimental data, Biomass and Bioenergy 28
(2005), pp. 307-320
23 Zhou H, Jensen, A. D., Glaborg, P., Jensen, P., A., Kavaliauskas, A.,
Numerical modeling of straw combustion in a fixed bed, Fuel 84 (2005),
pp. 389-403
24. Janse, A., M., C., de Jonge, H., G., Prins, W., van Swaaij, W., P., M.,
Combustion Kinetics of Char Obtained by Flash Pyrolysis of Pine Wood,
Ind. Eng. Chem. Res. 37 (1998), pp. 3909-3918
25. Dupont, C., Chen, Li., Cances, J., Commandre, J.-M., Cuoci, A.,
Pierucci, S., Ranzi, E., Biomass pyrolysis: Kinetic modeling and
expiremental validation under high temperature and flash heating rate
conditions, Journal of Analytical and Applied Pyrolysis 85 (2009), pp.
260-267
26. Babushik, V., I., Dakdancha, A., N., Global kinetic parameters for
high-temperature gas-phase reactions, Combustion, Explosion, and
Shock Waves 29 (1993), pp. 464-489
27. Matsumoto, K., Takeno, K., Ichinose, T., Ogi, T., Nakanishi, M.,
Gasification reaction kinetics on biomass char obtained as a by-product
of gasification in an entrained-flow gasifier with steam and oxygen at 900
– 1000 °C, Fuel 88 (2009), pp. 519 – 527
28 Yang, Y., B., Sharifi, V., N., Swithenbank J., Effect of air flow rate
and fuel moisture on the burning behavioursof biomass and simulated
municipal solid wastes in packed beds, Fuel 83 (2004), pp. 1553-1562
29. Horttanainen M., Saastamoinen, J., Sarkomaa, P., Operational Limits
of Ignition Front Propagation against Airflow in Packed Beds of
Different Wood Fuels, Energy & Fuels 16 (2002), pp. 676-686
17
Data-Based Uncertainty Modeling of MIMO Systems
Hamed Jafarian and Kurt-Erik Häggblom
Process Control Laboratory, Department of Chemical Engineering,
Åbo Akademi University, Åbo, Finland
Abstract
Many robust control design methods require a linear model consisting of nominal model augmented
by an uncertainty description. A general form for such a model is
10 21 11 12( )G G H I H H
−= + ∆ − ∆ (1)
where G is the transfer function of the true system, 0G is a nominal model, and ∆ is a perturbation causing uncertainty about the true system. Depending on the particular type of uncertainty
(additive, input or output multiplicative, inverse types of uncertainty, combinations of various types
of uncertainty), 11H , 12H and 21H can contain combinations of (known) constant matrices and the
nominal model.
Assume that we have information about the true system in the form of a number of possible transfer
functions kG , 1, ,k N=  . The nominal model and the perturbation k∆ associated with kG are
unknown, but they have to satisfy
10 21 11 12( )k k kG G H I H H
−= + ∆ − ∆ , 1, ,k N=  . (2)
How should 0G be determined? It has been shown that ∞∆ is a control relevant measure of the
distance between G and 0G for models of the form (1) and that the achievable stability margin by
feedback control is inversely proportional to this distance. For a given type of uncertainty model, this
suggests that 0G should be determined by solving the optimization problem
0
min max kG k ∞
∆ (3)
subject to the appropriate data matching condition (2). Obviously, the type of uncertainty model
giving the smallest minimum is the best one according to this measure.
If information about the system is obtain through identification, input-output data are available. An
attractive way of removing noise from the output is to fit a model kG to the data and to calculate a
noise-free output ky by k k ky G u= , where ku is the input in experiment k . Since the purpose of
the experiments in this context is to excite the system in various ways, the inputs do not tend to be
persistently exciting in all individual experiments. Thus, kG only applies to the particular input ku ,
and the relevant information is input-output data { },k ku y , 1, ,k N=  . This means that the model
matching condition (2) should be replaced by the input-output matching condition
10 21 11 12( )k k k k ky G u H I H H u
−= + ∆ − ∆ . (4)
It can be shown that the use of (4) instead of (2) results in a less conservative uncertainty model.
18
Our modeling approach is to model 0G in the frequency domain using sampled frequency responses
of the input-output data. Because of the availability of kG , these are easy to calculate for standardized inputs. The available information is thus { }( j ), ( j ),k ku yω ω ω ∈Ω , 1, ,k N=  , and we solve
the optimization problem frequency-by-frequency, i.e.
0
2( j )
min max ( j )kG kω
ω∆ s.t. (4), ω∀ ∈Ω . (5)
The uncertainty k∆ is assumed to be unstructured.
For some types of uncertainty, the optimization problem can easily be formulated as a convex
optimization problem. Additive uncertainty, for example, which in its basic form is described by
0k k k ky G u u= + ∆ , 1, ,k N=  (6)
results in the optimization problem
0
min
G
γ s.t. 0
0
0
( )
k k
k k k k
I y G u
y G u u u
γ
∗ ∗
− 
 
− 
 , 1, ,k N=  , ω∀ ∈Ω (7)
which is a convex optimization problem. Here A∗ denotes the complex conjugate transpose of A
and 0P denotes that P is positive semidefinite.
Many types of uncertainty descriptions do not readily give a convex optimization problem. For
example, an output multiplicative uncertainty described by
0 0k k k ky G u G u= + ∆ , 1, ,k N=  (8)
results in the optimization problem
0
min
G
γ s.t. 0
0 0 0
0
( )
k k
k k k k
I y G u
y G u u G G u
γ
∗ ∗ ∗
− 
 
− 
 , 1, ,k N=  , ω∀ ∈Ω (9)
which is non-convex due to the appearance of 0 0G G
∗ . An iterative solution by keeping 0 0G G
∗ fixed
during each iteration tends to produce local minima which are non-global. However, we show how
this optimization problem, and similar ones for some other types of uncertainty, can be reformulated
as a convex optimization problem.
For control design, 0G is needed as a transfer function or a state-space model. In principle, we
should determine such a model by replacing 0G in the appropriate consistency relations, like those
appearing in (7) and (9), by a suitable parameterization of 0G . However, so far we have not been
able to obtain a satisfactory solution in that way. Instead, we have fitted a model to the calculated
frequency responses 0( j )G ω , ω ∈Ω . A drawback of this approach is that 2min ( j )ω∆ will
increase, usually also min ∞∆ , sometimes even drastically. We have studied various approaches of
reducing the effects of this drawback.
19
 Modeling for Control of Beer Quality 
D.R. Warnasooriya
1
, P.G. Rathnasiri
1
, Bernt Lie
2
1Department of Chemical and Process Engineering, University of Moratuwa, Kattubedda, Moratuwa, Sri Lanka. 
 E­Mail; dilantha1981@gmail.com, ratnasiri@cheng.mrt.ac.lk 
2Telemark University College, P.O. Box 203/Postboks 203, N­3901 Porsgrunn, Norway. 
 E­Mail; Bernt.Lie@hit.no 
Abstract 
Beer is the most common alcoholic drink around the world. When talking about the beer quality, flavor of the beer 
is  more  concerned. Most  of  the  brewers  in  Sri  Lanka  are  using  traditional  methods  to  brewing  beer.  Most  of 
brewers  using  pre  identified  recipe  to  produce  mass  production  of  beer.  Therefore,  beer  quality  i.e.  flavor  is 
varying brand by brand.  It  is  important to study  the variation of temperature how will effect to the  final alcohol 
production  and  the  flavor  compound  formation.  Beer  manufacturing  industry  can  be  used  this  knowledge  to 
increase the production efficiency and the product quality.  
It is very important to know about dynamics of forming flavor compounds. In this work the fermentation process is 
concerned  since  all  the  flavor  compounds  are  formed  during  the  fermentation.    The  mechanistic  model  is 
developed  by  based  on  the  knowledge  of  biochemical  processes  in  the  yeast  cell  and  previously  developed 
mathematical models which  are  available  in  the  literature.  There  are many  beer models  can  be  found  such  as 
Engasser  et  al.,(1981);Growth  kinetic model,    Gee  at  al.,(1988),  Phisalaphong  et  al.(2006);Growth  kinetic model 
and effect on temperature, W.Fred Ramirez and Jan Maciejowski.,(2007);Optimal beer fermentation, etc. The beer 
fermentation process is modeled and simulated in MATLAB environment.  
Growth model,  nutrient model,  and  the  flavor model  are  considered  and  developed. Growth model  consists  of 
sugar  consumption,  biomass  growth  and  ethanol  formation  models.  Those  models  are  developed  with 
temperature dependant  parameters  to observe  the effect of  temperature.  Three amino acids which are Valine, 
Leucine  and  Iso  Leucine  are  considered  for  the  Nutrient  model.  Consumption  of  these  three  amino  acids  is 
considered during fermentation. Flavor model is developed based on the growth model and  the nutrient model. 
Flavor  compounds  are  categorized  into  three  groups  which  are  Fusel  alcohols,  esters  and  vicinal  diketones. 
Altogether  nine  parameters  are  considered  as  flavor  compounds  and  the  effects  of  temperature  on  those  are 
simulated  with  MATLAB.  Industrial  temperature  profile  is  obtained  and  applied  for  the  developed  model  and 
simulated in MATLAB and the results are analyzed.  
PI controller  is applied to get  identified temperature profile  to obtain optimal  flavor  formation and  the dynamic 
model is used for find suitable controller parameters for best control.  
20
      
               A two phase MPC and its application to a grinding process 
Alexey Zakharov*, Alexandre Boriouchkine*, Sirkka-Liisa Jämsä-Jounela* 
* Aalto University, P.O. box 16100, 00076, Aalto, Finland. e-mail: alexey.zakharov@ tkk.fi 
Abstract:  The  growing  complexity  of  the  control  systems  and  the  increased  use  of  nonlinear models 
cause a dramatic increase in the computational requirements of MPCs. Therefore, more computationally 
efficient MPC are needed. This paper presents a two-phase MPC approach for decreasing computational 
demand  without  sacrificing  its  efficiency.  The  first  phase  of  the  MPC  treats  the  input  variables  as 
independent decision variables of the objective optimization, since the largest part of the objective value 
arises  from  a  few  earliest  sampling  intervals.  In  contrast,  the  second  phase  combines  input  variables, 
defining  the  rest  of  the  MPC  objective  value,  in  an  open-loop  control  which  is  specified  by  a  few 
independent decision variables. The method is compared against the traditional Quadratic Programming 
implementation of an MPC for the Grinding Plant control problem. The two-phase MPC demonstrates a 
better performance compared with the traditional controller with the same control horizon.  
Keywords: MPC, dynamic optimization, grinding process, industrial application, stability 
1. INTRODUCTION 
Current  economic  conditions  has  set  new  challenges  for 
productivity  and  keeping  the  operational  conditions  of 
processes within required boundaries. These challenges have 
resulted  in  the  growing  complexity  of  industrial  control 
systems. Consequently,  the  increasing complexity of control 
systems requires more efficient advanced level (higher-level) 
control  strategy  which  almost  universally  utilizes  model 
predictive  controllers  (MPCs).  However,  the  use  of  MPCs 
can  be  computationally  heavy  because  of  several  factors, 
such as complex process models, as well as a greater number 
of  manipulated  and  controlled  variables,  and  constraint 
management.  Nevertheless,  a  limited  computational  power 
forces  to  make  a  compromise  between  efficiency  and 
computational time of a control strategy. However, in several 
applications  the  control  strategy  efficiency  is  critical.  To 
shorten  the  computational  time  without  losing  efficiency,  a 
modification  must  be  done  to  existing  MPC  techniques. 
Several options exist for improving the efficiency of an MPC. 
One of such techniques is the dynamic programming, since it 
provides  many  useful  insights  into  the  MPC  performance 
problems.  However,  dynamic  programming  reduces  the 
problem  to Hamilton-Jacobi-Bellman  (HJB) equation  that  in 
most  cases  can  not  be  solved  analytically,  because  it  is  a 
partial  differential  equation,  and  solving  HJB  equation  is 
computationally unattractive due  to  the high dimensionality. 
For  this  reason,  Dynamic  Programming  is  not  applied 
directly  to  practical  control  problems,  but  instead,  different 
simplified methods are used.  
In  the  60s,  the  development  of  Dynamic  Programmingemploying  control  techniques  led  to  the  invention  of  the 
linear  quadratic  regulator  (LQR).  The  LQR  is  a  rare  case 
when the HJB equation can be solved analytically. The LQR 
was found to be useful for many practical applications and it 
might  be  considered  as  the  direct  predecessor  of  modern 
MPCs (so called ‘zero generation’ of MPC). 
Unfortunately,  the  linear-quadratic  regulator  is  limited  by 
linear dynamics, quadratic objective function and absence of 
constraints,  thus  leaving many  industrial problems out of  its 
scope.  Moreover,  it  is  well  acknowledged  that  economic 
operating  points  in  typical  process  units  often  lie  at  the 
intersection of constraints. As a result, a successful industrial 
controller must keep the system as close to the constraints as 
possible  without  actually  violating  them.  Thus,  the  next 
generation  of  MPC  appeared  having  the  following  main 
features: linear process constraints, a linear process model, a 
quadratic objective and a finite time horizon (see Richalet et 
al. (1978), Prett et al. (1982)). The finite horizon was used to 
approximate  the  infinite  horizon  problem, which  hardly  can 
be  solved.  Since  in  the  presence  of  constraints  even  the 
solution of the finite horizon optimization problem cannot be 
derived  analytically,  the  quadratic  programming  was 
employed to perform the optimization.  
In  the  early  90s,  it  was  discovered  that  the  constrained 
optimization can cause feasibility problems, especially, when 
large disturbances appear. Therefore, the most of the modern 
MPC software products have been enforced to use soft output 
constraints (Qin and Badgwell 2003).  
On the other hand, because of the finite horizon formulation, 
MPC  faced  stability  problems. Attempts  to  achieve  stability 
included different prediction and control horizon approaches 
and the introduction of a terminal cost to the MPC objective. 
These methods were criticized in the study of Bitmead et al. 
(1990)  as  ‘playing  games’,  because  there  were  no  clear 
conditions  to  guarantee  stability.  Thereby,  the  stability  of 
MPC was studied actively during the early 90’s (Keerthi and 
Gilbert  (1988), Mayne and Michalska  (1990) are among  the 
first  papers  exploring  this  question)  and  a  comprehensive 
review  of  these  studies  is  provided  in Mayne  et  al.  (2000). 
21
      
Briefly, the stability is almost universally established through 
the use of the value function of MPC as a natural Lyapunov 
function.  On  the  other  hand,  the  Dynamic  Programming 
provided some useful insights concerning the MPC stability. 
One  example  of  that  is  the  ‘inverse  optimality  principle’, 
which is used to ensure the stability of MPC by utilizing the 
fake HJB equation (for details, see for example Bitmead et al 
1990, Magni and Sepulchre 1997). 
It is well known that the performance of MPC depends on the 
quality of underlying model: an MPC is as good as its model. 
For that reason during the last decade the focus was moved to 
the  nonlinear  MPC  utilizing  of  a  more  accurate  nonlinear 
process model.  Basically,  the  implementation  of  such MPC 
cannot  be  based  on  QP  anymore.  Therefore,  the  convex 
optimization techniques are employed instead of QP.  
Consequently,  the  nonlinearity  of  the  models  along  with 
other  factors,  such  as  the  complexity  of  control  systems, 
increases  the  computational  requirements  for  MPCs. 
However,  the  computational  requirements  of  MPCs  are 
critical  for many  applications,  especially,  for  large  and  fast 
processes.  Therefore,  many  researchers  have  concentrated 
their efforts on reducing the number of on-line computations 
(Bemporad  et  al.  2002,  Pannocchia  et  al.  2007,  Rao  et  al. 
1998).  
In  contrast  to  computational  requirements  and  stability, 
another  important  property  of  an  MPC,  namely,  optimality 
did not attract so much attention in the literature, even though 
the finite-horizon MPCs do not provide  the optimal solution 
of problems.  In general,  the  researchers do not  focus  on  the 
exploration of optimality because of the idea that a close-tooptimal solution may be found through increasing the control 
horizon. However, an  example  is  given  in Di Palma, Magni 
(2007),  where  MPC  performance  is  not  a  non-decreasing 
function  of  the  optimization  horizon.  In  addition,  a  longer 
control  horizon  also  requires  more  computations  and  a 
compromise  must  be  made  between  the  close-to-optimal 
properties of the controller and its computational demands.   
Simultaneously  with  traditional  MPC  development,  some 
attempts were made to estimate the solution of HJB equation 
indirectly.  For  example,  an  iterative  approach was  proposed 
in Sardis  and  Lee  (1979). Unfortunately,  until  today,  highly 
efficient  methods  based  on  HJB  equation  have  not  been 
developed  and  the  conclusion  was  made  in  Cannon  (2004) 
that  ‘compared with  conventional NMPC  the  computational 
burden of currently available methods for the HJB successive 
approximation approach remains prohibitive’.  
Even  HJB  equations  are  unattractive  for  numerical 
implementation,  the  Dynamic  Programming  appears  still  to 
provide  a  useful  insight  into  the  MPC  optimality.  One 
example  is  presented  in  the  work  of  Grune  and  Pannek 
(2009),  where  HJB  equation  was  employed  to  estimate  the 
‘degree  of  suboptimality’  of  MPC  solutions,  which  was 
further used for adaptive determining of the MPC horizon. 
Another  idea  risen  from  Dynamical  Programming,  (which 
present  research  is  focused  on)  is  the  desire  to  have  the 
terminal  cost  of MPC  as  close  to  the  value  function  of  the 
infinite horizon problem as  possible  (Mayne  et  al.  2000).  If 
the  value  function  is  employed  as  a  terminal  cost,  MPC 
provides  the  optimal  solution  even  with  the  time  horizon 
being  unity.  In  particular,  the  solution  of  Riccati  equation, 
which  is  the  value  function  of  the  unconstrained  infinite 
horizon  problem,    is proposed as  the  terminal  cost  for MPC 
objective  in  many  papers  (see  Chmielewski  and 
Manousiouthakis  (1996),  Sznaier  and  Damborg  (1987)). 
Although  the  stability  is  attained  within  the  approach,  in 
general,  it  is  not  possible  to  expect  that  a  ‘good’ 
approximation  of  the  value  function  can  be  found.  In 
particular,  if  the  MPC  setpoint  lies  on  the  border  of 
constraints,  quadratic  functions  cannot  capture  the  essential 
asymmetry of the value function.  
In the present paper, the emphasis is moved on the estimation 
of  the value  function of  the  infinite horizon problem, which 
provides  close-to-optimal  behaviour  of  the  controller  even 
with  a  short  control  horizon.  Thus,  the  method  achieves  a 
decrease  in  computational  demand  without  sacrificing  its 
efficiency.   An industrial application (a model of a Grinding 
process) is used to test the developed method.  
The  paper  is  organized  as  follows.  Section  2  contains  a 
description of the proposed MPC controller, and Section 3 a 
description of  the grinding process.  In Section 4,  the  results 
are presented and compared against a QP implementation of 
MPC, and Section 5 contains the conclusion. 
2. DESCRIPTION OF THE TWO-PHASE MPC  
2.1 The idea of the two-phase MPC 
In this section, the two-phase method will be presented for a 
simple linear discrete state space dynamics: 
)1()1(
)()()1(
+=+
+=+
kCxky
kBukAxkx
        (1) 
where  ),...,,( 21 nxxxx =   is  the  vector  of  the  current  state  of 
the  system,  ),...,,( 21 myyyy =   is  the  vector  of  the  system 
outputs,  and  ),...,,( 21 luuuu =   is  the  vector  of  the  input 
variables. For the sake of simplicity, it  is assumed that  there 
is no noise in neither, the dynamics nor in the measurements, 
and the state of the system is exactly known. In addition, the 
process is assumed to have M  linear constraints: 
MiqkyP ii ,...,2,1,)( =≤ .        (2) 
Under dynamics (1) and constraints (2),  the optimal setpoint 
y*  is  usually  defined  by  a  higher  level  of  the  control 
hierarchy (for example at the real time optimization layer). At 
this setpoint the steady state of the system x* and the optimal 
steady state control u* are defined using dynamics (1). 
A  typical  objective  function  of  a  MPC  with  the  control 
horizon equals  N  has the following form: 
))1(),(())(),(()),0((
1
0
−+Δ=�

=
NuNxFkukyluxJ
N
k N   (3) 
22
      
where  different  forms  of  ),( uyl Δ  may  be  used  in  different 
controllers and the terminal cost ),( uxF is needed to stabilize 
the  controller.  In  fact  the  widely  used  approach,  which 
spreads  the  control  action  at  control  horizons N until  the 
prediction  horizon K ,  employs  the  terminal  cost  of  the 
following form: 


=
=−
1
)0),(())1(),((
K
Nk kylNuNxF       (4) 
Another popular option is the solution of Riccati equation  is 
used as the terminal cost. 
On the other hand, dynamic programming theory provides the 
ideal candidate for the role of the terminal cost, which is able 
to guarantee both stability and optimality of the solution even 
with the control horizon equals one. Indeed, the finite horizon 
formulation  is  just  a  simplification  of  the  original  infinite 
horizon optimization problem with dynamics (1), constraints 
(2) and the following objective: 
  �

=
∞ Δ=
0
))(),(()),0((min
k kukyluxJ .    (5) 
According  to  dynamic  programming,  the  optimal  control 
action may be found as  
( )))0(),1(())0(),0((minarg
))1(),0((
)0(
*
uxVuyl
uxu
u +Δ=
=−
    (6) 
where  ))0(),1(( uxV   is  the  value  function  introduced  as  the 
optimal value of the infinite horizon optimization problem: 


=
Δ=
1
))(),((min))0(),1((
k u kukyluxV .    (7) 
In  fact,  MPC  and  Dynamic  programming  derive  current 
control  through  minimizing  items  (3)  and  (6)  respectively, 
and the value function plays the same role in Equation (6) as 
the terminal cost plays in Equation (3). Thus, MPC approach 
may  be  considered  as  an  implementation  of  dynamic 
programming  ideas,  but with  the  ‘inaccurate’ approximation 
of the value function. Thus a control horizon longer than one 
must  be  employed  to  obtain  the  satisfactory  performance  of 
MPC even though the terminal cost inaccurately estimates the 
value function of the infinite horizon problem.  
If the open-loop control  *u , which is optimal in state  x , was 
known, function  ))1(),(( −NuNxV  could be easily estimated 
to any reasonable accuracy. However,  the optimal control  is 
unknown  and  a  set  U of  second  phase  open-loop  controls 
must be used to get a relatively good estimation of the value 
function as follows:  


=

Δ=−
1
))(),((min))1(),((
K
Nk Uu U
K kukxlNuNxV ,   (8) 
here  K is  the  second  phase  horizon.  In  particular,  the 
common MPC with different prediction and control horizons 
uses  set  U consisting  of  a  single  open-loop  control  which 
expands  the  values  of  the  input  variables  at  the  control 
horizon until the end of the prediction horizon.  
A set  of  second phase open-loop controls U   containing  the 
optimal  control  ))1(),0((* −uxu   allows  to  get  the  accurate 
result:  
))1(),((lim))1(),(( −=−
∞→
NuNxVNuNxV UK
K
.  (9) 
Therefore the set of control strategies U  must be ‘divorce’ in 
a  sense  that  at  any  point  )1(),( −NuNx   a  close-to-optimal 
open-loop control can be found in U . On the other hand, the 
computational  complexity  of  the MPC  controller  grows  for 
wider  sets.  Thus,  if  the  set  of  second  phase  open-loop 
controls  is  chosen  as  a  parametric  family  of  functions,  the 
number of parameters must not be very high in order to avoid 
the high computational complexity of the method.  
Since for any set U  consisting of more  than one open-loop 
control  the computation of  the  terminal  cost  (8)  involves an 
optimization  of  the  system  dynamics  after  the  control 
horizon, this optimization is called ‘the second phase’ of the 
proposed MPC.  
2.2 Two sets of second phase open-loop controls  
Let us consider the following one-parametric set of functions 
presented in Figure 1: 
)exp()1()exp()( 21 kckckg −−+−= ααα ,    (10) 
where  coefficients  1c   and  2c are  fixed.  In  order  to define  a 
open-loop control for the whole MPC, it  is needed to define 
an  individual  control  for  each  input  variable.  This  can  be 
done in the following way:   
( ) liukguNukNu iiiii ,...,1,)()1()1( *)(* =+−−=−+ α ,  (11) 
here  *u  is the steady state optimal control. Thus the system 
open-loop  control  is  defined  by  vector ),...,,( 21 lαααα = , 
where  every  element  defines  the  control  for  the  respective 
manipulated variable.  
The proper choice of constants  1c  and  2c may be a problem if 
the  described  above  set  of  open-loop  controls  is  used.  In 
order  to  avoid  it,  another  set  of  second  phase  open-loop 
controls is introduced by adding a time scale parameter  β  as 
follows:  
)exp()1()exp()( 21, tctctg βαβαβα −−+−= .     (12)  
The open-loop controls of the MPC are constructed similarly 
to Equation (11): 
( ) liukguNukNu iiiii ,...,1,)()1()1( *),(* =+−−=−+ βα   (13) 
23
      
Here  different  parameters  iα are  used  for  different  input 
variables  of  the MPC,  but  a  single  parameter  β   is  used  to 
define  the  time  scale  for  all  input  variables.  In  the  present 
paper, the set of second phase open-loop controls defined by 
Equation  (13)  is  used  for  two-phase  MPC  implementation 
and testing.  
0 2 4 6 8 10 12 14 16 18 20
-1.5
-1
-0.5
0
0.5
1
1.5
2
Time
G
a (t
Figure 1. One-parametric set of functions  )(kgα with c1 = 0.2 
and c2 = 0.1. 
3. DESCRIPTION OF THE GRINDING PLANT 
Communition is a huge consumer of electrical power because 
crushing  rocks  into  powder  requires  a  lot  of  energy. 
According  to  Pomerleau  et  al.  (2000),  grinding  typically 
accounts for almost 50% of the costs of a concentrator and, as 
a  result,  the  optimization  of  grinding  mills  is  an  extremely 
important  research  topic. The aim of economic optimization 
is to maximize the feed rate or to achieve the desired particle 
size distribution, thus making production more profitable. An 
overview  of  the  control  methods  of  grinding  plants  is 
provided in Hodouin, Jämsä-Jounela et al. (2001). 
Figure 2. Grinding circuit, Lestage et al. 2002. 
The  ore  is  fed  to  the  rod mill  and  then  discharged  into  the 
pump sump. The slurry is then fed to a hydrocyclone, where 
it is separated into the overflow product and a recycled part, 
which  is  fed  back  to  the  ball-mill  (for  more  details,  see 
Lestage et al. 2002). The whole circuit is presented in Figure 
2. 
There are two manipulated variables available in the model: 
•   u1: the rod-mill feed (t/h) 
•   u2: the pump sump water addition (m
3/h) 
and four output variables: 
•   y1: the hydrocyclone overflow density (% solids) 
•   y2: the fraction of particles smaller than 325 mesh 
(47Am) in the product (%) 
•   y3: the tonnage through the ball mill (t/h) 
•   y4: the pump sump level (%) 
Typically  the  grinding  process  can  be  described  by 
means  of  a  transfer  function  of  second  order.  The  transfer 
function  presented  in  Lestage  et  al.  2002  is  used  to  test  the 
MPCs: 
u1      u2 
y1,  G11(s) =  )7501)(53001(
)56001(0255.0 600
ss es s
++
− −
   G12(s) =  )601)(32001(
)40501(14.0
ss s ++
+−
y2,  G21(s) =  )7501)(52001(
)9001(2.0
ss s ++
−−
  G22(s) =  )501)(44001(
)395001(012,0
ss s ++
+
y3,   G31(s) =  )4001)(57001(
8.13
ss ++   G32(s) =  )51)(50001(
)7001(2.4
ss s ++

y4,  G41(s) =  )2101)(55001(
5,749
ss ++   G42(s) =  )47001(
1,962
s+  
The  model  is  converted  into  the  discrete  time  state  space 
form  (1)  with  a  sample  time  equals  300s.  The  system  is 
described  with  15  state  variables  and  four  controlled 
variables.  
The  constraints  are  defined  as  product  specifications:  the 
hydrocyclone  overflow  density  (y1)  must  be  above  48%  to 
meet  the  flotation  requirements,  and must  be  below  52%  to 
avoid sedimentation problems. Product specification fineness 
(y2) is defined as 47% of the particles smaller than 47 �m. In 
order not  to overload  the ball mill,  the  throughput  (y3) must 
not  exceed  820t/h.  The  pump  sump  level  (y4)  must  remain 
between  15%  and  85%.  The  constraints  are  defined  in  the 
following order: lower and upper constraints on y1, lower and 
upper constraints on y2, upper constraint on y3, and lower and 
upper  constraints  on  y4.  Thus,  matrices  P  and  q  take  the 
following form: 
y1     y2      y3    y4 

























=
1000
1000
0100
0010
0010
0001
0001
P
,  
q = [52%, -48%, 47%, -47%, 820t/h, 85%, -15%]T.   
4. COMPARISON OF THE TWO-PHASE MPC 
AGAINST THE QP IMPLEMENTATION 
24
      
Next,  the  two-phase  MPC  method  is  compared  against  the 
soft-constrained QP implementation of MPC. 
In  order  to  achieve  smooth  trajectories,  the  following 
objective function is used for both controllers: 

��

=
= =
=
−′−+
+−
+−−′−−=
K
k K
k M
i iii
K
k ykySyky
qkyPQ
kukuRkukuxF
1
1 1
2
0
*))(()*)((
))0,)((max(
))1()(())1()(()(
,  (14) 
where diagonal matrixes R, Q and S are defined as follows: 
05,0,01,0,0,5
1,25,100
,
100
025
44332211
7654321
====
=======
��


��


=
SSSS
QQQQQQQ
R
.  (15) 
To  test  the  controller’s  ability  to  follow  changing  operating 
conditions, the constraints are varied as shown in Table 1. In 
every  case,  the  optimal  steady  states,  presented  in  Table  2, 
are obtained by maximizing the throughput of the plant. 
Table 1. Constraints 
Time 
period 
(h) 
Overflow 
solids (%) 
Particles < 
than 47�m 
(%) 
Ball mill 
through t/h 
Pump sump 
level (%) 
  min  max  min  max  max  min  max 
0…0.25  48  52  47  47  820  15  85 
0.5…5  48  52  49  49  820  15  85 
5…10  50  52  49  49  820  15  85 
10…15  50  52  47  47  820  15  85 
Table 2. Setpoints 
Time 
period 
(h) 
Overflow 
solids (%) 
Particles  < 
than 47�m 
(%) 
Ball mill 
throughput 
t/h 
Pump 
sump 
level (%) 
0…0.25  51,83  47,0  820,0  62.06 
0.25…5  48,0  49,0  811,95  64.5 
5…10  50,0  49,0  739,32  30,87 
10…15  51,83  47,0  820  62,06 
The  prediction  horizon  K in  QP  formulation  of  MPC  is 
always  10  steps  longer  than  the  control  horizon N .  For  the 
two-phase MPC  the  second phase  length  is  also  taken  to be 
10 steps. QP MPC is tested with control horizons equal to 2 
and  10  and  two-phase  MPC  is  tested  with  control  horizon 
equal 2.  
The  second-phase  open-loop  control  set  described  by 
Equation  (13)  is  used  for  the  two-phase  MPC 
implementation.  Since  there  are  only  two  input  variables,  a 
second-phase  control  is  defined  by  three  parameters: �1  and 
�2  select  the shape of  the open-loop controls of  the first and 
the second input respectively, while a parameter � is used for 
time scaling. Discrete values of �1 and �2 are considered with 
a  step equal 0.4 while �  is  taken  in  the following  form: � = 
0.9k with k having only integer values.  
Two  controlled  variables  are  compared  in  Figure  3  and  the 
manipulated variables are presented in Figure 4. In general it 
is clear that the two-phase MPC with control horizon  2=N  
demonstrates  behaviour,  which  is  closer  to  QP  MPC  with 
longer  control  horizon  10=N   rather  than  the  same  control 
horizon 2=N .  
The  efficiency  of  the  presented  methods  may  be  evaluated 
using  the  objective  function  values  of  every  method 
presented  in  Table  4.  Again,  two-phase  MPC  achieves  the 
same value of the objective function as the QP MPC with the 
longer  prediction  horizon  10=N does.  QP  MPC  with  the 
short  control  horizon    2=N   demonstrates  13%  bigger 
objective value.  
Table 4. Objective for QP MPC and two-phase MPC  
  QP MPC,     
N  = 2 
QP MPC,      
N  = 10 
MPC 2 phase    
N  = 2 
Constraint 1  971  978  993 
Constraint 2  1 845  1 358  1 392 
Constraint 3  0  0  0 
Constraint 4  0  0  0 
Setpoint 1  496  446  468 
Setpoint 3  1 326  1 106  1 139 
Setpoint 4  1 039  889  922 
Control 1  827  886  781 
Control 2  1 037  1 003  968 
Objective   7 543  6 669  6 665 
0 5 10 15
47.5
48
48.5
49
49.5
50
50.5
51
51.5
52
Time, hours
H
y d ro c y c lo n e  o
v e rf lo w  d
e n s it y
%
MPC, N=2
MPC, N = 10
MPC 2 phase, N = 2
0 5 10 15
46.5
47
47.5
48
48.5
49
49.5
Time, hours
O
v e rf lo w  p
a rt ic le  <
 4
7
 u
m  %
MPC, N = 2
MPC, N = 10
MPC 2 phase, N = 2
Figure  3.  Controlled  variables:  a  –  Hydrocyclone  overflow 
density % solids, b – % of particles < 47 �m  
25
      
0 5 10 15
125
130
135
140
145
150
Time, hours
R
o d  M
ill
 F
e e d ra te
t/ h MPC, N = 2
MPC, N = 10
MPC 2 phase, N = 2
0 5 10 15
75
80
85
90
95
100
105
110
Time, hours
W
a te r  to  P
u m p  S
u m p
m 3
/h MPC, N = 2
MPC, N = 10
MPC 2 phase, N = 2
Figure  4.   Manipulated  variables:  a  –  rod mill  feedrate,  b  – 
water to pump sump 
5. CONCLUSION 
A two-phase MPC controller is described in the paper, where 
the  best  second  phase  open-loop  control  is  chosen  from  a 
predefined  set  of  open-loop  controls. Thus,  a more  accurate 
estimation of the value function is used as a terminal cost in 
comparison  with  the  QP  formulation.  The  numerical 
comparison  against  the  QP  formulation  of MPC  has  shown 
that  similar  performance  might  be  achieved  with  a  shorter 
control horizon. Thereby a progress in the  trade-off between 
the  performance  and  computational  demands  of  MPC  is 
made.  
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229-237. 
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58, pp 161-168, 
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constrained  infinite-time  linear–quadratic  optimal 
control, System and Control Letters, 29, pp. 121-129. 
Sznaier  M.,  Damborg  M.J.  (1987),  Suboptimal  control  of 
linear  system  with  state  and  control  inequality 
constraints, Proceedings of 26th Conference on Decision 
and Control, Los Angeles, CA, December 1987. 
Pomerleau A., Hodouin B., Desbiens A., Gagnon E.  (2000), 
A  survey  of  grinding  circuit  control  methods:  from 
decentralized PID controllers to multivariable predictive 
controllers, Powder Technology, 108, pp. 103-115. 
Hodouin D.,  Jämsä-Jounela  S.-L.,  Carvalho M.T.,  Bergh  L. 
(2001),  State  of  the  art  and  challenges  in  mineral 
processing control, Control Engineering Practice, 9, pp. 
995-1005. 
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real-time  optimization  of  a  grinding  circuit  using 
steady-state  linear  programming  supervisory  control, 
Powder Technology, 124, pp. 254-263. 
26
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28
Tuning of ARX-based Model Predictive Control for
Offset-free Tracking.
Jakob Kjøbsted Huusoma, Niels Kjølstad Poulsenb,
Sten Bay Jørgensena and John Bagterp Jørgensenb
aCAPEC, Department of Chemical and Biochemical Engineering
bDepartment of Informatics and Mathematical Modelling
Technical University of Denmark
Keywords: Model Predictive Control, ARX-model, Controller Tuning, Recursive estimation
Model Predictive Control (MPC) is a control technology that uses a model of the system to predict the
process output over some future horizon. The controlled input signal is determined by solution of an open-loop
optimization problem using the model of the system. The first part of the optimal control sequence is implemented
on the process. Feedback is obtained by repeating this procedure as new measurements become available.
Advanced control strategies such as Model Predictive Control have gained wide spread interest in many areas in
the chemical industries, due to fast algorithms, a well established theory and growing number of successful
industrial implementations. The main feature is that the optimal control signal is determined as a constrained
optimization which utilizes future predictions of the plant behaviour. Hence the controller has a plant model
embedded for state estimation. The achieved closed loop performance is therefore dependent on the quality of
the future predictions. The performance of the state estimator is on the other hand dependent on the accuracy of
the process and the noise model. In this contribution, we discuss closed loop performance of MPC based on ARX
models when applied to systems with unmeasured step disturbances.
The majority of industrial MPC applications are today still based on linear models. ARMAX models
(AutoRegresive Moving Average model with eXogenous input) can be identified using standard tools from time
series analysis and systems identification. However, for multivariable systems it is difficult to select a structure for
the ARMAX model. Furthermore, identification of the parameters in ARMAX models constitutes a non-linear nonconvex optimization problem. If the input-output model is simplified to an ARX model (AutoRegresive model with
eXogenous input), the estimation problem becomes a convex optimization problem. Furthermore the MIMO
system can be handled as easily as SISO systems. The discrete time ARX model structure is given as
A(q-1)yk = B(q-1)uk + εk εk~N(0,σ2)
where A(q-1) and B(q-1) are polynomials in the back shift operator q-1.
Unmeasured step disturbances are common in the process industries and appear for instance when a
feed source changes. The composition of crude oil in refineries may change significantly when feed is changed
from one well to another. Similar unknown steps may occur in a cement plant when raw minerals are changed.
Unmeasured steps may also be used to represent the inevitable model-plant mismatch. To reject such
disturbances the basic MPC formulation needs to be expanded, such that model errors are introduced in order to
include integrators or augment the system with disturbance models. Alternatively the model predictive control
algorithm can be combined with a recursive disturbance estimation algorithm. The closed loop performance of the
system will depend on the nature of the disturbance and how the disturbance rejection is facilitated by the control
algorithm.
29
This contribution will analyse a range of methods for achieving offset-free tracking in ARX-based model
predictive control. Considerations regarding pre-tuning of the free parameters in the controller are also provided
based on the infinite horizon, unconstrained model predictive controller, i.e. the LQG control design. The tuning
aims at balancing the ability to reject unmeasured disturbances vs. noise sensitivity while keeping both the closed
loop input and the output variance small. This type of trade off is illustrated on a small example. A series of closed
loop simulations of the MPC has been performed using the third order ARX model as the true system. Three MPC
implementations are tested on a fixed scenario. The total simulation horizon is 250 samples. Between time 50 and
100 a step is introduced in the reference and between 150 and 200 an unmeasured step disturbance is acting on
the system. The input will be constrained between ±1. The models used by the controller are the ARX model, the
∆ARX model and finally the E∆ARX model. The ∆ in model name indicates that the noise is modelled as
integrated white noise. The E∆ARX model use an extended model formulation of the noise and has a free
parameter α which can vary between 0 and 1 in order to span the range between the two other models.
Deterministic simulations
Noisy simulations
ARX-MPC ∆ARX-MPC E∆ARX-MPC(α=0.7)
It is clearly seen on the figure that only the ∆ARX and the E∆ARX models can give offset free tracking. The output
trajectories resulting from these two models are very similar but the E∆ARX model give a less aggressive control
action.
Current research involves extensions and benchmarking of the framework using ARX-based, offset-free
MPC algorithms for a series of examples relevant for the process industry involving multivariable and time delay
systems.
30
1
Comparison of Decentralized Controller and MPC in Control
Structure of a CO2 Capturing Process
Mehdi Panahi, Sigurd Skogestad
Department of Chemical Engineering, Norwegian University of Science and Technology(NTNU), 7491
Trondheim, Norway
Abstract
In a previous study [1], we did the control structure design for a post combustion CO2 capturing process
using Skogestad’s method [2]
Based on the designed structure, in this paper we investigate the performance of the control structure with
large disturbances where some of the manipulated variables saturate.
To handle this kind of disturbances, a MPC has been designed and implemented on parts of the process and
the results show that we achieve good performance. On the other hand, if we implement decentralized PID
controllers, one needs to use the reverse pairing of what would be desired for good performance in order to
avoid instability when we have saturation.
Keywords: Process control, Plantwide control, Self-optimizing method
Introduction
Absorption/Stripping CO2 capturing processes are used in post-combustion part of power plants to remove
CO2 from flue gas streams. A simple typical flowsheet of such processes has been shown in Fig.1
Fig.1- Absorption/Stripping CO2 capturing process [2]
Energy requirements of CO2 capturing processes are relatively very high compare to the net generated
power in the main plant and operating of this process in the lower operation costs is important to save more
energy. To keep this process in the minimum level of energy requirement, a robust control structure is
needed.
We have designed a control structure using Skogestad’s method to omit the necessity of reoptimization of
the process when disturbances happen with an acceptable loss for the energy requirement of the process.
Fig.2 shows the proposed control structure. [1]
31
2
In this structure, recycle amine flowrate is manipulated to remove 90% of the CO2 content in the feed flue
gas and temperature control of tray no.4 (counting from the top) in the stripper (20 trays) has been found as
the best self-optimizing variable to be controlled by using the only unconstrained degree of freedom that is
reboiler duty of the stripper. [1] The other control loops are level controllers or are the loops to control the
variables that have been active during self-optimizing stepwise approach. [3]
Fig. 2- Process flowsheet with proposed control loops using self-optimizing method [1]
One of the main disturbances in this process is changing of the feed flowrate into the absorber. In this paper
we try to investigate and compare the performance of MPC and traditional PID controllers for stability of
the proposed control structure. The MPC controller is designed and implemented using Honeywell
technology that is named as Robust Model Predictive Control Technology (RMPCT) in UniSim process
simulator. [4]
Performance of PID loops and MPC for stabilization of the process
The proposed control structure using decentralized PID controllers gives a stable system as long as the
change in flowrate of the flue gas is less than +15% though saturation of reboiler duty happens when
flowrate of flue gas reaches to +10%. But, when the accession of flowrate passes +15% then temperature of
tray no.4 decreases significantly (Fig.4) and consequently the amount of CO2 at the bottom of stripper starts
increasing and with recycling to the absorber, CO2 is accumulated in the process and makes the plant
unstable. (Fig.5)
32
3
(a)
(b)
(c)
Fig. 3- a: saturation of reboiler duty when the flowrates of flue gas increases to +10%, b: when reboiler duty is
saturated, by increasing feed flow to +15%, temperature of tray no.4 decreases significantly, c: instability of the process
when flowrate of flue gas increases +15%
To avoid instability in our control structure we have considered two different ways: Reverse pairing and
design a MPC.
+10% flowrate of flue gas
+5% flowrate of flue gas
+5% flowrate of flue gas
+10% flowrate of flue gas
+15% flowrate of flue gas
+5% flowrate of flue gas
+10% flowrate of flue gas
+15% flowrate of flue gas
33
4
Reverse pairing
We control the temperature of tray no.4 using recycle amine flowrate and give up controlling of CO2
content in the clean gas when reboiler duty has become saturated. (Fig. 6a and b) Fig. 6b shows that by
using recycle amine flowrate we can control tray temperature no. 4 in its setpoint value.
(a)
(b)
Fig. 6- By reverse pairing and giving up controlling the CO2 content in the clean gas when reboiler duty is saturated,
control structure is stable when flue gas increases +15%
MPC (RMPCT)
A MPC controller is used in order to achieve CO2 composition control in the clean gas stream and
temperature control of tray no.4 in the stripper. In order to keep track of the pressure drop over the stripper,
the difference between bottom stage pressure and top stage pressure is included as an auxiliary CV in the
identification routine. List of CVs, MVs and DVs for this controller are in table 1.
Table 1. List of CVs, MVs and DVs for design a MPC
CVs MVs DVs
CV1- CO2 content in clean gas
CV2- Temperature of tray no.4 in the stripper
CV3- pressure drop in stripper
MV1- Recycle amine flowrate to
absorber
MV2- reboiler duty of the stripper
DV1- change in
flowrate of flue
gas (feed)
Identification of the model and making the required data files for the MPC are made by Profit Design
Studio (PDS) [5] and then the generated files are loaded in UniSim to run the RMPCT.
+15% flowrate of flue gas
34
5
Fig. 7 shows the performance of RMPCT for controlling this process when disturbance with different
magnitude happens. As Fig. 6a shows in +15% change of flowrate of flue gas, RMPCT gives up controlling
of CO2 content in the clean gas while it control tray temperature in its setpoint.
(a)
(b)
(c)
Fig. 7- a: CO2 content in clean gas stream, RMPCT has given up controlling of CO2 content in +15% of the
disturbance, b: Temperature of tray no.4 in the stripper is always in its setpoint c: pressure drop of stripper
+15% flowrate of flue gas +10% flowrate of flue gas
+5% flowrate of flue gas
+5% flowrate of flue gas +10% flowrate of flue gas
+15% flowrate of flue gas
+5% flowrate of flue gas
+10% flowrate of flue gas
+15% flowrate of flue gas
35
6
Conclusion
Performance of RMPCT and decentralized PIDs for stabilization of a designed control structure by selfoptimizing method for a CO2 capturing process has been considered when large disturbances have
happened. Results show that using RMPCT, the control structure will be stable while to keep the process
stable using PID loops, reverse pairings of some MVs and CVs has to be done.
Acknowledge
We would like to acknowledge to Bjørn Einar Bjartnes from Honeywell Company for helping in
implementing of the RMPCT in UniSim.
References
[1] Panahi, M., M. Karimi, S. Skogestad, M. Hillestad, H. F. Svendsen, 2010, Self-Optimizing and Control Structure
Design for a CO2 Capturing Plant, presented in 2nd gas processing symposium, Doha, Qatar, Jan.12-14.
[2] Skogestad, S., 2004, Control Structure Design for Complete Chemical Plants, Computers and Chemical
Engineering, 28, 219-234
[3] Skogestad, S., 2000, Plantwide control: The search for the self-optimizing control structure, Process Control, 10,
487-507
[4] UniSim design R380, Honeywell company
[5] Profit Design Studio (PDS R310) , Honeywell company
36
Potential of Economic Model Predictive �ontrol
for Management of Multiple Power Producers
and �onsumers
Tobias Gybel Hovgaard�, John Bagterp Jørgensen��
�Danfoss A/S, Nordborgvej 81, DK-6430 Nordborg, Denmark
tgh@danfoss�com
��DTU Informatics, Technical University of Denmark, Richard Petersens
Plads, Building 321, DK-2800 Kgs Lyngby, Denmark
jbj@imm�dtu�dk
Keywords : Model Predictive Control, Optimization, Energy Systems, Refrigeration Systems
Economic Model Predictive Control is a receding horizon controller that minimizes an economic objective function rather than a weighted least squares objective function as in Model Predictive Control �MPC). We use Economic MPC
to operate a portfolio of power generators and consumers such that the cost of
producing the required power is minimized. The power generators are controllable power generators such as combined heat and power generators �CHP), coal
and gas fired power generators, as well as a significant share of uncontrollable
power generators such as parks of wind turbines. In addition, some of the power
consumers are controllable. In this paper, the controllable power consumers are
exemplified by large cold rooms or aggregations of super markets with refrigeration systems. We formulate the Economic MPC as a linear program. By
simulation, we demonstrate the performance of Economic MPC for a small conceptual example.
We consider a number of dynamically independent systems that have the
one thing in common that they all influence a quantity that has to meet a set
of constraints. To be more specific we have a portfolio of multiple power generators which delivers a total production for the entire portfolio. On the demand
side a cold room consumes power in order to keep the temperature within certain bounds. Furthermore a reference consumption �which is assumed to be
predictable) combining all other power consumers and non-controllable power
producers like farms of wind turbines is added to the demand side. By adding
the cold room to the optimization problem the potential savings gained by controlling flexible loads on the demand side are revealed. The idea is that the
thermal capacity in the refrigerated goods can be utilized to store ”coldness”
such that the refrigeration system can pre-cool when the energy is free �i.e.
there is an over production from the generators). Thereby a lower than normally required cooling capacity can be applied later, for a period of time when
the energy prices are above zero again. The demands to the temperature in the
cold room are not violated at any time since the same total cooling capacity is
applied though shifted in a more optimal way. We exploit the special property
of the refrigeration system that the dynamics of the temperature are rather slow
The 16th Nordic Process Control Workshop �NPCW’10)
Department of Automatic Control and the Process Industrial Centre, Lund University, Lund,
Sweden, August 25-27, 2010
37
�bstract
while the power consumption can be changed rapidly.
For the power generation and cooling problem we should select the cheapest
possible feasible trajectory of inputs. Since the cost is related to producing the
power, the problem can be stated as:
min
�uk}�k=�
φ =

