Large Scale Systems - Jamshidi - Free Download PDF Ebook

NORTH-HOLLAND SERIES INSYSTEM 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 i i ") (. I'. j ' \ " " \ '. ' Z-" I ~ - ; ' " ""', () < " ~ J (--; NORTH-HOLLAND, New York' Amsterdam' Oxrord / I Elsevier Science Publishing Co .. Inc. 52 Vanderbilt Avenue, New York, New York 10017 Sole distributors outside the United States and Canada: Elsevier Science Publishers B.V. P.O. Box 211. 1000 AE Amsterdam. The Netherlands ©1983 by Elsevier Science Publishing Co., Inc. Library of Congress Cataloging in Publication Data 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 'i! . :.' ' .. 1 I 1 ~ l' ~ " . . , ~ 'j ~ j 1 j " ;! I t j' To three who meant the most in my life My Mother My Wife lila My Brother Ahmad ! i I' ; ;, ' ~ ) . 1 ,j, ~ I ! : . \. 6 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. '. 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. " : ~ I ',r" ':J j. " J :' f r Problems 5 4. 5. 6. 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 large- scale 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 information (be it a priori or sensor information) and the Itself are localized at a given center, very often a geographical pOSItlOn. . A notable of most large-scale systems is that centrality fails to hoi? to 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 (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 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 . a given system into a number of subsystems for computa- efflclency and design simplification. The idea of "decomposition" was first treated theoretically in mathematical programming by Dantzig and (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" "decoupled." The coupled approach keeps the problem's structure and takes advantage of the structure to perform efficient computations. The "compact basis triangularization" and "generalized upper bounding" are methods (Ho and Mitter, 1976). The "decoupled" 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 I f ..: I ' i I i 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 "interaction" 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 highl y 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 "large- scale system." Many viewpoints have been presented on thi s 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 reasons (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 l11ultivari- ahle 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 represenl 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 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 domai n technique for the simplification of linear dynamic systems. lilt . 1. COlltr. 20:7 1-79. Rao, A. S., S. S., and Rao, S. V. 1978. On simplifi cation 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 functions. lilt. 1. COlltr. 23:403-408. Shamash, Y. 1974. Stable reduced-order models using Pade-type approximations. 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 approximation to allow retention of domi nant modes Jilt. 1. COlltr. 2 1 :257- 272. Shamash, Y. 1975e. Multivari:lble system reduction via modal methods and Pade approximation. 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 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 of dynamic sys tcms. JIII..!. COlli. 14:111 -- IIH. Sinha, N. K., ,md Berczani , (;. T. 197 1 b. Opti mum 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 hi gh order linear systems using low order models. Meas urelllelli Control 3:TI - T9. Wall, H. S. 1948. Analytic Thcory of Continued Fractions. Van Nostrand, Ncw York. Wright , D. J. 1973. The continued 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. COlll r. 18:455-460. Chapter 4 I-lierarchical Control of Large-Scale Systems 4.1 Introduction The notion of a large-scal e 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 or smaller subsystems serving particular functions and shared resources and governed by interrelat ed goa ls and constraint s (Mahmoud, 1977). Although interaction among subsystems can take on many forms, ()[11! of the most common one is hi erarchi cal, which appears somewhat natural in economi c, management , organi zational , and complex industrial sll ch as steel, oil , and paper. Within thi s hierarchi ca l st ructure, the terns are positioned on levcls with different degrees of hi erarchy. A s lIhs\ lstem at a given level control s or "coordinates" the unit s on the it and is, in turn, controll cd or coordinat ed by the unit on the h:vel illl - mediat ely above it. Figure 4. 1 shows a typic;t1 hierarchical (" multil evel" ) system. The hi ghest level coordinator, sometimes ca lled the sliprellllll coordinator , can be thought o f as the hoard of directors of a corporati on. whih: another level's coordinators may he the president , vice-presi dent s, direcl\) J' s, etc. The lower level s can be occupi ed by plant managas, shop mana gers, etc., whil e the large-sca le sys tem is the corporati on itself. In spite of thi s seemi ngly natural represen ta ti on of a hi erarchical struc ture, it s exac t hehavior has not been well understood mainl y due to the fact that very litlh: quantitative work has been done on these large-scale sys tems and Si mon, 1958). Mesarovic et al. (1970) presented one of the earliest formal quantitative treatments of hi erarchical (multilevel) systems. Since then, a great deal of work has been done in the fi eld (Schoeffler and Lasdon 1966' Beneveniste et aI. , 1976; Smi th and Sage, 1973; Geoffri on, 1970; 197 1; Pearson, 197 1; Cohen and lolland, 1976; Sandell et aI., 1978; Singh, 104 CONTROL ACTION Hierarchical Control of Large-Scale Systems A / '\ / '\ / "- / "- / \ / \ \ \ \ \ \ \ \\ DECISION-MAKING UNIT \ PERFORMANCE RESULTS '\ \ '\ \ INPUTS L-_ ________ ___ _ _ _ OUTPUTS Figure 4.1 A hierarchical (multilevel) control strategy for a large-scale system. 1980). For a relatively exhaustive survey on the multil evel 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 .and 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 iterati vely. 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 hi erarchical (multilevel) systems depending on the model parameters, decision variables, behavioral and en- Introduction lOS vironmental aspects, uncert ainties, 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 st ructure 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 optimi zation (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 multil ayer 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 suhprobl ems. 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 functional divi sion is possihle (Singh and Titli. 1978). In this divi sion a given subsys tem's functiollal optimization prohlcm is decoll1poseJ into a finite numher of singlc-pmallleter optimi7,ation problems at a lower level and results in a c(\llsidcrahlc reduction in computational e ffort. This seheme will be dlscllssed in Section 4.5 In connection with the hi erarchical control of dislTl,tc-tilllc sys tcms. Before th e of 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 erarchi cal systems is descrihed in Section 4.2. Section 4.3 IS concerned \\ilh the control nr continuous-time hierarchical systems where cO(1rdination betwcen two levels are considered. The closed-loop hierarchi- Cl i C(1 nlrol of large-scale is di scussed in Section 4.4, including the of " intnacti()n predict inn" and "structural perturhation." [n Scc- ti(1n 4.5. the structural perturbation and interaction prediction approaches are extended t(1 take 011 the discrete-time case with and without delay. ;\ mono 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 and Singh (1976a, 1977;). A number of numerical examples illustra te the various techniques presented. The near-optimllm design or linear and nonlinear hierarchical systems are di scussed in Chapter 6. Section 4.6 is devoted to further di scus-sion and the evaiuation of hi erarchi cal 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 erarchically 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 Illethod) and gO(l /-coorriil1(lliol1l11cl/rod (or "dual-feasible" method) (Mesarovic et aI., 1(69). In the next 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 cti on be decomposed into two subsystems, i.e., and (4.2.3) (4.2.4) where Xi, 1/ i, and 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 still interconnected. 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/=w i , i=I,2 (4.2.5) Under this situation the problem (4.2.1 )-( 4.2.2) may be divided into the following two sequential problems: First - Level Problem - Subsystem i Find KI(w) = minl,(xi. u i • Wi) x', u' (4.2.6 ) (4.2.7) Second - Level Problem minimize K( w) = KI (IV)+ K2 (w) IV (4.2.8) The above minimi za tions arc to be done, respectively, over the follO\ving feasible sets: i = I,2 = {Wi: K I ( 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 \I' 1,(11') = "I(W) + A' 2 (\\, ) J2--- '111111 i l(\I. II I.j , l) , ,/II <;; 1I!' \ed to ,I (\ 1, /11 , H, I, 1\ 1 ) -:=- U ------- ---- r'llt\ A 2 (\I') = mill i l ( X 2 . Ill, \\,1) ,,2, 11 2 s. ub,;d \0 1 ., /1 (\ ' , 11 _. \\ ' . \\ ,- ) () Figllre 4.3 A two-levcl soluti on of a sLatic optimization probl cm 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 are fi xed hy adding a constraint to the mathematical Ill ode\. this procedure is called model coordination. In other words, due to the fact that all intermediate 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 var- i;, 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 stati c optimi za tion problem (4.2.1) - (4.2.2). In the goal coordinati oll method the int eractions 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 dCllot ed by z'. Duc to the remova l pi' a,'1 lillks het\veen subsys tems. it is clcar that Vi =1= Z i. Under thi s condition. :' acts as an arhitrary variable and should be chosen by the nptillli zing subsystellls like .Y. 11 . and y . Moreover, the optimization problem cOllsidered in the previous section is completely decoupled into two subsys- telll 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 suhprohlems yield a solution to the original problem. it is necessa ry that the illlcmclioll -halollce prillciple be satisfied, 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-Ievel nrohlcm is to force the first -level subproblems to a for Coordination of Hierarchical Structures 109 which the interaction-balance principle holds. Mathematically, thi s multilevel formulation can be set up by introducing a weighting parameter 0: which penali zes the 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 minimi zcd over the sct So it rcsu lts a function, K(a) = min 1( x,u,y,z ,a) (4.2.15) .,", II .y.ZES o After expanding the penalty term a T (y -z ) = a;()lI- z l)+Ci;e y 2 -.:;2) and considering the relations e 4.2. 11 )- ( 4.2.13). the first-level probl em is formulated as Subsystem 1: min 1,exl, /1 1 , y', Z2)+ aryl - xl, u 1 .rl ,z2 - Subsystem 2: ( 4,2.16) (4.2.17) (4.2.1 8) ( 4.2. 19) The second-level probl em is to manipulate the coordinating variable 0: III order to derive the two-s ubsys 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 int eraction balance l11inJI(x l . 1/I .yl. Z2)+ "' i y l _ ",rZ2 subject 10 fl(x l , 1/ 1 , y l . z2) = 0 l11in h(x 2 , 1/ 2. y 2. z 1) - ", r zl +"'f / subjecl 10 f2(x 2 . 1/ 2 , y2. zl ) = 0 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) 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), ... uT(t) = (unt), unt) , . .. (4.3 .3) (4.3.4) The objective, in an optimal control sense, is to find control vectors u 1 ' u 2 ,,,,,U N such that a cost function J=G(x(t l ))+ 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;( t l )) + 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;(Zj(t) , t) , Xj(t o)= 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 (x j , 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)=Ax(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 rchical Control of Large-Scale Systcms and the h, X I intcraCli()ll vector .V ,:, (1) = L G;; ." / - ' I (4.3 . I 6) is a lincar l"Olllhination of the states of the other N -- I subsystems, and G ' / <I n ", X " , matri x (Singh. I 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.ing + .:,'(f),':), .::, (I)] ill} (4 .3. 17) \\' here <l, <l IT " , X " , positive semidefinite matri ces, 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 , ; = I ThL' phvs ic:li int erpretati on of the last term in the integrand of (4.3.17) is diffi cu lt at this point. In fact. the introduction of thi s term, as will be seen Liter. i, to avoid singu lar control s. The "goal coordination" or "interaction h: li ;l ll L"l'" approach of Mesarvic et al. (1970) as applied to the " linear- LJlladratic" prohlem by Pearson (1971) and report ed by Singh (1980) is now 11IL'SL'llt cd. In thi s decompositi on of a large interconnected linea r system the COllllllon l"\)upling factors among it s N subsys tems are the" interacti on" variables :, ( I). \\ hi ch. ahmg with (4.3. 15)- (4.3.16). constitut e 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 considered to Il(lld . The g. 1()h, d prohlem .1"(; is replaced by a family of N subprohlems ('('nplcd tog.et her through a parameter vector a=(a l .a 2 , ... . a N )"1 and de- Il(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\lnil y 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' pf <y '". the slIhsys tems .\( (\" * ). i 1,2, ... , N, y,jeld the ';(, hlli OIl 10 Sr . ' In terms of hi erarchi ca l control Ilotation. this imbedding ("ll ll l.'!: !)1 is nOlhing hut the notion of c(l() rdination . hut in mathema ti cal j1I(1gr:lIllining pmblell1 tCl"minology, it is dellot ed as the " master" prohlem «('eoi"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 flll])1 thl' ("()()rd inator (second-level hierarchy). solves .\', «<), and trall Slll it s Opcn-Loop Hi erarchi cal Control of Continuous-Time Systcms SECOND LEVEl.. Figure 4.5 Thc two-level goal-coordinati on structure for dynami c systems. 11 3 I: II(ST t.l'VEI.. (report s) some functi on y/ of its soluti on to the coordinator. The coordinator, in turn, evaluates the next updatcd value or IX, i.e. , (4.3.19) wherc 1/ is thc fth iterat ion step sizc, and the upda te ter m iI'. as wil l he seen shortl y, is commonl y taken as a functi on of "interaction error": N e;(a(/), / )=z,(a(/) ,t)- L G,/x,(a(/). 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 the pa ramet er vector a( t) scrves as a set of "dual" variabl es or Langrange Illul ti corresponding to in teraction equality constrain ts (4.3.16). The fund ament al concept behind thi s approach is to convert the original system's minimi zati on problem into an easier maximization problem whose soluti on can be obtained in the two-l evel iterative scheme di scussed 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 X/(f)Qr ·\(t) v ( [ , = 1 0 + l/ ;"( f ) R ,11 , (I ) + .: ;r( f ) S,.:, ( f ) (4.3.22) +2a;'(z,( f) - Gr j"Y/I))l cit) .I _. I J where the parameter vector (\' consists of k Lagr:lI1 ge Illultipliers. In Ihis way the original eOllstrained (subsystems interacti ons) op timi zation problem is ! 11 I Iicra rcliica! ( '() 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 Li as defined by (4.3,24) suh ject to (4,3. 15) and using the Lagrange lllulti pli Crs_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 tllleop- strained nptil11i za tion such ii'StTi e 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 thl' fact that the gradient of <f((t) is defined by N "7 n)I ",._ , .•.. Z,. _ \", b. \'" " L V,r\; = <'i' i = L 2,,,, (4 ,3.25 ) 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'(lll d lc\'elthe vector H is updated as indicated by (4,3,19) and Figure 4.), If a gr<1 di ent (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 Hierarchi cal Control of Continuous-Timc Systems 11 5 gradient defined by dl 'l (f)=el l t(t)+/ ' Idl(f), (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, minimi ze 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 normali zed form Error = ( [ /{ Z; Gi/X j If T {z; - G,jX j } 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(' I1i crarchical Control of Large-Sca le Systel1ls -.-.. -.--- ---] II ·/,'-'----... 1 r---+ l - 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. ( 'oll sidn:t 12Ih-(lrdcr sy.-;lc ll1 illtroduCl:d by Pearson (lIn I) and showlI 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 fllncl 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 ( i U .ll)) Open-Loop Hierarchical Control of Continuous-Time Systems 11 7 with where The sys tem out pu t veclor is given 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 hi erarcllical control strategy through the interacti on balance (goal coordination) approach. SOLUTION: From the schematic of the system shown in Figure 4.6 (clott ed 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 dott ed lines in Fi gure 4.6) constraints given by = [( ZI - XI I) ,(Z2 -XI) , (Z3-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 set o f four third- order matri x 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 ; positi ve-ddinite symmetri c Rieea ti ma tri x and = B, R;-I B/'. T he " integration-free," or " doubling," method of solving the differen t ial matri x Ri ccati equalion proposed hy Davison and Maki ( 1973) and ovcrvicwccl by Jamshidi (1980) was used on an HP-45 comput cr and a BASI C source program. The subsys tcms' stat 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 _ .. _ .•. _.. _ .. . .. .• _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 cti 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 interpolation (Hewlett-Packard, 1979 ) t(1 cvnlllnte appwpriate nUllleri cal integrals. The step size was chosen 10 he (\{ ,- n. 1 as in earlier treatments of this example (Pearson, 1971: Singh, llJ XO) . The 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 report ed 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 l o.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 hiochell1i c<1 l oxygCll demand (BO \)) , * and x 2 is the COIl - c(' ntrati(lll (If di ssolved 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\(O) = (I I - I 1)1. Open-Loop lIierarchieal Control of Continuous-Time Systems 11 Y SOLUTION: The two first -level problems are identical. amI a scconu-oruer matri x Riecati equation is solved by integrating (4.3.33) using a fourth-oruer RUl1ge- Kutta method for 6.t = 0.1. The interaction error for thi s example reduced to about 10 - 5 in 15 iterations, as shown in Figure 4.8. The optimum BOD and DO concentrations of thc two reaches of the river are shown in Figure 4.9. 4.3.2 Int eractio/l Prediction Method An alternative approach in optimal control or hierarchical systcms which has both open- and closed-loop forms is the interacti on predi cti on method hased on the initial work of Takahara (1 965), which avoids second-level gradient-type iterations. Consider a large-scale linear int erconnected sys lem which is decomposed in to N subsystems, each of which is described by 5:; (t) =A, \' ,(t)+B;II ;(t) +C;z Jt), x; (O) ='\;0' i = l,2,. . ., N (4.3 .36) where the interaction vector z; is N z; (t) = L G;r\"j (t ) ./ = 1 (4 .3.37) The optimal control problem at the first level is to find a control [/ , (1) which Figure 4.8 Int eraction error behavior [or river pollution system of Exalllple 4,3,2, \ 50 o __ l __ __ __ __ __ I 3 4 5 r. 7 x <) 10 \1 12 13 14 \) ITERATIONS 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 .. - -: . 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 si'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 Illulti- pliers 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 = - iJllilax i = - Q,x i - A;p, + I: G/-ai(r) p, ( t j ) = 0 (i x;" ( r j ) Q i Xi ( I j ) ) I iJx;( 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) =xjn (4.3.45) N jJ,(r)=-Qjx,(/)- /t;p,(/)+ I: Gjj(X,(t), pj(rj)=Qjx,(l j ) (4.3.46) 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 coeffici ents 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;(l f ) 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 (open- loop) 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 equations 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).', rant'- 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\(' 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) (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 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 (4.J. SX ) It must be noted that depending on the type of digital computer amI 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'\llJd- level 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= __ _ __ -jx+ (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 control for (j = I. SOl tll JON: The system was divided into two second-order subsystems, and steps oUllined in Algorithm 4.2 were applied. At the first step, two independent differcntial matrix Riccati equations were solved 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 --- 1.261 2 O.(}9+0J)(17I -- 0.0271 2 A , (I) = I 2.X7 - 5 .Hllt .- _ - 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 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 \\ere '(l ln'd. At the s<:cond level. the interaction vectors [0: 11 (1), and 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 erations. as shown in Figure 4. 10. The optimum outputs for ( ', = (i I) control signals \vere obtained. Next, for the sake of compari- S(ln , the original system (4.].59) was optimi7.ed by solving a fourth-order tillll' -V; Hving nwtrix Riccati equat ion by hack ward int egration and solved fnrY, (I), i = l.2.],4: and lI , (t), ;=1.2. The output s and control sign;J1s f(lr buth hierarchic;J1 and exact centrali zed cascs arc shown in Figure 4. 11 , Note the relativel y close correspondence betwecn the output s for the (lriginal C\)upled and hierarchical decouplcd systems. However, as one \voldd ,'-"pccl. the two control s are different. When the decoupled the COlltwl signals 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 \ \ \ ____ ____ 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 .. __ __ 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 -- 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 - , _ '------- -- .. -- -l 1 f- Z o u 2 3 / / 4 - - ___ / .. _ . . -lIt ---. - - 112 .-.-112 5'---__ -'- 1 _ _ -'---_---'- 1 _ _ __ __'_ 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=R2 =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, a 2 (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 lJ -r: 0:: I Ii crarcllical COlltrn\ of Large-Scale Systems L'-l r- ::::: III " - () .___________________.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 ction (4.3.17) to avoid singularities. To see thai thi s is ill fact the case, let us reconsider 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 , + Opcn-Loop Hicrarchical Control of Continuous-Timc Systcms 129 As one or the necessary 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 singular 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 singularities at the first leve l. The following example illustrates the two approaches. Example 4.3.5. Consider the following system: ;'1 = - XI + x 2 -I- /II' xl(O) = X ln i: 2 = - X 2 + u 2 ' X 2 ( 0) = X 20 (4.3.68) It is desired to find (u l , u 2 ) such that (4.3.68) is sati sfied while a q uadratic cost function J = 1/2 t ( x + xi + II f + /I n dt o (4.3.69) is minimi zed via thc goal coordination method. SOLUTION : From (4.3.68)- (4.3.69) it is seen that the system can be decolllposed int o two first-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 following Hamilt onian : 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 211 2 (4.3.70) (4.3.7 1) (4.3.73) in which the intera ct ion variable appears linearl y. 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 linearl y in (4.3.73). The followin g system reformu- lations 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 \' ) lIicrarchi cal Control of Larg,c-Scalt: SyS ll' Ill S \yi1ich \\'( )\lIt1 givc the foll (l wing necessa ry conditions for optimalit v <II tl,e fir sl k n' l: t) lI lIl / nll l II, 1 PI ' () = iJ lIl / (I :' 1 = 2HZ I + /) 1 -, j) I ill 'I / il x I co .\ , - PI ' /) I ( I ) = () \' , iIJJI / ()p, = - XI + li l + " I ' .\1(0) = x II) I' ll ' fir st sUhSYS tl" 1l () = = 11 2 1- /) 2 - jJ 2 = (J ll,/ilx 2 =x2 - P2' P2(1) = 0 -' 2(O) =.X20 (4.3. 75 ) (4.3.76) 1'(1(' tlie sl.'l'Ond suhsys tem. Arl er thc introduction of the Ri cca 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 (/)) X 2 (1), X 2 (0) = X 20 ( 1) -- k ( 1 )\' 2 ( 1 ) (4 .3.77 ) (4 .3.n) " ' I! nl' k, ( 1) is the ilil time-varying Ri cca ti Illatri x. The ('( l(lrdin;lli PIl at the sccolld level is achi eved through thc following iteratiol1s: H' I I ( 1 ) = ttl ( 1 ) I- fill I ( e( 1 )) <'(I ) = (4.3 .79 ) (hi s rdorillul at ioll w(luld avoid singul ariti es hut makes the second-level it erati (l l1 s Cl1 nvcrgcnce vcry slow. 4 .. Reformulation 2 Singh cl ;d, (I CI7(,) sug.gcst ,111 alternative formulati on whi ch not onl y avoids sill gll\ :,!ilics hilt a\sll gives gO(ld Cl1n vergcllcl'. The procedure is hased Oil <;f )"'ill g tn! \' ill terms (11' the int eraction vector z and it into the C(ls t fllllL'ti(ll1 : i.e .. Z G ill hc written in general as (4 .. UO) \\ here (; «,Slimed to be nOllsingular and the rdormulated Hamiltonian is 11(·) = 1,,' (/)V\2I1z(I) + \1I 1 (/)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 singular, but " Open-Loop Ili cra rchical Control o f Continuous-Tilllc SyslClll s solution ca n still he found. J-Jere the Hamiltonian is 1I( . ) = + + az , - IXX 2 - PI" ; -I- PIli , + p, ", + lz;- + - P2X2 + P211 2 with first-I evcl subsystem problems bcing 0= aH/Oll 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/Oll 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 subsystem 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 equati on, which would, in erkct , mcan that Lh e solution of a Ri eca ti equation has been avoided for thi s particular example. For the first subsystem, however, following the formul ati 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 evaluated. For thi s exa mple, thc first subsystem problem is .\ I ( 1 ) = _. ( I -I- 2k I ( 1 ) ) X 1 ( 1 ) - 2 g, ( t ) - n ( 1 ) , .Y , (0) = .