LQ tracking and disturbance rejecting with invariant zeros on the unit circle

Abstract

A solution of the problem of optimal linear-quadratic (LQ) tracking and disturbance rejecting with invariant zeros on the unit circle of the plant is given, under a quite general assumption. For that purpose, we transform this problem to a problem of LQ control of an unstabilisable plant by augmentation, and then deal with weakly stabilising controls, defined as the controls such that the unstable modes of the closed-loop system are at most the unstabilisable modes of the augmented pair (A, B).Then we solve the transformed problem by the newly introduced minimal rank weakly stabilising solution of the most general discrete-time algebraic Riccati system (DARS), associated with the system given by matrix quadruple (A, B, C, D), with unstabilisable matrix pair (A, B).We show and illustrate by examples that there is a class of LQ tracking problems in the presence of disturbances, which cannot be solved by the existing methods, but can be solved by the introduced minimal rank weakly stabilising solution of the DARS.