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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

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

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

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

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

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

Mehdi Panahi and Sigurd Skogestad

Potential of Economic Model Predictive Control for Management of

Multiple Power Producers and Consumers

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

Johannes P. Maree and Fernando R. Camisani-Calzolari

Diagnosis of Oscillation Due to Multiples Sources Using Wavelet

Transforms

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

Vesa-Matti Tikkala, Alexey Zakharov, and

Sirkka-Liisa Jämsä-Jounela

16.10 End of sessions

19.00 Workshop dinner

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

Torsten Wik, Per Rutqvist, and Claes Breitholtz

Application of Optimal Control Theory to a Batch Crystallizer using

Orbital Flatness

Steffen Hofmann and Jörg Raisch

The Setpoint Overshoot Method: A Super-fast Approach to PI

Tuning

Mohammad Shamsuzzoha, Sigurd Skogestad, and Ivar J. Halvorsen

Comparing PI Tuning Methods in a Real Benchmark Temperature

Control System

Finn Haugen

10.10 Break

10.30 Session 5: Poster session

Tentative Dependence Analysis of Process Variables in a Circulating

Fluidized Bed Boiler

Laura Lohiniva and Kimmo Leppäkoski

Automated Controller Design using Linear Quantitative Feedback

Theory for Nonlinear systems

Roozbeh Kianfar and Torsten Wik

Optimal Controlled Variable Selection for Individual Process Units

in Self Optimizing Control with MIQP Formulations

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

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

Jukka Kortela and Sirkka-Liisa Jämsä-Jounela

Convex Approximation of the Static Output Feedback Problem with

Application to MIMO-PID

Henrik Manum and Sigurd Skogestad

11.30 Lunch 7

12.30 Session 6: Optimization

Modeling and Optimization of Grade Changes for Multistage

Polyethylene Reactors

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

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

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.

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.

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.

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

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

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

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

TTvk

x

k x

t

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

sin

ax s cond TeIx

k σ−=

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.

s x s cond Tex

k σ=

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

t

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:

HcombHcombCOcombCOcomb

sfpconvffbf

f

TTvkhv

xt h

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

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

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

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]:

, ...)///30exp(103.1

gCOcomb TmolekcalR

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]:

2, ...)/6900exp(1096.2

gOHcomb TKR

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.

scomb

TmolekJR

...))/(/125exp(103.5

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]:

sCOCOgasi

TmolekJPR

)1ln(1011

///9.93exp1024.2 322.0 22,

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]:

s OHOHgasi

TmolekJ

)1ln(1011

...///136exp

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

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

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.

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

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Characteristics of wood cylinders for conditions pertinent to fixed-bed

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Chemical Engineering Science 44 (1989), pp. 2861-2869

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McGraw-Hill 2004, 542 p.

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Char, Energy & Fuels 17 (2003), pp. 1251-1258

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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

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(2001), pp. 37-58

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J., Mathematical modelling of MSW incineration on a traveling bed,

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Fixed Bed, Combustion and Flame 121 (2000), pp. 167-180

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model predictions with experimental data, Biomass and Bioenergy 28

(2005), pp. 307-320

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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

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Pierucci, S., Ranzi, E., Biomass pyrolysis: Kinetic modeling and

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– 1000 °C, Fuel 88 (2009), pp. 519 – 527

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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

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

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

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

min max kG k ∞

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

It can be shown that the use of (4) instead of (2) results in a less conservative uncertainty model.

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.

2( j )

min max ( j )kG kω

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

min

γ s.t. 0

k k

k k k k

I y G u

y G u u u

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

min

γ s.t. 0

k k

k k k k

I y G u

y G u u G G u

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.

Modeling for Control of Beer Quality

D.R. Warnasooriya

, P.G. Rathnasiri

, Bernt Lie

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.

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

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).

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.

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.

