كتاب Introduction to Control Engineering Modeling Analysis and Design

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Introduction to Control Engineering Modeling Analysis and Design

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1.0 Introduction to Control Engineering 1
1.1 The Concept of Feedback and Closed Loop Control 2
1.2 Open-Loop Versus Closed-Loop Systems 2
1.3 Feedforward Control 7
1.4 Feedback Control in Nature 9
1.5 A Glimpse of the Areas where Feedback Control Systems have been
Employed by Man 10
1.6 Classification of Systems 10
1.6.1 Linear System 11
1.6.2 Time-Invariant System 11
1.7 Task of Control Engineers 13
1.8 Alternative Ways to Accomplish a Control Task 14
1.9 A Closer Look to the Control Task 15
1.9.1 Mathematical Modeling 16
1.9.2 Performance Objectives and Design Constraints 17
1.9.3 Controller Design 19
1.9.4 Performance Evaluation 19
2.0 The Laplace Transform 21
2.1 Complex Variables And Complex Functions 21
2.1.1 Complex Function 21
2.2 Laplace Transformation 22
2.2.1 Laplace Transform and Its Existence 23
2.3 Laplace Transform of Common Functions 23
2.3.1 Laplace Table 26
2.4 Properties of Laplace Transform 27
2.5 Inverse Laplace Transformation 31
2.5.1 Partial-Fraction Expansion Method 32
2.5.2 Partial-Fraction Expansion when F(s) has only Distinct Poles 32
2.5.3 Partial-Fraction Expansion of F(s) with Repeated Poles 34
xii CONTENTS
2.6 Concept of Transfer Function 35
2.7 Block Diagrams 36
2.7.1 Block Diagram Reduction 39
2.8 Signal Flow Graph Representation 42
2.8.1 Signal Flow Graphs 42
2.8.2 Properties of Signal Flow Graphs 43
2.8.3 Signal Flow Graph Algebra 43
2.8.4 Representation of Linear Systems by Signal Flow Graph 44
2.8.5 Mason’s Gain Formula 45
2.9 Vectors and Matrices 48
2.9.1 Minors, Cofactors and Adjoint of a Matrix 49
2.10 Inversion of a Nonsingular Matrix 51
2.11 Eigen Values and Eigen Vectors 52
2.12 Similarity Transformation 53
2.12.1 Diagonalization of Matrices 53
2.12.2 Jordan Blocks 54
2.13 Minimal Polynomial Function and Computation of Matrix Function Using
Sylvester’s Interpolation 55
MATLAB Scripts 57
Review Exercise 58
Problems 60
3.1 Introduction 65
3.2 System Representation in State-variable Form 66
3.3 Concepts of Controllability and Observability 69
3.4 Transfer Function from State-variable Representation 73
3.4.1 Computation of Resolvent Matrix from Signal Flow Graph 75
3.5 State Variable Representation from Transfer Function 77
3.6 Solution of State Equation and State Transition Matrix 81
3.6.1 Properties of the State Transition Matrix 82
Review Exercise 83
Problems 85
4.1 Time-Domain Performance of Control Systems 89
4.2 Typical Test Inputs 89
4.2.1 The Step-Function Input 89
4.2.2 The Ramp-Function Input 90
4.2.3 The Impulse-Function Input 90
4.2.4 The Parabolic-Function Input 90
4.3 Transient State and Steady State Response of Analog Control System 91
CONTENTS xiii
4.4 Performance Specification of Linear Systems in Time-Domain 92
4.4.1 Transient Response Specifications 92
4.5 Transient Response of a Prototype Second-order System 93
4.5.1 Locus of Roots for the Second Order Prototype System 94
4.5.1.1 Constant wn Locus 94
4.5.1.2 Constant Damping Ratio Line 94
4.5.1.3 Constant Settling Time 94
4.5.2 Transient Response with Constant wn and Variable δ 95
