A Poset Framework to Model Decentralized Control Problems

Parikshit Shah and Pablo A. Parrilo Laboratory for Information and Decision Systems Massachusetts Institute of Technology, Cambridge, MA 02139.

Abstract— In a previous paper [10], these authors showed 1) We study systems with time delays. It had been that posets provide a useful modeling framework for a reason- shown in a previous paper [9], that subject to certain ably large class of decentralized control problems. In this paper conditions on the delays between subsystems (namely we show more connections between posets and decentralized control. Firstly, we show that previously known results about the triangle inequality), the resulting problem was systems with time delays follow naturally using the poset frame- quadratically invariant (and thus amenable to con- work. We also mention extensions of this to infinite-dimensional vex optimization). We show that there is a natural spatio-temporal systems. Secondly, we study conditions under poset associated with systems with time delays with which the property known as quadratic invariance and poset this subadditivity property, and that the computational structure are equivalent. Through this paper, we hope to convince the reader of the important role that posets seem tractability is simply an algebraic consequence of this to play in much of the current theory of decentralized control. underlying poset. 2) We describe an extension of the preceding to spatially distributed systems. It was shown by Bamieh that spa- I.I tially invariant systems with a “funnel causal” impulse response was amenable to convex optimization. Using It is well-known that in general decentralized control is our poset approach, we are able to generalize the a hard problem. Blondel and Tsitsiklis [3] have shown that funnel causality condition. certain instances of such problems are in fact intractable. 3) We study the relationship between posets and quadratic On the other hand, Voulgaris [13], [14] presented several invariance. We show that quadratic invariance can be cases where decentralized control is tractable. In a previous naturally interpreted as a transitivity property, and paper [10], the authors were able to generalize these results that under certain natural settings, poset structures in an appealing framework using partially ordered sets. This and quadratic invariance are exactly equivalent. We paper includes extensions of that work to other settings. introduce the notion of a quoset, which is a poset Rotkowitz and Lall [9] have presented a criterion known as modulo an equivalence relation. We show that un- quadratic invariance that characterizes a class of problems der similar but somewhat more general conditions, in decentralized control that have the property that problems quadratic invariance is equivalent to quosets. become convex in the Youla parameter. Our results are Posets are very well-studied objects in combinatorics. and related to this property and we show the connection to their have been used in engineering and computer science (see work in our paper. We also show how some preexisting [10] and the references therein). results on quadratic invariance of networked systems with The rest of this paper is organized as follows. In Section II time delays can be interpreted as results about underlying we introduce the order-theoretic and control theoretic prelim- partially ordered sets. inaries that will be used throughout the paper. In Section III In a previous paper [10], we developed a framework for we study systems with time delays. In Section IV we decentralized control problems using posets. We argued that study an extension of the results of Section III to spatially posets provide the right language and technical tools to talk invariant systems. In Section V, we study the connection about a more general notion of causality (also referred to between quadratic invariance and posets/incidence algebras. as hierarchical control in the literature) among subsystems. In Section VI we conclude our paper. Associated to the notion of a poset (which is a combinatorial II. P object) is the notion of an incidence algebra [6], an algebraic object. This algebraic structure allowed us to convexify A. Order-theoretic Preliminaries the problem. We showed that some interesting examples of Definition 1: A partially ordered set (or poset) P = decentralized control that had been shown to be tractable (P, ) consists of a set P along with a binary relation  in the literature were in fact specific instances of this poset which has the following properties: paradigm. In this paper, we extend this poset framework to 1) a  a (reflexivity), other cases, for instance systems with time delays. 