Table of Contents
PREFACE xiii
CHAPTER 1 DYNAMIC-SYSTEM MODELS AND SIMULATION 1
SIMULATION IS EXPERIMENTATION WITH MODELS 1
1-1 Simulation and Computer Programs 1
1-2 Dynamic-System Models 2
1-3 Experiment Protocols Define Simulation Studies 3
1-4 Simulation Software 4
1-5 Fast Simulation Program for Interactive Modeling 5
ANATOMY OF A SIMULATION RUN 8
1-6 Dynamic-System Time Histories Are Sampled Periodically 8
1-7 Numerical Integration 10
1-8 Sampling Times and Integration Steps 11
1-9 Sorting Defined-Variable Assignments 12
SIMPLE APPLICATION PROGRAMS 12
1-10 Oscillators and Computer Displays 12
1-11 Space-Vehicle Orbit Simulation with Variable-Step Integration 15
1-12 Population-Dynamics Model 17
1-13 Splicing Multiple Simulation Runs: Billiard-Ball Simulation 17
INRODUCTION TO CONTROL-SYSTEM SIMULATION 21
1-14 Electrical Servomechanism with Motor-Field Delay and Saturation 21
1-15 Control-System Frequency Response 23
1-16 Simulation of a Simple Guided Missile 24
STOP AND LOOK 28
1-17 Simulation in the Real World: A Word of Caution 28
References 29
CHAPTER 2 MODELS WITH DIFFERENCE EQUATIONS, LIMITERS, AND SWITCHES 31
SAMPLED-DATA SYSTEMS AND DIFFERENCE EQUATIONS 31
2-1 Sampled-Data Difference-Equation Systems 31
2-2 Solving Systems of First-Order Difference Equations 32
2-3 Models Combining Differential Equations and Sampled-Data Operations 35
2-4 Simple Example 35
2-5 Initializing and Resetting Sampled-Data Variables 35
TWO MIXED CONTINUOUS/SAMPLED-DATA SYSTEMS 37
2-6 Guided Torpedo with Digital Control 37
2-7 Simulation of a Plant with a Digital PID Controller 37
DYNAMIC-SYSTEM MODELS WITH LIMITERS AND SWITCHES 40
2-8 Limiters, Switches, and Comparators 40
2-9 Integration of Switch and Limiter Outputs, Event Prediction, and Display Problems 43
2-10 Using Sampled-Data Assignments 44
2-11 Using the step Operator and Heuristic Integration-Step Control 44
2-12 Example: Simulation of a Bang-Bang Servomechanism 45
2-13 Limiters, Absolute Values, and Maximum/Minimum Selection 46
2-14 Output-Limited Integration 47
2-15 Modeling Signal Quantization 48
EFFICIENT DEVICE MODELS USING RECURSIVE ASSIGNMENTS 48
2-16 Recursive Switching and Limiter Operations 48
2-17 Track/Hold Simulation 49
2-18 Maximum-Value and Minimum-Value Holding 50
2-19 Simple Backlash and Hysteresis Models 51
2-20 Comparator with Hysteresis (Schmitt Trigger) 52
2-21 Signal Generators and Signal Modulation 53
References 55
CHAPTER 3 FAST VECTOR–MATRIX OPERATIONS AND SUBMODELS 57
ARRAYS, VECTORS, AND MATRICES 57
3-1 Arrays and Subscripted Variables 57
3-2 Vector and Matrices in Experiment Protocols 58
3-3 Time-History Arrays 58
VECTORS AND MODEL REPLICATION 59
3-4 Vector Operations in DYNAMIC Program Segments: The Vectorizing Compiler 59
3-5 Matrix–Vector Products in Vector Expressions 61
3-6 Index-Shift Operation 63
3-7 Sorting Vector and Subscripted-Variable Assignments 64
3-8 Replication of Dynamic-System Models 64
MORE VECTOR OPERATIONS 65
3-9 Sums, DOT Products, and Vector Norms 65
3-10 Maximum/Minimum Selection and Masking 66
VECTOR EQUIVALENCE DECLARATIONS SIMPLIFY MODELS 67
3-11 Subvectors 67
3-12 Matrix–Vector Equivalence 67
MATRIX OPERATIONS IN DYNAMIC-SYSTEM MODELS 67
3-13 Simple Matrix Assignments 67
3-14 Two-Dimensional Model Replication 68
VECTORS IN PHYSICS AND CONTROL-SYSTEM PROBLEMS 69
3-15 Vectors in Physics Problems 69
3-16 Vector Model of a Nuclear Reactor 69
