1 Introduction 1
1.1 Introduction 1
1.2 Introduction to Control Systems 1
1.3 Definitions 6
1.4 Historical Background 8
1.5 Digital Control Development 12
1.6 Mathematical Background 13
1.7 General Nature of the Engineering Control Problem 15
1.8 Computer Literacy 16
1.9 Outline of Text 16
2 Writing System Equations 19
2.1 Introduction 19
2.2 Electric Circuits and Components 21
2.3 Basic Linear Matrix Algebra 25
2.4 State Concepts 28
2.5 Transfer Function and Block Diagram 34
2.6 Mechanical Translation Systems 35
2.7 Analogous Circuits 41
2.8 Mechanical Rotational Systems 41
2.9 Thermal Systems 46
2.10 Hydraulic Linear Actuator 48
2.11 Liquid-Level System 53
2.12 Rotating Power Amplifiers 54
2.13 DC Servomotor 56
2.14 AC Servomotor 57
2.15 Lagrange's Equation 59
2.16 Summary 63
3 Solution of Differential Equations 64
3.1 Introduction 64
3.2 Standard Inputs to Control Systems 65
3.3 Steady-State Response:Sinusoidal Input 66
3.4 Steady-State Response:Polynomial Input 67
3.5 Transient Response:Classical Method 70
3.6 Definition of Time Constant 73
3.7 Example:Second-Order Svstem—Mechanical 74
3.8 Example:Second-Order System—Electrical 76
3.9 Second-Order Transients 77
3.10 Time-Response Specifications 80
3.11 CAD Accuracy Checks(CADAC) 82
3.12 State-Variable Equations 82
3.13 Characteristic Values 84
3.14 Evaluating the State Transition Matrix 85
3.15 Complete Solution of the State Equation 88
3.16 Summary 89
4 Laplace Transform 91
4.1 Introduction 91
4.2 Definition of the Laplace Transform 92
4.3 Derivation of Laplace Transforms of Simple Functions 92
4.4 Laplace Transform Theorems 94
4.5 CAD Accuracy Checks:CADAC 97
4.6 Application of the Laplace Transform to Differential Equations 97
4.7 Inverse Transformation 98
4.8 Heaviside Partial-Fraction Expansion Theorems 99
4.9 MATLAB Partial-Fraction Example 106
4.10 Partial-Fraction Shortcuts 107
4.11 Graphical Interpretation of Partial-Fraction Coefficients 109
4.12 Frequency Response from the Pole-Zero Diagram 113
4.13 Location of Poles and Stability 116
4.14 Laplace Transform of the Impulse Function 117
4.15 Second-Order System with Impulse Excitation 119
4.16 Additional Matrix Operations and Properties 120
4.17 Solution of State Equation 126
4.18 Evaluation of the Transfer-Function Matrix 128
4.19 Summary 129
5 System Representation 131
5.1 Introduction 131
5.2 Block Diagrams 131
5.3 Determination of the Overall Transfer Function 136
5.4 Standard Block Diagram Terminology 138
5.5 Position Control System 140
5.6 Simulation Diagrams 144
5.7 Signal Flow Graphs 149
5.8 State Transition Signal Flow Graph 154
5.9 Parallel State Diagrams from Transfer Functions 158
5.10 Diagonalizing the A Matrix 160
5.11 Use of State Transformation for the State Equation Solution 168
5.12 Transforming a Matrix with Complex Eigenvalues 169
5.13 Transforming an A Matrix into Companion Form 172
5.14 Summary 175
6.2 Routh's Stability Criterion 176
6 Control-System Characteristics 176
6.1 Introduction 176
6.3 Mathematical and Physical Forms 182
6.4 Feedback System Types 183
6.5 Analysis of System Types 185
6.6 Example:Type 2 System 190
6.7 Steady-State Error Coefficients 192
6.9 Use of Steady-State Error Coefficients 196
6.8 CAD Accuracy Checks:CADAC 196
6.10 Nonunity-Feedback System 198
6.11 Summary 199
7 Root Locus 200
7.1 Introduction 200
7.2 Plotting Roots of a Characteristic Equation 201
7.3 Qualitative Analysis of the Root Locus 204
7.4 Procedure Outline 207
7.5 Open-Loop Transfer Function 208
7.6 Poles of the Control Ratio C(s)/R(s) 209
7.7 Application of the Magnitude and Angle Conditions 211
7.8 Geometrical Properties(Construction Rules) 215
7.9 CAD Accuracy Checks(CADAC) 225
7.10 Examples 225
7.11 Example l:MATLAB Root Locus 231
7.12 Performance Characteristics 234
7.13 Transport Lag 238
7.14 Synthesis 240
7.15 Summary of Root-Locus Construction Rules for Negative Feedback 241
