Simulation of Dynamic Systems with MATLAB and Simulink

Simulation of Dynamic Systems with MATLAB and Simulink
اسم المؤلف
Harold Klee and Randal Allen
التاريخ
17 أغسطس 2023
المشاهدات
364
التقييم
(لا توجد تقييمات)
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Simulation of Dynamic Systems with MATLAB and Simulink
Third Edition
Harold Klee and Randal Allen
Contents
Foreword . xiii
Preface xv
About the Authors .xix
Chapter 1 Mathematical Modeling .1
1.1 Introduction .1
1.1.1 Importance of Models 1
1.2 Derivation of A Mathematical Model .4
1.3 Difference Equations . 10
1.4 First Look at Discrete-Time Systems 19
1.4.1 Inherently Discrete-Time Systems . 19
1.5 Case Study: Population Dynamics (Single Species) 22
Chapter 2 Continuous-Time Systems 29
2.1 Introduction .29
2.2 First-Order Systems .29
2.2.1 Step Response of First-Order Systems .30
2.3 Second-Order Systems 36
2.3.1 Conversion of Two First-Order Equations to a Second-Order
Model 41
2.4 Simulation Diagrams . 45
2.4.1 Systems of Equations . 51
2.5 Higher-Order Systems .54
2.6 State Variables . 57
2.6.1 Conversion from Linear State Variable Form to Single
Input–Single Output Form . 62
2.6.2 General Solution of the State Equations 63
2.7 Nonlinear Systems .66
2.7.1 Friction .68
2.7.2 Dead Zone and Saturation 71
2.7.3 Backlash .72
2.7.4 Hysteresis .72
2.7.5 Quantization . 76
2.7.6 Sustained Oscillations and Limit Cycles .77
2.8 Case Study: Submarine Depth Control System .85
Chapter 3 Elementary Numerical Integration . 91
3.1 Introduction . 91
3.2 Discrete-Time System Approximation of a Continuous First-Order
System . 92
3.3 Euler Integration 98
3.3.1 Explicit Euler Integration .99
3.3.2 Implicit Euler Integration .100
3.4 Trapezoidal Integration . 104viii Contents
3.5 Discrete Approximation of Nonlinear First-Order Systems . 112
3.6 Discrete State Equations 116
3.7 Improvements to Euler Integration 127
3.7.1 Improved Euler Integration 127
3.7.2 Modified Euler Integration . 131
3.7.3 Discrete-Time System Matrices . 132
3.8 Case Study: Vertical Ascent of a Diver . 146
Chapter 4 Linear Systems Analysis 155
4.1 Introduction . 155
4.2 Laplace Transform . 155
4.2.1 Properties of the Laplace Transform 156
4.2.2 Inverse Laplace Transform . 163
4.2.3 Laplace Transform of the System Response 164
4.2.4 Partial Fraction Expansion . 166
4.3 Transfer Function . 173
4.3.1 Impulse Function 173
4.3.2 Relationship between Unit Step Function and Unit Impulse
Function 173
4.3.3 Impulse Response . 175
4.3.4 Relationship between Impulse Response and Transfer Function 179
4.3.5 Systems with Multiple Inputs and Outputs 182
4.3.6 Transformation from State Variable Model to Transfer Function . 190
4.4 Stability of Linear Time Invariant Continuous-Time Systems . 194
4.4.1 Characteristic Polynomial 195
4.4.2 Feedback Control System .200
4.5 Frequency Response of LTI Continuous-Time Systems 206
4.5.1 Stability of Linear Feedback Control Systems Based
on Frequency Response 216
4.6 z-Transform 222
4.6.1 Discrete-Time Impulse Function 226
4.6.2 Inverse z-Transform 232
4.6.3 Partial Fraction Expansion . 233
4.7 z-Domain Transfer Function 242
4.7.1 Nonzero Initial Conditions .243
4.7.2 Approximating Continuous-Time System Transfer Functions .245
4.7.3 Simulation Diagrams and State Variables 250
4.7.4 Solution of Linear Discrete-Time State Equations .256
4.7.5 Weighting Sequence (Impulse Response Function) . 261
4.8 Stability of LTI Discrete-Time Systems 267
4.8.1 Complex Poles of H(z) 271
4.9 Frequency Response of Discrete-Time Systems 280
4.9.1 Steady-State Sinusoidal Response 280
4.9.2 Properties of the Discrete-Time Frequency Response Function . 282
4.9.3 Sampling Theorem .287
4.9.4 Digital Filters .293
4.10 Control System Toolbox 300
4.10.1 Transfer Function Models 301
4.10.2 State-Space Models 302
4.10.3 State-Space/Transfer Function Conversion 303Contents ix
4.10.4 System Interconnections 305
4.10.5 System Response 307
4.10.6 Continuous-/Discrete-Time System Conversion 309
4.10.7 Frequency Response . 311
4.10.8 Root Locus . 313
4.11 Case Study: Longitudinal Control of an Aircraft 319
4.11.1 Digital Simulation of Aircraft Longitudinal Dynamics . 333
4.11.2 Simulation of State Variable Model . 335
4.12 Case Study: Notch Filter for Electrocardiograph Waveform . 338
4.12.1 Multinotch Filters . 339
Chapter 5 Simulink .349
5.1 Introduction .349
5.2 Building a Simulink Model .349
5.2.1 The Simulink Library .349
5.2.2 Running a Simulink Model 353
5.3 Simulation of Linear Systems . 357
5.3.1 Transfer Fcn Block . 357
5.3.2 State-Space Block . 363
5.4 Algebraic Loops 371
5.4.1 Eliminating Algebraic Loops . 373
5.4.2 Algebraic Equations . 375
5.5 More Simulink Blocks .380
5.5.1 Discontinuities 385
5.5.2 Friction .386
5.5.3 Dead Zone and Saturation 387
5.5.4 Backlash . 389
5.5.5 Hysteresis . 389
5.5.6 Quantization . 391
5.6 Subsystems 394
5.6.1 PHYSBE . 395
5.6.2 Car-Following Subsystem 396
5.6.3 Subsystem Using Fcn Blocks . 398
5.7 Discrete-Time Systems 402
5.7.1 Simulation of an Inherently Discrete-Time System .403
5.7.2 Discrete-Time Integrator 406
5.7.3 Centralized Integration .409
5.7.4 Digital Filters . 412
5.7.5 Discrete-Time Transfer Function . 414
5.8 MATLAB and Simulink Interface 422
5.9 Hybrid Systems: Continuous- and Discrete-Time Components 431
5.10 Monte Carlo Simulation 435
5.10.1 Monte Carlo Simulation Requiring Solution of
a Mathematical Model 439
5.11 Case Study: Pilot Ejection .448
5.12 Case Study: Kalman Filtering . 453
5.12.1 Continuous-Time Kalman Filter 453
5.12.2 Steady-State Kalman Filter 454
5.12.3 Discrete-Time Kalman Filter . 454
5.12.4 Simulink Simulations . 455x Contents
5.12.5 Summary 468
5.13 Case Study: Cascaded Tanks with Flow Logic Control 469
Chapter 6 Intermediate Numerical Integration . 475
6.1 Introduction . 475
6.2 Runge–Kutta (RK) (One-Step Methods) . 475
6.2.1 Taylor Series Method . 476
6.2.2 Second-Order Runge–Kutta Method . 477
6.2.3 Truncation Errors . 479
6.2.4 High-Order Runge–Kutta Methods 484