3-17 Linear Transformations and Rotation Matrices 70
3-18 State-Equation Models of Linear Control Systems 72
USER-DEFINED FUNCTIONS AND SUBMODELS 72
3-19 Introduction 72
3-20 User-Defined Functions 72
3-21 Submodel Declaration and Invocation 73
3-22 Dealing with Sampled-Data Assignments, Limiters, and Switches 75
References 75
CHAPTER 4 EFFICIENT PARAMETER-INFLUENCE STUDIES AND STATISTICS COMPUTATION 77
MODEL REPLICATION SIMPLIFIES PARAMETER-INFLUENCE STUDIES 77
4-1 Exploring the Effects of Parameter Changes 77
4-2 Repeated Simulation Runs Versus Model Replication 78
4-3 Programming Parameter-Influence Studies 80
STATISTICS 84
4-4 Random Data and Statistics 84
4-5 Sample Averages and Statistical Relative Frequencies 85
COMPUTING STATISTICS BY VECTOR AVERAGING 85
4-6 Fast Computation of Sample Averages 85
4-7 Fast Probability Estimation 86
4-8 Fast Probability-Density Estimation 86
4-9 Sample-Range Estimation 90
REPLICATED AVERAGES GENERATE SAMPLING DISTRIBUTIONS 91
4-10 Computing Statistics by Time Averaging 91
4-11 Sample Replication and Sampling-Distribution Statistics 91
RANDOM-PROCESS SIMULATION 95
4-12 Random Processes and Monte Carlo Simulation 95
4-13 Modeling Random Parameters and Random Initial Values 97
4-14 Sampled-Data Random Processes 97
4-15 “Continuous” Random Processes 98
4-16 Problems with Simulated Noise 100
SIMPLE MONTE CARLO EXPERIMENTS 100
4-17 Introduction 100
4-18 Gambling Returns 100
4-19 Vectorized Monte Carlo Study of a Continuous Random Walk 102
References 106
CHAPTER 5 MONTE CARLO SIMULATION OF REAL DYNAMIC SYSTEMS 109
INTRODUCTION 109
5-1 Survey 109
REPEATED-RUN MONTE CARLO SIMULATION 109
5-2 End-of-Run Statistics for Repeated Simulation Runs 109
5-3 Example: Effects of Gun-Elevation Errors on a 1776 Cannnonball Trajectory 110
5-4 Sequential Monte Carlo Simulation 113
VECTORIZED MONTE CARLO SIMULATION 113
5-5 Vectorized Monte Carlo Simulation of the 1776 Cannon Shot 113
5-6 Combined Vectorized and Repeated-Run Monte Carlo Simulation 115
5-7 Interactive Monte Carlo Simulation: Computing Runtime Histories of Statistics with DYNAMIC-Segment DOT Operations 115
5-8 Example: Torpedo Trajectory Dispersion 117
SIMULATION OF NOISY CONTROL SYSTEMS 119
5-9 Monte Carlo Simulation of a Nonlinear Servomechanism: A Noise-Input Test 119
5-10 Monte Carlo Study of Control-System Errors Caused by Noise 121
ADDITIONAL TOPICS 123
5-11 Monte Carlo Optimization 123
5-12 Convenient Heuristic Method for Testing Pseudorandom Noise 123
5-13 Alternative to Monte Carlo Simulation 123
References 125
CHAPTER 6 VECTOR MODELS OF NEURAL NETWORKS 127
ARTIFICIAL NEURAL NETWORKS 127
6-1 Introduction 127
6-2 Artificial Neural Networks 127
6-3 Static Neural Networks: Training, Validation, and Applications 128
6-4 Dynamic Neural Networks 129
SIMPLE VECTOR ASSIGNMENTS MODEL NEURON LAYERS 130
6-5 Neuron-Layer Declarations and Neuron Operations 130
6-6 Neuron-Layer Concatenation Simplifies Bias Inputs 130
6-7 Normalizing and Contrast-Enhancing Layers 131
6-8 Multilayer Networks 132
6-9 Exercising a Neural-Network Model 132
SUPERVISED TRAINING FOR REGRESSION 134
6-10 Mean-Square Regression 134
6-11 Backpropagation Networks 137
MORE NEURAL-NETWORK MODELS 140
6-12 Functional-Link Networks 140
6-13 Radial-Basis-Function Networks 142
6-14 Neural-Network Submodels 145
PATTERN CLASSIFICATION 146
6-15 Introduction 146
6-16 Classifier Input from Files 147
