Trajectory-based Robustness Analysis for Nonlinear Systems

TL;DR

This paper introduces a method for analyzing the robustness of uncertain nonlinear systems along finite-horizon trajectories using linearization, IQCs, and differential inequalities, demonstrated on a robotic arm example.

Contribution

It presents a novel computational approach combining linearization, IQCs, and Riccati equations to assess robustness without heuristics.

Findings

Provides a DLMI-based robustness bound for nonlinear systems.

Avoids heuristic time gridding in solving DLMIs.

Demonstrates effectiveness on a robotic arm example.

Abstract

This paper considers the robustness of an uncertain nonlinear system along a finite-horizon trajectory. The uncertain system is modeled as a connection of a nonlinear system and a perturbation. The analysis relies on three ingredients. First, the nonlinear system is approximated by a linear time-varying (LTV) system via linearization along a trajectory. This linearization introduces an additional forcing input due to the nominal trajectory. Second, the input/output behavior of the perturbation is described by time-domain, integral quadratic constraints (IQCs). Third, a dissipation inequality is formulated to bound the worst-case deviation of an output signal due to the uncertainty. These steps yield a differential linear matrix inequality (DLMI) condition to bound the worst-case performance. The robustness condition is then converted to an equivalent condition in terms of a Riccati Differential Equation. This yields a computational method that avoids heuristics often used to solve DLMIs, e.g. time gridding. The approach is demonstrated by a two-link robotic arm example.