CONTENTS

PREFACE                                                                                                                        2

CONTENTS                                                                                                                    5

CHAPTER 1. INTRODUCTION TO DIGITAL CONTROL SYSTEMS                       11

1.1.                General introduction                                                                    11

1.2.                Brief historical overview of control systems                            12

1.3.                The basic structure of digital control systems                          13

1.4.      References                                                                                       16

CHAPTER 2. THE Z-TRANSFORM                                                                                 22

2.1.       Introduction                                                                                      23

2.2.       The basic discrete-time control signals                                       23

2.3.       The Z-transform                                                                                26

2.3.1.                      Introduction to Z-transform                                                    26

2.3.2.                      Properties and theorems of Z-transform                              29

2.4.       The inverse Z-transform                                                                 37

2.5.       Certain interesting Z-transform application examples           40

2.6.       References                                                                                       45

2.7.       Problems                                                                                           45

CHAPTER 3. DESCRIPTION AND ANALYSIS OF DISCRETE-TIME

AND SAMPLED DATA SYSTEMS                                                          46

3.1.       Introduction                                                                                      46

3.2.       Description and analysis of discrete-time systems                    48

3.2.1.                      Properties of discrete-time systems                                    48

3.2.2.                      Description of linear time-invariant

discrete-time systems                                                          49

 3.2.3.     Analysis of linear time-invariant

discrete-time systems                                                           53

 3.2.4.     Description and analysis of linear

time-varying discrete-time systems                                      64

 3.3.  Description and analysis of sampled-data systems                     65

3.3.1.                      Introduction to D/A and A/D converters                        65

3.3.2.                      Hold circuits                                                                           66

3.3.3.                      Conversion of G(s) to G(z)                                                    69

3.3.4.                      Conversion of differential state-space equations

to difference state-space equations                                      75

3.3.5.      Analysis of sampled-data systems                                       79

3.4.       Determination of closed-loop system transfer   

        functions                                                                       82

3.5.       References                                                                                       85

3.6.       Problems                                                                                           86

CHAPTER 4. STABILITY, CONTROLLABILITY AND OBSERVABILITY      90

4.1.    Introduction                                                                                      90

4.2.    Definitions and basic theorems of stability                       90

4.2.1.      Stability of linear time-invariant

discrete-time systems                                                          91

4.2.2.      Bounded-input bounded-output stability                                92

 4.3.   Stability criteria                                                                                94

4.3.1.      The Routh criterion using the Mobius
transformation                                                                                94

4.3.2.      The Jury criterion                                                                   96

4.3.3.      Stability in the sense of Lyapunov                                        101

4.3.4.      Influence of pole position on the transient

response                                                                              103

 4.4.   Controllability and observability                                                   105

4.4.1.        Controllability                                                                        105

4.4.2.        Observability                                                                        108

4.4.3.        Loss of controllability and observability

due to sampling                                                                   110

4.4.4.        Geometric considerations of controllability                          111

4.4.5.        Geometric considerations of observability                          113

4.4.6.        State vector transformations -

state space canonical forms                                               115

4.4.7.     Kalman decomposition                                                        120

4.5.   References                                                                                       121

4.6.   Problems                                                                                           123

CHAPTER 5. CLASSICAL DESIGN METHODS                                                            125

5.1.    Introduction                                                                                      125

5.2.    Discrete-time controllers derived from
continuous-time controllers                                                                  126

5.2.1.      Discrete-time controller design using

indirect techniques                                                                126

5.2.2.      Specifications of the transient response

of continuous-time systems                                                 127

5.3.    Controller design via the root-locus method                              139

5.4.    Controller design based on the frequency response                147

5.4.1.                         Introduction                                                                           147

5.4.2.                         Bode diagrams                                                                     150

5.4.3.                         Nyquist diagrams                                                                  158

5.5.    The PID controller                                                                           159

5.5.1.                        Proportional controller                                                           159

5.5.2.                        Integral controller                                                                  159

5.5.3.                        Derivative controller                                                              160

5.5.4.                        The three term PID controller                                               160

5.5.5.                        Design of PID controllers using

the Ziegler-Nichols method                                                   161

5.6.    Steady-state error                                                                           164

5.7.    References                                                                                      168

5.8.    Problems                                                                                          169

CHAPTER 6. STATE-SPACE DESIGN METHODS                                                      172

6.1.     Introduction                                                                                     172

6.2.     The pole-placement design method                                            173

6.3.     Deadbeat control                                                                            185

6.4.     State observers                                                                              189

6.4.1.                      Direct state vector estimation                                              189

6.4.2.                      State-vector reconstruction using a Luenberger
           observer                                                                                 194

