Input-to-State Safe Backstepping: Robust Safety-Critical Control with Unmatched Uncertainties

TL;DR

This paper introduces a new control framework ensuring safety in nonlinear systems with unmatched disturbances by generalizing input-to-state safety and constructing appropriate barrier functions for specific system classes.

Contribution

It extends the input-to-state safety framework to handle unmatched uncertainties using Optimal Decay CBFs and provides a systematic procedure for certain nonlinear system classes.

Findings

Successfully applied to inverted pendulum simulation.

Validated on planar quadrotor dynamics.

Demonstrates robustness against unmatched disturbances.

Abstract

Guaranteeing safety in the presence of unmatched disturbances -- uncertainties that cannot be directly canceled by the control input -- remains a key challenge in nonlinear control. This paper presents a constructive approach to safety-critical control of nonlinear systems with unmatched disturbances. We first present a generalization of the input-to-state safety (ISSf) framework for systems with these uncertainties using the recently developed notion of an Optimal Decay CBF, which provides more flexibility for satisfying the associated Lyapunov-like conditions for safety. From there, we outline a procedure for constructing ISSf-CBFs for two relevant classes of systems with unmatched uncertainties: i) strict-feedback systems; ii) dual-relative-degree systems, which are similar to differentially flat systems. Our theoretical results are illustrated via numerical simulations of an inverted pendulum and planar quadrotor.