# Online Feedback Optimization and Singular Perturbation via Contraction Theory

arXiv: https://arxiv.org/abs/2310.07966 · HTML: https://mlsanity.com/paper/2310.07966/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Oct 2023.

## Built on by (most cited first, 2 of 2)

- [O(1/k) Finite-Time Bound for Non-Linear Two-Time-Scale Stochastic Approximation](https://mlsanity.com/paper/2504.19375/) (2025)
- [Similarity Matching Networks: Hebbian Learning and Convergence Over Multiple Time Scales](https://mlsanity.com/paper/2506.06134/) (2025)

## Abstract (arXiv)

In this paper, we provide a novel contraction-theoretic approach to analyze two-time scale systems, including those commonly encountered in Online Feedback Optimization (OFO). Our framework endows these systems with several robustness properties, enabling a more comprehensive characterization of their behaviors. The primary assumptions are the contractivity of the fast sub-system and the reduced model, along with an explicit upper bound on the time-scale parameter. For two-time scale systems subject to disturbances, we show that the distance between solutions of the nominal system and solutions of its reduced model is uniformly upper bounded by a function of contraction rates, Lipschitz constants, the time-scale parameter, and the variability of the disturbances over time. Applying these general results to the OFO context, we establish new individual tracking error bounds, showing that solutions converge to their time-varying optimizer, provided the plant and steady-state feedback controller exhibit contractivity and the controller gain is suitably bounded. Finally, we explore two special cases: for autonomous nonlinear systems, we derive sharper bounds than those in the general results, and for linear time-invariant systems, we present novel bounds based on induced matrix norms and induced matrix log norms.