- Email: [email protected]
Hierarchical control of weakly-coupled systems
0005-I098/82/040467-05503.0010
Autumatica, Vol. 18, No. 4, pp. 467-.471, 19~2
Pergamon Press Ltd. © 1982 International Federation of Automatic Control
Printed in Great Britain.
Brief Paper Hierarchical Control of Weakly-coupled Systems* DOUGLAS
P. LOOZEt
a n d N I L S R. S A N D E L L ,
JR~
Key Words--Hierarchical systems; decomposition; weakly-coupled systems; optimization; largescale systems; control theory.
Abstract--The usual concept of weakly-coupled systems is generalized to provide a definition in terms of the decomposition used. The definition includes a measure of the strength of the coupling. The main result of the paper is a local convergence result for natural decompositions (decompositions which exploit system structure) of systems which are sufficiently weakly coupled.
approximation is used. This is seen in the bounds on the accuracy of the approximate solution obtained by Milne (1965), and in the suboptimality bounds of approximate linear quadratic controls developed by Bailey and Ramapriyah (1973). In the context of hierarchical control, the dependence of the concept of weak coupling on the problem is manifested in the ability of the decomposition to exploit the weak coupling structure of the problem. Thus it is necessary to define weak coupling in terms of the system, the cost, and the decomposition used. Section 3 presents this definition using a local decomposition framework outlined in Section 2 (see also Looze and Sandell, 1981). This definition includes a natural measure of the coupling strength of the decomposed problem. Another direct consequence of the definition is that the hierarchical iteration defined by a weakly-coupled system-decomposition pair will exhibit local convergence to the solution. Although this definition of weak coupling is in terms of both the problem and the decomposition, in most cases one would like to define the decomposition to exploit any inherent problem structure. A class of decompositions which use the structure of the problem as a guide to form the decomposition is defined in Section 4. The main result of the paper demonstrates that this class of natural decompositions possesses a very desirable property: whenever the problem structure on which the decomposition is based is sufficiently weakly coupled in the usual sense (independent of the decomposition), the hierarchical iteration will converge locally. Thus, we show an important, intuitive connection between weakly coupled problems and hierarchical control design. The interaction prediction method (Cohen, 1977; Singh, Hassan and Titli, 1976) is analyzed in this framework in Section 5. It is shown that this algorithm is a natural decomposition for subsystems with weak interactions. The resulting analysis provides an interesting insight into the degree of success one might expect when applying the interaction prediction algorithm. Section 6 presents an application which illustrates the concepts of Sections 2-4. The following notation will be used throughout the paper. Script capital letters (e.g. ~ will denote Banach spaces. Partial Fr6cbet derivatives of functions will be denoted by 0d (with respect to the ith argument) or fx (with respect to the argument x). The superscript * (e.g. A*) denotes the adjoint of the indicated linear operator. Finally, the notation p{.} denotes the spectral radius of the argument.
1. Introduction THE TOPIC of hierarchical control has developed into an important area of large-scale systems theory over the past two decades. Until recently, however, the theory has consisted mostly of extensions of decomposition results of mathematical programming to the type of optimization problems considered by control theorists (Takahara, 1965; FindeiSen, 1968; Pearson, 1971; Singh and Titli, 1975). As such, a particular structural property of the problem formulation-the additive separability of the cost and constraint functions--was exploited. Recently it has been demonstrated (Cohen, 1978) that the fundamentals of these algorithms can be generalized to define decompositions and hierarchies independent of the system structure. However, it is impossible to guarantee either global or local convergence in completely unstructured problems. Hence it is still important to consider system structure when specifying a decomposition or hierarchical algorithm. One such structural form which will be investigated in this paper is that of weak coupling. A weakly-coupled system is usually interpreted to be a decomposition of subsystems whose interactions are small. This structure has been used in various manners in control theory. Milne (1965) analyzed approximate solutions of weakly-coupled differential equations and provided a quantitative measure of the weak coupling. The exploitation of weak coupling in a system is important in aggregation (Aoki, 1978). Weakly-coupled structures have been used to obtain simplified control configurations (Kokotovic, 1972; Bennett and Baras, 1980). An important observation which results from examining these works is that the concept of small (i.e. the degree of coupling) depends not only on the dynamical system but also in the particular problem for which the
*Received 17 February 1981; revised 14 September 1981. The original version of this paper was presented at the 2nd IFAC Conference on Large Scale Systems, which was held in Toulouse, France during June 1980. The published proceedings of this IFAC meeting may be ordered from Pergamon Press Ltd, Headington Hill Hall, Oxford OX3 0BW, U.K. This paper was recommended for publication in revised form by editor A. Sage. l'Coordinated Science Laboratory, University of Illinois, 1101 W. Springfield Avenue, Urbana, IL 61801, U.S.A. ~Alphatech Inc., 3 New England Executive Park, Burlington, MA 01803, U.S.A. §Obviously, not every control problem can be expressed in this form. However, this assumption is sufficiently general to include a large number of such problems. In particular, differentiable optimization problems (including those with equality constraints), Riccati equations, and Lyapunov matrix equations are of this form.
2. Preliminaries For the purposes of this paper, it will be assumed that the problem to be solved can be expressed in the form§ f(x) = 0
(1)
where ~ is a reflexive Banach space and f: ~ is a continuously Fr~chet differentiable function. Decompositions of (1) will be defined using the framework introduced by Looze and Sandell (1981). Let f0: ~ × ~ ' - * ~ be a continuously Fr6chet differentiable function and define f~: ~ x ~ - - * ~ by f ( x ) = fo(x, y) + fl(x, y),
467
V x, y ~ ~.
(2)
468
Brief Paper
The decomposed problem is then to solve
fo(xk+,, xk) = -f,(x~, xk)
Proof. Since the induced norm of an operator is at least as large as its spectral radius, we have (3) 3, ---< to.
for xk+~ given xk. Equation (3) can be interpreted as a hierarchical solution method in two ways. First, (3) can be regarded as the subproblem (or the collection of subproblems) while the supremal problem is to simply transfer information. The more common interpretation assumes a particular structure for f and fo which results in a hierarchical organization within (3) (see Looze and Sandell, 1981 for details). The following theorem provides conditions under which (3) exhibits local convergence to a solution x* of (1) and gives the asymptotic rate of convergence.
Theorem 1. Let f, f0, and f~ be defined as above, and let x* be a solution to (1). If (i) 01fo(x*, x*) is n o n s i n ~ l a r (ii) 3' = p{-Oifo(x*, x*)-'alfl(x*, x*)} < I (4) then there exists an open neighborhood N C ge of x* such that for any xo E N there is a unique sequence {xk}~, C ~r satisfying (3). Moreover
Hence (4) holds, and (5) can be replaced by (8). [] Thus, we see that a weakly-coupled problem-decomposition pair (as defined above) always exhibits local convergence to the problem solution. In addition, the normalized coupling strrength provides an upper bound on the asymptotic convergence rate of the iteration.
