Theory of Hierarchical, MuItilevel, Systems M. D. MesarouiE, D.Macko, and Y.Takahara SYSTEMSRESEARCH CENTER CASEWESTERNRESERVEUNIVERSITY CLEVELAND, Orno
1970
A C A D E M I C P R E S S New York and London
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1970, BY ACADEMIC PRESS, INC.
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CONTENTS
xi xiii
Preface Acknowledgments
PART I : HIERARCHICAL SYSTEMS I. Examples 1. Hierarchical Systems in LargeScale Industrial Automation 2. Relevance of Multilevel Systems to Organization Theory 3. Hierarchical Order of Structures in Nature
4 14 30
II. Conceptualization 1. 2. 3. 4.
What Is a Multilevel, Hierarchical Structure? Basic Types of Hierarchies Interlevel Dependence and Coordination Why Hierarchical Structures?
34 37 56 63
III. Formalization 1. Introduction
2. Mathematical Concept of a General System 3. Decision-MakingSystems 4. Formalization of the Hierarchical Concepts V
67 69 72 79
Contents
vi
IV. Coordination General Description of a Two-Level System Decomposition of the Subsystems Coordinability The Consistency Postulate Coordination Principles Various Aspects of the Coordination Problem
1. 2. 3. 4. 5. 6.
85 88 93 96 98
102
PART I1 : A MATHEMATICAL THEORY OF COORDINATION V. General Theory of Coordination for Unconstrained Optimizing Systems 1. 2. 3. 4. 5. 6. 7. 8.
Introduction A Two-Level Optimization System Coordination Principles and Supremal Decision Problem Conflict Resolution in a Two-Level System Infirma1 Performance Generation and Modification Coordinability by the Balance Principle Coordinability by the Interaction Prediction Principle The Coordination Process
109 113 117 119 135 146 155 165
VI. Optimal Coordination of Dynamic Systems 1. 2. 3. 4. 5.
Introduction Preliminaries Application of the Balance Principle Application of the Prediction Principle Iterative Coordination Processes
178 182 194 215 225
VII. Coordination of Constrained Optimizing Systems 1. 2. 3. 4. 5.
Introduction Applicability of the Coordination Principles Coordination of Convex Programming Problems Coordination of Linear Programming Problems A Note on the Dantzig-Wolfe Decomposition
230 232 237 243 252
VIII. On-Line Coordination : Coordination for Improvement 1. 2. 3. 4.
Introduction On-Line Coordination Concepts in General Systems Theory Application to Linear Systems Sequential Adaptive Coordination
256 262 274 278
Contents
vii
Appendix I General Background References
283 287
Index
291
The book contains the presentation of a theory of large scale systems. The basic assumption is that the system under consideration consists of a family of hierarchically arranged subsystems. Although not every large scale system is of this type, there is little doubt that systems of this kind qualify to be considered as being large scale or complex. A distinction should be made between a system being large and a theory which claims to be developed for the study of large scale systems; the first is a value judgement. What is a large system for a psychologist might very well be viewed as nothing but a simple unit, component, by an economist. Occasionally, by a stroke of luck or out of necessity, one describes a system, unquestionably large (say, economy of a state) by’ a (mathematical) model which has a relatively simple structure; say, a second order differential equation. One can proceed then to draw the inferences about the real system by studying such a simple model. Although there is no point in arguing whether the system one is then investigating is really a large one, there is no question that one is using relatively simple theory. We shall take the position that for a (mathematical) theory to claim to be dealing with large scale, complex systems the complexity of the real system has to be reflected in the structure of the model which is the subject of theoretical investigations. Models in which it is explicitly recognized that the system consists of a family of hierarchically arranged (decision-making) subsystems is no doubt complex enough. Hence our claim that the theory presented in this book is a theory of large scale systems. ix
X
Preface
The book has two parts. In the first part, various hierarchical systems from several fields are discussed and it is shown that the description of these systems can be given in terms of three basic underlying concepts: levels of abstraction (or significance in modeling), levels of complexity of decision-making, and levels of priority (of action) in a multiunit decision system. All three concepts are formalized in the framework of mathematical theory of general systems. The first part concludes with formulation of a specificallyhierarchical problem which appears to be of central importance for hierarchical systems ; namely, that of coordination. In the second part a mathematical theory of coordination is developed. A two-level system is considered with n decision (control) units on the first level and a single decision unit on the second level. Coordination is defined as the decision problem of the second-level unit with the objective of influencing the first level so as to promote a goal defined for the entire system. Coordination strategiesare developed on the basis of what we call coordination principles. A detailed theory of coordination is then developed for the systems described in abstract as well as in more specific mathematical frameworks. Considerable portions of the mathematical theory deal with optimizing systems. This is done only for mathematical convenience; in order to be able to concentrate on the main issue-the relationship between the units on different levels-we found it convenient to assume that the decision problems of the subsystems are relatively well defined and simple; otherwise, we might have been bogged down with the intricacy of individual unit behavior and not have the chance to study the proper hierarchical questions at all. However, the very fundamentals of the theory, the coordination principles, are applicable in more realistic circumstances when decision making is under conditions of uncertainty as it is explicitly recognized in Chapter VIII and also indicated by the development of a “ second-level feedback” structure in Chapter IV. Presentation of mathematical development is done in a precise rigorous format by stating propositions and giving their proofs. This format is adopted in the first place to establish precisely the validity of the statements; paradoxically, the need for such a format seems to be even greater in the abstract theory where proofs are short and often almost trivial, yet because of the lack of structure the intuition is a poor guide and one has to proceed carefully in small steps. We have combined the proposition format with discussions of the meaning of the statements illustrated on occasion with the examples so that the implications of the propositions might be understood without necessarily going through the proofs themselves. A rigorous presentation of the theory then only helps the reading of the book since the formal deductions are isolated so that on one hand one might grasp the content of the results without following the deductive constructions (at least in the first
Preface
xi
reading) while on the other hand one can study each of the deductions (proofs) separately. Choosing the methodology of presentation for this book is also aimed at making one more point which is not directly related with the subject matter itself; we have tried to give an example of the application and usefulness of mathematical general systems theory. Some time ago a mathematical theory of general systems had been proposed [4], and its usefulness in systems engineering methodology, in particular for large scale systems studies, has been argued; it was stated that by using such a theory a possibility exists to formulate precise statements regarding systems which are very complex or have poorly defined structure : a system which otherwise could have been described only verbally. From such a starting point one can first of all investigate the structural problems of the systems involved and then using the same basis develop a deeper mathematical theory in the desired direction. We have tried to demonstrate for the first time such a methodology in action. Finally, the book is so organized that two parts could be read almost independently. If one is interested only in hierarchical systems and the foundations for a mathematical theory of such systems the first part would suffice. The reader interested in coordination only could proceed to the second part with occasional reference to Part I (Chapters I11 and IV) for some definitions and concepts. Furthermore, we have tried to make the chapters in Part I1 as self-contained as possible. If the reader is interested only in on-line coordination, Chapter VIII could be read independently of Chapters V-VII with occasional reference to Chapter IV for some basic definitions. Similarly, the reader interested in the control of dynamic systems or programming problems could read Chapter VI or VII, respectively, without going through Chapters V and VIII.
Multilevel systems have been the subject of research in the Systems Research Center of Case Western Reserve University over a number of years (ever since the publication of [33]), and not all of the results and investigations could have been reported in this book. Many of the findings, even if not considered explicitly here, have most certainly influenced our thinking in writing this book. We are most grateful to our colleagues and co-workers for continuing discussion on the subject, and to make sure that credit is given as much as possible where it is due, we have included in the References reports and papers by our co-workers at Case, which are not explicitly mentioned in the General Background given in Appendix I. The material presented in this book is the result of,research supported from many sources. In particular, we are grateful to the Office of Naval Research who offered support and encouragement of the research on multilevel systems at Case since its very beginning in 1961, when only the desire to work on the problem and the general strategy of how to proceed were available. Research was also supported by N.S.F., NASA,and Case Institute of Technology, at times when the funds were not as plentiful as was needed for the project. Finally, we would like to express our appreciation to Mrs. Mary Lou Cantini and Miss Linda Cottrell for transcribing, typing, and retyping the manuscript again and again. We all felt at times as if the book were going to have 3000 rather than 300 pages. Anyway, iteration had to stop even if the optimum was not reached. The state we arrived at is presented on the pages which follow.
xiii
PART I : HIERARCHICAL SYSTEMS
Chapter I
EXAMPLES
Multilevel hierarchical systems were not the subject of a mathematical theory or even a detailed theoretical investigation. This is not to say that scientists and philosophers were not concerned or very much aware of the importance of "hierarchical" order, and that a fair amount of discussion was not devoted to the subject. Rather, it suggests that, historically, consider~ ations were not directed toward and did not result in a theoretical, or even less, a mathematical framework for the investigation of the phenomena of hierarchies. As result, there is a lack of understanding among theoretical investigators of the issues in hierarchical systems. One can even say that the importance and omnipresence of multilevel systems is not sufficiently well recognized. Our objective in this chapter is to describe in as brief a form as possible some ofthe existing multilevel systems for which the theory presented in this book is intended. The prime purpose of presenting these examples is to motivate the interest of the reader by showing how widespread multilevel systems really are. In a true sense, we present a potpourri of cases from diverse fields, selected to facilitate and enhance the subsequent conceptualization and mathematical formalization. The reader not interested in a description of existing systems, but convinced of the need for a multilevel systems theory and interested only in the theory per se, would do well to proceed directly to Chap. II or III. Part II of this book with perhaps some reference to Chap. IV is sufficient for the reader interested only in the coordination problem. Without further ado we begin our informal discussion of various hierarchical systems. 3
I. Examples
4
1. HIERARCHICAL SYSTEMS IN LARGE SCALE INDUSTRIAL
AUTOMATION
We present three cases from the steel, petrochemical, and electric power industries. In order to avoid some pitfalls due to specific constraints in the design and implementation of particular installations, we shall as a rule describe hypothetical cases synthesized from instances reported in the technical literature.
Steel Industry It has been recognized for quite some time that in order to improve the economies of the steelmaking operation a more integrated automation is needed. Some of the reasons which have prompted development of integrated computer control of steel plants are the following: (i) Size, Complexity, and Diversity. At a large steelmill there “may be 500,000 tons or more of materials in the works, in store, or being processed, at any one time to fulfill hundreds of different orders” [l]. Information processing, supervision, and control, presently done by plant management, clerks, and operators is indeed a formidable job. Use of equipment (like computers) to facilitate that job and improve the performance by increasing the capability of both information processing and rational decision-making, is obviously a very attractive proposition. (ii) Broad Range of Systems Response Time. The entire steelmillviewed as a dynamic system is subjected to external effects with a very broad frequency spectrum; the entire plant has usually a weekly workload prepared several weeks in advance, while a strip mill is “running at speeds of up to 4oaO fpm,” so that “every second counts” [l]. In such a system there are bound to be discrepancies between a plan prepared in advance and its execution. Indeed, a lack of coordination can lead either to the orders not being completely filled (because of unacceptable variations in tolerances) or to an inventory which is too large. (iii) Increased Emphasis on Projitability. The cost of steel production has become increasingly important because of both increased competition and demands for higher quality products with closer tolerances. Steel industry income and finished tons shipped per ingot tons have been steadily declining
1. Hierarchical Systems in Large Scale Industrial Automation
5
as is well illustrated in Fig. 1.1 [2]. The solution is being sought in improved automation. The automation in the steel industry is built on the concept of linking together all information processing and control functions in a single system, from customer orders to drive-motor and temperature control. This results in a physical link between production scheduling and plant floor operation, a very desirable feature to both operation and management.
1959
1960
1961
1962
1963
1964
FIG.1.1 (a) Steel industry sales and (b) finished tons shipped per ingot ton.
A block diagram of an integrated steelwork is shown in Fig. 1.2 as a multilevel system with an organizational-type hierarchy. The total task of running the plant is specified on three strata; that is, the total system is a stratified system as defined in Chap. 11. The system has a very large number of units
planning
Finished prOduClS
Row moier moiei
F ~ G1.2 . Block diagram of a steel-making integrated control system.
6
I. Examples
and tasks to perform. (One of such installations is to involve nine computers, when completed [3].) We shall describe, however, only the main functions. From the total systems viewpoint, three main functions to be performed by the system are: production planning, scheduling and coordination of operation, and process control. They provide the framework for a hierarchical arrangement of the subsystems. The highest level unit accepts customer orders, then groups and arranges them so as to improve profitability of production within constraints imposed by the delivery times; the outputs from this level are weekly workloads. The weekly workloads are prepared several weeks in advance, and last minute adjustments are made on the basis of feedback information about the plant operation in reference to what actually has been accomplished in the intermediate period. The intermediate level units accept production schedules and break them down into local instructions for individual processes. They check actual production programs against the master schedule. They receive production and quality data and have the option of initiating complete rescheduling throughout the plant, if necessary. Their main function is coordination. Production is continuous (and in some subsystems with a high speed indeed!), so that the operation has to be coordinated on-line, continuously to avoid bottlenecks, slowing down production or expansive waste. It is precisely these types of processes which need fairly advanced coordination methods and have prompted the development of the concepts and theory reported in Part I1 of this book. The lowest level units are concerned with control of the process itself; they supervise and control the actual physical production of various subunits comprising the complex. This level includes optimization of some subprocesses (relative to minimizing the production cost) ; supervisory control. as well as direct digital process controls, etc. Input-output equipment and measuring and display devices are also included in this level. Several remarks should be added to this brief description: (i) Each of the levels as indicated in Fig. 1.2 can contain several sublevels, so that the “ three-level ” subdivision corresponds to a convenient first breakdown or vertical decomposition of the total task. (ii) The system has considerable autonomy of operation. It has been claimed that it makes such decisions as “when more steel should be ordered due to excessive rejects, whether material in excess of actual requirements is being processed and if so whether this can be applied to other orders ” [3]. The entire system is described as being adaptive in the sense that it provides for automatic rescheduling and can even reject certain orders if they cannot be fit within the accepted modes of operation.
I . Hierarchical Systems in Large Scale Industrial Automation
7
Petrochemical Industry Another good example of a multilevel system in the automation of large industrial complexes can be found in refineries with computer assited operations. Many arguments for the introduction of integrated control in refineries are similar for introducing such control in steel processing: bulk capacity and continuity of production are such that even small improvements in operation lead to considerable financial savings. Furthermore, such improvement looks quite plausible in view of the complexity of the total installation and the presently employed operations and management practices. Among the arguments favoring a hierarchical approach to the control problems in this type of industry are:
(i) Gap between Planning and Operation. To improve their competitive position, many companies are already using computers as an information handling and decision-making tool on the company level. In particular, the production target for a refinery is determined by computer simulations on the basis of market analyses and a linear programming model of the refinery. Computers are also used in the refineries for the control of physical processes such as direct digital control of temperature, pressure, etc. In between these two levels, human operators are using manual supervision and control, a rather outdated mode of operation. It is only natural to close this gap using modern decision-making and information-processing procedures. (ii) Strong Interactions and Increased Capacity. Strong interactions can exist within the refinery as a whole, as well as within the refinery processes themselves. One consequence of this is the existence of bottlenecks which limit the production capacity. Through improved scheduling, the capacity of the entire complex can be increased either by eliminating serious bottlenecks or by operating in a fashion that simultaneously saturates as many processes as possible. (iii) Production Cost Reduction. Parallel operations using common feed, heating, recycles, etc., require a careful allocation to minimize production costs, either for the individual processes or the entire complex. (iv) Response to the Market. Integrated automation allows relatively fast response to changing market conditions; there is a great variety of output products with different prices requiring a flexible operation of the system.
As a specific example, consider briefly an ethylene-producing naphthacracking plant with an integrated control system. The main production layout of the plant is given in Fig. 1.3 [4]. It consists of three main subprocesses:
I. Examples
8 Steom
Nophtho
Corn pression
De-eth.
Seporotion
Return lower
C,
products
‘2
H4
products
0 C, products
VCrocked gosoline
FIG.1.3 An ethylene-producing,naphthacracking process.
cracking furnaces, compression, and separation. Cracked gasoline from parallel furnaces is fed into a single primary fractionator, compressed, and then cooled for further fractionation which takes place in two parts: the low temperature part for the recovery of high purity ethylene products and the high temperature part for gasoline fractions. The entire system, including both the production and the control part, is shown in Fig. 1.4. It is apparently a multilevel system. The overall design of the system is based on two notions of hierarchy defined in Chap. 11. It is a multiechelon, organizational-type, system since the decision-making subunits are in a hierarchical, supremality-type, arrangement. Furthermore, it is a multilayer-type system since the overall task is vertically decomposed in a
1. Hierarchical Systems in Large Scale Industrial Automation
9
Naphtha and steam
FIG.1.4 Block diagram of an ethylene-cracking integrated control system.
multilayer fashion. There are three main layers although each layer can be realized by several sublayers. Production target specification is determined on the highest layer by profit maximization based primarily on outside (market) considerations rather than on the details of operating the complex. On the intermediate layer the products are separated at minimal cost. In addition to local cost minimization there are some adaptive functions on the intermediate layer; in particular, there is the so-called updating function, to determine coefficients for optimization of overall column efficiency and to predict reflux ratios, parameters related to enthalpies, specific heats, etc. On the first layer are supervisory and regulatory control. Supervisory control determines the set points for primary regulatory control, using information from the higher levels to calculate instant reflux rate and reboiling rate of distillation columns, taking into account the rate and enthalpy of the feed. Several remarks are of interest in connection with this multilayer hierarchy :
(i) The first layer has direct digital but classical-type control structures, while the higher layers involve more sophisticated optimization procedures. (ii) Only on the very first sublayer are actual physical variables the control variables. All other control layers and sublayers are basically adjusting parameters and set-points defined in a general sense; the ultimate concern is restructuring or adapting the overall control system in which there is an exchange of information between the layers.
10
I. Examples
(iii) The control on all three layers is on-line, but there is a basic difference in the times of their responses, characterized by the duration of their respective decision periods. First layer control acts on a continuous basis. Short range control on the second layer responds every one to five minutes, while the long-range program has a cycle time of 15-25 min. Finally, the highest layer control responds on a daily basis, except in contingent situations. It is of interest to mention another similar computer control system (200,000 ton ethylene plant) on which more data about the performance is available, although there is no information on the actual layout of the control system [5]. The control system receives information about the ethylene and ten major byproducts with sales statistics and derives an hourly profit rate. In case of major demand changes, it can bring the plant to maximum efficiency within a few hours at the new production level. It also provides, on an hourly basis, complete information concerning production and other conditions in the plant. Safety and reliability features have also been claimed to be quite good; in particular, shutdown of production of nine furnaces can be accomplished by ‘‘ pressing a switch ” while without computer control “the operator would have to try to adjust 45 controls manually ” [5].
Electric Power Systems Electric power systems with their diversity of equipment and complex relationships between many subsystems are naturals for application of the multilevel approaches. Electric power systems have grown in size over a period of time by a stepwise process of interconnecting existing systems into larger pools; at each step the tying of two subsystems offered apparent economic gains while there seemed to be no major technical or operational problems. This resulted in tightly coupled power systems of enormous size and complexity, requiring fast on-line decisions with sharply increased requirements on safety and reliability of the system as a whole, since malfunctioning can have significanteconomical, social, and political consequences. This requires a new look at how such large systems should be structured. A multilayer type of approach is presently being developed [6]. However, we limit our discussion here to the more classical problem of real power dispatch in a pool, which is perhaps the most classical example of a multilevel industrial system in operation. An interconnected system with n separate areas is diagrammed in Fig. 1.5. Customarily, boundaries of the areas are defined so that each area represents one company. In each of the areas there are of course a number of generating
I
,
1. Hierarchical Systems in Luge Scale Industrial Automation
11
r -----------__________ Air;
I I I
;Real
p+
Real power loss
ted]
Load
I
I
I L
I I
I I
I I
_ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _JI Net power interchange
------_-_ Flectric power pool
--1 I
I
I I
I
I I
I
I
L - -- - - - - - - -FIG.1.5 Interconnected electric power systems.
stations and various consumers, but since we are interested in inter-area exchange we assume each area is characterized by: ci the (total) load in the area i x i the (real) power generated by the units in area i y i the loss of (real) power in area i u i the net power interchange through the tie lines between area i and other areas. For the given load, the losses in the ith area depend on how much power is generated in that area and how much is exchanged: y i is a function of x i and u i ;y i = Pi(xi,ui). The balance of power in the area is then given by the equation Pi(Xi, Ui)
+ ci - U i + xi = 0.
(1.1)
I. Examples
12
There are n of these equations. Furthermore, the exchange of power between areas also has to be balanced; 241
+ . - -+ 24, = 0.
(1 -2)
Equation (1.1) represents the subprocesses, while Eq. (1.2) gives their interconnections. The problem of dispatching the (real) power in a pool consists of finding how much power should be generated in each area and how much should be exchanged through tie lines so that the cost of power generation is at a minimum. To that end, there is a cost function Fi(xi) association with the power generation in each area; the total cost of power generation is then F(x1,
...,x,,) =
Fl(X1)
+ + * * *
F,,(X,,).
(1.3)
The optimal dispatch problem is to minimize the total cost with respect to x = (xl, .. .,x,,) and u = (ul, ..., u.) under the power balance constraints of Eqs. (1.1) and (1.2). Notice that because of Eq. (1.2) there are only (n - 1) undetermined net exchanges. This problem can be solved either by a single computer (so-called single area approach) or by a two-level hierarchy of computers. In the two-level hierarchical approach, each area has its own dispatch computer, and in addition there is a pool dispatch computer. The arrangement is shown in Fig. 1.6. The problem of finding an optimal allocation of power generation and exchange represents then a two-level organizational hierarchy. computer
FIG. 1.6 Two-level power dispatch within an electric power exchange pool.
The task of the entire hierarchy is to solve the overall minimization problem; the question is how should this task be divided between the area computers and the pool computer. This is precisely the type of question investigated in great detail in Part I1 of this book. Several schemes have been proposed [7], relying primarily on the structure of the particular problem at
I. Hierarchical Systems in Large Scale Industrial Automation
13
hand. However, there is a general method, based on the so-called interaction prediction principle (described in Chap. IV) which can be applied in a rather natural way to designate the tasks of the individual computers. The method is as follows: The ith area computer solves the problem: minimize Fi(xi)with respect to x i under the constraint (1.1) for a given value of the power exchange, u:, specified by the pool computer. The pool computer task is then to specify the power exchanges u,", . .., u,". If the power exchange is as specified, the area optimizations yield the total optimum. When the discrepancy between the reference power exchange specified by the pool computer and the actual power exchangd is outside some prescribed limit, a new reference exchange can be derived by an iterative process (described in Chap. IV) involving both pool and area computers. We shall not pursue this problem here except to indicate that the entire body of the coordination theory, as developed in Chap. IV and Part 11, is applicable to these particular kinds of problems. Although we have only indicated the use of the so-called interaction prediction principle, the interaction balance principle and the interaction estimation principle can also be used. The practicality of applying any of these methods has to be assessed, of course, in the particular context. There are a number of reasons, technical, economical, and operational, why a multi-area arrangement (i.e., a two-level hierarchical system) is preferable to a single-area approach in the real power dispatch problem. Prominent among them are the following: (i) Dispatch can be performed faster (it requires less computation time) and with less total commitment of the computer capacity to the dispatch function than in the single-area approach. This, of course, depends upon the size of the system and the skill in specifying the task of the pool computer which acts as a coordinator. (ii) The system is less sensitive to changes in configuration that can occur on a regular basis or in contingencies. For example, a change in one area's conditions would require changes only in the local problem. In the centralized, single-area approach, the change cannot be localized and would require, in general, a change in the solution program for the entire system (the entire system is viewed as a single area). Also, membership in the pool can easily be changed. (iii) Computers involved in other technical and operational tasks might already exist in the area centers. The pool computer is then simply added for dispatch coordination and for other jobs on the pool level, such as billing of the actual power exchange between the areas, etc. It should also be recalled
I. Examples
14
that the areas usually belong to different companies, and there might exist some company-specific administrative and legal reasons why the two-level arrangement is preferable. Finally, let us mention that the hierarchical arrangement can be extended to the formation of superpools, as illustrated in Fig. 1.7.
Fig. 1.7 Formation of superpools from “2-area’’ pools for electric power dispatch.
2. RELEVANCE OF MULTILEVEL SYSTEMS TO
ORGANIZATION THEORY
Similarities between (human) organizations and the multilevel hierarchical decision systems of concern in this book are apparent from mere inspection of a multilevel system diagram as shown in Fig. 1.1 and the organizational chart in Fig. 1.8. Although there are quite a few different definitions of an organization, depending on the aspects of the organization chosen for emphasis, they all agree that the organization consists of a family of interacting, hierarchically arranged, decision-making units. Hence the relevance of our model. The importance of organizations cannot be overemphasized. As Arrow has pointed out [8]: Among man’s innovations the use of organization to accomplish his end is among both his greatest and his earliest. If we had no other evidence, we would know that complex organizations were necessary to the accomplishment of great construction tasks-planned cities like Nara or Kyoto, or monuments like Pyramids. For less material ends we know of organization of the Inca Empire of Peru where a complex and far-flung state was administered in a highly systematic manner with a technology so poor as to include neither writing nor the wheel.
15
2. Relevance of Multilevel Systems to Organization Theory
directors
a President
I
Vice president research
I
Vice president finance
"
I
Controller research
I
I
president morketin
FIG.1.8 Organization chart of a company.
The importance of organizations in the modern society is evident by the omnipresence and omnipotence of large organizations in the modern technological era. Galbraith argues that a modern state should be considered more appropriately as the industrial state because of the power and overriding importance of (large) industrial organizations 191. One of the principal features of modern organizations is the degree of specialization; for the men charged with running these organizations, Galbraith states with usual candor that " it takes a specialist in organization to run an organization that results from specialization." Clearly today's managers need all the help they can get to do their job properly. Management is becoming more of a social function and to a lesser degree has only limited implication for a small group of individuals, as was the case in the past. In view of such importance, it is somewhat surprising how little is known about organizations and how slow progress has been in developing organization theory. This is not intended to be a criticism of the field or the scientists working in it; rather the intention here is only to point out a problem area, its importance, and challenge. No doubt the lack of progress is due to the complexity of the subject, but it is also due to a lack of attention evident by
16
I. Examples
the scarcity of researchers in the field. One would assume that at present there are more researchers worrying how to " optimally " adjust parameters in feedback control systems (a problem which at best can bring marginal commercial improvement) than there are researchers worrying about quantitative aspects of control and communication processes in organizationaltype systems. Our research in multilevel systems was motivated towards making systems theory relevant in a more direct way to the theory of organizations. An organization can be viewed, of course, as a system, by simply considering the outside stimuli (inputs) and the associated responses (outputs). If this relationship is described as dynamic, as in so-called industrial dynamics, useful inferences can be drawn regarding changes in time that takes place through the history of an organization. However, the predictability of an overall input-output approach (which is sometimes also referred to as total systems approach) is limited by discrepancies between the structures of an inputoutput model and the organization itself. This remains true even if one introduces feedback interconnections and represents the organization as a singlelevel, although multivariable, feedback control system as is usually the case in industrial dynamics. If the principal issue of organization theory were to explain the growth of an organization and perhaps predict its evolution in the immediate future, then under limited circumstances such a model could turn out to be adequate. However, one of the overriding issues in organization theory is to assist managers, the men responsible for the well-being of an organization. Managers are themselves members of organizations and primarily need instructions on how to affect the organization from within so as to improve its operation. For this purpose, the model representing an organization from the overall input-ouput viewpoint is not adequate. These considerations indicate a need for a multilevel structure in a systems theoretic model of an organization. The structure should reveal the most prominent characteristics of an organization which are : (i) an organization consists of interrelated decision-making subsystems and (ii) these subsystems are arranged hierarchically. Therefore, a systems theory model of an organization is precisely a multiechelon system as described in Chap. 111. We proceed now to indicate the relevance of multilevel systems theory to organization theory. Our limited objective is to establish a bridge, even if tentative, between the formalism we are proposing and the problems of organizations and, thereby, to provide motivation for further theoretical studies in the area. The detailed and explicit application of the theoretical results to organization theory is a separate study and requires, of course, considerably more time and space than can be allotted in this book.
2. Relevance of Multilevel Systems to Organization Theory
17
Role of the Multilevel Systems Approach in the Spectrum of Organization Theories There is no single theory claiming to be the sole interpreter of an organization’s functioning; rather, there is a whole spectrum of theories with organizations as their subject matter. This is quite natural. A complex thing such as an organization has many facets, and knowledge from many different disciplines has to be brought to bear for its exploration. Furthermore, there are many diverse kinds of organizations as to size and purpose; mere talk of “ the organization ” is a gross simplification. To establish what new viewpoint is introduced by multilevel systems theory, one has to contrast this theory with some of the dominant trends. We shall do that rather briefly in order to provide a background (not to be construed as a review) for assessing the multilevel systems approach within the spectrum. Organization theories can be categorized as being classical (structural) behavioral (motivational), and systems oriented. We first contrast these orientations and then indicate the contribution of the multilevel systems approach in reference to : (i) how the individual members, participants, in the organization are viewed; (ii) how the structure of an organization is represented ; (iii) what research methods and tools are used.
Participants Classical theory tends to view a participant as an “ instrument” merely carrying out an assigned task. It is tacitly assumed that the participant is persuaded, by carrot or stick, to play the assigned role; his problem is merely to accomplish the task in the most efficient manner. Typically, using time and motion studies one investigates sequences of events constituting the necessary steps to accomplish the specified job. Such an approach is apparently suitable for studies concerning working teams in an automated assembly line. It is not adequate for a descriptive or normative study of participants (such as managers and top executives) on the higher levels of the organization. Behavioral theory takes special interest in the participants. It is concerned with the reaction of a participant to various pressures and influences from within a formal organization as well as from other groups to which he simultaneouslybelongs. To a large extent, it dealswith a member of an organization regardless of where he is placed in the organization; in short, it emphasizes his personality rather than his role in the organization. In the systems-oriented approaches, such as industrial dynamics, there is by and large no explicit recognition of the participants. Functioning of the overall system is describedin dynamicterms(by means of appropriate equations
I. Examples
18
or simulation models) and evolution of the system is investigated without explicit recognition of the human element. Structure The structure of an organization in classical theory is viewed as static. An organizational chart (Fig. 1.8) is used to represent the hierarchy and
specific goals are assigned to the different units represented on the chart. As a guide to goal and role assignment, several principles based on “ good management practice have been formulated. They are supposed to take into account the power and limitation of the participants. For example, the “ span of control ” principle limits the number of participants any member can supervise; the “ delegation principle states that authority should be delegated to match responsibility. In the behavioral approach emphasis is on the participants as members of an informal group, remindful of “group dynamics.” The hierarchical structure is considered only indirectly and is tacitly assumed to play a secondary role. In systems oriented approaches the hierarchical structure is again relegated to a secondary place; emphasis is on overall time evolution. The system is represented by means of some specific units such as “ time delays,” amplifications,” etc., or, more generally, with whatever structure is needed to match the input-output relationship. ”
”
“
Methodology The classical approach is based on observations of the actual processes and the acquired experience in performing some specific tasks in real organizations. The framework and language is largely that of classical mechanics. The behavioral approach is based on observations of the psychological reactions and performances (behavior) of the participants. In essence, much is borrowed from group dynamics, which is closer t o informal than formal organizations. Matching of explanantions with the actual observations from (formal) organization studies is a stated goal not always achieved. Systems and computer studies are largely computer simulations based on some observations selected as relevant for the overall response of the organization. To investigate the dynamics successfully, the variables have to be restricted to relatively few and, therefore, reflect the interest and bias of the investigator. The above mentioned categorization is not assumed to partition the field of organization theories ; there are approaches which combine the characteristics of two or more categories.
2. Relevance of Multilevel Systems to Organization Theory
19
Most notable, is the approach of Simon [lo]. The participant is viewed as a decision-maker (this notion was never formalized however) and concepts from modern psychological theories of human problem solving are used. Although the approach is often classified as behavioral, it has a good deal of the systems approach in it, as evident in simulation studies done by Bonini [l 11. The hierarchical structure, however it actually appears in the organization and exemplified on the organizational charts, is downgraded as in the behavioral approach. The participants are studied in terms of " motivation to produce " or " motivation to participate " without direct reference to their position in the hierarchy. A concept of hierarchy is used in the discussion of the overall task of the organization, but it is a multilayer type of hierarchy which does not correspond to the multiechelon hierarchy actually existing in the organization. Although there is a normative value in using multilayer hierarchies in the design or restructuring organizations, its use in descriptive studies might lead to confusion, since an actual organization (which surely has a multiechelon hierarchy) might not have been structured so as to make a multilayer decomposition of the overall task meaningful. The need for distinguishing which notion of heirarchy is used is quite evident. Another approach having the flavor of two categories is the application of game theory. It primarily belongs to the third category in terms of methodology; its emphasis on the bargaining process within the organization disregards the hierarchical structure. A similar conclusion holds for the theory of teams [12]. Position of Multilevel Systems Theory We argue that the position of multilevel systems theory in the spectrum cuts across all three categories : (i) It emphasizes the hierarchical structure in the sense of the organizational charts of the classical theory; it views the hierarchical arrangement of decision-making units as one of the primary characteristics of the organization. (ii) It views the participant as a decision-making (or goal-seeking) system in the sense of modern behavioral or, more specifically, motivational approaches. Levels of satisfaction and discrepancies between the actual and operational goals are explicitly recognized concepts. (iii) It recognizes, explicitly, that an organization invariably consists of an interconnection of decision-making subsystems. The proposed approach can be considered as belonging to a separate category : the application of mathematical systems theory to organizations. Reference to mathematical constructs might seem too restrictive ; it should,
20
I. Examples
however, be noticed that the approach is based on the general, mathematical theory of systems and provides a framework capable of formalizing weakly structured statements derived from actual observations. Such a general theory can accommodate behavioral constructs and yield formal mathematical counterparts without introducing restrictions. Indeed, one might almost state that nothing is lost by such formalization while the main benefit is increased precision in describing actual phenomena.
Potential Benefits of Multilevel Systems Theory Among the most immediate potential benefits of applying multilevel systems theory to organization research, the following are worth mentioning: (i) It provides a unified basis for different approaches through a framework in which the various approaches can be compared, contrasted, and used to complement each other. (ii) It offers mathematical precision in defining both concepts and issues. (iii) It provides a starting point for mathematical and computer simulation studies of various issues and problems both in analysis and design of organizations. The approach being based on the mathematical methods is primarily concerned with structural considerations such as communication, control, command, coordination, etc. However, it should be emphasized that the basic building block, a decision-making unit, is closer to the formalization of Simon’s “satisficing” man rather than the more classical optimizing man; it is therefore an “administrative” rather than an “economic” man [13]. Multilevel Systems Theory Formalizationsof Concepts from Organization Theory
To illustrate statements made in the last section about the potential of multilevel systems theory in organization studies, we briefly discuss how some of the behavioral-type concepts are reflected in the formalism. As the basis, we use March and Simon’s classical treatise on organizations [lo]. Representation of a Participant The concept of a decision-making unit as presented in Chap. 111 includes Simon’s administrative, “ satisficing” man as a special case. It is capable, therefore, of incorporating a number of behavioral concepts; for example, the tolerance function provides for the “ aspiration level,” while. the “evoked set” corresponds to the set of alternatives at a given decision time. More
2. Relevance of Multilevel Systems to Organization Theory
21
generally, the decision-making units which are used as building blocks of multilevel systems possess what is called “ discretion ” of a decision-maker: the response of the unit is specified by a satisfaction problem, the solution of which is generally not unique; selection of the actual action depends,therefore, upon the discretion of the unit. The formalism provides, more specifically, for all four types of “discretion” as defined by Simon. Apparently, sufficient latitude is provided to represent the decision-making (problem-solving) behavior of human participants as envisioned in modern psychology.
Supervisor and Subordinate Consider the relation between a supervisor and the subordinate or, in our terminology, the supremal and infimal units. Simon’s analysis indicates that a participant can be influenced by his supervisor through: (i) factors related to his goals, (ii) factors related to the expectations of the consequences of his decisions, and (iii) factors related to the set of alternative actions available at the moment of decision. In our formalism these correspond respectively to goal intervention, image (model) intervention, and constraint intervention.
Coordination Simon states that an organl’zation reacts to conflict either by using analytical processes ” or “ bargaining processes and that the analytical processes involve “ problem solving ” or ‘‘persuasion.” All these notions have counterparts in the formal theory of coordination developed in Part 11. In the problem-solving process of conflict resolution, Simon [9] assumes “ the objectives are shared among coordinator and subordinates and the decision problem is to identify a solution that satisfies the shared criteria.” This corresponds in our formalism to image intervention: the performance evaluation functions of the supremal and infimal units, if in harmony, remain unchanged, and coordination is accomplished by affecting the infimal outcome functions or models. When image intervention takes on the form of predicting interactions, it corresponds to the Simon’s concept of “ assembling information ” (about the future interactions). “ Evoking new alternatives” is recognized here as constraint intervention; the outcome functions are affected through constraints on the action sets. Analogously,persuasion corresponds to goal interventionin our formalism. Simon assumes that conditions for coordination by persuasion exist when “the individual goals may differ within the organization but the goals need not be taken as fixed.” Furthermore, he states that “implicit in the use of persuasion is the belief that at some level objectives are shared and that disagreement over subgoals can be mediated by reference to common goals.” Indeed, in the coordination theory developed in Part 11, agreement of objectives is achieved on the supremal level and intervention in the infirmal goal “
”
22
I. Examples
functions is done in reference to the overall goal “Testing the subgoals for consistency” corresponds to testing whether the necessary goal properties (interlevel or infirmal harmony) exist. “ Different new criteria ” generated to enable coordination, correspond to different performance functions which are set through goal intervention. Our formalism is broad enough to encompass the formalization of Simon’s idea of analytical conflict resolution by either problem solving or persuasion. The resulting sharpening of the concepts and issues involved is reward enough for the formalization. But, let us note in passing, that since the formalization is mathematical, a basis is provided for deduction and a comparison between the problem-solving and persuasion methods of coordination; furthermore, using results from Part 11 one can develop new procedures which specify how to actually carry out the coordination process. The formalism; therefore makes a definite step toward a normative theory of organization. Any specific normative proposition is based on more detailed, specific structures of a system and is, therefore, less general than verbal statements, which by necessity are often too vague; it is, however, only by being more specific and precise that one can arrive at logically defensible normative propositions. It appears that a theory of coordination using the bargaining approach could also be developed using the same framework. This has not been attempted even though it certainly represents an important and interesting avenue for new research. One could naturally expect that game theory is of considerable use in the bargaining approach. In fact, game theory has already been used to study bargaining-typesituations between organizations in an economy or a society. It should be noticed, however, that we are talking here about bargaining as a method for coordination within an organization. This implies a special role for the suprema1 with, probably, the power to change the rules so as to induce the participants to arrive at the desired equilibrium. A study of such an approach has been reported in [14]. The Hierarchy Finally, let us look at the hierarchy in Simon’s sense. Because of the limited decision capacity of the building blocks, (referred to as “ boundary of rationality”) the overall goal of the organization which reflects the purpose of the organization as a whole is broken down into a sequence of subgoals, so that the solution of the overall goal is replaced by the solution of the family of subgoals. Apparently, this concept corresponds to the layer concept introduced in Chap. 11. Such a hierarchy of goals can be used for both the design of a multilevel system and as a strategy for solving complex problems. Let us again call attention to the important distinction between the hierarchy of
2. Relevance of Multilevel Systems to OrgdniZatiOn Theory
23
goals and hierarchy of decision units as will be later pointed out in Chap. 11. The structural concepts such as line and staff distinction, span of control, etc., in organization hierarchies can be incorporated in our formalism in an obvious fashion. Actually, an organization is represented in multilevel systems theory in an organization-chart manner with (dynamic) decision-making units at the decision nodes.
Specialization (Decentralization)and Coordination A most prominent structural aspect of an organization is specialization inevitably followed by coordination. The purpose of this section is to demonstrate that the theory of coordination developed in Part I1 is relevant to coordination problems as viewed in human organization theory. Some of the arguments supporting this statement have already been presented in the preceding section ;we indicate here the specific relevance of the coordination principles based on the interactions between the infimal units. Specialization is one of the principal features of an organization; in fact, the organization results from isolating special jobs and assigning those jobs to individual specialized units. Broadly speaking then, specialization results in departmentalization and functionalization and in line and staff units. Line tasks reflect the purpose for which the organization is created. Staff activity is a supporting activity. The line units in general determine what will be done and when it will be done, while the staff units advise on how it can be done best. Specialization of any kind requires another feature of paramount importance in organizations; to accomplish the overall task, the Apecialized operations have to be coordinated. Coordination, which is also referred to in organization theory as control, divides itself quite naturally into two parts: the establishment of operating rules instructing the members of the organization how to act and the enforcement of these rules within the organization. The first is referred to as control-in-the-large and the second is control-in-thesmall [S]. Control-in-the-large corresponds in our formalism to the selection of appropriate infimal performance functions and, more generally, the selection of the modes of coordination. Control-in-the-small corresponds to the selection of the coordination input. As a rule we refer only to control-in-the-small as coordination. It is recognized in organization theory [lo] that one of the central variables in the theory of departmentalization is the degree of self-containment of the (several) organization units. A unit is self-contained to the extent and degree that the conditions for carrying out its activities are independent of what is
24
I. Examples
done in other organization units. The variables representing the degree of self-containment translated into our formalism are the interactions between the infimal unites. The problem of coordination is then primarily concerned with resolving the injimal unit interactions. A solution of this problem is developed in Part I1 by means of the so-called coordination principles. Basically, they specify strategies the coordinator could use to compensate for the fact that the individual units (departments or functional divisions) operate as if they are “ self-contained.” The principles and associated coordination methods provide a whole family of normative solutions to the problem of “ control in an organization.” They not only indicate how to coordinate specialized units, but, even more, they indicate the possibilities for new kinds of specializations by offering new methods of coordination. Let us repeat the fact that the proposed solutions are normative, and their value for any particular type of organization has to be assessed separately. The coordination principles and coordination methods developed in Part I1 have also a descriptive value. It is apparent that any specialization, as discussed before, represents some degree of decentralization. The classical method of decentralization is in terms of the profit centers. Departments in a corporation are viewed as more or less independent units charged with the responsibility of operating in the best possible manner so as to maximize profit under the given constraints imposed by the corporate management. The problem of decentralization is essentially how to design and impose constraints on the department units so that the well-being of the overall corporation is assured. The standard way to coordinate decentralized organizations is by means of the pricing mechanism; coordination is designed by analogy with the operation of a free market or competitive economy. Exchange of products between departments is allowed, and internal prices are specified for the exchanged commodities; the problem of effective decentralization reduces then to the selection of the internal prices. Now we shall show how the pricing mechanism of “ the economic-type decentralization” can be derived by application of one of the coordination principles, in particular, the interaction balance principle formulated later in this book. With this, we point out the descriptive value of the coordination theory developed in Part 11. For simplicity, we consider the original version of the decentralization argement for a welfare economy developed in the form presented by Arrow [15]. The argument, of course, is as old as Adam Smith’s theory of the “invisible hand,” but for our purpose Arrow’s form is most convenient. Consider a process P decomposed into n + 1 local processes as shown in Fig. 1.9. The subprocesses Pi,1 Ii In, are of the same form: each P i has an input mi and two kinds of outputs y i f and y i , . The subprocess P,,,i interconnects the subprocesses P i , 1 I i < n, as diagrammed, again discriminating
2. Relevance of Multilevel Systems to Organization Theory
i
m1
9
r--;
_o(
,7f
L----J
1
25
r
----1
Ylf p
i
y,w
I
4
45, I L----J I
I
I
el
I
I
-----
I
i mn
-
II 1
Cf
:---,a
II
0
ynf
r---- 1I
i
YI
Pnw
i I ,
Ym1.f
L----..I
I
F
-
h V i
between two kinds of outputs. The interactions between the subprocesses are then the inputs to P,,,l given by
ui/
= Kj/(Y1/ 9
ujw= Ki,(yl,
..
- 9 ~ n / ) 7
, ...,y,,,),
1 I i I k, 1 5 i I s.
Economic interpretation of the process P is as follows: the subprocesses P i , 1 I i n, represent the production processes or corporations in an economy, while the subprocess Pn+ represents the consumer sector. The
I. Examples
26
inputs to the subprocesses P i , 1 Ii In, represent the scales in the economy at which the processes are run, and the outputs are produced commodities. The input to the subprocess P,,,, represents demand. The subprocess P,,,, consists of the part P,,+ representing the relationship between consumer’s demand and the produced commodities, and the part P,,+l , w representing the exchange of the commodities between the production processes. Distinction between two types of outputs of the production processes is clear: ylf , .. , ynf are produced because there is a demand for those commodities, while y l w , ...,ynw are produced because they are needed for the functioning of the system as a whole; they are the intermediary commodities necessary for technological functioning of the individual processes. We assume there exists a single performance (utility) function G that can be used to evaluate the functioning of the entire economy. Justification for this assumption can be found in standard texts on mathematical economics. We assume the whole economy is consumer oriented, so that the performance function depends only upon the outputs Y , , + ~of, ~the consumer unit. The objective of the overall system’s operation is to maximize the utility ,f) over m,, ...,m,,+,.Such an overall problem is usually function G(Y,,+~ referred to as the problem of allocation of resources in an economy for a given state of technology. This problem can be easily solved in principle by a centralized authority. The following question, however, arises. Assume the production units, as well as the consumer, are regulated by their own decision-making units. Decision units D ifor P i , 1 Ii 5 n, are (production) managers, while the decision unit D,,+l forP,,,, is the “helmsman” [15]. D,,+l represents the government action which influences the consumer behavior. Is it possible to define decision problems for the decision units D i so that these units will select the optimal values mi relative to the overall welfare performance G ? In other words, can the managers and helmsman, while pursuing their own interests, achieve the social optimum relative to the welfare function G ? Consider the system as a two-level system with the managers and helmsman as the infimal units and the optimal welfare problem as the overall problem. A block diagram of the system is given in Fig. 1.10. The coordination theory development in Part I1 is applicable to.this kind of situation. Assume that decoupling coordination is applied and the suprema1 (coordination) problem is defined by the interaction balance principle (see Chaps. IV and V for the definition of these concepts). Notice that no performance function is specified, as yet, for the infimal units. In order to provide for coordination, we utilize the modification method of generating the infimal performance functions by means of the goal-interaction operators as developed in Part 11. ,f
.
2. Relevance of Multilevel Systems to Organization Theory
27
SupremoI unit
Coordinolion parameters P
-------- - - - - m1
FIG.1.10 A two-level hierarchical representation of an economy.
First of all, we make the following additional assumptions:
. ..
(i) The output y i f has k components, y i f = (yi’/, y&), while yiw has ...,yrw). The integers k and s are the same for all subprocesses. (ii) The input of the consumer’s section has k components, m,,+l = (nz,,1 + 1, ..,m%+1) ;furthermore, P,, + 1, and P,, + ,w are defined by the equations
s components, y i w = (y/w,
.
+ mi+l+ yi+l,w= ujw + mi+l + cwi7
yi+l,f = u j f
cfj,
1 Ij Ik, 1 Ij I s.
(iii) The interactions are given by the equations n
.
ujf = C y k , i= 1 n
l l j l k ,
.
U ~ , = , ~ Y ! ~ , ,
11j1s.
r=l
To generate infimal performance functions which can be coordinated by the suprema1 unit, we use the linear approximations of the goal-interaction operators as developed in Chap. VI. This gives the performance functions
I. Examples
28 for the n managers, while the consumer performance is
The supremal unit can then assume decoupling between the subprocesses and use the interaction balance principle to coordinate them. In order to achieve the overall optimum according to the decoupling coordination mode, the infimal units have to maximize their respective performances over both local controls and interactions, while the supremal and ’ Bw’ so that the interactions has to select the coordination parameters /?, are balanced : n
where tiix and
Qiw
n
are the optimal values selected by the helmsman, while
9jf and jfware the optimal values selected by the managers. This coordination process has a desirable economic interpretation. The managers’ performance function (derived by using projections and interaction operators) can be viewed as representing the profit resulting from production, with the coordination parameters being the price of the product. The managers therefore maximize their profits. The performance function of the helmsman represents the difference between the social benefit and the cost of the production. This is again maximized for the well-being of the economy. The supremal selected the coordination variables so as as to balance supply and demand for the considered commodities. The supremal therefore represents the market mechanism and the coordination parameters pi , 1 s i In, are nothing but prices. Coordinability of the system then implies that there exists a set of prices so that the welfare optimum is achieved. The conditions for coordinability of this type of system (i.e., when the market mechanism does work) are explored in detail in Part 11. For the simple case of a static system and convex G the conditions correspond to the wellknown welfare theorems in econometrics [I 51. Application of the balance principle therefore yields as a special case the classical description of the optimum behavior of the market mechanism. It should be noticed that the correct economic interpretation also requires that the following inequalities are satisfied :
As shown in Chap. VII, this does not change the applicability of the interaction balance principle.
2. Relevance of Multilevel Systems to Organization Theory
29
The generality of the coordination theory and its potential usefulness for designing new coordination methods can be now assessed. To this end the following remarks are pertinent: (i) In the theory of decentralized operation of an organization, questions are raised as to the optimality of the decentralized operation and the process arriving at the optimal coordination for the cases (quite usual in practice), when the relationship between the departments (production processes) is not strictly on a competitive basis, either because of external influences or because of additional constraints imposed by the concern for the corporate well-being. Many of these questions can be answered by the application of the general coordination theory from Part 11. (ii) The main problem in coordination is how to deal with the interactions (key variables according to Simon !). The coordination principles from Part I1 indicate many different ways in which this can be done. Apparently, the pricing mechanism is a very special case of the balance principle under rather restricted circumstances. There is no reason why decentralization in an organization cannot be based on other principles under conditions quite different than those provided by the analogy with the market mechanism. It is t0 be anticipated that the selection of the appropriate principle, and the form in which it is applied will depend upon the type of organization under consideration. (iii) It should be observed that the interaction estimation principle presented in Chap. IV is defined in terms of a satisfaction problem for the infimal decision units and, therefore, offers perhaps the most radical departure from the traditional method of decentralization in a firm. Let us emphasize again that these recommendations have at present only a normative value indicating how one ought to proceed under the specified conditions. However, the generality of the conditions when these principles are applicable suggests the possibility that they also .could have a descriptive value, as was shown to be the case for the market mechanism. Concluding Remarks The second half of the twentieth century has been characterized by the emergence of truly large organizations: industrial corporate enterprises and government bureaucracy. This poses a new challenge to the researchers of organizations not only because of their size, but also because of the change in the methods of operation and administration. This change is predicated on the new technology which is becoming available to the managers, not only to accomplish the technical task for which the organization was formed but also to improve and facilitate the job of managing. The best illustration of this
I. Examples
30
is the use of computers in management as, for example, illustrated by the increase usage of management information systems. To take advantage of these new developments, a new framework is needed. Since the computers increase both the information processing and the logic capability of the decision maker, the manager needs a new and more reliable basis to use these new tools. A general mathematical theory of multilevel systems hopefully can provide that, both because of its generality as well as its precision. The validity of this rather bold proposition can be established only by more detailed and elaborate research. If the present book encourages such a development, we shall consider our efforts well justified. 3. HIERARCHICAL ORDER OF STRUCTURES IN NATURE
Scientific inquiry is most successfully carried on in a compartmentalized manner in specialized disciplines, using specialized approaches and tools. Yet the phenomena of inquiry, man or his natural or social environment, are not compartmentalized as are our methods of analysis or specialized fields. What enables such a localized concern for isolated aspects to result in deep, revealing understandings is the existence in nature of a hierarchical order of structures, the multilevel aspect of natural phenomena. What constitutes a level and what are the main levels, are not definitely resolved and indeed depend to a degree upon the approach, interest, and methods of analysis. As Albert Szent-Gyorgyi has wryly put it: If you would ask a chemist to find out for you what a dynamo is, the first thing he would d o is to dissolve it in hydrochloric acid. A molecular biochemist would, probably, take the dynamo to pieces, describing carefully the helices of wire. Should you timidly suggest to him that what is driving the machine may be, perhaps, an invisible fluid, electricity, flowing through it, he will scold you as a “vitalist” [16].
Some of the levels, however, are well-recognized, and one might talk about hierarchical orders of structures in reference to these established levels: nuclear or atomic levels in physics ;molecular, cellular, supercellular, or organ levels in biology. We are interested here in a mathematical theory of hierarchies which is structural, formal, and independent of the field of application. Therefore, we shall not discuss significance, types, and implications of the hierarchies in nature, but, rather, restrict our comments to formal aspects. Among the most prominent features of a hierarchical order of structures are the following: (i) There is an order of magnitude difference in size of the units of concern on different levels.
3. Hierchical Order of Structures in Nature
31
(ii) What constitutes a unit on any particular level depends upon the interaction mechanisms operative on that particular level. In order for a scientific analysis to be successful it is necessary that the phenomenon under consideration is sufficiently isolated ; hence, the phenomenon has to encompass all strongly interacting aspects. For each level there is an appropriate concept of interaction which defines the unit emanable for analysis. The vagueness of these two criteria (“ order of magnitude ” and “ strength and type of interactions ”) indicates quite clearly the difficulty in establishing a hierarchy of structures which can stand the test of time even over a short period. Perhaps the only established fact is the existence of the hierarchical ordering, at least in the sense that the results of the analysis concerned with a localized unit on a given level can be confirmed experimentally. Existence of hierarchies in nature has been discussed by too many, too often, adding too little to what was already known: namely, that the hierarchical arrangement does exist. These considerations are usually met with a shrug of shoulders of the laboratory-bound experimental scientist; important as this fact is, it cannot help the experimentalist in his work (nor does it seem to hinder him, at the moment !), and there is nothing he can do about it, anyway. It seems that reappraisal of such a stand might be in order: (i) There is an increased number s f important scientific problems which need and indeed require multilevel analysis. An example of such a case will be presented in the sequel. (ii) To use multilevel analysis, the tools and methods of analysis on individual levels have to be sufficiently developed. These seem to be available now in many fields. Using molecular properties, for example, the problems involving both cellular and subcellular levels can be attacked. Particularly appropriate examples of a need for multilevel analysis can be found in biology; a good account of some multilevel experiments already performed is provided for the problem of cell cytodifferentiationin vitro [17]. We give that account only in broad strokes, omitting the specifics as irrelevant for our immediate aim. Study of the process of cytodifferentiation on the cellular level is not possible by observing gross morphological characteristics such as shape, pattern, etc. The sensitivity of detecting the changes is increased substantially by observing these changes on a lower, subcellular, molecular, level. Furthermore, it was found that cytodifferentiation takes place most readily if: (i) cell aggregates are beyond a minimum size, and (ii) they are in the presence of cells of unlike type. Therefore, suprecellular aspects play an important role. This experimental evidence leads to the conclusion that: “What we must do is to recognize that we have a multileveled phenomenon, and that part of our problem is to unravel the relationship between the levels” [17]. In other words, although the interest is focused on
32
I. Examples
the cellular level, one has to take into account both adjacent subcellular and suprecellular levels. One might consider the possibility of providing a proper environment which will interact with a cell in such a way that at least the suprecellular level can be eliminated. This, as yet, is only a theoretical possibility. However, even if this were possible for the study of cytodifferentiation, restricting observations to one level is not possible even conceptually in other areas, e.g., developmental problems; one of the important issues in the cleavage product is cohesiveness, which by definition is multicellular. Emergence of this suprecellular property from a single cell clearly requires simultaneous study on both cellular and suprecellular levels. In view of the apparent importance of the problem of hierarchies and the lack not only of a quantitative theory but even the concepts for the consideration of hierarchical systems, it is of interest to comment on possible reasons for such a state of affairs and perhaps suggest a remedy. Progress, indeed the beginning, of a theory (conceptual or mathematical) of hierarchical systems was prevented because the wrong questions were asked in the wrong framework; namely, the problem of hierarchical order in totality or the question of what is the best or actual hierarchical ordering is considered. Furthermore, these essentially scientific problems were couched in quasi-philosophical terms, thereby being concerned with the importance and meaning of hierarchical ordering. We believe that the beginning of a theory of hierarchical systems (for which the time seems to be ripe) should start by considering some specific, precisely defined, yet multilevel-type problems and investigating such problems in great detail. One such characteristically hierarchical problem is how to cross levels: how does the behavior of a system on one level affect the systems on the adjacent levels. Notice that the question does not deal with the hierarchy in general, but merely with the relationship between two adjacent levels. Also, the problem is not what are the properties that are recurrent on any or all of the levels, but rather, what are the interlevel properties. Only by making such admittedly small but specific steps can progress be made toward answering grand questions. Such a specific, hierarchical problem is that of coordination considered mathematically in detail in Part TI. Two additional comments are of interest in reference, in particular, to the detailed theory of two-level decision systems developed in Part 11. (i) Multilevel decision-making systems provide new types of models for the study of physiological problems. As an illustration, we can mention the problem of the eye-tracking, oculomotor, system. It is well established that the eye-tracking process has two distinct modes, continuous and saccadic, each of which can be used separately, or in combination if needed. To explain the entire range of possible behaviors, a two-level model is called for in which
3. Hierarchical Order of Structures in Nature
33
the second level, consisting of a coordinating unit, decides which of the two first-level control modes will be used, and in which way. There is reason to believe that the introduction of coordination models for modeling biological systems will have an impact not unlike the introduction of feedback control models. Some other studies in the cardiovascular area are also in progress. (ii) Coordination (in particular, the coordination principles) provides a possible mechanismfor achievingintegration. Namely, in a properly structured two-level system, the effort needed to achieve the overall goal is divided among the units on different levels, so that no unit is “in charge” of achieving the ultimate goal, yet the goal of each individual unit is such that the overall goal is achieved whenever each unit is functioning properly. Consider a twolevel system with n units on the first level, each having its own goal which depends on a coordination parameter specified by the coordinator. The coordinator has a goal different from the overall goal and selects the coordination parameter so as to promote its own goal. If the relationship between these goals is consistent, such a system might achieve the overall goal. From the overall viewpoint it is quite appropriate to consider that the entire system is pursing the overall goal, yet any attempt to find a unit in the system which has the task of achieving this goal will be frustrated. The integrated behavior is due to the action of the coordinator and his pursuance of his own goal. It is through coordination that the integration is achieved. Understanding integrative mechanisms involves finding the coordinator’s goal. One note of caution is appropriate here. We are talking about a coordinator and coordination process in the functional sense; a structurally separate coordinating unit is not necessary.
Chapter 11
CONCEPTUALIZATION
In this book we are interested in initiating a mathematical theory of multilevel systems. To that end we shall identify in this chapter the structural concepts which will be the subject of the mathematical studies. In view of the vastness of the subject of hierarchies and the limitations of our work, we have the following specific objectives in this chapter: (i) introduce some basic concepts for classification and studies of hierarchical systems in general; (ii) provide a conceptual foundation for the problem of coordination, which is the subject of extensive mathematical development in Part 11; and (iii) indicate some features of hierarchical systems which make them attractive for use in man-made systems and offer some explanation why they are so widespread in nature. 1. WHAT IS A MULTILEVEL, HIERARCHICAL STRUCTURE?
The concept of a multilevel, hierarchical structure cannot be defined by a short succinct statement. A glance through the cases presented in Chap. I can convince one that an all-embracing definition would amount to enumeration of possible alternatives. We therefore answer the question by pointing out some of the essential characteristics which every hierarchy has, in particular: vertical arrangement of subsystems which comprise the overall system, priority of action or right of intervention of the higher level subsystems, and dependence of the higher level subsystems upon actual performance of the lower levels. 34
.
1. What is a Multilevel, Hierarchical Structure ?
35
Vertical Arrangement Any hierarchy one considers contains a vertical arrangement of subsystems, in that the overall system is viewed as consisting of a family of interacting subsystems as shown in Fig. 2.1. By the term “ system ’’ or “ subsystem” we
1 I
Level n subsystem
-nI
I
output I ’
A
1
0
I
I
I
I I
I I
L - - - - - - - - - - - - - - - - - - - ~
FIG.2.1 . Vertical interaction between levels of a hierarchy.
simply mean a transformation of the input data into outputs; the transformation can be either dynamic evolving in actual time, an “on-line system,” or it can be a problem-solving procedure, in which case the decomposition is conceptual, in that each block represents an operation to be performed, not necessarily simultaneously, with the operations of other blocks, an “ of-line
36
II. Conceptualization
system.” Examples of both types of system will be given later. Both inputs and outputs can be distributed to all levels, although most often the exchange with the environment is taken place on the lower (or lowest) level. When talking about vertical arrangement, we refer to higher and lower level units, with the obvious interpretation. Let us also indicate that the interaction between levels does not have to involve only the adjacent levels, as shown for simplicity in Fig. 2.1, although, to a degree, this depends on what one considers as subsystems on a given level.
Right of Intervention
The operation of a subsystem on any level is influenced directly and explicitly from the higher levels, most often from the immediately superceding level. This influence is binding on the lower levels and reflects a priority of importance in the actions and goals of the higher levels; this influence will be referred to as intervention. In on-line systems, intervention usually appears in the form of changing parameters in the lower level subsystems. In off-line applications, intervention implies a sequential procedure in arriving at the solutions on different levels : the problem (or solution algorithm) on any level depends upon the solution of the problem on the higher level, in the sense that there exists an unspecified parameter in the former which represents (or is obtained by a transformation from) the solution of the latter; the problem on the lower level is well defined only after the higher level problem is solved. To emphasize the right of intervention, we refer to the higher level subsystems as supremal units while the subsystems on the lower levels are termed injimal units.
Performance Interdependence
Although priority of action is oriented downward in a command fashion, the success of the overall system, and indeed of the units on any level, depends upon the performance of all units in the system. Since priority of action tacitly assumes that intervention precedes the actions of the infimal units, success of a supremal unit depends upon that action, or rather the resulting performance of the infimal units. Performance can be viewed, therefore, as a feedback and a response to the intervention; it is orineted upward, as shown in Fig. 2.1. This performance interdependence is particularly apparent in the already mentioned case when exchange with the environment occurs primarily or exclusively on the lower levels of the system.
