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The generic number of invariant zeros of a structured linear system. (English) Zbl 0952.93056

For the linear system \(\dot x(t)= Ax(t)+ Bu(t)\), \(y(t)= Cx(t)+ Du(t)\) an invariant zero is defined as the complex number for which the matrix \(P(s)= \left( \begin{smallmatrix} sI-A &B\\ C &D \end{smallmatrix} \right)\) looses rank. In this article it is assumed that one has knowledge about the location of the zero elements in the matrices \(A\), \(B\), \(C\) and \(D\), and that the nonzero elements of these matrices are completely free. Under these conditions it is known that the number of invariant zeros is constant except on some proper algebraic variety. For the structured systems as described above the author present an expression for the (generic) number of invariant zeros. His proof is based on graph theory, and makes a fundamental decomposition of the directed graph representing the structured system. Using this he can make a simple block diagonal decomposition of the matrix \(P(s)\).

MSC:

93C05 Linear systems in control theory
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15A22 Matrix pencils
05C90 Applications of graph theory
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