- Email: [email protected]
Decentralized Control Design of Large-Scale Systems
Copyright © IF AC Distri buted I nle lli ge ll ce Systems, Va rna , Bul garia, 1988
MET HO DS FO R SYNTH ES IS OF CONTRO L SYSTEMS WI T H DIST RI BUTED INTE LLI GENCE
DECENTRALIZED CONTROL DESIGN OF LARGE-SCALE SYSTEMS L. Bakule I nstitute of Information Them) and Automation, Czechoslavak Academy of Sciences, 182 08 Prague, Czechoslovakia
Abstract. Some recent and new results on decentralized control design for large soale dynamic systems are surv'e yed. The decomposition of systems and the decentralization of the design task are characterized. The results concern the methods for the design of serially interconnected systems including complete decentralization, multi-controller configurations and simultaneous stabilization, sequential design, aggregation- and overlapping-decomposition design techniques, singularly perturbed systems, decentralized PI-control with integrity property and symmetric composite systems. The methods are presented with an emphasis on the robustness against model errors which are giv'e n by upper bounds. The design task comprises also the requirement on the I!O-behav'iour of the closed-loop system. Keywords. Large-scale systems; linear systems; state-space methods; decentralized control design; uncertainty; decomposition. robustness; system theory. gregation.
INTRODUCTION
Information structure eonstraints. Generally, no decision unit knows the complete system. It means that there is only a limited knowledge on the properties and control of some parts of a system.
The decentralization of the design process means that the whole de?ign problem is divided into weakly coupled subproblems that are solved independently. It differs from decentralized control systems, where the design process can be performed in centralized or decentralized way. Though the properties of decentralized control systems are intensively investigated, the decentralized control design problem is considered as a by-product problem which is connected with decentralized control systems.
Therefore, the analysis and synthesis cannot be performed in a single step as it is done by small scale systems. Suppose that a "large scale" system is simply a system which cannot be taokled by "standard" methods. To solve effectively large scale system control problems, several specific m,-', thodologies have been elaborated. They belong mainly to the following concepts:
Recent results (Bakule and Lunze, 1988. Findeisen, 1982; Jamshidi, 1983; Litz, 1983; Siljak, 1983; Voronov', 1985) indicate that the complexity of large scale control systems leads to severe difficulties that are encountered in the analysis, design and implementation of large scale system control strategies. These diffioulties arise mainly from the following reasons:
Decomposition. To reduce the time or memory complexity of the computations when analyzing and designing feedback systems, the global problem is decomposed into lower order subproblems which are solved separately and combining their solutions the global problem is solved. Notice that some kind of coordination technique can be inoluded when using this control design approach.
Dimensionality. The system has a high order. It simPIy means in praotice that the time of computations, which is not proportional to the system order but to a power of order greater than one, for higher order systems is unacceptable when using standard approaches.
Decentralization. The global problem is diVided into completely decoupled subproblem aims. Several different control sta-· tions have to solv'e problems which substitute the global problem. The quality of the solution is reduced because the subproblems cannot be generally completely separated from each other. No coordination is included here. Decentralization does not permit information exchange among the controllers. Decentralization occurs in connection with the control law, the design
uncertain~.The
overall system cannot be descri d by an exact model. Its model has primary uncertainties which result from an incomplete knowledge of the plant. Uncertainties are further caused by the information struoture constraints and approximations which can hav'e various reasons for iL H C.;J.6 -:, o"me kind of the model Simplification or ag-
133
134
L. Bakule
method, and with the implementation. These different types of decentralization can be combined. The majority of the results refer to the decentralization of the control law and extend the design principles of mu1tivariab1e feedback control law to structurally constraint feedbacks. Robustness. Uncertainties can be taken into consideration by using approximate model and the plant. Robustness is one of the -main issues in large scale system theory because uncertainty is a generic property by the large soa1e system modelling. It is closely related to the approximation, where the system equations are reduced to obtain reduoed order models. The main aim of the paper is to use the structural properties of the plant for the division of the overall problem into "weakly coupled" problems at the subsystem level. The different known methods are surveyed and recent and some new results are presented. THE DESIGN PROBLEM
Suppose that the servocompensator, i.e. the dynamic part of controller, is used. It means that the requirement (II) oan be omitted in further considerations. DECOMPOSITION OF SYSTEMS The goal of decomposition is the reduction of computational complexity by the solution of given analysis and synthesis problems. The submodels must be "weakly" coupled is some sense. The concept "weakly ooup1ed" is meant in differeht ways. Accordingly, different decomposition prinoiples have been established. Consider the deoomposition of the global system into a hierarchy of subsystems, into disjoint or overlapping subsystems and into subsystems with separated time scales. These concepts form the basis for the decomposition and decentralization of the design problem.
The design problem can be formulated as follows: C-onsider the overel1 system consisting of N subsystem which are described by the state space model xi- Aixi+Biui+Eisi' xi(O). x io ' Yi- Cixi+Diui+Fisi' i.1, •••
(1)
,N,
n m msi Pzi where XE:R i ,u i E R i ,sie: R ,z£R are vectors of the subsystem states, control inputs, interconnection inputs, oontro1 outputs, and interconnection outputs, respectively. Find a decentralized controller xri= Arixri+Briui+Erivi' u i - K1ixri+K2iYi+K3ivi'
(2) i=1, ••• ,N,
where x ,v denotes vector of the i-th contro11~r states,command inputs, respectively. Ai~ •••• K6i are constant matrices in Eqs. ,1),(27. The closed-loop system (1),(2) satisfies the following requirements: (I) The closed-loop system (1),(2) is asymptotically stable. (II) Asymptotic regulation occurs in the olosed-loop system, that is Yi (t)-v (t)--O i
as t __ 00,
where the oommand inputs v (t) and the disturbances Pi(t) ent~ring the system at arbitrary points belong to a class of admissible external signals. (Ill) The I/O-behaviour of the olosed-loop system is well-suited acoording to given reqUirements, for instance, on the step responses. (IV) The design requirements (I)-(III) are satisfied in spite of model uncertainties.
Hierarchy of Subsystems The difficulties in the analysis synthesis of interoonnected systems are caused mainly by the complete interdependenoe of the subsystems. Essential simplification oan be obtained if some subsystems have only a one-directional effeot on some others. In such situation the interconnection matrix is a lower triangular or can be made so by reordering the subsystems, see e.g. Bakule and Lunze (1985 - 1986), Bakule (1987), Hodzic and Siljak (1965),Pichai et al. (1983),Sezer and Siljak (1981). The strong coupling between two subsystems i,j is m~ ant in the digraph-theoretio sense. Defining the set of vertices by one-tc~one oorrespondence with the states and the set of edges, where an edge from the state k to the state 1 is defined if the corresponding element a kI of the matrix A is nonzero. The subsystems i and j are strongly coupled if there exist a path from the vertex i to vertex j and a path from vertex j to vertex i~ Using t bis digraph approach the subsystems with only one-directional interaotions can be detected. ~nd
Disjoint Decomposition The ooncept "weak" and ;'strong" coupling defined in the last subchapter refers to the presence of interactions among all parts of the system, whereas in this subchapter and the following ones this concept regards to the magnitude of present interaction lines. The motivation is based on the fact that the interactions are weak, i.e. they do not strongly influence the system perfor-
Dece ntralized Control Desig n o f Large-Sca le Systems
mance. Therefore, if the analysis or design is performed on the local independent subsystems, the difference between the global system and local subsystems is small. This difference is expressed for instance by linear-quadratic systems by the suboptimality index, i.e. its influence on the performance criteria values. Disjoint decomposition means that the overall system is divided into subsystems in the form xi= Aixi+Biui+~AijXj' Yi-
Cixi+Diu i , '''i
The isolated
(3)
i-1 •••• ,N.
transormation (8)
The systems S,Se satisfying Eqs. (7), (8) are ca~led a contraction, expansion, respectively. The properties of expansion-inclusion relations, or a systematic way for generating expansions for a given system are intensively investigated"see for instance Bakule (1985), Hodzlc and Siljak (1986), Ikeda and ~iljak (1980, 1984), Siljak (1980) •
subsystem has the form
Time-Scale Decomposition
xi- Aixi+Biu i ,
(4)
Yi- Cixi+Diui • This type of decomposition corresponds to numerous practical design problems, for instance in power systems, water managment and ecological systems, see e.g. Bakule (1978. 1979), Himmelblau (1973), Lunze (1980), Michel and Miller (1977), Sandell et al. (1978) ~iljak (1978), Singh and Titli (1978~, Singh (1981). Overlapping Decomposition Disjoint decomposi t ion means that the whole system is divided into disjoint parts. Such a decomposition may not be successful in case of large-magnitude interactions. This motivates the overlapping decomposition, where the resulting subsystems have joint parts-interconnections. These subsystems may be weakly coupled although disjoint subsystems are not. The overlapping property leads further to the effect of more reliable control of overlapped parts. A systematic way to overlapping decomposition begins with an expansion of the given system. The expanded system is decomposed into disjoint subsystems in the same way as described above. The question whether the original and e7.panded systems have the same trajectory is answered by the inclusion principle. Consider a pair of systems S, Se S:
135
x- Ax+Bu, y= Cx,
(6) y= Cexe' n where X€Rn , Xe€ Re, n~ne. Denoting x(t,x o )' xe(t,x eo ) the unique solutions of Eqs. (5),(6) for the initial time t-O and u(t)=O then a system S includes a system S if ther~ exists aneord~red pair of matrices (T,T ) such that TT -I for any x o€ X and it holds X(t,Xo)=TXe(t,T+Xeo ), (7) y[x(t)] = y[xe(t)] for all t, where T+ is a generalized inverse of T. The states x,x e are related by a linear
So far, the decomposit i on methods were based on the small-magnitude interconnection property. The following method is motivated by the consideration that the overall system consists of a fast and slow subsystem. Then the fast subsystem will arrive at its finel state before the slow subsystem had begun the essential part of its motions. The fast subsystem behaves as a static one from the point of view of the slow subsystem, and the slow seems to be quiescent in the time horizon of the fast subsystem. Their connection is weak. To formalize the time-scale decomposition consider the system in the form x- A11 x+A12 z+B 1u, x(O).x o ' ez= A21x+A22z+B2u, z(O)=a o '
(9)
y= C1x+C 2z, where e, is a small parameter E:>O. Using e. -0, the fast 8ll.bsystem is considered as infinitely fast and it is approximated by an algebraic equation. The degenerated system has the form
i-
A11x+A12z+B1u,
o.
