Disturbance Rejection Control under Nested Signal Temporal Logic Specifications: A Recursive Design Approach

TL;DR

This paper presents a recursive control barrier function design for uncertain systems to satisfy nested STL specifications, using a modified STL tree and quadratic programming, without prior disturbance knowledge.

Contribution

A novel recursive CBF design based on a modified STL tree enables explicit parameterized CBFs for nested STL specifications in uncertain systems.

Findings

Proposed recursive CBF guarantees STL specification satisfaction.

Reconstructed CBF approach works without prior disturbance knowledge.

Simulation validates the effectiveness of the method.

Abstract

This paper investigates the control synthesis for continuous-time uncertain systems under nested Signal Temporal Logic (STL) specifications containing nested temporal operators. Control Barrier Functions (CBFs) are utilized herein to encode STL formulas into system constraints. However, traditional CBF designs fail to encode nested STL formulas, whereas recent reachability analysis-based methods capable of handling such formulas are inapplicable to uncertain systems and suffer from a severe computational burden. To overcome these challenges, a novel recursive CBF design procedure based on a modified STL tree (sTLT) is proposed to yield explicit parameterized CBFs. Within this framework, sliding window variables are introduced to capture complex temporal relationships. Crucially, satisfying the resulting CBF constraints is proven to guarantee the fulfillment of the STL specifications. To render the proposed recursive CBF design applicable to systems subject to uncertain disturbance, a novel controller based on reconstructed CBF using quadratic programming (QP) is proposed, ensuring strict CBF constraint satisfaction under disturbances. In contrast to existing methods, the proposed reconstructed CBF approach requires no prior knowledge of the disturbances while relaxing initial safety assumptions. Simulation results validate the efficacy of the proposed approach.