k=0
c�uk �1)
with:
c = [c1� c2� . . . � cn� 0]
T u = [u1� u2� . . . � un� Te]
T �2)
ui is the input to power plant i, Te is the input to the cold room and ci is the
cost related to ui.
The problem in �1) is subject to:
- system equations:
xk+1 = Axk +Buk + Edk k = 0� 1� . . . � N
yk = Cxk +Duk k = 0� 1� . . . � N + 1
�3)
- limitations on the input:
umin � uk � umax k = 0� 1� . . . � N
Δumin � Δuk � Δumax k = 0� 1� . . . � N
�4)
- cooling capacity must be positive.
- production must be higher than demand at all times.
The last constraint is softened to allow the production to be lower than the
demand such that the we do not end up with an infeasible problem that cannot
be solved. However the penalty on underproduction is selected sufficiently large
such that power demands are met whenever possible.
The cost function and the constraints are formulated as a linear program
�LP) on the form:
min
x φ = g�x �5)
s.t.
Ax ≥ b �6)
where A is a block-angular constraint matrix. The LP in �5) and �6) can be
solved with standard solvers.
Simulations of a small case study with two power plants �a fast and expensive
and a slow but cheaper) and one cold room reveals a potential for significant
savings.
The 16th Nordic Process Control Workshop �NPCW’10)
Department of Automatic Control and the Process Industrial Centre, Lund University, Lund,
Sweden, August 25-27, 2010
2
38
Fault Detection for the Benfield Process using a Parametric Identification
Approach
Johannes P. Maree1, Fernando R. Camisani-Calzolari
Department of Engineering Cybernetics
Norwegian University of Science and Technology
7491 Trondheim, Norway
The Sasol group of companies specialises in diverse fuel, chemical and related
manufacturing and marketing operations. Sasol has interests in oil and gas exploration
and production, in crude oil refining and liquid fuels marketing. The efficient and
economical recovery of carbon dioxide (CO2), used in various processes at Sasol, is
accomplished by utilizing the Benfield process for CO2 extraction. The Benfield process
is a thermally regenerated cyclical solvent process that uses an activated inhibited Hot
Potassium Carbonate solution to remove CO2, H2S and other acid gas components.
The current operating philosophy for the Benfield process is to keep it
simultaneously hydraulic and loaded with CO2, as far as possible, to meet optimal profit
margins from the gas circuit operations. The hydraulic load is defined as the maximum
volume of CO2 that can be processed. Regeneration efficiency has been identified as one
of the major efficiency measures by UOP (Benfield technology licensor). Regeneration
efficiency is a measure of how much steam is required per unit volume of CO2 removed,
and gives an indication of the unit cost, the overall pressure drop in the regeneration
column, and the solution health. This regeneration is directly dependant on the CO2
absorption in the wash columns, which in its turn is again affected by the wash flowrates. Regular foaming and flooding diminishes the efficient CO2 absorption into the
Potassium Carbonate wash solution, resulting in inefficient regeneration. Foaming and
flooding is caused by abnormally high differential pressures on either the top, middle or
bottom bed. Bed differential pressures increase with increasing liquid and gas loads. High
differential pressures that change erratically indicate flooding or foaming. Abnormal high
and stable differential pressures indicate partial blockages in packed beds or liquid/gas
distributors. Abnormal differential bed pressures are classified as multiplicative faults in
abnormal process behaviour, which can be detected by monitoring parameter
fluctuations.
This work proposes a parametric model-based approach to Fault Detection (FD).
The proposed framework combines a parity space approach (subspace SID), to identify
initial process models, with a joint state and parameter estimation method (Extended
Kalman filter), to monitor parameter fluctuations used for FD. The motivation for
subspace SID methods is that these methods have proven to be computational efficient
where no a-priori process knowledge is necessary to estimate a system model. Subspace
SID methods thus allow the user to identify black box models, which can be used to
monitor processes. The challenge in fault detection with subspace methods comes in
how to monitor and evaluate the vast amount of system parameters efficiently and
elegantly. Re-identifying the process using subspace methods, necessary to track
parameter changes, is also not a feasible solution to fault detection. The reason is due to
the vast amount of data samples necessary, which must contain well-excited process
1 This research was conducted by Mr. Johannes P. Maree, for the fulfillment of the degree Master
of Engineering (Electronic Engineering), at the University of Pretoria, South Africa, under the
supervision of Prof. Fernando R. Camisani-Calzolari.
39
dynamics. An elegant solution to the FD problem, using subspace SID methods, would
thus be to identify an initial system process model, using a subspace method, where the
initial identified parameters of the subspace model are re-cursively updated as new data
becomes available. By updating the process parameters, without complete system reidentification, the user is able to track parameter changes, which contributes to the FD
and plant-model mismatch problem.
The proposed subspace SID method is based on two consecutive orthogonal
decompositions to identify a system in closed-loop. The subspace SID method
furthermore guarantees the estimation of stable system matrices by utilizing the shiftinvariant property of the extended observability matrix. Extended Kalman filtering is
used to recursively update a joint set of initial system states and parameters, using current
sampled process data and initial estimated parameters, obtained via the subspace method.
To detect a fault in the updated parameter set, it is necessary to compare the updated
parameter set, with the initial identified parameter set. This can be accomplished by
considering appropriate matrix measures, which will accentuate any fundamental
parameter matrix differences. The infinity matrix norm is proposed for detecting
discrepancies between the absolute difference of the initial identified, and recently
updated parameter set. The infinity matrix norm is used to reduce (averaging) the matrix
of differences to one value which is monitored against a threshold value.
40
Diagnosis of oscillation due to multiples sources using wavelet
transforms
Selvanathan Sivalingam and Morten Hovd
�bstract—Control loop oscillations are a common type of
plant-wide disturbance and the root-causes can be one or
more among poorly tuned controllers, process or actuator nonlinearities, presence of model plant mismatch and oscillatory
disturbances. The oscillations are the most prominent indications of deteriorated controller performance. This article
addresses the detection and diagnosis of oscillations in measurements due to multiple sources under the framework of internal
model control. A pattern recognition based approach using
cross wavelet transforms is proposed to pinpoint the source�s)
of oscillation in the control loop. The phase information in
wavelet domain between input and output signals is exploited
to diagnose the source of oscillation.
I. INTRODUCTION
It is well known that performance degradation in control
loops manifest as one or more of the following: (i) poor
set point (SP) tracking (ii) oscillations (iii) poor disturbance
rejection and (iv) excessive final control element variation.
Industrial surveys over the last decade indicate that only
about one-third of industrial controllers provide acceptable
performance and about 30� [1], [2], [3] of the control loops
exhibit oscillation. Since oscillations can lead to loss of
energy, isolating the root cause for oscillations is important
for improving the performance of the oscillating control
loops. Also, the presence of the oscillations in a plant
increases the variability of the process variables and thus may
cause loss of product quality. The oscillations (or vibrations),
in general, are a very drastic form of plant performance
degradation, which can, in many cases, be induced by the
feedback mechanism itself.
Oscillations are attributed to one or more among poor controller tuning, process or actuator non-linearities, presence
of model plant mismatch or oscillatory disturbances. A tool
to help the engineer should therefore automatically bring
oscillatory loops to his or her attention, characterize them
and highlight the presence of plant wide oscillations. Several
authors have addressed the detection of oscillatory measurements in process data. Early works appear in [4] followed by
[5], [6], [7], [8]. [4] proposed a technique to detect oscillating
loops “on-line” using the IAE criterion. This method does
not assume any particular shape for oscillation and only
requires the measurement to deviate significantly from the set
point. [4] also proposed a diagnostic procedure for finding
the source of oscillation and eliminating it. The diagnostic
Selvanathan Sivalingam and Morten Hovd are with Department of Engineering Cybernetics, Norwegian University
of Science and Technology, Norway 7491. Email addresses:
selvanathan�sivalingam@itk�ntnu�no and
morten�hovd@itk�ntnu�no
procedure is carried out by disconnecting the feedback (i.e.
switching the controller to manual mode). This approach is
simple and efficient and probably the most comprehensive
procedure available for diagnosing root cause for oscillations.
However, switching the controller to manual mode may not
always be allowed, especially if the loop is deemed critical.
Further, it will not be possible to apply this approach on
thousands of loops in a routine fashion. [5] presented an
offline technique for detecting oscillation using a regularity
factor. This method requires the user to specify the rootmean-square value of the noise and a thresholds a nontrivial
task when applied to hundreds of loops.
[5] and [9] proposed a set of procedures to detect and
diagnose oscillating loops using offline data. They combine
the techniques of controller performance assessment along
with operational signatures (OP-PV plots) and spectral analysis of the controller error for diagnosis. This technique,
though not completely automated, can distinguish the cause
of oscillation as one of the following: (i) poor tuning
(ii) nonlinearity or (iii) external disturbance. However, the
downside lies in manually inferring the loop signatures that
are based on spectral analysis or on a map of controller
output (OP) versus process variable (PV) and isolating the
oscillating portion from the entire data. [10] presented a
simple, practical approach to distinguish oscillating loops
that are caused by external disturbances and static friction.
This approach is based on cross-correlation between the
controller output (OP) and process output (PV). The crosscorrelation technique failed when the data had intermittent
oscillations and the set-point was also changing. [11] also
proposed a technique to identify stiction using nonlinear
filters. The method assumed that information such as mass
of stem, diaphragm area, and so on for each valve is readily
available. Since in a typical process industry facility there can
be thousands of control loops, it may be nearly impossible to
build/maintain a knowledge base of control valves, making
this technique difficult to implement.
[12] used higher order statistics for detecting nonlinearity
in data and have extended the method for diagnosing stiction
by fitting an ellipse of the OP-PV plot and inferring the
stiction from an assumed stiction model. The success of this
approach lies in correctly identifying the oscillation period
and its start and end point in the OP-PV data. [8] proposed
non-negative matrix factorization for detection and diagnosis
of plant-wide oscillations based on source separation techniques. As can be seen, the task of detecting stiction or
other nonlinearities in valves from routine operating data is
a challenging task. To summarize, data driven techniques
41
Fig. 1. Schematic representation of internal model control with actuator
that are presented in the literature till date are useful in (a)
assessing the performance of the controller by calculating a
figure of merits given that the cause of poor-performance is
only due to either an aggressive or sluggishly tuned controller
in pure feedback control, (b) detecting oscillating loops with
an user-specified parameter, and (c) limited diagnosis of
the cause of oscillation based on cross-correlation, power
spectral analysis, or OP-PV plots. The current approaches
lack (a) the capability to efficiently diagnose oscillations due
to multiple sources, (b) the ability to diagnose the causes
of time-varying oscillations and (c) an automated means of
oscillation diagnosis.
In this work, we have attempted to address some of
the aforementioned drawbacks by using wavelet and cross
wavelet transforms.
II. PROBLEM STATEMENT
Oscillations in model based control loops occur due to
either one of (i) valve stiction (ii) model plant mismatch, (iii)
external oscillatory disturbances or combination of any of
these. It becomes vital to diagnose the causes of oscillations
in order to take the appropriate remedial action. A procedure
based on pattern recognition techniques using cross wavelet
transform is devised in this article to diagnose the cause(s)
of the oscillation. The problem is setup in the internal model
control (IMC) framework (Figure 1).
III. PROPOSED METHODOLOGY
Cross wavelet transform of input and plant and that of
input and model output are computed and thereby a specific
pattern is sought for root cause diagnosis of oscillation
using the direction of wavelet phase difference between the
variables.
To illustrate the idea of cross-wavelet transform for an
input-output system, an open-loop process with Gp�s) =
1
10s+ 1
is considered. The process is simulated for a sinusoidal input having two frequencies and the time domain
plots of input and output are given in Figure 2. The cross
wavelet transform plot between two quantities u and y is
shown in Figure 3.
0 100 200 300 400 500 600 700 800 900 1000
−1
−0.5
0
0.5
1
O
u tp u t  (y )
Open loop system
0 100 200 300 400 500 600 700 800 900 1000
−1
−0.5
0
0.5
1
Time (samples)
In p u t  (u )
Fig. 2. Time domain behavior of input and output signals considered for
interpretation of wavelet analysis
Fig. 3. Cross wavelet transform between input and output signals
It is known from Figure 3 that the quantities u and y
show high common power at two frequencies between two
different time intervals (0.1 Hz, 0-511 and 0.2 Hz, 512-1024)
and the arrows indicate the direction of the wavelet phase
between u and y i.e.� u leads y by 90° (pointing down).
Based on the properties of cross wavelet transform, wavelet
phase difference and linear time invariant systems theory, the
following methodology is proposed to diagnose the source(s)
of oscillation in a control loop.
The quantities controller output (u), process output (y) and
model output (ym) of an oscillating control loop are obtained
either from simulation or from industry. The cross wavelet
transforms, Wuy�f� τ) and Wuy��f� τ) are computed. By
comparing the direction of wavelet phase, the following
conclusions can be drawn.
� If the oscillating source is only due to valve stiction, the
cross wavelet transform plots should not only exhibit
harmonics but also discontinuities.
� If the source is due to gain mismatch, the plots
Wuy�f� τ) and Wuy��f� τ) of should be identical since
the phase spectrum is independent of any changes
in gain. The arrows in the plots of Wuy�f� τ) and
Wuy��f� τ) will be in same direction.
� If the source is due to delay mismatch, the plots of
Wuy�f� τ) and Wuy��f� τ) will vary in phase direction
since the phase spectrum depends on delay changes.
42
IV. SIMULATIONS
A control system consisting of a process characterized
by the transfer function Gp =
Kp τps+ 1
e−�psand model
Gm =
Km τms+ 1
e−��s is simulated with IMC controller for
a unit step change in the set point. The different case studies
analyzed for the diagnosis of oscillation in a control loop
are (i) oscillation due to valve stiction (ii) oscillation due
to valve stiction and oscillatory disturbance (iii) oscillation
due to gain mismatch (iv) oscillation due to gain mismatch
and oscillatory disturbance and (v) oscillation due to delay
mismatch.
A simple yet efficient one parameter model proposed by
[4] is used to generate oscillations due to valve stiction. The
model is
x�t) =

x�t− 1) �u�t)− x�t− 1)� ≤ d
u�t) otherwise
(1)
Here u�t) and x�t−1) are present and past valve outputs,
u�t) is the present controller output, and d is the valve
stiction band. The valve stiction band is expressed in terms of
the percentage or fraction of valve movement corresponding
to the amount of stiction present in the valve. For instance, if
100 units of force are required to open the valve completely
from completely closed position and 10 units of force is
required to overcome the amount of static friction in the
valve, stiction band is 10� or 0.1. The stiction band of 0.1
is used in the simulation. Model plant mismatch is introduced
by changing the values of gain or delay appropriately in the
process. The sinusoidal disturbance of frequency 0.01 Hz
is considered for the simulation. The time domain plots of
controller output (u), plant output (y) and model output (ym)
for different simulation studies is shown in Figures 4, 5, 6,
7 and 8.
The cross wavelet transform computed between controller
output and plant output is compared with that computed
between controller output and model output. In the case of
oscillation due to valve stiction (Figure 4), the plots of cross
wavelet transform (Figures 9 & 10) not only show harmonics
but also discontinuities which are the characteristics of a
sticky valve. Figures 11 and 16 clearly indicate the presence
of the valve stiction as one of the sources of oscillation
between 800 and 1600 s and the other being the oscillatory
component of frequency 0.01 throughout.
If the oscillation is only due to MPM, there will be clearly
a single frequency in the cross wavelet transform plot. In the
case of gain mismatch (Figure 6), the plots of cross wavelet
transform between controller output and plant output and
controller output and model output (Figures 13&14) produce
identical plots since the phase spectrum is independent of the
changes in gain.
The control loop whose time domain trends are characterized by Figure 7 is diagnosed to have gain mismatch as one
of the sources of oscillation between 800 and 1600 s and
other being the oscillatory component of frequency 0.01 Hz
0 200 400 600 800 1000 1200 1400 1600 1800
0
0.5
1
1.5
P
la n t  o u tp u t  (y )
Valve stiction
0 200 400 600 800 1000 1200 1400 1600 1800
0
0.5
1
1.5
M
o d e l  o u tp u t  (y m )
0 200 400 600 800 1000 1200 1400 1600 1800
0
2
4
6
C
o n tr o ll e r  o u tp u t  (u )
Time (samples)
Fig. 4. Time domain behavior of plant� model and controller outputs for
the valve stiction as the source of oscillation.
(Figures 15 & 16). The presence of oscillatory component
can not be due to the presence of delay mismatch since the
presence of gain and delay mismatch at a same time will lead
to system instability. Hence, the loop can be said to have the
external oscillatory disturbance over the entire period and
gain mismatch between 800 and 1600 s as the sources of
oscillation.
The control loop whose outputs are given in Figure 8
is analyzed for diagnosing the source(s) of oscillations.
Figures 17 and 18 indicate the presence of a single frequency
component and a directional change in the phase difference.
The source of oscillation can be either oscillatory disturbance
or delay mismatch.
V. CONCLUSIONS
A pattern recognition technique for the diagnosis of
control loop oscillations in internal model control systems
due to multiple sources using cross wavelet transform of
two quantities has been developed. A diagnostic study of
oscillation due to either one of valve stiction, model plant
mismatch, oscillatory disturbance or combination of these
has been presented. The oscillations due to valve stiction
manifest as harmonics as well as discontinuities in the cross
wavelet transform plots whereas oscillation due to model
plant mismatch leaves distinct signatures in the phase information (arrows). If the oscillations are due to gain mismatch,
no change is observed in the phase spectrum computed
between controller output and plant output and controller
output and model output. On the other hand, oscillation due
to delay mismatch or oscillatory disturbance results in a
directional change in the phase difference computed between
controller output and plant output and controller output and
model output.
Further study to distinguish between the oscillatory disturbance and delay mismatch as sources of the oscillation
is currently underway. In a parallel study, it is found in
our preliminary extensions of this work that the results
demonstrated here can be applied to a more generalized
problem of the diagnosis of poor control loop performance.
43
0 200 400 600 800 1000 1200 1400 1600
0.9
1
1.1
1.2
P
la n t  o u tp u t  (y )
Oscillatory disturbance and valve stiction
0 200 400 600 800 1000 1200 1400 1600
0.9
1
1.1
1.2
M
o d e l  o u tp u t  (y m )
0 200 400 600 800 1000 1200 1400 1600
0.8
1
1.2
1.4
C
o n tr o ll e r  o u tp u t  (u )
Time (samples)
Fig. 5. Time domain behavior of plant� model and controller outputs for
the case oscillatory output and valve stiction as the sources of oscillation
0 200 400 600 800 1000 1200 1400 1600 1800
−1
0
1
2
3
P
la n t  o u tp u t  (y )
Gain mismatch
0 200 400 600 800 1000 1200 1400 1600 1800
−0.5
0
0.5
1
M
o d e l  o u tp u t  (y m )
0 200 400 600 800 1000 1200 1400 1600 1800
−5
0
5
C
o n tr o ll e r  o u tp u t  (u )
Time (samples)
Fig. 6. Time domain behavior of plant� model and controller outputs for
the case of gain mismatch as the source of oscillation.
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[2] L. Desborough and R. Miller, “Increasing customer value of industrial control performance monitoring: Honeywell’s experience,” Proc.
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[3] D. Ender, “Process control performance: not as good as you think,”
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[7] R. Rengaswamy, T. Hägglund, and V. Venkatasubramanian, “A qualitative shape analysis formalism for monitoring control loop performance,” Engineering Applications of Artificial Intelligence, vol. 14,
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[10] A. Horch, “A simple method for the detection of stiction in control
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control valves,” in On-line-fault detection and supervision in the
chemical process industry. Lyon, France: IFAC Workshop, 1998,
p. 4B.
0 500 1000 1500
−1
0
1
2
3
P
la n t  o u tp u t  (y )
Gain mismatch
0 500 1000 1500
−1
0
1
2
3
Oscillatory disturbance and gain mismatch
0 500 1000 1500
−0.5
0
0.5
1
M
o d e l  o u tp u t  (y m )
0 500 1000 1500
−1
0
1
2
0 500 1000 1500
−5
0
5
Time (samples)
C
o n tr o ll e r  o u tp u t  (u )
0 500 1000 1500
−5
0
5
Time (samples)
Fig. 7. Time domain behavior of plant� model and controller outputs for
the case of oscillatory disturbance and gain mismatch as the sources of
oscillation.
0 200 400 600 800 1000 1200 1400 1600 1800
−1
0
1
2
3
P
la n t  o u tp u t  (y )
Delay mismatch
0 200 400 600 800 1000 1200 1400 1600 1800
−1
0
1
2
3
M
o d e l  o u tp u t  (y m )
0 200 400 600 800 1000 1200 1400 1600 1800
−2
0
2
4
6
C
o n tr o ll e r  o u tp u t  (u )
Time (samples)
Fig. 8. Time domain plots of plant� model and controller outputs for the
case of delay mismatch as the source of oscillation.
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44
Fig. 9. Cross wavelet transform plot between � and yp when the oscillation
is only due to valve stiction.
Fig. 10. Cross wavelet transform plot between � and y� when the
oscillation is only due to valve stiction.
Fig. 11. Cross wavelet transform plot between � and yp when the
oscillation is due to oscillatory disturbance and valve stiction.
Fig. 12. Cross wavelet transform plot between � and y� when the
oscillation is due to oscillatory disturbance and valve stiction.
Fig. 13. Cross wavelet transform plot between � and yp when the
oscillation is gain mismatch
Fig. 14. Cross wavelet transform plot between � and y� when the
oscillation is gain mismatch
45
Fig. 15. Cross wavelet transform plot between � and yp when the
oscillation is due to oscillatory disturbance and gain mismatch.
Fig. 16. Cross wavelet transform plot between � and y� when the
oscillation is due to oscillatory disturbance and gain mismatch..
Fig. 17. Cross wavelet transform plot between � and yp when the
oscillation is due delay mismatch..
Fig. 18. Cross wavelet transform plot between � and y� when the
oscillation is due delay mismatch..
46
Availability Estimations for Utilities in the Process Industry
Anna Lindholm
Automatic Control, Lund
Hampus Carlsson
Perstorp AB
Charlotta Johnsson
Automatic Control, Lund
I. INTRODUCTION
An important performance measure of a plant is the
plant-availability. The higher availability the better, since a
high availability implies a possibility for a large production
volume and thereby an increased profit for the company. One
way of increasing the plant-availability is by eliminating,
or minimizing the effect of disturbances. The cause of a
disturbance can be personnel, material or equipment, where
material includes both raw materials and utilities.
The aim of this work is to increase the plant-availability by
decreasing the effects of plant-wide disturbances caused by
utilities. The first step is to determine the set of utilities that
can be present at an industrial site, what disturbances these
utilities can suffer, and how frequent and safety-critical these
disturbances are. A later step will be to determine the effects
on the plant-availability, and ways to decrease or eliminate
these effects.
The research is performed within the framework of PICLU (Process Industrial Centre at Lund University) supported
by the Foundation of Strategic Research (SSF).
II. UTILITIES
Utilities, in opposite to raw materials, are materials that
are used plant-wide and are crucial for plant operation but
are not part of the final product. Common utilities are
• Steam: The steam net is commonly used to supply
energy for distillation. Other uses are to supply energy
for endothermic reactions and to heat a reactor at startup. There could be several steam nets at the same site,
for example one net with high pressure steam and one
with low pressure steam.
• Cooling water: The cooling water system is used for
cooling at exothermic reactions and in the condensing
phase of distillation.
• Electricity: Electricity is needed in order for the instruments, e.g. pumps, to operate.
• Water treatment: A Water treatment utility is used for
purification of process water, precipitation and ground
water.
• Combustion of tail gas: A system for combustion of
tail gas, such as a flare, is a safety device needed at
unforeseen events.
• Nitrogen: Nitrogen is needed to maintain pressure in
vessels.
• Feed water: Feed water is used in boilers to produce
steam.
• Instrument air: Instrument air is needed for the pneumatic instruments to work.
• Vacuum system: Vacuum is used to lower the boiling
point of a liquid to facilitate distillation and to remove
gas produced in reactions.
A flowchart can be made for each of the utilities, showing
how the utility flows through the areas of the production site.
An example is showed in Figure 1.
Fig. 1. Example of a utility flowchart for steam. The site contains 6
areas, A-F, and has two steam nets producing high and low pressure steam
respectively.
A utility could suffer from different disturbances. For
example, a steam net could suffer disturbances such as too
high or too low steam pressure. One way of defining when
a disturbance on a utility occurs is to set limits, such that if
the parameter goes outside this limits, the disturbance will
have economical or safety consequences.
III. AVAILABILITY
The availability of a production unit is according to
ISO22400, draft 1, the fraction of the main usage time,
which is the producing time of the unit, and the planned
busy time, which is the time that the production unit is used
for the execution of a manufacturing order. The availability
of a utility can be estimated by taking the fraction of time
when the utility does not suffer a disturbance over the total
time. If measurements are available of the key parameters
that define the disturbance, the availability can be computed
directly from historical data.
IV. FUTURE WORK
When availabilities for the different utilities have been
computed, the consequences of each disturbance must be
evaluated. When both frequency and severity of all disturbances on utilities are known, focus on handling the
disturbance with highest severity×frequency-factor to improve availability of the entire production plant as much as
possible.
47
Detection and Isolation of Oscillations Using the Dynamic �ausal
Digraph Method
Tikkala� Vesa-Matti�; Zakharov� Alexey; Jämsä-Jounela� Sirkka-Liisa
�alto University School of Science and Technology
Department of Biotechnology and Chemical Technology
Abstract
This paper proposes a modification to the dynamic
causal digraph (DCDG) method in order to address
the detection and isolation of oscillations in a process.
The proposed detection method takes advantage of the
properties of residual signals generated by the DCDG
method by studying their zero-crossings. The method
is tested in an application to a board making process
and the results are presented and discussed.
1 Introduction
Demands to keep industrial processes running efficiently with a high rate of utilization are increasing
constantly due to the tightening global competition.
Since, the modern industrial processes are complex and
large-scale, operator-based monitoring cannot guarantee early enough detection and reliable diagnosis of the
faults and abnormalities. Therefore, the detection and
diagnosis of different abnormal and faulty conditions in
the processes have become increasingly important.
Common problems causing inefficient operation and
production losses in the process industry are oscillations. Oscillatory disturbances readily propagate in
the process and cause extensive variation in the process variables. The oscillations are usually originated
under feedback control, and they may have various
causes which have been categorized by Thornhill �
Horch (2007) into non-linear and linear causes. Nonlinear causes include for example extensive static friction in the control valves, on-off or split range control,
sensor faults, process non-linearities and hydrodynamic
instabilities. The most common linear causes are poor
controller tuning, controller interaction and structural
problems involving process recycles (Thornhill � Horch,
2007). According to Choudhury et al. (2008), valve
stiction is, however, the most common cause of these
oscillations in control loops.
The detection and diagnosis of oscillations have been
�Corresponding author: vesa­matti.tikkala@tkk.fi� Aalto
university� PL 16100� FI-00076 Aalto
previously addressed by data-based methods which
study, for example, the properties of controller error signals (Thornhill � Hagglund, 1997; Forsman � Stattin,
1999), the spectral properties (Thornhill et al., 2003)
or the nonlinearity of the measurement signals (Choudhury et al., 2004; Thornhill, 2005). Also, a variety
of multivariate methods, such as principal component
analysis (Thornhill et al., 2002) and non-negative matrix factorization (Tangirala et al., 2007), have been applied to solve this diagnosis task. A recent trend has
been to introduce process information into the diagnosis of plant-wide oscillations. Applications in which the
process connectivity information has been integrated
into data-based analyses to check hypotheses on the
fault origin, have been presented (Yim et al., 2007;
Jiang et al., 2009).
This paper aims at further development of the dynamic causal digraph (DCDG) method by addressing
the detection and isolation of plant-wide oscillations.
A detection algorithm, which is able to deal with oscillatory residuals, is proposed and integrated into the
DCDG method. In this paper, the modified DCDG
method is used to detect and isolate low-frequency oscillations caused by a valve stiction fault in a board
machine process.
The paper is organized as follows. In Section 2, the
dynamic causal digraph method and the new detection
algorithm are introduced. The process and the test environment are described in Section 3. The results of
the testing are presented in Section 4 followed by the
conclusions in Section 5.
2 Enhanced DCDG Method for the De­
tection and Isolation of Oscillations
The dynamic causal digraph method employs the process knowledge formalized as a causal digraph model in
order to perform the ordinary fault diagnosis tasks as
presented by Cheng et al. (2010); Cheng (2009). In the
enhanced DCDG, the detection of faults is performed
using the proposed method which observes the zerocrossings in the residuals generated by a comparison
48
of cause-effect models and the process measurements.
Next, the isolation is carried out by applying a set of
inference rules to the residuals in order to extract the
fault propagation path. Finally the arcs in the digraph
that explain the faulty behavior are identified. The enhanced DCDG method is described in more detail in
the following.
2.1 Fault detection
Fault detection is performed in two steps: residual generation and fault detection in the residuals using the
modified cumulative sum (CUSUM) algorithm.
2.1.1 Residual generation with dynamic models
The dynamic causal digraph produces two kinds of
residual to be used in fault detection and isolation:
global (GR) and local residuals (LR). The global residual is produced from the difference between the measurement and the global propagation value:
GR(Y ) = Y (k)− Ŷ (k)� (1)
where Y (k) is the measurement and Ŷ (k) is the global
propagation value obtained by
Ŷ (k) = fY

Û(k − 1)� Û(k − 2)� . . .

� (2)
where fY is a discrete-time model describing the causeeffect relationship from n predecessor nodes Ui to node
Y . Û(k−τ) = �û1(k−τ)� . . . � ûn(k−τ)} are the lagged
global propagation values from the predecessors with
time lags τ = 1� 2� . . . depending on the system order.
The local residuals are subcategorized into three
types: individual local residuals (ILR), multiple local
residuals (MLR) and total local residuals (TLR) (Montmain � Gentil, 2000).
The individual local residual is produced by taking
the difference between the measurement and the local
propagation value with only one measured input, while
all the others are propagation values from the parent
nodes:
ILRmY = Y − Ȳ � (3)
Ȳ (k) = fY

Ū(m� k − 1)� Ū(m� k − 2)� . . .


where
Ū(m� k − τ) =

ūi(k − τ)





ūi(k − τ) =

ûi(k − τ)� i �= m
ui(k − τ)� i = m
� 1 ≤ i ≤ n

� (4)
ûi(k) is the lagged global propagated value from the
predecessors, and ui(k − τ) is the measurement for the
i-th parent node.
Similarly, the MLR
P l

Y is produced as
MLR
P l

Y = Y − Ȳ � (5)
Ȳ (k) = fY

Ū(P lY � k − 1)� Ū(P
l Y � k − 2)� . . .


where
Ū(P lY � k − τ) =

ūi(k − τ)





ūi(k − τ) =

ûi(k − τ)� i /∈ P
l Y
ui(k − τ)� i ∈ P
l Y
� 1 ≤ i ≤ n

� (6)
P lY is the set of indices of the predecessors which use
the measurement as an input. The TLR(Y ) is produced
with P lY = PY , where PY is the set of indices of all the
predecessors of Y .
The residual generation scheme follows the DCDG
method developed in (Montmain � Gentil, 2000).
2.1.2 Fault detection using the modified CUSUM
method
The proposed detection method utilizes the cumulative
sum (CUSUM) method presented by Hinkley (1971),
by applying it to the detection of a change in the mean
and variance of the zero-crossings in the residual signals.
The CUSUM algorithm is defined for a positive change
as follows:
U0 = 0
Un =
n� k=1
d(k)− µ0 −
β
2
� (7)
mn = min
0≤k≤n
Uk�
where β is a user-specified minimum detectable change,
d(k) the observed signal with nominal mean value equal
to µ0. Whenever Un −mn > λ, a change is detected,
where λ is a design parameter, usually tuned according
to the requirements for the false alarm and missed alarm
rates.
The signal observed by the CUSUM algorithm, called
the detection signal, is defined as follows
d(k) = max

Δ̄t(k)�
σ2Δt(k)
Δ̄t(k)

� (8)
where max�·}-operator takes the maximum of its arguments, Δ̄t(k) and σ2Δt(k) are the mean and the variance of the time between consecutive zero-crossings in a
residual, respectively. Both Δ̄t and σ2Δt are calculated
in a moving window of length l: [e(k − l)� e(k)], where
e(k) is the residual.
In normal operation, when the residuals are assumed
to be zero-mean Gaussian noise, Δ̄t� σ2Δt(k) ≈ 2, since
49
Table 1: Fault isolation rules of the dynamic causal digraph
CU(GR(Y )) CU(TLR(Y )) CU(ILR� (m)) CU(ILR� (i)) CU(MLR� (P�)) CU(MLR� (P2)) Decision
0 0 0 0 0 0 No fault
1/-1 0 0� 1/-1� 0� 1/-1� Fault propagates from the parent
node m
1/-1 0 1/-1�� 1/-1�� 1/-1�� 0�� Fault propagates from the nodes with
subscript P2
1/-1 1/-1 1/-1 1/-1 1/-1 1/-1 Local fault on variable Y
� �i �= m� i ∈ P� �m ∈ P��m /∈ P2� P� is the set of subscripts of parent nodes of the node Y .
�� �i�m� i ∈ P� �m ∈ P� � �P�� P2 ⊆ P� .
the probability of e(t) having a different sign than
e(t − 1) is 0.5 for all t. Therefore, the nominal mean
value of the observed signal in (7) can be set as µ0 = 2
and β and λ are then tuned to obtain robust detection
with minimal false alarms. The window length l must
be selected to be larger than one half of the expected
period in the residual.
2.2 Fault isolation
2.2.1 Isolation of the fault propagation path
Fault isolation is performed recursively for the detected
nodes by using a set of rules. These isolation rules,
developed by Montmain � Gentil (2000), are converted
into a table for the convenience of implementation, as
shown in Table 1. After the isolation the nature of the
fault is determined by using rules in Table 2.
Table 2: Fault nature rules of the dynamic causal
digraph
CU(GR(X))� CU(TLR(X)) Fault nature
1/-1 1/-1 Local fault for that child node
1/-1 0 Process fault for the faulty node
0 1/-1 Measurement fault for the faulty
node

� is the subscript of any child node of the node Y .
2.2.2 Isolation of the faulty process component
In the case of a process fault, in addition to locating
the fault on the variables (nodes), locating it on the
arcs is also desirable. However, the MISO structure
of the digraph causes problems by generating multiple
possible results as 2n− 1� n ≥ 1, where n is the number
of input arcs of the fault origin node(s).
In order to decrease the number of possible results,
an inference mechanism between the arcs proposed in
(Cheng et al., 2008) is used. The inference mechanism
is based on an inter-arc knowledge matrix M defined
for node U as follows
MU (i� j) =



1� if inconsistency in arc �U� i�
causes inconsistency to �U� j�
0� otherwise�
(9)
where i and j refer to the matrix rows and columns,
respectively.
Next, each set of suspected arcs is tested in order to
determine whether the fault may be caused exactly by
the current set of arcs. In order to do it the matrix M
is multiplied with a vector representing the suspected
arc set, which is defined as follows
sv(i) =

1� if ARC(M� i) ∈ S� 1 ≤ i ≤ Na
0� otherwise�
(10)
where ARC(M� i) gives the arc corresponding to the
ith row in the matrixM. S is the set of suspected arcs.
If the number of non-zero elements of sv have changed,
the current suspected set of arcs must be excluded.
3 Description of the Process and the
Valve Stiction Faults
This test focuses on the stock preparation of the board
machine at Stora Enso’s mills in Imatra, Finland. The
simulation tests are run on a board machine simulator
model in the APROS simulation environment.
3.1 The Board Machine Process
The board making process begins with the preparation
of raw materials in the stock preparation section, as
shown in the flowsheet in Figure 1. Different types
of pulp are refined and blended according to a specific
recipe in order to achieve the desired composition and
properties for the board grade to be produced. The
consistency of the stock is controlled with dilution water.
The blended stock passes from the stock preparation
to the short circulation. First, the stock is diluted in
the machine chest to the correct consistency for web
formation. The diluted stock is then pumped with a
fan pump, which is used to control the basis weight of
the board, to cleaning and screening. Next, the stock
passes to the head box, from where it is sprayed onto
the wire in order to form a solid board web.
The excess water is first drained through the wire and
later by pressing the board web between rollers in the
50
Figure 1: Flowsheet of the stock preparation of the board machine process.
Table 3: Variables of the causal digraph model
for the stock preparation of the board machine.
Var. Description Type Unit
vb valve opening for the broke line A fb mass flow of the broke M kg/s
vbd dilution water valve opening for the broke line A fbd dilution water flow for the broke line E kg/s
cb broke consistency M �
rp pine pump rotation speed A �
fp mass flow of the pine stock M kg/s
vpd dilution water valve opening for the pine line A cp pine consistency M �
rc CTMP pump rotation speed A �
fc mass flow of the CTMP M kg/s
vcd dilution water valve opening for the CTMP line A cc CTMP consistency M �
vmcd dilution water valve opening for the machine
chest
A -
cmc consistency before the machine chest M �
pp pressure before the pine valve M kg/s
pc pressure before the CTMP valve M kg/s
ct consistency of the machine chest M �
A: Actuator signal, M: Measurement signal
press section. The remaining water is evaporated off in
the drying section using steam-heated drying rolls.
The variables used in the causal digraph model of the
stock preparation are listed in Table 3.
3.2 Valve Stiction Faults
A control valve is the most common final control element used in the process industry (Choudhury et al.,
2008). Therefore, the diagnosis of faults in valves is of
great importance. Stiction, short for static friction, is a
problem in control valves since it can cause significant
disturbances in the process variables. A valve suffering
from excessive stiction sticks when the control signal,
for example, changes the direction and does not move
until the force required to move the valve shaft exceeds
a certain limit. When the valve starts to move, it jumps
and then follows the control signal before it sticks again.
A sticking valve is likely to cause oscillations when it is
involved in a control loop.
The stiction in valves has been modelled and studied e.g. by Stenman et al. (2003) and Choudhury et al.
(2005). This paper considers a stiction fault in a pressure control valve which causes the control loop to oscillate and disturbs the operation of the plant.
4 Testing and Results
4.1 Simulation Environment and Fault Simula­
tion
The Imatra board machine model was developed by
Stora Enso and VTT in the APROS environment. It
was originally constructed on the basis of modeling
and simulation studies carried out during 1998–2002 for
Stora Enso’s Imatra mills. It has been previously used
for grade change simulations and in studies reported by
Lappalainen et al. (2003).
A valve stiction fault was simulated in the stock
preparation part of the board machine using the
APROS board machine model. The faulty valve is located in the CTMP line and is used to control the feeding pressure pc of the blend chest. The two-parameter
data-driven valve stiction model proposed by Choudhury et al. (2005) was implemented in the APROS simulation software for the simulation. The deadband and
slip-jump parameters of the stiction model were set to
S = 0.06 and J = 0.06 respectively. The fault was
evoked by a step change to the setpoint of pc.
The fault occurring at t = 200 causes an oscillation
51
0 100 200 300 400 500 600 700 800 900
ct cmc
cb cc cp vmcd
fb vb vbd
fc fp vb rc pc vcd
pp rp vpd
Figure 2: Normalized process variables during the
fault simulation.
with a period of approximately 120 samples, which affects most of the variables in the stock preparation.
Figure 2 shows the measured variables during the fault
simulation and it demonstrates clearly the effect of the
fault in the process.
4.2 Fault Detection and Isolation Results
First, the global residuals for all variables were produced by comparing the measured values of the variables and the estimates generated using the dynamic
causal digraph model. Then, the detection signals were
produced by calculating the mean and the variance of
zero-crossings in the global residuals. The proposed detection method was applied to analyse the residuals in
order to detect the faulty nodes. The parameters of
the modified CUSUM method were set to the following: β = 10, λ = 4 and l = 200. The global residuals,
detection signals and the detection results for variables
fc and cc are presented in Figure 3. The fault is detected in both signals GR(fc) and GR(cc). The change
in the detection signal dfc(k) was detected for the first
time at k = 203, three time instants after the fault
occurred. However, the detection result is not reliable
until k = 270. The global residual of cc is detected later
at k = 315.
Local residuals were generated in order to carry out
the inference to isolate the origin of the fault. The
local residual, the detection signal and the detection
results of cc are shown in Figure 4. The detection signal changes slightly after the fault occurrence, but no
detection is however made.
The performance of the proposed detection method is
satisfactory. The faults are detected with a reasonable
delay and no false alarms are generated. Detection in
variable cmc, based on the structure of the process and
the forecast of the fault propagation, was also expected.
0 50 100 150 200 250 300 350 400
cc fc G
lo b a l r
e si d u a ls Samples
0 50 100 150 200 250 300 350 400
0
50
Samples
D
e te ct io n  s
ig n a ls fc cc 0 50 100 150 200 250 300 350 400
0
0.5
1
Samples
D
e te ct io n  r
e su lt fc cc Figure 3: Global residuals GR(fc) and GR(cc),
detection signals and the detection results
0 100 200 300 400 500 600 700 800
−10
0
10
T
L
R
(c c) 0 100 200 300 400 500 600 700 800
2
3
4
Samples
D
e te ct io n  s
ig n a l cc 0 100 200 300 400 500 600 700 800
0
0.5
1
Samples
D
e te ct io n  r
e su lt cc Figure 4: Local residual TLR(cc), detection signal and the detection results
However, based on the simulation studies, it was found
out that the effect of the fault attenuates and therefore the change in the global residual of cmc becomes
undetectable.
The fault isolation rules presented in Table 1 were
applied in order to extract the fault propagation path
and the fault origin. The fault origin was located at the
node fc. The nature of the detected fault is diagnosed
as a process fault according to the rules presented in
Table 2.
Since the fault was a process fault, the structure of
the digraph model gave three possible sources for the
fault: vcd, pc and rc resulting (3
2 − 1) = 8 possible sets
of arcs explaining the faulty behaviour . The arc sets
were analysed using the process knowledge matrix M.
However the number of the suspected sets could not be
reduced in this case, since the input arcs to the node fc
are independent. If one input arc is faulty, it will not
cause inconsistency in other input arcs.
5 Conclusions
A method for detecting oscillatory residual signals was
presented in this paper. The method was integrated
into the DCDG fault diagnosis method and tested in
an application to a board making process.
52
The proposed method enables the detection and isolation of low-frequency oscillations caused by valve stiction faults in the process by exploiting the statistical
properties of the residual signals. The results show
that the proposed detection method is able to detect
the fault successfully and to provide the information
required for fault isolation.
The work presented in this paper represents the first
step in addressing the detection and isolation of faults
causing oscillatory behaviour in a process using the
DCDG method. In future, the aim is to generalize
the diagnosis methodology by developing new detection
methods that are able to cover a wider range of faults
occurring in industrial processes.
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53
Optimal Control of the Oil Reservoir Water­flooding process
Eka Suwartadi� NTNU; Stein Krogstad� SINTEF ICT; Bjarne Foss� NTNU Norway
Abstract
In the second phase of oil recovery water flooding is a common way to sweep remaining oil in the reservoir. The process can be
posed as a nonlinear optimization problem. This talk will address optimization of water flooding in the context of gradient based
optimization. The gradient is computed using the adjoint method. In the optimization problem there will be constraints on both
control inputs and state variables. The latter constraints are notoriously difficult to handle since they affect the efficiency of the
adjoint method adversely. We propose some methods to mitigate this. Further� we present a second order adjoint method to avoid
numerical problems which may arise in quasi Newton methods like the BFGS method.
54
State­Constrained Control Based on Linearization of the
Hamilton­Jacobi­Bellman Equation
Torsten Wik� Per Rutqvist and Claes Breitholtz
�bstract—For continuous time state constrained stochastic
control problems a method based on optimization is presented.
The method applies to systems where the control signal and
the disturbance both enters affinely, and it has one main
tuning paramater, which determines the control activity. If the
disturbance covariance is unknown, it can also be used as a
tuning parameter �matrix) to adjust the control directions in
an intuitive way. Optimal control problems for this type of
systems result in Hamilton Jacobi Bellman �HJB) equations
that are problematic to solve because of nonlinearity and infinite boundary conditions. However, by applying a logarithmic
transformation we show how and when the HJB equation can be
transformed into a linear eigenvalue problem for which there
are sometimes analytical solutions and if not, it can readily
be solved with standard numerical methods. Sufficient and
necessary conditions for when the method can be applied are
derived, and their physical interpretation is discussed. A MIMO
buffer control problem is used as an illustration.
I. INTRODUCTION
Consider a dynamical system described by
ẋ = f�x) + B�x)u +G�x)w� �1)
where x ∈ Ω ⊂ �n is the system state, u ∈ �m is the control
signal, f : Ω → �n, B : Ω → �n�m and G : Ω → �n�nw
are functions that describe the system dynamics, and w =
ω̇ ∈ �nw is a Gaussian white noise �where ω is a Wiener
process) having the covariance matrix W . The boundary ∂Ω
of the state space Ω defines the state constraints, i.e. the
system must be controlled such that x never leaves Ω.
The idea now is to formulate the control problem as
an optimal control problem for which a control policy
u�t� x) can be determined analytically or by straight forward
application of numerical methods. In general, the optimal
control problem for a system on the form �1) requires the
solution of a nonlinear partial differential equation �PDE),
the so-called Hamilton-Jacobi Bellman equation, that is both
nonlinear and will have infinite boundary conditions that
are difficult to handle numerically. However, by applying
a logarithmic transformation we show how and when the
PDE can be transformed into a linear eigenvalue problem for
which there are sometimes analytical solutions and if not, it
can readily be solved with standard numerical methods.
Rutquist et al. [7] used this logarithmic transformation
to linearize the HJB equation in the scalar one dimensional case. Concurrently, Itami [2] also studied the onedimensional case and used the transformation to make
T. Wik and C. Breitholtz are with Department of Signals and Systems, Chalmers University of Technology, SE 412 96 Göteborg, Sweden
tw@chalmers�se, claesbr@chalmers�se
P. Rutquist is with Tomlab Optimization AB, Västerås, Sweden
per�rutquist@gmail�com
a coupling between quantum mechanics �the Schrödinger
equation) and the Hamilton equation, which was earlier
pointed out also by Rosenbrock [4].
In a later study [5] the use of the transformation was
further developped to several dimensions for the special case
when the disturbance enters the system in the same way as
the control input, i.e.G = B. Here, we generalize the method
to the case whenG �= B, give necessary and sufficient conditions for when this linearization can be applied, and analyze
the implications of these conditions. When linearization can
be applied, it gives an intuitive control method for state
constrained systems with only one main tuning parameter
that determines the control aggression.
Two examples where the results are applied are presented
here. The first one shows how to deal with disturbances that
are not purely white. The other is a buffer example treating
the pumping of wastewater to a wastewater treatment plant
in Göteborg, Sweden.
II. THE OPTIMAL CONTROL PROBLEM
We define the problem as to find a feedback control policy
u�t� x) that minimizes
V �x�t)� t) =
E
�� tf
t
l�x�τ)) + uT �τ)Qu�τ)

dτ + Vf �x�tf ))|x�t)


