Control system design salgado goodwin graebe

12.14Using Continuous State Space Models 33813.4 Is a Special Theory of Digital Control Design Really Necessary?. Control Engineering plays a fundamental role in modern technological sys

CONTROL SYSTEM

DESIGN

Stefan F Graebe2Mario E Salgado3

Valpara´ıso, January 2000

1Centre for Integrated Dynamics and Control

University of Newcastle, NSW 2308 AUSTRALIA

2OMV Aktiengesellschaft

Department of Optimization/Automation

Schwechat, AUSTRIA

3Departamento de Electr´onica

Universidad T´ ecnica Federico Santa Mar´ıa

Valpara´ıso, CHILE

Dedicated, in thankful appreciation for support and understanding, to

Rosslyn

Alice Mariv´ı

CONTENTS OVERVIEW

vii

2.5 Prototype Solution to the Control Problem via Inversion 29

ix

2.6 High Gain Feedback and Inversion 32

4.7 Impulse and Step Responses of Continuous Time Linear Systems 74

Contents Overview xi

5.4 Closed Loop Stability Based on the Characteristic Polynomial 125

5.8 Relative Stability: Stability Margins and Sensitivity Peaks 141

7.3 PI and PID Synthesis Revisited using Pole Assignment 185

7.5 Summary 188

9.5 Poisson Integral Constraint on Complementary Sensitivity 252

10.2 Models for Deterministic Disturbances and Reference Signals 263

Contents Overview xiii

12.13Obtaining Discrete Models for Sampled Continuous Systems 335

12.14Using Continuous State Space Models 338

13.4 Is a Special Theory of Digital Control Design Really Necessary? 355

Contents Overview xv

15.5 Affine Parameterization for Systems having Time Delays 425

17.5 From Transfer Function to State Space Representation 488

18.6 Reinterpretation of the Affine Parameterization of all Stabilizing

18.7 State Space Interpretation of Internal Model Principle 538

18.9 Dealing with Input Constraints in the Context of State Estimate

19.6 Smooth Dynamic Nonlinearities for Stable and Stably Invertible Models560

Contents Overview xvii

22.5 Properties of the Linear Quadratic Optimal Regulator 66522.6 Model Matching Based on Linear Quadratic Optimal Regulators 669

22.17Problems for the Reader 715

24.7 Poisson Integral Constraints on MIMO Complementary Sensitivity 756

Contents Overview xix

26.6 Frequency Domain Constraints for Dynamically Decoupled Systems 852

The authors wish to thank the large number of colleagues and friends who haveworked with us in the area of control over the years This book is really a synthesis

of ideas that they helped us to formulate All three authors spent time together

in the Centre for Industrial Control Science at the University of Newcastle, tralia This was a fertile breeding ground for many discussions on the principles ofcontrol Financial support from the Australian Government for this centre underthe Commonwealth Special Centres program is gratefully acknowledged Also, fi-nancial and other support was provided by the Universidad T´ecnica Federico SantaMar´ıa covering, amongst other things, several visits to Chile by the first authorduring the writing of this book Many students and colleagues read drafts of thebook ranging over a five year period The authors accept full responsibility forthe views expressed in the book (and all remaining errors) Nonetheless, they wish

Aus-to particularly acknowledge suggestions from Thomas Brinsmead, Arthur Conley,Sam Crisafulli, Jose De Don´a, Arie Feuer, Jaime Glar´ıa, William Heath, Kazuo Ko-matsu, David Mayne, Trisan Perez, Mar´ıa Seron, Gustavo Vergara, Liuping Wangand Steve Weller The book was composed and typed by many people, includingthe authors However, in the final stages of producing the book, Jayne Disneygave considerable help Also, Tim Wylie and Adrian Bastiani kindly produced theEngineering Drawings shown in the text

xxi

Control Engineering plays a fundamental role in modern technological systems Thebenefits of improved control in industry can be immense They include improvedproduct quality, reduced energy consumption, minimization of waste materials, in-creased safety levels and reduction of pollution

