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An eigenvalue approach to feedback loop dominance analysis in non-linear dynamic models

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An eigenvalue approach to feedback loop dominance analysis in non-linear dynamic models
Mohamed Saleh
Information Science Department University of Bergen E-mail: mohamed.saleh@powersim.no
Pål Davidsen
Information Science Department University of Bergen E-mail: davidsen@ifi.uib.no
Abstract
The purpose of the System Dynamics method is to study the relationship between structure and behavior in non-linear, dynamic systems. In such systems, the significance of various structural components to the behavior pattern exhibited, changes as the behavior unfolds. Changes in structural significance, in turn modifies that behavior pattern, which, in turn, feeds back to change the relative significance of structural components. We develop a mathematical framework within which we can study the characteristics of this feedback between structure and behavior. This framework is based on a piecewise observation of the model over time, a characterization of the behavior pattern exhibited using eigenvalue analysis, and an identification of the relative contribution of each of the loop in the model to each of the eigenvalues that characterize the total behavior, and thus to the total behavior. This work is an extension of the work by Nathan Forrester and Christian Kampmann on the use of eigenvalue analysis in system dynamics. Our main contribution, in this paper, is embedding eigenvalue analysis in a broader analytic framework to capture the transient as well as long-term behavior of non-linear models in general. The mathematical framework developed has been implemented in the form of computer algorithms and tested in a variety of cases.
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Using this method, we can classify, at any point in time, the feedback loops in a system with respect to their relative significance to the system behavior. This allows us to offer a structural interpretation of the behavior exhibited. Moreover, the method is a key to managing such systems because it allows us to rank, at any point in time, the loops of a model with respect to their significance to the behavior of that model. Thus, as a basis for our management of the system, we may identify the loops that contribute most significantly to the model behavior in a favorable or in an unfavorable way, and, consequently, the loops to strengthen and weaken, respectively, while managing the system.
1 Introduction
System dynamics is the theory of the relationship between structure and behavior in dynamic systems. One of the most challenging tasks in system dynamics has been to understand how behavior emerges from the underlying structure, i.e. how behavior is created in non-linear models and how that behavior feeds back to change the relative significance of the various loops of the underlying model structure. In this paper, we will identify the units of analysis of the structure and the behavior of a non-linear model and develop a method by which we can identify the causal relationship between the two. More specifically, we want to attribute the properties of the model behavior, characterized by the
Behavior Pattern Indexes,
to properties of the underlying model structure, characterized by the
gains
of the loops in the model. As indicated in figure 1, we will demonstrate that the
eigenvalues
of the model constitute the link between model structure and behavior. On the one hand, they srcinate solely from the
loops
of the structure of the model and, on the other hand, they characterize the behavior of the model.
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Fig. 1: Structure drives behavior through the intermediate eigenvalues link.
We will illustrate the theory presented in this paper with several examples. A simple main example --example 1-- has been incorporated throughout the text to demonstrate the elements of the theory. In addition, we use other, more specific examples to illustrate particular issues raised. At the end of the paper, we present the “yeast cells generation” example to demonstrate all the steps of the procedure that we recommend be followed in the analysis of complex, dynamic systems. Our main example, example 1, described below, is inspired by a similar example given by Mojtahedzadeh (Mojtahedzadeh, 1996, p. 38) to demonstrate that a traditional eigenvalue analysis, alone, is not sufficient to explain the transient
behavior of even such a simple model. He argued that eigenvalue analysis could only be used to study long-term behavior. Our contribution, in this respect, is to demonstrate that eigenvalue analysis can be embedded in a broader analytic framework to capture the transient as well as long-term behavior of non-linear models in general. Example 1, presented in figure 2 and table 1, is a simple linear, second order model with state variables, Level_1 and 2, governed by the rates Slope_1 and _2, respectively.
BehaviorEigenvaluesStructure (loop gains)
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Fig. 2: Example 1, stock and flow diagram
init Level_1 = 7 flow Level_1 = Slope_1 init Level_2 = 3 flow Level_2 = Slope_2 Slope_1 = (-0.15*Level_1)+(-0.2*Level_2)+const_1 Slope_2 = (-0.2*Level_1)+(-0.15*Level_2) Const_1 = 0.1 Simulation Setup Parameters: Start Time=0; Stop Time =45; Simulation Time Step = 0.1
Table 1: Example 1, equations
Note: In this paper, we denote
scalars
using variables in small letters;
vectors
in bold, underscored, small letters; and
matrices
in bold, capital letters.
2 Behavior pattern indexes
2.1 Definitions and justification
The properties of the behavior that we will be focusing on are the slope (s), and the curvature (c) of each state variable (x). These are defined, in this paper, as follows:
- the slope, s, is defined as the (time) derivative,
x
&
, of that state variable, x; and
Level_1Level_2Const_1Slope_1Slope_2
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- the curvature, c, is defined as the double (time) derivative,
x
&&
, of that state variable. The convergence/divergence of a state variable at any instant of time is defined as the rate of change of the absolute value of the slope, s, of this state variable, i.e. d|s|/dt. If the state variable is in a convergent mode, then d|s|/dt will be negative, i.e. the abso-lute value of the slope, |s|, is decreasing with time. If the state variable is in a diver-gent mode, then d|s|/dt will be positive, i.e. the absolute value of the slope vector, |s|, is increasing with time. Note that |s|=0 is a partial condition for equilibrium. As the state of a model changes over time, so do the slopes associated with each of the state variables. Consequently, for a state variable x, we may express the new slope, i.e. s
x
new
, in terms of the srcinal slope, s
x current
, and the rate of change in the slope over the subsequent period of time, i.e. the curvature, c
x
. s
x
new
= s
x
current
+
∆
t * c
x
We may now investigate the properties of the ratio: c
x
/ s
x current
We want to illustrate that this ratio can be considered as a proxy for the d|s
x
|/dt indicator. Note that if c
x
and s
x current
have the same sign, then this ratio will be positive, and the absolute value of the slope will increase (i.e. d|s
x
|/dt > 0), - an indicator of divergent behavior. If, on the other hand, the two have opposite signs, then the ratio will be negative, and the absolute value of the slope will decrease (i.e. d|s|/dt < 0), - an indicator of convergent behavior. If c = 0, then the ratio will be 0, so that the absolute value of the slope will not change, i.e. d|s
x
|/dt = 0. In such a case, if c
x
changes its value from -
ε (ε−>0)
to +
ε,
then the ratio (c
x
/ s
x current
)
will change from negative to positive -- or the reverse, depending on the sign of s
x current
. As a consequence, the value of d|s
x
|/dt will also change from negative to positive -- or the reverse, - an indicator of a transition in the mode of behavior from convergence to divergence -- or the reverse. If s
x current
changes its value from -
ε
to +
ε,
then the ratio (c
x
/ s
x current
)
will change from -
∞
to +
∞
-- or the reverse, depending on the sign of c
x
. That would be an indicator of a discontinuity in d|s
x
|/dt, and thus, also in this case, a transition in the mode of behavior from convergence to divergence – or the reverse.

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