Incremental Gain Computation and Regulation of Discrete-time Positive Lur\'e Systems using Linear Programming

TL;DR

This paper develops linear programming methods to compute and regulate incremental  and  gains for discrete-time positive Lure9 systems, enhancing robustness analysis and control design.

Contribution

It introduces a novel approach to compute and regulate incremental gains for positive Lure9 systems using linear programming, with practical numerical verification.

Findings

Linear programming effectively computes upper bounds on incremental gains.

Regulation of  gain via state-feedback control is demonstrated.

Numerical examples confirm the approach's validity and robustness.

Abstract

This work approaches the problem of computing incremental $\ell_1$ and $\ell_\infty$ gains for discrete-time positive systems in \lure feedback with static memoryless nonlinearities, and regulating the $\ell_\infty$ gain through the design of a state-feedback controller. Finite incremental gains provide a quantitative measure of robustness for trajectories, and will ensure that all pairs of trajectories will converge to a fixed point or will diverge together in the absence of an applied input. Upper-bounds on these incremental gains can be computed through linear programming. Computation and regulation of the $\ell_1$ and $\ell_\infty$ incremental gains are verified by numerical examples.