However, a difficulty with the subject is that some of the more advanced aspectsdepend on sophisticated mathematical background Arguably, mathematical sys-tems theory is one of the most significant achievements of twentieth century science.However, its practical impact is only as good as the benefits it can bring Thus, inthis book, we aim to strike a balance which places strong emphasis on design

It was the author’s involvement in several industrial control system designprojects that provided part of the motivation to write this book In a typicalindustrial problem, we found ourselves investigating fluid and thermal dynamics,experiencing the detrimental effects of non-constant PLC scan rates, dealing withsystem integration and network communication protocols, building trust with plantoperators and investigating safe bumpless transfer schemes for testing tentativecontrol designs on potentially dangerous plants In short, we experienced the day-to-day excitement, frustration, set-backs and progress in getting advanced control

to contribute to a commercial company’s bottom line This is not an easy task.Moreover, success in this type of venture typically depends on the application of awide range of multidisciplinary skills However, it is rewarding and exciting workfor those who do it

One of the main aims of this book is to share this excitement with our readers

We hope to contribute to the development of skills and attitudes within readersand students that will better equip them to face the challenges of real-world designproblems The book is thus intended to contribute to the ongoing reform of theControl Engineering curriculum This topic is receiving considerable internationalattention For example, a recent issue of IEEE Control Systems Magazine features

an entire special section devoted to this theme

However, reforming the curriculum will not be done by books alone - it will bedone by people; by students, by teachers, by researchers, by practitioners, by publi-cation and grant reviewers; and by market pressures Moreover, for these efforts to

xxiii

be efficient and sustainable, the control engineering community will need to municate their experiences via a host of new books, laboratories, simulations andweb-based resources Thus, there will be a need for several different and comple-mentary approaches In this context, the authors believe that this book will havebeen successful if it contributes, in some way, to the revitalization of interest bystudents in the exciting discipline of control engineering.

com-We stress that this is not a how-to book On the contrary, we provide a

compre-hensive, yet condensed, presentation of rigorous control engineering We employ,

and thus require, mathematics as a means to model the process, analyze its ties under feedback, synthesize a controller with particular properties and arrive at a design addressing the inherent trade-offs and constraints applicable to the problem.

proper-In particular, we believe that success in control projects depends on two keyingredients: (i) having a comprehensive understanding of the process itself, gained

by studying the relevant physics, chemistry, etc.; and (ii) by having mastery of thefundamental concepts of signals, systems and feedback The first ingredient typi-cally occupies more than fifty per cent of the effort It is an inescapable component

of the complete design cycle However, it is impractical for us to give full details

of the processes to which control might be applied since it covers chemical plants,electromechanical systems, robots, power generators, etc We thus emphasize thefundamental control engineering aspects that are common to all applications andleave readers to complement this with process knowledge relevant to their particularproblem Thus, the book is principally aimed at the second ingredient of controlengineering Of course, we do give details of several real world examples so as toput the methods into a proper context

The central theme of this book is continuous-time control However we alsotreat digital control in detail, since most modern control systems will usually beimplemented on some form of computer hardware This approach inevitably led

to a book of larger volume than originally intended but with the advantage ofproviding a comprehensive treatment within an integrated framework Naturally,there remain specialized topics that are not covered in the book However, we trustthat we provide a sufficiently strong foundation so that the reader can comfortablyturn to the study of appropriate complementary literature

Thus, in writing this book we chose our principal goals as:

• providing accessible treatment of rigorous material selected with applicability

in mind,

• giving early emphasis to design including methods for dealing with

fundamen-tal trade-offs and constraints,

• providing additional motivation through substantial interactive web-based

support, and

• demonstrating the relevance of the material through numerous industrial case

studies

PREFACE xxv

Design is a complex process, which requires judgment and iteration The designproblem is normally incompletely specified, sometimes ill-defined, and many timeswithout solution A key element in design is an understanding of those factors whichlimit the achievable performance This naturally leads to a viewpoint of controldesign which takes account of these fundamental limitations This viewpoint is arecurring theme throughout the book

