Causality and C onjugate Variables in System Dynamics M odeling:
Enhancements or Impediments
by
Maciej Zgorzelski, Professor of Mechanical Engineering
Timothy M. Cameron, Associate Professor of Mechanical Engineering
Kettering University (formerly General Motors Institute), Flint, MI 48504
Abstract
Causality has been condemned by many system thinkers. It is generally perceived as a synonym of
reductionism, thinking in terms of simple cause-effect chains. System dynamics teaches that everything
operates in the form of interacting feedback loops, and in these it is inherently impossible to distinguish
between cause and effect. The balancing behavior of negative feedback, or the instability of positive
feedback, is a structural property of a system, irrespective of which variable is considered the input to start
the (computational) analysis. Classical system dynamics models, however, from Forrester and the early
Dynamo, impose causality. One may argue that causality is incorporated because of computational
requirements, nevertheless changing the input variable(s) requires creating a new model. Consequently, we
can only determine reactions of a system to specific inputs (causal). We are unable to draw behavioral
conclusions about a system based only upon its structure.
In the study of engineering systems these issues have been approached before. It is widely known
that an electrical circuit diagram (without current direction arrows!) is an acausal model, which will exhibit
oscillatory behavior (if it contains L and C elements), regardless of input choice. The technique of bond
graphs used for physical systems is also inherently acausal, and it also approaches dynamic system
modeling utilizing the concept of conjugate variables (such as voltage and current - the effort variable type
and the flow type). This is a powerful concept in modeling multi-disciplinary engineering systems. The
extensions to social and economic systems have never been fully explored. The present authors study the
bond graph technique for modeling the dynamics of social systems, focusing on the causality and
conjugate flows issues. This may lead to potential extensions of the current system dynamics modeling
techniques.
Introduction
Causality, from the philosophical point of view, is generally perceived as a
synonym of reductionism, of thinking about complexity of the world of systems in
oversimplifying terms of straightforward cause-effect chains. The undisputed father of
the General System Theory, Ludwig von Bertalanffy, wrote (back in 1955):
... In the world view called mechanistic, which was born of classical physics of the
nineteenth century, the aimless play of the atoms, governed by the inexorable laws of
causality, produced all phenomena in the world, inanimate, living and mental. No room
was left for any directiveness, order or telos. ... The only goal of science appeared to be
analytical, i.e., the splitting up of reality into ever smaller units and the isolation of
individual causal trains. ... Causality was essentially one way: one sun attracts one
planet in Newtonian mechanics, one gene in the fertilized ovum produces such and such
inherited character, one sort of bacterium produces this or that disease....We may state
as characteristic of modern science that this scheme of isolable units acting in one-way
causality has proved to be insufficient. ... We must think in terms of systems of elements in
mutual interaction.’
To quote a more contemporary source - Peter Senge says in his famous Fifth
Discipline:
... Reality is made up of circles but we see straight lines. Herein lie the beginnings of our
limitation as systems thinkers....In systems thinking feedback is a broader concept. It
means any reciprocal flow of influence. In systems thinking it is an axiom that every
influence is both cause and effect. Nothing is ever influenced in just one direction.
System dynamics teaches us thus that everything in complex systems operates in
the form of interacting feedback loops, and that in these it is inherently impossible to
distinguish between cause and effect. The balancing behavior of a negative feedback
loop, or the instability of the positive feedback, are both structural properties of systems,
irrespective of which one of the variables describing the system we will consider the
input to start the (computational) analysis.
Causality in Traditional System Dynamics M odeling
Classical system dynamics modeling techniques, however, since the time of Jay
Forrester and the early Dynamo, impose causality. One may argue that this is the type of
causality necessitated by the computational needs, and that this does not mean that
system dynamicists adopt the simple cause-effect philosophy. We have to notice,
however, that in classical system dynamics modeling, in order to change what we intend
to consider as the input variable(s) of a problem we have to create a new, different system
model. In consequence, we can only determine reactions of a model to specific inputs (i.e
all our modeling results are causal). In effect we still remain within the realm of causal
reasoning, the change we have made amounts only to replacing simple causal chains with
feedback loops, wherever we consider this appropriate. We are unable thus to draw
behavioral conclusions about systems based solely on their structure. We will return to
this issue shortly.