7.16 Summary 242
8 Frequency Response 244
8.1 Introduction 244
8.2 Correlation of the Sinusoidal and Time Responses 245
8.3 Frequency-Response Curves 246
8.4 Bode Plots(Logarithmic Plots) 247
8.5 General Frequency-Transfer-Function Relationships 249
8.6 Drawing the Bode Plots 250
8.7 Example of Drawing a Bode Plot 256
8.8 System Type and Gain as Related to Log Magnitude Curves 259
8.9 CAD Accuracy Check(CADAC) 262
8.10 Experimental Determination of Transfer Functions 262
8.11 Direct Polar Plots 263
8.12 Summary:Direct Polar Plots 269
8.13 Nyquist's Stability Criterion 270
8.14 Examples of Nyquist's Criterion Using Direct Polar Plot 278
8.15 Nyquist's Stability Criterion Applied to Systems Having Dead Time 281
8.16 Definitions of Phase Margin and Gain Margin and Their Relation to Stability 283
8.17 Stability Characteristics of the Log Magnitude and Phase Diagram 285
8.18 Stability from the Nichols Plot(Log Magnitude-Angle Diagram) 286
8.19 Summary 288
9 Closed-Loop Tracking Performance Based on the Frequency Response 290
9.1 Introduction 290
9.2 Direct Polar Plot 291
9.3 Determination of Mm and ωm for a Simple Second-Order System 292
9.4 Correlation of Sinusoidal and Time Responses 295
9.5 Constant M(ω)and α(ω)Contours of C(jω)/R(jω)on the Complex Plane(Direct Plot) 296
9.6 Constant 1/M and α Contours(Unity Feedback)in the Inverse Polar Plane 303
9.7 Gain Adjustment for a Desired Mm of a Unity-Feedback System:Direct Polar Plot 304
9.8 Constant M and α Curves on the Log Magnitude-Angle Diagram(Nichols Chart) 307
9.9 Generation of MATLAB(1992 Student Version)Bode and Nyquist Plots 309
9.10 Adjustment of Gain by Use of the Log Magnitude-Angle Diagram 312
9.11 Correlation of Pole-Zero Diagram with Frequency and Time Responses 312
9.12 Summary 317
10.1 Introduction to Design 319
10 Root-Locus Compensation:Design 319
10.2 Transient Response:Dominant Complex Poles 321
10.3 Additional Significant Poles 326
10.4 Root-Locus Design Considerations 329
10.5 Reshaping the Root Locus 331
10.6 CAD Accuracy Checks(CADAC) 331
10.7 Ideal Integral Cascade Compensation(PI Controller) 332
10.8 Cascade Lag Compensation Design Using Passive Elements 333
10.9 Ideal Derivative Cascade Compensation(PD Controller) 339
10.10 Lead Compensation Design Using Passive Elements 340
10.11 General Lead-Compensator Design 345
10.12 Lag-Lead Cascade Compensation Design 346
10.13 Comparison of Cascade Compensators 349
10.14 PID Controller 352
10.15 Introduction to Feedback Compensation 353
10.16 Feedback Compensation:Design Procedures 355
10.17 Simplified Rate Feedback Compensation:A Design Approach 355
10.18 Design of Rate Feedback 358
10.19 Design:Feedback of Second Derivative of Output 362
10.20 Results of Feedback Compensation Design 364
10.21 Rate Feedback:Plants with Dominant Complex Poles 364
10.22 Summary 365
11 Frequency-Response Compensation Design 367
11.1 Introduction to Feedback Compensation Design 367
11.2 Selection of a Cascade Compensator 369
11.3 Cascade Lag Compensator 372
11.4 Design Example:Cascade Lag Compensation 375
11.5 Lead Compensator 379
11.6 Design Example:Cascade Lead Compensation 381
11.7 Lag-Lead Compensator 385
11.8 Design Example:Cascade Lag-Lead Compensation 387
11.9 Feedback Compensation Design Using Log Plots 390
11.10 Design Example:Feedback Compensation(Log Plots) 392
11.11 Application Guidelines:Basic Minor-Loop Feedback Compensators 397
11.12 Summary 398
12.1 Introduction 401
12 Control-Ratio Modeling 401
12.2 Modeling a Desired Tracking Control Ratio 402
12.3 Guillemin-Truxal Design Procedure 406
12.4 Introduction to Disturbance Rejection 408
12.5 A Second-Order Disturbance-Rejection Model 409
12.6 Disturbance-Rejection Design Principles for SISO Systems 411
12.7 Disturbance-Rejection Design Example 415
12.8 Disturbance-Rejection Models 418
12.9 Summary 422
13 Design:Closed-Loop Pole-Zero Assignment(State-Variable Feedback) 423
