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

TL;DR

This paper develops a real-time feedback control method for linear systems to track solutions of time-varying convex optimization problems, ensuring stability and robustness through saddle-flow dynamics with projections.

Contribution

It introduces a novel projected primal-dual gradient flow approach for LTI systems, with stability guarantees tailored to time-varying strongly convex costs and polytopic constraints.

Findings

Guarantees exponential stability under certain parameter conditions

Ensures input-to-state stability with output-feedback control

Validated through a traffic ramp metering case study

Abstract

This paper investigates the problem of regulating in real 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 and 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.