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عدد المساهمات : 19002 التقييم : 35506 تاريخ التسجيل : 01/07/2009 الدولة : مصر العمل : مدير منتدى هندسة الإنتاج والتصميم الميكانيكى
| موضوع: كتاب Simulation of Dynamic Systems with MATLAB and Simulink الخميس 17 أغسطس 2023, 1:36 am | |
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أخواني في الله أحضرت لكم كتاب Simulation of Dynamic Systems with MATLAB and Simulink Third Edition Harold Klee and Randal Allen
و المحتوى كما يلي :
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 #ماتلاب,#متلاب,#Matlab,
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