Optimal control and learning for safety-critical autonomous systems

Optimal control of autonomous systems is a fundamental and challenging problem, especially when many stringent safety constraints and tight control limitations are involved such that solutions are hard to determine. It has been shown that optimizing quadratic costs while stabilizing affine control systems to desired (sets of) states subject to state and control constraints can be reduced to a sequence of Quadratic Programs (QPs) by using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs). Although computationally efficient, this method is limited by several factors which are addressed in this dissertation. The first contribution of this dissertation is to extend CBFs to high order CBFs (HOCBFs) that can accommodate arbitrary relative degree systems and constraints. The satisfaction of Lyapunov-like conditions in the HOCBF method implies the forward invariance of the intersection of a sequence of sets, which can then guarantee the satisfaction of the original safety constraint. Second, under tight control bounds, this dissertation proposes an analytical method to find sufficient conditions that guarantee the QP feasibility. The sufficient conditions are captured by a single state constraint that is enforced by a CBF and then added to the QP. Third, for complex safety constraints and systems in which it is hard to find sufficient conditions for feasibility, machine learning techniques are employed to learn the definitions of HOCBFs or feasibility constraints. Fourth, when time-varying control bounds and noisy dynamics are involved, adaptive CBFs (AdaCBFs) are proposed, which can guarantee the feasibility of the QPs if the original optimization problem itself is feasible. Finally, for systems with unknown dynamics, adaptive affine control dynamics are proposed to approximate the real unmodelled system dynamics which are updated based on the error states obtained by real-time sensor measurements. A set of events required to trigger a solution of the QP in order to guarantee safety is defined, and a condition that guarantees the satisfaction of the HOCBF constraint between events is derived. In order to address the myopic nature of the CBF method, a real-time control framework that combines optimal trajectories and the computationally efficient HOCBF method providing safety guarantees is also proposed. The HOCBFs and CLFs are used to account for constraints with arbitrary relative degrees and to track the optimal state, respectively. Eventually, an optimal control problem based on the proposed framework is always reduced to a sequence of QPs regardless of the formulation of the original cost function. Another contribution of the dissertation is to apply the above proposed methods to solve complex safety-critical optimal control problems, such as those arising in rule-based autonomous driving and optimal traffic merging control for Connected and Automated Vehicles (CAVs)