Archive of Issues
Russia Izhevsk
Section  Mathematics 
Title  To the property of consistency for fourdimensional discretetime linear stationary control systems with incomplete feedback of the special form 
Author(s)  Zaitsev V.A.^{a}, Maksimova N.V.^{a} 
Affiliations  Udmurt State University^{a} 
Abstract  We consider a discretetime 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 closedloop 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 timeinvariant. For the timeinvariant 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 timeinvariant 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, discretetime 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 fourdimensional discretetime linear stationary control systems with incomplete feedback of the special form, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 1, pp. 1931. 
References 

Full text 