4. Natural decompositions The preceding section defined a generalized measure of coupling stren~h for a problem-decomposition pair. However, no assumptions were made on the system structure itself. As such, Section 3 serves simply to delineate the weak coupling concepts. In this section, we introduce a structure to problem (1) in terms of a small parameter, (where the term 'small' will be made more precise later) and define a natural decomposition in terms of this structure. Let ~ and • be Banach spaces, and let f: ~ x . ~ ~,~' be continuously Fr6chet differentiable in its first argument and continuous in its second argument. Also, let f be such that 0if(x*, 0) is invertible. Then, equation (1) will be replaced by
lim Xk = X*
f(x; a) = O.
(9)
and for each • > 0 there exists an integer ko such that
IIx~ - x*ll-< (3' + ,Y,
v k ~ k0.
(5)
Proof. See Looze and Sandell (1981). 3. Weak coupling of decomposition algorithms The conditions of Theorem 1 can be used to develop the concept of coupling strength of a decomposition algorithm. The major nontechnical condition is the requirement that the asymptotic convergence rate 3' be less than unity. This can happen in several ways. First, the structural interaction of the two operators Omfo(x*,x*) -I and Omfm(x*,x*) can be such that the condition is satisfied. Secondly, the resulting operator O~fo(x*,x*)-~O~f'(x *, x*) can be a contraction operator. Finally, the products of the induced norms of the individual operators can be less than unity
IIO,/o(X*, x*)-'llUa,/,(x*,
x*)ll < 1.
(6)
It is obvious that the preceding conditions are not exclusive. The second is included in the first, and the third is a special case of the first two. However, it is the third condition (6) which is naturally interpreted as a weak-coupling condition. The reason for this interpretation is the following. The usual concept of a weakly-coupled problem is that the absolute size of the neglected interactions are not sufficient to greatly disturb the problem solution. With decomposition algorithms, the 'neglected' interactions are taken into account by the iteration. That is, fl represents these interactions. The quantity IlOd,tx*, x*)ll represents the size of the interaction appropriate to the decomposition algorithm. Thus, we interpret (6) as a weakly-coupling condition.
Definition. Problem (1) is (locally) weakly coupled with respect to the decomposition (2)-(3) if condition (6) holds. As a consequence of the preceding discussion, we define
to ~= IIo,fo(x*,
x*)-'lllla,/,(x*,
x*)ll
In (9), a is interpreted to be a parameterization of the original problem (1). This parameterization is to be interpreted as representing the problem structure in the following sense. If a = 0, then it will be assumed that the problem (9) possesses a structure which is more easily solved than for nonzero a. The particular case of interest in hierarchical control is when a is used to represent the interactions between subproblems. This assumption allows us to define a natural decomposition in terms of the parameterized system structure.
Definition. A natural decomposition of (9) is defined by any splitting function f0(x, y) such that fo(x,y)=f(x;O),
(10)
whenever a = 0. Note that this restricts the dependence of the splitting function on a to be determined by y. However, it does not uniquely define a natural decomposition. An examination of (10) leads to the following theorem.
Theorem 3. There exist open neighborhoods ag C ~¢ of 0 and 2¢ C • of x* such that Xo E 3f and a E ~ imply lira Xk = x*. k~
(ll)
Moreover, for every • > 0 there exists an integer k0 such that Ilxk - xdl < (to + c) k, v k -> k0.
(12)
Proof. By the implicit function theorem, x* [defined by (9)] varies continuously with a. Hence, both alf0(x*, x*) and Oil(x*, a) are also continuous i n a . By virtue of (10), we have for a = 0
(7)
as the normalized coupling strength. The following theorem is an easy consequence of Theorem 1.
¥x,y ~
0if(x*; O)- Otfo(x*, x*) =4)
(13)
llOmf(X*,0 ) I ~ M0 < co.
(14)
and
Theorem 2. If problem (1) is weakly coupled with respect to the decomposition (2)-(3) then the conditions and conclusions of Theorem 1 hold. In particular, the error estimate (5) can be replaced with Ilxk - x*l[ - (to + ~)k, V k -> ko.
(8)
Relations (13) and (14) together with the continuity properties of 01fo(x*,x*) and 01f(x*;a) imply there exists an open neighborhood d~ C ~ of zero such that
I[O~fo(x*, x*)-~ll <
Mo+ 1
(15)
Brief Paper
and
Equivalently, (19) can be considered from a local analysis point of view (recall the differentiability assumptions)
!
IIo,f(x*; a ) - O,fo(x*, x*)U< Mo'+ 1"
469
(16)
Combining (7) with (15) and (16) gives
J,(uk+l, K(uk)) + Jv(Uk, K(uk))Ku(uk) = 0.
This can be expressed within the decomposition frame.work of Section 2 by defining f(u) as the necessary conditions of (17) and (18)
,,, A IIO,fo(x*, x*)-'llllo,f(x*; ~)
f(u) = .l,(u, K(u)) + J~(u, K(u))K~(u) = 0.
- O,fo(X*, x*)ll < 1.
Theorem 2 then gives the desired conclusions. [] For an arbitrary problem which can be expressed according to the assumptions of Theorem 3, this result states that sufficiently weak coupling will result in a convergent algorithm. Thus, Theorem 3 emphasizes two points. First, it serves to reinforce the motivation behind the definition and choice of a natural decomposition. Because the natural decomposition is motivated by the ultimate weak-coupling situation, it is reassuring that such insight and intuition leads to a reasonable decomposition structure. Secondly, it emphasizes that the concept of weak coupling of decomposition algorithms is truly related to both the problem and the decomposition. However, the theorem shows it is possible to take advantage of any weak-coupling structure inherent to a problem by defining an appropriate decomposition. 5. The interaction prediction method The three preceding sections only considered abstract definitions of weak coupling and natural decompositions. This section attempts to illustrate these concepts as applied to a particular decomposition algorithm, the interaction prediction method (Cohen, 1977; Singh, Hassan and Titli, 1976). Although the original motivation for this decomposition was based on the separability of the implicitly defined cost, Cohen (1977) demonstrated that this assumption is not fundamental to the algorithm. We will show that the interaction prediction algorithm is a natural decomposition for weak interactions. Also, it will be demonstrated that a quadratic modification proposed by Cohen (1977) can be interpreted as modifying the decomposition in such a way as to lessen the importance of the interactions by strengthening the subproblems (i.e. by weakening the normalized coupling strength). This emphasizes the importance of the dependence of the coupling strength on the choice of the decomposition algorithm. Let ~ and ~" be Banach spaces and let J: 0//x ~'--+R and K: °I1-,~ be twice continuously Frtchet differentiable functions. Consider the following problem
(17)
u, v
fo(u, fA) = J~(u, K(li)).
