Periodic stabilizability of switched linear control systems.

*(English)*Zbl 1175.93192Summary: Stabilizability via direct/observer-based state feedback control for discrete-time Switched Linear Control Systems (SLCSs) is investigated in this paper. For an SLCS, the control factors are not only the control input but also the switching signal, and they need to be designed in order to stabilize the system. As a result, stabilization design for SLCSs is more complicated than that for non-switched ones. Differently from the existing approaches, a periodic switching signal and piecewise constant linear state feedback control are adopted to achieve periodic stabilizability for such systems. It is highlighted that multiple feedback controllers need to be designed for one subsystem. For discrete-time SLCSs, it is proved that reachability implies periodic stabilizability via state feedback. A necessary and sufficient criterion for periodic stabilizability is also established. Two stabilization design algorithms are presented for real application. Moreover, it is proved that reachability and observability imply periodic stabilizability via observer-based feedback for discrete-time SLCSs. Periodic detectability, as the dual concept of periodic stabilizability, is discussed and the stabilization design algorithms via observer-based feedback are presented as well.

##### MSC:

93D15 | Stabilization of systems by feedback |

93C55 | Discrete-time control/observation systems |

93C05 | Linear systems in control theory |

93B03 | Attainable sets, reachability |

93B07 | Observability |

##### Keywords:

switched linear control systems (SLCSs); periodic stabilizability; reachability; observability; periodic detectability; observer
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\textit{G. Xie} and \textit{L. Wang}, Automatica 45, No. 9, 2141--2148 (2009; Zbl 1175.93192)

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