Set-Invariance Methods for Time-Varying Constraints, Spatio-Temporal Logic and Coordination

In this thesis, we address control problems arising in constrained dynamical systems and in systems subject to high-level control tasks. Our primary focus is on the efficient control synthesis for systems with time-varying constraints. We further extend the insights developed in this setting to systems subject to spatio-temporal logic constraints, a class of high-level control specifications, as well as to coordination problems in distributed systems. In the first part of the thesis, we develop a systematic framework for synthesizing time-varying Control Barrier Functions (CBFs). While CBFs are a well-established tool for ensuring forward invariance, their design becomes challenging for time-varying constraints as their variations are, upon controller design, commonly not or only qualitatively known, e.g., in terms of their maximum rate of change. We address this challenge by decoupling the CBF synthesis into the design of a time-invariant value function and a time-varying transformation that captures constraint variations by uniformly shifting the CBF. This approach relies on a particular type of CBF, which we term shiftable CBF. It encodes the system’s dynamic capabilities with respect to a constraint, including its ability to react to constraint changes. As a result, the time-varying component can be adapted online without recomputing the CBF.We further introduce a predictive synthesis method for computing shiftable CBFs that allows the required time-varying capabilities to be specified explicitly as a design parameter. Moreover, to mitigate the computational cost of the CBF synthesis, we exploit equivariances in the system dynamics, showing how these can be leveraged to a simplified CBF computation and the construction of new CBFs from existing ones. Finally, we extend the decoupled design to more general time-varying transformations beyond uniform shifts using equivariance properties. Throughout this part, we employ nonsmooth CBFs defined in the Dini sense, which are less conservative than formulations based on generalized gradients and are well-suited for handling time-varying and logic-based constraints. In the second part, we build upon the notion of nonsmooth CBFs in the Dini sense and leverage them to address high-level control tasks expressed through spatio-temporal logic specifications. We explicitly allow for disjunctions (logic or) in the specifications, which cannot be accommodated by smooth CBFs without resorting to high-order controller design methods. Moreover, we propose a systematic approach to evaluating nonsmooth CBFs in the Dini sense, avoiding the need for numerical approximations of directional derivatives.In the third part of the thesis, we shift our focus to a distributed setting and investigate state-coupled CBFs for coordination within a case study on vehicle coordination. In the last part of the thesis, we depart from CBFs and present a parallelized distributed model predictive control (DMPC) scheme for systems subject to coupled state constraints. To guarantee constraint satisfaction even during the parallelized evaluation of the local optimal control problems, we introduce consistency constraints ensuring that the state trajectories remain within a neighborhood of a reference trajectory, both of which are known to neighboring subsystems. Thereby, the behavior of every subsystem remains predictable to its neighbors. Unlike existing approaches, we allow the reference trajectories to be updated at every time step. This design yields significantly lower computation times than sequential DMPC, while outperforming DMPC approaches based on fixed reference trajectories.