Online Optimization as a Feedback Controller: Stability and Tracking

• Published in: IEEE transactions on control of network systems Vol. 7; no. 1; pp. 422 - 432
• Main Authors: Colombino, Marcello, Dall'Anese, Emiliano, Bernstein, Andrey
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 01.03.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Computational geometry; Constraints; Control stability; Control systems; Control theory; Convexity; Design modifications; Design optimization; Dynamical systems; Error analysis; Feedback control; Heuristic algorithms; Lagrangian function; Linear matrix inequalities; Linear systems; Mathematical analysis; Optimization; Power flow; power system control; Power system stability; POWER TRANSMISSION AND DISTRIBUTION; Stability analysis; Steady-state; Time compression; Time-varying systems; Tracking control; Tracking errors; Trajectory optimization
• ISSN: 2325-5870; 2372-2533; 2325-5870
• DOI: 10.1109/TCNS.2019.2906916

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.