Disturbance localization and output deadbeat control through an observer in discrete-time linear multivariable systems

Abstract

This paper considers the problem of disturbance localization for the system x(i + 1) = Ax(i) + Bu(i) + Dd(i), y(i) = Cx(i), w(i) = Ex(i) , with disturbance d(i) , measurement output y(i) , and controlled output w(i) . It is shown that the problem is solvable by using an observer if and only if V^{\ast} \supset 2^{\ast} where V*is the largest ( A,B )-invariant subspace in \ker E and 2*is the least ( A,\ker C )-conditioned invariant subspace containing Im D . Also, it is shown that there exists a controller using an observer that achieves simultaneous disturbance localization and output deadbeat control if and only if the system is controllable modulo \ker E and, in addition, V^{\ast} \supset O^{\ast} where O*is the unknown input unconstructible subspace. A simple algorithm is proposed to design such a controller. This algorithm comprises those of designing the optimal output deadbeat state feedback controller and an unknown input observer.