2) a  b and b  a implies a = b (antisymmetry), The main contributions of this paper are the following: 3) a  b and b  c implies a  c (transitivity). Posets may be finite or infinite, depending upon the cardi- This research was funded in part by AFOSR MURI 2003-07688-1. nality of the underlying set P. Example 1: An example of a poset with three elements A standard corollary of this theorem is the following [12, (i.e., P = {a, b, c}) with order relations a  b and a  c Theorem 1.2.3]. −1 is shown in Figure 1. In the diagram (known as a Hasse Corollary 1: Suppose A ∈ IP is invertible. Then A ∈ IP. diagram), an up arrow indicates the relation . B. Control-theoretic Preliminaries b c In this paper we are interested in decentralized structures on linear time-invariant systems. We will not particularly emphasize the continuous or discrete time cases as our results a apply equally well to both the settings. We will consider Fig. 1. A poset on the set {1, 2, 3}. systems with the following description: u ∈ Rnu is the control input, y ∈ Rny is the plant output, w ∈ Rnw is the exogenous Definition 2: Let P be a poset. Let Q be a ring. The input, z ∈ Rnz is the system output. We will be interested in set of all functions representing our systems via transfer function matrices as f : P×P→ Q P (ω) P (ω) P(ω) = 11 12 , with the property that f (x, y) = 0 if x  y is called the " P21(ω) P22(ω) # incidence algebra of P over Q. It is denoted by IP(Q). If where P(ω) ∈ C(nw+nu)×(nz+ny) is the overall system transfer the ring is clear from the context, we will simply denote this function. Through the rest of this paper, we abbreviate by IP (we will usually work over the field of rational proper notation and define P22 = G. Furthermore, in several cases transfer functions, or related extended spaces). we will assume that system G has an equal number of inputs P When the poset is finite, the set of functions in the and outputs (i.e. nu = ny). In these cases, we will think of incidence algebra may be thought of as matrices with a G being composed of several subsystems, each subsystem specific sparsity pattern given by the order relations of the having one input and one output. While dealing with finite poset. dimensional LTI systems the signal and operator spaces will Definition 3: Let P be a poset. The function ζ(P) ∈ be the standard ones. In some sections we will be dealing IP(Q) defined by with systems with time-delays, in these cases the systems 0, if x  y are no longer finite-dimensional, and the relevant spaces will ζ(P)(x, y) = ( 1, otherwise need to be appropriately extended (see [9]). We denote by Rp the set of rational proper transfer functions. Given a is called the zeta-function of P. ny ×nu controller K ∈ Rp , the closed-loop system has transfer Clearly, the zeta-function of the poset is an element of the function: incidence algebra. −1 Example 2: The matrix representation of the zeta function f (P, K) = P11 + P12K(I − GK) P21. for the poset from Example 1 is as follows: We are interested in optimal controller-synthesis problems of 1 1 1 the form: ζP =  0 1 0  minimize k f (P, K) k  0 0 1    subject to K stabilizes P (2)   The incidence algebra is the set of all matrices in Q3×3 which K ∈ S, have the same sparsity pattern as its zeta function. where S is some subspace of the space of controllers. In this Given two functions f, g ∈ I (Q), their sum f + g and scalar nz×nw P paper, k · k is any norm on Rp , chosen to appropriately multiplication c f are defined as usual. The product h = f · g capture the performance of the closed-loop system. In this is defined as follows: paper S will represent different constraints on the controller K (for example sparsity, or delay bounds). It may be noted h(x, y) = f (x, z)g(z, y). (1) that for general P and S there is no known technique for Xz∈P solving problem (2). As mentioned above, we will frequently think of the func- Problem (2) as presented is a non-convex problem in K. If tions in the incidence algebra of a poset as square matrices the subspace constraint K ∈ S were not present, then several (of appropriate dimensions) inheriting a sparsity pattern techniques exist for solving the problem. One approach dictated by the poset. The above definition of function towards a solution to the problem is to write an explicit multiplication is made so that it is consistent with standard parameterization of all stabilizing controllers for the prob- matrix multiplication. lem. It is desirable to have the closed-loop