6.2.5 Linear Systems: Approximate Solutions Using RK Integration 486
6.2.6 Continuous-Time Models with Polynomial Solutions 488
6.2.7 Higher-Order Systems 490
6.3 Adaptive Techniques .500
6.3.1 Repeated RK with Interval Halving .500
6.3.2 Constant Step Size (T = 1 min) 505
6.3.3 Adaptive Step Size (Initial T = 1 min) 505
6.3.4 RK–Fehlberg 505
6.4 Multistep Methods . 512
6.4.1 Explicit Methods 513
6.4.2 Implicit Methods 515
6.4.3 Predictor–Corrector Methods 518
6.5 Stiff Systems 523
6.5.1 Stiffness Property in First-Order System .524
6.5.2 Stiff Second-Order System 526
6.5.3 Approximating Stiff Systems with Lower-Order Nonstiff
System Models . 529
6.6 Lumped Parameter Approximation of Distributed Parameter Systems 546
6.6.1 Nonlinear Distributed Parameter System 550
6.7 Systems with Discontinuities . 555
6.7.1 Physical Properties and Constant Forces Acting on
the Pendulum Bob 563
6.8 Case Study: Spread of an Epidemic 573
Chapter 7 Simulation Tools . 581
7.1 Introduction . 581
7.2 Steady-State Solver 582
7.2.1 Trim Function .584
7.2.2 Equilibrium Point for a Nonautonomous System . 586
7.3 Optimization of Simulink Models .596
7.3.1 Gradient Vector 605
7.3.2 Optimizing Multiparameter Objective Functions
Requiring Simulink Models .607
7.3.3 Parameter Identification . 610
7.3.4 Example of a Simple Gradient Search . 611
7.3.5 Optimization of Simulink Discrete-Time System Models .620
7.4 Linearization .630
7.4.1 Deviation Variables 631
7.4.2 Linearization of Nonlinear Systems in State Variable Form . 639Contents xi
7.4.3 Linmod Function 643
7.4.4 Multiple Linearized Models for a Single System .648
7.5 Adding Blocks to The Simulink Library Browser 659
7.5.1 Introduction 659
7.5.2 Summary 665
7.6 Simulation Acceleration 665
7.6.1 Introduction 665
7.6.2 Profiler 667
7.6.3 Summary 668
7.7 Black Swans .668
7.7.1 Introduction 668
7.7.2 Modeling Rare Events 668
7.7.3 Measurement of Portfolio Risk 669
7.7.4 Exposing Black Swans . 673
7.7.4.1 Percent Point Functions (PPFs) . 673
7.7.4.2 Stochastic Optimization . 673
7.7.5 Summary 676
7.7.6 Acknowledgements 676
7.7.7 References 676
7.7.8 Appendix—Mathematical Properties of the Log-Stable
Distribution . 676
7.8 The SIPmath Standard 677
7.8.1 Introduction 677
7.8.2 Standard Specification 677
7.8.3 SIP Details 678
7.8.4 SLURP Details . 678
7.8.5 SIPs/SLURPs and MATLAB . 679
7.8.6 Summary 680
7.8.7 Appendix 681
7.8.8 References 682
Chapter 8 Advanced Numerical Integration .683
8.1 Introduction .683
8.2 Dynamic Errors (Characteristic Roots, Transfer Function) 683
8.2.1 Discrete-Time Systems and the Equivalent
Continuous-Time Systems 684
8.2.2 Characteristic Root Errors 687
8.2.3 Transfer Function Errors 697
8.2.4 Asymptotic Formulas for Multistep Integration Methods .704
8.2.5 Simulation of Linear System with Transfer Function H(s) 708
8.3 Stability of Numerical Integrators . 714
8.3.1 Adams–Bashforth Numerical Integrators 714
8.3.2 Implicit Integrators . 722
8.3.3 Runga–Kutta (RK) Integration .726
8.4 Multirate Integration . 738
8.4.1 Procedure for Updating Slow and Fast States:
Master/Slave = RK-4/RK-4 . 742
8.4.2 Selection of Step Size Based on Stability 743
8.4.3 Selection of Step Size Based on Dynamic Accuracy . 745
8.4.4 Analytical Solution for State Variables 748xii Contents
8.4.5 Multirate Integration of Aircraft Pitch Control System . 750
8.4.6 Nonlinear Dual Speed Second-Order System 753
8.4.7 Multirate Simulation of Two-Tank System 760
8.4.8 Simulation Trade-Offs with Multirate Integration . 763
8.5 Real-Time Simulation 766
8.5.1 Numerical Integration Methods Compatible with
Real-Time Operation 769
8.5.2 RK-1 (Explicit Euler) 770
8.5.3 RK-2 (Improved Euler) . 771
8.5.4 RK-2 (Modified Euler) . 771
8.5.5 RK-3 (Real-Time Incompatible) . 771
8.5.6 RK-3 (Real-Time Compatible) 772
8.5.7 RK-4 (Real-Time Incompatible) . 772
8.5.8 Multistep Integration Methods . 772
8.5.9 Stability of Real-Time Predictor–Corrector Method . 774
8.5.10 Extrapolation of Real-Time Inputs . 776
8.5.11 Alternate Approach to Real-Time Compatibility: Input Delay 783
8.6 Additional Methods of Approximating Continuous-Time System Models 790
8.6.1 Sampling and Signal Reconstruction .790
8.6.2 First-Order Hold Signal Reconstruction 796
8.6.3 Matched Pole-Zero Method 796
8.6.4 Bilinear Transform with Prewarping .799
8.7 Case Study: Lego MindstormsTM NXT .803
8.7.1 Introduction 803
8.7.2 Requirements and Installation 805
8.7.3 Noisy Model .806
8.7.4 Filtered Model 810
8.7.5 Summary 815
References . 817
Index 821
Index
A
AB-m integrator, 515, 516
ACSL, see Advanced Continuous Simulation Language
Adams–Bashforth numerical integrators
characteristic root error formula, 715
method, 513–514
stability boundaries, 717–720
stability condition, 716
undamped second-order system, 719–722
z-domain transfer function, 714–715
Adams–Moulton implicit integrators, 519
chemical concentration, 724–726
stability boundaries, 723–724
z-domain transfer functions, 723
Adaptive step size, 505
Adaptive techniques
adaptive step size, 505
constant step size, 505
repeated Runge–Kutta with interval halving, 500–505
Runge–Kutta–Fehlberg method, 505–510
Advanced Continuous Simulation Language (ACSL), 349
Advanced numerical integration
continuous-time system models
bilinear transform, 799–801
first-order hold signal reconstruction, 796
matched pole-zero method, 796–799
sampling and signal reconstruction, 790–792
dynamic errors
asymptotic formulas, 704–708
characteristic root errors, 687–688
definition, 683
discrete-time and equivalent continuous-time
systems, 684–687
linear system simulation, 708–711
transfer function errors, 697–704
types, 683–684
Lego MindstormsTM NXT
feedback control systems, 804
filtered model, 810–815
IDE, 805
installation, 805–806
mechatronics, 803
noisy model, 806–810
software and hardware requirements, 805
multirate integration
aircraft pitch control system, 739
airframe dynamics, 739
analytical solution, state variables, 748–750
