Systems Analysis and Simulation 1988, I: Theory and Foundations. Proceedings of the International Symposium held in Berlin (GDR), September 12–16, 1988 [Reprint 2021 ed.] 9783112471760, 9783112471753

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Systems Analysis and Simulation 1988 I

Mathematical Research

Mathematische Forschung

Wissenschaftliche Beiträge herausgegeben von der Akademie der Wissenschaften der DDR Karl-Weierstraß-Institut für Mathematik

Band 46 Systems A n a l y s i s a n d S i m u l a t i o n

1988

Systems Analysis and Simulation 1988 I: Theory and Foundations Proceedings of the International Symposium held in Berlin (GDR), September 12-16,1988

edited by Achim Sydow Spyros G. Tzafestas Robert Vichnevetsky

Alcademie-Verlag Berlin 1988

Herausgeber: Prof. Dr. Achim Sydow, Zentralinsti tut für Kybernetik und prozesse der Akademie der Wissenschaften der DDR, Berlin Prof. Dr. Spyros G. Tzafestas, Computer Engineering Technical University of Athens, Athens P r o f . Dr. Robert Vichnevetsky, Universi ty, New Brunswick

Die T i t e l dieser Schriftenreihe Autoren reproduziert.

ISBN ISSN

Dept.

Division,

of Computer Science,

werden

vom Originalmanuskript

Informations-

National

Rutgers

der

3-05-500Ö54-2 0138-3019

Erschienen irn Akademie-Verlag Berlin,DDR-1086 Berlin,Leipziger Str. (c) Akademie-Verlag B e r l i n 1p88 Lizenznummer! 202.100/500/88 Printed in the German Democratic Republic Gesamtherstellung; VEB Kongreß- und Werbedruck, 9273 Oberlungwitz LSV 1095 Bestellnummer1 763 958 1 (2182/46) 05600

3-4

PREFACE The

present volume contains the papers which were accepted for

tation

at

the

3rd International Symposium for

Systems

presen-

Analysis

and

Simulation held in Berlin (GDR), September 12-16, 1986. It

is

already a tradition to meet a broad international

experts sium.

in systems analysis,

community

modelling and simulation at this

of

sympo-

This fact shows the requirements for a forum of presentation

and

discussion of new developments and applications of modelling and simulation in systems analysis. To

the great interest in this field one has to take into

con-

sideration the developed role of computer simulation as a powerful

realize

tool

of problem solving. been

More and more areas in sciences and production have

investigated by mathematical models and computer simulation.

Bio-

logical sciences and social sciences are even by now influenced by

this

trend. The

model

use on the computer has been very much improved in

decision

suppprt systems.

Parallel simulation will provide drastic shortening of

computing

Parallel simulation and model based

time.

decision

support

systems are brought in the focus of international activities. Numerical mathematics, algorithms or

systems theory and control sciences provide with

supporting the modelling process itself based on

analytic

modelling

methods.

and graphics for representing results are real model

systems. A

simulation

Such simulation systems equipped with tools

for

support

*

new important impact comes from artificial intelligence by

processing.

knowledge

Expert systems may help decision making in case of

mathematical models.

missing

Expert systems may also support teaching and using

simulation systems. New teria

application areas are investigated. control

problems

of

qualitative

Applications sciences

Complex systems with multicri-

problems are in the scope of the symposium analysis of small

in engineering sciences,

scale

as

nonlinear

economy and management,

well

as

systems. natural

and social sciences are examined but also mixed problems

from

different areas. The

state

efficient

of

computer technique and programming

conditions for simulations.

environment^

set

Personal computer are even

up used

for simulation more and more. The

symposium

reflects

the state of the art

and

trends

in

systems

analysis, modelling and simulation.

5

The 3rd International Symposium is organized by the Central Institute of Cybernetics and Information Processes of the Academy of Sciences of the GDR (ZKI) with cosponsorship of the -International Association for Mathematics and Computers in Simulation (IMACS), -International Federation of Automatic Control (IFAC), -International Institute for Applied Systems Analysis Laxenburg (IIASA), Scientific Society of Measurement and Automation (WGMA) in the Chamber of Technology (KdT) of the GDR, -Mathematical Society (MG) of the GDR. The papers included in these proceedings were not formally refereed. The authors themselves are fully responsible. The international Program Committee consisted of: W. Ameling (FRG), P. Borne (France), L. Dekker (The Netherlands), S. Deng (PRC), A. A. Dorodnicyn (USSR), K. H. Fasol (FRG), W. Findeisen (Poland), 0. I. Frankseri (Denmark), V. Hamata (Czechoslovakia), C. Hu (PRC), A. Javor (Hungary), K. Kabes (Czechoslovakia), V. V. Kalashnikov (USSR),- V. Kempe (GDR), E. J. H. Kerckhoffs (The Netherlands), R. Klotzler (GDR), R. Kulikovski (Poland), A. Kurzhansky (Austria, USSR), N. Levan (USA), A. H. Levis (USA), T. I. Oren (Canada), M. Peschel (GDR), P. D. Dieu (Vietnam), F. Pichler (Austria), K. Reinisch (GDR), W. Schirmer (GDR), B. Schmidt (FRG), V. V. Solodovnikov (USSR), F. Stanciulescu (Romania), J. M. Svirezhev (USSR), M. Thoma (FRG), I. Troch (Austria), S. G. Tzafestas (Greece), G. C. Vansteenkiste (Belgium), R. Vichnevetsky (USA) , A. Sydow (GDR). Many thanks should be given to the members of this committee for the very helpful cooperation. Special thanks are said to Prof. Dr. V. Kempe, Director of the ZKI, for his great support in preparing and performing the symposium. Furthermore, great gratitude is to express to Prof. Dr. R. Vichnevetsky (USA), IMACS-President, Prof. Dr. B. Tamm (USSR), IFAC-President, Prof. Dr # R. H. Pry (USA), IIASA-Director, Prof. Dr. H. Richter (GDR), Chairman of WGMA, Prof. Dr. R. Klotzler (GDR), Chairman of MG for help and encouragement. A lot of the hard preparation work was done by the Department for Systems Analysis and Simulation of the Central Institute of Cybernetics and Information Processes. The editor expresses his thanks to all col-

6

leagues and friends who were very much engaged in the research work and in the preparation. First of all I would like to name Dr. P. Rudolph and Dr. A. Wittmilß who helped to prepare the proceedings. Furthermore I thank these colleagues and Dr. K.Bellmann, Dr. W.Jansen, Dr. E.Matthäus, Dr. R. St.raubel and all the other colleagues for engaged cooperation for years in developing this research area. Mrs. Ch. Fröhlich and Mrs. J. Obretenov should be named for speedy service in preparing the manuscript. Mrs. S. Böttcher made an excellent job as organizer. Last not least I would give my thanks to the publishers, especially Mrs. R. Helle and Mrs. G. Reiher, for their assistance and cooperation. Finally I would like to express my expectation also on behalf of the coeditors Prof. Dr. S. G. Tzafestas and Prof. Dr. R. Vichnevetsky that also the third symposium will be a contribution to the further development in systems analysis, modelling and simulation as well as a place for cooperation and communication like the first both.

April 1988

Achim Sydow On behalf of the editors

7

TABLE OF CONTENTS 1. Methods and Fundamentals 1.1. Theory of Discrete Systems D.Garte; J.Haufe; St.Ruelke: A Language to Describe and to Simulate Digital Systems

15

J.Voeroes: The State Space Approach to Discrete Event Dynamic Systems

22

the

Analysis

of

1.2. Theory of Continuous Systems - Qualitative Theory G.-Jumarie /INVITED PAPER/: Catastrophe,Chaos,Synergetics and Thermodynamics. A Unified Approach via Information of Deterministic Maps

28

W.Metzler: A Route t'o Chaos

34

V.S. Anishchenko; T.E. Vadivasova; M.A. Safonova: Bifurcations of Two-Dimensional Tori and Chaos in Dissipative Systems

40

H.G.Bothe: riant Sets

42

Shape and Dimension of Certain Hyperbolic

Inva-

W.Jansen; U.Feudel: CANDYS/QA - A Software System for Qualitative Analysis of the Behaviour of the Solutions of NonLinear Dynamical Systems

45

W.Metzler; H.Krieger: Qualitative Behaviour of Ordinary Differential Equation Models Describing Forest Growth Under Air Pollution

48

1.3. Decision Support Systems L.Cserny: The Analysis of Decision Making Systems

53

P.Bronisz; L.fCrus: Interactive Procedures for Multi-Criteria Decision Support in Bargaining Problem

59

Z.Strezova: A Procedure for Decision Support Systems Design: Modelling and Simulation Environment

63

G.Kreiselmeier; R.Seidl: Textile Technology

67

Knowledge-Based

Decision Aid

in

R.Schmidt; B.Koch: A Multiobjective Decision Support System for the Top Management

71

1.4. Modelling K.-H.Schmelovsky /INVITED PAPER/: Modelling and Simulation

75

A.G.Ivakhnenko /INVITED PAPER/: Self-Organizing Methods in Modelling and Clustering: GMDH Type Algorithms

86

H. Schwarz; H.T. Dorissen; I.. Guo: Bi1inearization of Nonlinear Systems

89

Weijian Zhang: Analytical Analysis of a Stochastic Differential Equation

97

Partial

G. Bohlender: Is Floating-Point. Arithmetic Still Adequate?

105

R.H. Adams: Using Systems of Incomplete, Often Inconsistent, Models 109 W. Borutsky: Top-Down Modelling of Complex Systems by means of Word Bond Graphs 113 9

Yi Yunwen; Zhang Lu: An Approach of the Grey System Modelling and Simulation for Complex Systems 117 V. Wenzel; SONCHES

