An algebraic approach to the time-optimal output regulator†

Abstract

The time-optimal output regulator is considered for a special class of linear time-invariant discrete-time systems. It is shown that a simple state feedback law achieves time-optimal output regulator response in one sampling period. For a square system, the state feedback law always assigns the invariant zeros as closed-loop poles. For a non-square system, time-optimal regulator response is achieved even if the set of invariant zeros is empty. If the set of invariant zeros is not empty, the control law always assigns these zeros as closed-loop poles. The method of this paper, for the discrete system described by the state and output equations x(k + 1) = Ax(k) + Bu(k) and z(k) = Cx(k), does not require that A be invertible and that the open-loop system S(A, B, C) be state controllable. Furthermore, the only requirement is the existence of a right inverse for the matrix CB which is sufficient for output controllability.