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.