IEEE TRANSACTIONS ON AUTOMATIC COhTROL, JUNE 1977

T. L. Montgomery, J. M. Leavitt J. M. Crawford, and T. L. Gartrell. Control of ambient SO, concentration by noncontinuous emission limitation, large coal-fired

Engineers, Feb. 1973. power plants, in P ~ o c . 102nd A n m . Meering Amer. I n i f . Mtn.. Mer.. and Petrol.

T. L. Montgomery and J. W. Frey. Tall stacks and mtermlttent control of SO, emissions-TVA experience and plans. in Proc. Amer. M m n g Congr.. Oct. 1974.

Research metbodologS. for ax pollution control systems, IBM Scientific Center, R. del Llano. L. F. Escuder, F. Gomez-Pallete. J. Mantero, B. Orfila. and L. Serna.

Madrid. Spain. Rep. SCR-01.73. June 1973. J. B. Koogler er a/.. A multlvanable model for atmospheric dispenlon predict- ions.- 1. Air Poll. Conrr. Ass., vol. 17, pp. 21 1-214, Apr. 1967. 0. B. Turner, A diffusion model for an urban area.. 1. Appl. .Mereorol.. vol. 3. pp. 83-91. Feb. 1914. J. F. Clarke. A simple diffusion model for calculating point concentrations from multiple sources, J . Air Poll. Conrr. Ass., vol. 14. pp. 347-352. Sept. 1964.

hours in duration. Air Resources Laboratories Tech. Note. 39-ARL-3. Washmg- D. Slade. Estlmates of dispersion from pollutant releases of a few second to elght

ton, DC, 1966.

raphy, Armos. Entiron.. vol. 2. 1968. M. Hmo. Computer expenments on smoke dlffusion over a compllcated topolog-

L. J. Shieh. P. K. Halpern. 8. A. Clemens. H. H. Wan& and F. F. Abraham, Air quality diffusion model: Apphcatlon to New York Cltl. IBM J . Res. Dewlop.. vol.

0. G. Sutton, Theory of eddy dlffunon m the atmosphere. in Proc. Roy. SOC.. ser. 16, pp. 162-170, Mar. 1972.

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meteorology, J . Mefeorol.. vol. 9, pp. 252-259. Aug. 1952. F. N. Frenkiel, Appllcation of statistical theory of turbulent diffusion to micro-

C. C. Shir. Numerical investigation of the atmosphenc dispersion of stack effluents. IBM J . Res. Decelop.. vol. 16, pp. 171-179. Mar. 1972.

application to the study of SO, dlstributions In the St. LOUIS metropolltan area. J . C. C. Shir and L. J. Shteh, A generalizeJ urban a n pollutlon model and its

Appl. Mereorol., vol. 13. no. 2. pp. 185-201. Mar. 1974. H. G. Fortak. Mathematical modelling of urban pollut~on. in Proc. Inr. Symp.

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atmospheres: SO, in greater New York. In Proc. S v m p . Mulri-Source lrban Diff . L. J. Shieh. B. Davldson. and J. P. Friend. A model of dlffusmn In urban

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L,-Stability of Large-Scale Dynamical Systems: Criteria via Positive Operator Theory

MALUR K. SUNDARESHAN AND M. VIDYASAGAR, MEMBER, IEEE

Abstract-A new approach to the &-stability analysis of large-scale dpamical systems, described as interconnections of several subsystems, is presented. By conducting this study witbin the framework of positive operator theory, stability criteria, which impose explicit and distinct condi- tions on the subsystems and the interconnections, are obtained. Some strong points of the main result of the paper are: 1) it is simpler to apply than the existing results and 2) it specializes to the classical positivity theorem for the case of single-loop negative feedback systems. Modifica- tions of this result which permit the use of arbitrary multipliers are also outlined.

I. IKTRODUCXION

Decomposition methods in the stability analysis of large-scale systems are becoming increasingly popular in the recent times and criteria for both asymptotic stability (in a Lyapunov setting) and input-output stability (in a functional analysis framework) have been obtained by treating the original system as an interconnection of several simpler

Manuscript received July 31, 1975; revised April 5, 1976. Paper recommended by N. R. Sandell, Jr.. Past Chairman of the IEEE S C S Large Scale Systems, Differential Games Committee. This work was supported by the National Research Council of Canada under Grant A-7790.

