Techniques of predicting and/or controlling the dynamic behavior of large scale systems are discussed in terms of decentralized decision making. Topics discussed include: (1) control of large scale systems by dynamic team with delayed information sharing; (2) dynamic resource allocation problems by a team (hierarchical structure with a coordinator); and (3) some problems related to the construction of a model of reduced dimension

Aoki, M.

English

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SOME ASPECTS OF CONTROL OF
A LARGE-SCALE DYNAMIC SYSTEM*
Masanao Aoki
University of Illinois
Urbana, Illinois
INTRODUCTION
Problems of understanding and dealing effectively with dynamic systems of high dimension
(large- scale dynamic systems) have been objects of research for some time. The need for better
techniques of predicting and/or controlling dynamic behavior of large-scale systems, involving both
modeling and analysis, has been felt more intensely as demands have increased rapidly for designing
and managing large-scale systems (such as interconnected network of power systems, water
resources, computers, warehouses, or depots). Despite varied and intensive research, many questions
remain about large-scale dynamic systems for which we do not have adequate answers.
If we assume perfect and centralized information, that is, if all "relevant" data on systems and
environments1 and complete problem descriptions are available for a single decision maker, then
stabilization and control of large-scale systems are usually reducible to problems of constructing
algorithms for efficient information processing (such as decomposition algorithms in large-scale
mathematical programming) and associated problems of designing efficient information collection
and transmission structures. Existing dynamic systems with centralized information structure are
usually not "too" large and may have special structures because of the increasing cost of collecting
information and processing it by a single decision maker as systems become larger.
For large-scale systems, it is therefore likely that several decision makers exist who influence
the dynamic behavior of large-scale systems. These systems are called decentralized systems. If we
drop the assumption of centralized information pattern, we must explicitly recognize and cope with
the problem of interaction of decisions and information held by several decision makers.2 In
*Research reported here was supported in part by AFOSR Grants 1328-67 and 72-2319.
1
 We make the usual distinction of risk and uncertainty. The former is what is called purely stochastic and a
special case of the latter is called a parameter adaptive (ref. 1).
2
 Another aspect of research on large-scale systems becomes important if we retain the centralized information
pattern but drop the assumption of perfect information. This is the aspect of learning or adaptation in decision
making. In reality, neither assumption is likely to hold exactly. Most large-scale systems we encounter are likely to
be imperfectly modeled and are being acted on by more than one decision maker with nonidentical knowledge of
the system.
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https://ntrs.nasa.gov/search.jsp?R=19750021782 2020-03-22T20:58:05+00:00Z
decentralized systems, no single decision maker may have enough information needed for stable
operation of large-scale systems, to say nothing of "optimal" operation of systems.3
Thus, we characterize and somewhat limit the scope of our investigation on large-scale
dynamic systems as decentralized systems and as dynamic team problems. Sharing information
(such as choice of message alphabets and estimation and partial reconstruction of information
content held by other decision makers) and assigning control responsibility are some of the prob-
lems we have considered (see, for example (refs. 2-4)). Questions related to reliability, security,
design of good measurement and control subsystems, or situations such as oligopolistic competition
are not covered here.
In the next three sections, we summarize our findings in (1) control of large-scale systems by
dynamic team with delayed information sharing. (2) dynamic resource allocation problems by a
team (here we assume a hierarchical structure with a coordinator (central agent) who coordinates
decision making by lower-level (local) decision makers), and (3) some problems related to construc-
tion of a model of reduced dimension.
OPTIMIZATION BY A DYNAMIC TEAM WITH DELAYED INFORMATION SHARING
Consider a team composed ofN decision makers. Radner (ref. 5) proved that given a quadratic
performance index for a team
F = -u'Qu + 2n'u. u ' = ( u \ ..... u'N), M' = (v\, . . .
where Q and fi may contain random variables, the (person-by-person satisfactory) optimal decisions
are given as the unique solution of
where 5f| is the information o-field of decision maker /. This equation may be solved under a set of
suitable assumptions by adapting an iterative solution technique used by Aoki (ref. 6)(see also ch. 2
of ref. 7).