frame ratio, 741
master routine, 741
nonlinear dual speed second-order system,
753–760
simulation trade-offs, 763–764
slave routine, 741
slow and fast states procedure, 742–743
slow and fast subsystem interaction, 741
step size selection, 743–745
stiff system, 738
two-tank system, 760–763
real-time simulation
extrapolation, 776–783
high-fidelity-driving simulator, 769
HIL, 767–768
input delay, 783–786
predictor–corrector method, 772–776
RK-1 (explicit Euler), 770–771
RK-2 (improved Euler), 771
RK-2 (modified Euler), 771
RK-3 (real-time compatible), 772
RK-3 (real-time incompatible), 771–772
RK-4 (real-time incompatible), 772
two-pass numerical integration method, 769–770
vehicle ABS system, 768
stability
Adams–Bashforth numerical integrators, 714–720
Adams–Moulton implicit integrators, 722–726
Runge–Kutta (RK) integration, 726–736
Aircraft, longitudinal control, 319
altitude control system, 330
altitude from steady-state flight conditions, 328
angle of attack and forces, 321
body axis coordinates and Euler angles, 320
elevator response for open-and closed-loop, 332
linearized aircraft pitch response, 325
open-and closed-loop altitude response vs. time, 329
partial fraction expansion, 327
primary control surface, 321
short period and phugoid modes, 324–326
transfer function, 323, 327, 330
Aircraft longitudinal dynamics, digital simulation
of, 333–335
Aircraft pitch control system
block diagram of, 738–739
multiple integration of, 750–753
simulation diagram for, 56, 744
Algebraic constraint blocks, 378
Algebraic equations, 375–378
algebraic constraint blocks, 378
first-order autonomous system, 376
Algebraic loops, 371–373
circular nature, 372
eliminating, 373–375
equations, 375–378
Memory block, 373–375
submarine dynamics transfer function, 374
Algebraic manipulation, 373
AM-m integrator, 516, 732
Armature-controlled DC motor, 535
Asymptotic stability, 197
Autonomous nonlinear system, 80
B
Backlash, 72, 73
Backlash block, 389822 Index
Backward rectangular integration, 100
BIBO, see Bounded input-bounded output
Bode plot, 210
closed-loop frequency response functions, 213
for control system, 314
of discrete-time systems, 287, 298
for first-order system, 710
of frequency response function, 284
for marginally stable system, 219
of open-loop transfer function, 217–219
for second-order systems, 214–215
third-order Butterworth low-pass filter, 211
Bounded input–bounded output (BIBO), 197, 267–268,
271, 273–274
C
Car-following
models, 380, 381, 384
subsystem, 396–398
Cascaded tanks with flow logic control, 121–126
Centralized integration, 409–412
Characteristic root errors, dynamic errors
asymptotic formula, 689–690
complex pole relationship, 691–692
continuous and discrete-time unit step responses,
695–697
damping ratio error, 693, 694
equivalent system natural frequency, 695
exact and asymptotic fractional errors, 692–693
fractional error, 687–688
impulse responses, 697
responses, 690
step response, 695–696
trapezoidal integration, 690
z-domain transfer function, 690
Closed-loop depth rate control system, 363
Closed-loop transfer function, 362
Constant forces, physical properties and, 563–569
Constant step size, 505
Continuous-/discrete-time system
conversion, 309–311
poles, 276
Continuous System Modeling Program
(CSMP), 349
Continuous-time first-order system
discrete-time system approximation, 92
exact and approximate solution, 95–96
first-order, continuous-time systems, 92–93
improved Euler integration, 727
using trapezoidal integration, 288–293
Continuous-time Kalman filter, 453–454
Continuous-time system
bilinear transform
frequency response, 800–801
mapping, 800
prewarped transfer function, 801–802
dynamic systems with, 349
first-order continuous system, 92–93
n distinct integrations, 91–92
object’s velocity, 115
state derivative vector, 92
first-order hold signal reconstruction, 796
first-order systems
description, 29–30
step response, 30–36
higher-order systems
aircraft pitch control system, 56
feedback control system, 55
railroad cars, 57, 65
linear time invariant
frequency response, 206–216
stability, 194–206
matched pole-zero method
DC gains, 798
frequency response, 798–799
nonlinear systems
applied force vs. time, 69–70
backlash, 72, 73
coulomb friction, 68
dead zone, 71
first-order systems, 66
friction force vs. applied force, 68–69
hysteresis, 72–74
linear model approximation, 67
mechanical system, 81
progressive, 68
quantization, 76–77
saturation, 71–72
sustained oscillations and limit cycles, 77–80
temperature response, 74–76
time constant, 75
valve flow vs. current, 72
with polynomial solutions, 488–490
sampling and signal reconstruction
continuous-time system response, 790–792
frequency response function, 794–795
illustration, 791
piecewise constant function, 791
transfer function, 791
z-domain transfer function, 792
second-order systems
description, 36
first-order equation conversion, 41–42
mechanical system, 39–41
two-tank mixing system, 42–44
unit step response, 36–39
simulation diagrams
aircraft pitch control system, 56
description, 45
first-order system, 45–47
heat flows and temperatures, two-room
building, 51–52
room temperature model, 51–52
second-order system, 53–54, 58–59
state variables
dynamic system, 58, 61
interacting tank system, 61–62
linear state variable form conversion, 62–63
spring-mass-damper system, 57
state equations, 61–62
transition matrix, 63
submarine depth control system
block diagram, 85
controller and stern plane actuator, 89–90
difference equations, 88Index 823
discrete-time approximation, 89
simulation diagram, 86
state equations, 86–87
unit step response of, 277
Continuous-time system models, advanced numerical
integration
bilinear transform, 799–801
first-order hold signal reconstruction, 796
matched pole-zero method, 796–799
sampling and signal reconstruction, 790–792
Continuous-time system simulation languages
(CSSLs), 349
Control systems, 29
aircraft pitch, 739
components, 523
continuous-and discrete-time, 276
