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DYNAMIC MODELS FOR INTERNATIONAL STABILITY ANALYSIS P. Kopacek and F. Breitenecker r (,(/III/mf [ ' /I/l'l'n/l\"

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Abstract. There are three possible options far considering appropriate models ~or static and dynamic behaviour of nations and tor tha interactions between them, namely the 'macroeconomical' approach. the 'power'- approach and the 'ideological ' approach. Control engineering is frequently used to build up mathematical models. consisting e,g. of differential or difference equations, transfer functions. state space equations. etc. especially for preferably technical systems. Therefore in this paper an overview about mathematical models for systems of different complexity is given. These models are well known in control engineering and have to be adapted tor the purposes of international stability. Fundamentals of application of appropriate control alguriLloos will also discussed . Models of Simulation of Nations. j\eyword§.~ In ternational Stabi 1 i ty . Global Systems. Large Scale Systems. Contrvl and 0ptinti Zet ti':J!l.

MODELS

INTRODUCTION Most at the papers dealing with international stability or related topics today are more or less verbal descriptions or the phenomena arising. As pointed out at the first an second SWISS workshop and the last IFAC congress exact mathematical models for stud1es in international stability will be absolutely necessary in the nearest future. In our opinion an approach from the point of view of control engineering should be applicable.

For studying stability problems in a system the main question is the determination of an appropriate control algorithm. For this purpose a suitable mathematical model for the static as well as the dynamic behaviour of the system must be developed. While in technical systems the model parameters are well determined by physical laws in this case the model parameters must be calculated trom eg, stochastic time series.

The perhaps most interesting question is the question of stability within the relations of nations. For testing the stability in the senSe of control engineering a simple but sufficiently accurate model for the phenomena of the system under investigation will be necessary. In terms of control engineering one isolated nation might be described as a nonlinear multivariable dynamic system. For the description of this dynamic behaviour inputoutput relations. state space approaches and today other new methodological tools are developed.

But the first question is how to interprete ' measures' and 'quantities' which are able to describe the behaviour of a nation and the interrelations between nations, The question which immediately follows is which variables are the states (the variables which have their own dynamics and which can be influenced only on behalf of control variables), which are the control variables (variables which are able to influence states and which can be changed by an 'operator') and which are the output variables (states which can be 'observed' and consequently be measured for getting information for building up a certain control variable).

In the paper lirstly the problem at model building will be discussed and var10US approaches will be sketched briefly. for this reaS,:'ll three model philosopliies rlanl~ly

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As pointed out in earlier works (Breitenecke r, 1986; Chestnut , 1975; Kopacek. 1986) there exist three possible options for building up appropriate models for static and dynamic behaviour of the stability of a nation and the interrelations between nations. They depend mainly on the the modeller's point of view and will be discussed in the following.

' macroecononlical models ' . ' power and . id8vlugical H1ud81s ' ar8

investigated.

Eased

on

these

models

various c0ntr01 concepts seem to be usetul

tor

our purpose and are therefore shortly

reviewed.

s0cio -

A.s

appi1.cat.i0I1

political

~0n!lict5

a

ITIudtl

fur

is discussed.

The macroeconomical approach Bases of these types of models ara economic relations between various input-. output-, state- and control variables within a nation and/or betwen nations. These types of variables are well known from literature (e.g. Chestnut, 1982).

1..1:)

P. Kopacek and F. Breitenecker

146

This concept naturally leads to tht: 8onsideration of such important subprocesses as population dynamics, production factors, resources, environments, ecological systems (Peschel, 1983) .

The 'power ' - approach Basis of these types of models is tht: decomposition of the overall power within a nation and/or between natlons into special kind of powers: political power, mental

power,

The

power,

economic

power.

nature ) s

individual group ' s power, etc .

'ideological~~aGb

Bases of this ' phil os ophical' approach are interactions between almost undefinable variables and ' quantities ' like measures for ideology, society, structure, acceptance, pressure, tension lwithout specification),

etc .

