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Section
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Mathematics
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Title
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On the property of uniform complete controllability of a discrete-time linear control system
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Author(-s)
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Zaitsev V.A.a,
Popova S.N.a,
Tonkov E.L.a
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Affiliations
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Udmurt State Universitya
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Abstract
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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 $t\in[\tau,\tau+\vartheta)$.
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Keywords
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linear control system, discrete time, uniform complete controllability
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UDC
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517.977.1, 517.929.2
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MSC
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93B05, 93C05, 93C55
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DOI
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10.20537/vm140404
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Received
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15 August 2014
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Language
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Russian
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Citation
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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.
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References
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- 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).
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