Convex Methods for the Design of Structured Controllers: an Approach for Spatially Invariant Systems, with Extensions to General Interconnected Systems

This monograph deals with some control theoretic problems that arise from the study of systems composed by a collection of similar units. We seek convex conditions for controller design that explicitly impose structure on the controller, for example, forcing it to be decentralized, or alternatively allowing the controller to have access to observations from a limited number of neighboring units (a localized controller). Most of the monograph focuses on Spatially Invariant (SI) systems, but some results apply to more general interconnected systems. We show that if we force the Lyapunov function for the closed loop to be decentralized, then the restricted synthesis problem becomes convex both for stabilization and optimization with a quadratic performance (e.g, H2, Hinf). This method can be extended beyond SI systems, and as an application of this extension, we present a solution to a problem in the area of consensus over a network of agents. For SI systems, the restriction can be interpreted as an Integral Quadratic Constraint (IQC) that must be satisfied by each unit. This IQC is used to formulate a more general stability condition by relaxing the original one.