To emphasize the aspect of learning or adaptation, suppose M is imprecisely known. Suppose
further decision maker i observes dF/du^ (i.e., his differential influence on F) through an additive
noise. By sharing his observation with other decision makers with one period delay, the pbps
decisions are given as the solution of a set of equations such as equation (1). The decisions consist
of two parts:
"? = "M t Aut (2)
3
 We exclude from our consideration game situations, where decision makers may have contradictory objectives
or problems of incentive for cooperation.
196
where u£.j is the decision vector with the centralized information structure and Au. is the correction
term due to individual differences in the information set at time t.
Similar results were obtained independently by Sandell and Yoshikawa (refs. 8,9). The stan-
dard stochastic linear regulator problem with quadratic cost can be reformulated as a decentralized
control problem by a team with delayed information-sharing where decision makers observe dif-
ferent linear combinations of the state vector through noise. The structure of the decentralized
control is still of the form of equation (2).
It is also possible to obtain the decentralized optimal decision when some elements of Q are
not known precisely. In this case, however, it is not possible to characterize the decentralized
decision rules conveniently as in equation (2). (see ref. 7).
DYNAMIC RESOURCE ALLOCATION BY A TEAM
We now briefly describe our work which deals with a dynamic version of a stochastic resource
allocation problem discussion by Groves and Radner (refs. 10, 11). The problem is to allocate a
finite amount of resources to n subsystems to maximize some objective function cost for the whole
team. The amount available for allowcation depends on past decisions. Some parameters in the
objective functions are imprecisely known.4 This brings in the interactions between control and
information, the so-called "dual control effects."
Some approximations must be made to approximately evaluate contributions to the objective
function (cost) from the future (cost-to-go). The conflict arises between control and information
since the larger the control is, the smaller is the estimation error variance of the parameter value,
while the total amount of resource available for control is finite. Open-loop feedback, certainty
equivalent, and other schemes are compared (see ref. 2 for details).
AGGREGATION UNDER PERFECT INFORMATION
Another active area of research on large-scale systems is the following: construction of models
of similar structure and/or with smaller dimension and use of these models to stabilize and control
large-scale systems. Several reduced-order models may be used by a single decision maker or by
several decision makers in decentralized large-scale" problems (see, e.g., refs. 13 and 14).
In this section, we consider the feasibility of aggregating a collection of systems (called micro-
systems or subsystems) into larger systems (called macrosystems). One part of the problem may be
called the aggregation problem of optimal systems (or optimal configurations). It relates to the gross
characterization of optimal system performance characteristics. The other class relates to the possi-
bility of constructing a model, called a macromodel, or aggregates of microsystems and optimizing
such aggregates of systems via the macromodel.
'This problem has never been formulated in this manner so far as we can ascertain.
197
In many control problems of physical systems, systems are naturally divided into several
distinct subsystems because of the physical makeup of the problem. Here the word "physical"
should be interpreted broadly to include socioeconomic systems as well as the usual, truly physical
control systems. Loosely speaking, the concept of aggregation implies that two or more such
subsystems are combined to make a larger subsystem. One of the objectives of such aggregation is
usually to reduce the number of variables in the state space description of the problem. More often
the problem is already stated in an aggregated fashion because of the sheer necessity of limiting the
number of variables. The latter is true, for example, in describing the economic behavior of a
country by a dynamic macroeconomic model.
The concept of aggregation therefore complements that of decomposition. These two con-
cepts, taken together, form two sides of a coin. In applying the concept of decomposition to a
system to form a two-level structure, the subsystems in the lower level may be aggregated to a
smaller number of subsystems with some advantages, for example. The concept of decomposition,
with the resultant multilevel structure, is fairly well known and some computational algorithms
have been proposed. The results available to date, however, are of a mathematical nature and
somewhat of the nature of further developments of Lagrange multipliers and of generalized Kuhn-
Tucker conditions.
