NORTH-HOLLAND SERIES IN

SYSTEM SCIENCE AND ENGINEERING Andrew P. Sage, Editor

WiSIIIl'J' alld Chattergy Introduction to Nonlinear Optimization: A Prohlem Solving Approach

2 Suthl'J'land Societal Systems:

3 Siljak

4 Porter et a1.

5 Boroush et a1.

6 Rouse

7 Scngupta

Methodology, Modeling, and Management

Large-Scale Dynamic Systems

A Guidebook ror Technology Assessment and Impact Analysis

Tech nology Assessmen t : Creative Futures

Systems Engineering Models or Human- Machine Interaction

Decision Models in Stochastic Programming

8 Chankong and Haimcs Mulliobjective Decision Making: Theory and Methodology

9 Jamshidi Large-Scale Sys tems: Modeling and Control

Large-Scale Systems Modeling and Control

Series Volu111e 9

Mohanuuad J atushidi Department or Electrical and Computer Eng,incc'ring, The University or New Mexico, Albuquerque

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NORTH-HOLLAND, New York' Amsterdam' Oxrord

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Jamshidi, Mohammad. Large-scale systems. (North-Holland series in system science and engineering: 9)

Includes bibliographies and indexes. I. Large scale systems. 1. Title. II. Series.

QA402.J34 1983 003 82-815 II ISBN 0-444-00706-7 AACR2

Manufactured in the United States of America

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To three who meant the most in my life

My Mother My Wife lila

My Brother Ahmad

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Problem 1.2. (continued)

d. A home heating system. e. A radar control system.

Introduction to Large-Scale Systems

1.3. The allocation of water resources in any state is commonly the responsibility of the state engineers' office which checks for overall system feasibility by overseeing municipalities and conservancy districts, which work independently and report their programs to the state engineers' office. Consider a two-muni­cipality and three-district state and draw a block diagram representing the water resources system.

References Cruz. J . B., Jf. 1978. Leader-follower strategies for multilevel systems. IEEE TrailS .

Aut. Cont . AC-23:244-255.

Dantzig, G., and Wolfe, P. 1960. Decomposition principle for linear programs. Oper. Res. 8:101-111.

Haimes, Y. Y. 1977. Hierarchical Analysis of Water Resources Systems. McGraw-Hili, New York.

Ho. Y. c., and Mitter, S. K., eds. 1976. Directions in Large-Scale Systems, pp. v-x. Plenum, New York.

Mahmoud, M. S. 1977. Multilevel systems control and applications: A survey. IEEE Trans . Sys. Man. Cyb. SMC-7:125- 143.

Mayne, D. Q. 1976. Decentralized control of large-scale systems, in Y. C. Ho and S. K. Mitter, eds., Directions in Large Scale Systems , pp. 17-23. Plenum, New York.

Mesarovic, M. D.; Macko, D.; and Takahara, Y. 1970. Theory of Hierarchical Multilevel Systems. Academic Press, New York.

Saeks, R. , and DeCarlo, R. A. 1981. Interconnected Dynamical Systems. Marcel Dekker, New York.

Sandell, N. R., Jf.; Varaiya, P. ; Athans, M.; and Safonov, M. G. 1978. Survey of decentralized control methods for large-scale systems, IEEE Trans . Aut. Cont . AC-23 : 108- 128 (special issue on large-scale systems).

Singh, M. G., 1980. Dynamical Hierarchical Control, rev. ed . North Holland, Amsterdam, The Netherlands.

Varaiya, P. 1972. Book Review of Mesarovie et aI. , Theory of hierarchical multi-level systems. IEEE Trans . Aut. Cont. AC-17:280-28I.

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Chapter 2

Large-Scale Systems Modeling: Tilne Domain

2.1 Introduction

Scientists and engineers are often confronted with the analysis, design, and synthesis of real-life problems. The first step in such studies is the develop­ment of a "mathematical model" which can be a substitute for the real problem.

In any modeling task, two often conflicting factors prevail - " simplicity" and "accuracy." On one hand, if a system model is oversimplified, presum­ably for computational effectiveness, incorrect conclusions may be drawn from it in representing an actual system. On the other hand, a highly detailed model would lead to a great deal of unnecessary complications and should a feasible solution be attainable, the extent of resulting details may become so vast that further investigations on the system behavior would become impossible with questionable practical values (Sage, 1977; Siljak, 1978). Clearly a mechanism by which a compromise can be made between a complex, more accurate model and a simple, less accurate model is needed. Such a mechanism is not a simple undertaking. The key to a valid modeling philosophy is to set forth the following outline (Brogan, 1974):

1. The purpose of the model must be clearly defined ; no single model can be appropriate for all purposes.

2. The system's boundary separating the system and the outside world must be defined.

3. A structural relationship among different system components which would best represent desired or observed effects must be defined.

4. Based on the physical structure of the model, a set of system variables of interest must be defined. If a quantity of important significance cannot be labeled, step (3) must be modified accordingly.

lntroclucuon to Large-Scale Systems

DYNAMIC SYSTEM

1/1 YI liZ yz

CONTROLLER 1 CONTROLLER 2

1 VI f "z

Figure 1.2 A two-controller decentralized system

controllers (stations), which observe only local system outputs. In other words, this approach, called decentralized control, attempts to avoid difficul­ties in data gathering, storage requirements, computer program debuggings, and geographical separation of system components. Figure 1.2 shows a two-controller decentralized system. The basic characteristic of any de­centralized system is that the transfer of information from one group of sensors or actuators to others is quite restricted. For example, in the system of Figure 1.2, only the output YI and external input VI are used to find the control U I' and likewise the control U 2 is obtained through only the output Y and external input V 2 . 2

The determination of control signals U I and U 2 based on the output signals YI and Yl, respectively, is nothing but two independent output feedback problems which can be used for stabilization or pole placement purposes. It is therefore clear that the decentralized control scheme is of feedback form, indicating that this method is very useful for large-scale linear systems. This is a clear distinction from the hierarchical control scheme, which was mainly intended to be an open-loop structure. Further discussion of decentralized control and its applications for stabilization, robust controllers, etc., will be considered in Chapters 5 and 6.

In this and the previous two sections the concept of a large-scale system and two basic hierarchical and decentralized control structures were briefly introduced. Although there is no universal definition of a large-scale system, it is commonly accepted that such systems possess the following characteris­tics (Ho and Mitter, 1976):

I. Large-scale systems are often controlled by more than one controller or decision maker involving "decentralized" computations.

2. The controllers have different but correlated "information" available to them, possible at different times.

3. Large-scale systems can also be controlled by local controllers at one level whose control actions are being coordinated at another level in a "hierarchical" (multilevel) structure.

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Large-scale systems are usually represented by imprecise" aggregate" models. Controllers may operate in a group as a "team" or in a "conflicting" manner with single- or multiple-objective or even conflicting-objective functions. Large-scale systems may be satisfactorily optimized by means of sub­optimal or near-optimum controls, sometimes termed a "satisfying" strategy.

1.4 Scope

Since the subject of large-scale systems is rather new, and since the literature is still growing, it is difficult and pointless to attempt to cite every reference. On the other hand, if we were to confine the discussion to one or two subtopics and use only immediate references, it would hardly reflect the importance of the subject. In this text an attempt is made to consider primarily modeling and control of iarge-scale systems. Other important topics, such as stability, controllability, and observability, are discussed briefly. Most of our discussions are focused on large-scale linear, continu­ous-time, stationary, and deterministic systems. However, other classes of systems, such as discrete-time, time-delay, nonlinear, and stochastic large­scale systems, are also considered. Among control strategies, the main focus has been on hierarchical (multilevel) and decentralized controls. On the modeling side, aggregation and perturbation (regular and singular) are among the primary topics discussed. Other topics such as identification and estimation as well as large-scale systems control and modeling schemes, such as the Stackelberg approach (Cruz, 1978), component connection model (Saeks and DeCarlo, 1981), multilayer and multiechelon structures, and Nash games, are either considered very briefly or have not been discussed. The emphasis has been on the use of the subject matter in the classroom for students of large-scale systems in a simple and understand­able language. Most important theorems are proved, and many easily implement able algorithms support the theory and ample numerical exam­ples demonstrate their use.

Problems

1.1. Develop a multilevel (hierarchical) structure for a business organization with a board of directors, a chairman of the board, a president, three vice presidents (marketing-sales, research, technology), etc.

1.2. Explain whether the concept of "centrality" holds for each of the following systems. State your reasons in a sentence. a. An autopilot aircraft control system. b. A three-synchronous machine power system. c. A computer system involving a host computer and five terminals.

· 2 Introduction to Large-Scale Systems

The underlying assumption for all such control and system procedures has been "centrality" (Sandell et aI., 1978), i.e., all the calculations based ~pon sy~tem. information (be it a priori or sensor information) and the Inf~r~atlOn Itself are localized at a given center, very often a geographical pOSItlOn. .

A notable chara~teristic of most large-scale systems is that centrality fails to hoi? d~e to eIt~er the lack of centralized computing capability or centrahzed InformatlOn. Needless to say, many real problems are considered large-scale by nature and not by choice. The important points regarding large-scale systems are that they often model real-life systems and that their hi.erarc~cal (mul.tilevel) and decentralized structures depict systems dealing wlth soclety, bUSIness, management, the economy, the environment, energy, data networks, power networks, space structures, transportation, aerospace, water resources, ecology, and flexible manufacturing networks, to name a few. These systems are often separated geographically, and their treatment requires consideration of not only economic costs, as is common in central­ized systems, but also such important issues as reliability of communication links, value of information, etc. It is for the decentralized and hierarchical control properties and potential applications that many researchers through~ut the world have devoted a great deal of effort to large-scale systems III recent years.

1.2 Hierarchical Structures

One of the earlier attempts in dealing with large-scale systems was to ".decomp~s~" . a given system into a number of subsystems for computa­tIOna~ efflclency and design simplification. The idea of "decomposition" was first treated theoretically in mathematical programming by Dantzig and Wol~e (1960) by treating large linear programming problems possessing speCIal structure.s. The coefficient matrices of such large linear programs often have relatlvely few nonzero elements, i.e., they are sparse matrices. There are two basic approaches for dealing with such problems: "coupled" ~nd "decoupled." The coupled approach keeps the problem's structure I~tact and takes advantage of the structure to perform efficient computa­tions. The "compact basis triangularization" and "generalized upper bound­ing" are tw~ ~uch effici~n~ methods (Ho and Mitter, 1976). The "decoupled" appr~ach dIVIdes the ongInal system into a number of subsystems involving c.ertam values of parameters. Each subsystem is solved independently for a fIxed . value of the so-called decoupling parameter, whose value is subse­quently adjusted by a coordinator in an appropriate fashion so that the subsystems resolve their problems and the solution to the original system is obtained.

Perhaps the most active group in axiomatizing the decoupled approach has been Mesarovic, Lefkowitz, and their colleagues at the Case Western

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Decentralized Control

SECOND LEVEL

FIRST LEVEL SUJ3SYS1'EM N

Figure 1.1 Schematic of a two-level hierarchical system

Reserve University, who have termed it the "multilevel" or "hierarchical" approach (Mesarovic et aI., 1970). Consider a two-level system shown in Figure 1.1. At the first level, N subsystems of the original large-scale system are shown. At the second level a coordinator receives the local solutions of the N subsystems, s;, i = 1,2, ... ,N, and then provides a new set of "interac-tion" parameters, a;, i = 1,2, ... ,N. The goal of the coordinator is to arrange the activities of the subsystems to provide a feasible solution to the overall system.

The success of the hierarchical multilevel approach has been primarily in social systems (Mayne, 1976) and water resources systems (Haimes, 1977). The multilevel structure, according to Mesarovic et a1. (1970), has five advantages: (i) the decomposition of systems with fixed designs at one level and coordination at another is often the only alternative available; (ii) systems are commonly described only on a stratified basis; (iii) available decision units have limited capabilities, hence the problem is formulated in a multilayer hierarchy of subproblems; (iv) the overall system resources are better utilized through this structure; and (v) there will be an increase in system reliability and flexibility. There has been some disagreement among system-and control specialists regarding these points. For example, Varaiya (1972) has mentioned that the first three advantages are a matter of opinion, and there is no evidence in justifying the vther two. One shortcoming of most multilevel structures is that they are inherently open-loop structures, although closed-loop structures have been proposed (Singh 1980). Detailed discussion on the hierarchical (multilevel) method will be given in Chapter 4.

1.3 Decentralized Control

Most large-scale systems are characterized by a great multiplicity of mea­sured outputs and inputs. For example, an electric power system has several control substations, each being responsible for the operation of a portion of the overall system. This situation arising in a control system design is often referred to as decentralization. The designer for such systems determines a structure for control which assigns system inputs to a given set of local

Chapter 1

Introduction to Large-Scale SystenlS

1.1 Historical Background

A great number of today's problems are brought about by present-day technology and societal and environmental processes which are highly complex, "large" in dimension, and stochast;c by nature. The notion of "large-scale" is a very subjective one in that one may ask: How large is large? There has been no accepted definition [or what constitutes a "large­scale system." Many viewpoints have been presented on this issue. One viewpoint has been that a system is considered large-scale if it can be decoupled or partitioned into a number of interconnected subsystems or "small-scale" systems for either computational or practical reaso ns (J-Io and Mitter, 1976). Another viewpoint is that a system is large-scale when its dimensions are so large that conventional techniques of modeling, analysis, control, design, and computation fail to give reasonable solutions with reasonable computational efforts. In other words, a system is large when it requires more than one controller (Mahmoud, 1977).

Since the early 1950s, when classical control theory was being established. engineers have devised several procedures, both within the classical and modern control contexts, which analyze or design a given sys tem. These procedures can be summarized as follows:

1. Modeling procedures which consist of differential equations, input- output transfer functions, and state-space formulations.

2. Behavioral procedures of systems such as controllability, observability, and stability tests, and application of such criteria as Routh-Hurwitz, Nyquist, Lyapunov's second method, etc.

3. Control procedures such as series compensation, pole placement, opti­mal control, etc.

Marshall, S. A. 1074. Continlll:d-fracti on model-reduction technique for l11ultivari­ahle sys tems. Proc. lEE 12 I : 1032.

Nagarajan , R. 197 \. Optimum reduction of large dynamic sys tem. 11l1. 1. COlltr.

14:1164- 11 74. PaciC, l-I. 1 xn. Sur la represe nl ation approachec d'une function par dcs fractions

rationelks. Annalt's S'ci('nlijiques de l" Ecole Normale S lIpiL'lIre, Ser. :1 (SuPP!. )

0: 1-'n Parthasarathy, R ., and Singh, H. 1975. Comments on linear system reduction by

continued fraction expansior about a general point. Elect Letters. II: 102.

Paynter, H. M. 1956. On an ~malogy between stochastic process and monotone dynamic sys tems, in G. Muller, ed ., Regelullgstechllik iv[oc/ern Theoriell wul illre Venvendbarkeit. R. Oldenbourg-Verlag, Munich.

Rao, S. V., and Lunba, S. S. 1974. A new frequency domain technique for the simplification of linear dynamic systems. lilt . 1. COlltr. 20:7 1-79.

Rao, A. S., L~llba, S. S., and Rao, S. V. 1978. On simplification of unstable sys tems using Routh approximation technique. I EEE TrailS. Aut. COllt. AC-23:943 - 944.

Reddy, A. S. S. R. 1976. A method for frequency domain simplification of transfer func tions. lilt. 1. COlltr. 23:403-408 .

Shamash, Y. 1974. Stable reduced-order models using Pade-type approximations. J EEl~ TrailS. Aut. COlli . AC- 19:615-617.

Shamash, Y. 1975a. Model reduction using the Routh stahili ty criterion and the Pade approximation technique. lilt . 1. COlltr. 21 :475-4l\4.

Sham ash, Y . 1975b. Linear system reduction using Pade approxim ation to allow re tention o f dominant modes Jilt. 1. COlltr. 2 1 :257- 272.

Shamash, Y. 1975e. Multivari:lble system reduction via modal methods and P ade approximation . IJ;EJ~ TrailS. Alit. COlil . AC-20:XI5 - X17.

Shich, L. S., and Guadiano, r. F. 1974. Matrix continucd fraction cxpansion and inversion by thc g,eneralizcci matrix algori thm. lilt . J. COllII' . 20:727 - 737 .

Shieh, L. S., and Goldman, M. J . 1974. A mixed Caller form for linear sys tem reduction . I EEJ~ TrailS. Sl's . Mall. Cyl!. SMC-4:584- 5HX.

Shieh, L. S., and Wei, Y. J. 1975. A mixed method for multi-vari ahle systcm reduction . J EEL' TrailS. A III. COIlt. AC-20:429-432.

Singh, V. 1979. Nonuniqueness of model reduction using the Routh approach. I EEE TrailS. A ul. COllI. AC-24:65'J- 65 1.

Sinha, N. K ., and Pille, W. 19'7 1 a. A new method for rcduction o f dynamic sys tcms. JIII..!. COlli. 14:111 -- IIH.

Sinha, N. K., ,md Berczani , (;. T. 197 1 b. Optimum approximation of high-order ;,ysteI1ls by low-order modcis. Jill . J. Contr. 14:951 - 959.

Towill, D. R. , and Mehdi, Z. 1970. Prediction of transient response sensi tivity of high order linear systems using low order models. Meas urelllelli Control 3:TI - T9.

Wall, H. S. 1948. Analytic Thcory of Con tinued Fractions. Van Nostrand, Ncw

York. Wright, D . J. 1973. The con tinued fraction representation of transfer functions and

model simpli fication. fnt . J . COlllr. 18:449-454.

Zakian, V. 1973. Simpli fication of linear time-invariant systems by moment ap­proximations. Jill. J. COlllr. 18:455-460.

Chapter 4

I-lierarchical Control of Large-Scale Systems

4.1 Introduction

The notion o f a large-scale sys tem, as it was hriefly discussed in Chapter I, may be described as a complex system composed o f a numher o f C\ln­stituents o r smaller subsystems serving particular functions and shared resources and governed by interrelated goa ls and cons traint s (M ahmoud, 1977). Although interaction among subsystems can take on many forms, ()[11!

of the most common one is hierarchical, which appears somewhat natural in economic, management , organizational , and complex industrial systcm~

sll ch as steel, o il , and paper. Within thi s hie rarchica l st ructure, the ~ llh sys­terns are positioned on levc ls with different degrees of hierarchy. A s lIh s\ls­

tem at a given level control s or "coordinates" the unit s on the I ~vel belO\~' it and is, in turn , controll cd or coordinated by the unit on the h:vel illl ­mediately above it. Figure 4.1 shows a typic;t1 hierarchical (" multil evel" ) system. The highest level coord inator, sometimes ca lled the sliprellllll coordi­

nator , can be thought o f as the hoard of directo rs of a corporati on. whih: ano ther level's coordin ators ma y he the president , vice-presidents, direc l\) J's, etc. The lower levels can be occupied by p lant managas, shop mana gers, etc., while the la rge-sca le sys tem is the co rporation itse lf. In spite of thi s seemingly natural represen ta ti on of a hiera rchical struc ture, it s exac t hehav­ior has not been well unde rstood mainly due to the fact that ve ry litlh: quantitative work has been done on these la rge-sca le sys tems ( rvhtr~h and Simon, 1958). Mesarovic e t al. (1970) presented one of the earliest formal quantitative trea tments of hierarchical (multilevel) systems. Since then, a grea t deal of work has been done in the fi eld (Schoeffler and Lasdon 1966' Beneveniste et a I. , 1976; Smi th and Sage, 1973; Geoffri on, 1970 ; Sch ~) erfler', 197 1; Pearson, 197 1; Cohen and lolland, 1976; Sandell et a I., 1978; Singh,

104

CONTROL ACTION

Hierarchical Control of Large-Scale Systems

A / '\

/ '\ / "-

/ "-/ \

/ \ \

\ \

\ \

\ \\ DECISION-MAKING UNIT

~""--,A \ PERFORMANCE

RESULTS '\

\ '\

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LARGE~CALESYSTEM

INPUTS L-_ ________ ___ _ _ _ ~ OUTPUTS

Figure 4.1 A hierarchical (multilevel) control strategy for a large-scale system.

1980). For a relatively exhaustive survey on the multilevel systems control and applications, the interested reader may see the work of Mahmoud

(1977). . In thi s section, a further interpretation and insight of the notIon of

hierarchy, the properties and types of hierarchical pr~cesses, .and s~me reasons for their existence are given. An overall evaluatIOn of hIerarchical methods is presented in Section 4.6. . .

There is no uniquely or universally accepted set of properties aSSOCiated with the hierarchical systems. However, the following are some of the key properties :

I. A hierarchical system consists of deci sion making components struc­tured in a pyramid shape (Figure 4.1).

2. The system has an overall goal which may (or may not) be in harmony with all its individual components.

3. The various levels of hierarchy in the system exchange informa tion (usually vertically) among themselves iteratively.

4. As the level of hierarchy goes up , the time horizon increases ; i.e., the lower-level components are faster than the higher-level ones.

There are three basic structures in hierarchical (multilevel) systems de­pending on the model parameters, decision variables, behavioral and en-

Introduction lOS

vironmental aspects, uncertainties, and the existence of many conflicting goals or objectives.

1. Multistrata Hierarchical Structure: In this multilevel system in which levels are called strata, lower-level subsystems are assigned more specialized descriptions and details of the large-scale complex system than the higher levels.

2. Multilayer Hierarchical Structure: This structure is a direct outcome of the complexities involved in a decision making process. The control tasks are distributed in a vertical division (Singh and Titli, 1978), as shown in Figure 4.2. For the multilayer structure shown here, regula­tion (first layer) acts as a direct control action, followed by op timiza­tion (calculation of the regulators' set points), adaptation (direct adaptation of the control law and model) and self-organization (model selection and control as a function of environmental parameters).

3. Multiechelon Hierarchical Structure: This is the most general structure of the three and consists of a number of subsys tems situated in levels such that each one, as discussed earlier, c::.n coordinate lower-level units and be coordinated by a higher-level oae. This structure, shown

Figure 4.2 A multilayer control strategy for a large-scde system.

SELF-ORGANIZATION

STRUCTURE

ADAPTATION

PARAMETERS

OPTIMIZATION

SET POINTS

REGULATION

CONTROL MEASUREMENTS

LARGE-SCALE SYSTEM

INPUTS OUTPUTS

10(' Hierarchical Control of Large-Scale Systems

in Pigure 4. L consitl ers conflicting goals and objectives between deci­sion suhproblems. The higher-level echelons, in other words, resolve the conflict goals while relaxing interactions alllong lower echelons. The distrihution of control task in contrast to the multilayer s tructure descrihed in Figure 4.2 is horizontal.

111 add iti(1n to the vertical (multilayer) and horizontal (multiechelon) division of cnntwl task, a third division called a time or fun c tional divi sion is possihle (Singh and Titli. 1978). In this divi s ion a given subsys tem's functiollal optimization prohlcm is decoll1poseJ into a finite numher of s in glc-pmallleter optimi7,ation problems at a lower level a nd results in a c(\llsidcrahlc reduc tion in computational e ffort. This seheme will be dls­cllssed in Section 4.5 In connection with the hierarchical con trol of dislTl,tc-tilllc sys tcms.