Minnesota, Minneapolis, M N 55455. M. K. Sundareshan is with the Department of Electrical Engineering, University of

Montreal. P. Q., Canada. M. Vidyasagar is with the Department of Electrical Engineering, Concordia University,

subsystems. Some representative results in this area are those of Bailey [I] , Araki and Kondo [2], MichFl and his co-workers [3H5], Estrada [6], Willems [7], and Grujic and Siljak [8]. An interesting feature that is common to all these results is that the conditions for stability of the original large system are expressed as certain constraints on a test matrix T, which is constructed from boun-ds on the individual subsys- tems and the interconnections. Grujic and Siljak [8] call this an aggre- gation matrix, generated from norm bounds on the subsystem Lyapunov functions and the interconnection functions. Estrada 161 re- quires the construction of certain Lyapunov-type functions for the subsystems and generates the matrix T using their coefficients. Michel et a[.. [3H5] use the gains and sector bounds of the subsystems in develop- ing this matrix. Once the test matrix T is constructed, all these results proceed to demonstrate stability of the overall system by imposing suitable constraints on the eigenvalues (or the spectral radius) of T. As a consequence. it is often difficult to interpret them as corresponding conditions on the subsystems and the interconnections explicitly, except in special simple cases.

In contrast to the above, we shall present a new result in this paper for the input-output stability of large interconnected systems, which involve explicit positivity conditions on the subsystem operators and constraints on the interconnections. Our method of analysis draws motivation from principles of energy-bounding and seeks to answer the following question-How should an interconnection of positive (or passive) sub- systems be constrained to guarantee input-output stability of the overall system? A noteworthy feature of our main result is that for the special case of the single-loop negative feedback system, it reduces to the well-known positivity theorem due to Zames [9] and Sandberg [IO] and is hence, a generalization of the available results of this type for the single-loop systems. The philosophy behind the derivation of this result and also our desire to keep the constraints on the subsystems and the interconnections independent of each other yields a beneficial aspect that the present result can be more easily used in the stability design of large systems. Some modifications of this result which allow the use of arbitrary multipliers are also ou thed ; these permit a large body of available literature developed in the so called multiplier methods to be camed over to the stability analysis of interconnected systems in a straightforward manner.

11. PROBLEM FORMULATION

A . Notations and Definitions

Let Q and denote, respectively, the real and the nonnegative real numbers. It is assumed that the reader has a certain degree of f d a r i t y with the notions of the &-space, the extended space L,, and the properties of operators defined over these spaces. While [ 1 I] is suggested for details, a brief recapitulation of some important concepts is given below.

An operator G in L2(L,) is defined as a single-valued mapping of &(&) into itself. G is said to be causal if P,G= P,GP, V T E Q , P , being a projection on L, defined by P , x ( t ) = x ( t ) VtE[O, TI and zero otherwise. If G is a causal operator in be, then G is said to be positive (e) [strongly positive (e)] if the inequality ( P , x ( . ) , P,Gx(.)) > 6 ( P T x ( . ) , P , x ( . ) ) V x ( . ) E L z e , V T E S 1 holds with 6 > O [6>0]. Fi- nally, an operator G in he is said to have finite gain if

P,X(.)#O

where 1 1 . 1 1 indicates the L2-norm.

B. System Description

The system under consideration is a large-scale dynamical system 9 that may be described as an interconnection of n subsystems 9, , 9,, . . . ,9,

an exhaustive listing of all the published literature on the stability analysis of largescale It must be emphasized that the references mentioned here do by no means constitute

systems. which incidentally is too numerous to be mentioned here.

SHORT PAPERS

as

397

Now consider the m equations

In (l), Gi is an operator in describing the isolated subsystem 5; and h, 9. is a scala9 that specifies the interaction from the j t h subsystem 4, to the ith subsystem 4,. The n2 scalars h, may be regarded as elements of the nX n interconnection matrix H that specifies the interconnection pattern within the overall system 5. Further, ui(.) is the input to 5,. e;( .), its error signal andy,(.), its output. We shall assume that ui(.),ei(.),yi(.)ELze Vi=1,2;..,n and the operator equations (1) satisfy suitable well-posedness conditions to guarantee the existence of ei(.),yi(.)ELze Vi=1,2;-.,n, for every input set #,(.)EL,, i= 1,2;..,n [12], [13].