Causality in Physics and in Physical Systems Analysis
In the study of physical and engineering systems some of the philosophical
problems surrounding the causality issue have also been noticed. F. Cellier, H. Elmquist
and M. Otter, known authorities in the field of engineering systems simulation
techniques, write in a recent paper:
... Many engineers believe that physics is essentially causal in nature. Someone takes a
conscious decision to affect the world in a particular way, thereby causing the world to
react to his or her actions. Sir Isaac Newton followed the same line of reasoning when he
formulated his famous law about actio being equal to reactio. ... Yet, the distinction
between actio and reactio is a deeply human and moral concept, not a physical one.
There is no physical experiment in the world that can distinguish between actio and
reactio.... The relationship between voltage across and current through an electrical
resistor can be described by Ohm’s law: u=R*i - yet, whether it is the current flowing
through the resistor that causes the voltage drop, or whether it is the difference between
the electrical potentials on the two wires that causes current to flow is, from a physical
perspective, a meaningless question.... State-space models are written in assignment
statement form. There is always exactly one variable to the left of the equal sign, and the
model implies that the expression to the right of the equal sign is evaluated, and the
result of this evaluation is assigned as a new value to the left. Consequently, the modeler
needs two different models to describe an electrical resistor: a voltage-drop-causer model
and a current-flow-causer model. As mentioned earlier, from a physical perspective, this
makes no sense whatsoever.’
The authors of the above fragment touch upon some very critical questions here.
These are the issues of the role of conjugate variables (i.e. current and voltage in the
above example) in the analysis of physical dynamic systems, (which is very closely tied
to the causality question), and the issue of constitutive relations (such as the resistance R
of the resistor mentioned above) of the elements of a dynamic system, defining largely its
behavior.
Bond Graphs for Physical Systems Modeling
Roughly at about the same time and in the same place that Jay Forrester had been
working on the development of system dynamics modeling methodology for social and
economic systems, another MIT professor, Henry Paynter started the development of the
bond graph technique for modeling engineering systems’. This technique, further
developed by many worldwide, is inherently acausal, and it approaches dynamic system
modeling utilizing the concept of conjugate variables. This turned out to be a very
powerful idea, in particular for modeling mixed-domain physical systems. The conjugate
variables such as voltage and current in the electrical systems domain, force and velocity
in mechanics, or pressure and volume flow in hydraulics- in bond graph terminology, are
known as the effort variable and the flow variable. The product of effort and flow equals
the power flowing through a port (cabling, hydraulic hose, force application point in
mechanics) to (or from) the system element.
It is obviously impossible to summarize the wealth of concepts developed and
research done on bond graph modeling of dynamic systems in this short paper. Extensive
bibliography of this field can be found on the Internet’. Here, we present just a few basic
concepts needed for further discussion. In this exposition we follow Karnopp and
Rosenberg.°
The bond graph consists of bonds which carry power (represented by a half-
arrow, or harpoon, indicating the direction of power flow which we assume to be
positive, these arrows signify thus the power sign convention, not the causality). Power
manifests itself in the conjugate effort and flow variables. There are three basic dynamic
elements: a generalized resistor, a generalized capacitor and a generalized inertia, shown
respectively in Fig. 1, 2 and 3.
Resistance
constitutive
relationship
effort (nonlinear
in general)
flaw flow
@
(b)
vax
i x
a EL rng
F Fo Q—> 6
f <——_| [>F @
e FE P
—. =e wR
i B Vv Q
()
Figure 1.
effort
compliance
constitutive
relationship
— effort (nonlinear
flow in general)
@
g=displacement={ flow - dt
(6)
effort
flow
(a)
Figure 2.
flow
inertia
constit
ve
L
v,
se P. P=P\-Pi P,
ny) att 1 — Pa -
F P
vw a
Figure 3.
exeffort
C = capacitance
R= resistance
p= momentum @= displacement
Isinertia
fa
f=flow —
Figure 4.
Figure 4 shows the so-called tetrahedron of state, illustrating the relationships
between four possible state variables. Besides effort and flow we have here their
respective time integrals: a generalized momentum and a generalized displacement.
f at Wes dtp te
= >
en Z
1 et v apif a se {aes
1 2 ==
yh 12
; )
Figure 5.