Before th e ~c()pe o f the present chapter is given, based on the above disl'lISsinl1. one can make a tentative conclusion that a successful operation (11' hierarchical systems is best described by two processes known as deco/1/­[,!lsitinll alJd ('()(;rriillatioll. 1he coordination process as applicable to most hi era rchi ca l systems is descrihed in Section 4 .2. Section 4.3 IS concerned \\ilh the opel~-loop control nr continuous-time hierarchical systems where cO(1rdination betwcen two levels a re considered. The closed-loop hierarchi­Cl i C(1 nlrol of large-scale ~vs tel1l s is di scussed in Section 4.4, including the 11l\tiOll ~ of " intnacti()n predict inn" and "s tructural perturhation ." [n Scc­ti(1n 4.5. the s tructural perturbation and interaction prediction approaches are extended t(1 take 011 the discrete-time case with and without delay. ;\ mon o t he hi erarchical con tml strategies disclissed here are a three-l eve l l:'

time-dcl;l\' coordination algorithm hy Tamura (1974, 1975) and a "cos tate c(1ordinaiion" (1r "costate prediction" scheme due to Mahmoud et al. (1977) and lIa~sa n and Singh (1976a, 1977;). A number of numerical examples illu stra te the various techniques presented. The near-optimllm design or linear and nonlinear hierarchical systems are discussed in Chapter 6. Section 4.6 is devoted to further discus-sion and the evaiuation of hierarchical COil t rol tec h niqucs.

4.2 Coordination of Hierarchical Structures

[t was mentioned in the previous section that a large-scale sys tem can be hi era rchically controlled by decomposing it into a number of subsystems and then c;)ordinating the resulting subproblcms to transform a given intergrated system into a multilevel one. This trans fonnation can be achieved by a hos t of dirferent ways. However, most of these schemes are essentially a comhination of two distinct approaches: the model-coordinatiofllllethod (or "fL'a~i hle" Illethod) and gO(l /-coorriil1(lliol1l11cl/rod (or "dual-feasible" me thod) (Mesa rovic et aI., 1(69). In the nex t two scctions, these methods are

Coordination of Hierarchical Structures 107

described for a two-subsystem s tatic optimizatic!l (nonlinear programming) problem.

4.2.1 Model Coordination Method

Consider the following static optimization problem (Schoeffler, 1971) :

minimizeJ(x, u, y)

subject to f( x, Lt, y) = 0

(4.2. I)

(4.2.2)

whcre x = vector of sys tcm (state) variables, II = vector of manipulated (control) variables, and y = vector of interaction variables between subsys­tems. Let thc problem and its objective fun ction be decomposed into two subsystems, i.e.,

and (4.2.3)

(4.2.4)

where Xi, 1/ i, a nd f arc vectors of sys tcm, m;1I1ipulated, and interac ti on variables for ith subsystem, respectively. This decomposition has produced a pcrformance function for each subsys tem . However. through the vectors .1'1, i = 1,2, the subsys tems are s till interconnec ted . The objective of tile Ill odel coordination method is to convcrt the intcgrated problem (4.2.1)- (4.2.2) into a two-level problem by fixing the interacti on va riables / and)'2 at some value, say Wi, i = 1,2, i.e.,

Constrain/=wi, i=I,2 (4.2.5)

Under this situation the problem (4.2.1 )-( 4.2.2) may be divided into the followin g two sequential problems:

First -Level Problem - Subsystem i

Find KI(w) = minl,(x i. ui• Wi) x', u'

(4.2.6 )

(4.2.7)

Second - Level Problem

minimize K( w) = KI (IV)+ K 2 (w) IV

(4.2.8)

The above minimiza tions arc to be done, respectively, over the follO\ving feasible se ts:

S;={(X i, l/):r(X i, U i,W)=O~, i = I,2

S~ = {Wi: KI

( Wi) exists}, i = \. 2

(4 .2.9)

(4.2.10)

Figure 4.3 shows a two-level slnwtllre rnr Ihf'. nl().!f' l ('()nrrlin"linn nwth,,,,l

Hierarchical Control of Large-Scale Systems

[--·~osc \I' t O~' illillli7 C 1,(11') = "I(W) + A'2(\\, )

J2---~I: --~ _x~2,"2

[I\';~I\ ' ) = '111111 i l(\I. II I.j,l) , ,/II

<;; 1I!'\ed to ,I (\ 1, /11 , H, I, 1\ 1 ) -:=- U

------- ----

r'llt\ A2(\I') = mill i l ( X2. Ill, \\,1 )

,,2, 11 2

s.ub,;d \0 1 ., /1 (\ ' , 11 _ . \\ ' . \\ ,- ) ~ ()

---------------~

Figllre 4.3 A two-levcl soluti on of a sLatic optimization problcm using model l' tltlrdin :ltitln .

In this coordination procedure the variables Wi which fix interaction va riahles ri are termed coordillatillg variables. Moreover, since certain int ernal i'~leractiol1s are fi xed hy adding a constraint to the mathematical Illode \. this procedure is called model coordination. In other words , due to the fact that all intermed iate variables x, /1, and yare present, it is alt ernati vely termed " feasible decomposition method ." Therefore, a system can nperate with these intermediate values with a near-optimal performance. The fir st-level prohlems are constructed by fixing certain interacting var­i;, h1es in the original optimization prohlem, while assigning the task of detellllining these coordinating variables to the second level.

4.2.2 (;0(// Coordination Method

('onsidn the s tatic optimi za tion problem (4.2.1) - (4.2 .2). In the goal coordi­nati o ll me thod the interac tions are literall y removed by cutting all the links ;\I11 ong the su hsys lems. Let .I ,i be the outgoing variable from the ith subsvs tClll. while it s incoming variable is dClloted by z '. Duc to the remova l p i' a,'1 lillks het\veen subsys tems. it is clcar that Vi =1= Z i. Under thi s condition. :' acts as an arhitrary '~lallipulated variable and should be chosen by the nptillli zing subsystellls lik e .Y. 11 . and y . Moreover, the optimization problem cOllsidered in the previou s section is completely decoupled into two subsys­te lll s due to the fact that thei r interactions are cut and their objcctive fUllction s were already separated. III order to make sure the individual su hprohlems yield a solution to the original problem. it is necessa ry that the illlcmclioll -halollce prillciple be sa tisfied, i.e., the independently selected y' and Zi actually become equal (Mesarnvic et aI., 1969; Schoeffler, 1971).

Here again, the procedure is to decompose the problem into a number of denlllpkd subprohlems which constitute the firs l-Ieycl prohlem. The sec­olld-Ieve l nrohlcm is to force the first -level subproblems to a ~olution for

Coordination of Hierarchical Structures 109

which the interaction-balance principle holds. Mathematically, this multi­level formulation can be set up by introducing a weighting parameter 0:

which penalizes th e performance of the system when the interactions do not balance. Hence, to the objective function (4.2.3) a penalty term is added:

1 (x, U, y, z, a) = 11 (x I, U I, y' ) + 12 ( X 2 , Ii 2 , Y 2) + a'Tv - z)

(4,2.11)

where a is a vector of weighting parameters (positive or negative) which causes any interaction unbalance (y - z) to affect the objective function. By introducing the z variables, the system's equations an.: given by

fl(XI , Ul, y', Z2) = ()

f 2(X 2, u 2, y 2, Zl) = 0

The sct of allowable system variables is defined by

(4.2.12)

(4.2.13)

(4.2 . 14)

Once the objective function (4.2.11) is minimizcd over th e sc t So it rcsu lts a function ,

K(a) = min 1( x,u,y ,z ,a) (4.2.15) .,", II .y.ZESo

After expanding the penalty term aT(y -z ) = a;()lI- z l)+Ci;e y 2 -.:;2) and considering the relations e 4.2.11 )- ( 4.2.13). the firs t-level problem is formulated as

Subsystem 1: min 1,exl, /1 1, y', Z2)+ aryl - a~'z 2

xl, u 1.rl ,z2 -

Subsystem 2:

( 4,2.16)

(4.2.17)

(4.2.1 8)

( 4.2.19)

The second-level problem is to manipulate the coordinating variable 0: III

order to derive the two-subsys tems interaction error to zero, i.e.,

min e = min (y - z) (4.2.20) a a

It is clear frol11 the second-level problem (Equation (4.2.20)) that the coordinating variablc a is manipulated until the error e approaches zcro; i.e., the interaction balance is held by manipulating the objective functions of the first-Icvel problems (4.2.16) and e4.2.18) through variable 0:: hence. the name "goal coordination." Figure 4.4 shows the two-level solution via goal coordination . Thc rcader should compare the two structurcs in Fi gures 4.3 and 4.4.

'I"

' . . - . . .,.1'" Hierarchical Control of Large-Scale Systems

Choose Cc' to achi eve interaction balance

l11inJI(x l . 1/I .y l. Z2)+ "' i y l _ ",rZ2 subject 10 fl(x l , 1/ 1, y l . z2) = 0

l11in h(x2, 1/ 2. y 2. z 1) - ", r zl +"'f / subjecl 10 f2(x 2. 1/ 2, y2. zl ) = 0

Fig~re 4.4 A two-level solution of a static optimization problem using goal coordi­nation.

It will be seen later that the coordinating variable IX can be interpreted as a vector of Lagrange multipliers and the second-level problem can be solved through well-known iterative search methods, such as the gradient, Newton's, or conjugate gradient methods.

4.3 Open-wop Hierarchical Control of Continuous-Time Systems

In this section the goal coordination formulation of multilevel systems is applied to large-scale linear continuous-time systems within the context of open-loop control. In addition to the interaction-balance approach another scheme known as the interaction prediction method is also discussed.

Let a large-scale dynamic interconnected system be represented by the following state equation:

(4.3.1)

~here x and u are n X I and m X 1 state and control vectors, respectively. It IS assumed that the system consists of N interconnected subsystems s · i = 1,2, ... ,N, and the ith subsystem's state equation is given by "

x;=/;(x;.u;.t)+g;(x , t) , x;(tJ=x;n (4 .3.2)

where x, u, x;, u; are respectively n-, m-, n ;-, and m ;-dimensional, g; (.) represents the ith subsystem interaction and

xT(t) = (xnt),xi(t), ... , x ~ (t»)

uT(t) = (unt), unt) , . .. , u~(t»)

(4.3 .3)

(4 .3.4)

The objective, in an optimal control sense, is to find control vectors u1' u2,,,,,UN such that a cost function

J=G(x(tl ))+ F 1h(x(t),u(t),t)dt 10

(4 .3.5)

Open-Loop Hierarchical Control of Continuous-Time Systems III

is minimized subject to (4.3.1) and a feasible domain

u(t) E U( x (t) , t) = {ul v ( x (t) , u, t) ~ O} (4 .3.6)

Through the assumed decomposition of system (4.3.1) into N intercon­nected subsystems (4.3.2), a similar decomposition can be assumed to hold for the cost function constraint (4.3.6) and the interaction g;( x, t) in (4.3.2),

l.e.,

J = LJ; = L { G;( x;( tl )) + F 1h;(x;(t) , z;(t) , u;( t) , t) dt } I . Ito

v(x , u, t) = LVj(Xj , uj , t) j

g;(x , t) = Lg;j (Xj , t) j

(4 .3.7)

(4.3.8)

(4.3 .9)

where z;(t) is a vector consisting of a linear (or nonlinear) combination of the states of the N subsystems. Under the above assumption of separation, the large-scale system's optimal control problem (4.3.1), (4.3.5), and (4.3.6) can be rewritten as

minimize

LJ;=L{G;(X;(tl ))+{lh;( X; (t) ,Z;(t) , U;(t), t )dt } (4 .3.10) I I ' 0

subject to

x;(t)=!;( x;(t) , u;(t) , t)+t;(Z j(t) , t) , Xj(to)= x jo, i=I ,2, .. . ,N (4 .3.11)

t;(Zj(t) , t)=Lgij( X/t) , t) , i = l ,l ,oo . ,N j

L Vj (xj , uj (t) , t ) ~ 0 j

(4.3.12)

(4.3.13)

The above problem, known as a hierarchical (multilevel) control, was demonstrated for a two-level optimization of a static problem in the previous section. The application of two-level goal-coordination to large­

scale linear systems is given next.

4.3.1 Linear System Two -Level Coordination

Consider a large-scale linear time-invariant system:

x(t)=A x(t)+Bu(t), x (O) =xo

It is assumed that (4.3.14) can be decomposed into

Xj(t) = Ajx j(t)+ Bjuj(t)+ CjZj( t ), Xj(O) = Xjo

(4.3.14)

(4 .3.15)

11 2 II icra rchica l Control of Large-Scale Systcms

and the h, X I intcraCli()ll vec tor

.V

,:, (1) = L G;; ." / - ' I

(4.3 . I 6)

is a lincar l"Olllhination of the s tates of the o ther N -- I subsystems, and G ' /

i ~ <I n ", X " , matri x (Singh. I 9~0). The original sys tem's optimal control prohlem i, reduced to the optimization of N subsystems which collectively S:I ti sfy (4 . .1 . 15) .. (4.3 . 16) while minimil.in g

+ .:,'(f),':), .::, (I)] ill} (4 .3. 17)

\\' here <l, <l IT " , X " , positive semidefinite matrices, R; and S; are Ill, X III;

and " , X ", positi ve definite matrices with

,v

11 = L II ,.

r "" I

,v

III = L Ill; . ; ~ I

.'1'

k = Lie;, le r ~ Il , (4 .3. 1~) ; = I

ThL' phvs ic:li interpretation of the last term in the integrand of (4.3.17) is diffi cu lt at this point. In fact. the introduction of this term, as will be seen Liter . i, to avo id singu lar controls. The "goal coordination" or "interaction h: li ;l ll L"l'" approach o f Mesarvic et al. (1970) as applied to the " linear­LJlladratic" prohlem by Pearson (1971) and reported by Singh (1980) is now 11IL'SL'llt cd.

In this deco mpositi on of a large interconnected linea r sys tem the COllllllon l"\)up ling factors among its N subsys tems are the" interaction" variables :, ( I). \\ hi ch. ahmg with (4.3. 15)- (4.3.16). constitute the "coupling" COI1-

st r:li llts . Thi , rormul ation has been ca lled "global" by Benveniste et al. ( 1l)7(,) :llld is dl'll\)ted hy .\.(;. The foll owi ng assumption i ~ considered to Il(lld . Th e g.1()h,d prohlem .1"(; is replaced by a family of N subprohlem s ('('nplcd tog.ether throu gh a parameter vector a=(a l.a2 , ... . a N )"1 and de­Il(lt t.' d hv S, ( (v). i = 1. 2 ..... N. In other words, the globa I system problem .I·r;

is " illlhcdtkd" int(1 a L\lnily of subsystem pn)blcllls .1", ( (.\') through an illlkddillg l'ar:\llleler IY (S<J1H1e1l et ,d .. 197R ) in such a way that for :1

,);lIlil 'III ;II' valut' p f <y '". th e slIhsys tems .\( (\" * ). i ~ 1,2, ... , N, y,jeld the d es irL~d

';(, hlli OIl 10 Sr . ' In terms of hierarchica l con trol Ilotation. this imbedding ("ll ll l.'!: !)1 is nOlhin g hut the notion of c(l() rdination . hut in math ema ti ca l j1I(1gr:lIllining pmblell1 tCl"minology, it is dellot ed as th e " master" prohlem «('eo i"fri( ln. 1970 ). Figure 4.5 shows a two-level control structure or a l:iq.!.e-s(": d<.' system. Under this strategy, e:leh loca l controller i recei vcs <Y ~ flll])1 th l' ("()()rd inator (second-level hierarchy). solves .\',«<), and trall Slll it s

Opcn-Loop Hierarchical Control of Continuous-Time Systcms

SECOND LEVEl..

Figure 4.5 Thc two-level goal-coordination structure for dynamic systems.

11 3

I: II(ST t.l'VEI..

(reports) some functi on y/ of its solution to the coordinator. The coordina­to r, in turn, evaluates the next updatcd value or IX, i.e. ,

(4.3.19)

wherc 1/ is thc fth iteration step sizc, and the upda te term iI'. as wil l he seen shortly, is commonly taken as a fun ction of "interaction error":

N

e;(a(/), / )=z,(a(/) ,t)- L G,/x,(a(/). I) ;~I

(4.3.20)

The imbedded interaction variable Zr ( . ) in (4.3.20) can be considered as part of the control variable available to controller i. in which case th e pa rameter vector a( t) scrves as a se t of "dual" variabl es or Lan grange Illul ti plier~ corresponding to in teraction equality constrain ts (4.3.16). The fund amental concept behind this approach is to convert the original sys tem's minimi za­ti on problem into an easier maximization problem whose solution can be obtained in the two-level iterative scheme discussed above.

Let us in troduce a dual function

q(a) = min {L(x. u, z, u)} (4 .3 .21 ) X. Ii, Z

su bject to (4.3.15), where the Lagrangian L ( .) is defineu by

L(x,u,z,o:) = t ~xT( /f)Q;X' (/f)+~j'r X/(f)Qr ·\(t) v ( [ , = 1 0

+ l/ ;"( f ) R ,11 , (I ) + .: ;r( f ) S,.:, ( f ) (4.3.22)

+2a;'(z,( f) - ~ Grj"Y/I))l cit) .I _. I J

where the parameter vector (\' consists of k Lagr:lI1 ge Illultipliers. In Ihis way the original eO lls trained (subsystems interac ti ons) op timization problem is

! 11 I Iicra rc liica! ( '() l1lm! or I ,argl'-Scak Systems

t' il ,111 f',n ! (n ;111 lI11CollstrOlincd one. In pliler words, the cOllstraint (4.3. 1(1) is q li .'; i'i l' 11 hy rkterllllning a set of Lagrange Illuitiplicrs (Xi' i = 1,2, ... ,k. l ! nil e, .'lIch C;J':es, \\'liell the cOll straint s are convex, Geoffrion ( 1971 a, h) and Sil1?-h (llJXO) ha ve showil that

rV(;ni mi ze q( <t} ==:0 Minill1il.cJ (4 .3.23) I' II

in<iic;lling Ihat ll1inillli l;lti(l1l or J in (4.3 .17) subject to (4,3,15) - (4,3 .16) is l'qu i\ ',llcnt to m;1'\imi7.ing the dual function q(iX) in (4,3,21) with respect to n, Tn f:ll'ilitale the solution of this problem, it is observed that for a given set or I.agrange Illultipliers a = a*, the Lagrangian (4 ,3,22) can be rewritten <IS

(4.3.24)

whi ch reveals thaI the (kcc1l1lpositioll is carried on to the Lagrangian in sllch ;1 wav that a sub-Lagrangian exists for each subsystem, Each sll hsysle1l1 \\'oliid intend til Ill inimilt' its own suh-La~ran giaJ) Li as defined by (4.3,24) suh ject to (4 ,3. 15) and usin g the Lagrange lllulti pliCrs_a* which are trented as known funet iPlls at the first level of hinarc!J.y, The result of C<J ch s\ll'h 1l1illil11i 7.a tion would allow one to determine the rl ll al rllDc!iL1l1 q(x*) ilL (~. At the second level. where the solutions of all first-level subsystems are known, the value of q( a*) would be improved by a typi cal tllleop­strain ed nptil11i za tion such ii'StTie Newton's method, the gradient method or the cpnjllg;ill' gradiellt l11c1hod , The rcason I'm a gradient-type l11c1hod is dil l' t(l th l' fact that the gradient of <f((t) is defined by

N

"7 ~(I( n)I " ,._ , .•.. ~ Z,. _ \", b. \'" " L V,r\; = <'i' i = L 2,,,, (4 ,3.25 ) ; ~ t

is l1o!hing hut the suhsvstems' interaction errors, which are known through first · kycl soilltions, and \, f defines the gradient of f with respect tn .\'. At the sl'l'(llld lc\'elthe vector H is updated as indicated by (4,3,19) and Figure 4.), If a gr<1 dient (steepes t descent) method is employed, the vector d l in (4.3 . ILJ) is simply the Ith iteration interaction error (,1(1), However, a superior techniqlle from a computational point of view is the conjugate

Open -Loop Hierarchical Control of Continuous-Timc Systems 11 5

gradient defined by

dl 'l (f)=el l t(t)+/ ' Idl(f), O ~ t ~ /J (4,3,26)

where

[I( elt 1 (I) )f'e I 1 ' ( t) cll 1+1 -"-0 __ _._----- --

Y = [ /(el)Teld( o

(4 ,3.27)

and d O = eO, Once the error vector e(t) approaches zero, the optimum hierarchical control is resulted, Below, a step-by-step computational proce­dure for the goal coordination method of hierarchical control is given ,

A 19oritlzm 4.1. Goal Coordination Method

Step 1: For each first-level subsystem, minimize each sub- Lagrangian L; using a known Lagrange multiplier 0: = 0:*, Since the subsystems arc linear, a Riccati equation formulationi' can be used here, Store solu tions,

Step 2: At the second level, a conjugate gradient iterative method similar to (4 ,3,26)- (4,3,27) is used to update 0:*(/) trajectories like (4.3,19), Once the total system interaction error i!1 normalized form

Error = ( .~ [ /{ Z; -~ Gi/X j If T {z; - .~ G,jXj } dt) //:::' ( , = \ 0 J=\ J = \

(4,3,28)

is sufficiently small, an optimum solution has been obtained for the system, Here D.t is the step size of integration,

Two examples follow that help to illustrate the goal coordination or interaction balance approach, The first example. which was first suggested by Pearson (1971) and further treated by Singh (1980), is used in a modified form, The second example represents the model of a multi-reach river pollution problem (Beck, 1974; Singh, 1975), The overall evaluation of the multilevel methods is deferred until Section 4,6, and the treatment of nonlinear multilevel systems is given in parts in Section 4,5 and Chapter 6,

'i'Readers unfamiliar with the Riceati formulation can com,ult Section 4.3.2 on Interacti,)n Predict ion Method.

11(' I1icrarchica l Control of Large-Sca le Systel1ls

-.-.. -.---- - ~~)(-)~I~:·N /\T()·I~-- ---] II ; /-/~--· ·(:~ I ···· l '-;- ---::- JF~~

·/,'-'----... 1 ,'"",~,~ ·~· ~--~'~~~IBS~: 1 :-~; ~ 17'~[s'~~SY~; I : ~~~~~-;t- ' S 'UBSYS~-~J \~ r---+ l - ~:'r 1 I ' , I :., - q - -, - ; - , : I: /, -- ---.-- I I I " 1..:.... --· -· - ---- - ---- ---' I I I L ___ _ __________ ~ I L _____ _________ ___________ _______ I Figure .t.() A hlock diagralll [or :.;ys lcm of Example 4.3. 1.

V" :lIl1Pil' .I.J . I. ( 'ollsidn:t 12Ih-(lrdcr sy.-;lc ll1 illtroduCl:d by Pearso n (lIn I) and show lI in h gure 4.() with a stale equation

\ _.