4.5.2.1 Step Input Response 95
4.6 Impulse Response of a Transfer Function 100
4.7 The Steady-State Error 101
4.7.1 Steady-State Error Caused by Nonlinear Elements 102
4.8 Steady-State Error of Linear Control Systems 102
4.8.1 The Type of Control Systems 103
4.8.2 Steady-State Error of a System with a Step-Function Input 104
4.8.3 Steady-State Error of A System with Ramp-Function Input 105
4.8.4 Steady-State Error of A System with Parabolic-Function Input 106
4.9 Performance Indexes 107
4.9.1 Integral of Squared Error (ISE) 108
4.9.2 Integral of Time Multiplied Squared Error (ITSE) Criteria 108
4.9.3 Integral of Absolute Error (IAE) Criteria 108
4.9.4 Integral of Time Multiplied Absolute Error (ITAE) 109
4.9.5 Quadratic Performance Index 110
4.10 Frequency Domain Response 110
4.10.1 Frequency Response of Closed-Loop Systems 111
4.10.2 Frequency-Domain Specifications 112
4.11 Frequency Domain Parameters of Prototype Second-Order System 112
4.11.1 Peak Resonance and Resonant Frequency 112
4.11.2 Bandwidth 114
4.12 Bode Diagrams 115
4.12.1 Bode Plot 115
4.12.2 Principal Factors of Transfer Function 116
4.13 Procedure for Manual Plotting of Bode Diagram 121
4.14 Minimum Phase and Non-Minimum Phase Systems 122
MATLAB Scripts 123
Review Exercise 125
Problems 126
5.1 The Concept of Stability 131
5.2 The Routh-Hurwitz Stability Criterion 134
5.2.1 Relative Stability Analysis 139
5.2.2 Control System Analysis Using Routh’s Stability Criterion 139
5.3 Stability by the Direct Method of Lyapunov 140
xiv CONTENTS
5.3.1 Introduction to the Direct Method of Lyapunov 140
5.3.2 System Representation 141
5.4 Stability by the Direct Method of Lyapunov 141
5.4.1 Definitions of Stability 143
5.4.2 Lyapunov Stability Theorems 144
5.5 Generation of Lyapunov Functions for Autonomous Systems 147
5.5.1 Generation of Lyapunov Functions for Linear Systems 147
5.6 Estimation of Settling Time Using Lyapunov Functions 150
MATLAB Scripts 153
Review Exercise 154
Problems 155
6.1 Introduction 159
6.1.1 Poles and Zeros of Open Loop and Closed Loop Systems 159
6.1.2 Mapping Contour and the Principle of the Argument 160
6.2 The Nyquist Criterion 165
6.2.1 The Nyquist Path 166
6.2.2 The Nyquist Plot Using a Part of Nyquist Path 175
6.3 Nyquist Plot of Transfer Function with Time Delay 176
6.4 Relative Stability: Gain Margin and Phase Margin 177
6.4.1 Analytical Expression for Phase Margin and Gain Margin
of a Second Order Prototype 182
6.5 Gain-Phase Plot 183
6.5.1 Constant Amplitude (M) and Constant Phase (N) Circle 183
6.6 Nichols Plot 186
6.6.1 Linear System Response Using Graphical User Interface in
MATLAB 188
MATLAB Scripts 188
Review Exercise 189
Problems 190
7.1 Correlation of System-Roots with Transient Response 192
7.2 The Root Locus Diagram–A Time Domain Design Tool 192
7.3 Root Locus Technique 193
7.3.1 Properties of Root Loci 194
7.4 Step by Step Procedure to Draw the Root Locus Diagram 201
7.5 Root Locus Design Using Graphical Interface in MATLAB 211
7.6 Root Locus Technique for Discrete Systems 212
7.7 Sensitivity of the Root Locus 213
MATLAB Scripts 213
Review Exercise 214
Problems 217
CONTENTS xv
8.1 Introduction 218
8.2 Approaches to System Design 218
8.2.1 Structure of the Compensated System 219
8.2.2 Cascade Compensation Networks 220
