# Stability and performance analysis of linear positive systems with delays using input-output methods

## TL;DR

This paper demonstrates the effectiveness of input-output methods in analyzing the stability and performance of linear positive systems with delays, providing simple proofs and extending existing results.

## Contribution

It offers new, simplified proofs for stability conditions of delayed positive systems and extends these results to various delay types using input-output approaches.

## Key findings

- Input-output methods are non-conservative for certain positive systems.
- Provided simple proofs for stability of delayed positive systems.
- Derived performance bounds in L1, L2, and L∞ norms.

## Abstract

It is known that input-output approaches based on scaled small-gain theorems with constant $D$-scalings and integral linear constraints are non-conservative for the analysis of some classes of linear positive systems interconnected with uncertain linear operators. This dramatically contrasts with the case of general linear systems with delays where input-output approaches provide, in general, sufficient conditions only. Using these results we provide simple alternative proofs for many of the existing results on the stability of linear positive systems with discrete/distributed/neutral time-invariant/-varying delays and linear difference equations. In particular, we give a simple proof for the characterization of diagonal Riccati stability for systems with discrete-delays and generalize this equation to other types of delay systems. The fact that all those results can be reproved in a very simple way demonstrates the importance and the efficiency of the input-output framework for the analysis of linear positive systems. The approach is also used to derive performance results evaluated in terms of the $L_1$-, $L_2$- and $L_\infty$-gains. It is also flexible enough to be used for design purposes.