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This article is cited in 2 scientific papers (total in 2 papers)
MATHEMATICS
On the conditions of proportional local assignability of the Lyapunov spectrum of a linear discrete-time system
I. N. Banshchikovaa, E. K. Makarovb, S. N. Popovaa a Udmurt State
University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia
b Institute of Mathematics, National Academy of Sciences of Belarus, ul. Surganova,
11, Minsk, 220072, Belarus
Abstract:
We consider a problem of assigning the Lyapunov spectrum for a linear control discrete-time system
\begin{equation}
x(m+1)=A(m)x(m)+B(m)u(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n},\ u\in\mathbb R^{k},
\tag{1}
\end{equation}
in a small neighborhood of the Lyapunov spectrum of the free system
\begin{equation}
x(m+1)=A(m)x(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n},
\tag{2}
\end{equation}
by means of linear feedback $u(m)=U(m)x(m)$. We assume that the norm of the feedback matrix $U(\cdot)$
satisfies the Lipschitz estimate with respect to the required shift of the Lyapunov spectrum.
This property is called proportional local assignability of the Lyapunov spectrum of the closed-loop system
\begin{equation}
x(m+1)=\bigl(A(m)+B(m)U(m)\bigr)x(m),\quad m\in\mathbb N,\ x\in\mathbb R^{n}.
\tag{3}
\end{equation}
We previously proved that uniform complete controllability of system (1) and stability of the Lyapunov spectrum
of free system (2) are sufficient conditions for proportional local assignability of the Lyapunov spectrum
of closed-loop system (3). In this paper we give an example demonstrating that these conditions are not necessary.
Keywords:
linear discrete-time system, Lyapunov exponents, сontrollability, stabilizability.
Received: 22.07.2019
Citation:
I. N. Banshchikova, E. K. Makarov, S. N. Popova, “On the conditions of proportional local assignability of the Lyapunov spectrum of a linear discrete-time system”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 29:3 (2019), 301–311
Linking options:
https://www.mathnet.ru/eng/vuu683 https://www.mathnet.ru/eng/vuu/v29/i3/p301
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