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## Archive of Issues

Russia Izhevsk
Year
2014
Issue
3
Pages
13-27
 Section Mathematics Title Consistency of discrete-time linear stationary control systems with an incomplete feedback of the special form for $n=5$ Author(-s) Zaitsev V.A.a Affiliations Udmurt State Universitya Abstract We consider a discrete-time linear control system with an incomplete feedback $$x(t+1)=Ax(t)+Bu(t), \quad y(t)=C^*x(t), \quad u(t)=Uy(t), \quad t\in\mathbb{Z},\quad (x,u,y)\in\mathbb{K}^n\times\mathbb{K}^m\times\mathbb{K}^k,$$ where $\mathbb K=\mathbb C$ or $\mathbb K=\mathbb R$. We introduce the concept of consistency for the closed-loop system $$x(t+1)=(A+BUC^*)x(t), \quad x\in\mathbb K^n. \qquad (1)$$ This concept is a generalization of the concept of complete controllability to systems with an incomplete feedback. We study the consistency of the system $(1)$ in connection with the problem of arbitrary placement of eigenvalue spectrum which is to bring a characteristic polynomial of a matrix of the system $(1)$ to any prescribed polynomial by means of the time-invariant control $U$. For the system $(1)$ of the special form where the matrix $A$ is Hessenberg and the rows of the matrix $B$ before the $p$-th and the rows of the matrix $C$ after the $p$-th (not including $p$) are equal to zero, the property of consistency is the sufficient condition for arbitrary placement of eigenvalue spectrum. In previous studies it has been proved that the converse is true for $n <5$ and false for $n> 5$. In this paper, an open question for $n = 5$ is resolved. For the system $(1)$ of the special form, it is proved that if $n = 5$ then the property of consistency is a necessary condition for the arbitrary placement of eigenvalue spectrum. The proof is carried out by direct searching of all possible valid values of dimensions $m, k, p$. The property of consistency is equivalent to the property of complete controllability of a big system of dimension $n^2$. For the proof we construct the big system and the controllability matrix $K$ of this system of dimension $n^2\times n^2mk$. We show that the matrix $K$ has a nonzero minor of order $n^2 = 25$. We use Maple 15 to calculate the high-order determinants. Keywords linear control system, incomplete feedback, consistency, eigenvalue assignment, stabilization, discrete-time system UDC 517.977, 517.925.51 MSC 93B55, 93C05, 93C55, 93D15 DOI 10.20537/vm140302 Received 12 July 2014 Language Russian Citation Zaitsev V.A. Consistency of discrete-time linear stationary control systems with an incomplete feedback of the special form for $n=5$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 3, pp. 13-27. References Zaitsev V.A., Maksimova N.V. To the property of consistency for four-dimensional discrete-time linear stationary control systems with incomplete feedback of the special form, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2014, no. 1, pp. 19-31 (in Russian). Zaitsev V.A. Eigenvalue assignment in discrete-time systems, Proceedings of XII All-Russian Conference on Control, Moscow, Institute of Control Sciences, Russian Academy of Sciences, 2014, pp. 862-867 (in Russian). Lancaster P. Theory of matrices, New York-London: Academic Press, 1969. Translated under the title Teoriya matrits, Moscow: Nauka, 1978, 280 р. Zaitsev V.A. Consistency and pole assignment in linear systems with incomplete feedback, Proceedings of IFAC Workshop on Control Applications of Optimization. University of Jyvaskyla, Finland, Jyvaskyla, 2009, vol. 7, part 1, pp. 344-345. http://www.ifac-papersonline.net/Detailed/41934.html Zaitsev V.A. Spectrum control in linear systems with incomplete feedback, Differential Equations, 2009, vol. 45, no. 9, pp. 1348-1357. Zaitsev V.A. Control of spectrum in bilinear systems, Differential Equations, 2010, vol. 46, no. 7, pp. 1071-1075. Zaitsev V.A. Necessary and sufficient conditions in a spectrum control problem, Differential Equations, 2010, vol. 46, no. 12, pp. 1789-1793. Zaitsev V.A. Consistent systems and pole assignment: I, Differential Equations, 2012, vol. 48, no. 1, pp. 120-135. Zaitsev V.A. Consistent systems and pole assignment: II, Differential Equations, 2012, vol. 48, no. 6, pp. 857-866. Zaitsev V.A. Consistency and control over spectrum in linear systems with an observer, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2009, no. 3, pp. 50-80 (in Russian). Full text