** **Chapter 1 Introduction 1

** **Overview 1

Digital Control System 2

The Control Problem 5

Satellite Model 6

Servomotor System Model 8

Antenna Pointing System 10

Robotic Control System 11

Temperature Control System 12

Single-Machine Infinite Bus Power System 14

Summary 17

References 17 • Problems 17

** **Chapter 2 discrete-time systems and the z-transform 25

** **Introduction 25

Discrete-Time Systems 25

Transform Methods 27

Properties of the z-Transform 30

Addition and Subtraction 30

Multiplication by a Constant 30

Real Translation 31

Complex Translation 33

Initial Value 34

Final Value 34

Finding z-Transforms 35

Solution of Difference Equations 38

The Inverse z-Transform 41

Power Series Method 41

Partial-Fraction Expansion Method 42

Inversion-Formula Method 46

Discrete Convolution 47

Simulation Diagrams and Flow Graphs 49

State Variables 53

Other State-Variable Formulations 61

Transfer Functions 70

Solutions of the State Equations 74

Recursive Solution 74

z-Transform Method 76

Numerical Method via Digital Computer 77

Properties of the State Transition Matrix 78

Linear Time-Varying Systems 79

Summary 80

References and Further Readings 80 • Problems 80

** **Chapter 3 sampling and reconstruction 90

** **Introduction 90

Sampled-Data Control Systems 90

The Ideal Sampler 93

Evaluation of E*(S) 95

Results from the Fourier Transform 98

Properties of E*(S) 100

Data Reconstruction 103

Zero-Order Hold 104

First-Order Hold 108

Fractional-Order Holds 109

Summary 111

References and Further Readings 111 • Problems 112

** **Chapter 4 open-loop discrete-time systems 116

** **Introduction 116

The Relationship Between E(Z) and E*(S) 116

The Pulse Transfer Function 117

Open-Loop Systems Containing Digital Filters 123

The Modified z-Transform 126

Systems with Time Delays 129

Nonsynchronous Sampling 132

State-Variable Models 135

Review of Continuous-Time State Variables 136

Discrete State Equations 140

Practical Calculations 144

Summary 146

References and Further Readings 146 • Problems 146

**Chapter 5 closed-loop systems 157**

Introduction 157

Preliminary Concepts 157

Derivation Procedure 161

State-Variable Models 168

Summary 178

References and Further Readings 178 • Problems 178

** **Chapter 6 system time-response characteristics 188

** **Introduction 188

System Time Response 188

System Characteristic Equation 197

Mapping the s-Plane into the z-Plane 198

Steady-State Accuracy 205

Simulation 208

Control Software 213

Summary 213

References and Further Readings 214 • Problems 214

**Chapter 7 stability analysis techniques 220**

Introduction 220

Stability 220

Bilinear Transformation 224

The Routh-Hurwitz Criterion 226

Jury’s Stability Test 229

Root Locus 234

The Nyquist Criterion 238

The Bode Diagram 248

Interpretation of the Frequency Response 249

Closed-Loop Frequency Response 251

Summary 260

References and Further Readings 260 • Problems 260

** **Chapter 8 digital controller design 269

** **Introduction 269

Control System Specifications 269

Steady-State Accuracy 270

Transient Response 270

Relative Stability 272

Sensitivity 273

Disturbance Rejection 274

Control Effort 275

Compensation 275

Phase-Lag Compensation 277

Phase-Lead Compensation 284

Phase-Lead Design Procedure 285

Lag-Lead Compensation 293

Integration and Differentiation Filters 297

PID Controllers 299

PID Controller Design 303

Design by Root Locus 311

Summary 324

References and Further Readings 324 • Problems 325

** **Chapter 9 pole-assignment design and state

** **estImatIon 333

Introduction 333

Pole Assignment 333

State Estimation 342

Observer Model 342

Errors in Estimation 344

Error Dynamics 344

Controller Transfer Function 349

Closed-Loop Characteristic Equation 352

Closed-Loop State Equations 353

Reduced-Order Observers 354

Current Observers 359

Controllability and Observability 364

Systems with Inputs 368

Summary 373

References and Further Readings 374 • Problems 374

** **Chapter 10 system identification of discrete-time

** **systems 380

Introduction 380

Identification of Static Systems 381

Identification of Dynamic Systems 384

Black-Box Identification 384

Least-Squares System Identification 391

Estimating Transfer Functions with Partly Known Poles and Zeros 397

Recursive Least-Squares System Identification 399

Practical Factors for Identification 402

Choice of Input 402

Choice of Sampling Frequency 403

Choice of Signal Scaling 403

Summary 404

References and Further Readings 404 • Problems 404

** **Chapter 11 linear quadratic optimal control 408

** **Introduction 408

The Quadratic Cost Function 409

The Principle of Optimality 411

Linear Quadratic Optimal Control 414

The Minimum Principle 423

Steady-State Optimal Control 424

Optimal State Estimation–Kalman Filters 430

Least-Squares Minimization 436

Summary 436

References and Further Readings 437 • Problems 438

** **Chapter 12 case studies 444

** **Introduction 444

Servomotor System 444

System Model 446

Design 449

Environmental Chamber Control System 451

Temperature Control System 453

Aircraft Landing System 456

Plant Model 458

Design 458

Neonatal Fractional Inspired Oxygen 464

Plant Transfer Function 464

Taube’s PID Controller 466

MATLAB pidtool PIDF Controllers 467

Topology Identification in Electric Power System Models 474

References 478

Appendix I Design Equations 480

Appendix II Mason’s Gain Formula 481

Appendix III Evaluation of E*(s) 486

Appendix IV Review of Matrices 491

Appendix V The Laplace Transform 498

Appendix VI z-Transform Tables 512

Index 515