Distributed Asynchronous Discrete-Time Feedback Optimization

In this article, we present an algorithm that drives the outputs of a network of agents to jointly track the solution of a time-varying, strongly convex optimization problem. This algorithm is robust to asynchrony in the agents' operations, namely, first, computations of control inputs, second, linear measurements of network outputs, and third, communications of agents' inputs and outputs. We first show that our distributed asynchronous algorithm converges to the solution of a time-invariant feedback optimization problem in linear time. Next, we show that our algorithm tracks the solution of a time-varying feedback optimization problem within a bounded error dependent upon the movement of the minimizers and degree of asynchrony, which we make precise. These convergence results are extended to quantify agents' asymptotic behavior as the length of their time horizon approaches infinity. Then, to ensure satisfactory network performance we specify the timing of agents' operations relative to changes in the objective function that ensure a desired error bound. Numerical experiments verify these developments and show the utility of feedback optimization under asynchrony.