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An Error-Based Safety Buffer for Safe Adaptive Control (Extended Version)

Peter A. Fisher, Johannes Autenrieb, Anuradha M. Annaswamy

TL;DR

The paper addresses safe, real-time adaptive control for linear time-invariant plants with matched parametric uncertainties and state constraints. It introduces two schemes, an Error-Based Safety Buffer (EBSB) and an Error-Based Safety Filter (EBSF), to integrate adaptive control with Control Barrier Functions, guaranteeing stability and safety without requiring persistent excitation, and handling CBFs with relative degree $r \ge 1$. A data-driven enhancement via Set Membership Identification (SMID) tightens uncertainty bounds online, further reducing conservatism when excitation is present. Simulations demonstrate that EBSB and EBSF can closely track reference trajectories while safely navigating obstacles, with EBSF offering lower conservatism at the cost of tuning complexity and potential jitter, both mitigated by SMID.

Abstract

We consider the problem of adaptive control of a class of feedback linearizable plants with matched parametric uncertainties whose states are accessible, subject to state constraints, which often arise due to safety considerations. In this paper, we combine adaptation and control barrier functions into a real-time control architecture that guarantees stability, ensures control performance, and remains safe even with the parametric uncertainties. Two problems are considered, differing in the nature of the parametric uncertainties. In both cases, the control barrier function is assumed to have an arbitrary relative degree. In addition to guaranteeing stability, it is proved that both the control objective and safety objective are met with near-zero conservatism. No excitation conditions are imposed on the command signal. Simulation results demonstrate the non-conservatism of all of the theoretical developments.

An Error-Based Safety Buffer for Safe Adaptive Control (Extended Version)

TL;DR

The paper addresses safe, real-time adaptive control for linear time-invariant plants with matched parametric uncertainties and state constraints. It introduces two schemes, an Error-Based Safety Buffer (EBSB) and an Error-Based Safety Filter (EBSF), to integrate adaptive control with Control Barrier Functions, guaranteeing stability and safety without requiring persistent excitation, and handling CBFs with relative degree . A data-driven enhancement via Set Membership Identification (SMID) tightens uncertainty bounds online, further reducing conservatism when excitation is present. Simulations demonstrate that EBSB and EBSF can closely track reference trajectories while safely navigating obstacles, with EBSF offering lower conservatism at the cost of tuning complexity and potential jitter, both mitigated by SMID.

Abstract

We consider the problem of adaptive control of a class of feedback linearizable plants with matched parametric uncertainties whose states are accessible, subject to state constraints, which often arise due to safety considerations. In this paper, we combine adaptation and control barrier functions into a real-time control architecture that guarantees stability, ensures control performance, and remains safe even with the parametric uncertainties. Two problems are considered, differing in the nature of the parametric uncertainties. In both cases, the control barrier function is assumed to have an arbitrary relative degree. In addition to guaranteeing stability, it is proved that both the control objective and safety objective are met with near-zero conservatism. No excitation conditions are imposed on the command signal. Simulation results demonstrate the non-conservatism of all of the theoretical developments.

Paper Structure

This paper contains 36 sections, 11 theorems, 95 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Statement of the Problem
  3. Definitions and Notation
  4. Tangent Cones
  5. Statement of the Problem
  6. Safe Adaptive Control for Uncertain Unforced Dynamics
  7. Adaptive Control Design
  8. Error-Based Safety Buffer
  9. Intuition Behind $\Delta_{ebsb}$
  10. Safe Adaptive Control with an Uncertain Input Matrix
  11. Adaptive Control Design
  12. Error-Based Safety Filter
  13. Intuition Behind EBSF
  14. Data-Driven Performance Enhancement
  15. Set Membership Identification (SMID)
  16. ...and 21 more sections

Key Result

Lemma 1

Any closed, convex set $\mathcal{C} \subset \mathbb{R}^n$ with nonzero $n$-dimensional volume can be expressed as $\mathcal{C} = \{\mathbf{v} \in \mathbb{R}^n : \min\{g_1(\mathbf{v}), g_2(\mathbf{v}), \dots\} \geq 0\}$, where each $g_k : \mathbb{R}^n \to \mathbb{R}$ is continuously differentiable wi

Figures (8)

  • Figure 1: Trajectories resulting from aCBF taylor2020aCBFs, RaCBF lopez2021RaCBFs, EBSB, and EBSF for Problem \ref{['prob:unforced_dynamics']} compared to the ideal controller with all parameters known.
  • Figure 2: Reference inputs $\mathbf{r}_s$ from aCBF taylor2020aCBFs, RaCBF lopez2021RaCBFs, EBSB, and EBSF for Problem \ref{['prob:unforced_dynamics']}.
  • Figure 3: Trajectories resulting from aCBF taylor2020aCBFs, RaCBF lopez2021RaCBFs, and EBSF for Problem \ref{['prob:input_matrix']} compared to the ideal controller with all parameters known.
  • Figure 4: Reference inputs $\mathbf{r}_s$ from aCBF taylor2020aCBFs, RaCBF lopez2021RaCBFs, and EBSF for Problem \ref{['prob:input_matrix']}.
  • Figure 5: Trajectories resulting from aCBF taylor2020aCBFs, RaCBF lopez2021RaCBFs, EBSB, and EBSF with SMID for Problem \ref{['prob:unforced_dynamics']} compared to the ideal controller with all parameters known.
  • ...and 3 more figures

Theorems & Definitions (38)

  • Definition 1: $\mathcal{K}_{\infty,e}$ ames2019CBF
  • Definition 2: Control Barrier Function (CBF) ames2019CBF
  • Definition 3: Forward-Invariance ames2019CBF
  • Definition 4: Safety ames2019CBF
  • Definition 5: Relative Degree xiao2019HOCBF
  • Definition 6: Tracking
  • Lemma 1
  • proof
  • Definition 7: Tangent Cone
  • Remark 1
  • ...and 28 more