Eberlein, Robert L., "Linear Analysis and Model Simplification", 1985

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Linear Analysis and Model Simplification
Robert L. Eberlein, University of Alberta

ABSTRACT

There has been a great deal of work done in the simplification of linear dynamic
models. Given that most models that are in use are nonlinear this has restricted
the applicability of the available techniques. By concentrating on a particular
nonlinear phenomenon, in this case shifting loop dominance, it is possible to use
the techniques of linear analysis for the simplification of nonlinear models. The
theory for this is developed and it is shown how this can be applied to a model.
For purposes of exposition the market growth model is used and the results are en-
couraging. Though there is still a good deal of work to be done it seems feasible
to develop simplification techniques for nonlinear models that address directly
the nature of the nonlinearities.

INTRODUCTION

There has been a good deal of work done in the development of techniques for anal-
yzing and simplifying linear dynamic models. Though the results of this work are
important to the System Dynamics researcher the assumption of linearity is a
serious restriction. In this paper we will look more explicitly at nonlinear
models, though we will continue to use the tools of linear analysis. Since the
analysis of nonlinear models is obviously a large and difficult topic no attempt
will be made to deal comprehensively with the problem. Instead we will concen-
trate our attention on a particular type of nonlinear phenomenon, that of shifting
loop dominance. In this framework we will show how the techniques of linear
analysis with some extension can be useful.

We begin the paper with a brief review of the work that has been done in simplifi-
cation of linear models. The essential results along with their usefulness to the
modeler will be outlined. The techniques considered are strictly applicable only
to linear models, though they can be applied to nonlinear models through lineariz-
ation. Following this the issues of shifting loop dominance and their relation-
ship to the results of linear analysis are discussed. We use this discussion to
develop some theory for understanding and simplifying nonlinear models character-
ized by shifting loop dominance. These results are then applied to a relatively
simple model to illustrate their usefulness. Finally, some indications of areas
for future research are given.

LINEAR MODEL SIMPLIFICATION

We use the term model simplification to include a number of approaches to model
analysis. In many cases the approaches simply involve trying to get some under-
standing of what structures in a model determine certain of the behavior modes
generated by the model. It will not always be the case that an explicit simpli-
fied model will be built. Determining elements of the structure generating
behavior is, however, the equivalent of building an implicit simplified model.
The explicit process of simplification can be thought of as taking this sort of
model analysis one step further. We are not advocating explicit model simplifica-
tion in all cases, but simply pointing out that the processes of analysis and
simplification are closely related.

The results on model simplification are applicable to linear dynamic models
of the form
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x= Ax + Bute qa)

with x a vector of length N of endogenous variables, u a vector of exogenous vari-
ables and € a white noise error process. The process of model simplification
concentrates on the internally generated dynamics, essentially the A matrix. This
is consistent with stressing the endogenous point of view. Simplification con-
sists of finding a similar model with a smaller number of endogenous variables.
That is, a model of the form

(2)

Inte
a
E
+
2
+

with x a vector of length n<N. The difference between the original and simplified
model is simply one of size. The simplified model represents an explanation for
part of the original model on the basis of a subset of the variables of the
original model.

A simplified model will, of necessity, be different from the original model. In
order to carry out a simplification it is therefore necessary to concentrate on
certain characteristics of the model. The characteristics traditionally
considered, and the ones we consider, are a selected subset of the endogenously
generated patterns of behavior. This is a useful focus for simplification and,
given the emphasis in system dynamics on patterns of behavior, one that is usually
easy to implement.

The behavior modes are characterized by an eigenvalue and the associated right and
left eigenvectors. A simplified model that is good relative to a behavior mgde
will retain the eigenvalues and eigenvectors. Because x is of length N and x is
of length n, it is not possible to do this exactly. However, it may be possible
to retain elements of the eigenvectors_corresponding to components of x. This ~
allows for easy interpretation of the x vector as a subvector of x. ~

Early work on this includes that of Davison (1966) and Marshall (1966) who looked
at the problem of how to choose the A matrix to preserve certain apparently impor-
tant modes. The analysis here does not go into any detail on the choice of the
x's to include, but rather given that these have been chosen shows how to develop
a simplified model retaining the right eigenvectors. More recent work in this
area can be found in Gopal and Mehta (1982), Litz (1980) and Mahmoud and Singh
(1982). .