C. The Problem

The problem of interest is to determine conditions for the L,-stability of system 4 , i.e., to establish conditions on Gi and H that are sufficient to ensure

u,(.)EL, Vi=l,Z;..,n*e,(.), y j ( . ) L 2 Vi=1,2;..,n. (2)

111. AN L 2 - S ~ ~ ~ ~ ~ ~ CRITEFSOK

n

= u j ( . ) + hgj( . )+ 2 hg j ( . ) , i=1,2,- . . ,m. (6) m

j= 1 j = m + l

Since H can be partitioned as in (4) and rank ( I f b ) = min (m, n - m} = (n - m), there exists a nonzero minor of Hb of order (n - m) and hence it is possible to obtain an expression fory,(.), i=m+ l,m+2,... ,n in terms of ui(.), e;(.), andy,(.), i=1,2;-.,rn,as

vi( . )= x { ~ U ~ ( . ) + p U u , ( . ) + p ~ y j ( . ) } , i=m+l ,m+2; . . , n (7) where a,., u u p . , and pij are finite constants.

m

j - I

Now, from ( 5 ) m

For any specified integer m such that n/2 < m < n, let -x denote the where we have used (1) and (7) in the simplification. class of interconnection matrices H that satisfy the following properties: Since Y,(Gi) < co Vi= 1,2; . . ,m and ui(.) E & V i = 1,2; . . ,n, (8) can

be rewritten as

or equivalently - ( H + H T, is positive semidefinite, and 2) if H is partitioned in the form

the submatrix Hb is of full rank. We then have the following result. Theorem 1: If all subsystems G,, i= 1,2;..,n, are positive (e) and

there exists an integer m such that n/2 < m < n for which the subsystems Gi, i = 1,2,. . . ,m, are strongly positive ( e ) and have finite gain, then system 4 is &-stable for all interconnection matrices H E X.

Proof: For some T E %+ consider the sum of inner-products

m

2 sjIIpTei(-)112< C killPrei(.)ll+K, V T E ~ + (9)

where K and k,, i= 1,2,. . . ,m, are finite positive constants. It then follows simply that IIPTei(.)ll C ai(PTe,(.),Prq(.)) ( 5 )

require H + H T = O which satisfies (3) with equality and h s particular i = 1

consisting of n subsysiem operators interconnecte; in various ways through th> outputs. It is, however, true that the two subsystems in [9] and [IO] admit vector-valued inputs and outputs, which may be interpreted as a particular form of multiloop configuration

h are scalars. However, an extension of this to the case when hy are arbitrary operators true even if all our subsystems are defined on such vector-valued function spaces (in this 2For simplicity in the exposition of the basic result in this paper, we shall assume that (considered, for instance, in [6] ) ; but it is simple to observe that the present result holds

2 L, is straightforward. context, see also remark 3 in this section).

398 EEE TRANSACTIONS ON AUTOMATIC CONTROL, JUNE 1977

case admits a heuristic explanation, following Willems [7], that the interconnections do not contribute to any dissipation in the system. The rank condition (4) requires that it is possible to separate the subsystems into two groups, group A with m subsystems and group B with the rest of ( n - m ) subsystems, such that there exists sufficient interaction from DOUP B to group A . It is to be noted that only the subsystems in group A are required to be strongly positive and have finite gain by the theorem.

3) An &-stability criterion for multiloop systems has been derived earlier by Estrada [6] following a different method of analysis based on the construction of Lyapunov-type energy functions. However, it might be noted that the multiloop configuration treated in [6] is only a special form of the system (I) presently considered. Despite this generality, the present result compares favorably with the stability criteria of [6] in that 1) it is easier to apply, and 2) it imposes more relaxed conditions on the subsystems. Observe for instance that for the particular case of the single-loop feedback system, i.e., n = 2, h , , = - h,, = - 1, the main result (Theorem 2) of [6] requires that both the subsystems be strongly positive (e ) , thus not specializing to the positivity theorem of [9], [IO].