transfer function Theorem 1: Let P be a poset. Under the usual definition be an affine function in the parameter, so that the problem of addition, and multiplication as defined in (1) the incidence becomes convex. There are different approaches to perform algebra is an associative algebra (i.e. it is closed under the parametrization, for example the Youla parametrization addition, scalar multiplication and function multiplication). [9] and the so-called R-parametrization [4]. Proof: See [10]. Rotkowitz and Lall presented a notion called quadratic in- variance which gives a sufficient condition for reparametriz- applied at system j at time t1. We show next that if the delays ing a decentralized control problem in a way that is convex satisfy a triangle inequality then the relation  described in in the Youla parameter [9]. Definition 5 is a partial order relation. Definition 4: Given a system G and a subspace of Proposition 1: Suppose Dii = 0 (i.e. effect of input on constraints for the controller S , it is said to be quadratically output within same subsystems is without delay), Di j > 0 invariant with respect to G if for all K ∈ S , KGK ∈ S . (there is nonzero delay between distinct subsystems) and the Similar to quadratic invariance, systems with poset struc- Di j satisfy ture are amenable to convex reparametrization [10]. In the Di j + D jk ≥ Dik, (3) rest of this paper, we will not emphasize this aspect of for all i, j, k distinct. Then  is a partial order relation. reparametrization/convexification. The main thrust of the Proof: Since D = 0, by definition (i, t )  (i, t ). If paper is to show how posets, when chosen with insight, ii 1 1 (i, t )  ( j, t )and( j, t )  (i, t )) then t −t ≥ 0 and t −t ≥ 0 give nice classes of problems which are amenable to convex 1 2 2 1 1 2 2 1 (since delays are nonnegative), thus by definition t = t . reparametrization. 1 2 Since Di j > 0 for i , j it must be the case that i = j giving III. S  T D anti-symmetry. If (i, t1)  ( j, t2) and ( j, t2)  (k, t3), we have t ≤ t ≤ t . Further, t − t ≥ D and t − t ≥ D . By (3), As mentioned earlier, in previous work [10] these authors 1 2 3 2 1 ji 3 2 k j t3 − t1 ≥ D ji + Dk j ≥ Dki and hence (i, t1)  (k, t3), verifying showed that many examples of decentralized control prob- transitivity. lems studied in the literature can be modeled via posets. In Note that this triangle inequality structure on the delays is this section we provide another example of this, i.e. certain exactly the condition that appears in [9]. What is interesting structured time-delayed systems. It was shown by Rotkowitz here is that these delays actually give rise to a natural poset et al. [9] that systems involving time-delays which obey structure, as we have just pointed out (the poset is determined the triangle inequality have a property known as quadratic purely by the delays, the actual functional form of the invariance. In this section we show that this result is a impulse response does not matter). Furthermore, the set of direct consequence of underlying poset structure and its impulse responses gi j(t) which satisfy this delay structure associated incidence algebra. The emphasis here is on the actually forms an algebra of functions under convolution, as construction of the underlying poset and not the resulting the next proposition shows. quadratic invariance. We hope to convince the reader through Definition 6: Let Ψ = Di j be a given set of these and other examples of the fundamental role that posets 1≤i, j≤n delays. Let I denote the setn of (matrix)o impulse responses seem to play in much of the current theory on decentralized Ψ G(t) with the property that g (t) = 0 if ( j, 0)  (i, t). control. i j Intuitively g (t) = 0 means that the effect of an impulse In this section we consider LTI systems with time delays. i j at time t = 0 at subsystem j has not reached the output Given a decentralized plant with communication delays of subsystem i at time t. Thus I is precisely the set of between the different subsystems, we consider the task Ψ systems which obeys the delay structure prescribed by Ψ. of designing controllers for the subsystems which interact Given a set of impulse responses F = f (t) and G = g (t) according to a similar delay structure. It has been known [9] i j i j define F ∗ G to be the matrix of impulsen o responsesn witho that such communication structures are amenable to convex (F ∗ G) (t) = n f ∗ g (t). reparametrization due to their quadratic invariance. We will i j k=1 ik k j Proposition 2: Given a set of delays Ψ which satisfy show that posets arise naturally in this setup and that