Introduction to Modeling and Simulation

Preface xiii About the Companion Website xvii 1 Introduction 1 1.1 Introduction 1 1.1.1 Systems Engineering 1 1.1.2 The Input/Output Viewpoint 2 1.1.3 Some Examples 2 1.2 Model Classification 5 1.2.1 Static and Dynamic Systems 5 1.2.2 Linear and Nonlinear Systems 5 1.2.3 Distributed-Parameter Systems 6 1.2.4 Hybrid and Discrete-Event Systems 6 1.2.5 Deterministic and Stochastic Systems 7 1.2.6 Large-Scale Systems 7 1.3 Simulation Languages 9 1.4 Outline of the Text 10 Problems 11 2 Second-Order Systems 15 2.1 Introduction 15 2.2 State-Space Representation 19 2.3 Trajectories and Phase Portraits 22 2.4 The Direction Field 27 2.5 Equilibria 30 2.6 Linear Systems 33 2.7 Linearization of Nonlinear Systems 41 2.8 Periodic Trajectories and Limit Cycles 45 2.8.1 Relaxation Oscillators 45 2.8.2 Bendixson’s Theorem 49 2.8.3 Poincaré–Bendixson Theorem 51 2.9 Coupled Second-Order Systems 53 Problems 55 3 System Fundamentals 61 3.1 Introduction 61 3.2 Existence and Uniqueness of Solution 61 3.3 The Matrix Exponential 64 3.4 The Jordan Canonical Form 67 3.5 Linearization 71 3.6 The Hartman–Grobman Theorem 72 3.7 Singular Perturbations 73 Problems 79 4 Compartmental Models 83 4.1 Introduction 83 4.2 Exponential Growth and Decay 84 4.3 The Logistic Equation 87 4.4 Models of Epidemics 88 4.5 Predator–Prey System 95 Problems 97 5 Stability 101 5.1 Introduction 101 5.2 Lyapunov Stability 102 5.3 Basin of Attraction 109 5.4 The Invariance Principle 110 5.5 Linear Systems and Linearization 113 Problems 116 6 Discrete-Time Systems 119 6.1 Introduction 119 6.2 Stability of Discrete-Time Systems 123 6.3 Stability of Discrete-Time Linear Systems 124 6.4 Moving-Average Filter 126 6.5 Cobweb Diagrams 128 6.5.1 Cobweb Diagrams in Economics 130 6.5.2 The Discrete Logistic Equation 131 Problems 134 7 Numerical Methods 137 7.1 Introduction 137 7.2 Numerical Differentiation 138 7.3 Numerical Integration 141 7.4 Numerical Solution of ODEs 147 7.4.1 Euler Predictor–Corrector Method 150 7.4.2 Runge–Kutta Methods 152 7.5 Stiff Systems 155 7.6 Event Detection 160 7.7 Simulink 163 7.8 Summary 168 Problems 169 8 Optimization 173 8.1 Introduction 173 8.2 Unconstrained Optimization 177 8.2.1 Iterative Search 179 8.2.2 Gradient Descent 180 8.2.3 Newton’s Method 184 8.3 Case Study: Numerical Inverse Kinematics 187 8.4 Constrained Optimization 191 8.4.1 Equality Constraints 191 8.4.2 Inequality Constraints 196 8.5 Convex Optimization 200 Problems 204 9 System Identification 209 9.1 Introduction 209 9.2 Least Squares 209 9.3 Regression 212 9.4 Recursive Least Squares 217 9.5 Logistic Regression 220 9.6 Neural Networks 224 Problems 230 10 Stochastic Systems 233 10.1 Markov Chains 233 10.1.1 Regular and Ergodic Markov Chains 240 10.1.2 Absorbing Markov Chains 244 10.2 Monte Carlo Methods 249 10.2.1 Random Number Generation 250 10.2.2 Monte Carlo Integration 253 10.2.3 Monte Carlo Optimization 255 10.2.4 Monte Carlo Simulation 255 Problems 258 11 Feedback Systems 261 11.1 Introduction 261 11.2 Transfer Functions 263 11.3 Feedback Control 269 11.4 State-Space Models 273 11.4.1 Minimal Realizations 274 11.4.2 Pole Placement 280 11.4.3 State Estimation 283 11.4.4 The Separation Principle 285 11.5 Optimal Control 288 11.6 Control of Nonlinear Systems 289 Problems 292 12 Partial Differential Equation Models 297 12.1 Introduction 297 12.1.1 Existence and Uniqueness of Solutions 297 12.1.2 Classification of Linear Second-Order PDEs 298 12.2 The Wave Equation 299 12.2.1 The D’Alembert Solution 300 12.2.2 Initial-Value Problem 300 12.2.3 Separation of Variables 302 12.3 The Heat Equation 310 12.4 Laplace’s Equation 313 12.5 Numerical Solution of PDEs 315 Problems 319 13 Complex Networks 321 13.1 Introduction 321 13.1.1 Examples of Complex Networks 322 13.2 Graph Theory: Basic Concepts 324 13.2.1 Graph Isomorphism 327 13.2.2 Connectivity 327 13.2.3 Trees 331 13.2.4 Bipartite Graphs 332 13.2.5 Planar Graphs 333 13.2.6 Graphs and Matrices 335 13.3 Matlab Graph Functions 341 13.4 Network Metrics 343 13.4.1 Degree Distribution 343 13.4.2 Centrality 347 13.4.3 Clustering 350 13.5 Random Graphs 354 13.5.1 Erdős–Rényi Networks 354 13.5.2 Small-World Networks 358 13.5.3 Scale-Free Networks 360 13.6 Synchronization in Networks 362 Problems 366 Appendix A Linear Algebra 371 A. 1 Vectors 371 A. 2 Matrices 373 A. 3 Eigenvalues and Eigenvectors 375 Appendix B Real Analysis 379 B. 1 Set Theory 379 B. 2 Vector Fields 380 B. 3 Jacobian 381 B. 4 Scalar Functions 381 B. 5 Taylor’s Theorem 382 B. 6 Extreme-Value Theorem 383 Appendix C Probability 385 C.1 Discrete Probability 385 C.2 Conditional Probability 386 C.3 Random Variables 389 C.4 Continuous Probability 391 Appendix D Proofs of Selected Results 395 D. 1 Proof of Theorem 2.2 395 D. 2 Proof of Theorem 5.1 395 D. 3 Proof of Theorem 5.5 396 D. 4 Proof of Theorem 13.3 397 D. 5 Proof of Corollary 13.2 397 D. 6 Proof of Proposition 13.2 398 D. 7 Proof of Proposition 13.3 398 Appendix E Matlab Command Reference 399 References 403 Index 407