In practice a suitable model will be based on a mixed approach, which tends mainly to one of the three above sketched ' model philosophies' The ' ideological ' approach seems to be very suitable, but is difficult to use . Therefore the macroeconomical approach and the ' power'- approach will give results in a shorter time . Fo~

the purposes ot control englneering sUltable models tor the dynamlc behavl0ur are absolutely necessary, a static model is not sufficient . But these dynamic models have to be sufficient simple and accurate in order to be able to study the behaviour lstability, etc . ) of the system . Dynamic models assume that the whole system may be disaggregated into smaller subsystems which can be described by simpler but accurate enough dynamic models. There are two principal types of models which are used in control engineering and which will be used here, namely input-output models lexternal models) and state space models linternal models). On the one hand input -o utput models are suitable for simple linear processes with one input and one output lSISO-systems; single input single output) . On the other hand state spaCE: models give a deeper insight into the internal relations ot a system and are therefore commonly used :tor more complex systems with multiple inputs and multiple outputs lMIMO - systems). Both types of systems ISlSu systE:ms, MIMO systems I have various characteristics which are: A< nonlinear A<

linear

A< time invariant

* A<

*

(stationary) time varying ,instationary)

concentrated parameters distributed parameters

The hierarchy of systems based on these is shown in Fig. 1 characteristics ,Kopacek, 1978).

Systems with the highest complexity are nonlinear time varying systems with distributed parameters which are marked as stage IV lFig.1). Successive simplification of these models yields tinally to the simplest of linear time invariant systems with concentrated parameters. t'lodels tor the bE:haviour ut nation s alld inlE:ractions between nations are naturally ot nonlinear and timE: varying tYPE:· But this type uf models (stage III in Fig.11 is very cumplicatE:d to handlE:, for instance an amount at Itime varying) parameters is to be identif1ed in case of quantitative analysis and individual simulatiun.

ThE:rE:iurE: simpliticatiun to stage 11 u1' stage 1 is absolutely necessary, i. "'. tll'" models are almost eithE:r nonlinear or time varYlng or they haVE: distributE:d parameters.

As examplE:, thE: dynamics ot a state space t.i.mevaryiIlg a nunlinaar model 1 0 1' be given b y a system or tHl'10 - system may diflerential 8quat1. 0 !l.S

xlt) g l x, u, L)

It is to be notE:d, that this continuous model can be easily changed to a dis c ret", model by changing the ditferential quotient into the dittE:rence quotient.

une general difficulty in model building tor tor this purpose is how to definE: appropriate input, control, state and output variables and to d i stinguish between them. Depending on thE: spE:cial problem appropriate variables might bE: :

~

*

national income

productioll .,. capital stock

.+- consumption

*

*

* * *

overall resources

public well being prE:ssurE: tur rE:iorm resources for daien c e

employment foreign tract", A< population government servicE:s technological facts advanced technologies A< agricultural equipment government control * resourCE:S lar public wellfa1'e A< intrastruc turE: * manufacturing equipments * military E:quipment and fa c ili l iE:s

*

* * * * ~

*

private sector services

transportatiun equ1pmelrt .,. education

Depending un thE: aim ut the model and uf the investigations Itor instanCE: with tha same model I the outlined variables may be states for one investigation , controls' for another, etc.

Dynamic Models for International Stability Analysis

The third step is the simulation of the modelled process using the built up model (with known model parameters) . If the model and the model parameters are accurate enough, the simulation allows the prediction of the system behaviour for instance under certain possible controls. If the model and the model parameters are not accurate enough, the simulation gives different possible 'predictions'; for certain fixed assumptions on the actual model parameters (a so-called scenario) different predictions can be computed.

For a linear. time-independent MIMOsystem the state space equations reduce to the linear system x(t) y(t)

= A.x(t)

+ B.u(t)

C.x(t) + D. u(t)

From the viewpoint of control engineering models in form of e . g . linear differential equations of n-th order with constant coefficients are suitable for first investigations of a problem and often these investigations are sufficient. But the models necessary for studies in international stability have to be highly nonlinear MIMO-systems. Furthermore some of the system parameters (coefficients of the differential or difference equations) may vary in a stochastic manner.