2.1 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:

kCxky

kBukAxkx

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:

NuNxFkukyluxJ

k

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:

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:

))(),(()),0((min

k kukyluxJ . (5)

According to dynamic programming, the optimal control

action may be found as

( )))0(),1(())0(),0((minarg

uxVuyl

uxu

u

where ))0(),1(( uxV is the value function introduced as the

optimal value of the infinite horizon optimization problem:

))(),((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:

))(),((min))1(),((

Nk Uu

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

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)

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.

Time

a (t

Figure 1. One-parametric set of functions )(kgα with c1 = 0.2

and c2 = 0.1.

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

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:

ss es s

ss s

ss s

ss s

ss ++ G32(s) = )51)(50001(

ss s

ss ++ G42(s) = )47001(

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:

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

i iii

k ykySyky

qkyPQ

kukuRkukuxF

))0,)((max(

where diagonal matrixes R, Q and S are defined as follows:

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

Table 2. Setpoints

Time

period

(h)

Overflow

solids (%)

Particles <

than 47�m

Ball mill

throughput

t/h

Pump

sump

level (%)

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

MPC 2 phase

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

Time, hours

y d ro c y c lo n e

v e rf lo w

e n s it y

MPC 2 phase, N = 2

Time, hours

v e rf lo w

a rt ic le

m

MPC 2 phase, N = 2

Figure 3. Controlled variables: a – Hydrocyclone overflow

density % solids, b – % of particles < 47 �m

Time, hours

o d

ill

e e d ra te

t/ h

MPC 2 phase, N = 2

Time, hours

a te r to

u m p

u m p

m

/h

MPC 2 phase, N = 2

Figure 4. Manipulated variables: a – rod mill feedrate, b –

water to pump sump

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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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.

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

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.

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]

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)

(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

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%

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

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

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,

[4] UniSim design R380, Honeywell company

[5] Profit Design Studio (PDS R310) , Honeywell company

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

�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=�

k=0

c�uk �1)

with:

c = [c1� c2� . . . � cn� 0]

T u = [u1� u2� . . . � un� Te]

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

- limitations on the input:

umin � uk � umax k = 0� 1� . . . � N

Δumin � Δuk � Δumax k = 0� 1� . . . � N

- 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

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.

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.

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.

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

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.

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).

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) =

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.

u tp u t (y

Open loop system

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.

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

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

la n t o u tp u t (y

Valve stiction

o d e l o u tp u t (y m

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.

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.

la n t o u tp u t (y

Oscillatory disturbance and valve stiction

o d e l o u tp u t (y m

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

la n t o u tp u t (y

Gain mismatch

o d e l o u tp u t (y m

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.

[1] W. Bialkowski, “Dreams versus reality: a view from both sides of the

gap,” Pulp and Paper Canada, vol. 94, pp. 19–27, 1993.

[2] L. Desborough and R. Miller, “Increasing customer value of industrial control performance monitoring: Honeywell’s experience,” Proc.

AIChE Symp. Ser., vol. 98, pp. 153–186, 2002.

[3] D. Ender, “Process control performance: not as good as you think,”

Control Engineering, vol. 40, pp. 180–190, 1993.

[4] T. Hägglund, “A control-loop performance monitor,” Control Engineering Practice, vol. 3, pp. 1543–1551, 1995.

[5] N. Thornhill and T. Hägglund, “Detection and diagnosis of oscillation

in control loops,” Control Engineering Practice, vol. 5, pp. 1343–1354,

[6] K. Forsman and A. Stattin, “A new criterion for detecting oscillations

in control loops,” in CP8-3. Karlsruhe, Germany.: European control

conference, 1999.

[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,

pp. 23–33, 2001.

[8] A. Tangirala, J. Kanodia, and S. Shah, “Non-negative matrix factorization for detection and diagnosis of plant wide oscillations,” Industrial

and Engineering Chemistry Research, vol. 46, pp. 801–817, 2007.

[9] N. Thornhill, B. Huang, and H. Zhang, “Detection of multiple oscillations in control loops,” Journal of Process Control, vol. 13, pp.