E. Matthaeus; M. Flechsig: Generic Modelling in

121

G. Hertel: Statement and Tendencies of Models for Complicated Technical Systems 125 A.A. Lebedev: A Mathematical Model for Description of Random and Indefinite Factors from Unified Positions 130 G. Dzemyda: The Algorithms of Extremal Parameter Grouping

133

T. Lange: A New Approach for Structural Modelling

137

E. Apelt; D. Apelt: Problems of Qualitative Change of Parameters in Different Hierarchical Levels 141 L. Fortuna; A. Gallo; G. Nunnari: Studying the Interactions Among Model Reduction Algorithms via CAD Technique 146 J. Halawa; A. Trzmielak-Stanislawska: Determination of Simplified Models.by means of Chebyshev Polynomials 151 J. Halawa: A Note on Simplification of Large Dynamic Systems Using a Moment Technique 155 R. Boettner: Model Reduction and Stability Analysis of Nonlinear Dynamical Systems by means of Centre Manifold Theory 159 M. Kejak; P. .Javarsky: Method of Automated Construction of System Dynamics Models 163 S. Krueger; W. Mylius: A Modular Computer-Aided Modelling and Simulation System in Chemical Engineering 165 R. Funke: CANDYS/CM - A Dialogue System for Modelling Continuous Dynamical Systems with Chain Structure by Differentia] Equations 169 M.N. Krasilshchikov; V.I. Karlov: Control of the Observation Process by Probability Criterion 172 J. Markowski; M. Popkiewicz: Simulations Analysis of a Nonparametric Algorithm for Identification of Discrete-Time Hammerstein System 175 E. Jesierski: Remarks on Pole Assignment by Constant Output Feedback 179 P.S. Szczepaniak; A. Maiolepszy: On the Computational Solution ,of Differential Equations with Delay 183 Ch. Dahme: A Theory of Elementary Social Systems as a Basis for the Analysis and Modelling of Decision Situations 189 Th. Hager; Ch. Dahme: An Approach to the Development of Supporting Systems for Analysis and Construction/Influencing Social Systems 193 Z. Mital: Distributed Message Exchange System Modelling

196

E.-G. Woschni: Analysis and Optimization by means of Estimations in Measurement 200 M. Kejak: Simulation of Fuzzy System Dynamics Models 10

205

1.5. Multiobjective Optimization J. Ester /INVITED PAPER/: Multicriteria Fuzzy Decisions Anlan Song; Wei-Min Cheng: A Method for Multi-Criteria Decision Making

Multihuman

209 and

213

1.6. Methods of Optimisation and Control H.P. Schwefel /INVITED PAPER/: Evolutionary Learning Optimum Seeking on Parallel Computer Architectures 217 I. Troch /INVITED PAPER/: Control Design

Optimisation and Simulation

in

M. Peschel; H.-M. Voigt ; W. Mende; F. Breitenecker /INVITED PAPER/: System-Engineering Methodology for Simulation and Control of Dynamical Networks

226

232

J. Alder; K.J. Reinschke: Modelling of Large Processes Containing Continuously and Binarly Controlled Parts 237 Z. Emirsajlow: Integral Riccati Equations for a Feedback Solution of LQCP with a Terminal Inequality Constraint 243 J. Gondzio: Stable Variant of the Simplex Method for Solving Supersparse Linear Programs 247 •J. Fischer: Some Remarks on Optimizing Simulated Systems

251

Xu Kekang; Wang Zhenquan: D-Controllability and Strong D-Controllabi1ity and Control of Multiparameter and Multiple Time-Scale Singularly Perturbed Systems 255 J. Doleial: On-Line Optimal Control of Nonlinear Systems by Singular Perturbation Techniques 259 M. Schwaar: Design of Optimal Feedback Controllers for Some Classes of Nonlinear Systems 263 P. Javorsky: Systems

An Algorithm for Optimal Control of Nonlinear

J. Cretnik; S. Strrncnik: Design of a Combustion Controller

267 270

H.-M. Voigt; I. Santibanez-Koref: Solving Assignment Problems by Selection Pressure Controlled Replicator Networks 274 A. Grzech: Structures. viour

Local Area Networks with Different Topological Analysis of Qualitative and Quantitative Beha-

278

E. Szlachcic: Bicriterial Optimization of Structure of Complex Network 282 V. Sima: OPTPACK - An Interactive Package for Personal Computers 2.

Optimization

Software

286

Simulation Techniques

2.1. Simulation of Discrete Systems M. Aicardi; F. Davoli; R. Minciardi: Approximate Performance and Sensitivity Analysis of Closed Queueing Networks 290 S. Vincze: Computer-Aided Asynchronous Synthesis Procedure

294

K. Irmscher: Performance vices in CIM Environments

298

Evaluation of Communication Ser-

11

2.2. Simulation of Continuous Systems G.C. Vansteenkiste /INVITED PAPER/: ches to Ill-Defined Systems B. Schmidt /INVITED PAPER/: Simulation System SIMPLEX II

New Simulation Approa-

Methodological Basis

of

the

302 309

A. Dzieliriski: Real-Time Simulation of Nonlinear Quadratic Gaussian Adaptive Control Systems 319 V. Ceric: Simulation of Complex Real Systems: Practice 2.3. Parallel Simulation D.J. Evans; G.M. Megson: A Systolic Extrapolation

Theory

and 323

Design

32V

S. Szejko: SLA - A Language for Simulational Evaluation of Concurrent Systems Performance 331 2.4. Software Support K. Wang; 0. BjjSrke: The Off-Line Motion Planning via the Computer Graphics Simulation System

335

2.5. Simulation Environment E.J. Kerckhoffs /INVITED PAPER/: Mini-Supercomputers: Perspectives in Scientific Computation and Simulation

New

350

Duan Ping: The Simulated Performance of a Real-Time InterEvent-Driven processor Synchronization Algorithm Based on Method 356 M. Flechsig; E. Matthaeus; V. Wenzel: Simulation Environment in SONCHES 360 M. Marx; R. Czerner: A Program Generator for a Model-Based Simulation System 365 N.E. Madjarov; St.B. Maleshkov: Software Package for Linear Nonstationary Systems Analysis and Simulation 369 Th. Schulze: SIMPC - An Implementation of GPSS for Personal Computer 373 3.

Methods of Knowledge Processing for Systems Analysis S. Tzafestas /INVITED PAPER/: Expert Systems in CIM Operations: Key to Productivity and Quality 379 F. StSnciulescu /INVITED PAPER/: Construction of a Knowledge Base for Simulation and Control of Large Scale and Complex Systems. Applications 387 A. Javor /INVITED PAPER/: Knowledge Based Inference Controlled Logic Simulation 397 Dj.B. Petkovski /INVITED PAPER/: Knowledge-Based Systems for Distributed Decision-Making 406 A. Lehmann /INVITED PAPER/: Knowledge-Based Modelling and Simulation: Restrictions, Alternatives and Applications 412 B. Boehme; Control

R.

Wieland; U. Starke: Knowledge Based Process

A. Ligeza; M. Szymkat: Computer-Aided Modelling 12

A Symbolic-Numerical Support

for

419 423

LIST OF AUTHORS Adams, R. H. Aicardi, M. Alder, J. Anishchenko,V.S. Anlan Song Apelt.D. ApeIt,E.

109 290 237 40 213 141 141

Bjarke,0. Boehme, B. Boettner, R. Bohlender,G. Borutzky, W. Bothe, H. G. Breitenecker, F. Bronisz,P.

335 419 159 105 113 42 232 59

Öeri/ t h e r e is moreover the architectural level r e p r e s e n t e d by t h e P M S l a n g u a g e . T h i s l a n g u a g e is n o t y e t i n c l u d e d in the HB language For the purpose of structural and functional simulation of d i g i t a l s y s t e m s , a p a r t i c u l a r s i m u l a t o r of e a c h D D L ( B O O L E A N e q u a t i o n s or F 7 7 ) d e s i g n e r d e f i n e d f u n c t i o n i s g e n e r a t e d a n d l i n k e d t o s i m u l a t o r l^OSIM T h e c o m p l e t e s y s t e m is i l l u s t r a t e d i n f i g 1. The verification of every DDL unit is performed separately secause it is inefficient within the structural and functional simulator and because the internal f a c i l i t i e s of DDL unit are not directly visible For that reason, the generated particular DDL s i m u l a t o r is l i n k e d t o a simulator frame developed for it. The designer view for this simulator w a s d e v e l o p e d a n a l o g o u s l y to t h e d e s i g n e r v i e w of K O S I M s i m u l a t o r . The DDL simulator includes time controlled data input like the k O S I M simulator. T h e H B D - s y s t e m r u n s on the computer K1B40 from ROBOTRON and applies operating system SVP1800.

?..