The bond graph elements become connected using two types of junctions, known
as 0 (zero) and 1 (one) junctions, illustrated in Fig. 5, which enforce the generalized
Kirchhoff laws. There are several other types of elements: sources and sinks,
transformers and gyrators. Bond graphs can also accomodate pure signal flows (e.g. a
voltage signal for information transfer purposes) - we do need to get into any of these
issues for the purposes of this paper.
Modeling Dynamic Systems in Bond Graphs
ui
Cc \ x& R
3 c R common V 1
junction | 0]
q y
common V
T
-l
0 time T2
Figure 6. Figure 7.
Figure 6 illustrates the development of a complete bond graph model for a very
simple case of a damped mechanical oscillator, and Fig.7 - a typical result - velocity of
the common junction, as obtained using the bond graph (and control system) simulation
software 20-Sim.’
For comparison we show in Fig. 8 the system dynamics model of the same
damped oscillator problem, as developed in Vensim®, and the same simulated result. The
purpose of presenting both approaches here together lies not so much in comparing the
results (which are, as one would expect, identical) but rather in comparing the approaches
to model creation and underlying assumptions about causality, as well as about conjugate
variables and about generalized Kirchhoff laws (0 and 1 junctions in bond graphs).
Net force
Spring compliance C
3 Graph for velocity
A
‘Spring force Damper constant R
acceleration
nl
c Damper force 0
y Mass inertia M 05
velocity
Y INITIAL velocity ah
o 1 2 3 4 5 6 7 8 9 0 tt 1
Time (Second)
displacement
velocity : Curent. msec
INITIAL displacement
Fig. 8 System dynamics model of the damped mechanical oscillator. The same
parameters were used as in the bond graph model shown in Fig. 6 and 7
It seems important to notice here that the bond graph model does not require the
user to build feedback loops for the existing couplings between variables - the modeling
technique takes care of this problem, as each of the bonds represents the conjugate flow
(velocity in this case) and effort (mechanical force). The only things we have to specify
are the three elements (inertia, compliance and resistance) and their constitutve relations.
Here these are the linear laws: F=m*a for inertia, F=-q/C for spring (where C is the
spring’s compliance value - inverse of the so-called spring constant) and F=-R*v, where
R is the damping coefficient of viscous friction damper. (The relations used here are
linear, but the bond graph technique is by no means limited to linear cases.) Besides the
three elements and their constitutive laws, all we have to define are: the “1” junction,
assigning a common velocity link to all three bonds, and - obviously - the initial
conditions. Derivation of equations of state, and assignment of causality - this one
strictly for computational purposes - is done in modern bond graph systems by the
computer system. The modeling process, the resulting bond graph and the simulation
results would have been identical for, say, an electrical oscillating circuit consisting of a
capacitance, a resistance and an inductance.
Our intent is to show that many elements of the bond graph modeling technique
could be incorporated into system dynamics modeling of social and economic systems,
enhancing and - to a great extent - automating the modeling process, improving
consistency of the models, as well as modelers’ understanding of their inherent dynamics.
Approaches to Causality and C onjugate Variables
To illustrate the differences in approaches to causality and conjugate variables in
classical system dynamics modeling and in bond graph models we will discuss the
straightforward case of an inertia element - a simple mass M accelerated by force F,
resulting in acceleration a, which may be inegrated to yield velocity v and displacement
q. There will be two resulting different system dynamics models (Fig. 9) of this
mechanical problem, depending upon its assumed causality: model (a) shown on the left
assumes that force F is the input, velocity v the output, model (b) on the right assumes the
opposite. Of course, in accordance with what we said earlier, physics is essentially
acausal, and neither of these two models is more justified than the other.
What is modeled in classical system dynamics as a feedback mechanism is part of
the inherent physical behavior of the system. The conjugate variables in the bond graph
capture this. Consequently, in the bond graph technique, there is a useful distinction
between the “reciprocal flow of influence” inherent in the dynamics of the system, and
feedbacks that arise by human intervention, i.e. by design. Classical modeling techniques
fail to recognize this distinction, and it is in their use of causal feedbacks to model the
inherent dynamics - the reciprocal flow of influence between conjugate variables - that
requires a different model for different boundary conditions (inputs) despite the
constancy of the underlying system structure.
