Assigning controllability and observability Gramians in feedback control

Abstract

We wish to characterize the set of all X that may be assigned to the system by any controller. Such parameterization requires two steps: 1) deriving the necessary and sufficient conditions for the existence of some controller to assign X, and 2) finding the set of all controllers that assign X. The covariance assignment problem was first defined in Ref. 1 and solved for the state feedback case (see also Refs. 2 and 3). Subsequently, these results have been extended to the dynamic controller of any order for continuous systems, and all controllers that assign a specified state covariance to a continuous-time system are derived in Ref. 4. This paper extends the ideas of Ref. 4 to the deterministic problem of assigning prescribed matrix values to the closed-loop controllability and observability Gramians. The Gramian controllers are parameterized in terms of a matrix having physical significance (the controllability, observability Gramians). This gives a multiobjective flavor to the controller capability, because the n(n + l)/2 parameter values are assigned to the closed-loop system. Almost all robustness properties of linear systems can be related directly to properties of the controllability or observ-