Process Modelling, Identification, and Control

Ján Mikleš · Miroslav Fikar

Process Modelling,Identification, and Control

With 187 Figures and 13 Tables

ABC

Ján MiklesDepartment of Information Engineeringand Process ControlSlovak University of Technology in BratislavaRadlinského 9812 37 Bratislava 1Slovak Republicjan.mikles@stuba.sk

Miroslav FikarDepartment of Information Engineeringand Process ControlSlovak University of Technology in BratislavaRadlinského 9812 37 Bratislava 1Slovak Republicmiroslav.fikar@stuba.sk

Library of Congress Control Number: 2007928498

ISBN 978-3-540-71969-4 Springer Berlin Heidelberg New York

This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable for prosecution under the German Copyright Law.

Springer is a part of Springer Science+Business Mediaspringer.comc© Springer-Verlag Berlin Heidelberg 2007

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply,even in the absence of a specific statement, that such names are exempt from the relevant protective lawsand regulations and therefore free for general use.

Typesetting: by the author and Integra, India using a Springer LATEX macro packageCover design: Erich Kirchner, Heidelberg

Printed on acid-free paper SPIN: 11963554 5 4 3 2 1 0

To our wives, Eva and Zuzana, that showed so muchunderstanding during the preparation of the book

Preface

Control and automation in its broadest sense plays a fundamental role inprocess industries. Control assures stability of technologies, disturbance at-tenuation, safety of equipment and environment as well as optimal processoperation from economic point of view. This book intends to present modernautomatic control methods and their applications in process control in pro-cess industries. The processes studied mainly involve mass and heat transferprocesses and chemical reactors.

It is assumed that the reader has already a basic knowledge about con-trolled processes and about differential and integral calculus as well as aboutmatrix algebra. Automatic control problems involve mathematics more than itis usual in other engineering disciplines. The book treats problems in a similarway as it is in mathematics. The problem is formulated at first, then the the-orem is stated. Only necessary conditions are usually proved and sufficiencyis left aside as it follows from the physical nature of the problem solved. Thishelps to follow the engineering character of problems.

The intended audience of this book includes graduate students but canalso be of interest to practising engineers or applied scientists.

Organisation of the book follows the requirement of presentation of theprocess control algorithms that take into account changes of static and dy-namic properties of processes. There are several possibilities to capture thesechanges. As the control objects in process industries are mainly nonlinear andcontinuous-time, the first possibility is to use nonlinear mathematical modelsin the form of partial differential equations, or after some simplifications, non-linear differential equations. Usage of these models in control is computation-ally very demanding. Therefore, it is suitable to use robust control methodsbased on linearised models with uncertainties. However, strong nonlineari-ties can decrease performance of robust control methods and adaptive controlalgorithms can possibly be used.

The book presents process control using self-tuning (ST) controllers basedon linear models. There are several reasons for their use. ST controllers cantake into account changes of operating regimes of nonlinear processes as well as

VIII Preface

changes in physical process parameters. These changes can be caused by ageingor wearing of materials, etc. Finally, ST controllers can eliminate unmeasureddisturbances, process noise, and unmodelled dynamic properties of processes.

The last chapter of the book that discusses adaptive control makes useof all previous parts. Here, discrete and continuous-time ST control of singl-evariable and multivariable processes is investigated. For its implementationit is necessary to have in disposition an algorithm of recursive identificationthat provides an up-to-date model of the controlled process. This model isthen used in feedback process control based on either explicit or implicit poleplacement methods.

A basic requirement for a successful control is to have a suitable modelfor control. It is known that the process design in process industries involvesnonlinear models with states evolving in space and time. This corresponds tomodels described by partial differential equations. Automatic control basedsuch models is very difficult as it requires additional capacity of control com-puters and can be used only in some special cases. Self-tuning control requireslinear models only described by ordinary differential equations of low order.This is caused by the fact that higher order models can cause singularities incalculation of the control law.

The first chapter of the book explains basic concepts that can be encoun-tered in process control.

The second chapter describes principles of process modelling. Some princi-pal models as a packed absorption column, a distillation column, and a chem-ical reactor are developed. Mathematical model of a plug-flow reactor servesfor explanation of transport time delay that together with dynamic time delayforms a basic phenomenon in continuous-time process models. Description ofspecific process models is accompanied by assumptions when the models inthe form of ordinary differential equations can be valid. Finally, general pro-cess models are explained together with the method of linearisation and thenotion of systems and processes.

The third chapter explains briefly the use of the Laplace transform inprocess modelling. State-space process models obtained in the previous chap-ter are analysed. A concept of state is defined and the Lyapunov stabilityof continuous-time processes is investigated. The physical interpretation oftheoretical facts is explained on a model of a U-tube. Conditions of control-lability and observability are applied to a continuously stirred tank reactor.The state-space process models obtained from material and heat balances aretransformed into input-output models for the purposes of identification andcontrol design.

