Modeling and Simulation

Modeling and Simulation

Hartrnut Bossel UniversWit Gesamthochschule Kassel Kassel, Germany

AK PETERS

II Vleweg

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Copyright© 1994 by Springer Fachmedien Wiesbaden Urspriinglich erschienen bei A K Peters, Ltd. 1994

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission from the copyright owner.

Library of Congress Cataloging-in-Publication Data

Bosse!, Hartrnut [Modellbildung und Simulation. English] Modeling and simulation I Hartmut Bosse!.

p. em. Includes bibliographical references and index. ISBN 978-3-663-10823-8 ISBN 978-3-663-10822-1 (eBook) DOI 10.1007/978-3-663-10822-1 1. Mathematical models. 2. Digital computer simulation.

I. Title. T A342.B67 1994 003'.85--dc20

ISBN 978-3-663-10823-8

94-1847 CIP

Contents

Preface

1 SYSTEMS AND MODELS

1.0 Introduction

1.1 Tasks of Model Construction and Simulation 1.1.1 Why modeling and simulation? 1.1.2 How can we simulate behavior? l. 1.3 Applications of dynamic simulation models 1.1.4 Modeling and simulation for the study of development paths

1.2 Dynamic Models and Model Development 1.2.1 Spectrum of dynamic systems and models 1.2.2 Steps in the modeling and simulation process 1.2.3 Developing the model concept 1.2.4 Developing the simulation model 1. 2. 5 Simulation of system behavior 1.2.6 Performance evaluation, policy choice, system design 1.2.7 Analysis of the model system 1.2.8 Generic structures; systems zoo

1.3 Fundamental Properties of Systems 1.3.1 What is a system? System identity, integrity, purpose 1.3.2 Dynamic systems, system behavior, time period 1.3.3 System boundary, environment, inputs and outputs 1.3.4 When is a system observable? Behavior and state 1.3.5 State variables are memory variables 1. 3. 6 Elements and structure determine rates of change 1.3.7 Feedback generates system dynamics 1. 3. 8 System behavior: eigendynamics and forced dynamics 1.3.9 System and environment parameters determine response 1.3 .1 0 Subsystems and modularity 1. 3 .11 Superior systems and hierarchies 1.3 .12 System development: control, adaptation, evolution 1. 3 .13 Actors in the environment: orientation and interaction 1. 3.14 Unpredictability even for deterministic systems

xiv

1

1

3 3 4 5 6

8 8

12 14 14 16 18 19 20

21 22 23 24 24 26 26 27 28 28 29 30 31 33 34

vi

1.4 Fundamental Properties of Models 35 1.4.1 Using models: advantages and disadvantages 35 1.4.2 Model as a representation of limited validity 36 1.4.3 Model purpose determines system representation 36 1.4.4 Alternatives: Imitating behavior or modeling structure 37 1.4.5 Imitation of behavior 38 1.4.6 Explanation of behavior 39 1.4.7 Descriptive components in explanatory models 41 1.4.8 Different modeling approaches, and different data needs 42 1.4.9 Unterstanding future dynamics requires system understanding 44 1.4.10 Model validity: When can the model stand for the original? 44 1. 4.11 Scientific approach and model development 45

2 STRUCTURE 47

2.0 Introduction 47

2.1 Developing the Influence Diagram 48 2.1.1 A small global model: purpose, verbal model, and relationships 49 2.1.2 Logical deduction 51 2.1.3 The influence diagram 55

2.2 Differentiation of the Model Concept 60 2.2.1 Differentiation of the system components of the global model 60 2.2.2 Submodel"population" 64 2.2.3 Submodel"pollution" 71 2.2.4 Submodel "consumption" 75 2.2.5 Linking the submodels 77

2.3 Simulation and Model Behavior 83 2.3.1 Simulations with a simple simulation program 83 2.3.2 Validity of model formulation and simulation results 85 2.3.3 Comparison with a "real" world model 86

2.4 Summary of Important Results 89

vii

3 SYSTEM STATE 91

3.0 Introduction 91

3.1 System Elements and Basic System Structure 93 3.1.1 Input elements 93 3.1.2 State variables 94 3.1.3 Intermediate variables 97 3.1.4 Elementuy block diagram of a dynamic system 98

3.2 System State and State Computation 101 3.2.1 System state and state variables 101 3.2.2 Computing the system state 104