8.2.3 Design Concept for Lag or Lead Compensator in Frequency-Domain
224
8.2.4 Design Steps for Lag Compensator 226
8.2.5 Design Steps for Lead Compensator 226
8.2.6 Design Examples 226
8.3 Design of Compensator by Root Locus Technique 238
8.3.1 Design of Phase-lead Compensator Using Root Locus Procedure 238
8.3.2 Design of Phase-lag Compensator Using Root Locus Procedure 240
8.4 PID Controller 241
8.4.1 Ziegler-Nichols Rules for Tuning PID Controllers 242
8.4.2 First Method 242
8.4.3 Second Method 243
8.5 Design of Compensators for Discrete Systems 246
8.5.1 Design Steps for Lag Compensator 248
8.5.2 Design Steps for Lead Compensator 248
MATLAB Scripts 249
Review Exercise 252
Problems 253
9.1 Pole Assignment Design and State Estimation 255
9.1.1 Ackerman’s Formula 256
9.1.2 Guidelines for Placement of Closed Loop System Poles 258
9.1.3 Linear Quadratic Regulator Problem 258
9.2 State Estimation 259
9.2.1 Sources of Error in State Estimation 260
9.2.2 Computation of the Observer Parameters 261
9.3 Equivalent Frequency-Domain Compensator 264
9.4 Combined Plant and Observer Dynamics of the Closed Loop System 265
9.5 Incorporation of a Reference Input 266
9.6 Reduced-Order Observer 267
9.7 Some Guidelines for Selecting Closed Loop Poles in Pole Assignment
Design 270
MATLAB Scripts 271
Review Exercise 272
Problems 275
10.0 Why We are Interested in Sampled Data Control System? 276
xvi CONTENTS
10.1 Advantage of Digital Control 276
10.2 Disadvantages 277
10.3 Representation of Sampled Process 278
10.4 The Z-Transform 279
10.4.1 The Residue Method 280
10.4.2 Some Useful Theorems 282
10.5 Inverse Z-Transforms 286
10.5.1 Partial Fraction Method 286
10.5.2 Residue Method 286
10.6 Block Diagram Algebra for Discrete Data System 287
10.7 Limitations of the Z-Transformation Method 292
10.8 Frequency Domain Analysis of Sampling Process 292
10.9 Data Reconstruction 297
10.9.1 Zero Order Hold 299
10.10 First Order Hold 302
10.11 Discrete State Equation 305
10.12 State Equations of Systems with Digital Components 308
10.13 The Solution of Discrete State Equations 308
10.13.1 The Recursive Method 308
10.14 Stability of Discrete Linear Systems 311
10.14.1 Jury’s Stability Test 313
10.15 Steady State Error for Discrete System 316
10.16 State Feedback Design for Discrete Systems 321
10.16.1 Predictor Estimator 321
10.16.2 Current Estimator 322
10.16.3 Reduced-order Estimator for Discrete Systems 325
10.17 Provision for Reference Input 326
MATLAB Scripts 327
Review Exercise 329
Problems 331
11.1 Introduction 333
11.2 Optimal Control Problem 333
11.3 Performance Index 336
11.4 Calculus of Variations 336
11.4.1 Functions and Functionals 337
A. Closeness of Functions 338
B. Increment of a Functional 339
C. The Variation of a Functional 339
11.4.2 The Fundamental Theorem of the Calculus of Variations 342
11.4.3 Extrema of Functionals of a Single Function 343
11.4.3.1 Variational Problems and the Euler Equation 343
11.4.3.2 Extrema of Functionals of n Functions 346
11.4.3.3 Variable End Point Problems 347
CONTENTS xvii
11.4.4 Optimal Control Problem 352
11.4.5 Pontryagin’s Minimum Principle 354
11.5 The LQ Problem 357
11.5.1 The Hamilton-Jacobi Approach 358
11.5.2 The Matrix Riccati Equation 359
11.5.3 Finite Control Horizon 360
