Structure, Feasibility, and Explicit Safety Filters for Linear Systems
This work addresses a specific challenge in control systems for robotics or autonomous vehicles, but it is incremental as it builds on existing CBF and HOCBF methods.
The paper tackles the feasibility certification problem for safety filters based on control barrier functions in linear time-invariant systems with affine constraints, deriving explicit closed-form filters that avoid online optimization.
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.