The aim of the fourth chapter on dynamic behaviour of processes is toshow responses of processes to deterministic and stochastic inputs. Computersimulations explain how Euler and Runge-Kutta methods for integration ofdifferential equations can be used. Examples include MATLAB/Simulink, C,and BASIC implementations. Process examples in this chapter demonstratedifferences between original nonlinear and linearised models. The part dealing

Preface IX

with stochastic characteristics explains random variables and processes, thedefinition of the white noise and others needed in state observation and instochastic control.

Direct digital process control that constitutes a part of discrete-time adap-tive control needs discrete-time process models. Therefore, the fifth chapterdeals with the Z-transform, conversion between continuous-time and sampled-data processes. Also discussed are stability, controllability, observability, andbasic properties of discrete-time systems.

Both adaptive and non-adaptive control methods are usually based on amathematical model of a process. The sixth chapter is divided into two parts.In the first one identification of process models is based on their response tothe step change on input. The second part deals in more detail with recursiveleast-squares (RLS) methods. Although primarily developed for discrete-timemodels, it is shown how to use it also for continuous-time models. The RLSmethod described in the book has been in use for more than 20 years at theauthors place. Here we describe its implementation in MATLAB/Simulinkenvironment as a part of the IDTOOL Toolbox available for free in the Internet(see the web page of the book). IDTOOL includes both discrete-time andcontinuous-time process identification.

The aim of the seventh chapter is to show the basic feedback control con-figuration, open and closed-loop issues, steady state behaviour and controlperformance indices. The second part deals with the mostly used controllersin practise - PID controllers. Several problems solved heuristically with PIDcontrollers are then more rigorously handled in next chapters dealing withoptimal and predictive control.

The most important part of the book covers design of feedback controllersfrom the pole-placement point of view. Optimal control follows from the prin-ciple of minimum and from dynamic programming. At first, the problem ofpole-placement is investigated in a detail based on state-space process models.The controller is then designed from a combination of a feedback and a stateobserver. This controller can then also be interpreted from input-output pointof view and results in polynomial pole-placement controller design. The Youla-Kucera parametrisation is used to find all stabilising controllers for a given pro-cess. Also, its dual parametrisation is used to find all processes stabilised by agiven controller. The Youla-Kucera parametrisation makes it possible to unifycontrol design for both continuous-time and discrete-time systems using thepolynomial approach. This is the reason why mostly continuous-time systemsare studied. Dynamic programming is explained for both continuous-time anddiscrete-time systems. The part devoted to the Youla-Kucera parametrisationtreats also finite time control – dead-beat (DB) that describes a discrete-time control of a continuous-time process that cannot be realised using acontinuous-time controller. State-space quadratically (Q) optimal control oflinear (L) process models, i.e. LQ control with observer and its polynomialequivalent are interpreted as the pole-placement problem. Similarly, LQ con-trol with the Kalman filter (LQG) or H2 control can also be interpreted as the

X Preface

pole-placement control problem. An attention is paid to integral properties ofcontrollers and it is shown that optimal controllers are often of the form ofmodified PID controllers. The examples in the chapter make a heavy use ofthe Polynomial Toolbox for MATLAB (http://www.polyx.com).

The chapter nine extends the optimal control design with predictive con-trol approach. It is explained why this approach predetermines discrete-timeprocess control. At the beginning, the basic principles and ingredients of pre-dictive control are explained. The derivation of the predictors and controllersis then shown for both input-output and state-space process models. Thechapter also deals with incorporation of constraints on input variables anddiscusses the issue of stability and stabilisability of predictive closed-loop sys-tem. Finally, the chapter concludes with both singlevariable and multivariabletuning and show examples of applications in process control.

The last chapter combines results of previous chapters in adaptive control.We prefer to use the so-called explicit, or self-tuning approach to adaptivecontrol where process parameters are estimated on-line using recursive least-squares methods. Then, at each sampling instant, the available model is usedin control design.

The book has an ambition to integrate interpretations of continuous-timeand discrete-time control, deterministic and stochastic control with incorpo-ration of sets of all stabilising controllers. Notations used in separate parts ofthe books have been adapted to this aim.

Some of the programs and figures of the examples in the book marked inthe margin are freely available at the web page of the book:wwwhttp://www.kirp.chtf.stuba.sk/~fikar/books/mic/.

The program sources are for MATLAB with toolboxes Simulink, ControlSystem Toolbox, Polynomial Toolbox (http://www.polyx.cz), Multipara-metric Toolbox (MPT, http://control.ee.ethz.ch/~mpt) and, IDTOOL(from the web site). The examples are verified in both simulation and real-time test cases using Real-time MATLAB toolbox and the industrial systemSIMATIC STEP7.

