Modeling and Control of Engineering Systems

Cover......Page 1
Title Page......Page 4
Copyright......Page 5
Contents......Page 8
Preface......Page 20
Acknowledgments......Page 24
Author......Page 26
Further Reading......Page 28
Units and Conversions (Approximate)......Page 30
1.1 Control Engineering......Page 32
1.2 Application Areas......Page 33
1.3 Importance of Modeling......Page 35
1.4 History of Control Engineering......Page 36
1.5 Organization of the Book......Page 38
Problems......Page 40
2.1 Dynamic Systems......Page 42
2.2 Dynamic Models......Page 43
2.2.2 Model Types......Page 44
2.2.3 Types of Analytical Models......Page 45
2.2.4 Principle of Superposition......Page 46
2.2.5 Lumped Model of a Distributed System......Page 47
2.3.2 Mechanical Elements......Page 51
2.3.3 Electrical Elements......Page 54
2.3.4 Fluid Elements......Page 57
2.3.5 Thermal Elements......Page 63
2.3.6 Natural Oscillations......Page 68
2.4 Analytical Model Development......Page 70
2.4.3 State-Space Models......Page 71
2.4.4 Time-Invariant Systems......Page 76
2.4.5 Systematic Steps for State Model Development......Page 78
2.4.6 I/O Models from State-Space Models......Page 81
Problems......Page 84
3.1 Model Linearization......Page 94
3.1.1 Linearization about an Operating Point......Page 95
3.2 Nonlinear State-Space Models......Page 97
3.2.1 Linearization......Page 98
3.2.2 Reduction of System Nonlinearities......Page 99
3.3.1 Capacitor......Page 116
3.3.3 Resistor......Page 117
3.4.1 Torque-Speed Curves of Motors......Page 118
3.4.2 Linear Models for Motor Control......Page 119
Problems......Page 120
4.1.2 Sign Convention......Page 128
4.2.1 Single-Port Elements......Page 131
4.2.2 Two-Port Elements......Page 133
4.3.1 Compatibility (Loop) Equations......Page 138
4.3.2 Continuity (Node) Equations......Page 140
4.4 State Models from Linear Graphs......Page 141
4.4.2 Sign Convention......Page 142
4.4.4 General Observation......Page 143
4.4.5 Topological Result......Page 144
4.5.1 Amplifiers......Page 159
4.5.2 DC Motor......Page 161
4.5.3 Linear Graphs of Thermal Systems......Page 165
Problems......Page 169
5.1 Laplace and Fourier Transforms......Page 180
5.1.2 Laplace Transform of a Derivative......Page 181
5.1.4 Fourier Transform......Page 182
5.2 Transfer-Function......Page 183
5.2.1 Transfer-Function Matrix......Page 184
5.3.1 Frequency Transfer-Function (Frequency Response Function)......Page 190
5.3.2 Bode Diagram (Bode Plot) and Nyquist Diagram......Page 192
5.4.1 Significance of Transfer-Functions in Mechanical Systems......Page 193
5.4.2 Mechanical Transfer-Functions......Page 194
5.4.3 Interconnection Laws......Page 195
5.4.4 Transfer-Functions of Basic Elements......Page 197
5.4.5 Transmissibility Function......Page 201
5.5 Equivalent Circuits and Linear Graph Reduction......Page 206
5.5.1 Thevenin’s Theorem for Electrical Circuits......Page 207
5.5.2 Mechanical Circuit Analysis Using Linear Graphs......Page 210
5.5.3 Summary of Thevenin Approach for Mechanical Circuits......Page 219
5.6 Block Diagrams and State-Space Models......Page 220
5.6.2 Principle of Superposition......Page 222
5.6.3 Causality and Physical Realizability......Page 238
Problems......Page 239
6.1 Analytical Solution......Page 248
6.1.1 Homogeneous Solution......Page 249
6.1.3 Impulse Response Function......Page 250
6.1.4 Stability......Page 253
6.2 First-Order Systems......Page 254
6.3.1 Free Response of an Undamped Oscillator......Page 256
6.3.2 Free Response of a Damped Oscillator......Page 258
6.4.1 Impulse Response......Page 263
6.4.3 Step Response......Page 265
6.4.4 Response to Harmonic Excitation......Page 267
6.5 Response Using Laplace Transform......Page 272
6.5.1 Step Response Using Laplace Transforms......Page 273
6.5.2 Incorporation of ICs......Page 274
6.6 Determination of ICs for Step Response......Page 276
6.7 Computer Simulation......Page 284
6.7.1 Use of Simulink® in Computer Simulation......Page 285
Problems......Page 291
7.1 Control System Structure......Page 302
7.1.2 Feedforward Control......Page 303
7.1.3 Terminology......Page 307
7.1.4 Programmable Logic Controllers (PLCs)......Page 308
7.1.5 Distributed Control......Page 311
7.1.6 Hierarchical Control......Page 314
7.2 Control System Performance......Page 316
7.2.1 Performance Specification in Time-Domain......Page 317