2. Basic Types of Hierarchies
37
2. BASIC TYPES OF HIERARCHIES
We introduce here three types of hierarchical systems which in a sense represents a classification of hierarchical systems. At this stage of development, this is an important task, since it leads to clarification of the issues by bringing more precision to the statements about systems and the problems involved. As is almost often the case in science, the classification is not to be understood as a partition; it emphasizes the differences, rather than excluding the possibilities of a system belonging to more than one class. We introduce three notions of levels: (i) the level of description or abtraction, (ii) the level of decision complexity, and (iii) the organizational level. To distinguish between these notions, we use the terms “ strata,” “ layers,” and “ echelons,” respectively. The term “ level ” is reserved as a generic term referring to any of these notions when there is no need to emphasize the distinction. We point out that, in the description of an actual hierarchical system, all three notions might be involved; the case where only one notion is applicable is an exception rather than the rule
Strata: Levels of Description or Abstraction Truly complex systems almost, by definition, evade complete and detailed descriptions. The dilemma is basically one between the simplicity in description, one of the prerequisites for understanding, and the need to take into account a complex system’s numerous behavioral (i.e., input-output) aspects. A resolution of this dilemma is sought in a hierarchical description. One describes the system by a family of models each concerned with the behavior of the system as viewed from a different level of abstraction. For each level, there is a set or relevant features and variables, laws and principles in terms of which the system’s behavior is described. For such a hierarchical description to be effective, it is necessary that the functioning on any level be as independent of the functioning on other levels as possible. To distinguish this concept of hierarchy from others, we use the term “ stratijied system ” or “ stratijied description.” The levels of abstraction involved in a stratified description will be referred to as strata. Examples of stratified descriptions in the natural sciences are abundant. The hierarchical order of structures in sciences discussed in Chap. I is precisely such a case. On each strata in the hierarchy of structures, there is a different set of relevant variables, which allow the study to be confined largely to one stratum only. The decoupling of strata enables the study of systems behavior to be conducted more efficiently and in considerable detail ;
II. Conceptualization
38
however, a complete decoupling cannot be fully justified, and neglecting the cross-strata interdependence cannot but lead to an incomplete understanding of the systems behavior as a whole. Indeed, a restriction to, say, biologicaltype inquiries represents in itself an isolation, since apparently the system under consideration (man!) can also be described on the stratum of concern to chemistry or physics, on the one hand, and ecology and social science on the other. For illustration, let us give one or two examples of man-made systems which need a stratified description. Consider modeling an electronic computer. Its functioning is customarily described on at least two strata [Fig. 2.21:
Stratum : 2 Mothematical operotions :
Input
I output
I
Input Physical operation
I
I
FIG.2.2 A two-strata representation of an electronic computer.
one in terms of physical laws describing the functioning and interconnection of the constituent parts, and the other in terms of processing abstract nonphysical entities, such as digits or information sequences. On the stratum of physical laws, one is concerned with the proper functioning of the components. On the stratum of information-processing, one is concerned with such problems as computation, programming, etc., and the underlying physical basis of operetion does not come explicitly into consideration. Of course, a description of the system or some of its subsystems on other than these two strata may also be of interest; the stratum of atomic physics is of interest for some component design, and the so-called systems stratum is of interest for problems such as time-sharing.
2. Basic Types of Hierarchies
39
Other man-made stratified systems can be found in industrial automation, as discussed in Chap. I. A completely automated industrial plant is usually modeled on three strata: the physical processing of material and energy, the control and processing of information, and the economical operation in reference to efficiency and profitability (Fig. 2.3). Notice that on any of these
0
Performance feedback
Intervention
FIG.2.3 A three-strata diagram of an automated industrial operation.
Stratum 2: Information processing and control r
Performonce feedback
Control 0
Stratum I: Physical processes Material
Finished products
three strata one is dealing with one and the same item, the basic physical product. On the first stratum it is viewed as a physical object to be changed in accordance with the physical laws; on the second stratum it is viewed as a variable to be controlled and manipulated; and on the third stratum it is viewed as an economic commodity. There is a different description, a different model for each of these views of the system; however, the system is, of course, one and the same. How a single system can be described by a hierarchy of models is nicely illustrated by a machine which produces a spoken literary composition [ 181. There is only one output of the system: the actual, physical, utterance of a literary text. The operation of the system can be viewed, as shown in Fig. 2.4, in terms of at least four strata. The first stratum entails the generation of letters and describes the system as a sound producing machine. The second stratum deals with the composition of letters into sequences that are acceptable as words in the grammar of a given language: the system is a word producing machine. On the third stratum the system is considered, in reference to the construction of sentences, to express certain statements according to the given syntax and semantic rules. Finally, on the fourth stratum the system is assessed by some literary aestetic standards as to the quality and literary value of the composition.
II. Conceptualization
40 I
I I I
Text generating mochine
I
I
I I
I
I
I
I
I
I II
I Slrotum 3 :Sentences
I I I
I
I
Stratum 2 ' Words I
I
I
Strotum I : Sound I
FIG.2.4 A four-strata diagram of a text generating machine.
Many other examples of stratified systems can easily be given. The preceding examples suffice, however, to illustrate some of the general characteristics of a stratified description of a system. Selection of strata, in terms of which a given system is described, depends upon the observer, his knowledge and interest in the operation of the system, although for many systems there are some strata which appear as natural or inherent. This was already illustrated. In the electronic computer case, if one is not familiar with the purpose of the system as a calculating machine or how the machine can be used as such, he might be restricted to the stratum of physical laws; he can develop (given sufficient time) a very detailed and accurate description of the system without being aware of the calculating aspects of the machine. Conversely, one can have an informationprocessing model without any knowledge of the physical laws involved. If one does not know the language of the text producing machine, the best one can do is to recognize the sound producing stratum of that machine; if one knows the language but has no literary standards or background to evaluate the meaning and composition of the text, the higher strata will be lost.
2. Basic Types of Hierarchies
41
In general, stratification is a matter of interpretating the system’s operations. The context in which the system is observed and used determines which strata will be primary, and furthermore which will be used at all. However, there are almost always some other strata which, although inherent, might not be of interest. Take, for example, the text producing machine: between the stratum of sentences and the stratum of composition one might introduce a stratum of stylistic aspects of the composition. Similarly, by observing special sequences of calculations of a computing machine, one can recognize the strata of special tasks which the machine is programmed to perform. A special purpose computer may be designed to perform special functions such as process control, routine engineering, or business data processing and then modeled as an appropriate single stratum system such as a digital controller or guidance correction apparatus.
Contexts in which the operation of a system on diflerent strata is described are not, in general, mutually related; the principles or laws used to characterize the system on any stratum cannot generally be derived from the principles used on other strata. The principles of computation or programming are not derivable from the physical laws governing the behavior of the computer on the lower strata, and vice versa. Similarly, the economic principles and physical laws governing the process operation are not related, and the stratified description results precisely from the application of these two different viewpoints which describe one and the same system. There exists an asymmetrical interdependence between the functioning of a system on diferent strata. The requirements for proper functioning of a system on any stratum appear as conditions or constraints on the operations on the lower strata. This is in accordance with the priority of action postulate. For example, if a computer is to perform certain calculations the physical process has to evolve in a fashion suitable to carry out the necessary arithmetic and other operations. The generation of a given text depends on the operation of the system on the sound producing strakm, but which sequence of sounds is to be produced is determined by the requirements of the higher strata. Evolution of the actual process is specified by the desired behavior on the upper stratum; for a proper functioning of the system on a given stratum, all the lower strata have to function correctly. This is in accordance with the performance feedback characteristics of hierarchies. Each stratum has its own set of terms, concepts, and principles. What is an object on a given stratum becomes a relation on a lower stratum; what is an element becomes a set; a subsystem on a given stratum is a system on the stratum below. This relationship between strata is pictorially represented in Fig. 2.5. On any given stratum one studies the behavior of the respective
42
II. Conceptualization
I
I
Stratum i + 3
.-c c
0 0 c
n a *l 0 al
.0
c
0
0
e z
FIG.2.5 Relationship between strata: A system on a given stratum is a subsystem on the next higher stratum.
systems in terms of their internal operation and evolution, while the study of how these systems interact so as to form a higher stratum system is studied on that higher stratum. This is important since it indicates that the studies on the lower strata are not necessarily better, more fundamental, or more basic than those on higher strata. On the lower strata one concentrates on the operation of the subsystems, leaving the study of their interrelationships for the higher strata. This would not be so if on the lower strata one considers the entire system just as on the higher strata; this, however, is usually not the case, since the principles and methodology on any stratum in general are not suitable for this procedure. It should be noted that this object-system relationship between descriptions on various strata leads to a hierarchy of appropriate description languages. Since for each stratum there is given a different set of concepts and terms to be used for the description of the system on that stratum, there exists, in general, a different language. These languages form again a hierarchy, with a semantic relation between two consecutive members o f the hierarchy. Understanding of a system increases by crossing the strata: in moving down the hierarchy, one obtains a more detailed explanation, while in moving up the hierarchy, one obtains a deeper understanding of its significance. It
2. Basic Types of Hierarchies
43
can be argued that explanation in terms of the elements on the same stratum is merely a description, while for proper functional understanding the description should be given in terms of the elements on lower, more detailed, strata. As Bradley [19] points out: It is a serial process: The biologist explains transmission of heredity in terms of DNA replication; the biochemist explains the replication in terms of the formation of complementary nucleotide base pairs; the chemist explains base pairing in terms of hydrogen bonding; the molecular physicist explains hydrogen bonds in terms of intermolecular potential functions; the quantum mechanician explains potential functions in terms of the wave equation.
One is able, by referring to lower strata, to explain more precisely and in more detail how the system functions so as to carry out a certain operation. On the other hand, by moving up the hierarchy, the description becomes broader and refers to larger subsystems and longer periods of time. The meaning and significance of the subsystems’ operation is better interpreted in such a broader context. In the example of the text producing machine, one comprehends the significance of the system’s operation only on the highest strata of a literary composition. In summary, it can be stated that for a proper understanding of complex systems, the hierarchical approach through stratified modeling is quite fundamental. Initially, one can confine his interest to one stratum, depending upon his interest and knowledge, and then increase his understanding of the system’s significance or functioning by moving, respectively, up or down the hierarchy. Selection of the initial stratum is also affected by the simplicity of the description on that stratum. Layers : Levels of Decision Complexity
Another concept of hierarchy appears in the context of a complex decisionmaking process. There are two extremely trivial (so trivial that they are too often forgotten) but profoundly important features of almost any real-life decision-making situation : (i) When the decision time comes, the making and implementation of a decision cannot be postponed; any postponement of action simply means that no action or change of action has been selected as the most preferable among the alternatives. (ii) The uncertainties regarding the consequences of implementing various alternative actions, and the lack of suficierat knowledge of the relationships involved, prevents a complete formal description of the situation which is needed for a rational selection of a course of action. These two factors result in the fundamental dilemma of decision making: on one hand, there is a need to act without delay, while on the other, there is an equally great need to understand the situation better. In
44
II. Conceptualization
complex decision-making situations, the resolution of this dilemma is sought in a hierarchical approach. Essentially, one defines a family of decision problems whose solution is attempted in a sequential manner, in the sense that the solution of any problem in the sequence determines and fixes some parameters in the subsequent problem, so that the latter is completely specified, and its solution can be attempted; the solution of the original problem is achieved when all subproblems are solved. This arrangement is represented diagramatically in Fig. 2.6. Each block represents a decision-making unit.
FIG. 2.6 Multilayer hierarchy of a decision-making system.
The output of a unit, say D 2 , represents a solution, or consequence of a solution, of a decision problem which depends upon a parameter fixed by the input x 2 , which in turn is the output of a unit on a higher level. In such a way, the solution of a complex decision problem is substitut'ed by the solution of a family of sequentially arranged, simpler subproblems, so that the solution of all subproblems in the family implies the solution of the original problem. Such a hierarchy is referred to as a hierarchy of decision layers, and the entire decision making system, represented by D in Fig. 2.6, is referred to as a multilayer (decision) system. It is easy to give examples from everyday life of complex decision situations which are approached in a multilayer fashion. Indeed, personal goals
2. Basic Types of Hierarchies
45
or objectives are, as a rule, vague and have to be translated into what might be called operational objectives, which then provide a basis for the selection of a concrete course of action. For example, the goal of an individual might be to achieve happiness or a certain level of satisfaction, but that vague goal has to be translated into objectives which lead to some specific actions. An objective has to be selected which in turn leads to subgoals; very often only after a subgoal is achieved is one in the position to evaluate whether the original goal is being approached. Consider now two examples of automated, man-made, decision systems (one from the field of artificial intelligence and the second from the industrial control area) in which the layer hierarchy appears more explicitly. In the heuristic programming approach to theorem proving by computer proposed by Newel1 et al. [20], the process of proving a theorem in a specified branch of mathematics consists of the following: The statement of the theorem is represented as an equality between two mathematical expressions which can be changed by applying transformations from a given set of allowable transformations. Proving the theorem consists of changing the expressions on both sides of the equality until they become the same; in other words, the equality is transformed into identity. The process of transforming equality into identity is arranged hierarchically. For example in the propositional calculus a theorem is presented by the equality, say R A ( - P + Q ) = < Q v P ) h R, and the proof of the theorem consists of exhibiting a sequence of legal transformations that will transform the left and right sides so as to be identical. The general strategy of theorem-proving is represented by a multilayer system as shown in Fig. 2.7. The layers are defined in terms of differences which might exist between various expressions in the propositional calculus; the following differences are recognized :
A V denotes that there exists a variable in one expression which does not appear in another; AN denotes that a variable appears a different number of times; A T denotes that the difference is in the negation of some variables; AC denotes that different connectives are used; AG denotes that the grouping is different; and A P denotes that the position of variables is different. The differences are then ordered according to an assumed priority and assigned to separate decision layers starting at the top layer, which is concerned with the most important difference. The task of the unit on each decision layer is to eliminate the respective difference. Each decision unit has a set of transformations which are deemed useful in eliminating the
II. Conceptualization
46
corresponding difference. The theorem-proving process consists, then, in presenting a theorem to the topmost decision unit, which after eliminating the respective difference presents the modified equality to the next unit. If all layers are successful, the equality is transformed into identity, and the
Legal transformalions reducing AV Modified equality
I I
AC
AG
FIG.2.7 Multilayer strategy in theorem proving.
theorem is proven. I t should be noticed that the diagram in Fig. 2.7 indicates only the most essential structure of a theorem-proving system. The complete system is much more complex: it provides for moving up and down the hierarchy to avoid being deadlocked if no solution is reached on any layer in a given time period; the equality can be fed back to one of the preceding layers, or it can temporarily be passed forward to the next layer, with the provision that it will be returned to the higher layers if needed. At any rate, even this simplified description illustrates the need for a multilayer decision structure in complex decision situations. Another example is provided by what we term the functional hierarchy in decision making or control. This hierarchy emerges naturally, in reference to three essential aspects of a decision problem under true uncertainty: (i) the
47
2. Busic Types of Hierarchies
selection of strategies to be used in the solution process, (ii) the reduction or elimination of uncertainties, and (iii) the search for a preferable or acceptable course of action under prespecified conditions. The functional hierarchy as shown in Fig. 2.8 contains three layers:
I
(PG learning strategy)
I I I I
FIG.2.8 Functional multilayer decision hierarchy.
I
I
Learning and adoption
I I
(i) Selection Layer: The task of this layer is to select the course of action,
m. The decision unit on this layer accepts the outside data (information) and applies an algorithm (as specified by the higher layers) to derive a course of action. The algorithm might be defined directly as a solution map T, giving a solution for any set of initial data or indirectly by means of a search process. For example, suppose there is given an outcome function P and an evaluation function G, and the selection of the action, say A, is based on the evaluation function G in reference to P. Using set-theoretic (general systems theory) specification, the outcome function P is a mapping P : M x U Y where M is the set of alternative actions, Y is the set of outcomes, while U is the set of uncertainties which can reflect in an equivalent fashion all the ignorance in relationship between the action m and the outcome y . Similarly, the evaluation function G is a mapping G : M x Y + V , where V is the set of values that can be associated with the performance of the system. If U has a single element or is void, meaning that there are no uncertainties regarding the outcome for a given m, the selection can be based on optimization: find A in
48
ZZ. Conceptualization
M , such that the value G = G(&,P(&)) is smaller than the value v = G(m, P(m)) for any other action m in M . If U is a larger set, some other procedures have to be devised for the selection of the appropriate action, and other mappings in addition to P and G might have to be introduced. At any rate, to define the selection problem for the first layer, it is necessary to specify the uncertainty set U and the necessary relationships P, G, etc. These are provided by the units on the higher layers.
(ii) Learning or Adaptation Layer: The task of this layer is to specify the uncertainty set U used by the selection layer. It should be noticed that the uncertainty set U is viewed here as encompassing all the ignorance about the behavior of the system and as reflecting all the hypotheses about the possible sources and types of uncertainties. U is derived, of course, on the basis of observations and communication. The inherent goal of the second-layer activity is to reduce the uncertainty set U.If the system and environment are stationary, the uncertainty set can be reduced to a unit set which corresponds to perfect learning, as in a controlled experimentation. However, it should be emphasized that U represents the uncertainties as assumed by the decision system, rather than as they actually are. The second layer might very well need to change U altogether, increasing it if necessary, and thereby acknowledging that some of the basic hypotheses were incorrect. Still, the overriding goal of the second or learning layer is to reduce the uncertainty set as much as possible, and in this way to simplify the job of the selection layer. (iii) Self-organizing Layer: This layer must select the structure, functions, and strategies which are used on the lower layers, so that an overall goal (usually defined in terms which cannot be easily made operational) is being pursued as closely as possible. It can change the functions Y and G on the first layer if the overall goal is not accomplished, or it can change the learning strategy used on the second layer if the estimation of uncertainties turns out not to be satisfactory. The automated industrial processes described in Chap. I offer good examples of multilayer hierarchies. The avid objective of introducing automation is to maximize profit, improve efficiency, and minimize cost of operation. Such grandiose objectives cannot be translated into concrete actions in the face of changing economic and technoIogica1 conditions. In addition, the investment and operating cost of the automated system itself has to be taken into account, as well as the technological constraints such as available hardware and prevailing engineering practice. All this leads to a hierarchical, multilayer structuring of a complex automated system. It should be noted that the functional hierarchy as depicted in Fig. 2.8 is based on the conceptual recognition of the essential functions in a complex
49
2. Basic Types of Hierarchies
decision system. It provides only a starting point for a rational approach to assign proper functions to different layers. In practice, a function on any one layer can be implemented by further decomposition. For example, in industrial automation, the selection layer is usually realized by a regulatory or direct control and optimization. The task of the regulatory control is to keep (in the face of inherent variations) the respective variables near prespecified values, which in turn are determined as so-called set-points by optimization.
Multiechelon Systems:Organizational Hierarchies For this notion of hierarchy, it is necessary that: (i) the system consist of a family of interacting subsystems which are recognized explicitly, (ii) some of the subsystems be defined as decision (making) units, and (iii) the decision units be arranged hierarchically, in the sense that some of them are influenced or controlled by other decision units. A diagram of a system of this type is given in Fig. 2.9. A level in such a A I
I I
\
\ \
Process
FIG.2.9 Multilevel organizational hierarchy; multiechelon system.
50
II. Conceptualization
system is called an echelon. These systems will be referred to also as multiechelon or multilevel, multigoal, because various decision units comprising the system have, in general, conflicting goals. This conflict appears not only as the result of the evolution and composition of the system, but can also be shown to be necessary (to a degree and in a given sense) for efficient operation of the overall system. The most prominent and generic example of this type of system is formal human organizations. With this in mind, it is hard to overemphasize the importance of these types of hierarchies. Many examples of multilevel, multigoal hierarchical systems can be found in biology, as well as in other areas. One important characteristic of multilevel, multigoal systems which sets them apart from conceptually simpler (although technically quite complex) multivariable decision systems should be emphasized. Namely, it is in the very nature of the multilevel, multigoal systems that the higher level units condition but do not completely control the goal-seeking activities of the lowerlevel units. The lower level decision units have to be given some freedom of action to select their own decision variables; these decisions might be, but are not necessarily, the ones which the higher level unit would select. Such a freedom of action is noticeable in any social or biological multilevel system. In the man-made systems, use of the resources available for decision making can be economized only if such a freedom of action is provided at the lower levels. It an be shown that it is egsential f o r the epectiue usage of the multilevel structure that the decision units be given a freedom of action; a suitable division of decision-making effort among the units on different levels should be established. Only then can the existence of the hierarchy be justified. This reasoning leads to a conceptually important classification of decisionmaking systems; with respect to the hierarchical arrangements of the decision units comprising a system (Fig. 2.10), one can recognize the following categories of decision-making systems : single-level, single-goal systems; singlelevel, multigoal systems; multilevel, multigoal systems. In the first class, a goal is defined for the overall system, and all decision variables are selected so as to satisfy this goal. Technically, the solution of the decision problem that satisfies the overall goal can be quite complex, since the problem is multivariable, and optimization as well as prediction can be involved. Yet, the conceptual simplicity of the single-level, single-goal system should be noticed; in particular, the absence of conflict within the boundaries of the system. A system in the class of single-level, multigoal systems consists of a family of decision units, each with its own goal. The goals of the system are not necessarily conflicting: a subfamily of decision units can form a coalition.
2. Basic Types of Hierarchies
51
3-
Decision-making unit
Single-level single-goal system
I I
I
r
I
Process p
Single-level multigoal system
Multilevel multigoal system
FIG.2.10 A classification of decision-making (control) systems.
There might, however, be a conflict between the decision units, but none of them has the power to resolve the conflict. Finally, the class of multilevel, multigoal systems is characterized by the existence of the hierarchical relation between the decision units of the systems. The existence of a supremal (top level) unit is the principal characteristic of these systems; the decision problem for the supremal unit is a principal problem specific for multilevel systems. Keeping in mind the above classification, it becomes apparent that a new theory is needed to deal with the multilevel systems. It might be argued that control theory, as currently conceived, is dealing with single-level, singlegoal (although multivariable and rather complex) decision problems, while for the single-level, multigoal systems, we' have game theory and the theory of teams. None of these theories is complete, and much investigation is needed, for example, to develop practical methods (i.e., numerical algorithims) for the control of single-level, single-goal systems or to understand the nature and effects of the conflict in single-level, multigoal systems. However, the framework for developing a theory for at least certain classes of single-level systems has already been established in the course of the extensive research over the past two or three decades. Apparently a novel approach is needed to deal with the hierarchical, multiechelon systems; a framework for the development of a mathematical theory of such systems is one of the prime objectives of this book.
52
II. Conceptualization
Relationships between the Different Notions of a Level It is beneficial to make a clear distinction as to which notion of level one is using when describing a hierarchical system, since the three notions are introduced with a different purpose in mind: the concept of strata is introduced for the modeling purpose; the concept of layers is introduced in reference to the vertical decomposition of a decision problem into subproblems ; the concept of echelons refers to the mutual relationship between decision-units comprising a system. The distinction can, perhaps, be best illustrated by considering the interplay of these concepts in the description of a multilevel system. We shall consider three cases:
Design of Multiechelon System Assume that we are to build a multiechelon (organizational-type) system. The first problem in designing such a system consists in the assignment of the tasks or roles which various levels or individual units have to perform. A rational starting point is provided by taking the “total system viewpoint ” of the entire system and the task it is supposed to perform, and using the strata and layers concepts of hierarchies. A stratified model of the entire system is derived on one hand, while the overall systems task is decomposed into layers on the other. The tasks of the units comprising the multiechelon system are then defined in reference to models and decision problems on appropriate strata and layers (Fig. 2.11). In this connection it should be stressed again that there is no one-to-one correspondence between strata, echelons, and layers. Tasks for more than one echelon can be defined by using the model from the same stratum, while the decision problem on a given layer can be distributed through several echelons ; furthermore, the task for an echelon can contain elements of the problems from more than one decision layer.
Multilayer Units in a Multiechelon System The concept of a multilayer system is presented in reference to a given decision problem which does not have to be the overall problem in any sense, but can be simply the decision problem of a unit which is a member of a still larger system. As an example, a two-level system is shown in Fig. 2.12, in which each of the decision units is using the multilayer approach to solve its own, local subproblem. Multilayer hierarchies are imbedded, so to speak, in a multiechelon system.
Complex Decision Units in a Multilayer System Consider a multilayer decision system formed in reference to a family of subproblems whose solutions imply the solution of the original problem. Each of the subproblems can in itself be complex, and it might be found
Stratification of the overall system
Decision layers of the overall task
Self-organization
Learning and adoption
Control
FIG.2.11 Vertical task assignment for an organizational hierarchy.
r-------
- - - -- - -
FIG.2.12 Multilayer hierarchies within decision units of a multiechelon systems. 53
54
II. Conceptualization
advantageous to organize its solution by a multilayer approach (say, using a functional hierarchy) or even to form a separate multiechelon system charged with the solution of the particular subproblem, if the resources and time limitations favor such an approach. This is illustrated in Fig. 2.13. Specific examples of this kind can be found in corporate planning and structuring.
FIG.2.13 Decision units of a multilayer hierarchy presented as multilayer or multiechelon hierarchies.
In spite of the differences there are features common to all three concepts of hierarchies. Some of the most prominent have already been presented in Sect. 1 of this chapter. They were primarily structural relationships between the subsystems. In view of the more detailed discussion of hierarchies in this chapter, we can point out some of the additional common features which refer to the tasks and roles of the subsystems. (i) A Higher Level Unit is Concerned with a Larger Portion or Broader Aspects of the Overall Systems Behavior. In the multiechelon hierarchy, this is reflected in the fact that a higher level unit is supremal to two or more units and the decision of the supremal coordinates the infimals in accordance with a goal (objective) defined over the domains of all the units infimal to it. For the layer concept, this shows up in the higher level units' concern of the systems behavior over a longer period of time. To collect the information for the reduction of the uncertainty set, the learning layer should observe
2. Basic Types of Hierarchies
55
several decision periods of the first layer. To determine how to change the structure of the first and second layer decision strategies, the third, selforganizing layer has to observe the infimal layers for an even longer period of time: to evaluate the performance of the learning strategy the latter has to be tried in at least a few instances. Similary for the strata concept, the system on any level is constructed from subsystems on the levels below, and therefore the higher strata is concerned with a broader aspect of the overall systems behavior. (ii) The Decision Period of a Higher Level Unit is Longer than That of Lower Units. For the layer and strata concepts this is quite apparent. This holds also for the echelon concept. Namely, to evaluate the effect of coordination, the suprema1 unit cannot act more often than the infimal units, whose behavior is conditioned by the coordination. (iii) A Higher Level Unit is Concerned with the Slower Aspects of the Overall Systems Behavior. This holds for all three types of levels and almost follows from the fact that a higher level unit is concerned with the broader aspects of the overall systems behavior and has a longer decision period. The higher levels cannot respond to variations in either the environment or the process itself, which are faster than the variations of concern to the lower levels, since the latter are reacting faster and are concerned with more particular, local, changes. This characteristic is of particular interest for the formation of multilayer hierarchies [21]. Assume that all outside effects, for the system viewed as a whole, are described in terms of frequency spectra of their time variations. Let F represent the range of frequencies for the entire set. One can partition F into subsets Fl, . ..,F,, and form a multilayer hierarchy so that the unit on any layer, say the ith layer, has the task of responding to the external influences whose spectra are in the range F i . The units on the ith layer then proceed on the assumption that the upper and lower layers are functioning properly and have taken care of all outside effects except those within the frequency range F i . (iv) Descriptions and Problems on Higher Levels are less Structured, with More Uncertainties, and More Dificult to Formalize Quantitatively. Decision problems on the higher levels can be considered as more complex. Of course, an approximation can be used to arrive at the solution of a higher level problem, but accuracy is then reduced, and one has to be cautious in interpreting the results. In general, for any level there is a specific set of techniques suitable for the solution of the respective tasks. For example, for each layer in the multilayer hierarchy, there is a different set of methods and techniques: on the selection layer, feedback control and numerical optimization methods are used; on the adaptation layer, statistical or pattern recognition techniques
ZI. Conceptualization
56
prevail; on the self-organizing layer, one resorts to heuristics. The task of the highest layer cannot be defined in such a way that a simple numerical solution can be attempted, and one applies what might be called the “management by exception ” approach : the overall performance is evaluated, and change is made only if the performance deteriorates to a degree that warrants such a change; the change, however, has, in general, no a priori guarantee of really improving the performance. Time-complexity relationships between the levels can be also traced to the limitation of the decision-making capacity of the units used in building a system. Assume that one has to build a system to perform a task which is beyond the capacity of any available decision unit. One must then resort to a multiechelon system, so that the units on the higher echelons are concerned with the broader aspects of the task and, therefore, have a more complex decision problem than those on the lower levels. But, having limited decisionmaking capacity causes the higher echelon units to take a longer time in arriving at their decisions. 3. INTERLEVEL DEPENDENCE AND COORDINATION
As mentioned in the introduction, the progress in developing a theory of hierarchical systems is expected to be made by steps in which some specific problems characteristic for multilevel systems are solved. Such a characteristically hierarchical problem with which we shall be concerned explicitly in this book is the interlevel relationship and, in particular, coordination. Consider a two-level decision system as shown in Fig. 2.14 with one suprema1 and n infimal units. Such a system is of special interest for a theory of multilevel systems since (i) it is the simplest type of system that exhibits the most essential characteristic of a multilevel system, and (ii) more complex, Suprema1 decision unit suprema1
lnfimal decision units
Process
FIG.2.14 Two-level organizational hierarchy.
57
3. Interlevel Dependence and Coordination
manylevel systems can be built using two-level systems in a modular fashion. We shall investigate, therefore, the interlevel relationship in the context of a two-level system. The situation here is reminiscent of the relation between two-person, zero-sum games and more general n-person games ; a zero-sum, two-person game is certainly but a special case in the general class of games, yet it contains sufficiently important ingredients to provide the basis for the introduction of some basic concepts and the development of a framework for the theory of games as such. With an analogous objective in mind, we concentrate on the theory of two-level systems as the first step toward a theory of multilevel systems in general.
Level Interdependencies The relationship between the supremal unit and one of the infimal units is such that the action (success) of one depends upon that of the other, as illustrated in Fig. 2.15. Since both are decision-making units, this means that, Suprema1 decision unit lnfluente of infimal decision unit
lnfimal decision units 1
FIG.2.15 Interaction between the supremal decision unit and an infimal decision unit.
in general, the decision problem of the infimal unit depends upon the action of the supremal unit as represented by a parameter; conversely, the decision problem of the supremal unit depends upon the action (or response!) of the infimal unit. Apparently, there is a deadlock here. This dilemma is resolved by the priority of action of the supremal unit. However, in general, one has to take into account the dynamics involved: the relation between the supremal and infimals units is dynamic and changes in time.
Intervention Times In reference to the infimal decision time, the time instant when the infimal decision is made and its implementation has begun, there are basically two time instants when the supremal unit can communicate its coordinating decision to the infimal units. The supremal unit can attempt to coordinate the infimal units before they arrive at their own decisions. We refer to this type of action as pre-decision
58
ZZ. Conceptualization
intervention, This is a fundamental and mandatory type of coordination since, due to the priority of action, the infimal decision problems are not well defined until the coordination terms are specified. Pre-decision intervention is based on a prediction of the behavior of the overall system, as well as the environment. Furthermore, the supremal unit in pre-decision intervention specifies the infimal performance functions and, in this way, indicates how the infimal units will share in the success of the overall system. After the infimal units apply their decision, and a certain amount of time has elapsed (e.g., at the end of the so-called decision period), the supremal must again communicate with the infimals. The supremal unit should correct the preceding instructions sent to the infimal units if the assumptions on which these instructions are based turn out to be incorrect. Furthermore, at the end of the decision period the supremal unit has either to confirm or change the pre-decision intervention plans for sharing in the success of the overall system. These actions of the supremal unit are referred to as postdecision intervention or, more descriptively, correction intervention and reward intervention. When considering post-decision intervention, it should be kept in mind that the objective is not to improve the goal-seeking of the infimal units but to further the goal of the supremal unit. The objective of post-decision intervention, in general, is not to correct the assumptions which the infimal units have about the behavior of the rest of the system or to give them a larger share in goal-achievement as indicated in the pre-decision intervention, byt rather to influence the infimal units so as to improve the behavior of the overall system as viewed by the supremal unit. In this, the supremal unit has also to consider the behavior of the entire system as dynamic, and to take into account the consequences of post-decision intervention on future performance. Namely, in a dynamic system, post-decision intervention represents the first phase of the pre-decision intervention for the subsequent decision period. The discrepancy between the pre-decision and post-decision intervention has to be viewed in this light. Too large a discrepancy might negatively affect the future performance of the infimal units. In the subsequent development, we shall be concerned solely with the pre-decision intervention (except for some occasional remarks).
Relationship of the Suprema1 Unit There are two signals connecting the supremal unit with the infimal units. The downward signal representing intervention specifies the decision problems for the infimal units, while the upward signal furnishes information about the lower level to the supremal unit.
3. Interlevel Dependence and Coordination
59
Znter vention
Due to priority of action, the supremal unit has the broad responsibilities of first, instructing the infimal units how to proceed and second, influencing them to change their actions if needed. The first responsibility corresponds in the organization theory to “ the control-in-the-large” and involves the selection of rules and procedures to be followed under various anticipated circumstances. In our formalism this represents the problem of selecting the structure for the supremal-infimal relationship; we refer to this as the seZection of a coordination mode. The second responsibility corresponds in organization theory to “control-in-the-small” and involves adjusting rules so as to improve the performance. In our formalism this corresponds to the selection of the actual intervention or coordination variable; we refer to this broadly as coordination. How a given infimal unit will communicate with other infimal units and which aspects of the infimal decision problem are available for change, if needed, to improve the overall performance, determines a coordination mode. These two factors are interwoven and depend upon the type of problems which the infimal units and the overall system are set up to solve. We indicate some of the main categories, keeping in mind the analysis in Part 11. The relationship of a unit with others on the same level can be characterized by its action and the response of the rest of the system as it influences that unit. This influence is referred to as the interface input. The principal question is therefore how the given infimal unit will consider, or take into account, the interface input. In this respect the following options are available to the supremal unit. (i) Interaction Prediction Coordination: The supremal unit specifies the interface input, and the infimal units proceed to solve their local decision problems on the assumption that the interface input will be exactly as predicted by the supremal unit. (ii) Interaction Estimation Coordinations: The supremal unit specifies a range of values for the interface inputs, and the infimal units treat the interface inputs as disturbances which can assume any value in the given range. (iii) Interaction Decoupling Coordination: The infimal units treat the interface input as an additional decisionvariable ;they solve their decisionproblems as if the value of the interface input could be chosen at will. (iv) Load-Type Coordination: The infimal units recognize the existence of other decision units on the same level; the supremal unit provides the infimal units with a model of the relationship between its action and the response of the system.
60
II. Conceptualization
(v) Coalition-Type Coordination: The infimal units recognize the existence of other decision units on the same level; the supremal unit specifies what kind of communications are allowed between them. This leads to a coalition or a competitive (game-theoretic-type) relationship between the infimal units. A remark about the usefulness and effectiveness of the above approaches is appropriate. Obviously, the last approach is the most sophisticated and comes closest to the actual situation in human organizations. However, it is the most complicated and can lead to rather complex infimal decision problems. In the extreme, each infimal unit would have to solve the decision problems of all the other infimal units in the system. The efficiency of such an approach could, consequently, be extremely poor. One reason for using a multilevel structure is to utilize decision units whose decision-making capacity is insufficient t o solve the overall problem by defining subproblems, so that through group action the overall problem can be solved. The art of synthesizing a multilevel system consists precisely of defining simplified problems for the infimal and supremal units. The first three approaches apparently have the advantage of simplicity over the latter two. In Part I1 we shall investigate in considerable detail the first three modes. Their common characteristic is that the infimal units consider the interface inputs as signals; the rest of the system is considered simply as an environment, and no causal relationship is assumed between the action of the infimal units and the response of the system. For illustration, we shall briefly indicate below some issues which arise in conjunction with the other two modes. (i) To allow or to prohibit communication between the infimal units has been shown [22] to depend on how the infimal problems are defined. It was shown that when the infimal performances are linear functions, communication between the infimal units is detrimental from the overall systems viewpoint, while for other forms of infimal performances (say, quadratic) communication channels between the infimal units can be beneficial. It is important to emphasize that we are also speaking of the semantic aspects of communication; the supremal has to decide not only on establishing communication channels, but also what type of information is to be exchanged. The analysis indicates that excessive communication between units on the same level might have the same effect as a lack of communication and lead to deterioration of the overall performance. (ii) In an analysis [I41 of whether the infimal units can use a gametheoretic approach it has been shown that the supremal unit can construct rules of conduct so that use of such an approach on the first level results in the overall optimum. This has implications for the problem of implementing an optimal solution in the context of social systems. The supremal unit might know the optimal solution, yet it cannot impose its executions; rather, it has
3. Interlevel Dependence and Coordination
61
to establish the relationship between the infimal units so that they themselves arrive at such a solution; a mathematical analysis [I41has shown that this is indeed possible using a game-theoretic approach on the first level.
Coordination Coordination, as a complex decision problem in itself, has two aspects: the self-organizing aspect (changing the structure) and the control aspect (selecting the coordination input for the fixed structure). We assume that a coordination mode is already determined and that self-organization refers to the changes in the functions and relationships used in the coordination process. We refer to these changes as modijications. In a very broad sense, any decision problem is defined by a goal and an image of the decisionmaking situation; in this respect, there are two kinds of modifications: goal modijication and image modijication (for a given coordination mode!) For example, the interaction decoupling mode can be specified (the control-in-the-large decision !) but the overall performance still might not be satisfactory. The supremal unit can then change the character of the infimal performance function : it can modify the infimal performance functions from a given form, such as
Gi=mi2+y:+Pimi-Piui, into another form, such as
Gi=rn~+y~+~imi"-Piu~. Similarly, the supremal unit can make image modifications by changing the structure of the infimal models or the constraints imposed on the infimal decisions. Finally, when the mode of coordination is selected and the structure is fixed, the supremal unit has the control-in-the-small problem of selecting the coordination inputs. We refer, for simplicity, to this problem as the coordination problem, admittedly using the term in a narrow context. The total job of the supremal unit then consists of: (i) selecting the coordination mode, (ii) modifying the infimal strategy functions, if needed, and (iii) selecting the coordination input after all other decisions are made. Both modification and coordination problems will be investigated in detail for various coordination modes in Part 11. Information Gathering The information needed by the supremal unit about the lower levels depends upon the supremal decision problem itself and the image (model) needed to solve its decision problem. These two factors are, however, interwoven since the type of decision problems that can be formulated for the
62
II. Conceptualization
supremal unit depend upon the kind of information available to it. What is needed is a model or image of the lower level behavior. We mention three approaches to the supremal model building problem. The most trivial solution of this problem assumes that the coordinator has a precise representation of the lower level behavior; the coordination problem then reduces to the classical type of control problem: everything is assumed known, and the problem is that of selecting the best decision. Before discarding this approach altogether, it should be noticed that, in the context of social systems, the problem of coordination is not only that of finding the best coordination term but also of finding ways to implement it. The most serious problems in the coordination of social systems are largely due to the difficulties in implementing a solution which is found to be good on the basis of technical, economic, and other considerations. The availability of a good solution in such situations is only a necessary first step in the coordination process. A more meaningful approach to model building for the supremal unit is to attempt a simplified description of the lower level subsystems. This, essentially, is the classical approach considered extensively in control theory. ,In the context of large scale systems theory, such approximate models can be developed by the so-called aggregation of variables, so that the supremal goal, in general, does not depend upon the complete set oflower level variables. Actually, the supremal performance function can often be an explicit function of only a few key variables such as the performance of the infimal units. There is no need then for the supremal unit to have a detailed knowledge of the behavior and time evolution of the infimal units. The success of this approach clearly depends on what and how many variables appear explicitly in the supremal performance function. The main conceptual drawback of such an approach as far as multilevel applications are concerned is that it does not take into account the decomposition of the system into subsystems on the lower level. Still, such an approach has been shown to have merit for multilevel systems applications [23]. The inspiration for some of these studies is based on earlier investigations in econometrics. We do not pursue this approach in this book, because it essentially does not represent a multilevel problem, but rather an application of the more classical control and programming approaches. The selection of the model for the supremal unit should not be based on a direct simplification (by aggregation or otherwise) of the lower level. Rather, it should rest on the recognition of the fact that the controlled process for the supremal unit consists of a family of interacting subsystems and, furthermore, that these subsystems are themselves goal-seeking systems. This is why the supremal unit has to coordinate, rather than control, the infimal units. The model for the supremal unit should then be based on the interactions between the infimals; just how depends on how the infimal units take
4. Why Hierarchical Structures ?
~
63
into account in their decision problems the interactions between each other. Actually, one again achieves some kind of aggregation ;however, each infimal unit aggregates its local variables into variables pertinent to the supremal unit’s needs. For example, in the performance balance coordination mode, the supremal unit requires information about the local performances, but local performance is nothing more than an aggregate variable. At any rate, it is the recognition of the decision-making functions on the lower levels and the resulting shifting of the decision-making burden to the lower levels, which is characteristic of effective multilevel systems.
Relationship of the Infimal Units to the Supremal Unit The infimal units can affect the supremal unit directly by providing it with whatever information is requested, or indirectly through the selected decision, since the ultimate success of the supremal unit depends upon the performance of the lower level systems. In the pre-decision exchange of information the supremal unit has supremality over infimal units and can request the type of information it wants. Usually, this information refers to the decision which the infimal units are about to make, or, more generally, it refers to an evaluation of the infimal decision processes. The infimal units, in turn, can use the information sent to the supremal unit as an additional decision variable to secure better conditions for themselves. However, if the process is dynamic, and post-decision intervention is considered, the infimal units have to take into account that a gross discrepancy between the information provided in the pre-decision period and the actual subsequent event can lead to an unfavorable response by the supremal unit. Typically, an infimal can communicate to the supremal unit in the pre-decision communication process that information which maximizes its potential return; the infimal unit should, however, observe the constraint that it might be required to give a rational explanation for any discrepancy which might occur in the postdecision period.
4. WHY HIERARCHICAL STRUCTURES?
A question which repeatedly comes up in discussions of hierarchical systems is why they appear so often in nature, or why one should select such a structure when designing a system. An apparent disadvantage of a multilevel system is its complex operation or behavior: functioning of the system is not easy to analyze or comprehend, nor can the system be controlled and influenced so easily. In particular, in designing systems anew, there exists a body of knowledge and concepts which can be used in various stages for
64
II. Conceptualization
systems design and operation if a single-level approach is used. However, for the multilevel approach quite a few new developments have to take place. The question, then, is whether there is any advantage in using a multilevel as opposed to a completely integrated and centralized approach. We have provided the answer to this question, at least implicitly, in the discussion throughout this chapter. However, it is useful to isolate and make more explicit some of the reasons favoring a multilevel approach.
Integration A hierarchical system often evolves in the process of restructuring or redesigning an existing system to improve its overall performance. In constrircting complete control (also referred to as “ integrated control ”) of an industrial complex, the possibility of completely redesigning and rebuilding the total complex in a most rational fashion is rarely, if ever, allowable, due to economic, technical, and human constraints. Essentially, one has to start from a given system which contains the process and lower level control, and add higher level control so as to integrate the total operation of the system: The situation here is not unlike the separation between the design of a process and its associated control system. It is, by now, very well recognized that one should design the system as a whole, rather than design the process first and then simply add the necessary controls. Although examples can be given in which the existence of control subsystems are taken into account in the process design, a “total system” approach, where no separation is made, has not been achieved as yet. Quite analogously, integrated control system design starts from the given processes and lower level controls with the objective of coordinating the interacting subsystems, so as to promote higher level objectives encompassing ever-increasing portions of the complex.
Stratification Descriptions or models of complex systems are often available only on a stratified basis in reference to physical subsystems, management and economic aspects, and the like. Also, the overall task which the integrated control system is supposed to perform is specified in practical terms by means of a hierarchy of jobs and subtasks to be performed.
Limitation of Building Modules Assume the task to be performed is such that it cannot be done by using any of the available decision units. In on-line situations, one then forms a multiechelon system; in off-line applications one uses the multilayer approach
4. Why Hierarchical Structures ?
65
or a multiechelon type of decomposition where a single decision unit is used to solve the subproblems sequentially. An obvious example of this is the problem of optimization by a computer with such memory limitations that only the decomposed subproblems can be programmed. It should be emphasized that the multilevel approach to solving complex problems represents, in general, an important method in systems engineering. One starts with the complex overall problem, forms a multilayer hierarchy of subproblems, and solves the subproblems one at a time, possibly using a single problem solving unit. Drew et al. [24] give an interesting account of the application of this approach to the design of transportation systems.
Better Utilization of Total Resources Total resources are better utilized when solving large scale complex problems if a multilevel approach is adopted. This, of course, is a disputable point and, in general, depends upon one’s skill in selecting the multilevel structure. However, a relatively simple analysis revealed that this can indeed be the case for a broad class of systems, if the higher level decision problems can be sufficiently simplified [25]. The analysis is based on the assumption that the decision-making effort is a convex function of the number of control variables and a linear function of the number of observed variables. After a certain point (depending upon the actual form of the decision-effort function), the total effort is reduced by decomposing the problem and using a two-level structure. The approach is justified if the coordination problem can be sufficiently simplified, with respect to the overall problem, so that the total effort of the two level system is smaller than the integrated system achieving the same performance level.
Flexibility and Reliability In a multilevel, decentralized, system the changes in the decision procedure necessitated by changes in the operation of a subprocess can be localized and, therefore, accounted for with less cost and in a shorter time. In general, the system adapts faster. For example, in multiarea dispatch of electric power, changes in one power generation area would only require changes in the loss formulae matrix for that area: nothing else has to be changed in the system; in single-area dispatch, the entire loss matrix has to be recalculated. Similarly, malfunctioning of a part does not propagate so readily through the entire system. This statement, of course, must also be qualified since, in general, this depends upon the particular system and the type of failure that can occur. However, the potential for increased reliability is certainly there.
Chapter III
FORMALIZATION
Our objective in this chapter is to present in the framework of mathematical systems theory various concepts of hierarchies as introduced in the preceding chapter. That is, we shall formalize in a set-theoretic framework various notions of systems, subsystems, and their interrelationships. The main points presented in this chapter are the following: (i) A functional system is a mapping S : X -+ Y between abstract sets X and Y representing, respectively, the set of inputs x and the set of outputs y. More generally, a system is a relation S c X x Yon abstract sets X and Y. A system as a relation on abstract sets is a formal, mathematical representation of the intuitive notion of a system. (ii) When a system is such that its input fixes the free parameters of a decision problem, and for any given input the resulting output is a solution of that decision problem, the system is decision making. (iii) The above concepts are used to formalize the different kinds of hierarchies discussed in the preceding chapter. The rationality of a system as a relation and the notion of a decisionmaking system are best illustrated by examples such as those given in Sects. 2 and 3 of this chapter. The reader who accepts (i) and (ii), and is interested only in the theory of coordination, can omit this chapter and proceed directly to the next chapter or even Chap. V. 66
1. Introduction
67
1. INTRODUCTION
There are three primary reasons for an abstract mathematical formalization of multilevel systems. (i) Precision: In such a formalization, the concepts are defined in a precise manner in reference to structural properties which are divorced from interpretation in specific fields and, therefore, capable of application in whatever context or discipline such a structural relationship appears. Furthermore, due to the abstract setting, this is accomplished by introducing hardly any constraints. (ii) Mathematical Theory: A foundation is provided for a more detailed mathematical study of hierarchical systems by introducing additional mathematical structure whenever needed. The desirability of such studies need not be elaborated here. (iii) Structural Studies : Through a mathematical investigation of the various aspectsof hierarchical systemsin an abstract yet conceptuallymeaningful setting, we can illuminate some of the most essential structural features of such systems. If one proceeds from the intuitive concepts straight into a deep and specific mathematical structure (e.g., systems described by linear differential equations), several questions will always arise: How much does a particular conclusion depend upon the specificity of the mathematical framework in which it was derived? How can the results be generalized? In which direction should the results be generalized? By investigating the problems in the framework of general systems theory, the question of how “ general ” is a given conclusion, is answered implicitly by the mathematical structure necessary both to formalize and analyze the problem. Selecting the “proper” level of abstraction reflects to some extent the intended application, viewpoint, or even aesthetics of the investigator. Yet, the methodology of working on as abstract a level as possible seems to have considerable conceptual advantages. As an illustration, we point to the applicability of the interaction-balance coordination principle described in Chap. IV. It can be shown quite readily that the principle is applicable in a two-level system having static linear processes with quadratic performances on the first level, when the overall performance is given as the sum of the first-level performances. However, does the principle apply to linear or nonlinear dynamic processes? Do the first-level performances have to be quadratic? What other overall performances are allowable ? In short, how general is the principle? These questions can be answered precisely, as shown in
ZZI. Formalization
68
Chap. V, in an elegant and extremely simple manner, when the general systems theory approach is taken. Analysis indicates that the only requirement for the applicability of the principle is a monotonic relationship between first-level and overall performances. This, indeed, shows the generality of the principle; it would, therefore, not be surprising to find the principle widely used. Of course, for more specific questions, such as those pertaining to coordinability as presented in Chap. V, a deeper structure is needed. However, wide applicability of the principle gives encouragement for its use in solving various problems in specific situations. Another example, illustrating the benefits of a general investigation, is the study of first-level performance modification through interaction operators which result in nonlinear modifications. The role of a mathematical theory of general systems in systems engineering methodology is illustrated in Fig. 3.1. Starting from a verbal description
VerboI description
0
block diagram representat ion
I
kvestigation
FIG.3.1
I
k e o r y framework 1
I
Methodolpgy for problem formulation and analysis.
of the concepts, one first constructs a block diagram to indicate the mutual arrangements and interrelationships between the subsystems; next, one formalizes the block diagram representation together with the descriptive statements accompanying the diagram to derive a general systems model ; finally, additional mathematical structure is introduced and the system’s behavior is studied analytically or by computer simulation. We shall not go into the details of the development of general systems theory, but shall only introduce those concepts which will be used later in this work.
2. Mathematical Concept of a General System
69
2. MATHEMATICAL CONCEPT OF A GENERAL SYSTEM
We start with the following simple notions: (i) A (general) system S is a relation on abstract sets X and Y,
s c x x Y. (ii) If S is a function, S : X + Y, it will be called a functional system. For simplicity we shall simply refer to a system without stating whether it is functional when this property is not needed, or when the fact is clear from the context. The component sets of a system S E X x Yare referred to as its objects: X is the input object or input set; Y is the output object or output set; the elements of X and Yare the inputs and outputs, respectively. Representation of a system as a relation is, therefore, an input-output representation. If a given system is a functional system, its inputs can be considered as causes and its outputs as effects; hence, the input and output sets may be referred to, respectively, as cause and effect objects. This terminology is in reference to the modeling of a cause-effect phenomenon. If the system is a relation, the cause represents a given input initial-state pair. However, we shall make special effort to use functional subsystems instead of proper relations in the general system development. We shall still use the term input set for the cause object, implying that the subsystem under consideration is at a given initial state. This will hardly limit the generality of the derived results ; nevertheless, the significant notational simplification in using functions, rather than proper relations as the subsystems, provides sufficient justification. In lieu of a lengthy discussion of why this notion of a system is introduced, we shall give several examples showing how it covers various specialized structures. Example 3.1 :A diflerence equation system Let us consider the difference equation Yk
= 2Yk- 1
+ xk
as describing certain observations over a discrete time period T = { 1, ...,n}. For a given initial condition yo = a, there corresponds to any given n-tuple x = ( x l , . . ., x,) in R",a unique n-tuple y = (y,, . . ., y,,) in R" which satisfies Eq. (3.1) for each k = 1, .. ., n. Therefore, there is defined a mapping S, : R" + R" such that for all x in R" the image y = S,(x) is the unique solution of Eq. (3.1), with the given initial condition yo = u. If a set Yo c R of
III. Formalization
70
initial conditions is allowed, we obtain a relation S c R" x R" such that S= S,. Therefore, in general, the above equation specifies a system S c R" x R", and, in particular, it specifies a functional system S, when an initial condition y o = a is given.
ulxeYo
Example 3.2: A sequential machine Let us consider a vending machine which accepts five and ten cent coins for a fifteen cent cup of coke and gives change, if any is due. Let A = ( 5 , lo} represent the set of coins acceptable to the machinekt B, = {4, coke} where 4 represents "no coke," and let B, = (0, 5 ) represent the set of coins the machine can give as change. The set B = B, x B, is then the set of outputs that can occur at any time. Let us introduce the set Q = { q o , q l , q2} of " states" of the machine. Now, the state transition function f:A x Q + Q and the output function h :A x Q + B are specified by the table: a=5 f(a,40) f(a,41) f(a,q2)
a=lO
41
42
42
42 40
qo
Ma, 40) 41) h(a, q d
a=5
a=lO
(4,O)
(490) (coke, 0) (coke, 5 )
(4,O) (coke, 0)
Consider the case of n trials. Let A" and B" be, respectively, the set of all n-tuples from the sets A and B. Then it is easy to see that, for a given initial state q = qi, there corresponds to any x in A" a unique y in B". In other words, there is defined a mapping S,: A" + B" such that for all x in A", the image y = S,(x) is the unique outcome corresponding to x and the initial state q = qi. Then, the vending machine is represented by the system S E A" x B" such that S = S , . Sometimes we can get a cup of coke for five or ten cents, but when in normal operation the initial state is qo, hence, the vending machine can be considered as the functional system S, where q = qo .
UqEQ
Example 3.3: A diferential equation system Let us consider the simple dynamic system illustrated below :
k Spring
Let the spring constant of the weightless spring be k . Let the displacement of the body, whose mass is m, from equilibrium at time t be y(t). Let x(t) be an external force acting on the body at time t . Suppose there is no friction. The relation between x(t) and y(t) is then given by the differential equation my(t) = x(t) - ky(t).
(3.2)
71
2. Mathematical Concept of a General System
Suppose we observe x(t) and y ( t ) on the time interval T = [0, a).Let X be the set of all integrable real valued functions on T, and let Y be the set of all real valued functions on T. Then, for a given initial condition a = (y(O),j ( O ) ) , there corresponds to any x in X a unique y in Y, such that for all t in T y(t> = y(0)cos cot
+ y(0)w sin wt +
~ sin ~w(to- z) * x(z> dz,
(3.3)
0
where w = (k/m)’”. A unique mapping S,: X - t Y is therefore specified by the above equation. When A E R x R is a set of allowable initial conditions, the above spring-mass system is represented by the relation
s= U a E s, A cxx
Y.
Example 3.4: A partial diflerential equation system Consider the one-dimensional heat conduction situation as shown : Insuloiion Heat source
Heof sink 0 oc
XOC
,
Insulation
0
n
gu
In general, if there is a heat source in the slab, the temperature distribution function 8(t, u) is given by the partial differential equation
aelat = a(a28/8u2)+ bz(t, u), where z(t, u) represents the heat source in the slab, and a and b are constants; for the sake of simplicity, we assume, however, that z = 0. Suppose O(0, u) = f(u) is given for u in [0, x ] , and at t = 0 a heat source of xo C is attached to the slab. Let Y be the set of real valued functions on [0, co) x [0, x ] . Then, when the Fourier expansion of the initial distribution f exists, there corresponds to eachxin R a unique 8 in Y,for the givendistributionf. For instance, when f = 0 m
o(t, u ) = x
- xu/x - C (2x/nk) exp (- kt2) sin ku. &= 1
Let this correspondence be represented by a mapping S,: R -+ Y, where f is the given initial distribution. Let F be the set of real valued functions defined on [0, x ] which have a Fourier expansion. Then the relation
is a representation of the above physical system.
11I. Formalization
72 3. DECISION-MAKING SYSTEMS
It is tacitly assumed in all examples considered in the preceding section that associated with any system there is a “ mechanism ” which relates the inputs and outputs. As a rule, a set ofequations is a very convenient mechanism for defining or specifying a given system. However, the existence of a causeeffect relationship, and therefore a system, is not necessarily preconditioned by the existence of a defining procedure or constructive specification, such as that offered by a set of equations. Actually, in some rather important considerations of structural interdependence between the subsystems, the defining procedures of the subsystems, even if given, are of no consequence and do not have to be explicitly taken into consideration. This is the case in the general theory of multilevel systems developed later in this work. Furthermore, the constructive specification can be given as an implicit function, i.e., the response (output) of the system for a given initial condition is defined as an implicit function of the input. A particularly convenient way to define a system is by means of a decision problem; that is, one defines a system S E X x Y such that the pair (x,y ) belongs to S only if y is a solution of the decision problem specified by x. A precise meaning of a decision problem will be given later in this section. We shall often assume that a system S so defined is functional The system S can then be considered as embodying a solution algorithm and will be referred to as decision-making or a decision-maker. Example 3.5: Optimizing system Perhaps the most straightforward example of a decision-making system is one which operates so as to minimize a given performance function. Let X be the set of continuous real valued functions on the interval [O, 11, and for each x in X let there be given the optimization problem g X minimize :
I
1
~ ( xm, , y)=
[(y - x12
+ cm21 dt,
0
under the constraint j
= ay
+ bm,
y(0) = a, where a, 6 , and c > 0 are given constants, and m and y are square integrable functions on [0, 11. The solution of the problem gXis given by the solution of the equations j = ay - b2p/2c,
Y O = a,
p = - a p + 2 ( x - u),
P ( 1 ) = 0,
m
=
-bp/2c.
(3.4)
(3.5)
3. Decision-Making Systems
73
Apparently, for every x in X there is a unique y in g Z [ O , I], given by Eqs. (3.4) and (3.5) which satisfies the problem gX. There is then defined a system S E X x g2[0, 11, such that for any x in X and y in Y,[O, 11, the pair (x, y ) is in the system S iffy is a solution of gXfor some initial condition y(0) = M. Example 3.6: Hamilton’s principle
Consider the simple spring-mass system in Ex. 3.3. For each external force history x over T = [0, GO), the resulting displacement y over T is, according to Hamilton’s principle, a solution of the decision problem gX: find y such that
6 ~ o m [ + m ( j+ ) 2( x y - S.ky2)]dt = 0 whete 6 denotes the first variation. The solutions of this problem are given by Euler’s equation :
which yields the equation of motion, Eq. (3.2). Hence, the solutions of the decision problem BXinclude the displacement histories y given by Eq. (3.3). In other words, the unique mapping S,: X + Y specified by Eq. (3.3) and the initial condition M = (y(O),j(0)) is such that, for any x in X , the image y = S,(x) is a solution of the decision problem Bx.Apparently, the system S in Ex. 3.3 can be specified in two different, but equivalent, ways: one way is by the constructive specification furnished by the differential Eq. (3.2), and x E X, the other way is in terms of the family of decision problems gX, provided by Hamilton’s principle. To define a decision-making system, we first need a concrete notion of a decision problem. Two are considered: a general optimization problem and a general satisfaction problem. General Optimization Problem Let g : X + V be a function from an arbitrary set X into a set V, linearly or partially ordered by the relation 1.A general optimization problem is the following problem: given a subset X* E X , find 2 E Xf such that for all x in Xf 9(3 g(4. (3.6) The set X i s the decision set, while the set X* is the set of feasible decisions.*
* The set X z is also referred to as the feasible domain of a set of allowable decisions.
74
III. Formalization
The function g is the objectivefimction,and the set V is the value set. A general optimization problem is specified by a pair (9, Xf,.An element 9 of Xf satisfying Eq. (3.6) for all x in Xf is a solution of the optimization problem specified by the pair (9, Xf). Often the function g is specified by two functions, P: X + Y and G : xx Y+V: g(x) = G(x, P W .
In this case, the function P is referred to as an outcome function or model of a controlled process, and the function G is referred to as a performance or evaluationfunction; an optimization problem may then be specified by a triplet (P, G , Xf)or a pair ( P , G), if Xf = X . Reference to P as a " model of the controlled process " implies that the optimization problem specified by the triplet (P, G , Xf)is a control problem; the decisions affect a process whose output in turn affects the performance or payoff. In general, nothing has to be assumed about the relationship between the model P and the actual process; even the existence of the actual process as an entity might be just an assumption used to define an optimization problem so as to be able to specify a system as a decision-making unit.
General Satisfaction Problem
Let X and R be arbitrary sets; let g be a function from X x Q into a set V, linearly or partially ordered by 5 , and let z be a function from R into V ; that is, g : X x R + V and z: R V . A satisfaction problem is the following decision problem: given a subset Xf G X , find 2 in Xf such that for all w in R --f
+4.
g(X, w)
(3.7)
The set R is referred to as an uncertainty set, z is a tolerance function, and the inequality Eq. (3.7) over R is the satisfaction criterion. The other elements of the satisfaction problem have the same interpretation as in the optimization problem. A quadruple (g, z, Xf,0)specifies a satisfaction problem, and any R in Xf satisfying Eq. (3.7) for all w in R is a solution of the satisfaction problem specified by ( 9 , z, Xf,R). The uncertainty set R is sometimes referred to as a disturbance set, for it represents the set of all possible effects which can influence the performance. When the objective function g is given in terms of an outcome function P: X x R .+ Y and a performance function G : X x R x Y --f V, g(x, 4 = G(X7
0
7
P(x, W)),
3. Decision-Making Systems
75
then the set R represents the set of all possible effects which can influence the outcome of a given decision x . Notice that, due to the level of abstraction, R covers both so-called parametric and structural uncertainties. The function z then specifies an “ upper limit ” of the tolerable or acceptable performance. A decision 2 is considered satisfactory if it yields a performance which does not exceed the specified tolerance z(w) for any outcomes of the uncertainty o in the given set R. Example 3.7 Let P:R x R + R be defined by the equation y
= m3
+ w = P(m, w),
+
where R is the set { - 1, l} of only two elements. Let the tolerance function z: f 2 + R be a constant, z(w) = 1, for all w in R, while the performance is measured by the absolute value of the output, C(m, Y ) = IYI. The satisfaction problem is then to find a control h in R,such that
IP(W 4 1
1,
for all w in R. Such a control is “ satisfactory.” An apparent satisfactory control is h = 0. It should be noticed, however, that the satisfactory control h = 0 is not optimal for any of the disturbances. Namely, if w = - 1, the control which minimizes IP(m,- 1)1 is h(- 1) = + 1, while for w = + 1, the optimal control is h(1) = - 1. This shows that a satisfactory solution is not optimal for any of the disturbances. Furthermore, none of the optimal solutions are satisfactory; for m = 1, IP(1,1)1 = 2, which violates the tolerance limit, while for m = - 1, IP(- 1, - 1)1 = 2, which again violates the tolerance. The distinction between an optimal and a satisfactory solution is quite clear in this case. Example 3.8: Detection function for a noisy channel Consider the problem of selecting a detection (decision) function for a noisy channel. Assume that the sender’s message set is the set A = { a l ,a,}, while the receiver’s message set is B = { b l , b, , b3} ,the difference between their messages being due to noise in the transmission channel. The detection problem is then the following: Assume a message b i is received; how can one determine, using statistical information about the channel, whether the message a, or a, was set at the other end? In other words, the problem is to select a function m:B --f A such that, if bi is received, the message a j = m(bi) is taken as the sender’s message.
III. Formalization
76
Let M be the set of all functions on B into A . Criteria must be specified to aid in selecting a particular m E M to be used as the detection function. One criterion can be formulated using a Baysian approach. Let p [ a i ] be the a priori probability that the message a, was sent; assume p [ a , ] = w , and therefore p [ a z ] = 1 - w . Let p [ b i I a j ] be the conditional probability of receiving message bi given that aj was sent. The conditional probabilities are given in the following table :
For convenience, let Fii denote the negation of a given detection function m ; if m(b,) = a , , then Fii(bi)= az , etc. The probability of making a wrong decision is, then, p[E(bi)].Using the Baysian approach, the conditional probability of making a wrong decision if a message bi is received is
We can now define a loss function 3
g(m, 0) = C PCWbi) I bi ; 0 1 . w(E(bi)), i= 1
where w :A + R assigns a penalty to the wrong choice. The problem of selecting a detection function can now be formulated as follows: Given the a priori probability w , find uiz in M such that the loss function is minimized. For the case when w = and the penalty function u’is such that w(aJ = 1 and w(az) = 2, the optimal detection function A is A@,) = a, , A&) = a 2 , A(&) = a 2 . Assume now that the a priori probability is not known with certainty, but it is assumed that w =p(al ) is within a given range; specifically, let w be in the interval 0 = [ f , $1. Then, if we define a tolerance, say E , where E is a constant number, we can specify the selection of the detection function as a satisfaction problem ; namely: Find A in M such that for all w in Q g(A,
0 ) IE.
