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## Archive of Issues

Russia Izhevsk
Year
2014
Issue
1
Pages
19-31
 Section Mathematics Title To the property of consistency for four-dimensional discrete-time linear stationary control systems with incomplete feedback of the special form Author(-s) Zaitsev V.A.a, Maksimova N.V.a Affiliations Udmurt State Universitya Abstract We consider a discrete-time linear control system with an incomplete feedback $$x(t+1)=A(t)x(t)+B(t)u(t), \quad y(t)=C^*(t)x(t), \quad u(t)=U(t)y(t), \quad t\in\mathbb{Z}.$$ We study the problem of control over the asymptotic behavior of the closed-loop system $$x(t+1)=(A(t)+B(t)U(t)C^*(t))x(t), \quad x\in\mathbb K^n, \qquad (1)$$ where $\mathbb{K}=\mathbb{C}$ or $\mathbb{K}=\mathbb{R}$. For the above system, we introduce the concept of consistency, which is a generalization of the concept of complete controllability onto systems with an incomplete feedback. The focus is on the consistency property of the system $(1)$. We have obtained new necessary conditions and sufficient conditions for the consistency of the above system including the case when the system is time-invariant. For the time-invariant system $(1)$, we study the problem of arbitrary placement of eigenvalue spectrum. The objective is to reduce a characteristic polynomial of a matrix of the stationary system $(1)$ to any prescribed polynomial by means of the time-invariant control $U$. For the system $(1)$ with constant coefficients of the special form where the matrix $A$ is Hessenberg, the rows of the matrix $B$ before the $p$-th and the rows of the matrix $C$ after the $p$-th are equal to zero (not including $p$), the property of consistency is the sufficient condition for arbitrary placement of eigenvalue spectrum. It has been proved that the converse proposition is true for $n<4$ and false for $n>5$. In present paper we prove that the converse proposition is true for $n = 4$. 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/vm140102 Received 22 December 2013 Language Russian Citation 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, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 19-31. References Lancaster P. Theory of matrices, New York-London: Academic Press, 1969. Translated under the title Teoriya matrits, Moscow: Nauka, 1978, 280 р. Popova S.N., Tonkov E.L. Control over the Lyapunov exponents of consistent systems. I, Differential Equations, 1994, vol. 30, no. 10, pp. 1556-1564. Popova S.N., Tonkov E.L. Control over the Lyapunov exponents of consistent systems. II, Differential Equations, 1994, vol. 30, no. 11, pp. 1800-1807. Popova S.N., Tonkov E.L. Control over the Lyapunov exponents of consistent systems. III, Differential Equations, 1995, vol. 31, no. 2, pp. 209-218. Popova S.N., Tonkov E.L. Uniform consistency of linear systems, Differential Equations, 1995, vol. 31, no. 4, pp. 672-674. Popova S.N., Tonkov E.L. Consistent systems and control of Lyapunov exponents, Differential Equations, 1997, vol. 33, no. 2, pp. 226-235. Makarov E.K., Popova S.N. Local controllability of characteristic Lyapunov exponents of systems with simple exponents, Differential Equations, 1997, vol. 33, no. 4, pp. 496-500. Zaitsev V.A., Tonkov Ye.L. Attainability, consistency and the rotation method by V.M. Millionshchikov, Russian Mathematics, 1999, vol. 43, no. 2, pp. 42-52. 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). 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 of linear stationary control systems with the observer of a special form, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2010, no. 1, pp. 22-47 (in Russian). Full text