Structure, Feasibility, and Explicit Safety Filters for Linear Systems

TL;DR

This paper develops explicit safety filters for linear systems with safety constraints, ensuring feasibility and avoiding online optimization by leveraging geometric insights and structured classes of constraints.

Contribution

It introduces explicit safety filters for LTI systems that guarantee feasibility and simplify implementation, extending control barrier function methods.

Findings

Explicit safety filters are derived for certain classes of linear systems.

Feasibility conditions are characterized based on the geometry of constraint normals.

Numerical examples demonstrate the effectiveness of the explicit filters.

Abstract

Safety filters based on control barrier functions (CBFs) and high-order control barrier functions (HOCBFs) are often implemented through quadratic programs (QPs). In general, especially in the presence of multiple constraints, feasibility is difficult to certify before solving the QP and may be lost as the state evolves. This paper addresses this issue for linear time-invariant (LTI) systems with affine safety constraints. Exploiting the resulting geometry of the constraint normals, and considering both unbounded and bounded inputs, we characterize feasibility for several structured classes of constraints. For certain such cases, we also derive closed-form safety filters. These explicit filters avoid online optimization and provide a simple alternative to QP-based implementations. Numerical examples illustrate the results.