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Matrix scaling for large-scale system decomposition
Pergamon
PII:
Aukmarico, Vol. 32, No. 8, pp. 1177-1181, 1996 Copyright 0 1996 Elsevier Science Ltd Prinled in Greal Britain. All rights reserved OOOS-lC98/96 $lS.OO + 0.00
sooo5-1098(%)ooo18-0
Brief Paper
Matrix Scaling for Large-scale System Decomposition* JOHN D. FINNEYT and BONNIE S. HECK+ Key Words-Large-scale
systems; decentralized systems.
r-decompositions of the associated system, thus removing the need for a separate c-decomposition algorithm. Following a brief review of graphs and c-decomposition, the results concerning scaling are detailed. After a review of max-balancing, the max-balancing algorithm found in Schneider and Schneider (1991) is rewritten to reveal its applicability to e-decompositions. Finally, it is shown how max-balancing enhances the c-decomposition when used for decentralized control subsystem identification.
Abstract-Many large-scale systems exhibit the structure of weakly connected components. In such cases, proper identification of weakly coupled subsystems will add insight into large-scale system behavior, and aid in related tasks such as the design of decentralized control. e-decomposition is a well-known efficient graph-theoretic algorithm for achieving a complete set of nested decompositions of a large-scale system. This paper shows how system matrix scaling can affect these decompositions, and determines that a system is properly scaled for e-decomposition when it is max-balanced, a property associated with weighted directed graphs. Also, it is shown that an existing algorithm for max-balancing can be altered slightly to return the complete set of Edecompositions, thus removing the need for two separate algorithms. Finally, the advantages of max-balancing before decomposition are shown for the application of decentralized control subsystem identification. Copyright 0 1996 Elsevier Science Ltd.
2.
r-Decomposition
Let M be an n X n real matrix
M = [q], i, j = 1,2, . . . , n. Given a positive real number E >O, M is said to have an e-decomposition if there exists an n x n permutation matrix P such that h? g PTMP is an N x N block matrix, 1 < N 5 n, with ail elements in off-diagonal blocks of A less than E in size, i.e.
1. Introduction Epsilon decomposition (or c-decomposition) was first introduced by Sezer and Siljak (1986) as an efficient means of decomposing a large-scale dynamical system into weakly coupled subsystems. Such a decomposition is useful, since it is not only adds insight into the behavior of the system but may also result in several smaller and more manageable analysis and control design problems. A graph-theoretic algorithm, e-decomposition determines subsystem components by identifying the weak coupling links in the system; the graph formulation of the problem results in a rather fast and efficient algorithm. In addition to subsystem identitication for decentralized control (Siljak, 1991), c-decomposition has been used for large-scale stability tests (Siljak, 1991) and parallel computation of the solution to nonlinear equations (ZeEeviC and Siljak, 1994). As noted by Sezer and Siljak (1986) e-decomposition has a drawback when used with systems where scaling of quantities such as states or solution variables is allowed. Namely, different scaled versions of the same system may yield different e-decompositions. The question of what scaling is appropr$te or optimal was left open in Sezer and Siljak (19%) and Srljak (1991). In this paper we examine the scaling issue and find that the system is properly scaled for decomposition when an associated weighted direct graph is max-balanced. The properties of max-balanced grap and algorithms for max-balancing have been detailed in Schneider and Schneider (1991) and Rothblum et al. (1992). Further, we prove that the process of max-balancing an arbitrary graph yields all the necessary information to identify all
where
m,,
zoG
,...)
(vi, “,) 1W((“,,
E
“j))
u,],
(1)
E, = mj,.
That is, each directed edge corresponds in position and weight to a nonzero entry in M. Note that a nonzero diagonal element of M corresponds to a directed edge (u,, u,). Such an edge is called a loop. Let D = (V, E) be an (unweighted) directed graph. A subgruph of D is a digraph D’ = (V’, E’) where V’ c V and E’ GE. An induced subgraph of D is a subgraph D[V,]=(V,,(V,xV,)flE) where V,cV. That is, D[V,] consists of vertex set V, and all edges from E that connect elements of V,. A pair of vertices u, and u, are said to be connected if any of the following three conditions are true: (i) u, = vi, (ii) {(vi, vi), (vi, u,)} tl E Z 0 or (iii) 3u, E V such that vi and uli are connected and u, and uk are connected. If all vertex pairs
* Received 1 July 1994; received in final form 15 December 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor M. Ikeda under the direction of Editor A. P. Sage. Corresponding author Professor Bonnie S. Heck. Tel. +l 404 894 3145; Fax +l 404 894 4641. t ABB Power T&D Company Inc., Transmission Technology Institute, 1021 Main Campus Drive, Raleigh, NC 27606-5202, U.S.A. t School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250, U.S.A. 1177
Brief Papers in a digraph are connected, the graph is said to be connected. Otherwise, the digraph can be separated into connected subgraphs. Specifically, connectedness is an equivalence relation on V with corresponding partition P = {V,, V,, , VNc} such that vertices v, w E V, for some I if and only if v and w are connected. The induced subgraphs O[V,], I = 1,. . . , NC, are called the components of D: components may be thought of as maximal disjoint portions of the digraph. If all vertex pairs in a digraph are connected, the graph is said to be connected. A path in D from “, to “, is a sequence of adjacented directed edges (vi, vkl), (I+,, v,+), , (I+,, vi) where each vertex v,, vi, uI: , , vk, is distinct. vi is said to reach uj in D (or v, is rencha b le from vi) if there is a path in D from u, to V~ For the purposes of this paper, by default we say a vertex v, reaches itself. Two vertices “, and vi are said to be strongly connected if u. reaches v; and v; reaches v.. A cvcle is a sequence of ddges (vk,,ikZ), (&,, vk,), .: (I++,vi,) connecting distinct vertices, i.e. a ‘closed path’. If all vertex pairs in a digraph are’strongly connected, the digraph is said to be strongly connected. Sfrong connectedness is also an equivalence relation, which partitions V into disjoint sets. The induced subgraphs of these sets are called strong components of D. Note that strong connectedness implies connectedness, and thus a strong component is a suhgraph of a component. Given a digraph D = (V, E, w), let P = {V,, , V,} be any partition of V. Since the equivalence classes V, form a set, the partition can be viewed as a new vertex set. Define EP = {(V,, Q:vk E 5, u, E Vi, (vk, v,) E E, Vj# E}. Further define a+:Ep+R by w,((V;, ~))=max{w((~~,“,)):u, E V,, “/ e v,, (v~, v,) E E, 5 # t$}. Then the digraph Dp = (f, Ep, wp) is the condensation of D with respect to partition P. That is, Dp is the digraph formed by using each equivalence class in the partition as a vertex, with an edge (l$ 5) wherever the original graph had an edge from an element of V; to an element of l$. The weight of that directed edge is taken as the maximum over all such edges in the original graph. Fast, efficient algorithms to determine the cycles, components and strong components of graphs and digraphs are well known, and can be found for example in Tarjan (1972). 2.2. Nested e-decompositions. Following the development in Sezer and &ljak (1986), the e-decomposition of a square matrix A E R”“” can be found as follows: Form a matrix M = [q] i,j = 1,2, , n.
from A = [aJ] by m,,:= la,,/,
Form D = (V, E, w) from the matrix M; Form D’:=(V,
E’, w), where E’={e
E E:w(e)ze}.