A21x+A22z+B2u,
y-
z.
(10)
C1x+C 2 supossing A22 to be nonsingular we obtain A -1 A A- 1B z--A 22A21 x - 22 2u and the system (9) has the form ;..
"A
J\
X= Ax+Bu,
(11 )
(12)
y=Cx+Du, /\.
"""
1\
where 1
A
A-A11-A12A22A21,
(13)
1
A
B-B1-.A12A22B2' A
1
A
1
C-C1-C2A22A21' D--C 2A22 B2 • The relations of an original system and its approximation (12) has been studied mainly from the point of view of stability prop-
L. Ba kule
136
erties and the generalization of the two_ -time scale systems to multi-time scale systems, see for instance Chow and Kokotovic (1976), Jamshidi (1983), Kbalil and Kokotovic (1978), Voronov (1983).
Consider a plant Pi:
Xi· Aixi+Biui+FiPi_1' Yi- Cixi+Diui+HiPi_1'
Pi" Cpixi+Dpiui+HpiPi_1' P0 .. 0, i.1 ••••• N.
DECENTRALIZATION OF THE DESIGN: PROBLEM Decentralization concerns the information structure related to the solution of a decision problem. The decision process has two phase s: 1) Design phase, when an appropriate control law is determined on the basis of a-priori information about the model and control aim of the plant to satisfy the design requirements (I)-(IV). 2) Operating phase, when controls u are determined on the basis of the a!posteriori information about the actual state and command signal. Since the controller (2) represents local feedbacks of y towards u no crodd-coupling between ,iand YA' i.j, are produced. The decision process aas a decentralized information structure, it means tbat each control station receives only the local information about the output Yi and the command signal v • The existenee of decentralized fixed modes and the studies of their properties illustrate that the decentralization in the operating phase brings new problems. There are several motivating reasons to extend the decentralization also into the design phase: - Change of the system structure during operation 90 that certain autonomy of control stations is necessary. - Considering the overall system decomposed into weakly coupled subsystems, the isolated subsystems can be designed in the centralized way. This procedure simplifies the design process. - In overall systems with subsystems which are widely distributed in space or representing different authorities which are responsible for the performance of the subsystems. Therefore, decentralization of the design process means that the control stations are designed independently by means of different plant models and design speoifications. Several concepts of decentralization of the global design problem are described. The main difficulties arise here from the non-classical information structure of the decentralized design process, see for instance Bailey (1978), Bakule and Lunze (1988). DESIGN FOR SERIALLY INTERCONNECTED SYSTEMS Serial connection structure enables completely decentralize the design of decentralized controllers because of the band structure of interconnection matrix. Different controllers are designed independently. Such a decomposition enables to deal effectively with dynamical requirements on the closed-loop system. Serially connected systems are considered as a specific type of hierarchically connected systems. com¥lete Decentralization ot he besign
(14)
Suppose a controller in the form (15 ) Ci : u i • KiYi' Complete decentralization means that the decompOSition of dynamieal requirements and the inclusion of the model uncertainti es can be considered only at the subsystem level. Therefore. suppose a model of the plant Pi in the form •
A
'"
A
A-
A
A
A
A
Mi: ~i~Aixi+Biui+Eisi+FiPi_1' (16)
Yi-Cixi+Diui+GiSi+HiPi_1' zi=CZixi+Dziui+GZisi+HziPi_1' Pi=Cpixi+Dpiui+Gpisi+HpiPi_1' and ME i :
Isil = Vi*lzil+roi(xio,t),(17)
where Si' zi' Vi' r oi are vectors of interconnection input, interconnection output to the error model (17), impulse response matrix, free motion of the model (17). Eq.(17) describes an upper bound of the model error, * is the convolution operation. Supposing the system (16) controllable-observable 'and the closed-loop in the form
xi~ Aixi+Eisi+FiPi_1' (18)
Mci :
Yi~' Cixi+Gisi+HiPi_1' zi"
CZixi+Gzisi+~ziPi_1'
Pi= Cpixi+Gpisi+HpiPi_1' and Eq.(17). This system is analysed using the following lemma. Lemma 1. (I) A sufficient condition for the s,ability of Eqs.(17)1(18) is: a) The isolated systems (17),~18) are stable. b) It holds 00 00 ~ (19) A M[
Ivi dtJ
o
0
where ~M(') is a maximal eigenvalue of the matrix (.). (II) Tbe I/O-behaviour of Eqs. (17),(18) with si=O is characterized under the validity of (19) by the inequality
137
Dece ntrali zed Co ntrol Design of La rge-Scale Syste ms
(22) (20)
where
'"
~
Vicvi+Vi*IGzie
Ai t~
\ Ei ;l-Vi rJ
and y denotes the output of the model (18),ii.e. for s -0. Approximate model (16) is designediusing standard LQ or pole placement techniques, see Bakule and Lunze (1985,1988). Multi-Controller Design Consider the problem (14),(15), where complete decentralization is possible. The basic idea of decentralized multi-controller configuPation with passive redundancy is for the i-th subsystem, how to divide a controller C in more parallel controllers. SUpposin~ the simplest possibility, that is Ci·Ci1+Ci2' Cij =0.5C i , j.1,2. The motivation for using parallel controllers is the inclusion of the possibility of controller failures. Therefore, if one of controllers fails the second is still operating though it operates in non-optimal way. The design is performed in the follOwing way. First, design the controller C using Lemma 1. Then use instead of conttoller Ci the controller Cii=0.5C~ and test the stability using Lemma 1. If it is not satisfied modify C and repeat the whoie design procedure. T~e ideas of multicontroller configurations are elaborated tor instance in Bakule and Lunze (1988), ~iljak (1980). Simultaneous stabilization Consider a set of M serially connected chains, each with N subsystems. Denote Pik the i-th subsystem of the k-th chain. The chains are not connected among themselves. The motivation for such problem are for instance production complex systems with parallel structure of independently oprating processes with the same number of serially connected subsystems or a nonlinear system consisting of serially connected subsystems which operates at k different operating points with linearized models. An attempt is to use one controller stabilizing all models. Suppose a plant Pik in the form Pik: xik=Ai~ik+Bikuik+FikPi_1,k'
(21)
Y~k=Ci~ik+Dikuik+HikPi_1,k ' Pik·Cpi~ik+Dpikuik+HpikPi_1,k '
Pok=O, i=1, ••• ,N, k.'1., ••• ,M
where u ik ' Yik' Pik are mi -, r i -, qi- dim ensional vectors which do not depend on k and which have the same meaning as u i ' Yi' piin Eq.(14). Suppose a controller in the form
Denote analogously to Pi' Ci' Mi , ME i , Mci in Eqs. (14)-(18) Pik' Ci' Mik , ME ik , Mcik the plant (21), controller (22), model of the plant (21). error model of the plant (21) olosed-loop &ystemof the plant (21j, respectively. The problem can be formulated as follows. Problem. Consider a set Pi-lPik:k.1, ••• ••• ,MJ find a controller "(22) sa tisfying the requirements (I)-(IV) for all plants Pik' Solution. FormUlate a composite system
~i:
ii=!i!i+~i~i+!i~i-1 '
(23)
li-2i!i+Pi~i+~i~i-1 '
Then the controller has the form
where ~i.diag(Ki' •••• Ki). Constructe analogously to Eqs.(14)-(18) ~i,MEi' ~ci' where Miis the model of the plant (23). MEi the error model of the plant (23), ~ci the c ~ osed-loop system of the plant (23). respectively. Mci has the form !ci: ~iC~!i+ji~i+~iEi_1 •
Ii·fi!i+gi~i·~iPi-1
(25)
•
~i·§zi!i+gzi~i+~ZiPi-1
'
Pi·gPi!i+gpi~i+ffpiPi-1
•
and ME i :
I!il~ Yi*l~il+~oi(xio.t).