�2)
where t ∈ � is the current time, tf > t is the final time,
l : Ω → �� describes the �time independent) cost �nonsingular on Ω) associated with the state, Vf : Ω → � is the
final cost, and uTQu defines the cost of the control signal.
The stochastic HJB equation for the minimization of �2)
can be formulated as

∂V
∂t
= min
u {l + �∇V )�f +Bu) + uTQu
+
1
2
tr
�∇T∇V )GWGT

}�
�3)
where V : [t� tf ]×Ω→ � is the so-called cost-to-go function
�see for instance [1] for details on the derivation of this
equation) and the gradient ∇V is defined as a row vector.
Minimization of this quadratic expression with respect to
u gives the optimal control input
u = −
1
2
Q−1BT �∇V )T . �4)
55
Inserting the optimal u into �3) gives

∂V
∂t
= l −
1
4
�∇V )BQ−1BT �∇V )T + �∇V )f
+
1
2
tr
�∇T∇V )GWGT
� �5)
with infinite boundary conditions on ∂Ω due to the state
constraints. The solution to this equation inserted into �4)
gives the optimal control policy. The problem though, is that
this PDE is not readily solved because it is nonlinear in V
and the infinite boundary conditions are difficult to handle
numerically.
III. LINEARIZATION
Now, applying the transformation
V = −2κ logZ� �6)
where κ is an arbitrary real constant, gives
∇V = −

Z
∇Z
∇T∇V =

Z2
�∇Z)T∇Z −

Z
�∇T∇Z).
Equation �5) is then transformed into

∂Z
∂t
=zl−
κ2
Z
�∇Z)BQ−1BT �∇Z)T − 2κ�∇Z)f
+
κ
Z
tr
�∇Z)T �∇Z)GWGT

− 2κtr

�∇T∇Z)GWGT

.
�7)
If two matrices A and B have matching dimensions
tr[AB] = tr[BA] = tr

ATBT

, and the trace of a scalar
equals the scalar itself. Therefore we have that
�∇Z)BQ−1BT �∇Z)T = tr

�∇Z)T �∇Z)BQ−1BT

tr
�∇Z)T �∇Z)GWGT

= �∇Z)GWGT �∇Z)T �
which gives
∂Z
∂t
=
l
Z +
1
2Z
�∇Z)�GWGT − κBQ−1BT )�∇Z)T
− �∇Z)f − tr

�∇T∇Z)GWGT

.
�8)
Now, if
GWGT = κBQ−1BT �9)
the transformed HJB equation becomes a linear PDE:
∂Z
∂t
=
l
− �∇Z)f − tr

�∇T∇Z)GWGT

�10)
with boundary conditions
Z = 0� x ∈ ∂Ω �11)
and
Zf
= Z�tf ��) = exp

−Vf


. �12)
For the finite time case this PDE can be solved using
variable separation, i.e. Z�t� x) = T �t)φ�x) [6]. The time
dependent part T �t) has an analytical solution and the state
dependent part φ�x) becomes a �partial) linear differential
equation which may have an analytical solution or be readily
solved numerically using an eigenvalue solver.
For the stationary problem tf →∞ and the integral in �2)
will not converge. However, since the process is ergodic the
expected cost per unit time �λ) will eventually be the same
everywhere in Ω, i.e.
λ = −
∂V
∂t
=

Z
∂Z
∂t
� �13)
which gives us the linear eigenvalue problem
λZ = lZ − 2κ�∇Z)f − κtr

�∇T∇)GWGT

Z = 0 on ∂Ω�
�14)
where the solution with the least λ is sought since it
corresponds to the lowest cost.
Clearly, the problem of determining the optimal solution is
greatly simplified by the transformation �6) if �9) also holds
so that the problem becomes linear as well. In the following
we will state the necessary and sufficient conditions for
the linearization to be applicable, and an analysis of the
implications of �9) on the optimal control.
Preliminaries
A singular value decomposition �SVD) of B gives
B = U1ΛBU
T
2 .
where U1 �n×n) and U2 �m×m) are orthogonal matrices,
and the rB = rank�B) singular values on the upper left diagonal of ΛB are the square roots of the nonzero eigenvalues of
both BBT and BTB. Then the columns of U1 and U2 give
the orthonormal bases for all four fundamental subspaces [8]:
ER� = the first rB columns of U1�
ENT

= the last n− rB columns of U1�
ERT

= the first rB columns of U2�
EN� = the last m− rB columns of U2�
where ER� has columns that are the base vectors for the
column space R�B), ENT

for the left nullspace � �BT ),
ERT

for the row space R�BT ), and EN� for the nullspace
� �B) of B.
For a symmetric positive definite matrix, such as the
covariance matrix W , SVD is identical to diagonalization
with orthogonal eigenvectors, i.e.
W = UWΛWU
T
W �
where ΛW is a �nw × nw) diagonal matrix with the real
positive eigenvalues λW�i of W on the diagonal. We may
then define the square root of such a matrix as
W 1/2 = UWΛ
1/2
W U
T
W �
where Λ
1/2
W = diag�

λW�1� . . .

λW�nw ). Clearly, W
1/2 is
also symmetric and positive definite, and W 1/2W 1/2 = W .
Congruously,
W−1/2 = UWΛ
−1/2
W U
T
W �
where Λ
−1/2
W = diag�1/

λW�1� . . . 1/

λW�nw ).
56
�heorem
The HJB equation �3) can be linearized if and only if
R�G) ⊆ R�B), which is true if and only if
ETNT

G = 0� �15)
The optimal control policy is then given by the linearized
HJB �14) and
u = B�GWGT �B�)TBT
1
Z
�∇Z)T � �16)
where B� denotes the pseudoinverse of B. The corresponding cost matrix Q is implicitly given by
Q−1 =
1
κ

B�GWGT �B�)T + EN�ΘΘ
TETN�
+�EΔE
T
Δ


�17)
where κ > 0 is an arbitrary scalar, Θ is any full rank matrix,
and
EΔ = �Im − Φ�Φ
TΦ)−1ΦT )ERT


where Φ = B�GW 1/2, and �→ 0.
Remarks
1) The test �15) corresponds to checking that all directions
of Gw can be directly counteracted by Bu. It is noteworthy that this sufficient and necessary condition for
HJB linearization has this clear physical interpretation.
2) Q determines the cost of control action through uTQu
in �2). In the same way as for W the SVD of Q is
Q = UQΛQU
T
Q �
where ΛQ = diag�λQ�1� . . . λQ�m), determines the cost
in the directions given by UQ. Since the directions are
preserved in the SVD of Q−1:
Q−1 = UQΛ
−1
Q U
T
Q � �18)
where Λ−1Q = diag�1/λQ�1� . . . 1/λQ�m), Q
−1 determines the ”preferred” control action directions.
• The second term in �17) is in the null space of B
and therefore does not contribute to Bu.
• The first term, on the other hand, corresponds to
a complete matching of the directions of Gw �see
the proof).
• The last term, which corresponds to the possible
directions of the control input that cannot directly
counteract the directions of the disturbance, is only
needed for Q to be formally invertible. However,
by setting � to a small number and invert Q−1
it becomes clear from large elements in Q what
directions �combinations of input signals) that
should/will be avoided.
3) κ is the main tuning parameter that determines the
trade-off between cost on the states and cost of control
activity. The smaller κ the more agressive the control.
4) If the covariance matrix W has not been determined
from data, it is next to κ the remaining tuning parameter. If it is evident, when applying the control policy,
that we are never close to a boundary in one direction, the covariance in the corresponding disturbance
direction can be decreased. If the constraints in one
direction are violated, the corresponding covariance
should be increased.
5) For the case when we have input noise on all inputs,
�17) reduces to Q−1 = κ−1W , which is the case
studied in [5].
Proof
We need to show that if the required conditions are
fulfilled, �17) implies that the equality �9) holds when �→ 0.
Presuming Q is positive definite and symmetric, Q−1 is
also positive definite and symmetric according to �18), and
we may define the square root in the same way as shown for
W . Equation �9) can then be written
GW 1/2�GW 1/2)T = �

κBQ−1/2)�

κBQ−1/2)T .
Clearly, the HJB equation is linearized if and only if
GW 1/2 =

κBQ−1/2. �19)
Because Q−1/2 has full rank, the columns on the right hand
side can take any, and no other, vector values than those
spanned by the columns of B, i.e. R�B). So, if and only
if the columns of GW 1/2 are entirely in R�B) will �19)
have a solution. According to the fundamental theorem of
linear algebra [8] the left nullspace � �BT ) is the orthogonal
complement of R�B). Hence, if the columns of GW 1/2 are
entirely in R�B) all columns should be orthogonal to all
base vectors for � �BT ), which is exactly the test �15).
Now, regard the columns of Q−1/2 as vectors �in �m).
Each of these vectors can be written as a sum of one vector
in the rowspace R�BT ) of B and another vector in the
nullspace � �B) of B, since these two spaces are orthogonal
and span the entire �m. We may therefore split Q−1/2 such
that
Q−1/2 = Q
−1/2
RT
+Q
−1/2
N �
where Q
−1/2
RT
have columns entirely in R�BT ) and Q
−1/2
N
have columns entirely in � �B). Then
Q−1 = Q−1/2�Q−1/2)T
= Q
−1/2
RT �Q
−1/2
RT )
T +Q
−1/2
N �Q
−1/2
N )
T
= Q−1
RT
+Q−1N
since �Q
−1/2
RT
)TQ
−1/2
N = 0 because of the orthogonality.
The component of the columns of GW 1/2 in R�BT ) are
Q
−1/2
G�RT
=
1

κ
B�GW 1/2.
Thus, Q−1 = Q−1
G�RT
= Q
−1/2
G�RT
�Q
−1/2
G�RT
)T is enough to
satisfy �9) since BQ
−1/2
N = 0 by definition. However, if
B or G does not have full rank Q−1 = Q−1
RT
will not be
invertible and we have to add symmetric matrices to give
Q−1 full rank. If B is not full rank we add the basis for the
nullspace. From the SVD of B we have that the columns of
57
EN� are an orthonormal base for � �B). Hence, we can fill
the space using
Q−1N = EN�Θ�EN�Θ)
T = EN�ΘΘ
TETN� �
where Θ is an arbitrary full rank matrix.
Since
BQ−1N B = �BEN� )�ΘΘ
TETN� )B
T = 0
Q−1N will not contribute to �9). Further,
Q−1N B
T �∇V )T =

�∇V )BQ−1N
�T
= 0
because Q−1N is symmetric. Hence, Q
−1
N will neither have
an effect on the control signal �4) and nor on the cost �2).
This is important because it means that no control signals are
wasted in the null space of B, and nor are any control signals
generated that would not contribute to the cost function.
If the test �15) is fulfilled
Q−1 = Q−1
G�RT
+Q−1N
will still not have full rank if GW 1/2 does not fill QRT , i.e.
there are missing dimensions in R�BT ). These dimensions
are obtained if we project R�BT ) on Q
−1/2
G�RT :
EΔ = �I − Φ�Φ
TΦ)−1ΦT )ERT


where Φ = B�GW 1/2. The columns of EΔ corresponds
to the directions of the control input not needed to directly
counteract the disturbances. If we add
Q−1
B−G�RT = EΔ�EΔ)
T
to Q−1
G�RT
+ Q−1N , Q
−1 becomes invertible but �9) will
no longer hold. However, since the inversion of Q is not
explicitly needed for the calculation of the optimal u, we
may add �Q−1B−G�RT to get �17), i.e.,
Q−1 = Q−1
G�RT
+Q−1N + �Q
−1
B−G�RT
and then let �→ 0.
IV. EXAMPLES
First we illustrate with a very simple example how the
vector spaces affect Q. Then two applications of the results
are presented. The first one is the case when the disturbance
enters as an addition to the control input, but is not purely
white. The other application is a buffer example treating the
pumping of wastewater to a wastewater treatment plant in
Göteborg, Sweden.
A� Illustration of the components of Q−1
Let
B =


1 0 1
0 1 1
0 0 0

 and G =


2
0
0


Clearly B has a null space since the last row are zeros.
Further, we see that R�G) ⊆ R�B) since G equals twice
the first column of B. Hence, we may linearize the HJB and
all three terms of �17) are non-zero. In fact, there will be
one dimension in each term:
EN� =
1

3


1
1
−1

 and EΔ =


0 0
0.61 0.35
0.61 0.35

 .
Also, because G equals twice the first column of B we
realize that only u1 should need to be used. Calculation of
Q from �17) with κ = 1 and � = 10−4 gives
Q =


0.46 0.27 −0.27
0.27 500 500
−0.27 500 500


which confirms that neither u2 nor u3 will be used in the
optimal control.
B� Coloured disturbance
For physical reasons almost all disturbances are more or
less of low pass character. However, if the system dynamics
are slower than the time constants of the disturbances the
assumption of white noise is often an acceptable approximation. In some cases though, there may be slow components
that also need to be considered. Consider the following
system with input disturbance
ẋ = Ax+B�u+ v)
y = Cx+Du+ e�
�20)
where the disturbance v contains slow dynamics. From physical modeling, or spectral analysis, a model of the disturbance
dynamics has been derived:
ẋw = Awxw +Bww
v = Cwxw +Dww�
�21)
where w is white Gaussian noise with covariance W . Inserting this disturbance model into �20) gives
ẋ =Ax+Bu+BCwxw +BDww
=Ax+Bũ+Gw
�22)
where ũ = u+ Cwxw and G = BDw.
The disturbance state variables xw are not controllable but
a successful observer �for example a Kalman filter) giving
estimates x̂ and x̂w close to the real states is assumed to
be at hand. Assuming fully known states, x and xw, the
system �22) fits into �1) and because G = BDw we have that
R�G) ⊆ R�B), which implies that the condition �15) always
holds. Hence, we may apply the linearization and solve the
linearized HJB equation for �22) to obtain the control policy
ũ�x). The actually applied control signal is then
u = ũ− Cwx̂w.
The cost matrix Q is now acting on ũ through
ũTQũ = uTQu+ 2uTQCwxw + x
T
wCwQCxw.
The first term is the same as before and has the same
interpretation as before, and the last term has no effect on the
optimal control policy since xw is not controllable and does
not depend on u. The second term is likely to have only a
58
river
to WWTP
Q
S
Q
S�
Q
in Q

V
S V
�Southtunnel
system �orthtunnel
system
�orthshaftSouthshaft
Fig. 1. The Rya WWTPs buffer system for their influent pumping station
�24 m below ground level).
limited influence on u because E {xw} = 0. However, using
SVD the added cost can be written as
uTQCwxw = u
TUQΛQU
T
QCwxw�
where Λ = diag�λQ�1� . . . � λQ�2) defines the costs in the
directions of UQ. It can therefore be seen as an additional
cost for the influence of the slow dynamic disturbances
�Cwxw) in the directions of u already undesired because of
the direct term.
If the disturbance does not enter as an addition to the input
as in �20), i.e.
ẋ = Ax+Bu+ G̃v
the treatment is only slightly changed if there exists a matrix
M such that G̃Cw = BM . Then we get u = ũ − Mx̂w
and G = G̃Dw. The interpretation of the linearization
requirement remains the same, i.e. the direct terms from the
white noise must be possible to directly counteract by the
control signal.
C� A wastewater treatment buffert system
The Rya wastewater treatment plant �WWTP) treats the
wastewater from the Göteborg region.Wastewater is transported to the Rya WWTP through a large tunnel system that
can be considered as two separated systems of about the
same size, separated by the river Göta älv �see Fig. 1).
Simplified, and with the notation
x =

VN
VS

� u =

Qin
QSN

and w =

QN
QS


where all variables are deviations from the operating point,
the system can be described by
dx dt =

−1 1
0 −1
� �
u1 u2
+

1 0
0 1
� �
w1 w2
.
w1 and w2 are considered �within the time scale considered)
as white with a fairly strong correlation, since they are both
the results of figuratively the same human habits and rain
run off. A covariance matrix
W =

1 0.8
0.8 1

�23)
is assumed.
Pumping is one of the major costs for this plant and many
other WWTPs. For this plant the average flow �including runoff and infiltration) is about 4 m3/s and hence, every saved
meter of elevation corresponds to more than 340 MWh/year.
Remembering that we deal with deviations from an operating
point, the variable cost for pumping is assumed to be
proportional to h1�max−h1�VN �t)), where h1 depends on the
horizontal cross sectional area of the north shaft. However
the levels in the north and south shafts should always be
above a minimum level and below a maximum level to avoid
overflows and the pumps from running dry.
The stationary control problem can now be formulated as
min
u E

h1�max − h�x1) + u
TQu

s.t. 0 ≤x1 ≤ x1�max
0 ≤x2 ≤ x2�max
�24)
where x1 and x2 are the volumes above minimum level. We
may assume that the relations between height and volume of
the north shaft are roughly given by [3]
VN = 10
4 · h21
VS = 10
4 · �0.11h22 + 0.07h
3
2).
Thus, h1 = 0.01

x1, and for h1�max = 4 m and h2�max =
6 m we get x1�max = 1.6 ·10
5 m3 and x2�max = 1.9 ·10
5 m3.
For this example we have that both B and G have full
rank, so
Q−1 =
1
κ
B�GWGT �B�)T �
which gives
Q =

2.8 −5
−5 10

for κ = 1. Because the volume in the north shaft �x1), is
affected by both disturbance flows �through u2) the cost for
u1 is less than the cost for u2.
Solving the linearized HJB-equation �14) for κ = 1 and
calculating u gives the result presented in Fig. 2. As can
be seen, the control policy produces a control vector that
aims away from the constraints, and guarantees that we do
not violate them. However, the cost for control activity is
not sufficiently low to keep the level high in the north shaft,
which was desired in order to reduce the pumping costs. If
κ = 0.01 we allow more control activity and then it should
be possible to stay closer to the maximum level in the north
shaft. This is confirmed in Fig. 3. The average relative height
in the north shaft was increased from 3.0 m �0.89 · 105 m3)
to 3.40 m �1.15 · 105 m3). Note that the full buffer capacity
in the south shaft is used in both cases.
Now, assume we have misjudged the disturbance so that
the variance of the disturbance in the North shaft is in fact
only a tenth what we thought, i.e.
W =

0.1 0.25
0.25 1

�25)
if the correlation coefficient is unchanged. In Fig. 4 a
simulation of that situation is shown �κ = 0.01). Because
of the overestimated variance the controller keeps the level
in the north shaft unnecessarily low. To push the level in the
north shaft a little higher we decrease the variance of the
59
Z
V
S
 / 10
5
 m
3
V
N
 /
 1
0
5
m 3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Fig. 2. Contour of the solution Z and direction of the input Bu when
κ = 1. The stationary point where the system has its maximum probability
to be is inside the inmost contour.
Z
V
S
 / 10
5
 m
3
V
N
 /
 1
0
5
m 3
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Fig. 3. Contour of the solution Z and direction of the input Bu when κ =
���1. The stationary point where the system has its maximum probability
to be is inside the inmost contour.
influent flow to the north shaft �keeping the cross correlation
coefficient). As can be seen in Fig. 5, redoing the design with
the smaller and correct W means that we can stay closer to
the upper limit.
V. CONCLUSIONS
A method for state constrained control based on stochastic
optimal control has been presented. The method is a result
of an exact linearization of the HJB-equations, which has
implications on how the cost for control is defined. However,
the implications have been thoroughly investigated and found
to be in agreement with intuitive reasoning about suitable
cost matrices. There is one main tuning parameter, which
determines the control activity, and then the covariance
matrix of the white noise disturbance can also be used for
tuning the preferred �or undesired) directions of the control.
The method assumes white noise but it has been shown
how it can also be applied in cases when the disturbance
0 1 2 3 4 5
0
0.5
1
1.5
2
x 0 1 2 3 4 5
−10
−5
0
5
10
 t
 u
 V
N
 V
S
 V
N
 V
S
Fig. 4. Simulation of the controlled system with the policy calculated for
W given by �23) and simulated for the true W given by �25).
0 1 2 3 4 5
0
0.5
1
1.5
2
x 0 1 2 3 4 5
−10
−5
0
5
10
 t
 u
 V
N
 V
S
 V
N
 V
S
Fig. 5. Simulation of the controlled system with the policy calculated for
W given by �25) and simulated for the same �true) W .
is coloured. A buffer example, pumping of wastewater to
a wastewater treatment plant from two different shafts in a
sewer system, has illustrated the use of the method.
REFERENCES
[1] P. Dorato, T. C. Abdallah, and V. Cerone. Linear Quadratic Control:
An Introduction. Prentice-Hall, 1995.
[2] T. Itami. Nonlinear optimal control as quantum mechanical eigenvalue
problems. Automatica, 41:1617–1622, 2005.
[3] J. Lindqvist, T. Wik, D. Lumley, and G. Äijälä. Influent load prediction
using low order adaptive modeling. In 2nd IWA Conference on
Instrumentation� Control and Automation, Busan, South Korea, 2005.
[4] H.H. Rosenbrock. Physics letters, 110A:343–346.
[5] P. Rutquist, C. Breitholtz, and T. Wik. On the infinite time solution
to state-constrained stochastic optimal control problems. Automatica,
44:1800–1805, 2008.
[6] P. Rutquist, C. Breitholtz, and T. Wik. Finite-time state-constrained
optimal control for input affine systems with actuator noise. Automatica,
Submitted.
[7] Per Rutquist, Claes Breitholtz, and Torsten Wik. An eigenvalue
approach to infinite-horizon optimal control. In Proc. 16th IFAC World
Congress, Prague, Czech Republic, jul 2005.
[8] G. Strang. Linear Algebra and its Applications. Harcourt Brace and
Company, Florida, 3 edition, 1988.
60
Application of Optimal Control Theory to a Batch Crystallizer using
Orbital Flatness
Steffen Hofmann1 and Jörg Raisch1,2
1Technische Universität Berlin, Einsteinufer 17, 10857 Berlin, Germany
2Max-Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr.1, 39106
Magdeburg, Germany
Abstract
In this contribution we suggest an efficient application
of the Pontryagin Minimum Principle (PMP) to the
solution of an optimal control problem for a standard
moment model of a batch crystallizer. The application
is based on a time scaling that turns the moment model
differentially flat, and on a simplifying assumption. We
present efficient solutions and also consider uniqueness
of the solutions. We demonstrate ways to assess and
partially correct the error due to simplification. Finally, we use a case study to demonstrate the optimal
trajectories, numerical effort and the magnitude of the
error.
1 Introduction
Optimal control of batch crystallizers has been an active topic for a long time. Mullin and Nyvlt [1] recognized in 1971 that the final crystal size can be increased
by using a “programmed” crystallization temperature
trajectory rather than natural (exponential) cooling of
the solution. In the time following, specific optimal
control problems were formulated and solved using optimal control theory as well as numerical methods [2],
[3], [4], [5], [6]. Requirements on the final crystalline
product are commonly expressed in terms of the crystal size distribution (CSD) or of its moments. The field
remains active, with recent publications like [7]. Also,
a new focus on feedback control has emerged. Schemes
such as online-optimization or model predictive control
[8], [9], [10] demand for computationally fast solutions
or other means of efficient control algorithms.
By recognizing the orbital flatness property of a standard moment model describing the crystallization of
a single substance out of solution, with crystallization temperature T , or jacket temperature Tj , being
the control input, a way was opened up in [11] to approach some related control problems in an analytical
way. In particular, it was shown that the model can
be inverted analytically in order to achieve a desired
final CSD stemming from nucleation. Based on this,
an optimization scheme for a common setup based on a
parametrization of this final CSD was suggested. Here,
we show that the results in [11] allow the application
of the Minimum Principle (PMP) for optimal control
of the batch crystallizer in an efficient way if a suitable
assumption is made, requiring very little numerical effort.
2 Model and optimization task
We consider a standard moment model that describes
the population dynamics for the crystallization of a
single substance out of solution:
µ̇i = iG (·)µi−1, i = 1, 2, . . .
µ̇0 = B (·) ,
(1)
where G (·) and B (·) are the (crystal size independent)
growth and nucleation rate, respectively, which both
depend on other variables in the system. In fact, a
model including only the first four moments, µ0 to µ3,
can be closed via the mass balance equation
ml (t) = ml,0 − ρkV (µ3 (t)− µ3 (0)) , (2)
where ml denotes the mass of dissolved substance, ρ is
the density of the crystals, kV is a volume shape factor,
and ml,0 = ml(0). G (·) and B (·) are defined as in [11]:
G (·) = kg (S − 1)g (3a)
B (·) = kb (S − 1)b µ3 (t) (3b)
Only growth and secondary nucleation are assumed to
take place, and kg, kb, g and b are positive constants
that depend on the substances used and on design variables, like stirrer speed. Furthermore, it is assumed
that b > g. This assumption is justified for many practically important substances. The supersaturation S is
given by
S =
c csat
, c =
ml mW , (4)
with mW being the mass of the solvent. The solubility
relation is approximated as a polynomial
csat (T ) = a0 + a1T + a2T 2, (5)
with positive coefficients a0, a1 and a2, where csat is
the saturation concentration, and T is the temperature of the slurry, or crystallization temperature, and
is assumed to be spatially constant.
61
2.1 Orbital flatness and separation of
growth and nucleation
In [11] it is shown that a standard moment model of a
batch cooling crystallizer is orbitally flat, i.e. flat in an
appropriate, scaled time domain. A new, scaled time
τ is introduced, together with the transformation
dτ = G (t) dt (6)
This time scaling turns the moment model into a chain
of integrators, where growth and nucleation rate enter
only at the input of the first integrator. Here, we apply
the transformation to a model describing separately the
evoultion of the moments µi,s of the CSD of growing
seed crystals and the moments µi,n of the CSD of newly
nucleated (and growing) crystals. This split model (in
normal time) is also used e.g. in [12], [7]:
d dτ µi,s = iµi−1,s, i = 1 . . . 3 (7a)
d dτ µ0,s = 0 (7b)
d dτ µi,n = iµi−1,n, i = 1 . . . 3 (7c)
d dτ µ0,n =
B (µ3,s + µ3,n, T (τ))
G (µ3,s + µ3,n, T (τ))
(7d)
The crystallization temperature, or slurry temperature, T is commonly used as a manipulated variable.
This is signified in the equations by explicitly denoting
the dependency on τ . It is assumed here that it can
be adjusted precisely and arbitrarily fast. In practice,
it can be regulated using a low-level thermostat controller. An additional equation is added to keep track
of real time t, which is also an important process variable. Storing this variable allows the transformation of
computed trajectories from the τ - into the t-domain:
d dτ t =
1
G (µ3,s + µ3,n, T (τ))
(7e)
The initial conditions of the system (7) are:
µi,s (0) = µi0,s ; µi,n (0) = 0 ; t (0) = 0 (8)
where µi0,s, i = 0 . . . 3 are the (positive) moments of
the CSD of seed crystals. Note that, as indicated in
(7d) and (7e), we want to emphasize that B (·) and
G (·) are functions only of T and the total third moment, µ3 = µ3,s +µ3,n, which are obtained by combining (2) to (5). From (7a) to (7b) it can be seen that
the evolution of the moments of the grown seeds CSD
is decoupled from that of the moments of the nucleated
CSD and depends only on initial conditions and on τ .
The third moment of the grown seeds CSD is given by:
µ3,s (τ) = µ00,sτ3 + 3µ10,sτ2 + 3µ20,sτ + µ30,s (9)
In the following discussion, we consider µi0,s as constants and refer to µ3,s (τ) as a known function of τ .
Note that the case of monodisperse seed crystals (seed
CSD thought to be concentrated at one length L0),
which is often used in literature (e.g. [2]), leads to a
special case of (9).
2.2 Optimal control problem
When the crystallization temperature T is considered
as the manipulated variable, or control input, one
wants to find an optimal input T ∗ (t). Such an optimal control problem is usually formulated using a cost
function, constraints that must hold while the process
evolves and final time constraints. The basic requirement is to produce a large amount of crystalline product in a short time, which could be formulated in different configurations of cost function and constraints.
But, at least for the growth and nucleation models used
here, this problem setting leads to trivial solutions, like
keeping temperature always as low as possible (or allowed).
Other problems have also been treated. For example, in [2] the final length of growing seed crystals is
maximized. In [5], [11], the final ratio of crystalline
masses stemming from nucleation and growing seeds,
impf :=
µ3f,n
µ3f,s
, (10)
where µ3f,n = µ3,n (τf ) and µ3f,s = µ3,s (τf ), is minimized subject to a constraint on the final total crystallized mass, mf := ρkV (µ3f,s + µ3f,n). In [13] the
difference µ3f,n − µ3f,s is minimized. In [7] a number
of related optimization problems are considered.
These approaches are mostly motivated by the idea
that crystalline mass stemming from nucleation is undesirable, e.g. because these crystals do not have a lot
of time to grow and are too small (product requirements, filtration etc.) at the end of the batch. Especially, if one wants to achieve a very narrow final CSD
consisting of the shifted CSD of the seed crystals, any
nucleation is disturbing. In the following, we develop
a new method for the solution of the following optimal
control problem, similar to the problem in [11]:
min.
u(τ)
µ3,n (τf ) s.t.
µ3,s (τf ) = µ3f,s,c
t (τf ) ≤ tf,c
u (τ) ∈ U ∀τ ∈ [0, τf ]
(11)
where τf is the final scaled time and U is the compact
set of allowed input values. Note that, for simplicity, (11) uses an equality constraint on µ3,s (τf ), rather
than the economically meaningful inequality constraint
µ3,s (τf ) ≥ µ3f,s,c. However, numerical results indicate
that increasing the constraint µ3f,s,c will increase the
minimal cost when, at the same time, the constraint
on tf is maintained. Note also that, when the final
µ3f,s = µ3,s (τf ) is fixed, minimization of µ3f,n is equivalent to minimization of impf . One could also think
of variations of the problem (11) that, for example, involve constraints on µ3,n (τf ). The general goal is to
keep µ3,n (τf ) low compared to µ3,s (τf ) while producing “enough” of the desired µ3,s (τf ).
With T as the control input, i.e. u (τ) ∧= T (τ), U
could be defined as U = [Tmin, Tmax]. The bounds on T
can be determined by the capabilities of the thermostat
used for regulation, etc. Here, we define a different
control input, namely the inverse of the growth rate,
62
in which case u (τ) ∧= 1/G (τ), and define fixed bounds
on this variable:
U =
[
( 1G )min, (
1
G )max
]
(12)
With the growth law (3a), ( 1G )min and (
1
G )max can be
seen as upper and lower limits on supersaturation S,
respectively. These bounds could make sense in that
the process is operated in a metastable region, where
the model equations can be considered to be sufficiently
accurate. We will allow ( 1G )max to be chosen arbitrarily large, so that it is never hit. It will be shown in the
sequel that, under normal conditions, the crystallization temperature T that is needed to realize 1/G can
be uniquely computed.
3 Application of the PMP
For solving the proposed optimization problems, the
necessary conditions of optimality according to the
Pontryagin Minimum Principle [14], [15] (PMP) have
been stated several times in the respective literature,
but analytical solutions have been found to be difficult to obtain. Numerical methods have then been employed, for example control vector iteration is used in
[2]. In other contributions, a simplification is suggested
to make solutions easier. For derivation of a constant
growth rate trajectory in [1], and for optimization in
[3], and subsequently in [4], it is suggested to neglect
nucleation as it is supposed to be very low when the
process is operated within the metastable region.
In contrast to the latter publications, we do consider
nucleation as it is part of the problem (11). However,
since the goal is to suppress nucleation, we neglect the
feedback that it has on the crystallization process. This
feedback would normally be due to the mass balance
(feedback of µ3,n on concentration and thus supersaturation) and to secondary nucleation. The effects of
this simplification will be assessed later on. The resulting optimization problem has a particularly simple
structure when it is written in the transformed time(τ)domain.
In preparation for doing so, we relabel the state variables. Remember that the moments µi,s of the CSD of
growing seed crystals can be expressed as functions of
τ , especially µ3,s (τ) is given by (9). Therefore, it is not
necessary to regard these moments as state variables.
Hence, we choose the state according to
x1 ∧= µ3,n; x2
∧= µ2,n; x3
∧= µ1,n; x4
∧= µ0,n; x5
∧= t;
(13)
We arrive at a system of differential equations for a
model with five states and explicit time(τ)-depedency:
d dτ x1 = 3x2,
d dτ x2 = 2x3,
d dτ x3 = x4 (14a)
d dτ x4 =
B
G (µ3,s (τ) + x1, u (τ)) (14b)
d dτ x5 = u (τ) (14c)
where u (τ) ∧= 1G (τ) is the control input, and
B
G (·) shall
be understood as:
B
G (µ3,s (τ) + x1, u (τ)) := B
(
µ3,s (τ) + x1, T (τ)
)
u (τ)
(15)
where, for every instant of τ , T (τ) is the (positive)
solution of
G
(
µ3,s (τ) + x1, T (τ)
)
= (u (τ))−1 (16)
For the simple growth and nucleation laws (3a) and
(3b), the temperature T need not be computed explicitly. The required supersaturation S can be obtained
from (3a) and then directly be inserted in (3b). The
resulting expression for BG is
B
G (µ3,s (τ) + x1, u (τ))
= kbk
− bg
g u (τ)
g−b
g (µ3,s (τ) + x1) (17)
However, as T is the manipulated variable, it has to
be computed. It can be seen from (4), (5) that, as
a0, a1, a2 > 0, a real positive (inc. zero) solution exists
for T if and only if csat (τ) = c (τ) /S (τ) ≥ a0.
We now propose an idealization of the model that
will substantially simplify the solution of the optimal
control problem discussed in the following. We negelect
any feedback of the nucleated part of the CSD on the
crystallization process, i.e., we assume that
µ3 = µ3,s + µ3,n ≈ µ3,s (18)
As a consequence, we replace µ3 = µ3,s + x1 by µ3,s.
With this simplification, the total third moment that
affects the system does only depend on the value of τ ,
see (9). The system can then be written
d dτ x1 = 3x2,
d dτ x2 = 2x3,
d dτ x3 = x4 (19a)
d dτ x4 =
B
G (µ3,s (τ) , u (τ)) (19b)
d dτ x5 = u (τ) , (19c)
where µ3,s (τ) is given by (9). Nevertheless, it is important to track the third moment from nucleation,
µ3,n
∧= x1, as one process requirement is to keep it as
low as possible. Only its feedback on the system is
neglected.
The optimization is a Mayer type problem, because
the cost function in (11) is expressed in terms of the
final state and does not include an integral term, with
the Hamiltonian H (x, u,ψ, τ) = ψT ddτ x:
H (x, u,ψ, τ) = 3ψ1x2 + 2ψ2x3 + ψ3x4
+ ψ4BG (µ3,s (τ) , u) + ψ5u (20)
Next, we state the necessary conditions (NC) for an
optimal trajectory. The differential equations of the
adjoint system ψ are, in general, ddτψ = − ∂∂xH (·).
Hence,
d dτ ψ1 = 0 (21a)
d dτ ψ2 = −3ψ1, ddτ ψ3 = −2ψ2, ddτ ψ4 = −ψ3 (21b)
d dτ ψ5 = 0 (21c)
At each instant of τ , as long as the control is not singular, the optimal control input u∗ (τ) must satisfy
u∗ (τ) = argmin
u∈U
H (x, u,ψ, τ)
= argmin
u∈U
ψ4
B
G (µ3,s (τ) , u) + ψ5u (21d)
63
where U is the compact set of allowed input values
(12). Like said before, the upper limit ( 1G )max is only
auxiliary and shall be chosen as large as necessary so
that it is never hit. The boundary conditions for the
states are
x (0) = 0; x5 (τf ) ≤ tf,c (21e)
From (11) follows the cost function φ, and, as x1f is
not part of a constraint, the terminal condition for ψ1:
φ (xf , τf ) = x1f ;
∂φ (·)
∂x1f
= 1 ⇒ ψ1 (τf ) = 1 (21f)
The final values of the moments µ0f,n, µ1f,n and µ2f,n,
i.e. of x4 (τf ), x3 (τf ) and x2 (τf ), are neither part of
the cost function nor of constraints, so that
ψ2 (τf ) = ψ3 (τf ) = ψ4 (τf ) = 0 (21g)
From (21a) through (21c) and (21g) it can be seen
that, with knowledge of τf , the evolution of all adjoint
states can be readily computed (uniquely) up to two
multiplicative constants, k1 and k2:
ψ1 (τ) = k1 (22a)
ψ2 (τ) = 3k1 (τf − τ) (22b)
ψ3 (τ) = 3k1 (τf − τ)2 (22c)
ψ4 (τ) = k1 (τf − τ)3 (22d)
ψ5 (τ) = k2 (22e)
In order to find the optimal solution, the two constants
k1 and k2 have to be determined. Additionally, (21d)
has to be solved for u∗ at each τ between 0 and τf .
The final “time” τf is given by the equality constraint
µ3f,s,c and can be computed beforehand by solving (9)
(where all µi0,s > 0). Because of (21f), k1 = 1.
With (17), and the simplification, (21d) becomes:
u∗ (τ) = argmin
u∈U
ψ4kbk
− bg
g u
g−b
g µ3,s (τ) + ψ5u (23)
A candidate for an unconstrained minimizer, uo, if it
exists, is obtained by setting the derivative of H with
respect to u equal to zero:
∂H (·)
∂u
∣∣∣∣
uo = −ψ4 b− g
g kbk
− bg
g (uo)
− bg µ3,s (τ) + ψ5
!= 0
uo =
1
kg (
ψ4
ψ5
b− g
g kbµ3,s (τ)
) g
b (24)
For a real valued solution to exist, the term inside
the parantheses must be positive. Because of (22d),
with k1=1, it follows that ψ4 (τ) > 0, ∀τ ∈ [0, τf ) and
ψ4 (τf ) = 0. Note that (g− b)/g in (23) is negative because of the assumptions that g > 0, b > 0 and b > g.
Also, kb > 0, kg > 0 and µ3,s (τ) > 0. Hence, H (·) is
strictly monotonically increasing in u when ψ4 = 0 and
ψ5 > 0 and it is strictly monotonically decreasing in u
when ψ4 > 0 and ψ5 ≤ 0 or when ψ4 = 0 and ψ5 < 0.
The second derivative with respect to u is
∂2H (·)
∂u2
= ψ4
b− g
g b g kbk
− bg
g u
−b−g
g µ3,s (τ) (25)
and its sign is always equal to the sign of ψ4 (u > 0).
Thus, H (·) is convex in u when ψ4 is positive and linear
in u when ψ4 is zero. The limits for u→ 0 and u→∞,
of the part of H given in (23), named H(u), are
lim
u→0
H(u) (·) = lim
u→∞
H(u) (·) ={
0 ψ4 = 0
∞ ψ4 > 0