Our objective is not to explore the full depth of mathematical completeness butinstead to give enough detail so that a reader can begin applying the ideas as soon aspossible This approach is connected to our assumption that readers will have readyaccess to modern computational facilities including the software package MATLAB-SIMULINK This assumption allows us to put the emphasis on fundamental ideasrather than on the tools Every chapter includes worked examples and problemsfor the reader

The book is divided into eight parts A brief summary of each of the parts isgiven below:

Part 1: The Elements

This part covers basic continuous time signals and systems and would be suitablefor an introductory course on this topic Alternatively it could be used to providerevision material before starting the study of control in earnest

Part II: SISO Control Essentials

This part deals with basic SISO control including classical PID tuning This,together with part 1, covers the content of many of the existing curricula for basiccontrol courses

Part III: SISO Control Design

This part covers design issues in SISO Control We consider many of these ideas

to be crucial to achieving success in practical control problems In particular, webelieve the chapter dealing with constraints should be mentioned, if at all possible,

in all introductory courses Also feedforward and cascade structures, which arecovered in this part, are very frequently employed in practice

Part IV: Digital Computer Control

This part covers material essential to the understanding of digital control We

go beyond traditional treatments of this topic by studying inter-sample issues

Part V: Advanced SISO Control

This part could be the basis of a second course on control at an undergraduatelevel It is aimed at the introduction of ideas that flow through to multi-inputmulti-output (MIMO) systems later in the book

Part VI: MIMO Control Essentials

This part gives the basics required for a junior level graduate course on MIMOcontrol In particular, this part covers basic MIMO system’s theory It also showshow one can exploit SISO methods in some MIMO design problems

Part VII: MIMO Control Design

This part describes tools and ideas that can be used in industrial MIMO sign In particular, it includes linear quadratic optimal control theory and optimalfiltering These two topics have major significance in applications We also include

de-a chde-apter on Model Predictive Control We believe this to be importde-ant mde-ateride-albecause of the widespread use of this technique in industrial applications

Part VIII: Advanced MIMO Control

This final part of the book could be left for private study It is intended totest the readers understanding of the other material by examining advanced issues.Alternatively instructors could use this part to extend parts VI and VII in a moresenior graduate course on MIMO Control

Two of the authors (Goodwin and Salgado) have taught undergraduate andpostgraduate courses of the type mentioned above using draft versions of this book

in both Australia and South America

The material in the book is illustrated by several industrial case studies withwhich the authors have had direct involvement Most of these case studies werecarried out, in collaboration with industry, by the Centre for Integrated Dynamicsand Control (CIDAC) (a Commonwealth Special Research Centre) at the University

of Newcastle

The projects that we have chosen to describe include:

• Satellite tracking

• pH control

• Control of a continuous casting machine

• Sugar mill control

• Distillation column control

• Ammonia synthesis plant control

• Zinc coating mass estimation in continuous galvanizing line

• BISRA gauge for thickness control in rolling mills

• Roll eccentricity compensation in rolling mills

• Hold-up effect in reversing rolling mills

• Flatness control in steel rolling

• Vibration control

Many of the case studies have also been repeated, and further embellished, onthe book’s web page where Java applets are provided so that readers can experimentwith the systems in the form of a “virtual laboratory” A secondary advantage of

The following material appears on the book’s web site:

B.5 Poles and Zeros

B.6 Matrix Fraction Descriptions (MFD)

