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Structure, Feasibility, and Explicit Safety Filters for Linear Systems

Shima Sadat Mousavi, Max H. Cohen, Pol Mestres, Aaron D. Ames

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.

Structure, Feasibility, and Explicit Safety Filters for Linear Systems

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.

Paper Structure

This paper contains 13 sections, 10 theorems, 67 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Feasibility Domain Characterization
  4. General Feasibility Condition
  5. Structural Characterization
  6. Closed-Form Safety Filters
  7. Parallel Constraints
  8. Independent Interval Blocks
  9. Numerical Results
  10. Double-Integrator Example
  11. 2D Double Integrator with Independent Parallel Blocks
  12. Aircraft Roll–Yaw Control
  13. Conclusion

Key Result

Proposition 1

For the LTI system eq:dyn with affine safety constraints eq:affine_h, define, for each $i=1,\dots,p$, and let $\phi_i=\phi_{i,r_i}$. Then the HOCBF constraint associated with $h_i$ can be written as where In particular, $\ell_i\in\mathbb R^m$ is constant and $\beta_i(x)$ is affine in $x$. $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure E1: Double-integrator example. Top row: $\mathcal{C}$, $\mathcal{S}$, and the feasibility domains $\mathcal{X}_T^{\mathrm u}$ and $\mathcal{X}_T^{\mathrm b}$ with representative trajectories. Bottom row: nominal (dashed) and filtered (solid) inputs.
  • Figure E2: Closed-loop vector field over the feasibility domains. Left: $\mathcal{X}_T^{\mathrm u}$. Right: $\mathcal{X}_T^{\mathrm b}$.
  • Figure E3: Explicit filter versus QP solution. Left: representative bounded-input trajectory. Right: maximum input error over the simulated trajectories.
  • Figure E4: Two-dimensional double-integrator example. Top: phase-plane slices showing $\mathcal{C}_i$, $\mathcal{S}_i$, and the feasibility domains for the unbounded and bounded-input cases. Bottom-left: nominal and filtered inputs. Bottom-right: position with the safe box and waypoints.
  • Figure E5: Closed-loop constrained outputs for the aircraft example. The safety filter enforces the prescribed bounds during tracking.

Theorems & Definitions (23)

  • Proposition 1
  • proof
  • Example 1
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Proposition 3
  • proof
  • ...and 13 more