Il

Il

- .1

1\ II

Il

1\ ()

Il I)

I

I I

(\

II

1\

II

Il

Il

II

II

(\

I)

Il

1\

II

I)

I)

Il

I I I

II

(\

Il 2

tl ()

II

II

()

I)

1

1/

Il I

1 , 1

t) I I)

() ()

() - I

t) I

I) I

()

()

i\ ()

an d a ljll :ldra li c cosl flln cl ion

o

1\

. -' t)

()

1

(\

o I

() I 0 1 I ()

I - 2 I ()

() I ()

() t) I

() L. -: 1_ ()

I )

1

()

()

()

I

()

- 2

()

()

()

t) 0 0 I () 1 ()

- .. - - - - - - - - -o I

() I 0 I

() I - - -- - - - - - - - -o I

1 I 0 I

_--.3 _ 1 _ _

() I () ()

() I () (\ I () 1 _ 3 - 2 - I

(4 .3.2l1)

.I ("{ I'(t )( )I(/) I 1I'(t)!<II(/ )} dt '11

( iU .ll))

Open-Loop Hierarchical Control of Continuous-Time Systems 11 7

with

where

The sys tem out pu t veclo r is give n by

I 0 0 0 (l

0 1 0

0 I °t " v = Cx= () I 0 () o I (l ()

() I 0

0 0 0

0

0

()

I 0 0 I

o ()

(4.3.3 1)

It is desired to find a hierarcllical control strategy through the interaction balance (goa l coordination) approach.

SOLUTION: From the schematic of the system shown in Figure 4.6 (clo tted lines) and the s tate matrix in (4.3 .29), it is clcar that there are fou r third-order subsys tems coupled together through six equality (number of do tted lines in Figure 4.6) constraints given by

= [( ZI - XI I) ,(Z2 -X I) , (Z 3-X7 ),(Z4 -X5 ), (ZS- XX) ,(Z(,- X2)]

(4 .3.32)

where e;, i = J, 2, ... ,6, represents the interaction errors between the four subsys tcms. The first- Icvel subsystcm problcms were solved through a se t o f four third-order matrix Riccati equat ions

K;(t) =A;K,(t) + K,(t)A, - K;(t)V;K,(t) +Q;, K,( IO)=O

(4.3.33)

where K,(/) is a n 11 / X 11 ; positive-ddinite symmetri c Rieea ti ma tri x and v~ = B,R ;-I B/'. T he " integration-free," or " doubling," method of so lving the differen tial matri x Ricca ti equalion proposed hy Davison and Maki ( 1973) and ovcrvicwccl by Jamshidi (1980) was used on an H P-45 computcr and a BASI C source program. The subsys tcms' s tat e equations were solved by a sl :llld :JI'lI fllurl h-ortlcr Runge--Kulta method . while thc Sl'Cll IHI-le\'cl ilL'ra-

...... 1 ~ 1 . _ .. _ .•. _ .. _ .. . .. ~_ .• _l ; "._ . "" I .............. . . 1..1 '1 11) \

II X llierarchi cal Conlrnl of Large-Scale Systems

1.0

1(1

I[)

If) '1 ____ L ___ . 1 _____ .1 ___ 1._. __ . __ 1 _ _ .. _ L __________ .--.l ___ J ,1 (, 7 R l) 1 ()

("f IN.1l )(;,', II' ( ; RII 1>11 ; N J liT RATIONS

Jiil'.lJl"(· ·1.7 NOflll aii rnl intt'r;l c ti ol) \'IT\lr vs cOl1gugate g. radi ent iteratio1ls for LX;1111 -

I'k 4 .1. 1.

(:L ·\.2(,) (4 .. LJ.7 ) utili zing the cubic ~pline interpolation (H ewlett-Packard, 1979 ) t(1 cvnlllnte appwpriate nUlllerical integrals. The step size was chosen 10 he (\{ ,- n.1 as in earlier treatments of this example (Pearson, 1971: Singh, llJXO) . Th e conjugate gradient algorithm resulted in a decrease in error from 1 to about 10 ) in six iterations, as shown in Figure 4.7, which was in close agreemcnt with previously reported result s of a modified version of system (4 .J,29) bv Singh (1980). Now let us consider the second example.

F,\<lmple 4.3.2. COllsider a two-reach model of a rivcr poilu lion control pmh1em.

[ .

-- 1 .J2 II I 0 0 , -- (U2 - 1.2 I 0 0 x = ------- ~- ----- - -

(J ,90 0 I - 1.32 0 () (J .9 I -- 0 ,J2 - 1.2 lo.1 I 0 j

x+ ~_~~ _ u o I 0.1

_0 I 0

(4. J.34)

\\ hl'll' c; I\ 'h f'l'<1ch (suhsys tl'm) of the rlycr h<l s Iwo 5t,lte5 ---'\"1 is the w il een I r,1 tion of hioch ell1i c<1 l oxygCll demand (BO \)) , * and x 2 is the COIl ­

c('ntrati(lll (If di sso lved oxygell (DO) -- and its control III is the BOD of the dllucnt di sch<1rge into the river. For a quadratic cost function

(4.J.35)

with (J -- diag.{2 .4,2.4) alld N c= dia g. {2. 2}, it is desired to find an uptilllal cOlltrol whicl! lIlinimil',cs (4.J,35) subject lo (4,J.34) alld\(O) = (I I - I 1)1.

Open-Loop lIierarchieal Control of Continuous-Time Systems 11 Y

SOLUTION: The two first -level problems are identical. amI a scconu-oruer matri x Riecati equation is solved by integrating (4.3.33) using a fourth-oruer RUl1ge- Kutta method for 6.t = 0.1. The interaction error for this example reduced to about 10 - 5 in 15 iterations, as shown in Figure 4.8. The optimum BOD and DO concentrations of thc two reaches of the river are shown in Figure 4.9 .

4.3.2 Int eractio/l Prediction Method

An alternative approach in optimal control or hierarchical systcms which has both open- and closed-loop forms is the in teracti on predi cti on method hased on the initial work of Takahara (1 965), which avoids second-level gradient-type iterations. Consider a large-sca le linear interconnected sys lem which is decomposed in to N subsystems, each of which is described by

5:; (t) =A, \',(t)+B;II ;(t) +C;zJt), x; (O) ='\;0' i = l,2,. . ., N (4.3 .36)

where the interaction vector z; is

N

z; (t) = L G;r\"j (t ) ./ = 1

(4 .3.37)

The optimal control problem at the first level is to find a control [/ , (1) which

Figure 4.8 Interac tion error behavior [or river pollution system of Exalllple 4,3,2,

\ 50

o __ l __ l~ __ ~~ __ ~~~ __ ~~ __ ~~~ I 3 4 5 r. 7 x <) 10 \1 12 13 14 \)

ITERAT IONS

120

o Vl / o f= ..-c,; f­/ .:J L : / . ~ .

L·

.'. ;.:, ;: '--

1.0

()

.. I ·

6 -.(' -

x /

1.0 f

/

/

/

IIicrarchical Control of Large-Scale Systems

-- --. -- t .. - -: . ::3'::::--f~~"~=;;;-~';"':'~~--;---- TIME ( Sl' e)

2. --.. 3 4 5

X I HOI> REACII I

Xl DO RFAU I I

\") HOIl RI'I\CII 2

\" .1 Il() RI'A(,II 2

Fi[!IJr(' ,1.9 Optilnlllll !lUI) and I)U concClltrati()ns fpr thc two-reach Jl1udel (If a rin'r p!lllllli()n L'(mtrnl prohlem in Example 4J.L

s:l li s i'i cs (4,3,36) -(4,3 .37 ) while lllinil11i7jng a usual quadratic cost [ullction

Thi s prnhlel11 C:l1I he s()ln'd hy first introducing a set of Lagmnge Illulti­pliers n,(t) and costate vec tors p,(I) to augment the "interaction" equality c<l llstr,lint (4.3.37) and suhsystcm dynamic constraint (4,3 ,36) to the cost fUIll,tinn's integrand: i.c .. Ihc ith subsys!cm Ilallliltonian is defined by

(4,3.39)

Open-Loop l-uerarehicaJ Control of Continllolls-Time Systems

Then the following set of necessary conditioJ<s can be written: N

jJ i = - iJllilaxi = - Q,x i - A;p, + I: G/-ai(r) J ~ I

p, ( t j ) = 0 (i x;" ( r j ) Q i Xi ( I j ) ) I iJ x;( r j ) = Q i X, ( 1 j )

,\ ,( /) = (lll,/op, = Aixi(r)+ Billi(t) +C,::j(/). x,(O) = Xjo

0 = 8Hj8u, = Rju;{t)+ B/jJ,(/)

121

(4.3.40)

(4.3.41)

(4.3.42)

(4,3.43)

where the vectors u,( t) and Z,( t) arc no longer considered as un k nowll s at the second level. and in fact z,(t) is augmellted with nj(t) to constitute a higher-dimensional "coordination vector." which will be obtained shortly. For the purpose of solving the first-level problem, it suffices to assume (IY:-(/) : z7(/)) as known. Note that IIj(/) can be eliminated from (4.:1.4:\),

u, ( I) = - R, - 1 n /jl I ( 1 ) ( 4.3.44)

and substituted into (4.3.40)- (4.3.42) to obtain

x,(r) = A,x,(t) - SjP,(t)+Cjzj(t), xi(O) =x jn (4.3.45)

N

jJ,(r)=-Qjx,(/)- /t;p,(/)+ I: Gjj(X,(t), pj(rj)=Qjx,(lj ) (4.3.46) j ~ 1

which constitute a linear two-point bounda;'y-value (TPBV) problem <Jlld Sj ~ BjR j- 1B/- It can he seen that this TPl3V problem can be decoupled by introducing a matrix Riccati formulation. Here it is assumed that

(4.3.47)

where gj(/) is an II,-dimensional open-Ioo]) 'adjoint," or "compensation," vector. [f both sides of (4,3.47) are differentiated and iJ/r) and .\-,(/) from (4.3.46) and (4,3.45) are substituted into it, makicg repeated usc of (4.3.47) and equating coefficients of the first and zeroth powers of x i (/), the following matrix and vector differential equations result :

N

id!) =- (A,-SjKj( t))Tg,(/) -Kj (/)CjZ j( t)+ I: G/n;(r) (4,3.49) / = 1

whose final conditions K,(t j ) and g;(lf ) foliow froJll (4.3.41) and (4.3.47), I.e ..

(4.3.50)

Following this formulation, the first -lewl oplimal c(>I1lml (4.3.44) hccomcs

(4.3.5 I)

I Iicrarcilical C(lllirol of Large-Scale Svslems

\\ hich 11<ls ;\ partial feed hack (closed-h,op) tcrm and a feed forward (open­loop) terill. Two points arc made here. First. the solution of the differential IIl<1lrix Riccati equatipn which involves 11 / (11

1 + 1)/2 llonlinear scalar equa­

tions is independent of the initial state X;(O). The second point is that unlike K./(t), ,1; /(1) in (4.3.49), hv virtue of ZI(t), is dependent on x;(O). This property ,viII he used in Section 4.4 to obtain a completely closed-loop conlrol in a hierarchical structure.

The second ·· level problem is essentially updating the new coordination \ l'dor (II : (/) : ::1 (1)'- For Ihis purpose, define the additivcly separable 1.; l).', ran t'- i;1lI

I 2 /I: ( / ) /{ I II I ( /) I (l: ( r ) Z I ( r ) . L 1< ( I ) (i ll \ I ( r )

, ·- 1

I f1/(/)i - i,(t) I AI.Y,(r) I /J ,/I;(t)-lC, Z/(I)1)dr)

II\(' \ ; du \.'~ P\" IY / ( t) ;lllt! .:/ ( t) can he obt<lincd hy

\\ hich plll\·idc

() = ill ., ( . ) / iI Z I ( t ) = It, ( t ) -I (,/ f! ; ( I )

v

() .~ ill ., ( . )/ (/H/(r) = Z/(t) -- L U'I " ,{r) I · I

N

(X;(t) =- C'/p;(t), z;(t)= L Gj;.,)t) i ~ 1

(4.3 .52)

(4.3.S3)

(4 .3.54)

(4.3.SS)

The sCl·olld -bcl ('()ordinatioil procedure at the (I + I )th iteration is simply

I.

<.Y I ( I ) ]1 I 1

:: I ( r )

.. C/'I), ( r )

.v L (il;\' (I)

1 = 1

(4 .3.56)

J'll\' intl'1';lcli('n prediction method is formulated by the following algorithm.

AIRor;,fllll 4. L, Interaction Prcdiction Method for Continllolls-Time Systems

,\/cP I: Snhe N illdcpcndcn t di ffcrential Ill;) trix RilTa li equations (4.3.4X) with filial condition (4.3 .50) and store K,(t), i = 1,2" . "N.

,\rlp }. For indial «( r), z: ( t) solve the "adjoint" equation (4.3.49) with fin,11 condition 14.1')0\ FV;tiW11P ;l1lel .~I()rt~ p( t) i = J? N

Opcn-Loop llil'l'archical Control of Continuous-Timc Systems 123

Stcp 3: Solve the stale equation

.x;(t) = (AI - S;KI(t))xl(t)-Slg;(t)+C;z;(t) , .\;(0) = .\1"

(4.3.57)

and storex;(t), i = 1,2, .. . ,N.

Stcp 4: At the second level, use the results of Steps 2 and 3 and (4.3.56) to update the coordination vector

Step 5: Check for the convergence at the second level by evaluating the overall interaction error

T

e(t) = i~ll)'r{z;(t) -i~IGljx/t)} {z,(t) - J~IC;;jXJ}dt/j,1 (4.J.SX )

It must be noted that depending on the type of digital computer amI it~

operating system, subsystem calculations may be clone in parallel and that the N matrix Riccati equations at Step 1 are independent of x/(O), and hence they need to be computed once regardless of the number of seL'\llJd­level iterations ill the interaction prediction algoritbm (4.3.56). It is further noled that unlike the goal coordination methods, no ZI(t) term is needed in tbe cost function, which was intended, as is di scusscd in next section, to avoid singularities.

The interaction prediction method, originated by Takahara (196S), has been considered by many researchers who have made significant contribu­tions to it. Among them are Titli (1972), who called it the" mixed method" (Singh 1980), and Cohen et al. (1974), who have presented more refined proofs of convergence than those originally suggested. Smith and Sage (197J) have extcnded the scheme to nonlinear systems which will be considered in Chapter 6. A genuine comparison of the interaetion prediction and goal coordination methods has been considered by Singh et al. (1975) which will be discussed in Section 4.6. The following two examples illustrate the interaction prediction method.

Example 4.3.3. Consider a fourth-order system

x= [~.~ __ -_(i~l_ ~ _ ~:~~ __ -_~~S -jx+ l~'2 ~ ~- -111 (4.3.59) 0.05 0 . 15 I 1 0.05 0 I O.S o - 0.2 I - 0.25 - 1.2 0 1 0.25

124 Hierarchical Control of Large-Scale Systems

\\ilb \"(0) =, ( -- l,O. l, I.O, -- O.S)/" and a quadratic cost functioll with (j =

diag(2, 1. 1. 2), R = diag( 1. 2) and no terminal penalty. It is desired to use the interaction prediction method to find an optimal contro l for (j = I.

SOl tll JO N : The system was divided into two second-order subsystems, and s teps oUllined in Algorithm 4.2 were applied. At the first step, two indepen­d ent differcntial matrix Riccati equations were so lved hy using hoth the douhling ;dg.()rithlll of Davisoll and Maki (1973) and the standard Runge K utta methods. The clements of the Riccati matrix were fitted in by qlladratil' polynomial in the Chebyschev sense (Newhouse, 1962) for com­putational con\enience:

/\ I (I) == I 4.44 + lU21 --- 1.2612

O.(}9+0J)(17I -- 0.0271 2

A , (I) = I 2.X7 - 5 .Hllt 2.42t ·~ .- _ - 0 .1+0.161 - 0.0541 2

0 .09+0 .0071 - 0.0271 2]

0.5 -Hl.0341 --- 0.141 t 2

- 0.1 +0.1(11 .- 0.0541 1]

O.73+0.IIXI -- O.X31 2

(4 .3 .60)

;\ t the fir st level. a set of two secoJld-order adjoint equations of the form (4_3.49) and two subsystem state equations as in Step 3 of Algorithm 4.2 u ~ in g. the fourth -order Runge - Kutta method and initial values

(\" (I) cc \ - () .5] I ().5 ' (4.3.61)

\1l.75]

(\,(1 ) = 0.75 ' z2(1) = G2I X I (O) = l = ~:~~5] \\ere '(l ln'd. At the s<:cond level. the interaction vectors [0: 11 (1), (~1 2 (t).

'~II(I ) , '~I,(I)I' and [1Y .' 1(t)'(~: .1 (t), Z2 1(t) ,Z22 (l)f w<:r<: predicted using the reCllrsi\'l' rdations (4.]. 56). and at <:ach infurmation exchange it<:ration the t('t :J1 interactiull error (4 .3.5X) was eva luated for 6.1 = 0 . 1 and a cubic spline inteq)(\iator program. Th<: interaction error was reduc<:d to 3.5113456 X 10 (>

ill six it e rations. as shown in Figure 4. 10. The optimum outputs for ( ', = (i I) ~l11d control signals \vere obtained. Next, for the sake of compari­S(ln , the original system (4 .].5 9) was optimi7.ed by so lving a fourth-order

tillll'-V;Hving nwtrix Riccati equat ion by hack ward integration and so lved

fnrY, (I), i = l.2.],4: . 1 ~ (t) and lI , (t), ;=1.2. The outputs and control sign;J1s f(lr buth hierarchic;J1 and exact centralized cascs arc shown in Figure 4 . 11 , Note the relativel y close correspondence betwecn the output s for the (lriginal C\)up led and hierarchical decouplcd systems. However, as one \voldd ,'-"pccl. the two control s are different. When the decoupled the COlltwl signa ls are partially dosed loop and partially open loop,

(4.3.62)

For the overall system, a complete feedback structure results :

u(I) =- F\(I) (4.H13)

10

10" _

c:r: ~ 10- 2

c:r: w :.<: o f: '-.J ..,' ~

~ 10- 3

6:

10-- 4

\ \ \

IO- 6~--~----~ ____ ~ ____ IIL-__ -L _____ 11 ____ -L __ _ 234 6 7

ITERATIONS

125

Figure 4.10 Interaction error vs iterations for the inter;'etion prediction in Example 4.3 .3.

j"b Hlcrarclucal Control of Large-Scale Systems

(/]

w (/]

z

1.0

o

~ -- 1.0 ,-(/]

w 0:: f­:::> r:: :::> o

-2.0

- 3.0

',", ~-"~.:-- ----------

--.." -- -- "- . - .. - .. -~.- .. - .. --

. )' I

.\'1

, .\'2

- ··-Y2

__ ... _L _ .. _L ---.l .. __ ---L-_.L--_L-.L~ __ ~_L_.

o 0.1 0.2 0 ,3 0.4 0.5 0.6 0.7 0,8 0.9 1.0

TIME (a)

Figure 4.11 The optimal (centralized) and suhoptimal (interaction prediction) rc­sponscs for Example 4,3.3: (a) outputs, (0) controls.

Now let us consider the second example.

Example 4.3.4. Consider an eighth-order system

- 5 0 0 0 0.1 - 0.5 - 0.009 3 () - 2 0 0 - 0.29 0 - 0.3 0.48

- O .O~ -- 0.11 - 3.99 - 0.93 0 0.1 0 0

.X = 0 ()

0 () 1.32 - 1.39 - I - 0.4 0 0

- 0.1 - 0.4 - 0 ,2 0 () 0 x

() 0 0 0 0 - 0,17 0 0 () 0 0 0 0 0 0.5 0 0 0 0 0 0 0.01 0 - 0,11

() 0 0 0 10 0

+ 0 ()

() 4 II (4.3.64)

0 0 0 0 0 0

Open-Loop Hierarchical Control of Continuous-Time Systems 127

4

3

2

'- .- . ------ --(/)

Z o - , _ ~ '-------~ 0r====±====~====t====!====~· -===·~/==·=-,~~·~~~~.~;--~--::,~::~~ ~ -- .. ---l

~ 1 f­Z o u

2

3 /

/ 4

--___ /l~

/ .. _ . . -lIt

---. - - 112

.-.-112

5'---__ -'-1 _ _ -'---_---'-1 _ _ .~L--.J __ __'_I __ --,-I ~ _ _ -'

o 0.1 0.2 0.3 0.4 0.5 0.6 0. 7 0,8 0,9 1.0

TIME

(0)

It is desired to use interaction prediction approach to find u*.

SOLUTION: The sys tem was decoupled into two fourth-order sub­systems and (/=2, t1t=O.I, QI=Q2= i 4 , RI=R 2 =1 were chosen. The initial values of a;'(t), i = 1,2, and state x(O) were assumed to be ar(t) = (0.5 I -I ol, a2(t)=(1 0 -\ O)T, and x(O)= ( - I - 0.5 0 .5 I - I 0 .5 0.5 )T. The convergence was very rapid, as shown in Figure 4.1 2. In just four second-level itera ti ons the interaction error reduced to 2x 10 - 4 . In fact there, was excellent conver­gence for a variety of x(O) and afU).

More applications of the interaction predicti(;ll method arc given in the problems section.

4.3.3 Goal Coordination alld Singularities

When the goal coordination method was discussed earlier in (4 .3.15)-( 4.3.17), it was mentioned that the posi live dcfini te matrices S; 'were

'/ o f ·~

lJ -r: 0::

I Ii crarcllical COlltrn\ of Large-Scale Systems

L'-l r-::::: III

" -

._ 2 X l0 · ~ ~-

() . ___________________ .1 =====--- L ~---- ='" I 2 J 4

ITI'.RAT1()NS

Fi!!l1H' ·U 2 The in Icract inn eIT(lr liS i tera t inns fnr (he eig,h (·order sys telll of Exalll · "il- ,U .I\

illl1(H.llIl"l'd ill th e C(lst fun c tion (4.3 .17) to avoid singularities. To see thai thi s is ill fact the case, let us reconsid er the problem of minimi l.ing L , 111

(4.3. 24) suhj ec t to (4 .:1. 15). Lel the ith subsys [clll's Hamiltonian he

I I, (' " II" .~" p" n; ) =, 1/ 2\: ( I) Q,-'", (I)

·1 1/ 211 : (I)R ,II ,(I)-1- 1/ 2:' /(I)S, Z,(I)

N

-I IY;\ -- L 1\ ; 1(;"X, I· I),/( ;I,x, + n,lI , + (', .~, ) .i ~ 1

Opcn-Loop Hicrarchical Control of Continuous-Timc Sys tcms 129

As one or the necessa ry equations for the sol LI tion of the i th su bsystel11 problem at the first lcvel, we have

(4.3.66)

or

Z; (t) = - S; - I ( C/p; (() + (x: (t)) (4.3.67)

whcrc a s ingular solution arises if the Z/C /)S;Z; (t) term docs not appear in the cost function. Here two alternative approaches are given to avoid s ingularities at the first leve l. The following example illustrates the two approaches.

