Introduction to control engineering modeling, analysis and design

Cover
......Page 1
Preface......Page 8
Contents......Page 12
1.0 Introduction to Control Engineering
......Page 22
1.2 Open-Loop Versus Closed-Loop Systems
......Page 23
1.3 Feedforward Control
......Page 28
1.4 Feedback Control in Nature
......Page 30
1.6 Classification of Systems
......Page 31
1.6.2 Time-Invariant System
......Page 32
1.7 Task of Control Engineers
......Page 34
1.8 Alternative Ways to Accomplish a Control Task
......Page 35
1.9 A Closer Look to the Control Task
......Page 36
1.9.1 Mathematical Modeling
......Page 37
1.9.2 Performance Objectives and Design Constraints......Page 38
1.9.4 Performance Evaluation
......Page 40
2.1.1 Complex Function......Page 42
2.2 Laplace Transformation
......Page 43
2.3 Laplace Transform of Common Functions
......Page 44
2.3.1 Laplace Table
......Page 47
2.4 Properties of Laplace Transform
......Page 48
2.5 Inverse Laplace Transformation
......Page 52
2.5.2 Partial-Fraction Expansion when F(s) has only Distinct Poles
......Page 53
2.5.3 Partial-Fraction Expansion of F(s) with Repeated Poles
......Page 55
2.6 Concept of Transfer Function
......Page 56
2.7 Block Diagrams
......Page 57
2.7.1 Block Diagram Reduction
......Page 60
2.8.1 Signal Flow Graphs
......Page 63
2.8.3 Signal Flow Graph Algebra
......Page 64
2.8.4 Representation of Linear Systems by Signal Flow Graph
......Page 65
2.8.5 Mason’s Gain Formula
......Page 66
2.9 Vectors and Matrices
......Page 69
2.9.1 Minors, Cofactors and Adjoint of a Matrix
......Page 70
2.10 Inversion of a Nonsingular Matrix
......Page 72
2.11 Eigen Values and Eigen Vectors
......Page 73
2.12.1 Diagonalization of Matrices
......Page 74
2.12.2 Jordan Blocks
......Page 75
2.13 Minimal Polynomial Function and Computation of Matrixfunction Using Sylvester’s Interpolation
......Page 76
MATLAB Scripts
......Page 78
Review Exercise
......Page 79
Problems
......Page 81
3.1 Introduction
......Page 86
3.2 System Representation in State-Variable Form
......Page 87
3.3 Concepts of Controllability and Observability
......Page 90
3.4 Transfer Function from State-Variable Representation
......Page 94
3.4.1 Computation of Resolvent Matrix from Signal Flow Graph
......Page 96
3.5 State Variable Representation from Transfer Function
......Page 98
3.6 Solution of State Equation and State Transition Matrix
......Page 102
3.6.1 Properties of the State Transition Matrix
......Page 103
Review Exercise
......Page 104
Problems
......Page 106
4.2.1 The Step-Function Input
......Page 110
4.2.4 The Parabolic-Function Input......Page 111
4.3 Transient State and Steady State Response of Analog Control System......Page 112
4.4.1 Transient Response Specifications
......Page 113
4.5 Transient Response of a Prototype Second-Order System
......Page 114
4.5.1.3 Constant Settling Time......Page 115
4.5.2.1 Step Input Response......Page 116
4.6 Impulse Response of a Transfer Function
......Page 121
4.7 The Steady-State Error
......Page 122
4.8 Steady-State Error of Linear Control Systems
......Page 123
4.8.1 The Type of Control Systems......Page 124
4.8.2 Steady-State Error of a System with a Step-Function Input......Page 125
4.8.3 Steady-State Error of a System with Ramp-Function Input......Page 126
4.8.4 Steady-State Error of a System with Parabolic-Function Input......Page 127
4.9 Performance Indexes
......Page 128
4.9.3 Integral of Absolute Error (IAE) Criteria
......Page 129
4.9.4 Integral of Time Multiplied Absolute Error (ITAE)
......Page 130
4.10 Frequency Domain Response
......Page 131
4.10.1 Frequency Response of Closed-Loop Systems
......Page 132
4.11.1 Peak Resonance and Resonant Frequency
......Page 133
4.11.2 Bandwidth