11.5.4 Linear Regulator Design (Infinite-time Problem) 362
11.6 Optimal Controller for Discrete System 363
11.6.1 Linear Digital Regulator Design (Infinite-time Problem) 365
MATLAB Scripts 367
Review Exercise 367
Problems 369
12.1 The Concept of Fuzzy Logic and Relevance of Fuzzy Control 371
12.2 Industrial and Commercial Use of Fuzzy Logic-based Systems 373
12.3 Fuzzy Modeling and Control 373
12.3.1 Advantages of Fuzzy Controller 374
12.3.2 When to Use Fuzzy Control 375
12.3.3 Potential Areas of Fuzzy Control 375
12.3.4 Summary of Some Benefits of Fuzzy Logic and Fuzzy Logic Based
Control System 376
12.3.5 When Not to Use Fuzzy Logic 377
12.4 Fuzzy Sets and Membership 377
12.4.1 Introduction to Sets 377
12.4.2 Classical Sets 378
12.4.3 Fuzzy Sets 379
12.5 Basic Definitions of Fuzzy Sets and a Few Terminologies 379
12.5.1 Commonly Used Fuzzy Set Terminologies 381
12.6 Set-Theoretic Operations 384
12.6.1 Classical Operators on Fuzzy Sets 384
12.6.2 Generalized Fuzzy Operators 386
12.6.2.1 Fuzzy Complement 386
12.6.2.2 Fuzzy Union and Intersection 387
12.6.2.3 Fuzzy Intersection: The T-Norm 387
12.6.2.4 Fuzzy Union: The T-Conorm (or S-Norm) 388
12.7 MF Formulation and Parameterization 388
12.7.1 MFs of One Dimension 389
12.8 From Numerical Variables to Linguistic Variables 391
12.8.1 Term Sets of Linguistic Variables 393
12.9 Classical Relations and Fuzzy Relations 394
12.9.1 Cartesian Product 394
12.9.2 Crisp Relations 394
12.9.3 Fuzzy Relations 395
12.9.4 Operation on Fuzzy Relations 396
xviii CONTENTS
12.10 Extension Principle 402
12.11 Logical Arguments and Propositions 403
12.11.1 Logical Arguments 403
12.11.2 Modus Ponens 407
12.11.3 Modus Tollens 407
12.11.4 Hypothetical Syllogism 407
12.12 Interpretations of Fuzzy If-then Rules 407
12.12.1 Fuzzy Relation Equations 409
12.13 Basic Principles of Approximate Reasoning 410
12.13.1 Generalized Modus Ponens 410
12.13.2 Generalized Modus Tollens 410
12.13.4 Generalized Hypothetical Syllogism 411
12.14 Representation of a Set of Rules 411
12.14.1 Approximate Reasoning with Multiple Conditional Rules 413
MATLAB Scripts 416
Problems 417
13.1 The Structure of Fuzzy Logic-based Controller 419
13.1.1 Knowledge Base 420
13.1.2 Rule Base 421
13.1.2.1 Choice of Sate Variables and Controller Variables 421
13.1.3 Contents of Antecedent and Consequent of Rules 422
13.1.4 Derivation of Production Rules 422
13.1.5 Membership Assignment 423
13.1.6 Cardinality of a Term Set 423
13.1.7 Completeness of Rules 423
13.1.8 Consistency of Rules 424
13.2 Inference Engine 424
13.2.1 Special Cases of Fuzzy Singleton 426
13.3 Reasoning Types 427
13.4 Fuzzification Module 428
13.4.1 Fuzzifier and Fuzzy Singleton 428
13.5 Defuzzification Module 429
13.5.1 Defuzzifier 429
13.5.2 Center of Area (or Center of Gravity) Defuzzifier 430
13.5.3 Center Average Defuzzifier (or Weighted Average Method) 431
13.6 Design Consideration of Simple Fuzzy Controllers 432
13.7 Design Parameters of General Fuzzy Controllers 433
13.8 Examples of Fuzzy Control System Design: Inverted Pendulum 434
13.9 Design of Fuzzy Logic Controller on Simulink and MATLAB Environment 441
13.9.1 Iterative Design Procedure of a PID Controller in MATLAB
Environment 441
13.9.2 Simulation of System Dynamics in Simulink for PID Controller
Design 444