[10] A. Horch, “A simple method for the detection of stiction in control

valves,” Control Engineering Practice, vol. 7, pp. 1221–1231, 1999.

[11] A. Horch and A. Isaksson, “A method for detection of stiction in

control valves,” in On-line-fault detection and supervision in the

chemical process industry. Lyon, France: IFAC Workshop, 1998,

p. 4B.

la n t o u tp u t (y

Gain mismatch

Oscillatory disturbance and gain mismatch

o d e l o u tp u t (y m

Time (samples)

o n tr o ll e r o u tp u t (u

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.

la n t o u tp u t (y

Delay mismatch

o d e l o u tp u t (y m

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.

[12] M. Choudhury, S. Shah, and N. Thornhill, “Detection and quantification of control valve stiction.” Boston, USA: DYCOPS, 2004.

[13] D. Bloomfield, R. McAteer, B. Lites, and P. Judge, “Wavelet phase

coherence analysis: Application to a quiet-sun magnetic element,” The

Astrophysical Journal, vol. 617, pp. 623–632, 2004.

[14] C. Torrence and G. Compo, “A practical guide to wavelet analysis,”

Bulletin of the American Meteorological Society, vol. 79, pp. 61–78,

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

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..

Availability Estimations for Utilities in the Process Industry

Anna Lindholm

Automatic Control, Lund

Hampus Carlsson

Perstorp AB

Charlotta Johnsson

Automatic Control, Lund

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).

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.

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.

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.

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

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:

where Y (k) is the measurement and Ŷ (k) is the global

propagation value obtained by

Ŷ (k) = fY

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:

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

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

P l

Ȳ (k) = fY

l

where

Ū(P lY � k − τ) =

ūi(k − τ)

ūi(k − τ) =

ûi(k − τ)� i /∈ P

l

ui(k − τ)� i ∈ P

l

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:

Un =

n� k=1

d(k)− µ0 −

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)

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

Table 1: Fault isolation rules of the dynamic causal digraph

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�

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) =

0� otherwise�

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

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

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

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.

cc fc

lo b a l r

e si d u a ls Samples

Samples

e te ct io n

ig n a ls fc cc

Samples

e te ct io n

e su lt fc cc Figure 3: Global residuals GR(fc) and GR(cc),

detection signals and the detection results

(c c)

Samples

e te ct io n

ig n a l cc

Samples

e te ct io n

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.

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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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.

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.

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.

We define the problem as to find a feedback control policy

u�t� x) that minimizes

V �x�t)� t) =

t

l�x�τ)) + uT �τ)Qu�τ)

dτ + Vf �x�tf ))|x�t)

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

= min

u {l + �∇V )�f +Bu) + uTQu

tr

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 = −

Inserting the optimal u into �3) gives

= l −

tr

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.

Now, applying the transformation

V = −2κ logZ� �6)

where κ is an arbitrary real constant, gives

Equation �5) is then transformed into

=zl−

tr

If two matrices A and B have matching dimensions

tr[AB] = tr[BA] = tr

, and the trace of a scalar

equals the scalar itself. Therefore we have that

tr

which gives

l

Now, if

the transformed HJB equation becomes a linear PDE:

l

with boundary conditions

and

Zf

= Z�tf ��) = exp

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.

which gives us the linear eigenvalue problem

Z = 0 on ∂Ω�

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

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�

= the last n− rB columns of U1�

= 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 ),

for the row space R�BT ), and EN� for the nullspace

For a symmetric positive definite matrix, such as the

covariance matrix W , SVD is identical to diagonalization

with orthogonal eigenvectors, i.e.

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

where Λ

W = diag�

λW�nw ). Clearly, W

1/2 is

also symmetric and positive definite, and W 1/2W 1/2 = W .

Congruously,

where Λ

W = diag�1/

λW�nw ).

