Digital Control Engineering Analysis and Design M. Sami Fadali Antonio Visioli AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Academic Press is an imprint of Elsevier
Contents CHAPTER 1 Introduction to Digital Control 1 1.1 Why Digital Control? 2 1.2 The Structure of a Digital Control System 2 1.3 Examples of Digital Control Systems 3 Resources 6 Problems 7 CHAPTER 2 Discrete-Time Systems 9 2.1 Analog Systems with Piecewise Constant Inputs 9 2.2 Difference Equations 11 2.3 The z-transform 12 2.4 Computer-Aided Design 31 2.5 z-transform Solution of Difference Equations 32 2.6 The Time Response of a Discrete-Time System 33 2.7 The Modified z-transform 38 2.8 Frequency Response of Discrete-Time Systems 40 2.9 The Sampling Theorem 46 Resources 50 Problems 51 Computer Exercises 53 CHAPTER 3 Modeling of Digital Control Systems 55 3.1 ADC Model 55 3.2 DAC Model 56 3.3 The Transfer Function of the ZOH 57 3.4 Effect of Sampler on Transfer Function of a Cascade 58 3.5 Transfer Function for the DAC, Analog Subsystem, ADC Combination 62 3.6 Systems with Transport Lag 70 3.7 The Closed-Loop Transfer Function 72 3.8 Analog Disturbances in a Digital System 75 3.9 Steady-State Error and Error Constants 77
vi Contents 3.10 MATLAB Commands 80 Resources 81 Problems 82 Computer Exercises 85 CHAPTER 4 Stability of Digital Control Systems 87 4.1 Definitions of Stability 87 4.2 Stable z-domain Pole Locations 89 4.3 Stability Conditions 89 4.4 Stability Determination 97 4.5 Jury Test 100 4.6 Nyquist Criterion 104 Resources 118 Problems 118 Computer Exercises 120 CHAPTER 5 Analog Control System Design 123 5.1 The Root Locus 123 5.2 Root Locus Using MATLAB 128 5.3 Design Specifications and the Effect of Gain Variation 128 5.4 Root Locus Design 131 5.5 Empirical Tuning of PID Controllers 154 Resources 158 Problems 158 Computer Exercises 159 CHAPTER 6 Digital Control System Design 161 6.1 z-domain Root Locus 161 6.2 z-domain Digital Control System Design 164 6.3 Digital Implementation of Analog Controller Design 176 6.4 Direct z-domain Digital Controller Design 196 6.5 Frequency Response Design 202 6.6 Direct Control Design 210 6.7 Finite Settling Time Design 215 Resources 225 Problems 225 Computer Exercises 227 CHAPTER 7 State-Space Representation 229 7.1 State Variables 229 7.2 State-Space Representation 232 7.3 Linearization of Nonlinear State Equations 237
Contents vii 7.4 The Solution of Linear State-Space Equations 240 7.5 The Transfer Function Matrix 256 7.6 Discrete-Time State-Space Equations 258 7.7 Solution of Discrete-Time State-Space Equations 261 7.8 z-transfer Function from State-Space Equations 268 7.9 Similarity Transformation 270 Resources 274 Problems 275 Computer Exercises 279 CHAPTER 8 Properties of State-Space Models 28i 8.1 Stability of State-Space Realizations 282 8.2 Controllability and Stabilizability 290 8.3 Observability and Detectability 301 8.4 Poles and Zeros of Multivariable Systems 307 8.5 State-Space Realizations 313 8.6 Duality 326 Resources 327 Problems 328 Computer Exercises 333 CHAPTER 9 State Feedback Control 335 9.1 On State and Output Feedback 335 9.2 Pole Placement 337 9.3 Servo Problem 349 9.4 Invariance of System Zeros 353 9.5 State Estimation 355 9.6 Observer State Feedback 362 9.7 Pole Assignment Using Transfer Functions 370 Resources 374 Problems 374 Computer Exercises 378 CHAPTER 10 Optimal Control 379 10.1 Optimization 379 10.2 Optimal Control 384 10.3 The Linear Quadratic Regulator 389 10.4 Steady-State Quadratic Regulator 399 10.5 Hamiltonian System 406 Resources 409 Problems 410 Computer Exercises 413
viii Contents CHAPTER 11 Elements of Nonlinear Digital Control Systems 415 11.1 Discretization of Nonlinear Systems 415 11.2 Nonlinear Difference Equations 424 11.3 Equilibrium of Nonlinear Discrete-Time System 425 11.4 Lyapunov Stability Theory 426 11.5 Stability of Analog Systems with Digital Control 439 11.6 State Plane Analysis 442 11.7 Discrete-Time Nonlinear Controller Design 447 Resources 452 Problems 452 Computer Exercises 455 CHAPTER 12 Practical Issues 457 12.1 Design of the Hardware and Software Architecture 457 12.2 Choice of the Sampling Period 461 12.3 Controller Structure 468 12.4 PID Control 471 12.5 Sampling Period Switching 481 Resources 494 Problems 494 Computer Exercises 495 APPENDIX I Table of Laplace and Z-Transforms 497 APPENDIX II Properties of the Z-Transform 499 APPENDIX III Review of Linear Algebra 501 A.1 Matrices 501 A.2 Equality of Matrices 502 A.3 Matrix Arithmetic 502 A.4 Determinant of a Matrix 508 A.5 Inverse of a Matrix 509 A.6 Eigenvalues 512 A.7 Eigenvectors 513 A.8 Norm of a Vector 516 A.9 Matrix Norms 517 A.10 Quadratic Forms 518 A.11 Matrix Differentiation/Integration 520 A.12 Kronecker Product 522 Resources 523 Index 525