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Root m-tissues: Systems under an action of the m-parameters. (English) Zbl 0584.93016

This paper considers the root loci of the characteristic polynomial of degree n with m parameters \[ \det (sI-F)=\sum_{m_ 1,...,m_ n}\delta^{1,...,n}_{m_ 1,...,m_ n}(\delta_{1,m_ 1}s- F_{1,m_ 1})...(\delta_{n,m_ n}s-F_{n,m_ n}) \] where the summation with respect to \(m_ 1,...,m_ n\) is over all permutations on 1,...,n, and \(\delta^{1,...,n}_{m_ 1,...,m_ n}=1\) (-1) for the even (odd) permutation \(\left( \begin{matrix} 1,...,n\\ m_ 1,...,m_ n\end{matrix} \right)\). Let \(K_ i\) \((i=1,...,m)\) be parameters. Then, the roots \(s_ 1,...,s_ n\) of the characteristic equation have the structure of the action of the additive group \(\{K_ 1,...,K_ m\}={\mathbb{R}}^ m.\)
The image \(s_ i=T(K_ 1,...,K_ m)\) of the linear mapping \(T: {\mathbb{R}}^ m\to {\mathbb{C}}\) is called the m-tissue at the point \(s_ i\). The m-tissues go through every root. The group action is used to compute the root differentials. Integrating them gives root loci. This method is illustrated by three numerical examples. In an example it is shown that m-tissues constitute 2m-gons under the simultaneous change of m- parameters. Accordingly, the robustness can be checked by using them. For the case that \(F_{i_ 1,j_ 1}=K\) is a simple parameter of F, Theorem 3 states that if the root loci have no multiplicy at \(\infty\), then there exist no root loci with odd number of even multiplicities.
Section 4 shows how to find the feedback gains \(K_ 1\) \((i=1,...,\rho)\) to shift the poles to the desired place. This is called the inverse problem. By integrating the equations for the gain differentials, the gains are obtained. This method is illustrated by an example.
Reviewer: M.Kono

MSC:

93B35 Sensitivity (robustness)
93B55 Pole and zero placement problems
93C35 Multivariable systems, multidimensional control systems
93D99 Stability of control systems
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References:

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