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Zbl 1193.34014
de la Sen, M.; Ibeas, A.
Stability results of a class of hybrid systems under switched continuous-time and discrete-time control.
(English)
[J] Discrete Dyn. Nat. Soc. 2009, Article ID 315713, 28 p. (2009). ISSN 1026-0226; ISSN 1607-887X/e

This paper addresses the stability problem for a switched system formed by a finite family of linear subsystems $$\dot x= A_ix+ B_iu\quad (i= 1,\dots, N),\quad x\in\bbfR^n,\ u\in\bbfR^m.$$ The authors assume that for each $i$, there exists a stabilizing linear feedback $u= K_ix$ for the $i$th subsystem, and that the resulting family of closed-loop subsystems admits a common quadratic Lyapunov function. The control law for the switched system is defined by means of a sequence of switching times $\tau_0= 0<\tau_1<\tau_2<\cdots$\ . On each interval $[\tau_j, \tau_{j+1})$ the control law may be indifferently defined as either $u(t)= K_ix(t)$ (where $i$ is the active index for the $j$th interval) or the sampled value $u(t)\equiv K_i x(\tau_j)$. The authors prove that asymptotic stability is guaranteed provided that the mesh size $\max\{\tau_{j+1}- \tau_j\}$ is sufficiently small. The paper contains also a sufficient condition for the existence of a common Lyapunov function, and extensions of the result to systems with static output feedback and/or centralized/decentralized control.
[Andrea Bacciotti (Torino)]
MSC 2000:
*34A36 Discontinuous equations
34D20 Lyapunov stability of ODE
34H05 ODE in connection with control problems
34H15

Keywords: switched systems; sampled data control; Lyapunov functions

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