A satisfactory solution for E = 5 is then A@,) = a,, A(b2)= a 2 ,A(b3)= a2. Notice that the satisfactory and optimal solutions are the same in this case.
3. Decision-Making Systems
77
In the general form of the satisfaction problem, the relation I in (3.7) may be replaced by any desired relation R c V x V. A general satisfaction problem is to find 2 in Xf such that for all o in R
(3.8)
g(2, o)Rz(@,
where R is a given relation in the value set V; that is, a decision 2 in Xf is satisfactory or solves the problem, if for any w in R the performance g(2, w ) is R-related to the tolerance z(o).
Decision-Making System We can now define quite readily what is meant by a decision-making system. Figure 3.2 illustrates the notion.
II X
I I
Decision unit d
z
o
Implementation
r
I
I
I
Y
0
A system S G X x Y is a decision-making system if there is given a family x E X , with the solution set 2 and a mapping of decision problems gX, T: 2 -,Y such that for any x in X and y in Y, the pair ( x , y ) is in the system S only if there exists z E 2 such that z is a solution of gXand T(z)= y . In most of our considerations (but not always!), the output itself is a solution of the given decision problem; 2 = Y and T is identity. In conclusion, let us mention the following remarks regarding the notion of a decision-making system : (i) A constructive specification in terms of some equations might be given for a decision system, particularly if there exists an analytical solution to the given decision problem, in the sense that for any input x in X there exists a analytical algorithm that determines the output y = S(x). However, such an algorithm need not always exist. We shall, indeed, require only that for any decision system there is a well-defined decision problem ; however, we make no requirement on the existence of an algorithm in one form or another to obtain a solution. (ii) Any input-output system can be described as a decision system, and conversely. A system may be viewed as a decision system purely on the grounds
I8
III. Formalization
of expediency in studying its behavior. Actually, both input-output and decision-making specifications can be used alternatively, depending upon one’s interest. A notable example of this can be found in classical physics where a given phenomenon can be described either on the basis of a law or a variational principle as in Example 3.6. In fact, in Chap. IV, a two-level system is described first in terms of decision-making units, and then as an inputoutput system. Let us emphasize that our interest is in systems which have subsystems that are primarily decision-making units-subsystems which are most naturally interpreted as such decision-making units. It is only with this distinction in mind that a multilevel system can be distinguished from systems which are interconnections of arbitrary subsystems. (iii) An optimization problem is apparently a special case of the satisand define ~ ( 0as ) the faction problem; for example, let IR be a unit set (0) Conversely, one might argue that the satisminimum of g over Xf x (0). faction problem can be reformulated as an optimization problem by a proper selection of a new performance function. This, of course, is to some extent a question of interpretation and aesthetics. For conceptual reasons which we shall not elaborate here, we prefer, however, to make a distinction between the two kinds of problems. The technical differences resulting from this distinction can be easily seen by glancing through Chaps. VI and VIII, where optimization and satisfaction problems are considered respectively. In the conceptual discussions of hierarchies in Chap. I1 and in the formalization in the following section, we refer to a goal and the goal-seeking activity of a system, as well as to a goal-seeking system. These notions are of course related to the concepts of decision-making and decision-making units considered in this section. Actually, the distinction between decisionmaking and goal-seeking depends on the viewpoints and definitions one adopts. The approach we shall take is the following: The.concept of a goal and a goal-seeking activity in the most general sense will be left unformalized; we refrained from describing purposeful behavior where both a precise meaning of the goal and the processes in terms of which it can be achieved are not explicitly stated. We assume, however, that the state of a goal being achieved is recognizable at least by the system itself. An example from psychology might help here. Reaching the “ state ” of “being happy ” might be a goal for man, yet its meaning is probably known only to the person involved, and the methods of achieving it are not known to him beforehand. In pursuing such a goal, one usually tries strategies which have some prospects of leading to fulfillment of the goal. He can try to get more education, make a fortune, get married, or try any other combination of strategies, but none of these strategies will necessarily lead him up to the goal. He can very
4. Formalization of the Hierarchical Concepts
79
well learn that his goal-seeking behavior has failed only after several attempts have been made. Not all goal-seeking situations are nonformalizable. If such a formalization is possible, it will inevitably lead to a decision-type problem. We shall assume therefore, that formalized goal-seeking is the same concept as decisionmaking. Therefore, we shall also talk about a (formalized) goal being defined by a decision problem, either an optimization problem or a satisfaction problem, and a goal being achieved when the corresponding decision problem is solved. It should be noticed that the goal-seeking problem can be formalized without giving a solution method for the associated decision problem.
4. FORMALIZATION OF THE HIERARCHICAL CONCEPTS
Here, we present formalizations of the more prominent conceptsintroduced in Chap. I and elaborated in Chap. 11. We abstract the basic notions of stratified systems, multilayer hierarchies and multiechelon (organizational) hierarchies.
Stratified Systems Stratification involves three most essential characteristics of hierarchies : vertical decomposition, priority of action, and performance dependence, as was described in Chap. I. A stratified system is illustrated in Fig. 2.1. The starting point for stratification of a given system S: X + Y is the assumption that the set X of outside stimuli and the set Y of responses are both representable as Cartesian products; it is assumed there are given two families of sets X i , 1 Ii In, and Yi, 1 5 i In, such that
X = XI x
x
X,,
and
Y = Yl x
x Y,.
(3.8)
This assumption corresponds to the ability of partitioning the input stimuli and responses into components. Given that the sets X and Y are representable as in (3.8), each pair ( X i , Yi),1 < i 5 n,is assigned to a particular stratum. The ith stratum of the system S is a system represented as a mapping Si: (i) Si:X i x W i--+ Yi if i = n, (ii) Si:X i x Vi x W i--f Yiif 1 < i < n, (iii) S i : X i x V i + Y i i f i = l .
(3.9)
A family of such systems Si , 1 < i < n, is a stratijication of S if there exist two
80
111. Formalization
families of mappings h i : Yi -+ “ly-i+l,1 < i < n, and ci: Yi+ Wi...l, 1 < i < n, such that for each x in X , and y = S(x): (3.10) The image set Yirepresents the responses of the ith stratum S , . The sets Wi and W i represent the sets of stimuli respectively from the strata immediately above and below the ith stratum. The mappings hi and c i are referred to as the ith stratum injormation function and decision function respectively; they tie the strata together as in (3.10) to form the system S. In reference to the mappings hi and ci, we can formally describe what is meant by a successful stratification; in so doing, we recognize several degrees of stratification. The system S is completely stratifed if any stratum Si,1 I i I n, is such that for any pair ( y i , ui)in Wi x W i and , any two elements xi and xi’in Xi : hi(Si(xi >
Y8 >
COi))
ci(Si(xi
~i
U’i))=
9
9
= hi(Si(xi’9 Y i 7
ui)),
ci(Si(xi’, Y i ,
ui)).
(3.11)
This simply means that for a given intervention y i and feedback u i , the response of the subsystem Si to any change in the stimuli xi will be such that there is no change either in intervention y i - l or feedback u / ~ + in ~ ; other words, the response is confined to the ith stratum only. Notice that complete stratification depends not only on the transformation Sibut upon the mappings c i and h, as well; the stratification depends upon interstrata relationships as well as the strata themselves. To require that the response of each stratum be completely localized is, of course, a strong condition. A weaker concept is that of a stable stratification in which localization is achieved for some but not all intervention-feedback pairs. In this, the prominence of the top stratum is acknowledged. The topmost stratum has its own set X , of outside stimuli, and its response depends upon the entire hierarchy below. The requirement which the top level places upon the hierarchy can then be represented in terms of the feedback information l ~, since ’ ~ this is the only input from below. A stable stratijication is characterized, therefore, by the following condition: there exists some x in X and y = s ( ~and ) , for each i, 1 < i < n, a pair (yi, mi) such that u’i = h i - l ( Y i - 1 ) 9 ~i
=Ci+l(Yi+l),
y i = Si(xi,y i , mi),
l
and, moreover, for all x’ in X and i, 1 5 i < n, condition (3.1 1) holds.
4. Formalization of the Hierarchical Concepts
81
Distinction between complete and stable stratification is that, in the latter, it is not required that the strata be independent for every interventionfeedback pair, but rather only that there exist such “states of the entire system,” so to speak, in which the responses are localized. Of course, there is the question of how a stable state in a hierarchy can be reached, but this is a question which needs deeper structure for the analysis. Both complete and stable stratifications represent only idealized cases and, therefore, approximations to the hierarchies in the actual real-life systems. There are a number of ways in which the conditions can be weakened leading to less completely stratified systems. We shall not proceed to formalize such weaker, but more realistic, notions but shall restrict ourselves to several comments. Condition (3.11) need not be satisfied for all stimuli in X but rather only those which represent “ normal ” conditions under which the system operates. Also, to realize a stable stratification or even a complete stratification under a restriction on the stimuli, it might be necessary to merge some adjacent strata into one stratum. It could very well turn out in some extreme cases that such mergers result in a single stratum, thereby eliminating stratification altogether. Stratification implies a reduction in information sent up the hierarchy : many lower strata stimuli look alike to the higher strata units. This “ upward reduction of information” has a variety of interesting consequences, one of them indicating a need for organizational-type multiechelon hierarchy. Consider the case where the stimuli are entering the system only on the lowest level. Because of the reduction of information, the higher levels have to process a smaller amount of information. This has two important consequences : (i) When the system is built out of building blocks which have a limited decision-making capacity, the higher strata will apparently be composed of a fewer number of units. (ii) Reduction of information can be achieved in many ways, one of them being aggregation. Aggregation, as discussed before, leads to the partitioning of a family of variables into the subfamilies, such that each subfamily is represented by a single “ aggregated variable. This effectively represents a decomposition o i a lower stratum into subsystems. Indeed, the information feedback can very well contain precisely the interaction variables between the subsystems, as in the case of the coordination by interaction prediction or balance. In short, reduction of information leads in a natural way to a horizontal decomposition of a stratum into subsystems. On each stratum, the decision units are concerned primarily with the operation of the subsystems, for the most part neglecting the interactions between the subsystems on the same stratum. The decision units on the higher strata are concerned then only with ”
III. Formalization
82
the interactions and interdependence between the subsystems, assuming each subsystem is functioning properly. This reasoning clearly leads to an organizational-type multiechelon hierarchy.
Layer Hierarchies The layer hierarchy consists of vertically arranged decision-making subsystems as shown in Fig. 3.3. Each subsystem Si is, in the first place, a
FIG.3.3 A layer hierarchy.
I
mapping Si:g i %‘i-l which is furthermore a decision-making unit: there exists a family of decision problems g i ( y i ) , yi E V i , and a transformation Ti such that for any input y i the output yi-l = Si(yi)is given as yi-, = Ti(xi), where x , is a solution of the decision problem gi(yi). The inputs y i from the unit immediately above apparently play the role of a parameter in the decision problems of S i ; hence, the outputs yi-l obtained by the transformation T i , are, in general, parameters for the unit immediately below. It should be emphasized that the multilayer hierarchy as given in Fig. 3.3 is of a simplified form. Two additional aspects might play quite an important role : --f
(i) Feedback interconnections between layers can exist. They may or may not be conditional, in the sense that the feedback effect can occur only in certain circumstances. For example, a time limit can be set for a particular layer so that if no solution of its specified subproblem is reached in the
4. Formalization of the Hierarchical Concepts
83
allotted time, a feedback signal is sent up, and a new subproblem is defined. These feedback relationships can be quite intricate ; we shall not, therefore, attempt to formalize them on such a general level. (ii) Exchange with the environment can occur on many layers simultaneously. The selection of layers which will interact with the environment depends upon the type of the subproblems assigned to the layers and, of course, the information potentially available from the environment. This will be particularly apparent in the case of a functional decision hierarchy. A functional decision hierarchy as described in Chap. I is of sufficient importance and generality that it deserves to be formalized explicitly. The starting point is the (general) decision-making problem under conditions of uncertainties specified as a satisfaction problem by a quadrude (9,z, Xf,$2): find x in Xf such that for all o in Q:
g(x, 45
w,
where I is a given relation. As already discussed, this apparently leads to three functional layers [Fig. 2.61, each of which can be represented as a mapping, although, more generally, they are proper relations. We represent the (first) selection layer by a mapping:
s,:W l x v, x Q + M , where W l is the set of elements representing feedback information from the controlled process (environment), V1 is the set of inputs from the third layer which specify the structure of S,, and 4Y is the set of inputs from the second layer specifying the uncertainty sets for the first layer. In the context of the satisfaction problem, an element from V, specifies the first three elements g,z, and Xf of the problem; the elements of 4 specify the last element R of the problem. The (second) learning layer is represented by a mapping: S,:W2 x
V2+4,
where the elements of the set W 2represent information from the environment, while the elements of the set 4 specify the uncertainty sets for the first layer. V , is the set of parameters which determine the structure of the learning layer just as Wgl determines the structure of the selection layer. Finally, the (third) self-organizing layer is represented by a mapping :
s,: W 3 + V , x v2, where the elements of the set W , represent feedback information available to the self-organization layer.
ZZZ. Formalization
84
Multiechelons (Organizational) Hierarchy The principal feature of the formalization of the organizational hierarchy is the need to define more precisely vertical arrangements between the subsystems. In the strata and layer hierarchies, there was formally one unit on any particular level. A specific characteristic of the echelon hierarchy is that, as a rule, there are many units on a given level. The starting point, therefore, is to recognize vertical position of the units in the system in reference to priority of action. If 9is a (finite) family of systems Si, iE I, where I is a finite index set, and >) is a hierarchy of if > is a strict partial ordering of Z, we then say (9, systems. If (9, >) is a hierarchy of decision-making systems, and the ordering > is such that i >j iff Sihas priority of action over S j ,then we say (9,>) is a decision-making hierarchy. Echelons in a decision-making hierarchy (9, >) are easily recognized in terms of the strict partial ordering > representing priority of action. The the family first echelon units are the minimal units of 9;
9'= {si:i E Zl} is the first echelon, where I, = { i : i is a minimal element of I}. The ith echelon units are the minimal units of 9 when all lower echelons are omitted; the family 9" = {Sk: k E Ii} is the ith echelon, where
Zi= {k: k is a minimal element of I- [Il u I , u
u Zi-']>. If each echelon consists of at most a single unit, we then have a multilayer hierarchy, provided the ordering > is appropriately defined. Finally, we define multiechelon hierarchies as a subclass of decision>) is a multiechelon making hierarchies. A decision-making hierarchy (9, hierarchy if for all i and j in Z, there is at most a unique k in I such that for all 1 in I I > i and I > j * I > k . This condition might be interpreted as any member of 9 having at most one unit of the immediately higher echelon which has priority of action over it. There is an interesting interpretation of a multiechelon hierarchy (9,>) when the relation > is defined as i > j iff the system S j is a subsystem of Si . Immediately, we almost have a stratified system in the sense that the lower level systems are subsystems of the higher level systems.
Chapter IV
The coordination problem in a multilevel system can be presented with sufficient generality within the framework of a two-level system. Our objective in this chapter is to set the coordination problem in a framework in which it can be analyzed mathematically. We choose a two-level system because of its relative simplicity, and the fact that it can be used as a basic module for the synthesis of multilevel systems in general. We shall first give a general systems description of a two-level system having n infimal control systems subordinate to a single supremal control system. We regard a decision unit as a system whose output satisfies a given decision problem, while we consider a control or decision system as consisting of a decision unit and a means of implementing the output decisions. We shall then pose the coordination problem within this framework and introduce the “ consistency postulate” and coordination principles as a basis for the solution of the coordination problem. The material presented here will be essentially descriptive: mathematical analysis will follow in the subsequent chapters. 1. GENERAL DESCRIPTION OF A TWO-LEVEL SYSTEM
A two-level system is diagrammed in Fig. 4.1. The blocks represent the subsystems, while their arrangement reflects the hierarchical structure of the overall system. The diagrammed system has n 2 basic subsystems: the supremal control system C , , the n infimal control systems C , , . . ., C,, and the controlled process P.Notice the two kinds of vertical interaction between
+
85
IV. Coordination
86
L
__--_-------------_------_- -I
FIG.4.1 A two-level system with n infimal control systems and a single supremal control system.
the subsystems. One is a downward transmission of command signals; the signals from the infimal control systems to the process will be termed control inputs, while the signals from the supremal to the infimal control systems will be termed coordination inputs or interventions. The other kind of vertical interaction is an upward transmission of information or feedback signals to the various control systems of the hierarchy. These signal transmissions are represented by dashed lines in the diagram. The simplest way to describe the subsystems of a two-level system is in terms of their terminal variables: inputs and outputs. It is convenient to describe the subsystems as being functional, in the sense that the inputs give unique outputs; one might view this as the situation in which the present state is given. Each of the blocks in Fig. 4.1 therefore represents a mapping. When we describe the subsystems represented by these blocks, we will introduce appropriate names for various objects in order to indicate their conceptual role in the operation of the two-level system. Consider first the process P, the system subject to the control action of the infimal control systems. It has two inputs: a control or manipulated input m from a given set M , called the control object, and an input w from a given set Q, called the environment or disturbance object. It also has an output y in a given set Y, referred to as the output object. The process P is assumed to be a mapping P:MxQ-,Y.
1. General Description of a Two-Level System
87
In reference to the existence of n infimal control systems, the control object M of the process P will be represented as the Cartesian product of n components, M = M I x ... x M,,,
so that the ith infimal control system has the responsibility of selecting the ith component m iof the control input m and applying it to the process. Next consider the ith infimal control system Ci. Here we shall regard it simply as an input-output system. As such it has two inputs: the coordination input y provided by the supremal control system from a given set V , and the feedback information input zi coming from the process. The output of C iis the (local) control mi selected from the set M i . The system itself is assumed to be a mapping Ci: V x b i + M i , where 6, is assumed to be the set of feedback information inputs zi . We shall refer to the set G2 as the coordination object and its elements y as coordination inputs, since it is through these inputs that the supremal control system influences the infimal control systems. Each infimal control system may interpret a given coordination input y differently from the others; in fact, the coordination inputs y in V can be n-tuples (yl, . . . , y,,) so that the ith infimal control system receives only the ith component, y i . The supremal control system C , will be referred to also as the coordinator, since its output is a coordination input y selected from the coordindtion object G2. We shall consider only one input to the system C , , namely the , from the infimal control systems, which it uses to feedback information Z arrive at the coordination input y. The system itself is then assumed to be a mapping
c,: W + V , where W is the set of feedback information inputs ((2. To complete the description of the two-level system we must specify the nature of the feedback information. The feedback information zi to the ith infimal control system contains information concerning the behavior of the process P;we shall therefore assume that it is a function of the control input m, the disturbance input o and the output y, given by a mapping f i : M x SZ x Y - t b i .
Similarly, the feedback information LO received by the supremal control system contains information concerning the behavior of the infimal control systems and is therefore assumed to be given by a mapping f,:V x 3 x M + W ,
88
ZV. Coordination
where 6 = 3, x * * x 6,;it is a function of the coordination input y, the feedback information z = (zl,.. ., z), received by the infimal control systems, ,.., m").In Fig. 4.1, the feedback information is and their output m = (ml, represented as an n-tuple w = (lo1,..., m,) where m iis the feedback information concerning the ith infimal control system. Two remarks should be made in connection with the indicated interrelationships between the subsystems: (i) There is no explicit provision for direct communication between the infimal control systems.This reflects our preoccupation with only the relationships between adjacent levels of the hierarchy. However, some of the results obtained in later chapters may be interpreted in terms of communication between infimal control systems through the supremal control system. In such cases, the supremal unit acts as a transmitter of information between the infimal control systems. (ii) There is no apparent interaction directly between the supremal control system and the process. This, however, is only superficial, for any one of the infimal control systems may transmit to the supremal the information received from the process. Indeed, in the subsequent chapters, we consider the case in which the supremal control system receives information not only from the infimal control systems, but also directly from the process. 2. DECOMPOSITION OF THE SUBSYSTEMS
Each of the n + 2 subsystems of the two-level system shown in Fig. 4.1 can be further decomposed. The most important decomposition is that of the process P. As far as control units are concerned, they need be decomposed only if the outpcts are not solutions of the associated decision problems,.but rather are obtained as a transformation of the corresponding solutions. Subprocesses The process P is the source of interaction between the infimal control systems and, hence, the reason for the existence of a coordinator, the supremal control system. The process P as shown in Fig. 4.2 may be viewed as representing n subprocesses each of which is under the control of a specific infimal control system. Let us assume, then, that there are given n subprocesses, the ith of which is a mapping P i : M ix uix R -+ Yi, where the set Ui is the set of inputs uithrough which the subprocess Pi is
2. Decomposition of the Subsystems
89
r-------,
I------
I
IP
IH
----I
I
L----J
FIG.4.2 Decomposition
coupled with the other subprocesses. Each subprocess is formally represented as being affected by the same external disturbance o from Q; however each subprocess may be affected differently than the others ; in fact, the external .. ., on),so that the ith subprocess disturbances o in Q can be n-tuples (o,, is affected only by the ith component of o. For each i, 1 Ii I n, we assume there is given a mapping Hi:M x Y + Ui, which couples the subprocesses. Often H i is simply a projection mapping, but it need not be so! We shall refer to the sets Ui as the interface objects, and their elements as the interface inputs; the mappings Hi will be referred to as the subprocehs couplingfunctions. The relationship between the process P and its subprocess Piis as follows. Let U = U, x x Un and define the functions H on M x Y and P on MxUxQas e . 0
H(m, Y) = (Hl(m, Y), .. * , Hn(m, Y)),
W, u, w) = (pl(ml, ~
o),.- ,pn(mn, u,
1 ,
3
0)).
(4.1) (4.2)
90
ZV. Coordination
P then represents the uncoupled subprocesses, while H represents the couplings. Now, the process P represents the coupled subprocesses iff the condition Y = P(m, H(m, Y ) , O)-Y = P(m,w), (4.3) holds for all (m,y, o)in A4 x Y x R ; that is, the solution of the (simultaneous) equations Y = a ( m , u, 01,
(4.4)
= H(m, Y ) ,
exists for any given control input m in M and disturbance input o in R, and Figure 4.3 shows this relationship between the yields the output y = P(m, 0). process P and the subprocesses as collectively represented by P.
w m
I
D
I I
I
I
I
D
I -
P
D U
H
-
1
TI
OY
I
I
I
12
1
I
0
I
I
-
a Y
D
I I I
I
FIG.4.3 Relationshipbetween the process P and the uncoupled subprocess represented by P.
From condition (4.3) it should be noticed that the interface inputs
u = (ul, . . . , u,) to the subprocesses can be expressed as a function of the
control inputs m and the disturbance inputs o.Precisely, u is given by the mapping K:MxR+U defined by the equation K(m, o)= H(m, P(m, 0)). (4.5) We shall refer to K as the subprocess interaction function. We may now see the process P as given in terms of the subprocesses and K by the equation p ( m , 0)= P(m, K(m, 01, o), as shown in Fig. 4.4.
(4.6)
91
2. Decomposition of the Subsystems
rn w
:ibfl - - - - - - -- - -- --1 I
U
‘I I
I
I I
L
K
I
I PI
_ _ _ _ _ _ _ - _ - -J _ - -
FIG.4.4 The relationship of the subprocess interaction function K to the subprocess coupling function H .
Some remarks concerning the process P and its representation or decomposition into subprocesses have relevance in the subsequent developments regarding coordination. Ci) Each infimal control system has primary interest only in some aspect of the process, although the end result of its action depends on the entire process. We denote this “local” interest by associating with the ith infimal control system the ith components of the control input m and output y: the ith infimal control system is interested in the relationship between the control input mi and the output y i , which is expressed as the ith subprocess P i . (ii) The subprocess coupling functions Hidetermine how the process is decomposed and, as a rule, one should choose them so that they are as simple as possible. In most cases, the coupling functions H iwill be projection mappings: the interface inputs u iwill consist of components of the process’ terminal variables m and y. For example, if m = ( m l , m 2 , m3), and y = ( y l ,y 2 ,y3), the interface input uI might be the pair ( m 2 ,y3).
IV. Coordination
92
(iii) The interaction function K embodies the overall process P,since for any control input m and disturbance input o,the value of K, u = K(m,o),is the interface input u = H(m, P(m,0)).K can be considered as the subprocess which generates the interactions of the subprocesses Pi.It will not matter in our analysis whether we use K or the coupling functions H ithemselves; it will be more convenient however to use the interaction function K, or its com2 + U i , rather than the functions H i . ponents Ki:M x f
Control Subsystems We shall regard a control system, as mentioned in Chap. 111, as a system composed of a decision unit and an implementor connected in cascade. An implementor simply modifies the outputs of a decision unit so as to make them acceptable to another system. The decomposition of the supremal control system C,: W + V into a supremal decision unit do,
do: W + X o , and implementor co , co: w x
x, v, --f
is shown in Fig. 4.5a. There is assumed to be associated with the supremal w E W ,with the decision control system a family of decision problems go(&), set X o such that for any to in W the output xo = d 0 ( u ) is a solution of the
W
FIG.4.5 The control systems of the two-level hierarchy.
93
3. Coordinability
decision problem g0(cu). It is allowed that the output of co depend upon w as well as upon do(tu). Hence, for any given feedback information in W , the coordination input produced by the supremal control system is y = co(tu, d,(u)) = CO(CL’).
Consider now the ith infimal control system C i :% x bi+ M i . The decomposition of C i into an infimal decision unit di , d i : G$ x bi-+ Xi,
in cascade with an implementor ci , ci:9i x Xi+Mi,
is similar to that of the supremal control system and is shown in Fig. 4.5b. With the infimal decision unit di there is assumed to be associated a family of decision problems g i ( y , zi), y E V and zi E bi, with the decision set X i , such that for each pair (y, z i ) in V x bi the output x i = di(y, z i ) is a solution of the problem g i ( y , zi). In most of our analysis, except in some cases in Chap. VIII, the decision problems Qi(y, z i ) will depend only on the coordination input y. The implementor ci generates the control inputs for the ith subprocess as a function of the decision x i and possibly the feedback information z i ; hence, the control input applied to the ith subprocess is
mi = ci(zi di(y, Z i ) ) where y is the given coordination input.
3. COORDINABILITY
To coordinate the subsystems is to influence them so that they function or act harmoniously, such as one would coordinate the activities of individuals or groups within an organization. To make this idea of coordination operational, one has to define more explicity what is meant by “ acting harmoniously.” In general, this is done in reference to a goal or objective; parts of an organization are coordinated relative to an overall objective, so that the whole achieves the stated objective. Coordination is the task of the supremal control system, in which it attempts to cause a harmonious functioning of the infimal control systems. The success of the supremal in its task of coordination is judged relative to a given overall goal of the two-level system. Since the infimal control systems operate so as to achieve their own individual goals, a conflict generally develops among them and results in a prescribed overall goal most likely not
ZV. Coordination
94
being attained. The task of a coordinator is precisely aimed at reducing the consequences of such an intra-organizational conflict, if not to eliminate it a1together. Achieving a goal, as pointed out in Chap. 111, can be represented as a solution of a decision problem, and each control unit in a two-level system is defined in terms of a family of decision problems. To formalize the concept of coordination, it is necessary, however, to introduce an additional decision problem, in terms of which the success of the coordinating action is evaluated. Such a problem is defined in terms of the entire system and, in particular, the overall process P,and will be referred to as an overall decision problem. We shall introduce two notions of coordinability in the context of a two-level system. One notion is coordinability relative to the supremal decision problem, while the other notion is coordinability relative to a given overall decision problem. To facilitate the presentation, let P(x, 9)be defined for all pairs (x, g), where 9 is an arbitrary decision problem, as the predicate P(x, 9)= x is a solution of 9.
(4.7)
Hence, the predicate P(x, 9) is true, iff 9 is a decision problem, and x is one of its solutions. Since we are interested at present only with the command aspect of a twolevel system, we assume that the feedback information to the supremal and infimal control system is fixed ; consequently, we assume the infimal decision problems are parameterized solely by the coordination inputs, and there is given a single supremal decision problem. Therefore, let 9, be the given supremal decision problem, and for each coordination input y in %? let !Bi(y) be the decision problem specified for the ith'infimal decision unit and B(y) be the set {g1(y), . . . , 9 " ( y ) } of these problems. We should point out here that B(y)is itself a decision problem consisting of the n independent problems g i ( y ) ; the solutions of g ( y ) are exactly those n-tuples (xl, . ..,xJ, such that i 5 n, x i is a solution of Qi(y). for each i, 1 I The two notions of coordinability are now given as follows: Coordinability Relative to the Suprema1 Decision Problem
Since the feedback information is assumed to be fixed, we can for simplicity assume (without loss in generality) that %? = X , ,and hence the outputs of the supremal decision unit are the coordination inputs sent to the infimal decision units. We then say the infimal decision problems are coordinable relative to the supremal decision problem iff the following proposition holds : ( 3 Y ) ( 3 x)CP(x, %Y)) and PbJ,9 0 ) l .
(4.8)
3. Coordinability
95
Coordiniability relative to the supremal decision problem requires therefore that the supremal decision problem have a solution, and for some coordination input y, which solves the supremal problem, the set a(y)of infimal decision problems also have a solution. It is convenient in subsequent analysis to represent condition (4.8) in a form which explicitly recognizes the dependence of the solution of the second level problem 9,upon the action of the infimal decision units. Namely, the supremal decision unit through coordination affects the infimal decision units, and whether or not a given coordination input solves the supremal decision problem can be expressed in terms of the outputs of the infimal decision units. The dependence of the supremal decision problem on the outputs of the infimal decision units is expressed formally as
P(Y,9 o ) * ( j x ) C Q o ( ~ , x>l,
(4.9)
where Q,(y, x ) is a given predicate defined for all pairs (y, x ) in V x X , where X is the Cartesian product of the infimal decision sets Xi:
X = XI x
*.-
x X,.
Condition (4.9) simply states that a given coordination input y solves the suprema1 decision problem iff there exists a corresponding infimal decision x so that the condition expressed by the predicate Qo(y, x ) is satisfied. The supremal decision problem is, therefore, to find y in V such that Q,(y, x) holds for the decision x which is the output of the infimal decision units. Now, by substituting (4.9) into (4.8), and constraining the variable x in (4.9) to be the same x as that in (4.8), we arrive at the proposition ( 3~)( 3x)CW g ( ~ and ) ) Qo
(4.10)
as expressing coordinability relative to the supremal decision problem. Specific forms of the condition Q , will be introduced in conjunction with various forms of coordination principles.
Coordinability Relative to a Given Overall Decision Problem The overall decision problem is defined as a rule in terms of the overall process, and therefore its decision set may be taken as the control object M . Now, with fixed feedback information, the control inputs applied to the overall process depend only on the decisions of the infimal units; therefore, let the control inputs be expressed by the mapping zM:X - t M . Then, we say the infimal decision problems are coordinable relative to a given overall decision problem 9, iff the following proposition holds : ( W (34CP(x7 W ) and P(n,(x),
9)l.
(4.11)
I V. Coordination
96
Coordinability relative to the overall decision problem simply means that the coordinator, the supremal control system, can indeed influence the infimal decision units so that their resulting action satisfies the overall decision problem. For the sake of simplicity, and in reference to the importance of the overall decision problem, we shall say the two-level system is coordinable if the infimal decision problems are coordinable relative to the given overall decision problem. We shall refer to the infimal decision units as being coordinable in a given sense, iff their associated decision problems are coordinable in the same sense.
4. THE CONSISTENCY POSTULATE
For the successful operation of a two-level system, it is essential that the goals of its subsystems be in harmony. In particular, the necessity for harmony between the overall goal of a hierarchical system and the goals of its subsystems has been recognized in many different contexts. It is interesting in this connection to quote Galbraith [9], who pointed out the need for a harmony between three kinds of goals that can be recognized in a society: The relationship between society at large and an organization must be consistent with the relation of the organization to the individual. There must be consistency in the goals of the society, the organization and the individual.
He refers to this as the principle of consistency and argues for its validity in the context of any social system and, in particular, in the so-called" industrial state." In a two-level system, there are three kinds of goals formally described by three types of decision problems: the overall, the supremal, and the infimal decision problems. Harmony between these goals and, therefore, the consistency principle, in this context, can be formalized as a logical proposition from the following observations. (i) The infimal decision units of a two-level system are the only subsystems in direct contact with the overall process. If the overall goal is to be attained, it has to be through the action of the infimal decision units; the infimal decision problems or infimal decision units must be coordinable relative to the given overall decision problem. Hence, proposition (4.1 1) must hold. (ii) The supremal decision unit, through coordination, affects the infimal decision units, but in a way which promotes its own objective: the coordinator selects a coordination input so as to promote its own goal. The infimal
4. The Consistency Postulate
97
decision problems must then be coordinable relative to the supremal decision problem. Hence, proposition (4.10) must hold. (iii) The overall problem is essentially outside the two-level system; no one of the decision units within the hierarchy is specifically entrusted with solving the overall problem and thereby pursuing the overall goal, although the problem is defined in terms of the overall process. For consistency of the decision problems and hence the goals within a two-level system, coordination of the infimal decision problems relative to the supremal decision problem must be appropriately related to the overall decision problem. The desired consistency relation is given by the following proposition ( VY)( Vx){CP(x, %Y))
and
Qo
S.
[P(x,
and P(7&),
9911, (4.12)
which will be referred to as the consistency postulate for a two-level system. If proposition (4.12) holds in a given two-level system, we will refer to the goals or decision problems associated with the system as being consistent. The postulate states that the infimal decision problems are coordinated relative to the overall decision problem, whenever they are coordinated relative to the supremal decision problem. If the decision problems of a given two level system are consistent, the overall objective is achieved when the supremal decision unit coordinates the infimal decision units relative to its own objective. It is of interest to point out that the consistency postulate is a metamathematical statement. As such, it does not depend on the type of decision problems presented. It is equally applicable in such areas as the control of systems described by differential equations, optimization, artificial intelligence, problem-solving, and other nonnumerical applications. We can combine consistency and coordinability to obtain a requirement for the supremal decision problem which provides a framework, to be used subsequently, for the synthesis of the supremal decision problem. The requirement is given by the following notion: a given two-level system is coordinable by the supremal decision problem iff both propositions (4.10) and (4.12) hold; in other words, the decision problems of the two-level system are consistent, and the infimal problems are coordinable, relative to the supremal problem. This is expressed by the proposition (v~)(vx){CP(x,B(Y)) and Qo(Y, 41
*
P(ndx),
%I,
(4.13)
which is logically equivalent to the consistency postulate. Therefore, we shall refer to decision problems as consistent if (4.13) is true.
98
ZV. Coordination
5. COORDINATION PRINCIPLES
Although the consistency postulate and the need for coordinability delineate what properties the suprema1 decision problem should have, the coordinator synthesis problem is far from being resolved. Actually, these requirements only help to formulate the synthesis problem as a structural problem ;they impose conditions on the strategies which the coordinator can use. We do not, nowever, know what information the coordinator should receive, nor how it should use that information to select the proper coordination input, that is, what is the actual strategy for the coordinator. It is instructive to see how similar structural problems were solved in the past. For a good analogy, we go back to the pre-feedback era in the control field, when the situation was essentially the following: Given a process P with two inputs, a control and a disturbance, how should the control be selected so as to counteract the effect of the disturbance. To arrive at a solution of this problem, a feedback control principle is introduced: the output should be compared with a desired reference and the error fed back via a transformation to effect changes in the control. Similarly, Bellman’s optimality principle [26] suggest that the selection of the control over the immeadiately following increment of time should be made under the assumption that the control over the remaining time period will be optimal. Once a principle has been proposed, the remaining problem deals with the specification of the resulting strategy (e.g., determination of parameters in the feedback loop, or the selection of a control for the initial time increment) and the analysis of the conditions under which a specific strategy is applicable. Notice that the first step is heuristic and involves what might be called innovation, while the second is a mathematical investigation. We shall proceed in a similar way. First we postulate certain coordination principles which specify various strategies for the coordinator (i.e., they determine the structure of the coordinator) then we analyze the domains of validity or application of these strategies. A prime reason for the existence of conflict in a two-level system is the subprocess interactions and a particular infimal decision unit’s ignorance of the actions to be taken by the other infimal decision units. The task of the coordinator, in a quite general sense, is to influence the infimal decision units so that the resulting interactions are desirable in some given sense. There are three ways in which the interactions can be handled: (i) Interaction Prediction. The coordination inputs may involve, among other possible items, a prediction of the interface inputs; each coordination input in % provides a prediction ay = (a,Y,. . . ,any)of the interface inputs that will occur in the system upon the application of a control input.
5. Coordination Principles
99
(ii) Interaction Decoupling. Each infimal decision unit is instructed in the solution of its own decision problem to treat the interface inputs as an additional decision variable free to be selected at will. Apparently, the infimal decision problems are defined in this case as if the infimal decision units and subprocesses were completely decoupled. The interface input u selected by the infimal decision units is then a part of decision x and will be given by a mapping nu: X -,U. (iii) Interaction Estimation. The coordinator, rather than predicting precise values of the interface inputs, specifies ranges over which they may c U; vary: each coordination input y in % specifies a set U y= U, x * * * x Urny the ith infimal decision unit then regards the Uiv as an estimated range of the disturbances. We now propose three coordination principles based on the consistency postulate as expressed in (4.13)and the way in which the interactions (interface inputs) are handled.
Interaction Prediction Principle Assume the coordinator predicts the interface inputs. It is rather natural to assume, then, that success in coordinating the infimal decision units depends upon the accuracy in predicting the interface inputs or, more generally, upon the effects of the prediction errors. The simplest form of the interaction prediction principle is given by the proposition
('dY)( Vx){CP(x, %Y)) and @%.Ax)) = a']
=.
P(%fW(x), -@I.
(4.14)
The principle simply states that the overall decision problem is solved by the control input m = nM(x)whenever the infimal decision problems are solved by x and the interactions are correctly predicted, i.e., ay is indeed the interface input occurring when the control input m = nM(x)is applied. Instead of comparing the predicted and actual interface inputs, one can compare the actual and predicted operation of the subsystemsin a more general way. The general form of the principle is given by the proposition ( VY>( W C P ( x ,
W )and d Y ,
= 4"(Y)1 =>
P(n,(x),
911
(4.15)
where 4" and q are given functions from W and W x X , respectively, into a common set and are used to determine correctness in prediction. Apparently, (4.14)is a special form of (4.15).The more general form of the principle will be referred to as simply the prediction principle.
IV. Coordination
100
Interaction Balance Principle Suppose interaction decoupling is used in coordination. Success in coordinating the infimal decision units can then be evaluated in reference to the discrepancies or imbalance between the actual interactions and those desired by the infimal decision units. The interaction balance principle is given by the propositon ('dY)(~X){CP(X,
%Y)) and mhdx))
=. "(X)l
* P(%f(X>,w>.(4.16)
This principle states that the control input m = n,(x) solves the overall decision problem whenever x solves the infimal decision problems, and the desired interface inputs uy = nU(x) balance with the actual interface inputs u = K(m), occurring when the control input m = n,(x) is applied to the process. The general form of the principle, referred to as the balance principle is expressed by the proposition
(VY)( W{CP(X, %J)) and 4(Y, XI =
aY,XI1
* P(n,(x),
WI
(4.17)
where 4" and q are given functions from W x X into a common set used to determine whether a balance, in an appropriate sense, has been achieved. . Interaction Estimation Principle Assume now the coordinator specifies ranges over which the interface inputs may vary, rather than predicting precise values. In keeping with the pattern of the preceding coordination principles, we assume that success in coordinating the infimal decision units can be evaluated in reference to the accuracy of the estimations. The interaction estimation principle is then expressed by the proposition
(VY)(V-~{CP(X,%Y)) and K(nM(4)E uyl * P(n,(x),
W>. (4.18)
This coordination principle may be viewed as an extension of the interaction prediction principle; in fact, if the estimated ranges U yare singleton (unit) sets we have the interaction principle. The condition K(m) E U yin (4.18) means that the actual interface input occurring upon application of the control input m falls within the estimated range Uy. The general form of the principle, referred to as the estimation principle, is expressed by the proposition ( VY>(VX){CP(X> B(Y))and 4(Y, 4 Eq"(Y)l => P(n,(x>, 911, (4.19) where q" is a given set-valued function on W,and q is a given function on
Wx
x.
5 . Coordination Principles
101
Coordinability Notions in Relation to the Coordination Principles A comparison of the consistency postulate as expressed by proposition (4.13) and a given coordination principle immediately indicates the predicate Qo(y, x ) in (4.9) and, hence, the supremal decision problem. For example, suppose the functionsq"and q are given for the prediction principle; the predicate Qo(y, x ) is, then, immediately given by (4.15) as (4.20) and, consequently, the supremal decision problem is to find y in %'suchthat
where x is the resulting decision of the infimal decision units. We shall refer to the predicate Qo(y, x ) in the consistency principle as a coordinating condition, and if Qo(y, x ) is defined with respect to a given coordination principle, as indicated above for the prediction principle, we shall refer to it as the coordinating condition of the principle. We have now the following notions : (i) A given coordination principle is applicable if the corresponding logical proposition expressing the principle is true. For example, (4.14) must be true for the applicability of the interaction prediction principle, while (4.16) must hold for the interaction balanc.: principle. (ii) A system is coordinable by a given coordination principle if the principle is applicable, and there exists a coordination input y E %' such that the corresponding coordinating condition, Q,(y, x), is satisfied. The notion of applicabilityis useful in the sense that it provides a guarantee that utilization of the principle will not lead to erroneous results. Applicability of a coordination principle does not guarantee coordinability by the principle. For example, suppose for each coordination input y at least one infimal decision problem fails to have a solution; then any coordination principle is applicable, but the system is not coordinable. On the other hand, suppose that in a given two-level system the coordinating condition of a given coordination principle is never satisfied; trivially then, the coordination principle is applicable, but again the system is not coordinable by the principle, although the system could very well be coordinable otherwise. As mentioned earlier, once a coordination principle has been adopted, the problem of coordination can be treated mathematically.Indeed, the second part of this book is concerned to a large extent with such a study.
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102
6. VARIOUS ASPECTS OF THE COORDINATION PROBLEM
Coordination as an activity of the supremal control system is in reference to three types of decision problems: the overall, the supremal, and the infimal decision problems. In this respect, four different situations emerge : (i) Coordinator Synthesis. Given the overall and infimal decision problems, find a supremal decision problem goso that the system is coordinable by 90.
(ii) Coordination Methods or Procedures. Given a two-level system that is coordinable by the supremal decision problem, find an efficient procedure to obtain a coordination input that will coordinate the system. (iii) ModiJication Problem. Given a two-level system not coordinable by the supremal decision problem (although the consistency postulate might hold), find a modification for the infimal decision problems so that the modified problems are coordinable by the supremal decision problem. (iv) Decomposition. Given only the overall decision problem, find both appropriate infimal decision problems and an appropriate supremal decision problem so that the two-level system is coordinable by the supremal decision problem. We shall now briefly discuss the relationship between these situations. Coordinator Synthesis Problem
Our approach to the coordinator synthesis problem is based on the coordination principles presented in the preceding section. Once a coordination principle is adopted, the supremal decision problem is specified. Two questions are then immediately of interest: (i) Is the consistency postulate satisfied? Is the chosen coordination principle applicable? (ii) If so, does there exist a coordination input which coordinates the system by the supremal decision problem specified by the coordinating condition of the coordination principle? In other words, is the system coordinable by the coordination principle ? In the second part of this book, we shall investigate mathematically the applicability and coordinability by the proposed coordination principles in various specific situations. The variety of cases considered and the generality of the analysis is a testimony to the breadth of application of the proposed principles.
6. Various Aspects of the Coordination Problem
103
Coordination Procedures Once a supremal decision problem is selected, the problem of solving it remains. There are, of course, a variety of ways in which to proceed; the most interesting appear to be the following: Of-Line and On-Line Iterative Procedures
Improvement of a given coordination input toward the solution of the supremal decision problem, and the satisfaction of a coordinating condition if the consistency postulate holds, may be achieved under suitable conditions by a joint effort of the decision units on both levels, in an iterative procedure that can be described as follows: Let y k and mk be the coordination and control input at the kth stage of an iteration; then, based on the system’s performance, the supremal decision unit derives a new coordination input yk+’, which it hopes is an improvement over y k ; now, using yk+’, the infimal decision units arrive at their decisions and produce a control input m k + l . This iterative procedure is repeated until the supremal decision problem is solved, or a desired state is reached. In off-line situations, the control input does not have to be applied until the coordination input solves the supremal decision problem or coordinates the system. In such cases, the question arises as to whether or not the iterations converge to a desired state, and, if so, how fast is the convergence. In on-line situations, application of the control input often cannot be delayed, and, in the extreme case, the control input must be applied at each stage of the iteration, due to the fact that the system evolves in time simultaneously with the decision-making processes, hence some loss of performance can result from a delayed application of the control input. The objective of online coordination is to improve the system’s performance at each stage of the coordination process. Both off-line and on-line iterative coordination procedures will be considered in some detail in Part 11. Feedback Approach
Use of a coordination principle, if applicable, has the advantage that, if the coordinating condition is not satisfied (by a given coordination input and the resulting infimals’ decision) an “error ” signal can be detected and then used to improve the coordination input. For example, suppose the interaction prediction principle is applicable in a given two-level system, and the set U is a linear space. Then, for a given coordination input y the error could be given as &
= ay
- K(mY),
IV. Coordination
104
where aY is the predicted interface input, u = K(my) is the actual interface input occurring when the control input m y= c(xy) is applied, and x y is the infimals’ decision. A new coordination input y‘ can be obtained by applying an appropriate transformation F to the error 6, y’ = y
+ F(&) = y + F(aY - K(mY)).
The block diagram in Fig. 4.6 illustrates this approach in using the interaction prediction principle. Figure 4.7 gives an illustration of a similar approach when the interaction balance principle is employed. To define an error signal when employing the interaction estimation principle, we need a metric or norm in the set U in order to determine the distance of a point from a set.
.
FIG. 4.6 Application of interaction prediction in second-level feedback.
FIG. 4.7 Application of interaction balance in second-level feedback.
After a coordination principle is selected and the feedback error signal established, the next question is what kind of transformation should be put in the second-level feedback. This type of problem is solved, in general, by specifying a class of transformations and selecting one element of that class on the basis of an assumed criterion. Usually, the class of transformations is specified such that the problem reduces to the selection of parameters in the second-level loop. Since the task of the second-level feedback, in general, is to eliminate the error, the simplest approach would be to put a classical PID controller in the second-level loop; the problem is then one of selecting gains and other controller constants. In more elaborate approaches, one would select a more complicated feedback structure derived, perhaps, from an optimization approach. For example, in Chap. VI, the analytical solution for the optimal coordination input derived for the linear-quadratic case can be used to construct second-level feedback. This problem will, however, not be pursued here any further ; nevertheless, it certainly presents a fruitful direction for developing new interesting and practical results.
6. Various Aspects of the Coordination Problem
105
To put this approach in a proper perspective, the following should be noted : The stimulus for change in coordination comes from an error measured, so to speak, at the output of the process. There is, therefore, an inevitable delay between the occurrence of a change, the response of the second-level feedback, and the resu!ting correction of the error. In short, any strategy involving only feedback and no feedforward compensation is “ suboptimal.” Improvement, of course, can be made by combining such a feedback with a feedforward action, and research along this line should be rather fruitful. Second-level feedback derived from the coordination principles has the advantage of any standard negative-feedback system : sensitivity of the performance to disturbances is reduced. The actual benefit, of course, depends upon the type of system and the disturbances involved, but second-level feedback can offer a relatively simple solution of the on-line coordination problem. This point deserves considerably more detailed investigation, particularly in the context of specific applications where advantages of the known structure can be properly utilized. Modification The problem of modifying the infimal decision problems can be posed as follows: Suppose the consistency postulate is satisfied in a two-level system having the infimal decision problems g(y),y E%, but the system is not coordinable. The modification problem is then to find a new family of infimal decision problems g’(y),y E %‘ such that (i) G2 G %‘ and 9 ( y ) = g’(y),for all y in 5%‘; (ii) the validity of the consistency postulate is preserved ; (iii) the system is coordinable by the supremal decision problem. These conditions require the original infimal decision problem to be embedded in the new family of problems, the consistency postulate to hold, and the existence of a coordination input that coordinates the system by the supremal decision problem. Often a single decision problem may be specified for each of the infimal decision units with no apparent provision for coordination. Despite the validity of the consistency postulate, it might be necessary to expand the number of possible infimal decision problems as indicated by condition (i), so that the system becomes coordinable by the supremal decision problem. Modifications of the infimal decision problem might be necessary in other situations. For example, a given two-level system could be coordinable but not by the supremal decision problem or, alternatively, the consistency postulate is not satisfied. Or, the infimal decisions problems might not be
106
ZV. Coordination
coordinable relative to either the supremal decision problem or the overall decision problem. Modification of the infimal decision problems is the means by which these pathologies can be eliminated. We shall consider in Part I1 the problem of modification in some detail. Various kinds of modifications will be presented and analyzed. Decomposition By decomposition, we mean the following situation: Given the overall decision problem, find the infimal and supremal decision problems so that the consistency postulate holds. We argue that the decomposition problem reduces to the three preceding problems ; namely, the coordinator synthesis problem, the modification problem, and the coordination procedure problem. Indeed, there are many ways of decomposing a given overall decision problem into subproblems. The real question, however, is how to coordinate them; this requires first the selection of a coordination principle (a supremal decision problem), modification of the subproblems, and finally the design of a method to find a coordination input that coordinates them. In any case, the key problem is that of coordination, and this is the problem on which we shall focus our analysis in the succeeding chapters. It might be considered that, in general, any new method of coordination implies a decomposition technique. The generality of the developed theory of coordination should be noticed in this context. The term decomposition is often used in the context of optimization problems or, more specifically, in mathematical programming problems ; however, the concepts and theories of coordination we develop here are much broader; they apply to a vastly more general class of systems such as on-line situations where a satisfactory rather than an optimum level of performance is sought.
PART II : A MATHEMATICAL THEORY OF COORDINATION
Chapter V
GENERAL THEORY OF COORDINATION FOR UNCONSTRAINED OPTIMIZING SYSTEMS
We shall consider in this chapter a two-level system as described in the preceding chapter. Starting from such a general type of system, we shall introduce the following two assumptions: (i) There are no external disturbances, so that all decision problems can be defined as optimizations; (ii) There are x M, no constraints on the decision sets, or more explicitly M = MI x Apart from the above two assumptions, the infimal and overall problems are formulated and the coordination problem analyzed in the most general setting, in order to show the broad applicability of the proposed coordination principles. We restrict ourselves to unconstrained optimization problems in order to emphasize the main, structural aspects of the results, and to avoid entanglement in special considerations of a technical rather than a conceptual interest. A deeper theory in the unconstrained case will be presented in Chap. VI, while constraints will be considered later in Chap. VII. The effect of disturbances is the subject for Chap. VIII.
---
.
1. INTRODUCTION
The coordination principles are central to our approach to coordination, for they offer strategies for the selection of coordination inputs. To help intuition, we shall give here some simple examples to illustrate the basic features of the coordination principles. 109
V. Coordination of Unconstrained Optimizing Systems
110
Let us consider the simplest form of a multilevel system: a two-level system having one suprema1 decision unit and two infimal decision units. Let the subprocesses P , and P, be defined on R2 by the equations
+
y , = 2m, u1 = P,(m,, u,,), Assume that the equations
y,
u1 = Y , 7
= 2m2 - u,
u2
= P 2 ( m 2 ,u,).
(5.1)
= Y17
specify the subprocess couplings so that the overall process as shown in Fig. 5.1 is the mapping P : R2 + R2 defined by the pair of equations y , = m,
+ m,,
y , = -m,
+ m,.
Let the overall performance function G be the real-valued mapping defined on R2 x R2 as G(m,y ) = m i 2
+
+ ( y , - 1), + (y2 - l)’,
and assume that the overall decision problem is the optimization problem: find a control input riz in R2 such that G(h, P(riz)) I G(m, P(m)) for all other controls m in R 2 . Let P = (P,, p2) be a pair of real numbers, and for each such pair let the given infimal performance functions GI, and G,, be the real-valued functions Glg(m1, u1, Y l ) = m12 + (Y, - 1)’ G2,(m2
7
u2
9
~
2
=) mz2
+ (YZ - 1)’
+ P1u1’ - P2Y12, + P z ~ 2 -, P i ~ 2
~
9
on R x R x R. Assume for any given pair p that both infimal decision prob, lems are optimization problems: find the pair 2iin R2 such that G i p ( x iPi(xi)) IG i p ( x iP , i ( x i ) )for all other pairs x i in R 2 . The pairs P are coordination inputs. For each P, the solution of the infimal problems yields local control inputs m,(P) and m,(P). The coordination problem is then to find a pair B so that the corresponding local control
1. Introduction
111
inputs m,@) and rn,(B) are the overall optimal controls; i.e., the control input m(P) = (m,(P),m,(P)) is a solution of the overall optimization problem. Notice that the infimal problems require optimization with respect to not only the local control inputs, but also the interface inputs u1 and u, . For a given pair P, let ul(P) and u,(P) be the values of the interface inputs selected by optimization. This selection is made by regarding the subprocesses as decoupled; the interface inputs are considered free, manipulated inputs. Because of the subprocess couplings there is actually no such freedom, and, therefore, if the control inputs rnl(P) and m,(P) are applied to the overall process, the actual interface inputs obtained from P and the subprocess couplings are u1 = -m,(P) + 4 P ) , u2 = rnl(P) + m,(P), and generally do not agree with those selected by optimization. This discrepancy provides a basis for designing a coordination strategy. The interaction balance principle proposed in Chap. IV states for the example under consideration: the control input m(P) = (m,(P),m,(P)) is overall optimal whenever P is such that the interface inputs balance: '1
= u1(P>7
'2
= '2(/j)'
This balance condition requires the desired interface inputs to be precisely those which occur when the control inputs ml(P) and mz(P) are applied to the process. One can easily verify that the interaction balance principle applies to the example under consideration. If P is the pair (0, -$), the solutions of the infimal optimization problems are (m1(P)7
'l(P))
= 7'(
3)7
(n12(P)7 u Z ( P ) )
=(59
3).
Upon solving the overall optimization problem, we find that the control m(P) = (0,3) is an overall optimal control input and, moreover, if the control input m(P) is applied, the resulting interface inputs would be u(P) = (3, +), which agrees with those selected by optimization. Hence, interaction balance is achieved, and the local control inputs form an optimal solution of the overall problem. Use of the interaction balance principle reduces the coordination problem to finding the pair (P1, bz) so that interaction balance is achieved. To illustrate another coordination principle, namely the interaction prediction principle, consider again an example of a two-level system having only two infimal decision units. However, let the local processes Pl and P, be the real-valued mappings defined on the sets M , x R and M , x R, respectively, as y, = rn,
+ ($)u13 = Pl(ml , ul),
y 2 = m, - uZ3= P, (m,, u,)
(5.2)
112
V . Coordination of Unconstrained Optimizing Systems
where M ifor i = 1, 2 is the set of all mi in R such that Im,(I1. Let the subprocess couplings be as before: u1 = y 2 and u, = y l . The overall process is the mapping P:M , x M , -+ R2 obtained by eliminating the interface inputs u, and 2i2 from the subprocess equations. Let the infimal performance functions G, and G2 be given on M , x R and M, x R, respectively, as Gl(ml, y1) = m12 - (
3 h + yl,
G2(m2, Y,)
= 4%'
+ (%m2 + Y , ,
while the overall performance function G is given on M x R2 as
G(m,y ) = m12
+ 4m2' + y 1 + y , .
The overall and infimal problems are assumed to be optimization problems. We will attempt to coordinate the infimal problems by having the coordinator predict specific values of the interface inputs. The coordinator will predict the values a1 and a, for the interface inputs u1 and u,; the infimal decision units will then select local control inputs by optimizing with respect to the predicted values of the interface inputs. For a given prediction a = (al, a,) of the interface inputs, denote the solutions by ml(a) and m2(a). We contend that the system can be coordinated by predicting appropriate values of the interface inputs. Let a = (a1, a2) be a given prediction of the interface inputs, and suppose the resulting control input m(a) = (m,(a),m2(a))is implemented. The actual interface inputs are then (u2 >
4 = ( Y l , Y2) = m l ( a ) , m2(4>,
and, in general, do not agree with the predicted values a, and a 2 . The interaction prediction principle for the example under consideration states : the control input m(a) = (ml(a),m2(a))is overall optimal whenever the prediction a = ( a l , a,) is correct, u1 = sll,
u, = a,.
It is easy to verify that the interaction prediction principle is valid for the example under consideration. Indeed, if a1 = -3 and a2 = -4, the resulting local control inputs are
m,(a) =
-+,
m2(a)=
-3,
and m(a) = (- 3, - $) is overall optimal, while the actual interface inputs which occur upon application of m(a) are uI = -3 and u, = -f. Moreover, for any prediction a of the interface inputs, the resulting control input m(a) is overall optimal only if a is a correct prediction. The discrepancies between the predicted and actual interface inputs can therefore be used to judge the goodness of the selected coordination input;
2. A Two-Level Optimization System
113
the coordination problem then reduces to the problem of predicting interface inputs. Furthermore, the differences between the predicted and actual interface inputs can serve to evaluate how far from optimum the system is operating and as a guide in improving the predictions. These examples illustrate the two basic coordination principles analyzed in this chapter. We shall first investigate the conditions under which application of the principles leads to successful coordination ;we shall then explore how one might proceed in search of a coordination input which coordinates infimal optimization problems. Attention will also be given to a comparison of the principles, regarding their relative applicability.
2. A TWO-LEVEL OPTIMIZATION SYSTEM
We consider in this chapter a two-level system as described in Chap. IV, but with the following two more specialized assumptions added : (i) The effect of the environment (disturbances) is either nonexistent or is completely known beforehand ; the overall system operates under the conditions of certainty. (ii) The overall decision problem and the infimal decision problems are given as optimization problems. Except for these two specializations, the system considered in this chapter is as general as that in Chap. IV. The objective here is to concentrate on the structural problems and reveal the basic properties of a two-level system which are not dependent upon specific technical issues associated with more detailed descriptions. For this reason, we assume that the optimization problems are essentially unconstrained. Effects of more detailed descriptions of the system and constraints will be investigated in subsequent chapters. Since the disturbance is assumed known (fixed or nonexistent), the overall process becomes a mapping P:M + Y, with the interface input u given by H : M x Y + U or K : M 4 U,the latter defined in terms of P and H, and the subprocesses become the mappings Pi:M ix Ui-+ Yi. In connection with the second assumption, we shall use the general optimization or optimal control problem as defined in Chap. 111.
The Overall Optimization Problem The overall decision problem, denoted by 9,reflects the overall goal of the two-level system, and, as an optimization problem, is specified, in general, by a pair (9,M ) , where g is a given objective function. We shall
114
V . Coordination of Unconstrained Optimizing Systems
assume that g is defined in terms of the overall process P and a given overall performance function C: M x Y -+ V, g(m) = G(m, P(m)). A solution of the overall problem 9 is then a control input
(5.3) in M such that
g(&) = min g(m). M
(5.4)
Such a control input riz will be called overall optimal. Notice that, formally, the overall optimization problem is an unconstrained optimization problem: In fact, however, M could be a subset of a larger set and defined as those control inputs satisfying certain prescribed constraints; nevertheless, M is assumed to be the Cartesian product of n components, M = MI x * x M , .
--
The Infimal Optimization Problems We stated in Chap. IV that each coordination input y in %? specifies a decision problem for each infimal decision unit. Henceforth, let g i ( y ) denote the decision problem specified by y for the ith infimal decision unit. The infimal decision problems will be optimization problems; therefore, the ith infimal decision problem g i ( y ) , y E %?, is assumed to be specified by the pair (gip,Xi,), where giyis a given infimal objective function defined on a set Xiand X i , is a given subset of X i . We assume gi, is defined in terms of an outcome function P i , and an infimal performance function Gi, , Yiy(xi) = Giy(xi Piy(xi))* (5.5) A solution of the infimal optimization problem g i ( y ) is then an element x i y in Xi,such that giy(xiy)= min giy(xi). 3
Xi,
Such an element x i is a y-optimal infmal decision or simply an optimal infimal decision when y is understood from the context. There is no a priori reason why either P i , or Gi, have to be related to P and G ; however, in view of the structure of a two-level system presented in Chap. IV, it is reasonable to make’ some assumptions in this direction. Therefore, we make the following assumptions about each of the infimal optimization problems: (i) The outcome function P i , is the subprocess P i ;P i , = P i ; (ii) The infimal performance function Gi, is a mapping G i , : M i x Yix U i + V. It should be noticed that no specific relationship is assumed between the
2. A Two-Level Optimization System
115
overall performance function G and the infimal performance functions Gi, . The infimal optimization problem g i ( y ) is now specified by a triplet ( P i , G i y ,X i J , where Xi,is a subset of M i x U i . It is of some interest to indicate that assumptions (i) and (ii) made in specifyingthe infimal optimization problems are not too restrictive and actually arise in a natural way. First of all, let us consider assumption (i). Use of the local sets M iand Y i in the definition of the outcome function P i , needs no explanation. It is also only natural to assume that the ith infimal decision unit is aware that it cannot completely decide the outcome y i , and that there are some outside effects which, most simply, can be lumped together into an additional set eiin the domain of the outcome function, leading to a mapping from Mi x 9?li into Y i. The inability of the ith infimal decision unit to completely determine the outcome y i is due solely to the activities of other infimal decision units, which affect it only through the interactions. It is natural, therefore, to assume ai= U i . Finally, we come to the most restrictive assumption, namely P i , = P i . The actual significance of this assumption depends on how the decomposition is performed. In general, if no constraint is imposed on how the decomposition of P into the subprocesses is to be made, one might say that for almost any selection of Pi,there exists a decomposition of P and corresponding coupling functions H i such that Pi,= P i ;it all depends upon the selection of the coupling functions H i . The assumption Pi,= Piis, therefore, not too restrictive in general. However, in subsequent developments, we shall often take H a s a projection mapping or, more generally, as a linear mapping in an appropriate sense. This assumption will be made in order to be able to concentrate on the central issue of the coordination problem, namely, the relationship between the levels rather than accuracy of the firstlevel models which, although technically quite important, is not taken as the main issue here. Assumption (ii) concerning the infimal performance functions G , needs no elaboration since, as already mentioned, no relationship is assumed between them and the overall performance function G . Modes of Coordination and Coordinability There are two ways to influence the infimal optimization problems, namely, through the performance function G i , and the set Xi,of feasible decisions; there are, therefore, two ways in which the coordinator can influence the infimal decision units: one way, referred to as the goal coordination mode, consists of changing the infimal performance functions ; the other way involves changing the sets Xi, and is referred to as the constraint or image coordination mode since it involves changing P i , .