�heorem

The HJB equation �3) can be linearized if and only if

R�G) ⊆ R�B), which is true if and only if

The optimal control policy is then given by the linearized

where B� denotes the pseudoinverse of B. The corresponding cost matrix Q is implicitly given by

where κ > 0 is an arbitrary scalar, Θ is any full rank matrix,

and

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

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:

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

Clearly, the HJB equation is linearized if and only if

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

where Q

have columns entirely in R�BT ) and Q

have columns entirely in � �B). Then

since �Q

N = 0 because of the orthogonality.

The component of the columns of GW 1/2 in R�BT ) are

Thus, Q−1 = Q−1

)T is enough to

satisfy �9) since BQ

N = 0 by definition. However, if

B or G does not have full rank Q−1 = Q−1

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

EN� are an orthonormal base for � �B). Hence, we can fill

the space using

where Θ is an arbitrary full rank matrix.

Since

Q−1N will not contribute to �9). Further,

because Q−1N is symmetric. Hence, Q

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

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

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

to Q−1

−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.,

and then let �→ 0.

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

and G =

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:

and EΔ =

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

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�

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�

where w is white Gaussian noise with covariance W . Inserting this disturbance model into �20) gives

ẋ =Ax+Bu+BCwxw +BDww

=Ax+Bũ+Gw

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

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

river

to WWTP

in Q

�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

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 =

� u =

Qin

and w =

where all variables are deviations from the operating point,

the system can be described by

dx dt

u1 u2

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

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

h1�max − h�x1) + u

TQu

s.t. 0 ≤x1 ≤ x1�max

0 ≤x2 ≤ x2�max

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]

4 · h21

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

which gives

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.

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

m

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.

m

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.

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

x

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).

x

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.

[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,

[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.

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, . . .

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

c csat

, c =

ml mW

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.

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 =

(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 =

(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:

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

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

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,

in which case u (τ) ∧= 1/G (τ), and define fixed bounds

on this variable:

( 1G )min, (

G )max

With the growth law (3a), ( 1G )min and (

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

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 =

G (µ3,s (τ) + x1, u (τ)) (14b)

d dτ x5 = u (τ) (14c)

where u (τ) ∧= 1G (τ) is the control input, and

G (·) shall

be understood as:

G (µ3,s (τ) + x1, u (τ)) := B

µ3,s (τ) + x1, T (τ)

u (τ)

where, for every instant of τ , T (τ) is the (positive)

solution of

µ3,s (τ) + x1, T (τ)

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

G (µ3,s (τ) + x1, u (τ))

= kbk

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

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 =

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,

∧= 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

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

d dτ ψ2 = −3ψ1, ddτ ψ3 = −2ψ2, ddτ ψ4 = −ψ3 (21b)

d

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

G (µ3,s (τ) , u) + ψ5u (21d)

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 ;

∂x1f

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

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:

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

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:

uo = −ψ4 b− g

g kbk

g (uo)

− bg µ3,s (τ) + ψ5

uo =

kg

b− g

g kbµ3,s (τ)

) g

b

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

b− g

g b g kbk

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→∞

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 (

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

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

∗ (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

kg

b− g

g kbµ3,s (τ)

) g

b

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:

min

k

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) =

kg

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 +

( 1G )mindτ = tf,c + (

G )min (τs − τf )

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 =

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

Of course, µ′3,s (τ) = µ

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

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

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.

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:

G (µ3,s (τ) + x1, u (τ))

= kbk

g u (τ)

g−b

g µ3,s (τ) + kbk

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,

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

cm3

; kV = 1

Initial conditions:

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

µ′3f,n [mm

µScst3f,n [mm

µScst3f,n−µ

3f,n

εmax

εmax

µScst3f,n−µ

3f,n

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.

t [min]

[m in m m

(1/G)Scst

TScst

Figure 1: Control in t-domain

τ [mm]

[m in m m

(1/G)Scst

TScst

Figure 2: Control in τ -domain

t [min]

,s

,n [c m

µScst

3,s

µScst

3,n

3,s

3,s

3,n

3,n

Figure 3: Trajectories of the third moments

t [min]

,n [c m

µScst

3,n

3,n

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

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[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.

