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

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.