One of the key issues in the simplification process is the choice of the variables
to be included in the simplified model. In particular, which variables are impor-
tant in generating the behavior modes of interest. This is an important issue and
was explicitly addressed in Perez (1981) and in Perez, Schweppe, and Verghese
(1982a, 1982b). In these works the idea of a participation factor is introduced.
The participation factor measures the importance of a state (or level) variable in
generating a behavior mode. It takes into account both the amount that the vari-
able moves over the course of a behavior mode and the amount that changes in the
variable impact on the behavior mode.

Further discussion of which variables to include in a model is given in Forrester
(1982, 1983). In this work the more technical means for selecting variables are
considered in terms of the analysis of feedback loops. This allows for some
blending with more traditional approaches in the Systems Dynamics field.
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Along these lines further analysis is given in Eberlein (1984). This analysis
suggests that a somewhat broader based approach to model simplification and con-
siders explicitly such issues as the coupling of behavior modes through variables
of interest. Again though, the emphasis is on determining which variables are
important in generating given behavior modes.

The analysis in the literature discussed above is all strictly applicable only to
the case of a linear dynamic model in the form of equation 1. In general, the
models developed in system dynamics are nonlinear. This does not mean that the
available results are inapplicable, but simply that there has to be a good deal of
judgement as to their validity and usefulness. The methods for the analysis of
nonlinear models are quite weak and there are few general results. In order to
deal with the problem we concentrate on a very specific set of issues in nonlin-
earity, the concept of shifting loop dominance. Even so, the results developed
can be no more than imperfect tools useful, but certainly not yielding any pre-
packaged results.

SHIFTING LOOP DOMINANCE

Shifting loop dominance is an important and well recognized phenomenon in dynamic
systems. It is also an inherently nonlinear phenomenon. Basically, shifting loop
dominance is the result of changes in gains around different loops. A common
example of this is that of a positive loop that goes from growth to stabilization
or decay. This will occur as the gain around the loop goes from larger than one
to smaller than one. A very simple example of this is the population model shown
in Figure 1. Initially, the relative food availability is high and, as a conse-
quence, the net birth rate is also high. As drawn, the polarity of the loop is
determined by the sign of the net birth rate, which starts out positive implying a
positive loop. However, as the population increases relative food availability
will fall, this causes the death rate to rise causing the net birth rate to fall
and fall and eventually go to zero. As this occurs the negative loop through food
availability is said to dominate.

The effect of the shifting loop dominance is not to remove the originally dominent
loop, but rather to alter it so that its influence on the system changes. Essen-
tially the negative loop is controlling the positive loop - and thereby the system
behavior. This is different from controlling the system behavior directly since
it suggests that there is something important to be learned by considering the
changes over time in the positive loop.

The system show in Figure 1 has only one level and it is therefore easy to deter-
mine the eigenvalue. This is given by (RF-RM*f(RFA)) where f is the function
appearing in Figure 2. At low levels of relative food availability the death rate
becomes very high and the eigenvalue will become negative. Starting the model
with high relative food availability the loop through net birth rate is positive
and generates exponential growth. As this happens, however, the food availability
necessarily falls off and weakens this positive loop. At a sufficiently high
population the positive loop will become a negative loop as the death rate rises
above the birth rate. In this case the negative loop through food availability
controls the polarity and dynamic consequences of the loop through net births.

It is important to note that the behavior in the model is still very much a conse-
quence of the feedback through the net birth rate loop. The point is that the
nature of this feedback has been changed by another loop. The distinction between
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“a POP
"
‘
N '
POP - Population
NBR ~- Net Birth Rate
: es ay NF - Net Fertility
same ee RF - Reference
abe Fertility
_ RFA - Relative
Food Availa-
bility
NF = RF ~ £(RFA)*RM RM - Reference
Mortality

Figure 1-.A simple Population Growth Model

f£ (RFA)

RFA

Figure 2 The Effect of Food Availability on Mortality
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one loop altering another loop and that loop controlling the behavior of a model
may at first glance seem arbitrary. In termsof using linear model analysis tech-
niques on the model it is important. If we think of a loop as altering the eigen—
value resulting from another loop (or loops) then we can try to incorporate both
the mechanisms generating the eigenvalue and the loop altering these in the
simplified model.

