Online Optimization as a Feedback Controller: Stability and Tracking

Marcello Colombino, Andrey Bernstein, Emiliano Dall'Anese

Research output: Contribution to journalArticlepeer-review

76 Scopus Citations


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. This 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 rebalancing and tracking of the DC optimal power flow points.

Original languageAmerican English
Article number8673636
Pages (from-to)422-432
Number of pages11
JournalIEEE Transactions on Control of Network Systems
Issue number1
StatePublished - Mar 2020

Bibliographical note

Publisher Copyright:
© 2014 IEEE.

NREL Publication Number

  • NREL/JA-5D00-72225


  • control systems
  • heuristic algorithms
  • linear systems
  • optimization
  • power system stability
  • steady-state
  • time-varying systems


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