Thissen, Wil, "Guidelines and Tools for Understanding Dynamic Models", 1976

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GULDELINES AND TOOLS FOR UNDERS TAR DING

DYNAMIC MODELS

by

Wil Thissen
Project Globale Dynamica,
Systems and Control Engineering Group,
Eindhoven University of Technology
Eindhoven Netherlands

ABSTRACT

Starting from the aims and difficulties of social systems wuliling
this paper argues that a good understanding of dynamic mathonati-
cal models is indispensible. The author's background, and its 1e~
lation to System Dynamics is elucidated, and a number of definitions
are given of concepts and terms that will be employed. A set ut
general guidelines, and a list of strategies and tools for unl
standing follow. Most of the methods presented have been epplind
successfully in an extensive study of the World Nodels by Forester
and Meadows et al., and are commonly used in systems and control
engineering. The main emphasis is on techniques and points of view
that are generally unknown to researchers and practicians in the non=
~technical disciplines.

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zo

L

I,

11,

Vv.

Introduction

Background
IL,1 Systems and Control Engineering, and ‘i
IL.2 The Project: ‘Global Dynamics!

11.3 Concepts and Definitions

Guidelines and Strategies
IIL.t Guidelines
IIL.2 Strategies

Tools

IV.1 Flow Diagram Manipulation

1V.2 Investigation of the Effects of Changes and
Perturbations

IV.3 Tools for Model Simplification

IV.4 Detection of Fundamental Properties of (sul)

IV.5 Analysis of Linear Approximations

Concluding Remarks

References

Appendix -
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I. INTRODUCTION

One of the main purposes of social systems modeling is to help gain
insight into the working of some real system. The investigator's view
of the main components of the system and their interactions is described
in terms of a set of mathematical equations. While important insights *
are generated in the phase of model conceptualization, a further step in
the direction of insight into the corresponding real system is to study
the structure and assumptions of the model in an attempt to understand
the causes of its behaviour. Moreover, insight into the working of a

preliminary version of a model may be of great help to the modeler: it

may attract attention to model parts that need further improvement, or
even may lead to a total redesign. Once a model is considered to be
completed, a thorough understanding of its inner working is indispensi-
ble as a convincing basis for formulating policy starts, and to determine
“the best approach for possible further study. Moreover, in the uncertain
environment of social systems, qualitative insights about behaviour modes,
sensitivities, etc. are often more important and robust than quantitative
results. Such conclusions, based on a thorough understanding of a model's

working can be explained clearly and in simple terms, and offer possibili
ties for an adequate and convincing communication of the results and
conclusions of a modeling project to the clients as well as to the public.
Finally, knowledge about the mechanisms governing a model's behaviour can
be used for judging the validity of the model.

For these and other reasons, understanding models is an

issue important to all modeling studies. In general terms, a model is

understood if ita results and that of the whole study can be expressed
in words and/or simple diagrams”, and ve made quite reasonable to anyone
More specifically, one should be able to answer three types of questions
about a model's behaviour corractly without performing simulations, same.
~ What will happen if ....(a certain assumption is made)?

- Under what conditions will .... (a specified behaviour take place?!

* It might even be argued that simple patterns or diagrams are superior to
verbal description whea a moie}'s working has tu be explained. Diagrams
are able to transfer the pattern of simultane-us interactions in a dynamic
model directly to the human «iid, Since language is essentially sequential,
verbal description cannot d his.

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- Why does the model behave the way it does? 7

‘The first two questions are closely related, and require a general
understanding of the dynamics included in the main components of the
system, and of the interactions between these components. ‘The latter
question asks for explanation of the general dyriamic properties in

terms of the assumptions underlying the equations.

Whereas a good understanding of the working of complicated dynamic
models is essential, its acquisition remains a difficult and energy-
-consuming task. At present, model understanding seems to be an under-
developed part of systems analysis. The lack of effective, systematic,
and preferably simple tools that help gain insight is serious. This
paper attempts to meet in part the need for such techniques by pre-
senting a list of guidelines that came up and tools that were found to
be of use during an extensive analysis of the World2 |3| and World3 |6|

models, The paper is a continuation and partial extension of an article

by Rademaker |9|, upon which it has drawn freely. For the most part, the
guidelines and tools are well-known and used frequently, particularly
in control~ and systems-engineering circles. In Chapter II, some back-
ground information will be given, mainly with respect to systems and
control engineering and its relation to System Dynamics, and with res~
pect tot the aims and goals of the ‘Global Dynamics’ project group.
Chapter III presents a list of guidelines. A concise description of a
number of tools and techniques that may be of use for understanding
dynamic models follows in Chapter IV. Since some of the techniques start
from views that are not Eamiliar to all system dynamicists, part of
Chapter 11 is devoted to a description of a few underlying concepts and

definitions,

‘The major part vf this paper discuse2s the application and use
of various concepts and tools. Whether ¢u investigator will appreciate
a technique as being approoriate and easy to apply is mainly determined
by his background and experience, and by his personal preferences. any
judgement about the importance, utility, aporopriateness, simpleness,”

energy costs, ete. of an approach wuntains 4 subjective element. parti~

cularly if the same results can be obtained slong difterent ways.

Clearly, the reader should keep in mind that his uso applies eo thy

judgements put forward in this paper,

AL syst

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11, BACKGROUND

is AND CONTHUL ENGINEERING, AND SYSTEM DYNAMICS

‘The quick progression made in the field of aviation, electronics and
the area of chemical processes was, at least in part, due to the fast
evolution in the field of systems and control engineering in the
years before and during World War 11, Conversely, the need for
practical and feasible solutions to actual problems was stimulating
is called "classical" in control

the development of what, nowaday
theory. these prior analyses and theories dealt almost exclusively with

linear systems, and were heavily based on impiriciem and trial and error dexign,

and characterised by the lack of a fundamental theory, However, problems in

cal and aerospace engineering grew more and more complex, and a

more fundamental approach was urgently required. "Modern" control
theory was born in the late fifties, Rather than centering around fre~
quency-domain techniques and feedback loops, as the classical approach,
modern control theory focusses around general time-domain

descriptions of dynamic systems. But, developed and

presented mainly by applied and pure matheme:icians, it is quite in~
accessible to most control engineers and others who are interested.
Although modern control theory has unmeasurable qualitites, in many
practical (and simple) problems the need for the cld art has not been
eliminated yet.

Many fundamental ideas of System Dynamics as formulated by Forrester
in Industrial Dynamics |2| (particularly the emphasis on feedback loops)
ave based on the classical approach in conttol theory. But there are
many, differences, not only paradigmatic, or in the nature of the re~
yearch objects, but also in the tecanical field, such as the emphasis
on nunlinearity, the time-domain representation, «nd the abundant use
of computer simulation,

Still, System Dynamics and systens and con:rol theery deal with
similar problems, Practicians in the two fields attemoc to analyse and
understand dynamic systems to be able to improv. real system behaviour.
Also, in both disciplines there is « tendency te tackle mre and more

complex problems. Increasing complexity was one of the incentives to

12

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adopt @ more general and fundamental approach in control theory,
whereas - except for the extension of computer-simulation faci-

lities - no further development of tools for solving the complexity
problem seems to have been realized in the system-dynamics field.
Therefore, System Dynamics may grow up to a higher degree of maturity

if it would adopt and/or adapt ideas and techniques developed in

modern (and classical) control theory. Certainly, not all concepts

will be useful, because System Dynamics focusses on a rather different
field of application, but particularly in problem areas of high complexi-
ty, where a systematic approach is required, such as parameter estimation,
model analysis and understanding, and policy analysis, control theory
offers a number of powerful tools which probably can be adapted to

System Dynamics without too much trouble. A few examples have al~

ready been given by Peterson |7| (parameter estimation) and Sharp |12(
(systematic sensitivity analysis). Also in this paper it will - among
other things ~ be tried to demonstrate the utility of certain engineering
approaches (state-space concept, total linearisation).

THE PHOJECT': "GLOBAL DYNAMICS"

This paper is one of several outcomes of the project 'Global bynamics'
(see also ref. |8]). The project wae started in the course of 1972,
after the appearance of the first publications on World models by
Forrester |2|, and Meadows et al. |5|. In these publications it was

- much more than in most econometric and macro-economic models - focussed

on the dynamic properties of systems, including levels and feed-back Loops,

‘and causality was emphasized. This might be one of the reasons that the

attention of a number of system and control engineers was drawn. Another

reason, however, was that there was a strong feeling that only little,

if any, of the existing knowledge about and experience with the analysis

and control of mathematical models of dynamic systems was used. There-

fore, among other reasons, a project was started, the main aims and goals

were not to build new models or to criticize the assumptions made by

the M.L.T.-groups, buts

- to analyse the models from the control~ and systemsscience point of
view, in an attempt to gain a thorough understanding of their inner

working and structure,
11.3.1

- 691 ~

~ to examine the effects of various kinds of control, particularly
(stabilizing) feedback control and optimizing control (dynamic
optimization).