C.1 Introduction

C.2 Independence of path

C.3 Simply connected domains

C.4 Functions of a Complex Variable

C.5 Derivatives and Differentials

C.6 Analytic Functions

C.7 Integrals Revisited

C.8 Poisson and Jensen Integral Formulas

C.9 Application of the Poisson-Jensen Formula to Certain Rational tions

Func-C.10 Bode’s Theorems

D.1 Solutions of the CTDRE

D.2 Solutions of the CTARE

D.3 The stabilizing solution of the CTARE

D.4 Convergence of Solutions of the CTARE to the Stabilizing Solution

Part I THE ELEMENTS

1

Designing and operating an automated process so that it maintains specifications onprofitability, quality, safety, environmental impact, etc., requires a close interactionbetween experts from different disciplines These include, for example, computer-,process-, mechanical-, instrumentation- and control-engineers

Since each of these disciplines views the process and its control from a ent perspective, they have adopted different categories, or elements, in terms ofwhich they think about the automated system The computer engineer, for exam-ple, would think in terms of computer hardware, network infrastructure, operatingsystem and application software The mechanical engineer would emphasize themechanical components from which the process is assembled, whereas the instru-mentation engineer would think in terms of actuators, sensors and their electricalwiring

differ-The control engineer, in turn, thinks of the elements of a control system interms of abstract quantities such as signals, systems and dynamic responses Theseelements can be further specified by their physical realization, the associated model

or their properties (see Table 1)

Tangible examples Examples of mathematical

si-Systems process, controller,

sensors, actuators,

differential equations, ence equations, transfer func- tions, state space models,

differ-continuous time, pled, linear, nonlinear,

sam-Table 1 Systems and signals in a control loop

3

This book emphasizes the control engineer’s perspective of process automation.However, the reader should bear the other perspectives in mind, since they formessential elements in a holistic view of the subject.

This first part of the book is the first stage of our journey into control ing It gives an introduction to the core elements of continuous time signals andsystems, as well as describing the pivotal role of feedback in control system design.These are the basic building blocks on which the remainder of the developmentrests

engineer-Chapter 1

THE EXCITEMENT OF CONTROL ENGINEERING

This chapter is intended to provide motivation for studying control engineering Inparticular it covers:

• an overview of the scope of control

• historical periods in the development of control theory

• types of control problems

• introduction to system integration

• economic benefits analysis

Feedback control has a long history which began with the early desire of humans

to harness the materials and forces of nature to their advantage Early examples

of control devices include clock regulating systems and mechanisms for keepingwind-mills pointed into the wind

A key step forward in the development of control occurred during the industrialrevolution At that time, machines were developed which greatly enhanced thecapacity to turn raw materials into products of benefit to society However, theassociated machines, specifically steam engines, involved large amounts of power

and it was soon realized that this power needed to be controlled in an organized

fashion if the systems were to operate safely and efficiently A major development

at this time was Watt’s fly ball governor This device regulated the speed of asteam engine by throttling the flow of steam, see Figure 1.1 These devices remain

in service to this day

5

Figure 1.1 Watt’s fly ball governor

The World Wars also lead to many developments in control engineering Some

of these were associated with guidance systems whilst others were connected withthe enhanced manufacturing requirements necessitated by the war effort

The push into space in the 1960’s and 70’s also depended on control opments These developments then flowed back into consumer goods, as well ascommercial, environmental and medical applications These applications of ad-vanced control have continued at a rapid pace To quote just one example fromthe author’s direct experience, centre line thickness control in rolling mills has been

devel-a mdevel-ajor success story for the devel-applicdevel-ation of devel-advdevel-anced control idedevel-as Indeed, theaccuracy of centre line thickness control has improved by two orders of magnitudeover the past 50 years due, in part, to enhanced control For many companies thesedevelopments were not only central to increased profitability but also to remaining

in business

By the end of the twentieth century, control has become a ubiquitous (but largelyunseen) element of modern society Virtually every system we come in contactwith is underpinned by sophisticated control systems Examples range from simplehousehold products (temperature regulation in air-conditioners, thermostats in hotwater heaters etc.) to more sophisticated systems such as the family car (which hashundreds of control loops) to large scale systems (such as chemical plants, aircraft,and manufacturing processes) For example, Figure 1.2 on page 8 shows the processschematic of a Kellogg ammonia plant There are about 400 of these plants around