[5] S. M. Miller and J. B. Rawlings, Model identification and control strategies for batch cooling crystallizers. AIChE J. 40:8

[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.

[9] Z. K. Nagy and R. D. Braatz, Robust nonlinear model predictive control of batch processes. AIChE J. 49:7 (2003), pp.

[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.

[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.

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E. F. Mishchenko, The Mathematical Theory of Optimal Processes. John Wiley & Sons (1962).

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* 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

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

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.

In process control, a first-order process with time delay is a

common representation of the process dynamics:

-θskeg(s)=

τs+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

1c s =K 1+

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

k τ +θ

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

' c

kK 1 τK = =max 0.5,

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.

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.

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):

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

cK

Fig. 1. Scaled proportional and integral gain for SIMC tuning rule.

pyΔ

pt y∞Δ syΔ

t0t =

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

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)

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

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:

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

Here, the value of the loop gain kKc0 for the P-control

setpoint experiment is given from the value of b:

c0 bkK =

(1-b)

overshoot (fractional)

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:

bτ =2A θ

1-b

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

1-b

(2) Process with relatively small delay. For a lag-dominant

(including integrating) process with τ/θ>8 the SIMC rule

gives

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

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

Overshoot

t p

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.

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

"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.

time

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)

Fig. 6. Responses for PI-control of (5 1)sg e s−= + (E1).

time

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)

Fig. 7 Responses for PI-control of sg e−= (E2).

time

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)

Fig. 8. Responses for PI-control of sg e s−= (E3).

time

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⎡ ⎤= +⎢ ⎥⎣ ⎦

time

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).

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

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.

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

se s

E2 se−

E3 se

s

ss s s s

s s s

1 0.2 1s s+ +

1s s +

se s

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).

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

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

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

for analog I/O

Fan speed

adjustment

On/Off

switch

PC with

LabVIEW

USB cable

Electrical heater

Fan

Mains cable

3 x Voltage AI (Temp 1, Temp 2, Fan

indication)

1 x Voltage AO (Heating)

Air

Pulse Width

Modulator

indicator

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.

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,

• 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

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).

(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)

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

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) =

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 =

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

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) =

Ts+ 1

e−τs (9)

From the step response I found4

For the controller tuning I use (8):

The PI controller parameters are

Kc =

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.

Figure 5: Skogestad’s method: Closed-loop responses

The IAE indexes and the gain margin was

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:

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

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

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

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

The slope R can be calculated as the initial

slope of the step response. For a first order

system,

where we have

The PI settings becomes

Kc =

(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

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

performance given a requirement on robustness. The air heater looks like a “timeconstant with time-delay” process. Assuming the process model is

Hpsf (s) =

Ts+ 1

e−τs (27)

the PI controller settings according to Hägglund and Åstrøm are as follows:

Kc =

Ti = τ

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 PI settings become

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

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

= Ti

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)

The PI parameter values become

Ti =

Pu

15 s

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

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

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

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

Uhigh − Ulow

If ”large” oscillation amplitude is allowed,

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

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

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

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

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

The ultimate gain becomes, cf. (48),

Kcu =

The PI parameter values become

Ti =

Pu

18 s

Figure 15 shows control system responses

with the above PI settings.

Figure 15: Relay tuning method: Responses in

the control system

The IAE indexes and the gain margin was

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:

ymax − y (∞)

y (∞)− y0

Also calculate the relative steady-state

change of the process output:

b =

y (∞)− y0

∆ySP

Define the following parameters:

(F = 1 for ”fast robust control” corresponding to TC = τ in Skogestad’s SIMC method,

but use F > 1 to detune), and

The PI parameter settings are

Kc = Kc0

Ti = min

(0.86Atp

b 1− b , 2.44tpF )

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

Figure 16: Sham’s Setpoint method: The responses with a P-controller with gain Kc0 =

which gives a stable response and a reasonable overshoot.From the responses we find

the actual overshoot as

ymax − y (∞)

y (∞)− y0

The relative steady-state change of the

process output is

b =

y (∞)− y0

∆ySP

We read off the peak time as

tp = 14 sec (67)

