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Canonical forms of singular 1D and 2D linear systems. (English) Zbl 1051.93021
Singular 1D discrete-time linear systems are considered, having the state-space representation $Ex_{i+1}=Ax_{i}+Bu_{i}, \qquad y_{i}=Cx_{i}\tag{1}$ $$i\in\mathbb{Z}_{+}$$, $$x_{i}\in\mathbb{R}^{n}$$, $$u_{i}\in\mathbb{R}^{m}$$ and $$y_{i}\in\mathbb{R}^{p}$$, where $$\det E=0$$ and $$\det [Ez-A]\neq0$$ for some $$z\in\mathbb{C}$$. Two new canonical forms are proposed. The matrices $$A,B,C$$ have the first canonical form if $$E=\text{diag}(E_{1},E_{2},\dots,E_{m}),\;A=\text{diag}(A_{1},A_{2},\dots,A_{m})$$, $$B=\text{diag}(B_{1},B_{2},\dots,B_{m}),$$ $E_{i}=\left[\begin{matrix} I_{q_{i}}&0 \\0&0\end{matrix}\right] \in\mathbb{R}^{(q_{i}+1) \times(q_{i}+1)}, \quad A_{i}=\left[\begin{matrix} 0&{}&I_{q_{i}}\\ {}&a_{i}&{}\end{matrix}\right]\in \mathbb{R}^{(q_{i}+1)\times(q_{i}+1)},$ $$a_{i}=[a^{i}_{0}\dots a^{i}_{r_{i}-1}\;1\;0\dots0]$$, $$B_{i}=[0\dots0\;1]^{T}\in\mathbb{R}^{q_{i}+1}$$, $$i=1,\dots,n$$ for $$n=m+\sum^{m}_{i=1}q_i$$, $$C\in\mathbb{R}^{p\times n}$$. The matrices $$\widetilde{A}$$, $$\widetilde{B}$$, $$\widetilde{C}$$ have the second canonical form if $$\widetilde{E}=\text{diag}(E_{1},E_{2},\dots,E_{p})$$, $$\widetilde{A}=\text{diag}(A_{1},A_{2},\dots,A_{p})$$, $$\widetilde{C}=\text{diag}(C_{1},C_{2},\dots,C_{p}),$$ with $$E_{i}$$ as above, $$\widetilde{A}$$ like $$\widetilde{A}_{i}^{T}$$ and $$C_{i}$$ like $$B_{i}^{T}$$, $$i=1,\dots,p$$ and $$\widetilde{B}\in\mathbb{R}^{n\times m}$$.
A necessary and sufficient condition for the existence of nonsingular matrices transforming the singular 1D system (1) to its canonical form is expressed in terms of the full rank of some reachability and observability matrices. A procedure to determine the canonical forms is proposed.
Similar canonical forms are introduced for the singular 2D Roesser model. A method for the determination of 2D realizations in canonical form is provided for SISO systems, as well as a procedure for the transformation of the matrices of the singular 2D Roesser model to their canonical forms.
It is stated that the considerations presented for SISO systems can be easily extended to the MIMO singular Roesser model and to singular 2D Fornasini-Marchesini models.

##### MSC:
 93B10 Canonical structure 93C55 Discrete-time control/observation systems 93B17 Transformations 93C05 Linear systems in control theory 93C35 Multivariable systems, multidimensional control systems
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