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Some necessary and sufficient conditions for eigenstructure assignment. (English) Zbl 0592.93023
This paper deals with eigenstructure assignment by output feedback. Consider a linear multivariable system (1) $$\dot x=Ax+Bu$$, $$y=Cx$$. Here x(t), u(t), y(t) are vector functions of time with n, m, p components respectively and A, B, C are real matrices with appropriate sizes. By applying an output feedback control (2) $$u=-Ky$$, the closed-loop system becomes $$\dot x=(A-BKC)x.$$
Let $$\lambda_ 1,\lambda_ 2,...,\lambda_ n$$ be the desired eigenvalues of the closed-loop system. Further, assume that $$S_ 0$$ is the $$n\times q$$ matrix whose columns constitute the desired right eigenvectors with $$\lambda_ 1,...,\lambda_ q$$ and $$T^ T_ 0$$ is the (n-q)$$\times n$$ matrix whose rows constitute the desired left eigenvectors. The eigenstructure assignment problem is to give necessary and sufficient conditions for the existence of K such that A-BKC has the desired eigenvalues, right and left eigenvectors. This problem can be reformulated into the existence of a real matrix K such that (3) $$A- BKC=L$$ for some given $$n\times n$$ matrix L.
The concept of an inner inverse of a matrix is employed to obtain the main results.
Some simple numerical examples are given in order to illustrate the constructions.
Reviewer: M.Kono

##### MSC:
 93B55 Pole and zero placement problems 93C05 Linear systems in control theory 93C35 Multivariable systems, multidimensional control systems 15A18 Eigenvalues, singular values, and eigenvectors 15A24 Matrix equations and identities 93C99 Model systems in control theory
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##### References:
 [1] BRADSHAW A., Int. J. Systems Sci 9 pp 1331– (1979) · Zbl 0398.93031 · doi:10.1080/00207727808941780 [2] FLETCHER L. R., Int. J. Control 31 pp 1121– (1980) · Zbl 0458.93024 · doi:10.1080/00207178008961104 [3] GOLUB G. B., Matrix Computations (1983) · Zbl 0559.65011 [4] KAUTSKY J., Int. J. Control 41 pp 1129– (1985) · Zbl 0567.93036 · doi:10.1080/0020718508961188 [5] KIMURA H., I.E.E.E. Trans, autom. Control 20 pp 509– (1975) · Zbl 0309.93017 · doi:10.1109/TAC.1975.1101028 [6] KOLKA G. K. G., Ph.D thesis (1984) [7] NASHED M. Z., Generalized Inverses and Applications (1976) · Zbl 0346.15001 [8] PORTER B., Int. J. Control 25 pp 483– (1977) · Zbl 0347.93018 · doi:10.1080/00207177708922246 [9] PORTER B., Int. J. Systems Sci 9 pp 445– (1978) · Zbl 0373.93015 · doi:10.1080/00207727808941710
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