Chapter 1 Preliminaries 1
1.1 Why Control? 1
1.2 Feedback 4
1.3 The Scope ofthe Book 6
1.4 Problems 8
Further Readings 9
Chapter 2 The Laplace Transform 10
2.1 Overview 10
2.2 Laplace Transform 11
2.3 Fundamental Transforms 12
2.3.1 The Exponential Function 12
2.3.2 TheStepFunction 14
2.3.3 The Impulse Function 15
2.4 Properties of the Laplace Transform 16
2.4.1 Linearity 17
2.4.2 Laplace Transforms of Derivatives of a Function 17
2.4.3 Laplace Transform of the Integral of a Function 19
2.4.4 Laplace Transform of tf(t) 20
2.5 The Inverse Laplace Transform 21
2.6 The Remaining Transforms 25
2.6.1 Transform of te-at 25
2.6.2 Complex Roots 25
2.7 Additional Properties 30
2.7.1 Final ValueTheorem 30
2.7.2 Initial Value Theorem 31
2.7.3 Time Delay 31
2.7.4 Convolution 32
2.8 Reprise 34
2.9 Problems 34
Further Readings 38
Chapter 3 The Transfer Function 39
3.1 Overview 39
3.2 TheTransferFunction 40
3.3 Transfer Function of the dc Motor 45
3.4 Transfer Function of a Brushless dc Motor 48
3.5.3 Step Response of Simple Second-Order System 50
3.5.2 Simplified Transfer Function 50
3.5 Finding Transfer Functions Experimentally 50
3.5.1 Introduction 50
3.6 Implementing Transfer Functions 57
3.6.1 Passive Implementations 57
3.6.2 Operational Amplifiers 59
3.6.3 Implementing Transfer Functions with Opamps 62
3.7 Reprise 70
3.8 Problems 70
Further Readings 75
4.1 Overview 76
Chapter 4 Introducing Feedback 76
4.2 Basic Formulation 77
4.3 Routh Criterion 82
4.4 Transient Behavior and Steady State Error 88
4.5 Reprise 91
4.6 Problems 91
Further Readings 96
5.2 Exchanging Algebra for Geometry 97
Chapter 5 Root Locus Analysis 97
5.1 Overview 97
5.2.1 Polar Formulation 98
5.2.2 Graphical Representation 99
5.3 Rules of Root Locus 105
5.4 More Examples 119
5.5 Negative Gain(Complementary)Root Locus 127
5.6 Polynomial Factorization 129
5.7 Reprise 131
5.8 Problems 133
Further Readings 136
Chapter 6 Quantifying Performance 137
6.1 Overview 137
6.2 Normalized Second-Order System 138
6.3 Step Response of Tu2 140
6.4 Figures of Merit 143
6.4.2 Time to Peaktp 144
6.4.3 Percent Overshoot 144
6.4.1 Period of Oscillation Td 144
6.4.4 Settling Time 146
6.4.5 Rise Time 146
6.5 Figures of Merit—A Discussion 147
6.6 Steady State Accuracy 150
6.6.1 Closed-Loop Fornulation 150
6.6.2 Unity Feedback Formulation 152
6.7 Reprise 157
6.8 Problems 158
Further Readings 163
Chapter 7 Cascade Root Locus Design 164
7.1 Overview 164
7.2 Proportional Plus Derivative Compensation 165
7.3 Cascade Lead Compensation 170
7.4 Proportional Plus Integral Compensation 181
7.5 Lag Compensation 186
7.6 PID and Lead/Lag Compensation 187
7.7 Robust Stability 192
7.8 Reprise 197
7.9 Problems 198
Further Readings 211
Chapter 8 Motor Speed Control:A Case Study 212
8.1 Overview 212
8.2 A New Identification Procedure 213
8.3.1 Data Measurement 223
8.3.2 Transfer Function without Cylinder Attached 223
8.3 Identification of dc Motor without Cylinder 223
8.3.3 TransferFunctionwith Cylinder Attached 227
8.4 Compensator Design and Implementation 233
8.5 Integral Control 233
8.6 PI Control 237
8.7 Reprise 238
8.8 Problems 239
Further Readings 241
9.1 Overview 242
9.2 Steady State Response to Sinusoidal Inputs 242
Chapter 9 Frequency Response 242
9.3 Bode P1ots 245
9.3.1 Time Constant Form of G(s) 245
9.3.2 SimplePole 247
9.3.3 Simple Zero 249
9.3.4 Qualitative Vector Analysis 249
9.3.5 Composite Asymptotic Magnitude Plots 249
9.3.6 Bode Plot of Repeated Poles 254
9.3.7 Phase and Magnitude Plots of Complex Poles 256
9.4 Transfer Function Identification 261
9.5 The Effects of Feedback 265
9.5.1 The Effect of Feedback on Bandwidth 266
9.6 Sensitivity Analysis 267
9.7 Disturbance Rejection 270
9.7.1 Disturbances at the Output 270
9.8 Disturbance at the Input 278
9.9 Reprise 279
9.10 Problems 279
Further Readings 284
Chapter 10 Nyquist Criterion 285
10.1 Overview 285
10.2 The Nyquist Equation 286
10.3 The Nyquist Criterion 288
10.4 Evaluation of GH Along Ω 290
10.4.1 NoPoles on the lmaginary Axis 290
10.4.2 Poles on the Imaginary Axis 298
10.5 Gain Margin and Phase Margin 310
10.6 The Log Magnitude Plot 312
10.6.1 ConstantMLoci 313
10.6.2 Loci of Constant Phase 315
10.6.3 The Nichols Chart 316
10.7 Reprise 318
10.8 Problems 319
Further Readings 325
Chapter 11 Bode Design 326
11.1 Overview 326
11.2 Figures of Merit 326
11.3 Lag Compensator Design 331
11.4 Lead Compensator Design 337
