Simulation of dynamical systems with MATLAB and Simulink

Cover......Page 1
Title Page......Page 2
Copyright......Page 3
Contents......Page 6
Foreword......Page 14
Preface......Page 16
Authors......Page 20
1.1.1 Importance of Models......Page 22
1.2 Derivation of a Mathematical Model......Page 25
Exercises......Page 29
1.3 Difference Equations......Page 31
1.3.1 Recursive Solutions......Page 32
Exercises......Page 33
1.4 First Look at Discrete-Time Systems......Page 34
1.4.1 Inherently Discrete-Time Systems......Page 38
Exercises......Page 41
1.5 Case Study: Population Dynamics (Single Species)......Page 42
Exercises......Page 49
2.2 First-Order Systems......Page 52
2.2.1 Step Response of First-Order Systems......Page 53
Exercises......Page 57
2.3 Second-Order Systems......Page 59
2.3.1 Conversion of Two First-Order Equations to a Second-Order Model......Page 64
Exercises......Page 67
2.4 Simulation Diagrams......Page 68
2.4.1 Systems of Equations......Page 74
Exercises......Page 76
2.5 Higher-Order Systems......Page 77
Exercises......Page 79
2.6 State Variables......Page 80
2.6.1 Conversion from Linear State Variable Form to Single Input–Single Output Form......Page 85
Exercises......Page 86
2.7 Nonlinear Systems......Page 89
2.7.1 Friction......Page 91
2.7.2 Dead Zone and Saturation......Page 93
2.7.4 Hysteresis......Page 94
2.7.5 Quantization......Page 98
2.7.6 Sustained Oscillations and Limit Cycles......Page 99
Exercises......Page 103
2.8 Case Study: Submarine Depth Control System......Page 106
Exercises......Page 110
3.1 Introduction......Page 112
3.2 Discrete-Time System Approximation of a Continuous-Time Integrator......Page 113
Exercises......Page 115
3.3 Euler Integration......Page 117
3.3.1 Backward (Implicit) Euler Integration......Page 120
Exercises......Page 122
3.4 Trapezoidal Integration......Page 123
Exercises......Page 127
3.5.1 Discrete-Time System Models from Simulation Diagrams......Page 128
3.5.2 Nonlinear First-Order Systems......Page 132
3.5.3 Discrete-Time State Equations......Page 135
3.5.4 Discrete-Time State System Matrices......Page 139
Exercises......Page 140
3.6.1 Improved Euler Method......Page 143
3.6.2 Modified Euler Integration......Page 146
Exercises......Page 156
3.7 Case Study: Vertical Ascent of a Diver......Page 159
3.7.1 Maximum Cable Force for Safe Ascent......Page 165
3.7.2 Diver Ascent with Decompression Stops......Page 166
Exercises......Page 168
4.2 Laplace Transform......Page 172
4.2.1 Properties of the Laplace Transform......Page 174
4.2.2 Inverse Laplace Transform......Page 180
4.2.3 Laplace Transform of the System Response......Page 181
4.2.4 Partial Fraction Expansion......Page 182
Exercises......Page 188
4.3.1 Impulse Function......Page 189
4.3.2 Relationship between Unit Step Function and Unit Impulse Function......Page 190
4.3.3 Impulse Response......Page 192
4.3.4 Relationship between Impulse Response and Transfer Function......Page 196
4.3.5 Systems with Multiple Inputs and Outputs......Page 199
4.3.6 Transformation from State Variable Model to Transfer Function......Page 205
Exercises......Page 208
4.4 Stability of Linear Time Invariant Continuous-Time Systems......Page 210
4.4.1 Characteristic Polynomial......Page 211
4.4.2 Feedback Control System......Page 215
Exercises......Page 219
4.5 Frequency Response of LTI Continuous-Time Systems......Page 221
4.5.1 Stability of Linear Feedback Control Systems Based on Frequency Response......Page 231
Exercises......Page 234
4.6 z-Transform......Page 236
4.6.1 Discrete-Time Impulse Function......Page 242
4.6.2 Inverse z-Transform......Page 246
4.6.3 Partial Fraction Expansion......Page 247
Exercises......Page 254
4.7 z-Domain Transfer Function......Page 255
4.7.1 Nonzero Initial Conditions......Page 257