Consider a collection of subsystems whose structures are alike. They differ only in their
parameter values that specify the systems completely. The problem is to allocate a limited amount
of resources so that the overall performance (return) from these subsystems is optimized (maxi-
mized, to be definite). The interactions among these subsystems come therefore from the common
sources of resources. No interactions through dynamic interactions among the subsystems are
considered.
Consider, for example, a network of power-generating stations. Optimal performance charac-
teristics of each power station may-have the same functional dependency on the key parameter,
such as the amount of fuel available, load of the station, capacity, etc. (see refs. 15, 16). More
precisely, we consider a macrosystem consisting of TV subsystems so that the total return from this
system F is related to the individual return of the/th subsystem gj by
where F and g^ are scalar valued and where xl is an m vector. The term F is to be maximized over
the domain Dpj defined by
N . .
2 xl =x, xl> 0.
Among the class of problems with such separable criterion functions, we are interested particu-
larly in the special case
g/xi)=g(xi,ai), Ki<,N
198
namely, we are interested in problems where subsystems are all alike and the return from each
subsystem differs from each other only in its parameter so that
max [g(x l,a l)+g(x2,a2)\ = glx.hfa1 ,a2 )\ (3)
X1 + X2 — X
x
l
 >0. x2 >0
where h(a l,a2) is a known function of a1 and a2. This means that the functional form g is
reproduced after aggregating two subsystems, a very desirable feature indeed. Equation (3) can be
interpreted as follows: A controller is associated with each subsystem /, and its performance is
optimized subject to the constraint that the given amount of resource be xl. The optimal perform-
ance is denoted by g(x*,al), which are called the optimal performance characteristics by Kulikowskj
(ref. 15). Then the optimal performance of the aggregate of N such subsystemsg\x,h(a l,
is given by5
g x,h(al , . . . , f l ; = max
Conditions for aggregating microvariables pertaining to dynamic behavior of subunits to obtain
macrovariables for describing a set of subunits are usually rather restrictive. Since only a small class
of problems permits perfect aggregation, investigating the effects of approximate aggregation is
important. However, it is useful to know when microstate variables can be aggregated without error.
In the dynamic resource allocation of section 2, this type of aggregation is possible for a class of
problems. An example of perfect aggregation follows (ref. 14).
Consider a system composed of N subsystems with the optimal performance characteristic
functions of a common type given as
where Aj,....,Z^, a, ft...., u> and kj > 0 are given real numbers. Assume that these TV subsystems
are coupled by the conditions
N N _N
i i' i '' "" i '
where
Interpret A as the total cost of control B, C,...,Z are the total amount of resources allocated to the
N subsystems. They are constrained from below and above by B_. B, ...ZandZ. The problem is to
minimize A.
s
 -In the multilevel approach (decomposition approach), the subsystems are on the first-level and the second-
level controller assigns certain xl to the /th subsystem.
199
The optimal performance characteristics of the total system can be shown to be of the same
type as
where k is given by
fN
k =
and the optimal parameter value settings are given by
/? In ^"^ ~^~ 7 I 7 ^~ f if I If iMJ 2 I-Lf ^^ ... ^^ £-t J • / £-1 \ A. ]\Li\i/
I I I
where the starred variables indicate optimal values.
Actually, any number of subsystems can be combined or aggregated to form a larger subsystem
without destroying the common functional form of the optimal performance characteristics.
For example, subsystems 1 and 2 can be combined to give
where
B12 B! +B2
and where
*
A j
 2 ^ min (Al + A 2 )
Bl + B2 = BI 2
i + Z2 = Zj
A*/A*2 =B*
200
Thus, for this class of problems, instead of optimizing the system with respect to variables
Bj,Cj,...Zj, 1 < / < N, some or all of the N subsystems can be aggregated and optimized separately to
form a system with fewer subsystems without incurring any loss in the overall system performance.
This fact will be referred to as a perfect aggregation. Unfortunately, the class of functions that
permits this perfect aggregation is not large and must be essentially of the type given by equation
(4). Therefore, some sort of approximate aggregation procedure must be developed. One such
approximation may be to approximate optimal operating characteristics with exponential-type
functions.
