Stabilization of discrete-time switched systems with constraints by dynamic logic-based switching feedback

A switching feedback has more flexibility in stabilizing systems compared to a feedback being continuous in state. In practice, a system controlled by a state-triggered switching law exhibits more robustness than by a time-triggered one. However, designing an admissible switching law in the static feedback form is difficult or even impossible when the switching sequence is subjected to certain constraints, such as the minimum dwell-time constraints. This study proposes a class of switching feedbacks with dynamic switching logic for discrete-time switched systems with constrained switching. First, a logic dynamic generator (LDG) is constructed to generate all admissible switching sequences. By combining the switched system with the LDG, the original switched system is transformed into a switched system without constraint on switching. Then, the logic and the continuous states are merged by using the theories of the semi-tensor product (STP) of matrices and the vector-representation of logic. Using this technique, the problem considered is then transformed into a problem of designing a state-triggered switching feedback for a merged switched system without constraints, and then to a problem of solving a set of bilinear matrix inequalities. Examples show that the proposed switching law with dynamic logic can stabilize certain switched systems that cannot be stabilized by existing switching laws in the static feedback form.