......Page 135
4.12.1 Bode Plot
......Page 136
4.12.2 Principal Factors of Transfer Function
......Page 137
4.13 Procedure for Manual Plotting of Bode Diagram
......Page 142
4.14 Minimum Phase and Non-Minimum Phase Systems
......Page 143
Matlab Scripts
......Page 144
Review Exercise
......Page 146
Problems
......Page 147
5.1 The Concept of Stability
......Page 152
5.2 The Routh-Hurwitz Stability Criterion
......Page 155
5.2.2 Control System Analysis Using Routh’s Stability Criterion
......Page 160
5.3.1 Introduction to the Direct Method of Lyapunov
......Page 161
5.4 Stability by the Direct Method of Lyapunov
......Page 162
5.4.1 Definitions of Stability
......Page 164
5.4.2 Lyapunov Stability Theorems
......Page 165
5.5.1 Generation of Lyapunov Functions for Linear Systems
......Page 168
5.6 Estimation of Settling Time Using Lyapunov Functions
......Page 171
Matlab Scripts
......Page 174
Review Exercise
......Page 175
Problems
......Page 176
6.1.1 Poles and Zeros of Open Loop and Closed Loop Systems
......Page 180
6.1.2 Mapping Contour and the Principle of the Argument
......Page 181
6.2 The Nyquist Criterion
......Page 186
6.2.1 The Nyquist Path
......Page 187
6.2.2 The Nyquist Plot Using a Part of Nyquist Path
......Page 196
6.3 Nyquist Plot of Transfer Function with Time Delay
......Page 197
6.4 Relative Stability: Gain Margin and Phase Margin
......Page 198
6.4.1 Analytical Expression for Phase Margin and Gain Margin of a Second Order Prototype......Page 203
6.5.1 Constant Amplitude (M) and Constant Phase (N) Circle
......Page 204
6.6 Nichols Plot
......Page 207
MATLAB Scripts 
......Page 209
Review Exercise
......Page 210
Problems 
......Page 211
7.2 The Root Locus Diagram—A Time Domain Design Tool
......Page 213
7.3 Root Locus Technique
......Page 214
7.3.1 Properties of Root Loci
......Page 215
7.4 Step by Step Procedure to Draw the Root Locus Diagram
......Page 222
7.5 Root Locus Design Using Graphical Interface in MATLAB......Page 232
7.6 Root Locus Technique for Discrete Systems
......Page 233
MATLAB Scripts 
......Page 234
Review Exercise 
......Page 235
Problems 
......Page 238
8.2 Approaches to System Design
......Page 239
8.2.1 Structure of the Compensated System
......Page 240
8.2.2 Cascade Compensation Networks
......Page 241
8.2.3 Design Concept for Lag or Lead Compensator in Frequency-Domain
......Page 245
8.2.6 Design Examples
......Page 247
8.3.1 Design of Phase-lead Compensator using Root Locus Procedure
......Page 259
8.3.2 Design of Phase-lag Compensator using Root Locus Procedure
......Page 261
8.4 PID Controller......Page 262
8.4.2 First Method
......Page 263
8.4.3 Second Method
......Page 264
8.5 Design of Compensators for Discrete Systems
......Page 267
8.5.2 Design Steps for Lead Compensator
......Page 269
MATLAB Scripts 
......Page 270
Review Exercise
......Page 273
Problems
......Page 274
9.1 Pole Assignment Design and State Estimation
......Page 276
9.1.1 Ackerman’s Formula
......Page 277
9.1.3 Linear Quadratic Regulator Problem
......Page 279
9.2 State Estimation
......Page 280
9.2.1 Sources of Error in State Estimation
......Page 281
9.2.2 Computation of the Observer Parameters
......Page 282
9.3 Equivalent Frequency-Domain Compensator
......Page 285
9.4 Combined Plant and Observer Dynamics of the Closed Loop System......Page 286
9.5 Incorporation of a Reference Input
......Page 287
9.6 Reduced-Order Observer
......Page 288
9.7 Some Guidelines for Selecting Closed Loop Poles in Pole Assignment Design......Page 291
MATLAB Scripts 
......Page 292
Review Exercise
......Page 293
Problems
......Page 296
10.1 Advantage of Digital Control
......Page 297
10.2 Disadvantages
......Page 298
10.3 Representation of Sampled Process