HB

language

description

T h e H B d e s c r i p t i o n l a n g u a g e is based on the NBS34 and KOSIM network d e s c r i p t i o n l a n g u a g e e x t e n d e d for d e s i g n e r d e f i n e d functions. It c o n s i s t o f a s t r u c t u r a l p a r t f o r d e s c r i b i n g n e t w o r k o f e l e m e n t s a n d several designer defined u n i t s and a f u n c t i o n a l part for d e s c r i b i n g these e l e m e n t s and these designer defined units included the pin functions of the circuit. W i t h i n the n e t w o r k e a c h e l e m e n t or u n i t is f e a t u r e d by i t s n a m e , i t s t y p e a n d i t s t e r m i n a l s . Optionally you can indicate parameters for instance time delay for the outputs. The s t a n d a r d d e l a y is o n e t i m e u n i t . All these i n f o r m a t i o n s have to meet the type declaration in the functional part T h e t e r m i n a l s can be busses, too. 1) Central Institute K u r s t r a s s e 33, B e r l i n .

of Cybernetics 1080, GDR

and

Information

Processes,

15

Here is an e x a m p l e (Fig. 2): a c o n t r o l l e d 8 bit circle counter. It c o n s i s t s of 2 elements: AND1 of type A N D and 0R1 of type OR and 2 shifter r e g i s t e r units SRI and S R 2 of type SR. First, there is a c o n n e c t i o n d e s c r i p t i o n for AND1. OR1, SRI and SR2 with the name of e l e m e n t or unit, c o n n e c t i o n list and type of element or unit Moreover, the structural description of the functions for the pins INP, TOR and RING by a table c o n t r o l l e d g e n e r a t o r INPUT of type T A B S D and for the pin TAKT by the clock generator LPER is following The types T A B S D and LPF.R are s t a n d a r d types of the K O S I M - l i b r a r y . Second, the e l e m e n t d e c l a r a t i o n s are following beginning with 'E: ' for the types AND, OR, T A B S D and LPER and the unit d e c l a r a t i o n for the type SR beginning with 'F ', latter are continued with the language type the functional description 'DDL'. In the case of B O O L E A N e q u a t i o n s units is c o n t i n u e d with 'BOOLE', of F77 units with 'F77' The syntax of all unit t e r m i n a l s is equal to that of the elements Input, o u t p u t and b i d i r e c t i o n a l t e r m i n a l s are following in turn A n o t h e r m o d i f i e d D D L d e s c r i p t i o n - a 4 bit counter d e s c r i p t i o n - is shown in fig 3. B O O L E A N e q u a t i o n s and r e g i s t e r t r a n s f e r s can be declared outside automata (global declaration) or within the automata (local declaration) Moreover, they can be declared within the automaton o u t s i d e the state d e c l a r a t i o n or w i t h i n the state declaration. A c c o r d i n g l y , the header example is e q u i v a l e n t to SR: < T I > TKT < T E > EIN- TOR, OUT. in D D L language. 3. D D L model and

of SR d e c l a r a t i o n

in HB language

of

this

simulation

The r e g i s t e r t r a n s f e r language DDL is based on the m u 1 t i a u t o m a t o n model (see fig 4). For the s i m u l a t i o n this model is t r a n s f o r m e d in to a single a u t o m a t o n model (see fig 5) which c o n s i s t s of a set of B O O L E A N e q u a t i o n s and a set of r e g i s t e r transfers. The set of B O O L E A N e q u a t i o n s is solved by m e a n s of static simulation. Incorrect cycles in the BOOLEAN equations are indicated. F e a t u r e s the s e p a r a t e D D L s i mu1 a tor are. - The s e p a r a t e D D L simulator frame is based on the developed DDL debugger Its commands are formally i d e n t i c a l l y with the d e b u g g e r c o m m a n d s of the used operating s y s t e m SVP 1800 Its r e s u l t s shown a c o m p r o m i s e b e t w e e n the s y s t e m d e b u g g e r r e s u l t s and the K O S I M s i m u l a t o r results. - R e f e r i n g to its d e f i n e d t e r m i n a l s there are only inputs, o u t p u t s and internal terminals but no b i d i r e c t i o n a l t e r m i n a l s The latters are n e c e s s a r y to link D D L units in a circuit network Therefore, they have to be considered It d e p e n d s on a u t o m a t o n state w h e t h e r the b i d i r e c t i o n a l terminal o p e r a t e s like an input or like an output. As an input the terminal is on the r i g h t side of a r e g i s t e r t r a n s f e r s or c o n n e c t i o n s o t h e r w i s e as o u t p u t on the left hand side. In dependence on g i v e n internal c o n d i t i o n s the b i d i r e c t i o n a 1 terminal in the o u t p u t mode is set to lower i m p e d a n c e o t h e r w i s e to h i g h e r impedance. - The i m p l i c a t i o n of the D D L s y s t e m clock ( c l o c k . ) is shown in fig. 6 (see a l s o part 4. ). E s p e c i a l l y if no s y s t e m clock is d e f i n e d the D D L unit is started by an event of its inputs and works autonomously as long as no internal state changes occur This i n t e r p r e t a t i o n is based on the p r o c e s s model from ISPS In order to reduce the description predefined operators are developed as usual in K O S I M and ISPS. In the HB s y s t e m there is a library of these p r e d e f i n e d operators. T h e s e o p e r a t o r s can be called as macros.

16

4.

HB

model

and

simulation

T h e H B s i m u l a t o r is b a s e d o n t h e e v e n t o r i e n t e d simulator KOSIM also. T h e e v e n t o r i e n t e d c a l l of t h e D D L u n i t w i t h i n K O S I M s i m u l a t o r s e p a r a t e d in a B O O L E A N e q u a t i o n call a n d in a r e g i s t e r transfer call accordingly fig 5 is s h o w n in f i g . 6. H e r e f o r t h e e v a l u a t i o n of the c o m b i n a t o r i c part f r o m fig 5 t h e B O O L E A N c a l l is p r e s e n t a n d f o r t h e e v a l u a t i o n of t h e s e q u e n c e p a r t t h e r e g i s t e r t r a n s f e r call. T h e v i s i b i l i t y of developed DDL debugger

the internal DDL f a c i l i t i e s ( s e e a l s o p a r t 3. ).

5

simulation

Design

strategy

and

is

obtained

by

the

dialogue

T h e H B s y s t e m c a n b e u s e d by t h e t o p douin d e s i g n (see fig 7) started for e x a m p l e on the r e g i s t e r t r a n s f e r l e v e l and s t o p p e d on the logic g a t e level. On this o c c a s i o n the D D L units are t r a n s f o r m e d into a n e t w o r k of l o g i c a l s t a n d a r d g a t e f u n c t i o n s . The HB s u p p o r t it.

6

dialogue

commands

are

deduced

from

this

strategy

and

Examples

T h e f o l l o w i n g e x a m p l e s c h a r a c t e r i z e the a p p l i c a t i o n v o l u m e of HB language: - counters, shifters - asynchronous automata. - a u t o m a t a net, coupled automata. - processors, multi processor systems and cellular automata.

7 /l/

/2/

/3/

/4/

/5/

/6/

References B a r b a c c i , M. R ; S i e w i o r e c k , DP.: "The design and analysis of instruction set processes" McGraui Hill, 1982 D i e t m e y e r , D o n a l d L. 470 " L o g i c D e s i g n of D i g i t a l Systems" Allyn and Bacon. Inc. . A t l a n t i c A v e n u e . B o s t o n 1971 D o n a t h , U. ; S c h w a r z . P. ; T r a p p e , P. "Dynamische Logiksimu1 ation auf Bitund Wortniveau" 19.Fachko11oquium Informationstechnik Dresden 1986 Issel, U et al "NBS-84: A structural description language for VLSI design", Circuit theory and design 8 5 , P r o c e e d i n g s of t h e 1 9 8 5 E u r o p e a n Conference, p 62-5 Sc hwar z , F "A p r o g r a m for the m i x e d - l e v e l s i m u l a t i o n of digital integrated circuits" Proc E C C T D ' 8 5 , P r a g 1 9 8 5 . p. 133-136 Siewioreck, D P et al . " P r o p o s a l for r e s a e r c h on DEMETER — a design methodology and e n v i r o n m e n t " R e s e a r c h R e p o r t No. C M U C A D - P 3 - 5 , Jan. 1983, CMU

8.

Figures

¡HB d e s c r i p t i o n : i s t r u c t u r a l and ! functional part V H B - c omp i 1 e r !HB f u n c t i o n a l p a r t ! with DDL units, ! BOOLEAN equations ! and F77 units ! ! > 1 ! >

!HB s t r u c t u r a l p a r t 1 with NBS84 included ! single BOOLEAN equations ! I ¡initialization i i of r e g i s t e r s a n d ! i memories» ! !test pattern, ! ¡simulator ! ! commands

units

V

V simulator

with

debugger

V results Fig. 1:

18

View

of

the

of

system

simulation

V

C I R C L E COUNTER: 2 DDL S H I F T R E G I S T E R S INCLUDED S T A N D A R D AND INPUT E L E M E N T S W I T H K O S I M INPUT L A N G U A G E L I N K E D TKT TOR OR_L + + INP

OR

RING

A N D — ' -I— ' VI I + + I AND T

! •»

EIN1 +

! SR 1 SR 'DDL' (RI. . R4)

SR I SR 'DDL' ! 'a ,* '1,2..,n ,

with the notation

n 2

f10"1)

which characterizes how uniformly distributed is H ^ ^ j (f(. ,z) ;iî) around its mean value. Entropie Distance. Given a map 1(.), we shall identify it with the map f z (-) whene ver the following condition is satisfied, that is < H b,c(z) ( f z ( - ) ; "' D ) - HcU(.);fl>2

i

ko2f

where k denotes a positive constant the value of which is chosen via practical considerations. • 10.3 Application to Potential Functions These results apply directly to the potential functions V(x,u) of the

catastrophe

theory, and one can then envisage the exhaustive study of the latter in terms of

entropy

of deterministic maps. 11.