Idenfity components
of D’ with corresponding partition , V,}. The partition yields the number and makeup of the diagonal blocks of M (and hence A). V = {V,, V,.
The definition of e-decomposition in this paper differs slightly from that of Sezer and Siljak in that E’ as defined above allows w(e) = Q, as opposed to a strict inequality. Observe that e-decompositions are nested, that is if two vertices v, and v, are grouped together for li > 0, they remain toeether for all E < E,. Sezer and Siljak used this notion to “develop an efficient algorithm that- generates all possible e-decompositions. The nested c-decomposition algorithm of Sezer and Siljak is outlined as follows:
returning partition P, = Iv,,,, , v,.,,,}. That is, the elements of P, are the disioint subsets of connected vertices of D:? Step 3. Termination condition: If P, has only one element, set k = i and stop. Step 4. Condensation: Set 4+,:=(1/4+,, E,,,, wi+,) equal to the condensation of D, with respect to c, Step 5. Iterate: Set i:=i + 1; return to Step 1. The result of executing the algorithm is a sequence of coupling strengths l, > l2 > > lk and corresponding partitions Pi, Pz, , Pk. By tracing the path of the original vertices vi,. , v, as they are collected in the partitions, the k < n r-decompositions are apparent. The necessary bookkeeping has been omitted for simplicity; see Sezer and Siljak (1986) for details. 3. Sculing Many systems that can be decomposed with l decomposition will permit a scaling of the system variables (such as states or solution variables). In terms of the matrix description, this corresponds to replacing M by a new matrix: MS= diag(s))’ M diag(s), where s = [s, s2 s,], s, >O, i = 1, , n, is a length-n vector of positive real numbers, and diag (s) is a n X n diagonal matrix with the elements of s along the diagonal. If D = (V, E, w) is the weighted digraph associated with M then the weighted digraph for MS is given by R = (V, 6 wJ>
(3)
Note that the structure of the new graph remains the same; only the weights have changed. A simple example illustrates how scaling can affect e-decomposition. Consider the matrix M=
*
:e
+c
*
26 2~ i 86 &e * I
The weighted digraph is depicted in Fig. l(a) (the edges corresponding to the diagonal elements of M have been omitted). The two edges of weight 2e prohibit any e-decomposition. (Although it can be shown that an overlapping decomposition does exist for this example, this paper considers only disjoint decompositions.) Now examine the graph shown in Fig. l(b), corresponding to MS, where s = [4 4 l] is the scaling vector. Now all edges are of weight be, and an r-decomposition separates the system into three subsystems. Equation (4) shows that by selectively choosing the components s, and si of the scaling vector, the scaled weight of any one given’edge can be made as large or as small as desired. Since different scalings result in different decompositions, the issue is what scaling is appropriate for e-decomposition. In the nested e-decomposition algorithm above, at each stage the next decomposition is determined by the largest weight in the digraph. Three observations can be made:
INPUT: Weighted digraph D = (V, E, w). OUTPUT: k coupling levels E, > l2 >. . . > ck and corresponding vertex partitions Pi, P2, , Pk. Step 0. Initialize:
Set
i:=l.
Set
D, = (V,, El, w,):=
(V, E, ~1.
(a)
Step 1. Find Strongest Coupling: Set e, := max,, E, wi(e) Step 2. Decomposition:
Perform
q-decomposition
on Di,
(b)
Fig. 1. Effect of scaling on weighted digraph: (a) before scaling; (b) after scaling.
1179
Brief Papers Scaling can make any non-zero weight the largest; improper scaling will result in artificially high coupling in the digraph and misleading decompositions. On the other hand, scaling may be used to lower the maximum weight of the digraph. When the maximum weight is minimized, the set of vertices coupled by edges with this weight exhibit the strongest coupling in the graph, free of distortions from artificially high coupling. When decomposing a large-scale system into weakly coupled components, it is advantageous to identify large numbers of groups with low levels of coupling. To obtain a complete nested set of r-decompositions with low levels of coupling li, it is therefore important that the l, at each stage be as small as possible. Definition 1. A weighted digraph is said to be min-max scaled if the largest weight in the graph is minimized with
respect to all possible scalings, and the number of edges with this maximum weight is also minimized. That is, D = (V, E, w) is min-max scaled means that for all allowable scaling vectors s (i) w,,, g max,..(w(e))smax,.E(~~(e));
and
(ii) I{eE E: w(e) = w,,,}J 5 I{e E E: w,(e) = wmax, s E smin~l~ where Smin is the set of all scaling vectors achieving equaltiy in the previous condition, and I{.}]denotes the number of elements in the set {.}. The following lemma gives a necessary condition for a graph to be min-max scaled.
and sufficient
2. Let D = (V, E, w) be a weighted graph with maximum weight E,,, = maxeEE w(e). D is min-max scaled if and only if the components of D’mar are strongly connected. Lemma
Proof Suppose D4m= has a component that is not strongly connected. Then that component has two vertices vi and uj such that in Dfm=, v, is reachable from vi but not vice versa. Let W c V be the set of all vertices from which vi is reachable in DCmal. Define two sets e+(w) g {(“k, u,)
E
E:u*
E
w, u,
v-
E
W},
S-(W)~{(“,,“,).E:“kEV-W,u,EW}.
(5)
Note that S’(W) has at least one edge e, with w(e,) = emax. If C(W) is nonempty, set o = max,,,-(w, w(e); otherwise, set o = +cmax. Note that (2 < E,,,.~. Form the scaling vector s as follows:
sk= t
(“k WI, fs,,Ja)~” vE
(u,
The resealed graph D, = (v, w,(e)
= w(e)
1
E
W).
E, WJ has
if e 6 S+(W) U 6-(W), if e E S+(W)US-(W).
(7)
In particular, w,(e,) < w(e,) = emax, implying that D = (V, E, w) is not min-max scaled. Thus 3 is shown. (e) Given that the components of D’mua = (V, Em=, w) are strongly connected, assume D is not min-max scaled. Thus either the maximum weight in the digraph is too large or the number of edges with the maximum weight can be decreased. Then for some scaling s there exists at least one directed edge ez E {E: w(e) = emaX}- {E: w5(e) = emax} such that
yea;
w&2)
<
emax,
(8)
w&J
5
hax,
(9)
Because ez belongs to a strongly connected component, it is a directed edge on some cycle C = (e,,, with w&2)
e,2..
