Reach-Avoid Controllers Synthesis for Safety Critical Systems

In this article, we propose a framework for synthesizing reach-avoid controllers for deterministic systems modeled by ordinary differential equations and stochastic systems modeled by stochastic differential equations based on the notion of control guidance-barrier functions. We first consider deterministic systems and aim to synthesize a reach-avoid controller that modifies a nominal controller in a minimal way to enforce the reach-avoid objective. This objective ensures that the system enters a target set eventually while staying within an open safe set before the first target hitting time. Three control guidance-barrier functions and corresponding conditions for synthesizing reach-avoid controllers are developed. These conditions progressively weaken, facilitating optimal controller design. We then extend two of these functions for deterministic systems to stochastic systems to synthesize reach-avoid controllers in the probabilistic setting. Finally, we provide several examples to demonstrate the effectiveness of our proposed methods, using the semidefinite programming tool.