$L_1/\ell_1$-to-$L_1/\ell_1$ analysis of linear positive impulsive systems with application to the $L_1/\ell_1$-to-$L_1/\ell_1$ interval observation of linear impulsive and switched systems

TL;DR

This paper develops convex conditions for the stability and $L_1/ ext{l}_1$-gain analysis of linear positive impulsive systems, and constructs interval observers for impulsive and switched systems using sum of squares programming.

Contribution

It introduces a novel convex framework for stability and $L_1/ ext{l}_1$-gain analysis, and designs interval observers via sum of squares relaxation for impulsive and switched systems.

Findings

Conditions are formulated as infinite-dimensional linear programs solvable by sum of squares programming.

Constructive methods for designing $L_1/ ext{l}_1$-to-$L_1/ ext{l}_1$ interval observers are provided.

Examples demonstrate the effectiveness of the proposed analysis and observer design.

Abstract

Sufficient conditions characterizing the asymptotic stability and the hybrid $L_1/\ell_1$-gain of linear positive impulsive systems under minimum and range dwell-time constraints are obtained. These conditions are stated as infinite-dimensional linear programming problems that can be solved using sum of squares programming, a relaxation that is known to be asymptotically exact in the present case. These conditions are then adapted to formulate constructive and convex sufficient conditions for the existence of $L_1/\ell_1$-to-$L_1/\ell_1$ interval observers for linear impulsive and switched systems. Suitable observer gains can be extracted from the (suboptimal) solution of the infinite-dimensional optimization problem where the $L_1/\ell_1$-gain of the system mapping the disturbances to the weighted observation errors is minimized. Some examples on impulsive and switched systems are given for illustration.