6-17 Classifier Networks 147
6-18 Examples 149
PATTERN SIMPLIFICATION 155
6-19 Pattern Centering 155
6-20 Feature Reduction 156
NETWORK-TRAINING PROBLEMS 157
6-21 Learning-Rate Adjustment 157
6-22 Overfitting and Generalization 157
6-23 Beyond Simple Gradient Descent 159
UNSUPERVISED COMPETITIVE-LAYER CLASSIFIERS 159
6-24 Template-Pattern Matching and the CLEARN Operation 159
6-25 Learning with Conscience 163
6-26 Competitive-Learning Experiments 164
6-27 Simplified Adaptive-Resonance Emulation 165
SUPERVISED COMPETITIVE LEARNING 167
6-28 The LVQ Algorithm for Two-Way Classification 167
6-29 Counterpropagation Networks 167
EXAMPLES OF CLEARN CLASSIFIERS 168
6-30 Recognition of Known Patterns 168
6-31 Learning Unknown Patterns 173
References 174
CHAPTER 7 DYNAMIC NEURAL NETWORKS 177
INTRODUCTION 177
7-1 Dynamic Versus Static Neural Networks 177
7-2 Applications of Dynamic Neural Networks 177
7-3 Simulations Combining Neural Networks and Differential-Equation Models 178
NEURAL NETWORKS WITH DELAY-LINE INPUT 178
7-4 Introduction 178
7-5 The Delay-Line Model 180
7-6 Delay-Line-Input Networks 180
7-7 Using Gamma Delay Lines 182
STATIC NEURAL NETWORKS USED AS DYNAMIC NETWORKS 183
7-8 Introduction 183
7-9 Simple Backpropagation Networks 184
RECURRENT NEURAL NETWORKS 185
7-10 Layer-Feedback Networks 185
7-11 Simplified Recurrent-Network Models Combine Context and Input Layers 185
7-12 Neural Networks with Feedback Delay Lines 187
7-13 Teacher Forcing 189
PREDICTOR NETWORKS 189
7-14 Off-Line Predictor Training 189
7-15 Online Trainng for True Online Prediction 192
7-16 Chaotic Time Series for Prediction Experiments 192
7-17 Gallery of Predictor Networks 193
OTHER APPLICATIONS OF DYNAMIC NETWORKS 199
7-18 Temporal-Pattern Recognition: Regression and Classification 199
7-19 Model Matching 201
MISCELLANEOUS TOPICS 204
7-20 Biological-Network Software 204
References 204
CHAPTER 8 MORE APPLICATIONS OF VECTOR MODELS 207
VECTORIZED SIMULATION WITH LOGARITHMIC PLOTS 207
8-1 The EUROSIM No. 1 Benchmark Problem 207
8-2 Vectorized Simulation with Logarithmic Plots 207
MODELING FUZZY-LOGIC FUNCTION GENERATORS 209
8-3 Rule Tables Specify Heuristic Functions 209
8-4 Fuzzy-Set Logic 210
8-5 Fuzzy-Set Rule Tables and Function Generators 214
8-6 Simplified Function Generation with Fuzzy Basis Functions 214
8-7 Vector Models of Fuzzy-Set Partitions 215
8-8 Vector Models for Multidimensional Fuzzy-Set Partitions 216
8-9 Example: Fuzzy-Logic Control of a Servomechanism 217
PARTIAL DIFFERENTIAL EQUATIONS 221
8-10 Method of Lines 221
8-11 Vectorized Method of Lines 221
8-12 Heat-Conduction Equation in Cylindrical Coordinates 225
8-13 Generalizations 225
8-14 Simple Heat-Exchanger Model 227
FOURIER ANALYSIS AND LINEAR-SYSTEM DYNAMICS 229
8-15 Introduction 229
8-16 Function-Table Lookup and Interpolation 230
8-17 Fast-Fourier-Transform Operations 230
8-18 Impulse and Freqency Response of a Linear Servomechanism 231
8-19 Compact Vector Models of Linear Dynamic Systems 232
REPLICATION OF AGROECOLOGICAL MODELS ON MAP GRIDS 237
8-20 Geographical Information System 237
8-21 Modeling the Evolution of Landscape Features 239
8-22 Matrix Operations on a Map Grid 239
References 242
APPENDIX: ADDITIONAL REFERENCE MATERIAL 245
A-1 Example of a Radial-Basis-Function Network 245
A-2 Fuzzy-Basis-Function Network 245
References 248
USING THE BOOK CD 251
INDEX 253