4) In comparison with the results of other investigators, notably due to Michel and his co-workers [3]-[5], it may be noted that Theorem 1 enunciates simpler and more explicit conditions in terms of the positivity of subsystems. In contrast, the stability criteria in (41, [SI involve condi- tions on the gains and sectoricity of the subsystem operators and hence, are not easy to check except in very simple cases of linear convolution and memoryless nonlinear operators. The conditions on the interconnec- tions also turn out to be less restrictive in many cases; compare, for instance, with the results of [4] for the case that the subsystem operators are conic with sector (0, co). Since the constraints on the subsystems and interconnections are independent of each other, the present result can find very easy application in the stability design of large systems. Of greater importance is the fact that the present result is not of a weak coupling type in the spirit of [3H5] (and many of the others [I], [2], [6], [ 8 ] ) because couplings of arbitrarily large magnitudes can be tolerated so long as the interconnection matrix satisfies the appropriate negative definiteness conditions.

5 ) Theorem 1 can be interpreted as guaranteeing L,-stability of sys- tem 4 for a class of interconnection matrices if the subsystems satisfy appropriate positivity conditions. It is thus evident that the present result ensures &-stability of the system in the face of structural perturbations caused by the disconnection and reconnection of one or more subsys- tems from the rest of the system during operation, if these do not cause any violation of the interconnection constraints (3) and (4). Hence, Theorem I, in effect, proves a stronger concept of stability, connective L,-stability, that is an input-output analogue of the connective asymp- totic stability introduced by Siljak [14] in the Lyapunov framework. It must, however, be noted in comparison that the present method of analysis, which proceeds from the determination of explicit conditions on the interconnection pattern within the system, is simpler and intui- tively more appealing for the study of connective properties than some of the earlier attempts [8], [14], which seem to be unnecessarily com- plicated and overly restrictive.

6) Like all other stability criteria derived via positive operator theory, the scope of application of Theorem 1 can be vastly enlarged by the introduction of suitable multipliers. However, unlike the well-known simple way of implementation of this technique in the case of single-loop feedback systems, such modifications for the large interconnected system under consideration seem to be possible only under the imposition of more constraints on the interconnections. A representative result of this type will be given in the sequel. Let be a subclass of of interconnection matrices H which can be partitioned as in (4) such that Ha = 0 and Hd= 0. We can then state the following result.

Theorem 2: If there exists a causal linear operator M in L,, such that 1) M : L2+L,; 2) M - I exists as an operator in & and further M - I : L2+L2; and 3) for some m > n/2, the operator compositions MG,, i = l,2, ..., m are

all strongly positive (e) and have finite gain; and 4) the (n - m ) operator compositions G,M -, i = m + I, m + 2 , . . . , n are

all positive (e), then system 4 is L2-stable for all interconnection matrices H E v.

We shall omit the proof of this theorem in the interests of brevity, except to mention that the key idea is to make the transformation e;(-)=Me,(.), k = m + l , m + 2 , ..., n, in the system description (1). A noteworthy feature of this result is the correspondence between the requirements on the structural arrangement of the system and the use of multipliers, which does not seem to have been appreciated in the earlier development of multiplier theory for the single-loop negative feedback systems.

V. AN ILLUSTRATIVE EXAMPLE

At first glance, the conditions imposed by Theorem 1 on the intercon- nection matrix may seem rather restrictive. This is not, however, true since if the original interconnection matrix does not satisfy the required conditions, it is often possible to modify the system structure4 so as to result in an interconnection matrix of a desired form. We shall illustrate ths by considering an example in this section.

Consider the interconnection of two single-loop feedback (generalized Lurd) systems as shown in Fig. 1, where GI, G,; . . ,G, are operators in he. When GI, G,; . . , G, are considered as subsystem operators to conclude the 4-stability of the system from an application of the present result, the interconnection matrix H is

0 0 0 - 1 0 -1 0 0 - 1 0 - 1

0 1 0 1 0 0 0 0

which is not negative semidefinite and hence, is not a member of x. However, with elementary manipulations, it is possible to rearrange the system as in Fig. 2 (which has the same inputs and outputs as the original configuration). Now considering G1,G2, G3, G4=(G4- G,) and 6, = (G, - G3) as the subsystem operators, the modified interconnection matrix can be obtained as