CONTROL ALGORITHMS The first difficulty is the definition of the term 'controller' for this subject. A controller should be an 'operator' who (which) sets appropriate actions . This operator works with a set of prescribed control algorithms where the actions to be choosen depend on different informations. As well known today classical as well as advanced control concepts are available .

Consequently in order to build up an appropriate MIMO-model it should be the first step to determine the structure of the system and the model (subsystems, feedbacks (especially 'dialectic' one)) .

The classical concepts are only applicable for very simpilfied models which are only valid for a distinct working point. The controller can be arranged with and without feedback. Consequently the classical control concepts may be only succesful for a first level of investigations.

The second step is the determination of the model parameters (the elements of the matrices A, B, C. D). These model parameters usually 'follow' for a 'technical' system from the 'construction' parameters of the system . In some cases these parameters follow from common known statistical data from the past, in other cases they have to be determined by comparing measured dynamic and modelled dynamic behaviour.

I- DISTRIBUTED I I

I

PARAMETER SYSTENS

ri

rr=========t=W=N.L I NIEAR

,-I

· U

r--N-L--T-I-V--D-F-'''' .

--r---~.

'flV

.. , T

SYSTEMS

.L-'_I -.-----,

TV

r-------r-..J TV

147

I I

',. ~

NL

TV

......-~._._L. I --.~ -.-J

r- -:-

L :

r--

'T'-+-' I I

I •

,

time varying (TV) • time invariant (TIV) nonlinear (NL) - - - - __ linear (LIN) distributed parameters (DP) -- . -- .- concentrated parameters (CP) Fig. 1. Main characteristics of models

ACVII -F

'l

-·l-"'-'--T"'IV~....I.--'

----J---1 TIV

It IV

lIT

J!::::;=::;::==:::II

L-. _ _ .-~ ~

CP

I

+ 1I

+ I

P. Kopacek and F. Breitenecker

148

Modern control concepts are the most suitable tool for describing the systems under investigation. For instance, state space models are able to describe complex behaviours and complex phenomena, especially in using nonlinear relations between the states mentioned before. But in most cases these state variables are not available (measurable). But modern control theory solves this problem using for instance (nonlinear) observers. Another problem arises in the determination of the structure of the control algorithm and also in the ~alculation of optimal control parameters. Therefore (advanced) algorithms for parametrization of control variables and for optimization are to be used . The . controller' there has to be seen as person (institution ... ). which (who) ~hooses an appropriate structure or changes the control structure 8orresponding to different criteria. An advanced feature of control engineering, namely adaptive control can help in building up appropriate models and appropriate controls. Starting with a simple model with corresponding control algorithms the operator ('controller') gets for each conflict situation suggestions for actions. If the operator carries out these actions he has to report the result to the model. The operator has also the possibility to neglect these proposals and suggestions and t o act in own response. The own decisions together with exspected results have to be given also to the model. In this way the model will be able to learn personal conflict situation and after a distinct time span it will be able to reproduce the reality. For the operator the model then works like an expert system with many degrees of freedom for him . For the operator in s o me sense the model then will have a kind of intelligent behaviour based on adaptive control. EXAMPLE FOH MODELING In earlier works (Breitenecker. 19SH; Kopacek 1986) modelling and simulation of a socio-political model (Saeed, 1982) was discussed (describing the peace and the stability within a nation). The model iD based on economic well being and depreviation as measures for the stability (peace). Stability and inner peace are guaranted if a certain amount of economic well being does not decrease under a certain limit and a certain amount of depreviation does not increase beyond a certain limit.

