On the control of linear systems whose coefficients are functions of parameters

Abstract

The stabilization and regulation of linear discrete-time systems whose coefficients depend on one or more parameters is studied. For linear systems whose coefficients are continuous functions of real or complex parameters (respectively, analytic or rational functions of real parameters), it is shown that reachability of the system for all values of the parameters implies that the system can be stabilized using gains that are also continuous (analytic, rational) functions of the parameters. Closed-form expressions for a collection of stabilizing gains are given in terms of the reachability Gramian. For systems which are stabilizable for all values of the parameters, it is shown that continuous (analytic, rational) stabilizing gains can be computed from a finite-time solution to a Riccati difference equation whose coefficients are functions of the parameters. These results are then applied to the problem of tracking and disturbance rejection in the case when both the plant and the exogenous signals contain parameters.