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This article is cited in 1 scientific paper (total in 1 paper)
The criterion of uniform global attainability of linear systems
A. A. Kozlov Polotsk State University, ul. Blokhina, 29, Novopolotsk, 211440, Belarus
Abstract:
In this paper,
we consider a linear time-varying control system with locally integrable and integrally bounded coefficients
\begin{equation}
\dot x =A(t)x+ B(t)u, \quad x\in\mathbb{R}^n,\quad
u\in\mathbb{R}^m,\quad t\geqslant 0. \tag{1}
\end{equation}
We construct control of the system $(1)$ as a linear feedback
$u=U(t)x$ with a measurable and bounded function $U(t)$, $t\geqslant 0$. For the closed-loop system
\begin{equation}
\dot x =(A(t)+B(t)U(t))x, \quad x\in\mathbb{R}^n, \quad t\geqslant 0, \tag{2}
\end{equation}
the criterion for its uniform global attainability is established.
The latter property means the existence of $T>0$ such that for any positive $\alpha$ and $\beta$ there exists a $d=d(\alpha,\beta)>0$ such that for any $t_0\geqslant 0$ and for any $(n\times n)$-matrix $H$, $\|H\|\leqslant\alpha$,
$\det H\geqslant\beta$, there exists a measurable on $[t_0,t_0+T]$ gain matrix function $U(\cdot)$ such that $\sup\limits_{t\in [t_0,t_0+T]}\|U(t)\|\leqslant d$ and
$X_U(t_0+T,t_0)=H$, where $X_U$ is the state transition matrix for the system (2).
The proof of the criterion is based on the theorem on the representation of an arbitrary $(n\times n)$-matrix
with a positive determinant in the form of a product of nine upper and lower triangular
matrices with positive diagonal elements and additional conditions on the norm and determinant of these matrices.
Keywords:
linear control system, state-transition matrix, uniform global attainability.
Received: 01.07.2018
Citation:
A. A. Kozlov, “The criterion of uniform global attainability of linear systems”, Izv. IMI UdGU, 52 (2018), 47–58
Linking options:
https://www.mathnet.ru/eng/iimi360 https://www.mathnet.ru/eng/iimi/v52/p47
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