Toward Safe Control under Uncertainty: Adaptation and Prediction for Control Barrier Functions

Abstract

Control design for nonlinear dynamical systems is an essential field of study in a world growing ever more reliant on autonomous system technologies. In practical applications, it is often desirable for the state of the system to converge to some target point or region; however, it is critical that the system remain safe, i.e., that the system trajectories satisfy a collection of spatiotemporal constraints over the operational lifetime. While generally difficult for nonlinear systems, guaranteeing safety via control design is further complicated under uncertainty introduced by confounding phenomena such as inaccurate system information, exogenous disturbances, measurement noise, or the presence of sovereign, unpredictable agents. Offline tools for policy learning and system verification may account for some of these effects, but their utility is diminished by the tendency of real-world environments to change. It is of paramount importance, therefore, to investigate viable options for online safe control. Toward this objective, much attention in recent years has been paid to control barrier functions as a tool for state-feedback control that certifies the satisfaction of spatiotemporal constraints at all times. This dissertation studies the theory and viability of CBF-based safe control synthesis for nonlinear systems under various classes of system uncertainty.

First, the problem of encoding future state prediction into CBF-based control for nonlinear, control-affine, multi-agent systems is studied. A novel class of future-focused CBFs is developed for autonomous vehicle control under the assumption that vehicles seek to minimize unnecessary acceleration or braking. Centralized and decentralized control laws are proposed for multi-agent systems with varying degrees of communicability, and it is shown how satisfaction of these CBF conditions guarantees collision avoidance.

Then, the problem of safe control design is studied for a dynamical system subject to an additive, parameter-affine perturbation to the system dynamics. Parameter adaptation laws are proposed to learn the unknown parameters within fixed time, i.e., within a finite time independent of the initial parameter estimates, when a system identifiability condition is met, and to learn the true perturbation when it is not. It is shown that the proposed adaptation framework may be used to learn a more generic class of additive, unmodelled dynamics within fixed-time via application to Koopman operator theory, by which a nonlinear system admits an analogous, infinite-dimensional, linear representation. A robust, adaptive CBF controller is then proposed to guarantee spatiotemporal constraint adherence under parameter adaptation despite the considered model uncertainty.

Next, control design for probabilistic safety over a finite time interval is studied for a class of nonlinear, control-affine, stochastic systems, i.e., systems subject to additive Brownian motion noise. A novel form of risk-aware CBFs is developed, the use of which for control design results in the satisfaction of a user-specified upper bound on the probability that the system becomes unsafe within the considered (finite) time interval. Conditions are derived under which the proposed risk-aware CBF controller reduces conservatism introduced by an existing method.

Finally, the problem of online, certifiably safe control for nonlinear, control-affine systems is addressed under a collection of arbitrarily many spatiotemporal constraints and input constraints. An approach to synthesizing one consolidated CBF candidate from the collection of constraints is proposed, and online parameter adaptation laws are introduced to vary the relative weightings of the individual constraint functions such that, under certain assumptions, the consolidated CBF is rendered valid despite limited control authority.