CONCLUDING

REMARK

The present theory of entropy of deterministic maps provides new basic

approaches

to fuzzy sets, probabilistic sets, pattern recognition, logical inference, combination of evidences, approximate reasoning, and so on.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] 110] [11]

ACZEL, J; DAROCZY, Z; On Measures of Information and Their Characterizations, Acade mie Press, New York, 1975 GUY, A. G.; The scientific revolution of 1987. Abdication of Boltzmann's Entropy, Applied Physics Communication, Vol 7, No 3, 217-235, 1987 HAKEN, H.; Synergetics, Springer Verlag, New York, Berlin, 1978 HAKEN, H.; Advanced Synergetics, Springer Verlag, New York, Berlin, 1983 JUMARIE, G.; Subjectivity, Information, Systems. Introduction to a Theory of Relati vistic Cybernetics, Gordon and Breach, New York, London, 1986 JUMARIE, G.; New results on the information theory of patterns and forms,J. Systems Analysis, Modelling and Simulation, Vol 4, No 6, 483-520, 1987 JUMARIE, G.; A Minkowskian theory of observation. Application to uncertainty and fu zziness, Fuzzy Sets and Systems, Vol 24, No 2, 231-254, 1987 JUMARIE, G.; Relative Information. Theories and Applications, Springer Verlag, New York, London (to appear) ROSEN, R. ; On information and complexity, in Complexity, Language and Life: Mathematical Approaches; Biomathematics, Vol 16, 174-196, 1986 SCHUSTER, H.G.; Deterministic Chaos, Physik-Verlag, Weinhein 1984 THOM, R.; Structural Stability and Morphogenesis, translated by D.H. Fowler, Benjamin, New York, 1975

33

A ROUTE TO CHAOS Wolfgang Metzler Abstract. The route to chaos of bifurcation from a stable fixed Hopf bifurcation, quasiperiodic the iteration scheme tends to a Eiffel tower.

the coupled logistic map is studied. We observe a flip point to a stable period-2 orbit which is followed by a behaviour and periodic orbits. At the end of the route fascinating strange attractor looking like the

1. INTRODUCTION Chaotic behaviour of simple dynamical systems is today widely believed to model temporarily irregular phenomena in many fields of science [4, 8 ] , Driven by graphic computers, chaotic dynamical systems also deliver to us a large variety of new fantastic insights into mathematical structures (Julia sets, strange attractors; cf.[2,12,14,15,17]). This paper provides an example of a plane dynamical system where computer graphics have opened a wide field of interesting mathematical questions (stability, bifurcations and chaos) and helped to formulate hypotheses and to prove theorems. 2. THE MAP It is well known that the iteration scheme x

k+l

=

x

k

+

h x

k(1 " xk) .

h

> 0 .

(1)

which is Euler's method for solving the logistic equation x = x ( l - x ) , may be transformed into

V i = ruk0

-



Department of Mathematics, University of Kassel, P.O. Box 101380, D-3500 Kassel, F.R. Germany.

Relative to (4), M = {(x,y)e]R

| x = y} is an invariant manifold. Reduction of (4) onto

M results in the one-dimensional iteration scheme k+1

= x. + h (2x. -x.f)

(5)

which may be transformed into (2) by u^ = (h/ (1 + 2h) Jx^ and r = l + 2h. The dynamics of the coupled system (4) will be discussed in this paper. It is based on an unpublished note [1]

which has been the reason for a computergraphics film [16].

Related couplings of nonlinear oscillators have been studied by Hogg and Hubermann [9] as well as Waller and Kapral [18]. Kaneko [10] investigated the transition to chaos of a coupled logistic map with a linear coupling term. 3. PERIOD-DOUBLING Considering eq. (4) we find, as the parameter h is varied, that there are two fixed points given by E 1 = (0,0) ,

(6)

E 2 = (2.2).

Linearizing (4) in the neighbourhood of E p we obtain the eigenvalue equation DF h (0,0) - A I

1 + h-A

h

h

1+h-A

(7)

= 0 ,

where DF^ denotes the Jacobian matrix of the first partial derivatives of the'function F h (x,y) = [x + h ( x - x 2 + y),y + h ( y - y 2 + x)j defined by (4). Because of its roots

= 1 + 2h and

(8) = 1 the fixed point E^ is un-

stable for h>0. For E 2 we obtain the eigenvalue equation 1 - 3h - A

DF h (2,2)-Al

h

h 1 -3h - A

(2)

= 0

(9)

(2)

with the eigenvalues AJ '= l - 2 h and A£ '= l - 4 h . Obviously E^ is a sink only if 0 < h < 0 . 5 . For h>0.5 the period-1 cycle E^ is unstable. There is a critical situatic at h = 0.5. If h >0.5, then °h

:

= {

(

M

^

'

H

^

(10)

'

is a hyperbolic period-2 orbit. It is born in the equilibrium point E^ = (2,2) and moves with increasing h to E^ = (0,0) on a circle, which is given by (x - l) 2 + (y - l) 2 = 2. In particular we have 0 Q

5

= {(2,2)} and 0^ = {(0,0)}.

The first part of this statement on 0, follows by inserting x, h 3 1 1 \/ k y we get k = h~h ™ t o eq" k+i - M

and X

k+2 = y k+l = x k

aS Wel1

aS y

^

k+2 = X k+1 = y k

h

h (11) (12)

35

Furthermore, using (12), eq. (4) can e a s i l y be shown to be equivalent to 2 h y k + 2 h x k - hy k 2 - h x k 2 = 0

(13)

and t h i s proves the second part of the statement. Moreover, for 0 . 5 < h < 0 . 6 proved

i s a stable o r b i t . I f h > 0 . 6 , 0^ i s unstable. This can be

straightforward looking at F ^ x . y ) = x + h ( x - x 2 + y)

(14a)

+

(14b)

F2(x,y) = y

h(y-y

2

+ x)

and ^(x.y)

:= F 1 ( F 1 ( x , y ) , F 2 ( x , y ) )

F 2 ( x , y ) := F 2 ( F 1 ( x , y ) , F 2 ( x , y ) ) respectively, for x =

(15a) ,

(15b)

V2h-1 , y = £ - ^ V 2 h - l

.

A lengthy c a l c u l a t i o n y i e l d s F j i i . y ) = 5 - 1 0 h + 2h2 = ^ F 2 ( i , y )

^

(16)

and ^FjU.i)

= 4 h V 2 h - l - 2h + 2 h 2 ,

(17a)

^F2(x,y)

= - 4 h V 2 h - 1 - 2h + 2 h 2 .

(17b)

This leads to the c h a r a c t e r i s t i c equation 5 - lOh + 2 h 2 - A -4h V 2 h - 1 - 2h + 2 h

4h V 2 h - 1 - 2h + 2 h 2 2

5 - lOh + 2 h 2 - A

0,

(18)

equivalent to A 2 + (-10 + 20h - 4 h 2 ) A + (25 - lOOh + 100h 2 ) = 0. The eigenvalues of (18) are Ax

2

= (5 - lOh + 2 h 2 ) ± V 2 0 h 2 - 4 0 h 3 + 4h 4 2

3

(19) 4

Analyzing the roots of the radicant 20h - 4 0 h + 4 h , one can e a s i l y show that

and

|A.| < 1

if

0.5 < h < 0.6,

(20a)

IA.| = 1

if

h = 0.6,

(20b)

|A.| > 1

if

h > 0.6.

(21)

The considerations concerning ( y , x ) are analogous. Thus (20) and (21) prove the above s t a b i l i t y statements about the period-2 o r b i t 0^. 4. THE ROUTE TO CHAOS Now we i n v e s t i g a t e the system's behaviour in the range 0 . 6 s h s 0 . 6 8 4 . As shown in the section before, there e x i s t s a stable period-2 o r b i t f o r 0 . 5 < h < 0 . 6 . F i r s t we consider the b i f u r c a t i o n at h = 0.6. We know that the one parameter family F ^ : 3 F^ • F^ s a t i s f i e s

36

F h (x,y) = (x,y) for h > 0 . 5 with x = ^ + ^ V 2 h - l and y = ¿ - £ V 2 h - l . Its Jacobian h h trix D F ^ x . y ) has two non-real eigenvalues A^ and A^ satisfying |A^| < 1 for h < 0.6 and |A^| > 1 for or h > O.i 0.6. Also we have d |A^(h) dh

10 > 0 .

(22)

h = 0.6

These three properties of the map (4) and further numerical studies (cf.[14]) indicate a Hopf bifurcation at (see [7], p. 162) h = 0.6. Accordingly for (y,x).

Fig. 1. Route to cnaos (after [15]): (a) Stable loops (h = 0.614), (c) Period-26 attractor (h = 0.678),

(b) Overlapping loops (h = 0.66), (d) "Eiffel tower" (h = 0.684). 2.1

V V^

. .

\ ••

- •• v

V •'

3

1.9

2.1

1.9

Fig. 2. The attractor of the map (4) at h = 0.684 with a blow up of the squared region [1.9,2.1] x [1.9,2.1],

37

Next we outline some computational results about the map (4) in the range 0 . 6 s h s O . 6 8 4 . These calculations have been performed on a graphic computer with double precision arithmetic. First we iterate the map 10.000 times to avoid transients. For h > 0 . 6 the periocH2 orbit blossoms out into two stable loops (Fig. la). The loops grow by turn with various periods, overlap themselves (Fig. lb) by turn with other periods (Fig. lc) and finally reach the structure of the Eiffel tower (Fig. Id). A blow up of the Eiffel tower's topstar (Fig. 2b) delivers a fractal structure which indicates sensitive dependence on the initial conditions. Indeed, one computes a positive characteristic exponent (cf. [6])

for h > 0.651, as shown in Fig. 3. Increasing

the values of h, the chaotic attractor of the map (4) continuously changes its shape, interrupted by different periodic orbits neglected in Fig. 3. The attractor's final shape at h = 0.684 is shown in Fig. 2. It is followed by escape which can be observed for h > 0.686.

Fig. 3. Spectrum of the characteristic exponents lim 1/n log

( X q , ) | for

0.6165 < h S 0.685 and (x 0 ,y Q ) = (0.4,0.5). REFERENCES [ 1] Beau, W., W. Metzler and A. Überla: The Route to Chaos of Two Coupled Logistic Maps. Preprint (1986). [ 2] Beau, W., W.H. Hehl, W. Metzler: Computerbilder zur Analyse chaoserzeugender Abbildungen. Informatik Forsch. Entw. 2 (1987), 122-130. [ 3] Collet, P. and J.-P. Eckmann: Iterated Maps on the Interval as Dynamical A. Jaffe and D. Ruelle (eds.). Birkhäuser, Basel/Boston/Stuttgart 1980. [ 4] Cvitanovic, P. (ed.):.Universality in Chaos. Adam Hilger Ltd., Bristol

Systems.