- to communicate the information gathered and the insights obtained
to an audience ag wide as possible, and to develop and compile an

array of techniques for the analysis and control of dynamic models.

This paper forms part of an attempt to fulfill the last goal, but it
cannot be seen as the result of a separate study. The activities of the
group were centered around the study of the World2 and World3 models,
and only afterwards the various methods and techniques used were listed,
and their general usefulness for understanding was judged. This explains
why the application of most of the tools described in Chapter IV will
be illustrated using examples of the World models.

CONVERTS AND DERINIT'IONS

This section will start with a description of the state-variable point
of view and of its advantages when looking at a complex dynamic system.
‘The state-variable concept underlies the greater part of modern systems
and control theory, and also.many of the tools for understanding that

are presented in this paper.

‘The second part of this section will be devoted to the clarification
of a number of terms that are used frequently, such as 'time constant",
‘model structure’, and ‘stability’. It is observed that different persons
(or the same persons at different times) may. adhere different meanings
to each of these terms. To avoid misinterpretations, it will be tried

to make a clear distinction between the various meanings.

Statenspace repres:
In classical control theory, the feedback loop plays a central part
as a tool for influencing (mostly: stabilizing) as well as for ex~
plaining @ system's behaviour. Similarly, system dynamicists often

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see a dynamic model as a conglomeration of interacting feedback loops".
Hlowever, in modern control theory the so-called state-space view is

prevailing. This means that a dynamic system is seen as a conglomeration
of interacting state variables. For ease of discussion a system without

time-varying inputs is considered. Its state-space description has the

form:
”
& = £G,0), x(t.) = x, or a)
x) =x, + S' eGunae (2)
t
‘0

& is the vector of state variables, t, the initialization time,

and £(x,t) is the vector of algebraic (that means: Lag-free) functions
relating the rate of change of each state variable to the actual value

of all state variables, including itself. State variables are all those
variables the value of which is adjusted by integration of

the effect of one or more rates Thus, in system-dynamics models all levels
together with the sublevels included in the 'delays'*** constitute the

vector of state variables. If the number of state va

bles isn, x is
a vector in an n-dimensional space, called the state-space.

The point of view that a dynamic system consists of a restricted
number of interacting state variables has a number of advantages:

a. By nature, the stdte-variables approach focusses attention on those
elements that are most important from the dynemical point of view:
the levels. ‘the dynamic behaviour of a system finds its origin in
the inertia included in the integration process taking place in

the levels; without levels, no dynamic sehav'vur is possible at all!

in this context, it is amazing to see hox little use is made of the
feedback loop as an instrument for influexicing vehaviour in systen-
~dynamics policy analyses.

& means: derivative of x with respect te time,

In control theory, the word 'delay' is used for pure or pipeline delays
only, while nth order stable systems (such as SMOOTH and 'DELAY' in
DYNAMO) are called 'Lags'. Terminology in this paper, however, will con-
form to common system-dynamics practice.

- 693 -

b. Automatically, components that have similar dynamic effects are
treated in the same way. Think, e.g., of delays and levela, and of
the relations among the state variables? Parallel links that
connect two variables in the same direction may easily be lumped

together to form one influence.

c. Only two kinds of variables have to be distinguished! state varia~
bles and their rates of change. This reflects and exploits the fact
that in essence all coupling variables are functions of the state
variables only, and that the system is fully specified by the values

of its state variables.

d. The stute-variable description of a dynamic system is at the same
time complete, and irreducible. The number of independent state
variables” determines the dimension of the system. Elimination
‘of one or more of the independent state variables essentially ex-
cludes part of a system's dynamics. Particularly if large and complex
systems are considered, this is a great advantage. The maximum nun-

ber of links between the state variables is proportional to the
ible

square of the system's dimension, whereas the number of po:
feedback loops is much higher. One might argue that the number of
important feedback loops will not be much Larger than the number
of levels in a system, However, it is easily seen that in a system
in which all links between the state variables are of more or less
equal importance, the number of feedback loops that affect system
behaviour is much larger than the number of levels, and even than
the number of Links,

the state-variable representation

Uowever, Like each point of vi
has its shortcomings also: It cannot adequately deal with high-order

" If the set of state variables is dependent, at least one of the state
variables can be writtan as 4 linear function =f (some of) the
others, and thus the dimension of tae state space is less than the
total number of state variables.

11.3.2

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delays. An nt order delay would induce n state variables, thus blowing

up the dimension of the system. Mowever, most high-order delays (or

even pure delays) can be replaced by first-order lags without affecting
@ system's basic dynamic properties”. Different problems emecge

if the system includes elements that display discontinuous behaviour

as a function of time (e.g. CLIP-functions), or that contain hysteresis

effects. The former can be dealt with by introducing time explicitly in
the right hand term of (1), but many state-space~based techniques ig-
nore this possibility,

Finally; many system dynamicists (and others!) will make the ob-
jection that the transition to the state-space representation of a dyna
mic system will lead us too far away from the issue we are interested

in, namely understanding the forces and mechani

6 acting in the real”
system. Indeed, there is real danger of drifting away into mathemati-_
eal abstractions. But, on the other hand, the analyst should not feel
refrained of using any tool that can be helpful. Techniques starting
from the state-space principle (such as total linearization, see

Chapter 1V) systematically and efficiently uncover the main dynamic
Properties and the most important Links in a model. Such information

is very helpful to the analyst, Although it is not immediately related
to the basic assumptions, it considerably facilitates the further _
exploration and understanding of a model's behaviour in terms of those
underlying assumptions, and also in terms of the feedback-loop structure.

time vons tanta

The term 'time constant’, usually represented by the symbol t, is used
frequently to characterize the time-variability of a dynamic system.
Intuitively we feel that the order of magnitude of the time constants
included in a system determines the system's inertial properties: large
time constants involve slow change, whereas small time constants may give
tise to quick variations. However, it appears to be quite difficult to

give a precise definition of what a time constant is, covering all

meanings in which the concept is used. Upon closer inspection, it appears

. Jeni 7 Acne
The explanation is that the many Levels inciuded in high-order delays

are virtually dependent, so that they can be iumped together without
inducing major dynamic moc1fications-

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that the term "time constant' is mostly used to denote one out of

three different meanings that will now be defined tentatively.

a. (1: a characteristic of a single, lay-free feedback Loop,
‘the average lifetime of capital, and the adjustment time of a level
to its indicated or required value are examples of this kind of use
of ‘time constant’. In principle, it would be more correct to speak
of «1 as a characteristic of the behaviour of an isolated state
variable with only one active lag-free feedback loop, but this was
not done to conform with common parlance, and to avoid confusion
with the second meaning in which the term is used. If the rate K
(affecting the level L) is a function of L itself, a mathematical
definition of «1 ij

. @)

Thus, if R is a linear function of L, tI is the reciprocal of the
multiplier that defines the rate in terms of the level. If «1 is
positive, L will display exponential growth, while exponential approach
of an equilibrium value will be found if +1 is negative. However,
usually only the absolute value of 11 is taken into consideration,

since negative time has no meaning.

b, 12: a characteristic of the behaviour of a state variable.
More often than not, the rate of change & of a state variable x is a
function of - among other state variables - the value of x itself.
Analogously to the definition (3) of 11, 12 is defined as

a : a)

12 characterises the behaviour of x if, in a model, all other state

variables are frozen. It is the reciprocal of the gain of the combina:

of all lag-free feedback loops around x. If 12 is negative, it is

a measure of the speed at which the state variable x will attain steady
state after a small step-wise change, and therefore of ‘the lag impli-
cit in x, 1f 12 is positive, the net feedback of all loops around x is

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positive, and x is a source of autonomous growth in the system,
This interpretation of the term time constant is used frequently
in classical control theory. The definition of 12 fits into the
concept of a ‘time constant of a first-order delay’.

. 13: a characteristic feature of the overall behaviour of a system
The speed of the changes that actually take place in a system or
model can be characterised by some measure of time, but it is
quite difficult to give a simple and precise definition, except
for linear systems, It is a well-known result of Linear theory,
that the behaviour of each system variable can be described by the
sum of a Limited number (equal to or less than the dimension of
the system) of exponential functions of time (including the complex
exponentials). The reciprocals of the values of the exponential

can be calculated

coefficients(t! the eigenvalues of the

matrix describing the system in state-space notation) are con-
sidered the time constants of the system. The time constants .«3
of a non-linear system may be computed by Linearization of the
systems's equations, or by analysis of the behaviour of its variables
(study of bandwidth characteristics, fitting of behaviour to exponen-

tial functions).

