Feedback Stabilization of Discrete-Time Switched Systems Under Büchi-Constrained Signals

This letter studies the feedback stabilization problem for a class of discrete-time switched systems. The goal is the design, via semidefinite optimization techniques, of feedback control rules depending only on the current state variable and on the past values of the underlying switching sequence. The resulting control policy achieves uniform exponential stabilization over a pre-constructed class of switching signals. The overall construction generalizes known approaches for stabilization over arbitrary switching sequences, but it is able to stabilize systems for which none of the defining subsystems is stabilizable. This extension is obtained employing graph-theoretic tools, introducing the Büchi automata formalism in order to specify the considered classes of admissible sequences, seen here in the general setting of
$\omega $
-regular languages. The proposed construction is finally illustrated with the help of a numerical example.