Iterative, Small-Signal L2 Stability Analysis of Nonlinear Constrained Systems

TL;DR

This paper introduces an iterative semi-definite programming approach to analyze small-signal L2 stability of nonlinear control-affine systems within compact sets, ensuring stability and input bounds through barrier functions.

Contribution

It presents a novel method combining storage and barrier functions with semi-definite programs for stability analysis of nonlinear systems on compact sets.

Findings

Method successfully computes L2 gain bounds.

Ensures state remains within a safe set using barrier functions.

Validated effectiveness through a numerical example.

Abstract

This paper provides a method to analyze the small-signal L2 gain of control-affine nonlinear systems on compact sets via iterative semi-definite programs. First, a continuous piecewise affine storage function and the corresponding upper bound on the L2 gain are found on a bounded, compact set's triangulation. Then, to ensure that the state does not escape this set, a barrier function is found that is robust to small-signal inputs. Small-signal L2 stability then holds inside each sublevel set of the barrier function inside the set where the storage function was found. The bound on the inputs is also found while searching for a barrier function. The method's effectiveness is shown in a numerical example.