Each chapter includes bibliographic references related to the text. Besidesthe classical material on process dynamics or feedback control, there are someparts of the book that follow from the journal publications of the authors.

Acknowledgements

The book has been in preparation in the form of study texts for almost 10years and came into existence mainly at Department Information Engineeringand Process Control established at Faculty of Chemical and Food Technology,Slovak University of Technology in 1960.

The authors wish to express their gratitude to all present and past employ-ees of the department that helped to provide a creative environment and to allstudents, PhD students that commented the manuscript. Namely we wouldlike to thank to our colleagues L’. Cirka, S. Kozka, F. Jelenciak, J. Dzivak, T.Hirmajer, K. Calık, L. Dermısek, and to associated prof. M. Bakosova fromSlovak University of Technology for comments to the manuscript that helpedto find some errors and problems.

Authors also thank for inspiration and ideas to all with whom they haddialogues during numerous stays abroad at TU Darmstadt, Uni Birmingham,TU Lyngby, Uni Dortmund, and ENSIC Nancy.

We are very thankful for the constructive remarks in identification toprof. Prokop from T. Bata University in Zlın, in modelling and dynamic simu-lations of continuous-time processes to associated prof. Lavrin from TechnicalUniversity of Kosice and to prof. Alexık from University of Zilina. We thankfor comments in parts of identification and optimal control to prof. Dostalfrom T. Bata University in Zlın and to prof. Rohal’ from Faculty of Mechani-cal Engineering at Slovak University of Technology in Bratislava. The part onpredictive control has greatly been enhanced by suggestions of M. Kvasnicaand his work on MPT Toolbox and explicit predictive control during his stayat ETH Zurich.

We also thank to prof. Kucera and prof. Sebek from Czech Technical Uni-versity in Prague for many inspiring ideas and long cooperation that has con-tributed to the conception of the book. We have been involved in the projectEUROPOLY supported by European Union and have had the possibility touse the most recent versions of the Polynomial Toolbox for MATLAB.

We are thankful to Dr. Hippe from University of Nurnberg-Erlangen forcomments to the manuscript.

XII Acknowledgements

Parts of the book were prepared during the stays of both authors at RuhrUniversity in Bochum that were supported by the Alexander von HumboldtFoundation. This support is very gratefully acknowledged. In this context weparticularly thank to prof. Unbehauen from the Ruhr University in Bochumfor his support, cooperation, and comments to the book.

Bratislava, Jan MiklesFebruary 2007 Miroslav Fikar

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Topics in Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 An Example of Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Steady-State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.3 Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.4 Dynamical Properties of the Process . . . . . . . . . . . . . . . . . 41.2.5 Feedback Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.6 Transient Performance of Feedback Control . . . . . . . . . . . 61.2.7 Block Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.8 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Development of Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Mathematical Modelling of Processes . . . . . . . . . . . . . . . . . . . . . . 132.1 General Principles of Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Examples of Dynamic Mathematical Models . . . . . . . . . . . . . . . . 16

2.2.1 Liquid Storage Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Heat Transfer Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Mass Transfer Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.4 Chemical and Biochemical Reactors . . . . . . . . . . . . . . . . . 31

2.3 General Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 Linearisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5 Systems, Classification of Systems . . . . . . . . . . . . . . . . . . . . . . . . . 452.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Analysis of Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 The Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.1 Definition of the Laplace Transform . . . . . . . . . . . . . . . . . 513.1.2 Laplace Transforms of Common Functions . . . . . . . . . . . . 53

XIV Contents

3.1.3 Properties of the Laplace Transform . . . . . . . . . . . . . . . . . 573.1.4 Inverse Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 613.1.5 Solution of Linear Differential Equations

by Laplace Transform Techniques . . . . . . . . . . . . . . . . . . . 623.2 State-Space Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2.1 Concept of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2.2 Solution of State-Space Equations . . . . . . . . . . . . . . . . . . . 673.2.3 Canonical Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2.4 Stability, Controllability, and Observability

of Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . 713.2.5 Canonical Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.3 Input-Output Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.3.1 SISO Continuous Systems with Constant Coefficients . . 843.3.2 Transfer Functions of Systems with Time Delays . . . . . . 933.3.3 Algebra of Transfer Functions for SISO Systems . . . . . . . 963.3.4 Input Output Models of MIMO Systems - Matrix

of Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.3.5 BIBO Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.3.6 Transformation of I/O Models into State-Space Models 1043.3.7 I/O Models of MIMO Systems - Matrix Fraction

Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4 Dynamical Behaviour of Processes . . . . . . . . . . . . . . . . . . . . . . . . . 1174.1 Time Responses of Linear Systems to Unit Impulse and Unit

Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.1.1 Unit Impulse Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.1.2 Unit Step Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.2 Computer Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.2.1 The Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.2.2 The Runge-Kutta method . . . . . . . . . . . . . . . . . . . . . . . . . . 1274.2.3 Runge-Kutta Method for a System of Differential

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2.4 Time Responses of Liquid Storage Systems . . . . . . . . . . . 1354.2.5 Time Responses of CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . 137

4.3 Frequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.3.1 Response of the Heat Exchanger to Sinusoidal Input

Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1454.3.2 Definition of Frequency Responses . . . . . . . . . . . . . . . . . . . 1494.3.3 Frequency Characteristics of a First Order System . . . . . 1534.3.4 Frequency Characteristics of a Second Order System . . . 1564.3.5 Frequency Characteristics of an Integrator . . . . . . . . . . . . 1584.3.6 Frequency Characteristics of Systems in a Series . . . . . . . 159

4.4 Statistical Characteristics of Dynamic Systems . . . . . . . . . . . . . . 162

Contents XV

4.4.1 Fundamentals of Probability Theory . . . . . . . . . . . . . . . . . 1624.4.2 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1634.4.3 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1694.4.4 White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1764.4.5 Response of a Linear System to Stochastic Input . . . . . . 1784.4.6 Frequency Domain Analysis of a Linear System

with Stochastic Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1834.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1854.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

5 Discrete-Time Process Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1895.1 Computer Controlled and Sampled Data Systems . . . . . . . . . . . . 1895.2 Z – Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955.3 Discrete-Time Transfer Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 2005.4 Input-Output Discrete-Time Models – Difference Equations . . . 204

5.4.1 Direct Digital Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2055.5 State-Space Discrete-Time Models . . . . . . . . . . . . . . . . . . . . . . . . . 2075.6 Properties of Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . 211

5.6.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2115.6.2 Controllability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2125.6.3 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2125.6.4 Discrete-Time Feedback Systems – Control Performance 212

5.7 Examples of Discrete-Time Process Models . . . . . . . . . . . . . . . . . 2145.7.1 Discrete-Time Tank Model . . . . . . . . . . . . . . . . . . . . . . . . . 2145.7.2 Discrete-Time Model of Two Tanks in Series . . . . . . . . . . 2155.7.3 Steady-State Discrete-Time Model of Heat Exchangers

in Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2165.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2175.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

6 Process Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

6.1.1 Models of Linear Dynamic Systems . . . . . . . . . . . . . . . . . . 2236.2 Identification from Step Responses . . . . . . . . . . . . . . . . . . . . . . . . . 225

6.2.1 First Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2266.2.2 Underdamped Second Order System . . . . . . . . . . . . . . . . . 2276.2.3 System of a Higher Order . . . . . . . . . . . . . . . . . . . . . . . . . . 230

6.3 Least Squares Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2336.3.1 Recursive Least Squares Method . . . . . . . . . . . . . . . . . . . . 2356.3.2 Modifications of Recursive Least Squares . . . . . . . . . . . . . 2416.3.3 Identification of a Continuous-time Transfer Function . . 245

6.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2506.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

XVI Contents

7 The Control Problem and Design of Simple Controllers . . . . 2537.1 Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

7.1.1 Feedback Control Problem Definition . . . . . . . . . . . . . . . . 2557.2 Steady-State Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2567.3 Control Performance Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

7.3.1 Time Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2577.3.2 Integral Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.3.3 Control Quality and Frequency Indices . . . . . . . . . . . . . . . 2607.3.4 Poles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

7.4 PID Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2667.4.1 Description of Components . . . . . . . . . . . . . . . . . . . . . . . . . 2667.4.2 PID Controller Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 2747.4.3 Setpoint Weighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2757.4.4 Simple Rules for Controller Selection . . . . . . . . . . . . . . . . 2767.4.5 Practical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2777.4.6 Controller Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2947.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

8 Optimal Process Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2978.1 Problem of Optimal Control and Principle of Minimum . . . . . . 2978.2 Feedback Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3058.3 Optimal Tracking, Servo Problem, and Disturbance Rejection . 317

8.3.1 Tracking Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3188.3.2 Servo Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3208.3.3 LQ Control with Integral Action . . . . . . . . . . . . . . . . . . . . 321

8.4 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3228.4.1 Continuous-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 3228.4.2 Dynamic Programming for Discrete-Time Systems . . . . . 3298.4.3 Optimal Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

8.5 Observers and State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3358.5.1 State Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3358.5.2 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

8.6 Analysis of State Feedback with Observer and PolynomialPole Placement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3408.6.1 Properties of State Feedback with Observer . . . . . . . . . . . 3408.6.2 Input-Output Interpretation of State Feedback

with Observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3458.6.3 Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3518.6.4 Polynomial Pole Placement Control Design . . . . . . . . . . . 3548.6.5 Integrating Behaviour of Controller . . . . . . . . . . . . . . . . . . 3558.6.6 Polynomial Pole Placement Design for Multivariable

Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3618.7 The Youla-Kucera Parametrisation . . . . . . . . . . . . . . . . . . . . . . . . 365

8.7.1 Fractional Representation . . . . . . . . . . . . . . . . . . . . . . . . . . 366

Contents XVII

8.7.2 Parametrisation of Stabilising Controllers . . . . . . . . . . . . 3688.7.3 Parametrised Controller in the State-Space

Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3708.7.4 Parametrisation of Transfer Functions of the

Closed-Loop System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3718.7.5 Dual Parametrisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3728.7.6 Parametrisation of Stabilising Controllers for

Multivariable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3748.7.7 Parametrisation of Stabilising Controllers for

Discrete-Time Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3758.8 Observer LQ Control, State-Space and Polynomial

Interpretations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3808.8.1 Polynomial LQ Control Design with Observer

for Singlevariable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 3818.8.2 Polynomial LQ Design with State Estimation

for Multivariable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 3858.9 LQG Control, State-Space and Polynomial Interpretation . . . . . 388

8.9.1 Singlevariable Polynomial LQG Control Design . . . . . . . 3888.9.2 Multivariable Polynomial LQG Control Design . . . . . . . . 392

8.10 H2 Optimal Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3948.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3978.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400

9 Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4039.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4039.2 Ingredients of MBPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

9.2.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4049.2.2 Cost Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

9.3 Derivation and Implementation of Predictive Control . . . . . . . . 4079.3.1 Derivation of the Predictor . . . . . . . . . . . . . . . . . . . . . . . . . 4079.3.2 Calculation of the Optimal Control . . . . . . . . . . . . . . . . . . 4099.3.3 Closed-Loop Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4119.3.4 Derivation of the Predictor from State-Space Models . . . 4129.3.5 Multivariable Input-Output Case . . . . . . . . . . . . . . . . . . . . 4169.3.6 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4169.3.7 Relation to Other Approaches . . . . . . . . . . . . . . . . . . . . . . 4189.3.8 Continuous-Time Approaches . . . . . . . . . . . . . . . . . . . . . . . 418

9.4 Constrained Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4199.5 Stability Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

9.5.1 Stability Results in GPC . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219.5.2 Terminal Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4219.5.3 Infinite Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4259.5.4 Finite Terminal Penalty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

9.6 Explicit Predictive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4269.6.1 Quadratic Programming Definition . . . . . . . . . . . . . . . . . . 426

XVIII Contents

9.6.2 Explicit Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4279.6.3 Multi-Parametric Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . 429

9.7 Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4319.7.1 Tuning based on the First Order Model . . . . . . . . . . . . . . 4319.7.2 Multivariable Tuning based on the First Order Model . . 4329.7.3 Output Horizon Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4339.7.4 λ Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4349.7.5 Tuning based on Model Following . . . . . . . . . . . . . . . . . . . 4349.7.6 The C polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

9.8 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4369.8.1 A Linear Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4369.8.2 Neural Network based GPC . . . . . . . . . . . . . . . . . . . . . . . . 4389.8.3 pH Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

9.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4429.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

10 Adaptive Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44510.1 Discrete-Time Self-Tuning Control . . . . . . . . . . . . . . . . . . . . . . . . . 44610.2 Continuous-Time Self-Tuning Control . . . . . . . . . . . . . . . . . . . . . . 44710.3 Examples of Self-Tuning Control . . . . . . . . . . . . . . . . . . . . . . . . . . 447

10.3.1 Discrete-Time Adaptive Dead-Beat Control of aSecond Order System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448

10.3.2 Continuous-Time Adaptive LQ Control of a SecondOrder System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451

10.3.3 Continuous-Time Adaptive MIMO Pole PlacementControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456

10.3.4 Adaptive Control of a Tubular Reactor . . . . . . . . . . . . . . . 45910.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

List of Figures

1.1 A simple heat exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Response of the process controlled with proportional

feedback controller for a step change of disturbance variable ϑv 71.3 The scheme of the feedback control for the heat exchanger . . . 81.4 The block scheme of the feedback control of the heat exchanger 9

2.1 A liquid storage system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 An interacting tank-in-series process . . . . . . . . . . . . . . . . . . . . . . . 172.3 Continuous stirred tank heated by steam in jacket . . . . . . . . . . . 192.4 Series of heat exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5 Double-pipe steam-heated exchanger and temperature profile

along the exchanger length in steady-state . . . . . . . . . . . . . . . . . . 232.6 Temperature profile of ϑ in an exchanger element of length

dσ for time dt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7 A metal rod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.8 A scheme of a packed countercurrent absorption column . . . . . 262.9 Scheme of a continuous distillation column . . . . . . . . . . . . . . . . . 282.10 Model representation of i-th tray . . . . . . . . . . . . . . . . . . . . . . . . . . 292.11 A nonisothermal CSTR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.12 Classification of dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . 462.13 A cone liquid storage process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.14 Well mixed heat exchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.15 A well mixed tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.16 Series of two CSTRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.17 A gas storage tank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1 A step function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 An original and delayed function . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3 A rectangular pulse function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4 A mixing process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.5 A U-tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

