Alternating convex projection methods for discrete-time covariance control design

Abstract

The problem of designing a controller for a linear, discretetime system is formulated as a problem of designing an appropriate plant-state covariance matrix. Closed-loop stability and multiple-output performance constraints are expressed geometrically as requirements that the covariance matrix lies in the intersection of some specified closed, convex sets in the space of symmetric matrices. We solve a covariance feasibility problem to determine the existence and compute a covariance matrix to satisty assignability and output-norm performance constraints. In addition, we can treat a 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 can also treat inconsistent constraints, where we look for an assignable covariance which best approximates desired but unachievable output performance objectives; we call this the infeasible covariance optimization problem. All these problems are of a convex nature, and alternating convex projection methods are proposed to solve them, exploiting the geometric formulation of the problem. To this end, 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 using alternating projections is discussed, and a numerical technique using alternating projections is suggested for a solution, although convergence of the algorithm is not guaranteed in this case. A control design example for a fighter aircraft model illustrates the method.