\ 10 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 equati ons for ",(I) and ;:;1 (1) must be solved backward in time. Thus, whil e 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 reformul at ion would requirc thc solution of .k= Ax -Sp - GQ - '(p +n), .,(O) =x o jl = -- C/'a + AijJ , p(lj ) = O II = - R - 'B Tp , z = - G Q - '( p + IX) (4 .3 .86) which indicates tilat thc cos tat c vector p equation is deeouplcd fr0111 .\ and ca n hc solved backward in time (eliminating a Ri cea ti equati on) and subSt ituted in the lop equat ion to find x. Since the iI. }3, Q, and R matri ces 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 Vz (l» is separabl e 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 dealt with 0pcll -Ill(lp lli era rchi cal control of continuoll s- timc , \', tCI11' ill ,,''' ich tli t.: control 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 the int eraction l" ('" Ii di(l n Ill ,., th(ld whi ch n:slllted a partia ll y closed-loop strll ctlll' e wit h all opell -I(l(l !1 cOlll!1Pllent ",hich still depended nn ,Y(lII)' Although one may ,,1 " <lYS hl' ahlc to ll11:aSIIIT ,\,(1 0 ), hy the time all open-loop con trol is c: tl clIi alcd alld <lpplit.:d to tht.: system. the inili ,d stat e has IllOSt likel y ,·h;1I1 )2,nl. thll s resulting ill lillpredictahle and undesirahle responses, I t is Ihcn: rore \\'orthwhile to C(lllstrul'l closed-loop control la ws whi ch an.: indc- IWlldent ('I' Ihe initial stat c (Singh. 19XO). The 1110st likely case in whi ch such a c(l lltnli structure is p(1ssihl e. as in nonhierarchi cal systelll s. is thc lincar quadratic regulator probl em. Many auth()rs have considered the prohlem of dosed-loop cOlltrol of hierarchic;d sys tems. Mesarovic et al. (I nO) gL's ted a sllil uptill1al structure whi ch had an adaptive feature as the sys tem Sage 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 al. . 1972, 1974) have considered the l'f(lhlcm alld sugges ted either a " partial" feedback controll er or "complete" r" l'( lh;ll'k cnntrnls wh ich invol ve off-lin e calculations of the overall coupled I"rge,sl',ll c sys tem. An atlr;lct ive extension of the partial [eedhack algorithm "I' ('oil cll l' t al. (1 974) whi ch provides " complet e" closed-loop struct ure hased on nrf-line c,llculations or feedback gains and their on-l ine implemen- tati on i , due to Singh (19RO). Another attemp t along thi s line, due to Siljak (Jild SUlldareshan (Silj ,lk and Sundarcshan, 1974, 1976a, b; Sundares han, 19'77: Sil,iak. 197R). is bascd on (1htai ning local feedback controllers for each StlhS\,s t"lll (Ill the first level hy igllnring the int eractions: then a glohal l'olltr(llkr ;11 the sCl'(l nd lew l is applied tll minimi 7.e the interaction errors "" d i 1111'1 (l\\: the (1nrllrll1;lIl Ce. "'"" is l11 et hod elll phasi l.cs the st ru ct u 1' ,11 perturhations cat cgnri 7cd as " benefi cial. " " nonbendicial," and " neutral" illlCl' c(1 I1IllTti ons and their corn.:sponding loss of perrnrmance. III Ihi s sectinn Singh's (19XO) extension o[ the interaction predi ction apIHl l<1 ch and the feedhack C(1ntrol scheme of Sil jak and SUlldareshan (1974. I l 17 (' a. il) alld Sundaresllan (1977) hased 011 structural perturhation along " 'ith Il\I1l1l'J'i cal exampl es are prese nt ed. 4.4.1 ('/o.l' (' d - / ,(}(}fI Co//trol oia ///I emcfi()// Predicti u// 111 II1\' I)f('vi(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 lltroll er ;lI1d an onen- Ioon C( " "I)O- CloscJ -Loop Hierarchical Control or Continuous-Ti me Systems 133 mentioned ea rlier, that thc open-loop component depcnds on the initial state of the system and there is no apparent possibility [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 eliminate 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/( K j ( I ) Xj ( I ) - gi ( I ) ) (4.4. 1) ; = 1 which indicates that the vector K; (t) depends on the states of all the other suhsystems and hence the initial statc x (I,,) . Thc foll owi ng theorcm due to Si ngh (1 \)1)0) relat es the open-loop component (4.4 .2) to the state x ( I) for the overall system which ean be used in a hi erarchical structure of a regu lator problem. Theorem 4.1. Tize open-lool' adjoin 1 vecl ur g(l) rlnel slal e x ( I) ar e refuled Ihrough llze followi ng tranl/orlllation: g(I) = M(tj ,t)x(t) (4.4.3) PROOF: Rewriting the adjoint equations (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 solutions. 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 matri x consi sting of subsystems Ri cca ti matrices. i.e. , K(I) = cliag{ KI(I) , ... ,K;(I), .... K,v (I) }. However. for the composite sys tem, .\:(1) = I1x(I) + BII(/) (4.4.6) with stalldard quadrati c cost .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 ( (4.4.9) 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(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 (), t I) 1'1 [ I ( t I' T) ( K ( T) CG to (4.4.11) By moving the term I (to' t I) inside the integral sign and taking advantage of product property of transition matrices, (4.4. 11) is rewritten g(t o) = M(t l , to)x(t o ) = {{'I(l o , 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 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 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 = - 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 G 2I 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 -2.20 - 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 .66 -87.40 (4420) - 82.881 -104.2 .. 65.70 - 218.4 - 86 .70 1_\ (, I licrarchi eal Conlrol or Larg,c-Scak SySll' IlI S "nd Ihe (l\Tr,, 1\ apprnxi111'1k feedba ck gam matri x hased on hi erarchical u lnl rill l(l he 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(ll1tml or a hierarchi ca l sys tem di scussed here is 0 11 the fe ed hack control suggested hy Siljak and SI11HI :ll l'S\i;1I1 (\474. \Y7(' ;1.h) ;1I1d ex tensions by Sl1llllaresh;\1l (19Tl ) h;lscd (' 11 <;1 111 1."1'1 1;11 j1er tlilhati llll s due to the inl eractions ;lInong suhsystel1l 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 the h:1Sil' i1\ilial issue addn:ssl' d hy Silj,d, al1d SI111l1 ;l1 l's h:J 11 (197(1:1.h) is that assul1ling each suhsys k11l ha s its own independent l(l cal controller. how reliabl e the overall system performance will be in the 11I l'Sl' 1\ Ce of structmnl perturhati oll s. /\ classical example is a "power 1',,"1" ill \y hidl each pOlVer C\lllipany (suhsys tem) is 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 riations in l)(1wer among the companies which would affect tlie entire sys tem' s (wer:"1 performance. Each suhsystcm has a feedback C( 'I11TOI consisting of a " local" component obtained Ihrough sllhs\'stClll ca1culati(1ns :1I1c1 a "global" component which minimi zes int erac- ti (11l -c lIPrs all d possihl v improves the (lVerall syslcm performance. ( 'ol1 sid','r a large-sca le linear time-invariant syst em described by N su hsystems: .,,- \,(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 fil1ding a " local" controller {( ,'(I) which minimi zes an as>(Wi :l1 cd lju:Hlrat ic ("(lS i .I, (I " . . 1, (/ 0 ), {( , ( 1 0 )) = 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 (',/ set to I nn. 11 is " SSl JlIl"" IIi :ll :dtlinll gh (,:ll' h slIhsVs tc' 111 is il1dcpcl1dcllt , 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 complete decentralized informati on structure and each subsystem pair (A j' B,) is completel y controllable. For the decoupl ed subsystem, x ;C/) =Aj-'dt ) + B,uj(t) , i = 1,2, ... ,N (4.4.25) and cost (4.4.23), the decentrali zed optimal control is given by uj'( /) = - R,- iBTKjx,( t) = - PjXj (t) (4.4.26) where K, is the symmetric positive-definite soluti on 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 optimal performance function of the centrali zed 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 Sundarcshan (1977 ) point s out and as we will see shor tl y 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 subsystems with overall and decoupled performances 1"(.) and J*(.) , respectivel y, an interacti on G={g'i}' i,j=I,2, ... ,N, is said to be "nonbeneficial" if J*(- ) < J O ( .). - Definition 4.2. An interacti on G is said to be ir J*(.) > 1"(.). Definition 4.3, The interaction G is said to bc " neutral " if J*(.) = .f 0(' ) . 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 etri c sO/lIl iOIl of ilh slIhs)"slClI1 A/,l'iRE (4.4.27). Theil tli c (iDem/! S\'stCIII 'S Olitilll!ll I ,cr/iil'l)/(/I/ ('c i I/ric. , J" (lll d the 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 ctri c matrix, i .c., S = - ST. I Ilw l'I'<,,,f PI' til l'pr CIll. dill' to SlInd:lresi1:11l (1977) , is givl' ll he low. "I(( lor : I.l'l II " , f1 matri,\ ,. ' he the 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 .... 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 symmetri c positive-definite suilition of AMRE: (4.4.33 ) Np\\, if (4.4 . .12.) and (4.4.33) arc comparcd, it fnlh1ws that J( (4A. .'n ) ;l!ld is nllci sYlllmetric, Therdorc. (4.4.3()) i Ill!, li es tilnt .1"( . ) = 1/2\ / (, (I) Kx ( Ill) =, J " ( . ). ' [ he n.'Vt' rsl' is also true, If F is the symmetric positive-definit e sollition of Ihcll by comparing (4.4.31) and (4,4,33), F = K onl y if KG + ( i' () : hellce, /\(i = _. (,"/\' = ,I.;. a skew-symmetric nwtrix. If we choose ,<; -c, /\ ,\'/\, til l' desirable condition (4,4,30) follows. Q.E.D. rn Ihi s thl' oITI11. as will hc seell by the numerical examples. Illay be used to find :1 l' l;JsS of int cractioll matrix patterns for which the optimum overall SystC!ll C:tll hc achicved by applying decentrali/.ed controls ut (l), i.e" N .\·; (I) = (:1 ; - B,[';)X,(I) + L G;ixi (t) , I (4.4.34) 1'(11' . . . ,N. It is Iwtcd that under condition (4.4.30) the local contr"ller s Il"t (ll1ly optimize the overall sys tem but also stabilize it. II must be l? 111ph:t silCd that if the structural perturbation of the systcm is limited to (4 .4.31)), one ca n ignore the interactions Hlld solve N independent small- 'l': lIc l'whlcllls which pnlVitic both optimal and stahili7.ing control. llo\\, - ncr. in gencral. (; docs not satisfy (4.4.30). and the application of u;"(1) to thc ovnall sys tcm will result in a performance N )(10' X(lo)) = L ),(I n , x,(t o )) (4.4.35) whcrc )'(/0' .\,(1 0 » = J,UI)' X,(l o )' ut Uo)) for the composite system (4.4.34) is greater than or less than .1* , depenuing on whether the intcraction matrix (i is 11(lllhcndicial or hClldieial in the sensc of Definitions 4.1 and 4.2. SiIjak . :t ntl SIIIIlI:trcsl1an (llJ7(l<l .h) have dcterlllineu \)()unds on the resulting Closed-Loop IIicrarchical Control of Continuous-Time Systems 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) 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) (4.4.38) for i = 1,2, ... ,N. The corrective gain matrices fI'l arc obtained through BlI = B"P (4.4.3LJ) where P = diag{ P I ,P 2 , ... ,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{B I , [J2, ... ,13 N }, 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 perturbation matrix BI' is determined, a bound on performance J rcsults. The following theorem, uue to Sundarcshan (1977), provides the mechanism to achieve thi s. 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 LOCAL CON! HOI..LI :({ I LOCAL CONTROLLI'.R N Fil!Il!'p <I.U 1\ closed -loop Illultil evel sta lc regul ation sl rul'lure. 1 ' l'Ol'ir/cs (/ l,cr(orl1/(Tllcc il1(/C.\ • . _ I . 1' . '. . .1(1 0 .\(1 0 )) - :;.\ (t o )( K + k. ) .\(t o ) (4.4.44 ) fill' fh e III ' C/'{/ 1/ .ITs/ell l (4.4.40) . fl.foreovcr. lir e /lpper hO/lnd 01/ lir e {Ie,!or- /11,/1/(',' r/n' i(/Iion fl = (.i -- .1*) j.I' (4.4.45) il ,l! /J ' CI/ 11.\' \I'hl'/'(, A",( ' ) I/I/d A M (') represent Ihe I1lil/IIl1ll111 (Jlld l1/(/xil1ll1l1l oj lire ei,e.cl/llll/ues (1/ tlr eir associated lIIatri.\ argll/)/ clll s. PROO! ': Consider the following 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' ( If -I- n l' ) N I( H + 13 " )r. Clearly, the closed-loop control l/( 1) ..- R I( n -I res ults in a minimulll cost iXJ(/o)Fx(to) for the fn' dhad c(l iltrol sys te!n i ( I ) ( A + (i) \' ( I ) .. V I' h ( I ) (4.4.4X) Ilowl'\'er. since the desired closed-loop system is of the form (4.4.40) , it is seen that for to take on that form the following 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 matri x which sati sfics (4.4.41), sincc (F - K) = (S - KG)( II + G) . I is a symmetric matrix. Now if we let K = ( p - K), then. through 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)) = 1/2x J (I O )Fx(/o) = 1/2x T (/o)(K+ K) x (to). which is the desired cost (4.4.44) for the overall composite system (4.4.40). Moreover. it follows that the suboptimalit y index p satisfies (4.4.45) and is given by p A M (K) / A II1 (K) for al1 t o and x (t o) . Q.E.D. (4.4.52) An immediate corollary of thi s theorem is that for a perturbation matri x /]" 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 tbe proof of thi s corollary. see Probl em 4. 5. This combined global-local controll er extends the earlier work of Siljak and Sundareshan (1974) in the sense that (4.4.53) t,lkes advantage of possible beneficial aspect s of interconnection. The foll owing examplcs il- lus trat e the structural perturbation method. [7urther discussion on the method will be given in Section 4.6. Example 4.4.2. Considcr a simple second-order system, '\:1= 2x 1 +1I 1 , x l(O) = i x 2 = 4X2 + 11 2 , .\' 2(0) = 0.5 with a 2x2 interacti on matri x (4.4.54) (4.4.55) whose el ements 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 thc 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)+ = 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 b O ] (4.4.30) implies that gil = g22 = 0, gl2 = -1.915g 21 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 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 (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)+uC(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) -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= -6. 2 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 = 1 2 • 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 .5tl(, 5 rc.)r\,(O) · - ( 1 (4.4.7:1 ) 0.) )', i 7= 1, 2. The interaction matrix ror .. .. - ...--.--.-.------------ - --------- ------- ,1.14 linn' 1'C'p(1n:<cs [pr Ihc 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(l 1> 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 U O \ I \ . 1 - If\ .(1 2:' . \ \ \ \ \ \ \ " ___ 1' 1 145 . ..j.I__ --'-__ .....l... _ ___ L _ _ ___ LI __ -' 1.5 2 2.5 3.5 4 4.5 (sec ) Ih) \ \ \ \ " ---- '-.. ------ '-- -- ............ _----- - .05 L......J. __ ......J __ -----'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 hil: a l l"lJl lmlni' 1.argc· Sc: 1i r 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 ollhcll d ici:lI . III order t(l rind out \I hi "h ,': 1."'.' it we need 10 lI SC Theorem 4,2 to factor (, as in (4.4.]0). If wc I :,ke ( ; ddincd hy (4.4.7(, ) ;1Ild I( = Bl ock-di ag(K I • thCll the cnrrc- ,\ 111 :1 lri :\ hccoll1 cs () 0 () () () 0 () () (4.4 .n) 2.] (1.3 () () () .. \ 1. 1 f) 0 Iy hi l' h j " 11 (11 :1 SkCI\ ··svtlllll elri c l11:ltri x. il11pl ying lhal the inter:1cli o11 nla tri x. (4.4.7(,) i, !lll i ncutral. In ll nkr to fill d out whether (, in (4.4. 76) is beneficial Ill' 1](1 ll h ' lldicial. The(1 rCl ll 4.3 and Equali on (4.4.4 1) are used 10 fin d ,)' : :ll1 d Ih':11 hv virtue of (4.'1.42). f) . l m .- 0 () 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 11.7 19J7 fm\(O) ·c (I 0. 5 (l .5 )' , whi ch indi - (' :lt e" Ih:lI the (l ri ginal int erac ti on l11:1tri :\ is nonhell eri ci:ti . The (f.'l lrt et'l llr) controll er 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 __ ( ... 4.2 - 24. 2 tl .6] (4.4.XO) \\ ith ; - () .7 19.l7, the perf(l rlllall Ce slihoptillwlil y index f> = O.2.l)(i (J . and "",,,, ,· 1, "'111< 1 " r Il <. I ·n . The above nerfortll ance index IV" S also checked hv ll icrarchi cal Con trol of Di serete-Timc Systcms 147 the ori gin,,1 sys tem's cost functi on by solving a fourth-order J\ MRE, whi ch provided the overall optimal system performancc index as J " = 0.7132, indicat ing that the proposed decentrali zed-global (corrcctor) control pcrfor- Illance index is within less than 1 % of the overall centrali zed optimum cost. To check the ori ginal system's optimal control solution versus the combined decentrali zed global solution, the fourth-order syst em (4.4. 72) was solved llsing and U,, = II * + lI c wi th x (O) = (1 , 0.5, 1,0.5)T and output matri x 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 appl ying 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 cl ose. The slates and outputs of the fourth-order system utili zing two controls are undis- tingui shahly close. Figure 4.14 indi cat es that the structural pertu rba tion closed-l oop cont rol considered here is a good approximal ion. Further commcnts on this suboptimality are given in Sccl ion 6.5. 4.5 Hierarchical Control of Discrete-Time Systems Thus far, the multil evel hi erarchi cal control has been used for linear stati onary continuous- time systems. In practi ce, however, ma ny systems are ncither linear nor continuous-t ime. In fact, many pract ical engineering systems, such as traffi c control, wa ter rcsources, and manufacturing processes. are both di screte and delayed in naturc. Duc to the import ance of such appli calions, many researchers have made signi ficant contri butions in appl ying or extending hi erarchi cal control to both l1 0ndelay di screte-time and time-del ay di screte-t ime systcms. In this secti on the extensions of hi erarchi cal control to such sys tems in both linear and nonlinear for ms arc considered. 4.5.1 Three- Level Coordinat ion for Discretc -Ti me !:>)'sfcli/s Thi s secli on deals with a three-l evel goal-coord inat ion st rategy suitable for di screte-time sys tems and their time-delay modifi cat ions. Such a stra tc!!.v was first proposed by Tamura (1974, 1975 ) and trea tec! further by ( 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)=x o (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= "2 x ;"(K)Q;(K)x;(K)+"2 + 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) 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 \7 a q(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 Thp or<:>rlipnt "f n( R \ tn th P: r.nnti1111()lI s-fimp. 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)+ [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 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 ,-. origi nal large-scale system is obtained (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 ill ustrated by the ...... > r- followi ng example . 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 x 2 (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.. <: .., wi th a cost function .... N ;:l 0::; t) 2 {I 9 1 0 .5 J = "2 x;(lO)Q;(lO)X;(lO) + "2 [x;(k) Qi(k )x i (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", Ri(k) = 0.5InJ, SiCk) = I k , n l = 2, n 2 = 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 computer using .... .;3 BASIC and an initial condition x(O) = (2 -3 4)T. The results were generally .., ,-, .;3 very satisfactory, especially with respect to the interaction 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 opti miza- ...... ;:! "'< .., b - ... tion of large linear discrete-time systems with delays in both state and ...:t: N 0.. >-l control 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)+ .. . +A s x(k-s)+B o u(k)+B 1 u(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:{x T (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" time- delay 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 di scussed 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» = 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{ - BT>-. *(k + i)]} (4.5044) where the I th element of the saturation function Sat x( v) is { X max , { if v{>x max , { Sat x(v{)= v{ if xmin,{« v{«xmax.{ xmin, { 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) = Sat x{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) = [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 +[°1. 5 O][U\(k)]+[O 0.25][U\(k-I)] (4.5.51) ° u 2 (k) I I u2 (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 (4.S.S3) and Q(S) = diag(l , 2), Q(k) = 1 2 , and R(k) = diag(l,O.S). SOLUTION: The problem was solved for an error tolerance 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-l evel 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 = L; = 2. xJ'(k )Q;x;(k) + 1 u ;(k )R;u; (k) + )z;(k) N .... L + 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 A j • 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 -C T P (k + 1) j 1 [ ( )] 1+ 1 I I A; k N zj(k) = {Djjllj(t() + Gjjxj(k)) (4.5.64 ) for k = 1,2, ... ,N j , 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 H j (-) = 1X;(k) Qjx;(k) + 1U;(k) R;uj(k) N + >/i ( k ) z; ( k ) - ( k ) ( D J j U j ( k ) + GFA) k ) ) j=1 + pi( k + 1) [Ajx j (k) + Bju; (k) + CjZj (Ie)] (4.5.65) Then the necessary conditions for optimality would lead to p j ( k ) = () Hj (- ) / () x j ( k ) = Q j x j ( k ) + A rp J k + I) N T - , 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 ( 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 (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, ( N j ) = 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 (k) GjJ' j = 1 Recalling the bounds (4.5.57), the expressions for 11 , (k) i 11 (4.5.68) a nd .\, ( k + I) in (4.5.55) a ft er eliminating {J,(k + I) via (4.5.69), the followi ng expressions arc sugges ted for conlrol and stale vect ors: 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) + 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,(/.; + I) > .l: ,(k + I) (4. 5.75) where L,(k) = L,I(k)A" M,(k) =E, (4.5.76) The di screte-t ime 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 first level, i = 1, .. 'ON and k = 1, .. . ,N/. Hierarchical Control of Discrete-time Systems Step 2: At the first level solve N ma trix Riccati equations (4.5.70) and s tore. Step 3: Solve N adj oint equations (4.5.7 1) and store for k = I , ... ,N/. Step 4: Using the stored 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. , whether their left-hand sides are within E = 10- 6 o f zero. If not, use (4.5.64) to updat e A,(k) and z,( k) , increment 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 thi s section the optimal hierarchical control scheme via s tructural perturbation of Section 4.4.2 for continuous-time systems is extended for discrete-time sys tems. For the sake of discussion, the di screte quantiti es are represent ed 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 may consist or M subsystems: (4.5.77) for ) = 1,2, ... , M and i = 1,2, ... , N. In the above relati on, xl and tI ! arc 11 ,- and m-dimensional state and control vectors of the )th subsystem at i th interva'1. The G jk is the II j X II k interconnection matrix between .the )th and k th subsys tems. The relati ons of Ai' B i , and G i k matnces are gIven below: AI I G I 2 I __ i _ _ l _ _ _ G 2 1 I A2 I G n I _ _ L __ L __ L . I - - - - I AM (4.5.78) The optimal control problem would be to find cont rol vectors Llf, j = 1,2, ... , M, sllch thaI (4.5.77)- (4.5.78) are sati sfied while minimi zing a 164 Hierarchical Control of Large-Scale Systems quadratic cost function N Jl( x{" lin = L (xrQ;x( + Rj u; _ I) (4.5.79) i = 1 Qj and. R; are 11; X 11 ; and m; X Ill ; positive-semidefinite and positive- defll1lte The optimal control problem, in its large- scale composIte IS to find an m-dimensional control u T = (U IT , ... , U MT ) WhICh would satisfy the overall state equation X; + 1 = (A + C)x; + Bu; (4.5.80) A = I ' A 2 ,· .. · ,A,If ): B = Block-diag(B I' B 2 , ... , B M ), and [0,k J.J . k - 1, 2, . . . ,M, willie the cost M J( x", ul) ) = L Jl(x /" (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'+u g 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 = - pi x j 1 1 1 (4.5.84) where and K/ls the positive-definite solutIOn of the discrete matrix Riccati equation: Kj= Q +ATKj (I+SKj )- I A ( ; j .; ; + 1 j / + 1 ;' KN=O 4.5.85) = 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 C e . 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 T B) - 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 neutrali ze 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 algorithm. Algorithm 4.6. Structural Perturbation Approach [or Discrete-Time Systems Step 1: Input all matrices {Aj' B j , Qj' R j , C jk }, initial states 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. 4.5.3. COI?sider a three-region energy resources sys tem shown in Figure 4.19. For thiS system, let the foll owing discrete-time model hold: r IJ I : () : () j Xi + I = l qo?J x, + 1/, Hierarchical Control of Discrete-Time Systems 167 where J m 0.5 0.75 0.5 0.5 0.25 0.5 0 (4.5.92) 0 0.5 0.2 0.25 0.25 0 0.5 0.25 0.3 0.2 H rO 0 0 0 10 0.5 0.5 0 G 21 = l 0 1.5 0 0 0 The quadratic cost function has weighting matrices QI = diag(J 10 1), Q2 = 51 4 , Q 3 = 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 hierarchical control policies, three 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 usi ng an HP-9845 computer and BASIC. Based on the sol utions of these and vector equat ions. three local control functions were obtained. Using the interation prediction method, an acceptable (within 10- 4 ) convergence was reached wi th :line iterations. Using the structural perturbation method, Equation (4.5.89) was used to ohtain the global cont rol component and added to the vector of local controls to fi nd the overall control (4.5.82). The resulting states ami control function via the hierarchical control Algorithms 4.5 and 4.6 along with the optimum 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 optimum 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 computationally 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 - - .. _. - - - - .. _. - --- CTNTRALIZED TRAJECTORY ---- NEAR- OPTIMUM lll ERARCl1 l CAL TRA.1FCTORY (Algorith11l4.6) ! / -'-' - NEAR-OPTIMUM IIIERARCIIICAL TRAJFCTORY (Al gorithm 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 (Al gorilhlll 4 .(1) 0- - . -.- NEAR- OPTIMUM IIIERARClIlCAL TRAJECTORY (AlgorilhIll 4.5) '", 0 L---l __ __ __ _ _ __ 10 15 20 25 30 35 40 45 50 PERIOD. k (b) Hierarchical Control of Discrete-Time Systems 169 .95 .85 .75 .