they P the conditions of Proposition 1. If F = f (t) , G = g (t) describe the communication constraints in an intuitive way. i j i j such that F, G ∈I for 1 ≤ i, j ≤ n, thenn F ∗ Go ∈I .n o Consider a system with n subsystems (let N = {1,..., n}). Ψ Ψ Proof: Suppose ( j, 0)  (i, t). It suffices to show that Let the system be described by the transfer function matrix (F ∗ G)i j(t) = 0. Now, G where Gi j(ω) describes the frequency response between input of system j and output of system i. An equivalent way n (F ∗ G)i j(t) = fik(t − τ)gk j(τ)dτ. to describe the plant is to specify the impulse responses gi j(t). ZR+ Define the delay between the subsystems i and j (denoted Xk=1 , by Di j) as follows If (F ∗ G)i j(t) 0 then there must be some k, τ such that gk j(τ) , 0 and fik(t − τ) , 0. This in turn means that τ ≥ Dk j = = ≤ Di j sup τ : gi j(t) 0 for all t τ . and t − τ ≥ Dik. Thus ( j, 0)  (k, τ) and (k, τ)  (i, t). By n o Note that since all systems are assumed to be causal, the transitivity, ( j, 0)  (i, t), contrary to our assumption. Since the impulse responses form a convolutional algebra, delays Di j are nonnegative. We define a relation  on N × R as follows. the transfer function matrices F(ω) and G(ω) form a mul- Definition 5: We say that ( j, t )  (i, t ) if tiplicative algebra and are thus quadratically invariant. This 1 2 allows us to conclude the following proposition. t2 − t1 ≥ Di j. Proposition 3: Consider a set of delay constraints Ψ Since the systems we are dealing with are time invariant, such that they satisfy the triangle inequality (3). Given a plant what this condition means intuitively is that ( j, t1)  (i, t2) if G ∈ IΨ with same delay constraints, the set of controllers system i at time t2 is in the cone of influence of an impulse K ∈IΨ is quadratically invariant with respect to G. IV. S I S possible to endow X×T with a natural poset structure. Define a relation  on X× T as follows. Let (x , t )  (x , t ) if: In this section we provide one more example of poset 1 1 2 2 structure arising in decentralized control problems, namely t2 − t1 ≥ f (x2 − x1). (5) in the context of spatially distributed systems. It is possible to naturally extend the results of the preceding subsection Notice the similarity between inequality (5) and Definition 5. − to a class of infinite dimensional systems that are spatially Both inequalities say that t2 t1 should be greater than the distributed [11]. These results were proposed in [11] by these delay between the subsystems, thus making (5) the natural authors. Similar results were independently and simultane- extension of the poset definition to the spatial invariance case. ously developed by Rotkowitz et al. in [8]. These results Proposition 4: [11, Proposition 1] Suppose the support = , generalized in multiple directions the previous results of function is such that f (0) 0, f (x) > 0 for x 0 and  Bamieh and Voulgaris [2]. We briefly review our results in subadditive. Then is a partial order relation. this subsection, since they nicely complement the preceding Under these assumptions on the support function, the impulse results. responses form a convolutional algebra, the usual Youla We consider systems that evolve along spatial coordinates parametrizations are employed, and convexification follows. (x ∈ X) as well as temporal coordinates (t ∈ T). Much Our results hold for multi dimensional spatial coordinates like temporal invariance, we say that a system is spatio- (for example, norms are examples of subadditive support temporally invariant if the effect of an impulse at spatial functions in higher dimensions). More interestingly, subaddi- tive support functions are a strictly larger class of functions coordinate x1 at time t1 at another location x2 at time t2 that contain concave functions as special cases [11]. Figure depends only on x2 − x1 and t2 − t1. Such systems may be specified by their spatio-temporal impulse response h(x, t). 2 shows an example of a subadditive support function that This function describes the response of the system at location is not concave. For further details, we encourage the reader (x, t) under the influence of an impulse at (0, 0). Given to read [11]. a system h(x, t) one defines the support function f (x) as t follows: Support of impulse response h(x, t )