Note also that the subsystems of the class of systems considered by Kulikowski interact only
indirectly through the allocation of common resources. No direct couplings of the dynamics of the
subsystem through their inputs and outputs are considered. A network of power-generating stations
has been mentioned as a possible system providing a motivation for such a model.
Another important example is a network of computing centers. A computer center may be
taken to be the basic subsystem. A local network consisting of several such centers may be con-
sidered next. The allocation of total computing load has been considered for a computer center
consisting of several central processing units with perhaps a common memory. What is being
considered is not the detailed job assignment schedule but a gross characterization of optimal
operating condition. Some job assignment schedule is assumed to have been adopted and the
operating characteristics involving such variables as the description of the composition of the types
of computing jobs, average times of executions, characterization of computing speed, etc., are
assumed to be known.
What is being proposed is to characterize the operating characteristic of a network consisting
of two or more computing centers with similar operating characteristics in terms of similar macro-
variables when these computing centers obtain jobs for a common source. Optimal job allocation is
considered in terms of the given operating characteristics of the individual computer centers. Of
course, in applying the theory to be developed in the proposed work, a major problem is to identify
the important variables and obtain adequate characterizations.
When the assumption of perfect information is relaxed, we can no longer perform perfect
aggregation because of interactions between control decisions and information of various decision
makers.
201
REFERENCES
1. Aoki, M.: Optimization of Stochastic Systems. Academic Press, 1967.
2. Aoki, M. and Li, M. T.: Partial Reconstruction of State Vectors in Decentralized Dynamic Systems. IEEE Trans.
Aut. Control, AC-18, June 1973, pp. 289-292.
3. Aoki, M.: On Feedback Stabilizability of Decentralized Dynamic Systems. Automatica, 8, 1972, pp. 163-173.
4. Aoki, M., and Li, M. T.: Controllability and Stabilizability of Decentralized Dynamic Systems. Preprint, Joint
Automatic Control Conference, Columbus, Ohio, 1973, pp. 278-286.
5. Radner, R.: Teams, in Decision and Organization, McGuire, C. B. and Radner, R. (eds.). North Holland, 1971.
6. Aoki, M.: Decentralized Linear Stochastic Control Problems with Quadratic Cost. IEEE Trans. Aut. Control,
AC-18, 1973, pp. 243-250.
7. Toda, M: Dynamic Team Decision Problems Under Uncertainty. Ph.D. Dissertation, Department of System
Science, University of California, Los Angeles, 1974.
8. Sandell, N. R.; and Athans, M.: Solution of Some Nonclassical Stochastic Decision Problems. IEEE Trans. Aut.
Control, AC-19, April 1974, pp. 415-441.
9. Yoshikawa, T.: Dynamic Programming Approach to Decentralized Stochastic Control Problems, submitted to
1974 Decision and Control, Phoenix, Arizona.
10. Groves, T.;and Radner, R.: Allocation of Resources in a Team. J. Econ. Theory, 4,1972, pp. 415-441.
11. Groves, T.: Market Information and the Allocation of Resources in a Dynamic Team Model. Preprint 1972,
Joint Aut. Control Conf.
12. Aoki, M.; and Toda, M.: Parameter Adaptive Resource Allocation Problem for a Decentralized Dynamic System.
To be presented at 1974 Decision and Control Conference, Phoenix, November 1974.
13. Aoki, M.: Aggregation. Chapter 5 in Optimization Methods of Large-Scale Systems, D. A. Wismer (ed) McGraw-
Hill, 1971, New York.
14. Aoki, M.: Aggregation and Informationally Decentralized Linear Dynamic Systems. Unpublished manuscript.
15. Kulikowski, R.: Optimum Control of Aggregated Multi-Level Systems. 1966 IF AC Congress Section 40,
London, June 1966.
16. Kulikowski, R.: Decentralized Optimization of Large-Scale Dynamic Systems. 1969 IFAC Congress, Warsaw,
June 1969.
202 NASA-Langley, 1975


Some aspects of control of a large-scale dynamic system

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