In the simple case of one state variable and one loop, the value
of the time constant is independent of the choice of definition. Also,

it is evident that time constants are really constant in linear systems

(or, for 11, for linear loops) only. In the case of nonlinear relations..

the values of the time constants according to all three definitions
may change with the state of the system, and thus with time. The time
constants tI and 12 may be implicit in table functions, multiplier
relationships, etc., but can always be computed directly from the

equations by tracing and combining 1ag-free feedback loops. The rela-
tion between ti and 2 on the one hand, and +3 on the other, is more

obscure. Except for a few simple cases (such as first~ and second-order

linear systems) it cannot be established without solving the model

equations.

15.3

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Structure

‘The word ‘structure’ is used frequently with reference to a system or
model, but not always with the same meaning. The structure reflects
the pattern of relationships between the variables, but to what degree
o£ detail? In fact, the word is used to indicate one out of a whole
series of possible meanings. A lot of discussion might be clarified
if more precise definitions of ‘structure’ would be used. As a first
step, it is proposed to distinguish between the following three inter-
pretations.

a. dn influence diagram
The influence diagram shows the existence or non-existence of
relations between the main variables of a model or system. It shows
also which variables are endogenous, and which exogenous. In this
context, a structural change means a change in model boundary,
omission of an existing link, or addition of a new Link, but not
a change within a link. Figure | shows a simple example of such a

structure.

Figure 1: Influenve dtagran

b. A flow diagran
Often, the word ‘structure’ can be repi:ced by ‘flow diagram’. It
includes all single tynamic relations -s well as the nature of the
variables (level, rate, intermediate viriablc). According to this

interpretation, 2 hange .n stt's.ture veang * change in flow diagran,

11,364

~ 696 -

such as the addition or omission of a loop or a delay. A para-
meter change may have structural implications if its result is
to cut or to add a loop or relationship (e.g. if the change results

in a zero-value of a multiplier).

e. A picture of the actual working of the system

This meaning of 'structuret will henceforth be denoted as "operating
etructure". It reflects the mechanisms actually governing a system's
behaviour, and may involve different degrees of detail. The operating
structure of a iodel can, e.g., be illustrated by a simplified flow
diagram containing the most influential loops and variables only, or
by a picture showing the major interactions between the subsystems
(as shown in Figure 2 in this article).

It is nat unusual that a system that is structurally complex
according to the first two interpretations of ‘structure’ (influence
diagram, flow diagram), has a very simple operating structure. Whether
a model's operating structure is changing with tine during a simulation
depends on the level of detail considered, and on the degree of non-
linearity of the model”. clearly, changes in parameter values may
have implications for the operating structure of a model.

Stability

Often, a system or model is called stable or unstable, but the concept

of stability is not always well defined. It is generally agreed that

a system is stable if it returns to an equilibrium state upon

exogenous perturbations. tlowever, the concept is also used to indicate that
the effects of perturbations in a dynawic mode of behaviour will vanish.

A third interpretation of stability applies to the general mode of
behaviour of a system: If a system displays growth followed by decline,

or oscillations, rather than a gradual approach to an equilibrium value

In systems theory, systems in whic nonlinearities play an important
part, or which include explicitly time-dependent relations, are
often called 'variable-structure system

- 699 -

(like,e.g.,the World models), it may be called unstable.

In this paper, a (sub)system will be called stable for a given
state if, all exogenous inputs being constant, it has the inherent
tendency to approach an equilibrium starting from the particular
state. It will be called unstable if it displays the tendency to grow
exponentially under the same conditions. The definition draws upon
linear system theory, and is consistent with at least the first two of
the interpretations given above. However, the definition implies that
it may be difficult to give a single judgement about the stability of
@ nonlinear system.For a certain state a nonlinear system may display
a tendency to grow exponentially (e.g., the World models during the
growth phase, i.e. between 1900 en 2010), whereas the same system may
possess all properties of stability for a different state (e.g., the
decline phase in the same model). Moreover, a system (nonlinear as well
as linear) may be composed of stable and unstable parts or subsystems
at one and the same time @.g., the capital subsystem in the world mo~
dels during the growth phase (unstable), and the persistent pollution
subsystem (stable)). Also, the stability properties of a nonlinear
system may depend on the exogenous inputs.

The same problems occur in nonlinéar systems theory. No simple
measure of stability exists for nonlinear systems. Gibson |4| mentions
that, for nonlinear systems, more than 28 definitions have been pro-
posed and used by various investigators, such as asymptotic stability
(boundedness if time approaches infinity) and monotonic stability
(which requires a gradual approach to equilibrium).

JIL, GUIDELINES AND STRATEGIES

In this chapter a number of guidelines and strategies for understanding

dynamic models are presented. They attempt to transmit to the reader
part of the empirical knowledge and experience existing in the field
o£ model understanding. Although many are trivial or well-known, they
are too important to be ignored.

The first part of this chapter gives a list of guidelines that
primarily attempt to draw attention to certain research attitudes that

are desirable when the dynamic behaviour of a model has to be understocd.

TiL,1

= 700 ~

Subsequently, the importance and possible outcomes of a number of re~
search strategies will be discussed. The guidelines and strategies are
not cited in a particular order of importance, Lt should be emphasized
that they undoubtedly have been influenced considerably by the author's
background, and that the list is far from being exhaustive: there is
auch room for improvement in this field.

GUIDELINE:
1, Be aware of the fact that the only general rule in nonlinear systems
is that there are no general rules, except this one. Any approach

may be useful in certain conditions, but completely useless or even

misleading in other circumstances.

2, Do not limit yourself: Use any idea or approach you can think of,

Bach has ite om serits - and shortcomings.

3. Be not afraid of abstracting from reality. Consider a model as a
jnathematical structure only, but always return to the original

starting point (basic assumptions, reality) afterwards.

4, Always ask: "Why?", and do not rest before you have the correct answer,

and you are sure that it is correct.
5. Do not overlook obvious things: they may be of crucial importance,

6. Always be sceptic about obvious explanatiuns. It is tou easy and
tempting to explain phenomena in a wrong way (apparently strong Loops
may hardly affect system behaviour, positive loops do not necessarily
induce exponential growth),

7. Keep always thinking yoursel£! No tool, technique or trick will ge~
nevate insight. Their only conteivution is to provide information in
a meaningful, ordered manner, so that the generation of insight by

the analyst is facilitated.

8 Do not expect that any tec!nique will ger-rate unique information, ALL

system properties may be wivovercd in diferent ways, but, in each
particular case, certain approaches might be mere appropriate than

others.
LLL?

HL

HLL e8

- ToL -

STRATEGIES

Devonport tion

Particularly when the set of equations involved is large, decomposition

of a system is fruitful. Dissection facilitates the analysis, because the
resulting subsystems are smaller, and therefore often more comprehensible

than the overall model, and lend themselves better to further examinations.
Decomposition on the basis of partial understanding may enforce the under-
standing, and facilitate the explanation of the model's working in terms of

the original assumptions.

The dissection may be based on several criteria. Large models, for
example, can be decomposed into the interacting submodels that have been
built more or less separately. Moreover, the flow diagram itself may be
revealing: clusters of equations that display many interactions but have
only few links with the rest of the model can often be discerned. Other
methods of decomposition are based on the actual working of the system,

A well-known strategy is to distinguish between active and dormant parts.
‘The so-called "dynamic decomposition" implies making a distinction between
subsystems that possess a relatively large amount of inertia (i.e. include
relatively large time constants in the sense of t3), and subsystems con-
taining relatively low time constants”.

Generally speaking, there is no unique way of dividing a system.

For a given purpose, one division may be more opportune, whereas, for

another purpose, another division may be more appropriate.

Underatanding a system on different Levels

Understanding a system on different levels is a very simple procedure, and
not substantially remote from what we do ix everyday life. When people try
to understand a complicated system as a whLe, they ignore many details and
focus only on the interplay of the subsystems that together make up the
whole system. In turn, the internal working of each individual subsystem
is understood in the same way, and so on. At each level of

understanding, it should be observed that the interactions between

* which does not necessarily imply thut the suksystem's variables actually

display quick variations!