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-_ _ __ _ _ .L I __ L __ __ 35 40 45 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 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 ) + C 2 ( 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 CENTRALIZED TRAJECTORY \ '\ ___ NEAR-OPTIMUM HI ERARCHICAL TRAJ ECTORY (Algorithm 4.6) \ . _. _ . _ NEAR-OPTIMUM HIERARCHICAL 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 (Algorithm 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 u 3 (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 C 2 (-). Therefore, the problem (4.5.93)- (4.5.96) can be rewritten 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)+ 1u T (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, C I , C 2 , 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 , = 1 , = 1 +p/(k + I)[A,(. )x,(k)+ BJ· )u, (k)] + d,)} +cx[(k)[x,(k)-x i (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 Equati on (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 ) ) + ( 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* (4 .S.112) D x (') = 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 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 larly, use (4.S.111) to find H'(k). St ep 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 al, 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 Algo- rithm 4.7 converges uniformly over an interval (0, K - I) if the following 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 consi sting 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) - 0.05c 2 x, (k)sin x , (k) - 0.025c 3 sin 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) x 4 (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 - 0.05K 7 )xc, (k)+O.05Kc,u 2 (k) where ( C I ' (' 2' c1, c 4 , c s ) = (2.1656, 13.997, - 55.565,1.03, 4.049) , (KI' K 2 , ... , 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+QD( X3 (k) - xn )2 k - () + RII(1I1(k) - u/I)2 + R n (u 2 (k) - 1I 12 )2} (4 .5. 116) where QII = QJ] = 0.2 and RII = Rn = 1. Hierarchical Control of Di screte-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) -0.05c 2 x)( k) sin x7(k) - 0.025c J sin 2x7( k) + (0.05/ M )X5 (k) x 3 (x + I) = (4.5 . 117) x 4 (k + 1) = x s(k + 1) = (1 - 0.05Ks)xs(k)+0 .05K4 x! (k) x 6 (k + 1) = (1-0 .05K 7 )x 6 (k)+0.05K 6 u 2 (k) Additional equality constraints are x7(k) = xJk), i = 1, ... ,6 (4 .5.1 IS) This problem was simulat ed on an HP-9845 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) = 1I 12 -0.05K 6 P6(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(x 3 (k) -x/ 3 )+ P3(k + 1)(1 - 0.05c 4 )+ 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) = - 0.05c 2 P2(k + l)xr(k)cosx7(k)-0.05c 3 P2(k + 1) . cos2x'i(/..: ) -0.05c s PJ (k + 1 )sin x7Ud lX 2 (k) = 0.05(PI(k + 1) - K 2 P4(k + I)) lX3(k) = -0.05c 2 P2(k + l)sinxf (k) lX 4 (k) = 0.05K 4 Ps(k + 1) (4.5.121) lX s(k) = (0.05/M)P2(k + I) lX6(k) = 0.05PJ(k + I) (4. 5. 121) In order to compare the results of thi s 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 Perind . 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 Peri od .• 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 Peri od .• 0. 5 0.4 0.3 0. 2 0. 1 - D.D --- - J._ ... __ L __ L __ L __ J o I 2 3 4 5 Pcri od.k o - 0.0004 - 0.0012 o I 2 3 4 5 Peri od . • 0.003 112 0.001 - O. 00 10 L. ---'---..L 2 ---'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 z 7 Sz 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 experience of this author and others (Singh and' Hassan, 1976) ven.fy that the convergence at the second level is reasonably fast. the interaction prediction method compares very favor- a?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 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 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 through hierarchical control strategies for structural perturba- tIOns 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 costate- prediction 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 guage and an integration routine such as Runge-Kutta to the aSSOCI- ated 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 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) = 1 2 . 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) x 2 (k + 1) = - 2xI (k) - x 2 (k)+ X2 (k - I) + ul (k -I) -u2(k) - u2(k-l) x (k ) + u T (k ) ( 0 6 5 182 . Hierarchical Control of Large-Scale Systems Problems (continued) with initial conditions x(k)=u(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 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; = 1 2 , 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 =2x z +u z , x 2 (0)=1 J -.! ('Xl (2 z + z 2 Z ) - 2 J o X I x 2 + u l + 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 = 1 2 , 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 For a system with the matrices r -5 A = . 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= ;1 -1 0 0 with Q = diag(l, 1,2,2,2) and R = 1 2 , 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 = 1 2 , R = 0.51 2 , ~ .. ~ 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 References Arafeh, S., and Sage, A P. 1974a. Multi-level discrete time system identification in large scale systems. Int . J. Syst . Sci. 5:753-791. Arafeh, S., and Sage, A P. 1974b. Hierarchical system identification of states and parameters in interconnected power system. Int. J. Syst. Sci. 5:817-846. Bauman, E. J. 1968. Multi-level optimization techniques with application to trajectory decomposition, in C. T. Leondes, ed., Advances ill Control Systems, pp. 160-222. Beck, M. B. 1974. The application of control and systems theory to problems of river pollution. Ph.D. thesis, University of Cambridge, Cambridge, England. Benveniste, A; Bernhard, P.; and Cohen, A 1976. On the decomposition of stochastic control problems. IRIA Research Report No. 187, France. Cheneveaux, B. 1972. Contribution a I'optimisation hierarchisee des systemes dynamiques. Doctor Engineer Thesis, No.4. Nantes, France. Cohen, G.: Benveniste, A; and Bernhard, P. 1972. Commande hierarchisee avec coordination en ligne d'un systeme dynamique. Renue Francaise d'Automatique, Informatique et Recherche Operationnelle. J4:77-1O 1. Cohen, G.; Benveniste, A; and Bernhard, P. 1974. Coordination algorithms for optimal control problems Part I. Report No. A/57, Centre d'Automatique, Ecole des Mines, Paris. Cohen, G., and Jolland, G. 1976. Coordination methods by the prediction principle in large dynamic constrained optimization problems. Proc. IFAC Symposium on Large-Scale Systems Theory and Applications, Udine, Italy. Davison, E. J., and Maki, M. C. 1973. The numerical solution of the matrix Riccati differential equation. IEEE Trans. Aut. Cont. AC-18:71-73. Dorato, P., and Levis, A H. 1971. Optimal linear regulators: The discrete-time case. IEEE Trans. Aut. Cont. AC-16:613-620. Fallaside, F., and Perry, P. 1975. Hierarchical optimization of a water supply network. Proc. lEE 122:202-208. Geoffrion, A M. 1970. Elements of large-scale mathematical programming: Part I, Concepts, Part II, Synthesis of Algorithms and Bibliography. Management Sci. 16. Geoffrion, A M. 1971a. Duality in non-linear programming. SIAM Review 13 : 1-37. Geoffrion, A M. 1971 b. Large-scale linear and nonlinear programming, in D. A Wismer, ed., Optimization Methods for Large Scale Systems. McGraw-Hill, New York. References 185 Haimes, Y. Y. 1977. Hierarchical Analyses of Water Resources Systems. McGraw-Hill, New York. Hassan, M. F., and Singh, M. G. 1976. A two-level costate prediction algorithm for nonlinear systems. Cambridge University Engineering Dcpt. Report No. CUEDF-CAMS/TR( 124). Hassan, M. F., and Singh, M. G. 1977. A two-level costate prediction algorithm for nonlinear systems. IFAC J. Automatica 13:629-634. Hassan, M. F., and Singh, M. G. 1981. Hierarchical successive approximation algorithms for non-linear systems. Part II. Algorithms based on costate coordination. J. Large Scale Systems. 2:81-95. Hewlett-Packard Company, 1979. Interpolation: Chebyschev polynomial. System 9845 Numerical Analysis Library Part No. 9282-0563, pp. 247-256. lyer, S. N., and Cory, B. J. 1971. Optimization of turbogcnerator transient performance by differential dynamic programming. IEEE Trans. Power. Appal'. SySI. 90:2149-2157. Jamshidi, M. 1975. Optimal control of nonlinear power systems by an imbedding method. IFAC J. AU/olllatica. 11:633-636. Jamshidi, M. 1980. An overview on the solutions of the algebraic matrix Riccati equation and related problems. J. Large Scale Systems I: 167-192. J anlshidi, M., and Heggen, R. 1. 1980. A multilevel stochastic managemen tmodel for optimal conjunctive use of ground and surface water. Proc. I FA C SymposiulII on Water and Related Land Resources Systems. (C\eveland, OH). Jamshidi, M., and Merryman, 1. E. 1982. On the hierarchical optimization of retarded systems via costate prediction. Proc. American COllII'. Conf., pp. 899-904 (Arlington, VA). Kleinman, D. L. 1968. On the iterative technique for Riccati equation computations. IEEE Trans. Aut. Cant. AC-13:114-115. Mahmoud, M. S. 1977. Multilivel systems control and applications. IEEE Trans. Syst . Man. and Cybern. SMC-7:125-143. Mahmoud, M. 1979. Dynamic feedback methodology for interconnected control systems. Int. J. Contr. 29:881-898. Mahmoud, M.; Vogt, W.; and Mickle, M. 1977. Multi-level control and optimization using generalized gradients technique. lilt. J. Contr. 25:525- 543. March, J. G., and Simon, H. A 1958. Organizations. Wiley, New York. Mesarovic, M. D.; Macko, D.; and Takahara, Y. 1969. Two coordination principles and their applications in large-scale systems control. Proc. IFAC COllgr. Warsaw, Poland. Mesarovic, M. D; Macko, D.; and Takahara, Y. 1970. TheOlY of Hierarchical Multilevel Systems. Academic Press, New York. Mukhopadhyay, B. K., and Malik, O. P. 1973. Solution of nonlinear optimization problem in power systems. Int. J. Colllr. 17:1041-1058. Newhouse, A 1962. Chebyschev curve fit. C0Il1111 . ACM 5:281. Pearson, J. D. 1971. Dynamic decomposition techniques, in Wismer, D. A, cd., Optimization Methods for Large-Scale Systems. McGraw-Hill, New York. Sage, A P. 1968. Optimum Systems COlllrol . Prentice Hall, Englewood Cliffs, NJ. 186 Hierarchical Control of Large-Scale Systems Sandell, N. R., Jr.; Varaiya, P. ; Athans, M.; and Safonov, M. G. 1978. Survey of decentralized control methods for large scale systems. IEEE Trails. Aut. COI/t. AC-23: \08-128. (special issue on large scale systems). Schoeffler, J. D., and Lasdon, L. S. 1966. Decentralized plant control. ISA Trans. 5:175-183. Schoeffler, J. D. 1971. Static multilevel systems, in D. A Wismer, ed., Optimization Methods for Large-Scale Systems. McGraw-Hill, New York. Siljak, D. D. 1978. Large-Scale Dynamic Systems. Elsevier North Holland, New York. Siljak, D. D., and Sundareshan, M. K. 1974. On hierarchical optimal control of large-scale systems. Proc. 8th Asilomar Con/. Circuits Systems Computers, pp. 495-502. Siljak, D. D., and Sundareshan, M. K. 1976a. A multi-level optimization of large scale dynamic systems. IEEE TrailS. Aut. Cont. AC-21 :79-84. Siljak, D. D., and Sundareshan, M. K. 1976b. Large-scale systems: Optimality vs. reliability. Internal Report, University of California, Berkeley, Department of Electrical Engineering. Singh, M. G. 1975. River pollution control. lilt . J. Syst . Sci. 6:9- 21. Singh, M. G. 1980. DYllamical Hierarchical COlltrol, rev. ed. North Holland, Amsterdam, The Netherlands. Singh, M. G., and Hassan, M. 1976. A comparison of two hierarchical optimisation methods. lilt. J. Syst. Sci. 7:603-611. Singh, M. G., and Titli, A 1978. Systems : Decomposition, Optimizatioll and Control. Pergamon Press, Oxford, England. Singh, M. G.; Drew, A W.; and Coales, J. F. 1975. Comparisons of practical hierarchical control methods for interconnected dynamical systems. I FA C J. Automatica II :331- 350. Singh, M. G.; Titli, A; and Galy, J. 1976. A method for improving the efficiency of the goal coordination method for large dynamical systems with state variable coupling. Computers alld Electrical Engineering 2:339-346. Smith, P. G. 1971. Numerical solution of the matrix equation AX + XA T + B = O. IEEE Trans. Aut. Cont . AC-16:278- 279. Smith, N. H., and Sage, A P. 1973. An introduction to hierarchical systems theory. Computers alld Electrical Engineerillg 1:55-71. Sundareshan, M. K. 1976. Large-scale discrete systems: A two-level optimization scheme. Int. 1. Syst . Sci. 7:901-909. Sundareshan, M. K. 1977. Generation of multilevel control and estimation schemes for large-scale systems: A perturbation approach. IEEE Trans. Syst.Man . Cyber. SMC-7:144-152. Takahara, Y. 1965. A multi-level structure for a class of dynamical optimization problems. M.S. thesis, Case Western Reserve University, Cleveland, OH. Tamura, H. 1974. A discrete dynamic model with distributed transport delays and its hierarchical optimization to preserve stream quality. IEEE Trans. Sysl. Man. Cyber. SMC-4:424- 429. Tamura, H. 1975. Decentralized optimization for distributed-lag models of di scretc systems. IFAC J. Automatica II :593-602. References 10 1 Titli, A 1972. Contribution a l'etude des structures de hierarchisees en vue de l'optimization des processes complexes, These d' Etat. No. 495, Toulouse, France. Ti tli , A ; Singh, M. G.; and Hassan, M. F. 1978. Hierarchical optimization of dynamical systems using multiprocessors. Computers alld Electrical Engineering 5:3- 14. Varaiya, P. 1969. A decomposition technique for non-linear programming. Internal Report, University of California, Berkeley, Department of Electrical Engineering. 'i! ' .. . :.' To three who meant the most in my life My Mother My Wife lila My Brother Ahmad Elsevier Science Publishing Co.. Inc. 52 Vanderbilt Avenue, New York, New York 10017 Sole distributors outside the United States and Canada: Elsevier Science Publishers B.V. P.O. Box 211. 1000 AE Amsterdam. The Netherlands ©1983 by Elsevier Science Publishing Co., Inc. Library of Congress Cataloging in Publication Data 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 1 I 1~ l' ., " ~ ~. 'j ~j 1 j I j' ;! " t . \. 6 Introduction to Large-Scale Systems Problem 1.2. (continued) d. A home heating system. e. A radar control system. 1.3. Chapter 2 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. Large-Scale Systems Modeling: Tilne Domain 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. 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. ! i I' ; ;, ' ~) . 1 ,j, ~ '. I ! : lntroclucuon to Large-Scale Systems Problems " I 5 DYNAMIC SYSTEM :~ 4. Large-scale systems are usually represented by imprecise" aggregate" models. 5. Controllers may operate in a group as a "team" or in a "conflicting" ',r" 1/1 YI liZ yz ':J j. " J:' CONTROLLER 1 CONTROLLER 2 1 f r VI f "z manner with single- or multiple-objective or even conflicting-objective functions. 6. Large-scale systems may be satisfactorily optimized by means of suboptimal or near-optimum controls, sometimes termed a "satisfying" strategy. 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. 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. 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. 1.2. value of information. Consider a two-level system shown in Figure 1. and their colleagues at the Case Western 1. (1970). . an electric power system has several control substations. etc.. 1.e rtam values of parameters. i. 1976) and water resources systems (Haimes. (iv) the overall system resources are better utilized through this structure." The coupled approach keeps the problem's structure I~tact and takes advantage of the structure to perform efficient computations.erarc~cal (mul. ecology. N subsystems of the original large-scale system are shown.. and there is no evidence in justifying the vther two.N. The important points regarding large-scale systems are that they often model real-life systems and that their hi. transportation. each being responsible for the operation of a portion of the overall system. 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.. to name a few.. and (v) there will be an increase in system reliability and flexibility. i = 1.N. Needless to say. Varaiya (1972) has mentioned that the first three advantages are a matter of opinion. 1977). the economy. The designer for such systems determines a structure for control which assigns system inputs to a given set of local I I i I f. data networks.e. . 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. and their treatment requires consideration of not only economic costs. This situation arising in a control system design is often referred to as decentralization..1 Schematic of a two-level hierarchical system Reserve University.2. At the first level. i = 1. many real problems are considered large-scale by nature and not by choice. The coefficient matrices of such large linear programs often have relatlvely few nonzero elements. a given system into a number of subsystems for computatIOna~ efflclency and design simplification. and then provides a new set of "interaction" parameters.3 Decentralized Control Most large-scale systems are characterized by a great multiplicity of measured outputs and inputs. Each subsystem is solved independently for a fIxed . as is common in centralized systems. value of the so-called decoupling parameter.1. . 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.... power networks. The "compact basis triangularization" and "generalized upper bounding" are tw~ ~uch effici~n~ methods (Ho and Mitter. all the calculations based ~pon sy~tem. For example. although closed-loop structures have been proposed (Singh 1980).decomp~s~" . 1970). 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.e.. i. very often a geographical pOSItlOn. . These systems are often separated geographically. information (be it a priori or sensor information) and the Inf~r~atlOn Itself are localized at a given center. Detailed discussion on the hierarchical (multilevel) method will be given in Chapter 4.Introduction to Large-Scale Systems Decentralized Control The underlying assumption for all such control and system procedures has been "centrality" (Sandell et aI. The multilevel structure. management. There has been some disagreement among system-and control specialists regarding these points.2. The goal of the coordinator is to arrange the activities of the subsystems to provide a feasible solution to the overall system. has five advantages: (i) the decomposition of systems with fixed designs at one level and coordination at another is often the only alternative available. SECOND LEVEL FIRST LEVEL SUJ3SYS1' EM N Figure 1. and flexible manufacturing networks. a. At the second level a coordinator receives the local solutions of the N subsystems. For example. bUSIness. The "decoupled" appr~ach dIVIdes the ongInal system into a number of subsystems involving c. 1978). 1976).s. One shortcoming of most multilevel structures is that they are inherently open-loop structures.. the environment. but also such important issues as reliability of communication links.. space structures. they are sparse matrices. The success of the hierarchical multilevel approach has been primarily in social systems (Mayne.: ' i . s. Lefkowitz.tilevel) and decentralized structures depict systems dealing wlth soclety. (ii) systems are commonly described only on a stratified basis. . There are two basic approaches for dealing with such problems: "coupled" ~nd "decoupled. Perhaps the most active group in axiomatizing the decoupled approach has been Mesarovic. hence the problem is formulated in a multilayer hierarchy of subproblems. (iii) available decision units have limited capabilities. who have termed it the "multilevel" or "hierarchical" approach (Mesarovic et aI.2 Hierarchical Structures One of the earlier attempts in dealing with large-scale systems was to ". energy. aerospace. according to Mesarovic et a1. water resources. analysis. 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. both within the classical and modern control contexts. etc. and computation fail to give reasonable solutions with reasonable computational efforts. pole placement. 1976).output transfer functions. Nyquist. Behavioral procedures of systems such as controllability. design." Many viewpoints have been presented on thi s issue. engineers have devised several procedures.c by nature. Lyapunov's second method . which analyze or design a given sys tem. 3. 1977). Control procedures such as series compensation. and state-space formulations.Chapter 1 Introduction to Large-Scale SystenlS 1. . optimal control. etc. Another viewpoint is that a system is large-scale when its dimensions are so large that conventional techniques of modeling. when classical control theory was being established. and stochast. control. observability. These procedures can be summarized as follows: 1. One viewpoint has been that a system is considered large-scale if it can be decoupled or partitioned into a number of interconnected sub systems or "small-scale" systems for either computational or practical reaso ns (J-Io and Mitter.1 Historical Background A great number of today's problems are brought about by present-day technology and societal and environmental processes which are highl y complex. input. and stability tests. and application of such criteria as Routh-Hurwitz. a system is large when it requires more than one controller (Mahmoud. "large" in dimension. 2. In other words. Since the early 1950s. Modeling procedures which consist of differential equations. . 1974. 20:727 . A. 1975a. 1958).. D. AC-23:943 . The con tin ued fraction representation of transfer functions and model simpli fication. Regelullgstechllik iv[oc/ern Theoriell w ul illre Venvendbarkeit. 1970.md Berczani . Since then. I-lierarchical Control of Large-Scale Systems 4. A mixed method for multi-vari ahle sys tcm reduction . Sch~)erfler'. 21 :475-4l\4. S. 18:455-460. L~llba. .5HX. Meas urelllelli Control 3:TI . J. 1974. T. 1978.. and Guadiano. Jill. PaciC. COlllr. J. 1973. 1074. 14:111 -. COlltr. ed . ma nagement . Within thi s h ie rarchi ca l st ru cture. 1976. II: 102. Model reduction using the Routh stahility criterion and the Pade approximation technique.EJ~ TrailS. Multivari:lble system reduction via modal methods and P ade approximation . and Singh.19:615-617.Marshall. orga nizational . Prediction of transie nt response se nsi tivity of hi gh order linear sys tems using low order models. . Munich.959. 1xn. in turn . and paper. Sur la repre se nl ation approachec d'une function par dcs fractions rationelks. ()[11! of the most common one is hi erarchi cal. COlli. 14:1164. Y.ysteI1ls by low-o rder modcis.IIH . On simplification of unstable sys tems using Routh approximation technique. K . M. Although interaction amo ng subsystems can take on many forms. D .. S. sometimes ca lled the sliprellllll coordinator . I EEE TrailS. Shich. V. (1970) presented o ne of the earliest formal quantitative trea tments of hierarchical (multilevel) systems. Contr..X17. Muller. A. S.. Jill . and Pille. . R. S.65 1. 1975b. 1. H. J. Pearson.T9. Matrix continu cd fraction cxpansion and inversion by thc g. 1976. lE E 12 I : 1032. Oldenbourg-Verlag. Y . In spite of thi s seemi ngly natural represen ta ti on of a hi era rchical struc ture. controll cd or coo rdinat ed by the unit on the h:vel illl mediat ely above it. A III. J. Y. which appears somewhat natural in economi c. Sham ash. Proc. 1. J . th e ~ llh sys terns are positioned o n levc ls with different d egrees of hi erarchy. Nonuniq ueness of model reduction using th e Routh approach. On an ~malogy between stochastic process and monotone dynamic sys te ms. I EEJ~ TrailS.eneralizcci matrix algori thm.944. AC-24:65'J. Singh. Aut. lilt .. R. A new m e thod for rcduction o f dynamic sys tcms. M. 1956. A mixed Caller form for linear sys tem reduction . V. K. 1976. V. The hi ghest level coord inator. COlli . and Mehdi. 1975e. R . A new frequency domai n technique for the simplification of linear dynamic systems. S. Shamash. 1973. ) Chapter 4 0: 1-'n P a rthasarathy. 1975. S. Zakian.. 1. Mall. ca n be thought o f as th e hoard of directo rs of a corpo rati o n. SMC-4:584.737 . I EEE TrailS. 1974. 1 shows a typic. Linear system reduction using Pade approxim ation to allow re tention o f domi n an t mod es Jilt. Stable reduced-order models using Pade-type approx imatio ns. lilt . Shieh. AC-20:XI5 . S.!. :1 (SuPP!. fnt . Van Nostrand. Paynter. Ncw York.. S. 1977). l-I. Aut. Annalt's S'ci('nlijiques de l" Ecole Normale S lIpiL'lIre. 1973. S. S. Shieh. lilt. COlil . 1. Cyl!. Rao.. S. Y . r. L. it s exac t hehavior ha s not been well und e rstood mainl y du e to the fact that ve ry litlh: quantitative work has been done on these la rge-sca le sys tems ( rvhtr~h and Si m o n. Reddy. Sl's . Figure 4. COllI. 1979. 1. a grea t d eal of work has been done in th e fi eld (Schoeffler and Lasdon 1966' Beneveniste et a I. S. Nagarajan . Geoffri o n. 197 1. S. Alit. and Rao. A ul. COlllr. Smi th an d Sage. Z. Sin gh. A. Ser. J. N. etc. H. and Lunba.. and Wei. COlltr. IJ. 1975. AC-20:429-432. Opti mum approximation of high-order . Optimum red uction of large dynamic sys tem. 1970 . V. lilt . shop mana gers.. 1974. S.. COlltr. Cohen and lolland. 1948. Sinha.t1 hierarchical (" multil evel" ) system . Y. (. W. while the la rge-sca le sys tem is the co rporati on itse lf. 197 1. COllII'. Towill.. Sha m ash. R. as it was hriefly discu ssed in Chapter I.11 74. 1978. Mesarovic e t al. Shamash. COIlt. . R. Wright. Simpli fication of linear time-invariant systems by moment approximations. vice-presi d ents. 20:7 1-79. JEEL' TrailS. L. 197 1b. o il . Wall. Elect Letters. may be described as a complex system composed o f a numher o f C\lnstituents o r smaller sub systems serving particular functions and shared resources and govern ed by interrelated goa ls and cons traint s (M ahmou d . 14:951 . AC. N. J. and Go ldman. JIII. in G . COlltr. whih: a no th er level's coo rdin ators ma y he th e president . H. 18:449-454. Continlll:d-fracti on model-redu ction technique for l11ultivariahle sys tems. J EEl~ TrailS. COlltr. direc l\) J's. 19'7 1a. Sand ell et a I. Comments on linear system reduction b y continued fraction expansior about a general point. Analytic Thcory of Con tinued Fractions. A s lIh s\lstem at a given level control s or "coordinates" the unit s o n the I ~vel belO\~' it and is. Rao.1 Introduction The notion o f a large-scal e sys tem. The lower level s can be occupied by p lant managas.272. 11l1. Sinha. 197 \. COllt. 23:403-408 .. 