higher-order systems
aircraft pitch, 56
feedback, 55
Lego MindstormsTM NXT, 804
linear, 216–219
Runge–Kutta (RK) method, see Runge–Kutta
(RK) method
ship heading, see Ship heading control system
stiff systems, see Stiff systems
toolbox, 300
continuous-/discrete-time system
conversion, 309–311
frequency response, 311–313
root locus, 313–316
state-space models, 302–303
state-space/transfer function conversion, 303–305
system interconnections, 305–307
system response, 307–309
transfer function models, 301–302
unity, 212
Coulomb friction, 68
CSMP, see Continuous System Modeling Program
CSSLs, see Continuous-time system simulation languages
Cylinder node temperatures, 550
D
Data logging of scope signals, 355
Dead zone block, 387–389
Dead zone nonlinearity, 71
Decompression Sickness (DCS), 146
Digital control system, for chamber temperature, 432
Digital filters, 293–297, 412–414
Digital simulation, of aircraft longitudinal dynamics,
333–335
Discontinuity functions, 385–386, 559, 568
Discrete event models, 3
Discrete state equations, 116, 126
discrete step response of circuit, 120
lead-lag network, 117–118
linear state equations, 116–117
predator-prey ecosystem, 125–126
simulation diagram for RC lead-lag network, 119
steady-state response, 121–122
tank level responses, discrete and continuous, 123–125
using explicit Euler integration, 117, 121
Discrete-time frequency response function, 282
Discrete-time impulse function, 226–228
Discrete-time signal, 222–226
Discrete-time systems, 402–403
block diagram of, 275
centralized integration, 409–412
digital filters, 412–414
impulse responses for, 274
integrators, 406–409
Kalman filter, 454–455
mathematical modeling
exact vs. approximate solutions, 13–14
inherent, 19–22
liquid tank continuous-time system, 19
liquid tank discrete-time system, 19
step size, 16–18
matrices, 132–146
damped natural frequency, 140
discrete and continuous responses, 136–139
discrete system matrix, 139
explicit numerical integrators, 133
improved or modified Euler, 133
interacting tanks, 134–136
nonlinear pendulum with damping, 141–143
nonlinear second-order system, 141
quasi exact solution, 142
step responses of a second-order system, 140
transition matrix, 132
output, 260
simulation of inherently, 403–406
transfer function, 414–418
Distributed parameter systems, 546–550
Dynamic errors
asymptotic formulas
Euler integrator, 705–707
numerical integrators, 704–705
z-domain transfer function, 704
characteristic root errors
asymptotic formula, 689–690
complex pole relationship, 691–692
continuous and discrete-time unit step
responses, 695–697
damping ratio error, 693, 694
equivalent system natural frequency, 695
exact and asymptotic fractional errors, 692–693
fractional error, 687–688
impulse responses, 697
responses, 690
step response, 695–696
trapezoidal integration, 690
z-domain transfer function, 690
definition, 683
discrete-time and equivalent continuous-time system
characteristic root, 688
continuous-time integrator, 686
step response, 685–686
linear system simulation
frequency response function, 708–709
RC circuit, 709–711
transfer function errors
continuous-and discrete-time integration, 702
explicit Euler and continuous-time integrator
outputs, 703
fractional error, 697–699824 Index
Dynamic errors (Continued)
frequency response functions, 697
phase angle plots, 701
time delay, 703–704
types, 683–684
E
ECRobot NXT Blockset, 806, 807
Elementary numerical integration, 91–92
discrete state equations, 116, 126
discrete step response of circuit, 120
lead-lag network, 117–118
linear state equations, 116–117
predator-prey ecosystem, 125–126
simulation diagram for RC lead-lag network, 119
steady-state response, 121–122
tank level responses, discrete and
continuous, 123–125
using explicit Euler integration, 117, 121
discrete-time system matrices, 132–146
damped natural frequency, 140
discrete and continuous responses, 136–139
discrete system matrix, 139
explicit numerical integrators, 133
improved or modified Euler, 133
interacting tanks, 134–136
nonlinear pendulum with damping, 141–143
nonlinear second-order system, 141
quasi exact solution, 142
step responses of a second-order system, 140
transition matrix, 132
discrete-time system, of continuous first-order
system, 92–98
Euler integration, see Euler integration
improved Euler integration, 127–131
modified Euler integration, 131–132
nonlinear first-order systems, discrete approximation
of, 112–117
trapezoidal integration, 104–111
area approximation, 104–105
continuous integrators, 105–106, 108
difference equation based on, 105, 107
discrete and continuous responses, 108–109,
110, 111
dynamics of sinking drum, 109–110
for first-order system, 106–107
integration step size, 104, 111
quadratic function, 108
vertical ascent of diver, 146–154
Epidemic model
baseline conditions, 575–577
fatal disease, 573
immigration and inoculation profiles, 574–575
sensitivity analysis, 578–579
S-I-R models, 573
state transition diagram, 574
symptoms, 573
Euler integration, 410, 539
area approximation, 98
comparison of explicit and implicit, 101
continuous-time signal, 103
discrete-time integrator, 102
explict, see Explict Euler integration
implicit, see Implict Euler integration
improvements to
accuracy, 128–131
improved state estimate, 128
new state using forward Euler integration, 127–128
inherent weakness of, 127
modified, 131–132
RC circuit, 102
tank flow, 103
Euler integrator (RK-1), 353, 479–481, 483, 486
Explicit Euler integration, 99–100, 133, 242, 244, 248,
283, 334, 411
damped pendulum response using, 141–143
discrete state equations, 117, 121, 142
numerical integrator, 100
undamped pendulum response using, 144
Explicit methods, 513–515, 519
F
Fcn blocks, 398–401
Feedback control system
block diagram, 200
characteristic polynomial, 201
closed-loop system
properties, 202
transfer function, 200
inverse Laplace transform, 202
ship heading response, 203–204
stability of linear, frequency response