0 0 -1 - 1 0 x_[: 0 0 - 1 H 0 -;I. 0 1

Observe that fi+ H r = O and further, fi admits a partitioning of the form (4) with m = 3. A straightforward application of Theorem 1 hence concludes the Lptability of the system in Fig. 2 (and consequently of the original system) if the following conditions are satisfied:

and 1) GI. G,, and G, are strongly positive(e) and have finite gain,

2) G4 = ( G4 - G3) and G, = ( G , - G,) are positivqe). Two interesting features of the above conditions are apparent. It is

simple to note that under the limiting condition that the two single-loop feedback systems get disconnected, which may be characterized by the condition G3=0, 1) and 2) above reduce precisely to the conditions of the positivity theorem applied individually to the two feedback systems. Secondly, by a simple relabeling of the subsystems, it is possible to interchange the positivity and strong positivity requirements imposed in 1) and 2) on two subsystems. For instance, it may be concluded from applying Theorem 1 that the system is also L,-stable if the following alternative conditions are fulfilled:

1) G2, G3 and 6,= (G4- G3) are strongly positive(e) and have finite gain and

2) GI and 6, = ( G , - G3) are positive(e).

decomposition of the original large system. This seems to provide a basis for the intuitive 40bserve that any wnstrain~~ on H are, in effecf conditions on the form of the

feeling that the key to a simple solution of any large-scale problem is a suitable decomposition of the problem.

SHORT PAPERS 399

Fig. I . Interconnection of two single-loop systems.

+Yt

VI. CONCLUSIONS

The major emphasis in this paper is put on the determination of the constraints on the interconnections among subsystems that may be modeled by positive operators, which ensure L2-stability of the overall system. Such results, as described in the main body of the paper, possess many interesting features, the most notable of which is their ease in application when compared with the existing results. The philosophy behind the present method of analysis yields a beneficial aspect that the conditions on the subsystems and the interconnections are independent of each other, thus leading to a true generalization of the positivity theorem to the multiloop case. Furthermore, important connective stability properties of the system can be easily concluded if the system decomposition leads to a total interconnection matrix that is the sum of several negative semidefinite matrices.

Several extensions and improvements of the present results appear to be possible through further investigation. Development of &-instability criteria which are counterparts of the present results, is quite straight- forward and may be found in [ 151, [16]. Since, as illustrated in Section V, a tradeoff between the constraints on the interconnections and the subsystems is apparent, one may try to relax the conditions on the subsystems at the expense of constraining the interconnection matrix more. An alternative, and more promising direction would be to obtain less stringent conditions on the interconnections by requiring more numbers of subsystems to be strongly positive, or by imposing additional norm-bounds on some of the subsystems.

ACKNOWLEDGMENT

M. K. Sundareshan wishes to thank Prof. D. D. Siljak of the Univer- sity of Santa Clara, California, for stimulating his interest in the stability analysis of large-scale systems during his visit to the University of Santa Clara.

REFERESCES

systems, S I A M J. Conrr.. vol. 3, pp. 4 4 3 4 2 . 1966. F. N. Bailey, The application of Lyapunovs second method to interconnected

systems. IEEE Trans. Auromar. Conrr., vol. AC-17. pp. 537-541, Aug. 1972. M. Araki and B. Kondo, "Stability and transient behaviour of composite nonlinear

A. N. Michel, Stability analysis of interconnected systems, S I A M J. Conrr., vol. 12, pp. 554579, Aug. 1974. D. W. Porter and A. N. Michel, Input-output stability of time-earying nonlinear multiloop feedback systems, IEEE Tram. Auromar. Conrr., vol. AC-19. pp. 422427, Aug. 1974.

IEEE Trans. Auromar. Conrr., vol. AC-21, pp. 84-89, Feb. 1976. E. L. Lasley and A. N. Michel, Input-output stability of interconnected systems,

Automar. Conrr., vol. AC-17. pp. 781-792, Dec. 1972. R. F. Estrada, On the stability of multiloop feedback systems, IEEE Trans.

