# Stability analysis of LPV systems with piecewise differentiable parameters

## TL;DR

This paper introduces a hybrid systems approach to analyze the stability of LPV systems with piecewise differentiable parameters, unifying existing criteria and providing tractable conditions via sum of squares programming.

## Contribution

It proposes a novel hybrid systems framework for LPV stability analysis, generalizing quadratic and robust criteria with finite-dimensional relaxations.

## Key findings

- Unified stability conditions for LPV systems with piecewise differentiable parameters.
- Relaxation via sum of squares programming yields practical stability tests.
- Illustrated effectiveness on literature examples.

## Abstract

Linear Parameter-Varying (LPV) systems with piecewise differentiable parameters is a class of LPV systems for which no proper analysis conditions have been obtained so far. To fill this gap, we propose an approach based on the theory of hybrid systems. The underlying idea is to reformulate the considered LPV system as an equivalent hybrid system that will incorporate, through a suitable state augmentation, information on both the dynamics of the state of the system and the considered class of parameter trajectories. Then, using a result pertaining on the stability of hybrid systems, two stability conditions are established and shown to naturally generalize and unify the well-known quadratic and robust stability criteria together. The obtained conditions being infinite-dimensional, a relaxation approach based on sum of squares programming is used in order to obtain tractable finite-dimensional conditions. The approach is finally illustrated on two examples from the literature.