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Controllability of 2-D systems. (English) Zbl 0753.93008
Autoregressive (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)

MSC:
 93B05 Controllability 93C35 Multivariable systems, multidimensional control systems 93B25 Algebraic methods
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