−∞ ψ5 < 0
0 ψ5 = 0
∞ ψ5 > 0
(26)
So, as ψ4 ≥ 0, and as long as not both ψ4 and ψ5
are zero, which by now we conclude is only possible
at τ = τf , (23) has always a unique solution which
is either ( 1G )min, (24) or (
1
G )max. Also, if a real valued solution exists for uo, it is really an unconstrained
minimizer (no maximizer or saddle point). Only when
both ψ4 and ψ5 are zero, H (·) is independent of u.
The fact that this is not possible for an extended period in (τ)-“time” prohibits the existence of solutions
with singular control intervals.
The remaining task is to find the value of k2 =
ψ5 (τ). The monotonicity considerations show that, as
k1 > 0, and in case k2 ≤ 0, u∗ would have to be ( 1G )max
at least for τ < τf . Looking at (19c), it is clear that for
any value of tf,c, ( 1G )max can be chosen large enough so
that the terminal constraint tf ≤ tf,c is violated. We
conclude that k2 > 0. Also, the fact that k2 6= 0 shows
that the constraint on tf cannot be removed, i.e. it has
to be active (otherwise the state x5 could have been removed from the problem, resulting in (23) without the
ψ5-term, i.e. ψ5 = 0). The constraint x5f ≤ tf,c can
be replaced by the new boundary condition
x5f = tf,c (27)
When the value of k2 that makes the NC hold is unique,
then the whole solution is unique. A solution is, however, not guaranteed. The minimum time tf possible
to produce the desired µ3f,s,c is obtained by setting
u (τ) = ( 1G )min ∀τ ∈ [0, τf ], leading to the maximum
possible µ3f,n. If this time is greater than tf,c, then
no solution is feasible. Also, if the increase in µ3 is
too large, i.e., concentration is reduced too much, then
the control u∗ (τ) can be computed, but it cannot be
realized by controlling T (see end of this section). We
assume that the process is operated in a region where
this is excluded.
In order to compute a solution to the NC, the only
remaining task is to find one value, k2, known to be
greater than zero. If an equation error can be defined which is strictly monotonic in this value, then
one way to numerically find an arbitrarily close approximation is to use bisection with a preceding extrapolation step. In fact, when k2 = ψ5 (τ) gets
smaller, then the unconstrained minimizer (24) gets
larger for any instant of τ . Then, ∀τ ∈ [0, τf ], the
constrained minimizer u∗ (τ) will increase or stay the
same. Near τf , when ψ4 (τ) → 0, there will always
be an interval where u∗ (τ) < ( 1G )max. When the case
u∗ (τ) = ( 1G )min ∀τ ∈ [0, τf ] has already been excluded
(this is only the optimal solution if the resulting tf is
64
exactly equal to tf,c), it is clear that at least for some
time u∗ will increase (when the constraints ( 1G )min and
( 1G )max are not active). This means that the final time
tf , which is the integral over u∗, will grow strictly
monotonically with decreasing k2.
This also shows that the value of k2 that makes the
NC hold is unique. Together with the unique solution
for u∗ at every τ (including τf as k2 = ψ5 (τ) > 0), a
unique solution to the necessary conditions is found (if
it exists), which must then be truly optimal.
The solution of the optimal control problem yields
not only u∗ (τ), but also the state trajectory x∗ (τ) and,
amongst others, µ∗3,s (τ) = µ3,s (τ) and supersaturation
S∗ (τ). Finally, to be able to realize the trajectory
u∗ (τ) = ( 1G )
∗ (τ) by manipulating T (t) in real time t,
two steps must be taken
1. T ∗ (τ) has to be computed from ( 1G )
∗ (τ) by using
(2), (3a), (4) and (5), τ ∈ [0, τf ].
2. T ∗ (τ) has to be converted to T ∗ (t). The trajectory t∗ (τ) = x∗5 (τ) can be used.
For the first step it has to be checked whether c∗sat (τ) =
c (τ) /S∗ (τ) ≥ a0 ∀τ ∈ [0, τf ]. As the concentration is
smallest and the supersaturation is at its upper limit
at τ = τf , it is enough to consider this instant.
3.1 Efficient solutions
For the given growth and nucleation laws, there exists
an alternative, and more efficient way to determine the
parameter k2. Recognizing from (24) that the unconstrained minimizer of the Hamiltonian will always tend
to zero towards the end of the batch, when ψ4 (τ) goes
to zero, it is clear that the inequality constraint on the
input will be active when τ approaches τf . It is then
assumed that the control consists of two arcs, one unconstrained (sensitivity seeking) arc at the beginning
and one constrained arc at the end. We denote the
switching time between these arcs τs. Equation (24)
can also be written
uo (τ) = ψ−
g b 5
1
kg (
ψ4 (τ)
b− g
g kbµ3,s (τ)
) g
b (28)
Note that no differential equations need to be integrated to compute uo (τ). At the switching instant
the unconstrained minimizer uo will be exactly equal
to the constrained minimizer ( 1G )min. This gives a relation between the switching instant τs and the constant
k2 = ψ5 required for the switching to happen at this
moment:
ψ5 (τs) = ( 1G )
− bg
min
k − bg
g ψ4 (τs)
b− g
g kbµ3,s (τs) (29)
By integrating (28), the evolution of real time t dependent on τ can be computed for the unconstrained arc,
up to ψ5 = k2, resulting in the real time ts corresponding to τs:
ts (τs) =
ψ5 (τs)
− gb
∫ τs
0
1
kg (
ψ4 (τ)
b− g
g kbµ3,s (τ)
) g
b dτ (30)
Because the term with ψ5 (τs) has been put outside the
integral, the integration can be done numerically in a
cumulative way, ∀τs ∈ [0, τf ]. It is not necessary to
integrate from zero to τs, for every value of τs. Knowing τf and the required final tf = tf,c, the real time
can also be integrated backwards from τf to τs. In this
case, the constrained control input has to be used:
ts (τs) = tf,c +
∫ τs
τf
( 1G )mindτ = tf,c + (
1
G )min (τs − τf )
(31)
Because (31) and (30) (where (29) is substituded for
ψ5 (τs)) have to be consistent, the switching instant can
be computed as the intersection of the corresponding
curves. This is equivalent to detecting the zero crossing
of (30)−(31). Finally, ψ5 can be obtained by again
using (29).
4 Analysis of the error
As mentioned earlier, because we consider T as the manipulated variable and the real time is t, T ∗ (t) has to
be computed from u∗ (τ) = ( 1G )
∗ (τ). However, there
exists a mismatch between the simplified model employed for finding the optimal control and the nonsimplified, detailed model, (1) to (5), where the third
moment from nucleation, µ3,n, has feedback on the system in two ways:
1. via (2), (3), (4): feedback on supersaturation via
the mass balance
2. via secondary nucleation, (3b), where really µ3 =
µ3,s + µ3,n
This mismatch will result in an error when the detailed model is simulated with T ∗ (t), which has been
computed using the simplified model, as the control.
By computing a new control, T ′ (t), we now propose a way to correct this error in the sense that
the constraints tf,c, µ3f,s,c are maintained. Therefore, the non-simplified model in τ -domain, (14), (17),
is simulated with u′ (τ) as the control, where we
set u′ (τ) = u∗ (τ), resulting in trajectories x′ (τ) =
[µ′3,n, µ

2,n, µ

1,n, µ

0,n, t
′]T (τ), µ′3,s (τ).
Of course, µ′3,s (τ) = µ

3,s (τ) = µ3,s (τ) , ∀τ ∈ [0, τf ].
From (14c) it is also clear that t′ (τ) = t∗ (τ) , ∀τ ∈
[0, τf ], whereas the positive feedback of x1 in (14a),
(14b), (17) causes µ′3,n (τ) > µ

3,n (τ) , ∀τ ∈ (0, τf ].
Like before, the control T ′ (τ) is obtained using
(2), (3a), (4) and (5), this time setting µ3 = µ′3 =
µ3,s + µ′3,n. Knowing t
′ (τ) = t∗ (τ), T ′ (τ) can be con-
verted to T ′ (t). In τ -domain, t∗ (τ) is a state trajectory. Changing the independent variable from τ to t,
one gets a corresponding state trajectory, τ∗ (t). Because t′ (τ) = t∗ (τ) in τ -domain, τ ′ (t) = τ∗ (t) in tdomain. Thus µ′3,s (t) = µ

3,s (t). From there it is clear
that both constraints will be maintained when the detailed model is simulated in normal t-time with T ′ (t)
as the control.
65
However, T ′ (t) is not, in general, the optimal control
for the detailed model, i.e., by solving the optimal control problem for the detailed model, one should achieve
a result µ∗∗3,n (τf ), which is less than µ

3,n (τf ), whilst
maintaining the constraints tf,c and µ3f,s,c. We use
the sub-optimal result µ′3f,n = µ

3,n (τf ) as an upper
bound for µ∗∗3,n (τf ). For finding a lower bound, we
consider (14a), (14b) and (17). In the detailed model
in τ -domain, the feedback of µ3,n
∧= x1 is only via (17),
which can be factorized:
B
G (µ3,s (τ) + x1, u (τ))
= kbk
− bg
g u (τ)
g−b
g µ3,s (τ) + kbk
− bg
g u (τ)
g−b
g x1 (32)
The second term, which must be zero or positive, is
only considered in the detailed model. Looking at the
linear chain of integrators, (14a), (14b), it is clear that
this additional nucleation term, as soon as it is greater
than zero, will cause an inevitable positive contribution
to µ3,n (τ). We conclude that optimal control of the
detailed model cannot yield less µ3,n (τf ) than optimal
control of the simplified model can do. Consequently,
µ∗3,n (τf ) ≤ µ∗∗3,n (τf ) ≤ µ′3,n (τf ) (33)
We define εmax := µ′3,n (τf )−µ∗3,n (τf ) as the maximum
amount of µ3,n that could be avoided by computing the
optimal control for the detailed model, over computing
the optimal control for the simplified model and applying the proposed error correction scheme.
5 Case study
The parameters and initial conditions for the case
study are adapted from [5], [11] and are given in Table
1. The final time constraint is set to tf,c = 100 min and
the control constraint is set to ( 1G )min = 5
min
mm . In order
to show the impact of the error caused by simplification, different values were identified for the constraint
µ3f,s,c, that let specific final ratios imp′f = µ

3f,n/µ3f,s,c
result from optimal control of the simplified model and
application of the proposed error correction scheme.
We were able to find these values for µ3f,s,c by repeating the optimization a number of times and using a
bisection scheme.
Figures 1 to 4 show optimal trajectories for the simplified model, as well as the results of the application
of the error correction scheme, for imp′f = 0.33. Furthermore, results are compared to those obtained with
constant supersaturation control, denoted by Scst. The
Table 1: Case study
Process parameters (KNO3, water):
kb = 3.47 · 104 1mm3min ; b = 1.78; kg = 6.97
mm min
; g = 1.32
a0 = 0.1286; a1 = 5.88 · 10−3 1◦C ; a2 = 1.721 · 10
−4 1
◦C2
ρ = 2.11 g
cm3
; kV = 1
Initial conditions:
fs (`, 0) = N0 ·δ (`− `0) ; `0 = 200µm; µi0,s = N0`i0
ms,0 = ρckvµ30,s = 0.5 g → µ30,s = 237.0 mm3
mW = 1650.1 g; c0 = 0.493 → ml,0 = 813.5 g;
Table 2: Results obtained by optimization and subsequent application of the error correction scheme ( ′ )
and with constant supersaturation control (Scst), with
error bounds εmax, for different ratios imp′f
imp′f [%] 1 5 10 20 33
µ3f,s,c [mm
3] 1344 2665 3768 5471 7251
µ′3f,n [mm
3] 13.4 133.2 376.8 1094.1 2392.8
µScst3f,n [mm
3] 15.3 148.3 415.7 1196.8 2603.6
µScst3f,n−µ

3f,n
µ′3f,n
[%] 13.7 11.3 10.3 9.4 8.8
εmax
µ′3f,n
[%] 0.05 0.33 0.73 1.58 2.73
εmax
µScst3f,n−µ

3f,n
[%] 0.38 2.92 7.08 16.87 30.99
constant S can be chosen so as to meet the constraints
tf,c and µ3f,s,c. The control TScst (τ), respectively
TScst (t), which realizes the constant S, can then be
determined with the help of the detailed model.
Table 2 contains the results for various imp′f . The
relative amount of µ3,n is given that can be avoided
by using the approximately optimal control T ′ (τ), respectively T ′ (t), over using constant supersaturation
control. The improvement is rather small, around 10
percent. The error bound εmax = µ′3f,n−µ∗3f,n is given
relative to µ′3f,n as well as relative to the improvement
over a constant supersaturation trajectory.
0 10 20 30 40 50 60 70 80 90 100
29
29.5
30
30.5
31
31.5
32
0
75
150
225
300
375
450
t [min]
1 G
[m in m m ]
T
[◦
C
]
(1/G)Scst
(1/G)′ = (1/G)∗
TScst
T′
T∗
Figure 1: Control in t-domain
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
29
29.5
30
30.5
31
31.5
32
0
75
150
225
300
375
450
τ [mm]
1 G
[m in m m ]
T
[◦
C
]
(1/G)Scst
(1/G)′ = (1/G)∗
TScst
T′
T∗
Figure 2: Control in τ -domain
66
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
4
5
6
7
t [min]
3
,s ,
3
,n [c m 3
]
µScst
3,s
µScst
3,n
µ′
3,s
= µ∗
3,s
µ′
3,n
µ∗
3,n
Figure 3: Trajectories of the third moments
99.5 99.6 99.7 99.8 99.9 100
1.2
1.6
2
2.4
2.8
t [min]
3
,n [c m 3
]
µScst
3,n
µ′
3,n
µ∗
3,n
Figure 4: Magnification of Figure 3
These results confirm that, the smaller the relative
amount of the third moment of the nucleated CSD gets,
the better the simplified model is suited for finding an
approximate solution to the optimal control problem.
Other tests have shown that the relative errors stay
very similar for a wide range of final time constraints
tf,c.
As in previous results, the computed optimal trajectories exhibit a steep drop in temperature at the end,
corresponding to a surge in supersaturation. This effect can also be seen on the curves in τ -domain, which
are “stretched” near τf , compared to the curves in tdomain. Crystals that nucleate as τ approaches τf do
not attain a high final length (only τf − τ). Therefore,
nucleation and growth can be greatly increased toward
τf . Also, in τ -domain, the interval with constrained
input u∗ = ( 1G )min is clearly visible.
The implementation was done in MatlabTM. It uses
equidistant discretization of the interval [0, τf ] into ten
thousand steps and trapezoidal integration. The computing time for a single solution of the optimal control
problem for the simplified model was about 0.04 s on
one core of an AMD AthlonTM 64 X2 Dual Core Processor 3800+ running at 1 Ghz. The application of
the error correction scheme took about 1 s when using ode45 for the numerical integration of the detailed
model.
6 Conclusion
We have presented a new, mostly analytic, solution for
the optimal control of a batch crystallizer. The solution is based on a time transformation that turns a moment model of the crystallizer flat, on the Pontryagin
Minimum Principle, and on an appropriate, simplifying assumption. It has been argued and shown in a
case study, that the error coming from simplification
is sufficiently small when the goal is to suppress nucleation. Current work is done on solving variations of
the optimal control problem as well as incorporating
constraints on temperature rather than on growth rate
or supersaturation.
We gratefully acknowledge funding by the German Research
Foundation under research grant DFG RA 516/7-1.
References
[1] J. W. Mullin and J. Nyvlt, Programmed cooling of batch crystallizers. Chem. Eng. Sci. 26:3 (1971), pp. 369-377.
[2] A. G. Jones, Optimal Operation of a Batch Cooling Crystallizer. Chem. Eng. Sci. 29 (1974), pp. 1075-1087.
[3] M. B. Ajinkya and W. H. Ray, On the Optimal Operation of
Crystallization Processes. Chem. Eng. Commun. 1:4 (1974),
pp. 181-186.
[4] M. Morari, Some Comments on the Optimal Operation of
Batch Crystallizers. Chem. Eng. Commun. 4:1-3 (1980), pp.
167-171.
[5] S. M. Miller and J. B. Rawlings, Model identification and control strategies for batch cooling crystallizers. AIChE J. 40:8
(1994), pp. 1312-1327.
[6] Y. Lang, A. M. Cervantes and L. T. Biegler, Dynamic Optimization of a Batch Cooling Crystallization Process. Ind.
Eng. Chem. Res. 38:4 (1999), pp. 1469-1477.
[7] J. P. Corriou and S. Rohani, A new look at optimal control of
a batch crystallizer. AIChE J. 54:12 (2008), pp. 3188-3206.
[8] G. P. Zhang and S. Rohani, On-line optimal control of a seeded
batch cooling crystallizer. Chem. Eng. Sci. 58:9 (2003), pp.
1887-1896.
[9] Z. K. Nagy and R. D. Braatz, Robust nonlinear model predictive control of batch processes. AIChE J. 49:7 (2003), pp.
1176-1786.
[10] E. Visser, B. Srinivasan, S. Palanki and D. Bonvin, A FeedbackBased Implementation Scheme for Batch Process Optimization. J. of Proc. Cont. 10:5 (2000), pp. 399-410.
[11] U. Vollmer and J. Raisch, Control of batch cooling crystallisers
based on orbital flatness. Int. J. of Cont. 76:16 (2003), pp.
1635-1643.
[12] Q. Hu, S. Rohani, D. X. Wang and A. Jutan, Optimal Control
of a Batch Cooling Seeded Crystallizer. Pow. Tech. 156:2-3
(2005), pp. 170-176.
[13] Q. Hu, S. Rohani and A. Jutan, Modelling and optimization
of seeded batch crystallizers. Comp. and Chem. Eng. 29:4
(2005), pp. 911-918.
[14] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and
E. F. Mishchenko, The Mathematical Theory of Optimal Processes. John Wiley & Sons (1962).
[15] D. E. Kirk, Optimal Control Theory. Prentice-Hall (1970).
67
* This work were presented and published at the IFAC Symposium on Dynamics and Control of Process Systems (DYCOPS)
in Belgium in July 2010.
The setpoint overshoot method: A super-fast approach to PI tuning
Mohammad Shamsuzzohaa,*Sigurd Skogestada, Ivar J. Halvorsenb
aNorwegian University of Science and Technology,
Trondheim, Norway, (shamsuzz@chemeng.ntnu.no), (*skoge@ntnu.no)
bSINTEF ICT, Applied Cybernetics, N-7465 Trondheim, Norway
Abstract: A simple method has been developed for PI controller tuning of an unidentified process using
closed-loop experiments. The proposed method is similar to the Ziegler-Nichols (1942) tuning method,
but it is faster to use and does not require the system to approach instability with sustained oscillations.
The method requires one closed-loop step setpoint response experiment using a proportional only
controller with gain Kc0. From the setpoint response one observes the overshoot, the corresponding time
to reach the peak (tp) and the steady-state change (b=y(∞)/ys). Based on simulations for a range of firstorder with delay processes, simple correlations have been derived to give PI controller settings similar to
those of the SIMC tuning rules (Skogestad, 2003). The controller gain (Kc/Kc0) is only a function of the
overshoot observed in the setpoint experiment whereas the controller integral time (τI) is mainly a
function of the time to reach the peak (tp). Importantly, the method includes a detuning factor F that
allows the user to adjust the final closed-loop response time and robustness. The proposed tuning method,
originally derived for first-order with delay processes, has been tested on a wide range of other processes
typical for process control applications and the results are comparable with the SIMC tunings using the
open-loop model.
Keywords: PI controller, step test, closed-loop response, IMC, overshoot
1. INTRODUCTION
The proportional integral (PI) controller is widely used in the
process industries due to its simplicity, robustness and wide
ranges of applicability in the regulatory control layer. On the
basis of a survey of more than 11 000 controllers in process
industries, Desborough and Miller (2002) have reported that
more than 97% of regulatory controllers utilise the PID
algorithm. A recent survey (Kano and Ogawa; 2009) from
Japan shows that the ratio of applications of PID control,
conventional advanced control (feedforward, ratio, valve
position control, etc.) and model predictive control is about
100:10:1. In addition, the vast majority of the PID controllers
do not use derivative action. Even though the PI controller
only has two adjustable parameters, they are often poorly
tuned. One reason is that quite tedious plant tests may be
needed to obtain improved controller setting. The objective of
this paper is to derive a method which is simpler to use than
the present ones.
To obtain the information required for tuning the controller
one may use open-loop or closed-loop plant tests. Most
tuning approaches are based on open-loop plant information;
typically the plant’s gain (k), time constant (τ) and time delay
(θ). One popular approach is direct synthesis (Seborg et al.,
2004) which includes the IMC-PID tuning method of Rivera
et al. (1986). The original direct synthesis approaches give
very good performance for setpoint changes but give sluggish
responses to input (load) disturbances for lag-dominant
(including integrating) processes with τ/θ larger than about
10. To improve load disturbance rejection, Skogestad (2003)
proposed the modified SIMC method where the integral time
is reduced for processes with a large value of the time
constant τ. The SIMC rule has one tuning parameter, the
closed-loop time constant τc, and for “fast and robust” control
is recommended to choose τc= θ, where θ is the (effective)
time delay. However, these approaches require that one first
obtains an open-loop model of the process. There are two
problems here. First, an open-loop experiment, for example a
step test, is normally needed to get the required process data.
This may be time consuming and may upset the process and
even lead to process runaway. Second, approximations are
involved in obtaining the process parameters (e.g., k, τ and θ)
from the data.
The main alternative is to use closed-loop experiments. One
approach is the classical method of Ziegler-Nichols (1942)
which requires very little information about the process.
However, there are several disadvantages. First, the system
needs to be brought its limit of instability and a number of
trials may be needed to bring the system to this point. To
avoid this problem one may induce sustained oscillation with
an on-off controller using the relay method of Åström and
Hägglund, (1984). However, this requires that the feature of
switching to on/off-control has been installed in the system.
Another disadvantage is that the Ziegler-Nichols (1942)
tunings do not work well on all processes. It is well known
that the recommended settings are quite aggressive for lagdominant (integrating) processes (Tyreus and Luyben, 1992)
and quite slow for delay-dominant process (Skogestad,
2003). A third disadvantage is of the Ziegler-Nichols (1942)
method is that it can only be used on processes for which the
phase lag exceeds -180 degrees at high frequencies. For
example, it does not work on a simple second-order process.
Therefore, there is need of an alternative closed-loop
approach for plant testing and controller tuning which avoids
the instability concern during the closed-loop experiment,
reduces the number of trails, and works for a wide range of
processes. The proposed new method satisfies the above
68
concerns: In summary, the proposed method is simpler in use
than existing approaches and allows the process to be kept
under closed-loop control.
An obvious alternative to the proposed method is a two-step
procedure where one first identifies an open-loop model from
the closed-loop setpoint experiment, and then obtains the PI
or PID controller using standard tuning rules (e.g., the SIMC
rules of Skogestad, 2003). This approach was used by
Yuwana and Seborg (1982). We found that this two-step
approach gives result comparable or slightly inferior
(Shamsuzzoha and Skogestad, 2010) to the more direct
approach proposed in this paper by using the SIMC method.
In addition, the proposed approach avoids the extra step of
obtaining the process parameters (k, τ, θ) and is therefore
simpler to use.
2. SIMC PI TUNING RULES
In process control, a first-order process with time delay is a
common representation of the process dynamics:
-θskeg(s)=
τs+1
(1)
Here k is the process gain, τ the dominant lag time constant
and θ the effective time delay. Most processes in the
chemical industries can be satisfactorily controlled using a PI
controller:
( ) c
I
1c s =K 1+
τ s
⎛ ⎞
⎜ ⎟
⎝ ⎠
(2)
The PI controller has two adjustable parameters, the
proportional gain Kc and the integral time τI. The ratio
KI=Kc/τI is known as the integral gain.
The SIMC tuning rule is widely used in the process industry
and for the process in Eq. (1) is given as:
( )c c
τK =
k τ +θ
(3)
{ }I cτ =min τ, 4(τ +θ) (4)
Note that the original IMC tuning rule (Rivera et al., 1986)
always uses τI = τ, but the SIMC rule increases the integral
contribution for close-to integrating processes (with τ large)
to avoid poor performance (slow settling) to load disturbance.
There is one adjustable tuning parameter, the closed-loop
time constant (τc), which is selected to give the desired tradeoff between performance and robustness. Initially, this study
is based on the “fast and robust” setting τc =θ, which gives a
good trade-off between performance and robustness. In terms
of robustness, this choice gives a gain margin is about 3 and a
sensitivity peak (Ms-value) of about 1.6. On dimensionless
form, the SIMC tuning rules with τc = θ become
'
c c
τK =kK =0.5
θ
(5)
' c
I
I
kK 1 τK = =max 0.5,
τ θ 16 θ
⎛ ⎞
⎜ ⎟
⎝ ⎠
(6)
The dimensionless gains Kc΄ and KI΄ are plotted as a function
of τ/θ in Fig. 1. We note that the integral term (KI΄) is most
important for delay dominant processes (τ/θ<1), while the
proportional term Kc΄ is most significant for processes with a
smaller time delay. These insights are useful for the next step
when we want to derive tuning rules based on the closed-loop
setpoint response.
3. CLOSED-LOOP EXPERIMENT
As mentioned earlier, the objective is to base the controller
tuning on closed-loop data. The simplest closed-loop
experiment is probably a setpoint step response (Fig. 2)
where one maintains full control of the process, including the
change in the output variable. The simplest to observe is the
time tp to reach the (first) overshoot and its magnitude, and
this information is therefore the basis for the proposed
method.
We propose the following procedure:
1. Switch the controller to P-only mode (for example,
increase the integral time τI to its maximum value or set the
integral gain KI to zero). In an industrial system, with
bumpless transfer, the switch should not upset the process.
2. Make a setpoint change that gives an overshoot between
0.10 (10%) and 0.60 (60%); about 0.30 (30%) is a good
value. Record the controller gain Kc0 used in the experiment.
Most likely, unless the original controller was quite tightly
tuned, one will need to increase the controller gain to get a
sufficiently large overshoot.
Note that small overshoots (less than 0.10) are not considered
because it is difficult in practice to obtain from experimental
data accurate values of the overshoot and peak time if the
overshoot is too small. Also, large overshoots (larger than
about 0.6) give a long settling time and require more
excessive input changes. For these reasons we recommend
using an “intermediate” overshoot of about 0.3 (30%) for the
closed-loop setpoint experiment.
3. From the closed-loop setpoint response experiment, obtain
the following values (see Fig. 2):
• Fractional overshoot, (Δyp - Δy∞) /Δy∞
• Time from setpoint change to reach peak output
(overshoot), tp
• Relative steady state output change, b = Δy∞/Δys.
Here the output variable changes are:
Δys: Setpoint change
Δyp: Peak output change (at time tp)
Δy∞: Steady-state output change after setpoint step test
To find Δy∞ one needs to wait for the response to settle,
which may take some time if the overshoot is relatively large
(typically, 0.3 or larger). In such cases, one may stop the
experiment when the setpoint response reaches its first
minimum and record the corresponding output, Δyu.
Δy∞ = 0.45(Δyp + Δyu) (7)
4. CORRELATION BETWEEN SETPOINT RESPONSE
AND SIMC-SETTINGS
The objective of this paper is to provide a more direct
approach similar to the Ziegler-Nichols (1942) closed-loop
method. Thus, the goal is to derive a correlation, preferably
as simple as possible, between the setpoint response data
(Fig. 2) and the SIMC PI-settings in Eq. (3) and (4), initially
with the choice τc=θ. For this purpose, we considered 15 firstorder with delay models g(s)=ke-θs/(τs+1) that cover a wide
range of processes; from delay-dominant to lag-dominant
(integrating):
69
τ/θ=0.1,0.2,0.4,0.8,1.0,1.5,2.0,2.5,3.0,7.5,10.0,20.0,50.0,100
Since we can always scale time with respect to the time delay
(θ) and since the closed-loop response depends on the
product of the process and controller gains (kKc) we have
without loss of generality used in all simulations k=1 and
θ=1.
'
cK '
IK
τ θ
Iτ τ= I 8τ θ=
Fig. 1. Scaled proportional and integral gain for SIMC tuning rule.
pyΔ
pt y∞Δ syΔ
t0t =
θ
uyΔ
syΔ
Fig. 2. Closed-loop step setpoint response with P-only control.
For each of the 15 process models (values of τ/θ), we
obtained the SIMC PI-settings (Kc and τI) using Eqs. (3) and
(4) with the choice τc=θ. Furthermore, for each of the 15
processes we generated 6 closed-loop step setpoint responses
using P-controllers that give different fractional overshoots.
Overshoot= 0.10, 0.20, 0.30, 0.40, 0.50 and 0.60
In total, we then have 90 setpoint responses, and for each of
these we record four data: the P-controller gain Kc0 used in
the experiment, the fractional overshoot, the time to reach the
overshoot (tp), and the relative steady-state change, b =
Δy∞/Δys.
Controller gain (Kc). We first seek a relationship between
the above four data and the corresponding SIMC-controller
gain Kc. Indeed, as illustrated in Fig. 3, where we plot kKc
(SIMC PI-controller for the corresponding process) as a
function of kKc0 for the 90 setpoint experiments, the ratio
Kc/Kc0 is approximately constant for a fixed value of the
overshoot, independent of the value of τ/θ. Thus, we can
write
c c0 K =A
K
(8)
where the ratio A is a function of the overshoot only. In Fig.
4 we plot the value of A, which is obtained as the best fit of
the slopes of the lines in Fig. 3, as a function of the
overshoot. The following equation (solid line in Fig. 4) fits
the data in Fig. 3 well,
A=[1.152(overshoot)2 -1.607(overshoot)+1.0] (9)
0 20 40 60 80 100 120
0
10
20
30
40
50
kK c kKc0
0.10 overshoot
kKc=0.8649kKc0
0.20 overshoot
kKc=0.7219kKc0
0.30 overshoot
kKc=0.6259kKc0
0.40 overshoot
kKc=0.5546kKc0
0.50 overshoot
kKc=0.4986kKc0
0.60 overshoot
kKc=0.4526kKc0
0 1 2 3 4 5 6
0
0.5
1
1.5
2
2.5
Fig. 3. Relationship between P-controller gain kKc0 used in
setpoint experiment and corresponding SIMC controller gain
kKc.
Actually, a closer look at Fig. 3 reveals that a constant slope,
use of Eq.(8) and (9), only fits the data well for Kc΄=kKc
greater than about 0.5. Fortunately, a good fit of the
controller gain Kc is not so important for delay-dominant
processes (τ/θ<1) where Kc΄<0.5, because we recall from the
discussion of the SIMC rules (Fig. 1) that the integral gain KI
is more important for such processes. This is discussed in
more detail below.
Integral time (τI). Next, we want to find a simple correlation
for the integral time. Since the SIMC tuning formula in Eq.
(4) uses the minimum of two values, it seems reasonable to
look for a similar relationship, that is, to find one value (τI1
=τ) for processes with a relatively large delay, and another
value (τI2 =8θ) for processes with a relatively small delay
including integrating processes.
(1) Process with relatively large delay. For processes with a
relatively large delay τ/θ<8 or θ>τ/8, the SIMC-rule is to use
τI = τ. Inserting τ = τI into the SIMC rule for Kc in Eq. (5) and
solving for τI gives:
I1 cτ =2kK θ (10)
As just mentioned, for processes with a relatively large delay
it is the integral gain KI=Kc/τI that matters most (Fig. 1) and
to avoid that any error in Kc originating from our correlation
Eq.(8) propagates into KI, we should in Eq. (10) use Kc =
Kc0A, where A is given as a function of the overshoot in Eq.
(9). In (10), we also need the value of the process gain k, and
to this effect write
kKc= kKc0.Kc/ Kc0 (11)
Here, the value of the loop gain kKc0 for the P-control
setpoint experiment is given from the value of b:
70
c0 bkK =
(1-b)
(12)
0.1 0.2 0.3 0.4 0.5 0.6
0.4
0.5
0.6
0.7
0.8
0.9
overshoot (fractional)
A
A=1.152(overshoot)2 -1.607(overshoot)+1.0
Fig. 4. Variation of A with overshoot using data (slopes)
from Fig. 3.
To prove this, the closed-loop setpoint response is Δy/Δys =
gc/(1+gc) and with a P-controller with gain Kc0, the steadystate value is Δy∞/Δys = kKc0/(1+kKc0)=b and we derive
Eq.(12). The absolute value is included to avoid problems if
b>1, as may occur for an unstable process or because of
inaccurate data.
In summary, we have derived following expression for τI for
a delay-dominant process:
( )I1
bτ =2A θ
1-b
(13)
One could obtain the effective time delay θ directly from the
closed-loop setpoint response, but this is generally not easy.
Fortunately, as shown in Fig. (5), there is a reasonably good
correlation between θ and the setpoint peak time tp which is
easier to observe. For processes with a relatively large time
delay (τ/θ<8), the ratio θ/tp varies between 0.27 (for τ/θ= 8
with overshoot=0.1) and 0.5 (for τ/θ=0.1 with all overshoots).
For the intermediate overshoot of 0.3, the ratio θ/tp varies
between 0.32 and 0.50. A conservative choice would be to
use θ=0.5tp because a large value increases the integral time.
However, to improve performance for processes with smaller
time delays, we propose to use θ=0.43tp which is only 14%
lower than 0.50 (the worst case).
In summary, we have for process with a relatively large time
delay:
( )I1 p
bτ =0.86 A t
1-b
(14)
(2) Process with relatively small delay. For a lag-dominant
(including integrating) process with τ/θ>8 the SIMC rule
gives
τI2=8θ (15)
For τ/θ>8 we see from Fig. (5) that the ratio θ/tp varies
between 0.25 (for τ/θ=100 with overshoot=0.1) and 0.36 (for
τ/θ=8 with overshoot 0.6). We select to use the average value
θ= 0.305tp which is only 15% lower than 0.36 (the worst
case). Also note that for the intermediate overshoot of 0.3, the
ratio θ/tp varies between 0.30 and 0.32. In summary, we have
for a lag-dominant process
I2 pτ =2.44t (16)
Conclusion. The integral time τI is obtained as the minimum
of the above two values:
( )I p p
bτ =min 0.86A t , 2.44t
1-b
⎛ ⎞
⎜ ⎟
⎜ ⎟
⎝ ⎠
(17)
0.1 0.3 0.5 0.6
0
0.1
0.2
0.3
0.4
0.5
Overshoot
θ/
t p
0.43 (τI1)
0.305 (τI2)
τ/θ=0.1
τ/θ=8
τ/θ=100
τ/θ=1
Fig. 5. Ratio between delay and setpoint overshoot peak time
(θ/tp) for P-only control of first-order with delay processes
(solid lines); Dotted lines: values used in final correlations.
5. ANALYSIS AND SIMULATION
Closed-loop simulations have been conducted for 7 different
processes and the proposed tuning procedure provides in all
cases acceptable controller settings with respect to both
performance and robustness. For each process, PI-settings are
obtained based on step response experiments with three
different overshoot (around 0.1, 0.3 and 0.6) and are
compared with the SIMC settings.
The closed-loop performance is evaluated by introducing a
unit step change in both the set-point and load disturbance i.e,
(ys=1 and d=1). To evaluate the robustness, the maximum
sensitivity, Ms, defined as sM =max 1/[1+gc(iω)] , is used.
Since Ms is the inverse of the shortest distance from the
Nyquist curve of the loop transfer function to the critical
point (-1, 0), a small Ms-value indicates that the control
system has a large stability margin.
The results for the 7 example processes, which include the
different types of the process mainly stable, integrating and
unstable plant dynamics, are listed in Table 1.
All results are without detuning (F=1). The complete
simulation results with additional examples are available in a
technical report (Shamsuzzoha and Skogestad, 2010). As
expected, when the method is tested on first-order plus delay
processes, similar to those used to develop the method, the
responses are similar to the SIMC-responses, independent of
the value of the overshoot. Typical cases are E1, E2 (pure
time delay) and E3 (integrating with delay); see Figs. 6-8.
For models that are not first-order plus delay (typical cases
are E4, E5 and E6, see Fig. 9 for E6 only), the agreement
with the SIMC-method is best for the intermediate overshoot
(around 0.3). A small overshoot (around 0.1) typically give
71
"slower" and more robust PI-settings, whereas a large
overshoot (around 0.6) gives more aggressive PI-settings. In
some sense this is good, because it means that a more
"careful" step response results in more "careful" tunings.
Also note that the user always has the option to use the
detuning factor F to correct the final tunings. Case E7 (Fig.
10) illustrates that the method works well for a simple
unstable process with delay.
0 20 40 60 80
0
0.25
0.5
0.75
1
1.25
time
O
U
TP
U
T
y Proposed method with F=1 (overshoot=0.10)
Proposed method with F=1 (overshoot=0.298)
Proposed method with F=1 (overshoot=0.599)
SIMC (τc=θ=1)
Fig. 6. Responses for PI-control of (5 1)sg e s−= + (E1).
0 6 12 18 24 30
0
0.5
1
1.5
2
time
O
U
TP
U
T
y Proposed method with F=1 (overshoot=0.10)
Proposed method with F=1 (overshoot=0.30)
Proposed method with F=1 (overshoot=0.60)
SIMC (τc=θ=1)
Fig. 7 Responses for PI-control of sg e−= (E2).
0 20 40 60 80 100
0
0.5
1
1.5
2
2.5
3
time
O
U
TP
U
T
y Proposed method with F=1 (overshoot=0.108)
Proposed method with F=1 (overshoot=0.302)
Proposed method with F=1 (overshoot=0.60)
SIMC (τc=θ=1)
Fig. 8. Responses for PI-control of sg e s−= (E3).
0 40 80 120 160 200
0
1
2
3
4
5
time
O
U
TP
U
T
y Proposed method with F=1 (overshoot=0.106)
Proposed method with F=1 (overshoot=0.307)
Proposed method with F=1 (overshoot=0.610)
SIMC (τc=θeffective=1.5)
Fig. 9. Responses for PI-control of ( )21 1g s s⎡ ⎤= +⎢ ⎥⎣ ⎦
(E6).
0 20 40 60 80
0
0.5
1
1.5
2
time
O
U
TP
U
T
y Proposed method with F=1 (overshoot=0.10)
Proposed method with F=1 (overshoot=0.30)
Proposed method with F=1 (overshoot=0.607)
Fig. 10. Responses for PI-control of (5 1)sg e s−= − (E7).
6. CONCLUSION
A simple and new approach for PI controller tuning has been
developed. It is based on a single closed-loop setpoint step
experiment using a P-controller with gain Kc0. The PIcontroller settings are then obtained directly from following
three data from the setpoint experiment:
• Overshoot, (Δyp - Δy∞) /Δy∞
• Time to reach overshoot (first peak), tp
• Relative steady state output change, b = Δy∞/Δys.
If one does not want to wait for the system to reach steady
state, one can use the estimate Δy∞ = 0.45(Δyp + Δyu).
The proposed tuning formulas for the proposed “Setpoint
Overshoot Method” method are:
c c0K = K A F
( )I p p
bτ =min 0.86A t , 2.44t F
1-b
⎛ ⎞
⎜ ⎟
⎜ ⎟
⎝ ⎠
where, 2A= 1.152(overshoot) - 1.607(overshoot) + 1.0⎡ ⎤⎣ ⎦
The factor F is a tuning parameter and F=1 gives the “fast
and robust” SIMC settings corresponding to τc=θ. To detune
the response and get more robustness one selects F>1, but in
special cases one may select F<1 to speed up the closed-loop
response.
The Setpoint Overshoot Method works well for a wide
variety of the processes typical for process control, including
the standard first-order plus delay processes as well as
72
integrating, high-order, inverse response, unstable and
oscillating process.
We believe that the proposed method is the simplest and
easiest approach for PI controller tuning available and should
be well suited for use in process industries.
REFERENCES
Åström, K. J., Hägglund, T. (1984). Automatic tuning of
simple regulators with specifications on phase and
amplitude margins, Automatica, (20), 645–651.
Desborough, L. D., Miller, R. M. (2002). Increasing
customer value of industrial control performance
monitoring—Honeywell’s experience. Chemical
Process Control–VI (Tuscon, Arizona, Jan. 2001),
AIChE Symposium Series No. 326. Volume 98, USA.
Kano, M., Ogawa, M. (2009). The state of art in advanced
process control in Japan, IFAC symposium ADCHEM
2009, Istanbul, Turkey.
Rivera, D. E., Morari, M., Skogestad, S. (1986). Internal
model control. 4. PID controller design, Ind. Eng.
Chem. Res., 25 (1) 252–265.
Seborg, D. E., Edgar, T. F., Mellichamp, D. A., (2004).
Process Dynamics and Control, 2nd ed., John Wiley &
Sons, New York, U.S.A.
Shamsuzzoha, M., Skogestad. S. (2010). Report on the
setpoint overshoot method (extended version)
http://www.nt.ntnu.no/users/skoge/.
Skogestad, S., (2003). Simple analytic rules for model
reduction and PID controller tuning, Journal of
Process Control, 13, 291–309.
Tyreus, B.D., Luyben, W.L. (1992). Tuning PI controllers
for integrator/dead time processes, Ind. Eng. Chem.
Res. 2628–2631.
Yuwana, M., Seborg, D. E., (1982). A new method for online controller tuning, AIChE Journal 28 (3) 434-440.
Ziegler, J. G., Nichols, N. B. (1942). Optimum settings for
automatic controllers. Trans. ASME, 64, 759-768.
Case Process model kc0 overshoot tp b kc τI Ms
2.75 0.10 3.60 0.733 2.338 7.240 1.50
4.0 0.298 3.049 0.80 2.494 6.538 1.56
5.75 0.599 2.705 0.852 2.592 6.030 1.60
E1
( )5 1
se s
+
SIMC - - - 2.50 5.0 1.59
0.10 0.10 2.0* 0.091 0.085 0.146 1.60
0.30 0.30 2.0 0.231 0.187 0.321 1.53
0.60 0.60 2.0 0.270 0.465 0.375 1.59
E2 se−
SIMC - - - kc/τI=0.50 1.59
0.59 0.108 3.976 1.0 0.495 9.702 1.67
0.80 0.302 3.282 1.0 0.496 8.008 1.70
1.10 0.60 2.909 1.0 0.496 7.098 1.72
E3 se
s
SIMC - - - 0.50 8.0 1.70
0.07 0.112 18.132 0.387 0.058 8.198 1.46
0.12 0.301 15.043 0.519 0.074 8.667 1.61
0.18 0.583 13.71 0.618 0.082 8.684 1.70
E4 ( ) ( ) ( )
( ) ( )
2 2
22
2 9 2 1 1 e
0.5 1 5 1
ss s s s
s s s
−+ + − + +
+ + +
SIMC - - - - - -
5.0 0.127 0.710 0.833 4.074 1.732 1.33
15.0 0.322 0.393 0.937 9.031 0.958 1.74
40.0 0.508 0.230 0.976 19.23 0.561 2.62
E5
( ) ( )
1
1 0.2 1s s+ +
SIMC - - - 5.5 0.80 1.56
0.32 0.106 8.985 1.0 0.270 21.923 1.51
0.58 0.307 6.188 1.0 0.357 15.10 1.75
1.15 0.610 4.492 1.0 0.516 10.961 2.30
E6
( )2
1
1s s +
SIMC - - - 0.330 12.0 1.76
3.10 0.10 4.647 1.476 2.636 10.54 2.12
4.0 0.30 3.671 1.333 2.487 7.852 2.33
5.30 0.607 3.164 1.233 2.379 6.475 2.67
E7
( )5 1
se s

