Continuous-time control synthesis under nested signal temporal logic specifications

TL;DR

This paper introduces a new control synthesis method for nonlinear continuous-time systems under nested signal temporal logic specifications, using signal temporal logic trees and control barrier functions to ensure system correctness.

Contribution

It develops a novel theoretical framework combining sTLT and CBFs to handle nested STL specifications, which was previously unaddressed in control synthesis.

Findings

Successfully applied to single-integrator models

Effective for unicycle models with nested STL

Ensures system correctness through explicit control barrier functions

Abstract

In this work, we propose a novel approach for the continuous-time control synthesis of nonlinear systems under nested signal temporal logic (STL) specifications. While the majority of existing literature focuses on control synthesis for STL specifications without nested temporal operators, addressing nested temporal operators poses a notably more challenging scenario and requires new theoretical advancements. Our approach hinges on the concepts of signal temporal logic tree (sTLT) and control barrier function (CBF). Specifically, 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.