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Russia Izhevsk
Year
2014
Issue
1
Pages
19-31
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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.
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