<p dir="ltr">In the context of a Human-in-the-Loop (HITL) system, the accuracy of reachability analysis plays a significant role in ensuring the safety and reliability of HITL systems. In addition, one can avoid unnecessary conservativeness by explicitly considering human control behavior compared to those methods that rely on the system dynamics alone. One possible approach is to use a Gaussian Mixture Model (GMM) to encode human control behavior using the Expectation-Maximization (EM) algorithm. However, relatively few works consider the admissible control input ranges due to physical limitations when modeling human control behavior. This could make the following reachability analysis overestimate the system's capability, thereby affecting the performance of the HITL system. To address this issue, this work presents a constrained stochastic reachability analysis algorithm that can explicitly account for the admissible control input ranges. By confining the ellipsoidal confidence region of each Gaussian component using Sequential Quadratic Programming (SQP), we probabilistically constrain the GMM as well as the corresponding stochastic reachable sets. A comprehensive mathematical analysis of how the constrained GMM can affect the stochastic reachable sets is provided in this work. Finally, the proposed stochastic reachability analysis algorithm is validated via an illustrative numerical example.</p>

<div>In this thesis, we explore different methods to enhance the safety and robustness for autonomous systems. We achieve this goal using concepts and tools from reachability analysis and control barrier functions. We first take on a multi-player reach-avoid game that involves two teams of players with competing objectives, namely the attackers and the defenders. We analyze the problem and solve the game from the attackers' perspectives via a moving horizon approach. The resulting solution provides a safety guarantee that allows attackers to reach their goals while avoiding all defenders. </div><div><br></div><div>Next, we approach the problem of target re-association after long-term occlusion using concepts from reachability as well as Bayesian inference. Here, we set out to find the probability identity matrix that associates the identities of targets before and after an occlusion. The solution of this problem can be used in conjunction with existing state-of-the-art trackers to enhance their robustness.</div><div><br></div><div>Finally, we turn our attention to a different method for providing safety guarantees, namely control barrier functions. Since the existence of a control barrier function implies the safety of a control system, we propose a framework to learn such function from a given user-specified safety requirement. The learned CBF can be applied on top of an existing nominal controller to provide safety guarantees for systems.</div>