Control Systems

Engineering FIFTH EDITION

I.J. Nagrath

M. Gopal

Ansh an

ANSHAN LTD 6 Newlands Road

Tunbridge Welis, Kent, TN4 9AT, UK

CONTENTS

Preface to the Fifth Edition v Preface to the Third Edition vii

1. INTRODUCTION 1-20

1.1 The Control System 2 1.2 Servomechanisms 6 1.3 History and Development of Automatic Control 10 1.4 Digital Computer Control 14 1.5 Application of Control Theory in Non-engineering Fields 18 1.6 The Control Problem 19

2. MATHEMATICAL MODELS OF PHYSICAL SYSTEMS 21-90

2.1 Introduction 22 2.2 Differential Equations of Physical Systems 24 2.3 Dynamics of Robotic Mechanisms 42 2.4 Transfer Functions 46 2.5 Block Diagram Algebra 54 2.6 Signal Flow Graphs 62 2.7 Illustrative Examples 72

Problems 83

3. FEEDBACK CHARACTERISTICS OF CONTROL SYSTEMS 91-129

3.1 Feedback and Non-feedback Systems 92 3.2 Reduction of Parameter Variations by Use of Feedback 93 3.3 Control Over System Dynamics by Use of Feedback 97 3.4 Control of the Effects of Disturbance Signals by Use of Feedback 100 3.5 Linearizing Effect of Feedback 102 3.6 Regenerative Feedback 103 3.7 Illustrative Examples 104

Problems 119

4. CONTROL SYSTEMS AND COMPONENTS 131-192 4.1 Introduction 132 4.2 Linear Approximation of Nonlinear Systems 133

CONTENTS

4.3 Controller Components 134

4.4 Stepper Motors 154

4.5 Hydraulic Systems 163

4.6 Pneumatic Systems 177

Problems 185

5. TIME RESPONSE ANALYSIS, DESIGN SPECIFICATIONS 193-268 AND PERFORMANCE INDICES

5.1 Introduction 194

5.2 Standard Test Signals 195

5.3 Time Response of First-order Systems 197

5.4 Time Response of Second-order Systems 199

5.5 Steady-state Errors and Error Constants 210

5.6 Effect of Adding a Zero to a System 214

5.7 Design Specifications of Second-order Systems 215

5.8 Design Considerations for Higher-order Systems 221

5.9 Performance Indices 223

5.10 Illustrative Examples 227

5.11 Robotic Control Systems 237

5.12 State Variable Analysis—Laplace Transform Technique 245

5.13 The Approximation of Higher-order Systems by Lower-order 248

Problems 261

6. CONCEPTS OF STABILITY AND ALGEBRAIC CRITERIA 269-295 6.1 The Concept of Stability 270

6.2 Necessary Conditions for Stability 275

6.3 Hurwitz Stability Criterion 277

6.4 Routh Stability Criterion 278

6.5 Relative Stability Analysis 287

6.6 More on the Routh Stability Criterion 290

6.7 Stability of Systems Modelled in State Variable Form 291

Problems 293

7. T H E ROOT LOCUS TECHNIQUE 297-343 7.1 Introduction 298

7.2 The Root Locus Concepts 299

7.3 Construction Foot Loci 302

7.4 Root Contours 327

7.5 Systems with Transportation Lag 332

7.6 Sensitivity of the Roots of the Characteristic Equation 334

7.7 MATLAB: Tool for Design and Analysis of Control Systems—Appendix III 340

Problems 340

CONTENTS XI

8. FREQUENCY RESPONSE ANALYSIS 345-376 8.1 Introduction 346 8.2 Correlation between Time and Frequency Response 347 8.3 Polar Plots 352 8.4 Bode Plots 355 8.5 All-pass and Minimum-phase Systems 366 8.6 Experimental Determination of Transfer Functions 367 8.7 Log-magnitude versus Phase Plots 370 8.8 MATLAB: Tool for Design and Analysis of Control Systems—Appendix III 371