Example 4.3.5. Consider the followin g system:

;'1 = - XI + x 2 -I- /II' xl(O) = Xln

i: 2 = - X 2 + u 2 ' X 2 ( 0) = X 20 (4.3.68)

It is desired to find (u l , u2 ) such that (4.3.68) is satisfied while a q uadratic cost function

J = 1/2 t ( x ~ + xi + II f + /I n dt o

(4.3.69)

is minimized via thc goal coordination method.

SO LUTION : From (4.3.68)- (4.3.69) it is seen that the system can be decolll­posed into two firs t-order subsys tems,

5; 1 = - XI + III + Z I' .\1(0) = X IO

}; 2 = - X 2 + u 2' X 2 (0) = .\ 20

with interaction constraint

The problem in its present form has the follow ing Hamilto nian :

II = (-ixf -I- ;lI f + lYZ I - (U 2 - PI XI + PIUI + PI:'I)

-' (I . 2 -I- I 2 . I- . ) T 2·\ 2 2 II 2 - P 2 '\ 2 - P 2 11 2

(4.3.70)

(4.3.7 1)

(4.3.73)

in which the intera ct ion va riable appears linearly. Thus, th e appli cation of goal coordination fo r the present formulati on would lead to a si ngu lar problem, since 2 1 appears linea rly in (4.3.73). The followin g system reformu­lations of the problem would avoid singularities.

4. ::I.::I .a Reformulation I

Bauman (1968) suggests rewriting the interact ion constraint (4.3.72) in ljuadratic fo rm,

1 \' ) lIicrarchical Control of Larg,c-Scalt: SyS ll' IllS

\y i1ich \\'( )\lIt1 givc th e foll (lwing necessa ry conditions for optimalit v <II tl,e fir sl k n' l:

t) ~ lI lIl / n ll l ~= II, 1 PI ' () = iJ lIl / (I :' 1 = 2HZI + /)1

-, j) I O~ ill 'I / il x I co .\ , - PI ' /) I ( I ) = ()

\' , ~ iIJJI / ()p, = - XI + li l + " I ' .\1(0) = x II)

I' ll ' fir st sUh SYS tl" 1l ~ 'lld

() = illl ~ / iJlI), = 11 2 1- /) 2

- jJ2 = (J ll,/ilx2 =x2 - P2' P2(1) = 0

' \·2= il n~/ (J p2= - .\ 2-1- U" -' 2(O) =.X20

(4.3.75 )

(4.3.76)

1'(1(' tlie sl.'l'O nd suhsys tem. Arl er thc introduction of the Ricca ti formulation (4 .. 1 75 ) alld (4.3.7(,) will lead to

\ , ( 1 ) ~. - { I -I (If I / It ) k I ( 1 ) } .\' I ( 1 ) , \' I (0) = \. )()

:. 1 ( I ) = - ( ~ (x) k , ( 1 ) .\ I ( 1 )

11 , (1) = - " 1( /)\ ,(1)

.\'2 (/) = - (I +k 2(/)) X2(1), X 2 (0) = X20

ll~ ( 1) ~= -- k ~ ( 1 )\' 2 ( 1 )

(4 .3.77 )

(4 .3.n)

" 'I!nl' k, ( 1) is the ilil ~ uhsys il:l1' ~ca lar time-varyin g Ri cca ti Illatri x. The ('( l(lrd in ;lli PIl at th e sccolld level is achi eved through thc following iteratiol1s:

H' I I ( 1 ) = ttl ( 1 ) I- fill I ( e( 1 ))

<'(I ) = z~ (I) - xH /) (4.3 .79 )

(hi s rdorillul at io ll w(luld avoid sin gul ariti es hut makes the second-level it erati (l l1 s Cl1 nvcrgcnce vcry slow.

4 .. ~. ,~ . h Reformulation 2

Sin gh cl ;d , (I CI7(,) sug.gcst ,111 alternative formulati on whi ch not onl y avoids sillgll\ :,!ilics hilt a\sll gives gO(ld Cl1n vergcllcl'. The procedure is hased Oil

<;f )"'ill g tn! \' ill terms (11' the int eraction vec tor z and .~uhs titutin g it into the C(ls t fllllL'ti(ll1 : i.e .. Z G ill hc written in general as

(4 .. UO)

\\ here (; i ~ «,Slimed to be nOllsin gular and the rdormulated Hamiltonian is

11(·) = 1,,'(/)V\2I1z(I) + \1I1(/)RII(I)+ly' i:

-- (rIG.\' + p'( ;Ix -I- fill + Cz ) (4.3.RI)

I'l l! Ill c exa mple under considerati on. the matri x. C; is sin gular, but "

Open-Loop Ili cra rchical Control o f Continu ous-Tilllc SyslCllls

solution ca n still he fo und. J-Jere the Hamiltonian is

1I( . ) = i x~ + i ll ~ + az , - IXX2 - P I" ; -I- PIli , + p,", + lz;- + ~ lI i - P2X 2 + P211 2

with first-I evcl subsys tem problems bcing

0= aH /O ll l = lI, + PI' 0 = fJ H/r7:. I =" , +a + p,

I ] I

- jl , = r11f /iJx , = x, - p" p,(I) = O (4 .3.83)

and

0 = OH/O ll 2 = lI 2 + P2

-- /1 2 = a H / (h 2 = - a - P2, P 2 ( I) = 0 (4 .3.R4)

.\: 2 = () If /0 P 2 = - X 2 -I- U 2 ' X 2 ( 0) = .\ 20

The second subsys tem can be solved immediately, since the P2-costate equa tion is decouplcd from X l and can be solved backward in time and substituted into the X 2 eq uation, which would , in erkct , mcan that Lh e solution o f a Rieca ti equation has been avoided for this particular example. For the first subsys tem, however, followin g the formul a ti on of first-I cvel problems in interaction prediction (4.3.40) - (4.3.5 1), both a Riecati and an opcn-loop adjoint (compensa ti on) vee tor equati on sll ch as (4.3.48) and (4.3.49) need to be eva lua ted. For this exa mple, thc first subsys tem prob-lem is

.\ I ( 1 ) = _. ( I -I- 2k I ( 1 ) ) X 1 ( 1 ) - 2 g , ( t ) - n ( 1 ) , .Y , (0) = .\ 10

i(, (I)= - 2kl(/) + 2k ~ (t)-I, k,(I) = O

gl( /) = (I +2k l(t))g,(t)+ k,(I) n(/). gl ( l )=O (4 .3.85)

UI(I) = - kl(I)X,(t)+ g,(t)

where two differential eq uations for ",(I) and ;:;1 (1) must be solved back­ward in time. Thus, while no auxiliary equati on needs to be solvcd for the second subsys tem. two such equati ons should be solved for the first subsys­tem. In general , this reformulation would requirc thc solution of

.k= Ax -Sp - GQ - '(p +n), .,(O) =xo

jl = -- C/'a + AijJ , p(lj ) = O

II = - R - 'B Tp , z = - G Q - '( p + IX)

(4 .3 .86)

which indicates tilat thc cos tatc vector p equation is deeouplcd fr0111 .\ and ca n hc so lved backward in time (eliminating a Ri cea ti eq uation) an d subSt ituted in the lop equat ion to find x. Since the iI. }3, Q, and R matri ces ; If'[~ hlnck -di ;H>nnal nrnhIPrn (41 X6) (' ;111 11<' ti ('('( ',mnnSt,d inln N sllh<: v<: l " lll

IIierarchical Control of Large-Scale Systems

I' r(1h1clIlS. prp\'idcd that the terlll ( X' (/)V FQ V z (l» is separable in z where I ' '7 (; I,

(·1.4 ( 'Iosell -I ,001' II i<'rarchkHI Contwl of COlllillllous-Timc Systellls

I il l' 1,Ist se,: tiPIl dea lt with 0pcll -Ill(lp lli era rchica l control of continuoll s- timc , \', tCI11' ill ,,''' ich tli t.: co ntrol depended Oil the sys tem's initial cond iti nll ,I ( 1,, ). 11 11,' SI:h': IlIl' Iwn lTs t tn a closed-loop structure was th e int eraction l" ('" Ii di(l n Ill ,., th(ld whi ch n:slllted a partia ll y closed-loop s trll ctlll'e wit h all

opell -I(l(l !1 cO lll!1Pllent ",h ich still depend ed nn ,Y(lII)' Alth ough one may ,,1 " <lY S h l' ahlc to ll11:aSIIIT ,\,(1 0 ), hy the time a ll open-loop con trol is c: tl clIi alcd alld <lpplit.:d to tht.: system . the inili ,d s tat e has IllOSt likely ,·h;1I1 )2,n l. thll s resulting ill lillpredictahle and undesirahle responses, I t is Ihcn: rore \\'o rthwhile to C(lllstrul'l closed-loop control la ws which an.: ind c­IWlldent ('I' Ihe initial stat c (Singh. 19XO). The 1110st lik ely case in which such a c(l lltnli s tru cture is p(1ssihle. as in nonhierarchi cal systellls. is thc lincar quadratic regulator problem. Man y auth()rs have cons idered the prohlem of dosed-loop cOlltrol of hierarchic;d sys tems. Mesarov ic et al. (I nO) ~ u g­

gL's ted a slliluptill1al structure whi ch had an adaptive feature as the sys tem c\ ' (l l ve~ . Sa ge and hi s associates (Smith and Sage, 1973; Arafch and Sage, 11.) 74:1. il) 1I< lye used a similar suboptimal technique for the [ilter problem. 01 hers (Cil ene\'caux. 1972: Cohen et a l. . 1972, 1974) have considered the l'f(lhlcm alld sugges ted either a " partial" feedback controller or "complete" r" l'( lh;ll'k cnntrnls wh ich in volve off-lin e calculations of the overall coupled I"rge,sl',ll c sys tem. An atlr;lct ive extens ion of th e partial [eedhack a lgo rithm "I' ('oil cll l' t a l. (1 974) whi ch provides " complet e" closed-loop structure hased on n rf-line c,llculations or feedback gains and their on-l ine imp lemen­tati on i , du e to Singh (19RO). Ano ther attemp t along thi s line, due to Siljak (Jild SUlldareshan (Silj ,lk and Sundarcshan, 1974, 1976a, b; Sund areshan, 19'77: Sil,iak. 197R ). is bascd on (1htai nin g local feedback controllers for each StlhS\,s t" lll (Ill the first leve l hy igllnring the int eractions: then a glohal l'o lltr(llkr ;11 the sCl'(lnd lew l is applied tll minimi 7.e the interaction erro rs "" d i 1111'1 (l\\: the (1nrllrll1;lIl Ce. "'"" is l11 et hod elll phasi l.cs the st ru ct u 1' ,11 pe rturhations catcgnri7cd as " benefi cial." " nonbendicial," and " neutral" illlCl'c(1 I1IllT ti ons and their co rn.:spondin g loss of perrnrmance.

III Ihi s sec tinn Singh's (19XO) extension o[ th e interaction prediction ap IHl l<1 ch and th e feedhack C(1ntrol scheme of Sil jak and SUlldareshan (1974. Il17 (' a. il) alld Sundares llan (1977) hased 011 structural perturhation a long " 'ith Il\I1l1l'J'i ca l exampl es are prese nted.

4.4.1 ('/o.l' ('d - / ,(}(}fI Co//trol oia ///I emcfi() // Predictiu//

111 II1\' I)f('v i(lli s sl'l'l i(lll . till' i ntcr,ll'lioll predict i(lll ill it s gcncra I se ns,' was ;"" ·",(II '·".! ", ;Ib :I n!lrl; :1) ,·In,,..d-)nnn (' l1 lltro ll er ;lI1d an onen- Ioon C( " "I)O-

CloscJ -Loop Hierarchical Control or Continuous-Ti me Sys tems 133

mentioned ea rlier, that thc open-loop component depcnds on the initial state of the system and there is no apparent poss ibility [or on-line imple­mentation. To see this dependency of the open-loop component. let us consider the differential equation for thc adjoint vector g;(t) in (4.3.49) and li se (4.3 .55 ) to elim inate D:,(t) and Z; (I ); i. e.,

~'

. T ' k.(I) = - (A, - S;K;(l)) g;(I) - K;{I )C, L G,;Xj (l)

j = 1

,\'

L Gj;C/( Kj ( I ) X j ( I ) - gi ( I ) ) (4.4. 1) ; = 1

which indicates that the vector K; (t) depends on the states of all the o ther suhsystems and hence the initial statc x (I,,) . Thc fo ll owi ng theorcm due to Si ngh (1 \)1)0) relates the open-loop compo nent

(4.4 .2)

to the sta te x ( I) for the overall system which ean be used in a hi erarchical s tructure of a regu lator prob lem.

Theorem 4.1. Tize open-lool' adjoin 1 vec lur g(l) rlnel slale x ( I) are refuled Ihrough llze following tranl/orlllation:

g(I) = M(tj ,t)x(t) (4.4.3)

PROOF: Rewriting the adjoint equatio ns (4.4.1), the overa ll system's adjoin t eq uation becomcs

g( t) = - (A - SK( I) + CG ) T g( t) - (K( t) CG + GrClK (t )) x (I)

g(l j ) = O (4.4.4)

which can be represented in terms of its homogeneous and particular so lutions.

g ( t) = (I) I ( / , /" ) g ( 1 () ) - 1'<1) I ( t , T ) ( K ( T ) C G 1- G 'e Tf{ ( T ) X ( T ) ) d T

(0

(4.4 .5)

where ({)I(I, (0 ) is the "state transition" matrix or (11 - SK( 1) - CG)T. Note al so that K(t) is a block-diagonal matrix consi sting of subsys tems Ricca ti matrices. i.e. , K(I) = cliag{ KI(I) , ... ,K;(I), .... K ,v (I) }. However. fo r the composite sys tem,

.\:(1) = I1x(I) + BII(/) (4.4.6)

with sta lld a rd quadratic cos t

.1 = .~ . . tty/(1 )Qx(t) + 1I1( I) NU(I)] ell ( 4.4 .7)

1"llcran.:JucaJ Control of Largc-Scale Systems

the dosed-loop optimal control system is well known:

x(t) = (A - SP(t))x(t)

or ( 4.4.~)

(4.4.9)

where P( t) is the n X n-dimensional time-varying Riccati matrix for the composite system and <1>2(1, to) is the state transition matrix corresponding to the feedhack system matrix (A - SK) and S = BR - IBT. Now if we substitute x(t) from (4.4.9) in (4.4.5),

g(t)=<lll(t,tO)g(to)- 1'<I>I(t,T)(K(T)CG to

(4.4.10)

U si ng the fi nal condi tion g(t I) in (4.4.4) at t = t ( and maki ng use of the properties of the state transition matrix, (4.4.10) can be used to solve for g(to ):

g ( to) = <I> I ( I (), t I) 1'1 [<I> I ( t I' T) ( K ( T) CG to

(4.4.11)

By moving the term <I> I (to' t I) inside the integral sign and taking advantage of product property of transition matrices, (4.4.11) is rewritten

g(to) = M(tl , to)x(to) = {{'<I>I(lo, T)(K( T)CG to

(4.4.12)

or

(4.4 .13)

which gives the desired relation. Q.E.D. •

A corollary of the above theorem can be stated as follows. For the time-invariant case as II -400 , constant A, B, C, G, Q, and R matrices and time-invariant g and K, M hecomes a constant transformation matrix (Sage, 1968). This corollary is used to find an approximate feedback law for the open-loop component.

The relation (4.4.13) is a sound theoretical property, but from an imple­mentation point of view, it is not very desirable for a large-scale system because the overall system's Riccati differential equation must be solved and that defeats the original purpose of finding a control via hierarchical control. Singh (19~0) has raised the point that for the time-invariant case, M can be computed easily, since near t = 0, M is constant while x and g are

Closed-Loop Hierarchical Control of Continuous-Time Systems 135

not. Therefore, it is suggested that if the first 11 = L~~ llli values of x(t,,) and g( t k) for k = 0, 1, ... ,11 are evaluated and recorded, M can be approximately given by

M=GX - I (4.4.14) where

G = [g ( (0) : g ( I I): .. . : g ( I II) ] , X = [ x ( lo) : x ( I I) : .. , : x ( t ,J ] (4.4.15)

and the inversion of X is done off-line. Note that if a time-varying M is desirable, it is possible to solve the problem wi th fl initial condi tions, i.e .. x(to),X(tO+I)"'" and form nXn time-dependent matrices G(t) and XCt) to find M(t) for each integration step. In summary, the resulting control for the composite system can be formulated by

u = - R - I B T Kx - R - I B~!!,

= - R - IBT(K + M)x = - Fx (4.4.16)

It is noted that the above gains are all independent of x(t o), and the matrices R, B, K, and M are obtained from decentralized calculations. The following example describes the above feedback law.

Example 4.4.1. Let us reconsider the system in Example 4.3 .3 with II = 4. G 12' and G2I matrices switched. It is desired to find a feedback gain matrix F.

SOLUTION: The decomposed system Riccati matrices at t = 0 are

K =[4.440 0.093] K2=[ 2.9067 - 0.1010] (4.4.17) I 0.093 0.498' -0.1010 0.7522

The problem was simulated on an HP-9845 cOinputer using pASIC. After six iterations of interaction prediction at the second level, G and X were determined to be

[

-0.78 G=IO - I -0.73

-0.69 -0.66

-2.13 -2.1~

-2.20 - 2.16

-1 .38 - ~ .20 -1.02 -0.85

1.241 1.28 1.29 1.24

[

1.0 -0.500 -1.00 0.100 J

X= 0.77 0.59 0.45

-0.510 - 0.516 -0 .516

the partial feedback matrix M to be

- 1.08 - I.I7 -1 .28

-210.14

-0.117 0.127 0.135

(4.4.18)

(4.4.19)

r

63.35

M=GX - I = 66.72 77.32

- 221.3 - 258.4

229.97 242.26 285.52 2~9 .66

-87.40 (4420) - 82.881

-104.2 .. 65.70 - 218.4 - 86 .70

1_\ (, I licrarchi eal Conlrol or Larg,c-Scak SySll' IlIS

"nd Ihe (l\Tr,, 1\ apprnxi111'1k feedba ck gam matri x hased on hi erarchical u lnl rill l(l he

F ~ N IB/( K -I- M) = [74 .S 27.5

- 232 .1 -- 9I.R9

254.2 102.0

-- 91.6 ] - 36 .X2

(4.4 .2 1)

4.-1.l ()nsl'd - Loo{l COll frn/I.' iu Sfrucfural Pertur /}(f(ioll

TI l\' sCl'I 'IHl1l1elhod of c l() ~ed - Ioop c(ll1tml or a hierarchi ca l sys tem di scussed here is h n~,'d 0 11 th e feed hack control ~ trl1clure suggested hy Siljak and SI11HI :ll l'S \i;1I1 (\4 74. \Y 7(' ;1.h) ;1I1d ex tension s by Sl1llllaresh;\1l (19Tl ) h;lscd (' 11 <;1 111 1."1'1 1;11 j1er tlilhati llll s du e to th e inl eractions ;lIno n g suhsys te l1l s in ;J

l;l1 12.<'-- s,, ;1I <_' Sy <; lelll . Slich 1'l'rlurhatill1\S 1ll ,lY ve ry well occur d,uring lhc Pl'n:l1illll pf tl1<.' svs lelll , <llld th e h:1Sil' i1\ilial issue addn:ssl'd hy Silj,d, al1d SI111l1 ;l1 l'sh:J 11 (19 7(1:1.h) is that assul1lin g each suhsys k11l ha s its own ind e­pendent l(lcal controller. how reliabl e the overall system performance will be in th e 11I l'Sl' 1\ Ce of structmnl perturhati o ll s. /\ classical ex ample is a "power 1',,"1" ill \y hidl each pOlVer C\lllipany (suhsys tem) is re~ p()ll s ihl e for the load, In'q uell l'Y rcgulati(1ll , and power generati on within its own regiun while tIHo\! £',h li c lines it can exch:mge power with other companies (subsystems) rcslIltill tc in va riation s in l)(1wer among the companies which would affect tlie entire sys tem's (wer:"1 performance. Each suhsystcm has a feedback C( 'I11TOI ~ In; l· ture consistin g of a " loca l" component obtained Ihrough sllhs\'s tClll ca1culati(1ns :1I1c1 a "global" component which minimi zes interac­ti (11l -c lIPrs all d poss ihlv improves the (lVerall sys lcm performan ce.

( 'ol1 sid','r a large-sca le linear time-invariant syst em described by N su hsys­

tems:

.,,-

\,(I) o=;l ,\,(/) -I- IJ,l(,(I)+ L (; ,/X/ (I), x,(fo) =XjO (4.4.22) / ,- I

1'(' 1 i I . . : .. __ , N. where all matrices aTC dcfi11l'd earlier. Each suhsys tem ha s :In illl l1l<.' .] iatc gLla l (If fil1din g a " local" controller {( ,'(I) which minimizes an as>(Wi :l1 cd lju :Hlrat ic ("(lS i

.I, (I " . . 1, (/ 0), {( , ( 10)) = ~ J OY; ( .\ ; ( I )Q,", (I) + U;(f )R,llj( ()) cit (4.4. 23 ) I n

\\ hilt- s: 11i s fvi ll g (4.4.22) ", ilh the intnactiol1 111:1lrin:s (',/ se t to I nn. 11 is " SSl JlIl"" IIi :ll :dtlinll gh (,:ll' h slIhsVs tc' 111 is il1d cpcl1dcllt , they have :1 "go:d h;lII1HlI1 Y" in mini11li;.ing the oVl'1all system cost functioll

N

.I ( I () .\ ( 10 ) , {( ( I (l )) = L .I, ( I () . . \ j ( I () ) . II j ( III ) ) ; ~ I

(4.4. 24 )

Closed-Loop II ie rarchical Conlrol of Continuous,Time Sys tems 137

Furthermore, it is assumed that the system possesses a co mplete decentral­ized information structure and each subsystem pair (A j' B,) is completely controllable . For the decoupled subsystem,

x ;C/) =Aj-'dt ) + B,uj(t) , i = 1,2, ... ,N (4.4.25)

and cost (4.4.23), the decentralized optimal control is given by

uj'( /) = - R,- iBTK jx,( t) = - PjXj (t) (4.4.26)

where K, is the symmetric positive-definite solution of the al gebraic matri x Riccati equation (AMRE)

(4.4.27)

where V; = I3J?" j- i/Jr, the va lue of the corresponding optimal cost, is given by

(4.4.20)

for i = 1,2, .. . , N. Let the op timal performance function of the centralized sys tem he denoted by J °(t IJ' x (lo»' The decoupled system's cos t

N

J *( /o' X( IO)) = L J /t. (lo, Xj(IIJ)) (4.4 .29) i = i

may, in general, be greater or less than 1"(to' x(r,,), depending on the type of interconnection among subsystems. In fact, as Sundarcsha n (1977 ) points out and as we will see shor tly through the followi ng theorem, there is a useful class of interconnections for which J *( .) == 1"( '). l3efnre we introduce the theorem. the following definitions are given:

Definition 4,1, For a large-scale system consist ing of N subsys tems with overall and decoupled performances 1"(.) and J*(.) , respectively, an interacti on G={g'i}' i,j=I,2, ... ,N, is said to be "nonbeneficial" if J*(- ) < J O

( .). -

Definition 4.2. An interaction G is sa id to be "b~ncfjcial " ir J*(.) > 1"(.).