(26)
T T T T T )T where ~i·(si1.···.siM) , ~i=(zi1.···.ziM • T T T !oi.(roi1 ••••• roiM) • This system is analyzed using the following lemma. Lemma 2. (I) A sufficient condition tor the stability of Eqs. (25).(26) is: a) The isolated system (25),(26) is stable. b) It holds (27) 00
00
'"
~M[JyidtJ(lgiIJ(t)+I.9zi/it¥il o
0
)dtJ<1 •
138
L. Bakule
(11) The I!O-behaviour of Eqs. (25).(26) with si-O characterizes under the validity of-\27) the inequality
is desi8Ded as the solution of the problem 00
JGiQi'ji+uiRiui )dt -- min
o
(28)
subject to Eq.{JQ). The robustness is evaluated for the closed-loop system in the form
Xi-
where ~
Ait", ~ Yi-!i+!i*IQzi e- ~i*!i and 1i denotes the output of (25) with si=O-;
()2)
Ai%i+Bivi+Eisi+FiPi_1 •
()))
~
The ideas stimulating simultaneous stabi_ lization can be found for instance by Minto and Vidyasagar (1986). SEQUENTIAL DESIGN Sequential design is an appropriate procedure to solve effectively systems with one-directional interconnection structures under assumption that the dynamical requirements on the system cannot be decomposed into the individual subsystems level. It means that by the i~th contr61ler design. i.e. for the i-th subsystem. the previous 1 ••••• i-1 controllers have been designed and this part of closed-loop systems is considered to influence the i-th subsystem as a known gisturbance. This idea has been used by Ozguner and Perkins (1978) for linear state regulator problem. Using this motivation this design has been extended to the serially connected systems including uncertainties by Bakule and Lunze (1986).Its basic steps follows. Consider a plant P in the form of subsystems described by Eq.(14). Find a controller Ci given by Eq.(15) satisfying the reqUirements (I)-(IV) and consider step signals wi. Because of requirement (11) PI-controllers are used (29) xei e i , e i = Yi-"'i' u i "' K1iei+~ixei' where Ki.(K1i'~i) is a constant matrix with det(I-K1iDi)~0. A complete decomposition of dynam~cal requirements is impos~ sible here. Considering the i-th controller design. the extended system is supposed in the form
~1-(~)t: :) (::](::)"1+Gr1
• ()O)
Y1-(f:t(!1 :) (:: )'(:1)"1. i
Then the controller
1i- Ci%i+DiYi+Gisi+HiPi_1 • zi= Czixi+DziV1-t4Zisi+HziPi_1 ' Pi= Cpixi+DpiVi+Gpisi+HpiPi_1 ' and Eq.(17). This system is analyzed wi th inputs v ,p and outputs Yi' Pi applyi~ te~ 1 with the tolerance band for Pi in the form
-
'" Pi+PiO"(t),
~ Pi-
where band,
.~ii ~
(4)
is the middle of the tolerance is a maximum width of the band. DESIGN USING DISJOINT DECOMPOSITION
The motivation for this type of decomposition is that the interconnections between subsystems are weak. Therefore, its influence on the system performance is small. General methodology is based on the two steps: the individual subsystems design and the iuclusion of the interconnection effect on the global system performance. These steps are performed either on the original system or on a simplified auxiliary system from its properties the properties of original system are evakuated, The first case concerns mainly exact models, for which the aggrega tiQn-decomposi tion was developed by Siljak see Michel and Miller (1977), ~lljak (1978),Voronov (198). This method concerns mainly the stability of the global system. The suboptimality index gives only a rough information on the I/O-behaviour. An another approach has been developed for the stability tests. It uses comparison systems. A comparison system is supposed in the form .. 00 r(t) =
JV(t-2')w(~)d't" ,
(5)
-00
where', r, ware output. input vectors, respectively. V is the impulse response matrix. The comparison system (35) for the linear system yet) .. G!tu(t) , G(t)::CeAtB is given by Eq.{J5) with
(6)
V(t)~IG(t)l.
Suppose a system x = Ax+Bu , Y .. ex
(7)
Dece ntra li zed Co ntro l Design of La rge-Scale Syste ms
with a controller u i '" KiYi.
(8)
The closed-loop system (37), (38) can be decomposed into xi= Aixi+Eisi+FiVi '
then 1 , A are in the expansion-contraction rilation.
(40)
Specific application of overlapping decompos1tion offer serially connected structures of subsystems, where some overlapped sUbsystems-interconnections belong both to the previous and to the following sUbsystems. Such type of decomposition is imposed by the feedback control structure. These structures are expanded, where expansions are of the disjoint-type decomposition character. This problem can be solved using the complete decentralization approach.
and where L is an interconnection matrix, s, z are interconnection outputs, inputs, respectively.Comparison systems of the subsystems (39) are IYi(t)l- Vl1~luil+Vi2itlsil ,
(41)
Izi(t) ~ Vi3*luil+Vi4~lsil with Vi1~lcie Is(t)\'I!O
Ai t
Ei l
L / z(t)i
etc., and with ~IL1.
(42)
~e
overall system (37), (38) is I/O-stable If all comparison systems are I/O-stable and the matrix P = (I-diagV. L) which represents the influence ofl~terconnec tions is an M-matrix. Using the comparison systems the I/O-behaviour can be evaluated see , Bakule and Lunze (1988), Lunze (1983).' DESIGN USING OVERLAPPING DECOMPOSITION Ov"erlapping decomposition leads to a complete decentralization of the design tasks because the stability of the expansion im-' plies the stability of the original system. Therefore, the stability of the whole system can be proved by means of stability of the lower order subsystems of the expansion. Concerning decentralized control design u nportant are the conditions stating the expansion-contraction relations for feedback systems. Consider the system (5) with a controller u=Kx and the system (6) with a controller u=K x and denote the c10sed-loop systems e e
A=
A+BK ,
Ae ..
Ae +Be Ke •
(43)
S~ppose that the rela tion between AC and A. has the form
.... + M ,
'" Ae .. TAT
(44)
where T is given by Eq.(8), M is some complementary matrix satisfying the relation T+r.1 i T .. 0
foral11.1, ••• ,n . • e
where the complementary matrix F satisfies the conditions (47) FMiT .. 0, FM i - 1N .. 0 for all i = 1 , ••• 'Ile-
(9)
Yi= Cix i , zi= Czix i s .. Lz ,
139
(45)
This restrict the class of gain matrices. If the relation between K and ~is supposed in the form (46)
An another application of overlapping decompositions is for reliable controllers design. It means that the part of the system is controlled by controllers with parallel structures. The problem how to divide a given controller into parallel controllers depends on the design requirements. It means for instance that supposing a given controller with reliability an attempt is made to design a parallel contr oller with desired reliability or a parallel controller satisfying some des1red dynamical requirements when given number of controllers fails. The design of such controllers for systems with uncertainties can be performed usi~ Lemma 1 which is applied to 1ndindual operation-failure structure~, for instance 2-out-of-3 structure etc. Though some results are available, the decentralized control design using overlapping is a promising area for further research aotivity, see e.g. Ikeda and §iljak (1980; 1984), Ikeda t §iljak and White (1984), ~iljak (1980, • DESIGN OF SINGULARLY PERTURBED SYSTEMS Deoentralized oontrol design for s1operturbed systems means that the multi-time scale properties are detected, the subsystems with one time scale are separated and looal feedback are designed inoluding the evaluation of erros of such approximations. The basio oase is the case of the system with two-time-soales. A two-time-soale property has been first characterized explioit systems by Chow and Kokotovic (1976a,1976b).Reoently, there are available results on the properties of implioit systems by Khalil (1984). Implioit singularly perturbed systems means that the following form of the system is considered gu~arly
:i:
a
A(€.)x ,
(48)
where A(O) is singular and A(g).is analytio at g . 0 that is A(£) .. ~lA • Using appropriate state transform a"\wo!time-scale property is deteoted. Deoentralized design means for instanoe for the pole plaoement problem th6t for a system in the
140
L. Bakule
form x t = A"x,+A'2X2+B,U •
(49)
x 2 = A2,X,+A22X2+B2U • where the matrices A satisfy the two-time-scale propertyi~ests, that the feedback control u=Kx 1s separated into the slow in fast feedback matrices Ks and Kf resulting in the control
This approximation of control is an O(~) Approximation for the eigenvalues. Analogous results are available for the LQ-s1ngularly perturbed systems, where further the 2control approximation (50) results in O(~ ) approximation for the criteria values. The two-time-scale systems results have been further extended to the case of multi-time scale systems or multi-parameter two-time-scale systems for instance see Jamshidi (1983), Khalil and Koko tovic (1978). DESIGN OF ROBUST PI-CONTROL Decentralized robust PI-control is motivated by the inclusion of uncertainty of decentralized controllers which is generated by individual controllers fai*ures. The closed-loop stability includes 2 operation conditions for N controllers corresponding to all possibi11ttes or failures. The problem can be formulated as follows.