V. Coordination of Unconstrained Optimizing System
116
Goal Coordination
The simplest way to describe the goal coordination modes is to start with n functions Gi, ,.. ., G,, , G,,: M i x Yi x Ui x 9 3 4 V
where 98 is a given set. We assign to each y in W a unique from G,, the infimal performance function Gi, ,
By in %3and obtain
Giy(mi 9 ui Yi) = Gidmi ,Y i > ui 7 By)* (5.7) To emphasize the dependence of Gi, on we replace the index y with B, so that G, = G,, , and hence gV = g i y when /?= By. We shall, on occasion, face the situation in which no provision is made for goal coordination. In such cases, we shall simply omit the index y (or fi) and assume that the infimal performance functions are mappings G,:M i x Ui x Yi V . We shall refer to these infimal performance functions as unmodijied or uncoordinated. 7
a
Image (Constraint) Coordination
For each y in V there is given a set Xi,of feasible decisions for the ith infimal decision unit which is, in general, a subset of M i x U idue to assumption (i) concerning the outcome functions. Hence, the sets Xi, as proper subsets of M i x U i represent constraints imposed on the infimal decisions. Since we are not explicitly considering constrained optimization problems, we shall impose a third assumption on the infimal decsision problems; namely, (iii) X i , = M i x Uiy, where U i yis a given subset of U i. Constraint coordination then reduces to the selection of appropriate subsets of the interface objects; in reference to the ways of handling interactions presented in Chap. IV, two modes will be considered in this chapter as appropriate for optimizing systems. The Interaction Prediction Mode
For each y in V, the specified set U yis a singleton set {my} and hence the ith infimal problem 9 i ( y ) is optimization over the set M i relative to the prediction cliy of the interface input ui . For each prediction cc = (q,. ..,c,) of the interface inputs, the infimal decision units apparently receive a new “image” of the remaining subprocesses; the image for the ith infimal decision unit is P:: M i Y i , P4(rni) = Pi(rni, cci).
3. Coordination Principles and Suprema1 Decision Problem
117
The Interaction Decoupling Mode For each y in W, the specified set U yis the whole set U, and hence X i y = Xi= M i x Ui for each i, 1 5 i 5 n. Each infimal optimization problem is defined completely independent of the other infimal optimization problems, and, more importantly, the infimal decision units are instructed to select in an optimal fashion, not only the local control inputs, but the local interface inputs as well. We point out that, even though we assume X i , = M ix Ui , our subsequent analysis is valid for the case when Xi,is any subset of M i x Ui containing all pairs (mi,Ki(m)),where m is in M .
Coordinability of Optimizing Systems For simplicity in presentation, we will henceforth use the following notation in this chapter: (i) r?t = an overall optimal control input; (5.8) (ii) x y = a pair (my,u y ) such that each pair (miY, uiy)is y-optimal. Now, by definition, the two-level system is coordinable iff the following proposition holds: ( 3y)( 3xy)(3 r ? t ) c x M ( x y ) =
(5.9)
where nM is the projection mapping from M x U onto M ; in other words, the system is coordinable iff there exists y in W and y-optimal infimal decisions (miY,uiY)for each i, 1 5 i 5 n, such that the control input m y = (mIY,...,mnY) is overall optimal, m y = A. A coordination input y which satisfies (5.9) will be referred to as an optimal coordination input. Hence, the two-level system is coordinable iff there exists an optimal coordination input. 3. COORDINATION PRINCIPLES AND SUPREMAL DECISION PROBLEM
Once we are given the overall optimization problem and the infimal optimization problems parameterized by the coordination inputs, the coordinator’s task is simply to find an optimal coordination input, provided one exists. The supremal decision problem should then be such that its solution is an optimal coordination input. The real question, though, is what this problem should be. Trivially, one might select the overall problem as the supremal decision problem; in such an approach, however, there is no genuine division of labor between the decision units on different levels. An
V. Coordination of Unconstrained Optimizing Systems
118
alternative approach is provided by the consistency postulate and coordination principles presented in Chap. IV; for a two-level system where all the decision problems are optimization problems, the prediction and balance principles of Chap. IV appear particularly suitable. The Balance Principle
For the balance principle, interaction decoupling is used as a coordination mode and, therefore, only goal coordination can be employed by the coordinator. We shall analyze two specific forms of the balance principle. Interaction Balance
This specific form of the balance principle has already been presented in Chap. IV; we can express it here by the following proposition: ( V y ) ( Vxy)(3h){[(m,u) = xy and
s- m = A}.
K(m) = u]
(5.10)
The principle states that an overall optimal control input is given by optimal infimal decisions whenever the interface inputs balance. Performance Balance
In this form of the balance principle, the infimal costs (performances) are compared, rather than the interface inputs themselves. For each y in %? let S,: M x U + V" be defined as
s&%4 = (91y(m,,
u1),
. . gn,(mn ufl)). . 7
(5.1 1)
9
The performance balance principle is then expressed by the proposition: ( V y ) ( Vxy)(3h){[(m,u) = x y and
gy(m,K(m))= g,(m, u)]
e-
m= (5.12)
which intuitively states that an overall optimal control input is given by optimal infimal decisions whenever the expected and actual infimal costs (performances) balance. The Prediction Principle
For a given coordination input y in %? let my denote a control input in M such that (my,a') = xy,where a y is the predicted value of the interface inputs. The interaction prediction principle defined by (4.14) may then be expressed by the proposition ( V y ) ( Vmy)(3 h ) { [ m= my and
K(m) = cry]
e-
m = A}.
(5.13)
4. Conflict Resolution in a Two-Level System
119
This special form of the prediction principle does not recognize whether or not goal coordination is used; it refers only to the correctness of the interface input predictions. The usefulness of the interaction prediction principle is very much reduced if goal coordination is not used; either the chances of a correct prediction are small, or some coirect predictions do not lead to an overall optimum. However, when goal coordination is employed, accuracy in predicting the interface inputs is not always sufficient to guarantee overall optimality. This suggests the need for a more general form of the principle. To explicitly recognize two modes of coordination (interaction prediction and goal coordination), we shall express each coordination input y as a pair (a,j?), where a = ay is the predicted value of the interface input, and j? = by is the parameter specifying the infimal performance functions G i y= G , . In this case, the coordination object V will be taken as a set d x 98, where d E U and 98 is a set of values for j?. Let q : M + 98 be a given mapping, and let 4,,(m)= (K(m),r ( 4 )
(5.14)
for all m in M . Then, for q,,,the prediction principle given by (4.15) is expressed by the proposition (Vy)( VmY)( 3A){[m= my and
q,,(m)= y ]
m = riz}.
(5.15)
To distinguish the use of a given mapping q, we shall refer to the (interaction) prediction principle as being relative to q . We remark here that, if a given two-level system is coordinable by the prediction principle relative to a given mapping q , the search for an optimal coordination input can then be restricted to the subset q,,(M) of the coordination object V.
4. CONFLICT RESOLUTION IN A TWO-LEVEL SYSTEM
Whether or not a given coordination principle is applicable, and whether its use results in an optimal coordination input, depends on the relationship between the overall objective function and the infimal objective functions. In general, the dependence is on the relationship between the overall and infimal problems; however, since we ape restricting our attention to unconstrained optimization problems, this dependency may be expressed in terms of the objective functions. Indeed, the infimal decision units when striving to minimize their own individual objective functions do not generally achieve the overall optimum. There therefore exists a conflict among the infimal
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V. Coordination of Unconstrained Optimizing Systems
decision units. The coordination principles state that detrimental conflicts are resolved when certain kinds of balances are secured among the infimal decision units. These balances, however, can be achieved more easily, or perhaps only if the system has certain properties. Our objective in this section is to introduce some basic systems properties and investigate their importance in conflict elimination. The results of the analysis will be directly applicable in the study of coordinability in the sebsequent sections of this chapter. There are, in general, two kinds of conflict within a two-level system: interlevel and intralevel conflict. Interlevel conflict is the conflict between two adjacent levels, or, generally, between any two different levels in a n-level system. In a two-level system, the overall goal might be to attain minimum overall cost, while the infimal goals might be to attain minimum local costs. If the overall goal and infimal goals are not compatible in the sense that attainment of minimum local costs prevents minimum overall cost, there is a conflict between levels. Intraleuel conflict is the conflict within a single level. Again, suppose the infimal goals are to attain minimum local costs. If the infimal goals are not compatible with each other in the sense that the attainment of minimum local cost by one infimal decision unit prevents another from attaining minimum local cost, there is a conflict within the given level. The properties introduced in this section indicate the existence or absence of these conflicts and may be used not only to characterize two-level systems and provide insight into the nature of the conflicts, but also to guide in the synthesis and alternation of two-level systems, as well as coordination. The properties are defined in terms of the interrelationships between the objective functions reflecting the various goals of a two-level system and will, therefore, be referred to as " goal-properties " of a two-level system. A starting point for defining the goal-properties is provided by the following auxiliary functions, defined in terms of the objective functions associated with the two-level system: (i) The overall cost (Objective)function g : M -+ V. (ii) The injimalcost functions hi,: M + V. For each y in %, and i, 1 5 i 5 n, the function hi, is given on M by the equation hiy(m) = giy(mi Ki(m))* 7
For any control input m in M , the value hi,(m) is the cost incurred by the ith infimal decision unit due to the local control input m i and the actual occurring interface input ui = Ki(m). (iii) The interlevel(performance)functions I),: V"+ V . For each coordination input y in W, there exists a unique relation Y yC _ V" x V, y y
=
W I y ( m ) ,..., k y ( 4 ) g(m)): , m
E MI,
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121
which relates, for any control input m in M , the overall cost g(m) to the actual infimal costs h,,(m). We refer to Y yas the interlevel (performance) relation for y. In general, the domain of Y yis a subset of V", meaning that there are some n-tuples (vl, . . . , v,) of infimal costs in V" which cannot occur for any control input. If Y yis a function, then d m ) = yy(hy(m), . . ., hny(m))
for all control inputs m in M , and hence the overall costs are a function of the actual infimal costs. Therefore, if Y yis a function, we say an interlevel (performance)function, denoted by $, ,exists for the given coordination input y, and we define it as an extension of Y yto the whole set V". (iv) The apparent-overall objective functions gw:W x M x U - t V. If for each coordination input y in V there exists an interlevel function $ y , we say an apparent-overall objective function exists and define it as the function gu on %? x M x U , g d Y , m, U) = $y(gly(m1, ul),
*.
.?
gny(mn
9
4).
(5.16)
The function g o , if it exists, then gives the overall cost as it appears to the infimal decision units; it does not always give the true overall cost, for it accepts all pairs (m, u) in it4 x U, some of which violate the constraint u = K(m) ; however for any y in V and m in M 9&, m, f4m)) = g(m),
and, therefore, gr(y, m, u) is the true overall cost whenever u = K(m).
The Monotonicity Property The existence and properties of an interlevel function reveal some important characteristics of a given two-level system. Of particular interest both conceptually and formally is the " monotonicity" property, which we introduce here and illustrate through some examples. First, recall the usual notion of a monotone function. A function f from a partially ordered set into another partially ordered set is (i) order-preserving or monotone iff f (x) sf(x'), whenever x I x', and (ii) strict order-preserving or strictly monotone iff f(x)
122
V. Coordination of Unconstrained Optimizing Systems
whenever f,(x) 5 f,(x'), and f , is strictly monotonically related to f2 iff fib)
-4
+
FIG.5.2
proper subset of a partially ordered set V , the function 4 can surely be extended to a function on V ; the extension, however, need be monotone only on 9 ( f 2 > . We now apply the above concepts of monotonicity to introduce a characteristic property or a two-level system. A given two-level system is referred to as having the monotonocity property if its interlevel relations are monotone functions. If a given two-level system has the monotonicity property, we shall assume, in view of the preceding remarks, that for each coordination input y in %' there exists a monotone function $7 from V" into Vserving as an interlevel function for that coordination input. An important aspect of two-level systems having the monotonicity property is expressed in the following:
4. Conflict Resolution in a Two-Level System
123
Proposition 5.1 Suppose a given two-level system has the monotonicity property. Let y be a given coordination input in %. Then g(m) = ming
whenever the control input m in M is such that
hi,(m) = min hi,, PROOF.
i = 1, ...,n.
Since the system has the monotonicity property, the equation = $y(hly(m), *
*
., hny(m))
holds for each control input m in 1 6 when $, is an interlevel function. Since $, is monotone, the conclusion follows immediately. If a two-level system has the monotonicity property, the overall objective function is monotonically related to the infimal cost functions; this describes a certain absence of interlevel conflict: a decrease in each of the actual infimal costs does not cause an increase in the overall cost. In fact, a decrease in each of the actual infimal costs would in turn cause a decrease in the overall cost whenever the monotonic relationship is strict; a steady decrease in the actual infimal cost would therefore cause the overall cost to approach a minimum.
Examp Ie 5.1 Let V = R, the set of real numbers, and for each y in %, let $, = $ 1 R" + R be an interlevel function for a given two-level system.
c:=
(i) Suppose $(v) = a, vi . If a, > 0 for each i, 1 2 i > n, then $ is monotone, and the system has the monotonicity property. If a, > 0 for each i, 1 I i In, then $ is strictly monotone, and the system has the strict monotonicity property. oil. Then, $ is not monotone; however, if (ii) Suppose $(v) = hi,2 0 for each i, 1 I i In, and all y in %, the system has the monotonocity property. x 0,. Then the system has the mono(iii) Suppose $(v) = vl x v2 x tonicity property if hi, > 0 for each i, 1 Ii < n, and y in %. (iv) In general, if qi: R" + R is monotone, i = 1, . . . ,p , and $o: RP-+ R is monotone; then $(u) = $o($l(v), ..., $,(u)) is monotone. (v) In general, if $ is differentiable, it is monotone iff all its partial derivatives are everywhere nonnegative.
Ic:=l
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V. Coordination of Unconstrained Optimizing Systems
Interlevel and Intralevel Harmony One might legitimately consider the decision units of a two-level system as being in some kind of harmony, if each of them can simultaneously achieve its own goal in spite of the interactions between them. In the context of optimizing systems, harmony would be characterized by the simultaneous achievement of their respective optima. If a given two-level system has the monotonicity property, then it has a desirable characteristic expressed by Prop. 5.1, but there is no guarantee of the existence of a control input which simultaneously minimizes each of the actual infimal costs. If such a control input exists, we would like to think of the system or the infimal cost functions hi, , 1 5 i 5 n, as possessing intralevel harmony. The monotonicity property is sufficient, but not necessary, for a given two-level system to have the characteristic expressed by Prop. 5.1; in fact, the result of the lemma does hold for some two-level systems which do not have the monotonicity property; hence, we would like to think of any given two-level system with this property as having interlevel harmony. Let us first introduce two notions of harmony in a rather abstract setting and then relate these notions to harmony in a two-level system. A family of functionsfi , 1 5 i 5 n, has harmony if there exists an element that is common to the domain of each f i and minimizes eachfi over its domain. Such an element will be referred to as harmonizing thefunctions f i or simply harmonizing, if the family of functions is understood from the context. The functionsfi need not have common elements, in which case the family could not have harmony; on the other hand, their domains could be the same singleton set, in which case the family would trivially have harmony. A family of functions fi, 1 5 i 5 n, is in harmony with a function f if every element harmonizing the functions f i is in the domain off and minimizes fover its domain. If the family does not have harmony, then, trivially, it is in harmony with any function. Example 5.2
Consider the functions f i whose graphs are given in Fig. 5.3. The family {f i , f 2 } has harmony (the harmonizing element in their domains is x,,) and is in harmony with f 3 . We now apply these abstract notions to describe the various forms of harmony which might exist in a two-level system. UnrestrictedHarmony
A coordination input y in V produces unrestricted infinial harmony if the corresponding family of infimal cost functions hi,, 1 Ii I n, has harmony. Hence, a coordination input y produces unrestricted infimal harmony iff
4. Conflict Resolution in a Two-Level System
125
FIG.5.3
there exists a control input in M that minimizes each infimal cost function hi, over M . A coordination input y in % produces unrestricted interlevel harmony if the corresponding family of infimal cost functions hi,, 1 I i I n, is in harmony with the overall objective function. We say a two-level system has unrestricted interlevel harmony if each coordination input produces that harmony. A two-level system has unrestricted interlevel harmony iff a given control input m in M minimizes the overall cost, whenever it simultaneously minimizes the infimal cost function hi,, 1 Ii I n, for some coordination input y in W.
Proposition 5.2
If a given two-level system has the monotonicity property, it also has interlevel harmony. The proof of this proposition is obvious. Its converse, however, is not in general true. A given two-level system might have interlevel harmony and yet not have the monotonocity property, as shown in the example below.
Example 5.3 Suppose a given two-level system is such that V = R and M = MI x M , = R2. Also, suppose
+
g(m) = 2 sinz(m12 m,')
+ k(m12+ mz2)
is the given overall objective function, where k > 0, and, for each y in %',the infimal cost functions hi, are
h,,(m)
= m12
+ BlYm,,
h,,(rn)
= mz2
+ Bzyml.
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V. Coordination of Unconstrained Optimizing Systems
If the coordination input y is such that ply = pZy= 0, then y produces infimal harmony, since m = 0 minimizes both h,, and h,, . No other y produces infimal harmony. The system does not have the monotonicity property; it does, however, have interlevel harmony, since g(m) takes on its minimum at m = 0, and m = 0 is the unique control input minimizing both hl, and h,, when y is such that ply = /j2, = 0. Restricted Harmony
The existence of a coordination input y that produces infimal harmony requires the existence of a control input in M which simultaneously minimizes each infimal cost function h i , , 1 I i I n, over the whole set M of control inputs. If no such control input exists for any coordination input of the system, the fact that the two-level system has interlevel harmony is of little use. However, we might be able to properly constrain the minimization so that infimal harmony is also achieved. This leads to some notions of restricted harmony within a two-level system, each differing primarily in how the subsets, on which the search for infimal optima is conducted, are restricted. We will be particularly interested in the restrictions generated by the interface inputs; for a given interface input ui we consider minimizing hi,(m) under the constraint Ki(m)= ui , since hi,(m) = giy(mi,ui), when Ki(m)= ui . For each m in M and i, 1 I i n, let [ m l K ibe the class of all m‘ in M equivalent to m, in the sense that Ki(m’)= Ki(m).Apparently, every m in M is contained in the sets [ m l K ,, ..., [ m l K ,which it generates, and, moreover, if K is one-to-one, it is the unique control input common to each of these sets. n, and m in M , let hi:”’) denote the restriction of hi, on For any i, 1 Ii I the set [ m l K , .We now introduce the following notions of restricted harmony in a two-level system: A coordination input y in %? produces restricted infimal harmony if there exists a control input m in M such that the family of restricted infimal cost n, has harmony. Apparently, if for a given m in M the functions II!;), 1 i i I family {hi:), ..., h$)} has harmony, then when K is a one-to-one function, the given rn is the unique control input that harmonizes the restricted infimal cost functions h!,”),1 < i 5 n. A coordination input y in %? produces restricted interlevel harmony if for all control inputs m in M , the family of restricted infimal cost functions hi,””, 1 5 i I n, is in harmony with the overall objective function. We refer to a two-level system as hatling restricted interlevel harmony if each coordination input of the system produces restricted interlevel harmony. There exists a relationship between unrestricted and restricted harmony. If the family of infimal cost functions hi,, 1 I i I n, has harmony, there
4. Conflict Resolution in a Two-Level System
127
exists, by definition, a control input in M that harmonizes the functions hi,, and, for any such control input m, the family of restricted infimal cost funciI n, has harmony. Therefore, any coordination input protions hl,"), 1 I ducing unrestricted infimal harmony also produces restricted infimal harmony. From this, we can conclude that a two-level system having restricted interlevel harmony also has unrestricted interlevel harmony. However, a system can have unrestricted interlevel harmony and yet not have restricted interlevel harmony as shown in the following example.
Example 5.4 Suppose a two-level system is given such that V = R and M = M , x M , = R2. Let the infimal objective functions g1 and 9, be defined on R2 as
+
gl(ml, ul) = m12 (m, + u1 g2(m2, u2) = m22+ (2m2 + u2 - 212, and suppose the interface inputs in the coupled system are u1 = 0 and u2 = m,.The corresponding infimal cost functions are then h,(m) = gl(ml,0) and h,(m) = g2(m2,ml). Suppose the overall objective function g is given as s(m> = h l ( 4 + h2(m). The system has the monotonicity property and, hence, unrestricted interlevel harmony. We shall show that this system does not have restricted interlevel harmony. Notice that the interaction functions Kl and K2 for this system are K, 3 0 and K,(m)= ml. Consider the control input 61= (+, $). The control input 6I harmonizes the functions hi6) and hi6), where
C W K , = R2,
CWK,
= {4} x R ,
and therefore the family {h,, h,} has restricted infimal harmony. Now upon minimizing g , we find that the unique overall optimal control input is fi = fi). The harmonizing control input 133 is, then, not overall optimal; this shows that the system does not have restricted interlevel harmony. Figure 5.3 is helpful in understanding the situation involved. Notice that the functionsfl and f 2 are in harmony with f3 ; the point xo is the unique point minimizing bothf, and f, over their domain [0, b ] , and xo minimizes f,,However, the restrictions of f, and f2 to the segment [0, a] are not in harmony withf,; the point xo' minimizes bothf, and f2 over [0, a] but does not minimize f3.
(A,
Conflict Resolution Relative to the Goal-Properties
We shall now present some of the more basic consequences of two-level systems having the monotonicity property and the various harmony properties referred to in general as goal-properties. Our objective is to indicate how
V . Coordination of Unconstrained Optimizing Systems
128
the existence or absence of a certain goal-property affects the possibility of resolving conflict among the overall and infimal goals of a given two-level system. Harmony and Conflict Resolution
For each coordination input y in V , let (my, u y )be a pair in M x UY,where * x Uny, such that for each i, 1 I i I n, the pair (miY,u i y ) is in M i x U i y and minimizes the given ith infimal objective function g i y over the set M i x Uiy. The existence, although not uniqueness, of a pair (my,u y ) will be assumed for each coordination input y in V. The following propositions give the characteristics of a two-level system having the various unrestricted harmony properties.
U y= U I yx
-
Proposition 5.3
If a given two-level system has unrestricted interlevel harmony and K ( M ) E U yfor each coordination input y in %, then m y is overall optimal whenever u y = K(my). PROOF. Let y be an arbitrary element of V and suppose uy = K(my); then for all m in M and each i, 1 5 i I n,
hiy(mY)= giy(miY,ui’) I giy(mi, Ki(m))= hiy(m>.
Then, by definition of unrestricted interlevel harmony, any m in M simultaneously minimizing the infimal cost functions hi?, 1 < i 5 n, over M also minimizes the overall objective function. This proves the proposition. Corollary
Suppose a given two-level system is such that K ( M ) G U yfor each coordination input y in %, Then, for a given coordination input y in V there exists a pair (my,u y ) such that uy = K(my)only if that y produces unrestricted infimal harmony. The next proposition gives stronger results regarding the unrestricted harmony properties. Proposition 5.4
Suppose a given two-level system is such that, in addition to K ( M ) c Uy, for each coordination input y in %, there exists for each i, 1 I i I n, a pair (miy,u i y )= ( m i ,Ki(m))for some m in M . Then: (i) Unrestricted interlevel harmony is both necessary and sufficient for m y to be overall optimal whenever u y = K(mY).
4. Conflict Resolution in a Two-Level System
129
(ii) The existence of a coordination input y in W producing unrestricted infimal harmony is both necessary and sufficient for the existence of a pair (my,u y )such that uy = K(mY). PROOF. Sufficiency in (i) is proven above. To prove necessity in (i) let y be an arbitrary element of V and let A minimize each infimal cost function hi,, 1 5 i In, over M . Then, from the hypothesis, there exists for each i, 1 I i I n, a pair (miy ,u i y )= ( A i ,Ki(A)). Hence rfi = m y and uy = K(mY). Now, if m y is not overall optimal, unrestricted interlevel harmony is denied. This proves necessity in (i). This argument also proves sufficiency in (ii). The proof is therefore complete.
Let us now consider the restricted harmony properties. Proposition 5.5
If a given two-level system has restricted interlevel harmony and U y= { a y } for each coordination input y in V, then m y is overall optimal whenever K(mY)= ay. PROOF. Let y be an arbitrary element of W and suppose a y = K(mY). Then, for each i, 1 I i I n, and each M in [ m Y l K,IKi(m) = aiY and
hiy(my)= giy(miy,aiy)I giy(mi,mi')
= gi,(m).
Therefore m y harmonizes the restricted infimal cost functions /I!,"") and, by definition of restricted interlevel harmony, is itself an overall optimal control input. Hence the proposition is proven. Corollary
Suppose for all coordination inputs y in W, the given two-level system is such that U y= {a'}. Then for a given coordination input y in V there exists m y such that K(mY)= ay only if y produces restricted infimal harmony. In Prop. 5.5 and its comllary, we did not recognize whether or not goal coordination was being used, although we did explicitly use the interactionprediction mode. If we recognize the two modes as being used independently, we can obtain stronger results regarding the restricted harmony properties. The following proposition is stated for the case when goal coordination is not used; however, it can easily be extended to cover the situation when goal coordination is used independently of interaction prediction. Proposition 5.6
Suppose the given two-level system is such that goal coordination is not employed; therefore, let the infimal objective functions be independent of the coordination inputs. For each y in W assume U y= { a y } , and for each
130
V. Coordination of Unconstrained Optimizing Systems
i, 1 i 5 n, there exsists miysuch that (miy,a,‘) = ( m i ,Ki(m))for some m in M . Furthermore, assume K ( M ) c { a y :y E U } . Then:
(i) Restricted interlevel harmony is both necessary and sufficient for my to be overall optimal whenever K(my)= ay. (ii) Restricted infimal harmony is both necessary and sufficient for the existence of a y and m y such that K(mY)= my. PROOF. Sufficiency in (i) is given by Prop. 5.5. To prove necessity, suppose m harmonizes the restricted infimal cost functions 1 I i I n, which are independent of the coordination inputs. Let 12= K(A), then since K(M) c { a y :y E %‘} there exists y in U such that aY= a. By the hypothesis, there exists miy = A i for each i, 1 I i I n. Hence my = A and K(mY)= ay. Now, if A is not overall optimal, restricted interlevel harmony is denied. This shows necessity in (i) and also proves sufficiency in (ii). The proposition is therefore proven.
Monotonicity and Conflict Resolution
Let us now leave the harmony properties per se and focus our attention on the more special goal-property : monotonicity. Proposition 5.2 relates the monotonicity property as being stronger than interlevel harmony. Monotonicity is a global property, whereas interlevel harmony is a point property. We should then expect more revealing results in the analysis of two-level systems having the monotonicity property. The following proposition gives a characteristic of a two-level system having an apparent-overall objective function but not necessarily the monotonicity property. Proposition 5.7
Suppose a given two-level system has an apparent-overall objective function g v and a coordination input y in %‘such that (i) the pair (my,u y ) exists, and U y contains zi = K ( A ) for some overall optimal control input A, and
(ii) m = m ywhenever gr(y, m, u ) = ga(y, my,uy),and (m, u) is in M x U”. Then m y is overall optimal if g&, my,u y )= niin g(m).
(5.17)
M
PROOF. By definition, gv(y, m, K(m)) = g(m) for any m in M . In particular, g%(y,A, K(A))= g ( A ) , where A is overall optimal and zi = K ( A ) is in U y . Hence, if ga(y, my,u y ) = g(m), we have from assumption (ii) that my = h. This completes the proof.
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131
The above proposition gives an equality condition (5.17) under which a coordination input y is optimal provided certain other conditions are satisfied. The conditions require the existence of optimal infimal solutions, satisfaction of the uniqueness assumption (ii), and inclusion of a so-called In order to test the equality, it overall optimal interface input in the set Uy. is necessary to have available the minimum overall cost 8. However, even if 0 were available, there is no indication of how to proceed in utilizing the apparent-overall objective function in search of an optimal coordination input. If a given two-level system has the monotonicity property, it then has certain characteristics which can be exploited in searching for an optimal coordination input.
Proposition 5.8 Suppose a given two-level system has the monotonicity property and, hence, an apparent-overall objective function 9%.Then for any y in V the inequality inf inf g&, rn, u ) I inf g(m) (5.18) Uv M
M
holds, whenever K ( M ) E U y or an overall optimal control & exists and ti = K(&) is in Uy. PROOF. Let y in G 3 be given, and suppose UY contains 6 = K(&) for some overall optimal control input &. Since the pair (A,ti) is in M x U yand gu(y, A,ti) = g(&), the inequality immediately follows. Now suppose K ( M ) E UY.Then, by definition of an apparent-overall objective function, inf inf gu(y, m, u ) I inf gr(y, rn, K ( m ) ) = inf g(m). Uv M
M
M
This completes the proof. The conceptual meaning of .(5.18) is that, if the infimal decision units are assumed decoupled and each one is freely selecting the interface inputs to its utmost advantage, the apparent-overall cost will be lower than, if not equal to, the minimum actual overall cost. This relationship is illustrated in Fig. 5.4; the surface v(y) = inf{g.(y, m,u ): (m, u ) E M x Uy}is always below the hyperplane v = 0. This immediately suggests the following corollary.
Corollary Suppose a given two-level system has the monotonicity property and, hence, an apparent-overall objective function g u . Assume for each y in % that either K ( M ) E U yor U ycontains ti = K ( h ) for some overall optimal control input &. Then sup inf inf g&, Y
Uv M
rn, u ) 5 inf g(m). M
V. Coordination of Unconstrained Optimizing Systems
132
This inequality is illustrated in Fig. 5.4. It means that the coordinator, by selecting y in % so as to maximize u ( y) , still keeps the minimum apparentoverall cost below the minimum actual cost. Proposition 5.7 gives an equality test (5.17) and conditions under which it can be used to determine the optimality of a coordination input. The corollary to Prop. 5.8 indicates how an optimal coordination might be sought. Together they lead to a sufficiency condition for coordinability.
Hyperplane v = j I
I
I
I
D Y
I
FIG.5.4 Relative location of the surface v(y) = inf{g,(y, m,u): (m,U ) E Mx Uy}and the hyperplane v = 8.
Proposition 5.9
Suppose a given two-level system has the monotonicity property and, hence, an apparent-overall objective function gv . Assume for each y E %' that either K E U y or U y contains ii = K(h) for some overall optimal control input h. Then, if the system's interlevel functions are strictly monotone, the equality max min min g&, I
Uy
M
m,u ) = min g(m).
(5.19)
M
is sufficient for coordinability. PROOF.
Suppose (5.19) holds, then there exists y in % such that min min g&, Uy
m,u ) = g ( h ) = g&,
&.K(h)),
(5.20)
M
where h is an overall optimal control input such that 6 = K(h) is in Uy. Since the interlevel functions are strictly monotone, a pair (my, u y ) exists, and for each i, 1 I i I n, giy(miy,u i y )I g i y ( h i ,Ki(h)). Suppose gjy(mjy,u j y ) < gjy(rfij,K j ( h ) ) for some j , 1 j 5 n. But, then, because of strict monotonicity, (5.20) would be denied. Therefore, for each i, 1 < i < n, the pair (hi,&(A)) minimizes giy over the set M i x Uiy. This shows coordinability and proves the proposition.
4. Conflict Resolution in a Two-Level System
133
There is still one drawback in addition to the requirement of strict monotonicity of the interlevel functions : the minimum overall cost might be unknown, and hence (5.19) could not be tested. However, if the apparent-overall objective function has a certain property given in the following proposition, we can use a saddle-type condition as a sufficiency condition for coordinability.
Proposition 5.10 Suppose a given two-level system has the monotonicity property, and its apparent-overall objective function gY satisfies the inequality
g(m)
SUP Sr(Y7 my u )
(5.21)
Y
for all (myu) in M x U. Then the following equality holds: inf g(m) = inf inf sup gY(y, rn, u). M
PROOF.
U
M
Y
If (5.21) holds for all (myu) in M x U, then inf g(m) I inf inf sup gr(y, m,u). M
U
M
V
However, for all m in M
g(m>= SUP g&,
w4).
m,
Y
Consequently, the inequality must be equality. This completes the proof. The following corollary to Prop. 5.9 and 5.10 gives the saddle-type condition.
Corollary Suppose a given two-level system satisfies the assumptions in Prop. 5.9. Assume the system’s interlevel functions are strictly monotone, and condition (5.21) holds for all (myu) in M x U. Then the equality max min min g&, 0
UV
M
m,u ) = min min max ga(y, m,u ) U
M
Y
is sufficient for coordinability. The inequality condition in Prop. 5.10 is illustrated in Fig. 5.5. It has the following important interpretation. If the coordinator maximizes over W for a given pair (myu), the apparent-overall cost appears to the infimal decision units as higher than the minimum actual cost. If the infimal decision units minimize the apparent-overall costs over M x U yfor a given y, the result will be lower than the minimum actual cost as illustrated in Fig. 5.4. When
134
V. Coordination of Unconstrained Optimizing Systems
I I
I
I
I I
I !
FIG.5.5 Relative location of the surfaces fi(m,u) = sup{g,(y, m, u): YE@} and the hyperplane v = 6 for fixed values of u. The values m', me, and m"' are such that u' =K(m'), u" = K(m"),and u'" = K(m").
these opposite actions balance, as expressed by the saddle-type condition, the system is coordinated, provided the assumptions of Prop. 5.9 are fulfilled. The above results apply mainly to the case in which interaction decoupling is used because of the assumption K ( M ) c U y . The alternate assumption, that U y contains a = K ( A ) for some overall optimal control input A, allows the use of interaction-prediction, provided the predictions are fixed at ay = ti.
To consider the case when the interaction-prediction mode is used, let there be given the sets d and 9J,where d is a subset of U such that K ( M ) E d and each p in 9J gives a family of infimal objective functions S i p , I 5 i 5 n. Assume %? = d x .Byso that each coordination input y is a pair (aY,By) where { a y } = U y and the infimal objective functions are g i y= g i p when
p = p'. Now, if the system in question has an apparent-overall objective function g, let the function ga be given on 9 3 ' x M x U so that
The function ga is an apparent-overall objective function for the system which exhibits the dependence on the goal coordination input p. The following proposition summarizes the above results as they apply to a two-level system in which the interaction-prediction mode is employed.
Proposition 5.11 Suppose a given two-level system has the monotonicity property, and hence an apparent-overall objective function ga , and %? = d x 9. Then sup inf min gas(/?,m, a) Iinf g(m). B.ef
M
M
5. Infmal Performance Generation and Modijication
135
Furthermore, if the system’s interlevel functions are strictly monotone, the equality max min min gI@, m, a) = min g(m) I
J
B
M
M
is sufficient for coordinability; and, in the case where (5.21) holds for all (m, u) in M x U,the saddle-type condition max min min ga(B, m, a) = min min max gW@, m, a) B
J
B
M
M
J
B
l
is sufficient for coordinability.
5. INFIMAL PERFORMANCE GENERATION AND MODIFICATION
It is desirable to have available some methods of modifying the infimal optimization problems or generating them from the overall optimization problem so that coordination can be achieved. Success in coordinating a two-level system depends on the infimal optimization problems and their interrelationships. A coordination strategy based on a given coordination principle can coordinate the system only if the relationships between the infimal units satisfy certain conditions. Modifications of the infimal optimization problems might, therefore, be necessary. We have limited our considerations in this chapter to goal coordination and either interaction-decoupling or interaction-prediction. Therefore, the freedom to modify the infimal optimization problems consists primarily, although not exclusively, in changing the infimal performance functions. Consequently, we shall be concerned with: (i) how infimal performance functions can be generated or modified in a simple yet natural way by starting with the overall performance function, (ii) how to modify already given infimal performance functions, and (iii) how goal-interaction operators can be used as a means of providing a systematic way of modifying the infimal performance functions.
Infimal Performance Function Generation If the infimal performance functions are not given beforehand, one can generate them from the overall performance function. The generation is trivial if an overall optimal control input is available. For example, if hjis the ith component of the overall optimal control input A, one can then
V. Coordination of Unconstrained Optimizing Systems
136
define the ith infimal performance functions as a real-valued function fi(mi, hi)2 0 such that fi(mi, hi)= 0 iff mi = h i . Barring this case, an apparent way of generating the infimal performance functions is by “isolating” those “parts” of the overall performance function which depend only on the local variables of the respective infimal decision units. There are two general ways of accomplishing this. Local Restriction
A straightforward generation of the infimal performance functions from a given overall performance function is by fixing the nonlocal variables. Let G be a given overall performance function. The ith infimal performance function generated from G by a given input-output pair (&, j j ) in M x Y is the function Gi given on M i x Yi by the equation Gi(mi,yi)= G(61,. . ., 6i-1, mi, f i i + l ? .
* *
,*n;.Y,,..
.,Pi-,
,Yi,jji+l,
9Yn)-
Apparently, Ci(mi, yi) = G(m, y ) for all (m, y) in M x Y for which mi = f i j and y , = j , for j # i. We can generate in this way a family of infimal performance functions Gi , 1 < i < n, from a given overall performance function. The infimal performance functions so generated can often be simplified: there usually are quantities in the overall performance function which have no effect on the selection of an ith local control input and may therefore be omitted from the ith infimal performance function. Example 5.5 Let M = Y = R2 and consider the function G given on M x Y as G(m,v) = m1y1 + (m2 + YI2) 1% m,.
For a given pair ( 6 , j )in M x Y, the infimal performance functions G , and G2 generated by restriction and after simplication are
Decomposition
Another way of generating infimal performance functions is by decomposing the overall performance function while taking into account the relation between the inputs and outputs of the overall process. For any given overall performance function G, there exists a family of functions G i , 1 < i < n, where Gi: M i x Yi x Wi + V, a family of functions 8 , , 1 1 i < n , w h e r e 8 , : M x Y + W i , a n d a m a p p i n g $ : V”+Vsuchthat G(m, V ) = $(G,(mi, Y , , O , ( ~ , Y)), ..., GAmn 3 Y n en(m, v)) 7
5 . Injimal Performance Generation and Modijkation
137
whenever y = P(m). We refer to the functions G i , 1 5 i < n, as infimal performance functions generated from G by decomposition, and the functions Bi , 1 5 i I n, as their interaction functions. Decomposition of a given overall performance function yields not only infimal performance functions, but also a mapping which can serve in some cases as an interlevel function. Generally, the infimal performance functions obtained by decomposition include a variable reflecting the nonlocal effects. If such a variable can be either eliminated or represent the subprocess interface inputs, we describe the overall performance as being " separable." More precisely, an overall performance function G is separable if there exists a family of infimal performance functions G i , 1 < i In, generated from G by decomposition such that subprocess coupling functions H i , 1 i < n, are their interaction functions; that is, G i :Mi x Yi x U i + V , 1 5 i 5 n, and G(m, Y ) = 1cI (Gl(m,
7
~ 1 ffl(m, , Y)), *
* - 7
Gn(mn
3
Y n2
Hn(m, Y ) ) )
whenever y = P(m). If a given overall performance function is separable, it can be decomposed in such a way that the generated infimal performance functions depend only on local variables and, moreover, the mapping $ : V" + V obtained in the decomposition process is an interlevel function for the system. Example 5.6
Let M
Y = RZ and consider the quadratic form G :
=
G(m, y ) = mTAm + yTBy,
on M x Y where A and B are given 2 x 2 matrices. Let the overall process P be given by the equations y,
=
-3m, - m , ,
y,
=
-4m1 - m , .
Suppose the subprocesses P , and P , are coupled according to the equations u1 = y 2 = Hl(m, Y ) ,
u2 = m,
+ y , = H2(m,y ) .
Because of the simplicity of this system, we can express the nonlocal variables of the ith subprocess as functions of its local variables. Therefore, G is separable : let Y,, ui) = aiimi2 + biiyi2 - a12m1(3m1 + ~
G1(m17
G2(mz Y z 7 Y
u2) =
i
+) b i z ~ i u , ,
a2zm22 + b22YZ2 - $a21m2(m2 + YZ)
+ &~2(4uz + m2 + ~ 2 ) .
V. Coordination of Unconstrained Optimizing Systems
138
Hence G(m, Y ) = Gdm1, Y1, H l ( 4 Y ) ) + G 2 b 2 7
Y2,
H,(m, Y ) )
whenever y = P(m). If B is diagonal, G, and G2 are independent, respectively, of u, and u,; if both A and B are diagonal, the decomposition is obvious. Example 5.7 Let M
=
Y = R 3 and consider the function G given on M x Y as
+ y12)lm2y21+ sin2 y 3 .
G(m,Y ) = (m,’
G is separable: let $: R3 -+ R such that $(UlY
u2
9
03)
= u1u2
+ u3
7
and let G,(m,, Y 1 ) = ,I2
+YI2,
G2@2 Y 2 ) = Im,y,l, Y
G3(m3
,Y3) = sin2 Y3
*
Infimal Performance Function Modification Modification of the infimal performance functions might be required for a number of reasons. In some instances, each infimal decision unit is given beforehand a single performance function with no apparent means of coordination: the entire first level is given, and the coordination problem is essentially that of coping with interactions. In another instance, each infimal decision unit is given a family of performance functions such that a given coordination principle is applicable, but the system is not necessarily coordinable by the principle. The problem of rendering the system coordinable by a given principle can be approached by modifying the infimal performance functions so that all modifications preserve the applicability of the principle, and, for at least one of them, the system is coordinable. Assuming an overall optimal control input is available, we present some general methods or approaches to modifying the infimal performance functions. Modijication by Input-Output Pairs If the infimal performance functions are generated by restriction on inputoutput pairs, it seems natural to use those input-output pairs as a means of modification and hence coordination. Let G be the overall performance function, and let the family of infimal performance functions G , , 1 i 5 n, be generated from G by input-output pairs. Let a = M x Y ; then for each pair /I in &?, the modified ith infimal performance function Gi, is generated from G by the pair /I.
5. Injimal Performance Generation and Modijication
139
A similar means of modification is available when the infimal performance functions are generated by decomposition, and the functions O,, 1 I i i n, are the corresponding interaction functions. Let the family of infimal performance functions Gi , 1 < i I n, be generated from G by decomposition. Again let W = M x Y. Then for each pair p in B, the modified ith infimal performance function G, is given on M i x Yi by the equation
The mapping $: V"-+ V obtained in the decomposition process is not generally a true interlevel function for the two-level system.
Balanced and Zero-Sum Modijications If the infimal performance functions are given beforehand without any apparent means of modification, it is not clear how to make the necessary modifications ; this situation arises when the infimal performance functions are generated by decomposition from a separable overall performance function. Is there, however, some general rule which one might follow in making the modifications? One such general rule refers to the strategy to be used for coordination. Assume that the infimal performance functions are such that a given coordination principle is applicable but the system is not coordinable; this can easily be the case, since a given coordinating condition is rarely satisfied without modification even if the associated principle is applicable. Modification of the infimal performance functions should then be such that applicability of the coordination principle is preserved. This leads to the notion of " balanced " modifications. Let B be the goal-coordination object of a two-level system; that is, for each p in B there is specified a family of infimal performance functions G, , 1 I i I n. We shall refer to the infimal performance functions specified by the elements of B as modijications or injimal performance modijications given by B. Also let 9 = {Gi : 1 5 i In} be a given family of infimal performance functions not necessarily included in the families specified by the elements of B. To facilitate the presentation, let gi and gip,1 I i I n and P E a, be the infimal objective functions corresponding, respectively, to the infimal performance functions Gi and G, . The infimal performance modifications given by B are $-balanced relative to the family 9, where $ is a given mapping $: V" + V, if the equation $(gl(ml, u1)7
* *
9
gn(mn
9
un))
= $(glp(ml?
ul),
* * * Y
gn/J(mn
9
un))
is satisfied for all (p, m,u) in B x M x U whenever u = K(m). If the infimal
140
V. Coordination of Unconstrained Optimizing Systems
performance modifications given by a are $-balanced relative to some family 9,they are $-balanced among themselves. Actually, we can use the infimal performance functions specified by any goal coordination input as a reference in determining whether or not the infimal performance modifications are $-balanced. If the infimal performance modifications in a two-level system are $,Ibalanced, and Ic/ is an interlevel function for some goal coordination input, then $ is an interlevel function of the system for every goal coordination input; in this case, we refer to the infimal performance modifications as being balanced. For a given mapping I) : V" -+ V ,letfbe the function defined on 2t3 x M x U by the equation
(5.22) gnp(mn 4). Then the infimal performance modifications are $-balanced iff the equality
f ( P , m, u) = $(gl&m,,
~ 1 1 3* * 9
9
f ( P , m, K(m)) =f(P', m, K(m)) holds for all B and P' in B and m in M . Hence, $-balanced injimalperformunce modijications have the property that the effects of the modijications cancel under $ whenever the interface inputs balance. Moreover, if the infimal performance modifications are $-balanced in a two-level system and $ is an interlevel function for some goal coordination input, then the function f defined as above for the system is an apparent-overall objective function for the system. If we are given a group operation in the evaluation object, we can give some general rules for obtaining balanced infimal performance modification. Let ( V , be an abstract group. Then, as an immediate consequence of ( V , .) being a group, there exists for each goal coordination input p in 9 and each i, 1 I i I n, a function pip defined on M i x Ui such that 0 )
giS(mi
9
U i ) = piP(mi 7
ui) * gi(mi
3
ui)
for all ( m i ,ui) in M i x Ui. Let the infimal objective functions g i pbe expressed as in (5.22). If a given mapping Ic/: V" -+ V satisfies certain conditions relative to the group operation in V , then $-balanced infimal performance modification can be expressed solely in terms of the modifying functions. Lemma If a given mapping $: V"-+ V is a homomorphism,* then the infimal performance modifications are $-balanced iff for all p in 99 and m in M p(P, m, K(mN = e
* The operation in V" is assumed to be induced by that in V.
5. Injimal Performance Generation and Modijkation
141
where p is defined on 98 x M x U by the equation p(P, m, u) = $(p1s(mi,
uA .
P " S ( ~ 3"un))
(5.23)
and e is the identity element of V. PROOF.
in 98. Let
Letfbe given on 98 x M x U by (5.22). Let 0 be a given element M x U be arbitrary. Since $ is a homorphism,
(B, m, u) in 98 x
f (P, m, 4 = P(P, m, u> * f(0,m, 4. Hence, the infimal performance modifications are $-balanced iff p(P, m, K(m)) = e. This completes the proof. If a given two-level system has an interlevel function for some coordination input, it is desirable to have the infimal performance modifications be balanced with respect to the interlevel function. It may be known that the infimal performance modifications are &balanced in the system, while another mapping $ is known to be an interlevel function for some coordination input. The following lemma gives a sufficient condition in terms of $ and I$ for the infimal performance modifications to be +-balanced when they are given to be +-balanced.
Lemma
+
Let $ and be homomorphic mappings from V" into V such that the kernel of $ includes the kernel of 4. Then the +-balanced infimal performance modification is also +-balanced. Let p J l and p L bbe defined on $8 x M x U respectively in terms of $ and by Eq. (5.23). Then ,u& m, K(m)) = e whenever p L ( P , m, K(m)) = e, since the kernel of $ includes the kernel of 4. This completes the proof. Notice that for given mappings $ and 4 satisfying the hypothesis of the above lemma there exists a homomorphism 8 : W(+)+ a(+)such that $(u) = O(+(u)) for all u in V" and, moreover, 8 is an isomorphism if $ and 4 have the same kernel. An important case is one in which the group operation in the evaluation object V is commutative and, hence, V is an additive group. If (V, +) is an additive group, the overall objective function is additive if PROOF.
+
S O = gl(m1, Kl(m)) + * *
+ gn(mn
9
Kn(m))
on M . Hence, the overall performance function of a given two-level system is additive iff the true overall cost is the " sum" of the true infimal costs.
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V. Coordination of Unconstrained Optimizing Systems
Infimal performance modifications in the case ( V , +) is an additive grotip can always be expressed in the form gip(mi ui) = gi(mi 9 ui) + Pi&mi ui) 7
9
where p i p is the modifying function. We say the infimal performance modifications are zero-sum if
for all (j?,m ) in x M . Zero-sum infimal performance modification is balanced if the overall performance function is additive. More generally, zero-sum infimal performance modification is balanced if there exists an interlevel function $ of the form $(u,,
where 8 : V
--f
. .. , O n )
= e(u,
+ * . . + u,,)
V is a homomorphism.
Example 5.8
Let M = Y = R2 and consider the overall process P and coupling functions H , and H z of the subprocesses P, and P, as given in Example 5.6. Incidentally, the subprocesses are Pl(ml, ul> = ml
+ ul,
Let 99 = RZ and for each pair
PAm, uz>= m2 + 2uz.
P = (p,, P,)
in B7 let the functions pip for
i = 1, 2 be given on R2 as
Plp(m1,
U l > = P1.l
- Pz(2m1 + ud7
Pzp(m2
7
u2)
= P z u2
- P l h , + 2uz)-
These modifying functions are zero-sum. To see this, we derive the subprocess interaction functions K, and K , : Kl(m) = Hl(m7P(m))= -4m, - m2 , K,(m) = H,(m, P(m))= - 2m, - m2 .
Now, kp(m1, u1) + Pz/?(m,>
- m.2 - 2%) + Pz(% - 2m1 - U l > * Therefore, if ti1 = Kl(m) = -4m, - m , and u, = K,(m) = -2m, - m,
4= P
Plp(m1, u1)
l h
+P z p h
Y
u2)
= 0.
If the infimal objective functions gi for i = 1,2 are such that gip(mi ui) = gi(mi > ui) + Pi/?(mi ui)7 where giis given, the infimal modifications are $-balanced for $(ul, u,) u1 uz. 9
+
7
=
5 . InJimal Performance Generation and Modijication
143
Example 5.9
Consider the system in the above example with M = Rf x R', where . R f is the set of positive real numbers. Let 8 = Ri and for each P in 8 let the functions p i p for i = 1 , 2 be given on R2 as lulp(m1, u1) = Pu1/(2m, + U1)Y
P2p(m2
9
u2) = (m2
+ 2u2)/Pu2.
Let the infimal objective functions gV for i = 1 , 2 be such that giB(mi ui) = gi(mi ui) * pip(mi > ui), 9
3
where g i is given. We claim that the infimal performance modifications are $-balanced for $(q,v 2 ) = u1v2.
Interaction Operators We have indicated above how given infimal performance functions can be modified and under what conditions the modifications are balanced. If the evaluation object is a group, modifications may be accomplished through the group operation. In case the evaluation object is an additive group, the given infimal performance functions may be modified by adding a modifying term. If an appropriate difference measure is available in the evaluation object, we can derive various kinds of interaction operators and use them to arrive at suitable modificating functions. A metric, or even the group operation in the case of the evaluation object is a group, may be used to define a difference measure (the difference measure in the evaluation object can be defined by subtraction, if the evaluation object is an additive group). The idea of the interaction operators is to give variations in performances and outputs due to input variations. We present here some so-called goal-interaction operators whose linear forms are used in the analysis in Chap. VI. In doing so, we use the following notation: Let 6 : V x V + V denote an appropriate difference measure available in the evaluation object. For a given element mo in M and mi in M i , let mO[mi]= (m,',
--
*
, ni:-'_,,mi m:i
-
*,
m,O).
That is, mo[mi] denotes the replacement of mi0 by mi in mo. Similarly, we define the element uO[ui] of U. We will also use the mapping P : M x U + Y,
P(m,U > = (P,(m,, . . ., P.(mfl, ufl)), to denote the decoupled subprocesses. By definition of the interaction K, P(m) = &,
so P explicitly shows how the interface inputs enter the system.
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V. Coordination of Unconstrained Optimizing Systems
Total Goal-InteractionOperators
Due to the interactions between the subprocesses of a two-level system, a change in any local control input will generally be transmitted throughout the entire system through the subprocess couplings. Hence, a change in a local control input can cause a change in the overall cost through nonlocaZ variables. A measure of this change in overall cost is given by the total goalinteraction operators which we define as follows: The ith total goal-interaction operator at a given point m" in My denoted by Ti(mo),is the mapping ri(mo):M i+ V , such that its value at any point m, in M i , denoted by ri(mO)mi, is given by the equation
ri(mO)mi= GIG(mo,P(m", K(mo[mi]))), G(m",P(m", K(m0)))3. (5.24) The quantity G(m",P(m", K(m")))is the (actual) overall cost for the given control input m", since P(mo, K(m"))= P(m"). Now, if mi0 is changed to m i ,the interface inputs will change from u" = K(mo) to u = K(mo[mi]). The quantity G(mo,P(mo,K(mo[mi])))is then the overall cost due to the change in mi0 as expressed only by the change in the interface inputs; it is not, in general, the resulting true overall cost since the equality K(mo)= K(m0[mi])need not hold. Partial Goal-InteractionOperators
The change in the overall cost due to a change in a local control input can also be measured in terms of how the change in the local control input affects the overall cost due to the resulting change in a given interface input; such a measure is only partial in contrast with the above measure of change which takes into account changes in all the interface inputs. A partial measure of the change in overall cost is given by the partial goal-interaction operators which we define as follows: The 0th partial goal-interaction operator at a given point m" in Mydenoted by Tij(mo),is the mapping r i j ( m O )M: i+ V, such that rij(mO)mi = GIG(mo,P(m", uo[Kj(mo[mi])])), G(moyP(m", u"))] (5.25)
for all mi in M i , where u" = K(mo). Again, the quantity C(m", P(m", u")) is the (actual) overall cost for the given control input ma. The change in the ith local control input from mi0 to mi causes the interface input to the j t h subprocess to change from u j o = Kj(mo)to uj = Kj(mo[mi]). Now, by changing only the interface input of the j t h subprocess from ujo to uj , we obtain the overall cost G(mo,P(mo, uo[Kj(mo[mi])])).
5. Infimal Performance Generation and Modijication
145
The partial goal-interaction operators are defined for all i and j , 1 5 i I n, 1 5 j I n. The partial goal-interaction operator rii(mO)at m0 gives a measure of the change in overall performance due to a change in the ith local control input, as the effect of this change is transmitted throughout the system and back to the ith subprocess. Interface Goal-ZnteractionOperators Rather than measure the changes in overall performance due to changes in a local control input as the effects are transmitted through the interface inputs, we can measure changes in overall performance due to changes in the interface inputs themselves. Such a measure is given by the interface goalinteraction operators which we define as follows: The ith interface goal-interaction operator at a given point m0 in M , denoted by Ai(mo),is a mapping Ai(mo):Ui + V such that
Ai(mo)ui= 6[G(mo,P(mo, uO[ui])),G(mo,F(mo, uO))]
(5.26)
for all ui in Ui, where uo = K(mo). These goal-interaction operators are quite similar to the partial goalinteraction operators as can be seen by comparing (5.25) and (5.26) and the discussion below. There are, of course, many other ways in which goal-interaction operators can be defined. For example, they can be defined using an apparent-overall objective function, if one exists for the system in question, rather than the overall performance function. If the overall performance function is additive, the goal-interaction operators can be defined directly in terms of the infimal performance functions rather than the overall performance function. Process interaction operators can be defined in the same manner as we defined the goal-interaction operators, provided a suitable difference measure is given in the output object. Relationships between the Goal-ZnteractionOperators By comparing (5.25) and (5.26) we see that the partial goal-interaction operators can be expressed in terms of the interface goal-interaction operators. Precisely, for any given m0 in in M and for any i and j , 1 I i I n, 1 5 j I n,
Tij(mO)mi= Aj(mo)Kj(mo[mi]) for all mi in M i .
V. Coordination of Unconstrained Optimizing Systems
146
The total goal-interaction operators and partial goal-interaction operators can be related here, if the overall performance function G is additive in the sense that
for all (m,y ) in M x Y. V must, therefore, be an additive group; hence, let 6 be subtraction in V. In this case, for any mo in M and i, 1 Ii 5 n, n
ri(rnO)mi=
{Gj(m;, Pj(m?, K j ( m o [ m i ] ) ) ) j= 1
- ~ j ( m j oPj(m;, , Kj(mo)))> while for allj, 1 I j In,
rij(mO)mi= Gj(mjo,Pj(mjo,Kj(mo[mi]))) - Gj(m;, Pj(m;, Kj(mo))). Hence, n
ri(nzo)mi=
C rij(mO)nii.
j= 1
Injimal Performance Modijications
We will now indicate how the goal-interaction operators might be used to modify the infimal performance functions. Suppose the overall performance function is additive, as given above. Let a = M , and for each P in B and i, 1I i In, let the modifying function pipbe given on M i x Ui as pib(mi ui) = ri(P)mi - Ai(P>ui 3
9
or, if interaction-prediction is used, we could just as well let pipbe such that pip(mi 9
ui) = ri(P)mi*
These two forms of the modifying functions will be considered in Chap. VI when the goal-interaction operators presented here are linearized.
6. COORDINABLLITY BY THE BALANCE PRINCIPLE
The analysis in this section is focused on the coordinability of a twolevel system by the interaction balance and performance balance principles given by (5.10) and (5.12), respectively. The results are given in reference to the absence of conflict in a two-level system as described by the monotonicity property and the unrestricted harmony properties.
6. Coordinability by the Balance Principle
147
First, let us recall some notation. Since only goal coordination will be employed (the balance principles are primarily for the case when interaction decoupling is used), we let the coordination object be the set W.For each jl in $39, then, xp denotes a pair (mp,up)in M x U,such that for each i, 1 Ii I n, the pair (m!, u!) is P-optimal; that is,
and hence (m!, u!) is an optimal solution of the infimal optimization problem gi(jl). The element A in M denotes an overall optimal control input. Briefly, the coordinability notions for the balance principles are as follows: (i) The interaction balance principle is applicable iff the following proposition holds: ( VP)( Vxa)( 3A){[(m,u) = xp and u = K(m)]
* m
= A}.
(5.27)
The principle is not applicable iff there exists P and (ma,us) = xp such that up = K(mP),and ma is not overall optimal. A two-level system is coordinable by the interaction balance principle iff the principle is applicable and there exists a coordination input jl and (mp,up)= xa such that up = K(mp). (ii) The performance balance principle is applicable iff the following proposition holds : ( VP)( Vxa)(3A){[(m,u) = xa and sa(m, K(m)) = sa(m,u)]
* m
= A}
(5.28)
where
Bp(m
= (91&9(m,,%)7
* * * Y
gnp(mn , un)>-
A two-level system is coordinable by the performance balance principle iff the principle is applicable and there exists jl in W and (ma,up) = xa such that ga(mP,K(mp))= gp(mp,up). We shall proceed in the analysis of coordinability by the balance principles from weaker to stronger requirements. There are two general properties, unrestricted interlevel harmony and the monotonicity property, which are relevant to the applicability of the principles, and one general property, unrestricted infimal harmony, relevant to the existence of an optimal coordination input. Interlevel harmony, a point property in contrast to the monotonicity property, is the weaker of the two; we therefore begin our analysis by considering systems having this property. *
* Henceforth in this section, we drop the modifier “unrestricted.”
148
V . Coordination of Unconstrained Optimizing Systems
Systems with Interlevel Harmony
We show here that interlevel harmony is sufficient for the two balance principles to be applicable, while the principles are themselves somewhat broader. A system need not have interlevel harmony in order that the balance principles be applicable; however, under rather mild conditions, interlevel harmony becomes necessary. We then show that a system in which a given balance principle is applicable must have infimal harmony if the system is to be coordinable by the principle. Yet, under the same mild condition for which interlevel harmony is necessary for applicability, infimal harmony is not only necessary, but also sufficient for coordinability by the principle. Our first analysis reveals a result which is very important conceptually.
Theorem 5.12 The interaction balance principle is applicable in a given two-level system iff the performance balance principle is applicable. PROOF. Let p be an arbitrary element in and suppose (ms, us) = X B exists such that ai(mP,up) = ai(mP,K(mP)).Then the pairs (m:, Ki(ms)) are /I-optimal and, moreover, (mP,K(ms)) satisfies the coordinating condition of the interaction balance principle. Consequently, there exists A such that ms = &I if the interaction balance principle is applicable. On the other hand, it is apparent that (ma, up) satisfies the coordinating condition of the performance balance principle; consequently, there exists riz such that mD = A if the performance balance principle is applicable. This. proves the theorem.
Theorem 5.13 A given two-level system is coordinable by the interaction balance principle iff it is coordinable by the performance balance principle. The proof of this theorem is contained in the proof of Theorem 5.12. The above theorems state that the two balance principles are mathematically equivalent; we can, therefore, restrict our analysis to the interaction balance principle. Conceptually, the two balance principles are quite different. Utilization of the interaction balance principle requires the actual interface inputs to be either measurable or calculable. If the actual interface inputs cannot be measured or calculated, the interaction balance principle cannot be used. Such difficulties can be avoided by utilizing the performance balance principle, since the apparent and actual infimal costs should always be available. We now proceed to analyse the coordinability of a two-level system by the interaction balance principle. Due to the above two theorems, the results are equally valid for the performance balance principle.
149
6. Coordinability by the Balance Principle
Proposition 5.14 The interaction balance principle is applicable in any two-level system having interlevel harmony. PROOF. We use Prop. 5.3. Suppose a given two-level system has interlevel harmony. Let p be an arbitrary element in 99, and suppose (ma, up)= xa exists. If the interface inputs balance, ua = K(ma); it then follows from Prop. 5.3 that ma = A. Hence, the interaction balance principle is applicable and the proposition is proven.
Suppose a given two-level system does not have interlevel harmony; then the balance principles are still applicable in the system provided that, for each coordination input not producing interlevel harmony, either the coordinating condition is not achieved or at least one infimal problem fails to have an optimal solution. This suggests a rather mild condition under which interlevel harmony is not only sufficient, but also necessary. Proposition 5.15 Suppose for each coordination input each i, 1 Ii I n, the equality
p of a given two-level system and
min gia(mi,u i ) = min gia(mi,Ki(m)) Mi X Uc
M
holds whenever the right side exists. Then the system having interlevel harmony is both necessary and sufficient for applicability of the interaction balance principle. PROOF. We need only prove necessity. Suppose the system does not have interlevel harmony. Then there exist p in 99 and m in M such that for each i, 1 Ii I: n, gia(mi,Ki(m)) is a minimum value of gia(mi’, Ki(m’))over M , and m is not overall optimal. From the hypothesis, there exists (ma, ua) = xB such that (m, K(m)) = (ma, up). The pair (ma, up) satisfies the coordinating condition of the balance principle, and ma = m is not overall optimal; this denies applicability of the principle. Necessity and the proposition are now proven. A condition under which the hypothesis of Prop. 5.15 is satisfied is that for each i, 1 I i In, Mi x Ui = { ( m i ,Ki(m)):m E M } .
This condition is satisfied if for all i, 1 Ii 5 n, the interface inputs ui given by Ki are independent of the control input m i , and Ui = K i ( M ) . Even though a given balance principle is applicable in a given two-level system, the system might not be coordinable by means of the principle. Coordinability by the principle fails if the system does not have a coordination input for which the coordinating condition is satisfied.