1983.

[ 5] Feigenbaum, M.: The Universal Metric Properties of Nonlinear Transformations. J. Stat. Phys. 21 (1979), 669-706. [ 6] Feit, S.D.: Characteristic Exponents and Strange Attractors, Commun. math. Phys. 61 (1978), 249. [ 7] Guckenheimer, J., P. Holmes: Nonlinear Oscillations, Dynamical Systems and Birfurcations of Vector Fields. Springer, New York/Berlin/Heidelberg/Tokyo 1986 2 . [ 8] Haken, H. (ed.): Evolution of Order and Chaos in Physics, Chemistry, and Biology. Springer, Berlin 1982. [ 9] Hogg, T. and B.A. Huberman: Generic Behavior of Coupled Oscillators, Phys. Rev. A 29 (1984), 275.

38

[10] Kaneko, K.: Transition from Torus to Chaos Accompanied by Frequency Lockings with Symmetry Breaking. Prog. Theor. Phys. 69 (1983), 1427. [11] Li, T.Y. and J.A. Yorke: Period Three Implies Chaos. Amer. Math. Monthly (1975), 958-992. [12] Mandelbrot, M.S.: The Fractal Geometry of Nature. Freeman, San Francisco 1982. [13] May, R.B.: Simple Mathematical Models with very Complicated Dynamics. Nature 261 (1976), 459-467. [14] Metzler, W., W. Beau, W. Frees, A. Überla: Symmetry and Self-similarity with Coupled Logistic Maps. Z. Naturforsch. 42a (1987), 310-318. [15] Metzler, W.: Chaos und Fraktale bei zwei gekoppelten nichtlinearen Modelloszi11atoren. PdN Physik 7/36 (1987), 23-29. [16] Metzler, W., W. Beau, A. Überla: A Route to Chaos. Computergraphies Film. Inst. 'f.d. Wiss. Film, C 1641, Göttingen 1987. [17] Peitgen, H.O., P.H. Richter: The Beauty of Fractals. Springer, Berlin/Heidelberg/ New York/Tokyo 1986. [18] Waller, H. and Kaprai, R.: Spatial and Temporal Structure in Systems of Coupled Nonlinear Oscillators. Phys. Rev. A 30 (1984), 2047.

39

BIFURCATIONS OF TWO-DIMENSIONAL TORI AND CHAOS IN DISSIPATIVE SYSTEMS V.S.Anishchenko, T.E.Vadivasova, M.A.Safonova Transition to dynamical chaos in different distributed and multidimensional systems is often preceded by a quasiperiodic motion bifurcations. In simplest case, chaos arises via distinction of two-dimensional torus (Tg) . This communication represents the results of computer and physical experiments on the investigation of torus distinction regularities, mechanisms of appearance of quasiattractors (CA.j) and their characteristics in different flow and diskrete systems. The following systems realising regim of quasiperiodic oscillations were investigated: driven generator with inertial nonlineority, two coupled generators and discrete system of coupled Feigenbaum maps. Numerical simulation was carried out with help of computer programs permitting to calculate the lines of limit cycle bifurcations on the parameter plane, various dynamical and statistical characteristics of oscillation regimes: Poincare section, probability distribution density, power spectra, Lapunov characteristic oxponents. Random noise generator was added in numerical scheme for fluctuation excitation simulation. Universal character of following bifurcation mechanisms of transition T CA^ predicted by torus distinction theorem was confirmed £2,3] 1. loss of smoothness and break-down of ergodic torus with soft appearance of torus-attractor. Smoothness loss effect can be preceded by the ergodic torus period-doubling, 2. Loss of smoothness of torus with resonance structure on it, its distinction on the resonance cycle stability loss line and appearance of chaos within looking region through Feigenbaum sequence of perioddoubling or through emergence of new torus and its following breakdown. 3. Hard appearance of torus-attractor via the saddle-node bifurcation on unsmooth torus or after its distinction. 4. Torus with resonance structure breaks also on tho line of homoclinic tangency of resonance saddle cycle manifolds. However, appearing hyperbolic subset of trajectories is nonattracting. Numerical experiments permitted to establish the role of natural fluctuations in situation under consideration. Noise influence on system dynamics in the regime of stable limit cycle with homoclinic structure in its neighbourhood leads to chaos arising. For example let us regard the results of numerical analysis of discrete system [3]

*~5 ' Saratov State University, Departaraent of Physics, SU-410601.Saratov, USSR 40

which simulates dynamics of two coupled generators [ 2 ]

.

Bifurcational diagram on parameter plane in neighbourhood of phase locking with winding number 0 = 2:5 is represented in Fig. 1.

Pig. 1. 1 0 - the Hopfe bifurcation line; li - the saddle-node bifurcation line; l* - the invariant manyfold homoclinic tangency line; lcr - the boundary of C A t - region; O - multipliJ cator of limit cycle. The digits on Fig. 1. indicate the motion directions on parameter plane where the bifurcational mechanisms mentioned above are realised. Regularities of transition "torus-chaos" predicted by one-dimensional circle mapping theory are regarded for flow systems. Quantitative relations in oscillation power spectrum at the T 2 -distruction moment are investigated in physical and computer experiments. Universal regularity in frequency distribution of spectral lines proved for model circle mapping are confirmed. Good agreement with one-dimensional theory results is obtained for fractal dimension of the set of irrational values of winding number near the torus break-down line. References [1] Anishchenko V.S.: Dynamical Chaos-Basic Concepts. Teubner-Verlag, Leipzig 1987. (2J Afraimovich V.S», Shilnikov L.P.: Invariant two-dimensional tori, their distruction and stochastisity. Methods of qualitative theory of differential equations. University Gorky, 1983, 3-25. [3] Anishchenko V.S.: Distruction of quasiperiodic oscillations and chaos in dissipative systems. Journal of Technical Physics, ¿6, (1986) 2, 225-237.

41

SHAPE AND DIMENSION OF CERTAIN HYPERBOLIC INVARIANT SETS H.G. Botho In many casas the evolution of a system whose state in each moment is determined by a point in a phase space P can be described by a dynamical system on P, i.e. by a family ^ l t f T of mappings vf" : P — > P depending on a time parameter t belonging to a set T which consists either of all real numbers, all non-negative real numbers, all integers or all non-nogative integers. These mappings satisfy . . -f S+t = Kf oSy t f O - id, where o denotes the composition of mappings. If at the beginning (i.e. at time 0) the system has the state corresponding to the point pQ in P, then its state at timo t corresponds to the point ^ (P 0 )' and the whole evolution of the system is described by the orbit W^Po^teT

startin

9

at

P0-

If a dynamical system •W^tj.T- o n a space P is given then it suggests itself to look for invariant sets A in P, i.e. for sets is an attractor, i.o. if for each satisfying (A) = A (t£T). If A point p in P sufficiently close to A we have

increases. Before analyzing this matter of fact, for the sake of completeness we compile all system parameters, constants and coefficients of (2.5) in Table 1. Via two pollution functions ^(ij)) and e[0,1.5] (df. [4], p.168), i.e., l=l(t,i|>) and a=a(t,t). The first quadrant D=IR*x]R* is an invariant set for the flow of (2.5). In D, (2.5) has two fixed points: (0,0) is asymptotically stable for all values of \|ie[0,1.5], and the nontrivial

49

c1 = 7[a]: average lifespan of needles Cj = 16[kg ASSl/(kg LEAF-a)]: optimal net photosynthesis rate c, = 241[kg H,0/(kg ASS1)]: transpiration coefficient c, - 2000[kg H,0/(kg ROOT-a)]: water supply coefficient cs = 6[t ASSI/(a • ha)]: growth of wooden biomass pollution functions: constants:

*!(*) = O.S^O.ô-O.A*): efficiency of needles = »(l+i»)" 1 : proportion of damaged needles • *c[0,1.5]: pollution parameter coefficients: c^jc,,"1*^ • (c^'CsO.Oô + c,'1! + yu = (1 - CjC^Oc^! - C!/6, Xi = 0.06y3c1"1c5, ». = -yiy,.

x, = -y^s,

*5 = yiy..

) + (1/6+ cs) =: y ^ y j .

y5 = », = -y,y5 .

*« = ys(y.-y,).

= y:y.-

Table 1. System parameters of (2.5), y 3 free for choice (cf. (2.3)) equilibrium ( l 1 , a 1 ) = (11(«|i),ai_(iti))1n the interior of D is an unstable saddle point for all parameter values with (1 x(4i),a1(4>))ED. After [5] there exists i|>0, 0> variations of the free parameter y 3 (as well as c2

and c 5 ) only change the position of

(l^ax).

1=0

Fig. 2. Decomposition of the phase space Fig. 3. Limiter functions Li (3.1) and L 2 of (2.5) for fixed E is attract((3.2), a = 0.5, b = 0.3) for ed by the origin (after [5]). xe[-0.5,1.5].

3. THE LIMITER-FUNCTION CONCEPT We will focuss now on another basic principle of forest ecosystems modelling turning out to be responsible for the distinct behavioural modes mentioned above. It is often useful to compare the resource demand of a specific internal process (e.g. assimilate demand for growth) with the appropriate supply (photosynthetic production). If the demand can be matched, the process will work unaffected. In case of insufficient supply the activity of the process must be diminished proportional to the supply demand ratio, while it doesn't make sense to increase the activity if the demand is oversupplied. Therefore, the supply demand ratio is restricted artificially to values lower than 1 before using 50

it to control the process. Additionally, the ratio is often restricted to positive values. The simplest approach to realize this kind of restriction is the LbniteJi-¿unatton M x )

:=

0 if x< 0 x if x e [ 0 , 1 ] 1 if x> 1

.