In using linear analysis on a nonlinear model it is necessary to linearize the
model. If the model is written as

x= gy, wy ©) 3)

with g a continuously differentiable function then the linearization of the model
will be given by

°
x=

2°

B(X0 U0 .0)(x-x0) + = B(X0 40 .0)(u-uo) + $e S(X0 80 DE (4)

The eigenvalues will be those of = g(x9,U9,0) and will clearly vary over time as

the derivative of g changes. However, as long as the derivative of & is continu-
ous the change in the eigenvalues will be continuous over time. As a consequence
it will be possible to trace the evolution of an eigenvalue associated with a par-
ticular feedback loop. This means that it is possible to trace the characteris-
tics (growth, oscillation, decay, etc.) of a feedback loop over time.

The determination of which feedback loop is associated with which behavior mode is
what the tools for linear model analysis are used for. In general this is not an
exact correspondence, but only an approximate one. In addition, for different
points in a simulation path the approximation may become worse. Stated alterna-
tively, at different times different feedback loops may become important in the
determination of a behavior mode. So that the approach we are taking is not by any
means exact. It represents an heuristic and useful method for analyzing the
determinants of behavior modes and using these to formulate simplified models.

LOCAL ANALYSIS

In order to analyze the problem we will make use of local analysis of the model.
This is clearly a limitation on the tools we will develop but is necessary to make
the issues tractable. The approach to this problem is relatively straightforward.
It is assumed that a pattern of behavior such as growth turning into decay has
been recognized and needs to be analyzed. In order to do this the growth behavior
mode must first be recognized and the important loop or loops identified. It will
be assumed that this has been done.

With the loops in hand we can talk of A, the eigenvalue associated with the mode.
The eigenvalue will depend upon the value of the state variables, that is we can
write A(x) where x is the set of endogenous variables. The eigenvalue A repre-
sents a mode of interest, for example a growth mode. The linear analysis gives us
the tools for analyzing this mode. In addition, though, the eigenvalue changes
over time as the state variables change. This is a nonlinear phenomenon and the

question we now pose what states are important in changing A.

There are two things required for a state to have an important influence on the
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eigenvalue of interest. First, a change in the value of the variable must change
the eigenvalue, and second the variable must change value. The second of these
sounds somewhat odd, but is very important. Clearly if a variable never changes
then its inclusion in a simplified model, or as an element of an explanation of
behavior, cannot be justified. More importantly though, because we are interested
in internally generated dynamics the variable must be changed because of the
influence of the behavior modes of interest.

This last point makes the determination of which variables to include in the
simplified model difficult. There will in general be a variety of different
feedback paths along which the variables important to an eigenvalue can influence
another variable. We will restrict our attention to only the most direct path.
That is, the movement of a variable over the course of the behavior mode of
interest. The influence of a behavior mode on the i'th state variable is given by
the corresponding component of the right eigenvector associated with the behavior

mode.

The influence of the state variable on the eigenvalue can be measured in terms of
the derivative of the eigenvalue with respect to the state variable. That is, we

consider a. A(x).
ox

If we let A denote the linearized dynamics matrix then we can write the above
derivative in terms of the element of A as

8 ace) = ee A Day (5)
ox 4 4 Oday Ox

If we assume that the eigenvectors are normalized to have inner product one then

_ = U4rj with 2 and r the left and right eigenvalues associated with h. Using
814

this we can rewrite equation 5 as

Ov 3;
Sea is L844, 6
Ox a : ANE (6)

To incorporate the influence of the behavior mode on the state we combine the
derivative of the eigenvalue with the corresponding component of the right
eigenvector. Writing things in matrix notation, with T denoting transpose, this
gives the mode sensitivity measure

oA
MS(xx) = 2T = erry. (7)
Ox_

Note that the mode sensitivity will be high when the mode does influence the state
variable substantially and the state variable also influences the mode. In other
cases either rm or the derivative of the eigenvalue will be small and so,
therefore, will be MS.

Relative mode sensitivities will not be influenced by the units of measurement of
different state variables. That is, the ratio of the mode sensitivities with
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respect to the ith and jth variables is independent of the units of measurement.
To see this note that changing the units of measurement of say x, from dollars to
cents will decrease the derivative of the eigenvalue by a factor of 100. At the
same time increasing the first component of the eigenvector by 100 will give an
eigenvector consistent with the newly measured state variable. These two changes
would suggest that there would be no change in the mode sensitivity. However, the
normalization for the right eigenvector is arbitrary and changing the normaliza-
tion will change all mode sensitivities proportionally.