Ksii between u i and Yi can be determined. It is known in advance and it is independent on other operating controllers. It suggests to use the model Xi = Ax i + Bi (1+k i )u i (55) Yi ., Ci%i • I kil:=: k i for the resulti~ plant of the i-th controller. where K is an upper bound for the model uncert!ity. Therfore. completely decentralized control design is performed in this way. see Bakule and Lunze (1ge8). Lunze (1985). DESIGN OF SYMMETRIC COMPOSITE SYSTEMS Supposing that the global system is composed of identical subsystems which are symmetrically interconnected. considerable simplifications of the design problem can be reached. Consider N subsystems each of which is described in the form (56) Yi~ CX i • zi= Czx i • where s • z are interconnection inputs. interco~ection outputs. respectively. The interconnection is described in a compactform by the equation s=Lz. where L=(L ij ). Lii=Ld • Lij=Lq for i~j. The aim is to design decentralized controller (29). The composite closed-loop system has the form (57)
•.(~; ~t"-1(::
Consider the plant X
Ax+Bu,
(51)
Cx under decentralized PI-control y
y=
(CC eT)' x • -co c
""'0
,..,
N
NO
(
)--'
where As.diag(As ••••• As ). Bii= N-' B. ";C.-...J t:!O.............,..,C Bij=-B for ifj. ~ .(B ••••• B). C =diag(C, "" ""0"'" : •• ~C). C =(C ••••• C). As=A+E~Ld:Lq~Cz' Ao =A+E(Ld +(N-1)L q )C z ' where A. B. Care the matrices of the closed the system (56).(29).Therefore. the reduced order system c~ '-' be"pon.lltI'J!cted only from the matrices As' Ao' B. C and it is derived that the command response of the system (57) can be represented by th~s reduced order model. This reduction offers ~e possibility for a completely decentralized control design. see Bakule and Lunze (1988). Lunze (1987). A.I
rv
a=(a" •••• ~)~a={a: a i e:fo.115 .(53) The scalar a i corresponds to the operation (a =1) or failure (a =0) of the i-th controlier. The decentralized controller can ensure the closed-loop stability only if the subsystems have certain autonomy. The design is performed for the given subsystem and the other controllers are included to act on this subsystem as model uncertainty. The plant (51) has then the following form for the i-th subsystem (54) Yi'" Cix , where Bi and Ci are the i-th hyper column and hyper row in B and C belonging to the 1-th input and output, respectively. The deviation of the behaviour of the model (54) depends on the controller parameters and therefore it cannot be given in advance. but in the static behaviour it depends only on the control stations which are in operation. The static reinforcement
AJ.-..J
CONCLUSION Some recent results and extensions for the decentralized control design are surveyed. The extension ccncerns the results for serially connected systems. i.e. complete decentralization. multioontroller configurations and simultaneous stabilization. se~u~ntial design. robust PI-control and symmetric composite systems. An emphasis is laid on the reduction of the complexity of the process. the information structure constraints and the uncertainties of the global system behaviour.
Dece ntrali zed Contro l Design of Large-Scale Syste ms
REFERENCES Bailey, F.N. (1978). Information problems in decentralized control systems. 18-th Asilomar Conference, Pacific Grove, 385-389. Bakule, L. (1978). Two-level control generation of multiarea electric energy systems. IMACS S~poSium on Simulation of Control ~stems.North-Holland, Amsterdam, 161-16 • Bakule, L. (1979). On decentralized large scale dynamic system control. Third Formator Symposium. Academia, Prague, 205-218.
Bakule, L. (1985). On the overlapping decomposition of dynamic systems. gYSi tem Analysis and Simulation. Aka em e -verlag, Berlin, Band 2771, 263-266. Bakule, L., and J. Lunze (1985). Completely decentralized design of decentralized centrollers for serially interconnected systems. 30. IWK TH Ilmenau, H.1, Ilmenau, 15-18. Bakule,L., and J. Lunze (1986). Sequential design of robust decentralized controllers for serially interconnected systems. Problems Control Inform. Theo~, Vol.15, No.2, 167-177. Bakuie, ~, and J. Lunze (1986a). RechnergestUtzter Entwurf von Regelunssystemen unter Verwendung der LQ-Regelung. msr, No.7, 292-298. Bakule, L. (~7). Robust control for uncertain large sparse systems. Wiss. Z. der TH Leipzig (in print). ----Bakule, L., and J. Lunze (1988). Decentralized design of feedback control for large-scale systems. Kybernetika, No.3-6, (in print). Chow, J.H., and P.V. Kokotovic (1976a). A decomposition of near-optimum regulators for systems with slow and fast modes. IEEE Trans. Automat.Control, Vol. AC-21, No.5. 701-(05. Chow~., and P.V. Kokotovic (1976b). Eigenvalue placement in two-time- . - scale systems. Proc. IFAC s~miosium tar,e Scale Systems Theory anpplicat ons,Odine, 321-326. Cohen, G•. and A. Benveniste (1974). Decentralization without coordination. Ricerche di Automatica, Vol.5, No.1, le-:!~.
Davison, E.J. (1 978). Decentralized robust control of unknown systems using tuning regulators. IEEE Trans. Automat. Control Vol.AC-23,No.2, 276-289. Findeisen, w. (1 g82). Decentralized and hierarchical control under consistency or disagreement of interests. Automatica,Vol.18, No.6, 647-664. Himmelblau, D. (1973). Decomrosition of larfe scale problems. Nor h-Holiand, AmS eraaJi. Hodzic, M. and D.D. Siljak (1985). Estimation and control of large sparse systems. Automatica, Vol.21, No.3, 277-292. Hodzic, M. and D.D. Siljak (1986). Decentralized estimation and control with overlapping information sets. IEEE Trane. Automat. Control, Vol. xc=)T, No.1, 83-86. Ikeda, M., and D.D. Siljak (1980). Overlapping decompositions, expansions and contractions of dynamic systems. Lar~e Scale Systems, Vol.1, No.1,
29- B.
Ikeda, M. r D.D. Siljak, and D.E. White (1984J. An inclusion principle for
dynamic systems. IEEE Trans. Automat. Control, VOI.AcJ29, No. 3, 244-249. Jamshidi, M. (1983). Lar~ Scale Systems, MOdeliK! an control. North-Holland, steraam/New York. Khalil, H.K., and P.V. Kokotovic (1978). Control strategies for decision makers using different models of the same system. IEEE Trans. Automat. Control, vol. AC-23, 289-298. Khal~.K. (1984). Time scale decomposition of linear implicit singularly perturbed systems. IEEE Trans. Automat. Control, vol. XO-29, 1054-1056. Levine, w.S., and M. Athans (1966). On the optimal error regulation of a string of moving vehicles. IEEE Trans. Automat. Control, Vol. Ac-l1,lo.4, 355-361. Litz, L. (1983). Dezentrale Regelung. Oldenoourg-verlag,
Mtihcnen.
Locatelli, A., F. Romeo, F. Scattolini, and N. Sch~Qvoni (1983). A parameter optimization approach to the design of reliable robust "decentralized regulators. 3-rd IFAC S*mposium Large Scale Systems: ~eory and A~~iications. Warsaw, 291-2 • Locatelli, A., R. Scattolini, and N. Schiavoni (1986). On the design of reliable robust decentralized regulators for linear systems. Large Scale Systems Vol.1Q. 10.2, 95.-113. Lunze, J. ( i 980). Ubersicht uber die Verfahren zum Entwurf dezentraler Regler fUr lineare Sysfeme. msr , Vol.23, No.6, 315-322. Lunze, J. 0.982). Entwurf robust~r Mehrgr03enregler bei beschrankter Modellunsicherheit. ZKI Informationen, No.4,AdW der DDR, Biriin, 1-118. Lunze, J.(1982). The design of robust feedback controllers for partly unknown systems by optimal control procedures. Internat. J. Control, Vol. 36,10.4, 611-630. Lunze, J. (1983). Evaluation of the I/O-behaviour of large scale dynamical systems by means of a generalized majorization principle. Fourth Formator Symposium. Academia, Prague 325-341. Lunze, J. l1984). The design of robust feedback controllers in the time domain. Intemat. J. Control, Vol. 39, No.6,
1243-1260.
Lunze, J. (1985). Zwei Ans&tze fUr den dezentralen Entwurf dezentraker Regler unter Berttcksichtigung der Teilsystemverkopplungen. !!! ,Vol.28,
No.9, 386-390.
Lunze, J. (1987). Stability analysis of large scale systems composed of strongly coupled similar subsystems. 10-th IFAC Congress ,Vol.6~ich.
14 1
142
L. Bakule
Minto, K.D., and M. Vidyasagar (1986). A state-space approach to simultaneous stabilization. Control-Theo~ and Advanced Technology, vol.2, 0.1, 39-64. Michel, A.N., and R.K. Miller (1977). Qualitative Analysis of Large Scale DYfUDhid sygCeMU. AcademIc PresY, BOW Ybrt. OZgUner, U., and W.R. Perkins (1978). Optimal control of multilevel large-scale systems. Internat. J. Control, Vol.28, No.6, 96't-980. Pichai, V., M.E. Sezer, and D.D. Siljak (1983). A graph-theoretical algorithm for hierarchical decomposition of dynamic systems with application to estimation and control. IEEE Trans. Systems Man Cybernet., VCI.SMC-l), NO.2, 19't-207. Sandell, Jr., N.R., P. Varaiya, M. Athans, and M.G. Safonov (1978). Survey of decentralized control methods for large scale systems. IEEE Trans. Automat. Control, Vol. AC-2), No.2, 108-128. Sezer, M.E., and D.D. Siljak (1981). On structural decomposition and stabilization of large-scale control systems. IEEE Trans. Automat. Control,Vol.AC-26, No.2, 439-444. Siljak, D.D. (1978). Large Scale DynamiC Systems: Stability and Structure. NOrth-Holland, New York. Siljak, D.D. (1980). Reliabie control using multiple control systems.Internat. J. Control, Vol.31, No.2, 303-329. Siljak, D.D. (1983). Complex dynamic systems: dimensionality, structure, and uncertainty. Large Scale Systems , Vol. 4, No.4,2'f9-294. S1ogh, M.G., and A. Titli (1978). Systems: Decomposition, ~timization and Con trol, pergamon ess, oxford. Sing~.G. (1981). Decentralized Control. North-Holland, Amsterdam. Tokumaru, H., N. Adachi, and T. Amemiya (1975). Macroscopic stability of interconnected systems. Proc. IFAC 6th World Congress, Boston. Vidyasagar, M., and N. Viswanadham (1985). Reliable stabilization using a multi-controller configuration. Automatice, Vol. 21, No.5, 599-602. Vidyasagar, M., and N. Viswanadham (1982). Algebraic design technique for reliable stabilization. IEEE Trans. Automat. Control, Vol. AC-27, No.5, 1085-1095. Voronov, A.A. (1985). Vvedenie v dinamiku slozhnych upravljajemych sistem. SaUka, Moscow.