3.3 Some Elementary Systems and Their Behavior 112 3.3.1 Memory-free system 112 3.3.2 Exponential growth and decay 113 3.3.3 Logistic growth 114 3.3.4 Exponential delay (exponential leak) 115 3.3.5 Linear oscillator 116 3.3.6 Bistable oscillator 119 3.3.7 Chaotic bistable oscillator 120

3.4 Dimensional Analysis 121 3.4.1 Checking dimensional validity 122 3.4.2 Finding correct conversion factors 123 3.4.3 Using dimensional analysis in formulating model equations 125

3.5 Model Development for the Rotating Pendulum 126 3.5.1 Problem statement, model purpose, and verbal model 126 3.5.2 Developing the influence diagram for the rotating pendulum 129 3.5.3 Quantities, dimensions, and relationships for the rotating pendulum 131 3.5.4 Model equations and simulation diagram for the rotating pendulum 134

3.6 Model Development for a Fishing System 135 3.6.1 System description, model purpose, verbal model, influence diagram 135 3.6.2 Quantities, dimensions, and relationships of fishery dynamics 139 3.6.3 Model equations and simulation diagram for fishery dynamics 140

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3.7 Nondimeosional Model Equations and System Similarity 141 3. 7.1 Derivation of generic state equations for the rotating pendulum 142 3.7.2 Condensing the fishery model to the generic predator-prey system 146 3.7.3 State equations using normalized state variables 147 3.7.4 Dimensionless state equations, normalized states, and normalized time 149

3.8 Summary of Important Results 151

4 BEHAVIOR 155

4.0 Introduction 155

4.1 Simulator for a Standard Programming Language: SIMP AS 158 4.1.1 Ways of using SlMPAS models 158 4.1.2 Use of compiled SlMPAS simulation programs 159 4.1.3 Developing a SlMP AS model unit 160 4.1.4 Using special functions in SlMPAS 162 4.1.5 Using SIMP AS functions: a sample program 169

4.2 Simulation of Rotating Pendulum Dynamics Using SIMP AS 172 4.2.1 Constructing the SI.MP AS model from the simulation diagram 172 4.2.2 Constructing the executable simulation program 175 4.2.3 Standard run and interactive model use 176 4.2.4 Changing parameters 183 4.2.5 Parameter sensitivity 186 4.2.6 Global response analysis 190 4.2.7 Linearization of the equations of motion 194 4.2.8 Summary of the observations with the rotating pendulum 195

4.3 Simulation of Fishery Dynamics with SIMP AS 196 4.3.1 Constructing the SlMP AS model from the simulation diagram 196 4.3.2 Setting-up the executable simulation program 199 4.3.3 Standard run of the fishery model 199 4.3.4 Response to parameter changes 200 4.3.5 Modification of the fishery model for density-independent harvest rate 204 4.3.6 Simulation results for density-independent fish harvest 205 4.3.7 Equilibrium points of the fishery model 208 4.3.8 Summary of observations using the fishery model 211

5

4.4 Graphic-interactive Simulation Environment: STELLA 4.4.1 Overview of the STELLA approach 4.4.2 Simulation of rotating pendulum dynamics using STELLA 4.4.3 Simulation of fishery dynamics using STELLA

4.5 Summary of Important Results

CHOICE AND DESIGN

5.0 Introduction

5.1 Criteria and Evaluation of System Behavior 5.l.l Orientors, indicators, and criteria 5.1.2 System behavior and orientation theory 5.1.3 Existence in the normal environmental state 5.1.4 Effectiveness in securing scarce resources 5.1.5 Freedom of action to cope with environmental variety 5.1.6 Security to protect from environmental variability 5.1.7 Adaptivity to deal with environmental change 5.1.8 Regard for other systems in the environment 5.1.9 Basic orientors, orientation, and evaluation of system behavior

5.2 Path Analysis 5.2.1 Introduction 5.2.2 System elements and simulation model for a miniworld 5.2.3 Criteria and indicators of system development 5.2.4 Scenarios and simulation runs 5.2.5 Comparative evaluation of the simulation runs

5.3 Policy Analysis 5.3.1 Introduction 5.3.2 Constraints and quality measures for the fishery optimization 5.3.3 Extension of the simulation model for optimization studies 5.3.4 Search for optimal investment when fishing without fish locators 5.3.5 Search for optimal investment when fishing with fish locators 5.3.6 Optimization over a time path, and using system orientors

ix

212 212 217 221

224

227

227

229 229 233 237 238 239 240 241 242 243

248 248 249 251 258 261

264 264 265 268 269 271 273

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6

5.4 System Design 5. 4.1 Introduction 5.4.2 Stabilization by a modified system structure: system equations 5.4.3 Simulation model for the stabilized pendulum system 5.4.4 Simulation runs and the search for "good" control parameters