150
V. Coordination of Unconstrained Optimizing Systems
Proposition 5.16 Suppose the interaction balance principle is applicable in a given two-level system. Then, in order for the system to be coordinable by the principle, the system must have infimal harmony. PROOF. Let a given two-level system be coordinable by the interaction balance principle. Then there exist a p in 98 and a pair (mp, up)= xp such that up = K(mp).From the corollary of Prop. 5.3, the system must have infimal harmony; hence, the proposition is proven.
The above proposition states that, in order for the coordinating condition of either of the two balance principles to be satisfied, the system must first have infimal harmony. However, for the same reason that interlevel harmony is not necessary for the balance principles to be applicable, infimal harmony itself does not generally guarantee satisfaction of the coordinating condition. Proposition 5.17 Suppose the interaction balance principle is applicable in a given twolevel system and, moreover, suppose the hypothesis of Prop. 5.15 is satisfied. Then, the system having infimal harmony is both necessary and sufficient for coordinability by the interaction balance principle. PROOF. We need only prove sufficiency; therefore, suppose the system has infimal harmony. Then there exist /3 in 93 and m in M such that for each i, 1 I i 5 n, gi(mi, Ki(m))is a minimum value of gip(mi’,Ki(m’))over M , and m = A. Consequently, from the hypothesis of Prop. 5.15, there exists (mD, up)= xp such that (m,K(m))= (ma, up). Hence (mp,up) satisfies the coordinating condition of the interaction balance principle. Sufficiency and the proposition are now proven.
Corollary 5.18 Suppose a two-level system satisfies the hypothesis of Prop. 5.15. Then the system having both interlevel and infimal harmony is necessary and sufficient for coordinability by the interaction balance principle.
In order to gain more depth in our analysis, we must now impose additional structure in the systems. We, therefore, proceed by considering more strongly conditioned systems, namely two-level systems having the monotonicity property. Systems Having the Monotonicity Property
If a given two-level system has the monotonicity property, it has, by definition, a monotone interlevel function for each coordination input of the system. There is, therefore, a definite relationship between the infimal costs
6. Coordinability by the Balance Principle
151
and the overall costs over the whole range of control inputs; this is not necessarily the case when the system has simply interlevel harmony. Consequently, we should obtain stronger results for systems having the monotonicity property. First, we have, from Props. 5.2 and 5.14, the following result:
Corollary 5.19 The interaction balance principle is applicable in any two-level system having the monotonicity property. We can now give the principal reason for introducing the idea of balanced infimal performance modification as in the preceding section.
Proposition 5.20 Suppose a two-level system has a monotone interlevel function $ for some coordination input, Then the interaction balance principle is applicable if infimal performance modification is $-balanced. PROOF. If $ is a monotone interlevel function of the system for some coordination input and infimal performance modification is $-balanced, then $ is an interlevel function for all coordination inputs of the system. Consequently, the system has the monotonicity property. Corollary 5.19 now proves the proposition.
In spite of their simplicity, or perhaps just because of it, the preceding two results compliment each other to provide a very powerful guide to achieving coordinability. The corollary states that a monotonic relationship between the overall and infimal performances is a good starting point to synthesize a coordinable system. Starting with a system having the monotonicity property, various balanced infimal performance modifications can be tried (without fear of loosing the applicability of the principle) in attempting to modify the infimal perforomnce functions so that the system is coordinable by the principle. The objecthe of infimal performance modification is to make a two-level system coordinable. In view of the two preceding results, we now analyze systems having the monotonicity property for conditions under which these systems are coordinable by the balance principles. We know infimal harmony is necessary, and in some cases sufficient, for coordinability; however, with the added property of monotonocity, we are able to obtain more practical conditions. If a given two-level system has the monotonocity property, let ga be an apparent-overall cost function of the system. Also, let g denote the system’s overall objective function.
152
V. Coordination of Unconstrained Optimizing Systems
Proposition 5.21
A two-level system having the monotonocity property is coordinable by the interaction balance principle only if the following equality holds : max min ga(P, m,u ) = min g(m) I
(5.29)
M
MXU
PROOF. Suppose a two-level system has the monotonicity property and is coordinable by the interaction balance principle. Then there exist in W and a pair (mp, up) = xp such that up = K(mB).Because the principle is applicable, ms = A.Therefore, because of the monotonicity property,
min g I ( P , m, u ) = ga@, mB,us> = gL(P, ms,K ( m p ) ) MxU
= g(mfl)= g ( h ) = min g ( m ) . M
The proposition is now proven by applying Prop. 5.8. Equality (5.29) is necessary, but not generally sufficient, for coordinability by either of the balance principles when the system has the monotonicity property. All the infimal problems in a given two-level system having the monotonicity property can fail to yield optimal solutions; the system is then definitely not coordinable, yet it is possible for Eq. (5.29) to hold. Proposition 5.22
Suppose the interlevel functions of a two-level system are strictly monotone. Then Eq. (5.29) is both necessary and sufficient for coordinability by the interaction balance principle. PROOF. The necessity part of this proposition is given by Prop. 5.21, while sufficiency is given by Prop. 5.9.
If the interlevel functions of a two-level system are strictly monotonic, then under a rather mild condition, the existence of a saddle-point of the apparent-overall objective function g I is necessary and sufficient for coordinability. Proposition 5.23
Suppose the interlevel functions of a given two-level system are strictly monotonic and, in addition, the inequality
(5.30) holds for all (m,u) in M x U. Then the existence of a saddle-point of gkB, namely the equality max min gL(& m, u ) = min max gL(p, m, u ) , L
MxU
MxU
L
153
6. Coordinability by the Balance Principle
is both necessary and sufficient for coordinability by the interaction balance principle. PROOF. Because of inequality (5.30) we obtain from Prop. 5.21 the equality inf g(m) = inf sup g&?, M
MXU
m yu ) .
9
Therefore, ga has a saddle-point iff Eq. (5.29) holds. The proof is now completed by applying Prop. 5.22.
A two-level system having the monotonicity property is coordinable by a given balance principle (the interaction balance or performance balance principle) iff there exists a coordination input such that the specified infimal optimization problems have an optimal solution which satisfies the coordinating condition of the given coordination principle. Let us refer to such a coordination input as compensating. The above results give conditions for the existence of a compensating coordination input in a two-level system having the monotonicity property. We will now characterize Compensating coordination inputs in such two-level systems. From Prop. 5.21, we have that, in a two-level system having the monotonicity property, a given coordination input ) is compensating (relative to either balance principle) only if B satisfies the inequality 9(&) g,
= A P , m,
4 - gSa(0, mY 4
(5.32)
on W x M x U.Now, if the group operation " * " in V preserves the ordering I in V , we have the following characterization of a compensating coordination input:
Proposition 5.24 Suppose in a given two-level system having the monotonicity property the group operation in the evaluation object is order-preserving. Then any compensating coordination B in B must satisfy the inequality (5.33) g(&) * [ga(0,m,41- s A P , m, 4 for all (myu) in M x U,where p is given on
W
x M x U by (5.32).
V. Coordination of Unconstrained Optimizing Systems
154 Corollary
Suppose the interlevel functions of a two-level system are strictly monotonic and the group operation in the evaluation object is order-preserving. Then, the compensating coordination inputs in B are precisely those coordination inputs in g which satisfy (5.33) for all (m, u) in M x U. The proofs of these two results follow immediately from the inequality (5.31). Finally, we consider the special case in which a two-level system has an additive overall performance function and zero-sum infimal performance modification. If a two-level system has an additive overall performance function, its evaluation object must be an additive group; therefore, its overall objective function g is, by definition, expressable as the sum n
g(m>= C gi(mi Ki(m)), 3
i=l
for some given infimal objective functions g l , .. . ,g n , and any of its infimal objective functions g i pis expressable as the sum of gi and a modifying function pip defined on M i x U i . The modifying functions provide a convenient means of characterizing a compensating coordination input when the infimal performance modifications are zero-sum. Proposition 5.25
Suppose a two-level system has an additive overall performance function with zero-sum infimal performance modifications, and suppose the addition in the evaluation object is order-preserving. Then, for a given coordination input of the system to be compensating, it is both necessary and sufficient that p satisfy the inequality n
for all (m, u) in M x U . PROOF. Because the overall performance function is additive and infimal performance modification is zero-sum, the addition mapping V" V is an interlevel function of the system for each coordination input. Since the addition is assumed order-preserving, it is then strict order-preserving; is therefore strictly monotone. Since the equality
1:
n
C gipOl?i i= 1
n >
ui) = C pip(mi i= 1
--f
n
7
ui)
+ iC gi(mi =l
ui)
holds on 98 x M x U , the proposition is proven by applying the corollary of Prop. 5.24.
7. Coordinability by the Interaction Prediction Principle
155
Figure 5.6 illustrates Eq. (5.34)for< = K(%) and a compensating coordination input fi. Notice that the surface pua(m,ii) = p i g ( m i ,iii) in the Y x M x U space intersects the ZJ = 0 hyperplane at the curve K(%) = u", since the modifications are zero-sum, and it is always a distance 6 = g ( h ) above the surface -gdm, ii> = -Cigi(mi, i i i ) .
xi
FIG. 5.6 Representation of the modifying term p@, u) = X I pli(ml, ul) for a compensating coordination input and p = K($, where the modifications are zero-sum, and where d m , u ) =&h(ml, ul).
B
7. COORDINABILITY BY THE INTERACTION PREDICTION PRINCIPLE
We consider in this section the coordinability of two-level systems when the coordination strategy is based on the interaction prediction principle. This principle is characterized by the proposition that an overall optimum is achieved by local action whenever the interface inputs are correctly predicted. Unfortunately, the proposition does not hold true for a broad class of twolevel systems, unless goal coordination of a certain kind is also used or the infimal performances are appropriately modified. We shall obtain in this section some general conditions for coordinability by the interaction prediction principle with and without goal coordination. Let us first recall some notation in addition to that used in the preceding section. To recognize explicitly the interaction-prediction mode and goal coordination, we let the coordination object be the set %? = d x 93, where d is a given subset of U such that K ( M ) c d ,and 93 is a given goal coordination object. For each coordination input y = (u, p), m y denotes a control input iI n, the local control input miy minimizes in M such that, for each i, 1 I Gis(mi, ai) over the set M i . If goal coordination is not used, the control
156
V. Coordination of Unconstrained Optimizing Systems
inputs my depend only on the interface input predictions a ; hence, in such cases we denote my by ma. Applicability of and coordinability by the interaction prediction principle depend on whether goal coordination is used and, if so, how it is used. We will consider two cases: (i) goal coordination is not used (only the interactionprediction mode is used), and (ii) goal coordination is made in accordance with the interface input predictions. These two cases are sufficient to cover most applications of the principle. Briefly, then, the pertinent coordinability notions are the following: (i) The interaction prediction principle is applicable (without goal coordination) iff the following proposition holds : (Va)( Vma)(3fi){[m= ma and
K(m) = a]
=>
m = &}.
(5.35)
A two-level system (without goal coordination) is coordinable by the principle iff the principle is applicable, and there exists a correct prediction a ; that is, for some a in d there exists ma such that K(ma)= a. A two-level system using only the interaction-prediction mode can easily fail to be coordinable by the interaction prediction principle (Example 5.9). Applicability fails whenever there is a correct prediction which does not result in overall optimal behavior. An overall optimum is achieved by a correct prediction a only if a is optimal in the sense that a = K ( A ) for some overall optimal control input f i . If the principle is applicable, coordinability fails when all predictions which are optimal in the above sense are not correct. These causes of coordinability failure can be avoided if appropriate infimal performance modifications are made in accordance with the predicted values of the interface inputs (Example 5.10). (ii) The (interaction) prediction principle is applicable relative to a mapping q : M -,93iff for any coordination input pair (a, j)such that j? = q(u), the following proposition holds: (Vy)( VmY)(3 f i ) { [ m= my and q&m) = y ]
=>
m = A},
(5.36)
where qn(m)= (K(m),q(rn)). A two-level system is coordinable by the principle relative to q iff the principle is applicable relative to q, and there exists a coordination input y for which there exists my such that q,(mY)= y. Coordinability without Goal Coordination
We first consider the case when goal coordination is not used. The relevant goal properties for coordinability are then the restricted harmony properties. The results more or less parallel those relating the (unrestricted) harmony properties to coordinability by the balance principles. The first result is an immediate consequence of Prop. 5.5.
7. Coordinubility by the Interaction Prediction Principle
157
Proposition 5.26 The interaction prediction principle is applicable without goal coordination in any two-level system having restricted interlevel harmony.
A given two-level system might not have restricted interlevel harmony and, yet, the interaction prediction principle could be applicable without goal modification. Such would be the case if the system did not have restricted interlevel harmony and all the interface input predictions were incorrect; the system, however, would fail to be coordinable. Under suitable conditions, however, restricted interlevel harmony is necessary when goal coordination is not used. Proposition 5.27 Suppose a two-level system is such that each a = K(m), m E M , satisfies the n equalities min gi(mi, ai) = Mi
gi(mi , Ki(m)),
min (m E M : K i ( m )= a t )
whenever the right sides exist. Then, the system having restricted interlevel harmony is both necessary and sufficient for applicability of the interaction prediction principle without goal coordination. PROOF. We need only prove necessity. Suppose the system does not have restricted interlevel harmony. Then there exists m in M such that for each i, 1 5 i n, the value gi(mi,Ki(m))is a minimum over the set of m‘, for which Ki(m’)= Ki(m), and, moreover, m is not overall optimal. From the hypothesis, there exists a in &‘ and masuch that a = K(m) and m = ma.This denies applicability of the principle. Necessity and the proposition are therefore proven.
The hypothesis of Prop. 5.27 is satisfied if each subprocess interaction function Ki , 1 5 i 5 n, is independent of the local control input m i . Without assuming the hypothesis of Prop. 5.27, we can still give a condition which is both necessary and sufficient for applicability of the principle without goal coordination. The condition is not in terms of the restricted harmony properties, and unfortunately, must be in reference to an overall optimal control input.
Proposition 5.28 For the interaction prediction principle to be applicable in a two-level system without goal coordination, it is both necessary and sufficient that every nonoverall optimal control input I%in M satisfy the inequality: inf gi(rni ,K i ( f i ) )< g i ( f i i , K i ( f i ) ) , Mi
for some i ,
1 I i I n.
V. Coordination of Unconstrained Optimizing Sygtems
158 PROOF.
Suppose the principle is applicable without goal coordination. Let
f i be an arbitrary nonoverall optimal control input in M and set a = K(fi). It must be the case that f i # ma for any ma;otherwise, applicability of the principle would be denied. Hence + satisfies i the inequality for some i, I I i I12.
Now, suppose the principle is not applicable without goal modification; then there exists c1 in d and a nonoverall optimal control input ma such that a = K(m"). However, ma fails to satisfy the inequality for any i, 1 5 i I n. This completes the proof. Applicability of the interaction prediction principle without goal coordination is useless if correct predictions cannot be made. If correct predictions are impossible, the principle is trivially applicable without goal coordination ; however, the system fails to be coordinable by the principle. We now show that restricted infimal harmony is necessary for coordinability by the principle without goal coordination. Proposition 5.29
If a two-level system is coordinable by the interaction prediction principle without goal coordination, the system must have restricted infimal harmony. PROOF. If the system is coordinable by the principle without goal coordination, then, by definition, there exist a in .dand ma such that a = K(m"). Now, from the corollary of Prop. 5.5, the system must have restricted infimal harmony; hence, the proposition is proven. Restricted infimal harmony is not sufficient for coordinability, for the same reason restricted interlevel harmony is not necessary for applicability. Hence, we have a condition under which restricted infimal harmony is both necessary and sufficient. Proposition 5.30 Suppose the interaction prediction principle is applicable without goal coordination in a two-level system satisfying the hypothesis of Prop. 5.27. Then, the system having restricted infimal harmony is both necessary and sufficient for coordinability by the principle without goal coordination. PROOF. We need only prove sufficiency. Let the system have restricted infimal harmony. Then there exists Ei in M such that for each i, I I i 5 it, the quantity g i ( f i i , K,(fi))is a minimum over the set of all m in M for which Ki(m)= K,(fi). Therefore, from the hypothesis of Prop. 5.27, there exists a in d and ma such that K(m")= a and m = ma.Since the principle is applicable, f i is overall optimal. This proves coordinability and the proposition. Corollary Suppose a two-level system satisfies the hypothesis of Prop. 5.27. Then the system having both restricted interlevel and restricted infimal harmony is
7. Coordinubility by the Interaction Prediction Principle
159
necessary and sufficient for coordinability by the interaction prediction principle without goal coordination. We have pointed out in the beginning of this section how a two-level system can fail to be coordinable by the interaction prediction principle without goal coordination. The causes for this failure can occur even if the sole source of interaction between the infimal decision units is through the process interface inputs. This seems surprising, but nevertheless, is demonstrated in the following example. Example 5.9
Let Mi = Yi = U i= R, the set of real numbers, for i = 1,2. Consider a two-level system having two subprocesses P, and P , given by the equations: Y , = m,
= Pl(ml, ul),
y 2 = 3m2 - u2 = P h , , u2);
and coupled according to the equations: u, = m2
= K,(m),
u2 = m,
= K,(m).
Let the infimal performance functions GI and G , be given on R2 as
and let the overall performance function be the sum of them, G(m, y ) = m,’
+ mZ2+ (yl - 1)’ + y Z 2 .
Suppose the interface inputs are predicted as a = (A,+). Then by minimizing the infimal costs relative to the given prediction, we obtain wila =
+,
mza = 2 2 0’
Hence, a is a correct prediction, but the unique overall optimal control input 3).The interaction prediction principle without goal coordination is rD1 = is therefore not applicable in this system. Moreover, this system possesses the property that a = K(rit) is not a correct prediction. The following reasoning may help to explain why the interaction prediction principle fails in the above example. Although the only interaction between the two infimal decision units is through the process interface inputs, selection of the local controls must be done in cognizance of the entire process. It is important that the local control selections be made in conjunction with each other, so that their relative contribution to the overall performance can be taken into account. This is not done in the example. Since the interface input to the subprocess P, has no effect on it, selection of the local control m, is completely independent of the selection of m2 .
(A,
160
V . Coordination of Unconstrained Optimizing Systems
Coordinability with Goal Coordination
A two-level system using only the interaction-prediction mode of coordination, no goal coordination, can easily fail to be coordinable by the interaction prediction principle. We have just seen an example of one such case. However, if we use goal coordination in conjunction with interactionprediction, it is possible to obtain coordinability by the principle where coordinability otherwise fails. Example 5.10
Consider the system given in Example 5.9 but with the infimal performance modifications for each p in 28 = RZ given as: Gi/j(mi Y i ) = Gi(mi 9
3
Yi)
+ P i mi.
For each prediction a = (al,a2) of the interface inputs, let P1 = 2(aZ
- hl),
P be given as:
P z = 0.
Then, for a given prediction a,the local optimal control inputs are:
mla = +(I - az + k,), m,' = L1 0a2 ' Now, using the subprocess couplings, u1 = m, and u2 = m,, we set m,' = a2 and m2' = a, and solve for a, and a, in the above expressions for mp. The g). only correct prediction is then B = ( B l , 2,) = (3, The system is coordinable by the interaction prediction principle relative to the mapping q : M -+ 28 given as: ql(m> = 2(ml - %),
qz(m) = 0.
We point out that -).rl(m) and -qz(m) are, respectively, linearizations of the total goal-interaction operators T,(m) and T2(m)at the point m. This example points out that goal coordination can be used in conjunction with interaction-prediction to make the system coordinable by the interaction prediction principle. This is, in general, true if sufficient freedom is allowed in making the infimal performance modifications. Therefore, the question of whether or not a system can be made coordinable by the prediction principle with goal coordination is meaningful only if the infimal performance modifications are given or limited to a specific class of modifications. Let 28 be a given set of goal coordination inputs; for each P in there is i I n. Conceptually, given a family of infimal objective functions g i p , 1 I we view the set 28 as specifyingthe infimal performance modifications (Example 5.10). The question of whether or not a system is coordinable by the prediction principle with goal coordination reduces, therefore, to the question of
7. Coordinability by the Interaction Prediction Principle
161
existence of a mapping q : M + %? so that (5.36) holds and, furthermore, there exists a coordination input y = (a, p) and control input m such that 4,(mY)= 7If there exists a mapping q : M + LB so that the system is coordinable by the prediction principle relative to q, and there exists a mapping 8 : d + LB such that
r ( 4= fw(m)>, then the appropriate goal coordination inputs /3 can be given directly as a function of the interface input predictions a ; p = O(a). Furthermore, in such a case, the coordinator need only be concerned with correctly predicting the interface inputs. Proposition 5.31
Suppose a two-level system is such that the subprocess interaction function K is one-to-one. Then, for the existence of a mapping q : M -+ LB such that the prediction principle to be applicable relative to q, it is both necessary and sufficient that each control input I% in M which is not overall optimal satisfies the inequality : inf gip(mi > Ki(m)) < gip(ffii> K i ( f i ) ) Mi
for some p in W and some i, 1 I i I n. PROOF. Suppose the prediction principle is applicable relative to some mapping q : M + LB. Let I% be an arbitrary input in M which is not overall optimal. Let y = (a, b), where a = K ( 2 ) and p = q(KZ). It must then be the case that 2 # my for any m y ;otherwise the applicability assumption would be denied. This establishes necessity. Suppose the prediction principle is not applicable relative to any mapping q : M + 3. Then there exists a in d such that for all p in %? there exists my, where y = (a, p), which is not overall optimal, but K(mY)= a. Since K is where a = K(KZ). Therefore, for all p in %? and each i, one-to-one, m y= 6, l s i l n ,
gip(fii, Ki(fi)) = min gip(mi ,Ki(m)). Mi
This establishes sufficiency and, hence, the proposition. Corollary
Suppose the interlevel functions of a two-level system are strictly monotonic and the subprocess interaction function K is one-to-one. Then a necessary and sufficient condition for the existence of a mapping q : M + %? such that
V. Coordination of Unconstrained Optimizing Systems
162
the prediction principle is applicable relative to q is that each control input A in M which is not overall optimal satisfies the inequality inf ssa(P, m, K ( W < g(W M
for some P in 98. If the subprocess interaction function K is not one-to-one, then the conditions expressed in the above proposition and its corollary are only necessary conditions; the assumption of K being one-to-one was required only in showing sufficiency. Let us now consider coordinability of a two-level system by the prediction principle, relative to a mapping q : M --f 9. Unfortunately, in the present general setting, we must assume the existence of goal coordination input /3 in 99 which coordinates the system in question by the interaction balance us)such that principle; that is, for that goal coordination input there exists (ms, K(mP)= up and for any (mP,us), ms is overall optimal whenever K(ms) = uB. This assumption is, however, stronger than the assumption of restricted infimal harmony.
Proposition 5.32 Suppose a two-level system has a goal coordination input P in 9 that coordinates it by the interaction balance principle. Then, if the prediction principle is applicable relative to some mapping q : M + 98, the system is coordinable by the principle relative to a mapping 9 : M + 98 not necessarily equal to q. PROOF. Let be a goal coordination input in 98 that coordinates the system by the interaction balance principle. Then there exists a pair (A, 6) = xB such that zi = K ( h ) and h is overall optimal. Let & = zi = K(A);then for each i, 1 Ii I n, the control input A i minimizes g i ( m i p 6,) , over M i .Hence A = m yfor some my, where y = (a, /5) and K(my)= B. Furthermore, any m y is overall optimal if K(m') = &. Now define 9 : M + 9 such that G(h)= /5 and i j agrees with q for any m # h. This completes the proof.
B
If the interaction balance principle is applicable in a two-level system, we can use this fact to advantage under suitable conditions.
Proposition 5.33 Suppose the interaction balance principle is applicable in a given two-level system, and suppose for each a in at there exists a /3 in 98 and a pair (mP,us) such that a = d . Then there exists a mapping q : M + 9 such that the prediction principle is applicable relative to q. Moreover, if the system is coordinable by the interaction balance principle, the system is also coordinable by the prediction principle relative to some mapping q : M 9. --f
7. Coordinability by the Interaction Prediction Principle
163
PROOF. Define 8: d + W such that for each u in d, 8(a) = p only if there exists a pair (mp,up) such that u = up. By assumption, such a mapping exists. Now define q : M -,W such that q(m) = B(K(m))for all m in M . We claim the prediction principle is applicable relative to q. If the system is coordinable by either balance principle, there exists jl in 99 and a pair (mp,up) such that up = K(ms); therefore, if 8(u) = fi for CL = up, the system is coordinable by the prediction principle relative to q. This completes the proof.
If the interaction balance principle is applicable and the condition K ( M ) E d is relaxed, we can always find a suitable subset d c U such that there exists a mapping 8: d -,W as constructed in the above proof. In some cases, however, the only suitable subset is the singleton set {a}, where = K ( h ) for some overall optimal control input h. This is demonstrated in the following example. Example 5.1 1 Let Mi = Yi = Ui = R, the set of real numbers, for i = 1,2. Consider a two-level system having two subprocesses Pi given by the equations : y i = mi
+ ui = Pi(mi, ui)
and coupled according to the equations:
u, =
m2 = H,(m, y ) ,
u2 = rn,
= H2(m,y ) .
Let the overall performance function G be given on M x Y as the quadratic form :
a m , Y ) = (m, - m2I2 + Y l Z + YZ2. The overall optimal control input is h = (0,0) and is unique. Let W = R2 and for each p = (plyp2) in W let the infimal performance functions Gpibe given as
Gip(mi,y i , ui) = m t
- aimi ui + y t + (pi - 1)ui2,
where a, = and a2 = 1. (These infimal performance modifications are not zero-sum, and it can be proven that the system does not have the monotonicity property). The system is coordinable by the interaction balance principle. Let (plyp2) be an arbitrary goal coordination input. If p1 2 and p2 2 then optimal solutions (mlp,ulp) and (mZp,uZs)of the infimal problems exist, namely ( m t , u!) = (0,O) for i = 1, 2. Otherwise, one of the infimal problems fails to have an optimal solution. The interaction balance principle is therefore applicable, and any p in W is compensating iff p, 2 &$$ and p2 2 +.
a=
+,
1 64
V. Coordination of Unconstrained Optimizing Systems
Therefore, there exists a mapping 8: d + W as constructed in the proof of Prop. 5.33 only if d is the singleton set((0, O)}.Furthermore, for any subset d E U such that d # ((0,O)},there is no mapping 8: d + W such that the prediction principle is applicable relative to 8: M + W where q(m) = 8(K(m)). This is shown as follows: For each coordination input y = (a,P) the infimal problems have unique optimal solutions miy and m Z v : mlY= -
*,, 8
m 2 Y = 4A 22 '
Notice that miyis independent of P ; therefore, if there exists one such mapping 8: d -+ B, then any mapping from d into W will suffice. Choose a, = 17 and a2 = 8. Then, regardless of P, m, = -8 and m 2 y= 2. These control inputs produce the interface inputs u, = +7-m2y= 17 and u2 = mly = - 8. Hence a is a correct prediction, but the control input my = (- 8,2) is not overall optimal. In fact, any a such that 17a2 = -8a, is a correct prediction but does not give an overall optimal control input. We claim that the system in this example is such that the prediction principle is not applicable relative to any mapping q : M + B. By straightforward calculations, it can be shown that the nonoverall optimal control input rii = (- 8,2) fails to satisfy the inequality in Prop. 5.31 for every P in W and each i = 1, 2. In Prop. 5.33, we concluded that, under suitable conditions, coordinability by either balance principle is sufficient for coordinability by the prediction principle relative to some mapping q : M -+ B. We show in the following example that coordinability by either balance principle is not necessary.
Example 5.12 Consider the system in Example 5.10. We have already displayed in the example a mapping q : M + B relative to which the system was coordinable by either balance principle; in fact, neither balance principle is applicable. For a given goal coordination input fi in B, the infimal optimal solutions are : =1 2P1,
m 2 @ =-1z P 2 ,
a,
ulp
arbitrary,
u2@=
-"P2
a,
2-
Now, up = K(mp) iff = -3P2. Now, if = - 3 P 2 , then mIp = 3mZB. 3)and, therefore, it is never The overall optimal control input is riZ = achieved when the balance condition uB= K(mp)is satisfied.
(A,
8. The Coordination Process
165
8. THE COORDINATION PROCESS
We are now in a position to approach the problem of synthesizing the coordinator of a two-level system. In the context of optimizing systems, coordination of a two-level system is achieved by applying an optimal coordination input to the infimal decision units; the synthesis problem is, therefore, to find a transformation which will take whatever information is provided by the infimal units (the infimal decision units and subprocesses) and generate an optimal coordination input. We shall consider two approaches to this problem. The first approach is based on the applicability of a coordination principle and leads to an iterative procedure involving the participation of the decision-making units of both levels. In the second approach, the suprema1 decision problem (the problem which the coordinator solves to arrive at an optimal coordination input) is defined as a;l optimization problem without any direct reference to the coordination principles. The control or decision-making hierarchy of a two-level system considered in this chapter is shown again in Fig. 5.7. The coordinator C, receives the information w from the infimal units, then selects a coordination input based on this information and applies it to each of the infimal decision units d , . Each infimal decision unit di once given the coordination input y, arrives at a decision (miy,u i y )= di(y), which is an optimal solution of the specified problem gi(y) provided such a solution exists, and transmits certain information back to the coordinator. The coordinator Co can be synthesized by specifying a coordination strategy so: %? x W + %? to use in generating an optimal coordination input. A coordination strategy depends not only on the mapping used to generate new coordination inputs, but also on the nature of the information provided by the infimal units.
FIG.5.7 Two-level decision-making hierarchy.
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V. Coordination of Unconstrained Optimizing Systems
Application of the Coordination Principles if a given two-level system is coordinable by one of the coordination principles, one can quite readily propose a coordination strategy which, under suitable conditions, gives an optimal coordination input. Suppose a two-level system is coordinable by a given coordination principle, and, furthermore, suppose that for any given coordination input it can be determined whether or not the specified infimal problems have optimal solutions and the coordinating condition is satisfied. if the coordinating condition is not satisfied for a selected coordination input, the coordinator can use the resulting inbalance in the system to modify or change the coordination input. For a given coordination input y of a two-level system, let m(y) and u(y) be, respectively, a control input from M and an interface input from U, such that for each i, 1 I i I n, the pair (mi(y),u,(y)) is an optimal solution of the specified ith infimal problem gi(y).The nature of the coordination input y depends on the modes of coordination employed in the system. The pair of transformations m and u represent collectively the infimal decision units of the two-level system. We should note that m(y) and u(y) might not exist for some coordination inputs. * Suppose the coordinator of a given two-level system specifies the coordination input y and either m(y) and u(y) do not exist or, if they exist, the coordinating condition of a given coordination principle is not satisfied. The coordinator should then modify or change y. The information w(y) provided by the infimal units, with y given, should be such that w(y) and y are sufficient for the coordinator to determine whether or not the infimal decisions are optimal solutions and satisfy the coordinating condition. If the strategy so: % x W +. % employed by the coordinator is such that so(y, w(y)) = y iff m(y) and u(y) exist and satisfy the coordinating condition of a given coordination principle, we then refer to so as being based on the given principle. Once the coordination strategy is specified, there is defined a transformation T : % +. %?, such that
Repeated application of the transformation T represents an iteration process involving the participation of the decision-making units of both levels. After k iterations, starting with the coordination input yo, the coordinator applies the coordination input yk to the infimal decision units. The infimal decision
* This change in notation is to emphasize that the control inputs and interface inputs selected by the infimal decision units are related (possibly functionally) to the coordination inputs.
8. The Coordination Process
167
units arrive at their decisions m(yk)and u(yk),if the specified infimal problems have optimal solutions, and provide the coordinator with the information o ( y k ) which depends on their decisions. The coordinator then computes a new coordination input y k + l = so(yk,w(yk)). The coordination strategy so should be such that the generated sequence of coordination inputs converges to or terminates in an optimal coordination input.
Example 5.13 Let M i = Yi = Ui = R for i = 1,2. Consider the two-level system having two subprocesses P , and P , given by the equations y1 = m, - u1 = P,(m,, a,),
y, = m2 - 2u,
= P,(m, , u2)
and coupled according to the equations
*
u1 = y 2 - m ,
+
u2 = y1 - m, = H2(m,y).
= Hl(m, y),
The overall process P : M + Y and subprocess interaction function K : M are then P(m) = Pm, K(m) = Krn,
+U
where P and K are the 2 x 2 matrices
Let the overall performance function G be given on M x Y as the sum G(m, y ) = G,(m,, y,) + G,(m2,y,) of the infimal performance functions G1 and G 2 , which are the quadratic forms
Gi(mi,y i ) = mi2
+ (yi - 1)2,
for i = 1,2. The overall optimal control input found by minimizing g(m) = G(m, P(m)) over M , is f i = (4,O) and is unique. Let the coordination object = R2, and, for each coordination input P in g, let the modified infimal performance functions G , be given as GI&&,
>
Gza(m2 7
u19
Yl) = G,(m1, Yl) + P l U l
u2 ? Y 2 ) =
G2(m, 7
+ +(P2 - 2 P l h 9 Y 2 ) + P 2 u2 + 3(P1 - Pz)m2.
The infimal performance modifications are zero-sum (balanced), and, therefore, the interaction balance principle is applicable ; furthermore, the system is coordinable by the principle. The infimal problems have unique optimal solutions for each coordination input P :
%(PI = 4 %(PI =
P 2 ,
-&Ply
Ul(P) =
-(I
u*(P) = -*(1
+ 3P1 + + P A + SP1 + 4P2).
(5.37)
V . Coordination of Unconstrained Optimizing Systems
168
The coordination input B = (-+, - 2) coordinates the system by the principle. The coordinator can arrive at B if its selection of the coordination input is successively adjusted according to the rule
P'
=
P + 'Cu(P> - Km(P>I,
(5.38)
where ;1is an appropriate diagonal matrix. Upon substituting the expressions for m(P) and u(P) into (5.38), we obtain P' solely in terms of P : P1'
=
- $'%l)Pl - ' 1 1 7
P 2 ' = (l - $'22)P2 - *'22
*
These two equations define a transformation T:29 + g.Since the optimal solutions of the infimal problems are unique and the system is coordinable by the interaction balance principle, T has a fixed point which is the optimal coordination input B, that is, T(B)= B. If we choose the diagonal matrix L and 0 < such that 0 < A,, < < 3, repeated application of T produces a sequence of coordination inputs which converges to the optimal coordination input B in the norm llxll = maxi IxJ. Convergence is independent of the initial selection of P. With the background provided by this example, we can proceed to discuss on an abstract level the coordinator synthesis problem in terms of coordination strategies based on the balance principles. A coordination strategy based on the interaction balance principle is a mapping so: g x U x U-29, where 29 is the goal coordination object, such that the condition Cso(P, u,
4 = PI *
cu = U'I
is satisfied on its domain, and the corresponding transformation T: satisfies the equation
(5.39) + 98
T(P)= S d P , u(P), K(m(P)))
on 29. The meaning of condition (5.39) should be obvious: if, for a selected coordination input P, the coordinating condition u(P) = K(m(P)) is satisfied, no adjustment is made. Therefore, if the interaction balance principle is applicable in the given system, any fixed point of T is an optimal coordination input, since
C W ) = PI
* C4P) = K(m(P))I.
Therefore, T has a fixed point only if the system is coordinable by the principle. Conversely, if the system is coordinable by the principle, T then has a fixed point,provided the optimal solutions of the infimal problems are unique. If the coordination object is provided with an appropriate structure, say a metric, the coordination strategy so should be such that T is a contraction mapping.
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8. The CoordinationProcess
A coordination strategy based on the performance balance principle takes on a form appropriate to the coordinating condition of the principle. If a given two-level system has the strict monotonicity property, then an example of a coordination strategy based on performance balance is a mapping so: B x V x V + B satisfying the condition Cso(P, u, 0’) = PI
=>
Cu = u’l
on its domain and for which the transformation T : B + B is given by the equation
where gs and g are, respectively, the system’s apparent-overall objective function and overall objective function. A coordination strategy based on the interaction prediction principle with goal coordination given by a mapping q: M + B must take into account correctness of not only the interface input predictions but also the goal coordination. The following examples show how this might be accomplished.
Example 5.14 Consider the system of Example 5.13, but now interaction prediction with goal coordination will be used as a means of coordinating the system. Then, for each coordination input pair (a, P), where a is a prediction of the interface inputs, the specified infimal problems have unique optimal solutions:
m,(a, P) = $(4
+ 4% + 2Pl - P Z ) ,
P) = g 4 + 8% - PI + 8 2 ) .
The prediction principle is applicable to the system relative to the mapping r,~: M + B defined as q(m) = O(K(m))where 8: JQ + B is given by the two equations = 4( -
+ 2~(,),
O,(a) = 4(a, - 4c(2 - 11,
found by setting a = u(P) in (5.37)and solving for in terms of a. The system is also coordinable by the prediction principle relative to q . It can be verified with some algebraic manipulation that, for any coordination input pair (a, P), the condition P = q(m(a,P)) and CL = Km(a, P) implies m(a, P) = m = (4, Furthermore, the pair (d, D), where d = (A,-&) and 8 = (-+, -2), is the unique pair of coordination inputs for which 8 = q(m(d, 8)) and d = Km(d, 8). The coordinator can arrive at the pair (d, 8) if its initial selection of a coordination input pair is successivelyadjusted according to the coordination strategy
a).
a’ = a
+ PIP - r,I(m(a,P ) ) I Y
P’ = P + 4. - m
a , PI1
V. Coordination of Unconstrained Optimizing Systems
170
where I and p are appropriate diagonal matrices. Now, by expressing the right sides explicitly in terms of a and P, we obtain the transformation T,,:d x @ . ? -+ d x G . 3 given by the equations a’, = M,
+ $pl1(16a1 - 24az + 15f11- 7pz 4- 4),
If I and p are chosen such that their norms are sufficiently small, repeated application of T,, produces a sequence of coordination input pairs which converges to the optimal coordination input pair (Sr, B). The coordination strategy in this example takes into account correctness of both predictions and goal coordination. It should be noticed that it is not generally the case that all coordination input pairs (a, p) in a sequence generated by the transformation T,, of the above example are such that j3 = O(cc). If we require all selected coordination input pairs to satisfy this constraint, we can still use the given rule for adjusting the predictions provided p is appropriately selected. Example 5.15
Consider the system of Example 5.14, and now suppose the coordinator is restricted to selecting only the coordination input pairs satisfying the constraint j3 = O(a). Since q is the composition of 8 and K, the coordinator need only adjust the interface input predictions according to the rule
where p = 8(a), and give the new goal coordination input P‘ = 8(a’). This adjustment of the predictions defines the transformation T,: d d given by the equations a,’ = a1
+ ,ul
1( -
1 8a1
+ 5 2 +~8),~
~ ( 2= ‘
+ pZ2(26al - 84aZ- 12),
found by using fl = 8(a) to eliminate p. If both plland p z z are sufficiently near zero, repeated application of T, produces a sequence of interface input predictions which converges to 61. This is so because 8 is a one-to-one function : if p = O(a) and j? = q(m(a,p)), then a = Km(a, P). If 8 were not one-to-one, convergence might then be to some a other than d .
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8. The Coordination Process
In general, a coordination strategy based on the prediction principle relative to a given mapping q : M + B is a mapping so: %‘ x B x U + %‘, where % = d x B, such that the condition [s0((a7
PI, P’, u) = (a7P)1 * CP’
=
P and u = ual
is satisfied on its domain, and the transformation T,,: % + %‘ from V into itself is given by the equation T,,(%P) = S(b7
PI, ?(m(aY P>)Y K(m(a7 P)))
on V. If the coordinator is required to produce only coordination input pairs satisfying the constraint P = O(a) when 8: d B is given such that q(m) = B(K(m)), the coordination strategy so should satisfy the additional condition [so((%
PI, P’, 4 = (a‘,P’)1 * CP’ = W)l
on its domain. If, in a given system, the prediction principle is applicable relative to some mapping q, any fixed point of the transformation T,, is an optimal coordination input pair and, consequently, if T,, has a fixed point the system is coordinable by the principle relative to q. Additional conditions should then be imposed on the coordination strategy so that T,, is a contraction mapping in the appropriate sense. The iteration processes presented here and based on the coordination principles will be studied in more detail in Chap. VI, where sufficient structure is available to discuss convergence. The solution of the coordination problem for a given two-level system coordinable by a given coordination principle may be obtained as the optimal solution of an appropriately defined optimization problem. To illustrate this, suppose there is defined a mapping G o : U x U + V satisfying the condition [Go(u, u’) = inf Go]
=>
[u = u ’ ] ,
on its domain. Now formulate the suprema1 optimization problem as follows: find p in %‘, the coordination object of the system, such that K ( m ( f ) ) )= min 4
GO(u(y),
K(m(y))),
assuming optimal solutions of the infimal problems exist for each coordinaThe outcome function for the coordinator is then the mapping tion input in %‘. P o : % + U x U given by the equation Po(y) = (u(y), K(m(y)))and encompassing all the infimal decision units as well as the subprocesses shown in Figure 5.8.
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V. Coordination of Unconstrained Optimizing Systems
r-
19 I I I I I
I
I I
I
If the interaction balance principle is applicable in the given system, let V = &?. Then any optimal solution of the supremal problem so formulated is an optimal coordination input. However, it is fair to say that the complexity of Po prevents this approach from being practical. The supremal optimization problem may be solved by an appropriate iteration process involving a coordination strategy based on the interaction balance principle, but these strategies are designed to solve the coordination problem directly and, therefore, there is no need to introduce a supremal performance function, much less an optimization problem. Viewing the coordination strategies as being designed to solve a supremal optimization problem, although correct, is redundant and might be confusing. Optimization Approach The supremal decision problem can be formulated as an optimization problem without any direct reference to the coordination principles. We shall describe two such approaches. The Supremal Problem ns the Overall Problem The most obvious choice for the supremal problem is the overall optimization problem of a given two-level system. However, some modification is necessary to take into account that the coordinator’s decision variables are
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8. The Coordination Process
coordination inputs and not control inputs. The performance function Go for the coordinator can, therefore, be the overall performance function G, but then the outcome function Po must be a mapping Po: V -+ M x Y. The supremal problem is then to find a coordination input in V such that GO(PO(9))= min Go(Po(Y)). Q
In order that such a selection of the coordination input actually coordinate the system, the outcome function Po should be given on its domain as (5.40)
It should be noticed that the outcome function Po is indeed quite complicated, since Po encompasses not only the interrelationships between the overall controls and the outputs given by the overall process P, but also the infimal decision units which select the controls that are applied to the process. In short, it is the composition of the overall process and all infimal decision units. Clearly, such an approach to the supremal decision problem is not very practical. It is of interest here only for comparison with other more realistic approaches. Namely, the infimal optimization problems are often specified in advance, and the task of the coordinator is, precisely, to induce the infimal decision units to produce a solution of the overall optimization problem, an optimal overall control input. If the supremal problem is the optimization problem defined above with the outcome function Po given by (5.40), the problem of finding an optimal coordination input is likely to be more difficult than even the overall problem. The source of difficulties in this approach is that all infimal units are viewed as a single process: sub-systems are not explicitly recognized. However, if one takes advantage of the existing sub-systems and “ distributes ” the load of acheiving the overall solution between them, a simplified problem for the coordinator can be formulated, and the overall problem can usually be solved by means of a supremal problem that is considerably simpler than the above formulation and, in particular, the overall problem. In the context of optimizing systems, the overall optimum is achieved with a considerable reduction of effort on the part of the coordinator if the conditions are appropriate for application of the coordination principles. Often, one of the main objectives of setting up a multi-level system is the reduction of .the total effort spent in problem-solving or control. It is only through the use of some ingenious approach such as that offered by the coordination principles that this becomes possible. We should point out again that saving of effort is not the only raison d’etre for a multilevel system. As already pointed out in Chap. I, the limitation in size and complexity of the sub-systems also plays a role in the selection
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V. Coordination of Unconstrained Optimizing Systems
of a system structure. In the context of social systems, even if an optimal decision is known to the manager or coordinator, a formidable problem of implementation still remains. A principal drawback, apart from complexity, in using the overall performance function G for the supremal optimization problem is that G does not explicitly depend on the coordination input y, the decision variable of the coordinator. It is, therefore, difficult to evaluate on the basis of G itself whether a particular choice of y is suitable and, if not, how to improve it. Consequently, it appears that, between two levels, there is no easy way to “divide” the effort spent in obtaining an overall optimum. If the overall problem is used on the second level, the coordinator must first solve the overall problem and then, from the solution, derive the required coordination input. If the infimal decision units are not mandatory for implementation, the coordinator having possession of an overall optimal control input could very well dispense of the entire hierarchical structure and simply apply the control input to the process P. This is also true when Po is used as the outcome function; although the decision variable is then a coordination input, the complexity of Po (as given in Eq. 5.40) makes the supremal optimization problem even more difficult than the given overall optimization problem.
The Suprema1 Problem Using an Apparent-Overall Objective Function Some of these difficulties can be alleviated by using for the supremal optimization problem a performance function which explicitly depends on the coordination inputs. Such is the apparent-overall objective function gs of the given two-level system because of the following reasons.
(i) ga is defined on all decision variables in the system: the value of gs depends on the selected goal coordination input p, the selected control input m, and the selected interface input u ; hence, the selection of p, m, and u may be made in reference to ga
.
(ii) ga is intimately related to the overall performance: by definition, ga(/?,m, u) = G(m, P(m)), whenever u = K(m). (iii) If the system has the monotonicity property, the optimal overall performance can, under suitable conditions, be obtained from gd by applying a sequence of extremization operators: if for some goal coordination input Po and control input m0 gs(fio, mo, K(mo))= max min min g(p, m,u), I
M
then m0 is an overall optimal control input.
U
(5.41)
8. The Coordination Process
175
Let ga be an apparent-overall objective function of a two-level system having the monotonicity property. If Eq. (5.41) holds for some goal coordination input Po and control input mo,we then have the two equalities: max s98V9 4 a
B)Y
if m(P) and u(B) exist for at least p
u(P)> = G(f% =
P(&)),
(5.42)
Po; and
if m(a, j?) exists for at least P = Po and ct = K(mo).The solution of the overall problem can then be defined in terms of extremization of the apparentoverall objective function and distributed among the two levels. Equation (5.42) indicates that, if the infimal decision units minimize their local performances over both local control and interface inputs, the coordinator should then choose 6 so to maximize gs . Likewise, Eq. (5.43) indicates that, if the infimal decision units minimize their local performance over only local control inputs, the coordinator should then choose. the predictions ct to minimize, while choosing p to maximize gg . There are some interesting observations regarding the formulation of the suprema1 optimization problem using an apparent-overall objective function ga of a two-level system having the monotonicity property. (i) Observe that the infimal decision units might quite legitimately consider ga as giving the actual overall cost: this is, indeed, the case whenever the selected interface inputs balance. Since the system has the monotonicity property, the infimal decision units minimize gg over the ranges of their respective decision variables. The coordinator, however, is not concerned with minimizing ga over the range of its decision variable P ; rather, it maximizes ga over @, as indicated above. The infimal decision units must then consider the coordinator as opposing their own interests, and, therefore, the interlevel relationship appears to be a two-person game : the infimal decisions units collectively versus the coordinator. Since both the infimal decision units and the coordinator are interested in minimizing the actual overall cost, the apparent competition between them is paradoxical. The explanation of this paradox lies in the fact that the infimal decision units consider the interface inputs as free control inputs and select them accordingly. This results in an apparent overall cost which is lower than, if not equal to, that actually obtainable when the subprocess coupling constraints are satisfied. The coordinator must then offset this action of the infimal decision units, for the fact that the apparent overall cost is lower than that actually obtainable indicates that the actual overall cost when the coupling constraints are satisfied might not be optimal. It is interesting to note that the action of the
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V. Coordination of Unconstrained Optimizing Systems
coordinator, “the boss,” appears contrary to the interest of the group, yet in the final analysis it is beneficial to all concerned. By maximizing over B, the coordinator recognizes the reality of the process interactions and ensures their compensation. (ii) Under appropriate conditions [strictly monotonic interlevel functions and (5.21)], Eq. (5.41) holds iff the apparent-overall objective function ga has a saddle value: there exists a triplet (p, h, 6 ) such that the inequalities, ga(P, A, 2) 5 ga
are satisfied on B x M x U. The infimal problems of minimizing local performances (and hence g) over both control and interface inputs, and the supremal problem of maximizing ga over the coordination inputs /3 can be considered “ dual problems. Namely, given a saddle point (B, h, fi) of ga , minimizing g,@, m,u) over M x U and maximizing ga(P, h, ti) over yield the same value for ga . Furthermore, we should note that, due to the relationship g(m) = G(m, P(m)), minimizing ga@, m, u) over the set ”
{(m,W4):m
E
W
is precisely the overall optimization problem. (iii) It might appear that the decision-making units on each level have their own separate decision problems. While this is true, it should be recognized that these problems are interdependent. In order for the coordinator to solve its problem of maximizing ga over B, it must be given the infimal but the infimal solutions cannot be produced optimal solutions m(P)and ,)@z without the coordination input. In order to derive an optimal coordination input by maximizing g,, the coordinator needs a transformation which will take a given coordination input p to infimal optimal solutions m(P) and u(jl). Such a transformation may be very complex, and, consequently, the same difficulties are faced here as in the direct approach using the overall performance function. The supremal “dual” problem may be solved by an iteration process involving techniques such as hill climbing. Even though the computation effort spent at each step of the iteration might be small, the total effort required can be considerable. The indirect approach using the apparent-overall objective functions (the so-called dual approach), therefore, appears to have more conceptual than practical value. (iv) Finally, the iteration processes based on the coordination principles should be distinguished from the above-mentioned iteration process. The latter is defined in terms of changes in the value of g, due to changes in the coordination inputs, while the former is defined in terms of the coordinating conditions without any reference to gss. There is, however, a relationship
8. The Coordination Process
177
between the two processes, in the sense that they both achieve the same result: under the right conditions, they both converge to or terminate in an optimal coordination input. One might then consider the coordinator as solving the “dual” suprema1problem. However, as we mentioned earlier in this section, this interpretation can be misleading if the coordinator’s strategy is, indeed, based on a coordination principle (interaction balance) rather than hill climbing on gr. It should be pointed out that the properties of gB affect the convergence of both kinds of iteration processes. The convergence of these iteration processes can therefore be examined in terms of the properties of ga and K ; this will be done for more structured systems in Chap. VI.
Chapter V I
OPTIMAL COORDINATION OF DYNAMIC SYSTEMS
In this chapter, we are concerned with the coordination of two-level systems as described in Chap. V, but with the additional assumption that the system’s objects are, in general, subsets of normed linear spaces. We shall focus our analysis on the application of the coordination principles to obtain a coordinated “state” of a two-level system. The results bear upon such properties as convexity, compactness, semicontinuity, and the like and are, therefore, more specific than those obtained in the preceding chapter. Of course the results obtained there in the general systems framework are applicable to the systems we consider here. In this chapter we explore not only the existence of optimal coordination inputs, but we also give some explicit expressions for them, in terms of linear approximations of the goal-interaction operators introduced in Chap. V, Sect. 5. Finally, some iterative techniques are given for solving the coordination problem.
1. INTRODUCTION
The class of systems considered in this chapter includes, among others, the class of dynamic multivariable systems described by ordinary vector differential equations. To introduce the reader to the application of the coordination principles to the problem of optimal coordination of dynamic systems, we give an example in the realm of linear dynamic systems with quadratic performance functions. 178
1. Introduction
179
Consider a two-level system having the overall process P described on the time interval [0,1] by the vector differential equation d
dt Y = A y + m,
Y(0)= 0,
where A is a 2 x 2 constant matrix [aij], and m is a pair of real-valued squareintegrable functions on [0, 11. Precisely, P is the mapping P:M - , Y,where M = Y = SZ[O, 11 x 9,[0, 11, defined by (6.1) for all m in M. Assume P consists of the subprocesses P i , i = 1,2, where Pi is described on [0,1] by the equation d
dt yi = aiiyi + mi + ui ,
yi(0) = 0,
(6.2)
and where the subprocess couplings are Ul = a1zyz
= HI(4 u),
u2
= az1y1
= Hz(m, v).
Hence, the ith subprocess Pi is the mapping P i : Mi x Ui + Y i , where M i = Ui = Yi = gZ[O, 11, defined by (6.2) for all ( m i , ui) in M i x U i . Let the overall performance function G be given on M x Y as the integral
where Q is a symmetric, positive semidefinite 2 x 2 matrix [qij], and r = (rl, rz) is a given vector. The overall (optimization) problem is to minimize G(m, P(m)) over the space M. Before addressing the coordination problem through application of the coordination principles, note that the overall problem has a unique optimal solution. A direct application of the maximum principle yields the overall optimal control input A; A = -+A,
where 1 satisfies the pair of vector differential equations d
1 = -AT1 + 2Q(r - y), dt d dt
-y on the interval [0, 11.
= Ay
- $A,
A(1) = 0,
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VI. Optimal Coordination of Dynamic Systems
Our first consideration will be to formulate the infimal (optimization) problems so that the system can be coordinated by application of the interaction balance principle. For simplicity, assume Q is diagonal, with qll> ail, and q22 > u f 2 . Then, for each pair /3 = (B1, p2) of functions in p2[0, 13, define the infimal performance functions G,, and G2, as follows:
+ B l U l - P 2 a2lYll dt, and 1
G2,(m2
9
u2
Y
Y2) =
J Cmz2 +
(u2
0
- a21rJ2 + 4;2(Y2
+ B2 u2 - /31a12Y,l
- r2)2
dt,
where qil = qll- u21 and qi2 = qZ2- a f 2 . By simple substitution, we see that G is additive: 2
G(m,Y ) = G1,(m1, a1zy2 Y1) + Gzp("22 a2lY1, YZ),
(6.4) for all goal coordination inputs /3. Given /3, the ith infimal problem is to minimize Gi,(mi, ui, P i ( m i ,ui)) over the space M i x Ui . Now, for each goal coordination input p = (pl, p2), both infimal problems have unique optimal solutions. Indeed, for the first infimal problem, we have from the maximum principle the optimal solution pair (mlS, u,,) : 9
9
d = - 4PlY % f l = a12r2 - 4B1-3P1, where p1 satisfies the pair of differential equations
d
Yl = ally1 - Pi + a12 1 2 - 3/31, dt
(6.5)
Yl(0) = 0,
on the interval [0, 11. The optimal solution of the second infimal problem is expressed by (6.5) and (6.6) with the indices 1 and 2 interchanged. The task of the coordinator is to find a goal coordination input /3, such that mB= (mls, mzs) is an overall optimal control input, the optimal solution of the overall problem. The interaction balance principle is applicable to this system because the system has the monotonicity property as shown by (6.4). Applicability can also be shown directly be solving (6.6) for p1 and pz under the constraints ul@= Hl(mfl,~ ( m s ) ) , u2fl = ~ ~ ( mP(mfl)). fi,
(6.7)
1. Introduction
181
Therefore, the coordinator utilizing the principle seeks a /3 for which (6.7)is satisfied. Such a coordination input exists: it is the coordination input 8 = (j,, S2),
where j = P ( h ) , and is the overall optimal control input. One can directly verify that for /3 = 8, the pair (m,, 14,) satisfies (6.7). Let us now formulate the infimal problems so that the system can be coordinated by application of the (interaction) prediction principle. We need make no assumptions concerning Q other than it be diagonal and at least positive semidefinite. For each pair B = (P,, P2) of functions in 9,[0,11, define the infimal performance functions G,, and G,, as 1
Gi,(mi 9 yi) =
J [mi2 + qii(yi - ri12 + 0
Pi Y
~ dt, I
for i = 1,2. Notice that, if optimization is over the spaces M i x U i , the infimal problems fail to have solutions. The system cannot then be coordinated by the interaction balance principle, even though the principle is applicable. However, if in addition to specifying the coordination input B, the coordinator also predicts the outcomes of the interface inputs and requires the prediction to be accepted as correct, coordination can be achieved. Let the coordination object be the space d x G?, where d and W represent the space U 2 [ 0 ,11 x S 2 [ 0 , 11. The elements a in d are viewed as predictions of the interface inputs, and the elements of G? determine the infimal performance functions G,, and G,, . Precisely, the ith infimal problem specified by y = (a, j?) is to minimize Gi,(mi,Pi(mi,ai))over the space M i ;the optimal solution is m i y : miy = - + p i ,
where p i satisfies the pair of differential equations
(6.9)
on the interval [0, 11. The interaction prediction principle can be applied in this example relative to the mapping q : M + G? given by the equations
Pi
= -%2m2
= v l h , m2),
P2 = - k z l m l = q 2 h 1 , m2).
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VI. Optimal Coordination of Dynamic Systems
If m y = (mIy,mzy) for a given y = (ay,By), we can directly verify that my is the overall optimal control input when aiy = H,(mY,P(m31,
(6.10) (6.1 1)
for i = 1,2. The coordinator need only search for a coordination input y such that (6.10) and (6.1 1) are satisfied. Such a coordination input is y = (a, where
B),
61 = WrA, P ( W ,
d = M9,
(6.12)
where h is the overall optimal control input. The fact that (6.12) gives the coordinating pair may be verified by directly substituting into (6.9), showing that (6.10) and (6.11) are satisfied.
2. PRELIMINARIES
Analysis of coordinability of two-level systems will be centered in this chapter on the coordination principles and those two-level systems whose objects are nonned linear spaces or, more specifically, sets of time functions. We present here briefly a realization of a normed linear space (vector space) as a space of time functions and then present the overall process and subprocess in a form suitable for analysis in normed linear spaces. We then give some preliminary results pertaining to the coordination problem.
Process Models The basic assumptions for the analysis in this chapter is that the systems objects are subsets of normed linear spaces: the overall process and subprocesses are assumed to be defined on subsets or normed linear spaces having, when appropriate, properties such as completeness, compactness, convexity, etc. Furthermore, in following standard procedures of functional analysis, the control and output objects of the subprocess models are defined by means of projection operators : bounded, linear, idempotent transformations of the space into itself. Hence, the local objects will be subsets of the overall objects. In following the development in this chapter, it might be helpful to have in mind some standard function space realizations of a normed linear space [27].
2. Preliminaries
183
Let Ibe the interval [a, b] of the real line R. The space U[.TT of all continuous functions f: F --+ W with the uniform norm.
llfll = sup{lf(tk t E a is a normed linear space. In fact, U [ F ] is complete in this norm and, hence, a Banach space. Also, for any integer n 2 1, the space U [ F ] " of all n-tuples of continuous functionals on F with norm
llfll =su~{lfi(t)l, t E Y, 1 I i I n> is a Banach space. Let N be a set of integers. The space RN of all functionsf: N -+ R with the norm
llfll = sup{lf(n)l, n E N > is a Banach space, and so is the space of all functionsf: N + R". Overall Process We assume the overall process P is given as an operator with domain a given normed linear space and range a subset of a given normed linear space Y.Hence, the overall process is a mapping P:ET -+ Y (as in Chap. V). The following examples point out some of the systems covered by this formulation of the overall process. Example 6.1. Dynamical Systems Let X be a topological space, R the real line, and B a set of functions on R closed under time shifts: for allfin B and t in R,the functionf, ,defined on R by the equationf,(z) =f(z + t), is also in B. We define a forced dynamical system in X x B as a mapping s: R x X x B + X satisfying the axioms: i. s(0, x , f ) = x, for all x in X and f i n B; ii. s is continuous in R x X, for allfin B; and iii. s(t t', x , f ) = s(t', s(t, x , f ) , f , ) , for all t , t' in R, x in X, and f in B.
+
Then, for any constant function f i n By the system sf: R x X - + X , given by the equation sf(t,x ) = s(t, x,f), is, according to Zubov [28], a dynamical system in X ; that is,
a. s,(O, x) = x , for all x in X; b. sf is continuous in R x X ; and c. sf(t + t', x ) = sf(t', sf(t,x)), for all t,
t'
in R and x in ,Y.
The above formulation of the overall process P covers at least the class of forced dynamical systems (i.e., dynamical systems with inputs) in X x B,
184
VI. Optimal Coordination of Dynamic Systems
where X and B are normed linear spaces. To show this let X and B be normed linear spaces, and let Y be the space of all continuous functions from R to X with the uniform norm. Now, define P : X x B + Y as follows:
P ( x , f ) ( t )= s(t, x, f),
for all t in R.
Then P is a mapping from the normed linear space = X x B into Y. Furthermore, if s is continuous in R x X x B, then P is continuous with respect to the uniform norm in Y. The set M c R of interest may be the set {x} x B, in which case the output trajectories y would be fixed at time t = 0 and varied at other times by applying different “controls” from B. On the other hand, M may be taken as the set X x {f},in which case the “control ” would then be fixed and the output trajectory y varied by changing the point through which it passes at time t = 0. Our formulation of P covers a much broader class of systems than the class of (forced) dynamical systems in normed linear spaces. In fact, the semigroup property (axiom iii) is not required. The process P may be any mapping from a subset of a normed linear space (of time functions) into normed linear space (of time functions) without any restrictions on its time evolution. The present formulation of P covers systems defined by differential and partial diffxential equations. Indeed, it has been shown [28] that the concept of a dynamical system includes systems described by ordinary differential equations, and the concept of a dynamical system in a function space includes systems described by partial differential equations, e.g., of the form aY -=a y + at
n
i=l
aY bi-, axi
where a and bi for i = 1, * . * ,n are given constants. Hence, our formalism also covers these. Apart from the overall process P being simply a mapping from one normed linear space into another, we also explicitly consider P as being defined by a set of differential equations as in the following example.
Example 6.2. Diferential Systems Let LT be the interval [0, 11 of the real line R. Let R be the space of all square-integrable functions m : LT -, 8,, where 8, is the euclidean p-space, and let Y be the space of all square integrable functions y : .F-,8,.Suppose the output y = P(m),for any m in M , is given by a vector-differential equation
on F. Under appropriate conditions on f, the above differential equation
2. Preliminaries
185
defines a mapping P: R is linear:
-+
Y. We consider, in particular, the case in which f
f(4 Y ( 0 , m(tN = AY(0 + Bm(t), where A and B are q x q and q x p matrices which may be time varying but bounded. The process P does not have to be defined only by a set of differential equations; in addition, a set of transformations giving the outputs as a function of the solutions of a set of differential equations might be required. The process P given as in the above example then represents only the state transition part of the system. However, notational and other resulting simplifications justify using the simpler form of P, and, moreover, for the type of problems we shall consider, any transformation on the outputs of P may be imbedded in the overall performance function. Subprocesses The overall control object M is assumed to be a subset of the normed linear space R.The local control objects are assumed to be subsets of the spaces M i, 1 I i I n, such that every m in R has a unique decomposition
(6.13)
m = m ,+ . . - + ~ z , , ,
with m iin d i . In other words, M is assumed to be the direct sum
R = dl0 - - O . dn.
