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Safety Filter for Robust Disturbance Rejection via Online Optimization

Joyce Lai, Peter Seiler

TL;DR

The paper tackles instability risks in disturbance rejection schemes that rely on online convex optimization when plant models are uncertain. It introduces a safety filter implemented as an adaptive FIR disturbance-rejection block whose gains are constrained by a scaled small-gain condition to guarantee robust finite-gain stability, while minimizing deviations from the unconstrained OCO command through an $\infty$-norm objective. The main contributions are (i) defining a safe gain set $\mathcal{F}_\beta$, (ii) proving an explicit online solution for the safety filter and a saturating corollary that avoids explicit coefficient computation, and (iii) demonstrating via an RLS-based AFDR example that robustness is preserved under model uncertainty with improved disturbance rejection. This approach enables online, provably safe disturbance rejection in high-precision control applications where OCO is used for learning disturbance characteristics, with practical saturation-based implementation.

Abstract

Disturbance rejection in high-precision control applications can be significantly improved upon via online convex optimization (OCO). This includes classical techniques such as recursive least squares (RLS) and more recent, regret-based formulations. However, these methods can cause instabilities in the presence of model uncertainty. This paper introduces a safety filter for systems with OCO in the form of adaptive finite impulse response (FIR) filtering to ensure robust disturbance rejection. The safety filter enforces a robust stability constraint on the FIR coefficients while minimally altering the OCO command in the $\infty$-norm cost. Additionally, we show that the induced $\ell_\infty$-norm allows for easy online implementation of the safety filter by directly limiting the OCO command. The constraint can be tuned to trade off robustness and performance. We provide a simple example to demonstrate the safety filter.

Safety Filter for Robust Disturbance Rejection via Online Optimization

TL;DR

The paper tackles instability risks in disturbance rejection schemes that rely on online convex optimization when plant models are uncertain. It introduces a safety filter implemented as an adaptive FIR disturbance-rejection block whose gains are constrained by a scaled small-gain condition to guarantee robust finite-gain stability, while minimizing deviations from the unconstrained OCO command through an -norm objective. The main contributions are (i) defining a safe gain set , (ii) proving an explicit online solution for the safety filter and a saturating corollary that avoids explicit coefficient computation, and (iii) demonstrating via an RLS-based AFDR example that robustness is preserved under model uncertainty with improved disturbance rejection. This approach enables online, provably safe disturbance rejection in high-precision control applications where OCO is used for learning disturbance characteristics, with practical saturation-based implementation.

Abstract

Disturbance rejection in high-precision control applications can be significantly improved upon via online convex optimization (OCO). This includes classical techniques such as recursive least squares (RLS) and more recent, regret-based formulations. However, these methods can cause instabilities in the presence of model uncertainty. This paper introduces a safety filter for systems with OCO in the form of adaptive finite impulse response (FIR) filtering to ensure robust disturbance rejection. The safety filter enforces a robust stability constraint on the FIR coefficients while minimally altering the OCO command in the -norm cost. Additionally, we show that the induced -norm allows for easy online implementation of the safety filter by directly limiting the OCO command. The constraint can be tuned to trade off robustness and performance. We provide a simple example to demonstrate the safety filter.

Paper Structure

This paper contains 10 sections, 4 theorems, 41 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Adaptive FIR Disturbance Rejection
  4. Example: Effect of Model Uncertainty
  5. Preliminaries
  6. Problem Formulation
  7. Background
  8. Main Results
  9. Application to RLS
  10. Conclusions

Key Result

Theorem 1

Let $T_{d\to y}(M,\Delta,F_\mathrm{LTV})$ denote the feedback interconnection in Figure fig:lft-uncertain with $F_\mathrm{SF}=1$. Assume $M:\ell_{pe}\to\ell_{pe}$, $\Delta:\ell_{pe}^{n_u}\to\ell_{pe}^{n_q}$, and $F_\mathrm{LTV}:\ell_{pe}^{n_w} \to\ell_{pe}^{n_r}$ are finite-gain stable systems of ap where $M_{11}$ and $M_{22}$ have dimensions $(n_u+n_w)\times (n_q+n_r)$ and $n_y\times n_d$, respec

Figures (4)

  • Figure 1: Feedback system with a baseline controller combined with an RLS-based adaptive FIR disturbance rejection controller.
  • Figure 2: RLS-based AFDR rejects the disturbance at the output for the nominal plant (top), but goes unstable for the uncertain plant (bottom).
  • Figure 3: Uncertain system $T_{d\to y}(M,\Delta,F)$ with disturbance, adaptive FIR filtering, and safety filtering.
  • Figure 4: RLS-based AFDR with safety filter improves the disturbance rejection at the output for both the nominal plant (top) and 100 uncertain plants (bottom).

Theorems & Definitions (7)

  • Definition 1: Nominal Finite-Gain Stability
  • Definition 2: Robust Finite-Gain Stability
  • Theorem 1: Scaled Small Gain
  • Lemma 1: Adaptive FIR Bounding Property
  • Theorem 2: Safety Filter Solution
  • proof
  • Corollary 1