(3.1)

As shown in Fig 3, it is also possible to apply the more smooth function L 2 (x) := 1/(1 + exp(-2*(x-a)/b))

(3.2)

with real parameters a,b. KRIEGER [2] shows that the complex forest ecosystem model described in [1] can be reduced to a two-dimensional, parameter-dependent model (3.3) containing only one simple-type limiter-function and showing qualitatively the same behavioural modes as the complex version (cf. Fig. 1(a),(b)) dl dr

=

- c 1 - » I - ( c 2 + » ; ) - l 2 + c 3 -(c u -t 1 -L 1 (r-(c 1 -art of the problems the decision situation is stochastic-like, i.e. some probability distribution can be assigned to each of the alternatives presuming a /subjective/ probability field (fi,B,P). For a given E the risk-event a €Q is the following 55

where a^j

is the a l t e r n a t i v e is the c o n d i t i o n a l

to be r e a l i z e d . T h u s

p. = P ( a . . , a ) = P ( l s . - s. .I>6 e' -j -jt J

P: A x Si c.)The

decision,

After having

> V = [ 0 , 1 ] £ R

the

solved

the a l t e r n a t i v e s ,

decision

should

would be his

decision.

If A is an o r d e r i n g X: Z x

(17)

make a

relation

/subjective/

T h e DM's

— — >

>

t h e set of

s t r a t e g i e s of t h e i - t h D M w h i c h m a y be u s e d in the c a s e of

o o 1 li o a e C !-• h- n> p B 1

w s t-1 M s ö g

•t) CÖ > "Li

l

65

As for the g e n e r a l technology, it concerns a class of r e a l - w o r l d problems i n the m a n a g e m e n t and p o l i c y analysis area. The specific DSS applications are c o n c e r n e d w i t h information and computer resource allocation p r o b l e m s . A n u m b e r of s u c h problems, r e l a t e d to analysis of r e g i o n a l (local) and sectoral p o l i c y of c o m p u t e r and i n f o r m a t i o n resource a l l o c a t i o n has b e e n performed. T o r s u p p o r t i n g all m o d e l l i n g , s i m u l a t i o n and v a l i d a t i o n functions of the

entity

of specific DSS a m o d u l a r software system has b e e n developed. SIMOS I consists of f o u r s t a n d a r d p r o g r a m packages

(discriminant analysis, ILP, G P S S , linear r e g r e s -

s i o n analysis), a centralized i n f o r m a t i o n b a s e , and a module of special p u r p o s e p r o g r a m s . T h e i n f o r m a t i o n b a s e includes d a t a and i n f o r m a t i o n c o n c e r n i n g the stated r e a l - w o r l d problem, model d e s c r i p t i o n d a t a , s i m u l a t i o n results

, validation data,

as w e l l as all m o d e l l e d and simulated alternatives, p r e s e n t e d explicitly. Designed i n such a way SIMOS I can b e considered as a step toward a n integrated m o d e l l i n g environment. A I - S U P P O R T IN MBDSS D E S I G N AND

IMPLEMENTATION

A r t i f i c i a l Intelligence (Al) is the subfield of the c o m p u t e r s c i e n c e w h i c h has p e r h a p s the greatest potential w i t h respect to d e c i s i o n m a k i n g . Expert i n c o r p o r a t e d i n MBDSS can assist a number of tasks. The so-called

systems,

"semi-expert"

systems /2/ combine the p o w e r of the expert system technology and the k n o w l e d g e a l r e a d y available - about problems w h o s e solution requires e x t e n d e d support, as this t e r m was described above. T h e p r a c t i c a l r e a l i z a t i o n of s u c h systems w i l l a l low to switch from the q u e s t i o n "what...if" to " w h a t ought to be" in areas w h i c h are challenge for technologies such as D S S , modelling, s i m u l a t i o n ajid A I . I n F i g . 1 are p o i n t e d out the stages of the procedure w h i c h c a n be supported b y a semi-experted s y s t e m (SES). S o m e SES components are included in SIMOS I. CONCLUSION The p r o c e d u r e for MBDSS, c o n s i d e r e d b r i e f l y above, is consistent w i t h the requirements of the concept of f l e x i b i l i t y of d e c i s i o n options and those of the v e r s a t i lity a p p r o a c h /1 / . If applied adequately this procedure c a n provide extended s u p port to ill-defined problems solving. REFERENCES 979)1. B o n d e r , S.: C h a n g i n g the future of operations r e s e a r c h . Opns Res 2J_ 2 . Keen, P.G.W.: D e c i s i o n support systems: the next d e c a d e . D e c i s i o n S u p p o r t S y s t e m s , ¿ (1987), 33. Keen, P.G.tf. and M.S. S c o t t Morton: D e c i s i o n S u p p o r t S y s t e m s : A n O r g a n i z a t i o nal P e r s p e c t i v e . A d d i s o n - W e s l e y , R e a d i n g , MA, 1978. 4 . S p r a g u e , R.H. and E.D. Carlson: B u i l d i n g Effective D e c i s i o n S u p p o r t S y s t e m s . P r e n t i c e Hall, Englewood Cliffs, N.J., 1982. 5. S t r e z o v a , Z.: T e c h n i c a l Reports IR 109, 111, 115. SC MIS, 1981-83. 'in B u l g . ) 6. S t r e z o v a , Z.: A n approximate a p p r o a c h to the d e s i g n of d e c e n t r a l i z e d m a n a g e m e n t systems structures. P r o c e e d i n g s of the 8th IFAC C o n g r e s s , v o l . 7, P e r g a m o n Press, 1982. 7. S t r e z o v a , Z.: V a l i d a t i o n in ill-defined problem m o d e l l i n g : resource a l l o c a t i o n p o l i c y analysis. Proceedings of the 5th IFAC/lFORS Dynamic M o d e l l i n g Conference. P e r g a m o n P r e s s , 1;)86.

66

KNOWLEDGE-BASED DECISION AID IN TEXTILE TECHNOLOGY *. ) Dr.rer.nat Gert Kreiselaeier

*. ) , Dr.-Ing.

Roland

Seidl

1. DECISION SITUATIONS IN TEXTILES Textile

technology

Technicians a

huaan

i n t e g r a t e s the skill of

» a n y special

disciplines.

and the o p e r a t o r s of t e x t i l e p l a n t s fit the d e f i n i t i o n

expert

co»pletely and provide

the

potential

for

textile

t e c h n o l o g y to take a d v a n t a g e of k n o u l e d g e - b a s e d c o » p u t e r p r o g r a m s . capture

a n d w i d e l y a p p l y the e x p e r i e n c e a n d e x p e r t i c e of t h i s

is the a i l of d e v e l o p i n g a u t o m a t i c d e c i s i o n To u n d e r s t a n d in

the

To

experts

aid.

in which r e s p e c t t h e s e c o a p u t e r p r o g r a a a s c a n be

field

of

of t e x t i l e s suppose uhat the f o l l o w i n g t a s k s

useful have

in

coaaon? d i a g n o s i n g fabric or a a c h i n e

faults,

c h o o s i n g an a p p r o p r i a t a a c h i n e to p r o d u c e a d e s i r e d

fable,

c o n f i g u r i n g the aany c o a p o n e n t s t h a t Bake u p a c l o t h i n g In

all

cases

singulare experts

decision

data do

is

nade

and i n f o r a a t i o n .

well.

in a c o a p l e x This

As

experts

s p r e a d out tions

This

Although

involved

the n e e d for soae d e c i s i o n s u p p o r t

is due to the fact that is rare,

is r e q u i r e d

coaputers

process

t h u s c o a p u t e r p r o g r a m s which do this c o n t i n u e

huaan expertise

huaen expertise

feu

people

thuab.

in the field of t e x t i l e s face a more c o a p l e x a n d

urgent and

aany a

Their reasoning

includes use of judgaent a n d r u l e s of

w o r l d w i t h i n which to o p e r a t e , becoaing

froa only

E x p e r t s are people who can do t h i n g s o t h e r

c a n n o t do b e c a u s e of t r a i n i n g a n d e x p e r i e n c e . in e a c h c a s e

situation

is a t a s k s that

for the c o n s i d e r e d

in h i g h d e a a n d a n d

in reaote

tools

in » a n y t e x t i l e p r o c e s s e s , e x p e r t s a b i l i t y to

there

become

often

indespensible

is c o n t i n u e d r e l i a n c e on

identify a n d s y n t h e s i z e d i v e r s e

fora j u d g e a e n t s , e v a l u a t e a l t e r n a t i v e s a n d to a i d

*. ) F o r s c h u n g s i n s t i t u t

applica-

expensive,

areas.

a n d 3 i a u l a t i o n a o d e l s have

huaan

is to

für T e x t i 1 t e c h n o l o g l e

factors,

the to

derisions.

Kar 1 - H a r x - S t a d t ,

DDR

67

the s t r a t e g y

it f o l l o w s

Is c a l l e d the control, s t r a t e g y .

is to scan t h r o u g h the r u l e s until one facts and

in the d a t a base, the

scanning

different

strategy

This

strategy

is f o u n d w h o s e c o n d i t i o n s

the r u l e is a p p l i e d ,

resuaes.

One

is

known

as

is to s e l e c t a goal to be a c h i e v e d a n d

t h e r e are no r u l e s to e s t a b l i s h the new s u b g o a l , for the n e c e s s a r y

strategy We

is known as

scan

that

over an e x c h a n g a b l e

the rule base at

a

hypothesis.

This

situation

general

characteristics.

If the

rule

This

on

interpreter

in

r u l e - b a s e a n d the b a s i c C13.

base.

