标题： 凸计算用于黑箱离散时间动力系统受控安全不变集 显示英文标题

标题： Convex Computations for Controlled Safety Invariant Sets of Black-box Discrete-time Dynamical Systems

摘要： 识别受控安全不变集（CSISs）在安全关键应用中是必不可少的。 本文解决了识别黑箱离散时间系统的CSISs的问题，其中模型未知，仅可访问有限的仿真数据。 传统上，CSIS被定义为一个安全集的子集，包括初始状态，对于这些初始状态存在控制输入，使得系统在下一步时间步内保持在该集合中——这被称为一步不变性性质。 然而，一步不变性的要求可以等价地转化为更严格的“始终不变性”条件，这意味着存在控制输入能够使系统无限期地保持在该集合中。 对于黑箱系统来说，这种条件可能过于严格或不切实际，因为在单个时间步之后或有限数量的时间步之后，预测可能会变得不可靠。 为了克服黑箱系统带来的挑战，我们将一步不变性性质以“可能近似正确”（PAC）的方式重新表述。 这种方法使我们能够在预定义的置信水平下评估存在控制输入使系统在下一步时间步内保持在CSIS中的概率。 如果系统成功在下一步时间步内保持在集合中，那么我们可以对新状态重新应用不变性评估，从而实现不变性的递归保证。 我们的方法使用了屏障函数和情景优化， resulting in a linear programming method to estimate PAC CSISs. 最后，我们的方法在几个示例中得到了验证。 显示英文摘要

摘要： Identifying controlled safety invariant sets (CSISs) is essential in safety-critical applications. This paper tackles the problem of identifying CSISs for black-box discrete-time systems, where the model is unknown and only limited simulation data is accessible. Traditionally, a CSIS is defined as a subset of a safe set, encompassing initial states for which a control input exists that keeps the system within the set at the next time step-this is referred to as the one-step invariance property. However, the requirement for one-step invariance can be equivalently translated into a stricter condition of ``always-invariance'', meaning that there exist control inputs capable of keeping the system within this set indefinitely. Such a condition may prove overly stringent or impractical for black-box systems, where predictions can become unreliable beyond a single time step or a limited number of finite time steps. To overcome the challenges posed by black-box systems, we reformulate the one-step invariance property in a ``Probably Approximately Correct'' (PAC) sense. This approach allows us to assess the probability that a control input exists to keep the system within the CSIS at the next time step, with a predefined level of confidence. If the system successfully remains within the set at the next time step, we can then reapply the invariance evaluation to the new state, thereby facilitating a recursive assurance of invariance. Our method employs barrier functions and scenario optimization, resulting in a linear programming method to estimate PAC CSISs. Finally, the effectiveness of our approach is demonstrated on several examples.