4.7.2 Approximating Continuous-Time System Transfer Functions......Page 259
4.7.3 Simulation Diagrams and State Variables......Page 265
4.7.4 Solution of Linear Discrete-Time State Equations......Page 269
4.7.5 Weighting Sequence (Impulse Response Function)......Page 274
Exercises......Page 278
4.8 Stability of LTI Discrete-Time Systems......Page 280
4.8.1 Complex Poles of H(z)......Page 284
Exercises......Page 290
4.9.1 Steady-State Sinusoidal Response......Page 293
4.9.2 Properties of the Discrete-Time Frequency Response Function......Page 295
4.9.3 Sampling Theorem......Page 299
4.9.4 Digital Filters......Page 305
Exercises......Page 310
4.10 Control System Toolbox......Page 313
4.10.2 State-Space Models......Page 314
4.10.3 State-Space/Transfer Function Conversion......Page 316
4.10.4 System Interconnections......Page 319
4.10.5 System Response......Page 320
4.10.6 Continuous-/Discrete-Time System Conversion......Page 323
4.10.7 Frequency Response......Page 324
4.10.8 Root Locus......Page 326
Exercises......Page 330
4.11 Case Study: Longitudinal Control of an Aircraft......Page 333
4.11.1 Digital Simulation of Aircraft Longitudinal Dynamics......Page 346
4.11.2 Simulation of State Variable Model......Page 348
Exercises......Page 350
4.12 Case Study: Notch Filter for Electrocardiograph Waveform......Page 351
4.12.1 Multinotch Filters......Page 352
Exercises......Page 359
5.2 Building a Simulink® Model......Page 362
5.2.1 Simulink® Library......Page 363
5.2.2 Running a Simulink® Model......Page 366
Exercises......Page 368
5.3 Simulation of Linear Systems......Page 370
5.3.1 Transfer Fcn Block......Page 371
5.3.2 State-Space Block......Page 374
Exercises......Page 383
5.4 Algebraic Loops......Page 384
5.4.1 Eliminating Algebraic Loops......Page 385
5.4.2 Algebraic Equations......Page 388
Exercises......Page 390
5.5 More Simulink® Blocks......Page 392
5.5.3 Dead Zone and Saturation......Page 398
5.5.4 Backlash......Page 400
5.5.5 Hysteresis......Page 401
5.5.6 Quantization......Page 402
Exercises......Page 403
5.6 Subsystems......Page 406
5.6.2 Car-Following Subsystem......Page 407
5.6.3 Subsystem Using Fcn Blocks......Page 410
Exercises......Page 413
5.7 Discrete-Time Systems......Page 414
5.7.1 Simulation of an Inherently Discrete-Time System......Page 415
5.7.2 Discrete-Time Integrator......Page 418
5.7.3 Centralized Integration......Page 419
5.7.4 Digital Filters......Page 423
5.7.5 Discrete-Time Transfer Function......Page 425
Exercises......Page 429
5.8 MATLAB® and Simulink® Interface......Page 432
Exercises......Page 438
5.9 Hybrid Systems: Continuous- and Discrete-Time Components......Page 441
Exercises......Page 444
5.10 Monte Carlo Simulation......Page 445
5.10.1 Monte Carlo Simulation Requiring Solution of a Mathematical Model......Page 449
Exercises......Page 455
5.11 Case Study: Pilot Ejection......Page 458
Exercises......Page 462
5.12.1 Continuous-Time Kalman Filter......Page 463
5.12.3 Discrete-Time Kalman Filter......Page 464
5.12.4 Simulink® Simulations......Page 465
5.12.5 Summary......Page 476
Exercise......Page 477
6.2 Runge–Kutta (RK) (One-Step Methods)......Page 478
6.2.1 Taylor Series Method......Page 479
6.2.2 Second-Order Runge–Kutta Method......Page 480
6.2.3 Truncation Errors......Page 482
6.2.4 High-Order Runge–Kutta Methods......Page 487
6.2.5 Linear Systems: Approximate Solutions Using RK Integration......Page 488
6.2.6 Continuous-Time Models with Polynomial Solutions......Page 490
6.2.7 Higher-Order Systems......Page 492
Exercises......Page 499
6.3.1 Repeated RK with Interval Halving......Page 502
6.3.3 Adaptive Step Size (Initial T=1 min)......Page 506
6.3.4 RK–Fehlberg......Page 507
Exercises......Page 511
6.4 Multistep Methods......Page 513
6.4.1 Explicit Methods......Page 514
6.4.2 Implicit Methods......Page 516