Definition 4.3, The interaction G is said to bc " neutral " if J*(.) = .f0(' ) .

Theorem 4,2, Let (Ill II X 11 block-diagollal matrix K = block-diag

{ K I' K 2 '"'' K,y}, where 1\" i = L 2,. '" N, is the positioe-defilli te srllllll etric

sO/lIl iOIl of ilh slIhs)"slClI1 A/,l'iRE (4.4.27). Th eil tli c (iDem /! S\'stCIII 'S Olitilll!ll

I ,cr/iil'l)/(/I/('c i I/ric. , J " (lll d th e dccollplei/ SI 'stelil optilll(l/ pcr/i'nJ11lI/cC .1 * (Ire

('1//((// il (ll/ri Ol/Z)" if the inleraction IIl!Ifri_\ C; = {g,,) , i, j = 1, 2, . . . , N, ('({/I he

jilctori:.cc/ liy -

G=SK (4.4.30)

II'/zere S is a skelV-,IYlIIlII ctric matrix, i .c., S = - ST.

I \~

Ilw l'I'<,,,f PI' Ihi ~; til l'p rCIll. dill' to SlInd:lresi1:11l (1977) , is g ivl' ll h e low.

"I(( lor : I.l'l II " , f1 matri,\ ,. ' he th e solution of the following /\MRE:

(A + (j) IF + 1-'( A + Ci) - FIlF + (l = 0 (4.4 .31)

wherc , . . ,. liR 1(11, " j ~ Block -diag(A I, A 2 , ... ,A N ), n = Block-di:l g(Il I,II:, .... /I\. ), J( = Block-diag(R I, J( l , .... R N ) , Q = Block-di ag(() I'(> " "" (!\) illld " ~ BI(lck-d i ag(VI,V" .... VN )' It is clear th;lt .f" ( '11' \( '0)) .-- 1/ 2 \ I ( t () ) /-'.\( II) ) i r F is positive-ddini te and sYlllmetric. I I' (; s:llislies (4.4.J()). (4.4.31) rcduccs to

(A'F + r >l- FVF +Q )+(FSK+KS 1 f') = O (4.4.32)

Si nce 1\ ,. i = I. 2. .... N. arc positivc .. definite and symmctric solutions or (,I An). tit l' n 1\ is thc symm etric positive-definite suilition of AMRE:

(4.4 .33 )

Np\\, if hl'.I < llioll ,~ (4.4 . .12.) and (4.4.33) arc comparcd, it fnlh1 ws that F c~ J(

s;l l i ,s fil'~ (4A. .'n ) ;l!ld is jl(l ~ ili vc-defillite nllci sYlllmetric, Therdorc. (4.4.3()) i Ill!, li es tilnt .1"( . ) = 1/2\ / (, (I) K x ( Ill) =, J " ( . ).

'[ he n.'Vt' rsl' is also true, If F is the symmetric positive-definit e sollition of ( '~.4.11) , Ihcll by comparin g (4.4.31) and (4,4,33), F = K onl y if KG + (i' /\. ~, () : hellce, /\(i = _. (,"/\' = ,I.;. a skew-symmetric nwtrix . If we choose ,<; -c, /\ ,\'/\, til l' desirable condition (4,4,30) follows. Q.E.D. rn

Ihi s th l'oITI11. as will hc see ll by the numerical examples. Illay be used to find :1 l' l;JsS of int cractioll matrix patterns for which the optimum overall SystC!ll C:tll hc achicved by applying decentrali/.ed controls ut (l), i.e"

N

.\·; (I) = (:1 ; - B,[';)X,(I) + L G;ix i (t) , ~- I

(4.4.34)

1'(11' i ~ l, 2.. . . . ,N. It is Iwtcd that under condition (4.4.30) the local con­tr"ller s Il"t (ll1ly optimize the overall sys tem but also stabilize it. II must be l? 111ph:t silCd that if the structural perturbation of the sys tcm is limited to (4 .4.31)), one ca n ignore the interactions Hlld solve N independent small­'l': lIc l'whlcllls which pnlVitic both optimal and stahili7.ing control. llo\\,­ncr. in gencral. (; docs not satisfy (4.4.30). and the application of u;"(1) to thc ovnall sys tcm will result in a performance

N

)(10' X(lo)) = L ),(I n, x,(to )) (4.4 .35) , ~ I

whcrc )'(/0' .\,(1 0» = J,U I)' X,(l o)' ut Uo)) for the composite system (4.4.34) is greater than or less than .1* , depenuing on whether the intcraction matrix (i is 11(lllhcndicial or hClldieial in the sensc of Definitions 4.1 and 4.2. SiIjak . :t ntl SIIIIlI:trcsl1an (llJ7(l<l .h) have dcterlllineu \)()unds on the resulting

Closed-Loop IIicrarchical Control of Continuous-Time Sys tems IJ'J

pcrfollnancc suboptimality based on the interactions and have established Illultilevel schemes fOf rninil1liz-i ng interaction errors and improving (In the performance. This is achieved by a so-called "corn:ctive" control 11;( t) in addi lion to the" local" decentralized control lIi( I), i.e.,

u;(t) = u;(t)+ u; (t) (4.4.36)

where /1; (1) is given by (4.4.26), while ur(t) is assul1leu to be

11'

ur(t) = - L HiiXi(t) ( 4.4.37) j ~ '

where Il; i is an 111 ; X n i gain matrix for the feedback signals from thel th subsystem to the ith subsystem. The application of ui(t) in (4.4.36) to (4.4.22) using (4.4.26) and (4.4.37) yields

N

,\:;(1) = (A, - B;P;)x;(t)+ L (G;I - FJJJiJ'j (t) J ~ '

(4.4.38)

for i = 1,2, ... ,N. The corrective gain matrices fI'l arc obtained throu gh

BlI = B"P (4.4.3LJ)

where P = diag{ P I ,P2 , ... ,PN)' f-! ={ f-!; ), i,.l = 1, 2, .. . ,N. and lJl' is an 11 X /JI perturbatioIl in B, Notc that matrix lJ is block-diagonaL i.e" B =

diag{BI, [J2, ... ,13N }, whilc [JP is not. By virtue of the ahove fOfmulation of a structu ral perturbation, (4.4.38) can be rcconstructed :

~\: ( t ) = ( A + G) x ( t ) - ( B + B I' ) Px ( t ) (4.4.40)

with A = diag {AI' A 2 " • . ,A N } , The closed-loop sys tem (4.4.40) is influenced through the two components, i.e., "local" BP.\ (I) and "corrector." or "global," BPpx(t). Figure 4.13 shows a schematic for the proposed multi­level control method. The local-global feedback control system (4.4.40) can be utilized to give an estimate on the performance index; i,e .. once the perturba tion matrix BI' is determined, a bound on performance J rcsults. The following theorem, uue to Sundarcshan (1977), provides the mechanism to achieve this.

Thcorcm 4.3, For a nOllsingular (A + G) lI1atri.\, let a SkC1F-,ITlI1l11clric /J/(/Irix S hc thc solution o/Ihc /ollowing lIIalrix LyaplillmHype equation:

(4AAI)

K = (S - KG)(A +G ) - I ( 4.4.42.)

bc such Ihat (K + K) is a positive-de/inile matrix. Then all inpul /Ilalrix perlllrhll/io/J J]I' gillen hI'

(4A.43)

14() I Iierarchical Conlrol of Large-Scale Systelll ~

LOCAL CON! HOI..LI :({ I LOCAL CONTROLLI'.R N

Fil!Il!'p <I.U 1\ closed -loop Illultil evel s ta lc regulation sl rul'lure.

1' l'Ol'ir/cs (/ l,cr(orl1/(Tllcc il1 (/C.\

• . _ I . 1' . '. . .1(10.\(10)) - :;.\ (to )( K + k. ) .\(to) (4.4.44 )

fill' fh e III 'C/'{/1/ .ITs/ell l (4.4.40) . fl.foreovcr. lire /lpper hO/lnd 01/ lire {Ie,!or­/11,/1/(',' r/n ' i(/Iion

fl = (.i -- .1*) j.I' (4.4.45)

il ,l! /J ' CI/ 11.\' I' ~ AM(f.. l/ A,,,(K) , \I'hl'/'(, A",( ' ) I/I/d AM (') represent Ihe

I1lil/IIl1ll111 (Jlld l1/(/xil1ll1l1l oj lire ei,e.cl/llll/ues (1/ tlr eir associated lIIatri.\

argll/)/cllls. rcsJle(' til ~e{r .

PROO! ': Consider the followin g perturbed system:

.\( I) = ( .1 + (;)X(/) + (fl + IJI')I/(I) (4.4.46 )

:Ind kt /I X II posi tive-definite and symllletric lll ;ltrix F he the solution of i\ 1\1 R 1 : ..

( / f l (i) /r + r( II I- G) - F/l I' 1-' I Q = () ( 4.4.47)

whne 1'1' ~.C ( If -I- n l' ) N I( H + 13 " )r. Clearly, the closed -loop control l/( 1)

~- .. - R I( n -I n l' )'j.~ \· (t) results in a minimulll cost iXJ(/o)Fx(to) for the fn'dhad c(l iltrol sys te!n

i ( I ) ~c ( A + (i) \' ( I ) .. ~ V I' h ( I ) (4.4.4 X)

Ilowl'\'e r. since the des ired closed-loop sys tem is of the form (4.4.40) , it is seen that for (4.4.4~) to take on that form the followin g relation SllOUld hold :

R I ( 11 + 1J ,, ) Jp = I' = R In/K (4.4.49)

Closed-Loop I J icrarchical Control of ContinLioLis-Time SystcIlIs 141

Since K is a positive-definite and symmetric solution of (4.4. 33), (4.4.49) can be used twice to rewri te (4.4.47):

(4.4.50)

Thc relatioll (4.4 .50 ) can hc rcwrittcn as

KG + (F-K)(A +G ) = S (4.4.51)

wherc S is a skcw-symmctric matrix which sa ti sfics (4 .4.41), sincc (F - K) = (S - KG)( II + G) . I is a symmetric matrix. Now if we let K = ( p - K), then. throu gh direct substitution. it can be seen that /J I' defined ill (4.4.43) in fact satisfies (4.4.49); therefore. thc minimum cost is J(tn. x(tn)) =

1/2xJ(I O )Fx(/o) = 1/2 x T(/o)(K+ K) x (to). which is the des ired cost

(4.4.44 ) for the overall composite system (4.4.40). Moreover. it follows that the suboptimality index p satisfies (4.4.45) and is given by

p ~ AM (K) / AII1

(K)

for al1 to and x (t o) . Q.E.D.

(4.4.52)

An immediate corollary of thi s theorem is that for a perturbation matrix /]" given by (4.4.43), the control law

u(t) = - (p + H )x(tl (4.4.53)

is a stabilizing control for the original large-scale system. For tb e proof of thi s corollary. see Probl em 4.5.

This combined global-local controller extends the earlier work of Siljak and Sundareshan (1974) in the sense that (4.4.53) t,lkes advantage of possible beneficial aspects of interconnection. The following examplcs il­lus trate the structural perturbation method . [7urther discu ssion on the method will be given in Section 4.6.

Example 4.4.2. Consid cr a simple second-order sys tem,

'\:1= 2x 1 +1I 1, x l(O) = i

x2 = 4X 2 + 11 2 , .\' 2(0) = 0.5

with a 2x2 interacti on matri x

(4.4.54)

(4.4.55)

whose elements arc kcpt variable for illustrative purposes. For a quadratic cos t function

J = ± to ( x ~ (t) + x ~ ( ( ) + u ~ ( I ) + u H t ) ) cll o

(4.4.56)

we would like to investigate th c effects of various interconnections.

"/' ,

Hierarchical Control of Large-Scale Systems

SOLUTION: To begin with, assume that the system is completely decoupled, i.e., G = 0, 'and hence the solutions of two independent scalar AMRE's provide KI = 4.24 and K2 = S.123 and optimal local controllers

[ur(t)] = [-4.24 ui(t) 0

( 4.4 .57)

with optimal decoupled performance index

2 I J* = E 1;* = 2{K l x?(0)+ K2X~(0)} = 3.1353

;=1 (4.4.58)

Next we consider the case of "neutral" interaction in which J= J*. Using K = diag{4.24,S.123} and an arbitrary skew-symmetric matrix

S = [_Ob bO]

(4.4.30) implies that

gil = g22 = 0, gl2 = -1.915g21

Thus, for any interaction matrix,

G = [~ - 6·915b ]

(4.4.59)

(4.4.60)

(4.4 .6 1)

where b is a nonzero constant, the performance index of the overall system does not improve any more. To see this we let b = 1 and solve a second-order AMRE for matrices,

(A+G)=[i -~.915], B=[~], Q = ~2 ' R=I2 (4.4.62)

with a solution

K O = [4.23671007 0.00143700

0.00143700] 8.12243780

(4.4.63)

resulting in a performance J=txT(O)KOx(O) = 3.13438, which corresponds to the value of J* in (4.4.58) after rounding.

Next let us consider a case of "beneficial" interaction by assuming an interaction matrix

G = [5.5 5.5

6 .2] 12

(4.4.64)

Using G, A, and K = diag{4.24, 8.123), the linear-matrix equation (4.4.41) is solved for S,

[ 0 00.62044]

S= -0.62044 (4.4.65)

and the corresponding K, K + K matrices follow from (4.4.42):

K=[=;:i =~:~~~], K+K=[_~:~~ -~:~~~] (4.4 .66)

Closed-Loop Hierarchical Control of Continuous-Time Systems 143

which are negative- and positive-definite, respectively. The value of J is J = 1X T(O)( K + k )x(O) = 0.4409 (4.4.67)

which is much smaller than J* in (4.4.58). The suboptimality index for this case is p = - 0.85933, with upper bound p ~ - 1.334. The control for this case is

, u(t) =u*(t)+u C(t) = u*(t)-BPPx

_ [ - 4.24 0 ] [x I (t) ] + [ - 12.34 o -8.123 x 2 (t) -12.64

- 11.88 ] [ x I ( t) ] -23 .82 x 2 (t)

( 4.4.68)

whose first part is local (decentralized) control component and the second part is the corrective control component.

The remaining possible structural perturbation to be discussed is "non be­neficial." Let us take an interaction matrix (or structural perturbation)

G=[_~ -6] (4.4.69)

Using this perturbation, the skew-symmetric matrix S from (4.4.41) becomes

S = [0 -4.64] (4.4.70) 4 .64 0

and the resulting K and K matrices become

K- [-0.1337 -0.1337] K+K= [ 4 .106 - 13.92 3.48' 13.92

-0.1337] 11.604

(4.4.71)

with performance index j = txT(O)(K + K)x(O) = 6.95. This value as com­pared with J* = 3.1353 indicates that (4.4.69) is a nonbeneficial perturba­tion. The performance suboptimality index p = i.22 and an upper bound p ~ 0.674.

The AMREs were solved by Newton's iterative formulation (Kleinman, 1968), while the Lyapunov-type equations were solved by an infinite series method given by Smith (1971). For a survey on the solutions of both equations, see Jamshidi (1980). Now consider a second example.

Example 4.4.3. Tllis example deals with a two-reach segment of pollution problem considered in Example 4.3.2. Consider

l-1.32 0 0 0 j l 0.1

x= -~:~2 -6.2 -~.32 ~ x+ ~ ' o 0.9 -0.32 -1.2 0

a flver

where each reach of the river is represented by two state variables, BOD (biochemical oxygen demand) and DO (dissolved oxygen). The remaining matrices were chosen as Q = diag( 1,2, 1,2) and R = 12 •

I

" i '

lIicrarc\}i Gil C(1ntrnl of Largc-Scale Systcms

SOII I IION: t Inlier dCC\ lup\cd c\lllditi(lIlS, the systeIll can he cOIl .,idcred as

(\\'t) (lll e·· reach subsyQc\1ls:

. [-.1 .. '\2 x, = , - . O .. ll.

(4.4 .73 )

\\ ith V, - ..ti :lg( 1. 2 ) and /~, "- I. For illter:lction 1l1<ltrix Ci '-7 0, the il1<.kpcll ­den t 111 :lt Ii X R in:a ti equa t iOlls lead to the rollowing solutions:

1\ _,_ /\ C7 1 ().40JX --- 0.10)(1 ,1 (4 .4.74) 1 l l - O.IOS() o)trn

:111\1 ,,kn:l1traii /C(i cOlltrollers

IIi' ( I )

,,;(1)

() 11 -\ I ( I ) 1 -- O.OI056\ 2(t)

with .I" -- tl .5 tl(, 5 rc.)r\,(O) · - ( 1

(4.4 .7:1 )

0 .) )', i 7= 1, 2. The interaction matrix ror

.. ----.--.-----.- -.-.--.---~ -----.. - ... --.--.-.------------ - ----------------

Fi~ lI1'" ,1.14 linn' 1'C 'p(1n:<cs [pr Ih c slrllclmal pl'IllII'bati(l1l in Example 4.4.3 show­in?, n:wl qllil1l1l!ll and appnl"\illlalc "a" "e" s"lulions: (a) states, (b) outputs, and ((' ) C(l1> I rtlls .

11

I I

.. L. _____ _ L . ________ ._1 _ ___ .. __ .L _______ ._ .. 1., ______ _ L _ __ ___ _ L . ______ 1 _____ J _ __ J 0:; I I.) :'.'i.l .I.'i 4 4.5

TIM!' Isec)

(a)

t o

14

1.3

12 --

.5 -

.4

.. 1 --

.1

o -. 1 _ ....J, __ --'-_

o .5

UO

\ I

\

. 1 -

If\

.(1 2:' .

\ \ \ \ \ \ \

" ___ 1' 1

145

. ..j.I __ --'-__ .....l... _ ___ L _ _ ~._.J, ___ LI __ -'

1.5 2 2.5 3.5 4 4.5

Tl~lE (sec )

Ih)

\ \ \ \

-;I~-----\-----

" ----'-.. ------

'-- --............ _------ .05 L......J. __ ......J __ -----'L... __ L.. __ ~ ___ ~.L, __ _' ___ ......J. __ _.l

o .5 1.5 2.5

TIME (sec)

(e)

3.5 4 4.5

I tl ,

(; =

()

()

0 .9 ()

ll irra rr h il: a l l"lJl lm ln i' 1.argc· Sc:1i r SY~ l r l ll ~

o o o 0.9

()

o ()

o

1.11 ()

(l

() -

(4.4 .76)

II I,i( h IIl av he ll!.' lI tr :. lI . h('ndici:lI , or ll o llhclld ici:lI . III o rder t(l rind out \I hi "h ,': 1."'.' it i ~ , we need 10 lISC Th eorem 4,2 to factor (, as in (4.4.]0). If wc I :,k e ( ; ddincd hy (4.4.7(, ) ;1Ild I( = Block -di ag(K I • !<.~ ), thCll the cn rrc-

~: I'(lndi ll ~ ,\ 111 :1 lri :\ hcco ll1 cs () 0 () ()

s ~ () 0 () ()

(4.4 .n ) 2.] (1.3 () ()

() .. \ 1.1 f) 0

Iy hi l'h j " 11 (11 :1 SkCI\ ··sv tlllllelri c l11:ltri x. il11pl yin g lha l the inter:1cli o 11 nla tri x. (4.4 .7(,) i, !lll i ncutral. In llnkr to fill d out wheth er (, in (4.4.76 ) is beneficial Ill' 1](1 ll h ' lldicia l. Th e(1 rCl ll 4.3 and Equali on (4.4.4 1) a re used 10 fin d ,)' :

:ll1 d Ih ': 11 hv virtue o f (4 .'1 .42).

f) . l m

.- 0 .()9~ () I X I

·- o.on 0 .5 I

.- () .2 ().I g I

- o.on

() -- (J .17 ()

(I .023X (UR

.- o. on x ()

- OJ lJ 7

- (J .042 0. 142 O.2]R - () .059

- (J . I22 (J.(}07 0 .30X 0.028

-- 0 .147 0 .142 1.07 - 0.059

- 0 .122 () A I I -- 0.077 O. 30X

(l .067 '1 -- lUX O.ll ] 7 o

-- 0 .055 0.3 17

- 0 .03 0

(J055 1 0.3 17

.. - 0.1 36 () .1\]3

(4.4. 7X )

(4.4.79 )

; 111<1 11 )1' \: tllI e of .l -~ 11.7 19J7 fm\(O) ·c (I 0. 5 (l .5 )' , whi ch indi -

(' :lt e" Ih :lI th e ~VS I l' lll 's (l ri gin al int erac tion l11:1tri :\ is nonhell eri ci:ti . Th e ~! IIlh : tI (f.'l lrt et'l llr) controller is givell by

fI ' (I) - n I ' f 'y = .... I (l ,I

- 3.55 1.77

-- -- - --- 15.2.

1.32

0 .93 I

- 4.(,:1 1

._ .. __ 1

-· 4 I

3.45 1

- 12.4 3.25 I n.R_. _ -::..0 :_2 __ ( ... ~·. I ._ ] 4.2 -- I . I \ .~

- 24. 2 tl .6] (4 .4.XO)

\\ ith ; - () .7 19.l7, the perf(l rlllall Ce slihop tillwlil y index f> = O.2.l)(i (J . and "",,,, ,· 1, "'11 1< 1 " r Il <. I ·n . Th e above nerfortll ance ind ex IV" S also checked hv

ll icra rchical Con trol of Diserete-Timc Systcms 147

th e ori gin,,1 sys tem's cost functi on by solvin g a fourth-ord er J\ MR E, which provid ed the overall optimal sys tem performancc index as J " = 0.7132, indicating that the proposed decentrali zed-global (corrcc tor) control pcrfor­Illance index is within less than 1 % of the overall centralized optimum cost.

To check the ori ginal system's optimal control solution versus the com­bined decentralized global solution, the fourth-ord er system (4.4.72) was solved llsing

and

U,, = II * + lI c

wi th x (O) = (1 ,0.5 , 1,0.5)T and output matrix

c = [6 0 .25 a

()

1

(4 .4.81)

(4 .4 .82)

() ] 0.25

(4.4.83 )

The rcsulting x ;(t), i = I, 2.3,4; yJ(t ), for j = 1, 2 applying exact optimal control (4.4. XI) and approximate oplimal cont rol (4.4.82) responses for o ~ t ~ 5.0 and a step size 61 = 0.1 were found to be very close. T he slates and outputs of the fourth-order sys tem utili zin g two controls are und is­tinguishah ly close. Figure 4.14 indicat es th at the structural pertu rba tion closed-l oop contro l considered here is a good app roximal ion . Fu rther commcnts on this suboptimality are given in Sccl ion 6.5.