SIMC - - - - - -
Table 1: PI controller setting for proposed method and comparison with SIMC method (τc=θeffective)
* For the pure time delay case (E4) use the end time of the peak (or add a small time constant to get tp in simulation).
73
Comparing PI tuning methods in a real benchmark
temperature control system
Finn Haugen∗
May 27, 2010
Abstract
This paper demonstrates a number of PI controller tuning methods being used to tune a
temperature controller for a real air heater.
Indexes expressing setpoint tracking and disturbance compensation, and stability margin
(robustness) are calculated. From these indexes and a personal impression about how
quick a method is to deliver the tuning result and how simple the method is to use, a
winning method is identified.
1 Introduction
The PI (proportional plus integral) controller
function is the most frequently used controller function in practical applications. The
PI controller stems from a PID controller
with the D-term (derivative) deactived. The
D-term is often deactivated because it amplifies random (high-frequent) measurement
noise, causing abrupt variations in the control signal. This paper assumes PI control
(not PID).
The continuous-time PI controller function is
as follows:
u (t) = Kce (t) +
Kc Ti ∫ t
0
e (τ) dτ (1)
where u is the control signal (the controller
output), e = r− y is the control error, where
r is the reference or setpoint and y is the
∗Telemark University College, Kjolnes
ring 56, 3918 Porsgrunn, Norway. E-mail:
finn.haugen@hit.no
process output variable (process measurement), Kc is the controller gain, and Ti is the
integral time. Kc and Ti are the controller
parameters which are to be tuned. In most
practical applications the continuous-time PI
controller is implemented as a corresponding
discrete-time algorithm based on a numerical approximation of the integral term. Typically, the sampling time of the discrete-time
controller is so small — compared to the dynamics (response-time or time-constant) of
the control system — that there is no significant difference between the behaviour of
the continuous-time PI controller and the
discrete-time PI controller. Concequently, in
this paper the sampling time is not regarded
as a tuning parameter.
This paper compares a number of methods
for tuning PI controllers using the following
measures:
1. Performance related to setpoint tracking
and disturbance compensation
2. Robustness against parameter changes
in the control loop
3. How quick and simple the method is to
use
Numerous studies about simulated control
systems exist (O’Dwyer, 2003) (Seborg et.al,
2004). However, in this paper only experiments on a physical system will be used as
the basis of the comparison of the tuning
methods. The system is a laboratory scale
air heater, cf. Section 2. I think it is particularly valuable to see various methods being applied to a physical system because a
1
74
physical system will always differ — more or
less — from the underlying model or assumptions of the controller tuning method. So,
applying a method to a physical system is
real testing. Of course, it would be nice to
accomplish such real tests with several different real processes, but that may be the topic
of future paper.
This paper contains the following subsequent
sections:
• Section 2: The experimental setup
• Section 3: The methods to be compared
Section 4: Measures to compare the tuning methods
• Section 5: Control tunings and results
• Section 6: Summary and discussion
• Section 7: Conclusions
2 The experimental setup
The physical system used in the experiments
is the air heater laboratory station shown in
Figure 1. The temperature of the air outlet is
controlled by adjusting the control signal to
the heater.1 The fan speed can be adjusted
manually with a potensiometer. Changes of
the fan speed is used as process disturbance.
The voltage drop across the potensiometer is
used to represent this disturbance.2
Figure 2 shows a block diagram of the temperature control system.
The nominal operating point of the system is
temperature at 34 oC and fan speed potensiometer position at 2.4 V (corresponding
1 The supplied power is controlled by an external voltage signal in the range [0 V, 5 V] applied
to a Pulse Width Modulator (PWM) which connects/disconnects the mains voltage (220 VAC) to
the heater. The temperature is measured with a
Pt100 element which in the end provides a voltage measurement signal. The National Instruments
USB-6008 is used as analog I/O device. Additional information about the air heater is available
at http://home.hit.no/~finnh/air_heater.
2 The potensiometer voltage is roughly in range
2.4 — 5.0 V, with 2.4 V representing minimum speed.
Temperature
sensor 1
Temperature
sensor 2
NI USB-6008
for analog I/O
Fan speed
adjustment
On/Off
switch
PC with
LabVIEW
USB cable
Electrical heater
Fan
Mains cable
(220/110 V)
3 x Voltage AI (Temp 1, Temp 2, Fan
indication)
1 x Voltage AO (Heating)
Air
Pulse Width
Modulator
(PWM)
PWM
indicator
AC/DC
converter
Pt100/
milliampere
transducer
Air pipe
Figure 1: The air heater lab station with NI
USB-6008 analog I/O device
Controller Process
Sensor
Reference
Process output
variable
(temperature )
Process
measurement
Control
signal
Filter
Filtered
(smoothed)
process
measurement
+
Measurement
noise
Process
disturbance
Control error
= Ref - Meas
r [oC]
ymf [
oC]
u [V] y [oC]
ym [
oC]
e [oC]
d n Figure 2: Block diagram of the temperature
control system
to a relatively low speed). The measurement filter is a time-constant filter with timeconstant 0.5 s. To demonstrate the setpoint
tracking the setpoint is changed from 34 to
35 oC, and — thereafter — to demonstrate the
disturbance compensation, the fan speed (air
flow through the pipe) is changed from minimum (i.e. indicating voltage of 2.4 V) to
maximum (5.0 V).
The temperature control system is implemented with National Instruments LabVIEW running on a PC.
75
3 The methods to be compared
In general, both experimental (model-free)
and model-based tuning methods are available. In this presentation methods of both
these classes will be tested, but among the
model-based methods only those methods
which that can be applied without automatic
system identification are compared. This is
because it is my view that system identification tools should not be used unless the
user has knowledge about the basic theoretical foundation of such methods and are able
to evaluate different estimated models, and
few control engineers have such knowledge.
In other words: The mathematical model to
be used in the tuning method must simple
and easy to estimate from experiments, e.g.
time-constant with time-delay models.
The methods which will be compared are the
following:
Open-loop methods, which are methods
based on experiments on the open-loop system (i.e. on the process itself, independent of
the controller, which may be present or not):
• Skogestad’s Model-based method (or:
the SIMC method — Simple Internal
Model Control) (Skogestad, 2003)
• Ziegler-Nichols’ Process Reaction Curve
method (or the Ziegler-Nichols’ OpenLoop method) (Ziegler and Nichols,
1942)
• Hägglund and Åstrøm’s Robust tuning
method (Hägglund and Åstrøm, 2002)
Closed-loop methods, which are methods
based on experiments on the already established closed-loop system (i.e. the feedback
control system):
• Ziegler-Nichols’ Ultimate Gain method
(or the Ziegler-Nichols’ Closed-Loop
method) (Ziegler and Nichols, 1942)
• Tyreus-Luyben’s method (which is
based on the Ziegler-Nichols’ method,
but with more conservative tuning),
(Luyben and Luyben, 1997)
• Relay method (using a relay function
to obtain the sustained oscillations as
in the Ziegler-Nichols’ method) (Åstrøm
and Hägglund, 1995)
• Sham’s Setpoint method (based on Skogestad’s SIMC method) (Shamsuzzoha
et. al., 2010)
• Good Gain method (Haugen, 2010)
Each of these methods are described in their
respective subsections of Section 5 of this paper.
The above list of tuning methods contains
well-known methods (i.e. often refered to in
literature), and also some methods which I
personally find interesting.
4 Measures for comparing
the tuning methods
The measures for comparing the different
methods of PI controller tuning are as follows:
1. Performance related to setpoint tracking and disturbance compensation:
(a) Setpoint tracking : The setpoint is
changed as a step of amplitude 1,
from 34 to 35 oC. The IAE (Integral
of Absolute Error) index, which is
frequently used in the litterature
to compare different control functions, is calculated over an interval
of 100 sec. The IAE is
IAE =
∫ tf
ti |e| dt (2)
where ti is the initial (or start) time
and tf is the final time. tf − ti =
100 sec. This IAE index is denoted
IAEs . The less IAEs value, the better control (assuming that the behaviour of the control signal has no
weight).
76
(b) Disturbance compensation: After
the temperature has settled at the
new setpoint, a disturbance change
is applied by adjusting the fan
speed voltage from 2.4 (min speed)
to 5 V (max speed). Again the IAE
index is calculated over an interval
of 100 sec. This IAE index is denoted IAEd .
2. Robustness against parameter changes
in the control loop is in terms of stability
robustness against parameter variations
in the control loop. An adjustable gain,
KL, is inserted into the loop (between
the controller and the process, in the
LabVIEW program). Nominally, KL =
1. For each of the tuning methods, the
KL that brings the control system to the
stability limit (i.e. the responses are sustained oscillations) is found experimentally. This KL value is then the gain
margin, ∆K, of the control loop.
It might be interesting also to insert an
adjustable time-delay, Tdelay, into the
loop (between the controller and the
process, in the LabVIEW program) and
find experimentally the time-delay increase in the loop which brings the control system to the stability limit. (This
is closely related to finding the phase
margin of the control loop in a frequency
response analysis.) However, it is assumed the gain margin is suffient to
express the stability robustness of the
loops in our case, and to simplify the
analysis, only the gain margin is considered.
3. How quick and simple a given
method is to use. It is necessary for a
tuning method to be attractive to a user
that it gives good results, but also that
it is not too complicated to use (i.e. requires lots of calculations) and that the
experiments does not take too long time
to accomplish. Both the quickness and
the simplicity of each of the methods are
evaluated with a number ranging from
10 (best) to 0.
5 Controller tunings and
results
The subsequent sections describes the controller tuning principle and the actual tuning
and results for each of selected tuning methods. The results are summarized in Table 3.
5.1 Skogestad’s method
Skogestad’s PID tuning method (Skogestad,
2003) (or: the SIMC method — Simple Internal Model Control) is a model-based tuning
method where the controller parameters are
expressed as functions of the process model
parameters. The process model is some
continuous-time transfer function. The control system tracking transfer function T (s),
which is the transfer function from the setpoint to the (filtered) process measurement,
is specified as a first order transfer function
with time delay:
T (s) =
ymf (s)
ySP (s)
=
1
TCs+ 1
e−τs (3)
where TC is the time-constant of the control
system which the user must specify, and τ is
the process time delay which is given by the
process model (the method can however be
used for processes without time delay, too).
Figure 3 shows the response in ymf after a
step in the setpoint ySP for (3).
Figure 3: Step response of the specified tracking transfer function (3) in Skogestad’s PID
tuning method
77
By equating tracking transfer function given
by (3) and the tracking transfer function for
the given process, and making some simplifying approximations to the time-delay term,
the controller becomes a PID controller or a
PI controller for the process transfer function
assumed.
The process transfer functions for which Skogestad has calculated tuning formulas includes time-constant with time-delay:
Hpsf (s) =
K
Ts+ 1
e−τs (4)
which is the model we will use. (The
other process models are given in (Skogestad, 2003).) For this process a PI-controller
is tuned as follows:3
Kc =
T
K (TC + τ)
(5)
Ti = min [T , c (TC + τ)] (6)
Originally, Skogestad sets the factor c to
c = 4 (7)
This gives good setpoint tracking. But the
disturbance compensation may become quite
sluggish (e.g. in integrator with time-delay
processes). In most control loops the disturbance compensation is the most important task for the controller. To obtain faster
disturbance compensation, you can try e.g.
c = 2. The drawback of such a reduction of
c is that there will be somewhat more overshoot in the setpoint step response, and that
the stability of the control loop will be somewhat reduced (the stability margins will be
reduced). Both values of c (4 and 2) will be
tried in this paper.
Skogestad suggests setting the closed-loop
system time-constant to
TC = τ (8)
Application to the air heater
To find a proper transfer function model, the
process was excited by a step change from 1.5
Figure 4: Skogestad’s method: Process step
response
to 1.8 V, see Figure 4. The response indicates
that a proper model is time-constant with
time-delay:
Hpsf (s) =
K
Ts+ 1
e−τs (9)
From the step response I found4
K = 5.7 oC/V; T = 60 s; τ = 4.0 s (10)
For the controller tuning I use (8):
TC = τ = 4.0 s (11)
The PI controller parameters are
Kc =
T
K (TC + τ)
=
60
5.7 · (4 + 4) = 1.3
(12)
Ti = min [T , c (TC + τ)] (13)
= min [60, 4 (4 + 4)] = 32.0 s (14)
Figure 5 shows control system responses with
the above PI settings.
3 “min” means the minimum value (of the two alternative values).
4 An exact value of the time-delay is not so easy to
determine from the response, but other experiments
indicate 4 sec or a somewhat less, so I set 4.0.
78
Figure 5: Skogestad’s method: Closed-loop responses
The IAE indexes and the gain margin was
IAEs = 12.5; IAEd = 27.2; ∆K = 2.4 = 7.6 dB
(15)
Figure 6 shows the responses with this gain
increase.
Figure 6: Skogestad’s method: Responses with
loop gain increase of 2.4
One interesting observation is that the actual time-constant (63% response time) as
seen from Figure 5 is approximately 5 sec,
which corresponds well with the specified
time-constant of 4 sec.
Finally, to try to obtain faster disturbance
compensation, let’s set
c = 2 (16)
Now we get
Ti = min [T , c (TC + τ)] (17)
= min [60, 2 (4 + 4)] = 16.0 s (18)
The controller gain is as before:
Kc = 1.3 (19)
Figure 7 shows control system responses with
the above PI settings.
Figure 7: Skogestad’s method: Closed-loop responses with c = 2
The IAE indexes and the gain margin was
IAEs = 18.1; IAEd = 18.3; ∆K = 2.2 = 6.8 dB
(20)
Thus, by setting c = 2 in stead of 4, the setpoint tracking is worse, but the disturbance
compensation is better. The gain margin is
only a little worse with c = 2. In most control loops the disturbance compensation is
the most important task for the controller.
5.2 Ziegler-Nichols’ Process Reaction Curve method
Ziegler and Nichols (1942) designed two tuning rules — known as the Ultimate Gain
method and the Process Reaction Curve
method — to give fast control but with acceptable stability. They used the following
definition of acceptable stability: The ratio
of the amplitudes of subsequent peaks in the
same direction (due to a step change of the
disturbance or a step change of the setpoint
in the control loop) is approximately 1/4.
The Ziegler-Nichols’ Process Reaction Curve
method ([9]) is based on characteristics of the
79
step response of the process to be controlled
(i.e. the open-loop system). The PID parameters are calculated from the response in
the (filtered) process measurement ymf after
a step with height U in the control variable u.
From the step response in ymf , read off the
equivalent dead-time or lag L and the rate or
slope R, see Figure 8.
Figure 8: Ziegler-Nichols’ open loop method:
The equivalent dead-time L and rate R read
off from the process step response. (The figure
is a reprint from [9] with permission.)
After this step response test, the controller
parameters are calculated with the formulas
in Table 1.
Kp Ti Td
P controller 1LR/U ∞ 0
PI controller 0.9LR/U 3.3L 0
PID controller 1.2LR/U 2L 0.5L =
Ti 4
Table 1: Ziegler-Nichols’ open loop method:
Formulas for the controller parameters.
Application to the air heater:
To tune the PI controller, I use the data from
the open-loop experiment recorded for Skogestad’s method, cf. Section 5.1. The timedelay is
L = τ = 4.0 s (21)
The slope R can be calculated as the initial
slope of the step response. For a first order
system,
R =
KU
T
(22)
where we have
K = 5.7 oC/V; T = 60 s (23)
The PI settings becomes
Kc =
0.9
LR/U
=
0.9
LKUTU
=
0.9
LK/T
= 2.4 (24)
Ti = 3.3 · L = 3.3 · 4 s = 13.2 s (25)
(Reading off R more directly from Figure 4
gives R = 0.025 oC/s, and Kc = 2.7.)
Figure 9 shows control system responses with
the above PI settings.
Figure 9: Ziegler-Nichols’ Process Reaction
Curve method: Responses in the control system
The setpoint response indicates that the stability is very poor. However, the disturbance
response indicated that the stability is ok.
The latter is due to the fact the increased fan
speed (increased air flow) reduces the process
gain and the process time-delay — thereby improving the stability of the control loop.
The IAE indexes and the gain margin was
IAEs = 19.5; IAEd = 8.6; ∆K = 1.2 = 1.6 dB
(26)
5.3 Hägglund-Åstrøm’s robust
tuning
Hägglund and Åstrøm (2002) have derived PI
controller tuning rules for “integrator with
time-delay” processes and “time-constant
with time-delay” processes giving maximum
80
performance given a requirement on robustness. The air heater looks like a “timeconstant with time-delay” process. Assuming the process model is
Hpsf (s) =
K
Ts+ 1
e−τs (27)
the PI controller settings according to Hägglund and Åstrøm are as follows:
Kc =
1
K
(
0.14 + 0.28
T
τ
)
(28)
Ti = τ
(
0.33 +
6.8T
10τ + T
)
(29)
Application to the air heater:
To tune the PI controller, I use the data from
the open-loop experiment recorded for Skogestad’s method, cf. Section 5.1:
K = 5.7 oC/V; T = 60 s; τ = 4.0 s (30)
The PI settings become
Kc = 0.76 (31)
Ti = 17.6 s (32)
Figure 10 shows control system responses
with the above PI settings.
Figure 10: Hägglund-Åstrøm’s Robust tuning
method: Closed-loop responses
The IAE indexes and the gain margin was
IAEs = 17.5; IAEd = 32.8; ∆K = 3.6 = 11.1 dB
(33)
5.4 Ziegler-Nichols’ Ultimate
Gain method (Closed-loop
method)
The Ziegler-Nichols’ Ultimate Gain method
is based on experiments executed on an established control loop (a real system or a simulated system): The ultimate proportional
gain Kċu of a P-controller (which is the gain
which causes sustained oscillations in the signals in the control system without the control signal reaching the maximum or minimum limits) must be found, and the ultimate
(or critical) period Pu of the sustained oscillations is measured. Then, the controller
is tuned using Kcu and Pu in the formulas
shown in Table 2.
Kc Ti Td
P controller 0.5Kcu ∞ 0
PI controller 0.45Kcu Pu1.2 0
PID controller 0.6Kcu Pu2
Pu 8
= Ti
4
Table 2: Formulas for the controller parameters
in the Ziegler-Nichols’ closed loop method.
Application to the air heater
Figure 11 shows the oscillations in the temperature response with the ulitmate gain
Figure 11: Ziegler-Nichols’ Ultimate Gain
method: Response with ultimate gain
Kcu = 3.4 (34)
The period of the oscillations is
Pu = 15 s (35)
81
The PI parameter values become
Kc = 0.45Kcu = 0.45 · 3.4 = 1.5 (36)
Ti =
Pu 1.2
=
15 s
1.2
= 12.5 s (37)
Figure 12 shows control system responses
with the above PI settings.
Figure 12: Ziegler-Nichols’ Ultimate Gain
method: Responses in the control system
The IAE indexes and the gain margin was
IAEs = 13.8; IAEd = 11.7; ∆K = 1.8 = 5.1 dB
(38)
5.5 Tyreus-Luyben’s tuning
method
The Tyreus and Luyben’s tuning method
(Luyben and Luyben, 1997) is based on oscillations as in the Ziegler-Nichols’ method, but
with modified formulas for the controller parameters to obtain better stability in the control loop compared with the Ziegler-Nichols’
method. For a PI controller they suggest
Kc = 0.31Kcu (39)
Ti = 2.2Pu (40)
Application to the air heater
Applying the same data as for the ZieglerNichols’ Ultimate Gain method, cf. Section
5.4, we get
Kc = 0.31Kcu = 0.31 · 3.4 = 1.1 (41)
Ti = 2.2Pu = 2.2 · 15 = 33 s (42)
Figure 13: Tyreus-Luyben’s method: Responses in the control system
Figure 13 shows control system responses
with the above PI settings.
The IAE indexes and the gain margin was
IAEs = 14.2; IAEd = 35.7; ∆K = 3.1 = 9.8 dB
(43)
5.6 Relay-based tuning method
Åstrøm-Hägglund’s relay-based method
(Åstrøm and Hägglund, 1995) can be regarded as a practical implementation of
the Ziegler-Nichols’ Ultimate Gain method.
In the Ziegler-Nichols’ method it may be
time-consuming to find the ultimate gain
Kcu. This problem is eliminated with the
relay-method of Åstrøm-Hägglund. The
method is based on using a relay controller
— or on/off controller – in the place of
the PID controller to be tuned during the
tuning. Due to the relay controller the
sustained oscillations in control loop will
come automatically. These oscillations will
have approximately the same period as if
the Ziegler-Nichols’ closed loop method were
used, and the ultimate gain Kcu can be
easily calculated, as explained below.
The parameters of the relay controller are
the ”high” (or ”on”) and the low (or ”off”)
control values, Uhigh and Ulow , respectively.
Once they are set, the amplitude A of the
relay controller is
A =
Uhigh − Ulow
2
(44)
If ”large” oscillation amplitude is allowed,
82
you can set (assuming that the control signal is scaled in percent)
Uhigh = Umax = 100% (typically) (45)
and
Ulow = Umin = 0% (typically) (46)
But there may be no relay controller in the
control system! You can turn the PID controller into a relay controller with the following settings:
Kc = very large, e.g. 1000; Ti =∞; Td = 0
(47)
With the relay controller in the loop, sustained oscillations comes automatically. The
ultimate gain of the relay controller can be
calculated as:
Kcu =
Amplitude of relay output
Amplitude of relay input
=
Au Ae
4A
π
E
=
4A
πE
(48)
where Ae = E is the amplitude of the oscillatory control error signal, and Au = 4A/π
is the amplitude of the first harmonic of an
Fourier series expansion of the square pulse
train on the output of the relay controller.
So, after the relay controller is set into action, you read off the ultimate period Pu from
any signal in the loop, and also calculate the
ultimate gain Kcu with (48). Finally, the
controller parameters can be calculated using
the Ziegler-Nichols’ formulas given in Table
2.5
Application to the air heater
The high and low control signals are, according to their physical limits:
Uhigh = Umax = 5 V (49)
5 I have experienced (at least with the PID Advanced controller in LabVIEW) that the period of the
oscillations are smaller than expected when the PID
controller is turned into a Relay controller by setting
Kcvery large, e.g. 1000, and Ti also very large. Probably this problem is due to the anti-windup function
combined with the P control action of the controller.
In the experiments (see below) accomplished for this
paper, I deactivated the anti-windup function by setting the max and min control signal limits to very
high values: 1000 and —1000, respectively. Doing this
I got the same amlitude and period of the oscialltions
as with an ideal relay function.
and
Ulow = Umin = 0 V (50)
Hence, the relay amplitude is
A = 2.5 V (51)
Figure 14 shows the oscillations in the tuning
phase. From Figure 14 we find the ultimate
Figure 14: Relay-tuning method: Responses in
the control system with relay controller
period
Pu = 18.0 s (52)
(which is quite equal to the period found
with the ultimate gain in Ziegler-Nichols’
method). The amplitude of the control error is appoximately
Ae = 0.9
◦C (53)
The ultimate gain becomes, cf. (48),
Kcu =
4A
πAe
=
4 · 2.5 V
π · 0.9 ◦C = 3.54 V/
◦C
(54)
The PI parameter values become
Kc = 0.45Kcu = 0.45 · 3.54 = 1.6 (55)
Ti =
Pu 1.2
=
18 s
1.2
= 15 s (56)
Figure 15 shows control system responses
with the above PI settings.
83
Figure 15: Relay tuning method: Responses in
the control system
The IAE indexes and the gain margin was
IAEs = 13.4; IAEd = 12.9; ∆K = 2.0 = 6.0 dB
(57)
5.7 Sham’s Setpoint method
Sham’s Setpoint method (Shamsuzzoha et.
al., 2010) is based on Skogestad’s SIMC
method. According to Skogestad himself, ”It
is simpler to use because it requires only one
setpoint experiment. It uses P-control — a
bit similar to Ziegler-Nichols — but is much
quicker because one does not need to crank
up the gain to get persistent oscillations.”
The method is as follows: Start by using a
P-controller with gain Kc0, and apply a setpoint change of amplitude ∆ySP . Kc0 should
be selected so that you get a proper overshoot
in the setpoint response (in the process output). A typical value is claimed to be 0.3.
From the setpoint response you read off the
maximum response, ymax, and the steadystate response, y (∞), and the time to reach
the peak, tp. Assume that the process output has value y0 before the setpoint change.
From these quantities you calculate the actual overshoot:
S =
ymax − y (∞)
y (∞)− y0
(58)
Also calculate the relative steady-state
change of the process output:
b =
y (∞)− y0
∆ySP
(59)
Define the following parameters:
F = 1 (60)
(F = 1 for ”fast robust control” corresponding to TC = τ in Skogestad’s SIMC method,
but use F > 1 to detune), and
A = 1.152 · S2 − 1.607 · S + 1.0 (61)
The PI parameter settings are
Kc = Kc0
A
F
(62)
Ti = min
[
(0.86Atp
b 1− b , 2.44tpF )
]
(63)
Application to the air heater
Figure 16 shows the closed-loop response to
a setpoint step change of amplitude ∆ySP =
1.0 oC with a P-controller with gain
Kc0 = 1.8 (64)
Figure 16: Sham’s Setpoint method: The responses with a P-controller with gain Kc0 =
1.8
which gives a stable response and a reasonable overshoot.From the responses we find
the actual overshoot as
S =
ymax − y (∞)
y (∞)− y0
=
35.25− 35.0
35.0− 34.1 = 0.28
(65)
The relative steady-state change of the
process output is
b =
y (∞)− y0
∆ySP
=
35.0− 34.1
1.0
= 0.9 (66)
84
We read off the peak time as
tp = 14 sec (67)
The PI parameter settings becomes
Kc = Kc0
A
F
= 1.8
0.64
1
= 1.2 (68)
Ti = min
[
(0.86Atp
b 1− b , 2.44tpF )
]
(69)
= min [(69.4, 34.2)] = 34.2 s (70)
Figure 17 shows control system responses
with the above PI settings.
Figure 17: Sham’s Setpoint method: The responses in the control system
The IAE indexes and the gain margin was
IAEs = 12.2; IAEd = 36.1; ∆K = 2.7 = 8.6 dB
(71)
5.8 Good Gain method
The Good Gain method6 (Haugen, 2010) is
a simple method based on experiments with
a P-controller, like in Sham’s method and
Ziegler-Nichols’ Ultimate Gain method. As
in Sham’s method, the system is not brought
into marginal stabililty during the tuning,
which is benefical. The procedure described
below assumes a PI-controller.
First, the process should be brought close to
the specified operation point with the controller in manual mode. Then, ensure that
the controller is a P-controller with Kc = 0
6 I am responsible for this name.
(set Ti = ∞ and Td = 0). Switch the controller to automatic mode. Find a good gain,
KcGG, by trial-and-error which gives the control loop good stability as seen in the response in the measurement signal due to a
step in the setpoint. I assume that a response
with a small overshoot and a barely observable undershoot (or the opposite, if the setpoint step is negative) represents good stability. A proper value of the integral time Ti
is (hopefully)
Ti = 1.5Tou (72)
where Tou is the time between the first
overshoot and the first undershoot of the step
response (a step in the setpoint) with the Pcontroller, see Figure 18.
TouSetpoint step
Step response
Figure 18: Reading off the time between the
first overshoot and the first undershoot of the
step response with P controller
Due to the inclusion of the integral term, the
control loop will get somewhat reduced stability than with the P-controller only. This
can be compensated for by reducing Kc to
e.g. 80% of the original value.
Here is the background of this method: Assume that the control loop with the Pcontroller behaves approximately as an underdamped second order system with the following transfer function model from setpoint
to process output:
y(s)
ySP (s)
=
Kω2
0
s2 + 2ζω0s+ ω20
(73)
It can be shown that with ζ = 0.6 the step
response is damped oscillations with an overshoot of about 10% and a barely observable
85
undershoot, as in the Good Gain tuning, and
that the period of the damped oscillations is
Pd =


1− ζ2ω0
=
2π√
1− 0.62ω0
(74)
=

0.8ω0
= PGG = 2Tou (75)
If the oscillations are undamped, as with the
Ziegler-Nichols’ Ultimate Gain method, the
period of the oscillations is
PZN =

