Dynamic Systems. Simulation of. with MATLAB and Simulink. Harold Klee. Randal Allen SECOND EDITION. CRC Press. Taylor & Francis Group

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1 SECOND EDITION Simulation of Dynamic Systems with MATLAB and Simulink Harold Klee Randal Allen CRC Press Taylor & Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business

2 Contents Foreword, Preface Authors X1U xv X1X Chapter 1 Mathematical Modeling Introduction Importance of Models Derivation of a Mathematical Model 4 Exercises Difference Equations Recursive Solutions 11 Exercises First Look at Discrete-Time Systems Inherently Discrete-Time Systems 17 Exercises Case Study: Population Dynamics (Single Species) 21 Exercises 28 Chapter 2 Continuous-Time Systems Introduction First-Order Systems Step Response of First-Order Systems 32 Exercises Second-Order Systems Conversion of Two First-Order Equations to a Second-Order Model 43 Exercises Simulation Diagrams Systems of Equations 53 Exercises Higher-Order Systems 56 Exercises State Variables Conversion from Linear State Variable Form to Single Input-Single Output Form General Solution of the State Equations 65 Exercises Nonlinear Systems Friction Dead Zone and Saturation Backlash Hysteresis Quantization Sustained Oscillations and Limit Cycles 78 v

3 vi Contents Exercises Case Study: Submarine Depth Control System 85 Exercises 89 Chapter 3 Elementary Numerical Integration Introduction Discrete-Time System Approximation of a Continuous-Time Integrator 92 Exercises Euler Integration B ackward (Implicit) Euler Integration 99 Exercises Trapezoidal Integration 102 Exercises Numerical Integration of First-Order and Higher Continuous-Time Systems Discrete-Time System Models from Simulation Diagrams Nonlinear First-Order Systems Discrete-Time State Equations Discrete-Time State System Matrices 118 Exercises Improvements to Euler Integration Improved Euler Method Modified Euler Integration 125 Exercises Case Study: Vertical Ascent of a Diver Maximum Cable Force for Safe Ascent Trial and Error Analytical Solution Diver Ascent with Decompression Stops 145 Exercises 147 Chapter 4 Linear Systems Analysis Introduction Laplace Transform Properties of the Laplace Transform Inverse Laplace Transform Laplace Transform of the System Response Partial Fraction Expansion 161 Exercises Transfer Function Impulse Function Relationship between Unit Step Function and Unit Impulse Function Impulse Response Relationship between Impulse Response and Transfer Function Systems with Multiple Inputs and Outputs Transformation from State Variable Model to Transfer Function 184 Exercises 187

4 Contents 4.4 Stability of Linear Time Invariant Continuous-Time Systems Characteristic Polynomial Feedback Control System 194 Exercises Frequency Response of LTI Continuous-Time Systems Stability of Linear Feedback Control Systems Based on Frequency Response 210 Exercises z-transform Discrete-Time Impulse Function Inverse z-transform Partial Fraction Expansion 226 Exercises z-domain Transfer Function Nonzero Initial Conditions Approximating Continuous-Time System Transfer Functions Simulation Diagrams and State Variables Solution of Linear Discrete-Time State Equations Weighting Sequence (Impulse Response Function) 253 Exercises Stability of LTI Discrete-Time Systems Complex Poles of H(z) 263 Exercises Frequency Response of Discrete-Time Systems Steady-State Sinusoidal Response Properties of the Discrete-Time Frequency Response Function Sampling Theorem Digital Filters 284 Exercises Control System Toolbox Transfer Function Models State-Space Models State-Space/Transfer Function Conversion System Interconnections System Response Continuous-/Discrete-Time System Conversion Frequency Response Root Locus 305 Exercises Case Study: Longitudinal Control of an Aircraft Digital Simulation of Aircraft Longitudinal Dynamics Simulation of State Variable Model 327 Exercises Case Study: Notch Filter for Electrocardiograph Waveform Multinotch Filters 331 Exercises 338 Chapter 5 Simulink Introduction Building a Simulink Model 341