2 1:257 . A method for frequency dom ain simplification of transfer fun c tions. and complex indu strial systcm~ sll ch as steel. F. L. etc. the properties and types of hierarchical pr~cesses. The various levels of hierarchy in the system exchange informa tion (usually vertically) among themselves iterati vely. a further interpretation and insight of the no tIon of hierarchy. i. In thi s section. adaptation (direct adaptation of the control law and model) and self-organization (model selection and control as a function of environmental parameters).n coordinate lower-level units and be coordinated by a higher-level oae. . "- \ \ \ \ \ \ \ \ \ \ DECISION-MAKING UNIT ~""--. the time horizon increases .2. However. and the existence of many conflicting goals or objectives. As the level of hierarchy goes up . For a relatively exhaustive survey on the multilevel systems control and applications.1). the lower-level components are faster than the higher-level ones. The control tasks are distributed in a vertical division (Singh and Titli.e. An overall evaluatIOn of hIerarchical methods is presented in Section 4.104 A Hierarchical Control of Large-Scale Systems Introduction lOS / / '\ '\ / / / / "- vironmental aspects. behavioral and enSET POINTS REGULATION CONTROL MEASUREMENTS LARGE-SCALE SYSTEM INPUTS OUTPUTS . There is no uniquely or universally accepted set of properties aSSOCiated with the hierarchical systems. There are three basic structures in hi erarchica l (multilevel) systems depending o n the model parameters. uncertainties. A hierarchical system consists of deci sion making components structured in a pyramid shape (Figure 4. Multilayer Hierarchical Structure: This structure is a direct outcome of the complexities involved in a decision making process. followed by op timization (calculation of the regulators' set points). as shown in Figure 4. the following are some of the key properties : SELF-ORGANIZATION STRUCTURE ADAPTATION PARAMETERS OPTIMIZATION I. shown Figure 4.and s~me reasons for their existence are given. 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. . 2. c::. The system has an overall goal which may (or may not) be in harmony with all its individual components. . For the multilayer structure shown here.1 A hierarchical (multilevel) control strategy for a large-scale system. 4. decision variables. Multistrata Hierarchical Structure: In this multilevel system in which levels are called strata. the interested reader may see the work of Mahmoud (1977). This structure. lower-level subsystems are assigned more specialized descriptions and details of the large-scale complex system than the higher levels. as discussed earlier. 2. regulation (first layer) acts as a direct control action. Figure 4. 3.6. 1980).2 A multilayer control strategy for a large-scde system.. 1978).A \ '\ PERFORMANCE RESULTS \ '\ ---~ CONTROL ACTION LARGE~CALESYSTEM \ INPUTS L-_ _ _ __ __ __ _ __ _ _ _ ~ OUTPUTS 1. )ordinating the resulting subproblcms to transform a given intergrated system into a multilevel one.2. 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. This trans fonnation can be achieved by a hos t of dirferent ways. Lt.1). 4.rriillatioll. including the 11l\tiOll ~ of " intnacti()n predict inn" and "s tructural perturhation .e.2) into a two-level problem by fixing the interacti on va riables / and)'2 at some value. In the nex t two scctions. over the follO\ving feasible se ts: S.5) Under this situation the problem (4.ation problems at a lower level a nd results in a c(\llsidcrahlc redu c tion in computational e ffort.2.1 )-( 4.2. u. u i• Wi) x'.10(' Hierarchical Control of Large-Scale Systems Coordination of Hierarchical Structures 107 in Pigure 4. 1he coordination process as applicable to most hi era rc hi ca l systems is descrihed in Section 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 numb er of subsystems and then c. say Wi.W)=O~.2 (4. i = \. Section 4.Subsystem i Find KI(w) = minl.2) may be divided into the followin g two sequential problems: First . one can make a tentative conclusion that a s uccessful operation (11' hierarc hical systems is best described by two processes known as deco/1/[. l/):r(X i.2 (4 .l\' coordination algorithm hy Tamura (1974. y) (4.2. A number of num erical examples illu stra te the various techniques presented. The near-optimllm design or lin ear and nonlinear hierarchical systems are disc ussed in Chapter 6. I) (4.2.2. 1971) : minimizeJ(x. Let thc problem and its objective fun c ti on be d ecomposed into two subsystems. m . 1(69). i=I.Level Problem .1'1.5. d escribed for a two-subsystem s tatic optimizatic!l (nonlinear programming) problem. 1/ i..2) subject to f( x.3) and (4. Section 4.l .!f' l ('()nrrlin"linn nwth. The objective of tile Ill odel coordination method is to convcrt the intcgrated problem (4.1 Model Coordination Method Consider the following static optimization problem (Schoeffler. However. in other words. the s tru c tural perturbation and interaction prediction approaches are extended t(1 take 011 the discrete-time case with and without d elay.6 is devoted to furth er discussion and the evaiuation of hi erarchi cal CO il t rol tec h niqucs.e. i = 1. Before th e ~c()pe o f th e present chapter is given. However.2.2.2..2.2 is horizontal. y) = 0 whcre x = vector of sys tcm (state) variables.8) The above minimiza tion s arc to be done. The higher-level echelons.2. . The closed-loop hierarchiCl i C(1 nlrol of large-scale ~vs tel1l s is di sc ussed in Section 4.dcl. L consitl ers conflicting goals and objectives between decision suhprobl ems. (1977) and lIa~ sa n and Singh (1976a. S~ = {Wi: K I ( Wi) exists}.!lsitinll alJd ('()(. nl(). the subsys tems are s till interconnec ted . (4.10) Figure 4." [n Sccti(1n 4.. respectively. i. a third division called a time or fun c tional divi sion is possihle (Singh and Titli.). U i. u' (4. mos t 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" m e th od) (M esa rovic et aI.4. 1977 . respectively. these methods are Second . and y = vector of interaction variables between subsystems.\ mon l:' t he hi erarchical con tml strategies disclissed here are a three-l eve l o tim e.(4. a nd f arc vectors of sys tcm..2.2. 1978). i = 1.2. 2 i = I.3 IS concerned \\ilh th e opel~-loop control nr continuous-time hierarchical systems where cO(1rdination betwcen two levels a re considered.2. 9) (4.2.. This sehem e will be dlscllssed in Section 4.(x i.7) 4.Level Problem minimize K( w) = KI (IV)+ K 2(w) IV (4. through the vectors . Constrain/=w i .. i. In this divi s io n a given subsys tem's functiollal optimization prohlcm is decoll1poseJ into a finite numher of s in glc-pmallleter optimi7.6 ) (4. 1975) and a "cos tate c(1ordinaiion" (1r "costate prediction" sche me du e to Mahmoud et al.2.1I1ipulated. and interac ti on variables for ith subsystem.4) where Xi. II = vector of manipulated (control) variables. This decomposition has produced a pcrformance function for each subsys tem .2. based on the above disl'lISsinl1.tc-tilllc sys tcms. 111 add iti(1n to the vertical (multilayer) and horizontal (multiechelon) division of cnntwl task.5 In connection with th e hi erarchical con trol of dislTl.={(X i.3 shows a two-level slnwtllre rnr Ihf'. u 2.j. a) = 11 (x I.2. Schoeffler. y. i. 11 . and y .I (\ s. to the objective function (4. the independently selected y' and Zi actually become equal (Mesarnvic et aI. 1969.2. while assigning the task of detellllining these coordinating variables to the second level. \\. Mathematically.20)) that the coordinating variablc a is manipulated until the error e approaches zcro.2.ub.2. \\ ' .." Therefore. Ill. By introducing the z variables. 1I!'\ed to .(4.4 shows the two-level solution via goal coordination .'1 lillks het\veen subsys tems.y.2. y' ) + 12 ( X 2.()lI. ) which the interaction-balance principle hold s..2.e. /1 . In th e goal coordinati o ll me thod the interac tions are literall y re moved by cutting all the link s . /11 .- ---------------~ fl(XI . /1.\I11 o ng the su hsys lem s.17) (4. Moreover. u 1. Let .3 and 4. min e a 0: III = min (y .u.(4. it is alt ernati vely termed " feasible d ecomposition method .13) Figllre 4. z. :' acts as an arhitrary '~lallipulated variable and should be chosen by the nptillli zing subsystellls lik e . it is clcar that Vi =1= Z i. Z2)+ aryl - a~'z 2 - ( 4.z) (4. Moreover.~I\ ' ) = '111111 . the procedure is to decompose the problem into a number of denlllpkd subprohlems which constitute the firs l-Ieycl prohlem .l) -:=- r'llt\ A 2(\I') = .) ~ () \0 1 .rl .2. H. this multilevel formulation can be set up by introducing a weighting parameter 0: which penali zes th e performance of the system when the interactions do not balance. y'.Hierarchical Control of Large-Scale Sys tems Coordination of Hierarchical Structures 109 J2--~I:--~_x~2. The secolld-Ieve l nrohlcm is to force the first -level subproblems to a ~olution for The second-level probl em is to manipulate the coo rdinating variable order to derive the two-s ubsys tems interaction error to zero. Once the objective function (4. while it s incoming variable is dCllot ed b y z '. i.3 A two-levc l soluti on of a sLatic optimization problcm using model l'tltlrdin :ltitln .4.: given by ------. the interaction balance is held by manipulating the objective function s of the first-Icvel problems (4.11 ).\\ . this procedure is called model coordination.2. I. where a is a vector of weighting parameters (positive or negative) which causes any interaction unbalance (y .2. U I. In other words . and yare present..16) 4. U. the name "goal coordination. Z2) = f 2(X 2..1) .1 8) ( 4./II l(\I..2. The fir st-level prohlems are constructed by fixing certain interacting vari. II . 1\ 1 ) /1 (\ ' . the optimization problem cOllsidered in the previou s section is completely d ecoupled into two subsyste lll s due to the fact that th ei r interactions are cut and their objcctive fUllction s were alread y separated. since certain int ernal i'~leractiol1s are fi xed hy adding a constraint to the mathematical Ill ode \. i.e y 2 -." Figure 4.e.(11') "I(W) [ i --·~osc tO~'illillli7C \I' = + A' 2 (\\.11) is minimi zcd over th e sc t So it rcsu lts a function .. .3) a penalty term is added: 1 (x.0(// Coordination M ethod Subsystem 2: (4. Zl) = 0 The sct of allowable system variables is defined by (4. 11 _ .12) (4. Ii 2.2.z . III order to make sure the individual su hprohlem s yield a solution to the original problem.2.2.13).z) to affect the objective function. 19) ('onsidn th e static optimi za tion problem (4.20) It is clear frol11 the second-level problem ( Eq uation (4.z2 1..Y."2 1. the system's equation s an.1 ) <. th e first-level probl em is formulated as Subsystem 1: min xl. Hence.y .2 .2.2 (... K(a) = min .2.z) a (4.15) After expanding the penalty term a T(y -z ) = a.i be the outgoing variable from the ith s ub svs tClll.2. a system can nperate with these intermediate values with a near-optimal performance.2.18) through variable 0:: hence.2) and considering the relations e4.2).z l)+Ci.:. Under thi s condition. 1 y'.16) and e4.I .e.2. Y 2) + a'T v . III. () (4.2. 11 2 mill i l ( X 2 . 1971). Here again.. y 2.11) [ I\'.a) (4...". it is necessa ry that the illlcmclioll -halollce prillciple be sa tisfied. Duc to the remo va l p i' a. Thc rcader should co mpare th e two structurcs in Fi gures 4. ..2 .. h1es in the original optimization prohlem. 14) In this coordination procedure the variables Wi which fix interaction va riahles ri are termed coordillatillg variables.d U 1. due to th e fact that all intermed iate variables x. Ul.ZESo 1( x.exl. y l.(t) . uj .3.n (4 . . u. Let a large-scale dynamic interconnected system be represented by the following state equation: (4. m-. .( t) . u~(t») (4. . g. and the ith subsystem's state equation is given by L Vj (x j . Newton's.2. u.. y2 .l .j (X j . It IS assumed that the system consists of N interconnected subsystems s · " i = 1.( tl )) + F 1 h. t) .2). or conjugate gradient methods.e.3. i=I .3 .5) Xj(t) = A jx j(t)+ Bjuj(t)+ CjZj( t ). Z2)+ "' i y l _ ".14) can be decomposed into (4. unt) . t) j 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. l.( x.3. t)=Lgij( X/t) ..(x. t) .u. .3. where z. t) in (4.(Z j(t) . .(x. Under the above assumption of separation..4 A two-level solution of a static optimization problem using goal coordi- j nation. such as the gradient.9) 4.7) v(x ..(tJ=x. u.(Zj(t) ...=/. .3.3 . l11inJI(x l . x. z2) = 0 l11in h(x 2 .12) (4.8) (4.3. The application of two-level goal-coordination to largescale linear systems is given next.N j (4. x (O) =xo It is assumed that (4. . y l . z..-. are respectively n-. and (4. t ) ~ 0 j x.3. t )dt } (4 .oo .. in an optimal control sense. U. u.6) can be rewritten as minimize 4.(t) .(t) . and m .4) 4. u.rZ2 fl(x l .N.(X.1 Linear System Two .1) ~here x LJ..3.t)dt 10 (4 . the large-scale system's optimal control problem (4. 1/I . .3) (4 . t) dt } I . xT(t) = (xnt). t)+t.3. i = l .Z. x ~ (t») uT(t) = (unt).( X. t) ~ O } (4 .2.U N such that a cost function x (t)=A x(t)+Bu(t)..Level Coordination Consider a large-scale linear time-invariant system: The objective.-.-dimensional.(tl ))+{lh. In addition to the interaction-balance approach another scheme known as the interaction prediction method is also discussed . a similar decomposition can be assumed to hold for the cost function constraint (4. (.(t) .(x .(t)=!. n . t) .6) Through the assumed decomposition of system (4.6) and the interaction g..( x.2) where x.15) . .(t) . r zl subjecl 10 f2(x 2. y 2.( x. t) = Lg. t) = LVj ( Xj .' .13) and u are n X I and m X 1 state and control vectors. is to find control vectors u 1' u2 . (t) .(x . . respectively... Hierarchical Control of Large-Scale Systems Open-Loop Hierarchical Control of Continuous-Time Systems III is minimized subject to (4.1) and a feasible domain 'I" Choose Cc' to achi eve interaction balan ce u(t) E U( x (t) .3. 1/ 2. I t o (4 .3.3 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.1'" .3.3. zl ) = 0 +"'f/ J = LJ.3. t) Fig~re (4.. = L { G.10) I I '0 subject to x. 1/ 2.14) J=G(x(tl ))+ F 1 h(x(t).N (4 .3. t) = {ul v ( x (t) .2). g.=L{G.3.1). known as a hierarchical (multilevel) control. u.3.xi(t).t)+g.) represents the ith subsystem interaction and The above problem. (4.(t).3.1) into N interconnected subsystems (4.u(t). uj (t) . was demonstrated for a two-level optimization of a static problem in the previous section.3.3. z 1) ... . Xj(to)= x jo.3..3.(t) is a vector consisting of a linear (or nonlinear) combination of the states of the N subsystems.11) t. Xj(O) = Xjo (4 ."..5).3. x. subject 10 1/ 1. (I)] ill} \\' here <l.( f) - (4. is replaced by a family of N subprohlem s ('('nplcd tog. In Ihis wa y the original eO llstrained (subsystems intera c ti ons) op timi zation problem is .u. ( 1l)7(.11 2 II icra rchica l Control of Large-Scale Systcms X Opcn-Loop Hi erarchi cal Control of Continuous-Time Systcms 11 3 and th e h. is co mmonly taken as a fun cti on of "interaction error": e.(a(/) .20) The imbedded interaction variable Zr ( .d . «<).'(f)..(a(/). positive se midefinite matrices.22) . i.\( (\" * ). R. . (1970) as applied to the " linearLJlladratic" prohlem by Pearson (1971) and report ed by Singh (1980) is now 1IL'SL'llt cd.\'.15).v II . constitute the "coupling" COI1st r:li llts .= I le r ~ Il .:. N. X ". hlli OIl 10 Sr .::. ( (. reduced to th e optimization of N subsystems which collectively S:I ti sfy (4 . this imb eddin g ("ll ll l. 197R ) in such a way that for :1 .t)- I ThL' ph vs ic:li int erpretati o n of the last term in the integrand of (4.20) can be co nsidered as part of th e control variable available to controller i. 17) Ill. in turn. rormul ation has been ca lled "globa l" by Benveniste et al. evaluates the next updatcd value or IX. e:leh loca l controller i recei vcs <Y ~ flll])1 th l' ("()()rd inator (second -level hierarchy). solves . 1 In thi s deco mpositi on of a large interconnected lin ea r sys tem the COllllllon l"\)up lin g factors among it s N subsys tems are th e" interacti on" variables :. ~ k = Lie.19) X III.. . + . (4. I ~ Grj"Y/I))l cit) J where th e parameter vector (\' consists of k La gr:lI1 ge Illultipliers. in which case th e pa ramet er vector a( t) scrves as a se t of "dual" variabl es o r Lan grange Illul ti plier~ corresponding to in teraction equality constrain ts (4. and S.3..1 ()h.\') throu gh an illlkddillg l'ar:\llleler IY (S<J1H1e1l et . th e globa I system problem . / )=z.3.e-s(": d<. Z (4 . as wil l he see n shortly. In other words.' d h v S..V .'1' Ill..2. F igure 4. The "goal coordination" or "interaction h: li . . matri x (Singh.3.(4. are and " . positi ve definite matrices with . I 6) is a lin car l"Olllhination of th e states of th e o ther N -.in g I: II(ST t. u)} X. th e slIhsys tems .z. (4 .3. i ~ 1. . 15 ). ) in (4.. (4. . .3.3.. ( I).3. is " illlhcdtkd" int(1 a L\lnil y of sub syste m pn)blcllls . ' In terms of hi erarchi ca l con trol Ilotation.16). G..3. Let us in troduce a dual function q(a) = min {L(x. Thi .:. as will be seen Liter.et her throu gh a parameter vector a=(a l."( f ) R .) :llld is dl'll\)ted hy . X " . .5 Thc two-level goal-coordinati on structure for dynami c systems.!.l ll L"l'" approach o f Mesarvic et al.' system. a nd the upda te ter m iI'.3 . N. u.. and G' / i ~ <I n ". Ii.'!: !)1 is nOlhin g hut th e notion of c(l() rdination .e. ( (v).(. X " .\. <l IT " .. z.l'VEI..(a(/). and trall Slll it s L . hut in math ema ti ca l j1I(1gr:lIllinin g pmblell1 tCl"minology.). .d prohlem .'(z.(I ) + . .1 . a N )"1 and deIl(lt t. 15).. 1970 ).. 16) while minimil. to avo id singu lar co ntrol s.3.3. (report s) so me functi o n y/ of its soluti o n to the coordinator.v 11 = (4 .II' valut' p f <y '". I intcraCli()ll vec tor . \\ hi ch. Under this strategy. i.11 . ( f ) +2a.(. th e introdu ction of thi s term. .: ." (4.. (1) = L / -' I SECOND LEVEl. Th e g. it is dellot ed as th e " ma ster" prohlem «('eo i"fri( ln. I) (4. The foll ow i ng assumption i ~ considered to Il(lld .1"(. I 9~0). where the Lagrangian L ( .X' (/f)+~j'r [ X/(f)Qr ·\(t) 0 + l/ ./x.21 ) su bject to (4.:. The fund a ment al concept behind thi s approach is to convert the original sys tem's minimi zati on prob lem into an easier maximization problem whose soluti on can be obtained in the two-l evel iterative scheme di scussed above. 1~) / wherc 1 is thc fth iteration step sizc.16). i = 1..5 shows a two-level control structure o r a l:iq.o:) = . . L r "" I III = L ..r( f ) S.3. Th e coordinato r..I _. Figure 4.jeld the d es irL~ d '.~I N G.17) is diffi cu lt at this point.I subsystems.. ahm g with (4.= 1 t v ( ~xT( /f)Q. The original sys tem's optimal control prohlem i.3 .) is defin eu by L(x. y . In fact.I·r. 2.3 .':).lIlil 'III .a 2 .1". in g the dual function q(iX) in (4. where the solutions of all first-level subsystems are known.r\. Each sll hsysle1l1 \\'oliid intend til Ill inimilt' its own suh-La ~ran g iaJ) Li as defined by (4. a Riccati equation formulationi' can be used here.• ~ . agrange Illultipliers a = a*.3. 1975).24) suh ject to (4 .1'\imi7. (4.3.111 f'.1 wav that a sub-Lagrangian exists for each subsystem. the optimum hierarchical control is resulted. The second example represents the model of a multi-reach river pollution problem (Beck. a conjugate gradient iterative method similar to (4 . Once the total system interaction error i!1 normalized form Error = ( .3.~ G.3.. using a known Lagrange multiplier 0: = 0:*. i'i l'11 hy rkterllllning a set of Lagrange Illuitiplicrs (Xi' i = 1..25 ) is l1o!hin g hut the suhsvstems' interaction errors.1.! 11 I Iicra rc liica! ( '()l1lm! or I.3 .3.. and \. a step-by-step computational procedure for the goal coordination method of hierarchical control is given . ~ t <'i' i = L 2. which was first suggested by Pearson (1971) and further treated by Singh (1980).111 lI11CollstrOlincd one. If a gr<1 di ent (steep es t descent) method is employed. Singh. Since the subsystems arc linear.3 . 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 \'" Step 2: At the seco nd level.3 .~ [ /{ -~ Z. and the treatment of nonlinear multilevel systems is given in parts in Section 4. minimize each sub. 15) and usin g the La grange lllulti pliCrs_a* which are trented as known funet iPlls at the first level of hinarc!J..5 and Chapter 6. .).a tion would allow one to determine the rl ll al rllDc!iL1l1 q(x*) ilL ( ~ . L " V. Here D.\'.y. f defines the gradient of f with respect tn . 1974.3.3. ...3.1(1). Tn f:ll'ilitale the solution of this problem. A 19oritlzm 4. Goal Coordination Method Step 1: For each first-level subsystem.jX j } dt) //:::'( (4.2 . The first example.27) and d O= eO. The result of C<J ch s\ll'h 1l1illil11i 7.3. 1(1) is q li . the value of q( a*) would be im proved by a typi cal tllleopstrain ed nptil11i za tion such ii'StTi e Newton's method. ~(I( n)I "". n ! (n . In pliler words.16) is l'qu i\'. At the second level.3. = . b.(4.= \ 0 .19). . ILJ) is simply the Ith iteration interaction error (. which are known through first · kycl soilltions.2 on Interacti.3.24) whi ch reveals thaI the (kcc1 l1lpositioll is carried on to the Lagrangian in sllch .argl'-Scak Systems Open -Loop Hierarchi cal Control of Continuous-Timc Systems g radient defined by 11 5 t'il . it is observed that for a given set or I.La grangian L. \\'liell the cOll straint s are convex.26) Sil1?-h (llJXO) ha ve showil that rV(.lling Ihat ll1inillli l. _ . Once the error vector e(t) approaches zero._ . 17) subject to (4. Gi/X j If T {z. Below. ' l ! nil e. an optimum solution has been obtained for the system.6. Store solu tions.llcnt to m.21) with respect to n.'lIc h C.k.ni mi ze q( <t} I' ==:0 Minill1il. (4 ..23) Y [I( elt 1 )f'e I 1 ' ( t) cll (I) 1+1 -"-_ __ . Geoffrion ( 1971 a. . Z.3.J':es. However. the vector d l in (4. - J=\ J = \ . Two examples follow that help to illustrate the goal coordination or interaction balance approach.26).19) and Figure 4.t is the step size of integration.27 ) is used to update 0:*(/) trajectories like (4.-0 -- = in<iic. is used in a modified form.cJ II (4 .ult Section 4. The overall evaluation of the multilevel methods is deferred until Section 4. where O ~ t ~ /J (4. h) and dl 'l (f)=el l t(t)+/ ' Idl(f). the gradient method or th e cpnjllg. _ \ .15) ..22) can be rewritten <IS [ /(el)Teld( o (4 . the Lagrangian (4 .28) is sufficiently small.. At th e sl'l'(llld lc\'elthe vector H is updated as indicated by (4.3..)n Predict ion Method.(4.3.lti(l1l or J in (4.3..3. the cOllstraint (4.ill' gradiellt l11c1hod .3. a superior techniqlle from a computational point of view is the conjugate 'i'Readers unfamiliar with the Riceati formulation can com. 2l1) Il K. ( 'oll sidn:t 12Ih -(lrdcr sy..I Il I) = [( ZI ... _ .ys lcm of Example 4.~~)(-)~I~:·N /\T()·I~-- ---] ~:'r ---.-. /-/~--· ·(:~ I ···· l '-. . . Il Il II I II II (\ II an d a ljll :ldra li c cosl flln cl ion . .. _ . "" I.' -' L _____ _________ __ ___ ___ ___ _______ I Figure .I (" { I'(t )( ) I(/) I 1I'(t)!<II(/ )} dt '11 ( iU .+ I ' l - I I I Th e sys tem o ut pu t veclo r is give n by I 0 0 1 0 0 0 I () 0 (l 0 0 () A hloc k diagralll [or :. v = Cx= V" :lIl1Pil' . .---' I L ___ _ __________ ~ where I I I r--.1 () 0 Il Il 1 I °" t I 0 o I () (l () I 0 0 0 o I I 0 0 I o () o I () t) I () 2 tl () ..JF~ ~ . .3...32) where e..2 I ..~'~~~IBS~:1 :-~..X2)] (4 .-.K.(t)V.. SOLUTION: From the schematic of the system show n in Figure () o () () I I I I () 1\ . ".... .() .. 2....:. i = J. 3 _ () () I I () () (\ () 4... -: 1_ () -2 () () _--.-.t..11(' I1icrarchica l Control of Large-Sca le Systel1ls Open-Loop Hierarchical Control of Continuous-Time Systems -.6 (clo tted lin es) an d the s tate matrix in (4.. (Z S..XX) .3..1 t) () I) () I I I \ _.29). I. --· -· ..-I B/'..tS'UBSYS~-~J \~ 1 .•.- 0 1 0 () (4. while th c Sl'CllIHI-le\'cl ilL'ra.-.-- 11 7 with ·'/-..0 _ II II Il () I) () ..• _l ..K. The first..(t) + K.order matri x Riccati eq uat ion s I) I II (\ Il 1\ II I) I) 1/ II 1\ II (4 .._ .. ~ ·~ ~. .. positi ve-ddinite sy mmetri c Rieea ti ma tri x and v~ = B.33) where K. T he " integration-free. (Z 3-X7 )..2 () I I I I () .I () -' t) () ....-. ~_ .3.6) constraints given by 1\ () I I) 1 (\ () 1 () () 1_ 3 .I I I 1_ i\ () t) I) I () t) 1 (\ () L. -" 1.(Z4 -X5 )." method of so lving th e differen tial matri x Ri cca ti equa lion prop osed hy Dav iso n and Maki ( 1973) and ovcrvicwccl by Jamshidi (1980) was used o n an H P-45 co mput cr and a BASI C so urce program.Icvel subsystcm problcms were solved throu gh a se t o f four third.q ..( IO)=O (4.3. ..ll)) _ . The subsys tcm s' stat e equation s were solved by a sl :llld :JI'lI fllurl h-ortlcr Run ge-... represents the interaction errors between the fo ur sub sys tcms.(Z(. . .1 '1 11) \ . ~17'~[s'~~SY~..(t) +Q." or " doublin g..1"".I : ~~~~~ -.(Z2 -X I) .- o 1 -.... 1..Kulta method . it is clcar that there are fou r third-order subsys tems coupled together thro ugh six equality (number o f d o tted lines in Fi gure 4.-.(t) =A....---::.. 1. I 1\ 0 () () () .. : I: /.K.lc ll1 illtroduCl:d by Pearso n (lIn I) and sho w lI in h gure 4.. _ .() with a stale equation Il Il .3 ... K. R .~..I..6.(t)A.II .(/) is a n 11 / X 11 ... 1 ~ 1.J .XI I) .3.---.. 1 () 0 . I :.3 1) It is desired to find a hi erarcllical control strategy through the interacti o n balance (goa l coordination) ap proac h. 34) I I 0._ 1 _ _ _ L __ _ . . For a quadratic cost function (4.J. 2}.) (4 .2. 1971: Singh.1. 1 _____ .1___ 1.7 NOflll aii rn l intt'r.--.1..5 in 15 iterations.33) usin g a fourth-oru er RUl1ge.'.J.9 . \ 50 -.35) with (J -.diag.1111 - Jiil'. RII 1>11 ./ = 1 G. 7 R l) 1() ("f IN.(t)+B.7 ) utili zing th e cubic 1979 ) ~ pline interpolation (H ewlett-Packard.2(.8. (1) The optimal control problem at the first level is to find a control which F.2.2. 7 x <) 10 \1 12 13 14 \) ITERAT IONS . N J liT RATIONS LX. (t) = L .0 llierarchi cal Conlrnl of Large-Scale Systems Open-Loop lIierarchieal Control of Continuous-Time Systems 11 Y SOLUTION: 1(1 The two first -level problems are id entical.1.r\"j (t ) (4 .1tion of hioch ell1i c<1 l oxygCll demand (BO \)) . Figure 4. * and x 2 is the COIl c(' ntrati(lll (If di sso lv ed oxygell (DO) -. as shown in Figure 4. (O) ='\.9 I -.z Jt).4. ________.37) [/ . {2.2._. is z.1.(t) +C.{2 .(J . 1 as in earlier treatments of this example (Pearson.n.\<lmple 4.3.1. it is desired to find an uptilllal cO lltrol whicl! lIlinimil'. II' ( . An alternative approach in optimal control or hierarchical systcms which ha s bo th open. .l c ti ol) \'IT\lr vs cOl1gugate g.cs (4.Kutta method for 6. The interaction error for thi s example reduced to about 10 .