. , e,,)
e, E Pm=, i = 1,2,. . , 1. Under Observe that the product < emax.
the scaling s, of edge weights
around any cycle remains unchanged after any scaling. Thus there must be some edge e on cycle C such that w,,(e) > Q,,,, contradicting (9) above. By contradiction, D must be min-max scaled. (An alternate proof to the one given here can be developed using the notion of level sets as detailed in Rothblum et al. (1992); we proceeded as above to avoid cl introducing new terminology.) For any weighted digraph D = (V, E, w), the definitions in (5) can be applied to any vertex subset W c V. Thus applied, the two sets represent the edges connecting the subgraph induced by W to the rest of D. In Schneider and Schneider (1991) these sets are used to define a special property of weigthed digraphs. Definition 3. A graph D = (V, E, w) is said to be mux-bafunced if for every W E V, S,&,(W) = 6&,(W),
where G,,(W)
G,,(W) p max,,6+(wj w(e) g max,,6-(w) w(e).
and
In Schneider and Schneider (1991) and Rothblum ec al. (1992) the properties of max-balanced digraphs are investigated, and efficient algorithms to find a scaling that max-balances a digraph are given. We now show that max-balancing is particularly appropriate for nested edecompositions. Theorem 4. In the nested r-decomposition algorithm each of the digraphs D,, i = 1,2,. , k, are min-max scaled if and only if the original graph D = (V, E, w) is max-balanced. Proof: (3) Assume D is not max-balanced. Then there is a W c V such that 6,&,(W) # 6&,(W). Without loss of generality, let 6,&,(W) > 6,&J W). Further, 3u, t W, uj E V - W such that w(ui, vi) = a;,,. It follows that in the nested r-decomposition algorithm for some 1, e, = 6&,,(W). In D;f,
the vertices representing the equivalence classes containing u, and uj are connected but not strongly connected, since 6&J W) < 6,. Thus, by Lemma 2, D, is not min-max scaled. ( G ) Assume D, is not min-max scaled for some 1. By Lemma 2, there exist equivalence classes uli and uli that are connected but not strongly connected in D;‘; without loss of generality, assume I+; reaches V,j in Df’ (but not vice versa). Let W be the union of all vertices in Df’ that reach Q. By construction, D has 6&J W) < 6,&,,(W) = l,, and thus D is not max-balanced. q This section has been primarily limited to the discussion of digraphs. Recall from (1) and (2) that square matrices and their weighted digraphs are interchangeable; hence we shall speak of their scaling and e-decomposition interchangeably. Also, because e-decomposition only involves the size of the matrix elements, we can assume nonnegative-valued matrices without loss of generality. Further, we shall now restrict our attention to digraphs that are strongly connected. A digraph that is not strongly connected corresponds to a matrix M that is reducible, i.e. a matrix that can be permuted to assume the form
M=[ii::
:,I.
Because of the zero submatrix, this system can be scaled to make M,, as small as desired, separating the system into two isolated components. System decomposition can then proceed separately on the two subsystems M,, and M,,. Thus, without loss of generality, we require all matrices to be irreducible, that is, all digraphs to be strongly connected. 4. Max-balancing algorithm In the previous section it was shown that the only way a digraph will be free of artificially high coupling for all c-decompositions is if it is max-balanced. Thus it appears that when system scaling is performed, two steps are needed: first max-balancing and then e-decomposition. It is shown now, however, that all the calculations required for generating the set of nested c-decompositions are an integral part of the max-balancing procedure. We shall do this by
Brief Papers carefully rewriting the original algorithm found in Schneider and Schneider (1991) in such a way that we can compare it conveniently with the nested e-decomposition algorithm seen earlier. The max-balancing algorithm found in Schneider and Schneider (1991) uses the concept of the mean of a cycle. For a cycle C in a weighted digraph D = (V, E, w), let ICI denote the number of edges making up C. Then we can define the (geometric) mean of C, denoted W(C), by G(C) p
(,g w(e))“‘c’.
WV
If D is not acyclic, we define the maximum cycle mean of D as mcm (D) = max{@(C):C E cycles(D)}, (11) where cycles (D) denotes the set of all cycles in D. If D is acyclic, we define mcm (D) = 0. The algorithm given below is essentially the same as that discussed in detail in Schneider and Schneider (1991). Our scaling is multiplicative, whereas the original algorithm was written for additive scaling; the formulas used below have been adjusted accordingly. Although the algorithm requires the digraph to have no loops, this does not present a restriction, since scaling does not affect the weight of loops. INPUT: Strongly
connected
weighted
digraph
D =
(V, E, w) (with loops removed).
OUTPUT:
Scaling vector s that max-balances D; k maximum cycle means a, > a2 >. . akr and k corresponding partitions II,, I&, . . . , lIk.
Step 0. Initialize: Set i := 1, D, := D = (V, E, w) and a0 = 0. Step 1. Find Maximum Mean Cycle: Using equations given in the Appendix, identify a cycle Ci with ai 4 W(C) = mcm (DJ. If more than one such cycle exists, choose one arbitrarily. Step 2. Rescale: If oi # ai_,, solve for the scaling vector s’ using equations given in the Appendix; rescale Di with scaling s’ to get Dd. If ai = a;_,, set D,i:= Dti Step 3. Partition: Form’a partition Iii where all vertices on the maximum mean cycle C, belong to one equivalence class and every other vertex is its own equivalence class. If Iii has only one element, proceed to Step 6. be the condensation Step 4. Condense: Let D,, respect to the partition I&.
of D, with
If oi # ai-, , set Di+, := Dtemp, increment and go to Step 1. Otherwise, set Di = Dtemp, and go to Step 1.
Step 5. Iterate: i:=i
+ 1,
Step6. Terminate: k:=i-1,
If aifui_,, and&:=&
set k:=i; Stop.
if (Y~=(Y,_,, set
The following lemmas, found with proofs in Schneider and Schneider (1991) describe properties of the algorithm. Lemma 5. At each iteration, the resealed digraph Dtemp has
the following properties: (i) each directed edge on the maximum mean cycle Ci assumes a weight exactly equal to the maximum cycle mean; (ii) no other edge in the digraph Dtemp has weight larger than the maximum cycle mean. Lemma 6. The max-balancing scaling vector s is unique a positive scalar.