ω0
(76)
Hence, the relation between the period of
the damped oscillations of the Good Gain
method and the undamped oscillations of the
Ziegler-Nichols’ method is approximately
PZN = 0.8PGG = 1.6Tou (77)
In the Ziegler-Nichols’ method we set
Ti =
PZN
1.2
=
1.6Tou
1.2
= 1.33Tou (78)
If we make the Ti setting somewhat more relaxed (to obtain better stability and better
robustness), we can increase Ti to
Ti = 1.5Tou (79)
In the Ziegler-Nichols’ method the controller
gain Kc of a PI-controller is 90% of the gain
of the P-controller. To compensate for the
inclusion of the integral term we can reduce
the original controller gain of the Good Gain
method to 90%, but to relax the setting even
more, let’s set
Kc = 0.8KcGG (80)
Application to the air heater
Figure 19 shows the closed-loop response to a
setpoint step change with a P-controller with
gain
KcGG = 1.5 (81)
The half-period is
Tou = 12 s (82)
Figure 19: Good Gain method: Response with
P-controller with gain KcGG = 1.4
The PI parameter values becomes
Kc = 0.8 ·KcGG = 0.8 · 1.5 = 1.2 (83)
Ti = 1.5 · Tou = 1.5 · 12 = 18 s (84)
Figure 20 shows control system responses
with the above PI settings.
Figure 20: Good Gain method: The responses
in the control system
The IAE indexes and the gain margin was
IAEs = 14.3; IAEd = 21.5; ∆K = 2.4 = 7.6 dB
(85)
6 Summary and discussion
Table 3 summarizes the results with the different tuning methods. Both the quickness
and the simplicity of each of the methods
are evaluated with a number ranging from
10 (best) to 0.
86
Kc Ti IA E s IA E d ∆K Q S
S 1 1 .3 3 2 .0 1 2 .5 2 7 .2 2 .4 1 0 9
S 2 1 .3 1 6 .0 1 8 .1 1 8 .4 2 .2 1 0 9
Z N -P 2 .4 1 3 .2 1 9 .5 8 .6 1 .2 1 0 8
H Å 0 .7 6 1 7 .6 1 7 .5 3 2 .8 3 .6 1 0 9
Z N -U 1 .5 1 2 .5 1 3 .8 1 1 .7 1 .8 5 6
T L 1 .1 3 3 .0 1 4 .2 3 5 .7 3 .1 5 6
R 1 .6 1 5 .0 1 3 .4 1 2 .9 2 .0 8 6
S S 1 .2 3 4 .2 1 2 .2 3 6 .1 2 .7 5 6
G G 1 .2 1 8 .0 1 4 .3 2 1 .5 2 .4 7 1 0
Table 3: Results for different PI controller tunings. (S1 = Skogestad original, with c = 4.
S2 = Skogestad with c = 2. ZN-P = ZieglerNichols’ Process Reaction Curve method. HÅ
= Hägglund-Åstrøm’s method. ZN-U =
Ziegler-Nichols’ Ultimate Gain method. TL =
Tyreus-Luyben’s method. R = Relay method.
SS = Sham’s Setpoint method. GG = Good
Gain method.
Methods which results in gain margin ∆K
less than 2.0 are here regarded as poor. So,
the Ziegler-Nichols’s methods are in trouble.
Also, a method should not give poor disturbance compensation, i.e. not too large
IAE index compared with other methods. In
this light, Tyreus-Luyben’s method, Sham’s
Setpoint method, and Hägglund-Åstrøm’s
method are in trouble, although the latter
gives a large stability margin. Also Skogestad’s method with the original value c = 4 is
doubtful. The Relay method works well, but
it is on the limit regarding robustness, and
it may be a little difficult to implement in a
practical system because the relay (on/off)
function is not immediately available. The
simple-to-use Good Gain method performs
ok, although it gives a little slow disturbance
compensation.
However, we want a winner, and Skogestad’s
method with c = 2 is the lucky one! It gives
good results and is simple to use.7 The model
that the method requires can be obtained
7 Additional benefits of Skogestad’s method are
application to processes without time-delay where
stability-based methods fail (as in level control), and
easy continuous adaptation of controller parameters
to possibly varying process model parameters.
from a simple step-response test.
7 Conclusions
This paper has demonstrated a number of
PI controller tuning methods being used to
tune a temperature controller for a real air
heater. Indexes expressing setpoint tracking and disturbance compensation, and stability margins (robustness) were calculated.
From these indexes and a personal impression about how quick a method is to deliver the tuning results and how simple the
method is to use, a winning method has been
identified (from the tests reported in this
paper), namely Skogestad’s method (with a
modified integral time tuning).
References
[1] Åstrøm, K. J, Hägglund, T., PID Controllers: Theory, Design and Tuning,
ISA, 1995
[2] Haugen, F., Basic Dynamics and Control, ISBN 978-82-91748-13-9, pp. 127129, TechTeach, 2010
[3] Hägglund, T., Åstrøm, K. J, Revisiting
the Ziegler-Nichols’ Tuning Rules for PI
Control, Asian J. Of Control, Vol. 4, 364,
2002
[4] Luyben, W. L., Luyben, M. L., Essentials
of Process Control, McGraw-Hill, New
York, 1997
[5] O’Dwyer, A., Handbook of Controller
Tuning Rules, Imperial College Press,
London, 2003
[6] Seborg, D. E., Edgar, Th. F., Mellichamp, D. A., Process Dynamics and
Control, Ch. 12, John Wiley and Sons,
2004
[7] Shamsuzzoha, M., Skogestad, S.,
Halvorsen, I. J., On-Line PI Controller
Tuning Using Closed-Loop Setpoint
87
Responses for Stable and Integrating
Processes, to be presented at IFAC
conference on dynamics and control of
process systems processes (DYCOPS),
Belgium, July 2010
[8] Skogestad, S., Simple analytic rules for
model reduction and PID controller tuning, Journal of Process Control, Vol. 13,
pp. 291-309, 2003
[9] Ziegler, J. G. and N. B. Nichols: Optimum Settings for Automatic Controllers,
Trans. ASME, Vol. 64, page 759-768,
1942
88
Extended Abstract for 16th Nordic Process Control Workshop, Lund, Sweden, 25-27th of August 2010
Tentative Dependence Analysis of Process Variables in a Circulating Fluidized Bed Boiler
Lohiniva, Laura & Leppäkoski, Kimmo
Systems Engineering Laboratory, Department of Process and Environmental Engineering
University of Oulu, Finland
laura.lohiniva@oulu.fi, kimmo.leppakoski@oulu.fi
1. INTRODUCTION
Ash-induced problems in boilers include deposit formation, accelerated corrosion and erosion of surfaces and
decrease in energy efficiency due to formation of insulating layers. Co-combustion of fossil fuels and biomasses of
various quality may even worsen the problems in unfavourable process conditions and fuel mixes. There are
various different approaches to investigating slagging and fouling of heat exchanger surfaces. There have been
various attempts to model, predict and monitor deposit formation (for example Henderson et al. 2006, Ma et al.
2008, Räsänen et al. 2006). If conventional process measurement variables could be used in monitoring, slagging
and fouling could be investigated during normal operation of the plant.
Preliminary data analysis, preceding a more accurate analysis and later control design, is used to study properties
of data and dependence among variables, but also in detecting errors. Methods can be quantitative or qualitative.
As computing power and data storage capacity have increased, effective utilization of soft computing methods has
become possible. The Self-Organizing Map (SOM) is one of the most popular methods used in data mining; it
performs a nonlinear dimensionality reduction on the data by using competitive unsupervised learning and then
produces a visualization of the results, usually on a 2-dimensional regular lattice (Kohonen 2001). In addition to
visualization and preliminary analysis, the SOM is also an efficient tool for modelling and data preprocessing
purposes as well as for reducing noise and computational load in clustering.
2. DATA AND METHODS
Ordinary process measurement data from a circulating fluidized bed boiler (CFB) was retrieved from the process
automation system of a power plant located in northern Finland. Main fuels used in the plant include peat (up to
80%), wood, bark and other wood-based materials. CFB boilers tolerate fuels of fluctuating quality and properties,
which makes them highly suitable for co-combustion. Data from the collection period, between September 2st and
December 31st 2008, was extracted from the data storage of the process automation system by an Excel
application. Missing values in data were replaced with mean approximations during collection. The plant normally
operates throughout the year. During the four collection months there were two shutdown periods (September
29th – October 10th, October 23rd – 29th) that were excluded from the analysis. The limited data consisted of 138
days, and since the data consists of minute averages, the number of sample rows was 198720. Two methods were
used in data analysis. Regular correlation coefficient calculation can be used to obtain numerical results. The SelfOrganizing Map (SOM) provides visual presentation of the results. Before analysis, data was further preprocessed
in several ways. Flue gas emissions were reduced to standard 6 % O2 content in order to obtain comparable and
usable values. Data was mean-centered and normalized to unit variance for SOM.
3. RESULTS AND DISCUSSION
Correlation coefficients (covariance of two variables divided by the product of their standard deviations, in the
range of [-1,1] and indication of linear dependency) were calculated for a group of variables. The results could be
compared to previous results (Lohiniva & Leppäkoski 2010) from bubbling fluidized bed boiler (BFB) data. The
differences in BFB and CFB processes, most importantly in the recirculation of bed material and unburnt fuel, and
the location of measurement points account for some of the differences.
The obtained coefficients showed that SO2 and NOX contents in flue gas had medium correlation (0.4647). In
previous studies, the two variables have been considered quite independent in the case of CFB boilers. SO2
content had medium correlation with flue gas temperatures in different parts of the flue. Flue gas oxygen contents
89
at economizer and before stack had surprisingly low correlation (0.5469). Flue gas pressure differences over three
parts of the flue gas duct had strong or almost linear correlation with each other. Flue gas temperatures did not
correlate with each other as strongly as expected. Temperatures and pressure differences had only medium
correlation (table 1). Most of the results differed from the results obtained from BFB data.
Table 1. Correlation coefficients between flue gas temperatures and pressure differences
T after superheaters T after economizer T after air preheaters
pd at superheaters 0.5695 0.6197 0.6256
pd at economizer 0.8735 0.8644 0.8705
pd at air preheaters 0.7401 0.7192 0.7345
The SOM results (map size [69,32], quantization error 1.590, topographical error 0.115) supported the results
from correlation calculation. Component planes effectively visualize, for example, the independence of SO2
content from other variables. The CO-O2-relation can be clearly seen. Steam flow, pressure difference and flue
gas temperature maps have similar variation, although the temperature planes are not alike. These differences
could be further investigated.
Figure 1. SOM component planes
Both methods provided fast and illustrative preliminary presentation of the data. The methods used are
inexpensive to compute and could be used in online monitoring.
REFERENCES
Henderson, P., Szakálos, P., Pettersson R, Andersson, C., and Högberg, J. (2006). Reducing superheater corrosion in
wood-fired boilers. Materials and Corrosion, 57 (2), pp. 128-134.
Kohonen, T. (2001). Self-Organising Maps, 3. p.. Springer Verlag, Berlin, Heidelberg, Germany.
Lohiniva, L. & Leppäkoski, K. (2010). Preliminary Dependence Analysis of Process Variable: A Case Study of a Bubbling
Fluidized Bed Boiler. IFAC Conference on Control Methodologies and Technology for Energy Efficiency, Vilamoura,
Portugal, March 29-31 2010. (CD)
Ma, Z., Iman F., Lu P., Sears R., Kong L., Rokanuzzaman A.S., McCollor D.P. and Benson S.A. (2008). A comprehensive
slagging and fouling prediction tool for coal-fired boilers and its validation/application. Fuel Processing Technology,
88 (11-12), pp. 1035-1043.
Räsänen, T., Kettunen, A., Niemitalo, E., and Hiltunen, Y. (2006). Self-refreshing SOM for dynamic process state
monitoring in a circulating fluidized bed energy plant. 3nd International IEEE Conference on Intelligent Systems.
September 2006, pp. 344-349.
90
Automated Controller Design using Linear Quantitative Feedback
Theory for Nonlinear systems
Roozbeh Kianfar and Torsten Wik∗
Abstract— A method to design simple linear controllers for
mildly nonlinear systems is presented. In order to design the
desired controller we approximate the behavior of the nonlinear
system with a set of linear systems which are derived through
linearizations. Classical local linearization is carried out around
stationary points but in order to have a better approximation
of the nonlinear system selected non-stationary points are taken
into account as well. This set of linear models are considered
as an uncertainty description for a nominal plant. Qunatitative
Feedback theory (QFT) may be used to guarantee specification
to be fulfilled for all linear models in such an uncertainty set.
Traditionally QFT design is carried out in a Nichols diagram
by loop shaping of the nominal linear plant. This task highly
depends on the experience of the designer and is difficult for
unstable systems. In order to facilitate this task an optimization
algorithm based on Genetic algorithm is used to automatically
synthesize a fixed structure controller. For illustration and
evaluation the method is succesfully applied to a Wiener system
and a nonlinear Bioreactor benchmark problem.
KEYWORDS: Nonlinear, QFT, loop shaping, linearization,
non-stationary point, genetic algorithm.
I. INTRODUCTION
In the process industry ease of implementation is without
doubt one of the most important aspects of automatic control.
Provided the performance is acceptable, fixed structure and
linear controllers, such as PID controllers, are therefore advantageous even though the process itself may be nonlinear.
In line with this, the aim of the work presented here is a
semi-automized method for determination of fixed structure
low order linear controllers for mildly nonlinear single input
single output (SISO) processes.
Depending on the character of a nonlinear process there
are many methods for designing nonlinear controllers, such
as Feedback linearization, Sliding control, Adaptive control
and Model predictive control (c.f. [9]). However, in many
control systems there is little support for these methods and
operators are untrained in their use. As a consequence, most
mildly nonlinear plants are controlled by linear controllers,
mainly PID controllers, which are either tuned experimentally or synthesized for a specific operating point. Because
of the nonlinearities the system will have deteriorating properties and may become unstable when operated too far away
from the design point. The idea here is to find a controller
parameterization that gives a robust system in the sense that
it has an acceptable performance in a large operating region.
Since there is an abundance of efficient methods for
synthesis of linear controllers from linear models, the use
∗Department of Signals and Systems, Chalmers University of Technology, SE-412 96 Göteborg, Sweden, E-mail: roozbeh@student.chalmers.se,
tw@chalmers.se
of a linear process model is in many cases motivated. One
way is then to use a linear model and treat the nonlinearities
as model uncertainties. Schweickhardt and Allgöwer [14],
[15], [16] use this to define the best linear model as the one
with the smallest gain of the uncertainty. In [15] they pursue
by determining the linear controller such that the Small gain
theorem can be used to guarantee stability. The drawbacks
are difficulties in the computation of the nonlinearity measure
(uncertainty gain), that the process needs to be stable and that
the use of the Small gain theorem introduces conservatism in
the resulting solution. Basically, the problem of conservatism
and its connection to the nonlinearity measure originates
from the fact that the gain is considered for signals that the
controllers might neither use nor apply. To some degree this
can be taken into account by bounding the input amplitude
[16].
Olesen et al. [13] also used the idea of treating the
nonlinearities as uncertainties and disturbances to show that
only a few linear controllers are needed in gain scheduling
control of the temperature in an exothermic tank reactor.
They use model linearizations to generate a set of transfer
functions that can be interpreted as a model uncertainty
description. This is then followed by a controller design using
Quantitative Feedback Theory (QFT) to guarantee robustness
specifications for all transfer functions in the set. By adding
non-stationary linearization points the robustness and operating window for each controller could be made significantly
larger. The use of off-equilibrium linearization has also been
shown to improve performance of gain scheduling control
when the controller parameters are interpolated [8].
QFT was originally developed for linear systems with
uncertainties (c.f. [7]), but there are also extensions to
nonlinear systems with uncertainties, which are based on
finding an equivalent linear model for the nonlinear system
(see [1] and references therein). However, this requires the
knowledge of what specific input signal that generates the
desired output, which currently limits its use.
Basically, standard linear QFT design is based on loop
shaping the nominal loop transfer function such that for each
frequency considered it does not violate frequency dependent
Horowitz Sidi (HS) bounds. A drawback of standard QFT for
linear systems is that the manual loop-shaping in the Nichols
chart highly depends on the experience of the designer.
During the last decade solutions to how to automate this
step have therefore been proposed. Basically, they rely on
optimization where the bounds constrain the search space.
The optimization problem, however, is in general nonconvex. Chait et al. [1] convexifies the HS-bounds and
91
solve the problem using linear programming. However, the
use is limited because the method requires that the closed
loop poles are known beforehand. Another approach is to
use a global optimization routine. Nataraj et al. [12], [11]
propose an interval analysis, and Chen et al. [2] use genetic
algorithm. To improve the accuracy for a given numerical
effort Fransson et al. [4] use a combination of a global
(DIRECT method) and a local optimization routine. It should
be noted that optimized control of uncertain linear systems
can also be determined using the structured singular value
for the constraints, as in [5] and [17]. However, for SISO
systems with less than 8 parameters to optimize the use of
HS-bounds can in general be recommended [17].
The method we present here for nonlinear processes is
based on the manual method used by Olesen et al. [13],
combined with an optimization using genetic algorithm,
which has the advantage that no initial guess is required
- a valuable property from an automation point of view.
Based on systematically selected simulations of the nonlinear
system new linearization point are added to the set of transfer
functions until performance and robustness no longer is improved. This method is then applied to a Wiener system [15]
and a nonlinear benchmark problem, an unstable bioreactor
[10]. Some solutions have been proposed for this problem
such as [3], though it appears as if no linear controller for
the process has been evaluated earlier. The PID controller
derived with the method presented here performs well over
the operating window and also compare well to the sliding
mode controller by Mehmed et al. [3].
II. METHOD
A. Quantitative Feedback Theory for design and analysis
QFT is a method for design and analysis of feedback
controlled uncertain system, originally developed by I.
Horowitz [7]. The uncertainties and closed loop specifications should be translated to frequency domain and one
arbitrary transfer function Pnom will be considered as the
nominal one. Instead of simultaneous design for all the loop
transfer functions defined by the uncertainties, the design
can then be carried out only for the nominal loop transfer
function, Lnom(jω) = Pnom(jω)G(jω), where G(jω) is
the controller (see Fig. 1). The uncertainties are taken into
account in a translation of the specifications into frequency
dependent boundaries (the Horowitz-Sidi bounds) which the
nominal loop transfer function should not violate.
Fig. 1. Two degree of freedom controller.
The design of the feedback compensator, is carried out in
the following steps:
• Define the uncertainties in the process by a set of transfer function Pj(s). One of these plant transfer functions
is selected to be the nominal one. Then we calculate
the so called templates for relevant frequencies, i.e, the
values of all Pj(jωi) for chosen relevant frequencies.
• Formulate closed loop specifications, such as servo
specification and sensitivity specifications.
• Use the templates to calculate the corresponding
Horowitz-Sidi bounds for the specifications.
• Given the nominal plant and the Horowitz-Sidi bounds
exploit the methods in classical control and loop shaping
techniques to shape the nominal loop transfer function
such that it satisfies all the Horowitz-Sidi bounds.
• Check the stability of the closed loop system for all
plants with the Nyquist criterion.
• If the system response is not within the acceptable servo
specification envelope, a prefilter F (s) is needed prior
the loop.
III. AUTOMATED CONTROLLER SYNTHESIS USING
GENETIC ALGORITHM
In practice QFT loop-shaping is carried out in a Nichols
diagram for a finite number of frequencies, Ω = {ωi}.
The idea in this work is to translate this problem into an
optimization problem. If we assume that the controller has a
fixed structure
G(s) =
θms
m + θm−1sm−1 + ... + θ0
sn + θm+nsn−1 + ... + θm+1
(1)
where θ is the parameter vector to be determined by optimization. The Horowitz-Sidi bounds at each frequency
ωi are denoted Bi(∠L0(jωi, θ), ωi). These bounds have
different shapes and may be single-valued or multiple valued,
depending on the specifications.
The objective here is to synthesize a controller such that:
• The Horowitz-Sidi bounds at each frequency ωi are not
violated.
• The nominal loop-transfer function is stable.
• The controller has low complexity.
In most QFT literature the aim is to minimize the high
frequency gain of controller. In this work we follow that
tradition by choosing a cost function J(θ) that is the high
frequency gain of the controller, However low frequency
disturbance rejection, as in [4], could equally well have been
used. The Horowitz-Sidi bounds are translated to nonlinear
constraint inequalities as below:
ubi(θ) = Bi(∠L0(jωi, θ), ωi)− |L0(jωi, θ)| ≤ 0 (2)
lbi(θ) = |L0(jωi, θ)| −Bi(∠L0(jωi, θ), ωi) ≤ 0 (3)
where ubi and lbi are upper and lower single-valued bound
constraints. Multiple valued bounds are split into one upper
and one lower bound. There are in general no analytical
functions for these bounds and in this work we derive them
numerically using the QSYN toolbox for Matlab [6]. The
nominal closed loop transfer function stability imposes one
more constraint to the problem: the roots λ of 1 + L0 = 0
92
should be in the LHP. Hence, the problem can now be
formulated as
min
θ
J(θ)
subject to: ubi(θ) ≤ 0 ∀i
lbi(θ) ≤ 0 ∀i
Re[λ(1 + L0)] ≤ 0
This problem is classified (to be) in the global optimization
category with nonlinear constraints, a class that classical
gradient based optimization methods are generally not suited
for. However, Genetic algorithm, which is a powerful evolutionary method with the ability to handle the nonlinear
constraints, is a good candidate to solve this problem.
Advantages of this method are: (1) the order of the
controller can be assigned beforehand. (2) There is no need
for an initial guess. (3) There is no need to determine
optimization variable search space in advance.In order to
illustrate the efficiency of this automated synthesis, a manual
design example from the QSYN-manual is compared with
the solution derived by optimization.
Example 1
An uncertain plant, with parametric uncertainty is given:
P (s) = K.
s + a
1 + 2ζs/ωn + s2/w2n
(4)
where K ∈ [2, 5], a ∈ [1, 3], ζ ∈ [0.1, 0.6] and ωn ∈ [4, 8],
and the structure of G(s) is
G(s) =
θ3s
3 + θ2s2 + θ1s + θ0
s4 + θ36 + θ5s2 + θ4
(5)
i.e. the same as the controller presented in the manual. The
design specifications are:
MT ≤ 0.1 (6)
Ts ≤ 10s (7)
‖S(jω)‖ = 1|1 + G(jω)P (jω)| ≤ 6dB (8)
where MT is the maximum overshoot of the closed loop
step response, Ts is the settling time and S(jω) is the
sensitivity function. First of all, the design specifications are
translated to frequency domain. In QSYN this task is carried
out through an approximation of the closed loop system by
a low order system, from which the correspondence between
time and frequency domain is used.
The objective function to be minimized is the high frequency gain of controller, i.e. J(θ) = θ3, and the optimization variables are the coefficients of G(s), i.e. θ =
[θ0, θ1, ...θ6]>. As can be seen from Fig. 2, the automatically
synthesized controller satisfies all specifications, that is, the
nominal loop transfer function for selected frequencies is
outside the sensitivity HS bounds and above the bounds for
the servo specifications. The nominal loop transfer function
L0(jω) is stable and the high frequency gain of the controller
−350 −300 −250 −200 −150 −100 −50 0
−30
−20
−10
0
10
20
30
40
50
60
12
6
3
1
0.5
0.25
0
−50
−40
−30
−20
−12
−6
−3
−1
M
ag [d B
]
Phase [degree]
Nichols Chart
0.20.5
1
20.20.5
1 2
10
20
50
0.2
0.5
1
2
10
50
0.2
0.5
1
2
5
10
20
505
20
Fig. 2. The green curve to the right is the nominal loop transfer function
(manual design). The blue curve to the left is derived from automated loopshaping.
in the automated design is less than the solution given in the
manual.
The foregoing example showed how Genetic algorithm can
be used to solve the loop-shaping problem for an uncertain
linear system. The Matlab Toolbox for genetic algorithm
was used to solve the problem with a population size of 60.
The time of optimization highly depends on the number of
optimization variables and population size, for this specific
example it is around 30 min.
The ultimate goal is to present a controller that not only
works in a small region around the equilibrium points but
also is robust to rather large deviations in initial points from
equilibria. Quantitative feedback theory is a useful method
for design and analysis of uncertain linear systems. There are
solutions to design QFT controller for nonlinear systems.
However, a so-called equivalent linear model needs to be
found then. To find this equivalent linear model it is required
to have a good knowledge about which input signal will yield
the desired output, and for an unstable system this is not a
trivial task (cf. the issues in finding the best linear model
[15]). Different methods of finding this equivalent linear
model have been discussed in [1]. In general it can be divided
into global and local approach. Because of some difficulties
to deal with the global approach, the local approach is our
interest here in this work.
For a nonlinear system:
ẋ = f(x, u) (9)
y = h(x, u) (10)
local linearization around (x̄, ū) gives:
∆ẋ = A(x̄, ū)∆x + B(x̄, ū)∆u
+R(x̄, ū) (11)
where A =
∂f
∂x
(x̄, ū), B =
∂f
∂u
(x̄, ū) (12)
and ∆x and ∆u are the deviations from x̄ and ū respectively.
In classic linearization this task is carried out only for
stationary points which are derived from:
f(x̄, ū) = 0 and ȳ = h(x̄, ū) (13)
93
which implies R(x̄, ū) = 0. Linearization around nonstationary points introduces a constant R(x̄, ū) and due
to this term, properties such as stability are meaningless,
because these points are reached only during transients.
Anyway (11) approximates the possibly transient dynamics
of the nonlinear system when the trajectory is close to
R(x̄, ū).Under the assumption that R(x̄, ū) is small enough,
R can also be considered as a process disturbance. The
nonlinear system is approximated by a family of LTI systems
and a set of process disturbances.
In this work the design is carried out through the following
steps:
1) Define specifications and cost function.
2) Determine equilibria and linearize around them to get
the initial set {P0i(jωk)}.
3) Determine the relevant non-stationary points in the
desired operating window and linearize around them
to get the new templates {Pi(jωk)}.
4) Translate the specifications into Horowitz-Sidi bounds
in the Nichols chart.
5) Decide the structure of controller.
6) Run optimization algorithm using GA.
7) Simulate the system with initial conditions in the
desired operating window.
8) If the response becomes unstable go back to step 3 and
repeat the algorithm again.
Looking at the simulation result and direction of transient
response (for the case of second order system such as our
problem looking at the phase plane) is informative and
might help to decide about new non-stationary points. The
design procedure will be explained in detail for a benchmark
problem in next section.
Example 2
In [15] an example is presented which compares the
capability of two controllers in rejecting disturbances. One
of the controllers is designed for the best linear model and
another one is designed for a linear model derived thorough
classic linearization. Here, we applied the proposed method,
based on QFT, on the same nonlinear system and compare
the result with the two previous ones.
The nonlinear model is a simple Wiener system. The
Wiener system is given by the series connection of the linear
system G(s) = 1(s+1)3 followed by a static nonlinearity
f(x) = x+x3. The control signal is limited to be u ∈ [−2, 2].
First of all, the nonlinear system is linearized around
stationary and relevant non-stationary points. The effect
of linearizing at different points is only in the DC gain
of the system. It can be interpreted that we can replace
the nonlinear system with an uncertain linear system with
uncertainty only in the DC gain. Then, in the next step the
servo and disturbance rejection specifications are translated
to frequency domain. Finally, Genetic algorithm is used to
synthesize a PID controller for this system such that the high
frequency gain of the controller is minimized.
The two controllers presented in [15], are the controller
designed for the best linear model
u = (0.2 +
0.12
s )e (14)
and the controller derived from local linearization:
u = (1 +
0.6
s )e (15)
The controller designed using QFT and linearization at
stationary and non-stationary points (to minimize the high
frequency gain of the controller are) is
u = (
0.397s2 + 0.562s + 0.334
s )e (16)
As can be seen in Fig. 3 and 4 adding non-stationary
linearization points improves the robustness and performance
considerably. The derived PID controller also compares well
to the controller based on the best linear model [15]. We also
used the following criterion
JLF = ‖S(s)‖∞ (17)
where S(s) is the sensitivity function.The foregoing criterion
is a measure of the system’s ability to compensate LF output
disturbances (see Fig. 4). The controller derived for this
criterion is:
u = (
0.588s2 + 0.614s + 0.336
s )e (18)
0 5 10 15 20 25
−1
0
1
t(sec)
ou tp ut Fig. 3. Closed loop responses to a step with height one as output
disturbance. The solid curve (blue) corresponds to the QFT controller, the
dashed curve (red) is the response corresponding to the best linear model,
and the dashed-dot curve (green) is the closed loop response with the
controller based on local linearization.
IV. BENCHMARK PROBLEM
A. MODELING
In order to further evaluate the performance and characteristics of our method, we have selected a Bioreactor
Benchmark problem as the plant to be controlled due to
its interesting characteristics [10]. Although this process is
rather simple and only has two state variables it is difficult
to control due to strong nonlinearity. The bioreactor is a
continuous flow stirred tank reactor (CSTR) with water and
cells (e.g., yeast or bacteria) which consumes nutrients (’substrate’) and produce products (both desired and undesired)
and more cells. The stated control problem is tracking a
desired amount of cell mass.
94
0 5 10 15 20 25
−2
−1
0
1
2
t (Sec)
y (o ut pu t) Fig. 4. Closed loop responses to the step with height two as the output
disturbance. The solid curve (blue) corresponds to the QFT controller
minimized respect to HF gain and the purple curve is the QFT controller
minimzed respect to LF disturbance rejection, the dashed curve (red) is
the response corresponding to the best linear model and the dashed-dot
curve (green) is the closed loop response with the controller based on local
linearization.
X
1
SF
8 S
ρ
Fig. 5. Bioreactor with ρ as input and x1 as output
The state space equations of the plant are:
Ẋ1 = −X1ρ + X1(1−X2)eX2/γ (19)
Ẋ2 = −X2ρ + X1(1−X2)eX2/γ 1 + ρ1 + ρ−X2 (20)
where X1 is dimensionless cell mass and X2 is nutrient
conversion, defined as (SF − S)/SF , where SF is the
concentration of nutrient in the feed to the reactor and S is
the concentration (of nutrient) in the reactor. The constraints
on the state variables are, X1 ≥ 0 and X2 ≤ 1. ρ is the
control signal, which is the flow rate through the reactor
(0 ≤ ρ ≤ 2). The constants β and γ determine the rate of
cell growth and nutrient consumption. From the equations
we may also deduce that cell growth in moderate nutrient
concentrations is faster than at very high or low conversion.
This system is a challenging benchmark because it is
highly nonlinear and for some values of ρ limit cycle is
unavoidable, see Fig. 6. The system is also unstable, as can
be seen in the phase portrait in Fig. 7. It can be noted that
the system has one stable and one unstable eigenvalue in this
area so the equilibrium points are saddle points. The system
response is very sensitive to parameter variation. It means
that a small error in the model can cause a large change in
the control problem.
B. CONTROL DESIGN
According to our design procedure we begin by linearizing
the system around its stationary curve. To obtain stationary
points we need to solve the (19) and (20) at a steady state,
which gives
ẋ1 = 0 ⇒ ρss = (1− x2)x2/ρ (21)
ẋ2 = 0 ⇒ x2 = 0 or x2 = x1 1 + β1 + β −X2 (22)
x ’ = − 1 x + y (1 − x) exp(x/.48) (1 + .02)/(1 + .02 − x)
y ’ = − 1 y + y (1 − x) exp(x/.48)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
X
2
X
1
Fig. 6. Limit cycle for ρ = 1
x ’ = − 1.26257 x + y (1 − x) exp(x/.48) (1 + .02)/(1 + .02 − x)
y ’ = − 1.26257 y + y (1 − x) exp(x/.48)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.05
0.1
0.15
0.2
0.25
X
2
X
1
Fig. 7. saddle point equilibrium for ρ = 1.26
From the plot in Fig. 8. we observe that in a steady state
we cannot achieve any value larger than 1+β4 = 0.255 for
cell mass. As mentioned, the main goal is to track a desired
cell mass (X1). We limit our design to 0 ≤ X1 ≤ 0.255 and
0 ≤ X2 ≤ 0.51. The nonlinear plant is linearized around
stationary points in this region and one of these plant is
selected as the nominal one. Then, the servo specifications
are translated from time domain into frequency domain.
Output disturbance rejection constraint is also applied to
the system in the form of a constraint on the sensitivity
function ‖S‖ ≤ 3. The Matlab toolbox Qsyn [6] is used to
calculate the corresponding Horowitz-Sidi bounds for these
specifications. In Fig. 9 the nominal plant together with the
Horowitz-Sidi bounds is portrayed for different frequencies.
Clearly, the nominal plant is unstable and violates all the
Horowitz-Sidi bounds. We also observe that the gain uncertainty of the template, especially for frequencies smaller
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
X
2
X
1
Fig. 8. Stationary points for bioreactor
95
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30
40
50
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−1
M
ag [d B
]
Nichols Chart
0.05
0.1
0.2
0.5
1
1
4105
0.05
0.1
0.2
0.5
1
4 10500.050.10.2
0.5
1
4
10
50
Fig. 9. The blue curve to the left is nominal loop transfer function before
design and the purple thick curve to the right is the loop transfer function
after design for only stationary points
10
−1
10
0
10
1
10
2
−20
0
20
rad/s
dB Fig. 10. The blue curve is nominal plant. Small circles, show the uncertainties defined by the template. The red envelope is servo specification.
than the bandwidth frequency, is larger than the tolerance
specification (see Fig. 10).Feedback control is needed to
reduce the uncertainty within the acceptable envelope. The
idea here is to automatically design a PID controller such
that the closed loop system becomes stable and fulfill the
specifications for all frequencies. The controller has the the
following (ideal) transfer function:
G(s) =
KDs
2 + KP s + KI
s (23)
The optimization variables are θ = [KD,KP ,KI ]>, and
the objective function to be minimized in this case is KD
(high frequency gain of controller) subject to the following
specifications:
• Servo specification
a(ω) ≤
∣∣∣∣
F (jω)G(jω)P (jω)
1 + G(jω)P (jω)
∣∣∣∣ ≤ b(ω) (24)
• Sensitivity specification
∣∣∣∣
1
1 + G(jω)P (jω)
∣∣∣∣ ≤ 3 (25)
Fig. 9. shows the nominal loop transfer function after design
of the PID controller. From the plot we can see that the
system becomes stable and the nominal loop transfer function
satisfies the specifications for all frequencies. However, we
cannot claim that it has the desired performance on the
original nonlinear system unless we test our design through
simulation. When we simulate the system from an initial
condition in a region close enough to the stationary points,
−350 −300 −250 −200 −150 −100 −50 0
−40
−30
−20
−10
0
10
20
30
40
50
12
6
3
1
0.5
0.25
0
−60
−50
−40
−30
−20
−12
−6
−3
−1
Ma g [d B]
Phase [degree]
Nichols Chart
0.05
0.1
0.20.5
1
10
14
10
50
0.05
0.1
0.2
0.5
1
4
10
50
Fig. 11. The blue curve is nominal loop transfer function after designing
the controller
the system response is satisfactory but for the larger perturbations in the initial condition from equilibrium points
the system becomes unstable. These new non-stationary
points are then added to the set of linearization points. This
imposes tougher boundaries on the nominal loop transfer
function L0(jωi) in the Nichols chart (see Fig. 11). The
problem is then solved with genetic algorithm once more.
The simulation results for this new controller, the former one
and a sliding mode solution is presented in the next section.
In Fig. 12 the gain extent of the closed loop system together
with the uncertainty in the template is portrayed. We see that
after designing the feedback the uncertainty is reduced to an
acceptable level. We also conclude that there is no need to
design a prefilter F (s).
10
−1
10
0
10
1
−20
0
20
rad/s
dB Fig. 12. The blue curve is nominal plant, the small circles show the
uncertainty in the template, and the red envelope is the servo specification.
V. SIMULATION RESULTS
Simulations were carried out in Simulink for different
initial values and different square waves as reference signal. First of all, the two controllers, one derived from
linearization around only stationary points and another one
from linearization around both stationary and non-stationary
points, are simulated for two different initial values. For an
initial condition close enough to the stationary curve both
controllers work, but as can be seen in Fig. 13 for an initial
condition x1(0) = 0.15 and x2(0) = 0.3 the controller
designed from linearization around stationary points only
results in large overshoots. In Fig.14 we perturbed the system
harder by giving an initial condition equal to x1(0) = 0.09
and x2(0) = 0.4 . For this rather large deviation from the
96
stationary curve the first controller gives an unstable response
but the second one is more robust and shows a satisfactory
response. In [3] a sliding mode controller is designed for this
0 100 200 300 400
0.05
0.1
0.15
0.2
0.25
0.3
T(sec)
X
1
Fig. 13. Red dashed line is reference signal, blue curve is for controller
designed using non-stationary points and the green response is the response
from controller designed for stationary points only.
0 100 200 300 400
0.05
0.1
0.15
0.2
0.25
0.3
T(sec)
X
1
Fig. 14. Red dashed is reference signal, blue curve is for the controller
designed with QFT for non-stationary points and the green response is the
response for the controller designed for stationary points only.
system. In Fig. 15 that sliding mode controller is compared
to the PID controller designed with the QFT method. The
minimum and maximum values of the square wave are close
to the maximum values that the system can reach. The
systems are simulated for an initial values that causes large
initial error in the control. As can be seen in Fig. 15 the
PID controller response has an acceptable response though
a significant overshoot. However, from an implementation
point of view the PID controller is clearly preferable.
VI. CONCLUSION
In this paper a method based on QFT is used to design
simple linear controllers for mildly nonlinear systems. The
design is based on local linearization of the nonlinear system.
In addition to classical linearizations around only equilibrium
points non-equilibruim points are taken into account as well.
Simulation results demonstrates that this method improve
both transient response and robustness of the controller.
In order to facilitate the design procedure loop-shaping is
carried out with using an optimization algorithm based on
Genetic Algorithm.
0 100 200 300 400
0
0.05
0.1
0.15
0.2
0.25
T(sec)
X
!
Fig. 15. Red dash line is the reference signal, the blue one is the
QFT controller and the green one is the sliding mode controller response.
x1(0) = 0.05, x2(0) = 0.3
REFERENCES
[1] A. Bafios, O. Yaniv, and F.J. Montoya. Nonlinear QFT based on local
linearization. International Journal of Control, 2003.
[2] W. Chen, D.J. Ballance, and Y. Li. Automatic loop-shaping in QFT
using genetic algrithm. 3rd Asia-Pacific Conference on Control and
Measurement, pages 175-179, China, Dunhuang, Sep 1998.
[3] Mehmet Onder Efe. Model refernce sliding mode control for bioreactor benchmark problem. International Workshop on Variable Structure
Systems, 2008.
[4] C. Fransson, T. Wik, B. Lennartson, M. Saunders, and P. Gutman.
Non-conservative robust control: Optimized and constrained sensitivity
functions. IEEE Transactions on Control Systems Technology, 17:298308, May 2009.
[5] C. M. Fransson, B. Lennartson, T. Wik, and K. Holmström. Multi
criteria controller design for uncertain MIMO systems using nonconvex global optimization. In 40th IEEE Conference on Decision and
Control, volume 1-5, pages 3976–3981, Orlando, FL, USA, Dec 2001.
[6] P.O. Gutman. Manual for the toolbox for Robust Control System
Design for use with Matlab, May 1996.
[7] I. Horwitz. Quantitative Feedback Design Theory. QFT Publications.,
1993.
[8] T. A. Johansen, K. J. Hunt, P. J. Gawthrop, and H. Fritz. Offequilibrium linearisation and design of gain-scheduled control with
application to vehicle speed control. pages: 167-180. Control enginnering practice, 1998.
[9] H. Khalil. Nonlinear systems. Prentice Hall., 2003.
[10] W. T. Miller, R. S. Sutton, and P. J. Werbos. Neural Networks for
Control. MIT Press, pp.387-402, 1990.
[11] P. S. V. Nataraj and M. M. Deshpande. Automated synthesis of
fixed structure QFT controller using interval constraint satisfaction
techniques. In Proceedings of the 17th IFAC World Congress, pages
4976–4981, Seoul, South Corea, July 6-11 2008.
[12] P. S. V. Nataraj and S. Tharewal. An interval analaysis algorithm for
automated controller synthesis in QFT design. Journal of Dynamic
Systems, Measurement, and Control, Vol. 129. pages:311-321, 2007.
[13] V. Olesen, C. Breitholtz, and T. Wik. Tank reactor temperature control
using quantitative feedback theory. In Proceedings of the 17th IFAC
World Congress, pages 4970–4975, Seoul, South Corea, July 6-11
2008.
[14] T. Schweickhardt and F. Algöwer. Linear control of nonlinear systems
based on nonlinearity measures. J. Process Control, 17:273–284, 2007.
[15] T. Schweickhardt and F. Allgöwer. A robustness approach to linear
control of mildly nonlinear processes. International Journal of Robust
Control, 17:1163–1182, 2007.
[16] T. Schweickhardt and F. Allgöwer. On system gains, nonlinearity
measures, and linear models for nonlinear systems. IEEE transactions
on automatic control, 54(1):62–78, 2009.
[17] T. Wik, C. M. Fransson, and B. Lennartson. Feedforward feedback
controller design for uncertain systems. In 42th IEEE Conference on
Decision and Control, pages 5328–5334, Maui, Hawaii, USA, Dec
2003.
97
Nordic Process Control
Aug 19-21, 2010
OPTIMAL CONTROLLED VARIABLE SELECTION FOR
INDIVIDUAL PROCESS UNITS IN SELF OPTIMIZING CONTROL
WITH MIQP FORMULATIONS
Ramprasad Yelchurua, Sigurd Skogestadb
a,b Department of Chemical Engineering
Norwegian Science and Technological University, Trondheim, 7032
Corresponding Author’s E-mail: skoge@chemeng.ntnu.no
Keywords: Optimal operation; selection of controlled variables; measurement combination;
plantwide control; Mixed Integer Quadratic Programming
EXTENDED ABSTRACT
Optimal operation aids in improved productivity and profitability of the process plants. To facilitate
the optimal operation in the presence of disturbances, the optimal control structure selection is
important. The decision on which variables should be controlled, which variables should be measured,
which input variables should be manipulated and which links should be made between them is called
control structure selection. Usually, control structure decisions are based on the intuition of process
engineers or on heuristic methods. This does not guarantee optimality and makes it difficult to analyze
and improve the control structure selection proposals.
This paper considers the selection of controlled variables (CVs) associated with the unconstrained
degrees of freedom. We assume that the CVs c s are selected as a subset or combination of all
available measurements y. This may be written as
c=Hy where ny ≥ nc;
ny: number of measurements, nc: number of CVs = number of unconstrained DOFs
where the objective is to find a good choice for the matrix H. In general, we also include inputs (MVs)
in the available measurement set y.
Skogestad and coworkers have proposed to use the steady state process model to find “selfoptimizing” controlled variable as combinations of measurments. The objective is to find ‘H’ such
that when the CVs are kept at constant set points, the operation gives acceptable steady state loss from
the optimal operation even in the presence of disturbances.
The theory for self-optimizing control (SOC) is well developed for quadratic optimization problems
with linear models. This may seem restrictive, but any unconstrained optimization problem may
locally be approximated by this. The “exact local method” of Halvorsen et al. (2003) handles both
disturbances and measurement noise. The problems of finding CVs as optimal variable combinations
(c=Hy, where H is a full matrix) are found to be difficult to solve numerically (Halvorsen, 2003), but
recently it has been shown that it may be reformulated as a quadratic optimization problem with linear
constraints (Alstad et al., 2009).
We consider three interesting problems related to finding ‘H’:
1) Selection of CVs as combination of best subset of n measurements. Where { : }n nu ny∈
2) Selection of CVs as combination of disjoint measurement subsets using all measurements
3) Selection of CVs as combination of disjoint measurement subsets using n measurement
subset. Where { : }n nu ny∈
98
Nordic Process Control
Aug 19-21, 2010
We consider the solution of these problems when applied to the exact local method formulation of
Halvorsen et al. (2003). Problem 1 has been solved by Kariwala and Cao (2009) and MIQP based
approaches by Yelchuru et al. (2010).
Problem 2 is more appealing for practical usage of the SOC concepts than Problem 1, as using MVs
of a process unit to control CVs as the measurement combinations (i.e. all/subsets of measurements)
of the same process unit in a process flow sheet. This improves the process controllability
significantly as the MVs and CVs are local than that of using CVs as combinations of all the
measurements of the process flow sheet. Even though Problem 2 makes intuitive sense it is still an
open problem. Following the QP formulation proposed by Alstad et al., 2009, we propose to solve
Problem 2 by solving a MIQP problem for a given measurement subset to find the CVs as
combinations of disjoint measurement subsets.
In this paper we propose a method to solve Problem 2 by reformulating the exact local method
problem formulation for a given measurement subset as MIQP problem. Further we extend the
formulation to solve Problem 3. The developed methods are evaluated on a toy problem and on a
distillation column case study with 41 trays. The developed MIQP based methods for solving Problem
2 and Problem 3 in SOC are generic and can easily be evaluated for any system.
References
1. S. Skogestad, Plantwide control: the search for the self-optimizing control structure. Journal of
Process Control, 10(5), 487 - 507.
2. J. Halvorsen, S. Skogestad, J.C. Morud, and V. Alstad., Optimal selection of controlled variables.
Industrial Engineering and Chemistry Research, 42, 14, 3273 – 3284, 2003
3. V. Kariwala and Y.Cao., Bidirectional branch and bound for controlled variable selection. Part II:
Exact local method for self-optimizing control, Computers and Chemical Engineering, 33, 8,
1402 – 1414, 2009
4. S. Skogestad and I. Postlethwaite., Multivariable Feedback Control: Analysis and Design. John
Wiley & Sons, Chichester, UK, 2nd edition, 2005.
5. V. Alstad, S. Skogestad, Eduardo S. Hori, Optimal measurement combinations as controlled
variables, Journal of Process Control, 19, 138 – 148, 2009.
6. Y. Cao and V. Kariwala, Bidirectional branch and bound for controlled variable selection Part I.
Principles and minimum singular value criterion, Computers and Chemical Engineering, 32,
2306 – 2319, 2008.
7. S. Skogestad. Dynamics and control of distillation columns – A tutorial introduction. Trans.
IChemE Part A, 75:539-562, 1997
8. R. Yelchuru, S. Skogestad and H. Manum, MIQP formulation for Controlled Variable Selection
in Self Optimizing Control, DYCOPS 2010, accepted for oral presentation.
99
MIQP formulation for Controlled Variable Selection in Self Optimizing Control
Ramprasad Yelchuru*, Sigurd Skogestad*, Henrik Manum*
*Department of Chemical Engineering
Norwegian Science and Technological University, Trondheim 7032.
_____________________________________________________________________
Abstract In order to facilitate the optimal operation in the presence of process disturbances, the optimal selection of controlled
variables plays a vital role. In this paper, we present a Mixed Integer Quadratic Programming methodology to select controlled
variables c=Hy as the optimal combinations of fewer/all measurements of the process. The proposed method is evaluated on a
toy test problem and on a binary distillation column case study with 41 trays.
Key words: Optimal operation, selection of controlled variables, measurement combination, plantwide control, Mixed Integer
Quadratic Programming
____________________________________________________________________________________
1. INTRODUCTION
To facilitate the optimal operation in the presence of
disturbances, the optimal control structure selection is
important. The decision on which variables should be
controlled, which variables should be measured, which
input variables should be manipulated and which links
should be made between them is called control structure
selection. Usually, control structure decisions are based on
the intuition of process engineers or on heuristic methods.
This does not guarantee optimality and makes it difficult to
analyze and improve the proposals.
This paper considers the selection of controlled variables
(CVs) associated with the unconstrained degrees of
freedom. We assume that the CVs c s is selected as
individual measurements or combinations of subset or all
available measurements y. This may be written as
c=Hy where ny ≥ nc;
ny : number of measurements;
nc: number of CVs = number of unconstrained
DOFs = nu ;
where the objective is to find a good choice for the matrix
H. In general, we also include inputs (MVs) in the
available measurements set y.
Skogestad and coworkers have proposed to use the steady
state process model to find “self-optimizing” controlled
variable as combinations of measurements. The objective
is to find H such that when the CVs are kept at constant set
points, the operation gives acceptable steady state loss
from the optimal operation in the presence of disturbances.
The theory for self-optimizing control (SOC) is well
developed for quadratic optimization problems with linear
models. This may seem restrictive, but any unconstrained
optimization problem may locally be approximated
suitably by this method. The “exact local method” of
Halvorsen et al. (2003) handles both disturbances and
measurement noise. The problems of finding CVs as
optimal variable combinations (c=Hy, where H is a full
matrix) are originally believed to be difficult to solve
numerically (Halvorsen, 2003), but recently it has been
shown that SOC problem may be reformulated as a
quadratic optimization problem with linear constraints
(Alstad et al., 2009). The problem of selecting individual
measurements as controlled variables (so H contains nc
number of columns with a single 1 and rest of the columns
are zero, mathematically HHT = I) is more difficult. The
maximum gain rule (Halvorsen et al., 2003) may be useful
for prescreening but it is not exact. Even though these
methods simplify the loss evaluation for a single
alternative, it requires evaluation of every feasible
alternative to find the optimal solution. As the number of
alternatives increase rapidly with the process dimensions,
resorting to exhaustive search methods to find the optimal
solution is computationally intractable. Kariwala and Cao
(2009) have derived effective branch and bound methods
for the exact local method. These branch and bound
methods require monotonicity property in the objective
function. Furthermore, branch and bound methods are
quite complex and they require derivation of good upper
and lower bounds. This motivates the need to develop
simple and efficient methods to find the optimal solution.
We consider three interesting problems related to finding
H:
1. Selection of best individual measurements as CVs
(select n = nc measurements)
2. Selection of CVs as combination of all ( ny )
measurements.
3. Selection of CVs as combination of best subset of
n measurements. Where { , }n nu ny∈
We consider the solution of these problems when applied
to the exact local method formulation of Halvorsen et al.
(2003). Problem 2 is the easiest one, Problems 1 and 3
involve structural decisions (discrete variables) and are
therefore more difficult to solve. Nevertheless, from a
practical point of view Problems 1 and 3 are important as
100
it is not wise to use more measurements than necessary to
get an acceptable loss.
To solve Problem 1, Cao and Kariwala (2008) has
developed bidirectional branch and bound methods to find
the best individual measurements as CVs using minimum
singular value criterion. To solve Problem 2, Alstad et al.
(2009) has reformulated the self optimizing control
problem as a constrained quadratic optimization problem.
To solve Problem 3, Kariwala and Cao (2009) developed
partial bidirectional branch and bound (PB3) methods to
find best subset of measurements. The methods developed
by Kariwala and Cao (2009) exploit the monotonic
property of objective function in SOC problem and these
methods are of limited/no use if the objective functions are
not monotonic.
In this paper we propose a different method to solve
Problems 1 and 3 by reformulating the exact local method
problem formulation as a Mixed Integer Quadratic
Programming (MIQP) problem. The MIQP formulation is
simple and can easily be extended to other cost functions.
The developed methods are evaluated on a toy problem
and on a binary distillation column with 41 trays. The
developed MIQP methods for SOC are generic and can
easily be evaluated for any system.
2. EXACT LOCAL METHOD FORMULATION
We here review the “exact local method” formulation from
Halvorsen et al. (2003) and its optimal solution from
Alstad et al. (2009). We want to operate the plant close to
optimal steady state operation, by using available degrees
of freedom { } { }all acu = u u∪ . The steady state cost
function J(uall,d) is minimized for any given disturbance d.
The possible process parameter variations are also
included as disturbances. Few of the available degrees of
freedom uac are used to implement optimally “active
constraints”, so that u contains only the remaining
unconstrained steady state degrees of freedom.
The “reduced space” unconstrainted optimization problem
then becomes
min ( , )
u J u d
(1)
In this work we want to find a set of nc = nu controlled
variables c, or more specifically optimal measurement
combinations
c = Hy (2)
such that a constant set point policy (where u is adjusted to
keep c constant) yields optimal operation (equation (1)), at
least locally. With a given d, solving equation (1) for u
gives Jopt(d) , uopt(d) and yopt(d) . In practice, presence of
implementations errors and changing disturbances makes
it impossible to have u = uopt(d) and results in deviation
from optimal operation and this deviation is quantified as
loss. The resulting loss (L) is defined as the difference
between the cost J, when using a non-optimal input u , and
Jopt(d) as in Skogestad and Postlethwaite (2005):
L = J(u,d) - Jopt(d) (3)
The local second-order accurate Taylor series expansion of
the cost function around the nominal point (u*; d*) can be
written as
[ ]
* *( , ) ( , )
1
2
T
u d
T
uu ud
T
ud dd
u J u d J u d J J
d J Ju u
d dJ J
Δ⎡ ⎤
= + ⎢ ⎥Δ⎣ ⎦
⎡ ⎤Δ Δ⎡ ⎤ ⎡ ⎤
+ ⎢ ⎥⎢ ⎥ ⎢ ⎥Δ Δ⎣ ⎦ ⎣ ⎦⎣ ⎦
(4)
where ∆u = (u - u*) and ∆d = (d - d*). nu and nd are sizes
of ∆u and ∆d. For a given disturbance (∆d = 0), the
second-order accurate expansion of the loss function
around the optimum (Ju = 0) becomes
2
1 ( ) ( )
2
1 1
2 2
opt T opt
uu T
L u u J u u
z z z
= − −
= =
(5)
where
1/ 2 ( )optuuz J u u−
In this paper, we consider a constant set point policy for
the controlled variables which are chosen as linear
combinations of the measurements as in equation (2).
The constant set point policy implies that u is adjusted to
give cs=c+nc where nc is the implementation error for c.
Here we assume implementation error is caused by the
measurement error i.e. nc = H*ny. Now we want to express
the loss variables z in terms of d and ny when we use a
constant set point policy.
The linearized (local) model in terms of the deviation
variables is written as
y ydy G u G dΔ = Δ + Δ (6)
dc G u G dΔ = Δ + Δ (7)
where yG HG= and yd dG HG=
For a constant set point policy (∆cs = 0) (Halvorsen et. al.
2003)
1opt
uu udu J J d
−Δ = − Δ
1( )opt y yuu ud dy G J J G d F d
−Δ = − − Δ = Δ (8)
The F in equation (8) is the disturbance sensitivity matrix
from disturbances d to measurements y at optimal
operating points. And this F can be evaluated directly from
optimal process operating data. For illustration, select the
process operating data close to optimal operation for the
possible process disturbances ∆d and for these
disturbances ∆yopt are known and disturbance sensitivity
matrix F can be calculated directly. And this obviates the
need to calculate yG , ydG and ,uu udJ J . The magnitudes of
the disturbances d and measurement error ny are quantified
by the diagonal scaling matrices Wd and Wny respectively.
And we write
101
dd W d ′Δ = (9)
y y y
n n W n ′= (10)
and by introducing the magnitudes of ∆d and ny, the loss
variables z in equation (3) can be written as
y y d n
z M d M n ′′= + (11)
where 1/ 2 1( )yd uu dM J HG HFW
−= − (12)
1/ 2 1( ) yyn uu nM J HG HW
−= − (13)
1
( )[( ) ]
y y
uu ud d d n ny ny ndY G J J G W W

× += − (14)
Using the equations (12), (13), (14) and (5) the loss can be
rewritten as
'
2'
1/ 2 11 ( ( ) )
2
y uu y
d L J HG HY
n
⎡ ⎤
= ⎢ ⎥
⎢ ⎥⎣ ⎦
(15)
The loss in equation (15) can be minimized with H as the
decision variable. Similar to Halvorsen et.al. 2003 the
norm of d’, ny’ is chosen to be constrained by '
'
1
y d n ⎡ ⎤
≤⎢ ⎥
⎢ ⎥⎣ ⎦
,
and the opitmization problem is formulated to minimize
the worst case loss and average loss as in Kariwala et al.
(2008).
1/ 2 1 2
1min ( ( ) )
2
y uuH
J HG HYσ − (16)
21/ 2 11min ( ( ) )
6( )
y uu FH
J HG HY
ny nd

+
(17)
For these SOC problems Kariwala et.al. (2008) proved that
the combination matrix H that minimizes the average loss
in equation (17) is super optimal and in the sense that the
same H minimizes the worst case loss in equation (16).
Hence solving the optimization problem in equation (17) is
considered in the rest of the paper. The scaling factor
1
6( )ny nd+
does not have any effect on the solution of
the equation (17) and hence it is omitted in the problem
formulation.
Lemma 1: The problem in equation (17) may seem nonconvex, but it can be reformulated as a constrained
quadratic programming problem (Alstad et al., 2009).
2
1/ 2
min
.
F
H
y uu HY
st HG J=
(18)
Proof: From the original problem in equation (17) the
optimal solution H is non-unique. If H is a solution then
H1 = DH is also a solution as
1/2 -1 1/2 -1
uu y uu 1 y 1(J (HG ) )(HY) = (J (H G ) )(H Y) for any
non-singular matrix D of nu x nu size. This means the
objective function is unaffected by the choice of D. One
implication is that HGy can be chosen freely. We can thus
make H unique by adding a constraint, for
example 1/ 2y uuHG J= . More importantly this simplifies
the optimization problem in equation (17) to optimization
problem shown in equation (18). End proof
The problem in equation (18) is a constrained quadratic
programming problem in measurement combination
matrix H. We further reformulate the problem in (18) by
vectorizing the decision matrix H to a vector x as
described in Alstad et al., (2009).
First X is introduced as TX H . The matrices X and
1/ 2
uuJ are split into vectors as
1/ 2
1 2 1 2[ ]; [ ];nu uu nuX x x x J J J J= = and we
further introduce the long vectors as below
1 1
2 2
( * ) 1 ( * ) 1
;
nu nunu ny nu ny
x J
x J
X J
x J
δ δ
× ×
⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥= =
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
and the large matrices
( * ) ( *( ))
( * ) ( * )
0 0 0 0
0 00 0 ;
0 00 0
T
T
T
y y T
y nu ny nu nu nd
nu nu ny nu
G Y
YGG Y
YG
δ δ
× +
×
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦
As 2
1
22 2
1 2
2 2
( ) ( )
T
T
T T T T
F nu F
T
nu F
T T T T
F F
x Y
x Y
HY HY HY x Y x Y x Y
x Y
X Y Y X X Y Y Xδ δ δ δ δ δ δ δ
⎡ ⎤= = =⎣ ⎦
= = =
and as Juu is symmetric positive definite matrix,
1/ 2
uuJ is
also symmetric positive definite
1/ 2T T T
y y y uuHG G H G X J= = = and as
1 2 1 2[ ]
T T T Ty y y y
n nuG X G x G x G x J J J⎡ ⎤= =⎣ ⎦
the constraint can be written as
1 1
22
( * ) 1
( * ) 1
T
T
T
y y T
y n nu nunu nu nu
G x J
JG x G X J
JG x
δ δ δ
×
×
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥= → =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦
.
In summary, the optimization problem (18) for finding the
optimal H can be written as a constrained quadratic
programming problem in the variables Xδ as follows.
102
min
.
T T
XN
T
X Y Y X
st G X J
δ δ δ δ
δ δ δ=
(19)
Note here that Xδ is a stacked vector of all the columns in
X or HT.
3. MIQP FORMULATION
The mixed integer quadratic programming (MIQP)
approach provides a different method to solve Problems 1
and 3 described in introduction. Note here that Problem 1
and Problem 2 may be considered as special cases of
Problem 3. The main advantages with the MIQP
formulation are that these are simple, easily extendable
and exact.
We start from the formulation given in (19) to find the
optimal loss for the exact local method. Then we address
this best measurement subset selection problem by
formulating the problem in equation (19) as a Mixed
Integer Quadratic Programming (MIQP) problem as
described below. Let 1 2, , nyσ σ σ { }0,1∈ be binary
variables and let rest of the variables be the same as in
equation (19). For the chosen measurement subset in the
ny measurements, the decision variables associated to that
binary variables are chosen to be bounded in a range of –M
to M. And these bounds are formulated as big-M
constraints. Thus the MIQP problem with big-M
constraints can be written as in equation (20).
( )
{ }
( 1)*
min
.
0 0 0 0
0 0 0 0
1,2, ,
0 0 0 0
0,1
aug
T
T
aug augx
y new aug
aug
i ny i
i i
nu ny i
i x Fx
st G x JN
Px n
xM M
xM M
fori ny
M Mx
σ σ
σ
+
− +
=
=
⎧ ⎫⎡ ⎤−⎡ ⎤ ⎡ ⎤
⎪ ⎪⎢ ⎥⎢ ⎥ ⎢ ⎥−⎪ ⎪⎢ ⎥⎢ ⎥ ⎢ ⎥≤ ≤⎪ ⎪⎢ ⎥⎢ ⎥ ⎢ ⎥ =⎨ ⎬⎢ ⎥⎢ ⎥ ⎢ ⎥⎪ ⎪− ⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦⎪ ⎪
⎪ ⎪∈⎩ ⎭
(20)
where
1
2
( * ) 1
aug
ny nu ny ny
X
x δ
σ
σ
σ
+ ×
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎣ ⎦
and
[ ( , )]; [ ( * , )];
[ (1, * ) (1, )]
TT y T
newF Y Y zeros ny ny G G zeros nu ny ny
P zeros nu ny ones ny
δ δ δ= =
=
and n is the measurement subset size.
In MIQP formulations, selections of a higher value for M
in big-M constraints guarantee optimal solution, when
bounds on decision variables are unknown. Note that each
binary variable σ in inequality constraints in equation (20)
provides bounds on nu number of elements in Xδ vector.
But higher M requires increased computational time in
finding the optimal solution. Hence to find the suitable M
value in finding optimal solution in an acceptable
computational time, the constrained QP problem in (19)
with ny measurements is solved. Based on the solution of
equation (19) M is chosen as 2 times the maximum
absolute value of the solution. Then MIQP problem in
equation (20) is solved for different values of n from nu to
ny. Later, the optimal measurement subset size n can be
selected for the concerned process.
Lemma 2: The best individual measurements in exact
local method (Problem 1) can be obtained from the MIQP
problem formulation (equation (20)) solution for
measurement subset size equal to nc.
Proof: As mentioned in the proof of Lemma 1, If H is a
solution then H1 = DH is also a solution for any nonsingular matrix D of size nuxnu as
-1/2 -1 -1/2 -1
uu y uu 1 y 1(J (HG ) )(HY) = (J (H G ) )(H Y) .Hence
the objective function is unaffected by the choice of D.
Let Hnc. be the optimal solution to this MIQP problem
(equation 20) for best nc measurements combination
matrix. Now by choosing
1
ncD H
−= and we find the best
indiviual measurements Him.(Solution to Problem 1) End
proof
Application to toy test problem. To illustrate the problem
formulation, consider the toy problem of Halvorsen et a.l.
(2003) which has two inputs ( )1 2
Tu u u= , one
disturbance d and two output measurements
( )1 2
Tx x x= . The cost function is
2 21 2 1( ) ( )J x x x d= − + −
where the outputs depended linearly on u , d as
x xdx G u G d= + with
1110 10
; ;
10 9 10
x x
dG G
⎡ ⎤ ⎡ ⎤
= =⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦
At the optimal point we have 1 2x x d= = and Jopt(d) = 0.
Both the inputs and outputs are included in the candidate
set of measurements y. For the example, the steady gain
matrix from y to u (Gy), steady disturbance gain matrix
from y to d ( ydG ), hessian of cost function with u , d
,uu udJ J and disturbance, noise weight
matrices dW , nW used are
1
2
1
2
11 10 10
10 9 10
; ; ;
1 0 0
0 1 0
1 0 0 0
244 222 198 1 0 0 1 0 0
; ; ; 0.01*
222 202 180 0 1 0 0 1 0
0 0 0 1
y y
d uu ud d n
y y C G G
u u J J W W
⎡ ⎤ ⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = =
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎣ ⎦
⎡ ⎤
⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥= = = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎢ ⎥
⎣ ⎦
103
The resulting optimal sensitivity matrix is computed as
follows
1
( )[( ) ]
y y
uu ud d d n ny ny ndY G J J G W W