MET HO DS FO R SYNTH ES IS OF CONTRO L SYSTEMS WI T H DIST RI BUTED INTE LLI GENCE
DECENTRALIZED CONTROL DESIGN OF LARGE-SCALE SYSTEMS L. Bakule I nstitute of Information Them) and Automation, Czechoslavak Academy of Sciences, 182 08 Prague, Czechoslovakia
Abstract. Some recent and new results on decentralized control design for large soale dynamic systems are surv'e yed. The decomposition of systems and the decentralization of the design task are characterized. The results concern the methods for the design of serially interconnected systems including complete decentralization, multi-controller configurations and simultaneous stabilization, sequential design, aggregation- and overlapping-decomposition design techniques, singularly perturbed systems, decentralized PI-control with integrity property and symmetric composite systems. The methods are presented with an emphasis on the robustness against model errors which are giv'e n by upper bounds. The design task comprises also the requirement on the I!O-behav'iour of the closed-loop system. Keywords. Large-scale systems; linear systems; state-space methods; decentralized control design; uncertainty; decomposition. robustness; system theory. gregation.
INTRODUCTION
Information structure eonstraints. Generally, no decision unit knows the complete system. It means that there is only a limited knowledge on the properties and control of some parts of a system.
The decentralization of the design process means that the whole de?ign problem is divided into weakly coupled subproblems that are solved independently. It differs from decentralized control systems, where the design process can be performed in centralized or decentralized way. Though the properties of decentralized control systems are intensively investigated, the decentralized control design problem is considered as a by-product problem which is connected with decentralized control systems.
Therefore, the analysis and synthesis cannot be performed in a single step as it is done by small scale systems. Suppose that a "large scale" system is simply a system which cannot be taokled by "standard" methods. To solve effectively large scale system control problems, several specific m,-', thodologies have been elaborated. They belong mainly to the following concepts:
Recent results (Bakule and Lunze, 1988. Findeisen, 1982; Jamshidi, 1983; Litz, 1983; Siljak, 1983; Voronov', 1985) indicate that the complexity of large scale control systems leads to severe difficulties that are encountered in the analysis, design and implementation of large scale system control strategies. These diffioulties arise mainly from the following reasons:
Decomposition. To reduce the time or memory complexity of the computations when analyzing and designing feedback systems, the global problem is decomposed into lower order subproblems which are solved separately and combining their solutions the global problem is solved. Notice that some kind of coordination technique can be inoluded when using this control design approach.
Dimensionality. The system has a high order. It simPIy means in praotice that the time of computations, which is not proportional to the system order but to a power of order greater than one, for higher order systems is unacceptable when using standard approaches.
Decentralization. The global problem is diVided into completely decoupled subproblem aims. Several different control sta-· tions have to solv'e problems which substitute the global problem. The quality of the solution is reduced because the subproblems cannot be generally completely separated from each other. No coordination is included here. Decentralization does not permit information exchange among the controllers. Decentralization occurs in connection with the control law, the design
uncertain~.The
overall system cannot be descri d by an exact model. Its model has primary uncertainties which result from an incomplete knowledge of the plant. Uncertainties are further caused by the information struoture constraints and approximations which can hav'e various reasons for iL H C.;J.6 -:, o"me kind of the model Simplification or ag-
133
134
L. Bakule
method, and with the implementation. These different types of decentralization can be combined. The majority of the results refer to the decentralization of the control law and extend the design principles of mu1tivariab1e feedback control law to structurally constraint feedbacks. Robustness. Uncertainties can be taken into consideration by using approximate model and the plant. Robustness is one of the -main issues in large scale system theory because uncertainty is a generic property by the large soa1e system modelling. It is closely related to the approximation, where the system equations are reduced to obtain reduoed order models. The main aim of the paper is to use the structural properties of the plant for the division of the overall problem into "weakly coupled" problems at the subsystem level. The different known methods are surveyed and recent and some new results are presented. THE DESIGN PROBLEM
Suppose that the servocompensator, i.e. the dynamic part of controller, is used. It means that the requirement (II) oan be omitted in further considerations. DECOMPOSITION OF SYSTEMS The goal of decomposition is the reduction of computational complexity by the solution of given analysis and synthesis problems. The submodels must be "weakly" coupled is some sense. The concept "weakly ooup1ed" is meant in differeht ways. Accordingly, different decomposition prinoiples have been established. Consider the deoomposition of the global system into a hierarchy of subsystems, into disjoint or overlapping subsystems and into subsystems with separated time scales. These concepts form the basis for the decomposition and decentralization of the design problem.
The design problem can be formulated as follows: C-onsider the overel1 system consisting of N subsystem which are described by the state space model xi- Aixi+Biui+Eisi' xi(O). x io ' Yi- Cixi+Diui+Fisi' i.1, •••
(1)
,N,
n m msi Pzi where XE:R i ,u i E R i ,sie: R ,z£R are vectors of the subsystem states, control inputs, interconnection inputs, oontro1 outputs, and interconnection outputs, respectively. Find a decentralized controller xri= Arixri+Briui+Erivi' u i - K1ixri+K2iYi+K3ivi'
(2) i=1, ••• ,N,
where x ,v denotes vector of the i-th contro11~r states,command inputs, respectively. Ai~ •••• K6i are constant matrices in Eqs. ,1),(27. The closed-loop system (1),(2) satisfies the following requirements: (I) The closed-loop system (1),(2) is asymptotically stable. (II) Asymptotic regulation occurs in the olosed-loop system, that is Yi (t)-v (t)--O i
as t __ 00,
where the oommand inputs v (t) and the disturbances Pi(t) ent~ring the system at arbitrary points belong to a class of admissible external signals. (Ill) The I/O-behaviour of the olosed-loop system is well-suited acoording to given reqUirements, for instance, on the step responses. (IV) The design requirements (I)-(III) are satisfied in spite of model uncertainties.
Hierarchy of Subsystems The difficulties in the analysis synthesis of interoonnected systems are caused mainly by the complete interdependenoe of the subsystems. Essential simplification oan be obtained if some subsystems have only a one-directional effeot on some others. In such situation the interconnection matrix is a lower triangular or can be made so by reordering the subsystems, see e.g. Bakule and Lunze (1985 - 1986), Bakule (1987), Hodzic and Siljak (1965),Pichai et al. (1983),Sezer and Siljak (1981). The strong coupling between two subsystems i,j is m~ ant in the digraph-theoretio sense. Defining the set of vertices by one-tc~one oorrespondence with the states and the set of edges, where an edge from the state k to the state 1 is defined if the corresponding element a kI of the matrix A is nonzero. The subsystems i and j are strongly coupled if there exist a path from the vertex i to vertex j and a path from vertex j to vertex i~ Using t bis digraph approach the subsystems with only one-directional interaotions can be detected. ~nd
Disjoint Decomposition The ooncept "weak" and ;'strong" coupling defined in the last subchapter refers to the presence of interactions among all parts of the system, whereas in this subchapter and the following ones this concept regards to the magnitude of present interaction lines. The motivation is based on the fact that the interactions are weak, i.e. they do not strongly influence the system perfor-
Dece ntralized Control Desig n o f Large-Sca le Systems
mance. Therefore, if the analysis or design is performed on the local independent subsystems, the difference between the global system and local subsystems is small. This difference is expressed for instance by linear-quadratic systems by the suboptimality index, i.e. its influence on the performance criteria values. Disjoint decomposition means that the overall system is divided into subsystems in the form xi= Aixi+Biui+~AijXj' Yi-
Cixi+Diu i , '''i
The isolated
(3)
i-1 •••• ,N.
transormation (8)
The systems S,Se satisfying Eqs. (7), (8) are ca~led a contraction, expansion, respectively. The properties of expansion-inclusion relations, or a systematic way for generating expansions for a given system are intensively investigated"see for instance Bakule (1985), Hodzlc and Siljak (1986), Ikeda and ~iljak (1980, 1984), Siljak (1980) •
subsystem has the form
Time-Scale Decomposition
xi- Aixi+Biu i ,
(4)
Yi- Cixi+Diui • This type of decomposition corresponds to numerous practical design problems, for instance in power systems, water managment and ecological systems, see e.g. Bakule (1978. 1979), Himmelblau (1973), Lunze (1980), Michel and Miller (1977), Sandell et al. (1978) ~iljak (1978), Singh and Titli (1978~, Singh (1981). Overlapping Decomposition Disjoint decomposi t ion means that the whole system is divided into disjoint parts. Such a decomposition may not be successful in case of large-magnitude interactions. This motivates the overlapping decomposition, where the resulting subsystems have joint parts-interconnections. These subsystems may be weakly coupled although disjoint subsystems are not. The overlapping property leads further to the effect of more reliable control of overlapped parts. A systematic way to overlapping decomposition begins with an expansion of the given system. The expanded system is decomposed into disjoint subsystems in the same way as described above. The question whether the original and e7.panded systems have the same trajectory is answered by the inclusion principle. Consider a pair of systems S, Se S:
135
x- Ax+Bu, y= Cx,
(6) y= Cexe' n where X€Rn , Xe€ Re, n~ne. Denoting x(t,x o )' xe(t,x eo ) the unique solutions of Eqs. (5),(6) for the initial time t-O and u(t)=O then a system S includes a system S if ther~ exists aneord~red pair of matrices (T,T ) such that TT -I for any x o€ X and it holds X(t,Xo)=TXe(t,T+Xeo ), (7) y[x(t)] = y[xe(t)] for all t, where T+ is a generalized inverse of T. The states x,x e are related by a linear
So far, the decomposit i on methods were based on the small-magnitude interconnection property. The following method is motivated by the consideration that the overall system consists of a fast and slow subsystem. Then the fast subsystem will arrive at its finel state before the slow subsystem had begun the essential part of its motions. The fast subsystem behaves as a static one from the point of view of the slow subsystem, and the slow seems to be quiescent in the time horizon of the fast subsystem. Their connection is weak. To formalize the time-scale decomposition consider the system in the form x- A11 x+A12 z+B 1u, x(O).x o ' ez= A21x+A22z+B2u, z(O)=a o '
(9)
y= C1x+C 2z, where e, is a small parameter E:>O. Using e. -0, the fast 8ll.bsystem is considered as infinitely fast and it is approximated by an algebraic equation. The degenerated system has the form
i-
A11x+A12z+B1u,
o.