It would, of course, be nice if there were a clearly interpretable absolute
measure of mode sensitivity. Unfortunately there is none as this would require a
precise definition of the degree of nonlinearity in the model. For a linear model
the modal sensitivities will all be 0. And if it is felt that the model is nearly
linear it is unlikely to be appropriate to make use of the modal sensitivities, or
to use anything but the linear model. The determination of the degree of non-
linearity in a model is a topic beyond the scope of this paper. However, if the
mode sensitivities are normalized to have their absolute values sum to one there
will be a conveniently usable measure.

The variables with the high mode sensitivities have to be included in the simpli-
fied model. These variables may include some variables already in the simplified
model as well as others that are not. In either case it should be recognized that
the mode sensitivities indicate that a nonlinear relationship incorporating the
variable is called for. The exact nature of this nonlinearity cannot be determin-
ed from what has been stated, but it should follow the original model structure.
The nonlinearity incorporated should also be such that, in the simplified model,
the derivative of the eigenvalue of interest with respect to the included variable
should be close to, or match, that in the original model.

APPLICATION TO THE MARKET GROWTH MODEL

In order to get some feeling for how useful this approach may be we have applied
it to the market growth model of Forrester (1968). In this model Forrester con-
siders the problem of market growth in a limitless market which responds only to
the level of the sales force and the delivery delay for the product. Forrester
identified a basic positive loop through the sales force, sales, and the budget
for sales responsible for growth (see Figure 3). Working against this is negative
loop that decreases the effectiveness of the salesforce as the delivery delay
rises. An example, of course, of shifting loop dominance.

We have rewritten the model, slightly changing the table functions for analytic
convenience. We linearized the model at the initial values, and along the simula-
tion path for the first year. During this time there is one growth root, implying
a rate of growth initially of close to 4% per month. Over the first year of the
simulation this falls slowly to imply a growth rate of about 3% per month at the
end of the first year. As time progresses, the root combines with another to form
a complex root, which implies growth and oscillation of a very long duration.

Using the technique of linear model analysis the basic elements of the growth loop
are clearly identified as backlog and the sales force. This is consistent with
what is shown in Figure 3. Because of the way the calculations are done only the
level variables are included when the elements required for the simplified model
are determined. It is necessary to incorporate additional variables to make the
simple model easily understood. Given a simplified model containing the sales-
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force and backlog elements we want to ask what causes the decreasing rate of
growth.

BL
|e
owe, .
B - Budget S - Salesmen
BL ~- Backlog SE - Sales Effectiveness

DR ~- Delivery Rate
IS - Indicated Salesmen
OB - Orders Booked

Figure 3 The Positive Loop Causing Growth

The second derivatives of the dynamics matrix were calculated at the beginning of
the simulation. When these were weighted according to the formula given in equa~
tion 7 the results obtained are reported in Figure 4. The mode sensitivities have
been normalized so that the absolute values add to one. Note that the sign of the
derivatives have been maintained.

Mode sensitivity of growth mode with respect to

Delivery Delay Recognized by Market -.29
Salesforce 224
Backlog 224
Delivery Rate Average -.14
Delivery Delay Traditional -.06
Production Capacity 204
Delivery Delay Recognized by Company 03

Figure 4 Mode Sensibilities

From this it appears that the delivery delay recognized by the market is the
important influence on the eigenvalue. This is consistent with the incorporation
of a negative loop as shown in Figure 5 and certainly goes along with intuition.
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BL - Backlog
DDI - Delivery Delay Indicated
DDRM - Delivery Delay Recognized by Market

DR - Delivery Rate

OB - Orders Booked

Ss - Salesmen

SE - Sales Effectiveness

Figure 5 The Negative Loop Retarding Growth

Following these guidelines we construct a simplified model that contains the
salesforce, backlog and delivery delay recognized by the market as its levels.

The simplified model has production capacity constant and removes some of the
structure that does not seem to be central to the behavior mode under investiga~
tion. A dynamo equation listing of the simplified model is given in the

appendix. The model is not, in itself, of great interest but serves to illustrate
how an understanding of the basic mechanisms generating a pattern of behavior can
be formalized.

The analysis of the market growth model has not offered any new insights into this
model. The model is itself simple enough and comes complete with sufficient
switches to generate the simplified model we present. What the analysis does
show, however, is that the tools that we are using do yield results that are con-
sistent with our previous understanding of this model. This is important, since
that is a necessary condition for their usefulness in a more general setting.