4.5 Hierarchical Control of Discrete-Time Systems

Thus far, the multilevel hierarchical control has been used for linear stationary continuous- time sys tems. In practi ce, however, ma ny sys tems are ncith er linear nor continuous-t ime. In fact, many pract ica l enginee ring systems, such as traffi c contro l, wa ter rcsources, and manufac turing processes. are bo th discrete and delayed in naturc. Du c to the importance of such app li ca lions, many researchers have made signi ficant contributions in appl yin g or ex tending hi erarchical co ntrol to both l1 0nd elay discre te-time and time-delay di screte-t ime sys tcms. In th is secti on the ex tensions of hi erarchica l control to such sys tems in both linear and nonlin ear forms a rc considered.

4.5.1 Three- Level Coordination for Discretc -Time !:>)'sfcli/s

Thi s sec li on deals with a three-l evel goal-coord inat ion st ra tegy suitable for di screte-time sys tems and their time-delay modifi cat ions. Such a s tra tc!!.v was first proposed by Tamura (1974, 1975 ) and trea tec! fur ther by S in~ll ( 1 l)~O ) .

148 Hierarchical Control of Large-Scale Sys tems

Consider a large-scale linear discrete-time system

x(k+l) = Ax(k)+Bu(k) , x(o)=xo (4.5.1)

where the usual definitions hold for x , u, A , and B. The optimal control problem is to find a sequence of discrete-time control vectors u( k) , k =

0, 1,2, ... ,K -I which minimizes a quadratic cost function

I I K - \

J = "2 xT(K)Q(K )x(K) +"2 L {xT(k )Q(k )x(k) + uT(k) R (k )u(k)} k=O

(4.5.2)

while (4.5.1) holds. Following the decomposition procedure discussed in Section 4.3, this problem can be reformulated by minimizing

N{l lK - l J= ;~l "2 x;"(K)Q;(K)x;(K)+"2 k~O[X;(k)Q;(k)X;(k)

+ z;(k)S;(k)z;(k)+ U;(k)R ;(k)U;(k)]} (4 .5.3)

while subsystem dynamic state equations

x;(k + 1) = A;x;(k) + B;u;(k) + C;z;(k), x;(o) = X;o ,

i=I,2 , . . . ,N, k=0, 1,2, . . . ,K-I

and interaction relations N

(4 .5.4)

z;(k)= L Gijxj (k) , k=O , I, . .. ,K - I, i=I,2, .. . ,N (4.5 .5) j- I

are satisfied. Following the decomposition of the Lagrangian and the duality of

two-level strategy formulated by (4.3.21 )- ( 4.3.22), let us define a dual function

q ( ex) = Min L ( x, u, z, ex) x, u.z

where N

L(x,u , z , ex)= L L;( x pu; , z; , ex;) ; = 1

(4.5.6)

(4 . ~. 7)

Hierarchical Control of Discrete-Time Systems 149

Through this decomposition, as in the continuous-time case, one can pro­ceed to apply the two-level iterative procedure to improve on the values of the Lagrangian multipliers ex; using the gradient of the dual function q( ex) as in (4.3.25):

N

\7aq(ex)la,=ai = z;(k)- L G;jxj(k) = e;(k) (4 .5.8) j = I

for i=I,2, ... ,N; k=0, 1,2, ... ,K-1. Although one may proceed to use (4.5.8) and a few iterations of conjugate gradient or steepest descent to obtain a feasible and optimum solution, our objective here is to present a three-level modification of the problem due to Tamura (1974, 1975).

The essential point in this modification is to recognize the fact that the "first-level" solutions of a two-level structure can be obtained by utilizing the concepts of duality and decomposition instead of solving N independent problems of minimizing L j defined by (4.5.7) and subject to (4.5.4)-(4.5.5). At this level, the sub-Lagrangian for every subsystem is further decomposed by the discrete-time index k, thereby reducing a "functional" optimization problem at the first level of a two-level structure to one of a "parametric" optimization at the first level of a three-level structure. This decomposition was mentioned in Section 4.1 under" time" or "functional" division of the control task. It can also be considered a "temporal" decomposition versus a "spatial" one in Section 4.3.

In order to determine the optimal strategy of this three-level structure, let us define a dual problem for minimizing L;(·) in (4.5.7) subject to (4.5.4) as follows :

p ( f3;) = Min L; ( x ;, U;, Z;, (X i , f3; ) } Xj, Uj , Z;

(4.5.9)

where

+ zT{k )Sj(k )z;(k)+ uj(k )R;(k) u;(k)

+ f3r(k)( A;x;(k) + B;u;(k) + C;z;(k) - x; (k + I))] N

+ exiT(k)z;(k)- L exf(k)Gjix; (k) (4.5.10) j = I

and f3; is the i th subsystem Lagrange multiplier vector corresponding to dynamic equality constraint of (4.5.4), and all the other variables and parameters are defined earlier. The last constraint to be determined is the initial condition, also given by (4.5.4). In other words, the dual problem to minimizing J in (4.5.3) subject to (4.5.4)-(4.5.5) is maximizing the dual function p(f3;) defined by (4.5 .9)-(4.5.10) and subject to (4.5.4) at a given () - ()* /, - 1 '") M Th p or<:>rlipnt "f n( R \ i ~ ~in1ibr tn th P: r.nnti1111()lI s-fim p.

150 Hierarchical Control of Large-Scale Systems

case, the error is satisfying equality constraint (4.5.4), i.e.,

'ilp,p (f3n = - x;(k + I) + A;x;(k) + B;u;(k)+ C;z,( k) (4.5 .11)

for k=O,I, ... ,K-I, and i=I,2, ... ,N, with x;(k) and u;Ck) being the state and control vectors obtained from minimizing L; in (4.5.7) subject to (4.5.4) for a given f3; = f3r Now let us define the Hamiltonian H;(·) of the i th subsystem:

H; (x; (k), u;(k), z;(k), k) = iX;(k )Q;(k )x;(k)+ iU;(k) R;u;(k)

1 N + 2z;(k)S;z;(k)+ajT(k)z;(k)- L aY(k)Gj;x;(k)

j= I

(4 .5.12)

for k = 0, 1,2, ... ,K -I, and i = 1,2, ... ,N. Note that without loss of general­ity R . and S. matrices are assumed to be constant. By regrouping the last

I I

term f3r(K - l)x;(K) inside the bracket in (4.5.10) with ix;(K)Q;(K)x;(K) and adding the term f3;( -1)x;(O) to the sum inside the bracket with f3;( -1) defined to be zero, the functionp(f3;) in (4.5 .9) can be rewritten in terms of H;(·), i.e.,

K - I

+ L {H;(x;(k), u;(k) , z;(k), k)- f3;*T(k -I)x;(k)} (4.5.13) k=O

The minimization problem at the first level is divided into three portions: for k = 0, k = 1,2, ... ,K -I, and k = K. For k = 0, the problem is defined as

MinimizeH;(x;(O), u;(O), z,(O),O) (4.5.14) uj(O) . z, (O)

subject to

x;(O) = x;o (4.5.15)

In view of H;(x;(k), u;(k), z;(k), k) in (4.5.12) for k = 0, the necessary conditions for the minimization problem (4.5.14)-(4.5.15) are

or

'ilu'(O)H;(·) = R;u;(O)+ BF/3;*(O) = 0 (4.5.16)

u;(O) = - Rj IBrf3;*(O)

z;(O) = - S;- I( Crf3;*(O) + al(O))

(4.5 .17)

(4.5 .18)

(4.5.19)

In a similar fashion, for the second portion k = 1, 2, ... , K - I, by virtue of (4.5 .13), the minimization problem is

Mi"irni7P (H(x ,(k),u,(k),z ,(k),k) - f3;*T(k-l)x;(k)} (4.5 .20)

Hierarchical Control of Discrete-Time Systems 151

which leads to the following relations (Singh, 1980; Tamura, 1974):

x;(k) = - Qj l(k){ A;f3;*(k)+f3;*(k -1)+ j~1 [ajT(k)Gj;r} (4.5 .21)

u;(k) = - RjIB!fJ;*(k) (4.5.22)

and

z;(k) = - S;- I( Crf3;*(k) + aj( k)) (4.5.23)

The final portion of the first-level minimization is defined by

Minimize {ixT(K )Q;(K )x;(K) - f3;*T(K - J)x; (K)} (4.5.24)

which results in

(4.5.25)

Therefore, by virtue of this three-level hierarchical structure, the large-scale optimal control problem defined by (4.5.3)- (4.5.5) can be solved by the following algorithm.

Algorithm 4.3. Three-Level Coordination of Discrete-Time Systems

Step 1: At the first level, for given Lagrangian mUltiplier ai(k), f3;*(k) sequences, (4.5.18)-(4.5.19), (4.5.21)- (4.5.22), and (4.5.25) can be used to find x;(k), u;(k), and z;(k) for k = 0, 1,2, ... , K and i=I,2, ... ,N.

Step 2: At the second level, x;(k), u;(k), and z;( k) of the first level can be used along with the gradient of p C f3;) in (4.5.11) to improve on the value of f3;, i.e.,

f3;*'+ I (k) = f3t' (k) + o/J' (k) ,

k=O,I, ... ,K, i=I,2, ... ,N (4.5.26)

where /' (k) is a function of gradient of p C f3n, and again a simple gradient or conjugate gradient method can be used. Steps I and 2 are repeated alternatively until the optimal f3;*(k), i = 1, 2, ... ,N, and k = 0, 1,2, ... ,K, are obtained.

Step 3: At level 3, the optimal f3;*(k) values can be used to improve on the values of aiCk) using the gradient of q(a),

ai'+ '(k) = ai'(k)-t- €;g{(k),

k = O,I , ... ,K, i=I,2 , ... ,N (4.5.27)

where g{Ck) is a function or v~q~a;) given by (4.5.8).

f:·t·

j"( 152

Hierarchical Control of Discrete-Time Systems 153

Q Once the above three steps are complete, the interaction errors defined by Z>-l b>-l (4.5.8) and (4.5.11) would become zero and the optimal control of the O~ ,-. tIl~

original large-scale system is ob tained (Singh, 1980). Figure 4.15 shows a u> ~ 0::;>

~~ ...... ~ schematic of the three-level modified coordination principle proposed by tIl >-l u...>-l

Q>-l ~

Tamura (1974). The above structure and algorithm is illustrated by the o::; ~ ...... > ~ r- followi ng example . ::r:~ 0-b>-l

~

"'< oj

Example 4.5.1. Consider a third-order system, .... ;:J

~ ~ [ x, ( k + I) H -0 I 005: 005 f (k) 1 [ 025: 0 1 [ u, ( k) 1

b >-.

.D x 2(k+ l ) - 0 -0 .2 1 0 x2(k) + 0 10 - ___ ~ 0 '"0

;3(k+l) -0.2--::"'0.1-1--=-3-- ;;(k) 0 - -1-0 u2(k) II .., </)

"'< 0 0.. 0

(4.5 .28) .... 0..

<: .., with a cos t function .... N ;:l

0::; t) 2 {I 9 1

0 ~ .5 J = ;~1 "2 x;(lO)Q;(lO)X;(lO) + k~O "2 [x;(k) Qi(k )xi (k) b II </)

...:t: "'< s:1

+ z;(k )S;(k )zi(k) + ui(k) Ri(k) [l i(k)] } ,~~ Z 0

2S ~ 'LJ

(4 .5.29) oj "". s:1 -0::; · :a 0 • · .... 0 0 where Qi(k) = 21", R i(k) = 0.5InJ, SiCk) = Ik , n l = 2, n2 = 1, /11 1 = /11 2 = 0 U <.> , , ,

0 OJ 1, kl = 2, k2 = 1. It is desired to find an optimum control strategy through

II > the three-level coordination Algorithm 4.3. "'< ~

oJ .., SOLUTION : The algorithm was simulated on an HP-9845 compu ter using ....

.;3 BASIC and an in itial condition x(O) = (2 -3 4)T. The results were generally .., ,-,

.;3 very satisfactory, especially with respect to the in teraction error between "'<

~ "-< levels 2 and 3. Typical resulting error and states are shown in Figures 4.16 0

II .~ and 4.17. The interaction error between levels I and 2 reduced from 1,000 to C;; "'< ~ 149 in some 50 iterations and the convergence was not as fast as the S ..,

second-third levels interaction errors . A <.> </)

-<t:: - 0 I/) 4.5.2 Discrete -Time Systems with Delays N ->-l - II -.i ...:t: In this section one of the more powerful algorithms for multilevel op timiza-...... ;:! "'< .., b - ...

tion of large linear discrete-time systems with delays in both state and ...:t: N ~

0.. >-l ~ con trol is presented. The problem was first introduced by Tamura (1974, til ...:t: 0::; 1975) and has been further applied by many others (Fallaside and Perry, 0

1975; Jamshidi and Heggen, 1980; Singh, 1980). 0.. ::8

Consider the following large-scale discrete-time system with delays ~ b

x(k + 1) = Aox(k)+ A1x(k - 1)+ A2X(k - 2)+ .. .

+Asx(k-s)+Bou(k)+B1u(k- l )+ ... +Bsu(k-s)

(4 .5.30)

154

\60

140

..., I

N 120 ....! w > w !::!, 100 t:t:: 0 t:t:: t:t:: 80 w z 0 f::: 60 u <t: t:t:: W I- 40 ~

20

0 I 2 3 4 5 6 7 8 9 10 II 12

ITERATIONS

Figure 4.16 Typical interaction errors of Example 4.5.1.

Figure 4.17 Typical state trajectories for the third-order system of Example 4.5.1.

4

"'" ~ k C/l IJ-l I-<t: 0 I-C/l I

I - I I

I I

-2 I I I

-3 I 2 3 4 G

DISCRETE TIME k

7

--- x2 (k)

- . - x )(k)

8 9 10

Hierarchical Control of Discrete-Time Systems 155

where A ; and B;, k = 0, 1, 2, . . . ,s, arc n X /1 and /1 X m matrices, respectively, and x and u are /1- and m-dimensional state and control vectors. The system (4.5.30) has 2s delayed terms; hence it requires 2s discrete-time initial functions, assumed to be zero without loss of generality:

x(k)=O , u(k)=O, -s ~ k < O, x(O) = xo (4.5.31)

Physical interpretation of initial functions in (4.5.31) for system (4.5.30) is that the system is operating at its steady-state and receives a disturbance at k = 0 and derives it to a known value xo' The system cost function is assumed to be quadratic:

1 K - I 1 J = 2:xT(K)Q(K)x(K)+ L 2:{xT(k )Q(k )x(k)+ uT(k )R(k) u(k)}

k = O

(4.5.32)

where Q(k) and R(k) are both assumed to be positive-definite. The optimal control problem is to find a sequence of control vectors u(O), u( 1), ... , u(K -1) such that (4.5.32) is minimized while (4.5.30)- (4.5.31) and a set of inequality constraints

X min ~ X ( k ) ~ X max

U min ~ U ( k ) ~ U max

(4.5.33)

(4.5.34)

are satisfied. It goes without saying that a solution to the problem (4.5.30)- (4.5 .34) in usual "centralized" methods by the application of the maximum principle, as it is demonstrated in Section 6.4, results in a TPEY (two-point boundary value) problem which involves both delay and advance terms, making the attainment of an optimum solution very difficult indeed, if not impossible. In the literature, there are several approximation tech­niques to deal with continuous-time systems with delay which will be discussed in Chapter 6. Here the objective is the application of hierarchical control via the interaction balance principle.

Following the formulation of discrete-time maximum principle (Dorato and Levis, 1971), let us define the Hamiltonian:

H(x(k), u(k), A(k), k) = 1{x T(k )Q(k )x (k)+ uT(k )R(k) u(k)} of

i = 0

(4.5.35)

where k = 0, 1, ... ,K -1, A(k) is a vector of Lagrange multipliers at k, and A (K), A (K + 1), '" are defined as zero vectors. In a manner similar to previous discussions regarding Equation (4.5 .13) for a given vector A = A* ,

156 Hierarchical Control of Large-Scale Systems

the Lagrangian can be defined as

L(x , u, >-.*, k) = t xT(K)Q(K)x(K)- >-.*T(K -1)x(K) K - l

+ L {H( x (k), u(k), >-.*(k) , k)- >-.*(k - 1)x(k» k = O

(4 .5.36)

Thus the optimization problem is to minImiZe (4.5.36) subject to (4.5.33)- (4.5.34). As before, this problem can be altered to that of maximiz­ing the minimum of L(·) with respect to >-.. The power behind the" time­delay algorithm" of Tamura (1974) is the decomposition of this problem into (K + I) independent minimization problems for a given >-. *, as in the three-level coordination formulation discussed in the last section, which reduces a "Junctional" optimization problem to a "parametric" one.

4 .S.2.a Problem k = 0 By virtue of (4.5.36) , definition (4.5.35), and constraints (4.5.33)- (4.5 .34), the optimization problem for k = 0 is

MinH(x(O) , u(O), >-'(0» = Min~{xT(O)Q(O) x (O)+ uT(O)R(O) u(O)} u(O)

s

+ L >-.*T(i)(Aix(O)+Biu(O») (4 .5.37) i = l

subject to

(4.5.38)

Now if R(O) is assumed to be a diagonal matrix, the necessary conditions for (4.5.37)- (4.5.38) lead to a set of m independent relations, each of which has an explicit solution given by setting JH(')/ Ju(O) = 0, i.e.,

(4.5.39)

where the" saturation" function Sat u(' ) is

(4.5 .40)

and the indexj represents thejth element of control tl j , j=1,2, .. . ,m.

(/

Hierarchical Control of Discrete-Time Systems

4.S .2.b Problem k = 1,2, .. . ,K-l

The intermediate problem is defined by

157

Min H( x (k), u (k), >-. *(k), k) - >-. *T( k - I)x(k) (4.5 AI) x (k ) , lI(k )

subject to

xmin « x(k) « x max (4 .5042) and

umin « u(k) « u max (4.5.43)

<?nce again, assuming that R(k) and Q(k) are diagonal, the partial deriva­tives JH(·)/ Jxi(k) and JH(-)/Juj(k) for i=I ,2, ... ,n andj=1,2, ... , 111

lead .to a. set of n + m independent one-parameter equations whose general solutIOn IS

x'Ck) ~ Sa'. ( - Q- 'Ck)[ - )..'Ck - 1) + ,to Ar)..' Ck+;)])

u*(k) = satu{ - R - l(k)[i~O BT>-.*(k + i)]} (4.5044)