× += −
These matrices are used to get the stacked vector Xδ, Jδ,
GδTand Yδ and the associated matrices in MIQP
formulation in equation (20) are
2 2 -18 18 0 0 0 0 0 0 0 0
2 2 -18 18 0 0 0 0 0 0 0 0
-18 -18 162 -162 0 0 0 0 0 0 0 0
18 18 -162 162 0 0 0 0 0 0 0 0
0 0 0 0 2 2 -18 18 0 0 0 0
0 0 0 0 2 2 -18 18 0 0 0 0
0 0 0 0 -18 -18 162 -162 0 0 0 0
0 0 0 0 18 18 -162 162 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0
F








=
⎣ 12 12×








⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥
⎢ ⎥⎦
412
11 10 1 0 0 0 0 0 0 0 0 0
10 9 0 1 0 0 0 0 0 0 0 0
0 0 0 0 11 10 1 0 0 0 0 0
0 0 0 0 10 9 0 1 0 0 0 0
Ty newG
×
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
4 1
11.6
10.5
10.5
9.6
nJ ×
⎡ ⎤
⎢ ⎥
⎢ ⎥=
⎢ ⎥
⎢ ⎥
⎣ ⎦
[ ]112
2
0 0 0 0 0 0 0 0 1 1 1 1
M
P ×
=
=
4. RESULTS
4.1 Toy problem
The minimized loss function with the number of
measurements used as CVs (i.e. the measurement
combinations) is shown in Figure 1. From Figure 1, the
loss is minimized as we use more number of
measurements to find the CVs as the combinations of
measurements. And the reduction in loss is very small
when we increase the measurement subset size from 3 to 4.
Figure 1. Optimal average loss with best measurement
combinations vs no. of measurements used.
Based on the Figure 1, we can conclude that using CVs as
combinations of 3 measurement subset is optimal for this
toy problem.
4.2 Binary distillation column Problem
The binary distillation column and the associated data are
taken from Skogestad (1997). The distillation column in
LV-configuration with 41 stages is used. The 41 stage
temperatures are taken as candidate measurements. Note
that we do not include the inputs in the candidate
measurements for this case study. The economic objective
J for the indirect composition control problem is
2 2
, ,
, ,
H H L L
top top s btm btm s
H L
top s btm s
x x x x
J X
x x
⎛ ⎞⎛ ⎞ ⎛ ⎞− −⎜ ⎟= Δ = +⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎝ ⎠
where ∆X is the root mean square of the relative steady
stae compostion deviation. , ,, , ,
H L H L
top btm top s btm sx x x x , L and
H denote the heavy component composition in top tray,
light component composition in bottom tray, specification
of heavy component composition in top tray, specification
of light component composition in bottom tray, light and
heavy key components respectively. We formulated the
MIQP problem for the distillation column with 41 trays to
find the 2 CVs as the combinations of 41 tray
temperatures. An MIQP is set up for this distillation
column with an M value of 1 in big-M constraints in
equation (20). We solved the MIQP to find the CVs as the
combinations of best measurement subset size from 2 to
41. The CPLX solver in Tomlab environment is used to
solve the MIQP problem. The same problem is solved by
downwards branch and bound, partial bidirectional branch
bound methods of Kariwala and Cao (2009). The
computational times (CPU time) taken by MIQP,
Downward BAB, PB3 method and exhaustiv|e search
method are shown in Figure 3. Note that exhaustive search
is not performed and an estimate of CPU time assuming
0.01 s for each evaluation is plotted. From Figure 3, it can
be seen that the MIQP finds optimal solution in 6 orders
faster than exhaustive search methods in computational
(CPU) time. MIQP method runs relatively quickly for
measurement subset size from 25 to 41, but it took fairly
longer time for subset sizes from 10 to 19. As these subset
sizes (10 to 19) have very high number of possibilities
(41C10 to 41C19), the longer time taken by MIQP method
is justifiable. But on an average basis MIQP methods are
slower by 1 order to PB3 and 0.5 orders slower than
Downwards BAB methods. In conclusion, even though the
MIQP methods are not computationally attractive to that
of Downwards BAB and PB3 methods; the variation in the
computational time by order of 1 is acceptable as these
optimal CVs selection problems are performed offline.
Despite these, MIQP method is valuable as the method is
simple and can easily be extended to any quadratic cost
functions to find optimal CVs in SOC framework. The
minimized loss function with the number of measurements
104
Figure 2. Optimal average loss using MIQP method with
best measurement combinations vs no. of measurements
used.
Figure 3. Comparsion of computation times
used for CVs (i.e. the measurement combinations) is
shown in Figure 2. From Figure 2, it can be seen that the
loss decreases rapidly when the number of measurements
increased from 2 to 14, and from 14 very slowly. Based on
the Figure 2, we can conclude that using CVs as
combinations of 14 measurements subset is optimal for
this 41 stage binary distillation column problem. MIQP
formulations are easy than the BAB methods and
structural constraints such as selection of certain number
of measurements from top section, selection of certain
number of measurement from bottom section can be done
easily.
5. CONCLUSIONS
Optimal CV selection as measurement combinations to
minimize the loss from the optimal operation is solved.
The CV selection problem in self optimizing control
framework is reformulated as a QP and CVs selection as
combinations of measurement subsets is formulated as an
MIQP problem. The developed MIQP based methods are
easier compared to the bidirectional branch and bound
methods reported in literature to find the CVs as
combinations of measurement subsets. And MIQP
methods cover wider spectrum of quadratic based
objective functions whereas bidirectional branch and
bound methods are limited to objective functions with
monotonic properties. MIQP based methods takes longer
time than bidirectional branch and bound methods, but this
is acceptable as the optimal CV selection problem is done
offline. MIQP problem formulations are easily extendable
for optimal measurement subset selection for systems with
few structural constraints.
REFERENCES
V. Alstad, S. Skogestad, Eduardo S. Hori, Optimal
measurement combinations as controlled variables, Journal
of Process Control, 19, 138 – 148, 2009.
Y. Cao and V. Kariwala, Bidirectional branch and bound
for controlled variable selection Part I. Principles and
minimum singular value criterion, Computers and
Chemical Engineering, 32, 2306 – 2319, 2008.
I. J. Halvorsen, S. Skogestad, J.C. Morud, and V. Alstad.,
Optimal selection of controlled variables. Industrial
Engineering and Chemistry Research, 42, 14, 3273 – 3284,
2003
V. Kariwala and Y.Cao., Bidirectional branch and bound
for controlled variable selection. Part II: Exact local
method for self-optimizing control, Computers and
Chemical Engineering, 33, 8, 1402 – 1414, 2009
V. Kariwala, Y.Cao and S. Janardhan., Local selfoptimizing control with average loss minimization,
Industrial Engineering and Chemistry Research, 47, 11501158, 2008.
S. Skogestad, Plantwide control: the search for the selfoptimizing control structure. Journal of Process Control,
10(5), 487 - 507.
S. Skogestad and I. Postlethwaite., Multivariable Feedback
Control: Analysis and Design. John Wiley & Sons,
Chichester, UK, 2nd edition, 2005.
S. Skogestad. Dynamics and control of distillation
columns – A tutorial introduction. Trans. IChemE Part A,
75:539-562, 1997
TOMLAB v7.1 - The TOMLAB Optimization
Environment in Matlab (1999)
105
NPCW 2010
Dynamic Characteristics of Counter­Current Flow Processes
Jennifer Puschkea, Heinz A Preisigb
 aRWTH Aachen, Templergraben 55, 52062 Aachen, Germany, 
Jennifer.Puschke@rwth­aachen.de
bChemical Engineering,, NTNU, N – 7491 Trondheim,  Norway,
heinz.preisig@chemeng.ntnu.no (corresponding author)
Abstract
In   industry   counter­current   flow   processes   are   common.  Although   these   processes   have   been  widely   studied   in 
literature,  relatively little has been published on their dynamic behaviour.  Two very common counter­current  flow 
processes  are heat  exchangers  and distillation columns. Ma’s  study based on dynamic models of heat  exchanger’s 
dynamic behaviour [1] reports an internal resonance effect, also earlier reported by Profos in 1943 [4] and Friedly in 
1972 [3]. Here the study is repeated with lumped models, first for heat exchangers and thereafter for simple distillation 
columns. Not unexpectedly, the dynamic properties change gradually as the number of lumps increases towards the 
distributed systems and for high frequencies similar internal resonance effects evolve and the envelopes show a very 
low­order behaviour, which though somewhat surprisingly is independent of the number of lumps. Finally we show that 
the eigenvalues of the normed system matrix lie on a circle in the complex plane. 
Keywords: Modeling, distributed/lumped model, Resonance effect, Frequency analysis
1. Introduction
An industrial process consists of a series of material transformations. Examples of such operations are batch reactors, 
compressors, heat exchangers and distillations. Many of these operations are based on two phases exchanging material 
and/or energy in the form of heat. The two phases are passing each other either in co­current or counter­current fashion 
often arranged in stages in each of which one drives the system towards equilibrium. 
Although counter­current flow processes have been widely studied in literature, little of it reports on their fundamental 
dynamic behaviour.  Common dynamic models   for  heat  exchangers  are 
simple first­order plus dead­time models. Exception are Profos (1943) [4] 
reporting   the   internal   resonance   effects,   Friedly   (1972)   [3]   derived 
reduced­order models and X H Ma [1] who derived a new set of high­
fidelity low­order models also confirming the internal  resonance effect, 
which  years   later  has  been  show to exist   in  an experimental  study by 
Grimm [2].
In a distributed model the temperatures on the inner and the outer tube of 
the heat  exchanger are considered  as continuous functions of   time and 
spatial coordinates yielding a set of partial differential equations (PDE’s). 
Ma’s   distributed  model   shows   the   presence   of   the   internal   resonance 
effect in the high frequencies domain. She splits the transfer function into 
a   resonance   and   a   non   resonance   part   assuming   a   linear   underlaying 
behaviour.   This   procedure   yields   high­fidelity   analytical   low­order 
models   being   the   envelopes   of   the   oscillating   transfer   function.   For 
distillation columns however, no such behaviour has been reported. Since 
standard   tray   columns   are   better   described   as   counter­current   staged 
processes,  Ma's study was repeated with lumped models. The model is 
constructed as a network of communicating capacities for each of which a 
mass and an energy balance is constructed. The energy balances are being 
transformed into the alternative state space of the intensive quantity temperature all of which forms a set of ordinary 
differential   equations   (ODE’s).   It   is   expected   that,   as   the   number   of   lumps   approaches   infinity,   the   solution 
approximates the solution of the distributed model hoping that the resonance effect shows also for low­order models. In 
a second step this is applied to a kind of a mass transport distillation column.
Figure 1: Model of the counter current double 
pipe heat exchanger with n equal three stages
106
J.Puschke, H.Preisig
2. Lumped Model of the Heat Exchanger 
Ma’s double pipe heat exchanger is analyzed approximating it as series of heat­exchanging, paired lumps on one side 
representing the hot stream and on the other the cold one. Since the result shall be compared with distillation models 
only the counter­current flow pattern is being considered. Ma’s work discusses different cases,  which are based on 
different sets of assumptions. Here we focus on Ma's case I. We start with a 3­stage process as shown in Figure 1. The 
assumptions are:
• The total volume of stream A (consisting of all equal­sized lumps ai) is the same as the total volume of stream B and 
both are constant: V A=V B=V=const
• The volume of each lump has the same size:  V a1=⋯=V an=V b1⋯=V bn=
V
n • The heat transfer area  Oi  with  i={1,2,⋯, n}  between two lumps with the same index  i  is constant and equal: 
Oi=
Ooverall
n ∀ i , with  Ooverall as the overall heat transfer area between stream A and stream B.
• Heat is only transferred between two lumps with the same index.
2.1. State Space Model Equations for n Stages 
The   energy   balances   is   drawn   up   for   each   lump   and   solved   for   the   temperature  T   resulting   in   the   state   space 
representation:   ẋ=AxBu    with the two matrices A and B being
A=
−τA−d A τ A dA
⋱ ⋱ ⋱
⋱ τA ⋱
−τA−d A d A
dB −τB−d B
⋱ τB ⋱
⋱ ⋱ ⋱
dB τB −τB−dB
 ; B= 0 0⋮ ⋮τA 00 τB⋮ ⋮0 0               (3)
And the  y=Cx  with the matrix  C=1 ⋯ 00 ⋯ 1 
With   τm=
Vm
V
n   ;   d m=
kmOi
ρmc pmV
,    m∈{A ,B }
The quantities are: km :: heat transfer coefficient of stream m, Oi :: heat transfer area between two lumps, ρm :: density of 
the stream m, cpm :: specific heat of stream m, V/n :: individual lump volume. 
The state is  x=T a1 ⋯ Tan Tb1 ⋯ Tbn T , the input is u=T α Tβ T and the output is  y=T γ T δ T .
2.2. Bode Plots
The dynamic behaviour of the models is depicted in Bode Plots of the model transfer functions. The transfer functions 
are derived by transforming the state space model into the frequency domain solving for the output y = x in dependence 
of u. The transfer function matrix is then simply:   G=C  sI−A −1 B (4)
The transfer functions of input   to the output   shows similar behavior as the one from the input   to the output  .α γ β δ  
Only one of the two down stream responses, namely  G11  from the input   to the output   is shown in Figure 2. Theα γ  
same applies to the cross stream transfer functions, where only the transfer function G12 from the input   to the output α δ 
is   plotted.  The   behavior   of   all   transfer   functions   approaches  Ma’s   distributed  model   as   the   number   of   stages  n 
approaches infinity which is also shown in Figure 2 as a reference.
107
Dynamic Characteristics of Counter­Current Flow Processes
Down Stream Response: The behavior of the transfer functions varies with the number of stages n. In the amplitude plot 
with an increasing number of stages  n    the slope of the amplitude decreases.  As the number of stages approaches 
infinity the slope approaches zero. The latter implies that there exists only one gain. In the phase plot an increasing 
number of lumps increase the negative phase shift. For an infinite number of stages the phase lags go to minus infinity, 
which indicates the existence of a dead time. But there is not a resonance 
effect in the amplitude or the phase.
Cross   Stream   Response:  The   Bode   plot   shows   the   resonance   effect   in 
amplitude and phase. Furthermore one observes that the curves show a first 
corner at the frequency of  =1Hz for the chosen set of parameters. Aboveω  
this corner frequency the slope in the amplitude plot of the resonance part is 
in average minus one. And in the phase plot the resonance part average is 
­90 degree.  The transfer  functions with the number of stages  being small 
than   infinite   show a  decaying   resonance  part  with   increasing   frequency, 
which finally disappears. The apparent length of the resonance part depends 
on the number of stages: With an increasing number of stages, the resonance 
part grows longer until the infinite case, where the resonance part does not 
decay anymore. Also the models with the number of stages being less than 
infinity, the final slope in the amplitude plot is ­2 and the final phase lag is 
­180 degree. Hence this transfer functions show a second corner frequency 
under which the resonance part decays. Both corner frequencies depend on 
the number of stages. 
2.3. Detailed Analysis of the Cross Stream Response
To get more information about the second­order behavior of the cross stream response, one needs the pole excess of the 
transfer function. Due to the structure of the matrices B and C only four entries of the matrix (Is­A)­1 are relevant for the 
transfer functions matrix and only two of these entries for the cross­stream transfer functions. The zeros of the transfer 
functions are the zeros of the adjoint matrix adj(Is­A). For the number of stages n = 3 or 4 it is easy to show that the 
respective adjoints have 2n­2 zero.  Since the poles are the eigenvalues of A, their number is 2n.  So the pole excess is 
Figure 2: Bode plots of the down stream transfer function G
11 
(left) and of the cross stream transfer function G
12
(right) with different numbers of stages n. The parameters are chosen to be d
A
=d B
=0.01; τ
A
=1;  τ
B
=1.
Figure 3: The normed eigenvalues of 
the system matrix A in the complex 
plane with n=100.
108
J.Puschke, H.Preisig
2n­(2n­2)=2, which explains the observed second­order behavior. In addition, by closer examination of the poles, one 
finds that the normed eigenvalues of the matrix A form a circle with radius one and the center at (­1,0) as shown in 
Figure 3.
3. Lumped Model of a Distillation Column 
Figure 4 depicts a distillation column and an abstraction 
there off which underlays the construction of the model 
equations [5].  
3.1. The State Space Model 
The  mass   balances   drawn   for   each   lump,   assuming   a 
linear transfer law making the mass transfer proportional 
to   the   composition   differentiate   and   solved   for   the 
concentration  c  of   the  lumps yields again  a  linear  state 
space model (A,B,C,D), with the state x , input u and the 
output y being
x= ca1 ⋯ can cb1 ⋯ cbn T ;        u=cα          and 
y=cβ cγ T with:
;
B=0 ⋯ 0 τα 0 ⋯ ⋯ ⋯ 0 T ;   C=1 0 ⋯ ⋯ 00 ⋯ 0 1 0 ⋯ 0  ;   D=0 0 T      with          τm= VmV  
and        τi2=τi3=τin−1 ,   i∈{a , b }
These sparse system matrices for the distillation column show a similar structure as the system matrices of the heat 
exchanger.
3.2. Bode Plots
The transfer functions in the following Bode plots are obtained in the same way as this was done in the analysis of the 
heat exchanger.
The curves in the magnitude plot of the transfer function from the input   to the output   in the Bode plots of Figure 5α β  
shows a  resonance  effect.  This  resonance  part  decays before  the curves   reach  a multiple corner   frequency,  which 
depends on the number of stages n. With a larger number of lumps the length of the resonance part is longer. If the 
number of stages go to infinity one could assume, that there is only a steady state gain with resonance, whereas in the 
phase shift plot the resonance part does not appear. The general behavior of the curves with the number of stages  n 
going to infinity suggests the existence of a dead time. The magnitude plot of the transfer function from the input   toα  
the output   (see Figure 5) shows a comparable response behavior as the transfer function from the input   to theγ α  
output  . The differences are in the amplitude of the resonance part and the corner frequency. But the phase plot of theβ  
of the transfer function from the input   to the output   is a resonance part.α γ
Figure 4: Process of the distillation column (left) and model 
of the distillation column (right)
A=
−τa1−τβ τ a1
τa2 −τ a1−d d
⋱ ⋱ ⋱
τan−1 −τan−1−d d
−τ an −τan−τγ 0
0 −τb1
d −τb2−d τb2
⋱ ⋱ ⋱
d −τ bn−1−d τbn−1
τbn −τbn

109
Dynamic Characteristics of Counter­Current Flow Processes
By closer examination of the poles in the complex plane, one finds again that the standardized eigenvalues of A form a 
circle with radius one and the center at (­1,0) as Figure 3 shows.
4. Conclusion
The dynamic characteristics of two counter­current processes are compared: a single tube heat exchanger and a staged 
distillation column. For both simple linear transfer models are assumed yielding linear systems that are of very similar 
structure. If normed, both show the same behaviour with respect to the system eigenvalues: they lay on a shifted unit 
circle in the complex domain. Both systems show resonance effects for some parts. Heat exchangers show it for cross­
stream transfer functions, but not for down­stream transfer functions, whilst in distillation one finds the resonance also 
in the down stream transfer function, at least in the amplitude. In both cases, the magnitude of the resonance effect is a 
function of the number of lumps or stages. 
In case of the heat exchanger the pole excess is 2, but the second corner frequency approaches infinity as the number of 
lumps approaches infinity. Thus for the distributed system the pole excess is only 1. This behaviour is also detected in 
the phase plot with a max phase shift of ­180 degrees for finite number of lumps and ­90 degrees for the distributed 
system. 
The cross  transfer   functions  for   the distillation column behaves   like a  dead  time for  high frequencies,   though the 
position of the multiple zeros shifts to higher and higher frequencies as the number of stages increases. 
References 
[1] Ma, X H. Dynamic Modelling, Simulation and Control of Heat Exchanger. PhD thesis, School of Chemical Engineering and 
Industrial Chemistry, University of New South Wales, Kensington, Sydney, Australia,1993.
[2] Grimm, R. Low­Order Modelling of the Dynamic Bahaviour of Heat Exchangers:Theory and Experimental Verification. Diploma 
thesis, 1999.
[3] Friedly, J C. Dynamic Behaviour of Processes. Prentice­Hall, Englewood Cliffs, NJ, 1972.
[4] Profos, P. Die Behandlung von Regelproblemen vermittels des Frequenzganges des Regelkreises. PhD thesis, ETH, Zuerich, 
1943.
[5] Dones, Ivan and Preisig, Heinz, Graph tehory and model simplifcation. Case study: distillation column. Comp & Chem Eng. 
Accepted for publication, 2009. 
Figure 5: Bode plots of the transfer function G
 αβ (left) from the input   α to the output   β and of the transfer function Gαγ  (right) from 
the input α to the output   γ with different numbers of stages n.
110
�bserver design for the activated sludge process
Marcus Hedegärd and Torsten Wik
Automatic Control� Department of Signals and Systems�
Chalmers University of Technology
SE-412 96 Göteborg� Sweden
The activated sludge process �ASP) is the most common process in biological wastewater
treatment. However, they are very costly to operate, with energy for aeration being their
largest cost. To optimize the process, a model is needed. The most widely used model
for modeling of ASP:s is the Activated Sludge Model NO.1 �ASM1) [1]. It is physically
based and a good compromise between accuracy and simplicity, and from this comes its
popularity. Unfortunately, the model contains several concentrations that cannot reliably
be measured online. Some of these are degradable dissolved and particulate organic matter,
and biomass concentration. Totally Suspended Solids �TSS) measurements give indication
of biomass concentration though. Online substrate analyzers have been available for many
years but have historically been considered unreliable. Under the assumption that all
relevant concentrations in the ASM1 were available online, the operation of ASP:s could
be optimized to an extent that is not possible today. Some of the inputs to be optimized are
the different input flowrates �water), TSS in the sludge recycle flow, the aerobic volume,
external carbon addition and the air flowrates in the aerobic compartments. Motivated
by this, an observer has been designed for the process based on the ASM1.
Observers based on the ASM1 have earlier been formulated by [3] and [4]. Both of these
are for one aerobic reactor in the benchmark model [2] and the biomass concentration is
assumed to be known and constant. A reduced order observer model is used in both cases.
In [3] an altered version of the ASM1 which only includes one kind of substrate is used as
the model for a nonlinear observer. In [4] an extended Kalman filter approach is taken.
Different sets of measurements are considered in these cases but in both it is concluded
that all relevant variables cannot be estimated at the same time. When lab analysis of
substrate in the input is fed to the observers they are convergent though.
In this work we try to estimate all concentrations including unknown inputs in a reduced
order model of two aerobic reactors with an extended Kalman filter. The success to
estimate all unkwnown inputs compared to [3] and [4] relies on the following features:
• Contrary to the work mentioned above, the process considered here is predenitrifying
with postnitrification in trickling filters. This means that nitrification stands for a
very small portion of the aerobic reactions in the ASP, which in turn leads to that less
111
concentrations in ASM1 need to be considered. The concentrations in the reduced
order model are: SO �oxygen), SS �readily biodegradable substrate), XS �slowly
biodegradable substrate) and X�H �heterotrophic biomass). Among these, only
oxygen is measured.
• By including two reactors, one additional measurement of oxygen is gained and
better coupling between the states is achieved. The process model holds totally 11
states.
• The assumption that the biomass concentration is constant can for most plants
in reality only be assumed on a very short time basis �hours). This can namely
vary fast with varitions in the ratio between the input flowrates and with TSS in
the return sludge flow. On the other hand, the composition of the sludge can be
assumed to be slowly time varying. Instead of estimating the unknown input of
biomass directly the observer estimates a parameter γX�H : the ratio of the return
sludge being heterotrophic biomass. The sludge concentration in the reactors can
be simulated from TSS measurements in the recycle flow. This approach allows for
larger variations in biomass concentration.
• The unknown input concentrations: SS , XS and the parameter γX�H are all modeled
as random walk processes.
Initially, it was considered to estimate γX�H in the upstream anoxic compartments by
using that some of the Monod expressions in the ASM1 model then can be assumed to
be saturated. The developed EKF gives close to unbiased estimates of all states. The
transient of the filter for three of the estimates of simulated variables are shown in Figure
�1). The parameter γX�H is shown for a longer time period because of its slower variation.
0 0.5 1 1.5 2 2.5 3
0
5
10
15
20
25
30
35
40
Time�days)
m g C
O
D
/
l �a) Readily biodegradable substrate at the inlet of the first reactor.
0 0.5 1 1.5 2 2.5 3
140
150
160
170
180
190
200
210
220
230
240
Time�days)
m g C
O
D
/
l �b) Slowly biodegradable substrate at the inlet to the first reactor.
0 2 4 6 8 10 12 14
0.35
0.4
0.45
0.5
0.55
0.6
Time�days)
g C
O
D
/
g T
S
S
�c) The parameter �X�H
Figure 1: Estimated �noisy) and true variables �smooth).
Unfortunately, the system can be very sensitive to model errors, especially for errors in the
�La �Oxygen transfer) function, which is known to be time varying. To make the filter
useful in an application it is most probably necessary to estimate the �La function on a
continuous basis. Most methods for estimating this function needs excitation of the air
flowrate. The purpose of the observer is to save money and excitation is costly. However
a �La estimation method where the air flowrate is excited with small amplitude has been
described in [5].
112
�eferences
[1] M. Henze, C.P.L. Grady, W. Jr., Gujer, G.V.R. Marais, T. Matsuo, Activated sludge
model no. 1� IAWQ Scientific and Technical Report No. 1, London, UK, 1987.
[2] B. Boulkroune a, M. Darouach, M. Zasadzinski, S. Gillé, D. Fiorelli, A nonlinear observer design for an activated sludge wastewater treatment process, Journal of Process
Control 19 �2009) 1558–1565
[3] F. Benazzi, K.V. Gernaey, U. Jeppsson and R. Katebi, On-line estimation and detection of abnormal substrate concentrations in WWTP:S using a software sensor: a
benchmark study, Environmental Technology, Vol. 28. pp 871-882
[4] J. Alex, J.F. Beteau, J.B. Copp, C. Hellinga, U. Jeppsson, S. Marsili-Libelli, M.N.
Pons, H. Spanjers, H. Vanhooren, Benchmark for evaluating control strategies in
wastewater treatment plants, in: European Control Conference 1999, ECC’99, Karlsruhe, Germany, August 31–September 3 1999.
[5] G. Olsson, B. Newell, Waste water treatment systems: modeling� diagnosis and control, Sweden and Australia, 1999.
113
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115
Model predictive control for plant-wide control of a
reactor-separator-recycle system
Dawid Jan Białas1, Jakob Kjøbsted Huusom1, John Bagterp Jørgensen2, Gürkan Sin
1
1 Department of Chemical and Biochemical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
2
Department of Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
ABSTRACT
Ethylene glycol is produced by reaction of ethylene oxide and water in continuous-stirred tank reactor. To
increase the overall conversion, the unreacted substrates are separated from the product in a distillation column
and recycled to the reactor. This process is an example of a reactor-separator-recycle system that is widely used
for manufacture of chemicals in industrial scale. We use a mathematical dynamical model of this process to
investigate Model Predictive Control structures for plant-wide control of reactor-separator-recycle system.
Previously, different regulatory control structures and strategies for plant-wide control of reactor-separator-recycle
system have been investigated. This work we investigate linear Model Predictive Control (MPC) for plant-wide
control of the ethylene glycol reactor-separator-recycle system. The MPC gives setpoints to the regulatory
controllers. The regulatory control structure is determined using a plant-wide control methodology such that
suitable trade-off between sensitivity to disturbances and agile tracking of setpoints is achieved. The MPC
coordinates the operation of each regulatory controller such that disturbances are rejected faster with less upset
of plant operation. Consequently, the MPC facilitates operations aimed to maximize the throughput, minimize
operating costs and ensure product quality.
In one plant-wide control strategy, the following controlled variables and manipulated variables were paired:
reactor feed composition vs. fresh water flow-rate, water fraction in the bottom vs. vapor flow-rate, glycol
composition in the top vs. recycle flow-rate. The essence of MPC is to optimize the predicted process behavior
over a future horizon by manipulating the inputs subject to a number of constraints (process and actuators). A
nonlinear process model was developed to simulate the ethylene glycol reactor-separator-recycle system. The
filtering and predictions in the MPC are based on various linear models (state-space and reduced input-output
models). The controller performances are investigated for scenarios with different disturbances entering the
process. The key contribution of this paper is a demonstration of linear MPC for plant-wide control of a reactorseparator-recycle system.
116
Fuel quality soft-sensor for control strategy improvement of the
Biopower 5 CHP plant
Jukka Kortela, Sirkka-Liisa Jämsä-Jounela
Abstract— This paper aims to present an enhanced method
for estimating fuel quality on a BioGrate combustion process
and its use in a control strategy improvement. The dynamic
model based method (DMBM) utilizes combustion power which
is calculated using the oxygen consumption of the furnace and
the energy balance of the boiler. The proposed method is tested
with data from the industrial scale Biopower 5 CHP plant
and compared with the method currently used in industry, and
finally the results are analyzed and discussed.
I. INTRODUCTION
Biomass fired power plants are usually controlled by
means of conventional feedback control strategies where
the main measurements are obtained from steam generation.
However, these strategies face challenges due to great delays
in control schemes associated to fuel feed and air supply
[4]. Disturbances in combustion should thus be detected as
early as possible before they have a significant effect on the
process and its operation.
The main disturbances to the process are caused by fuel
quality variations. Even for the same type of bio fuels, their
chemical properties may differ greatly; for example – on
account of harvesting, storing and transporting conditions
[11]. In order to control the combustion process in an optimal
way, it is essential to compensate variations in the fuel
quality.
Referring to the theoretical studies and practical tests
by Kortela & Lautala [3] for a coal power plant, the fuel
combustion power in the furnace can be estimated on the
basis of the measured oxygen consumption. The on-line
measurement of the oxygen consumed is used when a new
cascade compensation loop was built to optimally control
the fuel flow. In that control strategy, the set point of the
feed mainly depends on the output of the master controller,
which obtained the main reference signal from the drum
pressure. Taking into account the oxygen consumption and
total air feed, the combustion power was calculated. The fuel
feed setpoint was also modified as a direct result of making
use of this variable. It is reported that the amplitude and
the settling time of the response of the generator power
decreased to about one third of the original when this
cascade compensation loop was added to the present system.
Combustion power control (CPC) was implemented also in
peat power plants [6], where it was able to stabilize the
furnace. The control actions of the burning air flow decreased
when variations in the oxygen consumption were eliminated.
The control strategy could thus reduce the standard deviation
Corresponding author: jukka.kortela@tkk.fi, Aalto University,
PL 16100, FI-00076 Aalto
of the flue gas oxygen content and the air flow could be
lowered close to the optimal flow. As a consequence, the flue
gas losses were reduced. Furthermore, the stabilized steam
temperatures reduced thermal stress on superheaters and
associated pipes. In addition an integrated optimisation and
control system has been applied for minimization of (NOx),
(SO2) and (CO) emissions in the bubbling fluidized bed
boiler in [5], where the use of the combustion power control
algorithm made possible to stabilise the burning conditions
in co-combustion. This leads to the better control of flue
gas emissions when using the CPC together with the expert
system. Hence, the combustion power control was reported
cutting steam pressure deviation by 50 %.
However, there are still some challenges and objectives
in the combustion power method. For example, variations in
the moisture of fuel should be taken into account in order
to correct any estimation errors in the combustion power. It
is reported that temperature measurement in the furnace (or
in the boiler) and calculation of the steam enthalpy could
be used to estimate the fuel quality. However, temperature
measurements are sensitive to process and measuring disturbances and the system dynamics introduce a delay in the
method using enthalpy measurements [3].
The model-based predictive control was used by Havlena
& Findejs [2] to enable tight dynamical coordination between
air and fuel to take into account variations in power levels.
The results showed that this approach could be used to
increase boiler efficiency while considerably reducing the
production of (NOx) emissions. Similar results are also
reported on an application of a local model networks (LMN)
based multivariable long-range predictive control (LRPC)
strategy for a simulation of 200 MW oil-fired drum-boiler
thermal plant [10].
This paper sets forth to introduce an enhanced method for
fuel quality estimation and its use in a control strategy. The
paper is organized as follows. In Section 2, the biopower
plant process is presented. Section 3 presents an enhanced
control strategy, combustion power calculation for a BioGrate
process and dynamic models of a boiler. The process experiments with varying fuel quality and the diagnosis results are
given in Section 4 in order to demostrate the applicability of
the method, followed by the conclusions in Section 5.
II. DESCRIPTION OF THE PROCESS AND ITS CONTROL
STRATEGY
In the Biopower 5 CHP plant, the heat used for steam
generation is obtained by burning solid biomass fuel: bark,
sawdust and pellets, which are fed to the steam boiler
117
together with combustion air. As a result combustion heat
and flue gases are generated. The heat is then used in the
steam-water circulatory process.
Fig. 1 shows the boiler part of the Biopower 5 CHP plant.
The essential components of the water-steam circuit are an
economizer, a drum, an evaporator and superheaters. Feed
water is pumped from a feed water tank to the boiler. First the
water is led to the economizer (4) that is heated by flue gases.
The temperature of flue gases is decreased by the economizer
and the efficiency of the boiler is improved and thus further
optimized.
Fig. 1. 1. Fuel, 2. Primary air, 3. Secondary air, 4. Economizer, 5. Drum,
6. Evaporator, 7. Superheaters, 8. Superheated steam
From the economizer, heated feed water is led to the drum
(5) and along downcomers into the bottom of the evaporator
(6) tubes that surround the boiler. From the evaporator tubes
the heated water and steam return back to the steam drum,
where steam and water are separated. Steam rises to the top
of the steam drum and flows to the superheaters (7). Steam
heats up furthermore so it superheats. The superheated highpressure steam (8) is led to a steam turbine, where electricity
is generated.
A. Fuel composition and fuel quality
The composition and the quality of fuel have big effect
on its heat value. Thus fuel quality is playing a key role
when designing a control strategy of a biopower plant and
guaranteeing its optimal operation. Common elements to all
biomass fuels are carbon (C), hydrogen (H), oxygen (O) and
nitrogen (N ). In addition, biomass fuels contain substances
from soil, such as water, minerals, rock materials and sulphur
(S).
The actual volatile components of fuels are carbon, hydrogen and sulphur. Sulphur is an unwanted component, because
it forms harmful sulphur dioxide, when it is burned. Nitrogen
is also harmful; part of nitrogen reacts with oxygen and
forms nitrogen oxides.
Water in fuel requires heat for its evaporation. Because
of this, moisture decreases the heat value of fuel. Table I
lists the composition and typical moisture content of wood
fuels burned in the Biopower 5 CHP plant. The heat value
of fuel can be determined by the equation that has been
derived from heat values between combustible components
TABLE I
THE COMPOSITION OF WOOD FUELS BURNED IN THE BIOPOWER 5 CHP
PLANT
Fuel C H S O N Ash M
Wood 50.4 6.2 - 42.5 0.5 0.4 50
Pine 54.5 5.9 - 37.6 0.3 1.7 60
Spruce 50.6 5.9 - 40.2 0.5 2.8 60
and oxygen [1]. For solid fuel the following equation can be
used.
qf = 34.8 ·mC + 93.8 ·mH + 10.5 ·mS
+6.3 ·mN − 10.8 ·mO − l25 ·mH2O[MJ/kg] (1)
where m is the mass percent of a component and the sub
index C is carbon, H hydrogen, S sulphur, N nitrogen, O
oxygen and H2O the water content of fuel. In order to use
Equation 1, the composition of fuel has to be known.
B. Control strategy of the biograte process
The main aim of the control strategy of the biograte
process is to produce desired amount of energy by keeping
the drum pressure constant. At the highest level of the control
strategy, the drum pressure control defines the power of the
boiler. At a lower level, the needed boiler power is produced
by controlling the amount of combustion air and fuel.
The primary air flow is controlled by the set point that
comes from the pressure control. The fuel feed is controlled
to track the primary air flow measurement. The needed
amount of primary air and secondary air for diverse fuel
and power levels are specified by air curves that have been
calculated in a boiler design phase. The flue gas oxygen
controller acts as a master controller while the set point of
the secondary air controller is adjusted to provide the desired
amount of excess air for the combustion.
III. ENHANCED CONTROL STRATEGY FOR BIOPOWER 5
CHP PLANT
In the enhanced control strategy, (taking into account the
oxygen consumption and total air feed), the amount of fuel
burned is estimated and the fuel feed setpoint is modified
accordingly. The integrator in Fig. 2 removes steady-state
offset in the control loop. Furhermore, the enhanced control
strategy uses the dynamic model of the boiler to take into
account variations in the moisture content of fuel. Therefore,
it is possible to control the combustion process dynamically
preventing steam temperature and pressure oscillations.
The biograte process is characterized by large time constants and long time delays. Thus the drum pressure control
has to be tuned slow to maintain stability. The disturbances
in fuel quality and fuel feed have a strong effect and proportionate direct correlation on steam pressure. These pressure
and temperature disturbances settle slowly. To overcome this
limitation, a dynamic model has been developed using the
combustion power and energy balance of the boiler which
are described next.
118
Fig. 2. Enhanced strategy based on the oxygen consumption and the energy
balance of the boiler
A. Estimation of the combustion power
The combustion reaction in fossil fuel power plants occurs
mainly between carbon and oxygen. Therefore, a good
measure of heat generation in the furnace is the oxygen
consumption [4].
When the fuel composition and combustion reactions are
known, the combustion air and the composition of flue
gases can be calculated. This information can then be used
to conclude the completeness of the combustion and the
correctness of the fuel-air ratio. Table II gives moles per
unit of fuel from mass fractions of the fuel. The amount
TABLE II
MOLES OF THE COMPONENTS OF THE FUEL PER UNIT MASS
Component Mass wi () (g/mol) (mol/kg)
C wc(1− w/100) 12.011 wC · 10/nC
H wh(1− w/100) 2.0158 wH · 10/nH
S ws(1− w/100) 32.06 wS · 10/nS
O wo(1− w/100) 31.9988 wO · 10/nO
N wn(1− w/100) 28.01348 wN · 10/nN
Water w 18.0152 10/nW
of oxygen needed for fuel combustion can be determined
from the reaction equations. By summing up the oxygen
needed for different components and subtracting the amount
of oxygen in the fuel, the theoretical amount of oxygen
needed to burn completely one kilogram of the fuel is:
NgO2 = nC + 0.5 · nH2 + nS − nO2 [mol/kg] (2)
Air contains mainly oxygen and nitrogen. Argon is often
included in nitrogen portion, so there is 21% oxygen and
79% nitrogen in the air. Theoretical amount of dry air needed
is then:
NAir = N
g O2
· 1
0.21
= NgO2 · 4.76[mol/kg] (3)
In addition to combustion products, nitrogen N that comes
with the air, is included in flue gases. There is 3.76 times
more nitrogen compared with needed oxygen in flue gases.
Incombustible components for example water are included
as such. Flue gas flow for one kilogram of fuel is
Nfg = nC +nH2 +nS +3.76 ·N
g O2
+nN2 +nH2O[mol/kg]
(4)
Similarly, the flue gas losses per kilogram of fuel are
qgfg = (nCCCO2 + nSCSO2 + (nH2O + nH2)CH2O
+(3.76 ·NgO2 + nN2)CN + (FAir/(22.41 · 10
−3 ·mf )
−4.76 ·NgO2)CAir) · (tfg − t0)[J/kg] (5)
where Ci is specific heat capacity i (J/molT ), FAir is
total air flow (m3/s), mf is fuel flow (kg/s), tfg is temperature of the flue gas (◦C), and t0 reference temperature
(◦C).
The combustion power of the BioGrate boiler is estimated
using Equations 6 - 10.
The total oxygen consumption is
N totO2 = 0.21 · nAir −
XO2
100
· nfg[mol/s] (6)
where N totO2 is total oxygen consumption (mol/s), nAir is
total air flow (mol/s), XO2(t+ τ) is oxygen content of flue
gas (%), and nfg flue gas flow (mol/s).
The flue gas flow is
nfg = mf ·Nfg + nAir + 4.76 ·mf ·NgO2 [mol/s] (7)
On the other hand, the oxygen consumption can be presented in the form:
N totO2 = mf ·N
g O2
[mol/s] (8)
and thus the fuel flow is
mf =
(0.21− XO2100 )nAir
NgO2 +
XO2
100 (Nfg − 4.76 ·N
g O2
)
[kg/s] (9)
Finally, the net combustion power for a given fuel flow is
P = (qf − qgfg − qcr) ·mf [MW ] (10)
where qcr is convection and radiation losses (MJ/kg).
B. Dynamic models of the boiler and estimation of moisture
Much of the behaviour of the boiler is captured by global
mass and energy balances [12]. The heat released by the
combustion of fuel is transferred to the water and steam
of the boiler where each section can be considered as a
thermal system. Therefore, the model is developed using the
combustion power and the energy balances of the boiler to
detect fluctuations in fuel quality.
The global mass balance is
d dt (%sVst + %wVwt) = mf −ms[kg/s] (11)
where %s is specific density of steam (kg/m3), Vst is
volume of steam (m3), %w is specific density of water
(kg/m3), Vwt is volume of water (m3), mf is feed water
flow (kg/s), and ms steam flow rate (kg/s).
The global energy balance is
d dt (%susVst + %wuwVwt +mtCptm)
= Q+mfhf −mshs[kJ/s] (12)
where us is specific internal energy of steam (kJ/kg), uw
is specific internal energy of water (kJ/kg), mt is total mass
119
of the metal tubes and the drum (kg), Cp is specific heat of
the metal (kJ/kgK), tm is temperature of the metal (K) and
Q is heat transfer from metal walls to steam/water (kJ/s).
Since the internal energy is u = h−p/%, the global energy
balance is
d dt (%shsVst + %whwVwt + pVt +mtCptm)
= Q+mfhf −mshs[kJ/s] (13)
where hs is specific enthalpy of steam (kJ/kg), hw is
specific enthalpy of water (kJ/kg), and hf specific enthalpy
of water (kJ/kg).
and
Vt = Vst + Vwt[m
3] (14)
Multiplying Equation 11 by hw and subtracting the result
from Equation 13 gives
(hs − hw)
d dt (%sVst) + %sVst
hs dt + %wVwt
hw dt −Vt
dp dt +mtCp
dts
dt = Q−mf (hw − hf )−ms(hs − hw)[kJ/s] (15)
Equation 15 is used for economizer, evaporator, and superheaters subsections of the boiler and it is modified as
necessary. If the drum level is controlled well, the variations
in the steam volume are small and energy balance for a
subsection is [8],[9]
dh2
dt =
1
%V
(Q+mfhf −mshs)[kJ/(s · kg)] (16)
Energy balance for tube walls is
dtm
dt =
1
mtCp
(P −Q)[K/s] (17)
Heat transfer from metal walls to steam/water for convection heat transfer (superheaters) is
Q = αm0.8s (tm − T )[kJ/s] (18)
where α is conversion heat transfer coefficient and for boiling
heat transfer (water wall)
Q = α(tm − T )3[kJ/s] (19)
T = (T1 + T2)/2[
◦C] (20)
where T1 is input temperature (◦C), and T2 output temperature (◦C).
The temperature of a secondary superheater is kept constant at specific temperature. De-superheating spray is used
to achieve mixing between the superheated steam at the
outlet of the preceding component. Because the attemperator
has a relatively small volume, the mass storage inside it is
negligible. The steady state energy balance yields
minhin +mdshds = mouthout[kJ/s] (21)
In normal operation, the steam flow mout in the secondary
superheater is imposed by the load controller, the enthalpy
of primary superheater hin is determined by the upstream
superheater and enthalpy of de-superheating spray hds is
nearly constant.
The value for the moisture parameter w is obtained by
minimizing
min J(w) =
N∑
i=0
(h− ĥ) (22)
where N is prediction horizon, h is measured output
enthalpy of the boiler (kJ/kg), and ĥ estimated output
enthalpy of the boiler (kJ/kg).
IV. TEST RESULTS OF THE FUEL QUALITY SOFT-SENSOR
AND THE ENHANCED CONTROL STRATEGY
First, the performance of the fuel quality soft-sensor is
tested with real data obtained from the Biopower 5 CHP
plant. Next, the three different control strategies (the current
control strategy, the combustion power control strategy and
the enhanced control strategy proposed in this paper) are
evaluated on the Biopower 5 CHP plant simulator using
MATLAB simulation environment.
A. Performance of the fuel quality soft-sensor
Process tests were carried out in the Biopower 5 CHP
plant and ten hours of data were logged with 1 second as
the sampling time (benchmark interval). Two different fuels
were varied during the time period. In order to show the
performance of the fuel quality soft-sensor, the monitoring of
the most important variables is investigated to study the effect
of the fuel quality: drum pressure, the temperature of the
furnace, the temperature over the secondary superheater and
temperature of the flue gases. The value of the fuel moisture
parameter w in Fig. 3 agrees with the power of the boiler.
Although the fuel flow in Fig. 4 increases, the power of
the boiler drops. Therefore, the value of the fuel moisture w
captures the fuel quality well. The combustion power method
estimates the combustion power of the fuel flow well when
the fuel quality does not change significantly. Since the fuel
type was varying greatly, there is an error in the estimation
of the power in Fig. 5. The temperature of the furnace in
Fig. 5 drops when the value of the fuel moisture w increases.
However, the change in fuel quality is shown 20 minutes later
if compared with the fuel moisture parameter. Moreover, the
temperature of the furnace is disturbed by air flows that cool
the furnace. Fig. 6 shows the performance of drum pressure
control when the fuel type changes. Since the controller is
not tuned to handle variations in fuel quality it saturates
and the power of the boiler is not controlled. Moreover, the
drum pressure is characterized with a large time delay. The
temperature difference between the secondary superheater in
Fig. 7 correlates with the power of the boiler. However, the
temperature is greatly disturbed. On the other hand, the flue
gas temperature in Fig. 7 increases when moisture content
in fuel increases. Since more fuel and air are needed to
achieve the same amount of energy, more heat flows through
120
Fig. 3. The power of the boiler and the value of the fuel moisture parameter
Fig. 4. The estimated fuel flow and the stoker speed
Fig. 5. Combustion power and temperature of furnace
the gases, and therefore altering its temperature. Also the
measurement is greatly filtered. Therefore, it cannot be used
as a measure of fuel quality. The validity of the fuel quality
soft-sensor was tested by calculating the cross correlation
between different values in Table III. The positive delay
378 s means that the moisture value shows the change in
fuel quality earlier than any other value. Also value -0.74
means good correlation with the power of the boiler.
According to these tests, the fuel quality soft-sensor is
a promising way to measure the changing fuel quality and
could be used in a control strategy.
Fig. 6. Drum pressure control
Fig. 7. The temperature difference between the secondary superheater and
flue gas temperature
B. Comparision of control strategies.
Fig. 8 and 9 present disturbance situations in the fuel feed.
The power demand of the boiler is kept on 13.5 MW while
the moisture content of fuel is varied from 55 % to 65 %. The
settling time in the response of the current control strategy
is about 2 hours, whereas it is about 10 minutes when
using the enhanced control strategy. With the combustion
power control, there are minor oscillations. Fig. 10 and 11
Fig. 8. Disturbances in power and pressure.
present boiler load disturbances. With the control strategy
121
TABLE III
CORRELATIONS OF DIFFERENT FUEL QUALITY INDICATORS
Corr Delay (s)
Moisture content, Boiler power -0.74 378
Furnace temperature, Boiler power 0.92 -666
Drum pressure, Boiler power 0.81 -424
Temperature over superheater, Boiler power 0.64 298
Flue gas temperature, Boiler power -0.67 -1464
Fig. 9. Disturbances in air and fuel.
used currently, load disturbances cause strong oscillations.
Using the enhanced method no oscillations occur. Also it has
half of the settling time compared with the method based
only in oxygen consumption, because the fluctuating fuel
quality is taken into account already in combustion power
calculation.
Fig. 10. Load disturbances in power and pressure
The process simulation tests proved that the enhanced
control strategy is able to efficiently stabilize the combustion
process. The control strategy managed to keep the pressure
level by using air flow, oxygen content and fuel quality softsensor for the estimation of the fuel flow and the allocation
of fuel.
V. CONCLUSIONS
An enhanced method for estimating fuel quality on a
BioGrate combustion process, along with its use in a control
strategy improvement environment at the Biopower 5 CHP
Fig. 11. Load disturbances in air and fuel
plant was presented in this paper. Using the enhanced control
strategy, it is possible to control the combustion process
dynamically preventing steam temperature and pressure oscillations.
The fuel quality estimation method was tested with real
data. The enhanced control strategy was tested in the controlled simulation environment. The results of the tests
dramatically demonstrate that the enhanced control strategy
efficiently stabilizes the combustion process.
REFERENCES
[1] Effenberger, H. (2000) Dampferzeuger. Germany: Springer.
[2] Havlena, V & Findejs, J. (2005) Application of model predictive
control to advanced combustion control Control. Control Engineering
Practice, Vol 13, Issue 6, June 2005, pp. 671-680.
[3] Kortela, U. & Lautala, P. (1981) A New Control Concept for a Coal
Power Plant. Proceedings of the 8th IFAC World Congress, Kyoto,
Japan 1981.
[4] Kortela, U. & Marttinen, A. (1985). Modelling, Identification and
Control of a Grate Boiler. American Control Conference, 1985, Boston,
MA, USA 19-21 June 1985. pp. 544-549
[5] Kortela, U., Ikonen, I., Kotajärvi H. & Heikkinen, P. (1994) Modelling, Simulation and Control of Fluidized Bed Combustion Process.
International Journal of Power & Energy Systems, Vol 14, no. 3, pp.
92-97.
[6] Lehtomäki, K., Kortela, U., Wahlström, F. & Luukkanen, J. (1982)
New Control Methods for Combustion Stabilization in Peat Power
Plants. The Joint Soviet-Finnish Symposium on Automation in Process
Industries, Espoo, Finland 13-16 December 1982.
[7] Leppäkoski, K., Mononen, J. & Kovács, J. (2000) Integrated Optimisation and Control System to Reduce Flue Gas Emissions. Power
Plants and Power Systems Control 2000: A Proceedings Colume from
the IFAC Symposium Brussels, Belgium, 26-29 April, 2000. pp. 113118
[8] Lu, S. (1999) Dynamic modelling and simulation of power plant
systems. Proceedings of the Institution of Mechanical Engineers, Part
A: Journal of Power and Energy, Vol 213, Number 1/1999, pp. 7-22.
[9] Lu, S. Hogg, W. (2000) Dynamic nonlinear modelling of power plant
by physical principles and neural networks. International Journal of
Electrical Power & Energy Systems, Vol 22, Issue 1, January 2000,
pp. 67-78.
[10] Prasad, G., Swidenbank, E. & Hogg, B. W. (1998) A Local Model
Networks Based Multivariable Long-Range Predictive Control Strategy for Thermal Power Plants. Automatica, Vol 34, Issue 10, October
1998, pp. 1185-1204.
[11] Yin, C., Rosendahl, L. & Kær, S. (2008) Grate-firing of biomass
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122
Convex approximation of the static output feedback problem with
application to MIMO-PID
Henrik Manum and Sigurd Skogestad∗
Department of Chemical Engineering
Norwegian University of Science and Technology
N-7491 Trondheim
1 Introduction
In this contribution we derive convex approximations
to the static output feedback (SOF) problem. This may
seem like a restrictive formulation, but the problem of
finding an optimal multiple input − multiple output
(MIMO) proportional integral deriative (PID) controller
can be posed on this form. In the literature it is proved
that a problem closely related to SOF belongs to a class
of problems that cannot be solved by polynomial time
algorithms (NP-hard problems), and it is further conjectured that also the SOF problem is an NP-hard problem
[Blondel and Tsitsikilis, 1997]. For an overview of different approaches to this problem the reader is referred
to the survey paper by Syrmos et al. [1997].
2 Problem formulation
Consider a linear process on discrete form
xk+1 = Axk + Buk + dk,
yk = Cxk + nk,
(1)
where xk ∈ Rnx are the states, uk ∈ Rnu are the inputs
and yk ∈ Rny are the measurements. The H 2 infinite
horizon static output feedback (IH-SOF) problem is to
choose an output feedback uk = −Kyyk that minimizes
the objective function
JIH-SOF =


