Time-Varying Optimization of LTI Systems Via Projected Primal-Dual Gradient Flows

• Published in: IEEE transactions on control of network systems Vol. 9; no. 1; pp. 474 - 486
• Main Authors: Bianchin, Gianluca, Cortes, Jorge, Poveda, Jorge I., Dall'Anese, Emiliano
• Format: Journal Article
• Language: English
• Published: Piscataway IEEE 01-03-2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Constraints; Convexity; Cost analysis; Cost function; Costs; Cyber-physical systems; Dynamic stability; Dynamical systems; Feedback control; Gradient flow; Linear systems; networked control systems; optimal control; Optimization; Output feedback; Power system dynamics; Stability analysis; Time-varying systems; Trajectory; transportation networks
• ISSN: 2325-5870; 2372-2533
• DOI: 10.1109/TCNS.2021.3112762

This article investigates the problem of regulating, at every time, a linear dynamical system to the solution trajectory of a time-varying constrained convex optimization problem. The proposed feedback controller is based on an adaptation of the saddle-flow dynamics, modified to take into account projections on constraint sets and output feedback from the plant. We derive sufficient conditions on the tunable parameters of the controller (inherently related to the time-scale separation between plant and controller dynamics) to guarantee exponential input-to-state stability of the closed-loop system. The analysis is tailored to the case of time-varying strongly convex cost functions and polytopic output constraints. The theoretical results are further validated in a ramp metering control problem in a network of traffic highways.