XX List of Figures

3.6 Time response of the U-tube for initial conditions (1, 0)T . . . . . 763.7 Constant energy curves and state trajectory of the U-tube in

the state plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.8 Canonical decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.9 Block scheme of a system with transfer function G(s) . . . . . . . . 853.10 Two tanks in a series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863.11 Block scheme of two tanks in a series . . . . . . . . . . . . . . . . . . . . . . 883.12 Serial connection of n tanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.13 Block scheme of n tanks in a series . . . . . . . . . . . . . . . . . . . . . . . . 903.14 Simplified block scheme of n tanks in a series . . . . . . . . . . . . . . . 913.15 Block scheme of a heat exchanger . . . . . . . . . . . . . . . . . . . . . . . . . 923.16 Modified block scheme of a heat exchanger . . . . . . . . . . . . . . . . . 923.17 Block scheme of a double-pipe heat exchanger . . . . . . . . . . . . . . 973.18 Serial connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.19 Parallel connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.20 Feedback connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 983.21 Moving of the branching point against the direction of signals 993.22 Moving of the branching point in the direction of signals . . . . . 993.23 Moving of the summation point in the direction of signals . . . . 1003.24 Moving of the summation point against the direction of signals1003.25 Block scheme of controllable canonical form of a system . . . . . . 1063.26 Block scheme of controllable canonical form of a second

order system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1073.27 Block scheme of observable canonical form of a system . . . . . . . 108

4.1 Impulse response of the first order system . . . . . . . . . . . . . . . . . . 1194.2 Step response of a first order system . . . . . . . . . . . . . . . . . . . . . . . 1204.3 Step responses of a first order system with time constants

T1, T2, T3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.4 Step responses of the second order system for the various

values of ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234.5 Step responses of the system with n equal time constants . . . . 1244.6 Block scheme of the n-th order system connected in a series

with time delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.7 Step response of the first order system with time delay . . . . . . . 1254.8 Step response of the second order system with the numerator

B(s) = b1s+ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1254.9 Simulink block scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.10 Results from simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.11 Simulink block scheme for the liquid storage system . . . . . . . . . 1374.12 Response of the tank to step change of q0 . . . . . . . . . . . . . . . . . . 1384.13 Simulink block scheme for the nonlinear CSTR model . . . . . . . 1434.14 Responses of dimensionless deviation output concentration

x1 to step change of qc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

List of Figures XXI

4.15 Responses of dimensionless deviation output temperature x2

to step change of qc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1464.16 Responses of dimensionless deviation cooling temperature x3

to step change of qc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1474.17 Ultimate response of the heat exchanger to sinusoidal input . . 1494.18 The Nyquist diagram for the heat exchanger . . . . . . . . . . . . . . . . 1534.19 The Bode diagram for the heat exchanger . . . . . . . . . . . . . . . . . . 1544.20 Asymptotes of the magnitude plot for a first order system . . . . 1554.21 Asymptotes of phase angle plot for a first order system . . . . . . 1564.22 Asymptotes of magnitude plot for a second order system . . . . . 1574.23 Bode diagrams of an underdamped second order system

(Z1 = 1, Tk = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1584.24 The Nyquist diagram of an integrator . . . . . . . . . . . . . . . . . . . . . . 1594.25 Bode diagram of an integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1604.26 The Nyquist diagram for the third order system. . . . . . . . . . . . . 1614.27 Bode diagram for the third order system . . . . . . . . . . . . . . . . . . . 1614.28 Graphical representation of the law of distribution of a

random variable and of the associated distribution function . . . 1634.29 Distribution function and corresponding probability density

function of a continuous random variable . . . . . . . . . . . . . . . . . . . 1664.30 Realisations of a stochastic process . . . . . . . . . . . . . . . . . . . . . . . . 1704.31 Power spectral density and auto-correlation function of white

noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1774.32 Power spectral density and auto-correlation function of the

process given by (4.276) and (4.277) . . . . . . . . . . . . . . . . . . . . . . . 1784.33 Block-scheme of a system with transfer function G(s) . . . . . . . . 183