5 vjjj Contents Simulink Library Running a Simulink Model 345 Exercises Simulation of Linear Systems Transfer Fen Block State-Space Block 353 Exercises Algebraic Loops Eliminating Algebraic Loops Algebraic Equations 367 Exercises More Simulink Blocks Discontinuities Friction Dead Zone and Saturation Backlash Hysteresis Quantization 381 Exercises Subsystems PHYSBE Car-Following Subsystem Subsystem Using Fen Blocks 389 Exercises Discrete-Time Systems Simulation of an Inherently Discrete-Time System Discrete-Time Integrator Centralized Integration Digital Filters Discrete-Time Transfer Function 404 Exercises MATLAB and Simulink Interface 411 Exercises Hybrid Systems: Continuous- and Discrete-Time Components 420 Exercises Monte Carlo Simulation Monte Carlo Simulation Requiring Solution of a Mathematical Model 428 Exercises Case Study: Pilot Ejection 437 Exercises Case Study: Kalman Filtering Continuous-Time Kalman Filter Steady-State Kalman Filter Discrete-Time Kalman Filter Simulink Simulations Summary 455 Exercise 456

6 Contents '* Chapter 6 Intermediate Numerical Integration Introduction Runge-Kutta (RK) (One-Step Methods) Taylor Series Method Second-Order Runge-Kutta Method Truncation Errors High-Order Runge-Kutta Methods Linear Systems: Approximate Solutions Using RK Integration Continuous-Time Models with Polynomial Solutions Higher-Order Systems 471 Exercises Adaptive Techniques Repeated RK with Interval Halving Constant Step Size <T= 1 min) Adaptive Step Size (Initial T= 1 min) RK-Fehlberg 486 Exercises Multistep Methods Explicit Methods Implicit Methods Predictor-Corrector Methods 498 Exercises Stiff Systems Stiffness Property in First-Order System Stiff Second-Order System Approximating Stiff Systems with Lower-Order Nonstiff System Models 509 Exercises Lumped Parameter Approximation of Distributed Parameter Systems Nonlinear Distributed Parameter System 531 Exercises Systems with Discontinuities Physical Properties and Constant Forces Acting on the Pendulum BOB 543 Exercises Case Study: Spread of an Epidemic 552 Exercises 559 Chapter 7 Simulation Tools Introduction Steady-State Solver Trim Function Equilibrium Point for a Nonautonomous System 565 Exercises Optimization of Simulink Models Gradient Vector Optimizing Multiparameter Objective Functions Requiring Simulink Models 587

7 x Contents Parameter Identification Example of a Simple Gradient Search Optimization of Simulink Discrete-Time System Models 599 Exercises Linearization Deviation Variables Linearization of Nonlinear Systems in State Variable Form Linmod Function Multiple Linearized Models for a Single System 627 Exercises Adding Blocks to the Simulink Library Browser Introduction Summary 645 Exercise Simulation Acceleration Introduction Profiler Summary 647 Exercise 648 Chapter 8 Advanced Numerical Integration Introduction Dynamic Errors (Characteristic Roots, Transfer Function) Discrete-Time Systems and the Equivalent Continuous-Time Systems Characteristic Root Errors Transfer Function Errors Asymptotic Formulas for Multistep Integration Methods Simulation of Linear System with Transfer Function H(s) 672 Exercises Stability of Numerical Integrators Adams-Bashforth Numerical Integrators Implicit Integrators Runga-Kutta (RK) Integration 692 Exercises Multirate Integration Procedure for Updating Slow and Fast States: Master/Slave RK-4/RK-4 = Selection of Step Size Based on Stability Selection of Step Size Based on Dynamic Accuracy Analytical Solution for State Variables Multirate Integration of Aircraft Pitch Control System Nonlinear Dual Speed Second-Order System Multirate Simulation of Two-Tank System Simulation Trade-Offs with Multirate Integration 725 Exercises Real-Time Simulation Numerical Integration Methods Compatible with Real-Time Operation RK-1 (Explicit Euler) 734

8 Contents RK-2 (Improved Euler) RK-2 (Modified Euler) RK-3 (Real-Time Incompatible) RK-3 (Real-Time Compatible) RK-4 (Real-Time Incompatible) Multistep Integration Methods Stability of Real-Time Predictor-Corrector Method Extrapolation of Real-Time Inputs Alternate Approach to Real-Time Compatibility: Input Delay 746 Exercises Additional Methods of Approximating Continuous-Time System Models Sampling and Signal Reconstruction First-Order Hold Signal Reconstruction Matched Pole-Zero Method Bilinear Transform with Prewarping 763 Exercises Case Study: Lego Mindstorms NXT Introduction Requirements and Installation Noisy Model Filtered Model Summary 779 Exercise 779 References 781 Index 785

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