111.243 Undere tending sys tem h

. a

the separate parts are more important than the individual subsystems, in
fact, understanding a system on different levels is the only possibility,
since the capacity of human mind sets limits to the number of details
that can be considered at one and the same time,

rarchy

‘The interplay of the subsystems causes a system to be more than the sum
of its parts.Therefore, understanding the hierarchy of the subsystems is
essential to understanding the working of the whole,

cR
POP: Population subsystem

Capital and Resource subsystem

cR

PoP A

A + Agricultural subsystem

PPOL? Persistent Pollution subsystem
PPOL

Figure 2: Major interavtions among the subsystems of Worlds wider standard-
run oontitions, cleariy shoving the syatem's hierarchy

The interplay betweer the main parte of the World3 model under standard-
“run conditions provides a clear eaample. Figure 2 shows the hierarchy of
the model, Except for the influence of the agricultural sector (quite weak
under standard-run conditions), the capits! and resource subsystem behaves

quite autonomously, and sffects the sehaviour of all other model sectors,

- 103 ~

It is not exaggerated to regard this subsystem as the central pover
station of the world model. When it goes down too. The growth and decline
food production sooner or later have to go down. The growth and decline

viour is included in the capital and resource sector, and
ctors (see |14|). This insight has important

mode of bel

impressed on all other

implications for policy analyses: when controls are implemented that
effectively stabilize the capital and resource subsystem, all other
sectors will ultimately tend to equilibrium also! Because the nature
of the equilibrum of population is rather undesirable", a combination of
two modifications will have to be introduced to improve the model's
behaviour in a fundamental way: one resulting in a stable capital and
resource subsystem (e.g. by allocation of part of the industrial out~
put to resource conservation), and another resulting in more desirable
equilibrium possibilities for population (e.g. by reducing the number
of children per family).

This example clearly illustrates the importance of understanding
system hierarchy. It enables the analyst of locating the basic causes
of system behaviour, and of formulating policy starts that take ad-

vantage of the system's natural properties.

Modes of behaviour

For various reasons, it seems wise to start with a thorough investigation
of only one mode of behaviour, The techniques and methods that are
adopted, and the dissection that is found to be appropriate for under-
standing one particular mode of behaviour may be vf great use to detect
the model

properties under Largely different conditions. The under-
standing of a reference behaviour provides an excellent starting point
for answering such questions as: “Under what conditions will the con-
clusions dravn becone invalid?", “what will happen if the Limit Lo the
validity of certain simplifications is exceeded?", and "Which subsystem
‘The natural equilibrum of pepulation in Vorld3 is mainly caused by
starvation, A different kind of natural equilibrium may exist at rather
high levels of income (abest the present U.S. conditions), but it is - un-

der present conditions ~ extremly unlikely that such high income levels
can bé sustained for long tor th whole world population,

AIT 208

"11.2.6

L1L.2.7

o

= Toh -

will behave differently under alternative conditions, and which not?".
Model simplt Floutton

A simple, but expedient strategy is to try to simplify the original set
of model equations as much as possible, but without loosing the basic
behavioural characteristics. If the set of equations that remains cannot
be reduced any further without affecting behaviour, it must consist of
the fundamental assumptions leading to the model's overall behaviour,
and further research inté the causes of this behaviour can be directed
to these equations solely. Aleo, the knowledge that other assumptions are

unessential to behaviour may be valuable.

Listie modi fications
For model-testing purposes, all changes (in equations, or parameters) will
have to be limited to the set of more or less realistic possibilities.
However, when a model's operating structure is investigated, rigorous
modifications and falsifications are permitted or even required, since
their effects may be much more pronounced, and hence much easier to

.
interpret™,

Combinations of tools
Combination of different tools for gaining information about a dynamic
model is not unusual, but it seems desirable to draw attention to this
possibility. Particularly, sequential application of various techniques
and strategies may be very expedient, e.g., simplification of equations,
reformulation®* and analytical solution”.

Also, it should be emphasized that almost each technique can be
applied for at least two purposes: first, for the detection of information
‘on system properties, and, second, for the verification of hypotheses. If,

for instance, omission of a loop does not affect system behaviour, the

Probably because of their «motional involvement with the product of their
own efforts, most modelers are inclined to subject a model to realistic
modifications only. Therefore, it might be ergued that a modeler should
delegate the task of analysis te some.ne not closely involved in the
process of gathering dats and buildiry the model,

See the next chapter.
Wedd

= 105 -

suggestion is put forward that the particular loop is not influential,
‘The same suggestion, however, can come up as a consequence of another
experiment, and then cutting the loop can be used as a technique for

verification of this hypothesi

Iv, TOOLS

Various techniques that are helpful in the process of understanding
dynamic mathematical models are presented. The list has been divided

into groups. Each group consists of a number of tools that are par-

ticularly suited for achieving a specific sub-goal (e.g. simplifi~

cation of equations) or that have a similar character (e.g, model
modifications), It must be re-emphasized that, in fact, the distinction
is not so clear, and that most techniques can be applied in various

ways.

ELON DIAGRAM MANIPULATION

A flow diagram may serve many purposes: it may be used as an influence
diagram, as an illustration of the elementary assumptions made in a

model, as a source of information concerning the form of the equations,

as a means for conmunication of information, as a basis for discussion

and, last but not least, as a point of departure and a source of ideas

for the model analyst. For any purpose, it is required that a flow diagram
is clear and understandable. A few methods of improving DYNAMO flow diagraus Iv.2
for the purpose of model analysis and understanding follow below,

Addition of move information
It is helpful to the analyst if the model equations can be written down
directly from the flow diagram, But then, the diagram must contain more
information than DYNAMO flow dingroms inc] ide. Particularly, adding mulci-
plication, division, addition ani subteactson sigus, and introducing
different symbols for table functions and other algebraic relationships
facilitates the direct tran: ition from the diagram to a set of well-

defined equations.

- 106 -

Since diagrams may serve various purposes, they should be redesigned in

a different way for each goal. During the insight-generation process,

a diagram may be a powerful aid for tracing influences from one variable
to another, or for detecting feedback loops. A DYNAMO flow diagram
usually is a mixture of an influence diagram and a picture of the flow of
physical goods and information, As a consequence, it may be difficult

or even impossible to go around loops in the right direction, Out-
flow-rates, for example, are connected to the corresponding level

by arrows leaving the level, whereas their value is actually influencing
the level, Therefore, redesign of DYNANO flow diagrams so that the direction
of all arrows corresponds to the direction in which the variables actually
influence each other may be useful.

Lf a diagram has to distinguish as clearly as possible between
dynamic elements (state variables) and algebraic ones, it is wise to
redesign it in such a way that only two different symbols are used - one
for state variables and one for all other variables.

Finally, in all phases of model building and analysis, the modeler
should rearrange the diagram time and again in an attempt to bring out

the structure ,
possible, This may be achieved by emphasizing similarities in different
parts of the model, by separating individual subsystems, by avoiding

intersections by the influencing Lines as much as possible, and in many

n any sense) of the set of equations as lucidly as

other ways,

INVESTIGATION UF THE EFRECTS OF CHANGES AND PENPURBALIONS

Many techniques for model analysis and testing are based on the introduction of
exogenous perturbations or of one or more changes in the set of equations,
and comparison and explanation of the differences and similarities between

the outcome of the modified or perturbed model and the original one. The
introduction of perturbations and modifications can yield information

relevant to answering all three questions involved in model understanding
happen?),

(hat will happen, if...2, Why does it happen?, and: When will.

> 10T -

Again, the list of possibilities following below is far from being
exhaustive, It will be tried to put emphasisis on those techniques
that are simple and expedient, but not so well-known in system-

dynamics circles,

Modifications in enogenvus inputs

It is started from the point of view that parameters (and constants)
are in fact exogenous inputs that do not change in a relevant manner
during the simulations, Thus, modification in exogenous inputs in-

eludes changes in parameter values as well as in the behaviour of time~

8 best known as sensitivity analysis

~varying inputs. The technique
Simple applications consist in changing the value of one or a few
Parameters at a time, There are more complicated variants also, such
as Monte-Carlo tests, the direct calculation of sensitivity functions,
(see Sharp |12| and Tomovic |15]), and the so-called hill-climbing
methods. Au advantage of these methods is that they are systematic,
and include variations in a/Z parameters, Nowever, for the greater
Part the information they produce is restricted to the possibility
of a certain behaviour, or to the sensitivity of the parameters, and
therefore their utility is larger for testing than for understanding models.
If an extensive sensitivity analysis has to be performed, the best
bet is to first investigate how the parameters occur in the equations.
Often, several parameters perform in an exactly analogous way, for instance
if only the product or quotient of two coefficients occurs in the
equations. These cvefficients can be combined into groups of parameters,
only one of which must be varied to show the sensitivity of all. in [14[,
the use and detection of parameter groups has been illustrated for the
World3 capital and resource subsystem,
Sensitivity analysis may yield important information on model
hierarchy: usually, each of the subsystems contains sensitive as well
as insensitive parameters, but overall behaviour is affected only by
modifications in those subsystems that play a leading part in the model
m be borresed from the World3 model:

hierarchy, Again, an example

Changes that influence the behaviour of tne capital and resource subsystem

= 108 -

(ouch as a change in the value of the industrial capital output ratio
(1cor)* from 3 to 4) cause significant deviations in other parts

of the model, Conversely, if modifications are introduced elsevhere,
for instance in the persistent pollution sector (such as a change of
the assimilation half life in 1970 (ANL70) from 1,5 to 2), the effect
on the other subsystems is almost negligible. This result suggests a
dominant position of the capital and resource subsystem, and a minor part

for the persistent pollution sector in the model hierarchy.

