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

This paper considers the problem of disturbance localization for the systemx(i + 1) = Ax(i) + Bu(i) + Dd(i), y(i) = Cx(i), w(i) = Ex(i), with disturbanced(i), measurement outputy(i), and controlled outputw(i). It is shown that the problem is solvable by using an observer if and only ifV^{} ⊃ 2^{}where V*is the largest (A,B)-invariant subspace in Eand 2*is the least (A, C)-conditioned invariant subspace containing ImD. 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 Eand, in addition,V^{} ⊃ O^{}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.