Controllability of 2-D systems.

*(English)*Zbl 0753.93008Autoregressive (AR) 2-D systems are studied from the system theoretic viewpoint. The system is described by \(\sum=(\mathbb{Z}^ 2,\mathbb{R}^ q,{\mathcal B})\) for some integer \(q>0\), where \(\mathcal B\), called the behavior of the system, is a class of functions from \(\mathbb{Z}^ 2\) to \(\mathbb{R}^ q\) given by the kernel of a polynomial operator in the shifts. The concept of controllability is defined for \(\sum\) in terms of concatenation of a function \(w_ 1\in\mathcal B\) with another \(w_ 2\in\mathcal B\) relative to disjoint \(I_ 1,I_ 2\subset\mathbb{Z}^ 2\). It is shown that a controllable AR system is characterized by the system which can be described in input-output form by means of a 2-D transfer function. A state-space model for 2-D systems is derived and written by
\[
\begin{cases} S(\sigma^{-1}_ 2\sigma_ 1)x=0,\quad x\in\mathbb{R}^ n,\\ \sigma_ 1 x=A(\sigma^{-1}_ 2\sigma_ 1)x+B(\sigma^{-1}_ 2\sigma_ 1)v,\\ w=Cx+Dv,\quad w\in\mathbb{R}^ q,\end{cases}
\]
where \(S(s)\) denotes a polynomial matrix, \(\sigma_ 1\) and \(\sigma_ 2\) the left- and the down-shift, respectively, and \(A(s)=A_ 1 s+A_ 2\), \(B(s)=B_ 1 s+B_ 2\); \(A_ 1\), \(A_ 2\), \(B_ 1\), \(B_ 2\), \(C\), and \(D\) being real matrices. Then it is shown that a controllable AR 2-D system has the above state-space representation. Trimness, reachability, and observability for the state-space system are characterized in terms of the system representation.

Reviewer: T.Nambu (Kumamoto)