The ultimately weakly coupled problem for (17) and (18) would be a constant interaction K(u) = ~, Y u E cll.
(23)
A natural parameterization of this structure is to choose a as any continuous parameterization of the nonlinearity K such that a is zero whenever (23) is satisfied. Then, for a = 0 it is easily seen from (21) and (22) that fo(u,~) =f(u).
(24)
That is, f0 as defined by (22) is a natural decomposition. Theorem 3 implies that if the variation of K with u is: sufficiently small (reflected by a as defined above), the iteration defined by (20) will converge. This can also be confirmed by computing to directly and using Theorem 2. Define A a=J~(u*, K(u*)) B A Ku(u*)
C A r..(u*, u*) D A A~(u*, K(u*))
E a__Y~(u*, K(u*)) P a=Jr(u*, K(u*)). Then =
A
(25)
Oil(u*) - alfo(u*, u*) = B*DB + E*B + B*E + Cp. Hence, w as defined in (7) is
v = K(u).
(18)
In (17), the cost functional Y is interpreted as the cost (implicitly) defined by the original problem cost functional and the subsystem constraints. Equation (18) embodies the interactions between subsystems. The variable v is the interaction variable. The assumption of multiple subsystems is not germane to the following analysis except for the interpretation of the interaction constraint and interaction variables. Thus, we will assume only a single subsystem (keeping in mind the noted interpretations) to avoid the inherent additional notational complexity associated with several subsystems. The appfication of the interaction prediction principle to (17) and (18) defines the following iteration in u (Cohen, 1977) min {J(uk+l, K(uk)) + Jv(uk, K(u~))K,(u~)uk+l}.
A U T Vol. 18, No. 4=-G
(22)
(26)
subject to
llli+ i 15 ~ll
(21)
Then, (20) is produced by the splitting function
O]fo(u*, u*)
min J(u, v)
(20)
(19)
o, = IIA-'IIB*DB + E * B + B * E + cpll.
(27)
For K~(u*) and Kuu(u*) sufficiently small (i.e. for K approximately constant near u*), w will be less than unity and the iteration will converge. It is interesting to examine the quadratic modification suggested by Cohen (1977) within this context of weak coupling. The effect of this modification is to add a quadratic operator to M to (25) and subtract M from (26). Hence o, = I[(A + M)-1IIIIB*OB + E*B + B*E + Cp - MII. (28)
From (38), we can see that one useful criterion for choosing M might be to choose the operator in such a manner as to weaken the normalized interaction to. This can be done either by reducing the magnitude of the interaction or (more likely) by increasing the dominance of the subproblem solution.
470
Brief Paper
6. Example
(35)
y = Cx + O.
The purpose of this section is to illustrate the concepts presented in Sections 2-4 with a specific example. The problem of estimating traffic density and velocity on a freeway from density measurements will be considered. A dynamical model for freeway traffic behavior has been developed by Payne (197 1). This model describes traffic flow behavior in terms of aggregate, fluid analog variables which are interpreted as the spatial mean speed and spatial mean density on sections of the freeway. If a freeway is partitioned into N sections and indexed in the direction of traffic flow (i.e. section 1 is the furthest upstream section) then the evolution of the spatial average density p~ and velocity v~ on the ith link is given by
[~i i)i = -
piVi -- pi-lVi-I ~_ ~Pi li
The steady-state Kalman filter based on (34) and (35) is given by = A~ + H(y - CJ)
(38)
The Riccati equation (38) can be solved iteratively using the decomposition technique of Section 2. To illustrate this, consider the partition of (34) into two subsystems. The corresponding decomposition of the Riccati equation was first proposed by Sandell (1976). Let
(29)
rA,, ,,,:1
o],
A = LA21 A22J'
j -~[vl-v<,(pi)]
li
(37)
0 = A P + P A T + '~ - p C T O - I C P .
1 [vi-vi-i]_ 1
viL
(36)
H = p c T o -I
[P,,
e,2]
P = LP21 P22J +"T f±lrp,-p,+,l Lp, J L ~ J + f~'
(30)
where lr is the length of the ith section, v,~(p~)is desired steady-state velocity,f and T and v are which reflect the drivers' sensitivity to traffic The vector ~ = [~1~1 ... ~pN~oN]r is assumed to white noise process with
E{~(t)~r(~-)}= =.a(t
-
where the above partitioning corresponds to the partitioned system, and assume that the covariances are block diagonal
the drivers' coefficients conditions. be a vector
~).
":=[0'
0],
0=[00, 0].
The iterative equations are then (Sandell, 1975)
(31)
i
i
T
i
i-I
Equations (29) and (30) also apply to the initial and final links with
i
A H P II + P NAN + QI - P NSIP II + i-I
T
i-I
i-I
i-I
i-I
--
{AI2P21 +PI2 A I 2 - P l 2 $2P21 } - 0 i
i
T
i
(39)
i
A22P 22+ P 22A22+ Q2 - P z2SzP 22+ i I
i-I
T
{A21Plz + P21 A21-P21 SIPt2 } = 0
(40)
A
1)0= 1)I
( A t , - P',,S,)P'~2 + P ~2(A22- P~2S2}r + A,2P~2 + P ~tA~ = 0
povo ~- input flow (given) PN+! ~ PN"
Equation (29) represents the conservation of flow on the freeway. Equation (30) is obtained from a finite element approximation of a continuous flow model. Additional details may be found in Payne (1971). It will be assumed that a noisy density measurement is available for each section Yi =
Pi + 0i.
(32)
The vector O = [0~... 0~] T is assumed to be a white noise process with E = {o(t)or('~)}
= O~(t
-
.r)
(33)
where ~(t - ~) is the Kronecker delta function. The problem considered is to estimate the deviation of the traffic variables (density p~ and velocity v~) from their nominal values P~,omand Vi~om.One solution to this problem is to use a Kalman filter based upon the linearization of (29) and (30) around the nominal values P~,omand v~,om. Let
X
A [(Pl -- Plnom)(Pl
Y tt [ ( Y l - -
Plnom)...
