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Stability results of a class of hybrid systems under switched continuous-time and discrete-time control. (English) Zbl 1193.34014

This paper addresses the stability problem for a switched system formed by a finite family of linear subsystems
\[ \dot x= A_ix+ B_iu\quad (i= 1,\dots, N),\quad x\in\mathbb{R}^n,\;u\in\mathbb{R}^m. \]
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.

MSC:

34A36 Discontinuous ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34H05 Control problems involving ordinary differential equations
34H15 Stabilization of solutions to ordinary differential equations
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References:

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