(U2 () II .II X 1. (t) =A. x= [ I 0 0 ------.0' N i = l..2 I 0 0 x+ (J .3.1 0 \\ hl'll' c. .3.3.and its control III is the BOD of the dllucnt di sch<1rge into the river. 1. each of which is described by 5:..t = 0.J 2 . t(1 cv nlllnte appwpriate nUllleri cal integrals.1 ~_~~ _ I 0 o ju (4. Co nsider a large-sca le linear int erco nnected sys lem which is decomposed in to N subsystem s.2 l _0 o.1l )(.0 .1. x.----.2 Int eractio/l Prediction Method If) '1 ____ L ___ .. \'. llJ X . Th e conjugate gradient algorithm resulted in a decrease in error from O) 1 to about 10 ) in six iterations.35) subject lo (4.3. COllsider a two-reach model of a rivcr poilu lion control pmh1em .lJl"(· ·1.I\ 'h f'l'<1ch (suhsys tl'm) of the rlycr h<l s Iwo 5t. where the interaction vector z. as shown in Figure 4.29) bv Singh (1980).34 ) alld\(O) = (I I . Now let us consider the second example. radi ent iteratio1ls for I'k 4 .7.8 Interac tion erro r behavior [or river pollution system of Exalllple 4. N (4.and closed-loop forms is the in teracti o n predi cti on method ha sed on the initial work of Takahara (1 965). I[) 4.90 0 I .36) (:L ·\. which was in close agreemcnt with previously report ed result s of a modified version of system (4 . The optimum BOD and DO concentrations of thc two reaches of the river are show n in Figure 4.~. The step size was chosen 10 he (\{ .4) alld N c= dia g. which avoids second-level gradient-type iterations.. __ . amI a scconu-o ruer matri x Riecati equation is solved by integratin g (4.lte5 ---'\"1 is the w il ee n Ir.I 1)1. J..II .J.J.l ___ J .32 0 -.LJ. o I __ l __ 3 l~ 4 __ ~~__~~~__~~__~~~ 5 r.J2 .3 ..1 (. 3.c .43) p.50) (4.II 2 f which constitute a linear two-point bounda.5 I) .41) and (4.45) are substituted into it.3.44) L· . it suffices to assume (IY:-(/) : z7(/)) as known.37 ) while lllinil11i7jng a usual quadratic cost [ullction where gj(/) is an II.p./t.3.3.3.(l j ) (4. Note that IIj(/) can be eliminated from (4. and in fact z.9 Optilnlllll !lUI) and I)U concClltrati()ns fpr thc two-reach Jl1udel (If a rin'r p!lllllli()n L'(mtrnl prohlem in Example 4J. . 3 2." or "compensation.37) and suhsystcm dynamic constraint (4.47). .(t) is augmellted with nj(t) to constitute a higher-dimensional "coordination vector.." ( rj ) QiXi ( I j ) ) I iJ x.(/)+ I: j ~ 1 N Gjj(X.(/) -Kj (/)CjZ j( t)+ I: /= 1 N G/n. .(l f ) foliow froJll (4. . I · / u. xi(O) =x jn pj(rj)=Qjx.(A.47) and equating coefficients of the first and zeroth powers of x i (/).(/) o Vl / c. ---.(r) = A.'. makic g repeated usc of (4.:.3.(t j ) and g. .SjP.( /) = (lll.120 IIicrarchical Co ntrol of Large-Scale Systems 1.L s:l li si'i cs (4..: '-- / X I and substituted into (4..(4.3.(t)+Cjzj(t)..3.1 Il() jJ. 4 5 where the vectors u.\-.3.(O) = Xjo 0 = 8Hj8u.4:\).3.\ . I.(' . (4. = Aixi(r)+ Billi(t) +C.x i .:1.-SjKj( t))Tg. ::3':::: --f~~"~=.46) 6 -.3 .( t) arc no longer considered as un k nowll s at the second level.47) are differentiated and iJ/r) and .(t) and costate vec tors p.3. = Rju. + I: J ~ I N G/-ai(r) (4.42) (4.3. - -- -: .:J L: ~ .36) to the cost fUIll.47) Fi[!IJr(' .49) whose final conditions K. [f both sides of (4.(r)=-Qjx.tinn's integrand: i.3. the following matrix and vector differential equations result : Thi s prnhlel11 C:l1I he s()ln'd hy first introducin g a set of Lagmnge Illultipliers n.3. RI'A(. Ihc ith subsys!cm Ilallliltonian is defined by id!) =." vector. For the purpose of solving the first-level problem. f/ / f= .It can he seen that this TPl3V problem can be deco upled by introducing a matrix Riccati formulation.(t) .TIME ( Sl' e) --.46) and (4..3.1B/.(I) to augment the "interaction" equality c<l llstr.'y.3.42) to obtain HOI> REACII I DO RFAU I I x.t . the first -lewl oplimal c(>I1lml (4.. x.- o () -..x.0 Xl \") HOIl RI'I\CII 2 \" .3.-~'..45) (4.44) hccomcs (4..39) Following this formulation. Here it is assumed that (4.3.3./ x / 1.40) (4..3.0 Open-Loop l-uerarehica J Control of Continllolls-Time System s 121 Then the following set of necessary conditioJ<s can be written: jJ i = - iJllilax i = - Q.40)."':'~~--.( rj ) = Q i X.value (TPBV) problem <Jlld Sj ~ BjR j.1. ( 1j ) .3. . .3.e ..A.( t) and Z./op.1n/jl I ( 1) ( 4.(r) (4.41) (4.lint (4.p.(t).(/).3 . ( tj ) = 0 (i x.-dimensional open-Ioo]) 'adjoint.(/) from (4.3. . ( I) = - R.::j(/).{t)+ B/jJ.36) -(4." which will be obtained shortly. 6.(t)-lC.3. :: I ( r) L 1= 1 (il. ( .(t) IY / ( I AI. It must be noted that depending on the type of digital computer amI it~ operating system.(t) .~I()rt~ p( t ) i = J? N [~.54) I \\ hich plll\·idc (X..25 0. ( .KI(t))xl(t)-Slg.For Ihis purpose. z.2 I I 1 . .2 0 0 I 1 O.2. C/'I)..\.3.49). ·.\1" (4.49) with fin. and hence they need to be computed once regardless of the number of seL'\llJdlevel iterations ill the interaction prediction algoritbm (4.) trix RilTa li equations (4. SX ) T I f1/(/)i .3 .(t) .'~ P\" t) () = . ( t ) -I (..3. subsystem calculations may be clone in parallel and that the N matrix Riccati equations at Step 1 are independent of x/(O).)t) (4.3. ran t'. is dependent on x.Y.i.11 condition 14..\' (I) (4 . )/ (/H/(r) = Z/(t) -- L U'I " . as is di scusscd in next section. Z/(I)1)dr) II\(' \ .S3) () .Y I 1 .25 .(O).3. no ZI(t) term is needed in tbe cost function. which was intended./ f! . .05 .(r) I /J. Stcp 4: At the second level.57) and storex.3.3 .SS) The sC l·olld -bcl ('()ordinatioil procedure at the (I + I )th iteration is simply .(t)+C. Smith and Sage (197J) have extcnded the scheme to nonlinear systems which will be considered in Chapter 6.56).x./(1) in (4. to avoid singularities.(t)= L i ~ 1 N Gj. A genuine comparison of the interaetion prediction and goal coordination methods has been considered by Singh et al.~ ill .v I. i = 1.3. For indial «( r). "N.3.. the solution of the differential IIl<1lrix Riccati equatipn which involves 11 / (11 + 1)/2 llonlinear scalar equations is independent of the initial state X. 1')0\ FV.56) to update the coordination vector Step 5: Check for the convergence at the second level by evaluating the overall interaction error 2 I /I: (/)/{ III I( /) (l: (r ) ZI( r ) .{r) I · v (4 .I Iicrarcilical C(lllirol of Large-Scale Svslems Opcn-Loop llil'l'archical Control of Continuous-Timc Systems 123 \\ hich 11<ls . 15 . First.0.(t) -i~IGljx/t)} {z.1. has been considered by many researchers who have made significant contributions to it.2" .1 (4.(t) =.(O).(t). Two points arc made here. It is further noled that unlike the goal coordination methods.1.lcli('n prediction method is formulated by the following algorithm . (1975) which will be discussed in Section 4. hv virtue of ZI(t).tiW11P . z: ( t) solve the "adjoint" equation (4.(t).~ __ -_(i~l_ ~ _~:~~__ -jx+ l~'2 ~ ~.J.1 (4.C'/p. The second point is that unlike K. use the results of Steps 2 and 3 and (4. .52) e(t) = i~ll)'r{z.J~IC.z.:/ ( t) can he obt<lincd hy ill .1lI 1 Stcp 3: Solve the stale equation .3..56) J'll\' intl'1'. . ) / iI Z t ) = I ( It.4 to obtain a completely closed-loop conlrol in a hierarchical structure.(t) = (AI .\/cP I: Snhe N illdcpcndcn t di ffcrential Ill. The following two examples illustrate the interaction prediction method.4X) with filial condition (4.S 0. The second ·· level problem is essentially updating the new coordination \ l'dor (II : (/) : ::1 (1)'.l).3.viII he used in Section 4.l1lel . du \.i./(t). Among them are Titli (1972).3.\ partial feed hack (closed-h.S.50) and store K. L. (1974). i = 1. Consider a fourth-order system x= ..05 (4. This property . .fllll 4. and Cohen et al. ( I ) (4. define the additivcly separable 1.0. originated by Takahara (196S). The interaction prediction method..59) 0 . Example 4.N.jXJ}dt/j. who have presented more refined proofs of convergence than those originally suggested.'. L 1< (I )(i ll \ I(r ) I .3.(t).\rlp }.-111 -_~~S o 0. (r ) < I ( I ) ]1 . Interaction Prcdiction Method for Continllolls-Time Systems . who called it the" mixed method" (Singh 1980)./I.(0) = .op) tcrm and a feed forward (openloop) terill. AIRor.lllt! .3. . 2) and no terminal penalty. the interaction vectors [0: 11 (1). At the first step. SOl tll JO N : \ \ The system was divided into two second-order subsystems.0.2. Note the relativel y close correspondence betwecn the outp ut s for the (lriginal C\)up led and hierarchical decouplcd systems. Next.F\(I) (4. (t). 1. 11 . 10. two independ ent differcntial matrix Riccati equations were so lved hy using hoth the douhling .Hllt 2.4: .3.2 u ~ in g.62) For the overall system. a complete feedback structure results : u(I) =.]. as shown in Figure 4.61) \\ere '(l ln'd. The optimum outputs for ( '.' 3 ' (\. as one \voldd . for the sake of compariS(ln .() '-. l.2 . The clements of the Riccati matrix were fitted in by qlladratil' polynomial in the Chebyschev sense (Newhouse.2 were applied.ed by so lving a fourth-order tillll'-V.3.42t ·~ _ .(I)I' and [1Y .X7 .0541 1 ] O.X31 2 2. (4. 1962) for computational con\enience: /\ I (I) == 10" _ A . ( -. (I) ..J1s f(lr buth hierarchic. R = diag( 1. and at <:ach infurmation exchange it<:ration the t('t :J1 interactiull error (4 . 56). '~II(I ) .60) c:r: c:r: w :.dg.141 t 2 . When the decoupled the COlltw l sig na ls are partially dosed loop and partially open loop. Th<: interaction error was reduc<:d to 3.\ t the fir st level.5 .0.O. a set of two secoJld-order adjoint equations of the form (4 _ 3.1(11 .124 Hierarchical Control of Large-Scale Systems 10 125 \ \\ilb \"(0) = .1.' 1(t)'(~: .0 .3 .44 + lU 21 --.10 Interaction error vs iterations for the inter.75 ' \ 1l. the two co ntrol s are different.0341 --.0.5113456 X 10 (> ill six it e rations.<: ~ 10.l.5] ~ 106: ~ .5 9) was optimi7.0.09+0 . 1+0 .O. and s teps oUllined in Algorithm 4. the fourth -order Runge .O.1 ~ (t) and lI .3.- = I I 4. 1.]. (I).'etion prediction in Example 4. = (i I) ~l11d control signals \vere obtained. -. (~1 2 (t). i = l.5 -Hl.6~--~----~____~____ IIL-__-L_____11____-L___ 234 ITERATIONS 6 7 Figure 4. the original system (4 .S)/" and a quadratic cost functioll with (j = diag(2.1 (t).73+0.0541 2 (4 .0271 0 .(1 ) = 0.O. 10-.1 +0.0.]. '~I.0.Kutta method and initial values (\" (I) I cc \ . It is desired to use the interaction prediction method to find an optimal contro l for (j = I.3 .IIXI -. I.0271 2 ] 0. Z2 1(t) .'-"pccl.5X) was eva luated for 6. Hving nwtrix Riccati equat ion by hack ward integration and so lved fnrY .Z22 (l)f w<:r<: predicted using the reCllrsi\'l' rdations (4. At the s<:cond level. However.J o f: ().4 IO .J1 and exact centralized cascs arc shown in Figure 4 .5 .2. The outputs and control sign..49) and two sub system state equations as in Step 3 of Algorithm 4.0071 .3. 1 and a cubic spline inteq)(\iator program.H13) . .261 2 2 O.161 .(}9+0J)(17I -.()rithlll of Davisoll and Maki (1973) and the standard Runge K utta methods.75] z2(1) = G (O) 2I X I = l = ~:~~5] (4.=1. 2).1 = 0 . 01 . 7 0.2 0.0 - ··-Y2 / 4 ---L-_.3 Goal Coordination alld Singularities When the goal coordination method was discussed earlier in (4 .- (/] r:: :::> o -2. .4 0.. i = 1.0 .1 0 0 0 0 0 ...1 0.3 0. " .9 1.4 0 .I 0 ..ll method arc given in the problems section..4 .. _ .L~ .5 0. (0) controls.O .4 0. The convergence was very rapid.I ." 3 ---. u o 2 Z -0.5 0 .99 1.0 )' I .0 .17).29 0 .3. -~ ~ -l '.I . ..0.1 .5 TIME 0.. . _ -----. RI=R 2 =1 were chosen.2. o ~..0..2 .. O~ 0 -2 -. In fact there.3 0 0 () 0 0. + 0 10 0 () 0 0 0 () More applications of the interaction predicti(. t1t=O.--.\'2 ..5 0 0.64) 4.2 0 0 0 ..L--_L-.. .--~--::~ .3.93 .~ _ I I _ -' o 0. 5 TIME ___ /l~ / 3 / .. and x(O)= (.0 5'---_ _ -'-_ 1 _ -'---_---'-_ 1 _ . QI=Q 2= i 4 .6 0.0.. . 15)-( 4..-- ~ f- 1 .4 0 0 0 0..7 _ _ ~_L_..0 Open-Loop Hierarchical Control of Continuous-Time Systems 4 127 '.48 0 0 0 0 0 .I. l. 'were . The initial values of a.6 0. -~. (/) 2 (/] (/] w z Z o ~ -.3.5 )T.~L--. 3 0.-.'(t).32 .1 .5 I -I a 2(t)=(1 0 -\ O)T..3.3..9 1..1 12 _ _ .3.. -lIt ---...~~·~~~~.\'1 :::> 0:: f- w '..11 x sys tem was decoupled into two fourth-order sub systems and (/=2. .0.11 () .0 (a) Figure 4. Example 4.0. as shown in Figure 4.. - ..0.5 () .0..J _ ___'__ _--.__ 0 . II 0 0 0 4 0 0 0 (4. ol.0._ L o 0. was excellent convergence for a variety of x(O) and afU).3: (a) outputs. Consider an eighth-order system It is desired to use interaction prediction approach to find u*. 1 0. - - 112 .5 I .3.8 0.--.. .1. 0..0. SOLUTION: The ..4.0. and state x(O) were assumed to be ar(t) = (0.0. (0) Now let us consider the second example. 8 0.39 .5 0 3 0.~.009 . ::~~ -.._L _ .3.j"b Hlcrarclucal Control of Large-Scale Systems 1. :-"~ --.11 The optimal (centralized) and suhoptimal (interaction prediction) rcsponscs for Example 4.~ 0r====±====~====t====!====~-===·~ · /==·=-.. it was mentioned that the posi live dcfini te matrices S.0.".17 0 0.1. In just four second-level itera ti ons the interaction error reduced to 2x 10 .X = 0 0 () () () 0 0 0 0 () 0 0 .1 2. let us recons id e r the problem of minimi l. we have (4. 2 T 2·\ 2 P2 '\ I- . ) =.a Reformulation I Bauman (1968) suggests rewriting the intera ct ion constraint (4._____ ______________ . + n.lI .I Ii crarcllical COlltrn\ of Large-Scale Systems Opcn-Loop Hicrarchical Control of Continuous-Timc Sys tcms 129 As one or the necessa ry equations for the solLI tion of the i th su bsystel11 problem at the first lcvel.RAT1()NS .3.I.69) it is seen th at the system ca n be decolllposed int o two firs t-order subsys tems.7 1) () .3.68) X 20 lJ 0:: -r: L'-l ::::: r- It is desired to find (u l .3.='" J The problem in its present form has the follo w ing Hamilt o nian : Fi!!l1H' ·U 2 The "il. SO LUTION : " From (4.in g L .::I . (4.67) whcrc a s in gular solution arises if the Z/C /)S._ 2 X l0 · ~ with interaction constraint 4 ~L ~---.\ 20 X IO (4..II . ::I./( . 1 = - - }.1/ 2:' /(I)S.3. since 2 1 appears lin ea rl y in (4. 2 PI XI + PIUI + PI:'I) (4.5.3. 1(. Lel the ith subsys [clll's Hamiltonian h e in which the intera ct ion va riable appears linearl y. The following exa mple illu strates the two approaches. (' " II" . Z.1 I 2 =====--ITI'. - I ( C/p.3. I I. P2 11 2 ) illl1(H.73) -' (I .I\ in Icra ct inn eIT(lr liS i tera t inns fnr (he eig.\1(0) = X 2 + u 2' X 2 (0) = . (t) term docs not appear in the cost function.69) is minimi zed via thc goal coordination method.3.17) to a void singularities.\ -.3 . th e appli cation of goal coordination fo r th e present formulati on would lead to a si ngu lar problem.3. 2 = - + III + Z I' .68). u 2 ) such that (4.'1 '/ = .68) is satisfied while a q uadratic cost function III J = 1/2 to (x~ + xi + XI II f + /I n dt (4. The followin g system reformulation s of the problem would avoid singularities.70) (4. 1/ 2\: ( I) Q.x. 5.Z.L . Here two alternative approaches are given to avoid s in gularities at the first leve l. (I) ·1 1/ 211 : (I)R . (() + ( (t)) x: (4.lI f + lYZ I -II 2 2 II 2 - (U 2 .3. Consider the followin g system: . -I IY.-'".66) or Z.3.:1. To see thai 111 thi s is ill fact the case.3. .~.i ~ 1 N 1\ .(I) 4. ) . Thu s.(I)-1.U . 15). Example 4.73).h (·order sys telll of Exalll · II = (-ixf -I- .llIl"l'd ill th e C(lst fun c tion (4. + ('."X.~" p" n .(4.3.3. (t) = - S.XI X 2 + x 2 -I+ u2 ' /II' X 2( xl(O) = X ln o f ·~ i: 2 = - 0) = (4.72) in ljuadratic fo rm.3. I· I). 24) suhj ec t to (4 . 1 PI ' co .3.76) and 1'(1(' tlie sl.\ 10 <'(I ) = z~ (I) .UO) \\ here (.1 (1) mu st be so lved backward in time.\ 20 (4 .40) . in erkct.IXX2 .3. i ~ «.(rIG. and R matri ces .". (I CI7(.3. \ .. II . T hus. .111 1 > 1<' ti ('('( '.a .jl . If'[~ hlnck -di . ) = i x~ + i ll ~ + az . + p.xH /) i(.\: 2 lI 2 + P2 P2 ( I) X 2( = aH/ (h 2 = . -I. ( 1 ) ~.3.f )"'ill g tn! \' ill terms (11' the int eraction vec tor z and . since the P2-costate eq ua tion is deco uplcd fro m X l and can be so lved backward in time a nd sub stituted into the X 2 eq uati on.(I) n(/).k ~ ( 1 )\' 2 ( 1 ) (4 .P I" ./1 2 .\ I ( 1 ) (4 .C/'a + AijJ .P2. I =" . p. +a + p. z = .1 \' ) lIicrarc hi cal Control of Larg.'( p + IX) 11(·) = 1.'(/)V\2I1z(I) + \1I 1 (/)RII(I)+ly' i: -. For the first subsys tem.79 ) 1 ) = _.2k I ( 1 ) ) X 1 ( 1 ) .3. X 2 (0) = X20 ll~ ( 1) ~= -.3.3 . while no auxiliary equati o n needs to be solvcd for th e second subsys tem.3. ( t ) . In ge neral .c-Scalt: SyS ll' IllS Open-Loop Ili cra rc hical Control o f Co ntinu o us-Tilllc SyslCllls solution ca n still he fo und. . Th e procedure is hased Oil <.'l'O nd suhsys tem. \' I (0) = \.(I +k 2 (/)) X2 (1).~ uh s titutin g it into th e C(ls t fllllL'ti(ll1 : i.~ . gl ( l )=O UI(I) = - (4 .X20 (4.{ I -I (If I / :.85) kl(I)X.:.3.P 2' '\·2= il n~/ (J p 2= . which wo uld .3. - () = iJ lIl / (I :' 1= 2 HZI + /) 1 /) I ( I ) = () = 1I( .d ." 1( /)\ .. Th e ('( l(lrd in .RI) I'l l! Ill c exa mple under co nsid erati on. Z G ill hc written in ge neral as gl( /) = (I +2k l (t))g.gcst . 1 ( I ) = It ) 0 = OH/O ll 2 = -.(4. = x.lli PIl at th e sccolld level is achi eved throu gh thc followin g iteratiol1s: H' I I ( T he second subsys tem can be solved immediately. ( 1 ) .3. this reformulation would requirc thc so lution of .p" .(1) . h Reformulation 2 Sin gh cl . (0) = .77 ) 11 . ( I -I.e fir sl k n' l: t) ~ lI lIl / n ll l ~= II.Ix -I- fill + Cz ) (4.3. (hi s rdo rillul at io ll w(luld avoid sin gul ariti es hut makes the seco nd -level it erati (l l1 s Cl1 nvcrgcnce vcry slow.48) a nd (4. two such eq uati ons should be so lved fo r the first subsystem.5 1).\' + p'( .R .!ilics hilt a\sll gives gO(ld Cl1n vergcllcl'. For thi s exa mple. (I)= . C. j) I O~ ill 'I / il x I PI ' (4. -. = 11 2 1..'B Tp .111 alternative formulati on whi ch not onl y avoids sill gll\ :. is sin gular./ilx 2 =x2 .PIli .XI + li l + " I ' . J-Jere the Hamiltonian is I] I \y i1ich \\'( )\lIt1 givc th e foll (lwin g necessa ry co ndition s for optimalit v <II tl.\ I ( 1 ) = ttl ( 1 ) I- fill I ( e( 1 )) (4.8 3) .R4) k I ( 1 ) }.\ 2-1.3. ..e . followin g th e formul a ti on of first-I cvel problems in interaction prediction (4. mnn St.49) need to be eva lua ted. the ma tri x.75 ) x II) + lz.jJ2 = (J ll. both a Riecati and an opcn-loop adjoint (compensa ti o n) vee tor eq uati on sll ch as (4. . but " which indicates tilat thc cos tatc vector p equation is deeouplcd fr0111 .. )() = () If /0 P2 = - -I- U2' 0) ( ~ (x) k .(O) =x o (4 .(I) = O (4 . d inln N sllh<: v<: l " lll .n) " 'I! nl' k. ( 1) is the ilil ~ uh sys il:l1' ~c a lar time-varyin g Ri cca ti Illatri x. k. = . Arl er th c introduction of th e Ri cca ti formulation (4 .GQ .k= Ax -Sp .(I) = O .7(.\'2 (/) = .'(p +n). = r11f /iJx ./) 2 + PI' 0 = fJ H/r7:.\1(0) with first-I evcl subsys tem problems bcing I'll ' fir st sUh SYS tl" 1l ~ 'lld () = illl ~ / iJlI). Q. jl = -.P 2 X 2 + P2 11 2 \' . 4 . }3.) sug.GQ .1 75 ) alld (4..\' I ( 1 ) . thc first subsys tem problem is .~.Slimed to be nOllsin gular and the rdormulated Hamilton ian is p(lj ) = O = .(I) and .(t)+ k.(t)+ g. however. . ~ iIJJI / ()p.) will lead to \ . (1) = .H nnal nrnhIPrn (41 X6) (' .3 .86) (4 .2kl(/) + 2k ~ (t)-I. Since the iI. X 2 = 0 = . mcan that Lh e solution o f a Ri eca ti equation has been avoided for thi s particular exa mple.2 g .+ ~ lI i 0= aH /O ll l = lI.\ and ca n hc so lved backward in time (eliminatin g a Ri cea ti eq uati on) an d subSt ituted in the lop equ at ion to find x.U" P2(1) = 0 -' 2(O) =.n ( 1 ) .3.Y ...(t) where two differential eq uatio ns for ". lct ive extens ion of th e partial [eedhack a lgo rithm "I' ('oil cll l' t al.t)x(t) (4. 11 11.1. Tize open-lool' adjoin 1 vec l ur g(l) rlnel slal e x ( I) ar e refuled Ihrough llz e followi ng tranl/orlllation: g(I) = M(tj ." " nonb endicial. Theorem 4.e th e interaction erro rs "" d i1111'1 (l\\: the (1nrllrll1. the overa ll system's adjoin t eq uation beco mcs g( t) = - (A . du e to Singh (19RO). .: ind cIWlldent ('I' Ih e initial stat c (Sin gh.lIl Ce.4.3 . . .s t" lll (Ill th e first le ve l hy igllnrin g th e int eractio ns: then a glohal l'o lltr(llkr ./ ..C/( K .sl'. is ba scd on (1htai nin g local feedback controllers for each StlhS\.(I) = . 1972. e. \'.7) . prp\'idcd that the terlll ( X' (/)V FQ V z (l» is separabl e in z whe re I' '7 (.' was .d sys tems. Mesarov ic et al.SK( I) + CG ) T g( t) . T L j = 1 ' ~' G.1).l'('d .(I) .·h.3. ..e. Arafch and Sage.(t) in (4..(K( t) CG + GrClK (t )) x (I) (4." and " neutral" illlCl'c(1 I1IllT ti o ns and their co rn. tCI11' ill . .(II '·".(l)) g. . i.cs the st ru ct u 1'. (I ).Y(lII)' Alth ough on e may ..lk and Su ndarcshan.55 ) to elim inate D:. 1974) h ave consid e red the l'f(lhlcm alld sugges ted eith er a " partial" feedback controller or "complete" r" l'( lh.\.d-)nnn ('l1 lltro ll er .: syste m .Ib :I n!lrl.4 . :1) .: co ntrol depend ed Oil the sys tem's initial cond iti nll .4.: tiPIl dea lt with 0pcll -Ill(lp lli era rchi ca l co ntrol of continuoll s.(A.. .6) with sta lld a rd quadrati c cos t C( " "I)O- 111 II1\' I)f('v i(lli s sl'l'l i(lll .=1 j ( I ) Xj ( I ) - gi ( I ) ) (4.I ( 1.4) g(lj ) = O which can be represented in terms of its homogeneous and particular so lution s. Thc fo ll owi ng theorcm due to Si ngh (1 \)1)0) relat es the open-loop compo nent (4..001' II i<'rarchkHI Contwl of COlllillllous-Timc Systellls I il l' 1 se. hy th e tim e all open-loop con trol is c: tl clIi alcd alld <lpplit..4. K(I) = cliag{ KI(I) . il) 1I<lye used a similar suboptimal techniqu e for th e [ilter problem. let us consider the differential equation for thc adjoint vector g. Sund ares han.v (I) }.:d to tht. b. 197R ). 11.K. (t) dep end s on the states of all the o ther suhsystems and hence the initial statc x (I.1 " <lY S h l' ahlc to ll11:aSIIIT . K . the inili .Ioon .11 pe rturhatio ns cat cgnri 7cd as " benefi cial. k. 1973. as in nonhierarchi cal systellls..\' .iak. Ano ther attemp t along thi s lin e. (0 ) is the "state tran sition" matrix or (11 .SK( 1) ."" ·". .:spondin g loss of perrnrmance.5) where ({)I(I.K.''' ic h tli t.4 . thll s resultin g ill lillpredictahle and undesirahle respon ses. Man y auth()rs ha ve cons id ered the prohlem of dosed-loop cOlltrol of hierarchic..2) to the sta te x ( I) for th e overall system which ean be used in a hi erarchical s tructure o f a regu lator prob lem.t ty/(1 )Qx(t) + 1I1( I) NU(I)] ell ( 4. However. /" ) g ( 1() ) - 1'<1) I ( t . III Ihi s sec tinn Singh's (19XO) exten sion o[ th e interaction predi ction ap IHl l<1 ch and th e feedha ck C(1ntrol sch eme of Sil jak and SUlldareshan (1974.S.4.lI1d an onen. CloscJ -Loop Hierarchical Control or Continu ous-Ti me Sys tems 133 I. il) alld Sundares llan (19 77 ) hased 011 structural perturhation a long " 'ith Il\I1l1l'J'i ca l exampl es are prese nted.4.K.1 = ..d s tat e has IllOSt likely . i.! ".h ich still depend ed nn . The 1110st lik ely case in whi c h suc h a c(l lltnli s tru cture is p(1ssihl e. 1974. 1) whic h indicates that the vector K.n l. (1 974) whi ch provid es " complet e" closed-loop stru ct ure hase d on n rf-lin e c. 19XO).(}(}fI Co //trol oia ///I emcfi()// Predictiu // = I1x(I) + BII(/) (4.. 1972: Co hen et al.(t) and Z. I t is Ih cn: rore \\'o rthwhile to C(lllstrul'l closed -loop control la ws whi ch an.49) and li se (4. An atlr..IIierarchical Control of Large-Scale Systems I' r(1h1clIlS.. 19'77: Sil. T ) ( K ( T ) CG (0 1- G 'e Tf{ ( T ) X ( T ) ) dT (4.Xj (l) L Gj. till' i ntcr.. ).llculations or feedback gains and their on-l ine imp lementati on i.11 th e sCl'(l nd lew l is applied tll minimi 7.(1 0 ).. (·1.ll'k cnntrnls wh ich in vol ve o ff-lin e calculation s of the overall coupled I"rge.Ist .4 . fo r the co mposite sys tem.. Sa ge and hi s associates (Smith and Sage. du e to Siljak (Jild SUlldareshan (Silj .ll c sys tem.\:(1) 4.4 ( 'Iosell-I .(I). "'"" is l11 et hod elll phasi l. I l 17 (' a.CG)T.) 74:1.4.3) PRO O F: Rewriting the adjoint equatio ns (4. is th c lin car quadratic regulator probl em. th at th c open-loop co mponent depcnd s on th e initial state of the system and there is no app arent poss ibility [or on-line implementation.tim c . To see this dependency of the open-loop component.ll'lioll predict i(lll ill it s gc ncra I se ns. 01 hers (Cil ene\'caux. g ( t) = (I) I ( / .{I )C. (I nO) ~ u g gL's ted a sllil uptill1al structure whi ch had an adaptive feature as the sys tem c\' (l l ve~ .) .1 ('/o. . Note al so that K(t) is a block-diagonal matri x consi stin g of sub sys tems Ri cca ti matrices. 1976a.~. mentioned ea rlier.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 "..' SI: h':IlIl' Iwn lTs t tn a closed-loop structure was th e int eraction l" ('" Ii di(l n Ill.·In.1I1 )2. 1~ -2.4.4.IBT(K + M)x g ( to) = = - Fx (4.I7 -1 .20 . E.X(tO+I)"'" and form nXn time-dependent matrices G(t) and XCt) to find M(t) for each integration step.0 0.R .08 .2 . to) is the state transition matrix corresponding to the feedhack system matrix (A . 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. K.0.38 .16 -1 .28 0. M can be approximately given by (4. it is possible to solve the problem wi th fl initial condi tions.14) x ( lo) : x ( I I) : .14 . constant A. g(t o ) = M(t l . G 12' and G2I matrices switched.1.1"llcran. T)(K( T)CG to Example 4. Singh (19~0) has raised the point that for the time-invariant case. since near t = 0. = . .221.498' K2=[ 2.. but from an implementation point of view.02 -0.SP(t))x(t) ( 4. .35 M=GX I = 66.16) I( I ().40 -104. After six iterations of interaction prediction at the second level. (4. B.881 -87.~) g( t k) for k not.32 65.10) Usi ng the fi nal condi tion g(t I) in (4. This corollary is used to find an approximate feedback law for the open-loop component. G.I B T Kx .9) where P( t) is the n X n-dimensional time-varying Riccati matrix for the composite system and <1>2(1.24 (4..10) can be used to solve for g(to ): u = . • The problem was simulated on an HP-9845 cOinputer using pASIC.3 with II = 4. x(to).82.127 0.4.13) is a sound theoretical property.4.4 . the resulting control for the composite system can be formulated by g(t)=<lll(t.4.) and = 0. Let us reconsider the system in Example 4. 1968).4.4.IBT.135 (4.29 1. : x ( t .440 I 0. For the time-invariant case as II -400 .4 229.241 1.4.26 285.510 .1010 .4..1010] 0.45 -2.e ..4.2.4.100 J -0 .4.66 .11 are evaluated and recorded. Note that if a time-varying M is desirable. .0. tI) 1'1[ I( tI' T) ( K ( T) CG to (4.78 G=IO .5). to)x(t o ) = {{'I(l o . X = [ (4.117 0.66 X= 1. 