7.2.2 Simple Oscillator Model......Page 319
7.3 Control Schemes......Page 322
7.3.1 Feedback Control with PID Action......Page 325
7.4 Steady-State Error and Integral Control......Page 327
7.4.2 Manual Reset......Page 328
7.4.4 Reset Windup......Page 330
7.5 System Type and Error Constants......Page 331
7.5.2 Error Constants......Page 332
7.5.4 Performance Specification Using s-Plane......Page 336
7.6 Control System Sensitivity......Page 340
7.6.1 System Sensitivity to Parameter Change......Page 341
Problems......Page 344
8.1.1 Natural Response......Page 360
8.2 Routh–Hurwitz Criterion......Page 362
8.2.1 Routh Array......Page 363
8.2.2 Auxiliary Equation (Zero-Row Problem)......Page 364
8.2.3 Zero Coefficient Problem......Page 365
8.2.4 Relative Stability......Page 366
8.3 Root Locus Method......Page 367
8.3.1 Rules for Plotting Root Locus......Page 369
8.3.2 Steps of Sketching Root Locus......Page 374
8.3.4 Variable Parameter in Root Locus......Page 387
8.4 Stability in the Frequency Domain......Page 389
8.4.1 Response to a Harmonic Input......Page 390
8.4.2 Complex Numbers......Page 391
8.4.3 Resonant Peak and Resonant Frequency......Page 392
8.4.4 Half-Power Bandwidth......Page 396
8.4.5 Marginal Stability......Page 398
8.4.6 PM and GM......Page 400
8.4.7 Bode and Nyquist Plots......Page 401
8.4.8 PM and Damping Ratio Relation......Page 403
8.5 Bode Diagram Using Asymptotes......Page 404
8.5.1 Slope-Phase Relationship for Bode Magnitude Curve......Page 406
8.5.2 Ambiguous Cases of GM and PM......Page 414
8.5.3 Destabilizing Effect of Time Delays......Page 415
8.6 Nyquist Stability Criterion......Page 416
8.6.1 Nyquist Stability Criterion......Page 417
8.6.3 Steps for Applying the Nyquist Criterion......Page 418
8.7.1 Graphical Tools for Closed-Loop Frequency Response......Page 425
8.7.2 M Circles and N Circles......Page 426
8.7.3 Nichols Chart......Page 429
Problems......Page 431
9.1 Controller Design and Tuning......Page 440
9.1.1 Design Specifications......Page 441
9.2 Conventional Time-Domain Design......Page 442
9.2.1 Proportional Plus Derivative Controller Design......Page 443
9.3 Compensator Design in the Frequency Domain......Page 445
9.3.1 Lead Compensation......Page 446
9.3.2 Lag Compensation......Page 451
9.3.3 Design Specifications in Compensator Design......Page 454
9.4.1 Design Steps Using Root Locus......Page 458
9.4.2 Lead Compensation......Page 459
9.4.3 Lag Compensation......Page 462
9.5.1 Ziegler–Nichols Tuning......Page 467
9.5.2 Reaction Curve Method......Page 468
9.5.3 Ultimate Response Method......Page 470
Problems......Page 472
10.1.1 Computer Control Systems......Page 478
10.1.2 Components of a Digital Control System......Page 479
10.2 Signal Sampling and Control Bandwidth......Page 480
10.2.2 Antialiasing Filter......Page 481
10.2.3 Control Bandwidth......Page 482
10.2.4 Bandwidth Design of a Control System......Page 485
10.2.5 Control Cycle Time......Page 486
10.3 Digital Control Using z-Transform......Page 487
10.3.1 z-Transform......Page 488
10.3.2 Difference Equations......Page 489
10.3.3 Discrete Transfer Functions......Page 491
10.3.4 Time Delay......Page 492
10.3.5 s–z Mapping......Page 493
10.3.7 Discrete Final Value Theorem (FVT)......Page 495
10.3.8 Pulse Response Function......Page 497
10.4 Digital Compensation......Page 498
10.4.1 Hold Operation......Page 499
10.4.2 Discrete Compensator......Page 500
10.4.4 Causality Requirement......Page 504
10.4.5 Stability Analysis Using Bilinear Transformation......Page 505
10.4.6 Computer Implementation......Page 506
Problems......Page 507
11.1 Modern Control......Page 514
11.2.1 The Scalar Problem......Page 515
11.2.2 Time Response of a State-Space Model......Page 517
11.2.3 Time Response by Laplace Transform......Page 525
11.2.4 Output Response......Page 526
11.2.5 Modal Response......Page 527
11.2.6 Time-Varying Systems......Page 532
11.3.1 Stability of Linear Systems......Page 534
11.3.2 Stability from Modal Response for Repeated Eigenvalues......Page 538
11.3.3 Equilibrium......Page 539
11.3.4 Stability of Linear Systems......Page 540
11.3.5 Second Method (Direct Method) of Lyapunov......Page 544
11.4 Controllability and Observability......Page 548
11.4.1 Minimum (Irreducible) Realizations......Page 552
11.4.3 Implication of Feedback Control......Page 556
11.4.4 State Feedback......Page 557
