Advanced Dynamic-System Simulation: Model Replicat ion and Monte Carlo Studies, Second Edition

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