1. The relation (4. and R matrices and time-invariant g and K.1'I(t.18) -0.13 -2.. G and X were determined to be A corollary of the above theorem can be stated as follows.1.516 -0 .D . and M are obtained from decentralized calculations.258. and the matrices R.52 2~9 .13) which gives the desired relation.093 0.9067 -0. : g ( I II) ] .4.4. Q.00 .I.I -0.97 242.R - I B~!!. M hecomes a constant transformation matrix (Sage. Now if we substitute x(t) from (4.4.SK) and S = BR .19) the partial feedback matrix M to be 63.77 [ 0.093] 0.15) and the inversion of X is done off-line.4. B.20 -1. It is desired to find a feedback gain matrix F.~ .12) or K =[4.516 -1. it is suggested that if the first 11 = L~~ llli values of x(t.17) (4.4) at t = t ( and maki ng use of the properties of the state transition matrix.70 -210.7522 (4.R .9) in (4. M can be computed easily. i.218. Therefore. Q . The following example describes the above feedback law.85 1.4.28 1. . In summary.69 -0.T)(K(T)CG to (4.. .70 (4420) ..3 .73 [ -0.4 . C.86 .J ] G = [g ( (0) : g ( I I): .59 0.72 r 77.11) By moving the term I(to' tI) inside the integral sign and taking advantage of product property of transition matrices.4. (4.500 -0.3 . M=GX where I (4.11) is rewritten It is noted that the above gains are all independent of x(t o). SOLUTION: The decomposed system Riccati matrices at t = 0 are (4.:JucaJ Control of Largc-Scale Systems the dosed-loop optimal control system is well known: Closed-Loop Hierarchical Control of Continuous-Time Systems 135 x(t) or = (A . M is constant while x and g are -0.tO)g(to). 30) = .\1l (19Tl ) h. there is a useful class of interconnections for which J *( . ilh th e intnactiol1 111:1lrin:s ('. Eac h suhsystcm ha s a feedback C( 'I11TOI ~ In.25) and cost (4.) > 1"(. x. ~ N .(I)+ or a hierarchi ca l sys tem di sc ussed where K. . N. i.I. .20) for i = 1.<'-. lelll .. = Definition 4.23 ).Loo{l COll frn/I.. ( I )Q. is the positioe-defillite srllllll etric sO /lIl iOIl of ilh slIhs)"slClI1 A/.\ I j ( I () ) . (lo. \ I licrarchi eal Conlrol or Larg.9I. For a large-scale system consist ing of N sub sys tems with overall and decoupled performances 1"(.. The interactio n G is said to bc " neutral " if J*(. {( . ..22) ". N. ( 'ol1 sid'.27).. where 1\" i = L 2.' svs lelll .s:11i s fvi ll g (4. .4.y}.2.uj(t) .C/) =Aj-'dt ) + B.urin g lh c _ ' Sil' Pl'n:l1illll p f tl1<. where all matrices aTC dcfi11l'd earlier.22) 1'(' 1 i I . how reliabl e the overall system performance will be in th e 11I l'Sl' 1\Ce of structmnl perturhati o ll s. .1I< Sy <.29) may.:ll' h slIhsVs tc' 111 is il1d cpcl1d cllt .".\. i = 1.lscd (' 11 <. be greater or less than 1"(to' x(r. is given (4. l· ture co nsistin g of a " loca l" compon e nt obtained Ihrough sllh s\'s tClll ca 1culati(1ns :1I1c1 a "global" component which minimi zes int era cti (11l -c lIPrs all d poss ihl v improves the (lVerall sys lcm performan ce.l1 l's h:J 11 (19 7(1:1. .36 . is th e sy mmetric positive-definite soluti on of the al gebraic matri x Riccati equation (AMRE) (4.lY ve ry well occur d. = {g.1I1d ex tension s b y Sl1llllaresh.). i ..4. .) and J*(. 2."1 ill \y hidl eac h pOlVer C\lllipany (suhsys tem) is re~ p()ll s ihl e for the load. by = I3J?" j i/Jr. al1d SI111l1 ..l ()n sl'd .4. \ ( 10 ) ..4. II j ( III ) ) (4.4 .2.2. .1 111 1. j = 1.1.4.l . the following definitions are given: Definition 4.4.J l. " In'q uell l'Y rcgulati(1ll . .IYlIIlII ctric matrix.4.) .. it is assumed that the system possesses a co mplete decentralized informati on structure and each subsystem pair (A j' B.llj( ()) cit In (4.(fo) =X jO (4.iBTK jx.j=I.ST.-1.I.( 10 )) = ~ J OY. ( I () .cc/ liy G=SK II'/zere L .).\ .] iatc gLla l (If fil1din g a " loca l" controller {( .lIno n g suhsys tel1l s in . S is a skelV-. l3efnre we introduce the theorem.11 j1er tlilhati llll s du e to th e inl eraction s .' iu Sfrucfural Pertu r /}(f(ioll TI l\' sCl'I 'IHl1l1elhod of c l() ~e d . in general. ..: ..1. {( ( I (l ) ) = 4. __.Ioop c(ll1tml here is h n~..c-Scak SySll' IlIS Closed-Loop II ie rarchical Conlrol of Continuous. Let (Ill II X 11 block-diagollal matrix K = block-diag { K I' K 2 ' " ' ' K .' d 0 11 th e fe ed hack control ~ trl1clure suggested hy Siljak and SI11HI :ll l'S\i.1\ apprnxi111'1k feedba ck ga m matri x hased on hi erarchical u lnl rill l(l he F~ N IB/( K -I- M) = [ 74 .X2 (4.( t) = - PjXj (t) (4.I (.1_ (.) < J O ( . ..4 .h) is that assul1lin g each suhsys k11l ha s its own ind ependent l(lcal controller.(f )R.3.. the va lue of the corresponding optimal cost.) is completely controllable.IJ. J " (lll d th e dccollplei/ SI'stelil optilll(l/ pcr/i nJ11lI/cC .f 0(' ) . /\ classical ex ample is a "power 1'. a n interacti on G={g'i}' i.5 . 11 is " SSl JlIl"" IIi :ll :dtlinll gh (.in g the oVl'1all system cost functioll . \Y 7(' .l(. (I) + U. they ha ve :1 "go:d h..1 * (Ire ' ('1//((// il (ll/ri Ol/Z)" if th e inleraction IIl!Ifri_ C.1.6 ] . <llld th e h:1 i1\ilial iss ue addn:ssl'd hy Silj.'(I) which minimi zes an as>(Wi :l1 cd lju :Hlrat ic ("(lS i ir J*(. 23 ) \\ hilt. S .c. respectivel y.l1 12.4.0 -.cr/iil'l)/(/I/('c iI/ric.d./X/ (I)..91.. .l'iRE (4.\.26) 4./ se t to I nn. Let the op timal performance function of the centrali zed sys tem he denoted by J °(t IJ' x (lo»' The decoupled system' s cos t J *( /o' X( IO)) = L J /t. (. (/ 0 ).232 ..Time Sys tems 137 "nd Ih e (l\Tr.) Theorem .s.4.4.(/) -I. depe nding on the type o f interconnection among subsystems. is said to be "non beneficial" if J*(. For the decoupl ed subsystem. Slich 1'l'rlurhatill1\S 1ll. (I " . N.4. and power ge nerati o n within its own regiu n while tIHo\! £'. Definition 4. ('({/I he \ jilctori:.1 -.h li c lin es it can exch:mge power with oth er companies (sub systems) rcslIltill tc in va riation s in l)(1wer among the companies which would affect tlie entire sys tem' s (we r:"1 performance. Each suh sys tem ha s :In illl l1l<. 24 ) (4.S 27.2.2 1) Furthermore.1I1 (\4 74. In fa ct. Xj(IIJ)) i= i N (4. . . x .) .2. '" N. the decentralized optimal control is given by uj'( /) = - R .(I) o=. .. ) .'.27) where V.h) .I ( I () .'r a large-sca le linear time-invariant syst em described by N su hsystems: . .2 102. An interacti on G is sa id to be "b~ n cfjcial " L / .) == 1"( '). Th eil tli c (iDem /! S\'stCIII 'S Olitilll!ll I .N. as Sundarcs ha n (19 77 ) point s out and as we will see shor tly throu gh the follo wi ng th eorem .. R9 25 4."1'1 1.lII1HlI1 Y" in mini11li.N (4. 4 . The corrective gain matrices fI'l arc obtained throu gh '[ he n. i.3.I < llioll .A N ). I I' (.\( II) ) i r F is positive-ddini te and sY lllmetric.4. a skew-symmetric nwtrix .B.E.4.4 .11) . 2.4. . as will hc see ll by the numerical examples. () : hellce.. Therdorc.\. .4. (1) is given by (4.4 . J( l . :t ntl SIIIIlI:trcsl1an (llJ 7(l<l.PN)' perturbatioIl in B.40) can be utilized to give an estimate on the performance index. ..4. . Figure 4.. whilc [JP is structu ral perturbation.33 ) (4.40) L . "local" BP.P2 . i. Illay be used to find :1 l'l.'n ) .. [J2.35) whcrc )'(/0' ..3LJ) where P = dia g{ P I . If we choose . .'Vt' rsl' is also true.34) .ing control.4..s fil'~ (4A. . For a nOllsingular (A + G) lI1atri.4. llo\\. ~ I (4.FIlF + (l = 0 (4. ).4..26) and (4.h) have dcterlllineu \)()und s on the resultin g K = (S . /\ .\:.. (t) (4.e . once the perturb a tion matrix BI' is determined.\'/\.38) ~\: ( t ) = ( lJl' is an that matrix lJ is block-diago naL i. it fnlh1 ws that F c~ J( s. .42.4. . -c. matri.)x. F = K onl y if KG + ( i' /\. ).1/ 2 \ I ( t () ) /-'.N.4.B. . in ge ncral.\. This is achieved by a so-called "corn:ctive" control 11.4..36) wh ere /1.4. if hl'. til l'p r CIll. 2." or "global.\·... x.4. . II must be l?111ph:t silCd that if the structural perturbation of the sys tcm is limited to (4 .P.l!ld is jl(l ~ ili vc.37) yields N (4. (I) = (:1 ..4. BlI 11 X /JI = B"P (4. Q.I.D.33).. .1* .4 .4. let a SkC1F-.I \~ Closed -Loop IIicrarchical Control of Continuous-Time Sys tems IJ'J Ilw l'I'<. J " ( .. .(1 0 » = J. and A + G) x ( t ) . i = I.l = 1. (4.~ 1.. ~.KG)(A +G ) .) an d (4.38) N p\\. (I) K x ( Ill) =.30) the local contr"ller s Il"t (ll1ly optimize th e overall sys tem but also stabilize it.\ (I) and "corrector.JsS of int cractioll matrix patterns for which the optimum overall SystC!ll C:tll hc achicved by applying decentrali/. . .-.13 shows a schematic for the proposed multilevel control method. Thcorcm 4.N. rn Ihi s th l'oITI11.1 and 4.( B + B I' ) Px ( t ) (4. is g ivl' ll h e lo w.e" B = not. provides the mechanism to achieve this. Ihcll by comparin g (4. Q = Blockdi ag(() I'(> " "" (!\) illld " ~ BI(lck-d i ag(VI.(t) = wherc . docs not satisfy (4.37) Si nce 1 . 32 ) u. ). one ca n ignore the interactions Hlld solve N independent small'l': lIc l'whlcllls which pnlVitic both optimal and stahili7.12..f" ( '11' \( '0)) . X n i gain matrix for the feedback signals from thel th subsystem to the ith subsystem. .4. liR 1(11.. VN )' It is clear th.) bc such Ihat (K + K) is a positive-de/inile matrix. " j ~ Block -diag(A I. . a bound on performance J rcsults.e" .(I n.. Notc diag{B I.ITlI1l11clric /J/(/Irix S hc thc solution o/Ihc /ollowing lIIalrix LyaplillmHype equation: (4AAI) )(10' X(lo)) = L ). The closed-loop sys tem (4..I . "I(( lor : I. (4.(t o )) .31) rcduccs to (A'F + r >l..\ ..4.FJJJiJ'j (t) (4. while ur(t) is assul1leu to be 11' ur(t) = - L j ~ ' HiiXi(t) ( 4.lt .(1) = (A." BPpx(t). /\(i = _ .. uu e to Sundarcshan (1977).e.31)). It is Iwtcd that under condition (4.f PI' Ihi ~.ed controls ut (l).33) arc comparcd. The local-global feedback control system (4. . The following theorem.(l o )' ut Uo)) for the composite system (4.4.4.A N } ..30) follows.l l i . . .N. . Then all inpul /Ilalrix perlllrhll/io/J J]I' gillen hI' (4A.II:. tit l' n 1 is thc sy mm etri c positive-definite suilition of AMRE : \ or where Il.( t) in addi lion to the" local" decentralized control lIi( I).34) is greater than or less than . (. definite and symmctric solutions \ (.36) to (4.<.4.4. 2.43) .3()) iIll!.2..(I) + 1'(11' i ~ l."/\' = .J()). SiIjak .22) using (4.ix i (t) I (4.31) and (4.(t)+ for i = L J ~ ' (G.40) is influenced through the two components.V" . n = Blockdi:l g(Il I..['. . J( = Block-diag(R I. (4. ..U I)' X. ) = 1/2\ / (.26).13 N }. u.4. A 2 .4. s:llislie s (4.. i is an 111 .. ' he th e solution of the following /\MRE: (A + (j) IF + 1-'( A + Ci) . .FVF +Q )+(FSK+KS 1 f') = O (4. N. By virtue of the ahove fOfmulation of a can be rcconstructed : f-! ={ f-!. (. li es tilnt . .4.2.4.e.defillite nllci sYlllmetric. R N ) ..30).1"( . . /I\. If F is the symmetric positive-definit e sollition of ( '~. .. The application of ui(t) in (4. dill' f1 to SlInd:lresi1:11l (1977) . . . till' desirable condition (4.ncr.4..I An).)X.l'l II " .4.(t)+ u.4. i. i. arc positivc ."(1) to thc ovnall sys tcm will result in a performance N with A = diag {AI' A 2 " • . and the application of u. depenuing on whether the intcraction matrix (i is 11(lllhcndicial or hClldieial in the sensc of Definitions 4. i. .I ( 4. ~- N G.31) pcrfollnancc suboptimality based on the interactions and have established Illultilevel schemes fOf rninil1liz-i ng interaction errors and improvin g (In the performance. 51) wherc S is a skcw-sy mmctric matri x which sa ti sfics (4 . (4.50) Thc relatioll (4.53 ) t. ) .E.R N p~ AM (K) / A (K) II1 (4.foreovcr. 1' ..4 X) whose el ements arc kcpt variable for illustrative purposes. .53) lir e {Ie..46 ) :Ind kt i\ 1\1 R 1: . For a quadratic cos t function J= Ilowl'\'e r. then.4..\( I) = ( .40) . th c minimum cost is J(tn. therefore.4. x(tn)) = J 1/2x (I O )Fx(/o) = 1/2 x T (/o)(K+ K) x (to).4. /I X II posi tive-definite and symllletric lll .4.K).LI :({ I LOCAL CONTROLLI'.l! /J ' CI/ (.(K).55) i ( I ) ~c ( A + (i) \' ( I ) .\' 2(0) = 0.4.43).4.43) in fact satisfies (4.4..\' I' ~ AM(f.49) can be used twice to rewri te (4.\ (t o )( K + k.4. which is th e des ired cost (4. A".U 1\ closed -loop Illultil evel s ta lc regul atio n sl rul'lure./1/('..Scale Sys telll ~ Closed-Loop I Jicrarchical Control of ContinLioLis-Time SystcIlIs 141 Since K is a pos itive-definite and symmetric solution of (4.41).4.4. ) J p = I' = R In/ K (4.. \I'hl'/'(.5 (4.4.I' (4. it is seen that for (4.40).4.47): (4.4.i -. Q. 5.4.\ is a stabilizing control for the original large-scale system.4.4.4~) to take on that form the followin g relation SllOUld hold : R I ( 11 + 1J . it follows that the suboptimality index p satisfies (4.4. For tb e proof of thi s corollary. x l(O) = i /r + r( II I- G) .n l' ) N I( H + 13 " )r. ~ V I' h ( I ) (4.4.KG)( II + G) .\(1 0 )) .)X(/) + (fl + IJI')I/(I) (4. Example 4. '.. I is a sy mmetric matrix.ltrix F he the solution of ( / f l (i) '\:1= 2x 1 +1I 1.4.40) .cr(orl1/(Tllcc il1 (/C. Clearly.45) 11.52) Fil!Il!'p <I.44 ) for the overall composite system (4.. l/ A.\ An imm ediate co rollary of thi s th eorem is th at fo r a perturbation ma tri x /]" given by (4..54) whne 1'1' ~.1 + (.4. throu gh direct substitution.( ' ) I/I/d AM (') represent Ihe I1lil/IIl1ll111 (Jlld l1/(/xil1ll1l1l oj lire ei.4. T he foll owing examplcs illu s trat e th e structural perturbation method . . th e closed -loop control l/( 1) ~.lkes advanta ge of possible beneficial aspects of interconnection.F/l I' 1-' I Q = () ( 4.14() I Iierarchical Conlrol of Large.2.D. Moreover.:.R I( n -I n l' )'j.6. fl. Now if we let K = ( p .4 . . the control law _ I . sin cc (F .. Consid cr a simple seco nd-ord er sys tem. .45) and is given by LOCAL CON! HOI.4. for al1 t o and x (t o) . (1/ tlr eir associated lIIatri.49) ±tox ~ (t) + x ~ ( ( ) + u ~ ( ( o I) + u Ht ) ) cll (4.K) = (S .47) x2 = 4X 2 + 11 2 .cl/llll/ues argll/)/clll s.4.49).1 * ) j.!or- r/n ' i(/Iion fl = il .4.~ \· (t) res ults in a minimulll cost iXJ(/o)Fx(to) for th e fn'dhad c(l iltrol sys te!n ( with a 2x2 interacti o n matri x (4. it can be see n that /J I' defined ill (4.4..4.\(t o ) fill' fh e III 'C/'{/1/ .' • .4.e. see Probl em 4.C If -I. PROO! ': Consider the followin g perturbed system: . This combined global-local controll er extend s th e earlier work of Siljak and Sundareshan (1974) in the sen se that (4.1(1 0 .ITs/ell l (4.56) we would like to in vestigate th c effects of various interconnectio ns.. since the des ired closed-loop sys tem is of the form (4.44 ) 01/ u(t) = - (p + H )x(tl (4. [7urther discu ssion o n the method will be given in Section 4. (4. lir e /lpper hO/lnd /11. 33).50 ) ca n hc rcwrittcn as KG + (F-K)(A +G ) = S (4. 1' l'Ol'ir/cs (/ l.4. rcsJle(' til ~ e {r . 2] 12 (4.58) after rounding. Next let us consider a case of "beneficial" interaction by assuming an interaction matrix G = [5.4.48' K.67) provide KI = 4.85933.1353 l I .41) becomes [~ - 6· 915b ] (4. 0 S= [ -0.62044 and the corresponding 00.32 -6. see Jamshidi (1980).4. the skew-symmetric matrix S from (4. The remaining possible structural perturbation to be discussed is "non beneficial.41) is solved for S.4 . 0 2 -~.24 = E 1.70) where b is a nonzero constant.4.2 I "' i . Now consider a second example.92 (A+G)=[i with a solution -~. BOD (biochemical oxygen demand) and DO (dissolved oxygen).106 13. 8. The suboptimality index for this case is p = ..4. G= b] O (4. Thus.4.92 K+K= [ 4 .69) is a nonbeneficial perturbation.95.62) -0.0.64) Using G.and positive-definite. K + K matrices follow from (4.4.4.68) Next we consider the case of "neutral" interaction in which J= J*.4 .4.604 (4.1353 indicates that (4.1 ~' 0.9 -1. and K = diag{4.4.57) J = 1XT(O)( K + k )x(O) = 0. Tllis example deals with a two-reach segment of a flver pollution problem considered in Example 4.4.24 [ ui(t) 0 with optimal decoupled performance index J* 2 which is much smaller than J* in (4. 123 and optimal local controllers ( 4.:i =~:~~~]. Consider 5.2.334." Let us take an interaction matrix (or structural perturbation) gl2 = -1.71) K O = [4.82 x 2 (t) ( 4.2.6 1) [0 4 .* = 2{K x?(0)+ K2X~(0)} = 3.58). 'and hence the solutions of two independent scalar AMRE's which are negative. For a survey o n the solutions of both equations.58) _ [ . The performance suboptimality index p = i. The value of J is (4.e.60) G=[_~ S and the resulting K and = -6] -4.24.4.63) resulting in a performance J=txT(O)KOx(O) = 3.674. for any interaction matrix.=1 o 0 ] I (t) ] + [ .S. which corresponds to the value of J* in (4. 1968).32 -0.1337] 3.64] 0 (4.Hierarchical Control of Large-Scale Systems "/' . the performance index of the overall system does not improve any more.1337 13.24 and K2 = S. To see this we let b = 1 and solve a second-order AMRE for matrices.4.1337] 11.4 .12.59) whose first part is local (decentralized) control component and the second part is the corrective control component. K matrices become -0.5 with performance index j = txT(O)(K + K)x(O) = 6.915g21 (4. with upper bound p ~ .13438.1.4.123).62044] (4.88 ] [ x I ( t) ] -23 .4. The remaining matrices were chosen as Q = diag( 1.123} and an arbitrary skew-symmetric matrix S = [_Ob (4.00143700] 8. A .34 -8.23671007 0. the linear-matrix equation (4. The control for this case is .4.123 x 2 (t) -12. u(t) =u* (t)+u C = u*(t)-BPPx (t) (4.915].22 and an upper bound p ~ 0. i. This value as compared with J* = 3. Using K = diag{4.4. Example 4.24. respectively. Closed-Loop Hierarchical Control of Continuous-Time Systems 143 SOLUTION: To begin with. 1.12243780 R=I2 (4.4409 ur(t)] = [-4. B=[~]. while the Lyapunov-type equations were solved by an infinite series method given by Smith (1971).4. The AMREs were solved by Newton's iterative formulation (Kleinman.3.00143700 (4. G = 0.11.65) x= -~:~2 K. Q = ~2 ' 0.4.42): K=[=.4. assume that the system is completely decoupled.32 0 0 ~ j x+ l 0 0.64 [x .4. K+K=[_~:~~ -~:~~~] (4.4.3.66) where each reach of the river is represented by two state variables.2) and R = 12 • l o 1.69) Using this perturbation.30) implies that gil = g22 = 0.[-0.64 (4.5 6 . 2 ) and /~.74) .----------._ __. i 7= 11 \ I( I ) 1 ...tl ..lIicrarc\}i Gil C(1ntrnl of Largc-Scale Systcms t Inlier dCC\ lup\cd c\lllditi(lIlS.) )'.4 ..(O) · . ".1 -- .73 ) 12 -- \\ ith V.. n:wl qllil1l1l!ll and appnl"\illlalc "a" "e" s"lulions: (a) state s.L.___'___ ___ -. (lll e·· reach subsyQc\1ls: 145 14 SOII I IIO N: (\\'t) 1.1 o)trn (4 .4.. _ _ _L I _ _.I~-----\----- " '-...kn:l1traii /C(i cOlltrollers IIi' ( I ) () 0 ..ti :lg( 1. O..O.._.)r\.O.._ /\ 1 :111\ 1 l ___ 1' 1 " C7 1 l .--. the il1<.5 TIM!' Isec) (a) .. = .kpcll den t 111 :lt Ii X R in:a ti equa t iOlls lead to the rollowing solutions: 1\ _. the systeIll can he cOIl. 1.______ 1 _____ J _ __ J .. \ \ -..4 ..-.J. I I..________ . ._1 _ ___ .... For illter:lction 1l1<ltrix Ci '-7 0...... l ..10)(1 ..(1) -. .J. idcred as ..J_ _-----'L.. _____ _ L .. and ((' ) C(l1> I rtlls.ll._ ___L _ _ ~..5 4 4...J.~ --.---_----.. The interaction matrix ror .-----...0. [-.3 x. .-..l 1.. L. ----.I.( 1 1.__ o . 1 _ .4 with ..1'\2 .1 - \ \ \ \ If\ \ \ \ .5 4 4.7:1 ) ._ _L. 5 rc.-----....4 .L_______ .OI056\ 2(t) (4. --. (4..-.....-..5 .5 O 1. 2.(1 2:' .14 linn' 1'C 'p(1 n:<cs [pr Ih c slrllclmal pl'IllII'bati(l1l in Example 4.. '-...5 (sec ) U I \ Ih) I I \ \ \ ._ _ ~ ~.j.5 2.--- t o ..._ _--'-_ .J. .l...I.5 2 2.5 - .) :'...40JX --.. (b) outputs..05 L.5 Tl~lE 3.1 Fi~ lI1'" .5 ..--.I" -...IOS() ()._ L 0:.5 TIME (sec) (e) 3.--. 11 o -.. ' i .' o .'i 4 4...--.. ______ _ _ __ ___ _L .I_ _--'-_ _.5 tl(.1..--------. __ .3 showin?.. 028 0 .4.2.' lI tr:.. wa ter rcsources. Figure 4.82) (1. ma ny sys tems are ncith er lin ea r nor continuous-t ime.3 () () () .4.I ..2 -.': 1.() 9~ () I XI ·..7(. 55 3. however.4. T he slates and outputs of the fourth-ord er sys tem utili zin g two controls are und istingui shah ly close.0 .XO) -- fI ' (I) - n I ' f'y = ._ ] 2 -4 .25 ] (4.5 )' . and manufac turin g processes.R_.0..055 0. 5 (':lt e" Ih :lI th e ~ VS I l' lll 's (l ri gin al int erac ti on l11:1tri :\ is nonhell eri ci:ti . "".9 () () .. f) .. 4.(. Th e above nerfortll ance ind ex IV" S also checked hv n . ll irra rr h il: a l l"lJl lm ln i' 1. Th e ~! IIlh : tI (f. 72) was so lved llsing (4 .4 .067 ' lU X 1 O.077 0..\ 1.76 ) is beneficial 1 ll h 'lldicia l.l7. 111<1 (l . il11pl yin g lh a l th e inter:1 cli o11 nla tri x.. = II * + lI c (4 .argc· Sc:1i r S Y~ l r l ll ~ ll icra rchi cal Con trol of Diserete-Timc Systcms 147 () (. I(l I .I 4. In practi ce.6] 3..) i.:1 1 n.time sys tems.Level Coordination fo r Discretc -Time !:>)'sfcli/s T hi s sec li o n dea ls with a three-l evel goa l-coord inat io n st ra tegy suitab le for di screte-time sys tems and their tim e-delay modifi cat io ns. ) . "'11 1< 1 \\ ith .0 .() .3 17 . 0. (J055 1 0.3. Fu rth er commcnts on this suboptimality are given in Scclion 6.] th e ori gin..4. h('ndici:lI .03 0 . many researchers have made signi ficant contri butions in ap pl yin g o r ex tendin g hi erarchi cal co ntrol to both l1 0nd elay di scre te-tim e and time-del ay di screte-t ime sys tcms.OJ lJ 7 o -.0 and a step size 61 = 0. Du c to the im portance of such app li ca lion s.0.on 0 .2]R . Such a stra tc!!. 8 1) a nd U.0 .0.4. lI .1\]3 .24.5..93 I .~ -.o. 147 1.059 (J.'1 . In ll nkr to fill d out wheth er (.on ..042 .4 . 142 .4 1) a re used 10 fin d .(l . I · .023X (UR :ll1 d Ih ':11 hv virtu e o f (4 .4 . 83 ) () () -.1 sys tem 's cost fun cti o n by solvin g a fourth-ord er J\ MR E.07 .1 were foun d to be very cl ose. are bo th discrete and delayed in naturc. i = I.() .30X -. . the multil evel hierarchical control has bee n used for linear stati onary continuous.n ) s ~ 2. a nd " r Il <.25 () a 1 () 0.4. _ -::. many pract ica l enginee rin g systems.7 19J7 fm\(O) ·c (I 0.5 Hierarchical Control of Discrete-Time Sy stems (4. in (4. th e perf(l rlllall Ce slihop tillwlil y ind ex f> = O.(J .5)T and output matri x Iy hi l'h Ill' (4.0.\ 111 :1 lri :\ hcco ll1 cs () () () 0 () () () 0 (4.9 () o o o 0.76 ) o II I.7132.4.() .3. l m -. s uch as traffi c contro l. 2 a pplyin g exact optimal co ntrol (4..2 to fa ctor (.15. 1. indicating that th e prop osed decentrali zed-glo bal (corrcc tor) contro l pcrforIllan ce ind ex is within less than 1% of the overall centralized optimum cost. ~·. I22 0 .82) respo nses for o ~ t ~ 5.4. 142 . the fourth-ord er system (4.ll ] 7 (4.]0). 2 1.4.I tl .on x () (I .· 1.. or ll o llh cll d ici:lI . -· 4 I tl .5 .l -~ 11. I \ .k e ( ..12. III o rder t(l rind out \I hi "h .5.i( h IIl av he ll!.4.(J .4 . . I .(}07 0.0 :_ __ ( . To ch eck the ori ginal system's optim al control solution versus th e combin ed dece ntralized global solution.0 59 () A I I O. In th is secti on the ex tensio ns of hi erarchi ca l control to such sys tems in bo th linear and no nlin ea r for ms a rc considered.1Ild I( = Block -di ag(K I • !<. 17 .o. 1 f) 0 wi th x (O) = (1 .1 Three.7 19.o..45 1 .42).. Th e(1 rCl ll 4. 7X ) ..3 a nd Equali on (4.I g I O.4. yJ(t ).3 2 (4 . whi ch prov id ed the overall optimal sys tem performancc index as J " = 0.(J .'l lrt et'l llr) co ntroll er is givell by 0 . = () 0 . we need 10 lI SC Th eore m 4. thCll th e cn rrc~: I'(lndi ll ~ .5 I .)' : ](1 j " 11 (11 !lll i c= [6 0 .7(.1 36 () . ncutral.77 .2 ().0 . whi ch indi 11 )1' \:tllI e of . In fa ct.~ ). XI) a nd ap proxima te oplimal cont rol (4._ 4.(t). 122 -..4.14 indi cat es th at the stru ctural pertu rba tion closed-l oop contro l con sidered here is a good app ro ximal ion .' it i ~ . 2. If wc I:.3 17 .4 . 30X Th e rcsulting x . :1 SkCI\ ··sv tlllll elri c l11:ltri x. __1 1. as in (4. 1975 ) and trea tec! further by S in~ll ( 1 l)~O ) .79 ) T hus far."'.1 o 11 () (l () - (4. 25 . ddin cd hy (4.v was first prop osed by Tamura (1974. for j = 1.0 .4.l)(i(J.2. z.5. . In order to determine the optimal strategy of this three-level structure.(K)x. .=ai = z.5.(k)S. N. the dual problem to minimizing J in (4. (k) (4..u . . let us define a dual problem for minimizing L.jxj(k) = e. ~..(k)X. .) + zT{k )Sj(k )z. 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. The optimal control problem is to find a sequence of discrete-time control vectors u( k) . I.. .K-1.1) holds.(k) u.5.5.5. as in the continuous-time case.8) L {xT(k )Q(k )x(k) + uT(k) R (k )u(k)} k=O (4.3.(4.4)-(4. .(k)U.o .(k)= L j.nnti1111()lI s-fim p. k=0 . In other words.2 . using the gradient of the dual function q( ex) as in (4. .4) at a given () ()* /.(k)+ uj(k )R.5.(k) .K-I (4 .x. .5.4). Following the decomposition of the Lagrangian and the duality of two-level strategy formulated by (4.N (4.(K)+"2 k~O[ X . and all the other variables and parameters are defined earlier.K .21 ). . the sub-Lagrangian for every subsystem is further decomposed by the discrete-time index k.5.3.5).9)-(4.(k)]} while subsystem dynamic state equations N{l lK .K -I which minimizes a quadratic cost function J Through this decomposition. k = 0.5.2.u.3. this problem can be reformulated by minimizing J= .2) while (4.5..4) as follows : are satisfied.) } (4.22). .(o) = X. . .5. x(o)=x o (4."(K)Q. 1. . let us define a dual function (4.3) x.(k) + B.(k)- L j = I G. - 1 '") M Th p or<:>rlipnt "f n( R\ i ~ ~ in1ibr tn th P: r. Z. u.10) and subject to (4.x.(k) (4 .( x pu. is the i th subsystem Lagrange multiplier vector corresponding to dynamic equality constraint of (4.) defined by (4.1) where the usual definitions hold for x . also given by (4. 7) and f3. The last constraint to be determined is the initial condition.(k)R . 1. Uj .2. z . 1975). f3. .5) for i=I. .~l "2 x .5) is maximizing the dual function p(f3. It can also be considered a "temporal" decomposition versus a "spatial" one in Section 4.2. i=I.5...(k)- L j = I N exf(k)Gjix. 1. i=I.(k) + C.2.5. At this level. (Xi . Z .(k) + f3r(k)( A. .4)-(4.4).l (4 .(k) + B.(k + I))] + exiT(k)z.6) q ( ex) = Min L ( x. This decomposition was mentioned in Section 4.x..1 under" time" or "functional" division of the control task..3) subject to (4. our objective here is to present a three-level modification of the problem due to Tamura (1974.5.5.5. Although one may proceed to use (4.5..(k)z. z.(k)Q.25): N = "2 x T(K)Q(K )x(K) +"2 I I K .(k) + C.\ \7 aq(ex)la.7) and subject to (4. u .. .5 ..z. ( x . U.I.10) (4 . ex) x. A .3.7) subject to (4. x.u. u.3. and interaction relations N k=0 .9) where where L(x.(k).5. . ex)= L . 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.5.4) z. Following the decomposition procedure discussed in Section 4.5.) = Min L.2. k=O .N.8) and a few iterations of conjugate gradient or steepest descent to obtain a feasible and optimum solution. ex.(k) + z. one can proceed to apply the two-level iterative procedure to improve on the values of the Lagrangian multipliers ex.(k)+ U.148 Hierarchical Control of Large-Scale Sys tems Hierarchical Control of Discrete-Time Systems 149 Consider a large-scale linear discrete-time system x(k+l) = Ax(k)+Bu(k) .I Gijxj (k) . . .5 . .