Table Of ContentCONTRIBUTORS TO THIS VOLUME
MICHELE BRETON
SHAWN E. BURKE
WODEK GAWRONSKI
YACOV Y. HAIMES
ALAIN HAURIE
JAMES E. HUBBARD, JR.
JER-NAN JUANG
CHIN-PING LEE
DU AN LI
N. HARRIS MCCLAMROCH
GARYA.MCGRAW
BERC RUSTEM
MUIBIADE. SALAMI
GEORGES ZACCOUR
CONTROL AND
DYNAMIC SYSTEMS
ADVANCES IN THEORY
AND APPLICATIONS
Edited by
C. T. LEONDES
School of Engineering and Applied Science
University of California, Los Angeles
Los Angeles, California
College of Engineering
University of Washington
Seattle, Washington
VOLUME 36: ADVANCES IN LARGE SCALE SYSTEMS
DYNAMICS AND CONTROL
@
ACADEMIC PRESS, INC.
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CONTRIBUTORS
Numbers in parentheses indicate the pages on which the authors' contributions begin.
Michele Breton (75), GERAD and HEC, Montreal, Quebec, Canada
Shawn E. Burke (223), The Charles Stark Draper Laboratory Inc., Cambridge,
Massachusetts 02139
Wodek Gawronski (143), Jet Propulsion Laboratory, California Institute of
Technology, Pasadena, California 91109
Yacov Y. Haimes (1), Center for Risk Management of Engineering Systems, Uni-
versity of Virginia, Charlottesville, Virginia 22903
Alain Haurie (75), University of Geneva, Switzerland
James E. Hubbard, Jr. (223), The Charles Stark Draper Laboratory Inc., Cambr-
idge, Massachusetts 02139
Jer-Nan Juang (143), NASA Langley Research Center, Hampton, Virginia 23665
Chin-Ping Lee (107), Department of Electrical Engineering and Computer Sci-
ence, The University of Michigan, Ann Arbor, Michigan 48109
Duan Li (1), Center for Risk Management of Engineering Systems, University of
Virginia, Charlotte sville, Virginia 22903
N. Harris McClamroch (107), Department of Electrical Engineering and Com-
puter Science, The University of Michigan, Ann Arbor, Michigan 48109
Gary A.McGraw (27'5), The Aerospace Corporation, El Segundo,California 90009
Berc Rustem (17), Department of Computing, Imperial College of Science, Tech-
nology and Medicine, London SW7 2BZ, England
Muibi Ade. Salami (333), Hughes Aircraft Company, Los Angeles, California
90009
Georges Zaccour (75), GERAD and HEC, Montreal, Quebec, Canada
vii
PREFACE
In the 1940s techniques for the analysis and synthesis of systems were simplis-
tic, at best, and so were the system models. In fact, a paper was published in
Econometrica in 1949 which represented national economies as a simple second
order model primarily because the techniques available at the time would not al-
low otherwise. Nevertheless, this was an important paper because it put forward
conceptually and somewhat analytically the important notion of feedback in
quantitative economics. It was not until the latter part of the 1950s, not all that
long ago, really, that a foundation for multivariable system analysis and synthesis
began to be laid, and certainly Kaiman played a key role in this. In any event, the
rapid growth of computer power, coupled with the requirement to more effec-
tively deal with large scale systems in such areas as engineering, industrial, and
economic systems, resulted in and continues to result in advances in techniques
for dealing with large scale systems problems. Indeed, these trends were recog-
nized and manifestly treated at various times in earlier volumes in this series. For
example, Volumes 14 and 15 were devoted to techniques for modeling (the be-
ginning of the system design process) large scale systems such as jet engines,
electric power systems, chemical processes, industrial processes, water resource
systems, and the thyroid hormonal system of the human body, it is now time to
revisit the area of large scale systems in this series, and so the theme for this vol-
ume is "Advances in Large Scale Systems Dynamics and Control."
The first contribution, "Multiobjective Decision-Tree Analysis in Industrial
Systems," by Yacov Y. Haimes and Duan Li, deals with the fact that many large
scale systems involve multiobjectives, and thus multiobjective analysis needs to
be performed to provide the decisionmaker(s) with a set of policies with a resul-
tant set of trade-offs between the different objectives. Two substantive examples
are presented in this contribution, and by these examples the essential signifi-
cance of multiobjective decision-tree analysis in applied situations is clearly
demonstrated. Professor Haimes is probably the single most visible contributor
on the international scene in this area of major applied significance, and so this
contribution by him and a colleague constitutes a most appropriate beginning for
ix
x PREFACE
this volume. The next contribution, "Methods for Optimal Economic Policy De-
sign," by Berc Rustem, presents important techniques for one of the ultimate
challenges in large scale systems, the determination of robust economic policies
in the face of anticipated disturbances and in the presence of rival economic
models, of which there are a plethora. It was in 1980 that Professor Lawrence R.