......Page 299
10.4 The Z-Transform
......Page 300
10.4.1 The Residue Method
......Page 301
10.4.2 Some Useful Theorems
......Page 303
10.5.2 Residue Method 
......Page 307
10.6 Block Diagram Algebra For Discrete Data System
......Page 308
10.8 Frequency Domain Analysis of Sampling Process
......Page 313
10.9 Data Reconstruction
......Page 318
10.9.1 Zero Order Hold 
......Page 320
10.10 First Order Hold
......Page 323
10.11 Discrete State Equation
......Page 326
10.13.1 The Recursive Method
......Page 329
10.14 Stability of Discrete Linear Systems
......Page 332
10.14.1 Jury’s Stability Test 
......Page 334
10.15 Steady State Error for Discrete System
......Page 337
10.16.1 Predictor Estimator
......Page 342
10.16.2 Current Estimator
......Page 343
10.16.3 Reduced-order Estimator for Discrete Systems
......Page 346
10.17 Provision for Reference Input
......Page 347
MATLAB Scripts 
......Page 348
Review Exercise
......Page 350
Problems
......Page 352
11.2 Optimal Control Problem
......Page 354
11.4 Calculus of Variations
......Page 357
11.4.1 Functions and Functionals
......Page 358
A. Closeness of Functions
......Page 359
C. The Variation of a Functional
......Page 360
11.4.2 The Fundamental Theorem of the Calculus of Variations
......Page 363
11.4.3.1 Variational Problems and the Euler Equation
......Page 364
11.4.3.2 Extrema of Functionals of n Functions
......Page 367
11.4.3.3 Variable End Point Problems
......Page 368
11.4.4 Optimal Control Problem
......Page 373
11.4.5 Pontryagin’s Minimum Principle
......Page 375
11.5 The LQ Problem 
......Page 378
11.5.1 The Hamilton-Jacobi Approach
......Page 379
11.5.2 The Matrix Riccati Equation
......Page 380
11.5.3 Finite Control Horizon
......Page 381
11.5.4 Linear Regulator Design (Infinite-Time Problem)
......Page 383
11.6 Optimal Controller for Discrete System
......Page 384
11.6.1 Linear Digital Regulator Design (Infinite-Time Problem)
......Page 386
Review Exercise
......Page 388
Problems
......Page 390
12.1 The Concept of Fuzzy Logic and Relevance of Fuzzy Control
......Page 392
12.3 Fuzzy Modeling and Control
......Page 394
12.3.1 Advantages of Fuzzy Controller
......Page 395
12.3.3 Potential Areas of Fuzzy Control
......Page 396
12.3.4 Summary of Some Benefits of Fuzzy Logic and Fuzzy Logic Based Control System......Page 397
12.4.1 Introduction to Sets
......Page 398
12.4.2 Classical Sets
......Page 399
12.5 Basic Definitions of Fuzzy Sets and a Few Terminologies
......Page 400
12.5.1 Commonly Used Fuzzy Set Terminologies
......Page 402
12.6.1 Classical Operators on Fuzzy Sets
......Page 405
12.6.2.1 Fuzzy Complement......Page 407
12.6.2.3 Fuzzy Intersection: The T-Norm......Page 408
12.7 MF Formulation and Parameterization
......Page 409
12.7.1 MFs of One Dimension
......Page 410
12.8 From Numerical Variables to Linguistic Variables
......Page 412
12.8.1 Term Sets of Linguistic Variables
......Page 414
12.9.2 Crisp Relations
......Page 415
12.9.3 Fuzzy Relations
......Page 416
12.9.4 Operation on Fuzzy Relations
......Page 417
12.10 Extension Principle
......Page 423
12.11.1 Logical Arguments
......Page 424
12.12 Interpretations of Fuzzy If-Then Rules
......Page 428
12.12.1 Fuzzy Relation Equations
......Page 430
12.13.2 Generalized Modus Tollens
......Page 431
12.14 Representation of a Set of Rules
......Page 432
12.14.1 Approximate Reasoning with Multiple Conditional Rules......Page 434
MATLAB Scripts 
......Page 437
Problems
......Page 438
13.1 The Structure of Fuzzy Logic-Based Controller
......Page 440
13.1.1 Knowledge Base
......Page 441
13.1.2.1 Choice of Sate Variables and Controller Variables
......Page 442
13.1.4 Derivation of Production Rules
......Page 443
13.1.7 Completeness of Rules
