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Some basic structural properties of generalized linear systems. (English) Zbl 0727.93024
The paper deals with generalized linear systems, i.e. with systems of the form \(\dot x=Fx+Gu\), \(y=hx+J_ 0u+J_ 1\dot x+...+J_{\nu}u^{(\nu)},\) where \(x\in {\mathbb{R}}^ n\), \(u,\dot u,...,u^{(\nu)}\in {\mathbb{R}}^ m\) and \(y\in {\mathbb{R}}^ p\) and \(F,G,H,J_ 0...J_{\nu}\) matrices of appropriate size. Clearly \(u^{(k)}\) denotes the k-th time derivative of the input function and the above representation differs from the more common representation where \(J_ 1=...=J_{\nu}=0\). In this way the class of generalized linear systems includes the so-called (regular) descriptor systems or singular systems \(E\dot x=Ax+Bu\), \(y=Cx+Du\), where E is an (n,n)-matrix which is not necessarily nonsingular. The methods used in the paper stem from module theory and enable the author to study controllability, observability and the observer design problem for a generalized system.

MSC:
93C05 Linear systems in control theory
93B15 Realizations from input-output data
93B07 Observability
93C99 Model systems in control theory
Keywords:
time-dependent
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