A21x+A22z+B2u,
y-
z.
(10)
C1x+C 2 supossing A22 to be nonsingular we obtain A -1 A A- 1B z--A 22A21 x - 22 2u and the system (9) has the form ;..
"A
J\
X= Ax+Bu,
(11 )
(12)
y=Cx+Du, /\.
"""
1\
where 1
A
A-A11-A12A22A21,
(13)
1
A
B-B1-.A12A22B2' A
1
A
1
C-C1-C2A22A21' D--C 2A22 B2 • The relations of an original system and its approximation (12) has been studied mainly from the point of view of stability prop-
L. Ba kule
136
erties and the generalization of the two_ -time scale systems to multi-time scale systems, see for instance Chow and Kokotovic (1976), Jamshidi (1983), Kbalil and Kokotovic (1978), Voronov (1983).
Consider a plant Pi:
Xi· Aixi+Biui+FiPi_1' Yi- Cixi+Diui+HiPi_1'
Pi" Cpixi+Dpiui+HpiPi_1' P0 .. 0, i.1 ••••• N.
DECENTRALIZATION OF THE DESIGN: PROBLEM Decentralization concerns the information structure related to the solution of a decision problem. The decision process has two phase s: 1) Design phase, when an appropriate control law is determined on the basis of a-priori information about the model and control aim of the plant to satisfy the design requirements (I)-(IV). 2) Operating phase, when controls u are determined on the basis of the a!posteriori information about the actual state and command signal. Since the controller (2) represents local feedbacks of y towards u no crodd-coupling between ,iand YA' i.j, are produced. The decision process aas a decentralized information structure, it means tbat each control station receives only the local information about the output Yi and the command signal v • The existenee of decentralized fixed modes and the studies of their properties illustrate that the decentralization in the operating phase brings new problems. There are several motivating reasons to extend the decentralization also into the design phase: - Change of the system structure during operation 90 that certain autonomy of control stations is necessary. - Considering the overall system decomposed into weakly coupled subsystems, the isolated subsystems can be designed in the centralized way. This procedure simplifies the design process. - In overall systems with subsystems which are widely distributed in space or representing different authorities which are responsible for the performance of the subsystems. Therefore, decentralization of the design process means that the control stations are designed independently by means of different plant models and design speoifications. Several concepts of decentralization of the global design problem are described. The main difficulties arise here from the non-classical information structure of the decentralized design process, see for instance Bailey (1978), Bakule and Lunze (1988). DESIGN FOR SERIALLY INTERCONNECTED SYSTEMS Serial connection structure enables completely decentralize the design of decentralized controllers because of the band structure of interconnection matrix. Different controllers are designed independently. Such a decomposition enables to deal effectively with dynamical requirements on the closed-loop system. Serially connected systems are considered as a specific type of hierarchically connected systems. com¥lete Decentralization ot he besign
(14)
Suppose a controller in the form (15 ) Ci : u i • KiYi' Complete decentralization means that the decompOSition of dynamieal requirements and the inclusion of the model uncertainti es can be considered only at the subsystem level. Therefore. suppose a model of the plant Pi in the form •
A
'"
A
A-
A
A
A
A
Mi: ~i~Aixi+Biui+Eisi+FiPi_1' (16)
Yi-Cixi+Diui+GiSi+HiPi_1' zi=CZixi+Dziui+GZisi+HziPi_1' Pi=Cpixi+Dpiui+Gpisi+HpiPi_1' and ME i :
Isil = Vi*lzil+roi(xio,t),(17)
where Si' zi' Vi' r oi are vectors of interconnection input, interconnection output to the error model (17), impulse response matrix, free motion of the model (17). Eq.(17) describes an upper bound of the model error, * is the convolution operation. Supposing the system (16) controllable-observable 'and the closed-loop in the form
xi~ Aixi+Eisi+FiPi_1' (18)
Mci :
Yi~' Cixi+Gisi+HiPi_1' zi"
CZixi+Gzisi+~ziPi_1'
Pi= Cpixi+Gpisi+HpiPi_1' and Eq.(17). This system is analysed using the following lemma. Lemma 1. (I) A sufficient condition for the s,ability of Eqs.(17)1(18) is: a) The isolated systems (17),~18) are stable. b) It holds 00 00 ~ (19) A M[
Ivi dtJ
o
0
where ~M(') is a maximal eigenvalue of the matrix (.). (II) Tbe I/O-behaviour of Eqs. (17),(18) with si=O is characterized under the validity of (19) by the inequality
137
Dece ntrali zed Co ntrol Design of La rge-Scale Syste ms
(22) (20)
where
'"
~
Vicvi+Vi*IGzie
Ai t~
\ Ei ;l-Vi rJ
and y denotes the output of the model (18),ii.e. for s -0. Approximate model (16) is designediusing standard LQ or pole placement techniques, see Bakule and Lunze (1985,1988). Multi-Controller Design Consider the problem (14),(15), where complete decentralization is possible. The basic idea of decentralized multi-controller configuPation with passive redundancy is for the i-th subsystem, how to divide a controller C in more parallel controllers. SUpposin~ the simplest possibility, that is Ci·Ci1+Ci2' Cij =0.5C i , j.1,2. The motivation for using parallel controllers is the inclusion of the possibility of controller failures. Therefore, if one of controllers fails the second is still operating though it operates in non-optimal way. The design is performed in the follOwing way. First, design the controller C using Lemma 1. Then use instead of conttoller Ci the controller Cii=0.5C~ and test the stability using Lemma 1. If it is not satisfied modify C and repeat the whoie design procedure. T~e ideas of multicontroller configurations are elaborated tor instance in Bakule and Lunze (1988), ~iljak (1980). Simultaneous stabilization Consider a set of M serially connected chains, each with N subsystems. Denote Pik the i-th subsystem of the k-th chain. The chains are not connected among themselves. The motivation for such problem are for instance production complex systems with parallel structure of independently oprating processes with the same number of serially connected subsystems or a nonlinear system consisting of serially connected subsystems which operates at k different operating points with linearized models. An attempt is to use one controller stabilizing all models. Suppose a plant Pik in the form Pik: xik=Ai~ik+Bikuik+FikPi_1,k'
(21)
Y~k=Ci~ik+Dikuik+HikPi_1,k ' Pik·Cpi~ik+Dpikuik+HpikPi_1,k '
Pok=O, i=1, ••• ,N, k.'1., ••• ,M
where u ik ' Yik' Pik are mi -, r i -, qi- dim ensional vectors which do not depend on k and which have the same meaning as u i ' Yi' piin Eq.(14). Suppose a controller in the form
Denote analogously to Pi' Ci' Mi , ME i , Mci in Eqs. (14)-(18) Pik' Ci' Mik , ME ik , Mcik the plant (21), controller (22), model of the plant (21). error model of the plant (21) olosed-loop &ystemof the plant (21j, respectively. The problem can be formulated as follows. Problem. Consider a set Pi-lPik:k.1, ••• ••• ,MJ find a controller "(22) sa tisfying the requirements (I)-(IV) for all plants Pik' Solution. FormUlate a composite system
~i:
ii=!i!i+~i~i+!i~i-1 '
(23)
li-2i!i+Pi~i+~i~i-1 '
Then the controller has the form
where ~i.diag(Ki' •••• Ki). Constructe analogously to Eqs.(14)-(18) ~i,MEi' ~ci' where Miis the model of the plant (23). MEi the error model of the plant (23), ~ci the c ~ osed-loop system of the plant (23). respectively. Mci has the form !ci: ~iC~!i+ji~i+~iEi_1 •
Ii·fi!i+gi~i·~iPi-1
(25)
•
~i·§zi!i+gzi~i+~ZiPi-1
'
Pi·gPi!i+gpi~i+ffpiPi-1
•
and ME i :
I!il~ Yi*l~il+~oi(xio.t).