- - P l n o m ) • • • (IJN" - - VN'nom)] T
where Si --- CTO~-ICi. Equations (39) and (40) are Riccati equations which can be solved simultaneously for the subsystem covariances PN and P22 given the most recent estimate of the covariance between subsystems P~2. Equation (41) is a linear matrix equation for P~2 which can be solved once the solution of (39) and (40) is complete. This can be interpreted as a hierarchical iteration with the supremal computing subsystem interactions [equation (41)] and the infimal solving subsystem estimation problems [equations (39) and (40)]. This can be formulated as in Section 2 with .f0(P i, P~-~) given by the non-bracketed terms in (39)-(41) and ft(pi-i, p H ) given by the bracketed terms F AI2P21 + P12A~2- p12S2p21 fi(P'')=LA2iPi'+P2iAr-p2isiP'2].
(42)
It is easily verified that as IIA& approaches zero so does Pl*2 also approaches zero). Also, f0 is a block lower triangular operator with 8tf0(P*,P*) being nonsingular whenever the filter dynamics are stable. Hence, equation (6) will be satisfied for sufficiently small IlAl:[I and the iteration defined by (39)-(41) will converge. This algorithm was applied to a four-section linearized freeway model with
Ila,f,(P*, P*)II (since
Pi,om 74 veh./mile (46 veh'/km)l i = 1. . . . . 4. vinom= 55 mile/h (88.5 kin/h) J
( Y N - - P N n o m ) ] T'
The state vector and measurement are approximately described by the linear dynamical system = Ax + ~
(41)
(34)
tSee Looze (1975), p. 42 for an exact description of this curve.
These are reasonable values from moderate traffic on a threelane U.S. freeway. The desired velocity function vei(p~) corresponds to such a freeway and is defined as in Looze (1975). The driver reaction and sensitivity coefficients are T = 5 s (1.39x 10-~ h) v = 18.75 mileZ/h (48.56 kmZ/h).
Brief Paper To introduce an inhomogeneity into the system (and hence a physical motivation for the decomposition), the lengths of the links were taken to be l~ = 0.2 mile (0.32 kin), i = 1, 2, 4 13-= 0.5 mile (0.8 kin).
(43)
It can be seem from (29) and (30) that as 13--,oo the freeway equations decouple into two four-dimensional subsystems consisting of sections 1-2 and 3-4, respectively. Hence a natural decomposition of the Riccati equation (38) with respect to the structure induced by (43) defines an iteration of the form (39)-(41). The iteration (39)-(41) was used to solve the filtering Riccati equation (38) for the freeway system described in the preceding paragraphs.'t The iteration converged (i.e. lip1_ p ~-'IIF-< 10-6 with I['I[Fdenoting the Frobenius norm) in five iterations. The Riccati equation (38) was also solved using an algorithm developed by Lanb (1979). The difference between the iterative solution PI and the direct solution PD is IIPI - PolIF = 4 × 10 -1°.
7. Conclusions The main concept of this paper is that weak coupling in the context of hierarchical control is necessarily expressed in terms of both the problem structure and the decomposition used. Defined in this manner, weak coupling becomes a natural criterion for local convergence of the hierarchical iteration. This definition of weak coupling is used to motivate the class of natural decompositions, i.e. those decompositions which directly exploit the system structure. The main result of the paper was to show that natural decompositions will converge locally if the underlying system structure is sufficiently weakly coupled in the usual sense. This result is applicable to several hierarchical algorithms. In particular, it was shown that the interaction prediction algorithm is a natural decomposition. It is also directly applicable to decompositions of the form proposed in Looze and Sandell (1977), as is demonstrated by the example of Section 6. Acknowledgements--This research was supported in part by ONR Contract N00014-76-C-0346; in part by the U.S. Department of Energy, Division of Electrical Energy Systems under Grant ERDA-E(48-18)-2087 at the Massachusetts Institute of Technology; in part by the National Science Foundation under Grant NSF ENG-79-08778; and in part by the Joint Services Electronics Program under Contract N00014-79-C-0424 at the University of Illinois. ?The numerical data are available from the authors.
471
References Aokl, M. (1978). Some approximation methods for estimation and control of large scale systems. IEEE Trans Aut. Control, AC.23, 173. Bailey, F. N. and H. K. Ramapriyan (1973). Bounds on the suboptimality in the control of linear dynamic systems. IEEE Trans Aut. Control, AC-18, 532. Bennett, W. H. and 3. S. Baras (1980). Block diagonal dominance and design of decentralized compensators. Preprints of the IFAC Syrup. on Large Scale Systems: Theory and Applications, Toulouse, France, pp. 93-102. Cohen, G. (1977). On an algorithm of decentralized optimal control. J. Math. Analysis and Appl., 59, 242. Cohen, G. (1978). Optimization by decomposition and coordination: A unified approach. IEEE Trans Aut. Control, AC-23, 222. Findeisen, W. (1968). Parametric optimization by primal method in multilevel systems. IEEE Trans S.S.C., SSC-4, 155. Kokotovic, P. V. (1972). Feedback design of large linear systems. In J. B. Cruz, Jr. (Ed.), Feedback Systems, Chapter 4. McGraw-Hill, New York. Lanb, A. J. (1979). A Schur method for solving Riccati equations. IEEE Trans Aut. Control, AC-24, 913. Looze, D. P. and N. R. Sandell, Jr (1977). Decomposition of linear, decentralized, stochastic control problems. IFAC Workshop on Control and Management of Integrated IndustriafComplexes, Toulouse, France. Looze, D. P. and N. R. Sandell, Jr (1981). Analysis of hierarchical algorithms via nonlinear splitting functions. Y. Optimiz. Theory and Appl. 34, 3. Looze, D. P. (1975). Decentralized Control of a Freeway Tranic Corridor. S.M. thesis. Massachusetts Institute of Technology, Cambridge, MA. Milne, R. D. (1965). The analysis of weakly coupled dynamic systems. Int. J. Control, 3, 171. Payne, H. J. (1971). Models of freeway traffic and control. In Mathematical Models of Public Systems (Simulation Councils Proc., 1, No. 1). Pearson, J. D. (1971). Dynamic decomposition techniques. In D. A. Wismer (Ed.), Optimization Methods for Large Systems. McGraw-Hill, New York. Sandell, N. R., Jr (1976) Decomposition vs. decentralization in large scale system theory. 1976 IEEE Conf. on Decision and Control, Clearwater, FL. Singh, M. G., H. F. Hassan, and A. Titli (1976). Multilevel feedback control for interconnected dynamical systems using the prediction principle. IEEE Trans. Syst., Man, Cybern., SMC-6, 233. Singh, M. G. and A. Titli (1975). Closed-loop hierarchical control for nonlinear systems using quasilinearization. Automatica, 11,541. Takahara, Y. (1965). A multi-level structure for a class of dynamical optimization problems. M.S. thesis, Case Western Reserve University, Cleveland, OH.
