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Graph-theoretic characterization of fixed modes in centralized and decentralized control. (English) Zbl 0537.93009
Consider a system $$\dot x=Ax+Bu$$, $$y=Cx$$ with centralized output feedback $$u=Fy$$, where $$x\in R^ n$$, $$u\in R^ m$$, $$y\in R^ r$$ and A, B, C, F are matrices. A decentralized control problem is further specified: $$u_ i=F_ iy_ i$$, $$B=diag(B_ 1,...,B_ k)$$, $$F=diag(F_ 1,...,F_ k)$$, $$u=(u^ T_ 1,...,u^ T_ k)^ T$$, $$u_ i\in R^{n_ i}$$, $$y_ i\in R^{r_ i}$$, $$\sum_{i}n_ i=n$$, $$\sum_{i}r_ i=r$$, k is a number of channels. The correspondence of the coefficients of the characteristic polynomial $$\det(sI_ n-A-BFC)=s^ n+\sum^{n- 1}_{j=0}p_{n-j}s^ j$$ to the cycle families of the weighted digraph is proved (Theorem 1): The coefficients $$p_ i$$ are determined by the cycle families of width i, it means that the cycle family touches i state vertices. Each cycle family of width i corresponds to one summand of $$p_ i$$. The summand is numerically determined by the weight of the cycle family multiplied by a sign factor. Using this result, the necessary and sufficient condition for the existence of fixed modes of multiplicity $$h\geq 1$$ at $$s=\lambda$$ ($$\lambda$$ a complex number) is proved (Theorem 2). The paper is self contained and clearly written. It is supplied with numerous illustrative examples. The results contribute to structural control theory by exploitation of the concepts of cycles and cycle families to determine the existence of fixed modes and structurally fixed modes of M. E. Sezer and D. D. Šiljak [Syst. Control. Lett. 1, 60-64 (1981; Zbl 0476.93042)], see also K. Reinschke [Proc. Third Conf. on Syst. Theory, Trondheim (October 1983), Tapir Publ., Trondheim (Norway), 1984].
Reviewer: L.Bakule

##### MSC:
 93A15 Large-scale systems 05C20 Directed graphs (digraphs), tournaments 05C38 Paths and cycles 93C05 Linear systems in control theory 93C35 Multivariable systems, multidimensional control systems 93C99 Model systems in control theory 94C15 Applications of graph theory to circuits and networks
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