An Iterative Method for Computing Controlled Reach-Avoid Sets

This paper focuses on addressing the problem of computing controlled reach-avoid sets for continuous-time systems modeled by polynomial ordinary differential equations with control inputs in order to ensure the safety and reachability of safety-critical systems. In a controlled reach-avoid set (CRAS), there exists a feedback controller capable of driving the closed-loop system into a target set of desired states while avoiding hazardous regions. The computation of a controlled reach-avoid set can be transformed into a search for a guidance-barrier function in existing literature. However, existing guidance-barrier functions tend to produce conservative CRASs due to certain flaws. To overcome this issue, a new guidance-barrier function is introduced, which generates less conservative CRASs compared to existing ones. Given the inherent nonlinearity associated with searching for guidance-barrier functions in the control setting, an iterative computational approach is proposed. This approach modifies controllers from previous iterations to find these functions and expand CRASs by solving convex optimizations. Finally, numerical results demonstrate the efficiency of our method in generating less conservative CRASs.