5.1 A digital control system controlling a continuous-time process 1895.2 Transformation of a continuous-time to a discrete-time signal . 1905.3 Continuous-time signal step reconstruction from its

discrete-time counterpart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.4 A possible loss of information caused by sampling . . . . . . . . . . . 1925.5 An ideal sampler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1925.6 Spectral density of a signal y(t) . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.7 Spectral density of a sampled signal y∗(t) . . . . . . . . . . . . . . . . . . 1945.8 An ideal sampler and an impulse modulated signal . . . . . . . . . . 1955.9 Sample-and-hold device in series with a continuous-time system2015.10 Sampler and zero-order hold in series with a continuous-time

system G(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2025.11 Direct digital control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2055.12 Zero-order hold in series with a continuous-time system . . . . . . 2075.13 A feedback discrete-time system . . . . . . . . . . . . . . . . . . . . . . . . . . 2125.14 Series of heat exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6.1 Step response of the first order system . . . . . . . . . . . . . . . . . . . . . 226

XXII List of Figures

6.2 Measured step response of a chemical reactor using the inputchange Δu = 10 (left), approximation of step response bythe first order step response (right) . . . . . . . . . . . . . . . . . . . . . . . . 228

6.3 Step response of the second order underdamped system . . . . . . 2286.4 Measured step response of an underdamped system (left).

Normalised and approximated step response (right) . . . . . . . . . . 2306.5 Step response of a higher order system . . . . . . . . . . . . . . . . . . . . . 2316.6 Normalised step response of the chemical reactor with lines

drawn to estimate Tu, Tn (left), approximated step response(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

6.7 Simulink schema for parameter estimation of the secondorder discrete system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

6.8 Estimated parameter trajectories for the second orderdiscrete system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

6.9 Alternate Simulink schema for parameter estimation of thesecond order discrete-time system using blocks from libraryIDTOOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

6.10 Block scheme of identification of a continuous-time transferfunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

6.11 Simulink schema for estimation of parameters of the secondorder continuous-time system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

6.12 Estimated parameter trajectories for the second ordercontinuous-time system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

6.13 Alternate Simulink schema for parameter estimation ofthe second order continuous-time system using blocks fromlibrary IDTOOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

7.1 Detailed and simplified closed-loop system scheme . . . . . . . . . . . 2547.2 Open-loop system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2557.3 A typical response of a controlled process to a step change of

a setpoint variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2587.4 A typical response of the controlled process to a step change

of disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2597.5 Gain and phase margins for a typical continuous system . . . . . 2617.6 Bandwidth, resonant peak, and resonance frequency in

magnitude frequency response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2627.7 Suitable region for pole locations of a closed-loop system . . . . . 2647.8 Curves with constant values of ζ, ω0 (ζ1 < ζ2 < ζ3,

ω1 < ω2 < ω3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2657.9 Response of a heat exchanger to a step change of the d with

a P controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2697.10 Response of a heat exchanger to a step change of the w with

a P controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2697.11 Proportional control to the setpoint value w = 1 with various

gains ZR of the controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2707.12 Heat exchanger – disturbance rejection with the PI controller . 272

List of Figures XXIII

7.13 Heat exchanger – setpoint tracking with the PI controller . . . . 2727.14 PI control with setpoint w = 1 and several values of TI . . . . . . 2737.15 Graphical representation of derivative controller effects . . . . . . . 2737.16 PD control with setpoint w = 1 and several values of TD . . . . . 2747.17 Closed-loop system with a two degree-of-freedom controller . . . 2767.18 Windup phenomenon. (a) Problem, (b) Solution . . . . . . . . . . . . 2787.19 Anti-windup controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2787.20 Step responses of selected standard transfer functions . . . . . . . . 2827.21 The PID controller tuned according to Strejc . . . . . . . . . . . . . . . 2857.22 Heat exchanger control using pole placement design . . . . . . . . . 2877.23 Closed-Loop system with a relay . . . . . . . . . . . . . . . . . . . . . . . . . . 2897.24 PID control tuned with the Ziegler-Nichols method . . . . . . . . . . 2907.25 PID control using the Astrom-Hagglund method . . . . . . . . . . . . 293

8.1 Optimal trajectories of input and state variables of the heatexchanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

8.2 Trajectories of P (t) for various values of r . . . . . . . . . . . . . . . . . . 3108.3 Optimal feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3118.4 Two heat exchangers in series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3128.5 Simulink program to simulate feedback optimal control in

