Modern Control Design

Preface xiii

1. Introduction 1

1.1 What is Control? 1

1.2 Open-Loop and Closed-Loop Control Systems 2

1.3 Other Classifications of Control Systems 6

1.4 On the Road to Control System Analysis and Design 10

1.5 MATLAB, SIMULINK, and the Control System Toolbox 11

References 12

2. Linear Systems and Classical Control 13

2.1 How Valid is the Assumption of Linearity? 13

2.2 Singularity Functions 22

2.3 Frequency Response 26

2.4 Laplace Transform and the Transfer Function 36

2.5 Response to Singularity Functions 51

2.6 Response to Arbitrary Inputs 58

2.7 Performance 62

2.8 Stability 71

2.9 Root-Locus Method 73

2.10 Nyquist Stability Criterion 77

2.11 Robustness 81

2.12 Closed-Loop Compensation Techniques for Single-Input, Single-Output Systems 87

2.12.1 Proportional-integral-derivative compensation 88

2.12.2 Lag, lead, and lead-lag compensation 96

2.13 Multivariable Systems 105

Exercises 115

References 124

3. State-Space Representation 125

3.1 The State-Space: Why Do I Need lt? 125

3.2 Linear Transformation of State-Space Representations 140

3.3 System Characteristics from State-Space Representation 146

3.4 Special State-Space Representations: The Canonical Forms 152

3.5 Block Building in Linear, Time-Invariant State-Space 160

Exercises 168

References 170

4. Solving the State-Equations 171

4.1 Solution of the Linear Time Invariant State Equations 171

4.2 Calculation of the State-Transition Matrix 176

4.3 Understanding the Stability Criteria through the State-Transition Matrix 183

4.4 Numerical Solution of Linear Time-Invariant State-Equations 184

4.5 Numerical Solution of Linear Time-Varying State-Equations 196

4.6 Numerical Solution of Nonlinear State-Equations 204

4.7 Simulating Control System Response with SIMULINK 213

Exercises 216

References 218

5. Control System Design in State-Space 219

5.1 Design: Classical vs. Modern 219

5.2 Controllability 222

5.3 Pole-Placement Design Using Full-State Feedback  228

5.3.1 Pole-placement regulator design for single-input plants  230

5.3.2 Pole-placement regulator design for multi-input plants  245

5.3.3 Pole-placement regulator design for plants with noise  247

5.3.4 Pole-placement design of tracking systems 251

5.4 Observers, Observability, and Compensators 256

5.4.1 Pole-placement design of full-order observers and compensators 258

5.4.2 Pole-placement design of reduced-order observers and compensators 269

5.4.3 Noise and robustness issues  276

Exercises 277

References 282

6. Linear Optimal Control 283

6.1 The Optimal Control Problem 283

6.1.l The general optimal control formulation for regulators  284

6.1.2 Optimal regulator gain matrix and the riccati equation  286

6.2 Infinite-Time Linear Optimal Regulator Design  288

6.3 Optimal Control of Tracking Systems 298

6.4 Output Weighted Linear Optimal Control 308

6.5 Terminal Time Weighting: Solving the Matrix Riccati Equation 312

Exercises 318

References 321

7. Kalman Filters 323

7.1 Stochastic Systems 323

7.2 Filtering of Random Signals  329

7.3 White Noise, and White Noise Filters  334

7.4 The Kalman Filter 339

7.5 Optimal (Linear, Quadratic, Gaussian) Compensators  351

7.6 Robust Multivariable LQG Control: Loop Transfer Recovery  356

Exercises 370

References 371

8. Digital Control Systems 373

8. l What are Digital Systems?   373

8.2 A/D Conversion and the z-Transform  375

8.3 Pulse Transfer Functions of Single-Input, Single-Output Systems  379

8.4 Frequency Response of Single-Input, Single-Output Digital Systems  384

8.5 Stability of Single-Input, Single-Output Digital Systems   386

8.6 Performance of Single-Input, Single-Output Digital Systems   390

8.7  Closed-Loop Compensation Techniques for Single-Input, Single-Output  Digital Systems  393

8.8 State-Space Modeling of Multivariable Digital Systems  396

8.9 Solution of Linear Digital State-Equations 402 

8.10 Design of Multivariable, Digital Control Systems Using Pole-Placement: Regulators, Observers, and Compensators 406

8.11 Linear Optimal Control of Digital Systems 415

8.12 Stochastic Digital Systems, Digital Kalman Filters, and Optimal Digital Compensators 424

Exercises 432

References 436

9. Advanced Topics in Modern Control 437

9.1 Introduction 437

9.2 Hâ Robust, Optimal Control 437

9.3 Structured Singular Value Synthesis for Robust Control 442

9.4 Time-Optimal Control with Pre-shaped Inputs 446

9.5 Output-Rate Weighted Linear Optimal Control 453

9.6 Nonlinear Optimal Control 455

Exercises 463

References 465

Appendix A: Introduction to MATLAB, SIMULINK and the Control System Toolbox 467

Appendix B: Review of Matrices and Linear Algebra 481

Appendix C: Mass, Stiffness, and Control Influence Matrices of the Flexible Spacecraft 487

Answers to Selected, Exercises 489

Index 495