f (x) = sup {τ : h(x, t) = 0 for all t ≤ τ} . (4) 1 1- ǫ f(x) Notice that this definition naturally extends the notion of delay between subsystems introduced in section III via (3). -1- ǫ -1 1 1 + ǫ x The support function evaluated at x tells one the delay involved in the effect of an impulse at the origin to reach Fig. 2. Support of impulse response. x. For example, if the system under consideration were a wave, then the support function would be exactly the light cone centered at the origin. V. C      Bamieh and Voulgaris [2] considered spatially invariant   systems with impulse responses whose support functions In this section we want to study the connection between were nonnegative and concave (they called such functions quadratic invariance and posets. We have seen that poset “funnel causal”). They showed that such impulse responses structure implies that the problem is quadratically invari- were convolutionally closed i.e., if h(x, t) and g(x, t) are two ant. We are now interested in understanding the converse, systems with support function f (x), then their convolution i.e. “does quadratic invariance imply existence of poset- like structure?” Quadratic invariance is really a transitivity (h ∗ g)(x, t) = h(x − ξ, t − τ)g(ξ, τ)dτdξ property. As argued earlier, posets provide the right language ZR ZR+ to describe transitive relations. In what follows, we make this is also supported on f (x). As a consequence of this closure connection more concrete. Connections between quadratic property, the set of spatially invariant controllers with the invariance and partially nested structures as defined in a specified support function f (x) can be reparametrized as a team-theoretic setting by Ho and Chu [5] have been studied convex set in the Youla domain. and pointed out by Rotkowitz [7]. The team theoretic prob- Their result depends on two key assumptions: lem considers a scenario where there are multiple decision • The spatial coordinates have to be one-dimensional, makers who must each make a decision in some order. The • The support function must be concave. paper considers a scenario where the order in which decisions Using our poset framework we are able to generalize these are made satisfy certain precedence relations. (Though this results. It turns out that the key property is subadditivity terminology is not used in these papers, these precedence (i.e. f (x1 + x2) ≤ f (x1) + f (x2)) of the support function. relations are in fact closely related to partial order rela- (Note that in the previous section, we assumed that the delays tions.) The paper by Ho [5] shows that problems with this satisfy the triangle inequality. Subadditivity of the support precedence structure (called partially nested problems) are functions is the natural generalization to this case.) If the amenable to convex optimization, and moreover, that optimal support functions involved satisfy sub-additivity then it is controllers are linear. Rotkowitz shows that existence of these precedence relations is equivalent to quadratic invariance. 3) The problem is quadratically invariant.

Our results are similar in spirit, in fact Proposition 5 is Then there exists a poset P over ny elements such that S essentially contained in [7]. However, we provide a finer is the incidence algebra of P. (Recall that S is the set of characterization of quadratic invariance in terms of posets matrices that satisfies the sparsity constraints of J.) and quosets. Proof: Since both G and K are ny × ny matrices, it is Consider the problem of designing an optimal controller enough to construct a poset on ny elements and show that K ∈ S as described in problem 2. In this section we revisit the sparsity pattern of S exactly corresponds to the incidence the model where decentralization constraints are viewed as algebra of this poset. Let us define our candidate for the sparsity constraints in the controller. Let K ∈ Rny × Rnu . De- partial order  as follows: i  j if (i, j) < J. We need to fine a subset of indices of K via J ⊆ 1,..., ny ×{1,..., nu}. verify that this is indeed a partial order relation. Then the subspace constraint is definedn as Kio j = 0 for all Since (i, i) < J, we clearly have i  i thus verifying (i, j) ∈ J. Quadratic invariance reduces to the following reflexivity. If i  j and j  i then it must be the case that transitive property in this model [9, Theorem 26]: (i, j) < J and ( j, i) < J. However the second assumption in Theorem 2: The subspace S is quadratically invariant with the statement of the proposition excludes the possibility of respect to a specified plant G if and only if for all K ∈ S such i, j being distinct, thus i = j and we have anti-symmetry. and all i, j, k, l, Finally, suppose we have i  j and j  l (i.e. (i, j) < J and ( j, l) < J). Choose index k such that k = j and use quadratic K G K (1 − K ) = 0. (6) i j jk kl il invariance to conclude from equation (8) that (i, l) < J. Thus