Input-to-State Stability (ISS) is fundamental in mathematically quantifying how stability degrades in the presence of bounded disturbances. If a system is ISS, its trajectories will remain bounded, and will converge to a neighborhood of an equilibrium of the undisturbed system. This graceful degradation of stability in the presence of disturbances describes a variety of real-world control implementations. Despite its utility, this property requires the disturbance to be bounded and provides invariance and stability guarantees only with respect to this worst-case bound. In this work, we introduce the concept of ``ISS in probability (ISSp)'' which generalizes ISS to discrete-time systems subject to unbounded stochastic disturbances. Using tools from martingale theory, we provide Lyapunov conditions for a system to be exponentially ISSp, and connect ISSp to stochastic stability conditions found in literature. We exemplify the utility of this method through its application to a bipedal robot confronted with step heights sampled from a truncated Gaussian distribution.

翻译：输入到状态稳定性（ISS）是数学上量化系统在存在有界扰动时稳定性退化程度的基础概念。若系统满足ISS条件，其轨迹将保持有界，并收敛至未受扰系统平衡点邻域内。这种存在扰动时稳定性的渐进退化特性描述了多种实际控制实现。尽管该特性具有实用性，但其要求扰动有界，且仅能基于最坏情况边界提供不变性和稳定性保证。本研究提出“概率意义下输入到状态稳定性（ISSp）”概念，将ISS推广至受无界随机扰动的离散时间系统。借助鞅理论工具，我们给出了系统满足指数ISSp的Lyapunov条件，并将ISSp与文献中的随机稳定性条件建立联系。通过将该方法应用于面临截断高斯分布采样步高的双足机器人实例，我们展示了该方法的实用性。