Towards robustness guarantees for feedback-based optimization

Abstract

Feedback-based online optimization algorithms have gained traction in recent years because of their simple implementation, their ability to reject disturbances in real time, and their increased robustness to model mismatch. While the robustness properties have been observed both in simulation and experimental results, the theoretical analysis in the literature is mostly limited to nominal conditions. In this work, we propose a framework to systematically assess the robust stability of feedback-based online optimization algorithms. We leverage tools from monotone operator theory, variational inequalities and classical robust control to obtain tractable numerical tests that guarantee robust convergence properties of online algorithms in feedback with a physical system, even in the presence of disturbances and model uncertainty. The results are illustrated via an academic example and a case study of a power distribution system.

Model-Free Nonlinear Feedback Optimization

“…Hence, the control inputs can be effectively updated also for problems with complex dynamics, e.g., real-time optimal power flow in electrical networks [15], [16] and congestion control in communication networks [17]. Furthermore, the feedback structure contributes to robustness against unmeasured disturbances and model mismatch [18], [19], as well as autonomous tracking of trajectories of optimal solutions of time-varying problems [20]- [25]. Some works [20], [22], [23], [26] consider fast-stable plants that are abstracted as algebraic steady-state maps.…”

“…Recent works demonstrate that the sensitivity can be estimated based on system identification [19], recursive least-squares estimation [32], or data-driven methods that use past input-output data of openloop linear systems [29], [33]. Nevertheless, on the one hand, the estimation can be a highly nontrivial task accompanied with errors, thus causing the approximate optimality of FO [18]. On the other hand, the uncertainties in and the complexity of engineering systems (e.g., volatile renewable energy sources in large-scale power systems [34]) may render the sensitivity of the plant costly to compute or even impossible to formulate, let alone feasible to estimate.…”

Sampled-Data Online Feedback Equilibrium Seeking: Stability and Tracking

“…A projected dynamical systems approach is explored in [13]. The robustness of FO is investigated theoretically [14] and experimentally [15]. All these works assume that the plant is a static map and thus neglect dynamical interactions.…”

“…While we do not explicitly address the issue here, feedback optimization is known to be robust to modelling errors, see e.g.,[14],[15].…”

Learning-Based Repetitive Precision Motion Control with Mismatch Compensation

“…ILC uses the nominal model of an uncertain plant and provides robustness to disturbances and model mismatches by leveraging online measurements from the plant [5]. Recent related work on run-to-run control includes [6], where the authors use a bounded linear approximation of a power system to derive robust convergence to an approximated fixed point. In [7], the authors have adopted a zeroth-order approximation of a generic nonlinear plant response from a previous run to show local convergence (under bounded modeling error).…”