(6.14)
The local control object i , 1 5 i I n, is then the subset of X igiven as the projection of M on J i .In general, M c A l0 * * 0 A%’,,; however, we assume
M = A, 0
0 An.
(6.15)
The case when M is a proper subset is covered in Chap. VII. This construction of the local control objects allows us to treat the local control inputs as members of the same set as the overall control inputs. This simplifies the notation considerably. For convenience we use the projection operators niM: + which are uniquely defined by the decomposition (6.13) of the elements in R ;for every m in M, IIiMm =mi is the projection of m in d i . The space M may be an n-fold Cartesian product, and hence the overall x M,, (as in Chap. V) with the control object can be given as M = Ml x local control objects the sets Mi. The corresponding decomposition of m in is given by Eq. (6.13) where for each i, 1 I i In, m i= (0,
-
with mi the ith component of m.
* *
-
,0, mi ,0, * - ,0),
VI. Optimal Coordination of Dynamic Systems
186
Likewise, the local output objects q i ,1 I i I n, are assumed to be such that Y is the direct sum
Y = Yyl 0 * * * 0 Y n . For each i, 1 I i I n, then let ni,be the projection operator on Y such that H i , y = y i is the projection of y in Y i We are now ready to proceed with the decompositionof the overall process. First, we introduce an interface object u = x .-x on,where the local interface objects O i , 1 < j I n, are normed linear spaces. The next step is to decompose P into n subprocesses Pi
.
u1
p i :Ai x
Vi+Yi,
and their n coupling functions Hi
H i : R x Y+ui, such that n
C1gi(mi
i=
9
Hi(m, p(m)>>= P(m)
(6.16)
for all m in R. The above decomposition of P may be realized as in the following examples.
Example 6.3 For each i, 1 I i In, the interface object ui is the image of M under the and the coupling function Hi= (I- niM) where Z is operator (I- niM) the identity operator. The interface input ui to the ith subprocess then consists of nonlocal control inputs: ui = m i+ + mi-l'+ mi+l + .*.+ m,,where m is the overall control input. The ith subprocess Si is then defined on 3 x U i by the equation
...
S i ( r n i , Ui) = n i y P(mi
+ Ui).
Such a decomposition of P is referred to as being input-canonical and is always possible.
Example 6.4 For each i, 1 I i 5 n, the interface object ui is the image of Y under the operator (I- ni,) which is taken as the coupling function H i . The interface input ui to the ith subprocess then consists of nonlocal outputs: ui =y1 * . . +y/ci-1 + y i + 1 * * - y,, . The ith subprocess P i is then defined on M i x Ui so as to satisfy the equation
+ +
+
gi(mi
, ( I - ni,)P(m)) = n i y P(m),
2. Preliminaries
187
for all m in M. Such a decomposition of P is referred to as being outputcanonical. A case in which this decomposition appears naturally is when y = P(m) is defined as the solution of a differential equation 3 =f(y, m) passing through a given point, as in Example 6.2. The subprocess interaction function Kis, as before, the mapping K: M + B given as K(m) =
P(m>),
where H(m, U) = (HI(m, Y>,.*
The mapping P : % x
9
Hn(m, u))*
u + Y, (6.17)
then represents the uncoupled subprocesses, since from Eq. (6.16) we have, (6.18)
where Ki: + Bi is the ith component mapping of K. The subprocess interactions give, as we recall from Chaps. IV and V, the (actual) interface inputs to the subprocesses as a function of the control inputs.
The Performance Functions The only additional assumption required to complete the list of assumptions we impose on the two-level systems to be analyzed in this chapter concerns the performance functions. We assume all given performance functions, overall and infimal, are real-valued functions defined on the appropriate objects. An overall performance function is defined on the space & xI Y, while a given ith infimal performance function is defined on the space Xi x gix Oi unless otherwise noted. Additive Realization of Overall Performance Functions An overall performance function G is in harmony with another overall performance function G in a given two-level system if G admits overall optimal control inputs which are overall optimal with respect to G. The performance values measured by G and G may be different for the same overall control; all we require is that a given control input be overalI optima1 with respect to G whenever it is overall optimal with respect to E.
188
VI. Optimal Coordination of Dynumic Systems
For any given overall performance function G, we can always defke iniimal performance functions G i , 1 I i In, where G i : Ai x g i -+ R, so that G,
is in harmony with G . The problem is trivial. If G admits an overall optimal control input h, let the infimal performance Gi be generated from G by restriction on the input-output pair (h,j),where 9 = P ( h ) . If G does not admit an overall optimum, simply choose the infimal performance functions Gi so that (7 does not admit an overall optimum. In view of this remark, our interests here are to derive under more restrictive conditions an additive overall performance function G which is in harmony with a given overall performance function. Specifically, assuming a family of infimal performance functions G i , 1 Ii In, is given we are interested in finding real numbers ai 2 0, 1 I i In, such that the overall performance function G, n
G(m,Y> = C ai Gi(mi yi > Hi(m, Y>>, 3
(6.19)
i=l
is in harmony with a given overall performance function G. We refer to such a performance function G as an additive realization of G in terms of the given infimal performance functions Gi , 1 I i In. The existence of an additive realization is of interest for the following reasons among others. Suppose a two-level system has an additive overall performance function and the infimal performance modifications are zero-sum; i.e., for any goal coordination input B in a given set W, and m in M , the overall performance is the sum of the (actual) infimal performances, n
g(m) = C gip(mi 2 Ki(m)>, i=l
for all m in R.Let the infimal objective functions gis be given in terms of the subprocess .Pi and a given infimal performance function Gis. Assume the system is coordinable by a given coordination principle. Then for any real numbers ai 2 0, 1 I i I n, and overall performance function G.
the system is also coordinable by that principle. Furthermore, if G is any overall performance function having an additive realization G,
for some B in W, the system is again coordinable by that principle.
2. Preliminaries
189
Therefore, the theory of coordinability developed here for systems with additive overall performance function and zero-sum infimal performance modifications can be immediately extended to all systems whose overall performance function has an additive realization in terms of the given infimal performance functions. We now give some conditions under which an additive realization exists. In doing so, we proceed from stronger to weaker conditions and present the results in terms of the overall objective function g and infimal cost functions h i , 1 I i I n, which we recall are given on M by the equations: S(m) = G(m, P(m)),
hi(m) = Gi(mi 2 pi(mi 3 ui), ui),
where ui = Hi(m, P(m)), G is the given overall performance function, and Gi , 1 I i 5 n, is the given ith infimal performance function of the two-level system in question. Proposition 6.1
Suppose M is convex and (6.20)
S O = $(h(m), * * * > hn(m)),
on M where $: Rn+ R is strictly monotone. Furthermore, suppose g is Frechet differentiable * on R and has a minimum over My and for each i, 1 Ii I n, the function hi is Frechet differentiable on M and strictly convex on M. Then there exist real numbers ai 2 0, 1 I i I n, such that the given overall performance function has an additive realization G of the form (6.19). PROOF. Let hbe a point in M minimizingg over M . Since g is differentiable
at h, g(h
+ Am) - g ( h ) = g ’ ( h ) h + @(A)
2 0,
for any real A, 0 < L < 1, and m such that h + Am is in M ywhere g’(h) is the Frechet derivative of g at h; hence g’(h)Am 2 0. From (6.20), we have for any m in &i
where = &)/ahi at the point (h,(m), * - - , hn(in)) and hi’@) is the Frechet derivative of hi at m. Since $ is monotone, we have for each i, 1 5 i In, that $i’(m) 2 0. And since $ is strictly monotone, a(m) = C;=l $i’(m) # 0.
* Frechet derivatives are necessary since., in general, we are dealing with functions whose argumentsare elements of function spaces.
VI. Optimal Coordination of Dynamic Systems
190
Now since u(m) > 0 and g’(rft)Am’2 0 for any rft each i, 1 I i In,
+ Am‘ in M, we have for
n
lim m-vii
1 [g’(m)lm‘]= C ai[h{(fi)]Am’2 0, u(m)
i=l
where ai= lim m-ril
1 [$:(m)].
(6.21)
u(m)
a, = 1. Since each hi is Note that ai 2 0 for each i, 1 I i I n, and strictly convex on M , and each ai is nonnegative, aihi is also strictly convex, on M . rft is, therefore, a global minimum over M and, moreover, it is unique. Consequently, G given by (6.19) is an additive realization of G. This completes the proof.
E=l
We showed, in proving the above proposition, how an additive realization of a given overall performance function can be constructed. We compute the coefficients ai for each i, 1 I i I n, given by the limit (6.21) and form new infimal performance functions Gi = aiGi for each i, 1 S i 5 n. Then, if these coefficients exist and the conditions of the proposition are satisfied,and additive realization G of G is given by (6.19). Corollary
Suppose a given two-level system is such that all the conditions of Prop. 6.1 are satisfied, except possibly the differentiability and convexity of the functions hi. However, suppose for some i, 1 Ii 5 n, the coefficient ai given by the limit (6.21) is positive and hi is strictly convex on M. Then G given by (6.19) is an additive realization of the given overall performance function. PROOF. For each i, 1 s i I n, let ai be given by the limit (6.21). Since aihi is strictly convex for some i, 1 I i I n, we have that aihi is strictly convex. Hence, if r?z in M minimizes g over M, then it is the unique point in M minimizing C;= ai hi over M. This completes the proof.
z=
The existence of an additive realization of a given overall performance function can be proved for the more general case in which differentiability is relaxed. Proposition 6.2
Suppose (6.20) is satisfied, is strictly monotone, and M is convex. Assume g has a minimum on M , and for each i, 1 I i I n, the function hiis strictly convex and bounded below on M. Then the given overall performance function has an additive realization of the form (6.19).
2. Preliminaries
191
PROOF. Let h be the point in M minimizing g over M. Let subsets of Rn defined as follows:
V and Y be
B= {(x,, x,,): for some m in M , h,(m) Ix i for i = 1, ..-,n}, Y = { ( x l , .*.,xn): h i ( h ) > xi for i = 1, ...,n}. - * a ,
Then, since M is convex and the functionals hi are convex on M, both sets Band Y are convex. Furthermore, V and _V are disjoint. If not, there exists x = (xl, x,,) common to both V and Y ;consequently there exists m in M such that hi(m) I xi< hi(&) for each i, 1 < i 5 n, and hence, because I/I is strictly monotone, $(m)< t,b(h),which is a contradiction. Also, neither nor Yare empty, since the hiare bounded below on M. Invoking the separating hyperplane theorem, we obtain real numbers a and ai for i = 1, -.-, n such that CS=, lai/ > 0, and
2 a, xi2 a, I
,x,,) in C
for any x = (xl,
* * *
for any x = (xl,
.-.,x,,)
i= 1 n
C a i x i < a, i= 1
in
_v.
From the definition of Y, we have for each i, 1 5 i I n, that a, 2 0. Since ,hn(h)) is on the boundary on,_V, ai h i ( h ) < a ; therefore, since R is in a, hi(h) = a. Now, x = (hl(m), * * * ,h,,(m)) is in for any m in M , and consequently,
2 = (hl(h),
---
c:=
cy=l
n
2a, hi(&) = min 2a, hi(m>.
i= 1
I
M
i=l
Notice that aihiis strictly convex on M, since hi is strictly convex, and a, 2 0 for i = 1, * ,n and a, > 0. Therefore, h is the unique point in M minimizing aihiover M. The proof is now complete.
-
In proving this proposition, we showed that the vector a = (al, ...,a,,) of coefficients defining the separating hyperplane can be used to construct an additive realization (6.19) of the given overall performance function. Linear God-Interaction Operators
It is often necessary to modify the given infimal performance functions of a two-level system in order to make the system coordinable. A vehicle for making such modifications is provided by the goal-interaction operators defined in Chap. V. Indeed, we use them in this chapter, especially their linear and I Y are Banach spaces approximations ; therefore, we assume the space & and introduce here a linearization of the goal-interaction operators.
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VI. Optimal Coordination of Dynamic Systems
The goal-interaction operators are defined in Chap. V for a given twolevel system in terms of the overall performance function, the subprocesses, and the subprocess interaction functions.
The Linearized Total Goal-InteractionOperator Following the definition in Chap. V the ith total goal-interaction operator at a point 61in &Z is the operator ri(fi): AI --* R, such that for all m in
T i ( f i ,m) = G(6, P ( 6 , K ( f i + lliM m))) - G(6, P(6)). The linear ith total goal-interaction operator at a point f i in functional ri(6)on R defined for all m in R by the limit
El is the linear
1
r;(fi)m = lim - [ri(6,Am)], 1-0 1 provided the limit exists for all m in stituting (6.22) into (6.23) that
(6.22)
(6.23)
a. If ri'(rii)exists, we have upon sub-
r;(fi)m = ~ ; ( f iJ, ) P / ( ~i ,i ) ~ ' ( f iniMm, )
(6.24)
where J = P ( f i ) and J = K(6). The quantities G;(fi, j j ) , P,'(6, u"), and K'(k) are, respectively, the Frechet derivatives of G with respect to y at (6, J ) , of P with respect to u at (%, u"), and of K a t 3.
The Linearized Partial Goal-InteractionOperators tor
The ijth partial goal-interaction operator at the point 51in rij(6): --* R,such that for all m in R rij(fiy m) = G(iii, P(6ii,ii
where ii
is the opera-
+ X j ( 6 + niMm)))- G(FR,P(fi)),
= K ( f i ) , and
x j ( m )= (0, * * * ,0, Kj(m),0, * - * ,0). Then, the linear 0th partial goal-interaction operator at a point FE in R is the on R defined for all m in R by the limit linear functional r;j(fi) 1
r i j ( 6 ) m = lim - [rij(fi7 h)], a-ro
1
provided the limit exists for all m in R.If Tij(fi)exists, we have that
r;j(m)m = G;(G, jj)q,(nzj, cj)~,.ym) niMm, where again j j
= P(IR)and
iij = K j ( f i ) .
(6.25)
2. Preliminaries
193
The Linearized Interface Goal-Znteraction Operator The ith interface goal-interaction operator at the point rii in operator Ai(Ai): V i3 R, such that for all ui in Ui , Ai(A, ui) = G(A, P(A, ii
R
is the
+ ai)) - G(6, P(A)),
-
where ii = K(fi), and ai = (0, * ,0, ui ,0, * . .,0). Hence, the linear ith interface goal-interaction operator at a point A in R is the linear functional Ai'(fi) on Vi defined for all ui in Ui by the limit 1 Ai(A)ui = lim - [Ai(A, Lu,)], 1-0
L
provided the limit exists for all ui in Vi. If Ai(fi) exists, we then have that Ai'(A)ui = Gy'(A, jj)9f"(Ai ,iii)ui , where j j = P(6i) and u",
(6.26)
= Ki(A).
Some UsefulForms of the Linearized Goal-Znteraction Operators Since our analysis in this chapter is centered on systems with additive overall performance functions, let G be additive in the sense that n
G(m7 Y ) = C Gi(mi 9 gi Hi(m, Y>>* 9
(6.27)
i=1
Hence, for j j
= P(A)
and ii
= K(A),
we have
n
GM'(fiyj j ) = C [GiM(&;ii i= 1
,@i ,6 i ) n i M
+ Gfu(&i ,
@i
U"i)HiM(A7j j ) ] .
Since the functions H i are the subprocess coupling functions, Hi is independent of @,andmi ,and hence HiM(&jj)lliM= 0 and H&(A, jj)niy = 0. Furthermore, from (6.1 S), we have P'(fi) = Py'(A, 6 )
+ P$(A,
n
=
C [9;M(&i i=l
y
iii)rIiM
ii)K'(fi)
+ 9;u(&i
y
u",)&yfi)].
We now use these identities to obtain the following forms of the linearized goal-interaction used later in this chapter :
194
VI. Optimal Coordination of Dynamic Systems
Let g : fi% -+ R be the overall objective function defined in terms of P and G ,
s(m>= G(m,P ( n 4 , and for each i, 1 Ii I n, let gi: corresponding to G i ,
xi x U i + R be the infimal objective function
Si(mi 9 ui) = Gdmi 9 pi(%, 9 ui), ui).
We then have the following, assuming the existence of the Frechet derivatives at $I and u“ = K(6i): (i) A,’(rR)ui = giv(&i,Ci)ui,
(6.28)
(ii) r;j(rFi)m= g(iU(Gj,iij)Kj’(rR)ni,m
(6.29)
niMm,
= Aj’(fi)Kj‘(fi)
(6.30) = [g‘(fi)
- g;&&,
ui)]n,, rn.
These identities point out not only the relationship between the linearized goal-interaction operators, but also shed some light on their conceptual significance when the overall performance function is additive.
3. APPLICATION OF THE BALANCE PRINCIPLES
The results regarding the concepts of applicability and coordinability for the balance principles in the preceding chapter are directly applicable to two-level systems having dynamic subprocesses. One advantage in dealing with weakly-conditioned systems, such as those in Chap. V, is the broad applicability of the results. On the other hand, most of the conditions given in Chap. V for coordinability by the balance principles are rather difficult to test, and so by utilizing the added structure of dynamical subprocesses we shall derive more specific conditions here. Since the interaction balance and performance balance principles have precisely the same domains of validity (Theorems 5.12 and 5.13) we need only consider one of them; namely the interaction balance principle.
195
3. Application of the Balance Principles General Results
Before beginning the analysis, let us review some results from Chap. V. For a given two-level system, let g : M + R,the overall objective function corresponding to the given overall performance function G , g(m)= G(m,P(m)). Let W be a given set such that each p in W gives the infimal performance functions G , , 1 I i I n. Let gis: X i x Uj--t R be the infimal objective functions corresponding to G,, Sip(mi y uj) = Gip(mi > Si(mi 9
Ui),
ui).
(6.31)
Hence, the set W is a goal coordination object for the system and, for each Uj, while the overall problem is to minimize g over the set M. The following two results from Chap. V are particularly relevant to our analysis.
p in W,the ith infimal problem is to minimize gV over the set Ai x
(i) The interaction balance principle is applicable in any two-level system having the monotonicity property (Cor. 5.19). Recall that a two-level system has the monotonicity property if, for each p in W,the system has a monotone interlevel function and, hence, the overall performance is in a monotonic relationship with the infimal performances. Therefore, for any goal coordinaup) is such that for each i, 1 s i 5 n, the pair tion input @ in W,if (ms, (m!, u!) is an optimal solution of the ith infimal problem and the interface inputs balance, us = K(ma),then m pminimizes g over M and is thus an overall optimal control input. (ii) If a given two-level system has the monotonicity property, then the existence of a goal coordination input p in W such that min ga(p, myu ) = min g(nt),
MXU
(6.32)
M
where ga is an apparent-overall objective function for the system, is necessary for coordinability by the interaction balance principle (Prop. 5.21). Furthermore, if the interlevel functions of the system have the strict monotonicity property, the existence of a coordination input for which (6.32) holds is both necessary and sufficient for coordinability by the interaction balance principle (Prop. 5.22). It is easy to give examples of two-level systems having the monotonicity property. Some have already been given in Chap. V. For example, if the interlevel function of a given two-level system is multiplication, the system then has the monotonicity property, provided each infimal objective function is nonnegative, and, moreover, its interlevel functions are strictly monotonic
VI. Optimal Coordination of Dynamic Systems
196
if each infimal objective function is strictly positive. In fact, the interlevel functions of a two-level system having the monotonicity property may be any combination of addition and multiplication that is order preserving. We shall restrict our analysis here to those two-level systems whose interlevel functions are addition; that is, the overall cost is the sum of the infimal costs, in the sense that
for all in in and fl in 2.Such systems have the strict monotonicity property. In our analysis, the overall performance function for a given two-level system is assumed to be given, along with the infimal performance functions G i , 1 I i I n, such that n
(6.34) where the functional g i , 1 i I n, is given in terms of G iby (6.31); that is, the given overall performance function is additive. Now, if n
min MxU
2 gi(mi,ui)= min g(m), i=l M
(6.35)
then the system, as given, is coordinable by the interaction balance principle : the infimal problems have optimal solutions such that the interface inputs balance, and hence the local control inputs so chosen are overall optimal. If Eq. (6.35) is not satisfied, the system is not coordinable by the interaction balance principle, and, therefore, in order to use the principle, the given infimal performance functions must be modified. If the given overall performance function is additive, so-called additive zero-sum modifications appear to be most attractive. Infimal performance modifications are additive zero-sum if for each in B the modifying terms clip Y
have the property that n
(6.36) i= 1
for all m in M . Condition (6.36) means that the infimal modifications cancel whenever the interface inputs balance. Such infimal performance modifications do not alter in any way the sum of the actual infimal performances. With such modifications, Eq. (6.33) is therefore satisfied.
3. Application of the Balance Principles
197
Most of the infimal performance modifications considered in this section are additive zero-sum. Hence, the starting point for our analysis of coordinability by the interaction balance principle is provided by the following corollary.
Corollary 6.3 Suppose a given two-level system has an additive overall performance function which admits an overall optimal (minimizing) control input, and the infimal performance modifications are additive, zero-sum. Then, the existence of a goal coordination input p in B such that n
inf
C gip(mi,ui) = inf g(rn),
MxUi=l
(6.37)
M
is both necessary and sufficient for coordinability by the interaction balance principle. PROOF.The function gEB) , n
CSip(mi i=
ga(B, my U) =
9
ui),
1
is an apparent overall-objective function for the system. NOW,note that because g takes on a minimum over M ythe right side of (6.37) exists and is the minimum of g on M. Then, when (6.37) is satisfied, the minimizing point of g yields (&ii , Ki(h)), which minimizes gV Consequently, (6.32) holds iff (6.37) holds. Since the given system has the strict monotonicity property (this is a consequence of the hypothesis), the corollary is proven by applying result (ii) above.
.
Corollary 6.3 will be taken as fundamental for obtaining conditions which guarantee coordinability by the interaction balance principle. We shall proceed by assuming additional properties (such as convexity, compactness, linearity, etc.) and determine under what conditions we can realize the requirements of the corollary and hence coordinability by the interaction balance principle.
Linear Modification One way of realizing additive zero-sum infimal performance modification the modifying terms p i p = Gi - Gip, 1 < i 5 n, is to choose, for each fi in 9, such that pi,.t(mi9
ui
3
yi)
= ai(mi)
+ Bi(U3 + yi(Yi),
(6.38)
VI. Optimal Coordination of Dynamic Systems
198 where
cli
, pi , and y i are given functionals, depending on p, such that
on M . Infimal performance modification given by (6.38) is zero-sum iff, for each in g,Eq. (6.39) is satisfied on M . This follows from (6.36) and (6.38). Hence, the problem of constructing additive zero-sum modifications under constraint (6.38) reduces to the problem of finding solutions of (6.39). This, however, could be a formidable problem, and in some cases only the trivial solution exists. Therefore, let us restrict our attention to some particular cases in which solutions of (6.39), other than the trivial solution, exist and are easily obtained. This is the case when a two-level system is such that its subprocess coupling functions are separable: for all i, 1 Ii In, either
or n
(ii)
K i ( m )=
1 Kij(mj)
on R.
j=1
If the subprocesses of a two-level system are obtained by an output-canonical decomposition (Example 6.4), the system then has property (6.40i). More generally, a two-level system has property (6.40i) whenever the functions Hi are linear; linearity, however, is not necessary. On the other hand, (6.40ii) is a property of a two-level system whose subprocesses are obtained by an input-canonical decomposition (Example 6.3). If a given two-level system has property (6.40i) and the functionals pi for each i, 1 I i I n, are linear, then the solutions of (6.39) are given by the n equations n
ai(mi)
+ ri(yi) = - j= C1PjHji(+Hi Yi). 7
Therefore, additive zero-sum infimal performance modification is realized if 9? = Ul* x . * .x Un*, where the super script * denotes conjugate space, and n, the modifying term pipis for each p = (pl , * * * ,p,) in Gi? and each i, 1 I i I given as follows: (6.41)
199
3. Application of the Balance Principles
On the other hand, if a given two-level system has property [6.4O(ii)] and, again, the functionak pi for each i, 1 Ii I n, are linear, then the solutions of (6.39) are given by the 2n equations n
a.a =
-j C DjKji, - I
Yi
=O*
Hence, additive zero-sum modifications can be realized if &? = Ul* x -.-x Un* and for each p = (PI, ... ,D,,) in &? and each i, 1 5 i In, n
The two forms of modifications given by (6.41) and (6.42) are indeed zero-sum, and since the goal coordination inputs are linear functionals, we refer to these two forms as linear zero-sum. The modifying terms assigned to the ith infimal decision unit for linear zero-sum infimal performance modification may be interpreted as linear functions of the interface input ui it “demands” from the other infimal units, and the interface inputs uji = Kji(mi)or uji = Hji(mi,yi) it intends to supply to the other infimal units. The linear functionals pi , ...,finthen play the role of prices (as discussed in Chap. I). In view of this form of modification, coordination by the interaction balance principle may be interpreted as assigning prices so that supply and demand balance, while coordination by the performance balance principle may be interpreted as, again, assigning prices, but now so that the anticipated and actual net costs balance. Assuming infimal performance modification is linear zero-sum, we can, with appropriate convexity, continuity, and compactness assumptions, obtain more special conditions than those of Cor. 6.3, which guarantee coordinability by the interaction balance principle. Proposition 6.4
Suppose a given two-level system has an additive overall performance function which admits an overall optimal control input, and the infimal performance modifications are linear zero-sum. Furthermore, assume the following: (i) For each i, 1 I i In, the set -Hi x Ui is weakly compact and convex, and for each j? in Bythe infimal performance function gis is convex and lower semi-continuous there. (ii) There exists fi in B such that n
C &mi MXU i=l inf
n
C gia(mi,ui). MXUi=l
, ui) = sup inf .¶
Then, the system is coordinable by the interaction balance principle.
VI. Optimal Coordination of Dynamic Systems
200
PROOF. Because of Cor. 6.3, we need only display a /3 in 8 for which (6.37) is satisfied. Since the modifications are linear zero-sum, 8 = U,* x * . . x Un*. For convenience, let R, , 1 I i 5 n, be defined on M x 0 as either
,
j= 1
or
&(m, u) = Ki(m), (6.43)
depending on whether the modifications are given by (6.41) or (6.42), respectively. Hence,
is the apparent-overall objective function of the system. Now, sup C pi(ui - Ri(m, u ) ) = co, B
if uiz Ki(m, u ) for some i,
1 I i I n.
i
Because of condition (i), each gia has a minimum on Ai x Ui ,and therefore, since modification is zero-sum, inf sup gd(p, m, u ) = inf sup g&?, m, K(m))= inf g(m). MXU
B
M 9 B
(6.45)
M
Also, from condition (i), gg is convex and weakly lower semicontinuous on M x U for each p in 8 and concave on B for each (m,u) in M x U. Moreover, M and U are weakly compact. Hence, sup inf ga(/3, m, u ) = inf sup gB(p, m,u). B MXU MxU a Therefore, from (6.45) and (6.46), we have that satisfies (6.37). Hence, the proposition is proven.
fl
(6.46)
given in assumption (ii)
If the subprocesses P i ,1 I i I n, of a given two-level system are linear and continuous, we can replace the convexity and continuity assumption (i) in the above proposition with the assumption that, for each i, 1 < i < n, and each fi in 93,the modified infimal performance function G , is convex and lower semicontinuous on A ix U ix CYi. Of course, assumption (i) does not require the G i pthemselves to be convex. Now, if in addition to the subprocesses being linear and continuous, the coupling functions H i , 1 5 i 5 n, are also linear and continuous, then, assuming infimal performance modification is linear zero-sum, assumption (i) is fulfilled if each given infimal performance function Gi,15 i I n, is convex and lower semicontinuous on its domain.
3. Application of the Balance Principles
20 1
Perhaps the most difficult assumption in the above proposition to verify in application is assumption (ii). For each p in 9 let
assumption (ii) requires that go takes on a maximum value on W. Given that the sets M , U i , and Y are subsets of reflexive Banach spaces, we can, by strengthening assumption (i) and introducing an additional assumption, verify assumption (ii) and, hence, coordinability by the interaction balance principle.
Proposition 6.5 Suppose a given two-level system has an additive overall performance function which admits an overall optimal control input, and the infimal performance modifications are linear zero-sum. Furthermore, let R and U i,for each i, 1 I i In, be reflexive Banach spaces, and assume the following: (i) For each i, 1 I i I n, the sets Ai and Ui are closed, bounded, and convex, and each gV ,for p in ik?, is convex and lower semicontinuous on Ai x U i ; (ii) go(p) < go(0) on a sphere Y rin W of radius r > 0 about the origin; and (iii) Ri is bounded on M x U. Then, the system is coordinable by the interaction balance principle. PROOF. A closed, bounded, convex set in a reflexive Banach space is weakly compact, and therefore Ai x Ui for each i, 1 I i I n, is weakly compact and convex. Consequently, we need only show that go has a maximum on W; coordinability then follows from Prop. 6.4. To show this, we use the fact that g o has a maximum on 9when go is concave and continuous on 2il and go@) + - 00 as llpll + co. To show concavity, let j3 in W be arbitrary. Because of linear zero-sum and ga, given by (6.44), is linear in j3. modifications, W = Ul* x ..*x Un*, Then for any real number 1,
g&P
+ (1 - W , m,4 = lga(P, m,u) + (1 - 4ga(B’, m,4.
Now, for 0 I 15 1, min ga(@ MXU
+ (1 - 1)F,m,u ) 2 min lg,(B,
Hence, concavity is proven.
MXU
m,u)
+ min (1 - A)g,(F, MXU
m,u),
VI. Optimal Coordination of Dynamic Systems
202 Now, let p" = AP
+ (1 - A)P'
and 0 < A
1
s 1; we then obtain from (6.47):
SO(P> 5 ;iCS0(p">- SO(B'>l
+ SO(P'>.
Let /?'= 0, p # 0, and 1 be such that r = AllPIj ; then S"P> 5 (IIPlllr)Cs"(B")
- S0(O)1 + g0(0).
Now, P' = 0; hence, p" = AP and IIp"II = r. Therefore, from assumption (iii), go@") - go(0) < 0, and consequently go($) 3 - 00 as llPll + 00. (Notice that 1 5 l / A = IIPll/r < co.) Finally, we show that go is continuous on 33. To do so, we use the follow ing lemma : A convex functional f on a topological linear space X is continuous in a subset X' of X iff there exists a nonempty open set 0 in X' such that f is majorized on 0. Let llj?ll = llpill be the norm in 99. Then, since the sets Ui are bounded and Ri is bounded on M x U , there exists for any in 33 a real number a > 0 such that
ci
I Ci Pi(ui - Ri(m7 u>>I5 x
i
IPiuiI
+ x i IPiRi(m, #>I
5 a IIPII,
and, therefore,
. 3 and, from the Hence, -go is majorized in any bounded open subset of % above lemma, go is therefore continuous on 9.This completes the proof.
If the mappings Xj given in (6.43) behave appropriately on their domains, assumption (ii) of the above proposition can then be verified. Notice that, for a given control input m in M , the interface input u = (ul, ..., u.) satisfying the equations ui = Ri(m, u) for each i, 1 I i In, is the interface input which actually occurs when the subprocesses P i are coupled. This follows from the relationship between the subprocesses and overall process given by (6.16). The following proposition specifies the appropriate behavior of the mappings
Ki .
Proposition 6.6 Suppose a given two-level system has an additive overall performance function which admits an overall optimal control input and the infimal performance modifications are linear zero-sum. Furthermore, let KI and Ui, for each i, 1 5 i 5 n, be reflexive Banach spaces and for each i, 1 5 i 5 n, assume :
203
3. Application of the Balance Principles
and Ui are closed, bounded, and convex, and each gis, for p in 2,is convex and lower semicontinuous on A j x Ui and gi is bounded there. (ii) For some E > 0 and all ei in Bi with norm lleill = E, there exists (my u) in M x U such that ei = ui - &(my u). Then, the system is coordinable by the interaction balance principle. PROOF.It suffices to show that assumption (ii) of Prop. 6.5 holds. Since the gi are bounded, there exist finite numbers a and b such that for all (m, u)
in M x U,
u IC gi(mi, ui) Ib.
(6.48)
i
Since infimal performance modification is linear zero-sum, 39 = U,* x x U,* and for each fl in 4,
---
inf ga(B, m yu ) I b MXU
+ inf
MxU i
Bi(ui - Ri(m, u)),
(6.49)
where ga is given by (6.44). Now, because of assumption (ii), there exists 0 such that
E
=-
ci
llpll = IIpill. Let r = b - a + 1. Then for any j? in 2 such that ~llpll= r, we obtain from (6.48)-(6.50),
where
inf ga(B, myu ) I b - E llj?llIa - 1 < a Iinf ga(O, myi d ) . MXU
MXU
This completes the proof. Assumption (ii) of the above proposition is rather easy to realize when n, is linear zero-sum modifications are given by (6.42), and Ki(M), 1 I i I bounded.
Corollary 6.7 Suppose a given two-level system has an additive overall performance function which admits an overall optimal control input, and the infimal performance modifications are given by (6.42). Let M and Uiy for each i, 1 Ii In, be reflexive Banach spaces, and for each i, 1 s i I n, assume the following: (i) Each gis, for p in %?, is convex and lower semicontinuous on .,Hi x U i and gi is bounded there;
204
VI. Optimal Coordination of Dynamic Systems and U i are closed, bounded, and convex, and U i contains the set Ki(M) + Ei where Ei is a closed ball in U i about the origin, with radius E > 0.
(ii)
&fi
Then, the system is coordinable by the interaction balance principle. Perhaps the most crucial assumption of the above proposition is boundedness of the functionals g i on x Ui . However, boundedness usually follows from the continuity of g i and boundedness of &fi x U i
.
Applications of the Linear Goal-Interaction Operators The linear interaction operators defined in the preceding section of this chapter can be used under suitable conditions as a means of modifying the infimal performance functions so that coordinability is achieved. We shall show here that, under appropriate conditions (Prop. 6.8), the linear interaction operators give an optimal coordination input for linear zero-sum modification. To use the linear goal-interaction operators we must assume their existence at a particular point in M and, therefore, we must assume differentiability of the subprocesses P i , the given infimal performance functions G , , and the subprocess interaction functions Ki
.
Lemma Suppose a given two-level system has an additive overall performance function which admits an overall optimal control input f i in M and, in addition, assume the following:
(i) For each i, 1 5 i I n, the functions P i , G i , and Ki are Frechet differentiable in each of their arguments; (ii) M is convex. For a given i, I 5 i In, let Si be the functional
gi(u L i , ui) = gi(# t i , ui) - A,’(&Z)U~ + riyh)m i ,
(6.51)
where A,’(&) and ri’(fi) are the linear goal-interaction operators at h,given respectively by (6.26) and (6.24). Then,
(6.52-3) for all
&ii
+ #ni in 4 ,and all ii, f ui in U i .
20 5
3. Application of the Balance Principles
PROOF. Since the overall performance function is additive, the overall objective function g is given by (6.34). For all h + m in M, 0 Ig(h + m ) - g(h). Now, from assumption (i), g is differentiable at h, and, from assumption (ii),
1 0 Ig'(rfz)m = lim - [g(& 2-0 A
+ Am) - g ( f i ) ] ,
(6.54)
for all rA + m in M , where A > 0. Since the chain rule of differentiation is applicable to Frechet derivatives, n
g'(h)rn = =
Since M = Al Cj3
i= 1
i=l
--
[ g i M ( R i ,ai)mi
+ gfu(Ri ,ai)Ki'(h)rn]
[ g i M ( R i ,ai) +
j= 1
1
(6.55)
0,
(6.56)
& ( A j , aj)K;(fi)IIiM
#)ti.
0 An,we have from (6.54) and (6.55),
*
[siM(Ai
7
ail +
igsu(fij
a j ) ~ i ( f i ) n i M ] m i2
j=l
for all Ri + mi in Ai. Now, from (6.30), the summation in (6.56) is the linear and goal interaction operator r;(nz), Si'dmi 7 Ui) = g i d m i 7 Ui)
+ rf(m),
(6.57)
therefore &(Ri , ai)mi 2 0 for all Ri + mi in Ai, which proves (6.52). On the other hand, &(Ri
,
= [giu(R, ,ai) - 4 ( f i ) ] U i .
But from (6.28), we have giu(Ri,ai) = Al(h), hence &(hi, ai) = 0. This proves (6.53) and the lemma. Notice that in assumption (ii) in the lemma, m need not be an internal point of M , but in that case differentiability should be suitably defined (e.g., one-side differential).
.
If h is an internal point, then the point ( R i ,hi) is a critical point of Si Also, notice that S i takes on a local minimum at (hi,Oi)if g i is convex, and this local minimum is absolute if, in addition, Ai x Ui is convex. These observations lead us to the main result concerning the use of the linear interaction operators, when for each i, 1 5 i 5 n, the connecting functions Ki for the subprocesses of the given two-level system satisfy (6.40ii), and linear zero-sum modification is given by (6.42).
VI. Optimal Coordination of Dynamic Systems
206 Proposition 6.8
Suppose a given two-level system has an additive overall performance function which admits an overall optimal control input rh in Myand the differentiability assumption of the above lemma is satisfied. In addition, assume the following: (i) Infimal performance modification is linear zero-sum; for each i, 1 I i I n, and p in gig(mi
9
ui) = g i ( m i
9
ui)
+ Pi ui -j C P j Kji (mi); = 1
(ii) For each i, 1 5 i 5 n, the set Ai x Ui is convex, and each gis is convex there. Then, the system is coordinable by the interaction balance principle, and, moreover, if fl in U,* x ... x Un*, is such that for each i, 1 5 i 5 n, pi =
-A:(&),
(6.58)
fi is an optimal coordination input. PROOF. If for any i, 1 5 i I n, we establish that giBM(Ri,ai) = &(Ai ,a,) and giBu(Oi,tii) = &,(ai ,Oi), where Si is given by (6.51) and ai = Ki(&);we where A,'(&) is given by (6.26), then
can then show, by using assumption (ii), that (Ai ,tii) minimizes gig over Ai x u,. Let i, 1 5 i In, be arbitrary; then, for arbitrary mi in X i we have at the point (Ai,ti,), g;BM(Ai ,Qi)mj=
[
g;M(Ri
,ai) - f /IjK&%J]
mi.
j=1
Since the K j must satisfy (6.40ii), Kii(Ri)rni= Kj'(.;i)#9Zi,
and, therefore, from (6.58), g I!S M(a. 1 3 ti.)&. i
i
=
[
gLM(&ii
, d i ) -k
j= 1
A;(dZ)KJi(dZ)]mi.
(6.59)
Now, from (6.30), the summation in (6.59) is the linear goal interaction operator rf(dZ);hence, from (6.57), we have that giBM(Ri,tii) = &(Ai ,tii). On the other hand, it is apparent that giBu(Ri,tii) = &(Ri, Eri) E 0. 0 * * 0 An, M is convex because of assumption (ii). Consequently, giBM(@Gi, Oi)mi2 0 for all Ri mi in Ai. Now, because of assumption (ii), (Ai, tii) minimizes gig over Ai x Ui. This is sufficient to prove the proposition.
-
+
3. Application of the Balance Principles
207
The above proposition applies, of course, to the case in which the subprocesses P i and the subprocess interaction functions Ki are nonlinear. If the functions Kiare linear, we need only require that the functionals gi be convex. On the other hand, if both the subprocesses S i and the subprocess interaction functions Kiare linear, we need only assume convexity of the given infimal performance functions G i.
Application to Linear Differential Systems The preceding results of this section are directly applicable to a variety of two-level systems whose subprocesses are described by sets of differential equations. To illustrate the application, we consider a few linear cases and, in doing so, derive a more explicit expression for an optimal coordination input. Let R and Y be the space of all square-integrablefunctions on the interval LT = [0, 13 with values, respectively, in euclidean spaces gP,p 2 n, and 6’q,q2n. Assume the overall process P is given as an operator with domain and range a subset of Y, such that the output y = P(m) for any m in is given as the solution of the vector differential equation j ( t ) = Ay(t)
+ Bm(t),
y(0) = 0
R Al
(6.60)
on 9, where A and B are q x q and q x p matrices which may be time-varying but bounded on 9. Hence, for any m in R and t in 9P(m)(t) = fQ,(t, cr)Bm(ts) dts, 0
where Q, is the fundamental matrix of (6.60). The control object M is assumed to be given as the subset of R consisting of all and only those control inputs m such that m(t) is in a compact, convex subset Cl of 8,for almost all t in 9. To construct the wbprocesses P i , 1 I i I n, we use the projection operators lliM and l l i ydefined as nonzerop x p and q x q diagonal matrices, with 0 and 1 as their diagonal entries such that Xi lliMand X i l l j yare unit matrices. We understand the extension of these projection operators to the spaces R and Y as pointwise operators. We also use the matrices A , and Bii given as follows: Aij
= lliyAnjy ,
Bij
= niy BlljM.
Now, for each i, 1 Ii I n, define the ith subprocess Sias the operator with domain d i x g i and range a subset of %Yi such that the output
208
VI. Optimal Coordination of Dynamic Systems
=Bi(mi,u,) for any ( m i ,ui) in X i x Yi is given as the solution of the differential equation
yi
&(t) = Aii&(t)
+
+ u,(t),
(6.61)
yi(0) = 0, on Y. Hence, for any ( m i ,ui) in X i x GYi and t in Y, BiiMi(t)
+
Bi(uzi,ui)(t) = f@,(t, a)[Bii(o)mi(a) ui(o)] do, 0
where cDi(f,
0)
is the fundamental matrix of (6.61); in fact, cDi(f, 0 ) = r I i , @ ( t , o)niy.
The dependence of ui(t) on the nonlocal inputs and outputs is given by the equation ui(t) =
1[Bij
+ Aij ~ j ( t >=I Hi(m, y)(t),
mj(t>
j#i
which defines the connecting function Hi as a linear operator on R x Y . One can easily verify that Eq. (6.16) is satisfied for all m in R. For each i, 1 I i I n, let the given infimal performance functions Gi be given on Xi x Yi as the integral Gi(mi
3
ui > yi)
=
1
(6.62)
Oi(t, mzi(t), ui(t), yi(t)) dt,
.F
and let the overall performance function G be additive: n
G(m, Y ) = C Gi(mi 7 Hi(m, Y>,yi>* i= 1
We assume Oi for each i, 1 I i I n, is convex for each f in 9, and continuous. The local control objects A i, 1 I i In, are the projections of M on the , hence (6.14) is satisfied. To ensure that M = Al 0 @ .An, space J iand we assume R = R, 0 0 Rnwhere Oi = ni,(0), 1 Ii I n. Now to apply the preceding results of this section and show coordinability of the above given two-level system when linear zero-sum infimal performance modifications are used, the interface objects Ui, 1 Ii In, must be selected as appropriate subsets of the space Yi, and the selection must be made in accordance with the form of linear zero-sum modification employed. Because of the linearity assumptions, both forms may be used. We may use the space W = Y as the coordination object, since any bounded linear functional on an L , space has a unique representation in terms of the inner product. For each /3 in W = Y, we let pi be its projection in Yi . First, consider the case when the modifications are given by (6.41). Applying Prop. 6.6, we obtain the following result:
--
3. Application of the Balance Principles
209
Corollary 6.9 For each i, 1 I i I n, and p in &?, let the infimal performance functions Gi for the above given two-level system be given as
Suppose for each i, 1 I i I n, that Gi(mi , ui,yi)is bounded on A?,x U iand Ui is a closed, bounded, convex subset of Yi such that for some E > 0 it contains all ui= Hi(m,y ) e , for all (myy ) in M x Y and ei in Yi with norm lleill I E satisfying the equation on 9:
+
j ( t ) = Ay(t)
+ Bm(t) + ei(t),
y(0) = 0.
(6.64)
Then, the system is coordinable by the interaction balance principle. PROOF. It can readily be shown that the assumptions of Prop. 6.6, except assumption (ii), are satisfied. Therefore, it suffices to show that assumption (ii) holds for some E > 0. For arbitrary, i, 1 I i I n, let eibe any element in %Yiwith norm lleill = E , where E > 0 is given in the hypothesis. Let m in M be given and let y in Y satisfy (6.64) on F.Then ui= Hi(m,y ) + e , is in Ui, and for eachj, 1 <j I n and j # i, uj = Hj(m, y ) is in U j Furthermore, g j = P j ( m j ,uj) for each j , 1 < j I n. Therefore,
.
1
e, = u i- Hi(m, y ) = ui - Hi m, C Pj(mj,u j ) = ui- Ri(m, u), (
j n= 1
---
where u = (ul, ,u,,). This proves assumption (ii) is satisfied, and hence the corollary is proven. Now consider the case when the modifications are given by (6.42). Direct application of Cor. 6.7 then yields the following result, which requires no proof: Corollary 6.10 For each i, 1 I i I n, and p in &?, let the infimal performance functions Gi, for the above given two-level system be given as GiP(mi ui 9
9
yi)
= Gi(mi >
ui
9
yi)
+ (Pi
n
9
ui> - C
j= 1
9
Hji(mi Pmi)>* (6.65) 9
Suppose for each i, 1 I i In, that G iis bounded as in Cor. 6.9, and U iis a closed, bounded, convex subset of Y, such that for some E > 0 it contains all ui = Hi(m,Pm) ei for all m in M and ei in Yi with norm lleill I E. Then the system is coordinable by the interaction balance principle.
+
210
VI. Optimal Coordination of Dynamic Systems
We now specialize to the quadratic case. For each, i, 1 I i I n, assume the integrand 0, in (6.62) is the quadratic form ei(t, mi(t),
ui(t),yi(t>>= mi(tITQiMmi(t)
+ Cui(t) - ai(t)lTQiuCui(t) - ai(t)l + b d t ) - bi(t)l'QiyC~i(t) - bi(t)l, where the matrices Q i M ,Q , , , and Qiy are symmetric, at least positive semidefinite and possibly time-varying, but bounded on F, and where aiand bi are given functions in g,. Hence, the infimal performance functions G i may be expressed in terms of the inner product: GdMi >
ui ,yi) = (mi
9
QiMmi)
+ (Yi
+ (ui
- bi , Q i Ayi
- ai 9
Qi&i
- ai))
- bi)).
(6.66)
Now, due to the assumptions on the matrices QiM,Qi,, and Q i y,the infimal performance function Gi is bounded below by 0, and Gi(mi,ui ,y i ( m i ,u,)) is bounded above on the set A ix U iwhen U iis bounded. Furthermore, Bi is convex for each t in Y and G , is continuous. We can, therefore, apply the above results. Corollary 6.1 1 For each i, 1 I i I n, let the infimal performance functions G , be given by (6.66), and suppose the modifications are given by either (6.63) or (6.65) with the interface objects selected as in Cor. 6.9 or 6.10, respectively. Then the above given two-level system is coordinable by the interaction balance principle. The linear interaction operators can be used to obtain an optimal coordination input when the modifications are given by (6.65) and, in some cases, when they are given by (6.63). The following three cases are considered : (1) If the modifications are given by (6.65), Prop. 6.8 gives an optimal coordination input fl in terms of the linear interface goal-interaction operators Ail at an overall optimal point (control input) f i ; for each i, 1 5 i 5 n,
(Bi,ui)
= -Ai(fi)~i,
for all ui in Ui . To evaluate Ai'(h), 1 I i I n, we compute the derivatives (i)
C:,
= (2Qiu(iii - a,), ) where
iii = Hi(&, P A ) ;
(ii) Cly = <2QiY(Bi- bi), ) where@,= ITiyPr?z;and (iii) Piu = Qi, where ( @ , x ) ( t )
=
J
@,(t,c)x(c) da for x in y , . 0
211
3. Application of the Balance Principles
Now, compute g;, = Gf, + GIyP$ and, substituting these values into (6.28), obtain for each i , 1 Ii In, A:(h)ui = 2( Qiu(Z;i - ai), ui)
+ 2( Qiy@i
- bi), 'Di ui).
Let cPi* be the adjoint of cPi ,i.e., for x in gi, 1
(mi*x)(t) =
J @T(o,t)x(o) do, t
and let Bj
Then
=
-2[
Qiv(Z;i
- ai)
+ 'Di*Qiy($i - bi)].
(6.67)
B = ziBiis an optimal coordination input.
(2) If the infimal performance modifications are given by (6.63) and the interface inputs depend explicitly only on the nonlocal control inputs ; A i j = 0 for all i # j , and hence,
B
Then = CiBlis an optimal coordination input when is given by (6.67).
Bi for each i, 1 s i I n,
(3) If the modifications are given again by (6.63), but the interface inputs depend explicitly only on the nonlocal outputs; Bij = 0 for all i # j , and hence,
To obtain an explicit expression for an optimal coordination input, we assume for each i, 1 Ii s n, the existence of an inverse matrix of (BE Bii), such that (6.68)
Qi E (BZBii)-'.
We then consider the local outputs g ias the control inputs and m ias the local outputs, i.e., mi = Q i B f [ D - Aii]yi
- Qi B:ui ,
(6.69)
where D is the differential operator. Using this inversion, an optimal coordination input is then = Bi where
B
xi
for all ui in U i .
9
ui> =
- A I(P>ui
9
(6.70)
212
VI. Optimal Coordination of Dynamic Systems
To compute hi@), 1 Ii In, let Si be the functional
Si(ui yi) = Gi(mi ui 3 gi), where *ni is given by (6.69). Then, from (6.28), 9
9
&(j) = &,
(6.71)
where
Sfu = (2QidGi - ail, >-(2Bii
Qi Q i M ,
>.
(6.72)
Now, from (6.70)-(6.72) we obtain
(6.73) Bi = 2[Bii(BiBii)-'QiMd- Qirr(iii- ai)]. The coordination input fl = cifli is then an optimal coordination input. Nonlinear Infimal Performance Modifications We have developed, in the preceding sections, a rather detailed theory of coordinability by the interaction balance principle using linear modifications of the infimal performances. However, the general results given earlier point out a very broad applicability of the principle, and one would expect that the system could be made coordinable by some other, nonlinear modification. This would be of interest not only as an alternative way of making the system coordinable, but can also have advantages for the coordination process of finding the optimal coordination by iteration or second-level feedback. We shall not give this subject a full treatment but rather only indicate that, under fairly general conditions, nonlinear modifications which will make the system coordinable are, indeed, possible. Furthermore, the existence of linear modifications which yield a system coordinable imply, under appropriate conditions, the existence of certain nonlinear modifications which accomplish the same thing. This stresses the importance of the analysis in the preceding sections.
Example 6.1 Let M i = Ui = Yi = R, the set of real numbers, for i = 1,2. Let the overall process P be represented by two subprocesses P, and P z , defined by the equations
+
+
y 1 = 2m, u1 = P,(m,, u,), y 2 = 2m2 u2 = Pz(mz , uz), whose couplings are u1 = m2 and u2 = m,. Assume the overall performance function G is given on M x Y as
+
G(m,y ) = mI2 m2'
+ (yl - 1)' + ( y z - 2)'.
The overall optimal control input is then fi
= (+,
6).
3. Application of the Balance Principles
21 3
The system can be made coordinable by the interaction balance principle if the overall performance is decomposed in reference to the local variables, and then linear zero-sum modification is used. Let 9 = RZ,and for each P in L’d let the infimal performance functions GiBbe
+ 81Ul - 8 2 mz, uz) = mz2 + 012 - 2Y + P z uz - P1mz
GlB(m1Y Y l u1) = m12 + 011 Y
Gzdmz Y z Y
Y
*
From the analysis in the preceding section, it follows that the system is coordinable by the interaction balance principle ; the optimal coordination input is B = (-3, $), and the corresponding optimal infimal solutions are (mls, ul@)= (3,g!),and (mzs,uzs)= +). Now, starting from the same unmodified infimal performance functions (GiBfor P = 0), the system can be made coordinable by many different kinds of nonlinear modifications. For instance, if the infimal performances modifications are selected for each p in B so that
(A,
+ + P1 sin +ul - Pz sin +ml, Gzfl(mz y 2 ,uz) = mz2 + (y2 - 2)2 + P2 sin +uz - P1 sin :m2, Gl,(ml, y1 ,ul) = m12 (yl - I)2
the system is coordinable by the interaction balance principle ; indeed, the optimal coordination input fl is
leading, of course, to the same optimal infimal solutions as in the linear case. Other nonlinear modifications which will make the system coordinable are plB(ml,ul) = Plulz - P2mlZand ,uza(mz,uz) = P2 u2’ - Plmlz. We shall not dwell on this any further except to offer a rather general result. Let the overall performance function be additive, and suppose the infimal performance modifications are linear zero-sum ; hence,
Let U i , 1 _< i In, be a convex subset of a Banach space Bi,and let : U i -P Ui be given. The function fi is not assumed to be linear. Let
fi
We have then the following:
VI. Optimal Coordination of Dynamic Systems
214
Lemma For a given p in 9?let (mp,up)minimize g&, myu) over M x U.Suppose M and U are convex, and for each i, 1 I i 5 n, suppose the functions gi ,f i , and K i are Frechet differentiable in each argument at mi = m t and ui = u t and, furthermore, the Frechet derivativefi.(ut) of f t at ui has a continuous left inverse T i ;that is, Tifi’(u!) is identity. Then, if gp is convex in (m, u), there exists b in 9?,namely,
b = C61T1,..- ,Pa K)Y such that the pair (ma, up) minimizes g,(fl, my u) over M x U. PROOF. Since (ma,up) minimizes ga@, my u) over M x U, we have for any point (ma my up u) in M x U that
+
+
g&,(P, mp,up), 2 0,
g&&?, mp, up)u 2 0.
If gg is convex, we have that
g,
mp, ufl)u,
where
gh,
mp, up),=
5 [giM(mip,
u?) - fiifi.(uia)K:M(mfi, up)ml,
i= 1
@&,@, mp, up)u = C [g:,(m,a, u t ) + Pi fi.(u?){ui -Ef,(mp, up)u}]. i=l
Therefore, if
fl = (&TI, - - ,p,T,), we obtain g&,(b, ma, up)m = ghM(P,mp,ua)m2 0, *
&&mB, I, up)u = g&&,
mp, up), 2 0.
This completes the proof. If gB is not convex, the pair (mP,up) still satisfies a necessary condition to minimize g,@, my u) where b = (/31T’, p,, T,). The next step is to show that under some conditions fi(Hi(m,y ) ) can be separable, so that infimal performance modifications can be defined. (Notice that gB does not imply infimal performance functions.) In order to do so, we point out the following fact.
Lemma Let V i = Ui1x * x Din and let the subprocess coupling functions Hi, 1 I i 5 n, satisfy (6.40(i),
--
n
Hi(m, Y ) = C H i j ( m j 7 y j ) , j= 1
such that Hij(mzj,y j )is in
aij= {ui E Ui: uik = 0 for k # j } .
215
4. Application of the Prediction Principle
PROOF.
The result follows immediately from the fact that fi(ui) =
Cj”=lfij(uij) and Hi(m,y ) = cj”=l H i j ( m j ,y j ) .
From these two lemmas we have immediately: Proposition 6.12 Suppose the conditions of the two above lemmas are satisfied. Then, if the system with the linear zero-sum modification is coordinable by the interaction balance principle (with ) being the optimal coordination input), then the system with nonlinear additive modification, ” pib(mi 9 ui 3 yi>= Pi fi(ui)
-
2
Bjfji(Hji(mi9
j=l
yi)),
is also coordinable by the same principle (with the optimal coordination input = ( f l I ~ 1 *, * * 9 PnTn))*
B
us) satisfies the necessary condition Even if grsis not convex, the pair (ms, for (mp,us) to minimize ga. Hence, if we prove that (ms,us) is the only point which satisfies the necessary condition for the minimum and there is a minimum for ga, the above proposition can still be applied. Actually, many other cases of nonlinear modifications can be derived in this way.
4. APPLICATION OF THE PREDICTION PRINCIPLE
We are able to obtain here more detailed conditions than those given in Chap. V for applicability of and coordinability by the interaction prediction principle. Two cases will be considered: (i) coordination is only through interface input predictions, and (ii) coordination is through interface input predictions with the aid of goal-coordination. The interaction prediction principle is more likely to be applicable in the second case, when goal-coordination is made in conjunction with the predictions.
General Results For a given two-level system, assume there are given two sets 5;9 and a such that K(M) = 5;9 and each B in 1gives the infimal objective functions gis, 1 I i 5 n, which, for convenience, are initially assumed to be defined on
VI. Optimal Coordination of Dynamic Systems
216
the set Ai x d rather than Ai x Ui. For each coordination input pair y = (a,p) in d x g,the ith infimal problem, 1 5 i In, is to minimize gip(mi,a) over the set di.Let mi denote an optimal solution of this problem. Let g be the overall objective function, given on A in terms of the overall performance function G. The overall problem as before is to minimize g over M. For a given pair y = (a,p) in d x By the prediction a is referred to as being a correct prediction relative to p if a = K(my) where m y = m l - * * mnY.Now, if the (interaction) prediction principle is to be applied, we cannot allow those pairs y = (a,p) for which the prediction a is correct, but the control input m yis not overall optimal. It was mentioned in Chap. V that goal coordination should in some way correspond to the interface input predictions. A convenient way to establish the desired correspondence is by a mapping q : M -+ B. The advantage of such a mapping stems from the fact that the coordination problem is essentially reduced to predicting the interface inputs, especially when q satisfies a decomposition with respect to K, q(m) = O(K(m)),while still utilizing the advantages of goal-coordination. There are two key results which serve as a starting point in our analysis of coordinability by the interaction prediction principle :
+ +
(i) Suppose the subprocess interaction function K restricted to M is one-to-one. Then, for the existence of a mapping q: M + B such that the prediction principle is applicable relative to q , it is both necessary and sufficient that each nonoverall optimal control input $I in M satisfies the strict inequality, inf giS(mi Ki(*)) < gip(&i, Ki(*)), 9
Ai
for some p in B and some i, 1 I i 5 n (Prop. 5.31). (ii) Suppose the subprocess interaction function K restricted to M is one-to-one and the overall performance function admits an overall optimal control input. Then, there is a prediction a in d such that for any j3 in the control input my,if it exists for y = (a,p), is overall optimal if a = K(my). Such a prediction is d = K(h)where h is an overall optimal control input. These two results, apart from the requirement that the restriction of K to M is one-to-one, are weak but provide a basis for our analysis. To show applicability of the prediction principle, it is both necessary and sufficient to show that a correct prediction cannot occur except when the predicted interface input is given by an overall optimal control input. However, if the overall optimal control inputs are not known, then neither will it be known beforehand whether or not a given prediction is given by such a control input. The
217
4. Application of the Prediction Principle
choice of the goal coordination input must be such that a prediction is correct only if it is given by an overall optimal control input. Now if the restriction of K to M is not a one-to-one function, result (i) becomes only necessary and of little value, and result (ii) is no longer valid; we then face the possibility that a prediction could be correct and correspond to an overall optimal control input, and yet the control input obtained by solving the infimal problems need not be overall optimal. We start by assuming the subprocess interaction function K is one-to-one and the infimal performance functions Gi , 1 I i 5.n, are generated from the given overall performance function G by restriction to local variables. For each i, I Ii 5.n, let Gi be given on Ai x Yi x d as Gi(mi, a, gi) = G(mi
+(I-
I&M)ma, yi
+(I-
niy)va),
(6.74)
where ma = K-'(a), and y" = P(m"), and let g i be the functional gi(mi a) = Gi(mi 7 a7 3
gi(mi
7
ail),
(6.75)
x d . The first result towards applicability of the prediction principle on is the following:
Proposition 6. I 3
Suppose the infimal performance functions of a given two-level system are given by (6.74). Assume M is convex and gi , 1 Ii I n, is differentiable and convex in mi for each a in d. Then for any a in d, max min [ g i ( m i , a) + Pi(*:
xi*
- mi)] = g(ma),
"4fi
with the max-min occurring at (P;, mi?, where 84 = g$(m,ol, a)i s the. Freshet derivative of g i with respect to mi at (m:, a). PROOF. Let a in d be arbitrary. Since a = K(m9, we have from (6.74) and (6.75), that gi(m;, a) = g(ma). Hence, for any Pi in Ai*,
inf [gi(mi, a) "ui
+ &(m; - mi)]5.g(rna).
Now, since gi is differentiable and convex in mi , Pi"(mi
- mi7 5 gi(mi a) - gi(mi(l7 a), 3
for all mi in .Mi where Pi(l = giM (m;,a). Therefore, inf [gi(mi, a) 4
+ Br(m; - mi)]= gi(m;,
and the proof is complete.
a) = g(m7,
VI. Optimal Coordination of Dynamic Systems
218
If the infimal performance functions of a given two-level system are given by (6.74), let I = M* and for each p in W and i, 1 Ii I n, let gis be the functional
(6.76) + Bi(mi), on Ai x d where pi is the restriction of p on x i .Proposition 6.13 then gip(mi 9 a) = gi(mi, a)
-
states that, for any prediction a in d which does not correspond to an overall optimal control input, the corresponding p in I must be such that
pi # 84 = glM(mf,a),
for some i ,
15 i In,
(6.77)
otherwise a would be a correct prediction, and applicability of the prediction principle would be denied. Condition (6.77) is, however, only necessary; it is not sufficient for the applicability of the principle. The ith linear total goal-interaction operator rl, 1 -< i In, at the point 6 in R is the bounded linear functional on R given by (6.30) as
r;(tiqmi= g y f i ) m i - giM(Gi,qrn)lmi. (6.78) Ti' for each i, 1 < i I n, exists at the point f i , let r(fi)be the bounded
If linear functional on M given as
Assuming differentiability of the objective functions g and gi, 1 I i s n, we can use these linear total goal-interaction operators to provide an appropriate p in 9l corresponding to a given a in d . First, we assume the subprocess interaction function K is one-to-one; later we relax this condition.
Proposition 6.14 Suppose the infimal performance functions of a given two-level system are given by (6.74) and the modifications are given by (6.76). Assume M is convex, g is convex and Frechet differentiable,the restriction K is one-to-one, and g ifor each i, 1 5 i I n, is Frechet differentiable in mi for each ct in d. Let q : M + 9l be defined for all m in M as q(m) = r y m ) .
(6.79)
Then the interaction prediction principle is applicable relative to q : for all y = (a, p) in d x 97,the control input m y is overall optimal if a = K(my) and p = q(my). PROOF. Let y = (a, p) in d x @ . be arbitrary. We need to show that my is Supposea = K(my)and p = ?(my); overall optimal if a = K(my)and p = ?(my).