The simplified model as we have developed it loses a lot of what is of interest in
the market growth model. The expansion of production capacity and the effect of
declining goals on the overall growth are obviously important issues, if not the
key issues the model addresses. This shortcoming of the simplified model makes an
important point. Simplified models are not, by their nature, as interesting and
the models from which they derive. Richness and often realism are sacrificed for
a simpler modeling aimed at a specific pattern of behavior. If this is not the
only pattern of behavior of interest, then something will be missed.
Simplification does not represent a magic tool by which understanding can be
obtained. It can, however, be very useful when combined with hard work and other
techniques for analyzing models.
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NECESSARY EXTENSIONS

The approach discussed in this paper is highly experimental and has not yet been
applied to any large models. There are two major barriers to its application. One
is the lack of of suitable software for calculation of the mode sensitivities.
The other is the great deal of computation required. Taking the derivative of the
A matrix requires that derivatives be calculated. Even in our simple example
this would involve 1000 calculations (the model having 10 states). While it is
true that, because of the sparse nature of the A matrix, it will not be necessary
to calculate this many derivatives, the burden it still substantial. In our exam-
ple there were 26 nonzero entries in the A matrix, thus requiring 260 derivatives.
While this computational burden is not overwhelming it is severe, and it is likely
that some decrease in the required computation is possible. This is an area
requiring further investigation.

There is another issue in this discussion and that is the consideration of only
the level variables. While it is possible to explicitly allow for the inclusion
of auxiliary variables in calculating the different sensitivities (see for example
CCREMS 1983, Eberlein 1984, chapter 5) the use that can be made of these is not
clear. This is a criticism of the linear model simplification techniques as well
as the ones discussed in this paper. Because the auxilitary variables are an
important part of any model their explicit inclusion would be helpful, and it
would certainly make the construction of the simplified model easier.

CONCLUSIONS

The techniques of linear model analysis have often been applied to nonlinear
models. Because the techniques assume complete linearity their application re-
quires some caution and is not likely to succeed in the face of severe nonlineari-
ties. By concentrating on a particular type of nonlinear phenomenon, specifically
shifting loop dominance, it is possible to explicitly recognize the nonlinearity
in developing the simplified model. Since developing a simplified model is the
same as gaining an understanding from an existing model the tools develoepd are
potentially very useful in analyzing complex nonlinear models. The application of
the techniques yield results that are useful in analyzing models.

Though the results quoted in this paper are of a tentative nature there does seem
to be some potential for the development of tools to analyze nonlinear models.
The techniques discussed in this paper require further work before they can be
easily implemented. In addition it is desirable to develop techniques to deal
with other types of nonlinearities. If a collection of such techniques can be
brought together in a unified package for model analysis the potential for in-
creased model understanding and higher efficiency in model analysis is great.
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APPENDIX: SIMPLIFIED MODEL EQUATION LISTING
* SIMPLIFIED MARKET GROWTH MODEL

NOTE
NOTE THIS MODEL USES THE MNEMONICS IN FORRESTER (1268) AND

NOTE MAINTAINS AS FAR AS POSSIBLE THE MODEL STRUCTURE THEREIN

NOTE
NOTE FIRST THE POSITIVE SALES HIRING LOOP IDENTFIED B THE LINEAR

yaa ANALYSIS AS MOST CLOSELY ASSOCIATED WITH THE GROWTH ROOT.

E ea nee SALESFORCE (PEOPLE)

=4

A SH.K=(1S.K-S.K)/SAT SALESFORCE HIRING (PEOPLE/MONTH)

C SAT=20 SALESFORCE ADJUSTMENT TIME (MONTHS)
A 1S.K=B.K/SS INDICATED SALESFORCE (PEOPLE)

C $S$=2000 SALESFORCE SALARY ($/MONTH/PERSON)
‘oe oer BUDGET FOR SALESFORCE ($/MONTH)

NOTE REVENUE FROM SALES HAS BEEN CHANGED TO MAINTAIN THE ROOT OF
NOTE INTEREST.
REVENUE FROM SALES ($/UNIT)

C RS=11.77

A OR.K=PC*PCF .K DELIVERY RATE (UNITS/MONTH)

A PCF.K=TABHL(TPCF,DDM.K,0,5,.5) PRODUCTION CAPACITY FRACTION

T TPCFS0./.31/.45/.55/.63/.71/.77/.84/.89/ .95/1 (DIMENSIONLESS
A DODM.K=BL.K/PC DELIVERY DELAY MINIMUM

L BL.K=BL.u+DT*(OB.J-DR.u) BACKLOG (UNITS)

