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Model matching for linear systems with delays and 2D systems. (English) Zbl 0937.93007
The model matching problem (MMP) is solved for linear systems with delays.
A class of linear neutral delay-differential systems with commensurate delays is considered. It is shown that a delay system can be represented as the usual model for linear systems, i.e. $x=A(\nabla)x+B(\nabla)u\qquad y=C(\nabla)x+D(\nabla)u$ where the matrices $$A$$, $$B$$, $$C$$, $$D$$ are defined over the $$\text{PID }{\mathcal R}_u(\nabla)$$ of “realizable rational functions in $$\nabla$$ with real coefficients”, ${\mathcal R}_u(\nabla)=\{p(\nabla)/q(\nabla)\mid p,q\in\mathbb{R}[\nabla], q(\nabla=0)\neq 0\}.$ The model matching equation is solved over the ring $${\mathcal R}_c={\mathcal R}_s\cap {\mathcal R}_u$$ where $${\mathcal R}_s$$ is the ring of proper rational functions in $$s$$ with coefficients in $$\mathbb{R}(\nabla)$$ and $${\mathcal R}_u$$ is the ring of rational functions $$p/q$$, where $$p,q\in\mathbb{R}[s,\nabla]$$ and the leading coefficient of $$q$$, considered as an element of $$\mathbb{R}[\nabla][s]$$ is invertible over $$R_u(\nabla)$$.
It is shown that the ring $$\mathcal R$$ characterizes the notion of realizability of a transfer matrix $$T(s,\nabla)$$, namely $$T(s,\nabla)$$ is realizable iff it is defined over $${\mathcal R}_c$$.
The key of the method for solving the MMP is the fact that $${\mathcal R}_c$$ is the intersection of two PIDs, $${\mathcal R}_s$$ and $${\mathcal R}_u$$ (but $${\mathcal R}_c$$ is not a PID).
Necessary and sufficient conditions are given for the MMP to have a solution for the above class of linear systems with delays, firstly for the case of injective systems; then, by introducing some “decompositions”, the problem is solved for the surjective and for the general case. These conditions are constructive and they can be used to solve some classical control problems.
It is noticed that the results are also valid for 2D transfer matrices (due to the obvious correspondence of variables $$\nabla=1/z$$).

##### MSC:
 93A30 Mathematical modelling of systems (MSC2010) 34K40 Neutral functional-differential equations 93C23 Control/observation systems governed by functional-differential equations
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