219
4. Application of the Prediction Principle
then because K is one-to-one, m y = ma and fi = T'(m7. Let i, 1 I i 5 n, be arbitrary, then gi(m:
+ mi ,a) + r;(mu)(m: + mi) 2 gi(m:,
a)
+ ri'(rn%:,
and, hence, gi(m;
for m:
+ mi ,a) - gi(mt, a) + ri'(ma)mi2 0,
+ mi in Ai . On the otherhand, since gi is differentiable, gi(m; + Ami, a) - gi(m;, a) = &,(mf,a)Ami + @(A),
and, therefore,
[T;(m')
+ g&(m;, a)]mi 2 0,
(6.80)
+
for m: mi in Ai . Now (6.80) holds for each i, 1 Ii In, and from (6.78) and the fact that M is a direct sum, n
for ma + m in M.Therefore, ma is a local minimum of g. But g is convex, and M is convex; therefore ma is an overall optimal control input. This completes the proof. This proposition, in contrast to Prop. 6.13, shows how the modifications should be made so that the interaction prediction principle is applicable provided K is one-to-one. We did not assume the existence of an overall optimal control input, for such an assumption is not required to prove applicability; however, its existence is implied by the existence of a correct prediction a in d for B = q(m'). Now, if an overall optimal control input exists, we can show, under suitable conditions, coordinability by the interaction prediction principle relative to q. Proposition 6.15 Suppose a given two-level system satisfies all the assumptions of Prop. 6.14 and, in addition, assume there exists an overall optimal control input and gi for each i, 1 I i I n, is convex in mi for each a in d .Then the system is coordinable by the interaction prediction principle relative to the mapping q given by (6.79): the principle is applicable relative to q and there exists y = (a, 6) in d x 1such that a = K(my)and B = ?(my) for some my. PROOF. Applicability relative to q is given by Prop. 6.14; therefore, we such that a = K(my)and need only show there exists y = (a, B) in .et x B = ?(my)for some my. Let a in d be such that K-l(a) = ma = h, where h
VI. Optimal Coordination of Dynamic Systems
220
is an overall optimal control input. Since g is differentiable and M is convex, g'(ma)rn= g'(&)m 2 0 for all ma m in M . Let /3 = ?(ma); then, since g i , 1 I i I n, is differentiable in mi ,
+
+ r;(ma)]mi = gh(m4, a)mi + g'(rna)mi- giM(m;,.)mi
gisM(m;,a)mi = &(m;,
a)
= g'(ma)mi 2 0,
for each m; +mi in .,Mi.Therefore for each i, 1 I i I n, m; is a local minimum of gis Now, g i is convex in mi , and hence gis is convex in m i . Also .,Mi is convex. Therefore, for each i, 1 Ii I n, m: minimizes gis over -Ui . Therefore m y= ma; a = K(ma)= K(mY);P = ?(ma)= ?(ma).This proves the proposition.
.
We will now relax the condition that K be one-to-one. In doing so, we assume the given two-level system has an additive overall performance function. The infimal objective functions g i , 1 I i I n, will be defined on the space .Zi x Bi, gi(mi
3
ui) = Gi(mi
9
ui pi(mi 9 Ui)),
(6.81)
where G i is the given infimal performance function. The infimal performance modifications corresponding to those given by (6.75) are such that for each /3 in iC;i* and i, 1 Ii I n, gis(mi ui) = gi(mi 9
where Pi is the restriction of
to
9
ui)
+ Pimi
7
(6.82)
Ji.
Proposition 6.16 Suppose a given two-level system has an additive overall performance function and the modifications are given by (6.82). Assume M is convex, g is convex and Frechet differentiable, and g i , 1 Ii I n, given by (6.81) is Frechet differentiable in mi for each u i . Then the interaction prediction principle is applicable relative to q given by (6.79). PROOF.
Assume m yexists such that a = K(my)and = ?(my)for some pair P = T'(mY).Let i, 1 I i I n, be arbitrary, and let
y = (a,p) in a? x W.Then miy mi be in M i .Then,
+
gi(miy + m i , ai)
+ ri'(miy)(mY+ mi) 2 gi(miY, ai) + r,'(mY)miy,
and, hence, g,(miY
+
mi
, ail - gi(miy,ai) + r;(mY)mi 2 0.
(6.83)
4. Application of the Prediction Principle
221
Since g i is differentiable, gi(miy
+ Ami, ai) - gi(mntiy, ai) = giM(miy,ai)12mi+ @(A).
Now, from (6.78) or (6.30), we have that I';(mY)mi= g'(my)mi- giM(miy,ai)mi
.
(6.84)
Therefore, from (6.83)-(6.84),
+
g'(my)Ami= [r;(mY) g&&$,
ai)]2m,2 0,
(6.85)
for 2 > 0. Since (6.85) holds for each i, 1 Ii In, and m i y+ mi in d i ,we have (from the convexity of g on M , the convexity of M, and the fact that M is a direct sum) that m y is an overall optimal control input. This completes the proof. Proposition 6.17
Suppose a given two-level system satisfies all the assumptions of Prop. 6.16 and, in addition, assume there exists an overall optimal control input and the infimal objective function g i for each i, 1 I i In, is convex in mi for each u i . Then the system is coordinable by the interaction prediction principle relative to tj given by (6.79). PROOF. Applicability is given by Prop. 6.16. Let y = (c1, fl) in d x d be such that a = ti = K(h) and fl = I"(&),where h is an overall optimal control input. We show that m yexists and m y = h.Mis convex and g is differentiable; hence g'(h)m 2 0 for all rh + m in M. Therefore, since g i , 1 Ii s n, is differentiable, we have from (6.30) that
9ipM(hi,
+ r;(filmi =
= g i Y ( h i , ailmi
2 0,
for each Ri + mi in Ai.Therefore, for each i, 1 I i n, Ri is a local minimum of gip(mi,tii), and since gi is convex and d iis convex (A4is convex), this local minimum is an absolute minimum in M i . Hence h = my for some my. This completes the proof. Some remarks regarding the implications of the above results are in order here. (i) The infimal performance modifications are linear but not zero-sum. Contrast this with the modifications used for applying the balance principles, which as a rule were required to be linear zero-sum modifications. However, it can be shown that Prop. 6.16 and 6.17 remain valid when the modifications are given by (6.42) and the mapping tj : M -+ 9# is defined as t j b ) = (-
&'(m),
- - , - A,,'W), *
(6.86)
VI. Optimal Coordination of Dynamic Systems
222
where Ai‘(&), 1 5 i I n, is the ith linear partial goal-interaction operator on Ui given by (6.28). This may be seen as follows:For each p in B = Ul * x * * x un* and i, 1 5 i I n.
-
n
gifi(ezi 7 ui) = gi(”i
7
ui)
+ Biui -j=11 Bj Kji(mi).
Now, at a point h in M , n
g;M(&i
, iii)mi = g i M ( A i , Qi)mi- C Bj Kii(fi)mi, j= 1
and if
Bj = -A;(&)
for eachj, I Ij 5 n, we have from (6.30),
gifiM(Ri, tii)mi = giM(Ri,tii)mi
+ r;(filmi= gyk)mi.
(6.87)
Therefore, in view of (6.87), both Prop. 6.16 and 6.17 are true when the modifications are given by (6.42), and q : M B is given by (6.86). --f
(ii) In reference to Prop. 6.17 and the above remark, a given two-level system, with an additive overall performance function and linear zero-sum infimal performance modification given by (6.42) and coordinable by the balance principles, is, under appropriate differentiability and convexity assumptions, also coordinable by the interaction prediction principle, relative to q given by (6.86).
Application to Linear Differential Systems The results of this section may be applied to the case in which the subprocesses are defined by differential equations. We shall consider here, as in the preceding section, only the linear-convex case, and the most classical of them, the linear-quadratic case. 1 I i I n, are Assume the overall process P and the subprocesses Pi, defined on the interval LT= [0, I] by the linear differential equations
3(t) = AY(0 + Bm(t), 3i(t>= A i i v i ( t )
Y(0) = 0,
+ Bi,mi(t) + ui(t),
~ i ( 0= ) 0,
with ui(t) = Cj+iCBijmj(t)
+ Aijyj(t)I = Hi(m, Y K ~ ) ,
as in the preceding section. Let the infimal and overall performance functions be as given there.
4. Application of the Prediction Principle
223
We make the same assumption made in that section regarding the overall process P , the subprocesses Bi,and the functionals Oi . Therefore: (i) M and hence , 1 Ii I n, are convex and weakly compact. (ii) g(m) = G(m,P(m)) is convex and Frechet differentiable on R. (iii) gi(mi,mi) = Gi(mi,u i,Bi(mi,q)),1 I i In and a in d , is convex, weakly lower semicontinuous, and Frechet differentiable on A i . The conditions under which Prop. 6.16 and 6.17 may be applied are therefore fulfilled. Let 9# = Myand for each j? in 9# let the modified infimal performance functions G i , 1 Ii In, be given as Gig(mi ui gi) = Gi(mi > ui 9
where
9
9
gi)
+
mi),
9
Pi is the projection of j? in Ji.From Prop. 6.17, we then obtain:
Corollary 6.18 The two-level system described above is coordinable by the interaction prediction principle relative to q : R + % . ? given by the equations (qi(trt),
mi) =
r;(fii)mi ,
(6.88)
where qi(m) is the projection of q(m) in J i, that is, q(m) =
Ciqi(m).
It is, therefore, desirable to obtain an explicit expression for qi . We will do so under some simplifyingassumptions. Assume the integrands Oi ,1 I i In, are the quadratic forms given in the preceding section but with Q i , the zero matrix. Hence, G i , 1 5 i In, is Gi(mi 9 gi) = (mi
9
Assuming Q i M = IIiM Q r M = Q i M overall performance function G is
and
lliM
y
Qir(gi
Qiy =
- bi)).
IIiy Qiy
(6.89)
= Qiy I I i y ,
the
+ (Y - by Qr(Y - b)),
G(m, Y ) = (my Q
=xi
+
QiMmj)
M ~ >
=xi
+ +
where QM Q i M ,Q y Q i y , and b = b, * * * b,. To compute q i ( f i ) for a given f i in R,we need the derivatives of g and gi at with respect to m i . Expressing the overall process P and subprocesses Bi in the forms Pm = QBm,
+
Pi(mi,ui)= Qi[Biimi ui],
we have g'(fi)mi = 2(QM fi, H i M
m> + 2(Qr(E - b), @BHiM m>,
and giMM(ai 9 G)mi = 2(QiM ai
9
mi>
+ 2(Qi&i
- bi), @i Bii mi),
224
VI. Optimal Coordination of Dynamic Systems
where 9 = Ph and Ci = K h . From (6.87), (6.30), and the identity < Q M ~ ,UiM")
- (QiMRi, hi) =O,
we then have, qi(lc2) = 2UiM[B*@*Qy(9 - b) - BE@i*Qiy(&i- bi)],
where
* denotes adjoint. We can further simplify q,(&) by noting that Qi@i - bi) = Q i Y niY@i- bi) = Qid9 - b),
and B$@)i*QiY = IIiMB*IIiy@*IIiyQy.
Hence, qi(&) = 2lTiMB*[@* - IIiy@*II,y]Qr(9 - b).
(6.90)
or 1
a ( : @ qi(rit)(t) = 211iMBT [QT(a,t ) -,
t)]QY[$(o) - b(o)] do. (6.91)
t
The computation of qi(h)is then a straightforward integration, provided one has the fundamental matrix @(f, a) of (6.60). If the interface inputs depend explicitly only on the nonlocal outputs, that is, Bij = 0 for i # j, we can choose the coordination object 93 as the set Y and modify the infimal performance functions as follows: for each /Iin &? and i, 1 I i 5 n, Gi@(*i
3
yj) = Gi(*i
7
yi) +
3
$2
where Gi is given by (6.89) and pi is the projection of each i, 1 s i I n, the inverse of B i B i i exists, (BzBii)-
E
(6.92)
p in GYi. Now, if for
Qi ,
we can not only show coordinability by the interaction prediction principle relative to some mapping i j : 93 but we can also give i j explicitly. Consider yi as the control input and mi as the local output given by (6.69). Through this inversion, one can conclude
a+
Corollary 6.19
If the interface inputs depend explicitly only on the nonlocal outputs, the infimal performance modifications are given by (6.92), and Qi = (B:Bii)-' exists for each i, 1 I i I n, then the system is coordinable by the interaction prediction principle relative to i j given by the equations < i j i ( f i ) , 9.i) = ri'cP)gi 3
5. Iterative Coordination Processes
225
where 9 = Ph,Ti’ is the linear total goal-interaction operator of the inverted system, and ifi(&)is the projection of q(h) in g i . To compute f i ( h ) for a given element m in j = P ( h ) , which is given as
R,we first compute T,’(j),
ri’(j)gi = C Aj’WAij gi
9
j#i
where A,’@) is the interface goal-interaction operator obtained in the preceding section [(6.71)-(6.73)] as A,’(P)uj = - 2
Aj uj>,
QjM
9
noting that Qju = 0. Hence, ri‘(j)gi = -2C (Bji Q j
QjM
j#i
= - 2 C
Rj 9 Aji gi> QjM
j#i
Aj Y
yi>
Therefore, we obtain & ( h ) as iji(fi) =
-2CAzBjj(BjTi Bjj)-’ QjMmj, j#i
or iji(fi)(t)= -2C AjT,Bjj(BjTiBjj)-‘ QjMmj(t).
(6.93)
j#i
Notice in this case, the simplicity of the expression for ij@) as compared with that for ?,(A)given by (6.90) or (6.91). In view of this comparison, one should use the modification given by (6.92) when the interface inputs depend explicitly only on the nonlocal outputs and (BE Bii)-’ exists. 5.
ITERATIVE COORDINATION PROCESSES
Some general strategies were presented in Chap. V on how to find an optimal coordination input through an iterative process involving both the coordinator and infimal decision units. In these processes, the coordinator receives as feedback information a pair (uy,K(my))in U x U,and, on the basis of the previously derived coordination input y, generates an improved coordination y‘ by using a strategy so:% x 5 x 5 + %. Since both uy and u = K(mY)are derived on the basis of the given coordination input y, the strategy so defines a transformation T: V + %,
VI. Optimal Coordination of Dynamic Systems
226
Iteration then consists of repeated application of the transformation T , and, therefore, success of the iteration process depends on the properties of T as already pointed out in Chap. V. If the iterations converge and the strategy so is based on a coordination principle by which the system is coordinable, convergence is to an optimal coordination input. We consider here a choice of the strategy so for which the iteration process converges. A Strategy Based on Interaction Balance
Let us consider the case in which a given two-level system is coordinable by the interaction balance principle. Coordination in this case consists solely of the selection of a goal coordination input in W so that the mapping so and T becomes so: 9? x x U + g and T: 98 + W,respectively. Let the strategy so be based on the interaction balance principle; in particular, we shall assume that corrections in a given coordination input y are proportional (in an appropriate sense) to the error e8, ep = d
- K(mp),
between the desired interface input upand the actual interface input u = K(m8). We assume the system in question has an additive overall performance function and the infimal performance modifications are additive zero-sum. In particular, _ _ we assume the spaces U i , 1 I i In, are Hilbert spaces and 9? = U = U , x . * * x Un so that the apparent-overall objective function gse of the system is given as n
g.dP, m, U) =
C gi(mi i=
n
9
ui)
1
+ iC (Pi =1
9
ui - Ki(m)>.
(6.943
Let the strategy so be given by the n mappings s i : W x Ui x Ui + W i ,
si(P, ui ui') = P i 7
+ ni(P)Cui - ui'l,
where A@) is a real number, possibly depending on P. Hence, repeated application of the transformation T : 98 -,W given by the n transformations Ti:&?+gi,
Ti(P) = Pi + A i W C U i s - Ki(m31,
(6.95)
represents the iteration process. If the interation process converges, it will converge to a point where the interface inputs are balanced; therefore, if the interaction balance principle is applicable, convergence is to an optimal coordination input. We consider the iteration process defined by (6.95) in terms of accumulation points. We show, in particular, that under suitable conditions there
5 . Iterative Coordination Processes
227
exists Ai(B), 1 Ii I n, such that a sequence generated by T satisfiesthe balance condition whenever it has an accumulation point. We then show when a sequence generated by T has an accumulation point. Let a mapping d : B + M x U represent collectively the infimal decision units; that is, for each fl in 93 let d(B) = (ms, us). This requires, of course, the existence of infimal optimal solutions for each fl in g.Also, let go be the real-valued function on B such that SO@) = sr@,msY us>= sr(P,d(P))Y
for all /Iin B. Proposition 6.20
Assume the functions gr and dare Frechet differentiable, and for all p in 9i?, the pair (msyus) = d@) is an interior point of M x U.Then for each /Iin B and for each i, 1 Ii I n, there exists A,@) > 0 such that for any sequence
{Bk} generated by the transformation T, (i) any accumulation point of {B’} satisfies the balance condition, and (ii) the sequence {go(Pk)} is monotonically increasing and, if {p’} has an accumulation point, converges to the minimum overall cost. PROOF.
From the expansion of go, one obtains go@
+ Ah) - gO(B)2 ga’(P, m@,u9Ah + @(A),
for all B and h in B and real A, where gr’(B, ms, us)is the Frechet derivative of B at the point (Byms, us). From (6.94),
gr with respect to
n
grl(p, mp, u s ) ~ = h A
C (hi ,uP - K,(rns>>.
i= 1
Now, set h = us - K(ma)and for each i, 1 I i I n, choose A,@) > 0 such that n
n
(6.96)
(6.97) The sequence { go@’)} is therefore monotonically increasing and converges.
VI. Optimal Coordination of Dynamic Systems
228
Suppose p is an accumulation point of the sequence {pk}. The functions T and go are continuous in p; therefore, if p does not satisfy the balance condition, i.e., UF - K(mg)# 0, then n
go(T(P))- go@) 2
11 Ai@)
i=
Hence, for some pk sufficiently near
IIU? - Ki<mF)I12> 0.
p,
gO(TUk)) - S"i9
>
0 7
which denies the fact that { go((Dk)} increase monotonically. Therefore, fl satisfies the balance condition and {go(pk)}converges to the minimum overall cost. This completes the proof.
Proposition 6.21 Suppose the assumptions of Prop. 6.20 hold and the Ai(p)'s are selected according to (6.96). If a sequence {pk} generated by the transformation T is such that for each i, 1 I i I n,the sequence {Ai(pk)} is bounded, then {pk} has an accumulation point. PROOF. From Prop. 6.20, the sequence {g(Bk)}converges to some U < 00, and from (6.97);
Therefore,
For any integer k > 0,
Then, since the Ai(p)'s are bounded,
llflk+l - pkl1 0, as k + 00. Hence, the sequence {pk} has an accumulation point. This completesthe proof. --f
It should be noticed that, in practice, the existence of an accumulation point can almost always be safely assumed, because the search for an optimal coordination input p is carried out on a closed and bounded region of a. If the interaction balance principle is applicable, this leads to the optimal point.
5. Iterative Coordination Processes
229
A Strategy Based on Interaction Prediction The convergence of the iteration process for the interaction prediction case be be shown quite analogously. The iteration process is defined by the transformations Ti, and Ti,, 1 Ii In, on d x W given as Ti,(Y) = a,'
- Pi(Y)CVi(m") - P i " ] ,
Tia(r) = Pi'
+ Ur)CK(mY) - ~ i ' l ,
where y = (ct, j?) and Ai(y) and pi(y) are positive numbers. The apparentoverall objective function g, is assumed to be given by (6.93). The convergence is proven by means of the first order expansion of @a. Although the details of the derivation are different than for the interaction balance case, the logic is the same and, therefore, we shall not repeat it.
Chapter VII
COORDINATION OF CONSTRAINED OPTIMIZING SYSTEMS AND MATHEMATICAL PROGRAMMING PROBLEMS
The two preceding chapters were concerned primarily with the coordination of optimizing systems without constraints. We did not, however, explicitly rule out the possibility that the overall and infimal optimization problems might have constraints imposed on the search sets. Nevertheless, we did require the overall control object, the set of feasible control inputs, to have a special property: in Chap. V it was assumed to be a Cartesian product, while in Chap. VI it was a direct sum. The two-level systems we consider here are basically the same as those considered in the preceding two chapters, except now we explicitly take into account constraints imposed on the overall control inputs and infimal decisions. We shall consider the effects constraints have on the applicability of the coordination principles, and then we shall apply the principles to the problem of coordinating (linear and nonlinear) programming problems. 1. INTRODUCTION
The overall problem of a given two-level system is specified in terms of the overall objective function g defined on the overall control object M = M I x * x M , and a given set M fc M of feasible overall control inputs. The problem is to minimize g over the set Mf.The fact that M fis a subset of M implies that constraints may be imposed on the overall control inputs.
--
230
23 1
I . Introduction
Constraints in optimization problems are usually expressed in terms of conditions imposed on the inputs and outputs. For example, the input and output objects M and Y of a given system may be spaces of square-integrable functions on the interval [0, 11 with the set Mf of feasible control inputs consisting of those control inputs m in M for which IIm(t)ll I1 for almost all t in [0, 11 and the output y = P(m) passes through a given region at some time t in [0, 11. Constraints in mathematical programming problems are usually expressed by sets of equalities and inequalities such as m 2 0 and P(m) 2 b. The infimal problems can also be constrained optimization problems. Their constraints are reflected by specifying only subsets of the sets M i x U, as the sets of feasible infimal decisions. We assume, in general that, there are given two sets d and W such that each pair (c1, p) in d x W gives for each i, 1 Ii In, an infimal objective function gis defined on M i x Ui and a set Xi, E x of feasible decisions. The ith infimal problem, 1 Ii n, for a given pair (a, j?) in d x W is to minimize gia over the set Xi, of feasible decisions. The sets Xi, of feasible infimal decisions are, in general, relations reflecting the imposed constraints which may differ for each c1 in d . The specification of appropriate constraints, and hence the sets of feasible decisions, is in itself a method of coordination. However, we shall be concerned primarily with the application of the coordination principles ;therefore, we limit our analysis to the following cases: We assume there is given for each i, 1 Ii In, a set Xi E Mi x Oi and then (i) for the application of the balance principles, we assume the set of feasible infimal decisions is fixed for each infimal decision unit, Xi, = Xi; and (ii) for the application of the interaction prediction principle we assume d = U and Xi, = Mi,x { a i } where Mi, is the set of local controls mi such that (mi ,ai) is in X i . The basic question we face here is what should the sets of feasible infimal decisions be so that the coordination principles can be applied. Before proceeding to answer this question, we offer the following observations :
mi ui
(i) Let M { , 1 Ii 5 n, be the projection of the set Mf of feasible overall control inputs on the set M i : M! = {mi:mi = q ( m ) for some m in M'}, where 7ci is the projection mapping of M onto M i . Then, in general,
M f c Mlf x *
*
Contrast this to the special case M f= Mlf x
x M"'.
--
*
x Mnfpreviouslyconsidered.
232
VZZ. Constrained Optimizing Systems and Mathematical Programming
x M n f , the specification of the sets of (ii) In case M f = MIf x feasible infimal decisions is straightforward; namely for each i, 1 I i I n,
xi= Mif x uif, where Uif is any subset of Uicontaining all interface inputs to the ith subprocess that occur in the coupled system when feasible overall control inputs are applied, K i ( M f )E U i f . This case, however, has already been considered in the preceding two chapters. Note that in Chap. VI, the set M fwas represented as a direct sum rather than a Cartesian product.
2. APPLICABILITY OF THE COORDINATION PRINCIPLE
Constraints, whether imposed on the overall control inputs or infimal decisions, have an effect on the applicability of the coordination principles. Indeed, a given coordination principle, applicable in a given two-level system in the unconstrained case, might not be applicable when constraints are imposed, because of the added feasibility requirement. For example, suppose the system is coordinable in the unconstrained case by the interaction balance principle, with ih being the unique overall optimal control input; then the interaction balance principle is not applicable when constraints are imposed on the overall control inputs in such a manner that ih is not feasible. On the other hand, a coordination principle not applicable in a given two-level system in the unconstrained case could very well become applicable in that system when appropriate constraints are imposed on the infimal decisions. For example, let the constraints imposed on the infimal decisions be such that for some overall optimal control input A, each set Xi of feasible infimal decisions be the singleton set {(hi, Ki(h))}. There are two facets of the constraint problem. One is how the constraints affect applicability and coordinability of the coordination principles, and the other is what should the feasible infimal decisions be so that the coordination principles can be applied. Most of our considerations of these two aspects are centered around two-level systems having at least the monotonicity property. Constraints on the Overall Control Inputs Suppose the sets X i of feasible infimal decisions in a given two-level system are such, that for each i, 1 I i 5 n, Xi = Mif x Uif where K i ( M f )E U/ and Mif is the projection of M fin M i . In the application of a given coordination principle in the case M f# MIf x * * x Mnf,it could very well turn out that for some coordination input the coordinating condition is satisfied by
2. Applicability of the Coordination Principle
233
optimal feasible infimal decisions and yet the resulting control input is not overall feasible. The coordination principle, even if applicable in the case Mlf x x Mnf is considered as the set of overall feasible control inputs, is not necessarily applicable in the given system. Perhaps the simplest way to handle the overall feasibility question in application of the coordination principles is to introduce a feasibility check. In addition to testing whether or not the coordinating condition of a given principle is satisfied for given optimal infimal decisions, the coordinator must also determine whether or not the resulting control input is overall feasible and reject it if it is not. The following propositions regarding the applicability and coordinability of the coordination principles with a feasibilitycheck introduced are immediate and require no proof.
---
Proposition 7.1
If a given coordination principle is applicable in a given two-level system when Mlf x x Mnf is considered as the set of overall feasible control inputs, then the principle with a feasibility check included is applicable in the system.
---
Proposition 7.2
If a given two-level system is coordinable by a given coordination principle when Mlf x x Mnf is considered as the set of overall feasible control inputs, then ming(rn)= Mf
min (Mi1
X...
g(m)
XMJ)
is necessary for coordinabilityby the principle with a feasibilitycheck included. These two propositions are true regardless of how the sets of feasible infimal decisions are specified. This approach to applying a given coordination principle when a question of overall feasibility is involved is, indeed, very simple but presents difficulties in application : (i) To have coordinability, condition (7.1) must be satisfied; it is nevertheless reasonable to assume that this is so only in some special cases. (ii) Even if condition (7.1) is satisfied, coordinability does not necessarily follow, and the task facing the coordinator could very well be formidable. For example, if for a given coordination input, the optimal feasible infimal decisions satisfy the coordinating condition but the resulting control input is not overall feasible, the coordinator does not know, in general, whether or
234
VZI. Constrained Optimizing Systems and Mathematical Programming
not the system is coordinable: to determine this, it must test each coordination input for which the optimal feasible infimal decisions satisfy the coordinating condition and determine whether or not the resulting control input is overall feasible. (iii) Even if the system is coordinable by the principle with a feasibility check included, it is difficult to determine how to adjust the coordination inputs in order to arrive at one which coordinates the system. When a feasibility check is not required, the imbalance in the system, or error in the interface input prediction, can be used to make adjustments in the coordination inputs as indicated in Sect. V.8 and VI.5. However, if a feasibility check is required, it is not clear, except in some special cases such as linear programming, how to make the adjustments.
Balanced Constraints The above approach to handling constraints on the overall control inputs is quite simple conceptually but not very practical for the reasons mentioned above. We shall, therefore, seek a relationship between the sets of feasible infimal decisions and the set of overall feasible control inputs which allows the elimination of an overall feasibility check. The coordination principles themselves suggest what the relationship should be; namely, infimal feasibility and balance should imply overall feasibility. Let X be the set of all pairs (myu) in M x U such that for each i, 1 5 i < n, the pair (mi, ui)is in X i . The proposition
expresses the desired relationship between the sets of feasible infimal decisions and the set of overall feasible control inputs. For the application of either the interaction balance or interaction prediction principle without an overall feasibility check, it is sufficient that (7.2) hold for the given constraints. We should point out here that this does not guarantee applicability of the principle; it only means that the feasibility check is not required. Condition (7.2) is not, however, sufficient to allow removal of the feasibility check in the application of the performance balance principle. For up) exists, where example, for some goal coordination input p suppose (ma, for each i, 1 s i I n ythe pair (mt,u!) is an optimal feasible infimal decision, and
2. Applicability of the Coordination Principle
235
Now, even if condition (7.2) holds for the two-level system in question, the pair (mt,Ki(ms)) for some i, 1 5 i 5 n, need not be feasible, and hence we cannot conclude that ms is an overall optimal feasible control input. We shall say that the constraints in a given two-level system are baIanced if the proposition (W"(m,
K O )E XI * cm E MS11
holds. Balanced constraints have the following consequence in two-level systems having the monotonicity property:
Proposition 7.3 Suppose a given two-level system has the monotonicity property and its constraints are balanced, then the interaction balance principle is applicable in the system. PROOF: Let /Ibe a given goal coordination input and suppose (ms, us) exists and us = K(ms).Then because the constraints are balanced and the system has the monotonicity property, ms is overall feasible, and
s(m9 = ssV, ms,W m 9 ) = min ssa(b,m,K(m)) = min s(m>, MJ MJ where g and gr are, respectively, the overall objective function and apparentoverall objective function. This proves applicability of the principle and completes the proof. Given that the constraints of a two-level system are balanced, we can apply the results already obtained in the preceding two chapters to obtain conditions for coordinability by the interaction balance principle. Of course, the various assumptions involving the overall control object and infimal decision objects must then be made in reference to the set of overall feasible control inputs and the sets of feasible infimal decisions. Since these changes are straightforward, we will not restate the conditions for coordinability in the constrained case. We should point out, however, that the specific results in Chap. VI pertaining to the linear goal-interaction operators do not generally apply to the constrained case. Similarly, the conditions not involvingthe linear goal-interaction operators for applicability of the (interaction) prediction principle and coordinability by the principle in the unconstrained case may be directly applied to the constrained case, provided the constraints of the two-level system in question are balanced. We shall now investigate the nature of balanced constraints in order to characterize a class of balanced constraints in a meaningful way.
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Without any loss in generality, assume that the overall feasible control inputs are defined by a mapping z : M + Q and a target set Qo E Q :
M f= {m:~ ( mE) Q O } . Similarly, assume that for each i, 1 I i In, there is given a mapping ri: M i x Ui -+ Qi and a target set Qio c Q i such that X i is defined as X i = {(mi,u i): Ti(mi,ui) E Q:}.
Constraints in optimization problems are generally presented in this fashion. Without any loss in generality, further assume that the mappings z and z i , 1 I i I n, are onto their respective,codomains.We then have the following.
Proposition 7.4 Suppose there exists a mapping 8: Q , x
*
- - x Qn + Q such that
(i) r(m) = e(z,(m,, K,(m)),* .., ~,,(m,, ,~ , , ( m ) )for ) all m in (ii) x * * . x eno is the inverse image of Qo under 8.
elo
~
f
,
Then, the constraints are balanced. Let m be an arbitrary element of M. Then, (myK(m)) is in X iff Qio for each i, 1 I i I n, and because of assumptions (i) and (ii) this occurs iff z(m) is in Qo. Since z(m) is in Qo iff m is in Mf,the proof is complete. PROOF.
zi(mi,Ki(m)) is in
This proposition characterizes a class of balanced constraints. It is important in the sense that it suggests how one might proceed to specify the sets of feasible infimal decisions so that the implied constraints are balanced. Condition (i) of the hypothesis of the proposition essentially says that the family of mappings z i , 1 I i I n, is a decomposition of z with the mappings Ki , 1 Ii I n, representing their interconnections or interdependencies. Condition (ii) gives then a sufficient (but not necessary) relationship among the target sets for the constraints to be balanced. As an example, note that condition (i) is satisfied if z is the overall process P and for each i, 1 I i I n, the mapping z i is the ith infimal subprocess P i . The mapping 8 in this case would be identity, and the constraints would be imposed on the outputs of the process. Now to satisfy condition (ii), the target sets Qo E Y and Q: G Yi, 1 Ii I n, should be such that
although this condition is not, in general, necessary.
3. Coordination of Convex Programming Problems
237
3. COORDINATION OF CONVEX PROGRAMMING PROBLEMS
The methods of dealing with feasibility requirements presented in the preceding section can be immediately applied to two-level systems whose overall and infimal problems are given as so-called mathematical programming problems. We shall consider here a class of two-level systems whose overall problems and infimal problems are convex programming problems and, in doing so, illustrate how the results of Chaps. V and VI combined with the results on balanced constraints in the preceding section can be used to coordinate convex programming problems. The overall problem (as presented in the introduction of this chapter) of a given two-level system is to minimize a given overall objective function g over a given set M , which is referred to, in mathematical programming terminology, as the feasible domain. We assume g is a convex real-valued function defined on an euclidean space M = Ml x * x Mnwhere for each i, 1 I i I n, the space M i may be a multidimensional euclidean space. For each i, 1 I i I n, let Fi be a convex function from M to an euclidean space Yi and assume the feasible domain M fis given as the set of all m in M satisfying the constraints,
--
m20
i = 1, ..-,n.
Fi(m) 5 0,
(7.3)
The orderings in Yi and M should be understood as given in the conventional way. The overall problem is then the convex programming problem: minimize g(m) under the constraints (7.3). The infimal problems will also be convex programming problems, but in the spaces M i x Ui , 1 I i In,where Ui is an appropriately selected space. The constraints defining the feasible domains X i , 1 I i 5 n, of the infimal problems will be selected so that they are in balance with the overall constraints (7.3). To do this, we decompose the functions Fi as follows : For each i, 1 I i I n, Fi is decomposed into two mappings P i : M i x Ui + Yi and K i : M + Ui,such that (i) Ui is a euclidean space; (ii) Pi is convex on M i x U i ; (iii) Pi(mi,Ki(m))= Fi(m) for all m in M.
(7.4)
As an example of such a decomposition, let Ui = Mi x
. a *
x
Mi-1
x Mi+i x
*..
x M,,
and Pi(mi, Ui) = f’i(Ui1, ..*,Ui,i-l,
mi,
ui,i+l, *..,
uin)
(7.5)
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VII. Constrained Optimizing Systems and Mathematical Programming
where
ui = (uil,
* *
2
ui,i-I,
ui,i+l,* .
* 9
uin).
The mapping K i is then simply the projection mapping from M to Ui . Now, for the ith infimal problem, we define the feasible domain X i as the set of all pairs ( m i ,ui) in M i x Ui satisfying the constraints
mi 2 0,
Pi(mi,ui) I 0.
(7.6)
We shall assume the ordering 2 in M is such that m 2 m' iff mi 2 mi' for each i, 1 I i I n. Hence, it can be shown that the set of infimal constraints given for each i, 1 I i I n, by ( 7 4 , is in balance with the overall constraints (7.3). To show this balance of the constraints, we express the overall constraints (7.3) as
z(m) I 0, where z(m) = (- m,F,(m), ..* ,F,,(m)), and similarly for each i, 1 I i I n, we express the infimal constraints (7.5) as
zi(mi, U i ) I 0, where zi(mi,ui) = ( - m i , Pi(mi,ui)). Then, due to (7.4), we have for every m in M y
z(m) I 0,
iff zi(mi, Ki(m))I 0 for every i, 1 I i In.
The constraints are therefore balanced. Indeed, Prop. 7.4 may be applied, since there exists a mapping 8 for which the hypotheses of the proposition are satisfied. For each i, 1 I i I n, suppose there is given a convex real-valued function g i defined on M i x Ui so that the ith infimal problem (without modification) is to minimize g i over the feasible domain defined by the constraints (7.6). The ith infimal problem of the two-level system is then the convex programming problem: minimize gi(mi, ui) under constraints (7.6). To apply the coordination principles to the problem of coordinating the given two-level system, we assume the given infimal objective functions g i are related in a particular way to the given overall objective function g ; namely, we shall assume that g is additive,
If the infimal objective functions are to be obtained by a decomposition of the overall objective function, the decomposition of the overall objection function
3. Coordination of Convex Programming Problems
239
and the constraint functions Fi must be performed simultaneously, so that both (7.4) and (7.7) are achieved. In the above example illustrating how the constraint functions might be decomposed, we can construct the infimal objective function g i , 1 I i I n, as 1 gi(mi,ui)=-g(uil, . * . , u i , i - 1 , mi, ui,i+l, * . * , u i n ) , n and thus achieve both conditions (7.4) and (7.7). Since the constraints defining the feasible domains of the infimal problems are in balance with the constraints defining the feasible domain of the overall problem, we can apply the interaction balance or (interaction) prediction principle without an overall feasibility check, in an attempt to coordinate the infimal problems relative to the overall problem. This " balanced distribution" of the overall feasibility requirement allows us to apply the results of Chapters V and VI to the coordination problem directly. The following proposition is a direct consequence of Prop. 6.4 and 7.4. Proposition 7.5 Suppose the overall objective function g is additive, the overall problem has an optimal feasible solution, and the given infimal objective functions are modified by linear zero-sum modifications;that is, W = Ul x x Unand for each i, 1 I i In, and B in W
-
Furthermore, assume there exists a coordination input fi in W such that
5 inf gib(mi,ui>
= sup
i=l Xi
a
i=l
inf g i j ( m i ,ui), Xi
(7.9)
and for each i, 1 I i I n, the feasible domain Xi is compact and gV ,for each
B in W ,is continuous there. Then, the two-level system is coordinable by the interaction balance principle. The modifications given by (7.8) are zero-sum iff for each i, 1 Ii 5 n, the equality (7.10)
is satisfied for each m in M. If (7.10) is violated for some m in M and j , i Ij In, there would then exist a /3 in W such that n
C gij(mi i= 1
7
+
Ki(m)) d m ) .
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VZZ. Constrained Optimizing Systems and Mathematical Programming
For the decomposition of the constraint function Fi given by (7.5), there exist functions Kij , 1 I j I n, such that (7.10) is satisfied for each m in M. The existence of a coordination input satisfying (7.9) is required for coordinability. However, if other assumptions are imposed, we need not explicitly assume the existence of such a coordination input. Because of the balanced constraints in the two-level system under consideration, we may apply Prop. 6.6 and obtain the following result:
Proposition 7.6 Suppose the overall objective function is additive, the overall problem has an optimal feasible solution, the given infimal objective functions are modified by linear zero-sum modifications, and for each i, 1 I i 5 n, the feasible domain X i is compact, and gU ,for each /3 in 9,is continuous there. In addition, assume the origin is an internal point of the set E consisting of all e = u - K(m) for some pair (myu) in X . Then, the two-level system is coordinable by the interaction balance principle. Compactness of the feasible domains X i is rather restrictive, although required in the above proposition. This assumption, however, may be relaxed, but then to prove coordinability, we must assume that the interaction functions K i are linear. This, and another consequence of the interaction functions Ki being linear, is stated in the following proposition:
Proposition 7.1 Suppose the overall objective function is additive, the overall problem has an optimal feasible solution, and the infimal objective function modifications are linear zero-sum. Assume the feasible domain M f has an internal point, and for each i, 1 i 5 n, assume the interaction function Kiis linear and gi is continuous on the feasible domain X i . Then the two-level system is coordinable by the interaction balance principle. PROOF.
Let U = U, x
1 .
*
x Unand define the functionsf and L on M x U
as n
f(m, u ) = C gi(JtIi 3
u)= u
ui),
- K(m),
i= 1
where K(m) = (K,(m), * * * , K,(m)). Then for all /3 in U, n
C gia("i i= 1
9
ui)
=f(m, u)
Now define the subsets Vand 'of
-
+ B T W , u).
(7.11)
R x U as follows:
V = {(r, e): r E R and f ( x ) < r and e = L(x) for some x in X}
-V = {(r, 0):r E R and r
3. Coordination of Convex Programming Problems
241
where 2 = (A, K(A)) and A is an optimal feasible solution of the overall problem. Since A exists and Kislinear, the sets Vand Yare disjoint, nonempty, and convex. Furthermore, the set V has an internal point. This follows from the assumption that the feasible domain has an internal point, and for each i, 1 I i I n, Ki is linear and g1is continuous on X i . Therefore, the separating hyperplane theorem may be applied to obtain real numbers 01 and ro and a vector B in U such that ar + B'e 2 ro ,
for all (r, e) in V,
ar + bTe < r o ,
for all (r, e) in _V.
(7.12)
It is an immediate consequence of the definition of _V that 0120. Observe that df(2),0) is in _V, and (r, 0) is in Vfor all r 0. Furthermore, ro = af (2);if not, ro < af (2) and hence ( f ( 2 ) ,0) would be in _V, which is impossible. Now from the definition of Vand (7.12), we have
f(2) = min Cf(x) X
+ B'L(x)I,
and hence, from (7.11) and the fact that g is additive, we have
which is sufficient to prove the proposition. Example 7.1
Consider the overall problem of minimizing the objective function g , defined on Kf = R x R, as g(m) = 3@12
+ mz2)- (m,+ 2m2),
under the constraints (7.3), where n = 2, and
Fl(m)= 2ml
+ 3m2 - 6,
F,(m) = ml
+ 4m2 - 5 .
The optimal feasible solution of this problem is A = (fi,H). We now formulate two subproblems which depend on the parameter p = (/$, &) in R x R. The first subproblem is to minimize the objective function 91p("11,
u1) = h
1 2
- (1 + B2)ml + Au1,
under the constraints ml 2 0, u1 2 0, and Pl(ml, ul) 50,where
+
Pl(ml, ul) = 2m1 224, - 6.
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VII. Constrained Optimizing Systems and Mathematical Programming
The second subproblem is to minimize the objective function
under the constraints m, 2 0, u2 2 0, and P2(mz, u2) I 0,Where
P2(m2,u2) = 4m2
+ u2 - 5.
Under the interrelationship of these subproblems given by the equations m, = u, and m2 = ul, the overall objective function is additive, modification of the subproblem objective functions is linear zero-sum, and the constraints are balanced. The interaction balance principle is, therefore, applicable to the problem of coordinating the two subproblems relative to the overall problem. B2) In fact, the subproblems are coordinable by the principle. Let B = (Bi, where B1 = 0 and p2 = -&; then the optimal feasible solutions of the subproblems are
++,
By setting uIP= it is apparent that this value of the parameter B is an optimal coordination input. We should point out here that the optimal coordination input B in the above example is not that given by the linear interaction operator, as in Prop. 6.8. The optimal coordination input for the constrained case is generally not the same as that for the unconstrained case. The (interaction) prediction principle may be applied to the problem of coordinating the two-level system under consideration. In fact, if the overall objective function is additive and the constraints are balanced, we modify the infimal objective functions as follows: let 9?= M , and for each i, 1 I i 5 n, and p in 9? let
We cannot apply Prop. 6.15 or 6.16 in the present case because of the constraints on the infimal decisions and overall control inputs. However, we can draw on the fact that a given two-level system with linear zero-sum infimal performance modification coordinable by the interaction balance principle is also coordinable by the prediction principle relative to some mapping q : R -+ 9?.We will derive such a mapping in the following section, where we consider the problem of coordinating linear programming problems. The following example illustrates the application of the interaction prediction principle to the problem of coordinating the system in Example 7.1.
4. Coordination of Linear Programming Problems
243
Example 7.2 Consider the same overall problem as given in the preceding example. Let d = R2 where each a in d is an interface input prediction. Now formulate two subproblems which depend not only on the parameter /3 = (p,, /?,) in R2, but also on the interface input predictions. For a given pair (a, p), the ith subproblem is: minimize gis(mi)
under the constraints mi 2 0 and Pi(mi,ai) I0,
where P1 and P2 are given in the preceding example, and 92@2) = h2 - (2 + Pz>mz. gls(m1) = h2 - (1 + P,)m1, The interrelationship of these problems is through the interface inputs, as expressed by the equations u, = m2 and u2 = m,. We can show that there exists a in R2 such that for any a in d with a 2 0, the vector m y = (mlY,mZY),where m i y is an optimal feasible solution of the ith subproblem for the pair y = (a,p), is an optimal feasible solution of the overall problem, when the condition
B
a1 = m2Y
and
a2 = m l y
(7.13)
is satisfied; furthermore, there exists an a 2 0 such that (7.13) is satisfied. Let (A,0). Then for any a 2 0 in R2 the optimal feasible solutions m l y and m2yof the subproblems are
B=
L
5(6 - 3a1),
mly=
13
if 3(6 - 3a1) I #, otherwise,
m2y= f ( 5 - a,). Assuming (7.13) holds, we substitute the above values of hlyand hZy into (7.13) and solve for a, and a, under the constraints a1 2 0 and a2 2 0. The solution is unique; namely, a --1m 8 and a 2 = -1 3
,
Hence, the subproblems are coordinable by the interaction prediction principle relative to the constant mapping t,~: M + {(&, O)}. 4. COORDINATION OF LINEAR PROGRAMMING PROBLEMS
The results of the preceding section can be applied to obtain a decomposition of a large linear programming problem into several smaller but interacting subproblems. The subproblems formed by the decomposition are again linear programs which depend on a parameter that can be used to
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VII. Constrained Optimizing Systems and Mathematical Programming
coordinate them relative to the given problem, in the sense that optimal feasible solutions of the subproblems directly yield an optimal feasible solution of the given problem. Consider the overall problem as given in the preceding section, but now assume that the overall objective function g is the linear form n
g(m) = cTm =
1ciTini,
i=1
where ci , 1 I i In, is a given vector in M i , and the constraint function Fi , 1 I i 5 n, is of the form Fi(m) = bi - A i m = bi - (Ailml
+ ... + A,m,,),
(7.14)
where bi is a given vector in Y i , and A i j , 1 I j I n, is an appropriately dimensioned matrix. The overall problem is then the linear program minimize cTm under the constraints m20,
and
A,m2bi,
1I i l n .
(7.15)
The orderings 2 should be understood as in the preceding section. We shall now decompose the overall constraints (7.15) so that we can assign constraints to each of n subproblems which are in balance with the overall constraints. For each i, 1 I i I n, we decompose the function Fi given by (7.14) into the functions P i : M i x Ui + Yi and K i : M + Ui of the form P i ( m j ,ui) = bi - A i i m j- B i i u i , Ki(m) = Kilml
(7.16)
+ - - + Kinmn, *
where Ui is an appropriate euclidean space and Bii and K i j , 1 <j I n, are appropriately dimensioned matrices, such that for each j, 1 <j < n, B . . K . .= "
''
{2ij,
if j = i, otherwise.
(7.17)
Under condition (7.17), all the requirements of condition (7.4) are satisfied, and hence we can proceed to assign constraints to each of n subproblems which are in balance with the overall constraints. Since the overall objective function is linear, a reasonable selection of the objective function g i for the ith subproblem is the linear form
.
g,(mi) = ciTmi
(7.18)
4. Coordination of Linear Programming Problems
245
Now, assuming the matrices B, and K i j, 1 5 i In, and 1 Ij In, satisfy (7.17), we define the ith unmodified subproblem as the linear programming problem : minimize c:mi under the constraints mi 2 0,
+
A i i m i Biiui2 b , .
and
(7.19)
The constraints of these subproblems are indeed in balance with those of the overall problem and, as is evident from (7.18), the overall objective function is additive. Any number of decompositions can be proposed for a given overall problem. The real issue is whether or not the resulting subproblems can be coordinated. We consider this question for the class of decompositions covered by (7.16) through (7.17). The decomposition of the overall objective function and constraints is specified, and, therefore, the coordinability question centers around the means of modifying the given objective functions of the subproblems. Application of the Interaction Balance Principle The questions of applicability of the interaction balance principle to the above described two-level system and the coordinability of the system by the principle have already been answered in the preceding section. Now, since the overall problem and subproblems are linear programming problems in real euclidean spaces, the coordinability question can be answered under milder conditions: namely, the assumption that the feasible domain of the overall problem contain an internal point may be relaxed. The overall objective function is additive, and, therefore, to coordinate the subproblems by the interaction balance principle, we modify their given objective functions gi by linear zero-sum modifications as in the preceding section: for each jl in U = U, x * x U,,let the modified objective function of the ith subproblem be given as
--
+
giS(mi, ui) = ciTmi fiiTui -
fijTKjim i .
(7.20)
J= 1
The interaction balance principle is then applicable to the problem of coordinating the subproblems with these modified objective functions. Moreover, if the overall problem has an optimal feasible solution, these subproblems are coordinable by the principle.
246
VIZ. Constrained Optimizing Systems and Mathematical Programming
Proposition 7.8
If the overall problem of the above described two-level system has an optimal feasible solution, and for each coordination input /I in 8,the ith subproblem, 1 I i I n, is to minimize gis(mi, ui) given by (7.20) under the constraints (7.19), then the system is coordinable by the interaction balance principle. PROOF. This proposition is a direct consequence of Prop. 7.7 except that Mazur’s separating hyperplane theorem used there requires the feasible domain of the overall problem to have an internal point. However, in the case of linear constraints, the separating hyperplane theorem may be applied without the feasible domain of the overall problem having an internal point. The coordinability question is answered by this proposition, but there are two difficulties in the practical application of the interaction balance principle.
(i) For any coordination input /I in U,the subproblems as a rule do not have unique solutions, even if the overall problem has a unique solution. Therefore, in order to determine whether or not the coordinating condition is satisfied for a given coordination input, all combinations of all optimal feasible solutions of the subproblems specified by that coordination input must be examined. This problem is avoided if the subproblems have at most unique optimal feasible solutions. The possibility of the subproblems having unique solutions is improved if the feasible domains of the subproblems are further restricted; for example, the interface input ui might be constrained to be in the set K i ( M f )where Mf,as before, is the feasible domain of the overall problem. Another approach can be based on the fact that the set of all optimal feasible solutions of a given linear programming problem is a convex subset of a hyperplane and that this set of optimal feasible solutions is characterized by a finite set of “extreme points.” A balance of interactions can then be tested by combining the extreme points furnished by the infimal decision units. Practical methods can be designed for this test. (ii) The subproblems for some coordination inputs might not have optimal feasible solutions even if the overall problem has one. This is due to the way in which the constraints were assigned to the subproblems. The constraint A i m 2 b, was the basis of the constraint Aiimi Biiui2 bi assigned to the ith subproblem, and hence the feasible domain of the ith subproblem could very well be unbounded, even if the feasible domain of the overall problem is bounded. This difficulty, however, can be overcome by imposing additional constraints on the subproblem variables. For example, we can add to the constraints (7.19) of the ith subproblem then - 1 constraints of the form
+
Ajimi
+ Bjiui 2 b j ,
(7.21)
4. Coordination of Linear Programming Problems
247
where j # i and the matrices B i j are appropriately selected. Now if the matrices Bji are such that, for each k, 1 Ik In, BjiKik =
0, ( Ajk,
if k = i, otherwise,
the addition of the n - 1 constraints given by (7.21) to the constraints (7.19) results in subproblems whose constraints are in balance with the overall constraints. Furthermore, with such an addition, the complete set of overall constraints serves as the basis for the constraints of each subproblem. Before leaving this section on the interaction balance principle, we should note the relationship between the optimal coordination input and the optimal feasible solution of the overall dual problem. The overall dual problem, considering the overall problem as primal, is maximize bTy under the constraints
y20,
and
ATy
where b = (bl, ...,bn) and A is the matrix [ A i j ] . It is elementary that the primal problem has an optimal feasible solution iff the dual problem has an optimal feasible solution. Proposition 7.8 may be restated :if the overall dual problem has an optimal feasible solution, the subproblems are coordinable by the interaction balance principle. However, the important aspect of the dual problem is that the optimal coordination input (for the interaction balance principle) is explicitly expressable in terms of the optimal dual solutions. This is so stated in the following proposition. Proposition 7.9 Suppose the overall dual problem has an optimal feasible solution 9 and the subproblems are as given in Prop. 7.8. Then the system is coordinable by the interaction balance principle and the coordination input fi in U, where
fii = B z j i ,
for each i,
1Ii
< n,
(7.22)
is optimal. PROOF. Let f i and j be respectively optimal feasible solutions of the overall primal and dual problems. Then (&, j ) is a saddle point of the functional $(m, y) = cTm +yT(b - Am),
cT& + yT(b - A&) IcT& + jT(b- A&) s cTm + jT(b- Am),
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VZZ. Constrained Optimizing Systems and Mathematical Programming
for every m 2 0 and y 2 0. Now, because of the decomposition of the constraints yT(b - Am) =
n
1y;[(bi i= 1
- Aiimi - Biiui) + Bii(ui- K i m ) ] ,
and since j T ( b- A h ) = 0, we have n
c T h 5 cTm
+ C [jiT(bi- Aiimi - Biiui) + jiTBii(ui- Ki m)l, i=l
for every m 2 0 and ui without restriction. Therefore, if A i i m i + Biiui 2 b, for each i, 1 I; i 5 n, then
i= 1
j?Bii ui -
j= 1
j j * ~ K~~ , mi]
where Bi = B L j , for each i, 1 I i I n. Consequently, for each i, 1 5 i I n, the pair (hi,tii), where tii = K i h , is an optimal feasible solution of the ith subproblem for the coordination input given by (7.22). This completes the proof.
B
We should point out that each subproblem has an optimal feasible solution for the coordination input given by (7.22) if the overall problem has an optimal feasible solution. However, for other coordination inputs, the subproblems might fail to have optimal solutions.
B
Application of the Prediction Principle
Unfortunately, most of the previous results concerning the application of the prediction principle cannot be directly applied to the present case. This is easily understood if we observe that the objective functions g i given by (7.18) depend only on the local variable m i ; the linear goal interaction operators T; are consequently identically zero. Nevertheless, some kind of modification of the subproblem objective functions is necessary to coordinate the system. The interaction in the present case comes from the constraints and affects the performance only through the constraints, and, therefore, it is not clear
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4. Coordination of Linear Programming Problems
how to define suitable interaction operators to be used in modifying the subproblem objective functions. We assume the subproblem objective functions are of the form gis(mi)= c?mi
+ BiTmi,
(7.23)
where /I= Vl, - * . ,B.) is a given coordination input from the space 1= M. Since the values of are not given by the linear goal-interaction operators, we must fmd some way of selecting them. We shall do this by deriving a mapping q : M + 1. Assume the overall constraints are now of the form m20,
and
Aim=bi,
1l i l n .
Apparently, this change removes no generality but does result in a simpler presentation here. The constraints of the ith subproblem are then m 2 0,
and
+
A i i m i Biiai= bi ,
(7.24)
where ai is a given interface input prediction. For a given pair (a, p) the ith subproblem is to minimize the objective function gill given by (7.23) under the constraints (7.24). There are two clues as to what the mapping q should be so that these subproblems are coordinable by the prediction principle relative to q. The first clue is provided by the fact that a two-level system with linear zero-sum infimal performance modifications coordinable by the interaction balance principle is also coordinable by the prediction principle relative to some mapping. The second clue is provided by Prop. 7.9. It provides a basis for deriving such a mapping by expressing the optimal coordination input for the interaction balance principle in terms of an optimal feasible solution of the overall dual problem. This suggests that for the present subproblems, the coordination input should depend on the dual variables as follows: (7.25)
-
where y = (yl, * * , y,,) is a dual variable of the overall problem, or equivalently, y j is a dual variable of thejth subproblem. The relationship (7.25) defines a function c: Y + 9,given in terms of its components as (7.26) for all y = (yl,- - -y,,), in Y. The following proposition suggests how be used in the coordination problem.
5 can
250
VII. Constrained Optimizing Systems and Mathematical Programming
Proposition 7.10 Let y = (a, p) be a given pair in d x 99 and suppose my and y y are such that for each i, 1 Ii I n, miy and yiy are respectively optimal feasible solutions of the ith subproblem and its dual. Then m y is an optimal feasible solution of the overall problem if the conditions a=K(mY)
and
p=
5(yY),
(7.27)
are satisfied. Furthermore, if the overall problem has an optimal feasible in d x 9 such that (7.27) is solution, then there exists a pair y = (a, /I) satisfied for some m y and y y . PROOF. To prove the first part, assume y = (a, p) is given and condition (7.27) is satisfied for some my and yy. For each i, 1 I i I n, then
ci
(miy)'(ci
+
pi
- AiiYi y >-o ,
+ pi - A; yr) = 0,
and of course (7.24) is satisfied. Now, if a = K(my),we can conclude from the fact that the constraints are balanced that m y is overall feasible. Suppose then that p = [(yy), then using (7.26), we obtain for each i, 1 I i < n, n
(m:)T(~i -
fA;yjY j= 1
Hence, m y satisfies the necessary and sufficient conditions to be an optimal feasible solution of the overall problem. To prove the second part, let & and j be respectively optimal feasible solutions of the overall problem and its dual, and let (a,p) be such that a = K(A) and p = [(j).It should now be apparent that for each i, 1 5 i < n, f i i and jiare respectively optimal feasible solutions of the ith subproblem specified by (a, p) and its dual. The proof is now complete. We may view the subproblems as consisting of two problems: the given subproblem and its dual. In fact, if the simplex method is used to solve the subproblems, the optimal feasibledual variable is also furnished. Furthermore, the variables can be considered as the interaction variables (interface inputs) of the dual subproblems. We can therefore view the selection of a pair (a, p) from d x 93 as constituting a prediction of the interface inputs of both the subproblems and their dual problems. We shall now proceed to give a function q : M + relative to which the prediction principle is applicable. Let 8 be a given function from d into the
4. Coordination of Linear Programming Problems
251
set Y of dual variables, and for each i, 1 Ii < n, let q i be defined on M as the composition ri(m) = li(o(K(m))), (7.28) where
li is given by (7.26), and let q be the function ?W = Mm)Y * - - r m ) .
(7.29)
Y
The coordinator uses the functions t i ( a ) = Ci(8(a)) to specify for each a in d the ith subproblem as the problem:
+
minimize gi,(mi) = ciTmi ti(a)'mi
,
(7.30)
under the constraints (7.24). These subproblems are precisely the same as those given above, except now the pi appearing in the objective function is given as a function of a. The coordination problem, therefore, reduces to the problem of selecting an appropriate a from the set d. For convenience, let m; and y t be, respectively, optimal feasible solutions of the ith subproblem given by (7.30) and (7.24) and its dual. Also, let ma = (mi", . * * ,m,? and y" = Cyl", * ,y,"). When 8 satisfies the condition that 8(a) = y" for every a in d,the interaction prediction principle is applicable (relative to ?) to the problem of coordinating the subproblems. For a given a in d,suppose m; and y t exist. Then from Prop. 7.10 and condition (7.28), we have that m" is an optimal feasible solution of the overall problem if a = K(m"). For the subproblems to be coordinable by the prediction principle relative to q, more restrictions must be imposed on 8. Namely, from Prop. 7.10, it can readily be seen that 8 must be such that for some a in d
-
8(a) = j
and
a =K(h),
(7.31)
where h and j are respectively optimal feasible solutions of the overall problem and its dual. In summary then, we state the following proposition.
Proposition 7.11 Let the function 1 be defined by (7.28) and (7.29), where 8 is a given function from d into the set Y of dual variables such that 8(a) = y" for every a in d. Then, the interaction prediction principle is applicable (relative to q) to the problem of coordinating the subproblems; that is, for any a in d ,m" is an optimal feasible solution of the overall problem if the condition a = K(m")
(7.32)
is satisfied. Furthermore, if 8 satisfies condition (7.31) for some a in d, the subproblems are coordinable by the interaction prediction principle relative to q ; that is, there exits an a in d such that (7.32) is satisfied.
252
VII. Constrained Optimizing Systems and Mathematical Programming
5. A NOTE ON THE DANTZIG-WOLFE DECOMPOSITION
The Dantzig-Wolfe decomposition of linear programming problems, unlike the decomposition presented in the preceding sections, does not appear to be a complete decomposition or decentralization. In the Dantzig-Wolfe decomposition, an optimal feasible solution of the overall problem is obtained as a convex combination of optimal feasible solutions of the subproblems. The coordinator assumes the role of a central agency deciding upon the best overall action in terms of suggested actions supplied by the local agencies. The decomposition is tailored for linear programming problems whose constraints have a block angular structure. The Dantzig-Wolfe Decomposition Procedure
-
Consider the linear programming problem in the variable m = (m,,. * ,m,) : n
minimize cTm = C ciTmi,
(7.33)
i= 1
under the constraints
Am
(7.34)
= a,
Bimi=bi
for m i 2 0 , i = l , - * * , n ,
(7.35)
where m is selected from the euclidean space My
M=Ml x
* * a
x M,,
where each space M i may be multidimensional. The cost coefficients ci, 1 I i I i z , are given vectors in M i , and the constants a and bi , 1 I i I n, are given vectors in euclidean spaces Yo and Y i , respectively. The constraint coefficient matrices A i and B i , 1 Ii I n, are assumed to be appropriately dimensioned. We shall refer to the problem given by (7.33)-(7.35) as the overall problem and assume it is feasible. In the Dantzig-Wolfe decomposition, the angular structure of the constraints (7.34) is used to formulate the subproblems. Corresponding to the overall problem, we formulate n subproblems, the ith of which is: minimize c?mi
- .nTAimi,
under the constraints
mi>O,
and
Bimi = b i ,
where x is supplied as a coordination variable by the master program.
5 . A Note on the Dantzig- Wove Decomposition
253
The master program used in the Dantzig-Wolfe decomposition is the linear programming problem in p real variables Ai: minimize f i n ,
+
* * *
+f,A,,
(7.36)
under the constraints FIAl + .*.
+ F,A, A, + ... + A,
=a, = 1,
(7.37) for Ai 2 0, i = 1, * - - , p ,
(7.38)
where for each i, 1 I i I p , fi=cTAi,
and
Fi=AAi,
and {A’, ..-,A,} is the set of extreme points of the set M E , the set of all m in M satisfying the constraints (7.35). The master program given by (7.36)-(7.38) is “equivalent ” to the overall problem in the case where MB is bounded. If M , is bounded, it is equal to the convex hull of {A’, A,}. Furthermore, the feasible domain M of the overall problem is in the intersection of MB and the set of all m in R satisfying the constraint (7.34), M = M E n { m : Am = a}. Therefore, if I = (Al, * * -,A,) is a feasible solution of the master program, A satisfies the constraints (7.37) and (7.38), and m = AIA1 . - * ApApis a feasible solution of the overall problem. Moreover, a given vector A in &f is an optimal feasible solution of the overall problem iff there exists an optimal feasible solution 2 = (&, * ,2,) of the master program and A = &A1 + .. + &AP. If the set MB is not bounded, we would have to include in the set {A’, * ,AP}the extreme rays of M , and modify the constraint (7.38) so that the &‘s give a convex combination of extreme points plus a nonnegative combination of extreme rays of M,. For simplicity, however, let us assume M E is bounded. If the coefficient matrix of the master program has rank k, then a basic feasible solution of the master program requires only k extreme points of M , . When the simplex method is used to improve a basic feasible solution, the improvement is made by introducing a new extreme point of M,. The required new extreme point is given by the solution of the subproblems. If the new extreme point does not result in an improvement, the basic feasible solution is optimal. Let there be given a basic feasible solution of the master program with 9 the corresponding basis and f the corresponding vector of cost coefficients fi . To determine if the given solution can be improved, we need only check the inequality me.,
+ +
--
-
--
0 cfi - S i T I - I
(7.39)
VII. Constrained Optimizing Systems and Mathematical Programming
254
for each, i, 1 Ii 4p , where 2Fi is the ith column of the constraint coefficient matrix of the master program, and ll is the vector l7 = (2F-')'J If (7.39) holds for all i, 1 5 i Ip , no improvement can be made, and hence the basic feasible solution is optimal. The right side of (7.39) is the relative cost coefficient of Ai. The elements of l7 are known as simplex multipliers and are always made available by the simplex method. Partition l7 into the vector x and the scalar no so that (n,no) = n. Notice then, that
no 2 min (c' - nTA)m, ME
iff (7.39) is satisfied for all i, 1 5 i 4p . In the Dantzig-Wolfe decomposition, the coordinator acts as a central agency which uses the master program to (i) find that convex combination of given extreme points of M B which minimizes the overall objective function under the constraint Am = a, and (ii) determine the coordination variable n to be used in modifying the subproblem objective functions. The infimal decision units may be viewed as local agencies whose task is to solve the subproblems and thereby provide the coordinator with extreme points of the set M E . Iteration between the coordinator and infimal decision units proceeds as follows: (i) Suppose there are given a sufficient number of extreme points, say h', * , A', to obtain a basic feasible solution of the master program. A basic feasible solution, say (Al,... ,A,'), is obtained and along with it a vector ll = (n,no) of simplex multipliers. The vector n of multipliers is then sent to the infimal decision units.
