ALTERNATING CONVEX PROJECTION METHODS FOR COVARIANCE CONTROL DESIGN

The problem of designing a static state feedback of full order dynamic controller is formulated as a problem of designing an appropriate plant state covariance matrix. We show that closed loop stability and multiple output norm constraints imply that the plant state covariance matrix lies at the intersection of some specified closed convex sets in the space of symmetric matrices. We address the covariance feasibility problem to determine the existence and compute a covariance matrix to satisfy assignability and output norm performance constraints. We address the covariance optimization problem to construct an assignable covariance matrix which satisfies output performance constraints and is as close as possible to a given desired covariance. We also treat inconsistent constraints where we look for an assignable covariance matrix which 'best' approximates desired but non-achievable output performance objectives (we call this the infeasible covariance optimization problem). All these problems are of a convex nature and alternating convex projection methodologies are suggested to solve them. These techniques provide simple and effective numerical algorithms for a solution of non-smooth convex programs and their presentation in this paper is of particular importance since (to the authors' knowledge) this is the first time these methods have been used in control design, and they might find wide applicability in several aspects of computational control problems. Expressions for the required convex projections on the assignability and the performance constraint sets are derived. An example illustrates the methodology.