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.

翻译：在存在不匹配扰动（即无法通过控制输入直接抵消的不确定性）的情况下确保安全性，仍然是非线性控制领域的一个关键挑战。本文提出了一种针对具有不匹配扰动的非线性系统的安全关键控制构造性方法。我们首先利用最近提出的最优衰减控制屏障函数概念，为具有此类不确定性的系统推广了输入-状态安全框架，这为满足相关的类李雅普诺夫安全条件提供了更大的灵活性。在此基础上，我们概述了为两类具有不匹配不确定性的相关系统构造输入-状态安全控制屏障函数的步骤：i) 严格反馈系统；ii) 双重相对阶系统（类似于微分平坦系统）。我们的理论结果通过倒立摆和平面四旋翼飞行器的数值仿真得到了验证。