(26)
T T T T T )T where ~i·(si1.···.siM) , ~i=(zi1.···.ziM • T T T !oi.(roi1 ••••• roiM) • This system is analyzed using the following lemma. Lemma 2. (I) A sufficient condition tor the stability of Eqs. (25).(26) is: a) The isolated system (25),(26) is stable. b) It holds (27) 00
00
'"
~M[JyidtJ(lgiIJ(t)+I.9zi/it¥il o
0
)dtJ<1 •
138
L. Bakule
(11) The I!O-behaviour of Eqs. (25).(26) with si-O characterizes under the validity of-\27) the inequality
is desi8Ded as the solution of the problem 00
JGiQi'ji+uiRiui )dt -- min
o
(28)
subject to Eq.{JQ). The robustness is evaluated for the closed-loop system in the form
Xi-
where ~
Ait", ~ Yi-!i+!i*IQzi e- ~i*!i and 1i denotes the output of (25) with si=O-;
()2)
Ai%i+Bivi+Eisi+FiPi_1 •
()))
~
The ideas stimulating simultaneous stabi_ lization can be found for instance by Minto and Vidyasagar (1986). SEQUENTIAL DESIGN Sequential design is an appropriate procedure to solve effectively systems with one-directional interconnection structures under assumption that the dynamical requirements on the system cannot be decomposed into the individual subsystems level. It means that by the i~th contr61ler design. i.e. for the i-th subsystem. the previous 1 ••••• i-1 controllers have been designed and this part of closed-loop systems is considered to influence the i-th subsystem as a known gisturbance. This idea has been used by Ozguner and Perkins (1978) for linear state regulator problem. Using this motivation this design has been extended to the serially connected systems including uncertainties by Bakule and Lunze (1986).Its basic steps follows. Consider a plant P in the form of subsystems described by Eq.(14). Find a controller Ci given by Eq.(15) satisfying the reqUirements (I)-(IV) and consider step signals wi. Because of requirement (11) PI-controllers are used (29) xei e i , e i = Yi-"'i' u i "' K1iei+~ixei' where Ki.(K1i'~i) is a constant matrix with det(I-K1iDi)~0. A complete decomposition of dynam~cal requirements is impos~ sible here. Considering the i-th controller design. the extended system is supposed in the form
~1-(~)t: :) (::](::)"1+Gr1
• ()O)
Y1-(f:t(!1 :) (:: )'(:1)"1. i
Then the controller
1i- Ci%i+DiYi+Gisi+HiPi_1 • zi= Czixi+DziV1-t4Zisi+HziPi_1 ' Pi= Cpixi+DpiVi+Gpisi+HpiPi_1 ' and Eq.(17). This system is analyzed wi th inputs v ,p and outputs Yi' Pi applyi~ te~ 1 with the tolerance band for Pi in the form
-
'" Pi+PiO"(t),
~ Pi-
where band,
.~ii ~
(4)
is the middle of the tolerance is a maximum width of the band. DESIGN USING DISJOINT DECOMPOSITION
The motivation for this type of decomposition is that the interconnections between subsystems are weak. Therefore, its influence on the system performance is small. General methodology is based on the two steps: the individual subsystems design and the iuclusion of the interconnection effect on the global system performance. These steps are performed either on the original system or on a simplified auxiliary system from its properties the properties of original system are evakuated, The first case concerns mainly exact models, for which the aggrega tiQn-decomposi tion was developed by Siljak see Michel and Miller (1977), ~lljak (1978),Voronov (198). This method concerns mainly the stability of the global system. The suboptimality index gives only a rough information on the I/O-behaviour. An another approach has been developed for the stability tests. It uses comparison systems. A comparison system is supposed in the form .. 00 r(t) =
JV(t-2')w(~)d't" ,
(5)
-00
where', r, ware output. input vectors, respectively. V is the impulse response matrix. The comparison system (35) for the linear system yet) .. G!tu(t) , G(t)::CeAtB is given by Eq.{J5) with
(6)
V(t)~IG(t)l.
Suppose a system x = Ax+Bu , Y .. ex
(7)
Dece ntra li zed Co ntro l Design of La rge-Scale Syste ms
with a controller u i '" KiYi.
(8)
The closed-loop system (37), (38) can be decomposed into xi= Aixi+Eisi+FiVi '
then 1 , A are in the expansion-contraction rilation.
(40)
Specific application of overlapping decompos1tion offer serially connected structures of subsystems, where some overlapped sUbsystems-interconnections belong both to the previous and to the following sUbsystems. Such type of decomposition is imposed by the feedback control structure. These structures are expanded, where expansions are of the disjoint-type decomposition character. This problem can be solved using the complete decentralization approach.
and where L is an interconnection matrix, s, z are interconnection outputs, inputs, respectively.Comparison systems of the subsystems (39) are IYi(t)l- Vl1~luil+Vi2itlsil ,
(41)
Izi(t) ~ Vi3*luil+Vi4~lsil with Vi1~lcie Is(t)\'I!O
Ai t
Ei l
L / z(t)i
etc., and with ~IL1.
(42)
~e
overall system (37), (38) is I/O-stable If all comparison systems are I/O-stable and the matrix P = (I-diagV. L) which represents the influence ofl~terconnec tions is an M-matrix. Using the comparison systems the I/O-behaviour can be evaluated see , Bakule and Lunze (1988), Lunze (1983).' DESIGN USING OVERLAPPING DECOMPOSITION Ov"erlapping decomposition leads to a complete decentralization of the design tasks because the stability of the expansion im-' plies the stability of the original system. Therefore, the stability of the whole system can be proved by means of stability of the lower order subsystems of the expansion. Concerning decentralized control design u nportant are the conditions stating the expansion-contraction relations for feedback systems. Consider the system (5) with a controller u=Kx and the system (6) with a controller u=K x and denote the c10sed-loop systems e e
A=
A+BK ,
Ae ..
Ae +Be Ke •
(43)
S~ppose that the rela tion between AC and A. has the form
.... + M ,
'" Ae .. TAT
(44)
where T is given by Eq.(8), M is some complementary matrix satisfying the relation T+r.1 i T .. 0
foral11.1, ••• ,n . • e
where the complementary matrix F satisfies the conditions (47) FMiT .. 0, FM i - 1N .. 0 for all i = 1 , ••• 'Ile-
(9)
Yi= Cix i , zi= Czix i s .. Lz ,
139
(45)
This restrict the class of gain matrices. If the relation between K and ~is supposed in the form (46)
An another application of overlapping decompositions is for reliable controllers design. It means that the part of the system is controlled by controllers with parallel structures. The problem how to divide a given controller into parallel controllers depends on the design requirements. It means for instance that supposing a given controller with reliability an attempt is made to design a parallel contr oller with desired reliability or a parallel controller satisfying some des1red dynamical requirements when given number of controllers fails. The design of such controllers for systems with uncertainties can be performed usi~ Lemma 1 which is applied to 1ndindual operation-failure structure~, for instance 2-out-of-3 structure etc. Though some results are available, the decentralized control design using overlapping is a promising area for further research aotivity, see e.g. Ikeda and §iljak (1980; 1984), Ikeda t §iljak and White (1984), ~iljak (1980, • DESIGN OF SINGULARLY PERTURBED SYSTEMS Deoentralized oontrol design for s1operturbed systems means that the multi-time scale properties are detected, the subsystems with one time scale are separated and looal feedback are designed inoluding the evaluation of erros of such approximations. The basio oase is the case of the system with two-time-soales. A two-time-soale property has been first characterized explioit systems by Chow and Kokotovic (1976a,1976b).Reoently, there are available results on the properties of implioit systems by Khalil (1984). Implioit singularly perturbed systems means that the following form of the system is considered gu~arly
:i:
a
A(€.)x ,
(48)
where A(O) is singular and A(g).is analytio at g . 0 that is A(£) .. ~lA • Using appropriate state transform a"\wo!time-scale property is deteoted. Deoentralized design means for instanoe for the pole plaoement problem th6t for a system in the
140
L. Bakule
form x t = A"x,+A'2X2+B,U •
(49)
x 2 = A2,X,+A22X2+B2U • where the matrices A satisfy the two-time-scale propertyi~ests, that the feedback control u=Kx 1s separated into the slow in fast feedback matrices Ks and Kf resulting in the control
This approximation of control is an O(~) Approximation for the eigenvalues. Analogous results are available for the LQ-s1ngularly perturbed systems, where further the 2control approximation (50) results in O(~ ) approximation for the criteria values. The two-time-scale systems results have been further extended to the case of multi-time scale systems or multi-parameter two-time-scale systems for instance see Jamshidi (1983), Khalil and Koko tovic (1978). DESIGN OF ROBUST PI-CONTROL Decentralized robust PI-control is motivated by the inclusion of uncertainty of decentralized controllers which is generated by individual controllers fai*ures. The closed-loop stability includes 2 operation conditions for N controllers corresponding to all possibi11ttes or failures. The problem can be formulated as follows.