(k + 1) = A.z p ( f3. and B.= 1 N L.. .z.5.5. Xj.(·) in (4. 5 .. r } (4.(O)+ BF/3.p (f3n = .12) for k = 0.(k).11) to improve on the value of f3. the problem is defined as MinimizeH.N (4. the necessary conditions for the minimization problem (4. (H(x .(x.13) k=O The minimization problem at the first level is divided into three portions: for k = 0. K . k = 1.*T(k-l)x.. .5 .5. (4. and S. .(K)Q.21). u. For k = 0.(·) = R.*(O) z.5 .13).(k). u.*(k) values can be used to improve on the values of aiCk) using the gradient of q(a).K -I.) in (4..*(O) = 0 (4.(k)} (4.2.(k)+ajT(k)z.(k) j= I (4.25) Therefore.2.(k).I( Crf3.I .11) x.16) (4.5.5. 2.5 .O) uj(O) .17) or u..2.27) In a similar fashion.2. z. and k = 0. i. z.(k)+ iU. the large-scale optimal control problem defined by (4.( -1)x.u.k) .( -1) defined to be zero.l)x.K.(K) and adding the term f3. Tamura. Step 2: At the second level.(O). 1974): 'ilp.7) subject to (4.14) subject to x.19). Note that without loss of generality R I. .S. (4. Three-Level Coordination of Discrete-Time Systems Step 1: At the first level.N. 2.(k)} (4. i = 1. z.(O) to the sum inside the bracket with f3. k) in (4.18) (4.2.(O) = . ai'+ '(k) = ai'(k)-t..5.8).(k + I) + A.12) for k = 0. . . .(k )Q.) in (4. for the second portion k = 1. . 1..(·).(k) + B.5. .5.K. 1980.z.5.5. .. for given Lagrangian mUltiplier ai(k) . with x.€.3.*(k) j~1 [ajT(k)Gj.(k) . f3.4) for a given f3.15) k=O.23) 1 N + 2z. 1. .4).e. matrices are assumed to be constant.z. (4. .(K )x.(O) = .5.*(k -1)+ RjIB!fJ.u.(K) .. (K)} which results in (4. by virtue of (4.5.(k). (O) (4. .(k)..5 .(k) and u..(O).(k )x.(k).18)-(4. x. u. the functionp(f3.R j IBrf3.*(k)..K. which leads to the following relations (Singh..f3.*(k) + aj( k)) The final portion of the first-level minimization is defined by Minimize {ixT(K )Q. k) = iX.f3.(k).*'+ I (k) = K .5. Algorithm 4. f3.(k)+ C.5...) given by (4.(O).5..u.(4.10) with ix. are obtained.5 . and k = K. by virtue of this three-level hierarchical structure.(k). where /' (k) is a function of gradient of p Cf3n. .22) for k=O.N.. . .(k) z. (x. .5..*(O) + al(O)) (4.x.15) are 'ilu'(O)H.z .. .e. and i = 1. and z..(k) for k = 0.. i=I. 1. .*T(k -I)x..(k).K -I. the minimization problem is Mi"irni7P k = O.24) (4 .(k). . . i. k).e.9) can be rewritten in terms of H.x.(4.I.(k) = .150 Hierarchical Control of Large-Scale Systems Hierarchical Control of Discrete-Time Systems 151 case.2.Ck) being the state and control vectors obtained from minimizing L.. z. u.2.5. K and i=I.19) Step 3: At level 3.(K) inside the bracket in (4.5..5.25) can be used to find x.N (4.20 ) where g{Ck) is a function or v~q~a. . .S.14)-(4.5) can be solved by the following algorithm.21) (4.L aY(k)Gj.5.J)x.(x. . i.5.(k)..5.o f3t' (k) + o/J' (k) .f3.5. z.22)..(·) of the i th subsystem: = - H. and (4.(k)S.(k).*(k) sequences.26) (4. . in (4. (k).. = f3r Now let us define the Hamiltonian H. the error is satisfying equality constraint (4.5. .I. and again a simple gradient or conjugate gradient method can be used.5.u. the optimal f3. . i=I.( k) of the first level can be used along with the gradient of p Cf3.3).5.N.2 .(k) u.x. and i=I.*T(K .. .(K)x.(k).2. u.5. u.5 .(k).*(k)+f3. . Steps I and 2 are repeated alternatively until the optimal f3.(k) R.K-I.I + L {H. In view of H.I. and z..f3.g{(k).I( Crf3.5.(k)...5.N....5.( k) (4.(O) = x..(x.(k) and = - Qj l(k){ A. By regrouping the last I term f3r(K . . The above structure and algorithm is illustrated by the followi ng example ..:! -N 0 II "'< -<t:: I/) -.2--::"'0.2 1 0 -0.5. · • · 0 :a . +A sx(k-s)+Bou(k)+B 1 u(k. tIl~ Q>-l ::r:~ tIl >-l ~~ u> ~ ... SOLUTION : "'< ~ . 1975..000 to 149 in some 50 iterations and the convergence was not as fast as the second-third levels interaction errors ...17.2 Discrete -Time Systems with Delays In this section one of the more powerful algorithms for multilevel op timization of large linear discrete-time systems with delays in both state and con trol is presented..... R i(k) = 0. 1980).... 0.11) would become zero and the optimal control of the origi nal large-scale system is ob tained (Singh. .(k) Qi(k )x i (k) + z. (k+ l ..(lO)Q..> ~ u...(k )S. .:t: .-.. "'< .-1-0 1 1-u.30) ..2)+ . The interaction error between levels I and 2 reduced from 1. n l = 2.. .~ ~ A S . Example 4.1. 0 .o::. . k2 = 1. 0. Typical resulting error and states are shown in Figures 4. 0::...29) 2 {I 1 0 s:1 s:1 2S 0 0 U 0::. the interaction errors defined by (4. +Bsu(k-s) (4 ..(k) (k) [ 025: 0 x 2(k) + 0 10 0 .> </) N - >-l . with a cos t function J = t) 0 b . Consider the following large-scale discrete-time system with delays .:t: til .. . /11 = /11 = ..~1 "2 x.l )+ . 0 II C.. ~ >-l .___ 1 [ (k) u 2(k) (4. 0 0 <..5. b>-l "'< oj ..-.(lO)X. The results were generally very satisfactory... 1975) and has been further applied by many others (Fallaside and Perry..> "" - . oJ . 1980).5 ..5InJ.. The algorithm was simulated on an HP-9845 compu ter using BASIC and an in itial condition x(O) = (2 -3 4)T. 0 '"0 </) II "'< . SiCk) = I k ..3 . ~ ..16 and 4.3..5 'LJ oj II "'< ~ </) .5.5.(lO) + k~O "2 [x.. It is desired to find an optimum control strategy through the three-level coordin ation Algorithm 4...3 "-< . <: N 0::.. > ~ ~ 0~ r- Once the above three steps are complete. 0.i 4.1)+ A2X(k .1-1--=-3-- I 005: 005 f 9 .. especially with respect to the in teraction error between levels 2 and 3.:l [ x 2(k + I)) x.:t: ~ ~ 0 ::8 b x(k + 1) = Aox(k)+ A1x(k .5.(k )zi(k) + ui(k) R i(k) [li(k)] } (4 . Consider a third-order system. 0::..15 shows a schematic of the three-level modified coordination principle proposed by Tamura (1974)...j"( 152 Hierarchical Control of Discrete-Time Systems Q Z>-l O~ 153 b>-l .. b ....>-l ~ .. <.. The problem was first introduced by Tam ura (1974.. .:J . 2 1 2 1...3(k+l) H0 - 0 0 -0 .... 1980. kl = 2.8) and (4. Singh.~ II "'< OJ > ~ .D >-.. b ~ ~ . where Qi(k) = 21". .:t: Z ~ ... n = 1. Figure 4. .. Jamshidi and Heggen..5.28) 0. In the literature.16 Typical interaction errors of Example 4.31) and a set of inequality constraints X min U min ITERATIONS Figure 4.5.5. k = 0. -s ~ k < O.s.5 . 4 . '" are defined as zero vectors. and A(K). x(O) = xo (4.34) ~ Figure 4.5. u( 1). .5.. let us define the Hamiltonian: IJ-l I- IC/l <t: 0 H(x(k). .32) 1 k=O I- ~ 20 0 I 2 3 4 5 6 7 8 9 10 II 12 where Q(k) and R(k) are both assumed to be positive-definite..! 120 x(k)=O .x )(k) "'k" C/l ~ are satisfied.5. results in a TPEY (two-point boundary value) problem which involves both delay and advance terms. if not impossible.32) is minimized while (4..5 . ~ X( U( k) ~ k) ~ X max U max (4. .I f::: W u <t: t:t:: L 2:{x T(k )Q(k )x(k)+ uT(k )R(k) u(k)} (4... . .5. as it is demonstrated in Section 6.30).35) where k = 0. .K -1.. Here the objective is the application of hierarchical control via the interaction balance principle. N I . 1. 100 0 t:t:: t:t:: z 0 w 80 60 40 Physical interpretation of initial functions in (4.5. 1.33) (4.31) for system (4.. -3 2 3 4 G DIS CRETE TIME k 7 8 9 10 . Following the formulation of discrete-time maximum principle (Dorato and Levis.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 J = 2:xT(K)Q(K)x(K)+ K . assumed to be zero without loss of generality: .17 Typical state trajectories for the third-order system of Example 4.13) for a given vector A = A* .(4.30). The optimal control problem is to find a sequence of control vectors u(O).5.154 \60 140 Hierarchical Control of Discrete-Time Systems 155 where A . k) = 1{x T (k )Q(k ) x (k)+ uT(k )R(k) u(k)} I I I I of . making the attainment of an optimum solution very difficult indeed. A(k) is a vector of Lagrange multipliers at k..5..1.. A(K + 1).30) has 2s delayed terms. 1971).4.. It goes without saying that a solution to the problem (4. .5. A(k). arc n X /1 and /1 X m matrices.(4. hence it requires 2s discrete-time initial functions. u(k)=O. and B.x2 (k) .31) > w t:t:: w !::!. .I i= 0 -2 I I I I I (4. u(K -1) such that (4. 2...34) in usual "centralized" methods by the application of the maximum principle. there are several approximation techniques to deal with continuous-time systems with delay which will be discussed in Chapter 6. In a manner similar to previous discussions regarding Equation (4. u(k).1.5.and m-dimensional state and control vectors. and x and u are /1. respectively.5. The system (4.5. l(K)A*(K -I)} (4.*T(i)(Aix(O)+Biu(O») (4 .4...5.l(k)[i~O BT>-.(4.5 . k = 0. .39)-(4. u (k).5. { >-'(0» = Min~{ xT(O)Q(O) x (O)+ uT(O)R(O) u(O)} + where the I th element of the saturation function Sat x( v) is if v{>x max .39) where the" saturation" function Sat u(' ) is 4.5 .5 . .2. the necessary conditions for (4..38) Now if R(O) is assumed to be a diagonal matrix. *(k).*(k + i)]} X max .34). as in the three-level coordination formulation di scussed in the last section.5.(4.36) Min H( x (k).48) The above so-called" time-delay algorithm" can be summarizcd as follows: (4.c Problem k This problem is = K ~~~ subject to {± xT(K)Q(K )x(K) .5044) satu{ .2.a Problem k = 0 By virtue of (4..5. the partial derivatives JH(·)/ Jxi(k) and JH(-)/Juj(k) for i=I .36). As before.34).*T(K -1)x(K) K . 2..S .5042) (4.*. each of which has an explicit solution given by setting JH(')/ Ju(O) = 0.e. this problem can be altered to that of maximizing the minimum of L(·) with respect to >-.33). { if xmin.>-.>-.:::: . >-. u(O) and u min « x(k) « x max « u(k) « u max (4 .5. k). *. .5.33).47) x(K) = Satx{Q .36) subject to (4.5.'Ck .{ subject to (4.*(k .48) for a fi xed set of Lagrange multipliers A(k) = A*(k). 111 lead ..156 the Lagrangian can be defined as Hierarchical Control of Large-Scale Systems (/ Hierarchical Control of Discrete-Time Systems 4.'Ck)[ - ).2.5. .{ if v{ < x min . assuming that R(k) and Q(k) are diagonal.5.to Ar).(4. . j=1.38) lead to a set of m independent relations. k) = t xT(K)Q(K)x(K).5 AI) k=O xmin Thus the optimization problem is to minImiZe (4. { (4 . 1.37).5.46) x mln ~ x(K) ~ X max . n andj=1.1) + . . .{« v{«xmax.*(k) . *T( k .A*T(K -I)X(K)} x (K) (4.40) Algorithm 4. .5.2.5.m.5. k) .35). solve K + 1 analytic problems defined by (4. The power behind the" timedelay algorithm" of Tamura (1974) is the decomposition of this problem into (K + I) independent minimization problems for a given >-..' Ck+. which reduces a "Junctional" optimization problem to a "parametric" one. Time-Delay Algorithm Step 1: At level one.5. . u. set of n + m independent one-parameter equations whose general solutIOn IS x'Ck) ~ Sa'. . i. . u(k).I)x(k) (4. and the indexj represents thejth element of control tl j .)]) (4.R.b Problem k = 157 1... >-. >-. and constraints (4.5.5.43) <?nce again.. . lI(k ) + L {H( x (k). ( u*(k) = Q.44) and (4.:::: whose solution is similarly given by (4 .l The intermediate problem is defined by x (k ) .K-l L(x . .2.37) Sat x(v{)= { v{ x min . definition (4.5.. (4. u(O).K-1.1)x(k» subject to (4 . 2. .5.>-.5. the optimization problem for k = 0 is MinH(x(O) . 4 ..45) L i= l s >-.S.to a.S. S.. x(O)=(~:. are n. the value of "A*(k) is improved through a gradient-type iteration .(k + I) = A. x(k+I)=Ax(k)+Bu(k)+d(k). represents the interactions between the i th subsystem and the remaining (N -1) subsystems. u(k). Consider a simple second-order system 0 z u f:: f-< ~ .S).4 is .5. X I. x(O)=x o (4. and II X I state.5 . and d(k) are 11 X I.u.N and X.].50) which follows from our previous discussions.1 [::~:::~]=[-~ -~][::~:~)+[~ ~][::~:=:n +[°1.e.2.x(k .5. I I u 2 (k -I) with cost function J = "2 XT (S)Q(S)x(S)+"2 I I L k =O 4 {xT(k )Q(k )x(k)+ uT(k )R(k) u(k)) (4. The vector z. The algorithm converged in 49 iterations.Xr.(k) + C.. (k). Example 4. dr(k) = f(er(k)) = fC~o [A. to be defined shortly. i.3 Interaction Prediction Approach Consider the following linear discrete-time system in its state-space form. 5 O][U\(k)]+[O 0. and l1i X I state.S. control.O. X I.S3) The problem was solved for an error to lerance of 0. respectively.S4) (4. III X I.S.ll. and n. i.5 <t: p:: . SOLUTION: XiII (4..Xm.7 p:: 0 p:: p:: . n. The following example illustrates the time-delay algorithm.5. for i = 1. 2).9 (4. The optimal control problem can be stated as follows: (4.8 . u. and disturbance vectors of the i th subsystem.i)+ B.5 .5.S2) where x(k). .25][U\(k-I)] ° u 2 (k) 0 0 4 81216202428323640444850 ITERATIONS (4. respectively.x(k . The matrices A. as shown in Figure 4.1 O. Let the system be decomposed into N subsystems x. m.51) Figure 4.SS). respectively.S . Equation (4.. and R(k) = diag(l.e. [=~]<U(k)«~].S7) ..S6) 4. Several other initial x(O) and >"(0) were tried and the convergence was achieved in a similar fashion.158 Hierarchical Control of Large-Scale Systems Hierarchical Control of Discrete-Time Systems 159 Step 2: At level two.(k) + B.z.i)] .SS) (4.5.x.18 Interaction error vs iterations for the time-delay algorithm in Example 4.6 ~ where dr(k) is a function of the error er(k).2.Xlli' I1.30). control .. and disturbance vectors.00 I. Bi' and C. subject to (4. are.. xi(O) = constraints [~]<X(k)<[..18.3 . and d..S. The integer r. .(k). represents the number of incoming interactions to the ith subsystem.S. and >"(0) = (0.5.49) . a step size of 0.] and Q(S) = diag(l .2 . Q(k) = 12 .1 for the conjugate gradient iteration.x(k + I)} (4.( k) + d. = j~1 N N .(k ).60) Here it is assumed that the Lagrangian Lis additively separable for Z. However.5.5.(k)= L j = 1 {D..5. H j (-) = 1X. L i ~ 1 A~' (k)(D.5.5. and I is the iteration number.59) Then the necessary conditions for optimality would lead to k=O where weighting matrices Q.5. follow the usual regulatory conditions.68).5.57) for the time being.(k)} (4.. which was discussed in the previous section. solving for x. and A. (4.59): p j ( k ) = () H j (. .5 . The other was di scussed in Section 4..63) . Consider the Lagrangian of the problem (4. and A.5.5.1 without the bounds (4.5.1 is extended to discrete-time.x.x.j and G.z.j are the appropriate matrices between controlllj(k).(k) + 1u . ) )= 1 N T ( 4.rPi (k + I) - L (X~ (k) D. 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 (4.5. respectively. or conjugate gradient techniques. k )] ( 1+ 1 where D.5..5.58)-(4. (k) + B.x. Here again it is noted thaI the computational effort at the second level IS very small compared WIth that of the gradient.~ X~ ( k ) ( DJ j=1 N j Uj ( k ) + GFA) k ) ) (4.64 ) [A J. Two solutions of the above optimization problem and its modified forms have already been presented.(k) and 1 - ~ (A~(k)Gji) .69) and equating the coefficients of x/k) and zeroeth-order terms would lead to the following Riccati and adJo1l1t zj(k).lli(k)+GI.1 { k~O 2.jxj (k) } (4.2.5.57) and utilizing the three-l evel discrete-time application of goal coordination.( k) + d.(k) Qjx .5.~I L.(k)+ u{'(k )R. ( k ) .67 ) + p.'II (Nr. instead of utili zing those methods. Newton. which has not received much attention in literature.4.(k )R.(k)+G. and R .u. (k)] } ) (4. and the state x. )/ aII j ( k ) = R j U j ( k ) + B.68) where Let us assume that the costate p/k) and state xj(k) are related by a linea r vector equation (4.) / () x j ( k ) = Qj x j ( k ) + A N T rp J k + I) (4. (k + I) ..1 + pi( k + 1) [Ajx j (k) + Bju.65) J.~ (Dj j llj(k)+GjjXj (k)) =0 j = 1 (4.66) L = . the continuous-time interaction prediction method of Section 4.58) [ A. . pj(Nj )=O ) =1 .5.x.ju.Nj .(k) + 1U.5. the objective function (4. (k + I) + A. Moreover. This would imply that [or any given z. trajectories.69) Now eliminating p(k) in (4.(k + 1)[ .x. xJ'(k )Q. ( .u. (k) + A~(k )z.56) is assumed to be quadratic: zj(k) = for k = 1.5.(k) R . (k) + C.(k)) o= or c7 H.(k + I).62) N (4 . (k) + CjZ j (Ie)] {xi(k )Q. and substituting it in (4.Hierarchical Control of Discrete-Time Systems 160 161 Hierarchical Control of Large-Scale Systems The coordination at the second level for this interaction prediction scheme and N would be z. = ~ ~ N . One is based on the time-delay algorithm due to Tamura (1975).5 . Here again. we will ignore the bounds (4.5.uj(k) + >/i (k ) z.. For a known set of augmented interaction vectors *T( k) I Z*T( k) the dh subsystem Hamiltonian is l N j~1 {Djjllj(t() + Gjjxj(k)) T CI PI (k + 1) j 1 (4 .lI.5 .66) while ~tilizing (4. there are N independent maximization problems.61) which result in (4. state x.55). t he )th and k th subsys tems.l:.. . In the above relati o n.lIi(k) . i = I .5. M.I ( k ) A. The present development.l(k)r. B.5. g. . e. Step 3: Solve N adj oint equations (4.64) to update A. 5./{K. ( k) X I (k ) + M.(k)... (k + I) = { ~" (k + I) ·· · .T h.69). xl and tI! arc 11 .. find and store u. K.5.57). ( k ) (4.. i = 1.- I : At the seco nd level set I = I. (k) E. xl represents the)th subsys tem's state at ith interval.5. use (4. (4. IGi (k) - !:Ii ( k ) ~ LI i ( k ) ~ u.K i {k + I) Ei ..72) Step 6: Stop. . ( k.: ) { ~ R. anu pass them d own to first level.(k + I) x. (J.73) u.5.(k. .(k) (4 ..5./: [/ ..N/. If not.I (k) [R j N + e.5. the discrete quantities are represented by index i in a subscripted form and the subsys tems by index) in a superscripted form. . the expressions fo r 11 .(k) + /. r. N/.b (k ) ..\IC{J 2 __ i _ I_ l _ _ _ G2 1 I A 2 I Gn I _ _ L __ L __ L . .. Algorifhm 4.78) The op timal control problem would be to find co nt ro l vectors Llf. the fo llowi ng ex press ion s arc sugges ted for conlrol and stale vecto rs: 4.75 ) where L.2 . (k + I ) Step 4: Using the sto red values of K.]Xi(k + 1) . .( k) = LI.(k + I) (4.I' ( k) 1I.7 1) and store for k = Xi (k) = . l(k) l~ (4.2. 'ON and k = 1. G.z. . ( Ie ) <!:I. x · (Ie . (k + I) . and go to Step 3.(k) a Illi = B.5. ( Nj ) =0 ( 4.5..(4. + I) L. (N/ ) = 0 (4 .(k) and z.77) for ) = 1.2 for continuous-time systems is extended for discrete-time sys tems. . )= = I + 1'.78) are satisfied while minimizing a . . (4.5. (k) i11 (4..5. IB.\ . sllch thaI (4.5. (k. ' K . ( k ) = Q. ( k + I) in (4. foll ows the work of Sundareshan J. in part.4 Structural Perturbation Approach In this section the optimal hierarchical control scheme via s tructura l perturbation of Section 4.I(k)A" M.(k).5 5) a ft er eliminating {J. 5. (k)] (4.\.(k+ I) (4. . The Gjk is the II j X II k interconnectio n m a trix between . An application of this algorithm is given in Example 4.5.(4.62) .(k) where ( 1976). .5.I) .N/.(k) = L. (k + I) ~x.'i( ( k + I) Ei .e. increm en t I = I + I.t.(4.(/.5.5.N and Ie = I .(k + I) <..76). .and m-dimensional state and control vectors of the )th s ubsystem at i th interva'1. (k) u...2.74) Consider a large-scale di screte-time sys tem which m ay consist or M subsystems: . wiIere E.5.( k) . assumc initial values for z. M and i = 1. N.( k + I)] .(k»u .73).5 .6 o f zero.7 1) I . Interaction Prediction Approach for Discrete-Time Sys tem s .63 ) i..(k) =E. (k) + d.(k).R.( k) h. . j = 1.(k) and x. ) = L j = 1 (l\~ (k) GjJ' Recalling the bounds (4.5.(k + I) via (4. I I AI I G I - AM .c. The relations of Ai' Bi . (k) = .( k). .3 ."[K..(k + 1) + g.70) Step 2: At the first level solve N ma trix Riccati eq uatio ns (4.. . (k + I)· ·· x. .4.162 Hierarchical Control of Large-Scale Sys tems Hierarchical Control of Discrete-time Systems 163 K . Step 5: Check for the co nvergence of (4.(k = + I)b.) Ie.5.68) a nd . (k) = zt (k) and AI( k) = A.g.70) and s to re. x.5.76) Th e di scre te-time interaction prediction approach is summarized by th e following algorithm. a nd Gi k matnces are gIven below: x. K.5. . ~x.. + I) > .. For the sake of discussio n..::. .77).(/. wheth e r their left-hand sides are within E = 10 .. . + A .5.. (k ) g. ... In sequel.5.IBT. Step 3: Evaluate the local control components by solving (4. This control scheme is summarized by an algorithm.87) Step 5: Using the control u and Equation (4. .83) and each compon.5 (interaction prediction) and 4.65).84). . The optimal control problem.·.82) V=LTQ L+P/TRPJ.164 Hierarchical Control of Large-Scale Systems Hierarchical Control of Discrete-Time Systems 165 quadratic cost function Jl( x{" lin N = L i= 1 (xrQ. i = I.5. Using the local and global controls defined by (4.1. . K . B M ).5.77). X 11 . (4. (4...5.x( + U{~I Rj u. p. . The matrix r should be chosen such that the norm IIC . . IICe l1 = O.89) The above choice would in effect neutralize the interconnections. = lllock-diag(P. . in its largescale composIte ~orm.5.5. . The application of this hierarchical control to the large-scale interconnected discrete-time system would.'.85) for Riccati matrix K( . Algorithm 4.2.Rj IB! and all other matrices are defined by (4. + Bu.88) is subject to the rank condition.77) --(4. The solution (4.5. etc. are 11. 1976) : (4. .M..5..82).81) is minimized . res~ectiv~ly. X Ill . . 1971).1. . information from all the states. B + = (l3 TB) . IS to find an m-dimensional control u T = (U IT . Sundareshan (1976) has proposed a hierarchical control which consists of a local and a global component.80) ~~I:re~ A = Blo~k-diag(A I ' A 2 ..84) and (4. { IT 2T Ii.6. .5. Thus. rank(B) = rank(BC).ent u{ is the optimal control of the jth decoupled subsystem problem deflI1ed by (4.k J.. positive-semidefinite and positivedefll1lte mat~lces. i. =u'+u 1 1 (4 .5 .5.5.80).M.5.5.U.5. respectively.86). . willie the cost M J( x".5.Bfll is minimized. .. in general. Algorithms 4.If ): B = Block-diag(B I' B 2 .A.5. and often unnecessary or unavailable..5.j= Qj +ATKj+ 1(I+SK /j+ 1). the closed-loop composite system becomes (4. initial states x~. cause a deterioration of performance whose sUboptimality degree is p if the following inequality holds (Sundareshan. . j=I.5. · . and store..' .5.5. g u 1. J JJJ JI 1 H=(I+p)K+Q 1 . and a single central controller would become necessary which may require too much. the global control component gain matrix is obtained [rom (4.e...5. . . 1 he global component II f is given by (4. 2.(4.5.(4.B r) is termcd as the effective interconnection matrix C e . i. _ U.86) for th e global control component and form the overall control (4.84) where P/=.89) and use (4. Step 4: Solve (4. Structural Perturbation Approach [or Discrete-Time Systems pix 1j 1 Step 1: Input all matrices {Aj' Bj .N .5. . p/.M) and the tcrm (C . MT} (4 . ul) ) = L j =I Jl(x /" ll ~) (4.5 .79).88) where B+ is the generalized inverse. + 1 = (A + C)x.90) where wh~r~ Qj and.U.J . .. and m . .. .Rj-IBrKI+ I(I+~KI+ I) .e. R . Qj' R j .5.e.6 (structural perturbation) are applied to a three-region energy resources system. The solution of this prohlem requires the solution of a large-order Riccati equation (Dorato and Levis. . U MT ) WhICh would satisfy the overall state equation X. .5. Cjk }.86) where r is an 111 X 11 gain matrix yet to be determined. i=I .. The solution to this minimization problem is well known: (4.72).N. k .IAj and K/ls the positive-definite solutIOn of the discrete matrix Riccati equation: Step 2: For each sub systemj. find state x.78) and (4.I A .79) where P. i.91) where the local control is given by . The jth subsystem local control is given by u 1j = - The terms AIII(D) and AM(D) are.. (4. . In order to alleviate these problems.. and [0.85) \~here S~ = B. the minimum and maximum eigenvalues of matrix D . solve (4.79) which is obtained through a Riccati formulation as in (4.. j KN=O ( 4.5. _ I) (4.5. Hassan and Singh (1976.166 Hierarchical Control of Large-Scale Systems Hierarchical Control of Discrete-Time Systems 167 where (A"B'}~([~0.3 G.j=L2.5 and 4. + ~ .5.:= jk~ =~~ IJ I : () : () j 1/.20 and 4.5. respectively.3.5 and 4. th ree loca l control fu nctions were obtained.92).5) were closer to the op tim um than those for the structural perturbation (A lgorithm 4. The state and control responses for the interaction prediction (Algorithm 4.5 0 .92) ~ (A'B.70) or (4. let the fo ll owing discrete-time model hold: For the hiera rchical control policies. For thiS system.71) were solved using an HP-9845 computer and BASIC.25 0 ~mm 0 0 0 1.5 Costate PredictioJl Approach In this section a computa tionally effective approach for hierarchical control of nonlinear discrete-time systems due to Mahmoud et al.5 0 H rO G = 21 l~ 10 0 0 . Using the interation prediction method.5 0.)~ ( [ ~5 0 .4 ) convergence was reached with :line iterations. and R.21.5. Q 3 = diag(1O 110). the third subsystem's trajectories were closer than the other two.5 0 0.25 m (4.6 to find a hierarchical allocation policy for tbe energy-resources system of (4. an acceptable (within 10 .5 0 . Equation (4.5 . E~all1ple 4.[ 0 .=I. Q2 = 514.6 along with the op timum centralized trajectories which were obtained by solving a tenth-order discrete-time Riccati equation are shown in Figures 4.5.5 0 0. ~ [~ REGION 2 ~~5Wj} 0.19.5. (1977). 1977) is presented.82). However.}~{r~5 (A"B.89) was used to ohtain the global control component and added to the vector of local con trols to fi nd the overall control (4.6).5.5. 4.2 0. The scheme is appl ied to nonlinear .5 0 0 ~j The q uadra tic cos t function has weigh ting matrices QI = diag(J 10 1). A reason for this may be the fact tilat the third subsystem is comp letely decoupled .19 A three-region energy resources system. The resulting sta tes ami co n trol func tion via the hierarchical control A lgorithms 4..5 0. Using the structural perturbation method .25 0.5 0 . Xi + I = lqo?J ~ _~] ~ _~~ r~~l~i~2~ _O_ x.75 J 0.5.3. th ree matrix Riccati equations of the form in (4.25 0 0. COI?sider a three-region energy resources sys tem shown in Figure 4. Based on the sol utions of these and vector equat ions. SOLUTION: REGION 3 Figure 4.85) and three adjoint vector equations of the form in (4.2 ~75]. lt is desired to apply Algorithms 4. -'-_--. (k + 1. _ _ L ___ _ I . ( k + 1. 15 ~ -.5 . .~ + c(x../ ffLW <!..J 50 . (b) state x 1(t).7 5 ._....7 ·-R --.1FCTORY (Algorith11l4 . systems. u. 30 35 40 45 50 S 10 15 20 25 PERIOD.-: 0 _ __ OPTlMUM CENTRALIZED TRAJECT ORY _ _ _ NEAR-OPTIMUM HIERARCIIlCAL TRAJECTORY (Algorithm 4.1 .(1) ·.\:' -. (k + 1. = 0 . k ) ) 11 ( k ) (4.JI_ __L .. The procedure begins by rewriting the nonlinear state equation (4. .5.55 .(4..94) 1. -.95 .3 .-C) -'< x(k+1)=/(x. k)) u (k) (4 . Consider a nonlinear discrete-time system described by -o S _.45 .3: (a) state x2(t).6) _. and its linear extension is rather simple (see Problem 4.93) -.. --- Of'TJMlJ~1 CTNTRALIZED TRAJECTORY .5.5 ) '"..< LW 1/ 1/ -4 _ i/ . B(·) are block diagonal matrices and I7 .8).NEAR . k c ( x.75 . k )) = C I ( x.65 L -_ _ ~_ _~ _ _ ~I_~ III () .k)).