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. Introduction 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. Preliminaries 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.