N BL=8000

A OB.K=S.K*SE.K

NOTE SALES EFFECTIVENESS AND THE EFFECT OF DELIVERY DELAY ON THE
NOTE MARKET. IDENTIFIED AS THE CHIEF CAUSE OF CHANGE IN THE

NOTE GROWTH ROOT

A SE.K=SEDM.K*SEM SALES EFFECTIVENESS

NOTE (UNI TS/MONTH/ PERSON)
C SEM=400 SALES EFFECTIVENESS MAXIMUM
NOTE (UNITS/MONTH/ PERSON)

A SEDM.K=TABHL(TSEDM,DDRM.K,0,10,1) SALES EFFECTIVENESS FROM DELAY
MULTIPLIER (DIMENSIONLESS)

NOTE

Noe SPT OT BIT TS BST RL WY 28/1 SOR

L_DDRM.K=DDRM. JU+(DT/TDDRM)*(DDI.J-~DDRM.J) DELIVERY DELAY RECOGNIZED
NOTE BY MARKET (MONTHS)

N DDRM=DDI

C TDDRM=10 TIME FOR DELIVERY DELAY

NOTE RECOGIZED BY MARKET (MONTHS)

A DDI.K=BL.K/DR.K DELIVERY DELAY INDICATED (MONTHS)
C PC=12000 PRODUCTION CAPACITY (UNITS/MONTH)
NOTE

NOTE CONTROL CARDS

NOTE

SPEC DT=.5/LENGTH=100/PRTPER=0/PLTPER=3
PLOT S/BL/DDRM
RUN
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REFERENCES

CCREMS - “LIMO: Linear Model Analysis,” Technical Document DT-29, Center for
Computational Research in Economics and Management Science, MIT, Cambridge, Mass.,
1983.

Davison, E.J. "A Method for Simplifying Linear Dynamic Systems,” IEEE Transactions
on Automatic Control, AC-11, 1966, pp. 93-101.

Eberlein, RL. Simplifying Dynamic Models by Retaining Selected Behavior Modes.
PhD Thesis, Sloan School of Management, MIT: Cambridge, Mass., 1984.

Forrester, J.W. “Market Growth as Influenced by Capital Investment,“ Industrial
Management Review, Volume 9, Number 2, Winter 1968, reprinted in Collected Papers
of Jay W. Forrester, Cambridge, Mass.: Wright~Allen Press, 1975.

Forrester, N.B. A Dynamic Synthesis of Basic Macroeconomic Theory: Implications
for Stabilization Policy Analysis, PhD Thesis, Sloan School of Management, MIT,
Cambridge, Mass., 1982.

ennainin » "Eigenvalue Analysis of Dominant Feedback Loops," Presented at the 1983
International System Dynamics Conference," Chestnut Hill, Mass., 1983.

Gopal, M. and S.I. Mehta. “On the Selection of the Eigenvalues to be Retained in
the Reduced Order Models," IEEE Transactions on Automatic Control, AC-27, 1982,

pp. 688-690.

Litz, L. “Order Reduction of Linear State-Space Models Via Optimal Approximation
of the Nondominant Modes,” in A. Titli and M.G. Singh, eds. Large Scale Systems:
Theory and Applications, Proceedings of the IFAC Symposium, New York: Pergamon
Press, 1980, pp. 195-202.

Mahmoud, M.S. and M.G. Singh. Large Scale Systems Modelling. New York: Pergamon
Press, 1981.

Marshall, S.A. “An Approximate Method for Reducing the Order of a Large System,”
Control, Volume 10, 1966, pp. 642-648.

Perez Arriaga, J.I. Selective Modal Analysis with Applications to Electric Power
Systems. PhD Thesis, Electrical Engineering and Computer Science, MIT, Cambridge,
Mass., 1981.

Perez Arriaga, J.I., F.C. Schweppe and G.C. Verghese. “Selective Modal Analysis
with Applications to Electrical Power Systems,” Parts I and II, IEEE Transactions
on Electric Power Systems, POS-101, 1982a, pp. 3117-3134.

+ “Rational Approximation Via
Selective Modal Analysis,” Circuits, Systems and Signal Processing, Volume 1,
1982b, pp. 433-445.