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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

x ˙=A i x+B i u(i=1,,N),x n ,u m ·

The authors assume that for each i, there exists a stabilizing linear feedback u=K i x for the ith 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 τ 0 =0<τ 1 <τ 2 < . On each interval [τ j ,τ j+1 ) the control law may be indifferently defined as either u(t)=K i x(t) (where i is the active index for the jth interval) or the sampled value u(t)K i x(τ j ).

The authors prove that asymptotic stability is guaranteed provided that the mesh size max{τ j+1 -τ 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:
34A36Discontinuous equations
34D20Stability of ODE
34H05ODE in connection with control problems
34H15Stabilization (ODE in connection with control problems)