# Stability analysis and stabilization of LPV systems with jumps and (piecewise) differentiable parameters using continuous and sampled-data controllers

## TL;DR

This paper develops new stability analysis and stabilization methods for LPV systems with jumps and piecewise differentiable parameters, using Lyapunov functions, dwell-time conditions, and sum of squares programming.

## Contribution

It introduces a unified approach for stability and stabilization of hybrid LPV systems with jumps, including uncertain and impulsive systems, using continuous and sampled-data controllers.

## Key findings

- Unified stability conditions generalize quadratic and robust stability.
- New state-feedback controllers ensure stabilization under dwell-time constraints.
- Stability conditions are formulated as solvable semidefinite programs.

## Abstract

Linear Parameter-Varying (LPV) systems with jumps and piecewise differentiable parameters is a class of hybrid LPV systems for which no tailored stability analysis and stabilization conditions have been obtained so far. We fill this gap here by proposing an approach based on a clock- and parameter-dependent Lyapunov function yielding stability conditions under both constant and minimum dwell-times. Interesting adaptations of the latter result consist of a minimum dwell-time stability condition for uncertain LPV systems and LPV switched impulsive systems. The minimum dwell-time stability condition is notably shown to naturally generalize and unify the well-known quadratic and robust stability criteria all together. Those conditions are then adapted to address the stabilization problem via timer-dependent and a timer- and/or parameter-independent (i.e. robust) state-feedback controllers, the latter being obtained from a relaxed minimum dwell-time stability condition involving slack-variables. Finally, the last part addresses the stability of LPV systems with jumps under a range dwell-time condition which is then used to provide stabilization conditions for LPV systems using a sampled-data state-feedback gain-scheduled controller. The obtained stability and stabilization conditions are all formulated as infinite-dimensional semidefinite programming problems which are then solved using sum of squares programming. Examples are given for illustration.