I

I r ···

i

Name

The model itself may be increased for instance by adding additional feedbacks and other influences or it may be reduced by neglecting certain feedbacks. The model is a compromise between bottom-up technique and top-town technique for model building of the systems under investigation (Breitenecker, 1986; Kopacek 1986; Peschel 1984). The variables of the model based mainly on the power- approach are dissidence, threat to power, depreviation and economic well being, the variables with economic background are total resources, resources for public projects and resources for maintainance of power. The model was studied extensively by Saeed (1982) for the dynamics of political revolution and fundamental socio-political changes; the model is based on ideas sketched by Forrester (19741. where a lot of other interesting fa c ts and assumption on the investigated problem can be found. Due to the furementioned idea of decomposing the system into subsystems described by important feedback relations (showing for instance controlled competing dialectic behaviour) one may decompose the process into subprocesses governed by important (stabilizing) negative feedback loops. by important (destabilizing) positive feedback loo ps and by coupled positive and negative feedback loops (varying behaviour). This kind of modelling yields to a model type of stage 11 as mentioned before. In Fig.2 the variables of this model are listed. The decisi ons of the regime to sto p or limit the its control are governed by several pressures resulting from positive and negative feedbacks. Positive feedbacks (of subsystems) lead to unidirectional changes in the variables in the feedback loop resulting in instability . Negative feedback loops have a controlling influence against changes in the variables in the loop giving in some sense stability.

Meaning

Coercive Control

I

Public Well Being Dissidence Pressure for Reform Threat to Power Total Resources Public Resources Power Resources

The socio-political model can be seen mainly as an example for the forementioned 'power'- approach. But it should be noted, that the way how to define ' powers' is based on the structure and ideology of the society and nation under consideration, on the structure and ideology of the modeller's society and nation and on the modeller itself. Consequently this model has to be seen as an example for the 'power'approach influenced by the economical approach and by the 'ideological' approach.

Control of the regime against dissidence. threat to power ... Measure for the Well Being of the people Measure for the dissidence in the public meaning Measure for the presure for reform Threat to the regime's power Total resources of the nation Resources for public projects Resources for maintainance of the power

Fig. 2. Model variables

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I! I

..... J

Dynamic Models for International Stability Anal ysis CONCLUSION For a deeper insight in problems related to international stability investigations from the view of control engineering seems to be very successful. Therefore in this paper concepts on control theory and related model building based on previous works are presented. Starting with general remarks the ideas of mathematical models used in control engineering are adapted for the purposes of the investigated topic . The properties of the model determine the mathematical effort of the user's evaluation. Hence, a simple but sufficiently accurate model is absolutely necessary. One of the main problems consists in determining appropriate input-, output-, controland state variables, where suggestions are given . Finally a short outline on adequate contro l concepts and simulation is given.

149

REFERENCES Breitenecker, F. , P . Kopacek and M. Peschel (1986). Peace as a cybernetic control problem. In R . Trappl (Ed . ): Cybernetics and System Research, D. Reidel Publ., 465-472. Chestnut, H. (Ed . ), (1975). Modeling large ~cale systems as national and regional level~ Report to NSF , Wo rkshop at Brookings Institution, Washington DC . Chestnut, H. (1982). Methodologies useful for improving international stability. IEEE _. Transactions on Jiystem . .t1an ang Q'yberpetiQJL Vo l. St1C-12 , no.5. 714'121 . Forrester. J . ( 1974). World Dynamics. MIT Press . Kopacek, P. (1978) . Identification of Time Varying Systems . Vieweg, Wiesbaden. Peschel, M. , F . Breitenecker. M. Grauer and W. Mende (1983). System analysis based on Volterra equations . Proc . of SWISS - Worksh.9RL Pergamon Press, 277283 . Peschel M. and F. Breitenecker (1984). Socio-economic c onsequences of the Volterra approach f or nonlinear systems . In R.Trappl (Ed . ): CYbernetics and System Research vol.2, North Holland, 656-660 . Saeed, K. (1982). Political revoluti ons and fundamental socio-political c hange : a system dynamics analysis. Proc . of the Applied Modelling and Simulation Co nference L Paris , July 1982, 21-31.

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