Problems 374

9. STABILITY IN FREQUENCY DOMAIN 377-423 9.1 Introduction 378 9.2 Mathematical Preliminaries 378 9.3 Nyquist Stability Criterion 381 9.4 Assessment of Relative Stability Using Nyquist Criterion 394 9.5 Closed-loop Frequency Response 409 9.6 Sensitivity Analysis in Frequency Domain 417

Problems 420

10. INTRODUCTION TO DESIGN 425-511 10.1 The Design Problem 426 10.2 Preliminary Considerations of Classical Design 428 10.3 Realization of Basic Compensators 435 10.4 Cascade Compensation in Time Domain 440 10.5 Cascade Compensation in Frequency Domain 459 10.6 Tuning of PID Controllers 477 10.7 Feedback Compensation 483 10.8 Robust Control System Design 490

Problems 506

11. DIGITAL CONTROL SYSTEMS 513-568 11.1 Introduction 514 11.2 Spectrum Analysis of Sampling Process 517 11.3 Signal Reconstruction 519 11.4 Difference Equations 519 11.5 The 2-transform 521 11.6 The 2-transfer Function (Pulse Transfer Function) 531 11.7 The Inverse 2-transform and Response of Linear Discrete Systems 535 11.8 The 2-transform Analysis of Sampled-data Control Systems 538 11.9 The 2-and s-domain Relationship 548

XII CONTENTS

11.10 Stability Analysis 549

11.11 Compensation Techniques 558

Problems 564

12. STATE VARIABLE ANALYSIS AND DESIGN 569-640

12.1 Introduction 570

12.2 Concepts of State, State Variables and State Model 571

12.3 State Models for Linear Continuous-Time Systems 578

12.4 State Variables and Linear Discrete-Time Systems 596

12.5 Diagonalization 599

12.6 Solution of State Equations 604

12.7 Concepts of Controllability and Observability 617

12.8 Pole Placement by State Feedback 625

12.9 Observer Systems 632

Problems 634

13. LIAPUNOV'S STABILITY ANALYSIS 641-662 13.1 Introduction 642

13.2 Liapunov's Stability Criterion 646

13.3 The Direct Method of Liapunov and the Linear System 650

13.4 Methods of Constructing Liapunov Functions for Nonlinear Systems 652

Problems 660

14. OPTIMAL CONTROL SYSTEMS 663-712

14.1 Introduction 664

14.2 Parameter Optimization: Servomechanisms 665

14.3 Optimal Control Problems: Transfer Function Approach 673

14.4 Optimal Control Problems: State Variable Approach 684

14.5 The State Regulator Problem 688

14.6 The Infinite-time Regulator Problem 697

14.7 The Output Regulator and the Tracking Problems 702

14.8 Parameter Optimization: Regulators 704

Problems 707

15. NONLINEAR SYSTEMS 713-764

15.1 Introduction 714

15.2 Common Physical Nonlinearities 719

15.3 The Phase-plane Method: Basic Concepts 723

15.4 Singular Points 725

15.5 Stability of Nonlinear System 731

15.6 Construction of Phase-trajectories 734

15.7 The Describing Function Method: Basic Concepts 742

CONTENTS XIII

15.8 Derivation of Describing Functions 743 15.9 Stability Analysis by Describing Function Method 749 15.10 Jump Resonance 757

Problems 759

16. ADVANCES IN CONTROL SYSTEMS 765-815

16.1 Introduction 766 16.2 Adaptive Control 767 16.3 Fuzzy Logic Control 783 16.4 Neural Networks 801

Problems 812

APPENDICES

Appendix-I Fourier and Laplace Transforms and Partial Fractions 818

Appendix-II Elements of Matrix Analysis 827

Appendix-Ill MATLAB: Tool for Design and Analysis of Control Systems 836

Appendix-TV Final Value Theorem 848

Appendix-V Proof of a Transformation 849

Appendix-VI Answers to Problems 852

BIBLIOGRAPHY 869-886

INDEX 887-895