# Simple interval observers for linear impulsive systems with applications to sampled-data and switched systems

## TL;DR

This paper develops a constructive method for designing simple interval observers for linear impulsive systems with dwell-time constraints, applicable to sampled-data and switched systems, using sum of squares relaxations.

## Contribution

It introduces a new approach to design interval observers for impulsive systems with dwell-time constraints, extending to sampled-data and switched systems, with solutions derived from infinite-dimensional linear programs.

## Key findings

- Observer gains can be extracted from the optimization solutions.
- The approach is flexible and can include performance and uncertainties.
- Three illustrative examples demonstrate the method's effectiveness.

## Abstract

Sufficient conditions for the design of a simple class of interval observers for linear impulsive systems subject to minimum and range dwell-time constraints are obtained and formulated in terms of infinite-dimensional linear programs. The proposed approach is fully constructive in the sense that suitable observer gains can be extracted from the solution of the optimization problems and is flexible enough to be extended to include performance constraints and parametric uncertainties. In order to be solvable, the infinite-dimensional linear programs are relaxed using a method based on sum of squares which is known to be asymptotically exact in the present case. Three examples are given for illustration: the first one pertains on the interval observation of an impulsive system under a minimum dwell-time constraint, the second one is about the interval observation of an aperiodic sampled-data system and the last one is about the interval observation of a linear switched system.