up to
The following theorem shows why max-balancing already does all the work of the nested c-decomposition algorithm. Theorem 7. Let D = (V, E, w) be a weighted digraph. Let ai and I$, i = 0,. . , m, be the distinct maximum cycle means
and partitions respectively returned from max-balancing D. Let 6, and I$ j = 1,. , k, be the coupling levels and partitions respectively returned from subsequently performing the nested a-decomposition algorithm. Then k = m and for all i = 1,. , k, E, = ai and P, = II,. Proof Note that because scaling does not affect the cycle mean of any cycle, max-balancing a graph twice returns exactly the same cycle means and partitions the second time as the first. Thus to show that e-decomposition after max-balancing is redundant, it is enough to show that the partitions and lk from the nested e-decomposition algorithm can be found from max-balancing a max-balanced graph. Assume for some i that II_, = P,_,; clearly this is true for i = 1 when both are the discrete n-member partition of V. Observe that if D is max-balanced then any condensation of D is also max-balanced. Let D, be the max-balanced condensation from the max-balancing algorithm. Because the D, is max-balanced, the largest weight in the digraph is equal to the maximum cycle mean ai, and thus cmax= (Y,. From Lemma 2, each component of D’max is strongly connected, and is thus either a single vertex or a union of cycles, each with mean ai. If only one such cycle exists then the partition returned from this iteration of e-decomposition is exactly that returned from the max-balancing algorithm. If m > 1 cycles with weight emBX exist then the max-balancing algorithm will return the same partition as the nested c-decomposition algorithm after m iterations of the max-balancing algorithm. Thus Pi = II;, and the theorem follows by induction. 0
A comment can be made here about the computational cost of scaling versus not scaling. For an n X n matrix, max-balancing requires approximately on the order of n times more calculations than a full e-decomposition, depending on the sparsity of the system. Therefore a system that is known to be well scaled should perhaps not be max-balanced if computation time is critical. While it is true that other scaling techniques such as line-sum scaling (Osborne, 1960) or the technique used in Ze&evif and Sdjak (1994) may offer acceptable results for specific problems with shorter computation time, the fact remains that only max-balancing will minimize the threshold coupling levels at each stage in the nested c-decomposition algorithm. The next section demonstrates just how advantageous max-balancing can be with e-decomposition. 5. Applications At this point we have seen that by reducing artificially strong coupling at all levels of decomposition, max-balancing will aid a-decomposition in partitioning a matrix M into weakly coupled blocks. Thus whenever an e-decomposition is performed on a system that permits scaling, max-balancing offers the potential for enhanced results. In Zehvif and Siljak (1994) c-decomposition is used in the parallel solution of nonlinear algebraic equations, and the technique is applied in Sezer and Siljak (1986) to identify subsystems for large-scale system stability tests. In Finney and Heck (1995) it is demonstrated that for both these applications max-balancing can greatly improve the results obtained. We now show an additional application of e-decomposition that is improved by max-balancing. Consider the large-scale LTI system i=Ax+Bu,
(12)
y = cx
The first step in designing a decentralized control scheme for this system is to decompose the system into weakly coupled subsystems. In Siljak (1991) e-decomposition is shown to be effective in determining the system decomposition as follows: create a matrix M defined by M=
r ;
f
1 ,
(13)
and create from M a weighted digraph D = (II U X U Y, E, w), where U = {u,, u2, , u,), X = {.r~l,~2, . , x,)
1181
Brief Papers and Y = {y,, y2, . , y,,} are the vertices corresponding to the m inputs, n states, and p outputs of the large-scale system. Given a decoupling parameter c, c-decomposition in D will decompose the system into N isolated subgraphs D: = D[Ui U Xi U I’J, i = 1,2, . . . , N, corresponding to N isolated LTI systems. See Siljak (1991) for more information and examples. Because the subsystems identified by t--decomposition will be used for control design, it is important that the isolated subsystems be controllable. However, it is easy to show that standard e-decomposition can result in subsystems that are not structurally controllable. Structural controllability is a necessary condition for controllability, and it is shown in Siljak (1991) that the system described above is structurally controllable if and only if each state is input-reachable and the digraph D,, = (U U X, E) has no dilation. A digraph D = (U U X, E) is said to have a dilation if there is a subset X,sX such that the number of vertices in D that reach at least one vertex in X, is smaller than the size of X,. Fortunately, the input/output mappings of the system (12) are unaffected by state scaling. The following result shows a clear advantage of max-balancing.
gratefully acknowledge the support of the Department of Education for a Graduate Assistance in Areas of National Need Fellowship. References
BollabC, B. (1979). Graph Theory: An Introductory Course. Springer-Verlag. New York. Finney, J. D. and B. S. Heck (1995). Matrix scaling for large-scale system decomposition. In Proc. American Conrrol Conf, Seattle, WA, pp. 2918-2923. Harary, F. (1%9). Gruph Theory. Addison-Wesley. Reading, MA. Osborne, E. E. (1950). On preconditioning of matrices. J. Assoc. Comput. Machinery, 7,338-345.
Rothblum, U. G., H. Schneider and M. H. Schneider (1992). Characterizations of max-balanced flows. Discrete Appl. Maths, 39,241-261.
Schneider, H. and M. H. Schneider (1991). Max-balancing weighted directed graphs and matrix scaling. Maths. Oper. Res., 16, 208-222.
Sezer, M. E. and D. D. Siljak (1986). Nested Edecompositions and clustering of complex systems. Automatica 22,321-331.
Theorem
8.
Let D = (U U X, E, w) be a weighted digraph
is maxDx e D[X] = (X, (X X X) II E, w) such that balanced. Let D, = (Ui U Xi, E,, w), i = 1,2, , N, be the subsystems identified by e-decomposition of D. If each vertex set Ui is nonempty then each isolated subsystem D, is structurally controllable.
Siljak, D. D. (1991). Decentralized Control of Complex Systems. Academic Press, Boston. Tarjan, R. (1972). Depth-first search and linear graph algorithms. SIAM J. Comput. 1,146-160. Zecevif, A. I. and D. D. Siljak (1994). A block-parallel Newton method via overlapping epsilon decompositions. SIAM .I. Matrix Anal. Applies, 15,824-844.
ProoJI Because Dx is max-balanced, so also is the digraph D,,= (Xi, (Xi XX,) nE, w). Since ui is non-empty, at least one vertex in X, is input-reachable. By Lemma 2, Di, is
strongly connected, implying each x E X, is input reachable. Further, since every vertex in X, reaches every other, D, can have no dilation. Thus 0, is structurally controllable. 0 Thus when max-balancing is used, the desired structural controllability is assured as long as at least one input is grouped with each subsystem. Dual results about structural observability follow directly from the section above. 6. Conclusions While e-decomposition is an efficient, powerful tool for large-scale system decomposition, we have seen that scaling system variables will result in non-unique decompositions. It was shown that for proper identification of the weakly connected subsystems, minimum-maximum coupling is desired at each stage of the nested e-decomposition algorithm, and that such a scaling occurs if and only if the system is max-balanced. Further, it was shown that an existing algorithm for max-balancing can be rewritten to generate the entire nested decomposition structure without additional computation. Acknowledgements-The authors gratefully acknowledge the support of the National Science Foundation for Grant ECS-9058140 and for an NSF Fellowship. The authors also
Appendix
This appendix contains the formulas necessary to calculate the cycle means and scalings used in the max-balancing algorithms of Section 4. They are taken directly from Schneider and Schneider (1991); the original presentation in that work was for additive and not multiplicative scaling, and the formulas have been adjusted accordingly. The maximum cycle mean mcm (D) of the digraph D = (V, E, w) with n vertices is found using the following recurrences: h)(u) = 0
for v E V,
(A.1)
Fk+du)=p=;$pw + log w(e)} r(D) = (mn; { ,5$,
for k = 0, 1,2,. . , n - 1, { ‘(vn) :
;(“)))j,
(A.2) (A-3)
mcm (D) = exp [t(D)].
(A.4)
Now compute the components of scaling vector s as s, = exp [ ,,5y2;_, IF,(u) - kr(D)1]
for u
l
V
(A.5)
To identify a cycle with the maximum cycle mean, one can use a depth-first search as described in Tarjan (1972).
