Data-driven safe gain-scheduling control

Data-based safe gain-scheduling controllers are presented for discrete-time linear parameter-varying systems (LPV) with polytopic models. First, $\lambda$-contractivity conditions are provided under which the safety and stability of the LPV systems are unified through Minkowski functions of the safe sets. Then, a data-based representation of the closed-loop LPV system is provided, which requires less restrictive data richness conditions than identifying the system dynamics. This sample-efficient closed-loop data-based representation is leveraged to design data-driven gain-scheduling controllers that guarantee $\lambda$-contractivity and, thus, invariance of the safe sets. It is also shown that the problem of designing a data-driven gain-scheduling controller for a polyhedral (ellipsoidal) safe set amounts to a linear program (a semi-definite program). The motivation behind direct learning of a safe controller is that identifying an LPV system requires satisfying the persistence of the excitation (PE) condition. It is shown in this paper, however, that directly learning a safe controller and bypassing the system identification can be achieved without satisfying the PE condition. This data-richness reduction is of vital importance, especially for LPV systems that are open-loop unstable, and collecting rich samples to satisfy the PE condition can jeopardize their safety. A simulation example is provided to show the effectiveness of the presented approach.