Disturbance decoupling for periodic and multirate systems (Q2745599)

The discrete-time, finite-dimensional system studied is given by NEWLINE\[NEWLINEx(t+1)= A(t+1)x(t)+ B(t+1)u(t)+ Dv(t), \qquad y(kT)=Cx(kT),NEWLINE\]NEWLINE where \(k,t,T \in\mathbb{N}\), and \(A,B\) are periodic with period \(T\). The disturbance decoupling problem studied is to find a controller of the form \(u(jT+i)= F_ix(jT+i)\), \(0\leq i<T\), \(j \in\mathbb{N}\) such that the disturbance \(v\) has no influence on the output \(y\). The author solves this problem via geometric notions. In particular, he defines for his class of systems ``\((A,B)\)-invariance'', and proves that the set of \((A,B)\)-invariant subspaces contained in the kernel of \(C\) contains a maximal element \(V^*\). Next he proves that the disturbance decoupling problem is solvable if and only if Im\(D\subset V^*\cap A(T)^{-1}(V^*+\text{Im}B(T))\cap\cdots \cap (A(T)\dots A(2))^{-1} (V^*+\text{Im}B(T)+ \cdots+A(T) \cdots A(3)\text{Im}B(2))\).