Online Feedback Optimization and Singular Perturbation via Contraction Theory

Tue, 03 Mar 2026 00:00:00 +0000

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 subsystem 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.

Coupled catastrophes in systems with bidirectional feedback

Mon, 03 Nov 2025 00:00:00 +0000

Catastrophes are present across many disciplines, ranging from the extinction of populations in ecosystems to the collapse of prices in financial systems. Here, we focus on how interactions between systems influence these catastrophes. Specifically, we study two bidirectionally coupled sub-systems—each of which possesses an S-shaped bifurcation curve with saddle-node bifurcations—and explore how interactions between them can lead to simultaneous bifurcations. There are four types of such coupled catastrophes – synchronization, anti-synchronization, consensus, and anti-consensus. Which of these behaviors manifests depends both the intrinsic dynamics of the subsystems and ways in which they are coupled. In general, there are three possible coupling classes – cooperation, competition, and predation. We develop an analytic/graphical methodology to determine and visualize the locus of the coupled catastrophes in parameter space, and we show which classes of couplings support different types of coupled catastrophes. Finally, we discuss several potential applications areas from distinct domains.

Linear Stability Analysis | Lily Cothren

Online Feedback Optimization and Singular Perturbation via Contraction Theory

Coupled catastrophes in systems with bidirectional feedback