Continuous-time control synthesis under nested signal temporal logic specifications

Signal temporal logic (STL) has gained popularity in robotics for expressing complex specifications that may involve timing requirements or deadlines. While the control synthesis for STL specifications without nested temporal operators has been studied in the literature, the case of nested temporal operators is substantially more challenging and requires new theoretical advancements. In this work, we propose an efficient continuous-time control synthesis framework for nonlinear systems under nested STL specifications. The framework is based on the notions of signal temporal logic tree (sTLT) and control barrier function (CBF). In particular, we detail the construction of an sTLT from a given STL formula and a continuous-time dynamical system, the sTLT semantics (i.e., satisfaction condition), and the equivalence or under-approximation relation between sTLT and STL. Leveraging the fact that the satisfaction condition of an sTLT is essentially keeping the state within certain sets during certain time intervals, it provides explicit guidelines for the CBF design. The resulting controller is obtained through the utilization of an online CBF-based program coupled with an event-triggered scheme for online updating the activation time interval of each CBF, with which the correctness of the system behavior can be established by construction. We demonstrate the efficacy of the proposed method for single-integrator and unicycle models under nested STL formulas.