Klein received the Nobel Prize in economics for his work in econometrics and
optimal fiscal policy generation by the modern techniques of optimization. The
advances represented by the differences between the Econometrica paper of
1949, referred to above, and Professor Klein's work, as well as the current work
of other leading researchers in this important area, clearly manifest the substan-
tive advances in techniques in this broad area of large scale systems. Therefore,
Rustem's contribution in the area of robust optimal economic policies is a most
welcome addition to this volume. The next contribution, "Methods in the Analy-
sis of Multistage Commodity Markets," by Michele Breton, Alain Haurie, and
Georges Zaccour, presents significant advances in techniques in the large scale
systems area of networks of interacting oligopolies. Methods for the characteriza-
tion and computation of equilibrium solutions for such complex large scale sys-
tems problems are motivated by a significant applied example from which more
general results and techniques are developed. Because of the pervasiveness of
this category of large scale systems on the international scene, this contribution is
also a most welcome addition to this volume.
Another major large scale systems area is production control and the determi
7
nation of optimal production rates, which is a control variable in the stochastic
environment which production systems represent. The next contribution, "A
Control Problem for Processing Discrete Parts with Random Arrivals," by Chin-
Ping Lee and N. Harris McClamroch presents rather general and powerful results
for dealing with the rather complex problem. The formulation of the problem by
the authors has a number of significant advantages, including generality, mathe-
matical tractability, and a formulation which has the essential properties of a
closed-loop production. A computationally effective algorithm (a nonlinear
search routine) is then presented for the determination of the optimal production
rate. This contribution concludes with a substantive example to illustrate the de-
sign methodology. Because of the great significance of optimal production con-
trol on the international scene, this chapter also constitutes a most welcome addi-
tion to this volume on large scale systems. The next contribution, "Model Reduc-
tion for Flexible Structures," by Wodek Gawronski and Jer-Nan Juang, presents
many highly effective techniques for near optimal and robust model truncation
techniques. Very important, also, is the fact that the techniques for model reduc-
tion, while developed within the context of flexible structures, are, in fact, appli-
cable to any linear system. Even with the powerful advances in computer technol-
ogy there is still a great need for model reduction of systems which would other-
wise require a high order number of state variables for their "faithful" representa-
tion. What is really important is an adequately effective model representation of
high order complex systems. Because of the comprehensive treatment of this
PREFACE xi
important subject in this contribution, it represents an important element of this
volume on large scale systems.
The increasing emergence of large structural systems and modern system per-
formance requirements has resulted in an increasing utilization of active control
techniques. To put it somewhat simplistically, in the "era" of simple rigid body
control problem, the system design and control actuator problem was considera-
bly simpler. However, the requirements dictated by more complex flexible struc-
tures or systems, both for reasons of system performance and structural allevia-
tion, has brought with it a requirement for distributed control as well as the ac-
companying distributed sensors. This area of what could be described as en-
hanced control of systems is commonly referred to as active control systems. To
put it another way, if one wants to do more in the way of control for distributed
systems, for whatever reason, one has to systematically introduce additional sen-
sors as well as related additional control actuators. The next contribution, "Dis-
tributed Transducers for Structural Measurement and Control," by Shawn E.
Burke and James E. Hubbard, Jr., is a rather comprehensive treatment of this sub-
ject and also a most welcome contribution to this volume on large scale systems.
As in the case of the previous contribution, it should also be pointed out that, al-
though the techniques in this contribution are presented within the context of
large scale structural systems, the techniques are more generally applicable to lin-
ear systems with similar fundamental characteristics.
The next contribution, "Robust Adaptive Identification and Control Design
Techniques," by Gary A. McGraw is a comprehensive treatment of robust adap-
tive identification and control algorithms for disturbances and unmodeled system
dynamics, a highly significant broad area in large scale systems. There are two
basic approaches to coping with uncertainty in control system design. One ap-
proach is robust control, in which a single controller is designed to operate over
all possible parameter variations and unmodeled system dynamics. The other
approach is adaptive control, in which the structured parameters are estimated
and the controller updated accordingly. An adaptive control should only be con-
sidered after a fixed robust design has been shown not to meet the design objec-
tives. In this event, this contribution presents numerous powerful techniques
which are verified in both computer simulations and in the laboratory in this con-
tribution. In the final contribution, "Techniques in the Optimal Control of Distrib-
uted Parameter Systems," by Muibi Ade. Salami, the significant issue of practical
means (algorithm development) of optimal control of distributed parameter sys-
tems is treated. Examples of the applied significance of the techniques presented
in this chapter include such large scale systems areas as aerospace, defense,
chemical, environmental, and infrastructural industries. As a result, this contribu-
tion is a most fitting chapter with which to conclude this volume.
