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Robust Online Convex Optimization for Disturbance Rejection

Joyce Lai, Peter Seiler

TL;DR

Robust online convex optimization for disturbance rejection addresses learning-based disturbance rejection in discrete-time LTI plants under model uncertainty. The authors derive a robust stability condition via a scaled small gain theorem and embed it as an online constraint, enabling a constrained OCO (C-OCO) controller that bounds the learning dynamics in the ℓ∞-norm. They formulate an LFT-based analysis, relate the LTV learning dynamics to a time-varying gain M_t, and demonstrate through simulations that enforcing the stability bound β preserves closed-loop stability when the plant is imperfect. This framework enables stable, real-time disturbance learning for high-precision control applications facing unmodeled dynamics, balancing adaptability with provable robustness.

Abstract

Online convex optimization (OCO) is a powerful tool for learning sequential data, making it ideal for high precision control applications where the disturbances are arbitrary and unknown in advance. However, the ability of OCO-based controllers to accurately learn the disturbance while maintaining closed-loop stability relies on having an accurate model of the plant. This paper studies the performance of OCO-based controllers for linear time-invariant (LTI) systems subject to disturbance and model uncertainty. The model uncertainty can cause the closed-loop to become unstable. We provide a sufficient condition for robust stability based on the small gain theorem. This condition is easily incorporated as an on-line constraint in the OCO controller. Finally, we verify via numerical simulations that imposing the robust stability condition on the OCO controller ensures closed-loop stability.

Robust Online Convex Optimization for Disturbance Rejection

TL;DR

Robust online convex optimization for disturbance rejection addresses learning-based disturbance rejection in discrete-time LTI plants under model uncertainty. The authors derive a robust stability condition via a scaled small gain theorem and embed it as an online constraint, enabling a constrained OCO (C-OCO) controller that bounds the learning dynamics in the ℓ∞-norm. They formulate an LFT-based analysis, relate the LTV learning dynamics to a time-varying gain M_t, and demonstrate through simulations that enforcing the stability bound β preserves closed-loop stability when the plant is imperfect. This framework enables stable, real-time disturbance learning for high-precision control applications facing unmodeled dynamics, balancing adaptability with provable robustness.

Abstract

Online convex optimization (OCO) is a powerful tool for learning sequential data, making it ideal for high precision control applications where the disturbances are arbitrary and unknown in advance. However, the ability of OCO-based controllers to accurately learn the disturbance while maintaining closed-loop stability relies on having an accurate model of the plant. This paper studies the performance of OCO-based controllers for linear time-invariant (LTI) systems subject to disturbance and model uncertainty. The model uncertainty can cause the closed-loop to become unstable. We provide a sufficient condition for robust stability based on the small gain theorem. This condition is easily incorporated as an on-line constraint in the OCO controller. Finally, we verify via numerical simulations that imposing the robust stability condition on the OCO controller ensures closed-loop stability.
Paper Structure (16 sections, 3 theorems, 39 equations, 6 figures)

This paper contains 16 sections, 3 theorems, 39 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notation
  4. Model Uncertainty
  5. OCO Control
  6. Model Uncertainty Effects on OCO Control
  7. Main Result
  8. Linear Fractional Transformation
  9. Scaled Small Gain Theorem
  10. Bounding the LTV Dynamics
  11. Application to OCO
  12. Estimator Design
  13. OPGD on an Ideal Cost
  14. Robust OCO Control
  15. Numerical Results
  16. ...and 1 more sections

Key Result

Theorem 1

Consider the interconnection $F_U(P,\Gamma)$ where $P:\ell_{pe} \to \ell_{pe}$ and $\Gamma:\ell_{pe} \to \ell_{pe}$ are linear systems with finite induced $\ell_p$-norm. Partition $P$ as: where $\bar{p}:= \left[ \right]$ and $\bar{q}:= \left[ \right]$ are the inputs and outputs of $\Gamma$. The interconnection has finite induced $\ell_p$-norm, i.e. $\|F_U(P,\Gamma)\| < \infty$, if $\|P_{11} \|\

Figures (6)

  • Figure 1: Discrete-time feedback system with unknown disturbance $d$ and uncertainty $\Delta(z)$. OCO control is used to reject the disturbance $d$ without knowledge of the uncertainty $\Delta(z)$.
  • Figure 2: Block diagram representation of the OCO controller in a discrete-time feedback system with unknown disturbance $d_t$ and uncertain plant $\tilde{G}(z)$. The OCO controller is composed of a state-feedback gain $K$, an estimator $E(z)$, and an LTV system $M_{LTV}$.
  • Figure 3: Equivalent LFT $F_U(P,\Gamma)$ of original system separating LTI dynamics $P$ from uncertainty $\Delta$ and time-varying learning dynamics $M_{LTV}$.
  • Figure 4: Per-step cost (top) and disturbance estimate (bottom) of running U-OCO on a perfect (red dashed) and imperfect (blue solid) plant model. U-OCO is stable with a perfect model and unstable for an imperfect model.
  • Figure 5: Per-step cost (top) and disturbance estimate (bottom) of running C-OCO at $\beta=1.5$ on a perfect (red dashed) and imperfect (blue solid) plant model. C-OCO is stable for the perfect and imperfect models.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof