Table of Contents
1 Introduction 1.1 Dynamic Phenomena 1
1.2 Multivariable Systems 2
1.3 A Catalog of Examples 4
1.4 The Stages of Dynamic System Analysis 10
2 Difference And Differential Equations
2.1 Difference Equations 14
2.2 Existence and Uniqueness of Solutions 17
2.3 A First-Order Equation 19
2.4 Chain Letters and Amortization 21
2.5 The Cobweb Model 23
2.6 Linear Difference Equations 26
2.7 Linear Equations with Constant Coefficients 32
2.8 Differential Equations 38
2.9 Linear Differential Equations 40
2.10 Harmonic Motion and Beats 44
2.11 Problems 47 Notes and References 54
3 Linear Algebra
Algebraic Properties
3.1 Fundamentals 56
3.2 Determinants 62
3.3 Inverses and the Fundamental Lemma 66
Geometric Properties
3.4 Vector Space 69
3.5 Transformations 73
3.6 Eigenvectors 77
3.7 Distinct Eigenvalues 80
3.8 Right and Left Eigenvectors 83
3.9 Multiple Eigenvalues 84
3.10 Problems 86
Notes and References 89
4 Linear State Equations
4.1 Systems Of First-Order Equations 90
4.2 Conversion to State Form 95
4.3 Dynamic Diagrams 97
4.4 Homogeneous Discrete-Time Systems 99
4.5 General Solution to Linear Discrete-Time Systems 108
4.6 Homogeneous Continuous-Time Systems 113
4.7 General Solution to Linear Continuous-Time Systems 118
4.8 Embedded Statics 121
4.9 Problems 124
Notes and References 130
5 Linear Systems With Constant Coefficients
5.1 Geometric Sequences and Exponentials 133
5.2 System Eigenvectors 135
5.3 Diagonalization of a System 136
5.4 Dynamics of Right and Left Eigenvectors 142
5.5 Example: A Simple Migration Model 144
5.6 Multiple Eigenvalues 148
5.7 Equilibrium Points 150
5.8 Example: Survival Theory in Culture 152
5.9 Stability 154
5.10 Oscillations 160
5.11 Dominant Modes 165
5.12 The Cohort Population Model 170
5.13 The Surprising Solution to the Natchez Problem 174
5.14 Problems 179
Notes and References 186
6 Positive Linear Systems
6.1 Introduction 188
6.2 Positive Matrices 190
6.3 Positive Discrete-Time Systems 195
6.4 Quality in a Hierarchy-The Peter Principle 199
6.5 Continuous-Time Positive Systems 204
6.6 Richardson's Theory of Arms Races 206
6.7 Comparative Statics for Positive Systems 211
6.8 Homans-Simon Model of Group Interaction 215
6.9 Problems 217
Notes and References 222
7 Markov Chains
7.1 Finite Markov Chains 225
7.2 Regular Markov Chains and Limiting Distributions 230
7.3 Classification of States 235
7.4 Transient State Analysis 239
7.5 Infinite Markov Chains 245
7.6 Problems 248
Notes and References 253
8 Concepts Of Control
8.1 Inputs, Outputs, and Interconnections 254
Transform Methods
8.2 z-Transforms 255
8.3 Transform Solution of Difference Equations 261
8.4 State Equations and Transforms 266
8.5 Laplace Transforms 272
State Space Methods
8.6 Controllability 276
8.7 Observability 285
8.8 Canonical Forms 289
8.9 Feedback 296
8.10 Observers 300
8.11 Problems 309
Notes and References 314
9 Analysis Of Nonlinear Systems
9.1 Introduction 316
9.2 Equilibrium Points 320
9.3 Stability 322
9.4 Linearization and Stability 324
9.5 Example: The Principle of Competitive Exclusion 328
9.6 Liapunov Functions 332
9.7 Examples 339
9.8 Invariant Sets 345
9.9 A Linear Liapunov Function for Positive Systems 347
9.10 An Integral Liapunov Function 349
9.11 A Quadratic Liapunov Function for Linear Systems 350
9.12 Combined Liapunov Functions 353
9.13 General Summarizing Functions 354
9.14 Problems 356
Notes and References 363
10 Some Important Dynamic Systems
10.1 Energy in Mechanics 365
10.2 Entropy in Thermodynamics 367
10.3 Interacting Populations 370
10.4 Epidemics 376
10.5 Stability of Competitive Economic Equilibria 378
10.6 Genetics 382
10.7 Problems 389
Notes and References 391
11 Optimal Control
11.1 The Basic Optimal Control Problem 394
11.2 Examples 401
11.3 Problems with Terminal Constraints 405
11.4 Free Terminal Time Problems 409
11.5 Linear Systems with Quadratic Cost 413
11.6 Discrete-Time Problems 416
11.7 Dynamic Programming 419
11.8 Stability and Optimal Control 425
11.9 Problems 427
Notes and References 435
References 436
Index 441
