Introduction to Modeling and Simulation A Systems Approach

An essential introduction to engineering system modeling and simulation from a well-trusted source in engineering and education

This new introductory-level textbook provides thirteen self-contained chapters, each covering an important topic in engineering systems modeling and simulation. The importance of such a topic cannot be overstated; modeling and simulation will only increase in importance in the future as computational resources improve and become more powerful and accessible, and as systems become more complex. This resource is a wonderful mix of practical examples, theoretical concepts, and experimental sessions that ensure a well-rounded education on the topic.

The topics covered in Introduction to Modeling and Simulation are timeless fundamentals that provide the necessary background for further and more advanced study of one or more of the topics. The text includes topics such as linear and nonlinear dynamical systems, continuous-time and discrete-time systems, stability theory, numerical methods for solution of ODEs, PDE models, feedback systems, optimization, regression and more. Each chapter provides an introduction to the topic to familiarize students with the core ideas before delving deeper. The numerous tools and examples help ensure students engage in active learning, acquiring a range of tools for analyzing systems and gaining experience in numerical computation and simulation systems, from an author prized for both his writing and his teaching over the course of his over-40-year career.

Introduction to Modeling and Simulation readers will also find:

• Numerous examples, tools, and programming tips to help clarify points made throughout the textbook, with end-of-chapter problems to further emphasize the material
• As systems become more complex, a chapter devoted to complex networks including small-world and scale-free networks – a unique advancement for textbooks within modeling and simulation
• A complementary website that hosts a complete set of lecture slides, a solution manual for end-of-chapter problems, MATLAB files, and case-study exercises

Introduction to Modeling and Simulation is aimed at undergraduate and first-year graduate engineering students studying systems, in diverse avenues within the field: electrical, mechanical, mathematics, aerospace, bioengineering, physics, and civil and environmental engineering. It may also be of interest to those in mathematical modeling courses, as it provides in-depth material on MATLAB simulation and contains appendices with brief reviews of linear algebra, real analysis, and probability theory.

Mark W. Spong, DSc, is Professor of Systems Engineering and Professor of Electrical and Computer Engineering at the University of Texas, Dallas, where he also holds the Excellence in Education Chair. Professor Spong received his doctorate in systems science and mathematics in 1981 from Washington University in St. Louis, MO. He is a Life Fellow of the IEEE and a Fellow of IFAC. Among the numerous notable awards he has received are the 2011 Pioneer in Robotics Award from the IEEE Robotics and Automation Society, the 2020 Rufus Oldenburger Medal from the ASME, the 2018 Bode Lecture Prize from the IEEE Control Systems Society, the 2016 Nyquist Lecture Prize from the ASME, and the IEEE Transactions on Control Systems Technology Outstanding Paper Award.

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

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