J. C. WiUems, Qualitative behabiour of interconnected systems. Ann Syrr. Res., vol. 3, pp. 61-80. 1973. ~

systems, I E E E Trans. Auromar. Conrr.. vol. AC-18, pp. 643-645, Dec. 1973. L. T. Grujic and D. D. Siljak. Asymptotic stability and instability of large-scale

G. Zames, On the input-output stability of nonlinear time-varying feedback sys- temr-Pafl I: Conditions derived usmg concepts of loop gain, conicity and positiv- ity, IEEE Trans. Auromar. Confr., vol. AC-II, pp. 228-238, Apr. 1966. I. W. Sandberg, Some results on the theory of physical systems governed by nonlinear functional equatlons, Bell Sysr. Tech. 1.. vol. 44, pp. 871-898, Sept. 1965. M. K. Sundareshan and M. A. L. Thathachar, L stability of linear time-varying systems-conditions involving noncausal rnulti$iers, IEEE Tram. Auromar. Conrr., vol. AC-17, pp. -510, Aug. 1972. .I. C. Willems. The Analysis of Feedback Systems. Cambridge, MA: MIT Press, C. A. Desoer and M. Vidyasagar, Feedback Systems: Inpur-Ourpur Prqerrier. 1970.

New Y-ork: Academic, 1975. D. D. Siljak, Stability of large-scale systems under structural perturbations, IEEE Tram. Syrr., Man, Cybern., vol. SMC-2, pp. 657-563, Nov. 1972.

Fig. 2. A rearrangement of the system structure.

I151 M. K. Sundareshan, Large-scale dynamical systems: &-instability criteria, Dep. EIec. En&, Concordia Univ., Montreal, P. Q., Canada, Tech. Rep. EE-7505, Oct.

1161 M. Vidyasagar, &-stability criteria for interconnected systems, S I A M J . Conrr. 1975.

Oprimiz . , vol. 15, pp. 312-328, Feb. 1977.

Interactive Search for the Global Maximum of a Function of Many Variables

TAKA0 TSUDA

Absrract-This paper discusses a new algorithm to locate the global maximum of a function defied in a multidimensional rectangular domain. The number of dimensions is as large as, or even more tban, 5 or 10. There are two important elements in this algorithm. One is the transformation of the object function in such a way that its global maximum corresponds to infinity while other secondary maxima are reduced to zero. Actually there is some departure from the ideal transformation because of possible overflow on the computer. This portion of the algorithm precedes the interactive (or conversational) use of a graphic display system. This interactive part makes the other element of the algorithm. A multidimen- sional point is represented as a curve on the display screen. By projecting numerous points in the multidimensional space to similarly numerou curves on the screen of the graphic display device, the human eye can make overall recognition much more efficiently than computers. This fact is exploited to reduce the problem to that of a set of unimodal peaks. Once the supporting domain for each of these peaks is separated by visual aid, one may leave the computer to handle the rest of the problem for itself. A number of numerical experiments are done and discussed to provide evidenm regarding the feasibility of the proposed algorithm.

I. IPITRODUCIION

The algorithm to be presented in this paperL aims at locating the global (or greatest) maximum of a function defined in a bounded domain of k-dimensional Euclidean space. The object function may be multipeaked or multimodal. The case of finding the global (or least) minimum is similar to the present problem.

The motivation behind this work is that there exists no adequate means for finding the global maximum (minimum) of a function of many variables. Brent states in his recent monograph [l] that methods for finding global minima of functions of a moderate or large number of variables are difficult and that usually the best we can do is to use a good local minimizer and try several different combinations of starting positions, steplengths, etc., in the hope that the best local minimum found is the global minimum. A similar statement is made in [2], where 15 different unconstrained optimization techniques are compared, each against 15 minimization test problems. Reference [3] reviews the efforts directed to global techniques for constrained problems only. Reference [4] discusses the problems where the objective functions are assumed

by E. J. Davisoh Past Chairman of the IEEE S C S Computattonal Methods and Discrete Manuscript received September 28, 1976; revised January I , 1977. Paper recommended

Systems Committee. The author is with the Department of Electrical Engineering Hokkaido University,

Sapporo, Japan. The preliminary version concerning only the noninteractive portion of the algorithm

was presented at the Second USA-Japan Computer Conference, Tokyo, Japan, August 1975.