A partial order approach to decentralized control

Abstract

In this thesis we consider the problem of decentralized control of linear systems. We employ the theory of partially ordered sets (posets) to model and analyze a class of decentralized control problems. Posets have attractive combinatorial and algebraic properties; the combinatorial structure enables us to model a rich class of communication structures in systems, and the algebraic structure allows us to reparametrize optimal control problems to convex problems. Building on this approach, we develop a state-space solution to the problem of designing H₂-optimal controllers. Our solution is based on the exploitation of a key separability property of the problem that enables an efficient computation of the optimal controller by solving a small number of uncoupled standard Riccati equations. Our approach gives important insight into the structure of optimal controllers, such as controller degree bounds that depend on the structure of the poset. A novel element in our state-space characterization of the controller is a pair of transfer functions, that belong to the incidence algebra of the poset, are inverses of each other, and are intimately related to estimation of the state along the different paths in the poset. We then view the control design problem from an architectural viewpoint. We propose a natural architecture for poset-causal controllers. In the process, we establish interesting connections between concepts from order theory such as Mobius inversion and control-theoretic concepts such as state estimation, innovation, and separability principles. Finally, we prove that the H₂-optimal controller in fact posseses the proposed controller structure, thereby proving the optimality of the architecture.