5.5 Summary of Important Results

SYSTEMS ZOO

6.0 Introduction

6.1 Dynamic Systems with One State Variable M10 1 SIMPLE INTEGRATION: CHANGE OF STOCK AND CONCENTRATION

M 102 EXPONENTIAL GROWTH AND DECAY

M103 EXPONENTIAL DELAY OF FIRST ORDER

M I 04 TIME-DEPENDENT EXPONENTIAL GROWTH

M105 BIRTH AND DEATH: SIMPLE POPULATION DYNAMICS

M106 OVERLOADING A BUFFER

Ml07 LOGISTIC GROWTH WITH CONSTANT HARVEST RATE

Ml08 LOGISTIC GROWTH WITH PREY-DEPENDENT HARVEST

M 109 DENSITY -DEPENDENT GROWTH (MICHAELIS-MENTEN)

MllO DAILY PHOTOPRODUCTION OF VEGETATION

6.2 Dynamic Systems with Two State Variables M201 DOUBLE INTEGRATION AND EXPONENTIAL DELAY

M202 TRANSITION FROM ONE STATE TO THE NEXT

M203 LINEAR OSCILLATOR OF SECOND ORDER

M204 ESCALATION

M205 BURDEN SHIFTING AND DEPENDENCE

M206 PREDATOR-PREY SYSTEM WITHOUT CAPACITY LIMIT

M207 PREDATOR-PREY SYSTEM WITH CAPACITY LIMIT

M208 COMPETITION

M209 TOURISM A.l'ID ENVIRONMENTAL POLLUTION

M210 OVERSHOOT AND COLLAPSE

M211 fOREST GROWTH

M212 RESOURCE DISCOVERY AND DEPLETION

M213 TRAGEDY OF THE COMMONS

274 274 275 279 282

286

289

289

295 296 298 300 302 304 306 308 310 312 314

317 318 320 322 324 326 328 330 332 334 336 338 340 342

7

M214 SUSTAINABLE USE OF A RENEW ABLE RESotJRCE M215 DISTURBED EQUILIBRIUM: C02. DYNAMICS OF THE ATMOSPHERE M216 STOCKS, SALES, ORDERS M217 PRODUCTION CYCLE M218 ROTATINGPENDULUM M219 Oscillator with limit cycle (van der Pol) M220 Bistable oscillator M221 Chaotic bistable oscillator

6.3 Dynamic Systems with Three or Four State Variables M301 TRIPLE INTEGRATION AND EXPONENTIAL DELAY M302 POPULATION DYNAMICS WITH THREE GENERATIONS M303 LINEAR OSCILLATOR OF THIRD ORDER M304 MiNIWORLD: POPULATION, CONSUMPTION, AND POLLUTION M305 PREDATOR WITH TWO PREY POPULATIONS M306 TWO PREDATORS WITH ONE PREY POPULATION M307 BIRDS, INSECTS, AND FOREST M308 NUTRIENT CYCLING AND PLANT COMPETITION M309 CHAOTIC ATTR.ACTOR (ROSSLER) M310 HEAT, WEATHER, AND CHAOS (LORENZ SYSTEM) M311 COUPLED DYNAMOS AND CHAOS M312 BALAJ'ICING AN INVERTED PENDULUM

MATHEMATICAL SYSTEMS ANALYSIS

7.0 Introduction

7.1 State Equations of Dynamic Systems 7.1.1 System concepts 7.1.2 System quantities as vectors 7.1.3 General state and behavior equations 7.1.4 General system diagram for dynamic systems 7.1.5 State computation 7.1.6 Numerical integration of the state equation 7.1.7 Transformation to first-order state equations 7.1.8 Transformation of an n-th order differential equation 7.1.9 Transformation of an n-th order difference equation 7 .1.1 0 State equation and system dynamics

xi

344 346 348 350 352 354 356 358

361 362 364 366 368 370 372 374 376 378 380 382 384

387

387

388 388 388 389 390 391 391 392 392 393 394

xii

7 .1.11 Linearization of the state equation 395 7 .1.12 Perturbation approach 396 7 .1.13 Approximation by Taylor expansion 397 7 .1.14 Linearization of the state equation: Jacobi matrix 398 7 .1.15 Equilibrium points 400 7 .1.16 Equilibrium points of non-linear systems 401 7 .1.17 Equilibrium points of continuous linear systems 402 7. 1.18 Equilibrium points of discrete linear systems 402

