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Data-Driven Input-Output Control Barrier Functions

Mohammad Bajelani, Klaske van Heusden

TL;DR

The paper addresses safe control for constrained, unknown systems by synthesizing an input-output discrete-time CBF (IO-DCBF) directly from input-output data. It builds a data-driven IO representation using Hankel matrices, derives an extended-state formulation, and computes a maximal input-output safe invariant set to define an IO-DCBF $h(\xi) = \min(c - H \xi)$ that enforces safety through a decay condition. A minimally invasive safety filter is formulated as a quadratic program, with a tunable decay rate $\lambda$ that balances feasibility and smoothness, and the IO-DCBF is shown to be equivalent to a one-step horizon MPSF under certain design choices. The approach is demonstrated on a time-delay LTI system, illustrating recursive feasibility and the interaction between IO constraints and safety, and it highlights the potential for extensions to noisy data and nonlinear dynamics. This work enables data-efficient, model-free safety guarantees for IO-constrained control, integrating seamlessly with predictive safety concepts while reducing reliance on full-state estimation.

Abstract

Control Barrier Functions (CBFs) offer a framework for ensuring set invariance and designing constrained control laws. However, crafting a valid CBF relies on system-specific assumptions and the availability of an accurate system model, underscoring the need for systematic data-driven synthesis methods. This paper introduces a data-driven approach to synthesizing a CBF for discrete-time LTI systems using only input-output measurements. The method begins by computing the maximal control invariant set using an input-output data-driven representation, eliminating the need for precise knowledge of the system's order and explicit state estimation. The proposed CBF is then systematically derived from this set, which can accommodate multiple input-output constraints. Furthermore, the proposed CBF is leveraged to develop a minimally invasive safety filter that ensures recursive feasibility with an adaptive decay rate. To improve clarity, we assume a noise-free dataset, though the method can be extended using data-driven reachability to capture noise effects and handle uncertainty. Finally, the effectiveness of the proposed method is demonstrated on an unknown time-delay system.

Data-Driven Input-Output Control Barrier Functions

TL;DR

The paper addresses safe control for constrained, unknown systems by synthesizing an input-output discrete-time CBF (IO-DCBF) directly from input-output data. It builds a data-driven IO representation using Hankel matrices, derives an extended-state formulation, and computes a maximal input-output safe invariant set to define an IO-DCBF that enforces safety through a decay condition. A minimally invasive safety filter is formulated as a quadratic program, with a tunable decay rate that balances feasibility and smoothness, and the IO-DCBF is shown to be equivalent to a one-step horizon MPSF under certain design choices. The approach is demonstrated on a time-delay LTI system, illustrating recursive feasibility and the interaction between IO constraints and safety, and it highlights the potential for extensions to noisy data and nonlinear dynamics. This work enables data-efficient, model-free safety guarantees for IO-constrained control, integrating seamlessly with predictive safety concepts while reducing reliance on full-state estimation.

Abstract

Control Barrier Functions (CBFs) offer a framework for ensuring set invariance and designing constrained control laws. However, crafting a valid CBF relies on system-specific assumptions and the availability of an accurate system model, underscoring the need for systematic data-driven synthesis methods. This paper introduces a data-driven approach to synthesizing a CBF for discrete-time LTI systems using only input-output measurements. The method begins by computing the maximal control invariant set using an input-output data-driven representation, eliminating the need for precise knowledge of the system's order and explicit state estimation. The proposed CBF is then systematically derived from this set, which can accommodate multiple input-output constraints. Furthermore, the proposed CBF is leveraged to develop a minimally invasive safety filter that ensures recursive feasibility with an adaptive decay rate. To improve clarity, we assume a noise-free dataset, though the method can be extended using data-driven reachability to capture noise effects and handle uncertainty. Finally, the effectiveness of the proposed method is demonstrated on an unknown time-delay system.

Paper Structure

This paper contains 8 sections, 4 theorems, 25 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A Data-Driven Input-output Representation for constrained LTI systems
  3. Data-driven Input-output CBF
  4. Equivalence of IO-DCBF Safety Filter and Predictive Safety Filter
  5. Data-Driven Output Model Predictive Safety Filter
  6. Equivalence of design in IO-DCBFs and MPSFs
  7. Simulation Results
  8. Discussion and Concluding Remarks

Key Result

Lemma II.5

Let Assumptions Assumption: I-Assumption: II hold. Then, $[H_u^{\top} H_y^{\top}]^{\top}$ spans all the input-output trajectories of the system eq: LTI-system with the length of $L$.

Figures (3)

  • Figure 1: The maximal input-output safe set resulting from the set propagation method: (I) projection onto ${[y_{t-1}, y_{t-2}]}^\top$, (II) projection onto ${[y_{t-1}, y_{t-2}, y_{t-3}]}^\top$, and (III) projection onto ${[u_{t-1}, y_{t-1}, y_{t-2}]}^\top$.
  • Figure 2: The proposed control barrier function applied to the projection of the maximal input-output safe set onto $[y_{t-1}, y_{t-2}]^\top$, where $c_p$ and $H_p$ define the projected set.
  • Figure 3: Input and output of system \ref{['equ: sim-ss']}, the value of the proposed barrier function, and the decay rate.

Theorems & Definitions (10)

  • Definition II.1: System's Lag 9654975
  • Definition II.2: Persistently Excitation
  • Lemma II.5: Fundamental Lemma willems2005note
  • Definition II.6: Extended State and Extended Constraints
  • Definition II.7: Input-output safe control invariant set
  • Definition III.1: IO-DCBF
  • Proposition III.2
  • Theorem III.3: Proposed IO-DCBF function
  • Remark III.4
  • Theorem III.5