where the I th element of the saturation function Sat x( v) is

{

X max , { if v{>xmax , {

Satx(v{)= v{ if xmin,{« v{«xmax.{

x min , { if v{ < x min ,{

4.S.2.c Problem k = K

This problem is

~~~ { ± xT(K)Q(K )x(K) - A*T(K -I)X(K)} x (K)

subject to

x . ,:::: x(K) ,:::: X mln ~ ~ max

whose solution is similarly given by

x(K) = Satx{Q - l(K)A*(K -I)}

(4 .5.45)

(4.5.46)

(4 .5.47)

(4.5.48)

The above so-called" time-delay algorithm" can be summarizcd as follows:

Algorithm 4.4. Time-Delay Algorithm

Step 1: At level one, solve K + 1 analytic problems defined by (4.5 .39)-(4.5.44) and (4.5.48) for a fi xed set of Lagrange multipliers A(k) = A*(k), k = 0, 1, ... ,K-1.

158 Hierarchical Control of Large-Scale Systems

Step 2: At level two, the value of "A*(k) is improved through a gradient-type iteration

where dr(k) is a function of the error er(k), i.e.,

dr(k) = f(er(k))

(4.5.49)

= fC~o [A;x(k - i)+ B;x(k - i)] - x(k + I)} (4.5 .50)

which follows from our previous discussions, i.e., Equation (4.5.30).

The following example illustrates the time-delay algorithm.

Example 4.5.2. Consider a simple second-order system

[::~:::~]=[-~ -~][::~:~)+[~ ~][::~:=:n +[°1.5 O][U\(k)]+[O 0.25][U\(k-I)] (4.5.51) ° u 2 (k) I I u 2 (k -I)

with cost function

I I 4 J = "2 XT(S)Q(S)x(S)+"2 L {xT(k )Q(k )x(k)+ uT(k )R(k) u(k))

k =O

(4.S.S2)

constraints

[~]<X(k)<[;], [=~]<U(k)«~], x(O)=(~:;] (4.S.S3)

and Q(S) = diag(l , 2), Q(k) = 12 , and R(k) = diag(l,O.S).

SOLUTION: The problem was solved for an error to lerance of 0.00 I, a step size of 0.1 for the conjugate gradient iteration, and >"(0) = (0.1 O.ll. The algorithm converged in 49 iterations, as shown in Figure 4.18. Several other initial x(O) and >"(0) were tried and the convergence was achieved in a similar fashion.

4.5.3 Interaction Prediction Approach

Consider the following linear discrete-time system in its state-space form,

x(k+I)=Ax(k)+Bu(k)+d(k), x(O)=xo (4.5 .S4)

Hierarchical Control of Discrete-Time Systems 159

.9

.8

.7 p:: 0 p:: .6 p:: ~

z 0 .5 f:: u <t: .4 p:: ~ f-< is .3

.2

.1

0 0 4 81216202428323640444850

ITERATIONS

Figure 4.18 Interaction error vs iterations for the time-delay algorithm in Example 4.5.2.

where x(k), u(k), and d(k) are 11 X I, III X I, and II X I state, control, and disturbance vectors, respectively. Let the system be decomposed into N subsystems

x;(k + I) = A;x;(k) + B;u;(k) + C;z;( k) + d; (k), xi(O) = XiII

(4.S.SS)

for i = 1, ... ,N and X;, u;, and d; are n; X I, m; X I, and l1i X I state, control , and disturbance vectors of the i th subsystem, respectively. The vector z, (k), to be defined shortly, represents the interactions between the i th subsystem and the remaining (N -1) subsystems. The matrices A;, Bi' and C; are, respectively, n;Xlli' I1;Xm;, and n;Xr;. The integer r; represents the number of incoming interactions to the ith subsystem. The optimal control problem can be stated as follows:

(4.S.S6)

subject to (4.S.SS),

(4.S .S7)

160

and

Hierarchical Control of Large-Scale Systems

N

z;(k)= L {D;ju;(k)+G;jxj (k) } j = 1

(4.5.58)

where D;j and G;j are the appropriate matrices between controlllj(k), state x,(k ), and the state x;(k + I), respectively.

Two solutions of the above optimization problem and its modified forms have already been presented. One is based on the time-delay algorithm due to Tamura (1975), which was discussed in the previous section. The other was di scussed in Section 4.5.1 without the bounds (4.5.57) and utilizing the three-level discrete-time application of goal coordination. However, instead of utili zing those methods, the continuous-time interaction prediction method of Section 4.4.1 is extended to discrete-time, which has not received much attention in literature. Here again, we will ignore the bounds (4.5 .57) for the time being. Moreover. the objective function (4.5.56) is assumed to be quadratic:

N - 1

J; = ~ ~ {xi(k )Q;x;(k)+ u{'(k )R;u;(k)} k = O

(4.5.59)

where weighting matrices Q; and R ; follow the usual regulatory conditions. Consider the Lagrangian of the problem (4.5.55), (4.5.58)-(4.5.59):

N ,'II (Nr- 1 { 1 L = ;~I L; = j~1 k~O 2. xJ'(k )Q;x;(k) + 1u;(k )R;u; (k) + A~(k )z;(k)

N

.... L A~' (k)(D;;lli(k)+GI;x,(k)) i ~ 1

+ p;(k + 1)[ - x; (k + I) + A;x; (k) + B;lI; (k) + C;z;( k) + d; (k)] } )

(4.5.60)

Here it is assumed that the Lagrangian Lis additively separable for Z; and A; trajectories . This would imply that [or any given z; and A;, there are N independent maximization problems, each with a sub-Lagrangian L j • What is left is to find a mechanism for updating Zj and Aj • A necessary condition [or this is

which result in

N

zj(k)- ~ (Dj j ll j(k)+GjjXj (k)) =0 j = 1

(4.5.61)

(4.5.62)

(4.5 .63)

Hierarchical Control of Discrete-Time Systems 161

The coordination at the second level for this interaction prediction scheme would be

l -CT P (k + 1) j 1

[

( )]

1 + 1 I I

A; k N

zj(k) = j~1 {Djjllj(t() + Gjjxj(k)) (4 .5.64 )

for k = 1,2, ... ,Nj

, and I is the iteration number. Here again it is noted thaI the computational effort at the second level IS very small compared WIth that of the gradient, Newton, or conjugate gradient techniques.

For a known set of augmented interaction vectors [A *T( k) I Z*T( k) J, the dh subsystem Hamiltonian is

Hj(-) = 1X;(k) Qjx ;(k) + 1U;(k) R;uj(k) N

+ >/i ( k ) z; ( k ) - ~ X~ ( k ) ( DJ j U j ( k ) + GFA) k ) ) j=1

+ pi( k + 1) [Ajx j (k) + Bju; (k) + CjZ j (Ie)] (4.5.65)

Then the necessary conditions for optimality would lead to

p j ( k ) = () H j (- ) / () x j ( k ) = Q j x j ( k ) + A rp J k + I) N T

- ~ (A~(k)Gji) , pj(Nj )=O (4.5.66) ) = 1

and N T

o = c7 H, ( . ) / a II j ( k ) = R j U j ( k ) + B;r Pi (k + I) - L (X~ ( k) D,; ) ( 4.5 .67 ) )= 1

or (4 .5.68)

where

Let us assume that the costate p/k) and state xj(k) are related by a linea r

vector equation

(4.5.69)

Now eliminating p(k) in (4.5.68), solving for x; (k + I) , and substituting it in (4.5.66) while ~tilizing (4.5.69) and equating the coefficients of x/k) and zeroeth-order terms would lead to the following Riccati and adJo1l1t

162 Hierarchical Control of Large-Scale Sys tems

K, ( k ) = Q, + A ;'i( ( k + I) Ei - I ( k ) A, ' K, ( Nj ) = 0 ( 4. 5.70)

Xi (k) = ;/: [/ - Ki {k + I) Ei - l(k)r;]Xi(k + 1) - lIi(k) , g, (N/ ) = 0

wiIere (4 .5.7 1)

E, (k. ) = I + 1'; K, (k + I )

r; = B,R, IB,T

h,( k) = ;/{K,(k + I)b,(k) + /,(k.)

h, (k) = E,- I (k) [R j Ie, (k) + e,z, (k) + d, (k)] (4.5.72) N

.t; ( k. ) = L (l\~ (k) GjJ' j = 1

Recalling the bounds (4.5.57), the expressions fo r 11 , (k) i 11 (4.5.68) a nd .\, ( k + I) in (4.5.5 5) a ft er eliminating {J ,(k + I) via (4.5.69), the fo llowi ng ex press ions arc sugges ted for conlrol and stale vecto rs:

where

a Illi

{

J.I' ( k)

1I,( k) = ~ R,- IGi (k)

u, (k)

LI, ( Ie ) <!:I, (J.: )

!:Ii ( k ) ~ LI i ( k ) ~ u, ( k )

u,(k»u,(k) (4.5.73)

G,(k) = B,"[K,(k + I) x,(k + 1) + g,( k + I)] - e,(k) (4 .5.74)

{~" (k + I) ·· · .\ ,(k + I) <.::..,,(/.; + I)

. L, ( k) X I (k ) + M, (k ) g, (k + I) - b (k ) .. . x · (Ie - I) x, (k + I) = , - , ~x, (k + I) ~x,(k+ I)

x, (k + I)· ·· x,(/.; + I) > .l:,(k + I)

(4. 5.75 ) where

L,(k) = L,I(k)A" M,(k) =E, l(k) l~ (4.5.76)

The di scre te-time interaction prediction approach is summarized by the following algorithm.

Algorifhm 4.5. Interaction Prediction Approach for Discrete-Time Sys tems

,\IC{J I : At the second level set I = I, assumc initial values for

z, (k) = zt (k) and AI( k) = A;( k), anu pass them down to firs t level, i = 1, .. 'ON and k = 1, .. . ,N/.

Hierarchical Control of Discrete-time Systems

Step 2: At the firs t level solve N ma trix Riccati equations (4.5.70) and s to re.

Step 3: Solve N adj oint equations (4.5.7 1) and store for k = I , ... ,N/.

Step 4: Using the s to red values of K,(k), K,(k), (4.5.73)- (4.5 .76), find and store u,(k) and x,(k), i = I , ... ,N and Ie = I , . .. , N/.

163

Step 5: Check for the convergence of (4.5.62)- (4.5.63 ) i.c. , whethe r their left-hand sides are within E = 10 - 6 o f zero. If not, use (4.5.64) to update A,(k) and z,( k) , incremen t I = I + I, and go to Step 3.

Step 6: Stop.

An application of this algorithm is given in Example 4.5.3 .

4.5.4 Structural Perturbation Approach

In this section the optimal hierarchical control scheme via s tructura l per­turbation of Section 4.4.2 for continuous-time systems is extended for discrete-time sys tems. For the sake of discussio n, the discrete quantities are represented by index i in a subscripted form and the subsys tems by index) in a superscripted form; e.g., xl represents the)th subsys tem's state at ith interval. The present development, in part, foll ows the work of Sundareshan ( 1976).

Consider a large-scale di screte-time sys tem which m ay consis t or M subsystems:

(4.5.77)

for ) = 1,2, ... , M and i = 1,2, ... , N. In the above relation, xl and tI ! arc 11 ,­

and m-dimensional state and control vectors of the )th subsystem at i th interva'1. The Gjk is the II j X II k interconnectio n m a trix between .the )th and k th subsys tems. The relations of Ai' Bi , and Gi k matnces are gIven below:

AI I GI 2 I __ i _ _ l _ _ _ G2 1 I A 2 I Gn I _ _ L __ L __ L

. I - - - -

I AM

(4.5.78)

The op timal control problem would be to find contro l vectors Llf, j =

1,2, ... , M, sllch thaI (4.5.77)- (4.5.78) are satisfied while minimizing a

164 Hierarchical Control of Large-Scale Systems

quadratic cost function

N

Jl( x{" lin = L (xrQ;x( + U{~I Rj u; _ I) (4.5.79) i = 1

wh~r~ Qj and. R ; are 11; X 11 ; and m ; X Ill ; positive-semidefinite and positive­defll1lte mat~lces, res~ectiv~ly. The optimal control problem, in its large­scale composIte ~orm, IS to find an m-dimensional control uT = (U IT, ... , U MT ) WhICh would satisfy the overall state equation

X; + 1 = (A + C)x; + Bu; (4.5.80)

~~I:re~ A = Blo~k-diag(A I ' A 2 ,· .. · ,A,If ): B = Block-diag(B I' B2 , ... , B M ), and [0,k J.J . k - 1, 2, . . . ,M, willie the cost

M

J( x", ul) ) = L Jl(x /" ll ~) (4.5.81) j = I

is minimized . The solution of this prohlem requires the solution of a large-order Riccati equation (Dorato and Levis, 1971), and a single central controller would become necessary which may require too much, and often unnecessary or unavailable, information from all the states. In order to alleviate these problems, Sundareshan (1976) has proposed a hierarchical control which consists of a local and a global component, i.e.,

u .=u'+ug 1 1 1 (4 .5.82)

where the local control is given by

, _ { IT 2T MT} U; - Ii; ,U; , ... ,U; (4 .5.83)

and each compon.ent u{ is the optimal control of the jth decoupled subsys­tem problem deflI1ed by (4.5.77)--(4.5.78) and (4.5.79) which is obtained through a Riccati formulation as in (4.5.65)- (4.5.72). The jth subsystem local control is given by

u j = - pix j 1 1 1 (4.5.84)

where P/=- Rj-IBrKI+ I(I+~KI+ I) - IAj and K/ls the positive-definite solutIOn of the discrete matrix Riccati equation:

K j= Q +ATKj (I+SK j )- IA ( ; j .; ; + 1 j / + 1 ;' KN=O 4.5.85)

\~here S~ = B;Rj IB! and all other matrices are defined by (4.5.77)- (4.5.79). 1 he global component II f is given by

(4.5.86)

where r is an 111 X 11 gain matrix yet to be determined. Using the local and global controls defined by (4.5.84) and (4.5.86), the closed-loop composite system becomes

(4.5.87)

Hierarchical Control of Discrete-Time Systems 165

where P; = lllock-diag(P;', p/, .. . , p;M) and the tcrm (C - B r) is termcd as the effective interconnection matrix Ce . The matrix r should be chosen such that the norm IIC - Bfll is minimized. The solution to this minimization problem is well known:

(4.5 .88)

where B+ is the generalized inverse, i.e., B+ = (l3 TB) - IBT. The solution (4.5 .88) is subject to the rank condition, rank(B) = rank(BC). Thus, the global control component gain matrix is obtained [rom

(4.5.89)

The above choice would in effect neutralize the interconnections, i.e., IICe l1 = O. The application of this hierarchical control to the large-scale interconnected discrete-time system would, in general, cause a deterioration of performance whose sUboptimality degree is p if the following inequality holds (Sundareshan, 1976) :

(4.5.90)

where

V=LTQ L+P/TRPJ, H=(I+p)K+Q J JJJ 1 JI 1 ,

(4.5.91)

The terms AIII(D) and AM(D) are, respectively, the minimum and maximum eigenvalues of matrix D. This control scheme is summarized by an algo­

rithm.

Algorithm 4.6. Structural Perturbation Approach [or Discrete-Time Systems

Step 1: Input all matrices {Aj' Bj , Qj' R j , Cjk }, initial states x~, etc., j=I, .. . ,M.

Step 2: For each subsystemj, solve (4.5.85) for Riccati matrix K( , i = I, ... ,N - 1, and store.

Step 3: Evaluate the local control components by solving (4.5.84).

Step 4: Solve (4.5.89) and use (4.5.86) for the global control component and form the overall control (4.5.82).

Step 5: Using the control u and Equation (4.5.80), find state x;,

i=I ,2, ... ,N.

In sequel, Algorithms 4.5 (interaction prediction) and 4.6 (structural perturbation) are applied to a three-region energy resources system.

166

REGION 3

REGION 2

Hierarchical Control of Large-Scale Systems

Figure 4.19 A three-region energy resources system.

E~all1ple 4.5.3. COI?sider a three-region energy resources sys tem shown in Figure 4.19. For thiS system, let the fo llowing discrete-time model hold:

r ~~l~i~2~ _O_ IJ I : () : () j Xi + I = l qo?J ~ _~] ~ _~~ x, + ~ - := jk~ =~~ 1/,

Hierarchical Control of Discrete-Time Systems 167

where

(A"B'}~([~ J

~75].[ m 0.5 0.75 0 .5

(A'B,}~{r ~5 0 .5 0 .25

~~5Wj} 0.5 0 (4.5.92) 0 0.5 0.2 0.25 0.25 ~ 0

(A"B,)~ ( [ ~5 0.5

~mm 0 .25 0 .3 0.2

H rO 0

~j G,, ~ [ ~ 0 0 10 0 .5 0.5 0

G2 1

= l ~ 0 1.5 0 0 0

The quadra tic cos t function has weigh ting matrices QI = diag(J 10 1), Q2 = 514, Q3= diag(1O 110). and R;=I,j=L2,3. lt is desired to apply Algorithms 4.5 and 4.6 to find a hierarchical allocation policy for tbe energy-resources system of (4.5.92).

SOLUTION: For the hiera rchical control policies, th ree matrix Riccati equa­tions of the form in (4.5 .70) or (4.5.85) and three adjoint vector equations of the form in (4.5.71) were solved using an HP-9845 computer and BASIC. Based on the solutions of these and vector equat ions. th ree loca l control fu nctions were obtained. Using the interation prediction method, an accep­table (within 10 - 4 ) convergence was reached with :line iterations. Using the structural perturbation method, Equation (4.5.89) was used to ohtain the global control component and added to the vector of local con trols to fi nd the overall control (4.5.82). The resulting sta tes ami control func tion via the hierarchical control Algorithms 4.5 and 4.6 along with the op timum central­ized trajectories which were obtained by solving a tenth-order discrete-time Riccati equation are shown in Figures 4.20 and 4.21, respectively.

The state and control responses for the interaction prediction (Algorithm 4.5) were closer to the op timum than those for the structural perturbation (Algorithm 4.6). However, the third subsystem's trajectories were closer than the other two. A reason for this may be the fact tilat the third subsystem is

completely decoupled .

4.5.5 Costate PredictioJl Approach In this section a computa tionally effective approach for hierarchical control of nonlinear discrete-time systems due to Mahmoud et al. (1977), Hassan and Singh (1976, 1977) is presented. The scheme is appl ied to nonlinear

1611

4 .

3

2 -

.<

LW

~ -- .1

t;:; - 4 _

-. S

7 -

III ()

0 - I ___ "l

- 3 - 4 -o S -- -C)

-'< -- 7

·-R

" LW --- 9 f-

III <!; f- - · 1 I (/)

- \:' -- \. 1

- 14

- I S

···· 1(,

I 7 ·- IX - 19 - 20

--

..

_.

-

-

--..

_.

-

--- Of'TJMlJ~1 CTNTRALIZED TRAJECTORY

---- NEAR- OPTIMUM lll ERARCl1 lCAL TRA.1FCTORY (Algorith11l4.6) ! /

-'-' - NEAR-OPTIMUM IIIERARCIIICAL TRAJFCTORY (Algorithm 4.5 ) i / i / . / 1/ 1/

.-'-_--.JI ___ L -_ ___ '--_. __ J __ __ . ___ ._L ____ ._ I __ --'--I _. __ L ___ .J S 10 15 20 25 30 35 40 45 50

PERIOD. k

(a)

~ --- OPTIMUM C'I'NTRALlZI'IJ TRAJFCTORY .~

- - -- NI' AR-OPT1MUM IIlI'RARC'IIiCAL TRAJHTORY (Algorilhlll 4 .(1) 0-- .-.- NEAR- OPTIMUM IIIERARClIlCAL TRAJECTORY (AlgorilhIll 4 .5 ) '",

0 L---l __ ~ __ L-_~ __ -L_~~ _ _ ~ __ L--+~

10 15 20 25 30 35 40 45 50

PERIOD. k

(b)

Hierarchical Control of Discrete-Time Systems 169

.95

.85

.7 5

.65

.55

.45

.35 ..:,-:

.25 0

'" . I 5 LW f- .05 <!; f-en -.05

- . 15

- .25

-. 35

- -.45

- .55

-- .65

- .75

___ OPTlMUM CENTRALIZED TRAJECTORY

___ NEAR-OPTIMUM HIERARCIIlCAL TRAJECTORY (Algorithm 4.6)

_._._ NEAR-OPTlMUM HIERARCHICAL TRAJECTORY (Algorithm 4.5)

L-_ _ ~ __ ~ _ _ ~I_~ .LI __ ~I_ L __ ~_~I __ ~I

35 40 4 5 50 0 10 15 20 25 30

PERIOD, I; (c)

Figure 4.20 Optimum centralized and near-optimum hierarehical trajeetories for Example 4.5.3: (a) state x2(t), (b) state x 1(t), (e) state XlO(t) .

systems, and its linear extension is rather simple (see Problem 4.8). The hierarchical control of nonlinear continuous-time systems is considered in Chapter 6 (Section 6.3.1.) under the context of near-optimum design of large-scale systems.

Consider a nonlinear discrete-time system described by

x(k+1)=/(x,u,(k+1,k)), x(O) = x"

with a quadratic cost function K - I

J=~ L (xT(k)Qx(k)+uT(k)Ru(k)) 1; = 0

(4.5.93)

(4.5.94)

The procedure begins by rewriting the nonlinear state equation (4.5.93) as

x (k + 1) = A (x, u, (k + 1, k) )x( k) + B (:x, fl, (Ie + 1, k)) u (k)

+ c(x, u,(k + 1, k)) (4 .5.95)

where A(·), B(·) are block diagonal matrices and

c ( x, u, (k + 1 , k )) = C I ( x, u, ( k + 1 , k ) ) x ( k ) + C2 ( x, u, (k + 1, k ) ) 11 ( k )

(4.5.96)

It is notecl that the reformulation (4.5.95)- (4.5.%) of' the system (4.5 .93) is

/ 170

.,1.

-5

-10

-15

-20

-25

3: -30 :; o-l 0 -35 p:: b -40 Z 0 U

-45

-50

-55

-60

-65

-70 0

0 - 2.5

- 5

-7.5

- 10 - 12.5

- 15 :;: - 17.5

N -20 " o-l -22.5 0 p:: - 25 b

Z 0 U

-27.5

-30

-32.5

-35

-37.5

- 40

- 42.5

- 45

--=--:::- .--. ....... -.

.......... """'- --- .----............... ........ ........ --...... '-...

........ --......

'- " '-" ."'"'-'- "'"'-

'-.. " " "­"" '",

" '" "" '" " \ "-'\ \

\ \ OPTIMUM CENTRALIZ ED TRAJECTORY \ '\

___ NEAR-OPTIMUM HIERARCHICAL TRAJ ECTORY (Algorithm 4.6) \ .

_. _ . _ NEAR-OPTIMUM HIERARCH ICAL TRAJECTORY (Algorithm 4.5) "

0

\0 15 20 25

PERIOD, k

(a)

30

___ OPTIMUM CENTRALIZED TRAJECTORY

35 40 45

_ _ __ NEAR-OPTIMUM HI ERARCII ICAL TRAJ ECTORY (Algorithm 4.6)

_._._ NEAR-OPTIMUM HIERARCHICAL TRAJECTORY (A lgorithm 4.5)

5 10 15 20 25

PERIOD, k

(b)

30 35 40 45

50

50

P .

Hierarchical Control of Discrete-Time Systems 171

0

-. 1

-.2

-.3

:;: - .4 ~

" . o-l

-.5 0 p:: b Z 0 -.6 u

- .7

-.8

- .9

I 0

_ _ OPTIMUM CENTRALIZED TRAJECTORY

__ _ NEAR-OPTIMUM HIERARCHICAL TRAJECTORY (Algorithm 4.6)

._. _ NEAR-OPTIMUM HIERARCHICAL TRAJ ECTORY (Algorithm 4.5)

10 IS 20 25 30 35 40 45 50

PERIOD, k (c)

Figure 4.21 Optimal centralized and near-optimum hierarchical trajectories for Example 4.5.3: (a) control Ul(t), (b) control U2(t) , (c) control u3(t).

always possible. Moreover, for N blocks in A and B, it is assumed that matrices Q and R also have N blocks. The basic reason for the reformula­tion (4.5.95)- (4.5 .96) is to provide" predicted" state and control vectors x* and u* to fix the arguments in the nonlinear coefficient matrices A(·), B(·), C 1 (.), and C2 (-). Therefore, the problem (4.5.93)- (4.5 .96) can be rewrit-ten as

_ I K - I

min J =2 L (xT(k)Qx(k)+uT(k)Ru(k)) k ~ O

(4.5 .97)

subject to

x (k + I) = A (x* , u*, (k + I, k)) x (k) + B (x* , u* , (k + I , k)) u (k)

+ c(x*,u* ,(k +l , k)) (4.5.98)

x*(k) = x(k), u*(k) = u(k) (4.5.99) The modified problem can be solved by defining a Hamiltonian:

H( ·) = 1XT(k )Qx(k)+ 1uT(k )Ru(k)

+ pT(k + l){A(x*, u*, (k + 1, k »)x(k)

+ B (x* , u* , (k + I, k) u (k) + c (x*, u* , (k + 1, k))

+ exT ( k ) ( x ( k ) - x * ( k ) ) + f3 T ( k ) ( u ( k ) - u * ( k ) ) (4 .5 . 100)

172 Hierarchical Control of Large-Scale Systems

In view of the assumptions made on matrices A, E , CI

, C2

, Q, and R , it is clear that the Hamiltonian H(·) in (4.5 . 100) is additively separable for given x* and 11*. i.e.,

N N

H = L H, = L {~xJ'(k)QxJk)+~U;r(k)Rll,(k) , = 1 , = 1

+p/(k + I)[A ,(. )x,(k)+ BJ· )u,(k)] + d,)}

+cx[(k)[x,(k)-xi (k)]+f3,'(k)[u,(k)-U i (k)]} (4 .S. IOI)

The necessary conditions for optimality are given by

0 = oHjou,

x,(k + I) = oH,/op,(k + I) p,(k) = aHjox,(k)

O= r7 IJ/aH, 0 = () J/ / iJ!J

0 = rJll/ox*(k), 0 = olJ/ou*(k)

Rela tion (4.S.102) yields an expression for u,( k),

(4 .S.102)

(4.S.103)

(4.S.104)

(4 .S .IOS)

(4.S . 106)

u,(k) =- R ,:' I{B,'(x* , u*,(k+l,k»p,(k + I) + f3, (k)} (4 .S. 107)

and subs tituting 1I,(k) in Equation (4.S.103) yields a new expression for x,(k + I). i.e.,

.\Jk + I) = A,(x*, u*,(k + I, k»x,(k) - B,( x*, u*,(k + I, k» R I I{ B,'( x*, 11*. (k + I, k » pJ'( k + I) + f3 , (k )} + c, (x*, u*, (k + I, k »

(4.S . IOX)

with x, (O) = x'o' The condition (4.S .104) gives the costate equation

P, (k ) = Q,x, (k) + Ai( x*, 11*, (k + I, k » p, (k + I) + H, (k) , P, (K) = 0

(4.S.109)

and the necessary conditions (4 .S.IOS) lead to the equality constraints (4.S.99), i.e.,

x*(k)=x(k). u*(k) = u(k) (4.S.110)

The first of the two conditions in (4.S .106) leads to an expression for cy(k) , I.e ..

H ( k ) = {F\T ( X * , u * , x, (k + 1 , k ) ) + G~ ( x * , II * , u , (k + I , Ie ) )

+ D\T( x*, u*, (k + I, k »} p (k + I) (4.S .111)

Hierarchical Control of Discrete-Time Systems 173

where

~,(- ) = 0 Z (- ) /0 x* ~ 0 {A ( x* , u* , (k + I , k ) ) x ( k ») / 0 x*

Gx(-) =iJy (-)/iJx*~iJ{B(x*,u*,(k+l ,k» u(k»)/ox* (4 .S.112)

Dx (') = oc(x*, u*,(k + I , k»/iJx*

Finally, in order to obtain an expression for f3(k), the second condition in (4.S.106) yields

f3 (k) = {F,;r( x*, u*, x, (k + I, k » + G;;( x* . u* , u, (k + I, k » + D,;(x*, u*,(k + I , k»}p(k + I) (4.S.113)

where F,,('), G,,('), and D,,(') are derivatives of the expressions in the brackets of (4.S.112) with respect to u*. The following algorithm sum­marizes the costate prediction method.