7.2 Matrix Operations for Linear Dynamic Systems 403 7.2.1 Operations with matrices and vectors 403 7.2.2 Eigenvalues, eigenvectors, and the characteristic equation 405 7.2.3 Transformation of basis 407

7.3 Behavior and Stability of Linear Systems in Unforced Motion 408 7.3.1 Form of the general solution of the state equation 408 7.3.2 Linear dynamic systems 408 7.3.3 Solution for autonomous time-invariant discrete system 409 7.3.4 Solution using the diagonal eigenvalue matrix 409 7.3.5 Solution for the autonomous time-invariant continuous system 410 7.3.6 Solution using the diagonal matrix exponential 411 7.3.7 Stability of linear systems 411 7.3.8 General form. standard form, and normal form: transformations 412 7.3.9 Behaviorally equivalent systems: example 414 7.3.10 Behavioral modes of linear systems 416 7.3.11 Continuous systems 417 7.3.12 Discrete systems 418 7. 3.13 Stability behavior of a two-dimensional linear system 419 7.3.14 Stability test for linear systems 420 7.3 .15 Remarks on the behavior of linear continuous systems 421

7.4 Behavior of Linear Systems in Forced Motion 423 7.4.1 Linear system and the superposition principle 423 7.4.2 Representation of aperiodic input functions 424 7.4.3 Representation of periodic input functions 425 7.4.4 Solution for the forced linear time-invariant system 426 7.4.5 Diagonalization of the system and decoupling of the characteristic modes 426 7.4.6 Linear system response to periodic input functions (frequency response) 429 7.4.7 Representation of frequency response 432

7.5 Behavior and Stability of Non-linear Dynamic Systems 7 .5.1 Stability of nonlinear systems 7.5.2 Attractors of nonlinear systems 7.5.3 Structural dynamics of systems 7.5.4 Comparison of linear and nonlinear dynamic systems 7. 5. 5 Perturbation dynamics of the influence structure

7.6 Summary of Important Results

BffiLIOGRAPHY

APPENDIX: PROGRAM LISTINGS Prog. 2.1 GLOBSIM.PAS:

TurboPascal simulation program for a primitive global model Prog. 4.1 MODLSTYL.TXT:

Programming template for SIMP AS model units MODEL.PAS Prog. 4.2 PREDATOR.MOD:

Sample SIMP AS model unit of a predator-prey system Prog. 4.3 FCTNDEMO.MOD:

Model unit for demonstration of SIMP AS functions and simulation Prog. 4.4 PENDULUM.MOD:

SIMP AS model unit for rotating pendulum dynamics Prog. 4.5 FISHERY.MOD:

SIMP AS model unit for fishery dynamics Prog. 5.1 WORLDSIM.MOD:

SIMP AS model unit for the Mini world global model Prog. 5.2 WORLDV AL.MOD:

SIMP AS model unit for the Mini world with orientor assessment Prog. 5.3 FISHOPT.MOD:

SIMP AS model unit of fishery dynamics with quality criteria Prog. 5.4 BALANCER.MOD:

SIMP AS model unit for balancing control of inverted pendulum Notes on the computer programs on the accompanying diskette

INDEX

xiii

432 432 433 435 436 437

439

443

449

450

452

454

456

458

460

463

465

468

471 474

475

Preface

It is the everlasting and unchanging rule of this world that everything is created by a series of causes and conditions and everything disap­pears by the same ntle; everything changes, nothing remains constant. 17ze Teaching of Buddha, Bukkyo Dendo Kyokai, Tokyo 1966, p. 82

Our world is dynamic, but the human mind is ill-equipped for comprehending,

assessing, and predicting dynamic processes. We often fail to understand and

anticipate the dynamics of systems until it is too late-until previously available

options have disappeared and we are trapped on a road to disaster. The dynamics

of population growth, ozone depletion, carbon dioxide accumulation in the atmos­

phere, environmental deterioration, AIDS, civil wars, forest die-back, recession,

unemployment, crime etc. have caught us largely unprepared, and have often led

to totally inadequate responses. It is possible to construct computer simulation models of dynamic processes

and systems, and it is possible to use these models for gaining a better

understanding of a system and its dynamics, and of the options available for

coping with the ensuing problems. However, as in any other discipline, the

process of modeling and simulation requires a certain amount of knowledge and

skills: about systems and systems analysis, and about the approaches and tools

available for modeling and simulation. Modeling and Simulation is the English language version of a successful