Ksii between u i and Yi can be determined. It is known in advance and it is independent on other operating controllers. It suggests to use the model Xi = Ax i + Bi (1+k i )u i (55) Yi ., Ci%i • I kil:=: k i for the resulti~ plant of the i-th controller. where K is an upper bound for the model uncert!ity. Therfore. completely decentralized control design is performed in this way. see Bakule and Lunze (1ge8). Lunze (1985). DESIGN OF SYMMETRIC COMPOSITE SYSTEMS Supposing that the global system is composed of identical subsystems which are symmetrically interconnected. considerable simplifications of the design problem can be reached. Consider N subsystems each of which is described in the form (56) Yi~ CX i • zi= Czx i • where s • z are interconnection inputs. interco~ection outputs. respectively. The interconnection is described in a compactform by the equation s=Lz. where L=(L ij ). Lii=Ld • Lij=Lq for i~j. The aim is to design decentralized controller (29). The composite closed-loop system has the form (57)
•.(~; ~t"-1(::
Consider the plant X
Ax+Bu,
(51)
Cx under decentralized PI-control y
y=
(CC eT)' x • -co c
""'0
,..,
N
NO
(
)--'
where As.diag(As ••••• As ). Bii= N-' B. ";C.-...J t:!O.............,..,C Bij=-B for ifj. ~ .(B ••••• B). C =diag(C, "" ""0"'" : •• ~C). C =(C ••••• C). As=A+E~Ld:Lq~Cz' Ao =A+E(Ld +(N-1)L q )C z ' where A. B. Care the matrices of the closed the system (56).(29).Therefore. the reduced order system c~ '-' be"pon.lltI'J!cted only from the matrices As' Ao' B. C and it is derived that the command response of the system (57) can be represented by th~s reduced order model. This reduction offers ~e possibility for a completely decentralized control design. see Bakule and Lunze (1988). Lunze (1987). A.I
rv
a=(a" •••• ~)~a={a: a i e:fo.115 .(53) The scalar a i corresponds to the operation (a =1) or failure (a =0) of the i-th controlier. The decentralized controller can ensure the closed-loop stability only if the subsystems have certain autonomy. The design is performed for the given subsystem and the other controllers are included to act on this subsystem as model uncertainty. The plant (51) has then the following form for the i-th subsystem (54) Yi'" Cix , where Bi and Ci are the i-th hyper column and hyper row in B and C belonging to the 1-th input and output, respectively. The deviation of the behaviour of the model (54) depends on the controller parameters and therefore it cannot be given in advance. but in the static behaviour it depends only on the control stations which are in operation. The static reinforcement
AJ.-..J
CONCLUSION Some recent results and extensions for the decentralized control design are surveyed. The extension ccncerns the results for serially connected systems. i.e. complete decentralization. multioontroller configurations and simultaneous stabilization. se~u~ntial design. robust PI-control and symmetric composite systems. An emphasis is laid on the reduction of the complexity of the process. the information structure constraints and the uncertainties of the global system behaviour.
Dece ntrali zed Contro l Design of Large-Scale Syste ms
REFERENCES Bailey, F.N. (1978). Information problems in decentralized control systems. 18-th Asilomar Conference, Pacific Grove, 385-389. Bakule, L. (1978). Two-level control generation of multiarea electric energy systems. IMACS S~poSium on Simulation of Control ~stems.North-Holland, Amsterdam, 161-16 • Bakule, L. (1979). On decentralized large scale dynamic system control. Third Formator Symposium. Academia, Prague, 205-218.
Bakule, L. (1985). On the overlapping decomposition of dynamic systems. gYSi tem Analysis and Simulation. Aka em e -verlag, Berlin, Band 2771, 263-266. Bakule, L., and J. Lunze (1985). Completely decentralized design of decentralized centrollers for serially interconnected systems. 30. IWK TH Ilmenau, H.1, Ilmenau, 15-18. Bakule,L., and J. Lunze (1986). Sequential design of robust decentralized controllers for serially interconnected systems. Problems Control Inform. Theo~, Vol.15, No.2, 167-177. Bakuie, ~, and J. Lunze (1986a). RechnergestUtzter Entwurf von Regelunssystemen unter Verwendung der LQ-Regelung. msr, No.7, 292-298. Bakule, L. (~7). Robust control for uncertain large sparse systems. Wiss. Z. der TH Leipzig (in print). ----Bakule, L., and J. Lunze (1988). Decentralized design of feedback control for large-scale systems. Kybernetika, No.3-6, (in print). Chow, J.H., and P.V. Kokotovic (1976a). A decomposition of near-optimum regulators for systems with slow and fast modes. IEEE Trans. Automat.Control, Vol. AC-21, No.5. 701-(05. Chow~., and P.V. Kokotovic (1976b). Eigenvalue placement in two-time- . - scale systems. Proc. IFAC s~miosium tar,e Scale Systems Theory anpplicat ons,Odine, 321-326. Cohen, G•. and A. Benveniste (1974). Decentralization without coordination. Ricerche di Automatica, Vol.5, No.1, le-:!~.
Davison, E.J. (1 978). Decentralized robust control of unknown systems using tuning regulators. IEEE Trans. Automat. Control Vol.AC-23,No.2, 276-289. Findeisen, w. (1 g82). Decentralized and hierarchical control under consistency or disagreement of interests. Automatica,Vol.18, No.6, 647-664. Himmelblau, D. (1973). Decomrosition of larfe scale problems. Nor h-Holiand, AmS eraaJi. Hodzic, M. and D.D. Siljak (1985). Estimation and control of large sparse systems. Automatica, Vol.21, No.3, 277-292. Hodzic, M. and D.D. Siljak (1986). Decentralized estimation and control with overlapping information sets. IEEE Trane. Automat. Control, Vol. xc=)T, No.1, 83-86. Ikeda, M., and D.D. Siljak (1980). Overlapping decompositions, expansions and contractions of dynamic systems. Lar~e Scale Systems, Vol.1, No.1,
29- B.
Ikeda, M. r D.D. Siljak, and D.E. White (1984J. An inclusion principle for
dynamic systems. IEEE Trans. Automat. Control, VOI.AcJ29, No. 3, 244-249. Jamshidi, M. (1983). Lar~ Scale Systems, MOdeliK! an control. North-Holland, steraam/New York. Khalil, H.K., and P.V. Kokotovic (1978). Control strategies for decision makers using different models of the same system. IEEE Trans. Automat. Control, vol. AC-23, 289-298. Khal~.K. (1984). Time scale decomposition of linear implicit singularly perturbed systems. IEEE Trans. Automat. Control, vol. XO-29, 1054-1056. Levine, w.S., and M. Athans (1966). On the optimal error regulation of a string of moving vehicles. IEEE Trans. Automat. Control, Vol. Ac-l1,lo.4, 355-361. Litz, L. (1983). Dezentrale Regelung. Oldenoourg-verlag,
Mtihcnen.
Locatelli, A., F. Romeo, F. Scattolini, and N. Sch~Qvoni (1983). A parameter optimization approach to the design of reliable robust "decentralized regulators. 3-rd IFAC S*mposium Large Scale Systems: ~eory and A~~iications. Warsaw, 291-2 • Locatelli, A., R. Scattolini, and N. Schiavoni (1986). On the design of reliable robust decentralized regulators for linear systems. Large Scale Systems Vol.1Q. 10.2, 95.-113. Lunze, J. ( i 980). Ubersicht uber die Verfahren zum Entwurf dezentraler Regler fUr lineare Sysfeme. msr , Vol.23, No.6, 315-322. Lunze, J. 0.982). Entwurf robust~r Mehrgr03enregler bei beschrankter Modellunsicherheit. ZKI Informationen, No.4,AdW der DDR, Biriin, 1-118. Lunze, J.(1982). The design of robust feedback controllers for partly unknown systems by optimal control procedures. Internat. J. Control, Vol. 36,10.4, 611-630. Lunze, J. (1983). Evaluation of the I/O-behaviour of large scale dynamical systems by means of a generalized majorization principle. Fourth Formator Symposium. Academia, Prague 325-341. Lunze, J. l1984). The design of robust feedback controllers in the time domain. Intemat. J. Control, Vol. 39, No.6,
1243-1260.
Lunze, J. (1985). Zwei Ans&tze fUr den dezentralen Entwurf dezentraker Regler unter Berttcksichtigung der Teilsystemverkopplungen. !!! ,Vol.28,
No.9, 386-390.
Lunze, J. (1987). Stability analysis of large scale systems composed of strongly coupled similar subsystems. 10-th IFAC Congress ,Vol.6~ich.
14 1
142
L. Bakule
Minto, K.D., and M. Vidyasagar (1986). A state-space approach to simultaneous stabilization. Control-Theo~ and Advanced Technology, vol.2, 0.1, 39-64. Michel, A.N., and R.K. Miller (1977). Qualitative Analysis of Large Scale DYfUDhid sygCeMU. AcademIc PresY, BOW Ybrt. OZgUner, U., and W.R. Perkins (1978). Optimal control of multilevel large-scale systems. Internat. J. Control, Vol.28, No.6, 96't-980. Pichai, V., M.E. Sezer, and D.D. Siljak (1983). A graph-theoretical algorithm for hierarchical decomposition of dynamic systems with application to estimation and control. IEEE Trans. Systems Man Cybernet., VCI.SMC-l), NO.2, 19't-207. Sandell, Jr., N.R., P. Varaiya, M. Athans, and M.G. Safonov (1978). Survey of decentralized control methods for large scale systems. IEEE Trans. Automat. Control, Vol. AC-2), No.2, 108-128. Sezer, M.E., and D.D. Siljak (1981). On structural decomposition and stabilization of large-scale control systems. IEEE Trans. Automat. Control,Vol.AC-26, No.2, 439-444. Siljak, D.D. (1978). Large Scale DynamiC Systems: Stability and Structure. NOrth-Holland, New York. Siljak, D.D. (1980). Reliabie control using multiple control systems.Internat. J. Control, Vol.31, No.2, 303-329. Siljak, D.D. (1983). Complex dynamic systems: dimensionality, structure, and uncertainty. Large Scale Systems , Vol. 4, No.4,2'f9-294. S1ogh, M.G., and A. Titli (1978). Systems: Decomposition, ~timization and Con trol, pergamon ess, oxford. Sing~.G. (1981). Decentralized Control. North-Holland, Amsterdam. Tokumaru, H., N. Adachi, and T. Amemiya (1975). Macroscopic stability of interconnected systems. Proc. IFAC 6th World Congress, Boston. Vidyasagar, M., and N. Viswanadham (1985). Reliable stabilization using a multi-controller configuration. Automatice, Vol. 21, No.5, 599-602. Vidyasagar, M., and N. Viswanadham (1982). Algebraic design technique for reliable stabilization. IEEE Trans. Automat. Control, Vol. AC-27, No.5, 1085-1095. Voronov, A.A. (1985). Vvedenie v dinamiku slozhnych upravljajemych sistem. SaUka, Moscow.