11.4.5 Stabilizability......Page 558
11.5 Modal Control......Page 559
11.5.1 Controller Design by Pole Placement......Page 560
11.5.2 Pole Placement in the Multiinput Case......Page 565
11.5.3 Procedure of Pole Placement Design......Page 567
11.5.4 Placement of Repeated Poles......Page 569
11.5.5 Placement of Some Closed-Loop Poles at Open-Loop Poles......Page 570
11.5.6 Pole Placement with Output Feedback......Page 573
11.6.1 Optimization through Calculus of Variations......Page 575
11.6.2 Cost Function having a Function of End State......Page 586
11.6.3 Extension to the Vector Problem......Page 587
11.6.4 General Optimal Control Problem......Page 588
11.6.7 PontryaginŁfs Minimum Principle......Page 590
11.7.1 The Euler Equations......Page 591
11.7.2 Boundary Conditions......Page 592
11.7.3 Infinite-Time LQR......Page 593
11.7.4 Control System Design......Page 597
11.8.1 Nonlinear Feedback Control......Page 600
11.8.2 Adaptive Control......Page 602
11.8.3 Sliding Mode Control......Page 604
11.8.4 Linear Quadratic Gaussian (LQG) Control......Page 605
11.8.5 H_Control......Page 607
11.9 Fuzzy Logic Control......Page 608
11.9.1 Fuzzy Logic......Page 609
11.9.2 Fuzzy Sets and Membership Functions......Page 610
11.9.3 Fuzzy Logic Operations......Page 611
11.9.4 Compositional Rule of Inference......Page 614
11.9.5 Extensions to Fuzzy Decision Making......Page 615
11.9.6 Basics of Fuzzy Control......Page 616
11.9.7 Fuzzy Control Surface......Page 620
Problems......Page 624
12.1 Control System Instrumentation......Page 634
12.2.1 Cascade Connection of Devices......Page 636
12.2.3 Operational Amplifier......Page 638
12.2.4 Instrumentation Amplifiers......Page 640
12.3 Motion Sensors......Page 641
12.3.1 Linear-Variable Differential Transformer (LVDT)......Page 642
12.3.2 Signal Conditioning......Page 643
12.3.3 DC Tachometer......Page 644
12.3.4 Piezoelectric Accelerometer......Page 645
12.3.5 Digital Transducers......Page 647
12.3.6 Shaft Encoders......Page 648
12.3.7 Optical Encoder......Page 649
12.4 Stepper Motors......Page 650
12.4.2 Driver and Controller......Page 651
12.4.3 Stepper Motor Selection......Page 654
12.5 dc Motors......Page 661
12.5.1 Rotor and Stator......Page 663
12.5.3 Brushless dc Motors......Page 664
12.5.4 DC Motor Equations......Page 665
12.5.5 Experimental Model for dc Motor......Page 668
12.5.6 Control of dc Motors......Page 669
12.5.7 Feedback Control of dc Motors......Page 673
12.5.8 Motor Driver......Page 675
12.5.9 dc Motor Selection......Page 679
12.5.10 Summary of Motor Selection......Page 683
12.6.1 Experiment 1: Tank Level Display......Page 685
12.6.2 Experiment 2: Process Control Using LabVIEW®......Page 691
Problems......Page 697
A.1 Laplace Transform......Page 708
A.1.1 Laplace Transforms of Some Common Functions......Page 709
A.2 Response Analysis......Page 713
A.3 Transfer Function......Page 720
A.4 Fourier Transform......Page 722
A.4.1 Frequency-Response Function (Frequency Transfer Function)......Page 723
A.5.2 Application in Circuit Analysis......Page 724
B.2.1 Computations......Page 726
B.2.2 Arithmetic......Page 727
B.2.3 Arrays......Page 728
B.2.5 Linear Algebra......Page 729
B.2.6 M-Files......Page 730
B.3.1 Compensator Design Example......Page 731
B.3.2 PID Control with Ziegler–Nichols Tuning......Page 734
B.3.4 MATLAB® Modern Control Examples......Page 739
B.4.1 Graphical Editors......Page 747
B.4.3 Practical Stand-Alone Implementation in C......Page 748
B.5.3 Working with LabVIEW®......Page 750
B.6.1 Sound and Vibration Toolkit......Page 755
B.6.2 Signal Acquisition and Simulation......Page 756
C.1 Vectors and Matrices......Page 760
C.2 Vector–Matrix Algebra......Page 762
C.2.1 Matrix Addition and Subtraction......Page 763
C.2.3 Matrix Multiplication......Page 764
C.3 Matrix Inverse......Page 765
C.3.1 Matrix Transpose......Page 766
C.3.3 Determinant of a Matrix......Page 767
C.3.4 Adjoint of a Matrix......Page 768
C.3.5 Inverse of a Matrix......Page 769
C.4.2 Vector Space (L)......Page 770
C.4.5 Bases and Dimension of a Vector Space......Page 771
C.4.7 Norm......Page 772
C.5 Determinants......Page 773
C.6 System of Linear Equations......Page 774
C.7 Quadratic Forms......Page 775
C.9.1 Similarity Transformation......Page 776
C.10 Matrix Exponential......Page 777
C.10.1 Computation of Matrix Exponential......Page 778
Index......Page 780