Autumatica, Vol. 18, No. 4, pp. 467-.471, 19~2
Pergamon Press Ltd. © 1982 International Federation of Automatic Control
Printed in Great Britain.
Brief Paper Hierarchical Control of Weakly-coupled Systems* DOUGLAS
P. LOOZEt
a n d N I L S R. S A N D E L L ,
JR~
Key Words--Hierarchical systems; decomposition; weakly-coupled systems; optimization; largescale systems; control theory.
Abstract--The usual concept of weakly-coupled systems is generalized to provide a definition in terms of the decomposition used. The definition includes a measure of the strength of the coupling. The main result of the paper is a local convergence result for natural decompositions (decompositions which exploit system structure) of systems which are sufficiently weakly coupled.
approximation is used. This is seen in the bounds on the accuracy of the approximate solution obtained by Milne (1965), and in the suboptimality bounds of approximate linear quadratic controls developed by Bailey and Ramapriyah (1973). In the context of hierarchical control, the dependence of the concept of weak coupling on the problem is manifested in the ability of the decomposition to exploit the weak coupling structure of the problem. Thus it is necessary to define weak coupling in terms of the system, the cost, and the decomposition used. Section 3 presents this definition using a local decomposition framework outlined in Section 2 (see also Looze and Sandell, 1981). This definition includes a natural measure of the coupling strength of the decomposed problem. Another direct consequence of the definition is that the hierarchical iteration defined by a weakly-coupled system-decomposition pair will exhibit local convergence to the solution. Although this definition of weak coupling is in terms of both the problem and the decomposition, in most cases one would like to define the decomposition to exploit any inherent problem structure. A class of decompositions which use the structure of the problem as a guide to form the decomposition is defined in Section 4. The main result of the paper demonstrates that this class of natural decompositions possesses a very desirable property: whenever the problem structure on which the decomposition is based is sufficiently weakly coupled in the usual sense (independent of the decomposition), the hierarchical iteration will converge locally. Thus, we show an important, intuitive connection between weakly coupled problems and hierarchical control design. The interaction prediction method (Cohen, 1977; Singh, Hassan and Titli, 1976) is analyzed in this framework in Section 5. It is shown that this algorithm is a natural decomposition for subsystems with weak interactions. The resulting analysis provides an interesting insight into the degree of success one might expect when applying the interaction prediction algorithm. Section 6 presents an application which illustrates the concepts of Sections 2-4. The following notation will be used throughout the paper. Script capital letters (e.g. ~ will denote Banach spaces. Partial Fr6cbet derivatives of functions will be denoted by 0d (with respect to the ith argument) or fx (with respect to the argument x). The superscript * (e.g. A*) denotes the adjoint of the indicated linear operator. Finally, the notation p{.} denotes the spectral radius of the argument.
1. Introduction THE TOPIC of hierarchical control has developed into an important area of large-scale systems theory over the past two decades. Until recently, however, the theory has consisted mostly of extensions of decomposition results of mathematical programming to the type of optimization problems considered by control theorists (Takahara, 1965; FindeiSen, 1968; Pearson, 1971; Singh and Titli, 1975). As such, a particular structural property of the problem formulation-the additive separability of the cost and constraint functions--was exploited. Recently it has been demonstrated (Cohen, 1978) that the fundamentals of these algorithms can be generalized to define decompositions and hierarchies independent of the system structure. However, it is impossible to guarantee either global or local convergence in completely unstructured problems. Hence it is still important to consider system structure when specifying a decomposition or hierarchical algorithm. One such structural form which will be investigated in this paper is that of weak coupling. A weakly-coupled system is usually interpreted to be a decomposition of subsystems whose interactions are small. This structure has been used in various manners in control theory. Milne (1965) analyzed approximate solutions of weakly-coupled differential equations and provided a quantitative measure of the weak coupling. The exploitation of weak coupling in a system is important in aggregation (Aoki, 1978). Weakly-coupled structures have been used to obtain simplified control configurations (Kokotovic, 1972; Bennett and Baras, 1980). An important observation which results from examining these works is that the concept of small (i.e. the degree of coupling) depends not only on the dynamical system but also in the particular problem for which the
*Received 17 February 1981; revised 14 September 1981. The original version of this paper was presented at the 2nd IFAC Conference on Large Scale Systems, which was held in Toulouse, France during June 1980. The published proceedings of this IFAC meeting may be ordered from Pergamon Press Ltd, Headington Hill Hall, Oxford OX3 0BW, U.K. This paper was recommended for publication in revised form by editor A. Sage. l'Coordinated Science Laboratory, University of Illinois, 1101 W. Springfield Avenue, Urbana, IL 61801, U.S.A. ~Alphatech Inc., 3 New England Executive Park, Burlington, MA 01803, U.S.A. §Obviously, not every control problem can be expressed in this form. However, this assumption is sufficiently general to include a large number of such problems. In particular, differentiable optimization problems (including those with equality constraints), Riccati equations, and Lyapunov matrix equations are of this form.
2. Preliminaries For the purposes of this paper, it will be assumed that the problem to be solved can be expressed in the form§ f(x) = 0
(1)
where ~ is a reflexive Banach space and f: ~ is a continuously Fr~chet differentiable function. Decompositions of (1) will be defined using the framework introduced by Looze and Sandell (1981). Let f0: ~ × ~ ' - * ~ be a continuously Fr6chet differentiable function and define f~: ~ x ~ - - * ~ by f ( x ) = fo(x, y) + fl(x, y),
467
V x, y ~ ~.
(2)
468
Brief Paper
The decomposed problem is then to solve
fo(xk+,, xk) = -f,(x~, xk)
Proof. Since the induced norm of an operator is at least as large as its spectral radius, we have (3) 3, ---< to.
for xk+~ given xk. Equation (3) can be interpreted as a hierarchical solution method in two ways. First, (3) can be regarded as the subproblem (or the collection of subproblems) while the supremal problem is to simply transfer information. The more common interpretation assumes a particular structure for f and fo which results in a hierarchical organization within (3) (see Looze and Sandell, 1981 for details). The following theorem provides conditions under which (3) exhibits local convergence to a solution x* of (1) and gives the asymptotic rate of convergence.
Theorem 1. Let f, f0, and f~ be defined as above, and let x* be a solution to (1). If (i) 01fo(x*, x*) is n o n s i n ~ l a r (ii) 3' = p{-Oifo(x*, x*)-'alfl(x*, x*)} < I (4) then there exists an open neighborhood N C ge of x* such that for any xo E N there is a unique sequence {xk}~, C ~r satisfying (3). Moreover
Hence (4) holds, and (5) can be replaced by (8). [] Thus, we see that a weakly-coupled problem-decomposition pair (as defined above) always exhibits local convergence to the problem solution. In addition, the normalized coupling strrength provides an upper bound on the asymptotic convergence rate of the iteration.