......Page 444
13.2 Inference Engine
......Page 445
13.2.1 Special Cases of Fuzzy Singleton
......Page 447
13.3 Reasoning Types
......Page 448
13.4.1 Fuzzifier and Fuzzy Singleton
......Page 449
13.5.1 Defuzzifier
......Page 450
13.5.2 Center of Area ( or Center of Gravity) Defuzzifier
......Page 451
13.5.3 Center Average Defuzzifier (or Weighted Average Method)
......Page 452
3.6 Design Consideration of Simple Fuzzy Controllers
......Page 453
13.7 Design Parameters of General Fuzzy Controllers
......Page 454
13.8 Examples of Fuzzy Control System Design : Inverted Pendulum
......Page 455
13.9.1 Iterative Design Procedure of a PID Controller In MATLAB Environment
......Page 462
13.9.2 Simulation of System Dynamics in Simulink for PID Controller Design
......Page 465
13.9.3 Simulation of System Dynamics In Simulink for Fuzzy Logic Controller Design
......Page 467
Problems
......Page 470
14.1 Introduction
......Page 474
14.1.1 Some Phenomena Peculiar to Nonlinear Systems
......Page 475
14.2 Approaches for Analysis of Nonlinear Systems:Linearization
......Page 478
14.3 Describing Function Method
......Page 479
14.4 Procedure for Computation of Describing Function
......Page 480
14.5.1 Describing Function of an Amplifying Device with Dead Zone and Saturation
......Page 481
14.5.2 Describing Function of a Device with Saturation but without any Dead Zone
......Page 484
14.5.4 Describing Function of a Relay with Dead Zone and Hysteresis
......Page 485
14.5.6 Describing Function of Backlash
......Page 487
14.6 Stability Analysis of an Autonomous Closed Loop System by Describing Function......Page 489
14.7 Graphical Analysis of Nonlinear Systems by Phase-Planemethods
......Page 492
14.8 Phase-Plane Construction by the Isocline Method
......Page 493
14.9 Pell’s Method of Phase-Trajectory Construction
......Page 495
14.10 The Delta Method of Phase-Trajectory Construction
......Page 497
14.12 Singular Points
......Page 498
14.13 The Aizerman and Kalman Conjectures
......Page 502
14.13.2 The Generalized Circle Criteria
......Page 503
14.13.3 Simplified Circle Criteria
......Page 504
14.13.4 Finding Sectors for Typical Nonlinearities
......Page 505
14.13.5 S-function Simulink solution of Nonlinear Equations 
......Page 506
MATLAB Scripts 
......Page 510
Problems 
......Page 513
15.2.1 Realization of Transfer Function
......Page 514
15.2.2 Series or Direct Form 1
......Page 515
15.2.3 Direct Form 2 (Canonical)
......Page 516
15.2.4 Cascade Realization
......Page 517
15.2.5 Parallel Realization
......Page 518
15.3.1.2 Round-Off Quantizer......Page 521
15.3.1.3 Mean and Variance......Page 523
15.3.1.5 Overflow
......Page 524
15.3.2.1 Truncation Operation
......Page 525
15.3.2.3 Mean and Variance
......Page 526
15.3.2.5 Overflow
......Page 527
15.5 Very High Sampling Frequency Increases Noise
......Page 528
15.6.1 Propagated Multiplication Noise in Parallel Realization
......Page 529
15.6.2 Propagated Multiplication Noise in Direct form Realization
......Page 531
15.7.1 Sensitivity of Variation of Coefficients of a Second Order Controller
......Page 532
15.8 Word Length in A/D Converters, Memory, Arithmetic Unit and D/A Converters......Page 533
15.9 Quantization Gives Rise to Non-Linear Behavior in Controller
......Page 536
15.10.1 Pole Zero Pairing
......Page 538
15.10.3 Design Guidelines
......Page 539
MATLAB Scripts 
......Page 540
Problems 
......Page 541
Appendix A : Mathematical Models of Some Representative Components and Systems
......Page 543
Appendix B : Mapping from S to Z Domain
......Page 600
Appendix C : Discretization of Analog Controllers
......Page 606
Notes on MATLAB Use 
......Page 610
Bibliography
......Page 616
Index
......Page 622