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A Mathematical Introduction to Control Theory
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Contents Preface 1. Mathematical Preliminaries vii 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 2.1 Transfer Functions . 2.2 The Frequency Response of a System 2.3 Bode Plots . 2.4 The Time Response of Certain "Typical" Systems 2.4.1 First Order Systems .. 2.4.2 Second Order Systems. xi 35 35 37 40 42 43 44 3. Feedback-An Introduction 4. The Routh-Hurwitz Criterion xii 2.5 2.6 2.7 2.8 3.1 3.2 3.3 3.4 3.5 3.6 3.7 4.1 4.2 4.3 A Mathematical Introduction to Control Theory Three Important Devices and Their Transfer Functions 2.5.1 The Operational Amplifier (op amp) . 2.5.2 The DC Motor . 2.5.3 The "Simple Satellite" . Block Diagrams and How to Manipulate Them A Final Example. Exercises . Why Feedback-A First View Sensitivity . More about Sensitivity A Simple Example ... System Behavior at DC Noise Rejection. Exercises . Proof and Applications A Design Example Exercises . 46 46 49 50 51 54 57 61 61 62 64 65 66 70 71 75 75 84 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 111 5.10 Relative Stability. . . . . . . . . . . . . . 113 5.11 The Bode Plots 118 5.12 An (Approximate) Connection between Frequency Speci- fications 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-Pale-Zero Cancellation 159 6.4 More on the Behavior of the Roots of Q(s)j K + P(s) = 0 161 6.5 Exercises........................... 163 7. Compensation 7.1 Compensation-An Introduction 7.2 The Attenuator . 7.3 Phase-Lag Compensation 7.4 Phase-Lead Compensation 7.5 Lag-lead Compensation 7.6 The PID Controller ... 7.7 An Extended Example .. 7.7.1 The Attenuator 7.7.2 The Phase-Lag Compensator 7.7.3 The Phase-Lead Compensator 7.7.4 The Lag-Lead Compensator 7.7.5 The PD Controller. 7.8 Exercises . 167 167 167 168 175 180 181 188 189 189 191 193 195 196 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 Control- lable. . . . . . . . . . . . . . . . . . . . . 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 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 10.17 10.18 10.19 10.20 10.21 10.22 10.23 10.24 Contents Introduction . . . . . . . . . The Definition of the Z-Transform Some Examples . Properties of the Z-Transform Sampled-data Systems . . . . . The Sample-and-Hold Element The Delta FUnction and its Laplace Transform The Ideal Sampler . . . . . . . . . . . . . The Zero-Order Hold . Calculating the Pulse Transfer FUnction . . . Using MATLAB to Perform the Calculations. The Transfer FUnction of a Discrete-Time System Adding a Digital Compensator ... Stability of Discrete-Time Systems . A Condition for Stability The Frequency Response . A Bit about Aliasing. . . . . . . . . The Behavior of the System in the Steady-State The Bilinear Transform . . . . . . . . . . . . . . The Behavior of the Bilinear Transform as T ---- O. Digital Compensators . . . . . . . . . . . . . . When Is There No Pulse Transfer FUnction? An Introduction to the Modified Z-Transform Exercises . xv 251 251 252 253 257 258 260 261 261 262 266 268 269 271 273 276 278 278 279 284 285 288 289 291 11. Answers to Selected Exercises 11.1 Chapter 1 . . . . . . 11.1.1 Problem 1 11.1.2 Problem 3 11.1.3 Problem 5 11.1.4 Problem 7 11.2 Chapter 2 . 11.2.1 Problem 1 11.2.2 Problem 3 11.2.3 Problem 5 11.2.4 Problem 7 11.3 Chapter 3 . 11.3.1 Problem 1 11.3.2 Problem 3 295 295 295 296 297 298 298 298 299 300 301 303 303 304 xvi A Mathematical Introduction to Control Theory 11.3.3 Problem 5 11.3.4 Problem 7 11.4 Chapter 4 . . . . . . 11.4.1 Problem 1 11.4.2 Problem 3 11.4.3 Problem 5 11.4.4 Problem 7 11.4.5 Problem 9 11.5 Chapter 5 . 11.5.1 Problem 1 11.5.2 Problem 3 11.5.3 Problem 5 11.5.4 Problem 7 11.5.5 Problem 9 11.5.6 Problem 11 . 11.6 Chapter 6 . . . . . . 11.6.1 Problem 1 11.6.2 Problem 3 11.6.3 Problem 5 11.6.4 Problem 7 11.6.5 Problem 9 11.7 Chapter 7 . . . . . . 11.7.1 Problem 1 11.7.2 Problem 3 11.7.3 Problem 5 11.7.4 Problem 7 11.7.5 Problem 9 11.8 Chapter 8 . 11.8.1 Problem 1 11.8.2 Problem 3 11.8.3 Problem 5 11.8.4 Problem 7 11.9 Chapter 9 . 11.9.1 Problem 6 11.9.2 Problem 7 11.10 Chapter 10 ..... 11.10.1 Problem 4 11.10.2 Problem 10 . 11.10.3 Problem 13 . 304 305 305 305 306 307 307 309 310 310 311 311 312 314 315 316 316 316 318 319 320 322 322 324 326 327 330 332 332 335 336 337 337 337 338 339 339 339 340 Contents 11.10.4 Problem 16 . 11.10.5 Problem 17 . 11.10.6 Problem 19 . Bibliography Index xvii 342 343 343 345 347 |
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