German textbook Model/bildung und Simulation (Vieweg Verlag, Braun­

schweig!Wiesbaden, 1992; 2nd ed. 1994). The English version has been

considerably revised and improved compared to the German editions. The book evolved from courses on Modeling and Simulation taught to undergraduate and

graduate students, managers and planners at Kassel University and at training

seminars in Germany, China, and Malaysia. The book is meant for students of all disciplines-dynamic systems are found everywhere. A first course in differential

equations, linear algebra and matrix analysis, or control systems, and minimal

programming experience might be helpful for studying the subject, but it is not a

prerequisite. The system-graphical method used for model development opens the

subject to all with only a limited mathematical background.

Modeling and Simulation is a self-contained and complete guide to the

methods and mathematical background of modeling and simulation of dynamic

systems, covering basic system concepts, the modeling approach, computer simulation and its different uses, mathematical systems theory, example models and applications, and a unique collection of 50 elementary system models. A substantial amount of exercise material is provided, in particular in Chapter 6 SYSTEMS Zoo. The SIMPAS simulation software and the "Systems Zoo" are provided on the accompanying diskette.

Chapter 1 SYSTEMS AND MODELS introduces to the concepts of systems, models, and simulation.

Chapter 2 STRUCTURE focuses on the development of influence diagrams (causal loop diagrams) as the basis of systems analysis and demonstrates modeling and simulation by developing a small global model.

Chapter 3 SYSTEM STATE concentrates on the accurate description of system components and their functions using concepts of state space analysis and of dimensional analysis, and linking this information in a complete mathematical model. Several (mostly nonlinear) systems are introduced, and simulation models are developed for the rotating pendulum and a fishery operation.

Chapter 4 BEHAVIOR deals with the translation of simulation diagrams or model equations into running computer simulation models, using two different programming approaches: the SIMP AS simulator (provided on the diskette and based on TurboPascal) and the STELLA simulation software. The various simulation approaches to the mapping and understanding of behavior are demonstrated.

Chapter 5 CHOICE AND DEsiGN introduces to the three fundamental tasks of systems analysis: path analysis, policy analysis, and system design. The three previously developed models (global model, pendulum, fishery) are applied to these tasks. The basic orientors of system behavior are derived as a framework for performance evaluation.

Chapter 6 SYSTEMS Zoo provides a unique collection of dynamic system models covering (mostly) elementary (nonlinear) processes of fundamental importance (examples: exponential and logistic growth, escalation, competition, dependence, predation, overshoot and collapse, population dynamics, production cycle, ecosystems, oscillators, chaos, etc.). Each model is fully documented; the systems zoo program with 50 models is also provided on the accompanying diskette.

Chapter 7 MATHEMATICAL SYSTEMS ANALYSIS provides a comprehensive survey of the mathematical background of dynamic systems analysis based on (vector) state space analysis.

The simulation software SIMP AS for DOS computers offers many interactive features, good (textual, tabular, graphic) documentation of models and simulation results, as well as additional features for the study of parameter

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sensitivities and global system behavior. The user can generate his/her own stand­alone simulation model with sophisticated features by inserting the model equations into SIMP AS and compiling under TurboPascal.

This book, the SIMP AS simulator, and the Systems Zoo have grown out of my research and teaching at Kassel University, the ASEAN Institute of Forest Management and the Forest Research Institute Malaysia (both Kuala Lumpur), the Sabah Forest Department (Sandakan, Borneo), and the Environment and Policy Institute of the East-West Center, Honolulu. I am grateful for the moral support provided by my friends of the International Network of Resource Information Centers (INRIC) during the gestation period of the project, in particular by Donella Meadows and Dennis Meadows. The first draft of the English edition was very thoroughly and critically reviewed by Donella Meadows, and her helpful comments have led to substantial improvements. I am deeply thankful to her. as well as to Ulla Marquardt for processing the text and mathematical formulae, to my wife Rika for providing moral and logistic support, and to our son Kendrik for drawing the system diagrams, and programming substantial improvements of the SIMP AS simulator.

November 1993 Hartmut Bossel