block diagram, 217
Bode plot of open-loop transfer function, 217–218
closed-loop transfer function, 219
First-order autonomous system, 376
First-order differential equations, 490
First-order discrete-time system, low-pass filter in, 262–265
First-order systems
block diagram, 46
continuous-time models, 92–93
description, 29–30
difference equations, 10
exact vs. approximate solution, 13–15
LTI continuous-time systems, 208–210
nonlinear system, 66
simulation diagram
linear tank, 47
RC circuit, 47–48
step response of
graphs of, 30, 31
liquid storage tank model, 32, 34–35
RC circuit, 32–34
rule of thumb, 31
stiffness property in, 524–526
temperature-controlled chamber, 35–36
trapezoidal integration for, 106–107
Fishery system dynamics
block diagram, 593
equilibrium states, 593–594
growth rate and equilibrium points, 592
Simulink® diagram, 591
state derivative function, 589–590
state responses, 592
Forward rectangular integration, 100
Fourier coefficients, 423, 424–426, 428Index 825
Fourier Series expansion, 423–424
Frequency response
control system toolbox, 311–313
function, 287, 540
LTI continuous-time systems
Bode plot for second-order systems, 214–215
circuit with high-pass filter transfer function, 216
closed-loop frequency response functions, 213
first-order system, 208–210
Fourier integral, 207
linear feedback control systems, 216–219
step responses for second-order systems, 215–216
third-order Butterworth low-pass filter, 211
unity feedback control system, 212
LTI discrete-time systems, 280
digital filters, 293–297
properties of, 282–283
sampling theorem, 287–288
steady-state sinusoidal response, 280–282
Friction, 386–387
G
Global truncation error, 479
Gradient search algorithm, 611–619
Gradient vector, 605–607
Graphical user interfaces (GUIs), 349
H
Hardware-in-the-loop (HIL) simulation, 767–768
Hemispherical tank-filling simulation
gradient search algorithm, flow chart, 616
objective function contours, 618–619
objective function surface, 614–615, 627–628
Simulink diagram, 615
Heun’s method, 128
Higher-order systems, 490–496
Higher-order systems, continuous-time system
aircraft pitch control system, 56
feedback control system, 55
railroad cars, 57, 65
High-order Runge–Kutta methods, 484–485
HIL simulation, see Hardware-in-the-loop (HIL) simulation
Human circulatory system, 395
Hybrid systems, continuous-and discrete-time
components, 431–433
Hysteresis, 389–391
I
IDE, see Integrated development environment
Implicit Euler integration, 100–102, 133
of continuous model, 115
difference equation based on, 116
numerical integrator, 100
Implicit methods, multistep methods, 515–518
Impulse response, LTI systems, 175–179
spring-mass-damper system
differential equation model of, 175–176
Laplace transform, 177–178
and transfer function, 179–182
Impulse responses
for discrete-time systems, 274
function, 261–265
graphs of, 274
Inherently discrete-time system, 403–406
Integrated development environment (IDE), 805
Integration step, 475
Intermediate numerical integration, 475
adaptive techniques, 500
adaptive step size, 505
constant step size, 505
repeated Runge–Kutta with interval halving,
500–505
Runge–Kutta–Fehlberg method, 505–510
epidemic model
baseline conditions, 575–577
fatal disease, 573
immigration and inoculation profiles, 574–575
sensitivity analysis, 578–579
S-I-R models, 573
state transition diagram, 574
symptoms, 573
lumped parameter approximation, 546–550
nonlinear distributed parameter system, 550–555
multistep methods, 512–513
explicit methods, 513–515
implicit methodd, 515–518
predictor–corrector methods, 518–522
Runge–Kutta one-step methods, 475–476
continuous-time models with polynomial solutions,
488–490
higher-order systems, 490–496
high-order Runge–Kutta methods, 484–485
linear system models, 486–488
second-order Runge–Kutta method, 477–479
Taylor Series method, 476–477
truncation errors, 479–484
stiff systems, 523–524
lower-order nonstiff system models, 529–542
stiffness property in first-order system, 524–526
stiff second-order system, 526–529
systems with discontinuities, 555–563
case study, 573–578
physical properties and constant forces, 563–569
Internal heat flows, 547
Interval halving, repeated Runge–Kutta with, 500–505
Inverse Laplace transform, 163–164
Inverse z-transform, 232–233, 239–240
Inverted pendulum, algebraic loop, 374
Iodine distribution, human body
block diagram, 185
compartmental model for, 184
state equations, 184–185
state variable model, 190–192
steady-state iodine levels, 186–187
step response, 187–188
transfer function, 185–187
K
Kalman filtering, 453
continuous-time, 453–454, 457
discrete-time, 454–455
Simulink simulations, see Simulink simulations
steady-state, 454
Kinetic friction, 386826 Index
L
Laplace transform, 524–525
inverse, 163–164
one-sided, 155
pairs for elementary continuous-time signals, 157
partial fraction expansion, 166–172
properties of, 156–163
region of convergence, 156
spring-mass-damper system, 175–177
of system response, 164–166
Lego MindstormsTM NXT
feedback control systems, 804
filtered model
block diagram, 811
discrete-time Kalman filter subsystems, 811
filtered data, 812, 814
function-call subsystem, 810
MATLAB plot, 814
real-time workshop report, 809, 813
signal generation, 811
IDE, 805
installation, 805–806
mechatronics, 803
noisy model
block diagram, 806
ECRobot NXT Blockset, 806
function-call subsystem, 807
MATLABO plot, 808
noisy data, 808
real-time workshop report, 809
signal generation, 809
software and hardware requirements, 805
Linear discrete-time state equations, 256–261
Linear second-order system, 491, 493–494
Linear system analysis
aircraft, longitudinal control of
altitude control system, 330
altitude from steady-state flight conditions, 328
angle of attack and forces, 321
body axis coordinates and Euler angles, 320
elevator response for open-and closed-loop, 332
linearized aircraft pitch response, 325
open-and closed-loop altitude response vs.
