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

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.

💡 Research Summary

The paper tackles the problem of guaranteeing safety for nonlinear systems that are subject to disturbances which cannot be cancelled directly by the control input (unmatched disturbances). While existing control barrier function (CBF) approaches—robust, adaptive, and input‑to‑state safe (ISSf) CBFs—handle matched disturbances relatively easily, they struggle when the disturbance enters the dynamics through channels that are independent of the control input.

To address this gap, the authors first generalize the system model to
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