This

that

is

PROLOG

facts s u p p l i e d by the to

establish

the

basis

of

infor»ation

i n f o r m a t i o n will be a c q u i r e d

is is,

user

truth

p r o b l e m can be u s e d to c l a s s i f y an the

which

idea of w h i c h

The e s s e n c e of the p r o g r a m

is u s e d to e x a m i n e

conclusion,

event

or

goal.

the p r o g r a m a s k s

facts a n d e n t e r s t h e » in the d a t a

a c c o r d i n g to S t e r l i n g a n d S h a p i r o arrive

A the

backward-chaining.

constructed a backward-chaining

operates

base

forward-chaining.

rules to find t h o s e w h o s e c o n s e q u e n t a c t i o n c a n a c h i e v e the user

Batch

u p d a t i n g the d a t a

to

of

a

objekt,

about

its

sequentially

in

d i a l o g with the user. The

s y s t e m a s k s q u e s t i o n s of the user

which

cannot

information.

be found

about

in the k n o w l e d g e

"primitive"

information,

base or d e r i v e d

The user can r e s p o n d to such q u e s t i o n s

in two

from

by s u p p l y i n g the r e l e v a n t

i n f o r m a t i o n as an y e s / n o - a n s w e r

q u e r y , c h o o s i n g an a n s w e r

from a menue or t y p i n g a n u m e r i c a l

- ask the system, why this The into

latter o p t i o n

is useful

the s y s t e m s c u r r e n t

information

that the s y s t e m s q u e r y a p p e a r s the From

query

The user can ask

irrelevant,

would r e q u i r e a d d i t i o n a l

the system

is a s k i n g

that

information.

Suppose,

than 5 k n i t t i n g - s y s t e m s

for

kind

or

"why"

value

the

is worth the e x t r a e f f o r t of

r e q u i r e d ?".

may type

user.

informaobtaining

is a s k i n g

Then the user,

cases

answering

judge wether the

for e x a m p l e , the s y s t e m

insight

in

in c a s e s that

e f f o r t on the part of

the c o n s e q u e n c e s of a yes or no r e p l y , uhat

the

is needed.

the s y s t e m s e x p l a n a t i o n s the user will

tion

to

in order to e n a b l e the user to get

intentions.

other

ways:

"are

not yet

more

knowing

"why" r e p e a t l y to

of k n i t t i n g m a c h i n e the s y s t e m has c h o o s e n as its

see

actuall

h y p o s e s i s. O n c e the s y s t e m has come up with an a n s w e r to the users q u e s t i o n , user may answering

like to see h o w t h i s c o n c l u s i o n was r e a c h e d . 3uch

a how-question

supplied primitive

68

information

the

A p r o p e r w a y of

is to d i s p l a y the s u b g o a l s from w h i c h the c o n c l u s i o n was

and

user-

reached.

2. E X P E R T S Y S T E M S P U T E X P E R I E N C E IN T H E P R O D U C T I O N In recent years, research •ade many been

An

in the field of artificial

important successes.

the

"Expert

developement

"Expert

require

of powerful

system"

new computer

is a computer programm

procedures

to

significant

human expertise

the knowledge base of the

inference engine

the explanatory

that uses

for

important respects. pertinent

general general

solution.

rules

In a conventional

computer p r o g r a m s

computer program,

to the problem and means and methods Intermixed,

so that

for

modify the

knowledge about the problem and the methods for a p p l y i n g

the

In an expert system there

The systeat can be changed by adding

3. O N E POPULAR APPROACH FOR KNOWLEDGE

REPRESENTATION

researchers have worked out a variety of m e t h o d s

and U3ing knowledge research

expert knowledge

in computers. This

in AI circles.

for

representing

is still a matter of d e b a t e and

One popular approach

is to use situation-action

can be connected to each other to form rule

to

which

networks.

such networks provide a s u l t i b l e leans for

ting knowledge,

required by the textile expert to decide

situation.

set of rules

The

represent

or production r u l e s ,

Once assembled,

is often referred to as the

represen-

In a certain "rule-base"

part of the program that d e c i d e s which rule to apply as

interpreter" or task

in

know-

utilizing

it is difficult to

base.

The

to main

component.

knowledge to the problem.

the

The

reasoning

or subtracting rules in the knowledge

active

and

enough

is a clear separation of

program.

"rule

as

inter face(why/how-component)

this knowledge are all

and

knoun

knowledge

complex

their

expert system differs from more conventional

several

AI

has has

are:

facts and for

the knowlede acquisiton

the

software

solve poblems that are

components of an expert system

ledge

intelligence

Among the most significant of these

systems".

inference

An

PROCESS

of the

inference

the

engine.

inference engine

Is to decide which rules to

apply,

69

The

diagnosis-shell

WIDIHO is in practical use on 16-bit PC for

the

following problems: - selection of Suitable knitting machines on the basis of desired properties of # the textile fabric; - »achine fault diagosis on multi shed weaving looms on the basis of information about elemantary defects; - selection

of suitable standards for special textile testing

problems; - interpretation of faults in textile fabrics during finishing processes; and - predetermination of properties in yarn production. These

problems are complicated enough to justify developement and use

of expert systems.

4. DECISION AID FOR DIAGNOSING FAULTS ON WEAVING LOOMS One major problem In de3iging an expert system is to find a method for the

acquisition and modelling of expert knowledge and for effectively

overcoming

the

psychological

"knowledge engineering

and analysis of expertise are important. to

bottleneck".

aspects of knowledge ellcitatlon,

extract

knowledge

from

textile

In

this

field

interview techniques

We used interview techniques experts

and

formulated

und

structured this knowledge in knowledge rules about weaving looms. knowledge space,

has

been represented as a hlerachleally

organised

search

as an AND/OK tree. The body of this tree is formed from inter-

related rules with the facts occupying terminal positions. ledge

base

fault

occurs.

150

rules.

This know-

is consulted by WIDIMO so that advice can be given

if

At present the knowledge base contains 90 objects

This

fault

diagnostic system beats

the

human

tenance

can

be put into effect.

less skilled staff is available or if the plan,t Is working

KEFEKENCES £1] Sterling, L., E. Shapiro: The Art of Prolog. The HIT Press, Cambridge, Mass., London, 1987.

70

main-

It al3o can be effectively used

reduced personnel.

a and

textile

experts in speed and availability so that a new area of machine only

The

if with

A MULTI-OBJECTIV DECISION SUPPORT SYSTEM FOR TOP MANAGEMENT (DSS-CAPS)

THE

Dipl.-Ing. Rolf Schmidt,VEB Trafowerk "K.Liebknecht", M a r k e t i n g , 1 1 6 0 B e r l i n , W i l h e l m i n e n h o f s t r . 83-86 D r . - I n g . B e r n d K o c h . A d W d e r D D R , Z e n t r a l i n s t i t u t für K y b e r n . u n d I n f o r m a t i o n s p r o z e s s e , 1 0 8 6 B e r l i n , K u r s t r a P e 33 Introduction: The c o n t i n u o u s s a t i s f a c t i o n of d e m a n d s of the national economy and international market requires a higher quality of m a r k e t i n g . In c a s e o f s u p p l y m i s s i n g (material, cooperation) o f t e n the e x a c t l y o b s e r v a n c e of the production and sales p l a n e is n o t p o s s i b l e in a l l d e t a i l s . By m e a n s of the programsystem CAPS, an i n t e r a c t i v e c o m p u t e r a i d e d production and sales decision support system (DSS-CAPS) using the a l g o r i t h m of R E H / S t r a u b e l - 8 5 - 1 / , / S t r a u b e l - 8 6 - 1 / , it is p o s s i b l e to c o m p e n s a t e or b e t t e r to r e d u c e d e v i a t i o n s of the p r o d u c t i o n p l a n a n d its c o n s e q u e n c e s o n the f u l f i l m e n t of the o v e r a l l t a r g e t s of the e n t e r p r i s e . Description: The f u n d a m e n t a l a i m w a s to c r e a t e a s i m p l e and robust model of p r o d u c t i o n a n d sale. The computer based system ha6 to be e a s y by u n d e r s t o o d a n d a p p l i e d by the user and particularly by the (top) m a n a g e m e n t . Paet experience hase s h o w n us t h a t e v e n the b e s t m o d e l / s y s t e m w i l l n e v e r be used if it is n o t c o m p r e h e n s i b l e a n d if the user ( top management ) c a n n o t g r a s p its f u n c t i o n . The above mentioned system is u s e d for d e c i s i o n m a k i n g for the direction of p r o d u c t i o n a n d d e l i v e r y w i t h i n the c u r r e n t y e a r of planning. The s t a r t i n g p o i n t for the c u r r e n t y e a r is t h a t the production a n d d e l i v e r y p l a n s are f i x e d f o r the c o n s i d e r e d period of time. The r e l a t i o n c a n be i l l u s t r a t t e d thus: Balance = Plan = Contracts

Pic. Rl.l.

(delivery).

Modelpart I- Data bank system of

RABS1

71

The production plan as well as the stocks which result from that are fixed regarding time and the stocks are largly linked with contracts. The thus chosen solutions for production and delivery of goods present efficient alternatives from computer- aided and determined compromise solutions ( illustration Rl.l and R1.2). The respective complexes and the existing connections and their interdependences will not be described in this article. As for the modelpart RABS1 we refer to /Schmidt-871/. The applied basic model is elucidated in pictures Rl.l and R1.2.

Pic. R.1.2.