IV.2.2 Kalai fication of etate variables

In contrast to sensitivity analysis, thé technique of falsification of

state variables consists in a perturbation of the values of one or wore
ange in a rate or coupling variable

endogenows variables. Because any
can be explained as the result of a change in a parameter, the attention

is focussed on the state variables only. A simple application of the
technique is to augment the value of a state variable at the initialisation
point or any other point in time during the simulation, and to compare

the results with the outcome of the unperturbed simulation. [t is an
excellent method of isolating the behavioral impacts of the variations

that occur in the value of a particular state variable. Also, it may

yield information on the time constant (12) associated with a state
variable, and on its importance in the model hierarchy.

The foregoing generalization can be illustrated by reference to the
World3 model. If the value of industrial capital IC in 1970 is doubled.
the overall behaviour of capital and pollution chauges considerably, while
population is hardly affected, at least in the grovth phase (see Figure 3).
On the contrary, if the value of persistent pntlution in 1970 is doubled,
or even multiplied by 10, minor changes can be perceived only during the
first 10 years following the perturbation, Even pollution itself returns
quickly to its original order of magnitude, These reeults illustrate that

A list of letterscripta and thei: asaveixced meaning is given in an

Appendix,

IV. 2.8

=q0998
PoP : Population: scale 0 - 10'° peraons
IC + Industrial Capitals scale 0 - 1.5 10!9 ¢

9500 1950 2000 2050 2100
————> time (years)

Higure 3: Behaviour of Worlds if, during a standard-run simulation, the
value of industrial capital IC ie doubled in 1970. Thin linea
show the unperturbed standard-run resulta.

the capital and resource subsystem behaves quite autonomously (changes
in the values of state variables persist, or even become larger),

that its influence on persistent pollution is quite large, that sopu-
lation is hardly affected by the variations in capital (at leas! ia
the growth phase), that the behaviour of pollution is nearly complatetw
determined by the other sectors, and that the time conytant .2 of pe:

sistent pollution is relatively small (a few years sr less).

This method can be applied to inv: -vigace rhe denavroural impact of
single celationships or tcops, ov >F com-inai ions. The technique is to

freeze the value of one 0: wore ceasing -ariab)

» tuble functions or

rates from a certain momen" onwards, Lf the mode's overall behaviour is

PPOLX: Persistent Pollution relative to 1970; scale 0 ~ 20.

AV 2.4

- To ~

been frozen may be

not affected, the contribution of the link that hi
neglected if only the basic reasons for model behaviour are sought”. In
most cases, cutting links implies cutting one or more loops. Therefore,
various explanations of the sensitivity to freezing a single Link can
exist: If the link forms part of only one loop, this loop is probably
unimportant to behaviour, but if more than one loop is involved, the
working of the different loops might be compensating.

Ereesing gtate variables

Since the ‘state variables are the basic sources of dynamic behaviour ,
freezing the value of a state variable means excluding part of the
dynamics (ort making part of the dynamics exogenous), In terms of feed~
back loops, freezing a state variable means cutting all loops passing
through the state variable, The technique is simple but poverful since
it may show which part of the dynamics is crucial, and which not, and
thus may yield a wealth of information on the model's hierarchy and
operating structure,

Let me illustrate the application of the technique using the World2
‘model aa a vehicle. Hach of the five state variables included in the
model was frozen from 1970 on. Freezing pollution POL or capital-investment-
~in-agriculture fraction CIAF does not significantly affect the behaviour
of any of the other state variables. Freezing population POP from 1970 on
affects pollution, Leaves CIAF more or less constant, but the behaviour of
capital investrients CI and natural resources NR is virtually un-
changed. If, in turn, CI or NR are frozen from 1970 on, all other variables
will be affected as vell (as illustrated by Cuypers [1], freezing CI results
in constant PUP, CIAF and POL within a few years). The simple diagram of
Figure 4, showing the hierarchy in the model's operating structure around
1970 is the result. The diagram points to the central position of capital
investments CI and natural resources NR in the model. Moreover,the
experiment of freezing Cl shows that the other parts of the model are in-
berently stable (that is, they witl not grow exponentially of their oun

account).

The fact that a certain assumption docs not “tfect behaviour is also
part of the explanation ©! that behaviow

-TM- -T2-

xm) Cl + Capital Investments
WR + Matural Resources . Proper constant value changes mode} behaviour. If no important changes
POP + Population E occur, the relationship can be omitted, but not before it is under-
GIAF: Capital Investment-in-Agriculture stood why the range of behaviour is so narrow.
Fraction The capital utilization fraction CUP in Vorld3 is a clear example
POL : Pollution of a variable that is virtually constant during a whole simulation.
(~") Its value remains equal to 1.0 in the standard run because, during

almost the whole simulation, labor force exceeds the total number of jobs.
As a consequence, the so-called "job-sector" can be ignored for the

(us) (+e) anatysia of standard-run and similar behaviour.

we 4V.3.3 Contribution analysia
If a variable is influenced by more than one other variable, changes

Figure 4: Hierarchy of the World? model around 1970. in ite value can be explained as the net result of the different con-
tributions of the influencing variables. Comparison (for example

by freezing one link) of the order of magnitude of the different con-
tributions can reveal that some of the influences are more or less
negligible compared to the others.

i¥.5 POOLS FOR MODEL SIMPLIFICATION Again, a clear example is found in World3. The land erosion rate
A List of simple techniques for the detection of model areas that may and the land removal for urban-industrial use affect the level AL (arable
possibly be simplified under certain conditions is given. The back- land), but their contribution is only small compared to that of the land
ground idea is that it facilitates understanding a particular behaviour development rate, and’compared to the value of AL itself. Negligence of the-
when all equations and assumptions that do not contribute to that be- se tworates hardly affects the standard-run results of the model, but
haviour are omitted, and when the set of equations generating the be- : permits, as is illustrated in |13[, the construction of a considerably
haviour of interest is as simple as possible. less complex-looking diagram of the agricultural sector.
1.8.1 1V.3.4 Gor
ALL techniques presented in the previous section can yield information This technique is self-evident: if the value of a variable is calculated
on the importance of single assumptions or equations to model behaviour. from one other variable by means of several consecutive functional opera-
Changes that do not significantly affect behaviour point to areas of tions, combination of these fun:tions may simplify the equations, and
potential simplification. However, equations or variables should not be  ~ reflect more clearly the total effect of the influence.
omitted before the reasons why, and the conditions under which they do not
influence behaviour are well known, . 1¥.3.5 Combination of pavallel Links
‘The underlying philosophy is to combine parallel links between variables
IV.3.2 Observation of the range of behaviour of variables in order to simplify the flow diagram and to bring out more clearly the
The general strategy is to look for variables that are more or less total effect of parallel lag-free influences.
constant during the simulations. Because merely small deviations in An interesting application is the combination of all parallel, lag-

a variable may not necesearily be insignificant, the tigator must ~free feedback loops that relate the rate of change of a state variable

subsequently test whether repiacement of the time-varying relation by a to its own value. The resulting loop srovides information on the time
IV.3.6

- 13 -

constant 12 of the state variable, and on its stability properties: when
all other state variables are constant, the state variable will grow
exponentially if the loop is positive, but, if it is negative, an
equilibrium will be approached. Such knowledge may be very helpful for
the explanation of model behaviour.

The exponential growth of population in World2, for example, is
easily explained if the net feedback population on its own rate of
growth is positive. However, it turns out (seeg|I|) that, all other state
variables being equal, an increase in population reduces population growth,
so that population is inherently stable. Therefore the explanation for the
exponential growth of POP must be sought elsewhere in the model (in this
case in the net positive feedback of capital investment to its own rate
of growth causes capital, and also population to grow exponentially).

The use and shape of non-linear (table) functions

Often, the principal behaviour of a model depends upon only a small part of
the model's non-linear functions. If the portion of a function that is
behaviourally significant has a distinctive shape, the overall function
may be replaced by the simple expression as long as model behaviour
remains roughly the same. In particular situations application of the
technique may directly give rise to interesting insights.