Section 1.2 Motivation for Control Engineering 7

the world An integrated chemical plant, of the type shown in Figure 1.2 willtypically have many hundreds of control loops Indeed, for simplicity, we have notshown many of the utilities in Figure 1.2, yet these also have substantial numbers

of control loops associated with them

Many of these industrial controllers involve cutting edge technologies For

ex-ample, in the case of rolling mills (illustrated in Figure 1.3 on page 13), the controlsystem involves forces of the order of 2,000 tonnes, speeds up to 120 km/hour andtolerances (in the aluminum industry) of 5 micrometers or 1/500th of the thick-ness of a human hair! All of this is achieved with precision hardware, advancedcomputational tools and sophisticated control algorithms

Beyond these industrial examples, feedback regulatory mechanisms are central

to the operation of biological systems, communication networks, national economies,and even human interactions Indeed if one thinks carefully, control in one form oranother, can be found in every aspect of life

In this context, control engineering is concerned with designing, implementingand maintaining these systems As we shall see later, this is one of the mostchallenging and interesting areas of modern engineering Indeed, to carry out controlsuccessfully one needs to combine many disciplines including modeling (to capturethe underlying physics and chemistry of the process), sensor technology (to measurethe status of the system), actuators (to apply corrective action to the system),communications (to transmit data), computing (to perform the complex task ofchanging measured data into appropriate actuator actions), and interfacing (to allow

the multitude of different components in a control system to talk to each other in

Market globalization is increasingly occurring and this means that, to stay in ness, manufacturing industries are necessarily placing increasing emphasis on issues

busi-of quality and efficiency Indeed, in today’s society, few if any companies can afford

to be second best In turn, this focuses attention on the development of improvedcontrol systems so that processes operate in the best possible way In particular,improved control is a key enabling technology underpinning:

• enhanced product quality

• waste minimization

• environmental protection

• greater throughput for a given installed capacity

• greater yield

• deferring costly plant upgrades, and

• higher safety margins.

All of these issues are relevant to the control of an integrated plant such as thatshown in Figure 1.2

Figure 1.2 Process schematic of a Kellogg ammonia plant

All companies and governments are becoming increasingly aware of the need toachieve the benefits outlined above whilst respecting finite natural resources andpreserving our fragile environment Again, control engineering is a core enablingtechnology in reaching these goals To quote one well known example, the changes

in legislation covering emissions from automobiles in California have led car facturers to significant changes in technology including enhanced control strategiesfor internal combustion engines

manu-Section 1.3 Historical Periods of Control Theory 9

Thus, we see that control engineering is driven by major economic, political,and environmental forces The rewards for those who can get all the factors rightcan be enormous

We have seen above that control engineering has taken several major steps forward

at crucial times in history (e.g the industrial revolution, the Second World War,the push into space, economic globalization, shareholder value thinking etc.) Each

of these steps has been matched by a corresponding burst of development in theunderlying theory of control

Early on, when the compelling concept of feedback was applied, engineers times encountered unexpected results These then became catalysts for rigorousanalysis For example, if we go back to Watt’s fly ball governor, it was found thatunder certain circumstances these systems could produce self sustaining oscillations.Towards the end of the 19th century several researchers (including Maxwell) showedhow these oscillations could be described via the properties of ordinary differentialequations

some-The developments around the period of the Second World War were also matched

by significant developments in Control Theory For example, the pioneering work

of Bode, Nyquist, Nichols, Evans and others appeared at this time This resulted

in simple graphical means for analyzing single-input single-output feedback control

problems These methods are now generally known by the generic term Classical Control Theory.

The 1960’s saw the development of an alternative state space approach to trol This followed the publication of work by Wiener, Kalman (and others) onoptimal estimation and control This work allowed multivariable problems to betreated in a unified fashion This had been difficult, if not impossible, in the classical

con-framework This set of developments is loosely termed Modern Control Theory.