Example 8.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3158.6 LQ optimal trajectories x1, x2, u1 of heat exchangers . . . . . . . . 3158.7 Optimal LQ output regulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3188.8 Optimal trajectory in n-dimensional state space . . . . . . . . . . . . . 3238.9 Optimal trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3248.10 Time scale with time intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3308.11 Block diagram of the closed-loop system with a state observer 3358.12 Scheme of an observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3368.13 State feedback with observer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3418.14 Block diagram of state estimation of a singlevariable system . . 3478.15 Polynomial state estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3478.16 Interpretation of a pseudo-state . . . . . . . . . . . . . . . . . . . . . . . . . . . 3478.17 Realisation of a state feedback with an observer . . . . . . . . . . . . . 3488.18 State feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3498.19 Feedback control with one-degree-of-freedom controller . . . . . . . 3508.20 Closed-loop system with a pure integrator . . . . . . . . . . . . . . . . . . 3558.21 State feedback control with state observation and reference

input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3628.22 Feedback control system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3658.23 Block diagram of the parametrised controller . . . . . . . . . . . . . . . 3708.24 State feedback controller with observer . . . . . . . . . . . . . . . . . . . . . 3708.25 State parametrised controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3718.26 Closed-loop system response to the unit step disturbance on

input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3738.27 Parametrised controlled system in the closed-loop system . . . . 3748.28 Discrete-time feedback closed-loop system . . . . . . . . . . . . . . . . . . 376

XXIV List of Figures

8.29 Simulink diagram for finite-time tracking of a continuous-timesecond order system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

8.30 Tracking in (a) finite (b) the least number of steps . . . . . . . . . . 3808.31 Scheme optlq22s.mdl in Simulink for multivariable

continuous-time LQ control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3878.32 Trajectories of (a) output, and setpoint variables, (b) input

variables of the multivariable continuous-time processcontrolled with LQ controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388

8.33 Block diagram of LQG control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3908.34 Standard control configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395

9.1 Principle of MBPC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4049.2 Closed-loop response of the controlled system . . . . . . . . . . . . . . . 4159.3 Controller regions and the optimal control for the double

integrator example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4329.4 Increasing value of N2 leads towards mean-level control . . . . . . 4379.5 Increasing value of λ - detuned dead-beat control . . . . . . . . . . . . 4379.6 Trajectories of the oxygen concentration (the process output) . 4399.7 Trajectories of the dilution rate (the manipulated variable) . . . 4399.8 Neutralisation plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4409.9 Setpoint tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4419.10 Disturbance rejection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

10.1 Block diagram of a self-tuning discrete-time controller . . . . . . . 44610.2 Block diagram of a self-tuning continuous-time controller . . . . . 44710.3 Scheme ad031s.mdl in Simulink for dead-beat tracking of a

continuous-time second order system with a discrete-timeadaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

10.4 Trajectories of (a) input, output, and setpoint variables, (b)identified parameters of the discrete-time process modelcontrolled with a dead-beat controller . . . . . . . . . . . . . . . . . . . . . . 451

10.5 Scheme ad0313.mdl in Simulink for LQ tracking of acontinuous-time second order system with a continuous-timeadaptive controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452

10.6 Trajectories of (a) input, output, and setpoint variables, (b)identified parameters of the continuous-time process modelcontrolled with LQ controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455

10.7 Scheme ad035s.mdl in Simulink for tracking of amultivariable system with a continuous-time adaptive controller457

10.8 Trajectories of (a) output, and setpoint variables, (b) inputvariables, (c) identified parameters of the continuous-timeprocess controlled with a pole placement controller . . . . . . . . . . 459

10.9 Comparison of the dead-beat (db) and mean-level (ml)control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

10.10 Effect of step disturbances in inlet concentration . . . . . . . . . . . . 462

List of Tables

3.1 The Laplace transforms for common functions . . . . . . . . . . . . . . . 56

4.1 Solution of the second order differential equation . . . . . . . . . . . . . 1344.2 The errors of the magnitude plot resulting from the use of

asymptotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.1 Laplace and Z-transforms of some common functions . . . . . . . . . 198

7.1 Controller tuning using Strejc method . . . . . . . . . . . . . . . . . . . . . . 2847.2 Ziegler-Nichols controller tuning based on permanent

oscillations. PID1 sets smaller overshoot, PID2 is underdamped 2897.3 The Ziegler-Nichols controller tuning based on the process

step response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2897.4 PI controller tuning based on step response for stable

processes. Parameters are given as functions of τ of theform (7.90) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

7.5 PID controller tuning based on step response for stableprocesses. Parameters are given as functions of τ of theform (7.90) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

7.6 PI controller tuning based on step response for processes withpure integrator. Parameters are given as functions of τ of theform (7.90) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

7.7 PID controller tuning based on step response for processeswith pure integrator. Parameters are given as functions of τ ofthe form (7.90) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

7.8 PI controller tuning based on frequency response for stableprocesses. Parameters are given as functions of κ of the forma0 exp(a1κ+ a2κ

2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2937.9 PID controller tuning based on frequency response for stable

processes. Parameters are given as functions of κ of the forma0 exp(a1κ+ a2κ

2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293