Periodic solutions versus practical switching control for sensorless Piecewise Affine systems (PWA)

This work concerns the stabilization of general Piecewise Affine (PWA) systems without common equilibrium; the main objective consists in proposing a characterization of periodic solutions, by determining the critical parameters values of the cyclic behaviors. The proposed approach is based on the expansion of previous results on practical stabilization by switching. Due to the non-convex nature of general PWA synthesis problems, we primarily present a BMI formulation of the practical stabilization that is used to generate periodic solutions. More precisely, we characterize ω-limit sets as periodic trajectories of the global PWA system in terms of special invariant sets of the practical stabilization method. This will avoid a posteriori subset inclusion checking, since the underlying set belongs to the admissible state space part. This approach generalizes previous results to obtain invariance conditions and the set ω-limit points. A methodology and algorithms to compute periodic trajectories parameters are provided. Two illustrative examples are used for simulation, in particular the third order of Goodwin oscillator model is investigated as a non-uniform oscillatory complex system.