Modelpart II of DSS-CAPS

Picture Rl.l illustrates the flow of information within operative working of customers'orders. The results of this model form the bases (a priori information) for the decision support system CAPS. The production managemant. is faced with the task of making optimum use of the existing production capacity.That means to realise the plannend production targets , which are assigned by top management, with a minimal expenditure of material, working time and man-power. A necessery consequence of this is , among others, the reduction standstills of machineries and its most efficient employment especially with regard to optimal production quantity. The 6ales management is confronted with the task of satisfying the needs of the market in such a way that the position in the market can be guaranteed in future and its own share in the market can be increased lucratively and profitably. The application of the DSS-CAPS realises for the first time the coherence or better interdependence of targets of pro-

72

duction w i t h t h o s e of t h e s a l e s m a n a g e m e n t . The production realisation according to the g i v e n p r o d u c t i o n p l a n depends mainly on the r e a l p r o d u c t i o n c a p a c i t y at the respective period of time . Unintentional production deficits can be caused for e x a m p l e b y s u p p l y w h i c h h a s gone missing, been d e l a y e d o r w h i c h is of p o o r q u a l i t y b o t h f r o m t h e h o m e m a r k e t and abroad and by unintentional demage6. This i n f l u e n c e s o n the p r o d u c t i o n p r o c e s s c a n a f f e c t t h e temporal c o n d i t i o n s as w e l l as s h i f t s in the a s s o r t m e n t of t h e p r o d u c e d g o o d s . As a r e s u l t n e g a t i v e e f f e c t s c a n c o m e into existence as to the r e a l i s a t i o n of c o n t r a c t s and the t o t a l t u r n o v e r - p o l i c y o f the p r o f i t - s e e k i n g e n t e r p r i s e . The f u n d a m e n t a l r e q u i e m e n t of t h e e n t e r p r i s e h o w e v e r reads : f u l f i l l m e n t of the m o n t h l y a n d c u m u l a t i v e e c o n o m i c t a r g e t s b y a p p l i c a t i o n of the m i n i - m a x - m e t h o d . By m e a n s of C A P S it is p o s s i b l e to c o m p e n s a t e or better to r e d u c e d e v i a t i o n s of the p r o d u c t i o n p l a n a n d its c o n s e q u e n c e s on the f u l f i l m e n t of the o v e r a l l t a r g e t s o f the enterprise within the current p e r i o d of p l a n n i n g ( c u r r e n t year of planning). We proceed from the idea t h a t the production management is a b l e to o f f e r v a r i o u s a l t e r n a t i v e s of product i o n for the r e s p e c t i v e types o f p r o d u c t s ( e i t h e r - or ). The c h o i c e of an e f f i c i e n t p r o d u c t i o n a l t e r n a t i v e c a n n o t a n d must n o t d e c i d e d by the p r o d u c t i o n roanagemant. It h a s not got either the n e c e s s a r y s u r v e y n o r the insight into the complexity of marketing processes. From this follows the necessity for the p r o d u c t i o n m a n a g e m e n t as w e l l as the sale management to suggest v a r i a n t s of decisions to the top m a n a g e m e n t of the e n t e r p r i s e . The v a r i a n t s s h o u l d be p a r e t o optimal solutions of a set of c o m p r o m i s e p o i n t s which are calculated by m e a n s of CAPS o n the b a s e s o f the following targets : b a c k l o g of c o m m o d i t y p r o d u c t i o n c u m u l a t i v e a n d m o n t h l y (ad v a l o r e m ) backlog of p r o d u c e d g o o d s a c c o r d i n g to the production p l a n - b a c k l o g of s o l d g o o d 6 (ad valorem) b a c k l o g s of f u l f i l m e n t of c o n t r a c t s backlog of the sold commodity production c u m u l a t i v e u n t i l the e n d of the c u r r e n t l y y e a r difference between the d a t e of realisation fixed in the c o n t r a c t a n d the a c t u e l d a t e of delivery d i f f e r e n c e b e t w e e n the f i n a l s t o c k according to the p r o d u c t i o n p l a n a n d the a c t u a l stock availabel deviations from t h e a s s o r t m e n t a c c o r d i n g to the p r o d u c t i o n p l a n w h i c h c a n n o t bo s o l d or linked with contracts within the current p e r i o d of p l a n n i n g

> Min > Min > Min > Min > Min > Min > Min

What alternative is g i v e n p r i o r i t y by t h e top management depends on the subjective preference structure of the d e c i s i o n s m a k e r s a n d thu6 it is the r e s p o n s i b i l i t y of the t o p management himself. The d e c i s i o n m a k e r h a s t h e possibility for assigning to the c u s t o m e r s different priorities. The customers and following from that the contracts are

73

prioritised taking into consideration economical and commercial aspects. The procedure CAPS offers the possibility of involving s t a t e o r d e r s as restrictions too. But this a p p r o a c h is left to t o p m a n a g e m e n t ' s d i s c r e t i o n . A d e s c r i p t i o n in d e t a i l o f t h e a l g o r i t h m o f REH y o u f i n d in /Straubel-83-1/, /Straubel-85-1/, /Straubel-86-1/ and /Wittmü(3-83-l/. Experience/Conclusions: The DSS-CAPS realises the connection between the t a r g e t s o f p r o d u c t i o n a n d s a l e s m a n a g e m e n t with the a i m for r e a l i z i n g the h i g h e s t p o s s i b l e p r o f i t . Consequently CAPS is a multi-criterical DSS using the algorithm of R E H . B y m e a n s of C A P S t h e d e c i s i o n m a k e r and t h u s the t o p m a n a g e m e n t h a s the p o s s i b i l i t y to d e c i d e f o r o n e of the p a r e t o - o p t i m a l s o l u t i o n of a s e t o f c o m p r o m i s e p o i n t s r e l a t i v e to p r o d u c e d a n d s o l d g o o d s . In d e p e n d e n c e from the choosen alternative all target results are shown to the decision maker (record). The data b a n k s y s t e m R A B S 1 is u s e d in m o r e t h a n 50 enterprices. T h e D S S - C A P S w i l l b e i n t r o d u c e into t h i s e n t e r p r i c e s in 1989. A m o n g o t h e r t h i n g s one r e s u l t a n d s u c c e s s is the improvement in the field of management in q u a l i t y and quantity. References /Schmidt.-87-1/ S c h m i d t , R .

RABS1- rechnergestützter A b s a t z 1. S t u f e , N e u e T e c h n i k im B ü r o 4/87 VEB Verlag Technik

/ S c h m i d t - 8 7 - 2 / S c h m i d t , R . D a t a B a n k S y s t e m s in t h e M a n a g e m e n t N e u s e T e c h n i k im B ü r o 6/87 VEB Verlag Technik lecture.no publeshed manuscript / S t r a u b e l - 8 5 - 1 / R. S t r a u b e l , A. W i t t m ü ß , R. R o s e n m ü l l e r in Proceedings of the international symposium held in B e r l i n (GDR) 1985, e d i t e d by Achim Sydow, Manfred Thoma and Robert Vichnevetsky A k a d e m i e V e r l a g B e r l i n 1 9 8 5 , M a t h , r e s e a r c h 27 / S t r a u b e l - 8 6 - 1 / R. Straubel, A. Wittmüß, Das Programmsystem REH z u r r e c h n e r g e s t ü t z t e n E n t s c h e i d u n g s h i l f e ZKI I n f o r m a t i o n 2 / 1 9 8 6 Akademie der Wissenschaften der DDR / S t r a u b e l - 8 3 - 1 / R. S t r a u b e l , A. W i t t m ü ß , R. R o s e n m ü l l e r eine rechnergestützte Entscheidungshilfe makroökonomischen Planansatzrechnungen ZKI I n f o r m a t i o n 4 / 1 9 8 3 p. 106- 156 /Wittmüß-83-1/

74

bei

A. W i t t m ü ß , Ein sequentielles Entscheidungsverfahren zur Bestimmung einer mehrkriterialen "optimalen" Steuerung ZKI I n f o r m a t i o n 2 / 1 9 8 3 , p. 94- 113

MODELING AND OPTIMIZATION 1

Schmelovsky, Karl-Heinz

This

paper

is an

'

attempt to get

physical modeling

optimization somewhat closer together, state

using

the

and

mathematical

concept of

enlarged

space. Here, every influence, assumed to be neither exactly known

nor totaly random, is described by state variables. This leads automatically to markoff processes in that state space. Furthermore state models can

always

be

formulated so that observables

depend

only

on actual

state and - possible - on white time discrete noise.

,

?j-y(e)

*

• can

e

'

( & ) u j

y ,

-

~

mostly by neglected.

strategy its only necessary to find respective next

order

to get it,

steps. sequence

80

U

are defined similar to before and

_ A Dependance of 0Cf g on For

'

This

meens

step.

its necessary to regard its influence on that

not only the reaction of object

But

in

consecutive on

a

given

of control steps but the sequence itself must be extrapolated.

Dynamical

programming

solves this problem by a s s u m i n g ,

d e c i s i o n s are optimal

also.

The problem is,

"as good as possible" with than existing Aim of optimization

is here to minimal

consecutive

that "optimal" m e a n s realy

k n o w l e d g e and processing

total

(more exact the integral) of immediate

that

loss, expressed

means.

by the sum

losses

J*-' 17 JA Immediate

loss can mostly

be expressed

control

costs

(e.g.energy)

10

ffcjtl

i Uj)Sf(*j+1~

The

basic

use

introducing

>

the value of a state

gain obtainable starting this

problem, by

+

error

loss and

wanted

terminius

also,

G(z)

i.e.

the

a

sum

of immidiate loss and value

of

maximal

steps.

minimum

i. e. real value has opposite sign. V a l u e function

minimizing

influence

from this s t a t e in remaining

in spite of that we treat

direct

trajectory.

"deterministic" approximation n e g l e c t s statistical

in this task, total

by control

We loss

is obtained

consecutive

state

recursively [*?]•

G

j-1 (zj-i) = Min

{ 0. Then { ri„; n = 0, 1, 2, • • - } is an ONB in H_„. which is a Hilbert scale space [2] with generating operator A and inner product