Trajectory‐based robustness analysis for nonlinear systems

Abstract

This article 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, for example, time gridding. The approach is demonstrated by a two‐link robotic arm example.