13.9.3 Simulation of System Dynamics in Simulink for Fuzzy Logic Controller
Design 446
Problems 449
14.1 Introduction 453
14.1.1 Some Phenomena Peculiar to Nonlinear Systems 454
14.2 Approaches for Analysis of Nonlinear Systems: Linearization 457
14.3 Describing Function Method 458
14.4 Procedure for Computation of Describing Function 459
14.5 Describing Function of Some Typical Nonlinear Devices 460
14.5.1 Describing Function of an Amplifying Device with Dead Zone and
Saturation 460
14.5.2 Describing Function of a Device with Saturation but without any
Dead Zone 463
14.5.3 Describing Function of a Relay with Dead Zone 464
14.5.4 Describing Function of a Relay with Dead Zone and Hysteresis 464
14.5.5 Describing Function of a Relay with Pure Hysteresis 466
14.5.6 Describing Function of Backlash 466
14.6 Stability Analysis of an Autonomous Closed Loop System by Describing
Function 468
14.7 Graphical Analysis of Nonlinear Systems by Phase-Plane Methods 471
14.8 Phase-Plane Construction by the Isocline Method 472
14.9 Pell’s Method of Phase-Trajectory Construction 474
14.10 The Delta Method of Phase-Trajectory Construction 476
14.11 Construction of Phase Trajectories for System with Forcing Functions 477
14.12 Singular Points 477
14.13 The Aizerman and Kalman Conjectures 481
14.13.1 Popov’s Stability Criterion 482
14.13.2 The Generalized Circle Criteria 482
14.13.3 Simplified Circle Criteria 483
14.13.4 Finding Sectors for Typical Nonlinearities 484
14.13.5 S-function SIMULINK Solution of Nonlinear Equations 485
MATLAB Scripts 489
Problems 492
15.1 Introduction 493
15.2 Implementation of Controller Algorithm 493
15.2.1 Realization of Transfer Function 493
CONTENTS xix
15.2.2 Series or Direct Form 1 494
15.2.3 Direct Form 2 (Canonical) 495
15.2.4 Cascade Realization 496
15.2.5 Parallel Realization 497
15.3 Effects of Finite Bit Size on Digital Controller Implementation 500
15.3.1 Sign Magnitude Number System (SMNS) 500
15.3.1.1 Truncation Quantizer 500
15.3.1.2 Round-off Quantizer 500
15.3.1.3 Mean and Variance 502
15.3.1.4 Dynamic Range of SMNS 503
15.3.1.5 Overflow 503
15.3.2 Two’s Complement Number System 504
15.3.2.1 Truncation Operation 504
15.3.2.2 Round-off Quantizer in Two’s CNS 505
15.3.2.3 Mean and Variance 505
15.3.2.4 Dynamic Range for Two’s CNS 506
15.3.2.5 Overflow 506
15.4 Propagation of Quantization Noise Through the Control System 507
15.5 Very High Sampling Frequency Increases Noise 507
15.6 Propagation of ADC Errors and Multiplication Errors through the
Controller 508
15.6.1 Propagated Multiplication Noise in Parallel Realization 508
15.6.2 Propagated Multiplication Noise in Direct Form Realization 510
15.7 Coefficient Errors and Their Influence on Controller Dynamics 511
15.7.1 Sensitivity of Variation of Coefficients of a Second Order
Controller 511
15.8 Word Length in A/D Converters, Memory, Arithmetic Unit and D/A
Converters 512
15.9 Quantization gives Rise to Nonlinear Behavior in Controller 515
15.10 Avoiding the Overflow 517
15.10.1 Pole Zero Pairing 517
15.10.2 Amplitude Scaling for Avoiding Overflow 518
15.10.3 Design Guidelines 518
MATLAB Scripts 519
Problems 520
Appendex A 522
Appendex B 579
Appendex C 585
Notes on MATLAB Use 589
Bibliography 595
Index 601

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