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## Archive of Issues

Russia Izhevsk
Year
2014
Issue
4
Pages
53-63
 Section Mathematics Title On the property of uniform complete controllability of a discrete-time linear control system Author(-s) Zaitsev V.A.a, Popova S.N.a, Tonkov E.L.a Affiliations Udmurt State Universitya Abstract We study the property of uniform complete controllability (according to Kalman) for a discrete-time linear control system $$x(t+1)=A(t)x(t)+B(t)u(t), \quad t\in\mathbb{N}_0, \quad (x,u)\in\mathbb{R}^n\times\mathbb{R}^m. \qquad(1)$$ We prove that if the system $(1)$ is uniformly completely controllable, then the matrix $A(\cdot)$ is completely bounded on $\mathbb N_0$ (i.e. $\sup_{t\in\mathbb{N}_0}(|A(t)|+|A^{-1}(t)|)<+\infty$) and the matrix $B(\cdot)$ is bounded on $\mathbb N_0$. We prove that the system $(1)$ is uniformly completely controllable if and only if there exists a $\vartheta\in \mathbb N$ such that for all $\tau\in\mathbb N_0$ the inequalities $\alpha_1 I\leqslant W_1(\tau+\vartheta,\tau)\leqslant\beta_1 I$, $\alpha_2 I\leqslant W_2(\tau+\vartheta,\tau)\leqslant\beta_2 I$ hold for some positive $\alpha_i$ and $\beta_i$, where $$W_1(t,\tau)\doteq\sum_{s=\tau}^{t-1} X(t,s+1)B(s)B^*(s)X^*(t,s+1),\quad W_2(t,\tau)\doteq\sum_{s=\tau}^{t-1} X(\tau,s+1)B(s)B^*(s)X^*(\tau,s+1).$$ On the basis of this statement, we prove the following criterion for uniform complete controllability of the system $(1)$, which is similar to the Tonkov criterion of uniform complete controllability for continuous-time systems: the system $(1)$ is $\vartheta$-uniformly completely controllable if and only if the matrix $A(\cdot)$ is completely bounded on $\mathbb N_0$; the matrix $B(\cdot)$ is bounded on $\mathbb N_0$; there exists an $\ell=\ell(\vartheta)>0$ such that for every $\tau\in\mathbb{N}_0$ and for any $x_1\in\mathbb{R}^n$ there exists a control function $u(t)$, $t\in[\tau,\tau+\vartheta)$, which transfers the solution of the system $(1)$ from the state $x(\tau)=0$ to the state $x(\tau+\vartheta)=x_1$, and the inequality $|u(t)|\leqslant \ell |x_1|$ holds for all $t\in[\tau,\tau+\vartheta)$. Keywords linear control system, discrete time, uniform complete controllability UDC 517.977.1, 517.929.2 MSC 93B05, 93C05, 93C55 DOI 10.20537/vm140404 Received 15 August 2014 Language Russian Citation Zaitsev V.A., Popova S.N., Tonkov E.L. On the property of uniform complete controllability of a discrete-time linear control system, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 4, pp. 53-63. References Gaishun I.V. Sistemy s diskretnym vremenem (Discrete-time systems), Minsk: Institute of Mathematics of the National Academy of Sciences of Belarus, 2001, 400 p. Kwakernaak H., Sivan R. Linear optimal control systems, New York-London-Sydney-Toronto: Wiley-Interscience, 1972. Translated under the title Lineinye optimal'nye sistemy upravleniya, Moscow: Mir, 1977, 650 p. Kalman R.E. Contributions to the theory of optimal control, Boletin de la Sociedad Matematika Mexicana, 1960, vol. 5, no. 1, pp. 102-119. Horn R., Johnson C. Matrix analysis, Cambridge: Cambridge University Press, 1988. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989, 655 p. Lancaster P. Theory of matrices, New York-London: Academic Press, 1969. Translated under the title Teoriya matrits, Moscow: Nauka, 1978, 280 р. Demidovich V.B. On a criterion of stability for difference equations, Differ. Uravn., 1969, vol. 5, no. 7, pp. 1247-1255 (in Russian). Tonkov E.L. A criterion for uniform controllability and stabilization of a linear recurrent system, Differ. Uravn., 1979, vol. 15, no. 10, pp. 1804-1813 (in Russian). Full text