--
(ii) Taking the vector n, each of the n infimal decision units solves its subproblem. The result is an extreme point hik+' = (h!", * * * ,h:+')of M E, where &$+' is an optimal feasible solution of the ith subproblem, whose cost is
The extreme point hk+'and its cost vk+' are then set to the coordinator. (iii) The coordinator makes the comparison no 2 d"'. If no 2 ukf', the ,A,) is optimal, and h = A1h' &hk is basic feasible solution (Al, an optimal feasible solution of the overall problem. If no < vk+' the coordinator takes the new extreme point h"' and improves the basic feasible solution of the master program.
+ ... +
5. A Note on the Dantzig- Wove Decomposition
255
Comparison of the Dantzig-Wolfe and the Balanced Constraint Decompositions The decompositions which result in balanced constraints (as described in the preceding sections) as applied to linear programming problems, are decompositions of the total constraint coefficient matrix. For the constraints of the overall problem considered here, such a decomposition would be the n sets of constraints of the form Aimi
+ ui = a,
Bimi = b i ,
mi 2 0.
The Dantzig-Wolfe decomposition results in n sets of the constraints (7.40). Apparently, there is, indeed, a "physical " distinction between the decompositions giving balanced constraints and the Dantzig-Wolfe decomposition, since in the Dantzig-Wolfe decomposition, constraints causing interaction are not decomposed. Furthermore, the specification of the infimal problem for the application of the coordination principles is such that when the coordination input is optimal, the optimal infimal solutions coincide with an overall solution. In the Dantzig-Wolfe decomposition, the subproblem solutions do not directly yield an optimal feasible solution of the overall problem; an optimal feasible solution of the overall problem is given by a convex combination of the subproblem solutions. There is no such notion as an optimal coordination input or parameter in the Dantzig-Wolfe decomposition; what is important is a sequence of coordination inputs or parameters. Apparently, the coordination principles presented in Chap. V and analyzed so far cannot be seen in the Dantzig-Wolfe decomposition.
Chapter VIII
ON-LINE COORDINATION : COORDlNATION FOR IMPROVEMENT
In this chapter we shall consider the problem of coordinating a two-level system so as to improve the overall performance rather than obtain an optimal overall performance. The coordinator is to influence the infimal decision units in their selection of local control inputs so that, without any further intervention by the coordinator, the selected control inputs upon application to the process result in an improvement of the overall performance. One way to measure improvement is to compare the performance with that of a given reference control input, which can be the control input which is currently being applied to the process (and will continue to be applied if no coordinating action is taken). In the case where external disturbances are present, improvement is judged over a given range of disturbances. In the approach presented, the infimal problems are formulated as problems of obtaining a locally improved or satisfactory control, rather than an optimal control. 1. INTRODUCTION
Our departure from being overly concerned with strictly optimal overall control inputs and infimal decisions is only natural in the context of multilevel systems. True optimality cannot be achieved in actual situations for numerous reasons, most of which are centered around a lack of information concerning the factors affecting the outcomes of selected decisions or control 256
1. Zntroduction
257
actions. In classical control and decision-making situations, optimization algorithms are justified primarily on the grounds that they resolve some dilemmas regarding uncertainties in the control or decision-making situation. That is, by using a so-called optimization algorithm, one is able to select a course of action which is, indeed, very good if the information and hypotheses on which it is based are reasonably accurate. The coordination problem in a two-level system is based on the premise that the infimal decision units exist and are in charge of controlling the subprocesses, while the primary responsibility of the coordinator is to “harmonize ” these infimal decision units in respect to the overall effect of their actions. The justification of a twolevel system should be based on the potential improvement of the system’s overall behavior, and, therefore, the coordinator’s performance index should measure overall improvement rather than optimality. Even in cases where complete information is available and the overall problem is optimization, a true optimum might not be attainable due to limitations in decisionmaking capacity or decision time; therefore the justification of the coordinator’s action should indeed by based on improvement of the overall performance. The notion of coordination for improvement of overall performance is consistent with the general concept of coordination presented in Chap. IV. Recall that the consistency postulate presented in Chap. 1V only requires the coordinator to induce the infimal decision units to solve the overall problem, which, of course, could very well be given as requiring an improvement of performance rather than optimal performance. Actually, we consider in this chapter the overall problem as such a problem rather than an optimization problem. The notion or coordination for improvement of overall performance reflects realistic situations in many fields. For example, the question of decentralization in economic systems (as, for instance, presented in Chap. 11) is treated theoretically as the problem of achieving the optimum of a welfare economy. The advantage of decentralization is argued on the grounds that the overall optimum of the economy can be achieved by decentralized actions. However, as Simon [ll] points out, the value of a decentralized market or economy is based on the simple fact that “it works”; while every participant is working to improve “its own lot,” the performance of the overall economy is improved and (almost) everybody gains. It is this harmony of ,goals, as expressed in the consistency postulate, that is the real justification for decentralization. A similar situation exists in large-scale industrial automation. A first echelon of control is required for proper and safe operation of the basic processes and generally exists even before one starts thinking of imposing higher echelons of control, which are justified, in suitable economic terms, on the basis of improving the operation of the process.
258
VIIZ. On-Line Coordination: Coordination for Improvement
In this chapter, we consider two questions regarding the way in which coordination for improvement in overall performance can be implemented. (i) How should the coordinator influence the infimal decision units in their selection of local control inputs so that, once a coordination input is given to them, they arrive at local control inputs such that, without any further intervention on the part of the coordinator, application of these local control inputs to the process results in an improvement of the overall performance. (ii) Given a time interval over which the two-level system’s performance is to be observed and during which there are n time instants at which the coordinator can influence the infimal decision units, what should the coordinator’s strategy be so that, after each coordination instant, the coordinator’s action results in an improved overall performance, or, in the absence of adversites, the overall performance improves monotonically over the given time interval. The above two questions are intimately related: a solution of the first suggests the strategy sought in the second. Employment by the coordinator of a coordination strategy in which the coordination inputs are updated over a sequence of time instants will be referred to as sequential coordination. If after each coordination instant the infimal decision units apply their selected control inputs to the process without any further intervention by the coordinator, the coordination will be referred to as on-line coordination. The above two questions are focused primarily on on-line coordination. The term on-line coordination is used to emphasize the fact that the benefits of feedback from the infimal decision units to the coordinator is denied from the time the infimal units receive the coordination input from the coordinator until after their selected control inputs are applied to the process. The coordinator must then, at each coordination instant, face a decisionmaking problem under uncertainty. We point out here that, in order to satisfy the consistency principle, the coordinator will specify the problems of each infimal decision unit as decision-making problems under uncertainty, even if the uncertainties are only in respect to actions of the other infimal decision units. Before we proceed with presenting a theory for on-line coordination, let us briefly consider a simple example so as to illustrate‘some of the concepts involved. Consider a two-level system whose overall process P is described by the linear transformation y = Am = P(m),
where the control input m and output y are three-dimensional real vectors,
259
1. Introduction
and A is a 3 x 3 real matrix [ a i j ] .For simplicity, assume there are no outside disturbances. Assume the overall cost evaluation function G, G(m,v) = mTm + 0 - f i T Q - fi,
is given, where j j is a prescribed reference output. Suppose the overall process consists of three subprocesses, the ith of which is described by the equation y i = a,, mi
+ ui = Pi(mi,ui),
where the control input m i , the interface input ui, and output y i are real numbers. When these subprocesses are coupled and the control input m = (mi, m 2 , mJ is applied, the interface input ui occurring at the ith subprocessis ui = C aij mi
= Ki(m).
j#i
The coordination problem we are faced with is the following: for a given rii2 ,6i3), the coordinator is to influence the infimal reference control ri3 = decision units in charge of controlling the subprocesses so that they produce a control input A = (Al, A2, h3)which, when applied to the process, results in a lower cost than that of ri3 as evaluated by G : G(h, P ( h ) ) < G(fi, P(&)).
Two questions are involved here: first, how do the infimal decision units acting independently of each other arrive at their selections of the control inputs, and, second, what mechanism can the coordinator use in influencing the infimal decision units. We assume the infimal decision units make their selections of the control inputs so that, based on the information given them, their own performance will be improved. Therefore, assume the ith infimal decision unit is given a cost evaluation function Gid, Gidmi Yi 9
9
= 3m:
Pi)
+ Bimi + O l i - Yi)*,
(8.1)
which it uses to measure its own performance. We then define the ith infimal problem as follows: for a given reference control f i i , a given parameter fli, and a given range oi over which the interface input ui may vary, find a control input hi such that g i A h i ui 9
for all ui in the estimated range
9
Bi>
< 9i(hi
9
ui 3
oi, where
gi.dmi ui > Pi) = Gidmi 3 Pi(mi 9
fli),
9
ui), Pi).
These infimal problems are so-called satisfaction problems, in that they
VIII. On-Line Coordination: Coordinationfor Improvement
260
do not require an optimal performance but only a satisfactory level of performance over the given uncertainty sets, which are in this case the given estimated ranges oi of the interface inputs. Now, if the ith infimal decision unit succeeds in solving its problem and the estimated range oi contains the actual value of the interface input, then the control input selected will, indeed, improve its own local performance. The interaction estimation principle (4.18) introduced in Chap. IV, if it applies, states that the control input A = (Al,Az, A3), so selected by the infimal decision units, produces a decrease in overall cost if the interface inputs ul, uz , and u3 which occur in the coupled system when the control A is applied are, respectively, in the estimated ranges, , and :
ol,o2
Ki(A)
E
oi,
o3
for each i = 1,2, 3.
We can and will generalize the infimal problems to the extent that the parameters pi are not prescribed; only a range d i over which the parameter Pi may vary is given. In this case the ith infimal problem is the following: for over which the a given reference control input Gi and given ranges Oi and parameter Biand the interface input ui may respectively vary, find a control input Ai such that
oi
gi(hi 7 ui 7 P i )
< gi(fii
9
ui 3
Pi),
oi.
for all p i in Biand all uiin The correctness of the estimated ranges of the interface inputs is determined by the subprocess interaction functions K i giving the actual interface inputs which occur in the coupled system as a function of the applied control input. The parameters pi appearing in (8.1) can be considered as representing direct interactions between the infimal performances if given by an appropriately defined function qi as a function of the applied control input. Assuming this is so, the estimation principle (4.19) states that the control input m produces a decrease in the overall cost if Ki(A) E
oi,
and
qi(A)E g i ,
for i = 1 , 2 , 3 ;
(8.2)
that is, the estimated ranges giand gi for each i = 1,2, 3 are correct. The coordination problem, given a reference control G , is how to generate the estimated ranges oi and Bi so that the estimation principle can be applied. Let the matrix A be given as
I . Introduction
261
Let the reference output be
= (5,
- 5,2) and the reference control be
f i = (0,0,O). The interface inputs occurring in the coupled system when % is applied are G1 = G, = G3 = 0. We should notice that the infimal cost
evaluation functions Gi, given by (8.1) are derived from the overall cost evaluation function G as in Sec. VIII.3 and therefore satisfy (8.20). The appropriate choice of the functions q iis then given by (8.22):
+ 4m3 - 6 q2(ml, m 2 ,m3) = 14ml + 8m3 + 24 q3(mi, m 2,m3) = 6m1 + 4m2 + 10 ql(m,, m,, m3) = lOm,
We can generate the estimated ranges
Gi and di in two ways.
(i) Select a neighborhood N(k), N ( f i ) = {m:max Imi - fiil I i
3,
i = 1,2,3},
of the given reference control %. Now, using the functions Ki and qi ,generate the estimated ranges and gi:
Ui
oi
= { u i: ui = Ki(m),
gi= {pi: pi = qj(m),
for some m in N(%)},
(8.3)
for some m in ~ ( k ) } ,
(8.4)
for each i = 1 , 2 , 3 . Hence, the estimated ranges are the intervals of the interface inputs ui and parameters pi given by the inequalities
- 3 Iu, I +, - 1 3 I p p , 21,
- 3 I u2
I 4,
-lIu311,
13 5 p2 5 3 3 ,
5Ip3<15.
Each infimal decision unit solves its problem in reference to (8.3) and (8.4) and the given reference control % = (0,0, 0), thereby yielding the control input A = (+, -A, 4). Notice that f i is not unique. The coordinator then checks whether or not (8.2) is satisfied. It is, and therefore we conclude that A produces a lower cost than that of %. Indeed, G(A, P ( f i ) ) = 42, while G(fi, P(%)) = 54. (ii) An alternate way of generating the estimated ranges is to specify neighborhocds of Ki(%) and qi(fi), rather than a neighborhood of %, in terms of a norm. For each i = 1 , 2 , 3 , the coordinator selects a pair (ei, SJ of positive real numbers and generates the estimated ranges
-
U i = {ui: Iui - Ki(m)l I ei},
Oi = {pi: IBi - qi(m)I ISi).
262
VZZZ. On-Line Coordination: Coordinationfor Improvement
It turns out in this example, that tlie pairs (el, 6,) = ($, 7), (e2 ,6,) = (-3, 1l), and ( E , , 6,) = (1, 5) generate the same estimated ranges as those given by (8.3) and (8.4). This approach is a heuristic one and, given that the interaction estimation principle is applicable, success in coordinating the system so as to obtain improvement in the overall performance depends on the selection of the estimated ranges. It should be mentioned, however, that, whenever the interaction estimation principle is applicable, as in the above example, a simple “feedback” check on the second level as given by condition (8.2) is sufficient to conclude that an improvement in the overall performance is obtained without evaluating the overall performance.
2. ON-LINE COORDINATION CONCEPTS IN GENERAL
SYSTEMS THEORY
The notion of on-line coordination, as pointed out in the preceding section, involves an updating of the coordination inputs over a sequence of time instants, and, moreover, after each coordination instant, the infimal decision units implement their selected local control inputs without any further intervention by the coordinator. The objective of the coordinator at each coordination instant is to influence the infimal decision units so that their actions result in an improvement of the overall. performance. The overall problem of the two-level system at each coordination instant, is, then, an improvement or satisfaction problem, rather than an optimization problem. Basic Notion Let a two-level system be given as follows: The overall process under the control of the two-level hierarchy consisting of the coordinator and the infimal decision units is given as in Chap. IV as a mapping P:M x R -+ Y, where R is the environment interface object or, more abstractly, a given set of uncertainties. We assume there is given an overall cost function g : M x ZZ --f I/, where V is a linearly ordered set, such that, given a subset M fE M of control inputs, the overall improvement problem can be stated as follows: find a control input rh in M such that g(A, 0 ) < g(*,
4,
for all o in R, where is a given reference control input in M. The set M f is the set of feasible control inputs.
263
2. On-Line Coordination Concepts in General Systems Theory
The overall cost function g can be interpreted in terms of the overall process P and a given cost evaluation function G defined on M x Y:
g(m, 0 ) = G(m, P(m, 4). (8.5) This interpretation is not necessary, but may be intuitively helpful. In reference to the overall cost function g , we can already indicate a preference among the control inputs in terms of reducing the value of g over a. Precisely, g induces an ordering > in M, such that for all m and m' in M, m > m' iff g(m, w) < g(m', w), for all w in a. (8.6) We shall say m is preferred to m' if m > m'. Hence, the overall improvement problem is: given a reference control input 6,find a control input in M f which is preferred to 6. In general, there may be given, in lieu of a reference control input, a so-called tolerance function z: Q --t V which is used to specify a standard of performance. In this case, the overall problem is to find a control input h such that g ( h , w ) < z(w) for all w in Q. We refer to this type of problem as a satisfaction problem rather than an improvement problem, to emphasize the fact that the standard of performance need not be given by a reference control. The overall process P is assumed to represent n coupled subprocesses P i : M i x Ui -, Yi as described in Chap. IV, where M = M I x x M,, and Y = Yl x * * . x Yn. The interface inputs ui to the ith subprocess, 1 Ii In, are assumed to be expressable as a function of the overall control inputs and the environment inputs (or uncertainties) by the mapping K i : M x Q + U i . That is, if the control input m is implemented, and the input w from Q occurs, the interface input appearing at the ith subprocess in the coupled system is ui = Ki(m, w). Thus the interface inputs to the ith subprocess in the coupled system indeed represent the actions of its environment, which includes not only the other subprocesses of the two-level system, but also the two-level system's environment. For each i, 1 i 5 n, we assume there is given an infimal cost function g i a : M i x Ui x Bi + V , where Bi is a given set of parameters. The ith infimal decision problem will be defined later in terms of the given infimal cost function g i a and given subsets to the sets Ui x g i . The given subsets of Ui x Bi will be referred to as estimated ranges, since they will assume the role of uncertainty sets for the infimal problems. As in the case of the overall cost function, the infimal cost function g i a , 1 Ii In, can be considered as being given in terms of the subprocess Pi and a given cost evaluation function Cia defined in M i x Yi x Ui x g i ,
-
giidmi , ui 9 P i )
= Gidmi3
Pi(mi ui>,ui Pi). 3
3
(8.7)
264
VIII. On-Line Coordination: Coordinationfor Improvement
Again, as for the overall cost function, this intrepretation of g i r is not necessary, although it might aid one’s intuition. It should be apparent that the coordinator can influence the infimal decision units by specifying the estimated ranges for each of them. Therefore, let us assume there is given a set V of elements y, such that for each y and i, 1 5 i In, there,is given a subset UiY x aiy E Ui x g i .The set V will then be used as a coordination object and its elements referred to as coordination inputs: the coordinator selects a y in V and thereby specifies for each i, 1 5 i In, the estimated range U i yx Biy. The major distinction between the coordination modes employed here and those employed in the preceding chapters is that the estimated ranges Uiy x 9if; will vary with y , but will not in general be singleton sets, although the sets Biyare considered in a special case as singleton sets. Most of our analysis will be centered on the case in which the overall problem is specified as an improvement problem in terms of a prespecified reference control input. We shall, therefore, first introduce the on-line coordinability concepts for the reference control approach and then, subsequently, present more general notions. Coordinability Concepts in the Reference Control Approach
A given reference control input can be specified by the coordinator, or it can be the control input which is currently being applied to the process and which will continue to be applied if no coordinating action is taken. The objective in the latter case is to improve on the existing control input. Each coordination input in V , through the estimated ranges and infimal cost function, induces a preference relation in the set M. For each y in V let (>,,) be the relation in M such that for all m and m‘ in M, m(>y)m’,
iff for each i, 1 I i I n, Sidmi ui 3
9
Pi)
I gidmi’, ui
9
Pi),
holds over the estimated range U i yx Biy,while for some j , 1 ~j In, the strict inequality g j d m j uj 7
9
Pj)
< gjdmj’, uj
7
Pj),
holds over the estimated range U j yx Bjy. If m(>y)m’, we shall say m is y-preferred to m’. For a given control input f i in M, a coordination input y in V is riiacceptable in the two-level system, if there exists a feasible control input which is y-preferred to 2.
265
2. On-Line Coordination Concepts in General Systems Theory
We now present the concept of coordinability for the reference control approach. Let f i be a given reference control input in M , and let M I be the given set of feasible control inputs of the two-level system. Then the two-level system is coordinable at f i iff there exists some %-acceptable coordination input y in %' such that Cm(>,)fil* cm > fil,
for all m in ~ f . This notion of coordinability means that there is some coordination input in %' and some overall feasible control input which improves at least one of the infimal performances over the given estimated range and does not worsen the other infimal performances over the given estimated ranges, and all such overall feasible control inputs improve the overall performance. The infimal decision problems in the reference control approach are now apparent from the y-preference relations in M and the above notion of coordinability. Namely, they are improvement problems, the ith infimal problem, 1 Ii I n, being the following: for a given coordination input y in %? and a given reference control input 6,find a control input hi in Mi' such that S d h i 3 ui 3 83
SiAfiii 9 ui 3 83,
over the estimated range U i yx Biy,where Mif is the subset of M i consisting of the ith components of all overall-feasible control inputs. We will refer to the ith infimal problem so given as the problem specijied by the pair (y, fi). Clearly, for any pair (y, 6)in V x Mf,the control input f i i is a solution of the ith infimal problem specified by (7, fi), but f i is not y-preferred to itself since the relation is irreflexive. On the other hand, if a pair (y, f i ) in V x M f is such that y is fi-acceptable, then by definition of acceptability each ith infimal problem specifiedby (y, f i )has a solution hisuch that h = (Al, * ,hn) is feasible and y-preferred to 6 ;therefore, in respect to the above notion of coordinability, we shall assume that for each such pair ( y , f i ) in %? x Mf, the ith infimal decision unit, for each i, 1 i I n, arrives at such a solution to its specified problem. Most of our analysis in the reference control approach will be centered on the above notion of coordinability. However, we can extend it to a notion of coordinability over a given subset of M ; the two-level system is coordinable over a set 1Gi E M iff it is coordinable at every f i in A?. This extension of the notion is of interest, for, if the system is coordinable over a given set A?, the coordinator can select any f i in A? with the assurance that it can influence the infimal decision units so that they produce an overall feasible control input which improves the overall performance. Apparently, this suggests a sort of optimization problem for the coordinator; namely, the coordinator is to select a reference control input f i in A? so as to get the most improvement
--
266
VIII. On-Line Coordination: Coordinationfor Improvement
from the infimal decision units. However, the coordinator can be assured of some improvement regardless of which control input is selected from I@ as a reference. The difficulty in applying the above coordinability concept is that after a reference control input f i and coordination input y are given to the infimal decision units the coordinator must then have available the costs g(& o) and g(h, w), where h is the control input selected by the infimal decision units, and compare them for each w in SZ or that w which occurs in order to assess whether an improvement is obtained. Although this might not be quite as difficult a task as determining optimality in the case of optimization, it is still desirable to have a simple test that will indicate whether the two-level system is coordinated. Analogous to the optimization cases previously considered, we, therefore, consider a coordination principle which is based on the interaction between the infimal units after a selected control input is applied to the process. The interactions between the subprocesses are exhibited by the interface inputs, since the subprocesses are coupled through these inputs, and, therefore, these interactions are given by the mapping K: M x R + U whose components are the mappings Ki. Interactions between the infimal performances are due to the interactions between the subprocesses but may also involve (or be considered to involve)direct interactions manifested through the parameters p i appearing in the infimal cost functions. We can use these interactions to postulate a coordinating condition and hence a coordination principle. The appropriate coordination principle is the interaction estimation principle (4.1 S ) , which for our purposes we express here by the proposition ( V y ) ( Vm)(Vco){[m(>y)fi and K(m, w ) E Uy]* m > f i } ,
(8.8)
where U y = U I yx ... x Uny.We postulate an alternate form of the interaction estimation principle when there is given a mapping q : M x SZ + A9 where By= gIy x . - -x gnY.The ahernate form obtained from (4.19) is expressed by the proposition
-
x * * x BnY.In the where q,(m, w ) = (K(m, w), y(m, o)) and B y= BIy alternate form of the interaction estimation principle, we consider the parameters pi appearing in the infimal cost functions as though they are interactions given by the mapping ?. The coordinability-type notions for the interaction estimation principle in reference to a given two-level system with the set M f of overall feasible control inputs and a given reference control input f i are as follows:
2. On-Line Coordination Concepts in General Systems Theory
267
The interaction estimation principle is applicable at L% iff (8.8) is true, and the system is coordinable at 61by the principle iff the principle is applicable at % I and there exists (y, m) in %? x M ysuch that the proposition (8.10)
is true. The notions of applicability and coordinability of the alternate form of the interaction estimation principle are similarly defined in reference to a given mapping q : M x IR + 923, with the exception that (8.10) is replaced by ( 3 J ( 3rn)(Vw)[m(>JL% and q,,(m, o)E U yx
By].
(8.11)
Applicability of the interaction estimation principle in either of its forms simply means that the overall performance is improved whenever the interactions resulting from the application of a y-preferred overall feasible control input fall within the estimated ranges. Coordinability by the principle is stronger than the above notion of coordinability and requires, in addition to applicability, the existence of an acceptable coordination y and a y-preferred overall feasible control input for which the resulting interactions fall within the estimated ranges. If the principle is applicable, a simple test can be devised for the coordinator to determine whether the actions of the infimal decisions units result in an improvement of the overall performance; the test need not involve an actual comparison of the overall costs. The interaction estimation principle provides a basis for devising a sequential coordination strategy for arriving at an optimal coordination input in the two-level systems considered in the preceding chapters. If the two-level system is coordinable by the estimation principle at each coordination instant, the system can then be coordinated at each of these instants so that the improvement in overall performance is monotonic. This kind of iteration is practical for on-line application, since at any intermediate stage one is at least assured of an improvement relative to the preceding stage. There is a potential difficulty in the application of the interaction estimation principle, since the solution of the infimal improvement problems to obtain a y-preferred overall feasible control input could be difficult. The potential difficulty arises from the requirement of improving at least one infimal performance over the whole estimated range, while not worsening the others. If the estimated ranges are reduced to singleton sets, UY x 923" = {(uy, by)},the infimal performances can usually be improved, but then the coordinating condition, K(m, o)E U yor q,,(m, o)E U yx gY,is not likely to be satisfied. On the other hand, if the estimated ranges are too large, improvement of the infimal performances over the estimated ranges might not be possible.
VIII. On-Line Coordination: Coordinationfor Improvement
268
Some Conditions for Coordinability in the Reference Control Approach Applicability of the interaction estimation principle in a given two-level system depends primarily on the relationship of the infimal cost functions to the overall cost function and the given set of overall feasible control inputs. We shall present here some conditions for applicability of the principle in the reference control approach in the case where the infimal cost functions are in balance with the overall cost function, and the set of overall feasible control inputs is from a certain class of connected subsets of M ; in this case we will give a condition for coordinability of the system. We assume the infimal and overall cost functions are real-valued functions. The infimal cost function g i , , 1 I i I n, is said to be in balance with the overall cost function g , if there exist two mappings q i : M x R + Bi and hi: M x R -+R, such that Gia(mi Ki(m, w), qi(m, 0)) + hi(m, 0) = .dm7 w)7 9
(8.12)
for all m and w in R. Example 8.1 Consider a two-level system with the subprocesses P, and P2 defined on RxRas P1(m,, u1) = m12 + u1,
P2(m2 7 u2) = m2 u2, where the interface inputs are given by the equations u1 = m2
+ w = Kl(m,, m 2 , w),
u2 = m,
+ w = K2(m1,m 2 , w ) ,
and w is a real number representing the external disturbances. Suppose a m , v) = mTm + (v1 - 71) + (v2 - Y2) is the given overall cost evaluation function. To obtain infimal cost functions which are in balance with the overall cost function g , g(m, w ) = G(m,P(m, w)) = m12 m22 (m12 m2
+
+ + w -7,) + (m2m1+ m 2 0 -y2),
+
we define the infimal cost evaluation functions G , , and G,, as Gl,(ml
9
Y1
2
~ , , ( m ,,v2
'1
u2
7
+ (v1 - 71) + B l ( m l + '117 = mZ2 + (v2 - 7 2 ) + B 2 m2
B1) = m 1 2
82)
and the functions qi and hi such that for all m,, m2 , and w ql(ml, m2
9
0)= m2,
h,(m,, m2 w ) = 9
-y"2
9
and qz(ml,m 2 , 0) = 1,
h2(m17
m2 7
0) = 2m12 - ml
- Yl.
2. On-Line Coordination Concepts in General Systems Theory
269
One can easily verify for each i = 1,2 that the infimal cost function giragiven in terms of the subprocess P i and cost evaluation function C i a is indeed in balance with the overall cost function 9. The notion of balance between the infimal cost functions and overall cost function covers several notions already encountered in the previous chapters. For example, suppose the overall cost function is additive and the modifications are balanced ;
whenever the Pi)s are identical. In this case, (8.12) is realized by defining hi as the function
where
pi = qi(m, 0).Clearly, the infimal cost functions in the additive case
(8.13) are in balance with the overall cost function.
As another example, let the overall and infimal cost functions g and gia, 1 Ii 5 n, be defined respectively by (8.5) and (8.6) in terms of given cost evaluation functions G and Cia. Suppose Gi, is generated from G in the sense that there is given an appropriately defined function Bi from M x Y into Bj such that G i d m i Yi 3
9
Bi)
= G(m,Y),
when pi = Bi(m, y). The infimal cost function gia is then in balance with 9 ; the required function qi is qi(m, 0 ) = ei(m, P(m, 0>>,
and hi is identically zero. We should point out here that the conditions we obtain for applicability of and coordinability by the interaction estimation principle require the infimal cost functions to be in balance with the overall cost function. Moreover, it is necessary that we assume the mappings &, y i and hi are independent of m i , the ith component of the control inputs m. The assumption that the mappings Ki are independent of mi means that the subprocess interconnections are essentially input-canonical. To guarantee the applicability of the interaction estimation principle, we shall have to restrict the feasible control inputs to appropriate subsets of M. Let p be an integer valued function on M x M such that for any m and m' in M,the integer p(m, m') is precisely the number of components in which m and m' differ. If p(m, m') = 0, then rn and m' differ in no components, and hence m = m'; p(m, m') = k indicates that m and m' differ in exactly k
270
VIII. On-Line Coordination: Coordinationfor Improvement
components, apparently, the maximum value of p is n. The function p is indeed a metric (distance function) in M. It is apparent that p is nonnegative, p(m, m‘) = p(m’, m), and p(m, m’)= 0, iff m = m‘. It can be verified quite readily that p satisfies the triangle inequality. The first result concerning applicability of the interaction estimation principle in the reference control approach is then the following. Proposition 8.1 Suppose the infimal cost functions of the two-level system are in balance with the overall cost function; for each i, 1 I i I n,
+ hi(m, SiAmi Ki(m, w), Vi(m, 0)) 9
0)= d m ,
w),
(8.14)
and the mappings Ki ,qi , and hi are independent of mi. Let % be a given reference control, and let the set ~f
(8.15)
= {m: p(m, 6)II},
be the set of feasible control inputs. Then the interaction estimation principle expressed by (8.9) is applicable at 6.
To prove applicability of the principle, suppose there exists y in > %. Because (8.15), assume that the difference between h and I53 occurs at the ith component. Then PROOF.
%? and h in M fsuch that (8.11) is true. We need to show that h
ui Pi) < S i d f i i ui Pi), over the estimated range Uiy x W i y . For all w in s1 we obtain from (8.11) that K i ( h ,w ) is in U i yand qi(h, w ) is in W i y ,and therefore from (8.14), Sidhi 9
S(h,
0)= Sidhi 9
7
9
9
Ki(h5 a),V i ( h , 0)) + hi(&, 0)
< Sidfii Ki(& 01, Vi(h, 0)) + hi(&, 0) 7
+ hi(*,
= gia(%i, Ki(6, o),~ i ( 6 , 0 ) )
W)
= g(6, 0).
Hence h > I53, and the proposition is proven. The next result concerning applicability of the interaction estimation principle extends the above proposition and requires the notion of p-connectedness. A pair {myf i } of elements in M , with distance p(m, f i ) = k > 0, is p connectedin a subset M’ of M if there is a sequence {m’, * ,mk}in M‘, where m0 = m and mk = f i , such that
--
p(m’-’, mi)= 1,
for i = 1, -..,k.
Such a sequence is a p-connection of the pair {m, f i } .
2. On-Line Coordination Concepts in General Systems Theory
27 1
Proposition 8.2
Suppose the infimal cost functions of the two-level system are in balance with the overall cost function as in the preceding proposition. Let A be a given reference control, and for each y in % let the set of feasible control inputs be the set ay,
fiy= { m : {myA }
is p-connected in
MY},
(8.16)
where MY= { m : {K(m, 0): o E R} c U y and
{q(m, o):o E R} E Wy. (8.27)
Then the interaction estimation principle expressed by (8.9) is applicable at A. PROOF. Suppose there exists y in % and rfi in fiy such that (8.11) is true. We need to show that h > A. Let k = p(&, A). If k I1, the result follows from Prop. 8.1. Hence, assume k 2 2. Let {m', , m k - l } be a p-connection of the pair {h,A } in M Y .Assume the differencebetween rfi and A occurs in the first k components; for each i, 1 Ii Ik,
gia(hi 3 ui 9 P i ) I gia(Ai ,ui y
Pi)
over Uiy x
Wiyy
(8.18)
with strict inequality for at least one i, 1 I i Ik. Using the triangle inequality, we can show that p(m', A) = k - 1. Suppose the difference between hand m1 occurs at the ith component. Then mi1 = A i ; otherwise, since hj = mj' for j # i, we get p(m', A) = k, which contradicts p(ml, A) = k - 1. Furthermore, 1 I i 5 k,because of the above assumption, and therefore from mi1 = Ai and (8.18), gia(rfii, ui , Pi) Igia(mi', u i , Pi)
over Uiy x Wiy.
We can, therefore, conclude, from the proof of Prop. 8.1, that rfi 3 m1 where the relation means g(h, o)Ig(m', w) over R. Notice that the sequence {m', ,m"-'} is a .p-connection of the pair { m l , A}. The above argument can, therefore, be applied to mi for i = 1, -.-, k - 1, to obtain the relationship
+
-
h3 h 1
> . a -
>mk-' + A .
The relation 3 is, of course, transitive, and strict inequality (8.18) holds for some i, 1 I i I k. Therefore, rfi > A. This completes the proof. We should note in respect to (8.16) and (8.17) that A need not be in the set M Y , nor is it necessary that MYcontain any of the control inputs from
By.
272
VIII. On-Line Coordination: Coordinationfor Improvement
Proposition 8.3
Suppose the infimal cost functions of the two-level system are in balance with the overall cost function as in Prop. 8.1. Let & be a given reference control, let ~f
= { m : p(m, &) I 2)
be the set of feasible control inputs, and, for each coordination input y in V, assume the set M Y given by (8.17) contains &. Then the interaction estimation principle is applicable at 5. PROOF. Suppose there exists y in V and A in M f such that (8.11) is true. 5 I, the result A > f i follows from Prop. 8.1. Therefore, assume If p(A, 61) p(A, f i ) = 2. Now, there exists an m in R such that p(A, m) = p(m, &) = 1. Suppose A i# mi and mi # k j .Then
g i d h i ui 9
7
Pi)
5 gia(*ii
3
Pj)
= gja(Aj 9
9
ui , P i ) = gia(mi ui
9
Pi)
Over
uiYx aiY,
and g j d m j uj 9
uj
Pj)
I gja(6j uj
9
Pj)
Over
ujyx
Wjy,
with strict inequality for at least one of them. We now use the assumption that f i is in M Y and Prop. 8.1 to obtain A 3 m 3 &. We also obtain either rfi > m or m > 6.Hence A > & and the proof is complete. The following result does not pertain to the interaction estimation principle per se, but does give a condition for coordinability in the reference control approach. Proposition 8.4
Suppose the infimal cost functions of the two-level system are in balance with the overall cost function. Let & be a given reference control, and let the set of feasible control inputs be given by (8.15). Then the system is coordinable at % if there exists an %-acceptablecoordination input y in V such that f i is in the set M Ygiven by (8.17). The proof of this proposition can be seen immediately from the proof of Prop. 8.1. We now offer the following remarks regarding the application of the interaction estimation principle. When employing the interaction estimation principle, the coordinator’s objective, once a reference control is specified, is to select the estimated range U v x BY such that for all o in Q, K(A, o)is in U Y and ?(A, w) is in W Y , where h is the control input chosen by the infimal decision units. The difficulty in doing this is due to the fact that the infimals’ choice of a control input is not available to the coordinator beforehand. However, the reference control
2. On-Line Coordination Concepts in General Systems Theory
273
is available to the coordinator, and it may be used as a basis of generating the estimated ranges. For example, let ri3 be a given reference control, and suppose a norm is defined in the sets U and 99. For each y in W, the coordinator can select real positive numbers E , and 6, and then specify the estimated range U yx 98, as
-= E,,
for some o in Q},
By= {P E ~#7:118 - q(fi, o)ll < 6,,
for some o in Q}.
U7= {u E U : 1Iu - K(ri3, w)ll
The selection of the numbers E , and 6, could be based on past experience. We should note, however, that an increase in E , and 6, increases the possibility of y being coordinating, but at the same time decreases the " effectiveness" of y in the sense that the potential improvement of a y-preferred control input decreases. Therefore, in applications, a balance should be sought between the potential improvement and the assurance that y is coordinating. The p-connectedness of the set of feasible control inputs required in Prop. 8.2 is not a very serious restriction. For example, any pair of elements in M f is p-connected in M f , if M f is an n-fold Cartesian product, M f= M I f x * x M n f .Therefore, the interaction estimation principle is applicable over any coordination object V such that {K(m, o):m E M fand o E Q} E U y and {q(m, o):m E M f and o E !2} E Wv.However, for a given reference control f i in M, there might not exist any control input in M fthat is preferable to ri3 over the estimated range U vx Wy.
-
Coordinability Concepts in the Satisfaction Approach On-line coordinability and the interaction estimation principle for the reference control approach are defined in terms of improvement over a given reference control input. The explicit use of a reference control input to set a standard of overall and infimal performance can be relaxed so that the performance standards are given by tolerance functions. Let z: Q + V be a given overall tolerance function, so that the overall problem becomes that of finding a control input h in M f such that g(h, o)Iz(o) for all w in Q. Similarly, for each i, 1 Ii In, let ria: Ui x W i + V be a given tolerance function so that the ith infimal problem specified by y in V becomes that of finding a control input hi in M i such that Sia(hi
9
ui 7 P i )
over the estimated range Uiv x Wiv.
5 zitIl(ui
9
Pi),
214
VIII. On-Line Coordination: Coordinationfor Improvement
The coordinability notion and the notions of the interaction estimation principle for the reference control approach are extendable in a straightforward manner to the case in which the overall and infimal problems are given as above. All we need do is appropriately redefine the preference relations in M so that they are compatible with improvement being defined in terms of tolerance functions. Convenient choices of the tolerance functions, especially in the case where R is a singleton set, are constant functions. The standards of performance are then simply upper bounds on the costs. The coordinator then selects upper bounds k,, ,k, for the infimal costs and, of course, the estimated ranges, so that the resulting action of the infimal decision units produces an overall feasible control input with overall cost less than a prespecified value k. Then, ,k, , the overall performance is steadily by steadily decreasing k and k,, improved provided the estimated ranges at each stage of adjustment are chosen so that the system is coordinated. There are, of course, other possible choices of the tolerance functions. They may be chosen as nonconstant functions, and each coordination input could specify a different set of tolerance functions for the infimal decision units. Such is the case in the reference control approach.
---
3. APPLICATION TO LINEAR SYSTEMS
The conditions obtained in the preceding section for applicability of the interaction estimation principle and coordinability can be directly applied to a two-level system whose overall process is linear with a quadratic performance function. We shall illustrate here how the infimal cost evaluations functions can be derived so that the corresponding infimal cost functions are in balance with the overall cost function. In doing so, we shall give the required functions q i and hi explicitly. x M, and Y = Y, x x Y, be linear Let the sets M = M , x spaces, and assume the component spaces M iand Yi, 1 I i I n, are Hilbert spaces. Also, let the set R be a given subset of a linear space Let the overall process P be a linear operator with domain M x and range in Y given as
a.
P(m, o)= Am
+ Bo + yf, a,
where A and B are linear operators with domains M and respectively, and yf is a given element in Y representing the free undisturbed response of the process, yf = P(0,O).
3. Application to Linear Systems
275
Let the given overall cost evaluation function G be defined on M x Y as the quadratic form
where QiM and Q i , 1 I i I n, are self-adjoint, positive, bounded linear operators, and J = (yl, * ,Fn) is a given reference output. The overall process P can be represented component-wise by the n equations
--
yi = A i m
+ B i o + yf,
(8.19)
a,
the ith component where 1 I i I n; that is, for a given m in M and o in of the output y = P(m, o)is given by (8.19). The overall process then represents n coupled subprocesses, the ith of which is defined on M i x U i , where Ui = Y i ,by the equation
Pi(mi,ui) = Aiimi where A i m = Ailml process is then
+ ui + y z ,
+ - + Ainmn.The interface input ui to the ith sub* *
where K i i = 0 and K i j = A i j for j # i. Notice that Ki is independent of m,; the interface input appearing at the ith subprocess in the coupled system is independent of the control input applied to that subprocesses. We now derive the infimal cost evaluation functions Cia so that the corresponding infimal cost functions given by (8.7) are in balance with the overall cost function g(m, o)= G(m,P(m, 0)). In fact, for each i, 1 Ii I n, we define the infimal cost evaluation function Gia on Mi x Yi x M i as the functional
and show that the equality
VIII. On-Line Coordination: Coordinationfor Improvement
276 holds on M x equations
a when the function qi and hi are defined on M x a by the (8.22)
where for eachj, 1 < j
< n,
tji(m, 0 )= 1 [Ajk mk k#i
+ B j 0 + yjf -
yj].
(8.24)
Notice that the functions qi and hi so defined are independent o f mi. To show that (8.21) holds on M x ll for any i, 1 I i I n, when Gipdand the function qi and hi are defined as above, let m and o be arbitrary but fixed elements of M and ll, respectively, and for each i, 1 Ii I n, let yi
= Aim
+ B i o +yif,
z ~ ( o )= Biw
if - y i .
Now, for each j , 1 s j 5 n,
- Y j y Qj&j = (Aj m
- Yj>>
+ zj(o), Qjy(Aj m + Z j ( 0 ) ) )
= ( A i m , QjYAjm)
+
+ 2(Zj(O), Q j y A j m )
QjYZj(0)) mk
9
QjYAj,m,> -k 2 C
+ < Z j ( o > , QjYzj(W)>
k
(8.25)
3. Application to Linear Systems
277
Finally, (8.22), (8.23), and (8.28) yield
C C(Yj - Y j
9
QjAYj
j#i
= <mi9
Qi
- Bj)) + < m j
9
QjM
mj>l
mi> + (rti(m, w), mi> + M m , 0).
(8.29)
We can now conclude from (8.29) that
G(m,P(m,0 ) )= (mi
9
(QiM
+ Qi)mi>+ (Pi(mi
9
&(my
0))
- Bi QiY(Pi(mi Ki(m, w ) ) - Y i ) ) 3
9
+ (qi(m, 01, mi> + hi(m, 01,
a.
for all m in M and u in We can now apply the results of the preceding section on applicability of the interaction estimation principle and coordinabilityin the reference control approach and conclude that the system is coordinable when the infimal cost evaluation functions are given by (8.20). The estimated ranges Uiy x Wiy may be generated from given subsets M Yof M by using the mappings Ki and qi :
Uiy= {ui:ui = Ki(m, o),
for some m in M Y and some o in Q} =&(MY, a),
Biy= {pi: pi = qi(m,w),
for some m in MY and some o in Q} =qi(MY,Q).
It should be noted that, for the linear-quadradic case, the functions qi and hi can be given explicitly in terms of the system's characteristics, i.e., A,, , Q j M ,Q j y, etc. Also, there is a relationship between the functions qi given by (8.22) and the linear interaction operators based on the infimal cost given by the functionals
Gi(mi Vi) = (mi > QiMmi> + (Vi - Y i 9 9
QiYbi
- 93).
The linear total goal-interaction operator T i at (h,o)in M x ded linear operator given on M i by the equation
w is the boun-
ri(&, o)mi= 2( A; Qjy(Aj& + B j o + ~ i / j j ) , mi). j#i Now, from (8.24), we get
Ti(&, w)mi = 2(1A,*,Qjy5ji(t2,o),mi) + 2 1 ( A j * i Q j Y A j i h imi>. , j#i
j#i
278
VIII. On-Line Coordination: Coordinationfor Improvement
We can consider the infimal cost evaluation functions as derived from the functionals Gi by adding to it appropriate modification terms. The modifying terms, unlike those used in Chap. VI, consist of a linear term and a quadratic term. In view of (8.30), we can consider the quadradic term as representing the " second order " approximations of the performance interaction.
4. SEQUENTIAL ADAPTIVE COORDINATION
The coordination approach presented in Sect. 2 of this chapter can be applied quite readily to a sequential, multistage coordination procedure. At each stage of the coordination process, the system is coordinated so as to improve the overall performance. The reference control or performance standard at any given stage is, or is given by, the control input applied at the preceding stage. Success of the sequential coordination procedure depends on how the infimal problems are specified. If the infimal problems are improvement problems and the interaction estimation principle is applicable at a given stage, coordinability at the stage depends on how the estimated ranges are specified. If for each coordination input y the estimated ranges UiYx BiYare determined by a pair (ciy,d i Y ) of positive real numbers specifying a neighborhood in Ui and Biin terms of given norms, then by taking an acceptable y so that c i y and diy are comparatively large, the system can be made coordinable, but then the improvement in overall performance would probably be rather small. On the other hand, if the infimal problems are satisfaction problems requiring the infimal costs to be lower than a prescribed value k over the estimated ranges, the amount of improvement can be controlled by an appropriate selection of k. A too ambitious selection of k can prevent the system from being coordinable. At any rate, by repeated application of the same on-line coordination method, the overall performance can be improved, however slowly, until satisfactory. A sequential coordination procedure can be used as an iterative technique to reach any performance level within the systems capacity and even the optimal performance in the case when external uncertainties are absent, so that optimality can be defined. Consider now the application of a sequential coordination procedure over a sequence { t o , t,, , tf} of time instants at which the coordinator may intervene in the infimal decisions units. We shall denote by hi the control input derived by the infimal decision units and applied in the interval following the coordination time t i . At the next coordination time, t i + l , the reference control riii+' is then the restriction of hito the remaining interval [ti+l, t,].
---
4. Sequential Adaptive Coordination
279
Now, if the system is coordinable at fi'", there exist an fii+'-acceptable coordination input y such that the new control input Ai+' derived by the infimal decision units at time t i + l is y-preferred to fii+'. If the system remains coordinable over the sequence of coordination instants and there is no change in its environment, the overall performance of the system will steadily improve. We should note that the coordinator is not taking advantage of the time between coordination instants ti and ti+'. During this time it could try to reduce the uncertainties and hence improve the estimations of the internal interactions and outside disturbances. In the case where uncertainties (outside disturbances) are absent, the coordinator can use the intermediate period to improve the reference control by solving a simple problem, called the auxiliary supremalproblem. The problem is to find a control input which is preferred to the reference control input. For a linear system as described in the preceding section, the auxiliary supremal problem can be formulated as follows:
Let f i i and A' denote, respectively, as above, the reference control and the derived control input at the ith coordination stage. Let [ki,Ail be the line in M joining these two points: [fi', A'] = { m : m = Ifi'
+ (1 -I)fii,
for some I,
o I I I I}.
The auxiliary supremal problem at the ith coordination stage is then: find the real number I in the interval [0, 11 and the associated control mi = Afi' +(I - I ) h i such that
G(mi,P(mi))= min G(m, P(m)). [a',mi1
(8.31)
The simplification of the problem results, of course, from the fact that optimization is over the interval [0, I] of real numbers, rather than a space of time functions. Still further simplification can be achieved by relaxing (8.31) so that only an improvement is sought:
G(m', P(m')) < G(Ai, P(A')).
(8.32)
Regardless of whether (8.31) or (8.32) is used, the sequential coordination procedure as diagrammed in Fig. 8.1 is the following: (i) At the coordination instant ti, the coordinator specifies for the infimal decision units the reference control f i i and a coordination input y which gives the estimated ranges of the interactions or uncertainties. (ii) Each infimal decision unit seeks a control input which, in respect to the referencecontrol fii,improvesits own local performance over the estimated
280
VIII. On-Line Coordination: Coordinationfor Improvement
Coordinator I I
fair)
I
I I 0
I 0
I
-
I
Time t, lProcedures(i)ond(ii)
I
Timet,,
,
I Procedure ( i i i )
i
Time
t,,
I
I
FIG.8.1 A sequential coordination procedure for on-line application.
ranges given by y. Assume the coordination input y is %'-acceptable; therefore, there exists such a control input, say hi. (iii) The coordinator receives hi from the infimal decision units, solves the auxiliary suprema1 problem, and thereby arrives at a control input mi. The coordinator then determines the reference control %'+' = mi I [ti+l, f f ] to be used at the next coordination instant, t i + l .
If at each coordination instant ti the system is coordinable at %', the given reference control, we then have mi
3 hi
%i;
since 6ii+' = mi I [ti+l, t f ] , the performance improves monotonically. Example 8.2
Consider the two-level system given in Example 8.1 of this chapter, with the overall cost evaluation function G, G(m,u) = mTm + (Yl - FA + (uz - Fz),
and the infimal cost evaluation function C i a , Gia(m19 ~
Gia(mz
9
YZ
1 ~,
1
~2
9
9
+ ( ~ 1- F1) + Pl(m1 + ~ 1 1 , P z ) =mZ2 + 0 2 - Y J + Pzmz. P7I ) =
(8.33) (8.34)
28 1
4. Sequential Adaptive Coordination
For simplicity, assume there are no external disturbances, o = 0. We shall illustrate how the sequential coordination procedure can be applied to coordinate the system so that the overall performance is steadily improved. Suppose we are at the kth coordination stage. Assume the reference control is % = (?El, %,). The coordinator must then specify the estimated ranges U,? x a,?and U2Y x 0 2 Y for the infimal decision units so that % can be improved, provided % is not overall optimal. Let yl = y2 = 1. Then the infimal cost functions for the system are
92kS(m, Y u, 9 P i ) = mlz + (m12 + u1 - 1) + PI(% g2kS(m2
9
u2 Y P 2 )
= m22
+ (m2 u2 - 1) +
P2
m2
+ Ul),
3
found by eliminating y, and y 2 from (8.33) and (8.34), respectively. Each infimal decision unit is to find a control input which decreases its own cost relative to the given reference control over the specified estimated range. This involves finding a control input m iwhich satisfies the inequality
(8.35)
Sidmi 7 ui 9 Pi) - gia(%i ui 7 B i ) 5 0 3
over U i yx giY. For the first infimal decision unit, Eq. (8.35) is
+
2(m12- fi12) @,(m,- fil) I0, while, since q2(ml,m2) = 1, the inequality for the second infimal decision unit is (m22
- %22)
+ + l)(m, (u2
- %2) 5 0 .
Notice that the coordinator need only specify ranges for p1 and u 2 , since Pz 3 1 and u1 does not enter the problem of the first infimal decision unit. Suppose the control % = (0,O) is being applied to the process. The performance given by the overall cost function g,
g(m,, m2) = m12
+ m22 + (m12 + m2 - 1) -I-(m2m, - I)
is g(%) = - 2. This control input can be improved by the sequential coordination procedure over several stages. We will illustrate the procedure over the first two stages.
Stage 1 The control input % = (0,O) is specified by the coordinator as the reference control for this stage. In an attempt to coordinate the infimal decision units so as to improve upon %, the coordinator generates the estimated ranges
Ply= {P,: I?,(%)
+ Pll I l},
and
U 2 y= { u 2 : IK&)
+ u21 I 3}.
282
VIII. On-Line Coordination: Coordinationfor Improvement
Since 0 = ql(fi) = Kz(fi), we have BlY= [- 1, 11 and U 2 y= [-+,+I. The infimal decision units then arrive at the control inputs
rizl = 0,
and
riz2 =
-4.
-+
The control h = (Al, hz)is y-preferred to f i , and since ql(h) = is in BlY and = 0 is in Uzy, we conclude from the interaction estimation principle that riz produces a lower cost than f i . In fact, upon applying h to the process, we obtain the performance g ( h ) = -$, and
g ( h ) = -$ < - 2 = g ( f i ) . Stage 2 The coordinator, using the control inputs from the preceding stage, solves the auxiliary suprema1 problem obtaining the reference control as 6 = (0, -+), the control input applied to the process in the preceding stage. The coordinator specifies the estimated ranges
+
BIY= 181:I?l(fi)
+ 811 5 $1 = C-t, -$I,
uzy= {up: IK,(m)
+ u21 5 31 =
L-49
+I7
since ql(fi) = - and K2(fi)= 0. The infimal decision units now arrive at the control inputs
h l = + , and
h z =-12'
Again h = (4, -+) is y-preferred to f i = (0,-+), and, since the estimated ranges are correct, we conclude that the overall performance is improved still -+) to the process, we obtain further. Upon applying h = (i, g(h) =
- 147/64 < -9/4 < -2.
We should notice now that the minimum overall cost is - 16/7and the overall cost obtained differs from the minimum by 5/448. We chose to illustrate the sequential coordination procedure in a rather simple system. The procedure is the same in the case the subprocesses are described by difference or differential equations, although the computations then become quite cumbersome and difficult. The example illustrates how the sequential coordination procedure can be applied as an iterative method of optimal coordination. The advantage of such a coordination procedure is the monotonicity of the improvement in overall performance. This feature is, of course, desirable in on-line situations. We point out also that the sequential coordination procedure is quite suitable in the case where external disturbances are present, although, for simplicity, we neglected them in the example. Perhaps the major difficulty with the approach is the difficulty in generating appropriate estimated ranges and solving the infimal problems.
The present book is a monograph on a new subject: it represents the research of the authors and development of a theory ab nuouo, both in scope and content, as well as in details of mathematical theory per se. The subject is new, and it is not easy to give either a complete reference list or a historical account. If we were to confine the listings to immediate references used, the list would be too short and would hardly reflect the importance of the subject. If we were to list the references concerned in general with the hierarchies and largescale systems of the kind considered in this book, the list would be too long. We shall steer a middle course and therefore briefly mention related, especially theoretical, work in general, and then mention specificreferences in connection with the material presented in different chapters. GENERAL BACKGROUND
Our broad objective was to initiate a theory of multilevel decision-making systems, i.e., systems with organizational-type structures. Our first motivation stems from the area of automation and control of complex industrial systems. In this respect, the multivariable control theory can be viewed as an immediate predecessor of the current work; in particular, the present book is a follow-up to [29] in the natural evaluation of a long-term concern with the problem of large-scale systems control. Our second motivation comes from Organization Research. The VonNeuman-Morgenstern theory of games concerned with competitive situations and the more recent Marshak-Radner theory of teams concerned with cooperative situations, are dealing with single-level systems, while our concern is with the multilevel systems. We share, of course, the methodology with these theories :we are interested in the mathematical theory of organizational structure with all benefits and limitations entailed. 283
284
Appendix I
Last, but not least, we were motivated by the problems of control and communications, in general, whether biological, social, or man-made. In this respect, the theory is in the spirit of cybernetics in Wiener’s sense and in the spirit of general systems theory presented in [30] and [31]. CHAPTER I
Only a handful of papers or books are explicitly cited. The literature here is really extensive, and no attempt is made to review it. CHAPTER II
Conceptualization is based on the framework reported in [32] and in an earlier publication [33] (which represents the first publication of the Systems Research Center, Case Western Reserve University, and the source for much of the further research ideas). More recent accounts are given in [34] and [35]. Concepts of multilayer hierarchies are based on [21] and [36]. Much of the motivation for the study of multilevel systems in industrial automation is provided by the pioneering paper [37]. Hierarchical arrangement of the solution algorithms for complex problems can also be found in [20] and [26]. CHAPTER In
Formalization is performed in the approach to mathematical theory of general systems as reported in [30], [31], and [38]. Formalization of multiechelon hierarchy is from [39] and 1401. CHAPTERS IV AND V
Coordination principles are first reported explicitly in [41] and [42], although the idea originated in [32] and was used in the research presented in Chapters V and VI. The coordinability concepts are those from [42] and
WI. CHAPTER VI
The particular development here is based on [44] and [45]. However, the concept of interaction decoupling from [32] has been investigated in detail in a more specific simplified context of static optimization in [46] and [47].
Appendix I
285
This was further pursued in [48] and [49], and an attempt was made to generalize the approach to dynamic systems in [50] and [51] (although the basic results for dynamic systems were already reported in 1401) and partial differential equations in [52]. However, the framework of functional analysis and the concept of interaction operators introduced in [44] help resolve much earlier confusion, indicate the generality of the interaction balance principle, and show the irrelevance of such incidental similarities as, e.g., the fact that linear zero-sum modification generated by the interaction operator leads to an overall apparent performance, which is of the so-called Lagrangian form (quadratic modification can be derived just as easily as the linear one!) CHAPTER W
Development here shows clearly the applicability of the coordination principles to programming problems. No attempt is made to relate this to other decompositions, except that of Dantzig-Wolf [53], since at one stage of development, confusion was introduced on that point. It would be of interest, of course, to compare the decompositionsresulting from the coordination principles with other, particularly nonlinear-type decompositions. The interaction-estimation principle also seems worth investigating, for the integer programming problems. CHAPTER VIII
Material here is primarily based on [54]and [55]. The satisfaction approach used by decision units in the system is from [36], [56], and [57] and was motivated by early studies on the sensitivity of optimal control, [48] and
WI.
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INDEX
Additive overall objective function, 141 Additive overall objective function, 141 Additive realization, 187-191 advantages of, 188 existence of, 189-190 Apparent-overall objective function, 121, 174-175 Auxiliary suprema1 problem, 279 Balance principle, 100 Balanced constraints, 235, 236 Balanced infimal cost functions, 268, 274277 Balanced infimal performance modifications, 139 Coalition-type coordination, 60 Compensating coordination input, 153-155 Conflict in two-level system, 120 Consistency of decision problems, 97 Consistency postulate, 97 Constraints, balanced, 235, 236 Coordinability at fi, 265 notions of, 94-96, 101 265 over set saddle condition for, 133,135,152 of two-level system, 96-97
a,
sufficient conditions for, 132-1 33 Coordinating condition, 101 Coordination, 21-22, 24,29, 33 game theoretic approach to, 22,60 for improvement, 257 modes of, 59-60, 115 notions of, 59 on-line, 258 sequential, 258 Coordination inputs, 86, 117, 153-155, 264 Coordination object, 87 Coordination principles coordinability notions of, 101, 147, 156 descriptive values, 24 with feasibility check, 233 forms of, 99400,118 Coordination problem, 102 for optimizing systems, 171-177 Coordination strategies, 165-166, 168-169, 171,225-229 convergence of, 227-228 Coordinator, 87 Coordinator synthesis problem, 102 Dantzig-Wolfe decomposition, 252-255 Decision-making hierarchy, 84 Decision-making systems, 73, 77-78 classification of, 50-51
291
292 “discretion” of, 21 formal definition of, 77 hierarchies of, 84 Decision problems, 73, 76 consistency of, 97 improvement, 262 optimization, 73 satisfaction, 74, 76 Decomposition, 106 canonical forms of, 186-187 Dantzig- Wolfe, 252-255 generation of infimal performance functions by, 137 of overall performance functions, 137 of overall process, 186 of programming problems, 237-238,244245 Echelon, 50, 52, 84 Estimated ranges, 263-273 Estimation principle, 100 Functional system, 69 Goal-interaction operators forms of, 143-145 linearization of, 192-194 relationships between, 145-146 use of linearized forms, 205-212,218-225 Goal-modification, 61 Goal-seeking, 78-79 Harmony properties, 124-126 Hierarchial systems classification of, 37-56 features of, 7,13-14,35-36,5656 Image-modification,61 Improvement problem, 262 Infimal cost functions, 120 balanced, 268,274-277 Infimal decision problems communication between, 60 coordinability of, 94-96 modification of, 105 relationship to suprema1 decision unit, 57, 63 Infimal harmony restricted, 126 unrestricted, 124
Subject Index Infimal performance functions generation of, 136-137 modifications of, 139, 142, 199, 212 Interaction balance principle applicability of, 148-149, 151, 195, 235 application in welfare economy, 26-28 coordinability of, 148, 150, 152, 195, 197, 199-204, 206, 209-210, 239-240, 246-247 with nonlinear modifications, 212-215 coordination strategy for, 168, 226-228 economic intrepretation of, 28 Interaction decoupling coordination, 59 application in welfare economy, 26-28 Interaction estimation coordination, 59 Interaction estimation principle, 100, 266 270-272 applicability at 61, basis of sequential coordination, 267 coordinability at rii, 272 coordinability notions of, 101, 267 Interaction operators, see Goal-interaction operators Interaction prediction coordination, 59 Interaction prediction principle, 99 applicability relative to 7 , 161, 218,220, 242, 251 without goal-coordination, 157 application in electric power pools, 13 coordinability relative to 7, 162, 219, 221-224, 242, 251 without goal-coordination, 158-216 coordination strategy for, 171, 229 Interactions, see also Interface inputs resolution by coordination, 24 subprocess interaction function, 90, 92 treatment of, 98-99 Interface goal-interaction operator, 145, 193 Interface inputs, 89 Interface object, 89 Interlevel harmony restricted, 126 unrestricted, 124 Interlevel conflict, 120 Interlevel (performance) function, 120 Interlevel relationship, 175 Intervention, 36, 59, see also Coordination Intervention times, 57-58 Intralevel conflict, 120
Subject Index Iterative coordination, 103, 176 convergence of, 227-228 Layers, 9-10, 4648, 52 of functional hierarchy, 46-48, 53 Levels, 6, 37, see also Strata, Layers, Echelons Linearized goal-interaction operators, 192194,205-212,218-225 Linear zero-sum infimal performance modifications, 199 Load-type coordination, 59 &Acceptable coordination input, 264 Modifications infimal decision problems, 105 infimal performance, 139, 142, 199, 212 kinds of, 61 Monotonicityproperty, 122 Multiechelon systems concepts of, 49,52 electric power dispatch, 12-14 formalized description of, 84 petrochemical industry, 9-10 Multilayer systems concepts of, 44 embedded in multiechelon systems, 52 formalization of, 82 petrochemical industry, 8-10 Multilevel systems, 63-65, see also Multilayer, Multiechelon, Stratified systems approach in organization theory, 19-20 disadvantagesof, 63 reasons favoring, 64-65 steel-making, 5-6 Nonlinear infimal performance modifications, 212-215 Objective function, 73 On-line coordination, 258 Optimal coordination input, 117 expressed by linearized goal-interaction operators, 206,210-212 Optimal infimal decision, 114 Organizations, see Multiechelon systems Outcome function, 74 Overall decision problem, 94 Overall dual problem, 247
293 Overall objective (cost) function, 120 additivity of, 141 separability of, 137 Overall optimal control input, 114 Overall performance function, 74 additive realization of, 187-191 decomposition of, 137 Partial goal-interaction operator, 144, 192 Performance balance principle, 118 applicability of, 148 coordinability by, 148 coordination strategy for, 169 Postdecision intervention, 58 Predecision intervention, 57-58 . . Prediction principle, 99 relative to 7,119 Preference relation> ,263-264 Priority of action, 36, 57 Restricted iniimal harmony, 126 Restricted interlevel harmony, 126 p-Connection, 270 Saddle condition, for coordinability, 133, 135, 152 Satisfaction criterion, 74 Second-level feedback, 104-105 Separability of overall objective function, 137 of subprocess coupling functions, 198 Separable overall objective function, 137 Sequential coordination, 258 procedure for, 278-280 Strata, 37,4043, 79 Stratified systems concepts of, 37-38 formalization of, 79-81 reduction of information in, 81 Subprocess coupling functions, 89,91 separability of, 198 Subprocess interaction function, 90, 92 Suprema1 decision unit, 57, 61-63 System formal definition of, 69 objects of, 69 Tolerance function, 20, 74 Total goal-interaction operator, 144, 192
294
Subject Index
Two-level system, 85-93 conflict in, 120 coordinability of, 96-97
Unrestrictedinterlevel harmony, 124
Uncertainty set, 74 Unrestricted infimal harmony, 124
Zero-sum infimal performance modifications, 142
Value set, 74