PII:
Aukmarico, Vol. 32, No. 8, pp. 1177-1181, 1996 Copyright 0 1996 Elsevier Science Ltd Prinled in Greal Britain. All rights reserved OOOS-lC98/96 $lS.OO + 0.00
sooo5-1098(%)ooo18-0
Brief Paper
Matrix Scaling for Large-scale System Decomposition* JOHN D. FINNEYT and BONNIE S. HECK+ Key Words-Large-scale
systems; decentralized systems.
r-decompositions of the associated system, thus removing the need for a separate c-decomposition algorithm. Following a brief review of graphs and c-decomposition, the results concerning scaling are detailed. After a review of max-balancing, the max-balancing algorithm found in Schneider and Schneider (1991) is rewritten to reveal its applicability to e-decompositions. Finally, it is shown how max-balancing enhances the c-decomposition when used for decentralized control subsystem identification.
Abstract-Many large-scale systems exhibit the structure of weakly connected components. In such cases, proper identification of weakly coupled subsystems will add insight into large-scale system behavior, and aid in related tasks such as the design of decentralized control. e-decomposition is a well-known efficient graph-theoretic algorithm for achieving a complete set of nested decompositions of a large-scale system. This paper shows how system matrix scaling can affect these decompositions, and determines that a system is properly scaled for e-decomposition when it is max-balanced, a property associated with weighted directed graphs. Also, it is shown that an existing algorithm for max-balancing can be altered slightly to return the complete set of Edecompositions, thus removing the need for two separate algorithms. Finally, the advantages of max-balancing before decomposition are shown for the application of decentralized control subsystem identification. Copyright 0 1996 Elsevier Science Ltd.
2.
r-Decomposition
Let M be an n X n real matrix
M = [q], i, j = 1,2, . . . , n. Given a positive real number E >O, M is said to have an e-decomposition if there exists an n x n permutation matrix P such that h? g PTMP is an N x N block matrix, 1 < N 5 n, with ail elements in off-diagonal blocks of A less than E in size, i.e.
1. Introduction Epsilon decomposition (or c-decomposition) was first introduced by Sezer and Siljak (1986) as an efficient means of decomposing a large-scale dynamical system into weakly coupled subsystems. Such a decomposition is useful, since it is not only adds insight into the behavior of the system but may also result in several smaller and more manageable analysis and control design problems. A graph-theoretic algorithm, e-decomposition determines subsystem components by identifying the weak coupling links in the system; the graph formulation of the problem results in a rather fast and efficient algorithm. In addition to subsystem identitication for decentralized control (Siljak, 1991), c-decomposition has been used for large-scale stability tests (Siljak, 1991) and parallel computation of the solution to nonlinear equations (ZeEeviC and Siljak, 1994). As noted by Sezer and Siljak (1986) e-decomposition has a drawback when used with systems where scaling of quantities such as states or solution variables is allowed. Namely, different scaled versions of the same system may yield different e-decompositions. The question of what scaling is appropr$te or optimal was left open in Sezer and Siljak (19%) and Srljak (1991). In this paper we examine the scaling issue and find that the system is properly scaled for decomposition when an associated weighted direct graph is max-balanced. The properties of max-balanced grap and algorithms for max-balancing have been detailed in Schneider and Schneider (1991) and Rothblum et al. (1992). Further, we prove that the process of max-balancing an arbitrary graph yields all the necessary information to identify all
where
m,,
zoG
,...)
(vi, “,) 1W((“,,
E
“j))
u,],
(1)
E, = mj,.
That is, each directed edge corresponds in position and weight to a nonzero entry in M. Note that a nonzero diagonal element of M corresponds to a directed edge (u,, u,). Such an edge is called a loop. Let D = (V, E) be an (unweighted) directed graph. A subgruph of D is a digraph D’ = (V’, E’) where V’ c V and E’ GE. An induced subgraph of D is a subgraph D[V,]=(V,,(V,xV,)flE) where V,cV. That is, D[V,] consists of vertex set V, and all edges from E that connect elements of V,. A pair of vertices u, and u, are said to be connected if any of the following three conditions are true: (i) u, = vi, (ii) {(vi, vi), (vi, u,)} tl E Z 0 or (iii) 3u, E V such that vi and uli are connected and u, and uk are connected. If all vertex pairs
* Received 1 July 1994; received in final form 15 December 1995. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor M. Ikeda under the direction of Editor A. P. Sage. Corresponding author Professor Bonnie S. Heck. Tel. +l 404 894 3145; Fax +l 404 894 4641. t ABB Power T&D Company Inc., Transmission Technology Institute, 1021 Main Campus Drive, Raleigh, NC 27606-5202, U.S.A. t School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250, U.S.A. 1177
Brief Papers in a digraph are connected, the graph is said to be connected. Otherwise, the digraph can be separated into connected subgraphs. Specifically, connectedness is an equivalence relation on V with corresponding partition P = {V,, V,, , VNc} such that vertices v, w E V, for some I if and only if v and w are connected. The induced subgraphs O[V,], I = 1,. . . , NC, are called the components of D: components may be thought of as maximal disjoint portions of the digraph. If all vertex pairs in a digraph are connected, the graph is said to be connected. A path in D from “, to “, is a sequence of adjacented directed edges (vi, vkl), (I+,, v,+), , (I+,, vi) where each vertex v,, vi, uI: , , vk, is distinct. vi is said to reach uj in D (or v, is rencha b le from vi) if there is a path in D from u, to V~ For the purposes of this paper, by default we say a vertex v, reaches itself. Two vertices “, and vi are said to be strongly connected if u. reaches v; and v; reaches v.. A cvcle is a sequence of ddges (vk,,ikZ), (&,, vk,), .: (I++,vi,) connecting distinct vertices, i.e. a ‘closed path’. If all vertex pairs in a digraph are’strongly connected, the digraph is said to be strongly connected. Sfrong connectedness is also an equivalence relation, which partitions V into disjoint sets. The induced subgraphs of these sets are called strong components of D. Note that strong connectedness implies connectedness, and thus a strong component is a suhgraph of a component. Given a digraph D = (V, E, w), let P = {V,, , V,} be any partition of V. Since the equivalence classes V, form a set, the partition can be viewed as a new vertex set. Define EP = {(V,, Q:vk E 5, u, E Vi, (vk, v,) E E, Vj# E}. Further define a+:Ep+R by w,((V;, ~))=max{w((~~,“,)):u, E V,, “/ e v,, (v~, v,) E E, 5 # t$}. Then the digraph Dp = (f, Ep, wp) is the condensation of D with respect to partition P. That is, Dp is the digraph formed by using each equivalence class in the partition as a vertex, with an edge (l$ 5) wherever the original graph had an edge from an element of V; to an element of l$. The weight of that directed edge is taken as the maximum over all such edges in the original graph. Fast, efficient algorithms to determine the cycles, components and strong components of graphs and digraphs are well known, and can be found for example in Tarjan (1972). 2.2. Nested e-decompositions. Following the development in Sezer and &ljak (1986), the e-decomposition of a square matrix A E R”“” can be found as follows: Form a matrix M = [q] i,j = 1,2, , n.
from A = [aJ] by m,,:= la,,/,
Form D = (V, E, w) from the matrix M; Form D’:=(V,
E’, w), where E’={e
E E:w(e)ze}.
