×

A graph-theoretic characterization for the rank of the transfer matrix of structured system. (English) Zbl 0747.93030

This paper considers structured state space systems \((A,B,C)\) where the entries of the matrices \(A\), \(B\), \(C\) are either zero or independent real parameters. For these systems it is shown that the generic rank of the transfer function \(C(sI_ n=A)^{-1}B\) is equal to the maximum number of vertex disjoint paths from the set of input vertices to the set of output vertices in the associated (directed) system graph. The result is then applied to obtain a necessary and sufficient condition for the generic solvability of the almost disturbance decoupling problem (for structured systems with a structured disturbance input matrix).

MSC:

93C05 Linear systems in control theory
93B99 Controllability, observability, and system structure
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] W. K. Chen,Applied Graph Theory, North-Holland, Amsterdam, 1971. · Zbl 0229.05107
[2] S. Even and R. E. Tarjan, Network flow and testing connectivity,SIAM J. Comput.,4 (1975), 507–518. · Zbl 0328.90031
[3] J. C. Willems, Almost invariant subspaces: An approach to high gain feedback design–Part I: Almost controlled invariant subspaces,IEEE Trans. Automat. Control,26 (1981), 235–252. · Zbl 0463.93020
[4] J. C. Willems, Almost invariant subspaces: An approach to high gain feedback design–Part II: Almost conditionally invariant subspaces,IEEE Trans. Automat. Control,27 (1982), 1071–1085. · Zbl 0491.93022
[5] W. M. Wonham,Linear Multivariable Control: a Geometric Approach, Springer-Verlag, New York, 1985. · Zbl 0609.93001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.