A f~orit"l11 4.7. Costate Prediction Approach

Stcp 1: Set iteration index / = I and guess vectors 1'1, x*', u*'. and f31 .

Step 2: At the first-level, substitutcP'(k), x*'(k), u*'( k) , and (3'(k), k = O, ... ,K - I , in (4.S .107)- (4.S.108) to obtain u;(I.;,) and x;(I.;) for i = 1.2, ... ,N. Simi la rly, use (4 .S.111) to find H'(k).

Step 3: At the second-level, use xi(k), !liCk), and (4.S. 109)- (4.S.113) to update coordination vector, q' = (pi, x*', U*' , f3') , i.e .,

P:+ I(k) = Q,xi(k) + AJ'(x*, u*, (k + 1, k» pi+ 1(1-: + I) + a;(k.)

X*f.lI(k) = x'(k)

11 *' 1 l(k) = 1I '( k) (4.S .114)

f3 '+ I (k) = { F.,T( . ) 1«·" /I". x') + G,;( . )1(.\' ·,. ,,". 1/) + D,;(- ) /(.\" ', I'- ')}

xp(k+I)

Step 4: lfql+ l(k)=ql(k)fork=O,I, ... ,K-I,stopandu'+ I(k)isthe optimal control. Otherwise go to Step 2.

Before this algorithm is illustrated by a numerical example, a few com­ments on the costate-prediction method are due. First, since the costate vector P'(k) is a component of the coordination '/ector ql(k). the first-level problem (Step 2) involves simple substitution and docs not require the solution of a TPBV problem or even a Riccati equation. Moreover, the second-level problem (Step 3) is also ra ther trivi a l, requiring the substitution of x, U , and f3 vectors which have been obtained from the previolls iteration . Therefore, unlike the other multilevel formulations, such as the goal coordi-

174 Hierarchical Control of Large-Scale Systems

nation and interaction prediction approaches, both the first- and second-level problems are mere substitution problems. The only question remaining is the nature of and/or conditions for the convergence of costate prediction. Hassan and Singh (1981) and Singh (1980) have proved theorems by which an open interval of time is obtained to guarantee convergence of the algorithm for the continuolls-time case (see Problem 4.12). Simi larl y for the discrete-time case, Hassan and Singh (1976) in an unpublished report have given some conditions for the convergence. They have shown that Algo­rithm 4.7 converges uniformly over an interval (0, K - I) if the fo llowing conditions hold: (i) g, y , Z , and C are bounded functions of k ; and (ii) z, y, and c in (4 .5.112) are differentiable with respect to x* and u* for each o ~ k ~ K - I and their derivatives are also bounded functions of Ie (Singh and Titli, 1978). Further discussion on costate coordination is given in Section 4.6.

The following example deals with an open-loop power system consisting of a synchronous machine connected to an infinite bus through a trans­fonner and a transmission line. This system, in continuous-time form, is treated in detail in Section 6.3 and has been treated by many authors (Mukhopadhyay and Malik, 1973; Jamshidi, 1975 ; Hassan and Singh, 1977). For the sake of the present discussion, a discrete-time formulation of the original continuous-time model (lyer and Cory, 1971) considered by Singh and Titli (1978) is used.

Example 4.5.4. Consider a sixth-order nonlinear di screte-time model of the open-loop synchronous machine system :

xl(k + I) = x ,(k)+0 .05x 2(k)

x 2 (k + I) = (1-0.05c , )x 2 (k) - 0.05c2 x , (k)sin x ,(k)

- 0.025c3sin 2x I (k) + (0.05/ M )X5 (k)

x,(k + I) = (I -0.05(4) x3 (1e)+0.05x6(k)+0.05 cs cosxl(k) (4.5 .115)

x4(k + I) = (1 - 0.05K3)x4(k)+0 .05K2x 2(k) +0.05Klul(k)

xs(k + I) = (I -0.05Ks)xs(k)+0.05K4x4(k)

x(,(k + 1) = (1 - 0.05K7 )x c, (k)+O .05Kc,u 2(k)

where ( CI ' (' 2 ' c1, c4 , cs) = (2.1656, 13 .997, - 55.565,1.03,4.049) , (KI' K2 , ... ,

K 7) = (9.4429, 1.0198, 5,2.0408,2.0408, 1.5,0.5) and M = 1. Apply the costate prediction approach of Algorithm 4.7 to satisfy (4 .5.115) starting with an initial state X7(0) = (0.71050.04.20.80.80.5) while minimizing a quadratic cost fUll ction

1 <)

.I =± t { QII(xl(k) -X/ I) 2+Q D( X3 (k) - xn )2 k - ()

+ RII(1I1(k) - u/I)2 + R n (u 2(k) - 1I12 )2} (4 .5.116)

where QII = QJ] = 0.2 and RII = R n = 1.

Hierarchical Control of Discrete-Time Systems 175

SOLUTION: In order to apply Algorithm 4.7, system (4.5 .115) must be reformulated in the form of Equation (4.5.98), i.e.,

x,(k + I) = x ,(k)+0.05x ! (k)

x 2 (k + 1) = (1 - 0.05c , )X2 (k) -0 .05c2x)( k) sin x7(k)

- 0.025cJ sin 2x7( k) + (0.05 / M )X5 (k)

x 3 (x + I) = (1 - 0 .05c4)x 3 (k)+0.05x~(k) + O.05cs cosx l· (k) (4.5 .117)

x 4(k + 1) = (1-0.05K3) x4(k)+0.05K2 x ~ (k) + 0 .05Klul(k)

x s(k + 1) = (1 - 0.05Ks)xs(k)+0 .05K4 x! (k)

x 6(k + 1) = (1-0 .05K7 )x6(k)+0.05K6u2(k)

Additional equality constraints are

x7(k) = xJk), i = 1, ... ,6 (4 .5.1 IS)

This problem was simulated on an HP-9 845 computer using BASI C. The necessary conditions of optimality through the Hamiltonian formulatJ on leads to the following control, costate, and f3(k ) vector equations:

uJk) = u/ I -0 .05K ,P4(k + I)

1I 2(k) = 1I12 -0.05K6P6(k + 1)

P I ( k ) = 0 .2 ( x I (k ) - xII) + PI ( k + 1) + 0: I ( k )

P2 (k) = (I - 0.05c l ) P2 (k + 1) + 0: 2 (Ie)

P3(k) = 0.2(x3(k) -x/3 )+ P3(k + 1)(1 - 0.05c4)+ 0: 3(k)

P4 (k ) = (I - 0 .05K 3) P4 (k + 1) + 0: 4 (k)

Ps(k) = (I - 0.05Ks) Ps{k + 1)+ lX s (k)

Pc, (Ie) = (1 - 0 .05K 7) P6 (k + I) + lX6 ( k )

withp;(K) = O, i = I, . .. , 6, and

(4.5 .119)

(4 .5 .120)

lXl(k) = - 0.05c2P2(k + l)xr(k)cosx7(k)-0.05c3P2(k + 1)

. cos2x 'i(/..: ) -0.05csPJ (k + 1 )sin x7Ud

lX 2(k) = 0.05(PI(k + 1) - K2P4(k + I))

lX3(k) = -0.05c2P2(k + l)sinx f (k)

lX4(k) = 0.05K4Ps(k + 1) (4.5.121)

lX s(k) = (0.05/M)P2(k + I) lX6(k) = 0.05P J(k + I)

(4. 5. 121)

In order to compare the results of this algori tbm to those reported earlier (Hassan and Singh, 1977), the same values of parameters and initial values were chosen. Here p(k) = x*(k) = 0 was chm-:en, and the algorithm con­verged in 86 second-level iterations. Figure 4.22 shows the optimum

176

u

I 1 .

4

Pe rind . k

2.0

10

\"~ 0.0 -

.' .0 .. . ....... 1 _ _ .J _ ... _ I . ____ J

I) c.1 " 5 Period. I,

4 ' .-

3.4

1 ' - .. ---~ __ l __ . L ___ l __ J . " ' Il I 3 4 5

Period .•

D.R -

0.4

. O. C ----L---'-__ J'----.JIL---' () I 2 .J 4

Per iod. k

0.0 - O. I '::-0 --'-- ---'-2- ---.l---.J

4---l

Period .•

0. 5

0.4

0.3

0.2

0. 1 -

D.D --- - J._ ... __ L __ L __ L _ _ J o I 2 3 4 5

Pc ri od.k

o

- 0.0004

- 0.0012

--I_--'-_--1I_~_...J

o I 2 3 4 5 Peri od . •

0.003

112 0.001

- O. 00 10 L. ---'---..L2---'3---'4L....-.-J

5 Period , k

Figure 4.22 Optimum timl' responses for the opcn-Ioop power system of Example 4.5.4.

Discussion and Conclusions J77

responses for states and controls, which turned out to be identical to those reported earlier.

4.6 Discussion and Conclusions

In this section, the open- and closed-loop hierarchical control of continu­ous-time systems and hierarchical control of discrete-time systems will be discussed and compared. Some aspects, such as computer time, storage, information transfer requirements, and potential practical applicability. will be discussed and an attempt is made to point out advantages and disad­vantages of each scheme.

The first method considered in this chapter was goal coordination. based on the interaction balance principle. The computational effort in the hierarchical control of a large-scale system reduces to that of a set of lower-order subsystems and a coordination procedure. The computational requirements of this method at the first level are normally much less than those of the original large-scale composite system. However, the overall computations depend heavily on the convergence characteristics of the second-level linear search iterations, e.g., gradient, Newton's, conjugate gradient, etc., methods. Since between each two successive second-level iterations, N decoupled first-level problems must be solved, the slower the second-level problem's convergence is, the more times the N first-level problems must be solved. In practice, one may use multiprocessors [or the first level in an attempt to save computational time for the N subsystems computations. Although this seems to be a good proposition and has been suggested by others (Singh, 1980), not many in the published literature support it (Sandell et a!., 1978). An exception to this is due to Ti tli et a!. (1978), who have used multiprocessors in the hierarchical can trol of a distributed parameter and a traffic control system. A major improvement in computations can be made when the first-level problems are both smaller and simpler. This latter observation is illustrated for coupled nonlinear systems with and without delays in Chapter 6. From the computer storage point of view, no significant limitations exist for the method. The scheme can handle constraints o[ inequality type on both the state and control, and, furthermore, the coordination at the second level has been shown (Varaiya. 1969) to converge to the optimum solution. The main disadvantage of goal coordination stems from the numerical convergence at the second level and its adverse effects on the first-level calculations mentioned before. Another shortcoming is the inadequacy of methods to find a ncar-optimum control in an attempt to avoid too many second-level iterations. A near-optimum control without convergence at the second level would be an unfeasible solution. Yet another difficulty is the need to add a quadratic term z7Sz of the interaction vector z in the cost function to avoid singular solutions. This

178: Hierarchical Control of Large-Scale Systems

problem, as demonstrated in Section 4.3.3, can, however, be avoided in two different ways.

Another hierarchical control technique discussed was the interaction prediction method. Based on its development in Section 4.3.2, it is clear that the second-level problem is much simpler here than in the goal coordination method. Furthermore, no possibility for a singular solution exists and the computat~onal experience of this author and others (Singh and' Hassan, 1976) ven.fy that the convergence at the second level is reasonably fast. Comp~tatIOnally, the interaction prediction method compares very favor­a?ly wIth the goal coordination approach. First from the storage point of VIew, the two methods are very similar. Both can be simulated in such a way so as to save on sto.rage requirements by using a common block of memory for both levels. This was done for almost all illustrative examples of this chap.tee. . From . the computational viewpoint, for a number of reported appitcatlOns (SIngh and Titli 1978, Singh 1980), including the 12th-order system of Example 4.3. I, the interaction prediction method took on the order of one-quarter of the computer time of goal coordination. It should be mentioned, however, that the extent of computational benefits resulting from ~o~h the interaction prediction and goal coordination as compared to the ongInal large-scale composite system depends on the format of the decomposition process. For example, if the 12th-order system of Example 4.3 .. 1 had been decomposed into two sixth-order systems, the computational savIngs woul? not h~ve been that appreciable. Another important point concerns the InfOrmatIOn structure between subsystems. An argument which can go against both the goal coordination and interaction prediction meth­ods is the . need for subsystems' full state information in a feedback configuration. Still another argument against the hierarchical control meth­ods has been that they are insensitive with respect to modeling errors and component failure (Sandell et aL, 1978). Part of the latter difficulty can be ~andled through th~ hierarchical control strategies for structural perturba­tIOns developed by Siljak and Sundareshan (l976a, b).

The feedback control based on interaction prediction discussed in Section 4.4.1 has the advantages of being able to handle large interconnected systems regulator problem with a complete decentralized structure while the feedback gains remain independent of the initial conditions. The main disadvantage of this scheme is the extensive amount of off-line calculations for the finite-time case, as suggested in Section 4.4.1.

The feedback control based on structural perturbation presented in Section 4.4.2 is perhaps one of the first attempts to come up with a c!osed-loop control ' law for hierarchical systems. The method is relatively SImpler than both the goal coordination and interaction prediction methods in that no second-level iterations or even simple updatings are involved . Moreover, no large-scale AMREs must be solved, but rather only a

Discussion and Conclusions 11 ':1

Lyapunov-type equation, i.e., (4.4.41). Although the solution of a Lyapunov equation is much easier than a Riccati equation, when the system order is very large (n > 200), finding an efficient method with reasonable storage as well as computer time is questionable (Jamshidi, 1980). It has been argued by Sundareshan (1977) that all calculations are done off-line in contrast to the goal coordination which involves extensive on-line iterations on the first-level subsystems calculations. In this author's opinion, the best situa­tion would be a balance between on-line and off-line calculations. The most interesting outcome of feedback control based on structural perturbation is that it takes possible beneficial effects of interactions into account. The establishment of a class of interconnection perturbations, which provide a foreseen effect on the overall system cost function, is a very useful flexibility for the coordinator. In spite of these advantages, it becomes clear that when the interconnection matrix Gis nonbeneficial, the near-optimality index p in (4.4.45) may become very large. Under such conditions, it would .be desira­ble to have an additional global controller, such as those of Siljak and Sundareshan (1976a), which would neutralize the effects of the interactions. In all, the feedback control based on structural perturbation is an effective step in the right direction for closed-loop hierarchical control, although the total state information structure still remains the same.

In Section 4.5, the hierarchical control of discrete-time systems was considered. Five approaches were presented. In Section 4.5.1 the continu­ous-time version of goal coordination approach was extended to linear discrete-time systems. The computational behavior of this scheme is very similar to continuous-time goal coordination, which is heavily dependent upon the second-level problem'S iterations. Moreover, both methods require the quadratic term in the cost function to avoid singularities. The only advantage of discrete-time over continuous-time goal coordination (Section 4.3.1) is that its first-level problem is very simple, as is evident from (4.5 .21 )-(4.5 .23).

The time-delay algorithm discussed in Section 4.5.2 is perhaps the most appropriate goal coordination method for practical problems. The applica­tion of the algorithm to traffic control problems (Tamura, 1974, 1975) is excellent evidence of this. This algorithm as well as goal coordination have been used for the optimal management of water resources systems by many authors (Fallaside and Perry, 1975; Haimes, 1977; Jamshidi and Heggen, 1980). Perhaps the most striking benefit from the time-delay algorithm of Tamura (1974) is the fact that it avoids the solution of a TPBV problem, which involves both delay (in state vectors) and advance (in costate vectors) terms. This type of TPBV problem is a common difficulty with optimal control of time-delay systems by the maximum principle, which is discussed in Chapter 6, where a near-optimum solution is proposed. Further advantages of the method are the convenient handling of inequality con-

l80 Hierarchical Control of Large-Scale Systems

straints and avoiding the possibility of singular solutions, which is not always possible in goal coordination procedure.

In Section 4.5.3 the continuous-time version of the interaction prediction approach was extended to discrete-time systems for the first time in the author's best recollection. The first-level problem constitutes the solution of the discrete-time Riccati equation, which can prove to be rather time­consuming in a real-time situation. This scheme, tested numerically in Example 4.5.3, seems less effective than the costate prediction method of Section 4.5.5. The reasons for this are the convenient solutions of the costate prediction's first- and second-level problems. More specifically, for the system of Example 4.5.4, the first-level problem consists of straight substitu­tions on the right sides of 14 equations (six for states x, six for Lagrange multipliers (3, and two for controls u), while the second-level problem involves six substitutions for the costate p(k). The computational effort per iteration for this approach is a small fraction of that required for the other methods, such as discrete-time goal coordination (Section 4.5.1) and interac­tion prediction (Section 4.5.3). The costate prediction scheme can be easily extended to the linear case (see Problem 4.8) or continuous-time case (Problem 4.12). Jamshidi and Merryman (1982) have extended the costate­prediction method to take on continuous-time and discrete-time nonlinear systems with delays in state and control.

The other scheme considered in Section 4.5 was the structural perturba­tion approach of Section 4.5.4, which is computationally more attractive than goal coordination or interaction prediction due to its convenient scheme for obtaining a global (corrector) component for control vectors. Also, the solution of a discrete-time Riccati equation is sought only once for the N subsystems. In other words, this scheme, as in its continuous-time version (Section 4.3.1), does not involve any iterations between first- or second-level hierarchies. The main difficulty with almost all of these hierarchical control schemes is that their convergence is not yet a settled issue. The only exception is perhaps continuous-time goal coordination (Varaiya, 1969). More discussions on hierarchical methods are in Section 6.7.

Problems

C4.1. Consider a two-subsystem problem,

: . .

Problems 181

C4.2.

C4.3.

Use the two-level goal coordination Algorithm 4.1 to find an optimum control which minimizes

Q=diag(I,2,2,1), R=/2 , and !:l.t=O.l. Use your favorite computer la~­guage and an integration routine such as Runge-Kutta to s~lve the aSSOCI­ated differential matrix Riccati equation and the state equatIOn.

Algorithm 4.3 represents a goal coordination proccdure for the hicrarchical control of a large-scale discrete-time system. Consider the system

0.Q2 0 -0.2 0

o -I o 0

with a cost function

4 4

J=.!. L xi(5)QiXi(5)+~ L [xT(k)Q(k)x(k) + uT(k)R(k)u(k)] 2 i = 1 k=O

where Q(k) = 14 and R(k) = 12. Find the optimal hierarchical control of this system by decomposing it into two second-order subsystems.

For a second-order discrete-time delay system

where

XI (k + 1) = 2xI (k)+ XI (k -1)+ 111 (k )+0.5112 (k - I)

x2 (k + 1) = - 2xI (k) - x 2 (k)+ X2 (k - I) + u l (k -I)

-u2(k) - u2(k-l)

[~] ~ X(k)~(;]' (=:1~u(k)~(:] J=~xT(7)(~ ~]X(7)

~] x (k ) + u T (k ) ( 065

182 . Hierarchical Control of Large-Scale Systems

Problems (continued)

with initial conditions

x(k)=u(k)=O, -1~k~O, xT(O) = [1 I].

use the time-delay Algorithm 4.4 to find the optimal sequence u*( k), k = 0, I, ... ,6. It is possible to solve this problem analytically; however, a computer implementat~on is more desirable for higher-order systems.

C4.4. Consider an interconnected system

x = r ~ h -~ ~~ ~ ~ F --~-! --= q'j x + r H ~ j u 0.1 - 0.2 1 0 - 2 0 01 I 0.4 0.1 I - 0.5 0 -4 0 I I

with cost function

2

J= .L ~ [xT(2)Q;X;(2)+ l\xT(t)Q;x;(t)+ u;(t)R;u(t») dt] I - I 0

with Qi = diag(2, I, I, I, I), R; = 12 , and initial conditions XT(O) = [ I 0 0.5 - I 0], 6.t = 0.1. Use the interaction prediction Algorithm 4.2 to find the optimal control u*( I) for the above problem.

4.5. Prove that the control law (4.4.53) is a stabilizing controller for the large-scale system (4.4.22). [Hilll: F= K + k is a positive-definite solution of AMRE (4.4.47) for the perturbed system (4.4.46).]

4.6. Repeat Example 4.4.2 for the following system:

and a cost function

4.7. For the system

XI=XI+U I, xl(O)=1

x z =2xz +uz , x 2 (0)=1

J -.! ('Xl (2 z + z 2 Z ) - 2 Jo

X I x 2 + ul + U z dt

X I = X I + ax z + u I ' X I (0) = I

xz=xZ+bxl+u Z' xz(O)=O

Problems 183

and a quadratic cost function with weighting matrices Q = R = 12 , find regions in the (b - a )-plane where the interconnected syslem can be beneficial, neutral, and nonbeneficial.

4.8. Extend the costate prediction approach of Section 4.5 .5 to a composite interconnected linear discrete-time system

4.9.

C4.10.

x(k + I) = A(k)x(k)+ B(k)u(k)+C(k)z(k)

where A and B are block-diagonal and Cis block-antidiagonal and z(k) is the interconnection vector.

Use the costate prediction Algorithm 4.7 to find the optimum control (or

( ) [0 .9 0 .2 .. ] ( ) [ O. I

x k+1 = 0.1 O.I-O.lxl(k) x k + 0

wi th x (0) = ( 10 5) T and a cost function

1 20

J='2 L (0.lx~(k) + 0.2xi( k) +0. lu~(k)+0.2u~(k» ) k~O

For a system with the matrices

r

-5

A = ~.17 . 0 - I

0.2 -2

o - I

o

0.5 0.2

-I o

- 0.5

0 .1 o o

- 0.5 o

r1' B= r~ ;1 -1 0 0

with Q = diag(l, 1,2,2,2) and R = 12 , find a hierarchical control law based on the structural perturbation method of Section 4.4.2. What is the value of the cost function deviation p? Use initial state x (O) = [ 1 I I 1 If·

4.11. Usc Algorithm 4.6 on the structural perturbation approach for discrete-time systems to find the optimum control for

with x(O) = (1 I )T and quadratic cost matrices Q = 12 , R = 0.512 ,

~ ..

~ I' . .:.~. j' .~. I' •

I I

oj

,;1

:·'·'1

i I ,

'1.

184 Hierarchical Control of Large-Scale Systems

Problems (continued)

4.12. Extend the discrete-time system's costate-prediction method of Section 4.5.5 to the continuous-time system

x(t) = f(x(t), u(t), t)

J = i x T ( If) Qx ( t f) + i fol, [ X T ( t) Qx ( t ) + u T ( t ) Ru ( t )] dt

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