i=0
x′iQxi + u

iRui (2)
for all disturbances dk ∈ Rnx and noise-terms nk ∈ Rny .
Throughout the paper we assume that Q = Q′ ≥ 0 and
R = R′ > 0. A related problem is the finite horizon static
output feedback problem (FH-SOF), which is the same
problem as IH-SOF, except that the objective function
∗Corresponding author: skoge@chemeng.ntnu.no
to be minimized is finite:
JFH-SOF = x

NPxN +
N−1

i=0
(
x′iQxi + u

iRui
)
. (3)
The problems posed above are feedback problems and
as already mentioned they are conjectured to be NPhard. To derive new convex approximations, we do not
directly minimize the cost (3), but rather we minimize
the (worst-case) loss to an optimal open loop controller,
which in this case is the finite-horizon linear quadratic
controller (FH-LQR). More precisely, the loss we want
to minimize is
L(d) = JKy(d)− JFH-LQR(d), (4)
where JKy(d) is the cost of a particular (static output)
feedback implementation and JFH-LQR(d) is the cost of
a FH-LQR problem.
In the previous Nordic Process Control Workshop
(Porsgrunn, 2009) we proposed a convex approximation based on implementing the first move of a sequence
of open loop moves
[
u′0 u

1 . . . u

N−1
]′
= Ky0. In
this paper we present a new approximation, which
seems to get closer to the global optimum, but at the expense of some more computational effort. The resulting
problem is a quite large quadratic program (QP).
3 Solution methods
By using a result from Alstad et al. [2009] we pose the
following problem as a convex relaxation of the SOF
problem:
min
H
‖HF‖F
s.t. H on the form
[
Hy Hu
]
=
[
diag(Ky) I
] (5)
Here F = −(GyJ−1uu Jud −G
y d) is the optimal sensitivity
matrix from the disturbance d = x0 to the “measurements” y = (y0,y1, . . . ,yN−1,u0,u1, . . . ,uN−1) and Juu,
123
Jud , Gy and G
y d are derived from the objective function
and the linear model as shown below.
Derivation of Juu and Jud Consider the FH-LQR
problem, which is an open loop problem in the inputs
u = (u0,u1, . . . ,uN−1) defined by first writing the linear
model xk+1 = Axk + Buk as





x1 x2 ...
xN−1





︸ ︷︷ ︸
x =





A
A2
...
AN





︸ ︷︷ ︸
Gxx0
x0 +





B
AB B
...
. . .
AN−1B · · · · · · B





︸ ︷︷ ︸
Gxu
u (6)
By further defining Q̄ = diag(Q, · · · ,Q,P) and R̄ =
diag(R, · · · ,R) and let the disturbance d = x0 we can
write the FH-LQR problem as
J∗LQR(d) = minu
[
u d ]′[
Juu Jud
J′ud Jdd
][
u d ]
(7)
with
Juu = G
x u ′Q̄Gxu + R̄, (8)
Jud = G
x u ′Q̄Gxx0 , (9)
Jdd = G
x x0 ′Q̄Gxx0 . (10)
Derivation of Gy and Gyd Using the linear model
xk+1 = Axk + Buk, yk = Cxk + Duk, we have:
yk = CA
kx0 +
k−1

j=0
CA jBuk−1− j + Duk, (11)
which on matrix form can be written as
ym =







y0 y1 y2 ...
yN−1







=







C
CA
CA2
...
CAN−1







︸ ︷︷ ︸
Gymx0
x0 + . . .
+






0 0 ... 0 0
CB 0 ... 0 0
CAB AB ... 0 0
...
...
. . .
...
...
CAN−2B CAN−3B ... CB 0


+ diag(D)


u
︸ ︷︷ ︸
Gymu
.
(12)
We now have that
y =
[
ym u ]
=
[
Gymu
I
]
︸ ︷︷ ︸
Gy u+ [
Gymx0
0
]
︸ ︷︷ ︸
Gyd
x0 ︸︷︷︸
d . (13)
4 Application to MIMO-PID
In order to use static output feedback synthesis for this
problem we augment the plant output with the integrated output and derivative. The augmented plant can
be written as
[
xk+1
σk+1
]
=
[
A 0
C I
][
xk σk
]
+
[
B
0
]
uk

yPk
yIk
yDk

 =


C 0
0 I
1
Ts C(A− I) 0


[
xk σk
]
+


0
0
B

uk.
(14)
We can now use the method above to calculate a static
output feedback Ky for the augmented output vector.
Example: Distillation We have calculated MIMOPI and -PID controllers for “column A” in [Skogestad,
1997] and we found that in terms of closed-loop norms
did the convex approximations come quite close to the
true optimal SOF controllers (found by nonlinear search
starting from the convex approximations). In addition
we have tight bound in terms of the worst case loss from
the LQR controller, and we found that the loss was less
than 3.5% for both cases. This will be further illustrated
in the presentation.
5 Conclusions
A new convex approximation to the static output feedback problem has been given, and we have shown that
the approximation can be used to find MIMO-PID controllers for interesting chemical engineering cases, such
as distillation control.
References
V. Alstad, S. Skogestad, and E.S. Hori. Optimal measurement combinations as controlled variables. Journal of Process Control, 19(1):138 – 148, 2009.
V. Blondel and J.N. Tsitsikilis. NP-Hardness of some
linear control design problems. Siam Journal of Control and Optimization, 35(6):2118–2127, November
1997.
S. Skogestad. Dynamics and control of distillation
columns - a tutorial introduction. Trans IChemE, Part
A, 75:539–562, September 1997.
V.L. Syrmos, C.T. Abdallah, P. Dorato, and K. Grigoriadis. Static Output Feedback − A Survey. Automatica, 33(2):125–137, 1997.
124
Modeling and Optimization of Grade Changes for Multistage
Polyethylene Reactors
Per-Ola Larsson, Johan Åkesson, Staffan Haugwitz, Niklas Andersson
I. INTRODUCTION
Polyethylene reactors are today able to produce different
grades by manipulating inflows of raw material. It is imperative for polyethylene manufacturers to change product
grades to increase their profitability as market demands
change, but also due to market competition and raw material
pricing. During grade transitions it is of importance that
production of off-specification material, raw material and
time is minimized. We present an optimization procedure
for grade change of a Borstar R© polyethylene plant.
Pre-poly.
Loop GPR
Catalyst
Monomer
Diluent
Gas to
recovery
Flash
Polymer
Product
outlet
Monomer
Co-monomer
Diluent
Fig. 1. Reactor chain of a Borstar R© process: Pre-polymerization, Loop,
and Gas phase reactor (GPR).
II. PLANT MODELING
The Borstar R© polyethylene plant at Borealis AB incorporates two slurry reactors, pre-polymerisation and loop reactor,
and a gas phase reactor, see Figure 1. The model of the plant
includes both first principles, semi-empirical, and empirical
relations. A total of 12 inputs flows, denoted u, are available
at optimization and outputs such as masses of both fluid
and solid components, reaction rates, instantaneous and bed
averaged component concentrations, split factor, catalyst and
polymer properties, denoted y, can be used. Together with
algebraic variables w, the model can be written in the general
non-linear index 1 differential algebraic equation form
0 = F (ẋ,x,w,u)
y = g(x,w,u).
(1)
and contains approximately 70 differentiated variables, 180
algebraic variables and 250 equations.
Sponsored by the Swedish Foundation of Strategic Research in the
framework of Process Industry Centre at Lund University (PICLU).
P. Larsson and J. Åkesson are with the Department of Automatic Control,
Lund University, Lund, Sweden, {perola|jakesson}@control.lth.se.
S. Haugwitz is with Borealis AB, Stenungsund, Sweden,
Staffan.Haugwitz@borealisgroup.com.
N. Andersson is with the Department of Chmical Engineering, Lund
University, Lund, Sweden. niklas.andersson@chemeng.lth.se.
Modelica, a high level language for encoding of complex
physical systems, is used for plant modeling. The Optimica extension, see [1], gives constructs for cost functions,
constraints and mechanisms to select inputs and parameters
to optimize. Using JModelica.org, an open source project
targeted at dynamic optimization, see [2], the optimization
problem is translated into a non-linear programming problem
using collocation on finite elements and solved using the
large-scale NLP solver IPOPT [3].
III. OPTIMAL GRADE TRANSITION
The grade transition example will change conditions in
all three reactors and corresponds to two grades currently
produced at Borealis AB. The main objectives are to change
raw material concentrations and concentration ratios, split
factor, and production rates. At transition start and end time,
i.e., t1 and t2, the plant fulfills the static non-linear equations
0 = F (0,x◦,w◦,u◦)
y◦ = g(x◦,w◦,u◦),
(2)
which corresponds to Eq. (1) when all derivatives equal 0
and superscript ◦ indicate constant value. Initial and end
conditions of the transition for states, inflows and algebraic
variables are given by solving the non-linear equations in
Eq. (2), i.e., a DAE initialization problem is posed and
contains approximately 280 equality constraints and 290
variables, of which 180 are algebraic and 230 have both
upper and lower limits. Solving the NLP takes less than 10
seconds per grade.
A quadratic cost function that includes deviations from the
grade to be are used, giving the possibility to emphasize the
importance of different variables. Also the deviation from
inflows yielding the new grade in stationarity will be used,
removing too large over- and undershoots. Introducing the
deviation vectors
∆u = u − u2 ∆y = y − y2,
where u2 and y2 are inputs and outputs defining the new
grade solved for in the DAE initialization problem, the
dynamic grade transition optimization problem can be formulated as
min
u t2∫
t1

∆y
∆u


T 

Q∆y 0 0
0 Q∆u 0
0 0 Qu̇




∆y
∆u

 dt (3)
subj. to 0 = F (ẋ,x,w,u), y = g(x,w,u)
ymin ≤ y ≤ ymax, umin ≤ u ≤ umax
wmin ≤ w ≤ wmax, u̇min ≤ u̇ ≤ u̇max,
125
0 0.1 0.2 0.3 0.4 0.5 0.6
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6
0.95
1
1.05
1.1
1.15
0 0.1 0.2 0.3 0.4 0.5 0.6
1
2
3
4
u p 2
[-
]
u e 2
[-
]
u h 2
[-
]
Time [-]
Fig. 2. Scaled optimal inflows to loop reactor at grade transition – propane
up2, ethylene ue2 and hydrogen uh2.
where also a cost of inflow derivatives is added such that
smoothness of inflows can be controlled. The weights Q∆y,
Q∆u and Qu̇ are chosen diagonal for simplicity and the
initial state of the plant is defined by the solution of the
DAE initialization problem.
Over- and undershoots are accepted up to a certain limit
for the instantaneous concentrations and ratios. However, for
the bed average concentrations and ratios and the split S,
no over- or undershoots are accepted in the grade change.
The constraints on the algebraic variables w are for instance
limits on volumes, component masses, and pressure, while
constraints on inflows, both magnitudes and rates of changes,
concern physical limits such as e.g., pump capacities.
After discretization, the NLP problem contains about
20.000–200.000 variables depending on number of elements
and collocation points. Initial trajectories can be generated
in JModelica.org via simulation using SUNDIALS, see [4],
with inflows ramping from initial to end values found in
the DAE intialization problem. With an Intel R© CoreTM2
Duo CPU@3.00GHz, a solution is obtained in 5-90 minutes
depending on number of variables and initial values.
Figures 2–3 show the resulting optimal inflows, component concentrations, and production rate of the loop reactor
and the split factor between the loop reactor and GPR. Note
the scaling, i.e., the transition is 1 time unit and all variables
have initial value 1.
Since the production rate Q2 is to be increased, the inflow
of ethylene is increased in total and at the same time inflow
of the diluent propane is decreased as shown in Figure 2. This
results in a longer hold up time of the polymer and thus also
a larger mass of polymer in the loop. The concentrations of
ethylene and hydrogen in the loop are higher in the new grade
and the decrease of diluent is not enough for the hydrogen
specification to be met. Thus, the inflow of hydrogen is
increased and to reach the specification of the hydrogenethylene ratio rapidly, the inflow of ethylene is initially
0 0.1 0.2 0.3 0.4 0.5 0.6
0.98
0.99
1
0 0.1 0.2 0.3 0.4 0.5 0.6
0.9
1
1.1
1.2
0 0.1 0.2 0.3 0.4 0.5 0.6
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5 0.6
0.95
1
1.05
1.1
1.15
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.9
0.95
1
X
e 2
,

e 2
[-
]
X
p 2
,

p 2
[-
]
X
h e 2
,

h e 2
[-
]
Q
2
[-
]
S
[-
]
Time [-]
Fig. 3. Scaled key parameters for loop reactor and split factor at grade
transition. Bed averaged (solid) and instantaneous (dashed) ethylene conc.
X̄e2, Xe2, propane conc. X̄p2, Xp2, hydrogen-ethylene conc. ratios X̄he2,
Xhe2, production rate Q2, split factor S.
decrased. Note that both the inflow of ethylene and hydrogen
have their derivative constraints active in the beginning, seen
by the linear decrease and increase. From Figure 3 it is
seen that the over- or undershoot constraints on the averaged
concentrations and ratios are obeyed and the instantaneous
measures have over- or undershoots. The split, see Figure 3,
which indirectly depends on the production rates in both loop
and gas phase reactor, is decreased by lowering production
rate in the GPR, i.e., decreasing the ethylene inflow to the
gas phase reactor. The transition in loop reactor is completed
after 0.5 time units. Similar trajectories for key parameters
and inflows are available for the pre-polymerization and gas
phase reactor.
REFERENCES
[1] J. Åkesson, “Optimica—An Extension of Modelica Supporting Dynamic Optimization,” in In 6th International Modelica Conference
2008. Modelica Association, Mar. 2008.
[2] J. Åkesson, K.-E. Årzén, M. Gäfvert, T. Bergdahl, and H. Tummescheit,
“Modeling and optimization with optimica and jmodelica.orglanguages and tools for solving large-scale dynamic optimization problem,” Computers and Chemical Engineering, Jan. 2010,
doi:10.1016/j.compchemeng.2009.11.011.
[3] A. Wächter and L. T. Biegler, “On the implementation of an interiorpoint filter line-search algorithm for large-scale nonlinear programming.” Mathematical Programming, vol. 106, no. 1, pp. 25–58, 2006.
[4] C. f. A. S. C. Lawrence Livermore National Laboratory, “SUNDIALS
(SUite of Nonlinear and DIfferential/ALgebraic equation Solvers),”
2009, https: //computation.llnl.gov/casc/sundials/main.html.
126
Magnus Glosli Jacobsen:
Challenges in optimization of operation of LNG plants
Abstract for the Nordic Process Control Workshop 2010, Lund, Sweden
Keywords: LNG, optimal operation, self-optimizing control, simulation
As pointed out by Jensen [1], most of the open research on LNG plants focuses on process
design. A typical study will seek to maximize profit over the life span of a plant, by chosing
the best configuration. This includes how many pressure levels the process should have,
whether to use single component or multi component refrigerants, what kind of drives to use
for the compressors and so on. Often such optimizations take the form of mixed-integer
nonlinear problems (MINLP).
The final result depends on factors like expected gas prices, distance between gas field and
liquefaction facility, and climate at the plant location. For example, if temperatures vary
significantly between summer and winter, one must have a plant design which is robust to
changes in ambient temperature.
Once the process design is done, however, one should obviously seek to run the process as
close to optimal as possible. This means that disturbances need to be handled – for example
by using MPC controllers or by controlling variables whose optimal values are not very
sensitive to disturbances. The latter approach is called self-optimizing control [2] – i.e. one
seeks to minimize loss caused by disturbances, by choosing the best variables to control at
constant setpoints.
In order to do the latter, one needs to perform off-line optimization of the plant with given
plant data. One will inevitably have fewer degrees of freedom for optimization in this case,
since equipment size, driver configurations, coolant compositions and so on are fixed.
Typically one can vary pressures and flowrates in certain streams. One must optimize the
process for nominal conditions and for different disturbance scenarios.
The problem does usually not contain any binary/integer decision variables, all variables are
continuous. This means we are left with a constrained non-linear problem, which may be
solved using a sequential quadratic programming method. However, the problem is not
always easy. Since we operate with small temperature differences (especially in the cold part
of the plant – often as low as 1°C) we encounter problems in the models used to calculate the
objective function and constraints.
Since each calculation of constraints and objectives requires convergence of the steady-state
process model, it is critical that the model is robust enough to handle the steps taken by the
optimization method. If, for example, the independent variables are temperatures in both ends
of a heat exchanger, it is easy to specify a step in those variables which is physically
infeasible. When you combine flows that go in closed loops with small margins to constraints,
you are bound to have a difficult optimization problem. Examples of difficulties are:
127
• A small change in one flow might result in temperature crossover unless the model is
good enough
• A small change in one temperature might lead to another temperature becoming
infeasible – for example, the location of the smallest temperature difference might
move to the other end of the heat exchanger, giving a very different solution
• The active set (of constraints) changes, especially if disturbances occur
It is necessary to know the process well in order to handle, or possibly avoid, these problems.
The example process that the work is focused on is the Air Products C3-MR process [3],
which is the most widely used process for liquefaction of natural gas to date. The main
approach to optimization has been to model the process in Honeywell’s Unisim simulation
software [4], and using Matlab’s Optimization Toolbox [5] to carry out optimization. The two
are linked using the actxserver function, which makes Unisim a COM server for Matlab and
allows Matlab to specify variables in the Unisim model.
In this work we have investigated possibilities for simplification of the optimization problem
– this includes changing the set of specifications, reformulating the models and finally
splitting up the flowsheets and optimizing each part with respect to appropriately changed
objective functions. We have also sought to identify the parts of the process which are most
likely to produce problems for optimization.
References:
[1] Jensen, J.B., Skogestad. S: “Optimal operation of a simple LNG process”, International
Symposium on Advanced Control of Chemical Processes, Gramado, Brazil, 2006.
[2] Skogestad, S.: “Plantwide control: the search for the self-optimizing control structure”, J.
Proc. Control, 10, 487-507 (2000).
[3] Newton, C. L.; Kinard, G. E.; Liu, Y. N.: “C3-MR Processes for baseload liquefied natural
gas”. Liquefied Natural Gas VIII Volume 1, Sessions I & II, June 15-19 1986, Los Angeles,
California.
[4] http://hpsweb.honeywell.com/Cultures/enUS/Products/ControlApplications/simulation/UniSimDesign/default.htm
[5] http://www.mathworks.com/products/optimization/?BB=1
128
Production Optimization for Two-Phase Flow in
an Oil Reservoir
Carsten Völcker, John Bagterp Jørgensen, Per Grove Thomsen
Department of Informatics and Mathematical Modeling
Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
Erling H. Stenby
Department of Chemical and Biochemical Engineering
Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark
Keywords : Reservoir simulation/management, Runge-Kutta, ESDIRK, optimal control, nonlinear model predictive control, adjoint sensitivity
Petroleum reservoirs are subsurface formations of porous rocks with hydrocarbons (oil and/or gas) trapped in the pores. Initially a reservoir may be under
sufficient pressure to push the fluids to the surface. However, as the fluids are
produced the pressure declines and production reduces over time. When natural drive becomes insufficient, then the pressure can be maintained artificially
by injection of water. Conventional technologies for recovery leaves more than
50 % of the oil in the reservoir. Wells with adjustable downhole flow control
devices coupled with modern control technology offer the potential to increase
the oil recovery significantly.
The objective is to maximize production by manipulating the well rates and
bottom hole pressures of injection and production wells. Optimal control settings of injection and production wells are computed by solution of a large
scale constrained optimal control problem. We present a two-phase immiscible flow model and describe a gradient based method to compute the optimal
control strategy. An explicit singly diagonally implicit Runge-Kutta (ESDIRK)
method with adaptive stepsize control is used for computationally effecient solution of the model. The gradients are computed by the adjoint method. The
adjoint equations associated with the ESDIRK method are solved by integrating backwards in time. The necessary information for the adjoint computation
is calculated and stored during the forward solution of the model. The backward adjoint computation then only requires the assembly of this information
to compute the gradients.
We demonstrate the optimal control strategy on a simple waterflooding example
using one injector and one producer, which are divided into several individually
controllable inflow valves.
Acknowledgement
This research project is funded by the Danish Research Council for Technology
and Production Sciences. FTP Grant no. 274-06-0284
The 16th Nordic Process Control Workshop (NPCW’10)
AF-Borgen, Lund, Sweden, August 25-27 2010
129
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q�1
q�2
q�i
q��
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�2
switching
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const� ��������� ������ ��� ������� ���� �� �����\������ ��������
�� � ������ �� ��������� �������� ���� ��������� ������������
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����� �� ��� �����\������ ��1� �2)� ��� ������ �� ��� ������ qbi � i = 1� 2� · · · � η �� ��� ����� ������ ��� ��������� �������
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min
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η�1
i=1
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130
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i=1
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���� �� ���� ��� ��������� ����� ��� ���� ����� ������� �� ��� �� ����� ���� ��� ������� �� ti�qbi�1 � qbi) ������ �� ���
�������� ��������� qbi�1 ��� qbi �� � 4 × 4 ��������� ������� ���� ����� ����������� ����� ����������� ���� ��� � ������
��� λ4�i �� ��� ����
λ4�i = Cgi
gi1gi2
�gi2 − gi1)2

�qbi�2 − qbi�1�2)
�gi1
gi2�gi1 �qbi�1 − qbi�1�1)
2gi1�gi2
gi2�gi1 + �qbi�2 − qbi�1�2)
gi1�2gi2
gi2�gi1 �qbi�1 − qbi�1�1)
gi2
gi2�gi1

.
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2
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8
1�
12
optimal
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desired
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initial
shape
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[m m ]
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131
Comparison of two main approaches for operating Kaibel distillation
columns  
Maryam Ghadrdan, Ivar J. Halvorsen*, Sigurd Skogestad,
NTNU, Deparment of Chemical Engineering
*) SINTEF ICT, Applied Cybernetics
The divided-wall distillation column (DWC) realizes the fully thermally coupled Petlyuk column,
which is to separate the feed in a prefractionator-sidestream arrangement with a direct coupling
of vapor and liquid streams between prefractionator and main column, into a single shell. This
arrangement for separating the feed to 4 products is called Kaibel arrangement. This tight
integration makes it challenging to design and control the column, compared to the conventional
sequence of simple columns. The design challenges have been mostly solved, but operation
and control remains largely an open issue.
The objective of this paper is to study the Kaibel distillation column from operability point of
view. Also a qualitative study is done for finding proper self-optimizing control variables. Two
different objectives, namely minimizing energy consumption at fixed product purities and
maximizing product purities with a fixed boilup are considered. It is usually assumed that the
objective is to make products of given purity using the minimum energy. However, in practical
operation this is often not the issue, but rather to make the purest possible products with a given
energy.
The idea behind self-optimizing control is to find a variable which characterize operation at the
optimum, and the value of this variable at the optimum should be less sensitive to variations in
disturbances than the optimal value of the remaining degrees of freedom. Thus if we close a
feedback loop with this candidate variable controlled to a setpoint, we should expect that the
operation will be kept closer to optimum when a disturbance occur. Self-optimizing control is
when we can achieve an acceptable loss L with constant setpoint values c, for the controlled
variables (Skogestad 2000).
The steady-state model used for this purpose is developed in UNISIM. The model in UNISIM is
optimized and purturbed around the optimum from MATLAB. This is because of the more
powerful optimization tool provided in MATLAB. The feed stream contains the first four simple
alcohols (Methanol, Ethanol, 1-Propanol, 1-butanol) as the components. The model has six
degrees of freedom: boilup rate (V), reflux (L), side stream flows (S1, S2), liquid split (Rl) and
vapour split (Rv). Boilup rate is set as a constraint in the first approach.This constraint should be
met as the operating conditions change. After finding the optimal nominal case, we have
visualized the objective functions around the optimum. This can be used to get insight in column
behavior and as a basis for a systematic control structure design. Another issue is that how the
stage design affects this comparison. This is also considered in this study.
References
1. S. Skogestad, J. Process Control, 10 (2000) 487-507
132
   
PRODUCTION OF DISTRICT HEATING AT SÖDRA CELL MÖRRUM 
Karin Axelsson*, Veronica Olesen** 
*Department of Chemical Engineering, Lund University 
**Solvina AB, Gruvgatan 37, Västra Frölunda, Sweden 
1  INTRODUCTION 
Södra  Cell  Mörrum,  SCM  pulp  mill  produces 
steam  in  the  chemical  recovery  process.  Some  of 
the  steam  is  used  to  make  hot  water  for  district 
heating.  The  hot  water  for  district  heating  is 
transported to the local energy company Karlshamn 
Energi AB, KEAB.  
A  varying  flow  of  cold  water  is  sent  back  from 
KEAB  to  SCM.  The  water  is  heated  up  and  sent 
back  at  a,  by  KEAB,  predefined  temperature. 
However,  there  have  been  oscillations  in  the 
process  leading  to  variations  in  outgoing 
temperature  to KEAB.  In  a  project  at  Solvina  the 
source  of  these  oscillations  was  identified,  a 
dynamic  model  of  the  process  was  made  and 
measures of  how to eliminate the oscillations were 
suggested.  
2  PROCESS DESCIPTION 
Surplus  steam  from  the  recovery  process  is  fed 
through a condenser. The heat from the condensing 
steam  is  used  for  heating  up  water.  A  pair  of 
control valves on the condensate pipe regulates the 
condensate flow from the condenser. Changing the 
positions  of  these  valves,  changes  the  condensate 
level and hence the area available for condensation 
and thus incoming steam flow to the condenser. A 
controller  working  with  the  steam  flow  as  its  set 
point  regulates  the  condensate  valves.  The  steam 
flow set point is given by the total steam demand in 
the  process  and  is  in  this  application  an 
uncontrollable variable.  
To compensate for unpredictable changes in steam 
supply  a  heat  buffer,  an  accumulator,  is  used  to 
keep the outgoing  temperature at its set point. The 
accumulator  is  a  hot  water  reserve,  storing  hot 
water in the top and cold water in the bottom. Flow 
can go  in both directions through the accumulator, 
depending  on  whether  it  is  being  charged  or 
discharged with hot water. See Figure 1. 
As seen  in  the  figure,  there  is  also a possibility  to 
bypass  cold  water,  to  keep  the  outgoing 
temperature down. 
Figure  1:  Sketch  of  the  system,  control  loops  are 
marked in red 
3  THE MODEL 
The simulation of the system was performed using 
Extend.  This  modelling  tool  provides  a  platform 
where  all  components  can  be  modelled  as  blocks 
containing first-principle models. All blocks can be 
connected, forming a sequence to be solved in each 
time step. 
The resulting model was validated using data from 
SCM,  collected  during  half  a  year.  During  this 
period  of  time  the  need  for  district  heating  varied 
from very low in the summer to high in the winter. 
The  resulting model was  able  to  capture  the  large 
scale  dynamics  of  the  system.  However,  an  ideal 
model  of  the  system  did  not  capture  the 
oscillations.  
4  PROBLEM IDENTIFICATION 
The  oscillations  observed were  found  to  originate 
in  the condenser and were especially prominent  in 
the  following measured variables:  incoming  steam 
flow,  outgoing  water  temperature  from  the 
condenser,  condensate  level  and  output  from  the 
steam flow controller. Closer analysis of the graphs 
showing  steam  flow  and  output  from  the  steam 
flow controller revealed that they followed patterns 
typical  for  systems  with  valve  friction.  Hence  it 
was  concluded  that  the  oscillations  in  all 
probability  were  caused  by  friction  in  the  large 
condensate valve. See Figure 2 for details. 
133
   
Figure  2.  Data  for  the  large  condensate  valve. 
Green  line  is  the  steam  flow  set  point.  Blue 
curve  shows  the control  output. Black curve  is 
steam flow measured value. 
5  IMPROVED CONTROL STRATEGY 
The  steam  flow  to  the  condenser  is  regulated  by 
two  condensate  valves.  The  output  from  the  flow 
controller is converted into two control signals, one 
for  each  valve.  Prior  to  this  project,  split  range 
control was used to divide the control signal to the 
valve  pair.  The  small  valve  was  used  for  small 
demands  on  steam  flow.  At  higher  demands,  the 
small valve was fully open and the large valve was 
used  for  control  of  the  steam  flow.  This  strategy 
relies  on  both  valves  being  able  to  move 
continuously  and  hence  fails  when  friction  is 
present in the large valve. 
A  new  strategy  was  proposed  where  both  control 
valves work  simultaneously  in  the  entire  range  of 
controller  outputs.  The  control  signal  to  the  large 
valve  as  a  function  of  controller  output  was 
suggested  as  a  discrete  stepwise  function.  The 
small  valve  could  then  be  used  for  fine  tuning  at 
each discrete  level. This  strategy has  two apparent 
advantages;  it  does  not  rely  on  the  large  valve 
being able  to move continuously  and  it  allows  the 
small valve to operate in its optimal working span, 
i.e. near 50 % opened.  
6  RESULTS  
The  control  strategy  was  implemented  in  the 
control  system.  The  oscillations  in  outgoing 
temperature were  reduced  and  the  system  showed 
significant improvement in its ability  to follow the 
steam flow set point using the new strategy.  
As a positive side effect of  this project, all control 
loops  could  be  better  tuned  as  the  cause  of 
oscillations was had been found and adjusted for. 
0 10 20 30 40 50 60 70 80 90 100
0
10
20
30
40
50
60
u501-controller output
u 2
-c o n tr o l  s ig n a l  to  l
a rg e  v
a lv e TV501:2
Figure 3: Control signal to large valve as a function 
of controller output 
FjV Solvina AB
13,0
12,5
12,0
11,5
11,0
10,5
10,0
9,5
9,0
8,5
8,0
26 kg/ s
14,4
14,2
14,0
13,8
13,6
13,4
13,2
13,0
12,8
12,6
12,4
12,2
12,0
11,8
11,6
11,4
11,2
11,0
10,8
10,6
10,4
10,2
100 %
01:30
2009-11-10
01:45
2009-11-10
02:00
2009-11-10
02:15
2009-11-10
02:30
2009-11-10
02:45
2009-11-10
03:00
2009-11-10
03:15
2009-11-10
03:30
2009-11-10
2009-11-10 01:20:00 C1 2009-11-10 03:38:38 -3,83 m 2009-11-10 03:39:00
134
CONTROL OF AN HMR PRE-COMBUSTION GAS POWER CYCLE
Lei Zhaoa, Finn A. Michelsenb, Bjarne Fossa
aNTNU, Department of Engineering Cybernetics,NO-7491 Trondheim, Norway
b SINTEF ICT, Applied Cybernetics, NO-7491 Trondheim, Norway
Abstract: CO2 capture and storage is becoming an increasingly important part of any discussion on clean coal
and natural gas based power production. Statoil has recently developed and patented a pre-combustion gas
power cycle based on a hydrogen membrane reformer (HMR). This is a promising option for capturing CO2 in
natural gas based power generation plants.
The plant consists of a pre-reformer, an HMR reactor, a medium temperature and a low temperature
conventional water gas shift (WGS) stage, gas and steam turbines, a heat recovery system, a CO₂ separation
unit, and several heat exchangers, separation and mixing units. Steam methane reforming (SMR) is among the
most common technologies for converting hydrocarbons (methane) to hydrogen. A mix of natural gas and
steam is fed to one of the sides of the HMR and undergoes steam reforming. The produced gas is a hydrogen
rich syngas. Compressed air drawn from the gas turbine compressor is supplied to the other side of the HMR
reactor. Permeated hydrogen is combusted, consuming approximately all oxygen in the air stream. This gives
"CO₂ free" heat for the endothermic SMR reactions. Syngas with high concentrations of H₂, CO₂ and CO is fed
via several heat exchangers to the WGS stages converting CO to H₂. The outlet gas from the permeate side
contains mainly H₂O and N₂ and is used to dilute the hydrogen fuel recovered in the CO₂ removal process. CO₂
removal may be performed by using a conventional absorption unit. This process has shown higher efficiency
than other pre-combustion processes, and it has a potential for cost reduction compared to other precombustion processes.
For this type of reforming, high operability and robustness is required. This is partly achieved through an
understanding of the system dynamics and robust control structure design. The paper identifies important
dynamic features of the plant. Based on this analysis, the paper explores various options for conventional
control strategies for this plant, and suggests a reasonable control strategy based on realistic disturbance
scenarios.
135
Control
 of
 industrial
 chromatography
 steps
 
 
Jan
 Peter
 Axelsson,
 Karolinska
 Institute,
 
 
Department
 of
 Biosciences
 and
 Nutrition,
 141
 57
 Stockholm,
 Sweden.
 
 
There
 is
 an
 increased
 interest
 to
 decrease
 process
 variation
 and
 to
 reduce
 production
 
costs
 in
 the
 biopharmaceutical
 industry
 today.
 This
 trend
 is
 facilitated
 by
 the
 so-­‐called
 
PAT-­‐initiative
 from
 FDA
 a
 few
 years
 ago.
 In
 this
 talk
 I
 will
 describe
 two
 different
 ideas
 of
 
control
 of
 chromatography
 steps
 with
 industrial
 relevance.
 Both
 examples
 address
 the
 
fact
 that
 the
 incoming
 material
 may
 vary
 in
 quality
 and
 how
 to
 adapt
 the
 operation
 of
 a
 
chromatography
 step
 to
 these
 variations.
 In
 the
 first
 example
 the
 batch
 is
 divided
 into
 
several
 parts
 and
 after
 processing
 of
 the
 first
 sub-­‐batch
 product
 quality
 is
 measured
 and
 
the
 information
 is
 used
 for
 adjustments
 of
 the
 operating
 conditions.
 Regression
 of
 
historical
 data
 is
 used
 to
 obtain
 a
 simple
 linear
 control
 law.
 In
 the
 second
 example
 the
 
focus
 is
 to
 extract
 the
 long-­‐term
 trends
 of
 performance
 despite
 considerable
 variation
 in
 
the
 incoming
 material.
 
 A
 simple
 control
 law
 is
 obtained.
 Data
 from
 commercial
 pro-­‐
duction
 illustrate
 the
 benefits.
 
 
 
 
 
Figure
 1
 Production
 data
 from
 the
 first
 example
 illustrating
 the
 possible
 improvement.
 
In
 the
 left
 diagram
 variation
 during
 manual
 control
 is
 shown.
 The
 right
 diagram
 shows
 
the
 prediction
 error
 that
 set
 the
 limit
 of
 the
 control
 performance.
 During
 the
 gap
 in
 data
 
(observation
 135-­‐180)
 the
 production
 was
 operated
 differently
 and
 the
 prediction
 
model
 was
 not
 relevant.
 Scales
 are
 the
 same
 in
 the
 two
 figures.
 
 
 

136
Basic control of complex distillation columns
Deeptanshu Dwivedi, Ivar J. Halvorsen1, Maryam Ghadrdan, Mohammad Shamsuzzoha
and Sigurd Skogestad
Norwegian University of Science and Technology, Department of Chemical Engineering,
Trondheim, Norway
1) SINTEF ICT, Applied Cybernetics, Trondheim Norway
Email: IvarJ.Halvorsen@sintef.no
Introduction
Dividing wall distillation columns (DWC) have received considerable attention in the last
decades. Although the patent of DWC was submitted by Wright in 1946, and the basic theory
that outlined potential energy savings by fully thermally coupled columns was presented by
Petlyuk in 1965 the industry were reluctant. The breakthrough came with the work of Kaibel
1987 and several DWCs were realised within BASF through the last decade of 1900.
Theoretical expressions for minimum energy for 3-component ideal zeotropic mixtures were
presented by Fidkowski in 86. Extension to any number of components, sharp and non-sharp
splits, and the general extended Petlyuk arrangement was presented by Halvorsen (2001).
Several papers based on model realized in rigorous simulation tools have been presented in the
last decade. Typical savings are reported in the range from 15% to 35% compared to
conventional sequences.
However, not many papers have been published on operation. Wolff and Skogestad (1995)
showed that the setting of liquid and vapour splits are very important. Triantafyllou and Smith
investigated selection of manipulated variables both from simulations and by a pilot plant
column (1992). Halvorsen explored the steady state properties of the Petlyuk arrangement. A
recent paper by Lyuben (2009) discusses composition and minimum energy control.
The objective is to explore the behaviour of a complex column in real operation and propose
how it should be controlled to obtain the potential energy savings in industrial practice.
The Approach in this paper is based on both simulations and on results from a 4-product
Kaibel-type laboratory column at NTNU. Several issues are studied:
What happens when the feed composition is changed? What happens if the required liquid- and
vapour splits as not properly set? What happens if the product draws are not properly set?
How should a complex column best be started up and stabilized? How should we ensure real
minimum energy operation in practice? And how does suboptimal operation affect the column
profile and product purities, that is, how can we identify non-optimal operation? How do the
control loops interact?
Many of these questions arose when we were operating the laboratory column, and to answer
them we have mainly used dynamic simulation. Note that the dynamic model was adjusted to
match the experimental column.
References
Wright, R.O., “Fractional Apparatus”, US Patent 2471134, May 1946
Petlyuk, F.B. “Thermodynamically optimal method for separating multi-component mixtures”,
Int.Chem.Eng. Vol 5, No 3, pp 555-561, 1965.
Fidkowski, Z and Krolikowski, “Thermally Coupled system of distillation columns. Optimization
procedure.”, AIChE Journal Vol 32, No 4 , 1986
Kaibel, G. “Distillation columns with vertical partitions”, Chem. Eng. Tech. 10 (1987) 92-98.
Wolff, E and Skogestad, S. ”Operation of integrated 3-product (Petlyuk) distillation columns” Ind.
Eng. Chem. Res. 1995, 34, 2094-2103.
Triantafyllou, C. and Smith, R. “The design and operation of fully thermally coupled distillation
columns”, Trans. IChemE, 7-(Part A), 118-132, 1992.
Halvorsen, I.J. “Minimum Energy Requirements in Complex Distillation Arrangements”, PhD
Thesis , NTNU 2001:43 
137

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This website is focused on providing document in readable format, online without need to install any type of software on your computer. If you are using thin client, or are not allowed to install document reader of particular type, this application may come in hand for you. Simply upload your document, and Docureader.top will transform it into readable format in a few seconds. Why choose Docureader.top?

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  2. Cross-platform - no compromised when reading your document. We support most of modern browers without the need of installing any of external plugins. If your device can oper a browser - then you can read any document on it
  3. Simple uploading - no need to register. Just enter your email, title of document and select the file, we do the rest. Once the document is ready for you, you will receive automatic email from us.

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