Let me illustrate the foregoing using an example. In World3, the
non-renewable resource usage rate NRUR is calculated as the product of
population POP and the per capita resource usage multiplier PCRUM:

NRUR * POP * PCRUM, (5)

PCRUM is a table function depending on industrial output per capita

ToPé (equal to industrial output 10 divided by PUP). The shape of the function
is shown in Figure 5 (fat line). During the standard-run simulation, I0PC
remains below 500$ per head per year, and for this range of values, the
relation between PCKUM and IOPC is more ur less proportional (see dotted line
in Figure 5):

PCRUM ¥ a * 1OPC , (6)

where a is a constant. Substitutio: of (6, into (5) leads to an intersting

result:

LV. 3.7

- Th -
8 = PCRUM: Per Capital Resource
t Usage Multiplier.
PCRUM
TOPC + Industrial Output
(resource Per Capita.
units per
head,year:
ene) te)
0 ‘

B00 1600
———® 10PC ($/head year)

Figure 5: Table function showing PCRUM as a funetion of IOPC.
The dotted line illustrates a linear approximation.
Shading shows the part not used during the standard-
-run simulation é
NRUR Y POP * a * IOPC = a * 10. q)
This means that in World3 it has implicitly been assumed that, as
long as standard~run conditions are held, the usage rate of resources is
more or less proportional to industrial output, and virtually independent
of the size of population!
Similarly, given the quasi~linearity of two other table functions,
it can be shown easily that the direct influence of population on the
allocation of industrial output is also virtually nil (see |14|),

El

ination of emall Lage
The basic idea is that the influences of tags incorporated in stable

subsystems depend on the variations in input variables. Under static

conditions, the influence of the lags in a stable subsystem is nil,

and the subsystem behaves as if it were algebraic since no changes

through time can be delayed or accelerated. If the input variations

take place only slowly, the effects of relatively small lags might also

be neglected, The equations could be simplified by replacing the

original differential equation with the proper algebraic relation and

thus eliminating a state variable, However, the technique should be
applied with great care and, preferably, only by experienced investigators.
- 715 -

Yest calculations always have to be performed, because even a small lag Wed

may have a great influence, for example, if it helps to calculate the
average rate of change of a coupling variable. Moreover, lags that
are negligible for a particular behaviour might be very important
under different circumstances,

The approach is illustrated using the land-fertility subsystem
in World3: Land Fertility LFERT appears as a state variable, calculated
according to the equation:

LFERT = (ILF-LFERT)/LFRT - LFDR * LFERT. (8)

Inherent, land fertility ILF is a constant, and LFRT (land fertility 1V.4.1
regeneration time) and LFDR (la®d fertility degradation rate) are
functions of the input variables to the subsystem. In the standard
notation for a first-order system, (8) is converted into

LFERT = ~ LFERTS(LFDR+I/LERT) + ILF/LFRT, ()
which nieans that the two feedback Loops from LFERT to LFERT have
been combined. This equation specifies a system with the time constant 12:

12 = =(LEDR+I/LERT)”! @ “LERT/(1+LEDRALFRI) 10)

and the static function

LEERY Se aege ~ ELE/ (+L FDRALERT). ay) é
‘The latter expression can be used to compute directly the equilibrium

value of LFERT if LEDR and LFRT are constant. From an observation of the

standard-run values of LFDR and LFRT, «2 apparently has a value of about

5 years or less during the whole simulation. Because the main changes in

the input variables take place over a period of about 100 years, the

dynamic effect of the LFERT subsystem can therefore probably be neglected.

This hypothesis has been fully confirmed by a test calculation in which

the land fertility subsystem was replaced by the algebraic relation (11).

- T16 -

DETECL'ION OF FUNDAMENTAL PROPERTIES OF (SUB)SYSTEMS
This section presents techniques that help understanding the basic
properties of a (sub)system. They are particularly useful in answering
and "When will...happen?".

a model will be considered

the questions "What will happen, if...?

More often than in the preceding section
a set of mathematical relationships only. However, it is usually not
difficult to find the elementary assumptions underlying particular

once the latter have been isolated. Clearly, most of

ayetem properti
the techniques presented in the two preceding sections may also be help~

ful for the detection of fundamental properties.

Equilibriun analysie

The principle of equilibrium analysis is to investigate the conditions
under which d (sub)system will be in equilibrium, and the nature

of that equilibrium. Lt is started from the observation that a
dynamic system with a set of exogenous inputs u can be described by:

&- £0) +2), a2)
where £ and g denote vectors of algebraic functions. The system is in
equilibrium if & = 0, and hencet

£Qe) + g(u) = 0 a3)

Generally, equation (13) cannot be solved since the number of unknown
variables (state~ and exogenous variables) exceeds the number of
equations (equal to the number of state variables). Therefore the
equilibria are investigated under specified constant exogenous conditions,
which Leads to the equilibrium equation for a closed system:
£w +o aay
From this equation, possible equilibria, if any, may be computed
directly using techniques of numerical analysis.

1V.4.2

= ET =

Wowever, when subsystems with only a few identifiable input varia~
bles are under consideration, an analysis using simulation is often
simpler, and may yield wore information. It consists of Freezing the
value of all input variables to the investigated subsystem. If the
subsystem is stable under the existing conditions, its state variables
will gradually tend to equilibrium. By application of various constant
input values insight can be gained into the general relation between the
equilibrium outputs and the corresponding input values.

Let us take the population sector of World3 as an example: if
ite input variables 10 (industrial output), SO (service output), F
(food) and PPOLK (Persistent Pollution relative to 1970) are given fixed
values, population POP will gradually tend to equilibrium. Investigation
of the ultimate equilibrium value of POP for different combinations of
values of 10, SO, F and PPOLX shows that the equilibrium value of POP is
mainly determined by food (the explanation is obvious: because total food
production is kept constant, either population growth continues until
starvation causes the death rate to rise considerably, or, when the
initial value of POP is above its ultimate equilibrium, population declines
due to starvation until birth and death rates are equal). This result
suggests that population will not continue to grow exponentially unless
one or more of the input variables do. It also shows that when a policy
succeeds in a stabilization of 10, SO, F and PPOLX, ultimately population
will be stable too, but people in the model will have to live under

rather miserable conditions.

Dynamic tests of subsyatems

This technique is suited to investigate the dynamic properties of stable
subsystems with few input variables. It consists in imposing a special
test signal (such as a pulse, a step function, a ramp function or a
harmonic signal) on one or wore of the constant input values, By comparing
the reaction of the investigated subsystem with the reaction of simple,
well-known systems (for example, a first-order lag) to the same input
signal, a fairly good indication of the subsystem's most important
dynamic properties can be obtained. Particularly the step-function
method is simple and expedient.

oS

1.0 Scale Por: 0-10! persons

POP

Poy

400 200 300
——_____> time (yeare)
Figure 8: Response of population POP in Worlds if industrial output
10, service output $0, food production F and persistent
pollution ratio PPOLX are doubled at time=0 (original values:
ro=2 107, so=3 101, ¥=7.5 107! and PROLX=5)

Figure 6 shows the reaction of population in World3 if, starting
from an equilibrium situation (constant inputs and constant population)
the value of the inputs is suddenly doubled (step function). Population
rises slowly to reach a new equilibrium value. the dynamics of the sector
are, apparently, rather slow, and may be characterised by a time constant
(13) of about 70 years.

Figure 7 shows the reaction of two output variables of the
agricultural sector of World3 if a step function is imposed on popu-
lation, and all other inputs to the sector are kept constant. Almost
inmediately, the outputs reach nev, fairly constant values, clearly
showing that the dynamics of the agricultural subsystem are relatively
fast (the time scale is different from that of Figure 6!). Many more
examples could be given, but they would lead us beyond the spatial con~
straints of this paper.
-~19-
1.0
0.9
she | FLOAA
FIOAA: Fraction of Industrial ?
Output Allocated to ee
Agriculture. o.6 |
Scale:0-0.2. FIOAA
Food production, O85)
Scale:0-5 1012 walt No
0.4 J
Vegetable equivalent 0.3
kilograms/ycar. y
0.2
Oo.
0 30 100

~~ time (years)

Figure 7: The response of tx. output variables of the World3 agricultural

sector if population ia doubled at time=0 (original value:
Por=4 10°), :

of equations
Equations may be reformulated to bring out their characteristic
sumptions.

ego

properties as lucidly as possible, or to uncover implicit
llowever, the usefulness of reformulation is closely related to the
specific structure and properties of the set of equations to be exa-
mined. No general strategy can be formulated. Some experience with the
manipulation of mathematical equations is very convenient.

Rademaker ({9| and[10|) discusses
the observation that pollution absorption POLA in World2 is a function

clear example: he starts from
F* of pollution POL only:

POLA = POL/POLAT = F'(POL),
because the pollution absorption time POLAT is a nonlinear function
of pollution. Figure 8 shows the graph of POLA as a function of POL,

The curve shows that pollution absorption will rise when pollution rises
from zero to 10 times its 1970 level, that POLA has a fixed value for

(15)

W.4.4

= 120 -

POLA : Pollution absorption relative to

Pollution in 1970.
POLR : Pollution relative to 1970.