The PI parameter settings becomes

Kc = Kc0

Ti = min

(0.86Atp

b 1− b , 2.44tpF )

= 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

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)

s2 + 2ζω0s+ ω20

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

undershoot, as in the Good Gain tuning, and

that the period of the damped oscillations is

Pd =

= PGG = 2Tou (75)

If the oscillations are undamped, as with the

Ziegler-Nichols’ Ultimate Gain method, the

period of the oscillations is

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

In the Ziegler-Nichols’ method we set

Ti =

1.6Tou

= 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

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

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

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.

Kc Ti IA E s IA E d ∆K Q S

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,

[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,

[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,

[7] Shamsuzzoha, M., Skogestad, S.,

Halvorsen, I. J., On-Line PI Controller

Tuning Using Closed-Loop Setpoint

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,

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

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.

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.

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

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.

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,

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.

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.

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

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

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].

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.

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

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

should be in the LHP. Hence, the problem can now be

formulated as

min

subject to: ubi(θ) ≤ 0 ∀i

lbi(θ) ≤ 0 ∀i

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

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) =

3 + θ2s2 + θ1s + θ0

s4 + θ36 + θ5s2 + θ4

i.e. the same as the controller presented in the manual. The

design specifications are:

Ts ≤ 10s (7)

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

ag [d

Phase [degree]

Nichols Chart

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 =

(x̄, ū), B =

(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)

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 +

s )e (14)

and the controller derived from local linearization:

u = (1 +

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 = (

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

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 = (

s )e (18)

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.

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.

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.

Fig. 5. Bioreactor with ρ as input and x1 as output

The state space equations of the plant are:

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.

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)

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)

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

Fig. 8. Stationary points for bioreactor

ag [d

Nichols Chart

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

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

s

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)

• Sensitivity specification

1 + G(jω)P (jω)

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,

Ma g [d

Phase [degree]

Nichols Chart

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).

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.

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

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

T(sec)

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.

T(sec)

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.

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.

T(sec)

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.

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Nordic Process Control

Aug 19-21, 2010

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

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∈

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,

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,

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.

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

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

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

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.

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

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

u d

uu ud

ud dd

u J u d J u d J J

d J Ju u

d dJ J

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

opt T opt

uu

L u u J u u

z z z

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.

1opt

uu udu J J d

1( )opt y yuu ud dy G J J G d F d

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

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

1/ 2 1( ) yyn uu nM J HG HW

y y

uu ud d d n ny ny ndY G J J G W W

Using the equations (12), (13), (14) and (5) the loss can be

rewritten as

y uu y

d

n

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 '

y d n

and the opitmization problem is formulated to minimize

the worst case loss and average loss as in Kariwala et al.

1min ( ( ) )

y uuH

21/ 2 11min ( ( ) )

y uu FH

ny nd

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

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).

min

y uu

st HG J=

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

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

uuJ are split into vectors as

1 2 1 2[ ]; [ ];nu uu nuX x x x J J J J= = and we

further introduce the long vectors as below

nu nunu ny nu ny

x J

x J

x J

and the large matrices

y y

y nu ny nu nu nd

nu nu ny nu

As

F nu F

nu F

x Y

x Y

HY HY HY x Y x Y x Y

x Y

and as Juu is symmetric positive definite matrix,

uuJ is

also symmetric positive definite

y y y uuHG G H G X J= = = and as

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

y y

y n nu nunu nu nu

G x J

JG x G X J

JG x

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.

min

st G X J

Note here that Xδ is a stacked vector of all the columns in

X or HT.

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).

min

aug

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

where

aug

ny nu ny ny

x

and

TT y T

newF Y Y zeros ny ny G G zeros nu ny ny

P zeros nu ny ones ny

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

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

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

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

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

y y

d uu ud d n

y y

u u

The resulting optimal sensitivity matrix is computed as

follows

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

Ty newG

nJ

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

top top s btm btm s

top s btm s

x x x x

x x

where ∆X is the root mean square of the relative steady

stae compostion deviation. , ,, , ,

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

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.