Remark Let us interpret equation (6) in an intuitive way. Let i  l, verifying transitivity. us denote the constraint Ki j , 0 by i →K j (which denotes The incidence algebra of this poset is the set of elements that there is a path from i to j in the controller) and G jk , 0 such that Ki j = 0 if i  j, (i.e. (i, j) ∈ J) which is exactly by j →G k (i.e. that there is a path from j to k in the plant). the definition of S . Then the equation (6) states that: B. Existence of Quosets i →K j, j →G k, k →K l implies i →K l. (7) It is natural to ask: “to what extent does the preceding The transitive structure becomes more apparent now. What theorem generalize?” It turns out that one can in fact relax quadratic invariance is saying is that the overall graph of the second assumption (anti-symmetry). It is possible to have the closed loop (which is comprised of a combination of a more general notion of a partial order in the absence subgraphs of the plant and the controller) is transitively of anti-symmetry. In that setting, distinct elements can be closed. The condition means that if l is not allowed to equivalent, and the partial order is defined on the quotient communicate to i in the controller then there must exist no set modulo the equivalence. The resulting object is similar path from l to i around the closed loop (because such a path to a poset (called a quotient poset or quoset, sometimes it is would produce a way for l to communicate to i by going called a preorder in the literature). There is a corresponding once around the closed loop). algebraic object, analogous to the incidence algebra, called When the graph inside the plant and the controller is iden- the structural matrix algebra [1]. = - tical, quadratic invariance reduces to transitive closure of Definition 7: A quoset Q (Q, ) is a set Q with a - - this (identical) graph. We next show that in this scenario binary relation such that is reflexive and transitive. - quadratic invariance corresponds to existence of poset struc- Thus it is possible for distinct elements i, j to satisfy i j - ture. and j i (we will call such elements equivalent and denote this by i ≃ j). This notion of a quoset captures the intuition A. Existence of Posets that if i and j can communicate to each other and if they have the same level of information richness then they are Consider a plant G and a decentralized control problem with equivalent. One defines the analogue of an incidence algebra sparsity constraints K ∈ S such that Ki j = 0 for all (i, j) ∈ J as follows. for some index set J. Consider the square case i.e. ny = Definition 8: Let F be a ring and Q = (Q, -) be a nu. Let N = 1,..., ny . We say that a given decentralized quoset. Let the structural matrix algebra M be the set of control problemn is plant-controllero symmetric if the given functions f : Q × Q → F with the property that f (i, j) = 0 if plant also satisfies the sparsity constraints of the controller i  j for all i, j. (i.e. G ∈ S ). In this setup, notice that quadratic invariance We leave it as an easy exercise to the reader to verify that c (6) is equivalent to the fact that J is transitively closed, i.e. M is an associative algebra. Figure 3 shows an example of (i, j) ∈ Jc, ( j, k) ∈ Jc, (k, l) ∈ Jc ⇒ (i, l) ∈ Jc. (8) a quoset and the sparsity pattern of the associated structural matrix algebra. The analogue of Proposition 5 to quosets is Proposition 5: Consider a plant-controller symmetric the following. control problem. Suppose the following assumptions are true Proposition 6: Consider a plant-controller symmetric of the index set: control problem. Suppose the following assumptions are true 1) (i, i) < J of the index set: 2) For distinct i and j we have (i, j) < J ⇒ ( j, i) ∈ J. 1) (i, i) < J 1 2 3 4 VI. C 2, 3 4 1 ∗ ∗ ∗ ∗ 2 0 ∗ ∗ 0  We presented a poset based framework to study decentral- ized control problems. We were able to extend our previous 3 0 ∗ ∗ 0    work on decentralized control using posets to other settings. 4 0 0 0 ∗  1 We showed a way to interpret some preexisting results on systems with time delays using posets. We were able to extend these results to the case of spatially distributed Fig. 3. A quoset and the sparsity pattern of its associated structural matrix systems. We also studied the connection between quadratic algebra. Elements with a ’∗’ indicate possible nonzero elements. invariance and posets and showed that they were equivalent in certain settings. Under somewhat more general conditions, we showed that quadratic invariance is equivalent to the 2) The problem is quadratically invariant. existence of quoset structure.

