Abstract
This brief studies the stability of switched systems in which all the subsystems may be unstable. In addition, some of the switching behaviors of the systems are destabilizing. By using the piecewise Lyapunov function method and taking a tradeoff between the increasing scale and the decreasing scale of the Lyapunov function at switching times, the maximum dwell time for admissible switching signals is obtained and the extended stability results for switched systems in a nonlinear setting are first derived. Then, based on the discretized Lyapunov function method, the switching stabilization problem for linear context is solved. By contrasting with the contributions available in the literature, we do not require that all the switching behaviors of the switching system under consideration are stabilizing. More specifically, even if all the subsystems governing the continuous dynamics are not stable and some of the switching behaviors are destabilizing, the stability of the switched system can still be retained. A numerical example is given to illustrate the validity of the proposed results.
Highlights
S WITCHED systems, as an important class of hybrid systems, are composed of a set of continuous-time or discrete-time subsystems and a switching rule acted among subsystems
In a cellular mobile communication system, when the user moves from a base station coverage area to another base station coverage area, the previous occupied channel is immediately switched to the pool of available channel, which can be modeled as a switched system
A lot of efforts have been focused on the stability analysis and the control problem of switched systems [3]-[10]
Summary
S WITCHED systems, as an important class of hybrid systems, are composed of a set of continuous-time or discrete-time subsystems and a switching rule acted among subsystems. The paper [15] first studied the topic for switched linear systems with some unstable subsystems by using the average dwell time approach. When all the subsystems are unstable, the stabilization problem is much more challenging since the trade-off idea obviously fails To solve this issue, the state-dependent switching strategies such as the min-projection strategy and the largest region function strategy have been proposed. The input-to-state stability of switched delay systems with all unstable subsystems were retained, provided that the switching signal satisfies a dwell-time upper bound condition in [20]. In [22], stabilization of switched linear systems composed fully of unstable subsystems under dwell time switching was investigated. A function β : R+ × R+ → R+ is of class KL if β (·,t) is of class K for each t ≥ 0 and β (s, ·) is decreasing to zero for each s ≥ 0
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