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Uncertainty Quantification for Recursive Estimation in Adaptive Safety-Critical Control

Max H. Cohen, Makai Mann, Kevin Leahy, Calin Belta

TL;DR

The paper addresses online parameter estimation with uncertainty quantification for adaptive safety-critical control. It shows that the continuous-time RLS estimate is an affine transformation of the initial parameter, enabling efficient propagation of uncertainty via zonotopes and safe controller synthesis using control barrier functions. By unifying set-membership identification with concurrent learning, the authors derive both pointwise and set-valued parameter estimates and provide containment guarantees under FE and disturbance bounds, culminating in robust RaCBF-based safety control implemented via a QP. The framework is demonstrated on nonlinear systems with parametric uncertainty and disturbances, highlighting practical safety benefits and outlining avenues for integration with MPC and broader set representations.

Abstract

In this paper, we present a framework for online parameter estimation and uncertainty quantification in the context of adaptive safety-critical control. The key insight enabling our approach is that the parameter estimate generated by the continuous-time recursive least squares (RLS) algorithm at any point in time is an affine transformation of the initial parameter estimate. This property allows for parameterizing such estimates using objects that are closed under affine transformation, such as zonotopes, and enables the efficient propagation of such set-based estimates as time progresses. We illustrate how such an approach facilitates the synthesis of safety-critical controllers for systems with parametric uncertainty and additive disturbances using control barrier functions, and demonstrate the utility of our approach through illustrative examples.

Uncertainty Quantification for Recursive Estimation in Adaptive Safety-Critical Control

TL;DR

The paper addresses online parameter estimation with uncertainty quantification for adaptive safety-critical control. It shows that the continuous-time RLS estimate is an affine transformation of the initial parameter, enabling efficient propagation of uncertainty via zonotopes and safe controller synthesis using control barrier functions. By unifying set-membership identification with concurrent learning, the authors derive both pointwise and set-valued parameter estimates and provide containment guarantees under FE and disturbance bounds, culminating in robust RaCBF-based safety control implemented via a QP. The framework is demonstrated on nonlinear systems with parametric uncertainty and disturbances, highlighting practical safety benefits and outlining avenues for integration with MPC and broader set representations.

Abstract

In this paper, we present a framework for online parameter estimation and uncertainty quantification in the context of adaptive safety-critical control. The key insight enabling our approach is that the parameter estimate generated by the continuous-time recursive least squares (RLS) algorithm at any point in time is an affine transformation of the initial parameter estimate. This property allows for parameterizing such estimates using objects that are closed under affine transformation, such as zonotopes, and enables the efficient propagation of such set-based estimates as time progresses. We illustrate how such an approach facilitates the synthesis of safety-critical controllers for systems with parametric uncertainty and additive disturbances using control barrier functions, and demonstrate the utility of our approach through illustrative examples.
Paper Structure (6 sections, 7 theorems, 57 equations, 4 figures)

This paper contains 6 sections, 7 theorems, 57 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Formulation
  3. Recursive-Batch Least Squares Regression
  4. Recursive Set-Membership Identification
  5. Robust Adaptive Safety-Critical Control
  6. Discussion and Conclusions

Key Result

Lemma 1

Let $\mathcal{H}$ satisfy the FE condition and suppose that $\skew{3}{\hat{}}{\bm{\theta}}$ and $\bm{\Gamma}$ satisfy eq:theta-hat and eq:Gamma, respectively. Then, for all $t\in\mathbb{R}_{\geq0}$, the parameter estimation error eq:theta-tilde satisfies:

Figures (4)

  • Figure 1: Evolution of the parameter estimation errors (a) and value of $\gamma$ (b) from \ref{['eq:FE']} corresponding to Example \ref{['example:simple-example']}. The solid curves correspond to the estimates using $\mathcal{H}$ and the dashed curves to those that do not exploit $\mathcal{H}$.
  • Figure 2: Zonotopes generated by the RLS algorithm from Example \ref{['ex:zono']}. In each plot, the black dot denotes $\bm{\theta}$ and the change in opacity of the zonotopes represents the evolution of time, where more transparent shapes denote zonotopes at earlier times and less transparent shapes denote zonotopes at later times. In each simulation, the update laws are defined by a history stack with $M=20$ entries, which is filled by periodically storing values of $y(t)$ and $\bm{\phi}(t)$ until the stack is full.
  • Figure 3: Initial and final zonotopes ($T=60$) corresponding to the different cases from Example \ref{['ex:zono']}. Here, Fig. \ref{['fig:zono-final']}(a) corresponds to the zonotopes from Fig. \ref{['fig:zono-example']}(a) and Fig. \ref{['fig:zono-final']}(b) corresponds to the zonotopes from Fig. \ref{['fig:zono-example']}(c).
  • Figure 4: Trace of the pendulum's position (a) and resulting parameter estimation error (b) for the simulations from Example \ref{['ex:cbf']}.

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Example 1
  • Definition 3
  • Lemma 2: SadraCDC19
  • Theorem 1
  • ...and 12 more