Stabilization of Switched Affine Systems With Dwell-Time Constraint

This paper addresses the problem of stabilization of switched affine systems under dwell-time constraint, giving guarantees on the bound of the quadratic cost associated with the proposed state switching control law. Specifically, two switching rules are presented relying on the solution of differential Lyapunov inequalities and Lyapunov-Metzler inequalities, from which the stability conditions are expressed. The first one allows to regulate the state of linear switched systems to zero, whereas the second one is designed for switched affine systems proving practical stability of the origin. In both cases, the determination of a guaranteed cost associated with each control strategy is shown. In the cases of linear and affine systems, the existence of the solution for the Lyapunov-Metzler condition is discussed and guidelines for the selection of a solution ensuring suitable performance of the system evolution are provided. The theoretical results are finally assessed by means of three examples.

💡 Research Summary

The paper tackles the challenging problem of stabilizing continuous‑time switched affine systems when a minimum dwell‑time between switches is imposed. Two state‑feedback switching laws are proposed. The first one addresses purely linear switched systems and guarantees global exponential convergence to the origin together with an explicit upper bound on the H₂ performance index (J=\int_{t_0}^{t}z^{\top}z,d\tau). The second law extends the approach to switched affine systems, where each subsystem may have a distinct constant offset (b_i); in this case only practical stability of the origin is ensured, but a guaranteed bound on the average H₂ cost is still provided.

The technical core relies on a hybrid Lyapunov construction. For each mode (i) a time‑varying positive‑definite matrix (P_i(t)) is defined on the interval (