6.4.3 Predictor–Corrector Methods......Page 519
Exercises......Page 523
6.5 Stiff Systems......Page 524
6.5.1 Stiffness Property in First-Order System......Page 525
6.5.2 Stiff Second-Order System......Page 527
6.5.3 Approximating Stiff Systems with Lower-Order Nonstiff System Models......Page 530
Exercises......Page 543
6.6 Lumped Parameter Approximation of Distributed Parameter Systems......Page 547
6.6.1 Nonlinear Distributed Parameter System......Page 552
Exercises......Page 555
6.7 Systems with Discontinuities......Page 556
6.7.1 Physical Properties and Constant Forces Acting on the Pendulum BOB......Page 564
Exercises......Page 570
6.8 Case Study: Spread of an Epidemic......Page 573
Exercises......Page 580
7.1 Introduction......Page 582
7.2 Steady-State Solver......Page 583
7.2.1 Trim Function......Page 585
7.2.2 Equilibrium Point for a Nonautonomous System......Page 586
Exercises......Page 595
7.3 Optimization of Simulink® Models......Page 597
7.3.1 Gradient Vector......Page 606
7.3.2 Optimizing Multiparameter Objective Functions Requiring Simulink® Models......Page 608
7.3.3 Parameter Identification......Page 611
7.3.4 Example of a Simple Gradient Search......Page 612
7.3.5 Optimization of Simulink® Discrete-Time System Models......Page 620
Exercises......Page 626
7.4 Linearization......Page 631
7.4.1 Deviation Variables......Page 632
7.4.2 Linearization of Nonlinear Systems in State Variable Form......Page 640
7.4.3 Linmod Function......Page 644
7.4.4 Multiple Linearized Models for a Single System......Page 648
Exercises......Page 654
7.5.1 Introduction......Page 658
7.6.1 Introduction......Page 666
7.6.3 Summary......Page 668
Exercise......Page 669
8.2 Dynamic Errors (Characteristic Roots, Transfer Function)......Page 670
8.2.1 Discrete-Time Systems and the Equivalent Continuous-Time Systems......Page 671
8.2.2 Characteristic Root Errors......Page 674
8.2.3 Transfer Function Errors......Page 685
8.2.4 Asymptotic Formulas for Multistep Integration Methods......Page 690
8.2.5 Simulation of Linear System with Transfer Function H(s)......Page 693
Exercises......Page 698
8.3.1 Adams–Bashforth Numerical Integrators......Page 701
8.3.2 Implicit Integrators......Page 708
8.3.3 Runga–Kutta (RK) Integration......Page 713
Exercises......Page 721
8.4 Multirate Integration......Page 723
8.4.1 Procedure for Updating Slow and Fast States: Master/Slave=RK-4/RK-4......Page 727
8.4.2 Selection of Step Size Based on Stability......Page 728
8.4.3 Selection of Step Size Based on Dynamic Accuracy......Page 729
8.4.4 Analytical Solution for State Variables......Page 733
8.4.5 Multirate Integration of Aircraft Pitch Control System......Page 735
8.4.6 Nonlinear Dual Speed Second-Order System......Page 738
8.4.7 Multirate Simulation of Two-Tank System......Page 744
8.4.8 Simulation Trade-Offs with Multirate Integration......Page 746
Exercises......Page 749
8.5 Real-Time Simulation......Page 751
8.5.1 Numerical Integration Methods Compatible with Real-Time Operation......Page 754
8.5.3 RK-2 (Improved Euler)......Page 755
8.5.5 RK-3 (Real-Time Incompatible)......Page 756
8.5.8 Multistep Integration Methods......Page 757
8.5.9 Stability of Real-Time Predictor–Corrector Method......Page 759
8.5.10 Extrapolation of Real-Time Inputs......Page 761
8.5.11 Alternate Approach to Real-Time Compatibility: Input Delay......Page 767
Exercises......Page 774
8.6.1 Sampling and Signal Reconstruction......Page 775
8.6.2 First-Order Hold Signal Reconstruction......Page 780
8.6.3 Matched Pole-Zero Method......Page 781
8.6.4 Bilinear Transform with Prewarping......Page 784
Exercises......Page 786
8.7.1 Introduction......Page 788
8.7.3 Noisy Model......Page 790
8.7.4 Filtered Model......Page 794
Exercise......Page 800
References......Page 802