# Co-design of aperiodic sampled-data min-jumping rules for linear impulsive, switched impulsive and sampled-data systems

## TL;DR

This paper presents a co-design approach for jump rules and sampled-data control laws in impulsive and switched impulsive systems with aperiodic sampling, using Lyapunov-Metzler conditions and sum of squares relaxations.

## Contribution

It introduces a novel co-design framework with a new set of conditions reformulated as finite-dimensional optimization problems for complex impulsive systems.

## Key findings

- Sum of squares relaxation is non conservative at high polynomial degrees.
- Effective results achieved with low polynomial degrees in examples.
- Provides a practical method for co-design in impulsive systems.

## Abstract

Co-design conditions for the design of a jumping-rule and a sampled-data control law for impulsive and impulsive switched systems subject to aperiodic sampled-data measurements are provided. Semi-infinite discrete-time Lyapunov-Metzler conditions are first obtained. As these conditions are difficult to check and generalize to more complex systems, an equivalent formulation is provided in terms of clock-dependent (infinite-dimensional) matrix inequalities. These conditions are then, in turn, approximated by a finite-dimensional optimization problem using a sum of squares based relaxation. It is proven that the sum of squares relaxation is non conservative provided that the degree of the polynomials is sufficiently large. It is emphasized that acceptable results are obtained for low polynomial degrees in the considered examples.