4. Natural decompositions The preceding section defined a generalized measure of coupling stren~h for a problem-decomposition pair. However, no assumptions were made on the system structure itself. As such, Section 3 serves simply to delineate the weak coupling concepts. In this section, we introduce a structure to problem (1) in terms of a small parameter, (where the term 'small' will be made more precise later) and define a natural decomposition in terms of this structure. Let ~ and • be Banach spaces, and let f: ~ x . ~ ~,~' be continuously Fr6chet differentiable in its first argument and continuous in its second argument. Also, let f be such that 0if(x*, 0) is invertible. Then, equation (1) will be replaced by
lim Xk = X*
f(x; a) = O.
(9)
and for each • > 0 there exists an integer ko such that
IIx~ - x*ll-< (3' + ,Y,
v k ~ k0.
(5)
Proof. See Looze and Sandell (1981). 3. Weak coupling of decomposition algorithms The conditions of Theorem 1 can be used to develop the concept of coupling strength of a decomposition algorithm. The major nontechnical condition is the requirement that the asymptotic convergence rate 3' be less than unity. This can happen in several ways. First, the structural interaction of the two operators Omfo(x*,x*) -I and Omfm(x*,x*) can be such that the condition is satisfied. Secondly, the resulting operator O~fo(x*,x*)-~O~f'(x *, x*) can be a contraction operator. Finally, the products of the induced norms of the individual operators can be less than unity
IIO,/o(X*, x*)-'llUa,/,(x*,
x*)ll < 1.
(6)
It is obvious that the preceding conditions are not exclusive. The second is included in the first, and the third is a special case of the first two. However, it is the third condition (6) which is naturally interpreted as a weak-coupling condition. The reason for this interpretation is the following. The usual concept of a weakly-coupled problem is that the absolute size of the neglected interactions are not sufficient to greatly disturb the problem solution. With decomposition algorithms, the 'neglected' interactions are taken into account by the iteration. That is, fl represents these interactions. The quantity IlOd,tx*, x*)ll represents the size of the interaction appropriate to the decomposition algorithm. Thus, we interpret (6) as a weakly-coupling condition.
Definition. Problem (1) is (locally) weakly coupled with respect to the decomposition (2)-(3) if condition (6) holds. As a consequence of the preceding discussion, we define
to ~= IIo,fo(x*,
x*)-'lllla,/,(x*,
x*)ll
In (9), a is interpreted to be a parameterization of the original problem (1). This parameterization is to be interpreted as representing the problem structure in the following sense. If a = 0, then it will be assumed that the problem (9) possesses a structure which is more easily solved than for nonzero a. The particular case of interest in hierarchical control is when a is used to represent the interactions between subproblems. This assumption allows us to define a natural decomposition in terms of the parameterized system structure.
Definition. A natural decomposition of (9) is defined by any splitting function f0(x, y) such that fo(x,y)=f(x;O),
(10)
whenever a = 0. Note that this restricts the dependence of the splitting function on a to be determined by y. However, it does not uniquely define a natural decomposition. An examination of (10) leads to the following theorem.
Theorem 3. There exist open neighborhoods ag C ~¢ of 0 and 2¢ C • of x* such that Xo E 3f and a E ~ imply lira Xk = x*. k~
(ll)
Moreover, for every • > 0 there exists an integer k0 such that Ilxk - xdl < (to + c) k, v k -> k0.
(12)
Proof. By the implicit function theorem, x* [defined by (9)] varies continuously with a. Hence, both alf0(x*, x*) and Oil(x*, a) are also continuous i n a . By virtue of (10), we have for a = 0
(7)
as the normalized coupling strength. The following theorem is an easy consequence of Theorem 1.
¥x,y ~
0if(x*; O)- Otfo(x*, x*) =4)
(13)
llOmf(X*,0 ) I ~ M0 < co.
(14)
and
Theorem 2. If problem (1) is weakly coupled with respect to the decomposition (2)-(3) then the conditions and conclusions of Theorem 1 hold. In particular, the error estimate (5) can be replaced with Ilxk - x*l[ - (to + ~)k, V k -> ko.
(8)
Relations (13) and (14) together with the continuity properties of 01fo(x*,x*) and 01f(x*;a) imply there exists an open neighborhood d~ C ~ of zero such that
I[O~fo(x*, x*)-~ll <
Mo+ 1
(15)
Brief Paper
and
Equivalently, (19) can be considered from a local analysis point of view (recall the differentiability assumptions)
!
IIo,f(x*; a ) - O,fo(x*, x*)U< Mo'+ 1"
469
(16)
Combining (7) with (15) and (16) gives
J,(uk+l, K(uk)) + Jv(Uk, K(uk))Ku(uk) = 0.
This can be expressed within the decomposition frame.work of Section 2 by defining f(u) as the necessary conditions of (17) and (18)
,,, A IIO,fo(x*, x*)-'llllo,f(x*; ~)
f(u) = .l,(u, K(u)) + J~(u, K(u))K~(u) = 0.
- O,fo(X*, x*)ll < 1.
Theorem 2 then gives the desired conclusions. [] For an arbitrary problem which can be expressed according to the assumptions of Theorem 3, this result states that sufficiently weak coupling will result in a convergent algorithm. Thus, Theorem 3 emphasizes two points. First, it serves to reinforce the motivation behind the definition and choice of a natural decomposition. Because the natural decomposition is motivated by the ultimate weak-coupling situation, it is reassuring that such insight and intuition leads to a reasonable decomposition structure. Secondly, it emphasizes that the concept of weak coupling of decomposition algorithms is truly related to both the problem and the decomposition. However, the theorem shows it is possible to take advantage of any weak-coupling structure inherent to a problem by defining an appropriate decomposition. 5. The interaction prediction method The three preceding sections only considered abstract definitions of weak coupling and natural decompositions. This section attempts to illustrate these concepts as applied to a particular decomposition algorithm, the interaction prediction method (Cohen, 1977; Singh, Hassan and Titli, 1976). Although the original motivation for this decomposition was based on the separability of the implicitly defined cost, Cohen (1977) demonstrated that this assumption is not fundamental to the algorithm. We will show that the interaction prediction algorithm is a natural decomposition for weak interactions. Also, it will be demonstrated that a quadratic modification proposed by Cohen (1977) can be interpreted as modifying the decomposition in such a way as to lessen the importance of the interactions by strengthening the subproblems (i.e. by weakening the normalized coupling strength). This emphasizes the importance of the dependence of the coupling strength on the choice of the decomposition algorithm. Let ~ and ~" be Banach spaces and let J: 0//x ~'--+R and K: °I1-,~ be twice continuously Frtchet differentiable functions. Consider the following problem
(17)
u, v
fo(u, fA) = J~(u, K(li)).