11.5 Lead/Lag Design Procedure 344
11.6 General Design Procedure 345
11.7 Closed-Loop Analysis 360
11.8 Reprise 362
11.9 Problems 363
Further Readings 380
Chapter 12 Robust Control 381
12.1 Overview 381
12.2 Norms Without Tears 382
12.3 Norms for Robust Control 385
12.3.1 Overview 385
12.4 Sensitivity Functions 386
12.5 Robust Stability Margins 388
12.6 Disturbance Rejection 389
12.7 Noise Rejection 392
12.8 Unmodeled Plant Dynamics and Noise Suppression 394
12.9 Combining Uncertainties 395
12.10 Connections to the Past 398
12.11 Reprise 403
12.12 Problems 403
Further Readings 412
Chapter 13 Position Control:A Case Study 413
13.1 Overview 413
13.2 Model Identification 415
13.2.1 Frequency Analysis Identification 415
13.2.2 Ideal Impulse Identification 417
13.2.3 Finite Width Pulse Identiffcation 418
13.3 Pulse Identification of a Transfer Function 421
13.3.1 Turntable DataAnalysis 421
13.3.2 Finding the Third Pole 424
13.4 Lead Compensation 434
13.5 Reprise 436
13.6 Problems 437
Further Readings 438
Chapter 14 Discrete Systems 439
14.1 Overview 439
14.2 The Ideal Sampler 440
14.3 The Laplace Transform of x*(t) 441
14.4 The ？ Transform ofx(t) 442
14.5 ？ Transforms Useful in Control 443
14.5.1 The Discrete Delta Function 444
14.5.2 The Discrete Step Function 444
14.5.3 Discrete Form of e-at 445
14.5.4 DiscreteForm of x(t)=tl(t) 446
14.5.5 Discrete Fom of x(t)=A cos(ωt+φ) 446
14.6 Altemative Representation 447
14.7 Important Theorems 448
14.7.1 Linearity 449
14.7.2 Right Shifting Property 449
14.7.3 Left Shifting Property 450
14.7.4 Final Value Theorem 451
14.8 Transfer Function in the z Domain 451
14.9 The Inverse ？ Transform 452
14.10 The Solution of Difference Equations 456
14.11 Convolution versus Multiplication 458
14.12 Frequency Response 459
14.13 The Mapping esT 461
14.14 The Primary StriP 462
14.15 The Starring Rule 466
14.16 Sampled Data Systems 467
14.17 Finding G′p(z) 470
14.18 Nyquist in the ？ Plane 473
14.19 Reprise 476
14.20 Problems 477
Further Readings 481
Chapter 15 Digital Control 482
15.1 Overview 482
15.2 Important Mappings 482
15.2.1 Lines of Constant Damping Ratio 483
15.2.2 Curves of Constantωn 484
15.2.3 The Bilinear Mapping 487
15.2.4 The Ad Hoc(Pole/Zero)Mapping 490
15.3 Designinthe s Plane 491
15.3.1 Root Locus Design 492
15.3.2 Bode Design 497
15.4 Design in the z Plane 504
15.4.1 Direct Design 504
15.4.2 Root Locus Design in the z Plane 513
15.5 Reprise 519
15.6 Problems 519
Chapter 16 Aircraft Pitch Control:A Case Study 531
16.1 Overview 531
16.2 Plant Modeling and Identification 532
16.2.1 Plant Parameter Identification 534
16.2.2 Data Collection 536
16.2.3 Analyzing the Data 538
16.3 Control Design 540
16.3.1 TransferringPlanttothe z Plane 540
16.4 Laboratory Setup 542
16.5 Reprise 546
16.6 Problems 546
Further Readings 547
Chapter 17 The Transportation Lag 548
17.1 Overview 548
17.2 Transportation Lag in a Continuous System 549
17.3 Approximations to the Transportation Lag 554
17.4 Compensator Design for Continuous Systems 555
17.4.1 Root Locus Design 555
17.5 Sampled Data Systems 565
17.5.1 Bode s Plane Design 565
17.5.2 Design in the z Plane 568
17.6 Reprise 573
17.7 Problems 574
Further Readings 583
Chapter 18 The State Model 584
18.1 Overview 584
18.2 Relation to the Transfer Function 585
18.3 The Transition Matrix 586
18.4 Phase Portraits 590
18.4.1 Real Distinct Roots 591
18.4.2 Isoclines 596
18.4.3 Complex Eigenvalues 602
18.4.4 Repeated Real Roots 609
18.5 State Models of Higher Dimension 612
18.5.1 Distinct Roots 613
18.5.2 Repeated Eigenvalues 616
18.6 The Jordan Canonical Form 623
18.7 Invariance of Eigenvalues 630
18.8 Reprise 631
18.9 Problems 632
Further Readings 637
Chapter 19 Observability and Controllability 638
19.1 Overview 638
19.2 Discretizing the State Model 639
19.3 Equivalent Transfer Function 644
19.4 The Regulator Problem 647
19.5 Controllability 652
19.5.1 Definitions 652
19.5.2 Controllability Theorems 653
19.6 Controllable Canonical Form 657
19.7 Observability 666
19.8 Observable Canonical Form 671
19.9 Reprise 672
19.10 Problems 673
Further Readings 680
20.1 Introduction 682
20.2 Prediction Observer 682
Chapter 20 The Control ler/Observer 682
20.3 Separation Theorem 690
20.4 Current Observer 691
20.5 Equivalent Transfer Function 696
20.6 Reference Input 697
20.7 Equivalent Tc(z) 698
20.7.1 Model Following Control 708
20.8 Reprise 714
20.9 Problems 714
Index 721