· 1I J=~ L (xT(k)Qx(k)+uT(k)Ru(k)) (4. en -..OPTIMUM lll ERARCl1 lCAL TRA.20 Optimum centralized and near-optimum hierarehical trajeetories for Example 4. .IS ···· 1(. with a quadratic cost function K .85 ._._ I _ _--'-.65 ... .I x(O) = x" (4.14 ..4 . k) )x( k) + B (:x. (/) f- III .45 .1611 Hierarchical Control of Discrete-Time Systems 169 4 .5. (Ie + 1.NEAR._ NEAR-OPTlMUM HIERARCHICAL TRAJECTORY (Algorithm 4.%) of' the system (4.L_ _~I_ I L _ _~_~I_ _~I 0 10 15 20 25 PERIOD.25 '" .5._L ____.OPTIMUM C'I'NTRALlZI'IJ TRAJFCTORY . I5 . S . 0 . u.u. I.l_ _~_ _L-_~_ _-L_~~ _ _ ~_ _ L--+~ 0 10 15 20 25 30 35 40 45 50 PERIOD.96) It is notecl that the reformulation (4.55 -.\. u.5.NI ' AR-OPT1MUM IIlI'RARC'IIiCAL TRAJHTORY (Algorilhlll 4 . 35 7 - .___..3.25 -.OPTIMUM IIIERARClIlCAL TRAJECTORY (AlgorilhIll 4 ..93) is (b) .) under the context of near-optimum design of large-scale systems.6) -'-' NEAR-OPTIMUM IIIERARCIIICAL TRAJFCTORY (Algo rithm 4. ..19 . The hierarchical control of nonlinear continuous-time systems is considered in Chapter 6 (Section 6..IX _. k)) 0where A(·). u.5.05 .5..5.:..I ___ "l - . fl. k 30 35 40 45 (c) (a) Figure 4.20 L .1.95) . __ J _ __.-. k ) ) x ( k ) + C2 ( x._ ___'--_ . u.1 t..93) as ~ x (k + 1) = A (x. 05 -. 5 ) 3 2 - i / / ! .(k+1. (e) state XlO(t) .(k + 1.35 .5) . (k + 1.-.:.95) .9 f- LW " <!. k 30 35 40 45 P.6 . .5 N " 0 o-l p:: -20 -22.).5) always possible.5 -35 -37."'"'''-.. k)) u(k) + c(x*...5 . _..96) can be rewritten as _ I K.95).. Therefore. """'- -10 -15 -20 -25 ... '-..2 -."'"'. -.5 .5..I (4.(4.10 ..93). for N blocks in A and B .5.99) x(k). 100) .5.3 3: o-l :..5. it is assumed that matrices Q and R also have N blocks..: ..42..5 ../ 170 . _ \'\ ... k)) (b) + exT( k ) ( x ( k ) ... Hierarchical Control of Discrete-Time Systems 171 -5 --=--:::-.17. _ .5 ..5 ... The basic reason for the reformu lation (4..2.. Moreover.. (k + 1..12. -30 -40 -45 -50 -55 -60 -65 -70 0 0 -35 p:: b """ " "-'". The modified problem can be solved by defining a Hamiltonian: H( ·) = 1XT(k )Qx(k)+ 1uT(k )Ru(k) + pT(k + l){A(x*..5 .. . _ NEAR-OPTIMUM HIERARCHICAL TRAJ ECTORY (Algorithm 4... and C2 (-). --. u* . -. (k + 1.5 -30 -32.5 .. (k + I.. .5 . u* .5 . 5) \0 _ _ _ NEAR-OPTIMUM HIERARCHICAL TRAJECTORY (Algorithm 4.(k +l .6) .. k »)x(k) 50 5 10 15 20 25 PERIOD..5 . (k + I . . k) u(k) + c (x*..---.. 0 -.3: (a) control Ul (t).98) u*(k) = u(k) (4. :.7 -...9 I .8 _ _ OPTIMUM CENTRALIZ ED TRAJECTORY ~ p:: b u Z 0 \ OPTIMUM CENTRALIZ ED TRAJECTORY _. + B (x* .5 -7. (k + I. _ NEAR-OPTIMUM HIERARCH ICAL TRAJECTORY (Algorithm 4.15 :.. (c) control u 3(t).40 ..25 -27.... k)) x (k) + B (x* ... 0 ..... u*...: ..u* . k 30 35 40 45 50 (c) (a) Figure 4..5. u*.6) _ ...---.u*( k ) ) (4 ... " 50 _ _ _ NEAR-OPTIMUM HI E RARCHICAL TRAJ ECTORY (Algorithm 4.97) min J =2 (xT(k)Qx(k)+uT(k)Ru(k)) L k ~ O subject to U 0 Z b x (k + I) = A (x* .1.5 -. (b) control U2(t) .(4. u* .. C 1(... .45 0 _ _ _ OPTIMUM CENTRALIZED TRAJECTORY _ _ _ _ NEAR-OPTIMUM HI ERARCII ICAL TRAJ ECTO RY (Algorithm 4..4 0 Z U " """"- '" '\ '"\ \ \ 0 o-l ".--. the problem (4.6) \ . .5) 15 20 25 PER IOD .. 1 --... k)) x*(k) = (4.21 Optimal centralized and near-optimum hierarchical trajectories for Example 4.. _ NEAR-OPTIMUM HIERARCHICAL TRAJECTORY (A lgorithm 4. '- '-" " .x * ( k ) ) + f3 T( k ) ( u ( k ) . k 30 35 40 45 0 10 IS 20 25 PERIOD . B(·).96) is to provide" predicted" state and control vectors x* and u* to fix the arguments in the nonlinear coefficient matrices A(·)... N N + I .S .'( x*. Q.k»p. (x*.S. u*..\'·. !liCk).)} +cx[(k)[x.S .) and x.. (k ) = P:+ I(k) X*f.k» u(k»)/ox* Dx (') = oc(x*.K-I.N. use xi(k). (k + I.R .S.(4.(. (k + I.(k) = = oH. I'-')} Q.( k).IOS) lead to the equality constraints (4.) for i = 1.(k)-x i (k)]+f3. (k + I.. (O) = x'o' The condition (4..e . Moreover.(k + I) + f3.S.e.lI(k) = Q.( x*.(.\Jk (4.(k + I. . the second condition in (4.(') are derivatives of the expressions in the brackets of (4. u*. 1/) + D. i.e . )1(. 107) : and subs tituting 1I.. k ) ) + G~ ( x * .(k)+ BJ· )u.5 . St ep 3: At the second-level.S . u*. q' = (pi. Step 2: At the first-level.I.106) leads to an expression for cy(k).K .e.r( x*.(k + I).S.108) to obtain u. (K) = 0 (4.S . and R . (k + I. The following algorithm summarizes the costate prediction method.. (k + 1..(').104) (4 .(k+l .(k + I) aHjox. )x. u*.'(x* . (k + I . (k )} + c.111) to find H'(k).(k) 0 = () J// iJ!J 0 = where F. (k)} (4 . (k + I) + H. I.S.(I.x.. and (3'(k).. (4.T( . k»}p(k + I) (4.(k.. u . i. Therefore. u. 109).(k)] + d.I ./op.) = = x'(k) 11 *' 1l(k) 1I'( k) F. and the necessary conditions (4 . + I) = A. H( k ) = {F\T ( X * . it is clear that the Hamiltonian H(·) in (4.S. i.B.113) to update coordination vector. IOX) with x.2.112) H = L H. and D.(x*. . x*(k)=x(k). use (4 . f3') ..(k)-U i (k)]} (4 .(k) .stopandu' + I(k)isthe optimal control.113) 0 = oHjou.(4. the first-level problem (Step 2) involves simple substitution and docs not require the solution of a TPBV problem or even a Riccati equation. x.99). u*. k »} p (k + I) (4.(k + I. and f31 . requiring the substitution of x.102) (4. u*..(k + I . k»/iJx* +p/(k + I)[A . u* . II *. k»x.=1 {~xJ'(k)QxJk)+~U. G.(k+l.S.S.S.S .S.S . k » pJ'( k + I) + f3 . in (4.S . x. (k + I. k » + G. k » + D..104) gives the costate equation P. P. (k + I.106) yield s f3 (k) = {F.e.112) with respect to u*. Ie ) ) + D\T( x*. the second-level problem (Step 3) is also ra ther trivi a l. 11*...('). k» R I I{ B. since the costate vector P'(k) is a component of the coordination '/ector ql(k).( x* .\" '. U . u*.u*. and (4. k» pi + 1(1-: + I) + a.S.(k + I .109) xp(k+I) Step 4: lfql + l(k)=ql(k)fork=O. 106) x.(k) in Equation (4. E . 100) is additively separable for given x* and 11*. ) 1«·" /I".103) yield s a new expression for x.S. . olJ/ou*(k) Rela tion (4. IOI) The necessary conditions for optimality are given by Finally.' I {B.(I. 11*. C2 ...S. substitutcP'(k). U*' . u * .S.'(k)[u. (k) .111) Before this algorithm is illustrated by a numerical example.7. u*'. . Costate Prediction Approach Stcp 1: Set iteration index / = I and guess vectors 1'1.. in order to obtain an expression for f3(k). u* .S.S. (k + 1.114) x') f3 '+I (k) = { + G. i.. . and f3 vectors which have been obtained from the previolls iteration . x*'(k).(k + I) p. A f~orit"l11 4.103) (4. such as the goal coordi- . k ) ) x ( k ») / 0x* (4 .(. O= r7 IJ/aH.( .S. k » p. x*'. (4 . = L .110) The first of the two conditions in (4. First.. u*'( k) . x*'. 0 = rJll/ox*(k).. Simi la rly. u*. (k 173 In view of the assumptions made on matrices A.. u*(k) = u(k) (4.102) yields an expression for u..) /(. unlike the other multilevel formulations. . u*..=1 .S. a few comments on the costate-prediction method are due.IOS) (4. C I . u. (k) + Ai( x*.xi(k) + AJ'(x*. .(x*. k » .107).".S.r(k)Rll.172 Hierarchical Control of Large-Scale Systems Hierarchical Control of Discrete-Time Systems where ~.S .) /0 x* ~ 0 {A ( x* .(k) =.) = 0 Z (. Otherwise go to Step 2.(k) Gx(-) =iJy (-)/iJx*~iJ{B(x*.. k = O. and x.u 2(k) where ( C I ' (' 2 ' c1 .xn )2 (4 .05/M)P2(k + I) lX6(k) = .I) if the fo llowing conditions hold: (i) g. (KI' K 2 . i = I. 05Klul(k) (1 . Hassan and Singh (1976) in an unpublished report have given some conditions for the convergence. (k)sin x . 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. Apply the costate prediction approach of Algorithm 4. In order to compare the results of this algori tbm to those reported ea rlier (Hassan and Singh.5 .05x~(k) + O.0. 13 .0. The necessary conditions of optimality through the Hamiltonian formulatJ o n leads to the following control. Consider a sixth-order nonlinear di screte-time model of the open-loop synchronous machine system : SOLUTION: In order to apply Algorithm 4.80. 05K s) Ps{k + 1)+ lX s (k) (4.1 IS) Additional equality constraints are x7(k) = xJk). x.05K 4Ps(k + 1) (4.5 .(k = = = (I -0.u/I)2 + R n (u 2(k) . Z .P4(k + I) 1I12 -0.03. and f3(k ) vector equations: uJk) = 1I 2(k) = u/ I -0 . Jamshidi.05K 6u 2(k) i = 1.5.0..5.5. and the algorithm converged in 86 second-level iterations.0.5. .117) (1-0.0.05K 7 )xc.05Klul(k) (I -0. For the sake of the present discussion. .05K3) x 4(k)+0.5) and M = 1.0408.174 Hierarchical Control of Large-Scale Systems Hierarchical Control of Di screte-Time Systems 175 nation and interaction prediction approaches.0 . (k) P4 (k ) = (I . 1973. is treated in detail in Section 6.05 cs cosxl(k) + I) + lX6( k ) (1 . K .I =±t where QII 1 <) { QII(xl(k) -X/ I) 2+Q D( X3 (k) .: ) -0.05c 2 P2(k + l)sinx f (k) (4.98)..0.5 .997 . in continuous-time form. y .05Ks )xs(k)+0 .e.05c 2x)( k) sin x 7(k) x 3 (x + I) = x 4(k + 1) x s(k + 1) x 6(k + 1) = = = (1 .0.0.05Ks)xs(k)+0.116) 0. 1971) considered by Singh and Titli (1978) is used.5 . 1978). and c in (4 . .115) must be reformulated in the form of Equation (4.0 . Simi larl y for the discrete-time case. 1975 .112) are differentiable with respect to x* and u* for each o ~ k ~ K . both the first.05K4 x! (k) (1-0 .05x 2(k) . 1.05c .0198 .05K 7 )x 6(k)+0.2(x 3(k) -x/ 3 )+ P3(k + 1)(1 .6 (4 . Hassan and Singh.0.05x ! (k) .121) lX s(k) = (0. i. and (ii) z..05K4x4(k) lXl(k) = .7 converges uniformly over an interval (0.5) while minimizing a quadratic cost fUll ction lX 2(k) lX3(k) = = 0. 4.I and their derivatives are also bounded functions of Ie (Singh and Titli. c4 .. Example 4.0..05K . .05c . ..12).05(PI(k + 1) .05Kc.05K7) P6(k withp.(k)+0.4429. 05c4)x 3 (k)+0.04.5. 6.(k + I) = x .05c l ) P2(k + 1) + P3(k) = 0: 2 (Ie) 0: 3 (k) xl(k + I) = x .(K) = O.05c s cosx l· (k) (4.5.0 .05P J(k + I) k .05c 3P2(k + 1) .115) Pc. 121) lX 4(k) = 0.05K 6P6(k + 1) (4.05K3)x4(k)+0 . )x 2 (k) . y.K 2P4(k + I)) -0. (Ie) = (1 . and C are bounded functions of k .0408. The following example deals with an open-loop power system consi sting of a synchronous machine connected to an infinite bus through a transfonner and a transmission line.3 and has been treated by many authors (Mukhopadhyay and Malik.05c 2P2(k + l)xr(k)cosx7(k)-0.xII) + PI ( k + 1) + 0: I ( k ) P2 (k) = (I . 1. cs ) = (2.2 and RII = R n = 1. a discrete-time formulation of the original continuous-time model (lyer and Cory. K 7) = (9.20.05K3) P4 (k + 1) + 0: 4 (k) Ps(k) = (I .119) P I ( k ) = 0 ..2 ( x I (k ) . 1977).71050. the same values of parameters and initial values were chosen. Here p(k) = x*(k) = 0 was chm-:en.1. 1977).22 shows the optimum .049) .025c 3sin 2x I (k) + (0. costate.025c J sin 2x7( k) + (0.0.1I12 )2} = QJ] = 0. . )X2(k) -0 .55.05x 6(k)+0.4. Figure 4.05c 4 )+ (4 .115) starting with an initial state X7(0) = (0.05K 2 x ~ (k) + 0 .1656 . cos2x 'i (/.6. system (4.05K2x 2(k) +0. Further discussion on costate coordination is given in Section 4.7 to satisfy (4 .and second-level problems are mere substitution problems. This system. The only question remaining is the nature of and/or conditions for the convergence of costate prediction.5 .2.() + RII(1I1(k) .5. .05(4) x3 (1e)+0.05c 2 x .5. They have shown that Algorithm 4.565. (k)+0 . This problem was simulat ed on an HP-9 845 computer using BASI C.0. 5.80.2.05/ M )X5(k) 0. 5. .7.05c s PJ (k + 1)sin x7Ud + 1) = (1 . 120) x 2 (k + I) = (1-0.(k + I) x 4(k + I) xs(k + I) x(. (k)+O .05 / M )X5(k) x 2(k + 1) = (1 . responses for states and controls. 1980). 1978).O. not many in the published literature support it (Sandell et a!.4 112 0. " ' Il I _ _. 2 0... who have used multiprocessors in the hierarchical can trol of a distributed parameter and a traffic control system. (1978). . • 4 5 D. The first method considered in this chapter was goal coordination..D --.J _. I.00 10 . the open. and. __L _ _ L_ _ L_ _ 5 o o . 4.003 --I _--'-_--1I _~_.____ J c. An exception to this is due to Ti tli et a!. N decoupled first-level problems must be solved. information transfer requirements.l 4 Peri od . 2 5 Period .0.0 4 Pe rind . A major improvement in computations can be made when the first-level problems are both smaller and simpler... C ----L---'-_ _' ----. storage.0004 J 5 I 2 3 Pc ri od. L---'3---'4L.0 .4 1 ' ... k .-J L.. will be discussed and an attempt is made to point out advantages and disadvantages of each scheme... the coordination at the second level has been shown (Varaiya. 1 - .. The computational requirements of this method at the first level are normally much less than those of the original large-scale composite system. k Figure 4. I '::-0. Another shortcoming is the inadequacy of methods to find a ncar-optimum control in an attempt to avoid too many second-level iterations. In this section.' .3 0. the slower the second-level problem's convergence is. conjugate gradient. the more times the N first-level problems must be solved... which turned out to be identical to those reported earlier. k 2. no significant limitations exist for the method.22 Optimum timl' responses for the opcn-Ioop power system of Example 4.5..4 10 \"~ 0.-.6 Discussion and Conclusions 0.- ..• 0... This latter observation is illustrated for coupled nonlinear systems with and without delays in Chapter 6.0012 3.' . 5 0.k 4 4 ' .J I 2 3 Peri od . L___ l _ _ J 3 4 5 Peri od . gradient.. based on the interaction balance principle. and potential practical applicability. e. ---~ __ l . Some aspects..and closed-loop hierarchical control of continuous-time systems and hierarchical control of discrete-time systems will be discussed and compared. one may use multiprocessors [or the first level in an attempt to save computational time for the N subsystems computations.. _ I .J. O. the overall computations depend heavily on the convergence characteristics of the second-level linear search iterations. 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. Although this seems to be a good proposition and has been suggested by others (Singh.0 . methods. A near-optimum control without convergence at the second level would be an unfeasible solution._ .0.J.... In practice. etc..O.• o 0.JIL---' J () I 2 .... 1969) to converge to the optimum solution. " D. 1 _ _ .. I) . Since between each two successive second-level iterations.. From the computer storage point of view. However. such as computer time.1 Period.g.4. 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.001 . Yet another difficulty is the need to add a quadratic term z 7Sz of the interaction vector z in the cost function to avoid singular solutions.l---. Newton's..J 4 Per iod..' ----'-2---. This . The scheme can handle constraints o[ inequality type on both the state and control. furthermore.R - 0.0 - 0.176 Discussion and Conclusions J77 u I 1 . need for subsystems' full state information in a feedback configuration. In all. From . 1976) ven. as suggested in Section 4. as is evident from (4. but rather only a Lyapunov-type equation.. the near-optimality index p in (4. The establishment of a class of interconnection perturbations. I. Five approaches were presented. Singh 1980). This type of TPBV problem is a common difficulty with optimal control of time-delay systems by the maximum principle. The computational behavior of this scheme is very similar to continuous-time goal coordination. which provide a foreseen effect on the overall system cost function. In this author's opinion.1) is that its first-level problem is very simple.3. which is discussed in Chapter 6. which would neutralize the effects of the interactions.4.1.4.5 .. Further advantages of the method are the convenient handling of inequality con- . Based on its development in Section 4. for a number of reported appitcatlOns (SIngh and Titli 1978.23). Both can be simulated in such a way so as to save on sto. i.5.be desirable to have an additional global controller. the feedback control based on structural perturbation is an effective step in the right direction for closed-loop hierarchical control. 1975. 1980). H aimes.tee.21 )-(4. which is heavily dependent upon the second-level problem'S iterations. In Section 4. the best situation would be a balance between on-line and off-line calculations. (4. it would . the interaction prediction method took on the order of one-quarter of the computer time of goal coordination. the computational viewpoint. Comp~tatIOnally. The time-delay algorithm discussed in Section 4. This was done for almost all illustrative examples of this chap. if the 12th-order system of Example 4. as demonstrated in Section 4. it becomes clear that when the interconnection matrix Gis nonbeneficial. Jamshidi and Heggen. Moreover.. Although the solution of a Lyapunov equation is much easier than a Riccati equation. the hierarchical control of discrete-time systems was considered. The most interesting outcome of feedback control based on structural perturbation is that it takes possible beneficial effects of interactions into account. the interaction prediction method compares very favora?ly wIth the goal coordination approach. The main disadvantage of this scheme is the extensive amount of off-line calculations for the finite-time case.5 . where a near-optimum solution is proposed. 1975) is excellent evidence of this. however. The only advantage of discrete-time over continuous-time goal coordination (Section 4. 1978). which involves both delay (in state vectors) and advance (in costate vectors) terms. b). An argument which can go against both the goal coordination and interaction prediction methods is the . finding an efficient method with reasonable storage as well as computer time is questionable (Jamshidi. Another important point concerns the InfOrmatIOn structure between subsystems. 1977.3. 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.5.4.3.4.2. no large-scale AMREs must be solved. it is clear that the second-level problem is much simpler here than in the goal coordination method.2 is perhaps one of the first attempts to come up with a c!osed-loop control ' law for hierarchical systems. no possibility for a singular solution exists and the computat~onal experience of this author and others (Singh and' Hassan. For example. 1974. Furthermore. First from the storage point of VIew. In spite of these advantages.5. It should be mentioned. including the 12th-order system of Example 4. is a very useful flexibility for the coordinator.fy that the convergence at the second level is reasonably fast. The feedback control based on interaction prediction discussed in Section 4. Part of the latter difficulty can be ~andled through th~ hierarchical control strategies for structural perturbatIOns developed by Siljak and Sundareshan (l976a. This algorithm as well as goal coordination have been used for the optimal management of water resources systems by many authors (Fallaside and Perry. the computational savIngs woul? not h~ve been that appreciable. 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. 1980).2 is perhaps the most appropriate goal coordination method for practical problems. be avoided in two different ways.4. In Section 4. when the system order is very large (n > 200). 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 .3. Another hierarchical control technique discussed was the interaction prediction method. 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. however. Moreover.1 the continuous-time version of goal coordination approach was extended to linear discrete-time systems. although the total state information structure still remains the same.3. such as those of Siljak and Sundareshan (1976a).1 had been decomposed into two sixth-order systems. both methods require the quadratic term in the cost function to avoid singularities.3. Under such conditions.rage requirements by using a common block of memory for both levels.45) may become very large. The feedback control based on structural perturbation presented in Section 4. The application of the algorithm to traffic control problems (Tamura.178 : Hierarchical Control of Large-Scale Systems Discussion and Conclusions 11 ':1 problem.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 two methods are very similar.e. 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.41). can. This scheme.5.1).5112 (k .3. 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. seems less effective than the costate prediction method of Section 4.5. and !:l. C4. and two for controls u). Also.2. The other scheme considered in Section 4. In Section 4.3. + 1) = 2xI (k)+ XI (k . Algorithm 4. 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. More discussions on hierarchical methods are in Section Use the two-level goal coordination Algorithm 4. .2. The costate prediction scheme can be easily extended to the linear case (see Problem 4. which is not always possible in goal coordination procedure.!.2. while the second-level problem involves six substitutions for the costate p(k).t=O. The first-level problem constitutes the solution of the discrete-time Riccati equation.4. such as discrete-time goal coordination (Section 4. which can prove to be rather timeconsuming in a real-time situation. The reasons for this are the convenient solutions of the costate prediction's first.l80 Hierarchical Control of Large-Scale Systems Problems 181 straints and avoiding the possibility of singular solutions. x 2 (k + 1) = - x 2 (k)+ (k . The main difficulty with almost all of these hierarchical control schemes is that their convergence is not yet a settled issue.3 represents a goal coordination proccdure for the hicrarchical control of a large-scale discrete-time system. does not involve any iterations between first.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. . this scheme.5.l.12).4. for the system of Example 4. The computational effort per iteration for this approach is a small fraction of that required for the other methods.3).2 o o 0 -I 0 with a cost function J=. six for Lagrange multipliers (3.and second-level problems.5. tested numerically in Example 4.8) or continuous-time case (Problem 4.5.]' (=:1~u(k)~(:] J=~xT(7)(~ ~]X(7) 5 ~] x (k ) + uT (k ) ( 06 : . the first-level problem consists of straight substitutions on the right sides of 14 equations (six for states x.5. More specifically. R=/2 . as in its continuous-time version (Section 4. the solution of a discrete-time Riccati equation is sought only once for the N subsystems.Q2 0 -0.I) X2 6. 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.7. Consider a two-subsystem problem.1 to find an optimum control which minimizes Q=diag(I.or second-level hierarchies. The only exception is perhaps continuous-time goal coordination (Varaiya. Consider the system 0. For a second-order discrete-time delay system XI (k C4.1).I) + u l (k -I) -u2(k) .2xI (k) -1)+ 111 (k )+0.5 was the structural perturbation approach of Section 4.u2(k-l) Problems C4.1) and interaction prediction (Section 4.3. 2 L xi(5)QiXi(5)+~ L i = 1 4 4 [xT(k)Q(k)x(k) + uT(k)R(k)u(k)] k=O where Q(k) = 14 and R(k) = 12.5.5. where [~] ~ X(k)~(. In other words. 1969). Find the optimal hierarchical control of this system by decomposing it into two second-order subsystems.1. I 0..5 .2x i( k) +0. 4.2 for the following system: o o o 0. x 2 (0)=1 and a cost function ('Xl z J .4 to find the optimal sequence u*( k).(t)R. neutral.. Use the costate prediction Algorithm 4.22). -1~k~O.5.4.2 0 0 -4 01 I 0 I I with cost function x ( k+1 ) = J= . k xT(O) = [1 I].4.5 to a composite interconnected linear discrete-time system use the time-delay Algorithm 4. = 12 . It is possible to solve this problem analytically. J='2 L k~O 20 (0.4.7.lx~(k) + 0. I.2 -I 4. C4.! o Z u2 + U z ) dt l 4.0.I 0].1 0 0 XI=XI+U I . ] ( ) [ O. For the system XI = XI + ax z + u I ' XI (0) = I with x(O) = (1 I )T and quadratic cost matrices Q = 12 .9 [ 0.2.1.2 .6 on the structural perturbation approach for discrete-time systems to find the optimum control for 4.(t)+ u.5 .2 J (2 X Iz + x 2 + with Q = diag(l.5 . C4.7 to find the optimum control (or 0 . x(k)=u(k)=O.5 .4. What is the value of the cost function deviation p? Use initial state x (O) = [ 1 I I 1 If· -. R = 0.17 -2 . lu~(k)+0 . and initial conditions XT(O) = [ I 0 0.4. 1. 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. Hierarchical Control of Large-Scale Systems Problems 183 with initial conditions and a quadratic cost function with weighting matrices Q = R = 12 . Usc Algorithm 4.I-O.0.. [Hilll: F= K + k is a positive-definite solution of AMRE (4.6. Prove that the control law (4.10. xz=xZ+bxl+u Z' xz(O)=O .u(t») dt] I.5 o -1 r1' B= r~ .1 0 .2.46).2 to find the optimal control u*( I) for the above problem. 0 . and nonbeneficial.x. Extend the costate prediction approach of Section 4.53) is a stabilizing controller for the large-scale system (4. I.t = 0.5 0.1 0. find a hierarchical control law based on the structural perturbation method of Section 4.47) for the perturbed system (4. 6. a computer implementat~on is more desirable for higher-order systems. 1 I . .9.0. Use the interaction prediction Algorithm 4. I. = 0.0. xl(O)=1 x z =2x z +u z .2u~(k» ) For a system with the matrices 4. Repeat Example 4. find regions in the (b .a )-plane where the interconnected syslem can be beneficial. R.] A = -5 r .2 ~.8. I O.182 Problems (continued) .L ~ [xT(2)Q.X.11. I).4.lxl(k) x k + 0 Consider an interconnected system x= r~ 0. however.2. .2) and R = 12 . .1 .I o o 0.I 0 2 wi th x (0) = ( 10 1 5) T and a cost function with Qi = diag(2.2 1 0 0.6. I.4 h-~~~ ~ ~F--~-!--=q'j rHj ~ x + u 4.512 .(2)+ l\xT(t)Q.4. Prentice Hall. 247-256. M . T. P. AC-16:613-620.. 1980. Cohen. M. D. AU/olllatica. Advances ill Control Systems. Cambridge.. University of Cambridge. 187. and Jolland. J4:77-1O 1. 1973. A 1958. M. 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M. lEE 122:202-208. Int. and Bernhard. Multi-level control and optimization using generalized gradients technique.12. Multilivel systems control and applications. lyer. Two coordination principles and their applications in large-scale systems control. D. Mahmoud.5. Y.. Large-scale linear and nonlinear programming. Cheneveaux.. Large Scale Systems. 16. and Levis. J.. Centre d'Automatique. W. P. Geoffrion. A M... M. SIAM Review 13 : 1-37. Hassan.:. 1974. 1. B. Chebyschev curve fit. Hierarchical Control of Large-Scale Systems Extend the discrete-time system's costate-prediction method of Section 4. Power.. NJ. McGraw-Hill. Warsaw. IEEE Trans. IEEE Trans. t) t) Qx ( t ) + u T t ) Ru ( t )] dt ( J= i T(If) x Qx ( tf) + i fol. B. 2:81-95. 1975. IFAC COllgr. Syst. Optimization Methods for Large Scale Systems. Y. System 9845 Numerical Analysis Library Part No.. 1968. 1971. and Simon. 90:2149-2157. Management Sci.. Mahmoud. No. and Mickle. Y. Bauman. VA). 1974. 1977. 1970. New York. 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Large Scale Systems - Jamshidi

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