By the 1980’s these various approaches to control had reached a sophisticatedlevel and emphasis then shifted to other related issues including the effect of modelerror on the performance of feedback controllers This can be classified as the period

of Robust Control Theory.

In parallel there has been substantial work on nonlinear control problems Thishas been motivated by the fact that many real world control problems involvenonlinear effects

There have been numerous other developments including adaptive control, totuning, intelligent control etc These are too numerous to detail here Anyway,our purpose is not to give a comprehensive history but simply to give a flavor forthe evolution of the field

au-At the time of writing this book, control has become a mature discipline It isthus possible to give a treatment of control which takes account of many differentviewpoints and to unify these in a common framework This is the approach wewill adopt here

1.4 Types of Control System Design

Control system design in practice requires cyclic effort in which one iterates betweenmodeling, design, simulation, testing, and implementation

Control system design also takes several different forms and each requires aslightly different approach

One factor that impacts on the form that the effort takes is whether the system

is part of a predominantly commercial mission or not Examples where this is notthe case include research, education and missions such as landing the first man onthe moon Although cost is always a consideration, these types of control designare mainly dictated by technical, pedagogical, reliability and safety concerns

On the other hand, if the control design is motivated commercially, one againgets different situations depending on whether the controller is a small sub-component

of a larger commercial product (such as the cruise controller or ABSin a car) orwhether it is part of a manufacturing process (such as the motion controller in therobots assembling a car) In the first case one must also consider the cost of in-cluding the controller in every product, which usually means that there is a majorpremium on cost and hence one is forced to use rather simple microcontrollers Inthe second case, one can usually afford significantly more complex controllers, pro-vided they improve the manufacturing process in a way that significantly enhancesthe value of the manufactured product

In all of these situations, the control engineer is further affected by where thecontrol system is in its lifecycle, e.g.:

• Initial grass roots design

• Commissioning and Tuning

• Refinement and Upgrades

• Forensic studies

In this phase, the control engineer is faced by a green-field, or so called grass roots projects and thus the designer can steer the development of a system from

the beginning This includes ensuring that the design of the overall system takesaccount of the subsequent control issues All too often, systems and plants aredesigned based on steady state considerations alone It is then small wonder thatoperational difficulties can appear down the track It is our belief that controlengineers should be an integral part of all design teams The control engineer needs

to interact with the design specifications and to ensure that dynamic as well assteady-state issues are considered

Section 1.5 System Integration 11

Once the basic architecture of a control system is in place, then the control engineer’sjob becomes one of tuning the control system to meet the required performancespecifications as closely as possible This phase requires a deep understanding offeedback principles to ensure that the tuning of the control system is carried out in

an expedient, safe and satisfactory fashion

Once a system is up and running, then the control engineer’s job turns into one ofmaintenance and refinement The motivation for refinement can come from manydirections They include

• internal forces - e.g the availability of new sensors or actuators may open the

door for improved performance

• external forces - e.g market pressures, or new environmental legislation may

necessitate improved control performance

Forensic investigations are often the role of control engineering consultants Here

the aim is to suggest remedial actions that will rectify an observed control problem

In these studies, it is important that the control engineer take a holistic view sincesuccessful control performance usually depends on satisfactory operation of manyinterconnected components In our experience, poor control performance is as likely

to be associated with basic plant design flaws, poor actuators, inadequate sensors,

or computer problems as it is to be the result of poor control law tuning However,all of these issues can, and should be, part of the control engineer’s domain Indeed,

it is often only the control engineer who has the necessary overview to successfullyresolve these complex issues

As is evident from the above discussion, success in control engineering depends ontaking a holistic viewpoint Some of the issues that are embodied in a typical controldesign include:

• plant, i.e the process to be controlled

• objectives

• sensors

• actuators

• communications