ALTERNATING CONVEX PROJECTION METHODS FOR DISCRETE-TIME COVARIANCE CONTROL DESIGN

The problem of designing a controller for a linear, discrete-time syst em is formulated as a problem of designing an appropriate plant-state covariance matrix. Closed-loop stability and multiple-output performan ce constraints are expressed geometrically as requirements that the co variance matrix lies in the intersection of some specified closed, con vex sets in the space of symmetric matrices. We solve a covariance fea sibility problem to determine the existence and compute a covariance m atrix to satisfy assignability and output-norm performance constraints . In addition, we can treat a covariance optimization problem to const ruct an assignable covariance matrix which satisfies output performanc e constraints and is as close as possible to a given desired covarianc e. We can also treat inconsistent constraints, where we look for an as signable covariance which best approximates desired but unachievable o utput performance objectives; we call this the infeasible covariance o ptimization problem. All these problems are of a convex nature, and al ternating convex projection methods are proposed to solve them, exploi ting 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 alterna ting projections is discussed, and a numerical technique using alterna ting 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.