13.1 Introduction 423
13.2 Controllability and Observability 424
13.3 State Feedback for SISO Systems 431
13.4 State-Feedback Design for SISO Systems Using the Control Canonical(Phase-Variables)Form 433
13.5 State-Variable Feedback(Physical Variables) 436
13.6 General Properties of State Feedback(Using Phase Variables) 439
13.7 State-Variable Feedback:Steady-State Error Analysis 442
13.8 Use of Steady-State Error Coefficients 444
13.9 State-Variable Feedback:All-Pole Plant 448
13.10 Plants with Complex Poles 451
13.11 Compensator Containing a Zero 451
13.12 State-Variable Feedback:Pole-Zero Plant 453
13.13 Summary 460
14 Parameter Sensitivity and State Space Trajectories 462
14.1 Introduction 462
14.2 Sensitivity 462
14.3 Sensitivity Analysis 466
14.4 Parameter Sensitivity Examples 472
14.5 Inaccessible States 473
14.6 State-Space Trajectories 477
14.7 Linearization(Jacobian Matrix) 488
14.8 Summary 492
15 Digital Control Systems 493
15.1 Introduction 493
15.2 Sampling 494
15.3 Ideal Sampling 496
15.4 z-Transform Theorems 500
15.5 Synthesis in the z Domain(Direct Method) 500
15.6 The Inverse z Transform 503
15.7 Zero-Order Hold 504
15.8 Limitations 506
15.9 Tustin Transformation 507
15.10 Tustin Transformation Properties 509
15.11 Pseudo-Continuous-Time(PCT)Control System(DIG Method) 512
15.12 Analysis of a Basic(Uncompensated)System 514
15.13 Design of Digital Control Systems 519
15.14 Direct(DIR)Design Technique 520
15.15 Lead Controller(Compensator):DIR Design Method 521
15.16 Lag and Lag-Lead Controllers:DIR Design Method 523
15.17 Digitization(DIG)Design Technique 523
15.18 Summary 526
16 Entire Eigenstructure Assignment for Multivariable Systems 527
16.1 Introduction 527
16.2 Effect of Eigenstructure on Time Response 528
16.3 Entire Eigenstructure Assignment 530
16.4 Examples of Entire Eigenstructure Assignment for Regulators 531
16.5 MATLAB Eigenvectors 537
16.6 Uncontrollable Systems 539
16.7 Tracking Systems 541
16.8 Tracking-System Design Example 543
16.9 MATLAB Example of Tracker Design in Sec.16.8 546
16.10 Summary 551
17.1 Introduction 554
17 Design of Tracking Systems Using Output Feedback 554
17.2 Output Feedback Tracking System 555
17.3 Block Diagonalization 557
17.4 Analysis of Closed-Loop System Performance 559
17.5 Design Procedure for Regular Plants 562
17.6 Regular System Design Example 563
17.7 Irregular Plant Characteristics 566
17.8 Irregular System Performance 568
17.9 Design of the Measurement Matrix M 569
17.10 Irregular System Design Example 571
17.11 Tracker Simulation 574
17.12 Summary 578
18 Quantitative Feedback Theory(QFT)Technique 580
18.1 Introduction 580
18.2 Frequency Responses with Parameter Variations 582
18.3 Introduction to the QFT Method(Single-Loop System) 584
18.4 Minimum-Phase System Performance Specifications 586
18.5 Multiple-Input Multiple-Output(MIMO)Uncertain Plants 590
18.6 Plant Templates of P(s),？P(jωi) 592
18.7 U-Contour 595
18.8 Tracking Bounds Lm BR(jω)on the NC 597
18.9 Disturbance Bounds BD(jωi):Case 1[d2(t)=Dou-1(t),d1(t)=0] 600
18.10 Disturbance Bounds BD(jωi):Case 2[d1(t)=Dou-1(t),d2(t)=0] 605
18.11 The Composite Boundary Bo(jωi) 607
18.12 Shaping of Lo(jω) 608
18.13 Guidelines for Shaping Lo(jω) 614
18.14 Design of the Prefilter F(s) 615
18.15 Basic Design Procedure for a MISO System 617
18.16 Design Example 1 618
18.17 Design Example 2 629
18.18 Template Generation for Unstable Plants 630
18.19 Summary 632
Appendixes 636
A Table of Laplace Transform Pairs 636
B.1 Introduction 640
B Interactive Computer-Aided Design Programs for Digital and Continuous Control-System Analysis and Synthesis 640
B.2 Overview of ICECAP-PC and TOTAL-PC 641
B.3 Overview of MATLAB 645
B.4 QFT CAD Packages 647
B.5 Computer-Aided Design Accuracy Checks(CADAC) 647
B.6 Other Computer-Aided Design Packages 649
Problems 651
Answers to Selected Problems 722
Index 739