Then there exists a quoset Q over ny elements such that S ACKNOWLEDGEMENT is the structural matrix algebra of Q. The authors would like to thank an anonymous referee for Proof: Again we construct a candidate quoset and an insightful and detailed review. verify the associated properties. We say that i - j if (i, j) < J. The verification of the properties are very similar to that R of Proposition 5, we leave this routine step to the reader. [1] M. Akkurt, G. P. Barker, and M. Wild. Structural matrix algebras We have thus seen that condition (2) from Proposition 5 and their lattices of invariant subspaces. Linear Algebra and its can be relaxed, and that in the relaxed setting quadratic Applications, 394:25–38, 2005. [2] B. Bamieh and P. Voulgaris. A convex characterization of distributed invariance is equivalent to existence of quoset structure in control problems in spatially invariant systems with communication the problem. What happens when condition (1) is relaxed constraints. Systems & Control Letters, 54(6):575–583, 2005. (i.e. allow constraints K = 0 for some i)? We answer this [3] V. D. Blondel and J. N. Tsitsiklis. A survey of computational ii complexity results in systems and control. Automatica, 36(9):1249– in the next proposition. 1274, 2000. Definition 9: Given J, we call J¯ = J\ (i, i) the [4] S. P. Boyd and C. H. Barratt. Linear Controller Design: Limits of Performance. Prentice Hall, 1991. reflexive closure of J. This is simply the operationS of adding [5] Y.-C. Ho and K.-C. Chu. Team decision theory and information c the reflexive relation to the set J which may not a priori structures in optimal control problems-part I. IEEE Transactions on satisfy reflexivity. Automatic Control, 17(1):15–22, 1972. c [6] G.-C. Rota. On the foundations of combinatorial theory I. Theory of We will say that the set J possesses quoset structure if M¨obius functions. Probability theory and related fields, 2(4):340–368, the collection of relations (i, j) ∈ Jc satisfy the axioms of a 1964. quoset, i.e. [7] M. Rotkowitz. On information structures, convexity, and linear optimality. In Proceedings of the 47th IEEE Conference on Decision 1) (i, i) ∈ Jc and Control, 2008. c c c [8] M. Rotkowitz and R. Cogill. Convex synthesis of distributed con- 2) (i, j) ∈ J and ( j, k) ∈ J implies (i, k) ∈ J . trollers for spatio-temporal systems with subadditive support functions. Proposition 7: Suppose we have a plant-controller In Forty-Sixth Annual Allerton Conference on Communication, Con- trol, and Computing, 2008. symmetric control problem with a specified index set (of [9] M. Rotkowitz and S. Lall. A characterization of convex problems sparsity constraints) J. (The sparsity constraints are thus in decentralized control. IEEE Transactions on Automatic Control, 51(2):274–286, 2006. Ki j = 0 for (i, j) ∈ J.) The problem is quadratically invariant c [10] P. Shah and P. A. Parrilo. A partial order approach to decentralized if and only if (J¯) has a quoset structure. control. In Proceedings of the 47th IEEE Conference on Decision and Proof: We first note that taking reflexive closure of a Control, 2008. [11] P. Shah and P. A. Parrilo. A partial order approach to decentralized transitively closed set does not affect any of the relations control of spatially invariant systems. In Forty-Sixth Annual Allerton c between distinct elements. Define I = J . Define i - j if Conference on Communication, Control, and Computing, 2008. (i, j) ∈ I. Suppose we add the reflexive relations so that [12] E. Spiegel and C. J. O’Donnell. Incidence Algebras. Marcel Dekker, ′ ′ 1997. I = I ∪ i∈N {(i, i)} . Consider the transitive closure of I . [13] P. Voulgaris. Control of nested systems. In Proceedings of the The onlyS way new relations
can be added is by combining American Control Conference, 2000. transitive relations with the newly added reflexive relations. [14] P. Voulgaris. A convex characterization of classes of problems in - - control with specific interaction and communication structures. In Thus if i j and j k, we know that for distinct i, j, k we Proceedings of the American Control Conference, 2001. already have i - k. If j = i or j = k we get no new relations. Hence I′ is its own transitive closure. Suppose the reflexive closure is a quoset. We know that in the closure operation, no new relations between distinct elements were introduced, hence transitivity is unaffected. By (8) the problem is quadratically invariant. Conversely, if the problem is quadratically invariant, we know from (8) that I is transitively closed. Thus if we take the reflexive closure, by Proposition 6 the resulting set is a quoset.