An Algebraic Framework for Quadratic Invariance Laurent Lessard 1 Sanjay Lall 2 IEEE Conference on Decision and Control, pp. 2698–2703, 2010 Abstract In this paper, we present a general algebraic framework for analysing decentralized control systems. We consider systems defined by linear fractional functions over a com- mutative ring. This provides a general algebraic formu- lation and proof of the main results of quadratic invari- ance, as well as naturally covering rational multivariable systems, systems with delays, and multidimensional sys- tems. The approach extends to the extended class of internally quadratically invariant systems. 1 Introduction In decentralized control, multiple controllers are used to control a plant with many inputs and outputs. Each con- troller has access to different outputs (measurements), and controls a different subset of the inputs. Such sys- tems arise when the plant is itself distributed, and it would be infeasible or otherwise impossible to control it using a centralized computer. In the linear fractional transformation (LFT) formula- tion, the closed-loop map is given by f (P,K)= P 11 + P 12 K (I - P 22 K) -1 P 21 where the P ij are system parameters. We seek a con- troller K that minimizes a performance norm minimize   f (P,K)   subject to K ∈ S where S is the set of admissible controllers. For a general constraint sets S, finding the optimal controller is a hard problem. Indeed, the set f (P,S) need not be convex, and the optimal controller need not be linear [11]. However, decentralized problems with linear optimal controllers have also been identified. The largest such class known is defined by a property called quadratic invariance [8, 9]. Henceforth, we will use the abbreviation G = P 22 . Quadratic invariance is a simple algebraic condition. We 1 L. Lessard is with the Department of Aeronautics and As- tronautics at Stanford University, Stanford, CA 94305, USA. lessard@stanford.edu 2 S. Lall is with the Department of Electrical Engineering and Aeronautics and Astronautics at Stanford University, Stanford, CA 94305, USA. lall@stanford.edu 3 This work was supported by NSF grant number 0642839 say S is quadratically invariant (QI) with respect to G if KGK ∈ S for every K ∈ S. Roughly speaking, if this condition holds and S is a sub- space, the set f (P,S) is affine, and the controller syn- thesis problem can be posed as a convex optimization problem and thereby solved. This result holds in a very broad sense. Subject to some technical conditions, it holds when the plant and controllers are bounded linear operators from one Ba- nach space to another [8]. More generally, it holds for causal maps on extended spaces [9]. This encompasses continuous and discrete systems, stable or unstable, and even systems with delays. In both cases, the results are proven using tools from analysis. Since the maps in question are potentially infinite-dimensional, questions of convergence arise. One must also take care in defining appropriate topologies so that the notion of convergence is the appropriate one. Both results require S to be a closed subspace, which is problematic when we seek controllers expressible as ra- tional transfer functions. Given that the QI result holds in a very broad sense, and that the QI condition is algebraic in nature, it en- courages one to seek an algebraic framework in which the results can be expressed naturally. In this paper, we present such a framework. We consider plants and controllers to be matrices whose entries belong to a com- mutative ring. A similar framework was suggested [10], which generalizes the notion of a transfer function ma- trix, and applies it to feedback stabilization. In Section 2, we cover some preliminaries; results that hold in the general ring case. In Section 3, we consider a more specific ring; multidimensional rational functions. We prove both QI and internal quadratic invariance [4] results in this case. Finally, in Section 4, we discuss some applications. 2 Commutative Rings A commutative ring is a tuple (R, +, ·) consisting of a set R, and two binary operations which we call addition and multiplication, respectively. The following properties hold for all a, b, c ∈ R. First, (R, +) is an abelian group: i) Closure: a + b ∈ R ii) Commutativity: a + b = b + a 1