1. Mathematical Preliminaries 1
1.1 An Introduction to the Laplace Transform 1
1.2 Properties of the Laplace Transform 2
1.3 Finding the Inverse Laplace Transform 15
1.3.1 Some Simple Inverse Transforms 16
1.3.2 The Quadratic Denominator 18
1.4 Integro-Differential Equations 20
1.5 An Introduction to Stability 25
1.5.1 Some Preliminary Manipulations 25
1.5.2 Stability 26
1.5.3 Why We Obsess about Stability 28
1.5.4 The Tacoma Narrows Bridge—a Brief Case History 29
1.6 MATLAB 29
1.6.1 Assignments 29
1.6.2 Commands 31
1.7 Exercises 32
2. Transfer Functions 35
2.1 Transfer Functions 35
2.2 The Frequency Response of a System 37
2.3 Bode Plots 40
2.4 The Time Response of Certain "Typical" Systems . . . . 42
2.4.1 First Order Systems 43
2.4.2 Second Order Systems 44
xi
xii A Mathematical Introduction to Control Theory
2.5 Three Important Devices and Their Transfer Functions . 46
2.5.1 The Operational Amplifier (op amp) 46
2.5.2 The DC Motor 49
2.5.3 The "Simple Satellite" 50
2.6 Block Diagrams and How to Manipulate Them 51
2.7 A Final Example 54
2.8 Exercises 57
3. Feedback—An Introduction 61
3.1 Why Feedback—A First View 61
3.2 Sensitivity 62
3.3 More about Sensitivity 64
3.4 A Simple Example 65
3.5 System Behavior at DC 66
3.6 Noise Rejection 70
3.7 Exercises 71
4. The Routh-Hurwitz Criterion 75
4.1 Proof and Applications 75
4.2 A Design Example 84
4.3 Exercises 87
5. The Principle of the Argument and Its Consequences 91
5.1 More about Poles in the Right Half Plane 91
5.2 The Principle of the Argument 92
5.3 The Proof of the Principle of the Argument 93
5.4 How are Encirclements Measured? 95
5.5 First Applications to Control Theory 98
5.6 Systems with Low-Pass Open-Loop Transfer Functions . 100
5.7 MATLAB and Nyquist Plots 106
5.8 The Nyquist Plot and Delays 107
5.9 Delays and the Routh-Hurwitz Criterion Ill
5.10 Relative Stability 113
5.11 The Bode Plots 118
5.12 An (Approximate) Connection between Frequency Specifications
and Time Specification 119
5.13 Some More Examples 122
5.14 Exercises 126
Contents xiii
6. The Root Locus Diagram 131
6.1 The Root Locus—An Introduction 131
6.2 Rules for Plotting the Root Locus 133
6.2.1 The Symmetry of the Root Locus 133
6.2.2 Branches on the Real Axis 134
6.2.3 The Asymptotic Behavior of the Branches . . . 135
6.2.4 Departure of Branches from the Real Axis . . . 138
6.2.5 A "Conservation Law" 143
6.2.6 The Behavior of Branches as They Leave Finite
Poles or Enter Finite Zeros 144
6.2.7 A Group of Poles and Zeros Near the Origin . . 145
6.3 Some (Semi-)Practical Examples 147
6.3.1 The Effect of Zeros in the Right Half-Plane . . . 147
6.3.2 The Effect of Three Poles at the Origin 148
6.3.3 The Effect of Two Poles at the Origin 150
6.3.4 Variations on Our Theme 150
6.3.5 The Effect of a Delay on the Root Locus Plot . 153
6.3.6 The Phase-lock Loop 156
6.3.7 Sounding a Cautionary Note—Pole-Zero
Cancellation 159
6.4 More on the Behavior of the Roots of Q(s)/K + P(s) = 0 161
6.5 Exercises 163
7. Compensation 167
7.1 Compensation—An Introduction 167
7.2 The Attenuator 167
7.3 Phase-Lag Compensation 168
7.4 Phase-Lead Compensation 175
7.5 Lag-lead Compensation 180
7.6 The PID Controller 181
7.7 An Extended Example 188
7.7.1 The Attenuator 189
7.7.2 The Phase-Lag Compensator 189
7.7.3 The Phase-Lead Compensator 191
7.7.4 The Lag-Lead Compensator 193
7.7.5 The PD Controller 195
7.8 Exercises 196
xiv A Mathematical Introduction to Control Theory
8. Some Nonlinear Control Theory 203
8.1 Introduction 203
