Introduction to Modeling and Simulation 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.
Table of Contents:
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