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.

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,

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, 1150

S. Skogestad, Plantwide control: the search for the selfoptimizing control structure. Journal of Process Control,

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,

TOMLAB v7.1 - The TOMLAB Optimization

Environment in Matlab (1999)

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

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=

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=AxBu with the two matrices A and B being

dB −τB−d B

And the y=Cx with the matrix C=1 ⋯ 00 ⋯ 1

With τm=

n

kmOi

ρmc pmV

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.

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

(left) and of the cross stream transfer function G

(right) with different numbers of stages n. The parameters are chosen to be d

=d

Figure 3: The normed eigenvalues of

the system matrix A in the complex

plane with n=100.

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:

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)

τa2 −τ a1−d d

τan−1 −τan−1−d d

−τ an −τan−τγ 0

d −τb2−d τb2

d −τ bn−1−d τbn−1

τbn −τbn

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,

[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.

�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

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.

Time�days)

m g

l �a) Readily biodegradable substrate at the inlet of the first reactor.

Time�days)

m g

l �b) Slowly biodegradable substrate at the inlet to the first reactor.

Time�days)

g

g

�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].

�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.

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 Department of Chemical and Biochemical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

Department of Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

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.

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.

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.

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

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

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

Fuel C H S O N Ash M

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

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.

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.

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

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

= 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

+nN2 +nH2O[mol/kg]

Similarly, the flue gas losses per kilogram of fuel are

qgfg = (nCCCO2 + nSCSO2 + (nH2O + nH2)CH2O

−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

The combustion power of the BioGrate boiler is estimated

using Equations 6 - 10.

The total oxygen consumption is

N totO2 = 0.21 · nAir −

· 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

[mol/s] (8)

and thus the fuel flow is

mf =

NgO2 +

100 (Nfg − 4.76 ·N

g

[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

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

Multiplying Equation 11 by hw and subtracting the result

from Equation 13 gives

(hs − hw)

d dt (%sVst) + %sVst

hs dt + %wVwt

hw dt

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

(Q+mfhf −mshs)[kJ/(s · kg)] (16)

Energy balance for tube walls is

dtm

dt

mtCp

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)

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) =

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).

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

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

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.

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.

[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.

[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.

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Plants and Power Systems Control 2000: A Proceedings Colume from

the IFAC Symposium Brussels, Belgium, 26-29 April, 2000. pp. 113

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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.

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for heat and power production. Progress in Energy and Combustion

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36, Issue 3, March 2000, pp. 363-378.

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,

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

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:

NPxN +

i=0

x′iQxi + u

iRui

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

= 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

s.t. H on the form

Hy Hu

diag(Ky) I

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,

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

Gxx0

x0 +

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

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

Gymx0

x0 + . . .

+ diag(D)

Gymu

We now have that

y =

ym u

Gymu

Gy u+

Gymx0

Gyd

x0

d

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

xk

uk

yPk

yIk

yDk

Ts

xk

uk.

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

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.

Modeling and Optimization of Grade Changes for Multistage

Polyethylene Reactors

Per-Ola Larsson, Johan Åkesson, Staffan Haugwitz, Niklas Andersson

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).

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).

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].

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◦),

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

u̇

Q∆y 0 0

0 Q∆u 0

0 0 Qu̇

u̇

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,

u p

u e

u h

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

e

e

p

p

h e

h e

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.

[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.

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:

• 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

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

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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

Karin Axelsson*, Veronica Olesen**

*Department of Chemical Engineering, Lund University

**Solvina AB, Gruvgatan 37, Västra Frölunda, Sweden

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.

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

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.

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.

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.

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.

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.

u501-controller output

u

-c o n tr o l s ig n a l to

a rg e

a lv e

Figure 3: Control signal to large valve as a function

of controller output

FjV Solvina AB

26 kg/ s

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.

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.

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

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