The authors are all to be commended for their superb contributions, which will
provide, in this volume, a significant reference source for practicing professionals
and research workers in the broad area of large scale systems for many years to
come.
MULTIOBJECTIVE DECISION-TREE ANALYSIS
IN INDUSTRIAL SYSTEMS
YACOV Y. HAIMES and DUAN LI
Center for Risk Management of Engineering Systems
University of Virginia
Charlottesville, Virginia 22903
I. INTRODUCTION
The design and operation of industrial systems often lead to
sequential decisionmaking problems. Since Howard Raiffa published
his celebrated book on decision-tree analysis [1] in 1968,
decision-tree analysis has been successfully applied in solving a
general class of multistage decision problems with probabilistic
nature. The analysis of a decision tree aims at evaluating
quantitatively and iteratively how good each of its decision
sequences achieves its objective. The optimal decision (in the
single-objective sense) is commonly found by a method called
averaging out and folding back - - an approach that is grounded on
the concept of the expected value.
Today's industrial systems are becoming more and more
complex. It is generally realized that a single criterion (such
as cost minimization) is seldom an adequate measure of the
performance of such systems. Noneommensurable and conflicting
objectives characterize most real-world industrial design and
operation problems. Multiobjective optimization (see Chankong and
CONTROL AND DYNAMIC SYSTEMS, VOL. 36 1 Copyright © 1990 by Academic Press, Inc.
All rights of reproduction in any form reserved.
2 YACOV Y. HAIMES AND DUAN LI
Haimes [2]) thus needs to be performed in order to provide the
decisionmaker(s) with a set of noninferior policies and with
trade-offs between the different objectives.
The expected value of a system's measure -- the most commonly
used scalar representation of risk -- commensurates the system's
various possible outcomes and their associated probabilities.
Often, the expected-value approach is inadequate to provide
sufficient information for risk management. It is particularly
deficient for addressing extreme events, since these events are
concealed during the amalgamation of events of low probability
and high consequence and events of high probability and low
consequence. Multiple risk measures need to be introduced to
quantify the risk at different levels. Risk management thus
incorporates the selection of the most preferred option in a
multiobjective framework.
Recently, a multiobjective decision-tree method has been
developed by Haimes et al. [3] to extend the traditional
decision-tree method to incorporate multiple objectives and
various risk measures. In that paper, the multiobjective decision
tree is explained through an example problem of a flood warning
system. In this article, the usefulness and efficiency of
multiobjective decision-tree analysis will be shown through some
example problems in industrial systems.
II. DECISION TREE WITH MULTIPLE OBJECTIVES
Similar to a single-objective decision tree, a multiobjective
decision tree is composed of decision nodes, chance nodes, and
branches. At each decision node, designated by a square in the
tree, the feasible alternatives are shown as branches emerging to
the right side of the decision node; from among these branches
the decisionmaker decides on the best alternative to perform.
Each alternative branch may lead to another decision node, a
chance node, or a terminal point. A chance node, designated by a
MULTIOBJECTIVE DECISION-TREE ANALYSIS 3
circle in the tree, indicates that a chance event is expected at
this point in the process; i.e., one of the states of nature may
occur. The states of nature are shown on the tree as branches to
the right of the chance nodes. A branch representing the state of
nature may be followed by another chance node, a decision node,
or a terminal point.
There exist, however, two major differences between the
traditional single-objective decision tree and the multiobjective
decision tree presented in Haimes et al. [3]. The new features of
the multiobjective decision tree are (a) the vector-valued
performance measure and (b) multiple-risk measures. These
features enable us to make a significant extension of the
averaging-out-and-folding-back strategies.
An alternative at a decision node in a multiobjective
decision tree is evaluated by multiple criteria. Assume that we
are performing vector minimization with k objective functions,
f]_, f 2, ···, fk· An alternative a is noninferior if there does
not exist another feasible alternative ä such that
fi(a) < fi(a*) i = 1,2,·.·,k (1)
with strict inequality holding for at least one i. Clearly, a
noninferior solution is not unique; the vector optimization at
each decision node yields, in general, a set of noninferior
solutions. Consider a decision node n in the tree. Assume that
the set of k-dimensional vector-valued costs-to-go (the
cumulative vector-valued objective function to the terminal
points of the tree) is available at the end of each branch
emerging to the right side of this decision node. Denote the set
of vector-valued costs-to-go at the end of a branch associated
with alternative ΆΑ at decision node n by ^n+l(a-j) with elements
in the form of φ = [φ ,···,<£ ]' e φ (SLA) , where ' is the
transpose operator; the "local vector-valued cost" associated
with alternative ELA by r(a;) = [r]_(a^ ),···, r^Ca-;)]'; and the k-
dimensional binary operator which combines the vector-valued