6.4.3.                      Reduced-order observers                                                    200

6.4.4.                      Closed-loop system design using

state observers                                                                     204

6.5.    References

6.6.    Problems

CHAPTER 7. OPTIMAL CONTROL                                                                                217

7.1.    Introduction                                                                                      217

7.2.    Mathematic background for the study of optimal control
       problems of discrete-time systems                                                217

7.2.1.      Maxima and minima using the method

of calculus of variations                                                         217

7.2.2.      The maximum principle for discrete-time systems             220

7.3.    The optimal linear regulator                                                           225

7.4.    The special case of time-invariant systems                                 233

7.5.    References                                                                                       236

7.6.    Problems                                                                                           237

CHAPTER 8. DISCRETE-TIME SYSTEM IDENTIFICATION                                       239

8.1.     Introduction                                                                                      239

8.2.     OFF-LINE parameter estimation                                                    240

8.2.1.       First-order systems                                                               240

8.2.2.       Higher order systems                                                            245

8.3.     ON-LINE parameter estimation                                                      252

8.4.     References                                                                                       263

8.5.     Problems                                                                                           264

CHAPTER 9. DISCRETE-TIME ADAPTIVE CONTROL                                               267

9.1.      Introduction                                                                                      267

9.2.      Adaptive control with the gradient method (MIT rule)           270

9.2.1.                        Introduction                                                                           270

9.2.2.                        Results for the general case of linear systems                    274

9.3.      Adaptive control using a Lyapunov design                                 277

9.4.      Model Reference Adaptive control-Hyperstability design     282

9.4.1.             Introduction                                                                              282

9.4.2.             Definition of the model reference control problem            283

9.4.3.             Design in the case of known parameters                               286

9.4.4.             Hyperstability design in the case of unknown

parameters                                                                           290

9.5  Self-Tuning Regulators                                                                    305

9.5.1.             Introduction                                                                              305

9.5.2.             Regulator and tracking with minimum variance

control                                                                                   307

9.5.3.   Pole placement Self-Tuning Regulators                                  314

9.6  References                                                                                         328
9.7. Problems                                                                                             330

CHAPTER 10. REALIZATION OF DISCRETE-TIME CONTROLLERS

AND QUANTIZATION ERRORS                                                        334

10.1.       Introduction                                                                               334

10.2.       Hardware controller realization                                              334

10.2.1.                         Direct realization                                                          335

10.2.2.                         Cascade realization                                                     337

10.2.3.                         Parallel realization                                                        342

10.3.       Software controller realization                                               347

10.4.       Quantization errors                                                                  349

10.4.1.     Fixed point and floating point

representations                                                            349

10.4.2.                         Truncation and rounding                                              351

10.4.3.                         A stochastic model for the quantization

error                                                                              354

10.4.4.     Propagation of the quantization error

through transfer functions                                            357

10.4.5.     Loss of controllability and quantized

closed-loop poles due to quantization                         362f

                             10.5.    References                                                                                  3621

10.6.         Problems                                                                                362m

CHAPTER 11. AN INTELLIGENT APPROACH TO CONTROL:

FUZZY CONTROLLERS                                                                    363

11.1.                   Introduction to intelligent control                                        363

11.2.                   General remarks on fuzzy controllers                                 364

11.3.                   Fuzzy sets                                                                               366

11.4.                   Fuzzy controllers                                                                    370

11.5.                   Elements of a fuzzy controller                                              372

11.6.                 Fuzzification                                                                              374

11.7.      The rule base                                                                         374

11.8.      The inference engine                                                         375

11.9.      Defuzzification                                                                          385

11.10.  Performance assessment                                                  386

11.11.  Application example: kiln control                                   387

11.12.  References                                                                                391

11.13.  Problems                                                                                    393

APPENDIX A. MATRIX THEORY                                                                          395

A.I.  Matrix definitions and operations                                    395

A. 1.1.    Matrix definitions                                                             395

A. 1.2.    Matrix operations                                                            398

A.2. Determinant of a matrix                                                        399

A.3. The inverse of a matrix                                                        400

A.4.  Matrix eigenvalues and eigenvectors                           400

A.5. Similarity transformations                                                    403

A.6. The Cayley-Hamilton theorem                                            406

A.7.  Quadratic forms and Sylvester theorems                     408

APPENDIX B. Z TRANSFORM TABLES                                                              411

B.I.   Properties and theorems of Z transform                       411

B.2.  Z transform pairs                                                                     413

INDEX 416