———> por

Fig™ure 8: Pollution absorption as a function of pollution

pollution between 10 and 20 times the 1970 value, and that POLA will
even decline if pollution rises to higher levels, thus yielding a
possibility for a real explosion of pollution at high generation
levels. The original formulation shows not carly as clearly how
POLA depends on POL. Starting from this graph, the behaviour of pol~

lution as a function of pollution generation can be easily traced

under all circumstances, normal and abnormal, and without any further
computation. However, because of spatial constraints, the reader is
referred to the original reports [9| and [10] for a further elaboration,

Analytical treatment of subsysvems

Analytical solution of simple subsystems is very informative, and it
allows direct computation of 9 system's behaviour at any specified time.
However, for the greater part analytical solution is impossible because
of high dimensionality and non) inearity. Yet, more attention should be
paid to the possible rewards of studying small subsystems using simple
analytical tools, a strategy often overlooked iv computer~simulation
circles. Analytical treatment, even without explicit solution, can

yield far more fundamental and genetal information on the properties of
1V.4,5

- Ta -

a (sub)system than numerous simulation runs. However, a fruitful application
of the approach calls for some experience and an elementary knowledge of
mathematical analysis.

In fact, two of the examples that have already been presented to
illustrate various techniques have been handled analytically: the
elimination of the lag in the land fertility subsystem of World3 (section
1V.3.7), and the linearization of the table function PCRUM (section
concluded that the

IV.3.6). Let us continue the latter example. It we
non-renewable resource usage rate NRUR was, for standard-run conditions,
virtually proportional to the industrial output 10, Hence, it follows

for NR, the rate of change of non-renewable resources NR:
Ry -ast0, (16)
‘The 1900-value of NR is NRI. Integration of (16) yields: .
t
NR(t) - NRE= =a f 10(t)dt a7)
1900

Therefore, since NR(t) cannot be negative:

fs To(tjdt ¢ NRI/a (8)
19007) /

In other words, once NRI is given, the total amount of industrial
output over time cannot exceed NRI/a. This is the logical consequence of
the assumptions that the amount of resources is limited and cannot but

decline on the one hand, and, on the other, that each unit of industrial
output produced requires a fixed amount of resources. The expression (18)
explains why changes in the model that cause a faster growth of industrial
output also provoke an earlier system decline. Conversely, if moderated,
industrial growth may persist Longer. The same characteristic features can
also be found by performing several sensitivity simulations, but this
simple analytical expression directiy shows the fundamental reasons of

system behaviour.

State-space techniques
This technique is based on the notion that, if the number of independent

state variables is n, the state vector X corresponds to a vector or point

- Tee -

in the n-dimensioanl state space. Conversely, each point in the space
is associated with a state vector, and the rate of change of each
vector can be calculated directly from the system's equations (1).
Consequently, it can also be derived directly in what direction each

point in the state space will move. Thus, starting from different initial

positions, the various state-space trajectories x may follow through time
can be constructed. Therefore, what kind’of behaviour will follow from
any starting point can be found easily. Also, possible equilibrium points

can be isolated, and characteristic features such as‘ Limit cycling can

be recognized and explained.

Unfortunately, the approach can only be used easily for (sub)ayatems
containing not more than 2 or 3 independent state variables. Typically,
some mathematical insight and some energy are required, but the general
and fundamental nature of the insights that may be obtained completely
counterbalance the disadvantages.

Again, the capital and resource subsystem of World3 offers an elucidating
example: In |14|it is shown that,without affecting the basic properties of
the eubsyatem, the interaction between industrial capital IC and non-

“renewable resources NR for NR/NRI < 0.5 can be described by: :
Ic=b* iC *#NR + c* IC and (ig)
NR = d 6 IC» NR, (20)

where b, c and d are constant, with b>0, c<O0, and d<0. Lf the two-dimensional
IG-NR state space is considered, some important conclusions can be drawn

directly Erom equations (19) and (20): The subsystem remaine at rest for

Ic#0, irrespective of the value of NR. Moreover, IC and NR being positive

and d being negative, NR can only decline. The rise or decline of IC

depends on the value of NK only: I¢ will rise if beNR + c>0, and hence

NR>c/b, but decline if the reverse is true. Therefore, as shown in

Figure 9, the IC-NR plane can be divided into two parts (only values

for IC and NR > 0 are considered): If NR > -c/b, IC will rise and NR

decline; if NR < -c/b, both IC and NR will decline; and for Ic+0, the bi
system will be stable. Finally, for NR=0, IC will decline exponentially
to zero. The arrows in Figure 9 shaw the directions in which the state
vector will move. As a result, if the values of 1¢ and NR are given, the
analyst can inmediately deduce the type of behaviour that will follow.

Capital)

1c
(industrial t
'
i
at
f
i '
‘
3
\
i ~
t +
H
~c/b

> NR (Non-renewable Resources)
Figure 9: The IC-NR atate plane (Worid3). The fat, dotted curve shows a
possible trajectory, whereas the arrows indicate the direction
in which the system will move at the specific value-combinations
of IC and NR
For values of NR larger than ~c/b, IC will rise first, and then decline,
while in all other cases decline will set in immediately. Ultimately, IC
will always tend to zero.
In this particular case, the equation describing the trajectory
in the IG-NR space can be derived analytically from equations (19)
and (20):

TC/NR = b/d + e/(d=NR). ay

is is an expression for the slope of the trajectory in the IC-NR

plane as a function of NR. The expression for the trajectory can be

found as the solution of

ge = b/d + ¢/c4ene). (22)

The result is:

V5

~ Tah -

TC(t)-1C, = (b/d) ¥(NR(E)-NR,)+(c/d)#1n(NR(t)/NR,), @3)

where IC, and NR, are the starting values of IC and NR. The curve,
sketched in Figure 9, according to equation (23) (fat, dotted Line)
unused

shows that, unless NR,=O, the system will always leave resourc:
because IC will tend to zero before NRO. Equation (23) can facilate
further investigations, such as the direct calculation of NR(), or the

calibration of b, c and d if a certain amount of resources has to remain.

ANALYSIS OF LINEAR APPROXIMATIONS

Linearization of non-Linear models, and subsequent analysis of the linear
approximation is an extremely powerful technique which is widely utilized

in the analysis and control of many dynamic systems. Because the
linearization itself, and many of the techniques for study of linear

systems can easily be implemented on a computer, application of the techni~
que requires practically wo effort, even for the study of models including
numerous state variables. Linearization-enables the analyst of using all

the tools of linear system analysis and design.

Linear system thegry provides, among other things, tools for
studying the operating structure of linear models (decomposition according
to the actual working of the equations), for deriving model hierarchy, for
study of the basic stability properties of the whole model ad well as of
each subsystem, for performing model initialization in a correct way (see
also |14]),for systematic order reduction of linear models, and for com-
putation of the time constants (12 as well as 13) included in a system.
Moreover, when performed on a computer, Linearization is more systematic
than many other techniques since it automatically incorporates all relation-
ships of the model.

However, as with all techniques based on approximation or simplification,
the validity of the Linear mode! is limited, and ali conclusions dravn should
be verified in the original model. Moreover, many of the techniques for
analysis of linear models that clearly show basic model properties include
mathematical operations that obscure the relation between the properties
detected and the original model assumptions. Yet, the experience of many
investigators tells us that linear models, even of highly nonlinear originals,
can reveal many things. In negligible time, much information that may be
extremely helpful in studying the original no;

n-linear model can be generated.

i
=925= ae

Moreover, since computers are available, numerous linearization and
analyses can be easily performed for consecutive pointe in time, and

changes in a model's operating structure caused by non-linearities may
clearly come to Light.
Let us now consider the technical detaile more closely. It is

started from the state-variable description for a closed system" 1

£lx(t)), x(t, )x5° : (2)

‘A Linear model that reflects the basic properties of the original model
around a state x(T) can be deduced by a first-order Taylor approximation

of £(Qc(t)) around x(7)
H(t) = £(T) + A» (x(t) - xT). (as)
where A is a matrix the elements A,; of which are equal to

ao (26)

Yhus, each element A; 5 gives the sensitivity of %; to small changes in Xe
‘The linear model (25) can be analysed in different ways. The stability
properties and time constants 13 may be derived by calculation of the
eigenvalues of A. The. diagonal elements show the strength and sign of the
net lag-free feedback of each state variable to its own rate of growth.
The actual influence of x on #; in the linear model may be evaluated by
muLtiplication of the sensitivity A;; by the changes that actually teke
place in x; during a time period sT, This can be done for all element

and then it follows from (25):

For ease of discussion, it is sssumed that ail exogenous influences are
constant. Variations in exogenvus influcnces, however, can be treated in
a similar vay.