time, 329
partial fraction expansion, 327
primary control surface, 321
short period and phugoid modes, 324–326
transfer function, 323, 327, 330
control system toolbox, 300
continuous-/discrete-time system conversion,
309–311
frequency response, 311–313
root locus, 313–316
state-space models, 302–303
state-space/transfer function conversion, 303–305
system interconnections, 305–307
system response, 307–309
transfer function models, 301–302
frequency response, LTI continuous-time systems
Bode plot for second-order systems, 204–215
circuit with high-pass filter transfer function, 216
closed-loop frequency response functions, 213
first-order system, 208–210
Fourier integral, 207
linear feedback control systems, 216–219
step responses for second-order systems, 215–216
third-order Butterworth low-pass filter, 211
unity feedback control system, 212
frequency response, LTI discrete-time systems
digital filters, 293–297
properties, 282–283
sampling theorem, 287–288
steady-state sinusoidal response, 280–282
Laplace transform
inverse, 163–164
one-sided, 155
pairs for elementary continuous-time signals, 157
partial fraction expansion, 166–172
properties of, 156–163
region of convergence, 156
of system response, 164–166
models, 486–488
notch filter for electrocardiograph waveform
magnitude function, 339
magnitude squared function, 339
multinotch filters, 339–346
stability
LTI continuous-time system, 195–206
LTI discrete-time system, 267–280
transfer function
impulse function, 173
impulse response, 175–179
and impulse response, relationship, 179–182
multiple inputs and outputs, 182–189
transformation from state variable model to,
190–194
unit step and unit impulse function, 173–175
z-domain transfer function
approximating continuous-time system transfer
functions, 245–247
definition, 242
Euler integration, 242–244
linear discrete-time state equations, 256–261
monetary fund, 257–258
nonzero initial conditions, 243–244
relationship of impulse response to, 264
simulation diagrams and state variables, 250–256
trapezoidal integration, 249–251
weighting sequence (impulse response function),
261–265
z-transform
discrete-time impulse function, 226–228
discrete-time signal, 222–226
inverse, 232–233, 239–240
Laplace and, 227
partial fraction expansion, 233–234
properties of, 229
table for inverting, 236
Linear systems simulation
state-space block, 363–370
Transfer Fcn block, 357–363
Linear time invariant (LTI), continuous-time systems
frequency response
Bode plot for second-order systems, 214–215
circuit with high-pass filter transfer
function, 216
closed-loop frequency response functions, 213Index 827
first-order system, 208–210
Fourier integral, 207
linear feedback control systems, 216–219
step responses for second-order systems, 215–216
third-order Butterworth low-pass filter, 211
unity feedback control system, 212
stability
feedback control system, 200–206
polynomial characteristic, 195–200
Linear time invariant (LTI), discrete-time systems
frequency response
digital filters, 293–297
properties, 282
sampling theorem, 287–288
steady-state sinusoidal response, 280–282
stability
BIBO, 267
complex poles of H(z), 271–273
impulse response, 268
z-domain transfer function, 267–268
Local truncation error, 479, 500–501, 515
Logistic population growth model, 144
Lookup Table block parameters, 383
Lower-order dynamics model, 534
Lower-order nonstiff system models
RK-4 integrator, 531–532
second-order system, 529, 532
sensor dynamics, 530
Simulink diagram, 531
step response, 531–532
step size vs. step number, 530
third-order system, 530
Low-pass digital filters, 293–297
Lumped parameter approximation, 546–550
nonlinear distributed parameter system, 550–555
Lumped parameter system model, 2, 550
LUNGS subsystem, 395
M
Matched pole-zero method, 796–799
Mathematical modeling
derivation, open tank
dynamic behavior, 4
flow between tanks, 8–9
fluid resistance, 7
volume, liquid flow, 5–6
difference equations, 10–12
discrete-time systems
exact vs. approximate solutions, 13–14
inherent, 19–22
liquid tank continuous-time system, 19
liquid tank discrete-time system, 19
step size, 16–18
dynamic systems, 555
lumped parameter model, 2
population dynamics
discrete-time model, 24
logistic growth population, 26–28
observed, discrete-time and continuous-time
populations, 25
population data, 22, 23
simulation models, 3
stochastic models, 3
MATLAB, 422–428, 436, 457
control system, 309
Fourier Series, 422–423
function, 531, 559
optimization toolbox, 599–600, 630
second-order system, 356, 426
truncated Fourier Series, 424–425
Workspace, 354, 356, 383
Memory block, algebraic loops, 373–375
MIMO, see Multiple input-multiple output
Modified Euler integration, 131–132, 133–134
Monte Carlo simulation, 435–439, 629
hospital occupancy, 623–624
mathematical model, 439–445
Multinotch filters, 339–346
input and output of, 341, 342, 346
magnitude function, 340, 341, 344, 345
magnitude squared function, 339, 340
for removing fundamental frequency, 345
Multiple input–multiple output (MIMO) system, 363
electric circuit, 182
iodine distribution, human
block diagram, 185
compartmental model for, 184
state equations, 184–185
state variable model, 190–192
steady-state iodine levels, 186–187
step response, 187–188
transfer function, 185–187
Multirate integration
aircraft pitch control system, 739
analytical, Simulink, and multirate
responses, 751
Simulink and multirate integration, 751
airframe dynamics, 739
analytical solution, state variables
advantage, 748
total elevator deflection and its
components, 750
total pitch response and its components, 749
frame ratio, 741
master routine, 741
nonlinear dual speed second-order system
air pressure, 754
coefficient matrix, 756
eigenvalues, 756–757
linmod function, 757–758
Simulink diagram, 758
two tank system, 753–754
procedure, slow and fast states, 742–743
simulation trade-offs
cpu time, 763–764
total execution time, 763
slave routine, 741
slow and fast subsystem interaction, 741
step size selection
dynamic accuracy, 745–748
stability, 743–745
stiff system, 738
two-tank system, 760–763
Multistep methods, 512–513
explicit methods, 513–515
implicit method, 515–518
predictor–corrector methods, 518–522828 Index
N
Nonlinear algebraic equations, 377
Nonlinear distributed parameter system, 550–555
Nonlinear dual speed second-order system
air pressure, 754
coefficient matrix, 756
eigenvalues, 756–757
linmod function, 757–758
Simulink diagram, 758
two tank system, 753–754
Nonlinear first-order systems, discrete approximation of, 112
continuous model for sinking drum, 113–116
exact solution for depth, 114
implicit numerical integrators, 112–113
object falling in a viscous medium, 116
Nonlinear systems
continuous-time systems
applied force vs. time, 69–70
backlash, 72, 73
coulomb friction, 68
dead zone, 71
first-order systems, 66
friction force vs. applied force, 68–69
hysteresis, 72–74
linear model approximation, 68
mechanical system, 81
progressive, 68
quantization, 76–77
saturation, 71–72
sustained oscillations and limit cycles, 77–80
temperature response, 74–76
time constant, 75
valve flow vs. current, 72
Nonstiff control system models, step response, 533
Notch filter, for electrocardiograph waveform, 338
input and output of, 345
magnitude function, 339, 344
magnitude squared function, 339
multinotch filters, 339–346
input and output of, 341, 342, 346
magnitude function, 340, 341, 344, 345
magnitude squared function, 339, 340