The ultimately weakly coupled problem for (17) and (18) would be a constant interaction K(u) = ~, Y u E cll.
(23)
A natural parameterization of this structure is to choose a as any continuous parameterization of the nonlinearity K such that a is zero whenever (23) is satisfied. Then, for a = 0 it is easily seen from (21) and (22) that fo(u,~) =f(u).
(24)
That is, f0 as defined by (22) is a natural decomposition. Theorem 3 implies that if the variation of K with u is: sufficiently small (reflected by a as defined above), the iteration defined by (20) will converge. This can also be confirmed by computing to directly and using Theorem 2. Define A a=J~(u*, K(u*)) B A Ku(u*)
C A r..(u*, u*) D A A~(u*, K(u*))
E a__Y~(u*, K(u*)) P a=Jr(u*, K(u*)). Then =
A
(25)
Oil(u*) - alfo(u*, u*) = B*DB + E*B + B*E + Cp. Hence, w as defined in (7) is
v = K(u).
(18)
In (17), the cost functional Y is interpreted as the cost (implicitly) defined by the original problem cost functional and the subsystem constraints. Equation (18) embodies the interactions between subsystems. The variable v is the interaction variable. The assumption of multiple subsystems is not germane to the following analysis except for the interpretation of the interaction constraint and interaction variables. Thus, we will assume only a single subsystem (keeping in mind the noted interpretations) to avoid the inherent additional notational complexity associated with several subsystems. The appfication of the interaction prediction principle to (17) and (18) defines the following iteration in u (Cohen, 1977) min {J(uk+l, K(uk)) + Jv(uk, K(u~))K,(u~)uk+l}.
A U T Vol. 18, No. 4=-G
(22)
(26)
subject to
llli+ i 15 ~ll
(21)
Then, (20) is produced by the splitting function
O]fo(u*, u*)
min J(u, v)
(20)
(19)
o, = IIA-'IIB*DB + E * B + B * E + cpll.
(27)
For K~(u*) and Kuu(u*) sufficiently small (i.e. for K approximately constant near u*), w will be less than unity and the iteration will converge. It is interesting to examine the quadratic modification suggested by Cohen (1977) within this context of weak coupling. The effect of this modification is to add a quadratic operator to M to (25) and subtract M from (26). Hence o, = I[(A + M)-1IIIIB*OB + E*B + B*E + Cp - MII. (28)
From (38), we can see that one useful criterion for choosing M might be to choose the operator in such a manner as to weaken the normalized interaction to. This can be done either by reducing the magnitude of the interaction or (more likely) by increasing the dominance of the subproblem solution.
470
Brief Paper
6. Example
(35)
y = Cx + O.
The purpose of this section is to illustrate the concepts presented in Sections 2-4 with a specific example. The problem of estimating traffic density and velocity on a freeway from density measurements will be considered. A dynamical model for freeway traffic behavior has been developed by Payne (197 1). This model describes traffic flow behavior in terms of aggregate, fluid analog variables which are interpreted as the spatial mean speed and spatial mean density on sections of the freeway. If a freeway is partitioned into N sections and indexed in the direction of traffic flow (i.e. section 1 is the furthest upstream section) then the evolution of the spatial average density p~ and velocity v~ on the ith link is given by
[~i i)i = -
piVi -- pi-lVi-I ~_ ~Pi li
The steady-state Kalman filter based on (34) and (35) is given by = A~ + H(y - CJ)
(38)
The Riccati equation (38) can be solved iteratively using the decomposition technique of Section 2. To illustrate this, consider the partition of (34) into two subsystems. The corresponding decomposition of the Riccati equation was first proposed by Sandell (1976). Let
(29)
rA,, ,,,:1
o],
A = LA21 A22J'
j -~[vl-v<,(pi)]
li
(37)
0 = A P + P A T + '~ - p C T O - I C P .
1 [vi-vi-i]_ 1
viL
(36)
H = p c T o -I
[P,,
e,2]
P = LP21 P22J +"T f±lrp,-p,+,l Lp, J L ~ J + f~'
(30)
where lr is the length of the ith section, v,~(p~)is desired steady-state velocity,f and T and v are which reflect the drivers' sensitivity to traffic The vector ~ = [~1~1 ... ~pN~oN]r is assumed to white noise process with
E{~(t)~r(~-)}= =.a(t
-
where the above partitioning corresponds to the partitioned system, and assume that the covariances are block diagonal
the drivers' coefficients conditions. be a vector
~).
":=[0'
0],
0=[00, 0].
The iterative equations are then (Sandell, 1975)
(31)
i
i
T
i
i-I
Equations (29) and (30) also apply to the initial and final links with
i
A H P II + P NAN + QI - P NSIP II + i-I
T
i-I
i-I
i-I
i-I
--
{AI2P21 +PI2 A I 2 - P l 2 $2P21 } - 0 i
i
T
i
(39)
i
A22P 22+ P 22A22+ Q2 - P z2SzP 22+ i I
i-I
T
{A21Plz + P21 A21-P21 SIPt2 } = 0
(40)
A
1)0= 1)I
( A t , - P',,S,)P'~2 + P ~2(A22- P~2S2}r + A,2P~2 + P ~tA~ = 0
povo ~- input flow (given) PN+! ~ PN"
Equation (29) represents the conservation of flow on the freeway. Equation (30) is obtained from a finite element approximation of a continuous flow model. Additional details may be found in Payne (1971). It will be assumed that a noisy density measurement is available for each section Yi =
Pi + 0i.
(32)
The vector O = [0~... 0~] T is assumed to be a white noise process with E = {o(t)or('~)}
= O~(t
-
.r)
(33)
where ~(t - ~) is the Kronecker delta function. The problem considered is to estimate the deviation of the traffic variables (density p~ and velocity v~) from their nominal values P~,omand Vi~om.One solution to this problem is to use a Kalman filter based upon the linearization of (29) and (30) around the nominal values P~,omand v~,om. Let
X
A [(Pl -- Plnom)(Pl
Y tt [ ( Y l - -
Plnom)...