8.2 The Describing Function Technique 204
8.2.1 The Describing Function Concept 204
8.2.2 Predicting Limit Cycles 207
8.2.3 The Stability of Limit Cycles 208
8.2.4 More Examples 211
8.2.4.1 A Nonlinear Oscillator 211
8.2.4.2 A Comparator with a Dead Zone 212
8.2.4.3 A Simple Quantizer 213
8.2.5 Graphical Method 214
8.3 Tsypkin's Method 216
8.4 The Tsypkin Locus and the Describing Function Technique 221
8.5 Exercises 223
9. An Introduction to Modern Control 227
9.1 Introduction 227
9.2 The State Variables Formalism 227
9.3 Solving Matrix Differential Equations 229
9.4 The Significance of the Eigenvalues of the Matrix 230
9.5 Understanding Homogeneous Matrix Differential Equations 232
9.6 Understanding Inhomogeneous Equations 233
9.7 The Cayley-Hamilton Theorem 234
9.8 Controllability 235
9.9 Pole Placement 236
9.10 Observability 237
9.11 Examples 238
9.11.1 Pole Placement 238
9.11.2 Adding an Integrator 240
9.11.3 Modern Control Using MATLAB 241
9.11.4 A System that is not Observable 242
9.11.5 A System that is neither Observable nor Controllable
244
9.12 Converting Transfer Functions to State Equations . . . . 245
9.13 Some Technical Results about Series of Matrices 246
9.14 Exercises 248
10. Control of Hybrid Systems 251
Contents xv
10.1 Introduction 251
10.2 The Definition of the Z-Transform 251
10.3 Some Examples 252
10.4 Properties of the Z-Transform 253
10.5 Sampled-data Systems 257
10.6 The Sample-and-Hold Element 258
10.7 The Delta Function and its Laplace Transform 260
10.8 The Ideal Sampler 261
10.9 The Zero-Order Hold 261
10.10 Calculating the Pulse Transfer Function 262
10.11 Using MATLAB to Perform the Calculations 266
10.12 The Transfer Function of a Discrete-Time System . . . . 268
10.13 Adding a Digital Compensator 269
10.14 Stability of Discrete-Time Systems 271
10.15 A Condition for Stability 273
10.16 The Frequency Response 276
10.17 A Bit about Aliasing 278
10.18 The Behavior of the System in the Steady-State 278
10.19 The Bilinear Transform 279
10.20 The Behavior of the Bilinear Transform as T -» 0 284
10.21 Digital Compensators 285
10.22 When Is There No Pulse Transfer Function? 288
10.23 An Introduction to the Modified Z-Transform 289
10.24 Exercises 291
11. Answers to Selected Exercises 295
11.1 Chapter 1 295
11.1.1 Problem 1 295
11.1.2 Problem 3 296
11.1.3 Problem 5 297
11.1.4 Problem 7 298
11.2 Chapter 2 298
11.2.1 Problem 1 298
11.2.2 Problem 3 299
11.2.3 Problem 5 300
11.2.4 Problem 7 301
11.3 Chapter 3 303
11.3.1 Problem 1 303
11.3.2 Problem 3 304
xvi A Mathematical Introduction to Control Theory
11.3.3 Problem 5 304
11.3.4 Problem 7 305
11.4 Chapter 4 305
11.4.1 Problem 1 305
11.4.2 Problem 3 306
11.4.3 Problem 5 307
11.4.4 Problem 7 307
11.4.5 Problem 9 309
11.5 Chapter 5 310
11.5.1 Problem 1 • 310
11.5.2 Problem 3 311
11.5.3 Problem 5 311
11.5.4 Problem 7 312
11.5.5 Problem 9 314
11.5.6 Problem 11 315
11.6 Chapter 6 316
11.6.1 Problem 1 316
11.6.2 Problem 3 316
11.6.3 Problem 5 318
11.6.4 Problem 7 319
11.6.5 Problem 9 320
11.7 Chapter 7 322
11.7.1 Problem 1 322
11.7.2 Problem 3 324
11.7.3 Problem 5 326
11.7.4 Problem 7 327
11.7.5 Problem 9 330
11.8 Chapter 8 332
11.8.1 Problem 1 332
11.8.2 Problem 3 335
11.8.3 Problem 5 336
11.8.4 Problem 7 337
11.9 Chapter 9 337
11.9.1 Problem 6 337
11.9.2 Problem 7 338
11.10 Chapter 10 339
11.10.1 Problem 4 339
11.10.2 Problem 10 339
11.10.3 Problem 13 340
Contents xvii
11.10.4 Problem 16 342
11.10.5 Problem 17 343
11.10.6 Problem 19 343
Bibliography 345
Index 347