for removing fundamental frequency, 345
square wave noise
components of ECG signal, 342
noise-corrupted ECG signal, 343
Numerical integration methods, 520
Nyquist frequency, 288
O
One-sided Laplace transform, 155
One-step methods, 475–476, 515
Optimization, Simulink
discrete-time system models, 620–625
gradient vector, 605–607
ground vehicle performance, 596
MATLAB optimization, 599–600, 630
minimum separation, 604
multiparameter objective functions, 607–610
optimum firing angle, 600–601
parameter identification, 610–611
projectile firing angle, 598–599
separation distance vs. time, 603
simple gradient search, 611–619
target and projectile system, 597–598
target speed sensitivity analysis, 602
P
Parameter Estimation, 581
Parenthesis, 547
Partial differential equation models, 1–2
Partial fraction expansion
coefficients, 260
Laplace transform
complex roots, 169–172
real and at least one multiple root, 167–169
real and distinct roots, 166–167
z-transform, 233–234
Pendulum bob dynamics, 564, 565
drag force, 566
physical properties and constant forces, 563–569
simulation of, 560
velocity, 565, 566
Periodic signals, 158
PHYSBE, 395–396
Physical models, 1
Pilot ejection, 448–452
diagram, 448
Simulink diagram, 451
trajectory of, 449
Pitch control system transfer function, 746
Plot of discontinuity functions, 567
Polynomial characteristic
asymptotic stability, 197
bounded input-bounded output, 197
higher order LTI system, 198
MIMO systems, 198–199
poles, natural modes, and stability, 197
stability of second-order linear system, 196
Population dynamics
discrete-time model, 24
logistic growth population, 26–28
observed, discrete-time and continuous-time
populations, 25
population data, 22, 23
Posteriori covariance subsystem, 466
Posteriori state subsystem, 466
Predator-prey
ecosystem, 125–126
model, 583–584, 595–596
Predictor–corrector methods, 518–522
Priori covariance subsystem, 465
Priori state subsystem, 464
Progressive nonlinearity, 68
Public safety organizations, 1
Q
Quadratic interpolation, 562
Quantization block, 391–392, 391–394
Quantization nonlinearity, 76–77
R
Real-time HIL simulation, 767–768
Real-time predictor–corrector method, 774–776Index 829
Real-time simulation
extrapolation
fractional error, 779
ideal extrapolator, 778–779
linear, 778
magnitude and phase plots, 780
uses, 777
input delay
fraction gain error, 785
phase angles, 786
phase error, 785
thermal system, 786–789
uses, 783
z-domain transfer function, 784
Repeated Runge–Kutta with interval halving, 500–505
Response Optimization, 581
RK-Fehlberg method, 505–510
boat crossing, 507–510
RK-1 integrators, 479–481, 483, 486
RK-2 integrators, 479–481, 483, 486, 528
RK-3 integrators, 484–485, 486
RK-4 integrators, 485, 486, 491, 505–506, 525–526, 567
RK-5 integrators, 486, 505–506
RK-6 integrators, 486
RK method, see Runge–Kutta (RK) method
Root locus, control system toolbox, 313–316
Runge–Kutta integration, 358
Runge–Kutta (RK) method
characteristic root errors, 730–731
modified Euler integration, 727–728
one-step methods, 475–476
continuous-time models with polynomial solutions,
488–490
higher-order systems, 490–496
high-order Runge–Kutta methods, 484–485
linear system models, 486–488
second-order Runge–Kutta method, 477–479
Taylor Series method, 476–477
truncation errors, 479–484
polynomials, 730
speed control system
analytical and RK-2 simulation, 734
analytical and RK-4 simulation, 735
analytical step response and RK-3 simulated
response, 736
block diagram, 733
RK-3 stability boundary, 734, 735
Simulink diagram, 734
z-domain transfer function, 729–730
S
Sampled sinusoid, aliasing of, 288
Sampling theorem, 287–288
Saturation block, 387–389
Saturation nonlinearity, 71–72
Second-order continuous-time, P-I control of, 275
Second-order RLC circuit, 527
Second-order Runge–Kutta method, 477–479
Second-order systems, 526–529, 555
Adams–Bashforth numerical integrators, 720–722
Bode plot, 214–215
characteristic polynomial, 196
description, 36
first-order equation conversion, 41–42
mechanical system
block diagram, 39
damping ratio and natural frequency, 39–40, 42
position and velocity response, 41
steady-state gain, 39, 42–43
transient period, 40
nonlinear dual speed
air pressure, 754
coefficient matrix, 756
eigenvalues, 756–757
linmod function, 757–758
Simulink diagram, 758
steady-state operating levels, 757
two tank system, 753–754
oscillatory step response, 38
phase angle term, 37
poles, natural modes, and stability, 197
response, 355
simulation diagrams, 53, 58–59
step responses, 215–216, 352
two-tank mixing system, 42–45
unit step response, 36–39
z-domain transfer function, 273–277
Second-order truncated Taylor Series method, 479
Ship heading control system
block diagram, 608
control parameters, 608
feedback control system, 200–206
objective function, 608–609
optimal parameter settings, 611
Simulink block diagram, 610
Simulated response, 569
using Euler integration, 539
Simulation diagrams
airframe dynamics, 739
continuous-time systems
aircraft pitch control system, 56
description, 45
first-order system, 45–48
heat flows and temperatures, two-room
building, 51–52
room temperature model, 51–52
second-order system, 48–49, 53–54, 58–59
fast subsystem, 741
nth-order continuous-time system, 252, 253
for RC lead-lag network, 119
second-order system
trapezoidal integration, 251
state variables and, 250–256
third-order system, 180
Simulation models, 3
Simulation tools
iterative procedure, 582
optimization, Simulink
discrete-time system models, 620–625
gradient vector, 605–607
ground vehicle performance, 596
MATLAB optimization toolbox, 599–600, 630
minimum separation, 604
multiparameter objective functions, 607–610
optimum firing angle, 600–601, 625–627
parameter identification, 610–611
projectile firing angle, 598–599830 Index
Simulation tools (Continued)
separation distance vs. time, 603
simple gradient search, 611–619
target and projectile system, 597–598
target speed sensitivity analysis, 602
steady-state solver
equilibrium point, nonautonomous system,
586–589
nonlinear state model, 582
predator-prey model, 583–584
trim function, 584–586
Simulink, 349
algebraic loops, see Algebraic loops
blocks, 380–385
acceleration response, 381
backlash, 389
car-following models, 380, 381–382, 384
dead zone and saturation, 387–389
discontinuities, 385–386
friction, 386–387
hysteresis, 389–391
lead and following vehicles, 380, 384–385
Lookup Table block parameters, 383
quantization, 391–392, 391–394
continuous-and discrete-time components, 431–433
diagram, 508
arrow and target simulation, 441
capacitive transducer, 588
car-following system, 383, 397, 398
cascaded tanks, 471
closed-loop depth rate control system, 363
continuous-time Kalman filter, 457
digital control system for chamber temperature, 432
explicit Euler integration, 411
first-and second-order models, 532
fishery system dynamics, 591
hemispherical tank-filling simulation, 615
hospital occupancy, 621
inverted pendulum, 400, 644
loan repayment, 405
low-pass filters, 417
lumped parameter system model, 550
nonlinear two-tank system, 652
nonlinear vs. linearized models, 646
notch filter, 413
pendulum dynamics, 564
PHYSBE model, 396
pilot ejection, 451
Relay block for thermostat, 391
second-order system, 357, 410, 532
ship heading step response, 610
simulating stiff control system dynamics, 531
for simulation of nonlinear and linearized
system, 636
solving algebraic equations, 376
