Polyhedral Estimation of L-1 and L-infinity Incremental Gains of Nonlinear Systems

TL;DR

This paper introduces a linear programming method to compute and optimize incremental L-1 and L-infinity gains of nonlinear systems using polyhedral Lyapunov functions, with applications to controller design.

Contribution

It develops novel dissipativity conditions for incremental L-1 gain bounds and extends existing L-infinity results, linking the two through system adjoints, and provides a computational algorithm for gain estimation and control.

Findings

The LP-based algorithm yields sharper gain bounds with more iterations.

The method can be used to design linear feedback controllers optimizing incremental gains.

Numerical examples demonstrate the effectiveness and limitations of the approach.

Abstract

We provide novel dissipativity conditions for bounding the incremental L-1 gain of systems. Moreover, we adapt existing results on the L-infinity gain to the incremental setting and relate the incremental L-1 and L-infinity gain bounds through system adjoints. Building on work on optimization based approaches to constructing polyhedral Lyapunov functions, we make use of these conditions to obtain a Linear Programming based algorithm that can provide increasingly sharp bounds on the gains as a function of a given candidate polyhedral storage function or polyhedral set. The algorithm is also extended to allow for the design of linear feedback controllers for performance, as measured by the bounds on the incremental gains. We apply the algorithm to a couple of numerical examples to illustrate the power, as well as some limitations, of this approach.