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 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.