- - P l n o m ) • • • (IJN" - - VN'nom)] T
where Si --- CTO~-ICi. Equations (39) and (40) are Riccati equations which can be solved simultaneously for the subsystem covariances PN and P22 given the most recent estimate of the covariance between subsystems P~2. Equation (41) is a linear matrix equation for P~2 which can be solved once the solution of (39) and (40) is complete. This can be interpreted as a hierarchical iteration with the supremal computing subsystem interactions [equation (41)] and the infimal solving subsystem estimation problems [equations (39) and (40)]. This can be formulated as in Section 2 with .f0(P i, P~-~) given by the non-bracketed terms in (39)-(41) and ft(pi-i, p H ) given by the bracketed terms F AI2P21 + P12A~2- p12S2p21 fi(P'')=LA2iPi'+P2iAr-p2isiP'2].
(42)
It is easily verified that as IIA& approaches zero so does Pl*2 also approaches zero). Also, f0 is a block lower triangular operator with 8tf0(P*,P*) being nonsingular whenever the filter dynamics are stable. Hence, equation (6) will be satisfied for sufficiently small IlAl:[I and the iteration defined by (39)-(41) will converge. This algorithm was applied to a four-section linearized freeway model with
Ila,f,(P*, P*)II (since
Pi,om 74 veh./mile (46 veh'/km)l i = 1. . . . . 4. vinom= 55 mile/h (88.5 kin/h) J
( Y N - - P N n o m ) ] T'
The state vector and measurement are approximately described by the linear dynamical system = Ax + ~
(41)
(34)
tSee Looze (1975), p. 42 for an exact description of this curve.
These are reasonable values from moderate traffic on a threelane U.S. freeway. The desired velocity function vei(p~) corresponds to such a freeway and is defined as in Looze (1975). The driver reaction and sensitivity coefficients are T = 5 s (1.39x 10-~ h) v = 18.75 mileZ/h (48.56 kmZ/h).
Brief Paper To introduce an inhomogeneity into the system (and hence a physical motivation for the decomposition), the lengths of the links were taken to be l~ = 0.2 mile (0.32 kin), i = 1, 2, 4 13-= 0.5 mile (0.8 kin).
(43)
It can be seem from (29) and (30) that as 13--,oo the freeway equations decouple into two four-dimensional subsystems consisting of sections 1-2 and 3-4, respectively. Hence a natural decomposition of the Riccati equation (38) with respect to the structure induced by (43) defines an iteration of the form (39)-(41). The iteration (39)-(41) was used to solve the filtering Riccati equation (38) for the freeway system described in the preceding paragraphs.'t The iteration converged (i.e. lip1_ p ~-'IIF-< 10-6 with I['I[Fdenoting the Frobenius norm) in five iterations. The Riccati equation (38) was also solved using an algorithm developed by Lanb (1979). The difference between the iterative solution PI and the direct solution PD is IIPI - PolIF = 4 × 10 -1°.
7. Conclusions The main concept of this paper is that weak coupling in the context of hierarchical control is necessarily expressed in terms of both the problem structure and the decomposition used. Defined in this manner, weak coupling becomes a natural criterion for local convergence of the hierarchical iteration. This definition of weak coupling is used to motivate the class of natural decompositions, i.e. those decompositions which directly exploit the system structure. The main result of the paper was to show that natural decompositions will converge locally if the underlying system structure is sufficiently weakly coupled in the usual sense. This result is applicable to several hierarchical algorithms. In particular, it was shown that the interaction prediction algorithm is a natural decomposition. It is also directly applicable to decompositions of the form proposed in Looze and Sandell (1977), as is demonstrated by the example of Section 6. Acknowledgements--This research was supported in part by ONR Contract N00014-76-C-0346; in part by the U.S. Department of Energy, Division of Electrical Energy Systems under Grant ERDA-E(48-18)-2087 at the Massachusetts Institute of Technology; in part by the National Science Foundation under Grant NSF ENG-79-08778; and in part by the Joint Services Electronics Program under Contract N00014-79-C-0424 at the University of Illinois. ?The numerical data are available from the authors.
471
References Aokl, M. (1978). Some approximation methods for estimation and control of large scale systems. IEEE Trans Aut. Control, AC.23, 173. Bailey, F. N. and H. K. Ramapriyan (1973). Bounds on the suboptimality in the control of linear dynamic systems. IEEE Trans Aut. Control, AC-18, 532. Bennett, W. H. and 3. S. Baras (1980). Block diagonal dominance and design of decentralized compensators. Preprints of the IFAC Syrup. on Large Scale Systems: Theory and Applications, Toulouse, France, pp. 93-102. Cohen, G. (1977). On an algorithm of decentralized optimal control. J. Math. Analysis and Appl., 59, 242. Cohen, G. (1978). Optimization by decomposition and coordination: A unified approach. IEEE Trans Aut. Control, AC-23, 222. Findeisen, W. (1968). Parametric optimization by primal method in multilevel systems. IEEE Trans S.S.C., SSC-4, 155. Kokotovic, P. V. (1972). Feedback design of large linear systems. In J. B. Cruz, Jr. (Ed.), Feedback Systems, Chapter 4. McGraw-Hill, New York. Lanb, A. J. (1979). A Schur method for solving Riccati equations. IEEE Trans Aut. Control, AC-24, 913. Looze, D. P. and N. R. Sandell, Jr (1977). Decomposition of linear, decentralized, stochastic control problems. IFAC Workshop on Control and Management of Integrated IndustriafComplexes, Toulouse, France. Looze, D. P. and N. R. Sandell, Jr (1981). Analysis of hierarchical algorithms via nonlinear splitting functions. Y. Optimiz. Theory and Appl. 34, 3. Looze, D. P. (1975). Decentralized Control of a Freeway Tranic Corridor. S.M. thesis. Massachusetts Institute of Technology, Cambridge, MA. Milne, R. D. (1965). The analysis of weakly coupled dynamic systems. Int. J. Control, 3, 171. Payne, H. J. (1971). Models of freeway traffic and control. In Mathematical Models of Public Systems (Simulation Councils Proc., 1, No. 1). Pearson, J. D. (1971). Dynamic decomposition techniques. In D. A. Wismer (Ed.), Optimization Methods for Large Systems. McGraw-Hill, New York. Sandell, N. R., Jr (1976) Decomposition vs. decentralization in large scale system theory. 1976 IEEE Conf. on Decision and Control, Clearwater, FL. Singh, M. G., H. F. Hassan, and A. Titli (1976). Multilevel feedback control for interconnected dynamical systems using the prediction principle. IEEE Trans. Syst., Man, Cybern., SMC-6, 233. Singh, M. G. and A. Titli (1975). Closed-loop hierarchical control for nonlinear systems using quasilinearization. Automatica, 11,541. Takahara, Y. (1965). A multi-level structure for a class of dynamical optimization problems. M.S. thesis, Case Western Reserve University, Cleveland, OH.