Idenfity components
of D’ with corresponding partition , V,}. The partition yields the number and makeup of the diagonal blocks of M (and hence A). V = {V,, V,.
The definition of e-decomposition in this paper differs slightly from that of Sezer and Siljak in that E’ as defined above allows w(e) = Q, as opposed to a strict inequality. Observe that e-decompositions are nested, that is if two vertices v, and v, are grouped together for li > 0, they remain toeether for all E < E,. Sezer and Siljak used this notion to “develop an efficient algorithm that- generates all possible e-decompositions. The nested c-decomposition algorithm of Sezer and Siljak is outlined as follows:
returning partition P, = Iv,,,, , v,.,,,}. That is, the elements of P, are the disioint subsets of connected vertices of D:? Step 3. Termination condition: If P, has only one element, set k = i and stop. Step 4. Condensation: Set 4+,:=(1/4+,, E,,,, wi+,) equal to the condensation of D, with respect to c, Step 5. Iterate: Set i:=i + 1; return to Step 1. The result of executing the algorithm is a sequence of coupling strengths l, > l2 > > lk and corresponding partitions Pi, Pz, , Pk. By tracing the path of the original vertices vi,. , v, as they are collected in the partitions, the k < n r-decompositions are apparent. The necessary bookkeeping has been omitted for simplicity; see Sezer and Siljak (1986) for details. 3. Sculing Many systems that can be decomposed with l decomposition will permit a scaling of the system variables (such as states or solution variables). In terms of the matrix description, this corresponds to replacing M by a new matrix: MS= diag(s))’ M diag(s), where s = [s, s2 s,], s, >O, i = 1, , n, is a length-n vector of positive real numbers, and diag (s) is a n X n diagonal matrix with the elements of s along the diagonal. If D = (V, E, w) is the weighted digraph associated with M then the weighted digraph for MS is given by R = (V, 6 wJ>
(3)
Note that the structure of the new graph remains the same; only the weights have changed. A simple example illustrates how scaling can affect e-decomposition. Consider the matrix M=
*
:e
+c
*
26 2~ i 86 &e * I
The weighted digraph is depicted in Fig. l(a) (the edges corresponding to the diagonal elements of M have been omitted). The two edges of weight 2e prohibit any e-decomposition. (Although it can be shown that an overlapping decomposition does exist for this example, this paper considers only disjoint decompositions.) Now examine the graph shown in Fig. l(b), corresponding to MS, where s = [4 4 l] is the scaling vector. Now all edges are of weight be, and an r-decomposition separates the system into three subsystems. Equation (4) shows that by selectively choosing the components s, and si of the scaling vector, the scaled weight of any one given’edge can be made as large or as small as desired. Since different scalings result in different decompositions, the issue is what scaling is appropriate for e-decomposition. In the nested e-decomposition algorithm above, at each stage the next decomposition is determined by the largest weight in the digraph. Three observations can be made:
INPUT: Weighted digraph D = (V, E, w). OUTPUT: k coupling levels E, > l2 >. . . > ck and corresponding vertex partitions Pi, P2, , Pk. Step 0. Initialize:
Set
i:=l.
Set
D, = (V,, El, w,):=
(V, E, ~1.
(a)
Step 1. Find Strongest Coupling: Set e, := max,, E, wi(e) Step 2. Decomposition:
Perform
q-decomposition
on Di,
(b)
Fig. 1. Effect of scaling on weighted digraph: (a) before scaling; (b) after scaling.
1179
Brief Papers Scaling can make any non-zero weight the largest; improper scaling will result in artificially high coupling in the digraph and misleading decompositions. On the other hand, scaling may be used to lower the maximum weight of the digraph. When the maximum weight is minimized, the set of vertices coupled by edges with this weight exhibit the strongest coupling in the graph, free of distortions from artificially high coupling. When decomposing a large-scale system into weakly coupled components, it is advantageous to identify large numbers of groups with low levels of coupling. To obtain a complete nested set of r-decompositions with low levels of coupling li, it is therefore important that the l, at each stage be as small as possible. Definition 1. A weighted digraph is said to be min-max scaled if the largest weight in the graph is minimized with
respect to all possible scalings, and the number of edges with this maximum weight is also minimized. That is, D = (V, E, w) is min-max scaled means that for all allowable scaling vectors s (i) w,,, g max,..(w(e))smax,.E(~~(e));
and
(ii) I{eE E: w(e) = w,,,}J 5 I{e E E: w,(e) = wmax, s E smin~l~ where Smin is the set of all scaling vectors achieving equaltiy in the previous condition, and I{.}]denotes the number of elements in the set {.}. The following lemma gives a necessary condition for a graph to be min-max scaled.
and sufficient
2. Let D = (V, E, w) be a weighted graph with maximum weight E,,, = maxeEE w(e). D is min-max scaled if and only if the components of D’mar are strongly connected. Lemma
Proof Suppose D4m= has a component that is not strongly connected. Then that component has two vertices vi and uj such that in Dfm=, v, is reachable from vi but not vice versa. Let W c V be the set of all vertices from which vi is reachable in DCmal. Define two sets e+(w) g {(“k, u,)
E
E:u*
E
w, u,
v-
E
W},
S-(W)~{(“,,“,).E:“kEV-W,u,EW}.
(5)
Note that S’(W) has at least one edge e, with w(e,) = emax. If C(W) is nonempty, set o = max,,,-(w, w(e); otherwise, set o = +cmax. Note that (2 < E,,,.~. Form the scaling vector s as follows:
sk= t
(“k WI, fs,,Ja)~” vE
(u,
The resealed graph D, = (v, w,(e)
= w(e)
1
E
W).
E, WJ has
if e 6 S+(W) U 6-(W), if e E S+(W)US-(W).
(7)
In particular, w,(e,) < w(e,) = emax, implying that D = (V, E, w) is not min-max scaled. Thus 3 is shown. (e) Given that the components of D’mua = (V, Em=, w) are strongly connected, assume D is not min-max scaled. Thus either the maximum weight in the digraph is too large or the number of edges with the maximum weight can be decreased. Then for some scaling s there exists at least one directed edge ez E {E: w(e) = emaX}- {E: w5(e) = emax} such that
yea;
w&2)
<
emax,
(8)
w&J
5
hax,
(9)
Because ez belongs to a strongly connected component, it is a directed edge on some cycle C = (e,,, with w&2)
e,2..