F
—+|M
= 51
, =| displacement
_— displacement qj velocity v P q
—¥ vel <TIME STEP>
Force F —sacceleration a ~/. ,
momentum p Force F
Mass M. Mass M od
last value of p
F fa
ee at
1 P — >] E
meeral ye
e=F
— a> 1:M
f=v
Figure 9. Two different models are needed in traditional system dynamics modeling
methodology, depending on the causality of the problem. Block diagrams, used extensively in
process automation, illustrated above are also causal. There is, however, only one bond graph
model.
Application of Bond Graphs to Social and Economic Problems
It remains to show that bond graphs can in fact be applied to social and economic
systems modeling. Some interesting results in that direction were obtained by Brewer.”
He proposed using unit price p ($/unit) of a specific product (or service) as the effort
variable, and the flow of orders (units of product/unit time) as the bond graph flow f
variable. Brewer's tetrahedron of state is shown in Fig. 10. Their product p*f is the
equivalent of power in engineering systems and obviously represents cash flow. The time
integrals of f and p have interesting meanings: the integral of f is the accumulation of
orders (“economic displacement”, which, when negative, represents simply inventory of
products), and the integral of p is something not recognized normally in economics, that
Brewer names the economic impulse. The three basic bond graph elements - compliance
C, inertia I and resistance R can be identified as representing, respectively: C - an
inventory (also, as Brewer shows, a natural resource), I - is obviously the effect
associated with the build-up of production (I represents the investment cost needed to
increase production by one unit), and finally R - represents market effects. The other
crucial element of the bond graph technique, the “0” and “1” junctions, represent the
economic equivalent of Kirchhoff laws, Walras’ law.
Effort:
unit price
Ly \
dt
f Cc
Y R q
Investment Neg-inventory
Impulse / or orders
Sat
f—~
Flow:
rate of flow of orders
Figure 10.
The present authors believe that introducing some of these concepts into the
practice of system dynamics modeling may greatly enhance the modeling efforts. Bond
graphs use conjugate variables instead of feedbacks (obviously not eliminating feedbacks
from where they really do belong: from economic control systems, i.e. company
policies). This immediately leads the modeler to the conclusion that he/she must include
financial considerations into the study of dynamics of economic systems: an inventory
cannot be any more considered strictly as an accumulating level of products, monetary
value must be assigned to it (the constitutive relationship of a compliance requires that).
Similarly, build-up of production in an enterprise cannot be considered as a simple time
delay: the conjugate variable of the cost of this build-up arises immediately. Finally, bond
graphs are inherently acausal, so that changing the assumed input does not change the
underlying model structure.
Besides the above outcomes, the inherent consistency of bond graph models
(which assures the conservation of financial and material flows in a system), automated
derivation of state equations by modern software, and automated (or, at least semi-
automated handling of causality issues) make, in the present authors’ opinion an attempt
to use bond graphs in economic and social system modeling - worthwhile undertaking.
' Ludwig von Bertalanffy: General System Theory - Foundations, Development, Applications; George
Braziller, New Y ork, 1995.
2 Peter M. Senge: The Fifth Discipline - The Art and Practice of the Learning Organization; Doubleday,
New Y ork,1994
3 Francois E. Cellier, Hilding Elmqvist, Martin Otter: Modeling from Physical Principles.
“ Henry M. Paynter: Analysis and Design of Engineering Systems; MIT Press, Cambridge, MA, 1961.
5 Bond graph bibliography is available at http://www.ece.arizona.edu/~cellier/bg.html
® Dean Karnopp, Ronald Rosenberg: System Dynamics: A Unified Approach; Wiley, New York, 1975.
7 20-Sim, University of Twente, Controllab Products, P.O.Box 217, 7500 AE Enschede, The Netherlands
® Vensim DSS ver. 3.0, Ventana Systems Inc., 60 Jacob Gates Road, Harvard, MA 01451
° (a) John W. Brewer: Structure and Cause and Effect Relations in Social System Simulations, IEEE Trans.
on Systems, Man and Cybernetics, June 1977, pp. 468 - 474.(b) John W. Brewer, Paul P. Craig, Mont
Hubbard, Kenneth E.F. Watt: The Bond-Graph Method for Technological Forecasting and Resource Policy
Analysis; Energy Vol.6, No.6, pp. 505-537, 1982. (c)John W. Brewer: Progress in the Bond-Graph
Representations of Economics and Population Dynamics; J. of the Franklin Institute, Vol. 328, No. 5/6, pp.
675-696, 1991
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