= 126 =

a
4 tar), = di (4; jel; (TeaTI (DDD), 27)
where n= the dimension of the system under consideration.
Now, if
Ag 7 Ag gels Teat)-x,(T), (28)

At, shows the amount of change in x, in the time period AT oving
to the variations in x; during the sane tine period, Comparison of

all elements of each row of A teaches which couplings in the linear model
ave important and which not. Subsequently, A’ can be simplified by
neglecting all weak couplings, and a hierarchical structure of state
variables may be derived automatically. Cuypers |1| and Schmidt |1!1{
have demonstrated the utility of the approach in an application to For-
rester's model. Recently, the technique has been extended, implemented
and tested on a computer using the World3 model as a vehicle. Since

it does not affect model properties, all third-order lags were replaced
by first-order lags with the appropriate time constant (12) to reduce
the number of state variables. The matrix A was calculated numerically
using a difference approximation of (26), Routines have been implemented
that rescale A according to (28), derive a Boolean matrix indicating

the major couplings, reorder it hierarchically, and provide a graphical
output of the results. Figure 10 shows the computer output obtained by
linearizing World3 around its 1970 state. The matrix was rescaled
according to (28) using the changes in each state variable that actually
take place in the nonlinear model during the period 1970-2000 (At=30
years)". The corresponding hierarchical diagram is shown'in Figure 11.

. 7 marctery
A test calculation has shown that the outcome of the linear model initialised

in 1970, hardly differs from the original World} model for about 30 years.
- 721 -
1 1c 1-165 4H

2 sc 1-55 4

3 at 1-10 0 # *

4 BHSPC 2-20.00 « *

5 Alec 2 -3.0 * * the influences
6 pierce 2 -20.0 * «

7 NR 2 week

6 PAL 2 -15.7 * « ae

9 AL 2-380 * e

10 UIL 2-100 * * 2

11 PPAPR 2-200 * * *

12 FCFPC 3-200 *¥* .

13 PI 3-70 * *

1a ‘PPL 3-228 * *

15 P2 4-223 * * *

16 LFERT A -2.9 “* .
17 PLE 5-20.00 * ## so *

18 P3 5-140 * + 8
19 PA 6 1204 * oe
20 PFR 6 -20 * * “+ *
21 LUFD 7-20 #* * ee ke

Figure 10: Computer output of total Lineartnation progran applied to
the World3 standard run between 1970 and 2000, The first
column gives the rov muber, the seound the letterseript
indicating the vartable's nane (aee Appendix for explanation),
the third the hierarchical level, and the fourth the time con-
stant 12 aseociated vith the particular variable, expressed in
years. The asterike show the major couplings in matrix form

‘The results are in full agreement with the knowledge about the model's
working that had been obtained in many other, much more time and
energy consuming ways. The capital and part of the agricultural system
is virtually autonomous, and influences - directly and/or indirectly ~
the behaviour of all other variables.

Similarly, linearizations have been performed for other points in
time during the Yorld3 standard run. The results clearly show the changes
in operating structure after 2020 owing to the nonlinearities in the
capital and resource subsystem coming into play: resources rise also to
the first level in the hierarchy, lowever, no further fundamental
changes in operating structure are found, which illustrates that the
capital and resource subsystem plays a leading part in the decline phase
also.

Further elaboration of the technique of linearization, and dis-
cussion of conclusions drawn from ice application is possible, but it
would lead us tuo far beycnd the scope of the present papers

Direction of

ior, by 10, At, Este, 2,

S) Inthe by Ady Ay PL, 2, BD.

, O66 an

Figure 11: Influence diagvan showing the hierarchy of the state varia~
bles in the World3 standard run between 1970 and 2000,
the letterseripte ave explained in the Appendix.

+ Influenced by 1C only.

V. CONCLUDING REMARKS

Various methods of acquiring insight into the working of dynamic models
have been presented. Yet, the list is far from being complete. Clearly,
there is much room left for improvement in the field of understanding.
For the time being, experience and intuition will be at least equally
important as technical skills. Therefore, rash application of the tools
described in this paper is discouraged since many of them may be use~

less or even dangerous in the hands of the inexperienced.
The author is greatly indebted to Mrs. J. Smulders, who managed
to type and re-type the manuscript in almost negligible time.
ll

13]

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tel

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fut

- 729 -

KEFERENCES

Cuypers, J.G.M. and Rademaker, 0,
An Analysis of Forrester's World Dynamics Model.
Automatica, Vol.10, Number 2(p.195-201), 1974.

{ral
Forrester, ‘J.W.

Industrial Dynamics.

The M.1.T, Press, Cambridge, Massachusetts 1961.
Forrester, J.W. [ral
World Dynamics.

Wright-Allen Press Inc., Cambridge, Massachusetts 1971.

Gibson, J.E.

Nonlinear Automatic Control.

McGraw-Hill Book Company, New York 1963. [as]
Meadows, D.ll. et al.

The Limits to Growth.

Universe Books/Potomac Associates, New York 1972,

Meadows, D.L. et al.
Dynamics of Growth in a Finite World.
Wright Allen Press Inc., Massachusetts 1974.

Peterson, D.!
Statietical Tools for System Dynamics.
Proceedings 1976 International Conference on Syatem Dynamics,
8-15 August 1976, Geilo, Norway.

Rademaker, O.et al.
Project Group Global Dynamic:
1972-1976."

Progress Reports numbers | to 5, ,

Rademaker, 0.
On Understanding Complicated Models: Simple Methods
Presented to the American-Soviet Conference on Methodological
Aspects of Social Systems Simulation, Sukhuni (U.S.S.R.),
October 24-26, 1973.

Rademaker, 0.
The Behaviour of the Pollution in Forrester's World Model.
Proc. IFAC/UNESCO Workshop on Systems Analysis and Modelling
Approaches in Environment Systems,
Zakopane, Poland, Sept. 17-22, 1973.
Schmidt, G. and Lange, B.
Evolution of the Significant Signal Connections in Forrester's
Model of the World Dynamic (in 3erman).
Regelungstechnik, vol. 23, number 52 (p.145-149), 1975.

* available on request from 0. Rademaker, P.0.Box 513, Eindhoven, Netherlands. _

- 730 -

Sharp, J.A.
Sensitivity Analysis Methods for System Dynamics Model
Proceedings 1976 International Conference on System Dynamics,
8-15 August 1976, Geilo, Norway.

Thissen, W. and De Mol, C.
An Analysis of the World3 Agricultural Submodel.
Presented at the Third I.1,4.S.A, Globgl Modelling Conference,

September 22-25, 1975, Baden, Austr:

Thissen, W.
Investigations into the World3 Model: The Capital and Resource
Subsystem.

IEEE Transactions on Systems, Man, and Cybernetics,

Vol. SMC~6, number 7 (p.455~466), 1976.

R, and Vukobratovic, R.
General Sensitivity: Theory.
Elsevier, New York, 1972.

Tomovic,

Ler
_—-Availabhe_on request from_O, Rademaker. ?.0.tax.513...Eindhoven, Netherlands.

- 7B -

APPENDIX : List of letterecripte, and their associated meaning.

A Agricultural subsystem
AIIL70 Assimilation Half-Life in 1970
AL + Agricultural Inputs

ALOPC + Average Industrial Output Per capita

AL + Arable Land

cr + Capital Investments

CIAF 3 Capital~Investment~in-Agriculture Fraction

cR : Capital and Resource subsystem

cur : Capital Utilization Fraction

DIOPC + Delayed Industrial Output Per Capita

guSPC + Effective Health Services Per Capita

E + Food

ECEPC : Fertility Control Facilities Per Capita

FIOAA : Fraction of Industrial Output Allocated to Agriculture

1c + Industrial Capital

ICOR : Industrial Capital-output Ratio

ILE + Inherent Land Fertility

OPC : Industrial Output per Capita

LEDR 3 Land Fertility Degradation Ratio

LEER! + Land Fertility

LER : Land Fertility Regeneration Time

LUED ": Labor Utilization Fraction Delayed

‘NR + Non-renewable Resources (World3); Natural Resources
(World2)

NRE + Non-renewable Resources Initial

NRUR 1 Non-renewable Resource Usage Rate

PL : Population, Ages 0-14

P2 ® + Population, Ages 14-44

P3 2 Population, Ages 45-64 .

PG : Population, Ages 65°

PAL + Potentially Arable Land

PCRUM. : Per Capital Resource Usage Multiplier

PER + Perceived Food Ratio

PLE : Perceived Life Expectancy

Por + Population (subsystem)

~ 132 -

Pollution (World2)

Pollution Absorption (World2)
Pollution Absorption Time (World2)
Population Initial

Persistent Pollution Appearance Rate
Persistent Pollution (World3)
Index of Persistent Pollution
Service Capital

Service Output F

Time

Time constant (loop)

Time constant (state variable)
Time constant (overall behaviour)
Urban-Industrial Land