Alternating convex projection methods for discrete-time covariance control design

Abstract

The problem of designing a controller for a linear discrete-time system is formulated as a problem of designing an appropriate plant state covariance matrix. Closed loop stability and multiple output covariance inequality constraints are expressed geometrically as requirements that the covariance matrix lies in the intersection of some specified closed convex constraint 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 covariance inequality constraints. We address the covariance optimization problem to construct an assignable covariance matrix which satisfies covariance inequality constraints and is as close as possible to a given desired covariance. We also treat inconsistent constraints where we look for an assignable covariance which best approximates desired but unachievable output performance objectives. Analytical expressions for the projections onto the covariance assignability and the output covariance inequality constraint sets are derived. Finally, the problem of designing low order dynamic controllers is discussed and a numerical technique using alternating projections is suggested for a solution. >