. , e,,)
e, E Pm=, i = 1,2,. . , 1. Under Observe that the product < emax.
the scaling s, of edge weights
around any cycle remains unchanged after any scaling. Thus there must be some edge e on cycle C such that w,,(e) > Q,,,, contradicting (9) above. By contradiction, D must be min-max scaled. (An alternate proof to the one given here can be developed using the notion of level sets as detailed in Rothblum et al. (1992); we proceeded as above to avoid cl introducing new terminology.) For any weighted digraph D = (V, E, w), the definitions in (5) can be applied to any vertex subset W c V. Thus applied, the two sets represent the edges connecting the subgraph induced by W to the rest of D. In Schneider and Schneider (1991) these sets are used to define a special property of weigthed digraphs. Definition 3. A graph D = (V, E, w) is said to be mux-bafunced if for every W E V, S,&,(W) = 6&,(W),
where G,,(W)
G,,(W) p max,,6+(wj w(e) g max,,6-(w) w(e).
and
In Schneider and Schneider (1991) and Rothblum ec al. (1992) the properties of max-balanced digraphs are investigated, and efficient algorithms to find a scaling that max-balances a digraph are given. We now show that max-balancing is particularly appropriate for nested edecompositions. Theorem 4. In the nested r-decomposition algorithm each of the digraphs D,, i = 1,2,. , k, are min-max scaled if and only if the original graph D = (V, E, w) is max-balanced. Proof: (3) Assume D is not max-balanced. Then there is a W c V such that 6,&,(W) # 6&,(W). Without loss of generality, let 6,&,(W) > 6,&J W). Further, 3u, t W, uj E V - W such that w(ui, vi) = a;,,. It follows that in the nested r-decomposition algorithm for some 1, e, = 6&,,(W). In D;f,
the vertices representing the equivalence classes containing u, and uj are connected but not strongly connected, since 6&J W) < 6,. Thus, by Lemma 2, D, is not min-max scaled. ( G ) Assume D, is not min-max scaled for some 1. By Lemma 2, there exist equivalence classes uli and uli that are connected but not strongly connected in D;‘; without loss of generality, assume I+; reaches V,j in Df’ (but not vice versa). Let W be the union of all vertices in Df’ that reach Q. By construction, D has 6&J W) < 6,&,,(W) = l,, and thus D is not max-balanced. q This section has been primarily limited to the discussion of digraphs. Recall from (1) and (2) that square matrices and their weighted digraphs are interchangeable; hence we shall speak of their scaling and e-decomposition interchangeably. Also, because e-decomposition only involves the size of the matrix elements, we can assume nonnegative-valued matrices without loss of generality. Further, we shall now restrict our attention to digraphs that are strongly connected. A digraph that is not strongly connected corresponds to a matrix M that is reducible, i.e. a matrix that can be permuted to assume the form
M=[ii::
:,I.
Because of the zero submatrix, this system can be scaled to make M,, as small as desired, separating the system into two isolated components. System decomposition can then proceed separately on the two subsystems M,, and M,,. Thus, without loss of generality, we require all matrices to be irreducible, that is, all digraphs to be strongly connected. 4. Max-balancing algorithm In the previous section it was shown that the only way a digraph will be free of artificially high coupling for all c-decompositions is if it is max-balanced. Thus it appears that when system scaling is performed, two steps are needed: first max-balancing and then e-decomposition. It is shown now, however, that all the calculations required for generating the set of nested c-decompositions are an integral part of the max-balancing procedure. We shall do this by
Brief Papers carefully rewriting the original algorithm found in Schneider and Schneider (1991) in such a way that we can compare it conveniently with the nested e-decomposition algorithm seen earlier. The max-balancing algorithm found in Schneider and Schneider (1991) uses the concept of the mean of a cycle. For a cycle C in a weighted digraph D = (V, E, w), let ICI denote the number of edges making up C. Then we can define the (geometric) mean of C, denoted W(C), by G(C) p
(,g w(e))“‘c’.
WV
If D is not acyclic, we define the maximum cycle mean of D as mcm (D) = max{@(C):C E cycles(D)}, (11) where cycles (D) denotes the set of all cycles in D. If D is acyclic, we define mcm (D) = 0. The algorithm given below is essentially the same as that discussed in detail in Schneider and Schneider (1991). Our scaling is multiplicative, whereas the original algorithm was written for additive scaling; the formulas used below have been adjusted accordingly. Although the algorithm requires the digraph to have no loops, this does not present a restriction, since scaling does not affect the weight of loops. INPUT: Strongly
connected
weighted
digraph
D =
(V, E, w) (with loops removed).
OUTPUT:
Scaling vector s that max-balances D; k maximum cycle means a, > a2 >. . akr and k corresponding partitions II,, I&, . . . , lIk.
Step 0. Initialize: Set i := 1, D, := D = (V, E, w) and a0 = 0. Step 1. Find Maximum Mean Cycle: Using equations given in the Appendix, identify a cycle Ci with ai 4 W(C) = mcm (DJ. If more than one such cycle exists, choose one arbitrarily. Step 2. Rescale: If oi # ai_,, solve for the scaling vector s’ using equations given in the Appendix; rescale Di with scaling s’ to get Dd. If ai = a;_,, set D,i:= Dti Step 3. Partition: Form’a partition Iii where all vertices on the maximum mean cycle C, belong to one equivalence class and every other vertex is its own equivalence class. If Iii has only one element, proceed to Step 6. be the condensation Step 4. Condense: Let D,, respect to the partition I&.
of D, with
If oi # ai-, , set Di+, := Dtemp, increment and go to Step 1. Otherwise, set Di = Dtemp, and go to Step 1.
Step 5. Iterate: i:=i
+ 1,
Step6. Terminate: k:=i-1,
If aifui_,, and&:=&
set k:=i; Stop.
if (Y~=(Y,_,, set
The following lemmas, found with proofs in Schneider and Schneider (1991) describe properties of the algorithm. Lemma 5. At each iteration, the resealed digraph Dtemp has
the following properties: (i) each directed edge on the maximum mean cycle Ci assumes a weight exactly equal to the maximum cycle mean; (ii) no other edge in the digraph Dtemp has weight larger than the maximum cycle mean. Lemma 6. The max-balancing scaling vector s is unique a positive scalar.
