Flexible-Step MPC for Switched Linear Systems With No Quadratic Common Lyapunov Function

In this article, we develop a systematic method for constructing a generalized discrete-time control Lyapunov function for the flexible-step model predictive control (MPC) scheme, recently introduced in (Fürnsinn et al., 2025), when restricted to the class of linear systems. Specifically, we show that a set of linear matrix inequalities (LMIs) can be used for this purpose, demonstrating its tractability. The main consequence of this LMI formulation is that, when combined with flexible-step MPC, we can effectively stabilize switched control systems, for which no quadratic common Lyapunov function exists.