Online Optimization as a Feedback Controller: Stability and Tracking

TL;DR

This paper introduces feedback-based online optimization algorithms for LTI systems that ensure stability and accurate tracking of time-varying solutions, demonstrated through power system applications.

Contribution

It develops a novel primal-dual feedback control method with LMI-based stability guarantees for tracking solutions of time-varying convex optimization problems in LTI systems.

Findings

Guarantees global exponential stability under certain conditions.

Achieves bounded tracking error with sufficient time-scale separation.

Effectively applied to power transmission systems for real-time control.

Abstract

This paper develops and analyzes feedback-based online optimization methods to regulate the output of a linear time-invariant (LTI) dynamical system to the optimal solution of a time-varying convex optimization problem. The design of the algorithm is based on continuous-time primal-dual dynamics, properly modified to incorporate feedback from the LTI dynamical system, applied to a proximal augmented Lagrangian function. The resultant closed-loop algorithm tracks the solution of the time-varying optimization problem without requiring knowledge of (time-varying) disturbances in the dynamical system. The analysis leverages integral quadratic constraints to provide linear matrix inequality (LMI) conditions that guarantee global exponential stability and bounded tracking error. Analytical results show that, under a sufficient time-scale separation between the dynamics of the LTI dynamical system and the algorithm, the LMI conditions can be always satisfied. The paper further proposes a modified algorithm that can track an approximate solution trajectory of the constrained optimization problem under less restrictive assumptions. As an illustrative example, the proposed algorithms are showcased for power transmission systems, to compress the time scales between secondary and tertiary control, and allow to simultaneously power re-balancing and tracking of DC optimal power flow points.