Hybrid and nonhybrid control barrier functions for constraint satisfaction in dynamical systems: Applied to safe motion control of autonomous vessels

Abstract

Control barrier functions (CBFs) enable constraint satisfaction in controlled dynamical systems, by mapping state constraints into state-dependent input constraints. CBFs may then be synthesized with any nominal control law by solving an optimization problem that finds the safe input that is closest to the nominal control input (by some appropriate measure). First-order CBFs are applicable for systems where the control input appears in the first time derivative of the controlled output. High-order CBFs (HOCBFs) extend the notion of CBFs to systems of any order, following a procedure reminiscent of the recursive design of a control Lyapunov function in backstepping. Augmenting CBFs with logic variables that may change value instantaneously result in hybrid CBF formulations. This thesis studies hybrid and nonhybrid control barrier functions (CBFs) for continuous-time systems, with contributions to both theory and applications. Two distinct robustness results for CBF-based control strategies are established: robustness towards bounded disturbances, and robustness towards arbitrarily small perturbations when employing discontinuous safeguarding control laws. Robustness of discontinuous CBF-based control strategies for control-affine systems is established, in the sense that the CBF-induced stability properties are retained under Krasovskii regularization. This result alleviates the need of establishing continuity properties of optimization-based control laws frequently employed together with CBFs. The result on robustness towards bounded disturbances is stated in the form of sufficient conditions for input-to-state stability (ISS) of nonhybrid HOCBFinduced systems with respect to the safe set, i.e., the subset of the state space where safety constraints are satisfied. This ISS result is established by constructing a vector comparison system from the worst-case evolution of the HOCBF along the disturbed system. Sufficient conditions for uniform global asymptotic stability (UGAS) of the safe set follow as a corollary from the sufficient condition for ISS. Both the ISS and UGAS results apply to closed, but not necessarily compact, safe sets. The distinction between compact and noncompact sets is important in context of HOCBFs, since the safe set defined by HOCBFs is often noncompact. Hybrid CBFs enable solving control problems that are not solvable by continuous control. Such control problems are frequently encountered in obstacle avoidance problems, since decisiveness with respect to turning direction of the vehicle is required either for task completion or for safety. This thesis proposes to construct hybrid CBFs by combining multiple CBF-like functions defining partially overlapping safe sets. Among the main contributions is a recursive design procedure for constructing hybrid HOCBFs, thereby extending hybrid CBFs to systems with high-order safety constraints. The theoretical results are complemented by several novel CBF-based control designs: Two hybrid CBF designs for safe motion control are proposed. The first ensures robust safety for vessels required to maintain a nonzero forward speed, whereas the second robustly resolves deadlocks that arise for CBF-based control strategies applied to obstacle avoidance. A dynamic maneuvering guidance scheme for path-following and obstacle avoidance is also proposed. The guidance scheme reactively generates a safe trajectory for autonomous for autonomous vessels to follow. Modifying the desired path itself, as opposed to forcing vessels to deviate from the path, avoids integral windup in adaptive control schemes that rely on integrating the tracking error. The thesis also motivates the use of CBF-based control strategies for constraint satisfaction in oscillatory mechanical systems, using a wave energy converter as a case study.