up to
The following theorem shows why max-balancing already does all the work of the nested c-decomposition algorithm. Theorem 7. Let D = (V, E, w) be a weighted digraph. Let ai and I$, i = 0,. . , m, be the distinct maximum cycle means
and partitions respectively returned from max-balancing D. Let 6, and I$ j = 1,. , k, be the coupling levels and partitions respectively returned from subsequently performing the nested a-decomposition algorithm. Then k = m and for all i = 1,. , k, E, = ai and P, = II,. Proof Note that because scaling does not affect the cycle mean of any cycle, max-balancing a graph twice returns exactly the same cycle means and partitions the second time as the first. Thus to show that e-decomposition after max-balancing is redundant, it is enough to show that the partitions and lk from the nested e-decomposition algorithm can be found from max-balancing a max-balanced graph. Assume for some i that II_, = P,_,; clearly this is true for i = 1 when both are the discrete n-member partition of V. Observe that if D is max-balanced then any condensation of D is also max-balanced. Let D, be the max-balanced condensation from the max-balancing algorithm. Because the D, is max-balanced, the largest weight in the digraph is equal to the maximum cycle mean ai, and thus cmax= (Y,. From Lemma 2, each component of D’max is strongly connected, and is thus either a single vertex or a union of cycles, each with mean ai. If only one such cycle exists then the partition returned from this iteration of e-decomposition is exactly that returned from the max-balancing algorithm. If m > 1 cycles with weight emBX exist then the max-balancing algorithm will return the same partition as the nested c-decomposition algorithm after m iterations of the max-balancing algorithm. Thus Pi = II;, and the theorem follows by induction. 0
A comment can be made here about the computational cost of scaling versus not scaling. For an n X n matrix, max-balancing requires approximately on the order of n times more calculations than a full e-decomposition, depending on the sparsity of the system. Therefore a system that is known to be well scaled should perhaps not be max-balanced if computation time is critical. While it is true that other scaling techniques such as line-sum scaling (Osborne, 1960) or the technique used in Ze&evif and Sdjak (1994) may offer acceptable results for specific problems with shorter computation time, the fact remains that only max-balancing will minimize the threshold coupling levels at each stage in the nested c-decomposition algorithm. The next section demonstrates just how advantageous max-balancing can be with e-decomposition. 5. Applications At this point we have seen that by reducing artificially strong coupling at all levels of decomposition, max-balancing will aid a-decomposition in partitioning a matrix M into weakly coupled blocks. Thus whenever an e-decomposition is performed on a system that permits scaling, max-balancing offers the potential for enhanced results. In Zehvif and Siljak (1994) c-decomposition is used in the parallel solution of nonlinear algebraic equations, and the technique is applied in Sezer and Siljak (1986) to identify subsystems for large-scale system stability tests. In Finney and Heck (1995) it is demonstrated that for both these applications max-balancing can greatly improve the results obtained. We now show an additional application of e-decomposition that is improved by max-balancing. Consider the large-scale LTI system i=Ax+Bu,
(12)
y = cx
The first step in designing a decentralized control scheme for this system is to decompose the system into weakly coupled subsystems. In Siljak (1991) e-decomposition is shown to be effective in determining the system decomposition as follows: create a matrix M defined by M=
r ;
f
1 ,
(13)
and create from M a weighted digraph D = (II U X U Y, E, w), where U = {u,, u2, , u,), X = {.r~l,~2, . , x,)
1181
Brief Papers and Y = {y,, y2, . , y,,} are the vertices corresponding to the m inputs, n states, and p outputs of the large-scale system. Given a decoupling parameter c, c-decomposition in D will decompose the system into N isolated subgraphs D: = D[Ui U Xi U I’J, i = 1,2, . . . , N, corresponding to N isolated LTI systems. See Siljak (1991) for more information and examples. Because the subsystems identified by t--decomposition will be used for control design, it is important that the isolated subsystems be controllable. However, it is easy to show that standard e-decomposition can result in subsystems that are not structurally controllable. Structural controllability is a necessary condition for controllability, and it is shown in Siljak (1991) that the system described above is structurally controllable if and only if each state is input-reachable and the digraph D,, = (U U X, E) has no dilation. A digraph D = (U U X, E) is said to have a dilation if there is a subset X,sX such that the number of vertices in D that reach at least one vertex in X, is smaller than the size of X,. Fortunately, the input/output mappings of the system (12) are unaffected by state scaling. The following result shows a clear advantage of max-balancing.
gratefully acknowledge the support of the Department of Education for a Graduate Assistance in Areas of National Need Fellowship. References
BollabC, B. (1979). Graph Theory: An Introductory Course. Springer-Verlag. New York. Finney, J. D. and B. S. Heck (1995). Matrix scaling for large-scale system decomposition. In Proc. American Conrrol Conf, Seattle, WA, pp. 2918-2923. Harary, F. (1%9). Gruph Theory. Addison-Wesley. Reading, MA. Osborne, E. E. (1950). On preconditioning of matrices. J. Assoc. Comput. Machinery, 7,338-345.
Rothblum, U. G., H. Schneider and M. H. Schneider (1992). Characterizations of max-balanced flows. Discrete Appl. Maths, 39,241-261.
Schneider, H. and M. H. Schneider (1991). Max-balancing weighted directed graphs and matrix scaling. Maths. Oper. Res., 16, 208-222.
Sezer, M. E. and D. D. Siljak (1986). Nested Edecompositions and clustering of complex systems. Automatica 22,321-331.
Theorem
8.
Let D = (U U X, E, w) be a weighted digraph
is maxDx e D[X] = (X, (X X X) II E, w) such that balanced. Let D, = (Ui U Xi, E,, w), i = 1,2, , N, be the subsystems identified by e-decomposition of D. If each vertex set Ui is nonempty then each isolated subsystem D, is structurally controllable.
Siljak, D. D. (1991). Decentralized Control of Complex Systems. Academic Press, Boston. Tarjan, R. (1972). Depth-first search and linear graph algorithms. SIAM J. Comput. 1,146-160. Zecevif, A. I. and D. D. Siljak (1994). A block-parallel Newton method via overlapping epsilon decompositions. SIAM .I. Matrix Anal. Applies, 15,824-844.
ProoJI Because Dx is max-balanced, so also is the digraph D,,= (Xi, (Xi XX,) nE, w). Since ui is non-empty, at least one vertex in X, is input-reachable. By Lemma 2, Di, is
strongly connected, implying each x E X, is input reachable. Further, since every vertex in X, reaches every other, D, can have no dilation. Thus 0, is structurally controllable. 0 Thus when max-balancing is used, the desired structural controllability is assured as long as at least one input is grouped with each subsystem. Dual results about structural observability follow directly from the section above. 6. Conclusions While e-decomposition is an efficient, powerful tool for large-scale system decomposition, we have seen that scaling system variables will result in non-unique decompositions. It was shown that for proper identification of the weakly connected subsystems, minimum-maximum coupling is desired at each stage of the nested e-decomposition algorithm, and that such a scaling occurs if and only if the system is max-balanced. Further, it was shown that an existing algorithm for max-balancing can be rewritten to generate the entire nested decomposition structure without additional computation. Acknowledgements-The authors gratefully acknowledge the support of the National Science Foundation for Grant ECS-9058140 and for an NSF Fellowship. The authors also
Appendix
This appendix contains the formulas necessary to calculate the cycle means and scalings used in the max-balancing algorithms of Section 4. They are taken directly from Schneider and Schneider (1991); the original presentation in that work was for additive and not multiplicative scaling, and the formulas have been adjusted accordingly. The maximum cycle mean mcm (D) of the digraph D = (V, E, w) with n vertices is found using the following recurrences: h)(u) = 0
for v E V,
(A.1)
Fk+du)=p=;$pw + log w(e)} r(D) = (mn; { ,5$,
for k = 0, 1,2,. . , n - 1, { ‘(vn) :
;(“)))j,
(A.2) (A-3)
mcm (D) = exp [t(D)].
(A.4)
Now compute the components of scaling vector s as s, = exp [ ,,5y2;_, IF,(u) - kr(D)1]
for u
l
V
(A.5)
To identify a cycle with the maximum cycle mean, one can use a depth-first search as described in Tarjan (1972).