sub depth control, 358
submarine depth rate., 361
third-order control systems, 532
truncated Fourier Series, 424
vehicle response traveling, 367
vehicle rolling down incline, 408
discrete-time systems, 402–403
centralized integration, 409–412
digital filters, 412–414
integrators, 406–409
simulation of inherently, 403–406
transfer function, 414–418
interface, 422–428
Kalman filtering, 453
continuous-time, 453–454
discrete-time, 454–455
Simulink simulations, see Simulink simulations
steady-state, 454
MATLAB, see MATLAB
model, 349, 353–355, 357
data logging of scope signals, 355
dialog box for configuring, 353
Euler integrator, 353
inverted pendulum with “Memory” block, 374
for RLC circuit, 528
running Simulink, 353–355
scope output, 354
screen capture, 354
second-order system response, 355
simulating coffee pot, 554
Simulink library, 349–353
model optimization
discrete-time system models, 620–625
gradient vector, 605–607
ground vehicle performance, 596
MATLAB optimization toolbox, 599–600
minimum separation, 604
multiparameter objective functions, 607–610
optimum firing angle, 600–601, 625–627
parameter identification, 610–611
projectile firing angle, 598–599
separation distance vs. time, 603
simple gradient search, 611–619
target and projectile system, 597–598
target speed sensitivity analysis, 602
Monte Carlo simulation, 435–439
mathematical model, 439–445
pilot ejection, 448–452
simulation of linear systems, 357
state-space block, 363–370
Transfer Fcn block, 357–363
subsystems, 394–395
car-following, 396–398
Fcn block, 398–401
PHYSBE, 395–396
Simulink library
blocks, 349–350
Browser, 350, 385
Discontinuities, 387
second-order system step response, 352
step response of second-order system, 352
Simulink optimization, hospital-patient occupancy
block diagram, 621
daily arrivals and departures, 620
daily net patient input, 621, 623
input and output relationship, 620
Monte Carlo simulation, 623–624, 629
objective function, 624
patient profiles, 621–622
Simulink simulations, 455–468
actual subsystem, 456
continuous-time Kalman filter, 456–457
discrete-time Kalman filter, 462, 464Index 831
Kalman gain subsystem, 465
plot of
acceleration, 459, 462, 468
range, 458
range error vs. time, 459, 463, 468
range estimates, 461, 467
velocity, 458, 462, 467
velocity error vs. time., 460, 463, 469
posteriori covariance subsystem, 466
posteriori state subsystem, 466
priori covariance subsystem, 465
priori state subsystem, 464
steady-state Kalman filter algorithm, 461
Simulink’s stiff integrators, 526
Single input-single output (SISO), 363
Spring-mass-damper system, 57
differential equation model of, 175–176
impulse response, 177–178
Stability, linear time invariant
continuous-time system
characteristic polynomial, 195–200
feedback control system, 200–206
discrete-time systems
BIBO, 267
complex poles of H(z), 271–273
impulse response, 268
z-domain transfer function, 267–268
linear feedback control systems, 216–219
State derivative function, 475
State-space block, 363–370
moving vehicle and suspension system model, 365
vehicle cab displacement, 368
State-space models, 302–303
State variable model, simulation of, 335–337
State variables, simulation diagrams and, 250–256
Steady-state Kalman filter, 454
Steady-state solver
equilibrium point, nonautonomous system, 586–589
nonlinear state model, 582
predator-prey model, 583–584
trim function, 584–586
Step response of second-order system, 352
Stiff control system models, step response, 533
Stiff integrators, 529
Stiffness property in first-order system, 524–526
Stiff second-order system, 526–529
Stiff systems, 523–524
lower-order nonstiff system models, 529–542
stiffness property in first-order system, 524–526
stiff second-order system, 526–529
Stochastic models, 3
Submarine depth control system, 358
block diagram, 85
closed-loop transfer function, 362
controller and stern plane actuator, 89–90
difference equations, 88
discrete-time approximation, 89
simulation diagram, 86
state equations, 86–87
state-space models, 303–305
Submarine dynamics transfer function, 374
Subsystems, 394–395
car-following, 396–398
Fcn block, 398–401
PHYSBE, 395–396
Tire Model, 395
vehicle dynamics model, 395
System interconnections, 305–307
System response, 307–309
Systems with discontinuities, 555–563
case study, 573–578
physical properties and constant forces, 563–569
T
Taylor Series method, 476–477, 479, 480, 483, 488–490
Tire Model, 395
Transfer Fcn block, 373
command and actual submarine depth rates, 359
second-order system, 357, 358
submarine depth rate control system, 358, 360
Transfer function, 414–418
conversion, 303–305
errors
continuous-and discrete-time integration, 702
explicit Euler and continuous-time integrator
outputs, 703
fractional error, 697–699
frequency response functions, 697
phase angle plots, 701
time delay, 703–704
of linear systems analysis
impulse function, 173
impulse response, 175–179
and impulse response, relationship, 179–182
multiple inputs and outputs, 182–189
transformation from state variable model to,
190–194
unit step and unit impulse function, 173–175
models, 301
Trapezoidal integration, 104–111, 249–251, 254, 255
area approximation, 104–105
continuous-and discrete-time, 251
continuous integrators, 105–106, 108
continuous-time first-order system in, 288–293
difference equation based on, 105, 107
discrete and continuous responses, 108–109, 110, 111
discrete integrators, 105–106, 108
dynamics of sinking drum, 109–110
for first-order system, 106–107
integration step size, 104, 111
of nonlinear time-varying system, 107–108
of second-order system, 255
state equations for, 255
of underdamped second-order system, 256
Trim function, 584–586
Truncated Fourier Series, 424–425
Twente University of Technology Simulator (TUTW), 349
U
Undamped pendulum response, using explicit Euler, 144
Unit impulse function, 173–175
Unit step function, 173–175
Unit step responses
first-and second-order system models, 538
unstable second-order model, 539
weighting sequences and, 264832 Index
V
Variable capacitance transducer
circuit diagram, 586
dynamic system with equilibrium conditions, 589
mathematical model, 586–587
Simulink diagram, 588
Vehicle dynamics model, 767–768
Vehicle response traveling, 367
Vertical ascent of diver, 146
air, 146
cable forces, 146
discrete differential pressure responses, 152
discrete-time system
equilibrium state, 149
outputs, 148, 151
state equation matrices, 148
state variables, 150
diver’s internal body pressure, 146
drag force, 146
dynamic system, 146–147
initial cable force, 148–149
maximum cable force, 152–153
net cable force, 147
second-order differential equation, 147, 153
third order linear dynamic system, 147
W
Weighting sequence (impulse response function), 261–265
Z z
-domain transfer function
approximating continuous-time system transfer
functions, 245–247
definition, 242
Euler integration, 242–244
linear discrete-time state equations, 256–261
monetary fund, 257–258
nonzero initial conditions, 243–244
relationship of impulse response to, 264
simulation diagrams and state variables, 250–256
trapezoidal integration, 249–251
weighting sequence (impulse response function), 261–265
z-transform
discrete-time impulse function, 226–228
discrete-time signal, 222–226
inverse, 232–233, 239–240
Laplace and, 227
partial fraction expansion, 233–234
properties of, 229
table for inverting

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