Stability and stabilization for LPV systems based on Lyapunov functions with non-monotonic terms

This paper presents new conditions for stability analysis, static output-feedback and state-feedback control design for discrete-time linear parameter-varying systems. The proposed methodology is based on the combination of quadratic “Lyapunov-like” terms such that individually each one is not necessarily monotonically decreasing along the state trajectories. Firstly, a new necessary and sufficient stability analysis condition based on the use of non-monotonic terms in the Lyapunov function is proposed. Concerning the stabilization problem, a novel strategy for the static output-feedback control design is proposed and, unlike most approaches in the literature, no structural constraints on the output matrix are imposed, which it is an extra feature advantage of the proposed method as well. Besides that, a new gain-scheduling state-feedback control design solution is derived. All proposed conditions are presented in the form of Linear Matrix Inequalities and their feasibility implies the existence of a Lyapunov function that is monotonically decreasing along trajectories. Numerical experiments illustrate the potential of the proposed techniques.