Distributed Stochastic Extremum Seeking for Optimal Resource Allocation over Networks with Time Delays

This paper investigates optimal resource allocation in multi-agent networks subject to simultaneous measurement and communication delays. We propose a distributed stochastic extremum seeking (SES) algorithm for connected undirected graphs with heterogeneous constant communication delays. The algorithm uses only delayed pointwise measurements of the local objective values; neither gradient nor Hessian information is fed to the controller. It combines a proportional–integral (PI) consensus protocol, which enforces the supply–demand balance, with a one-stage sequential predictor that compensates for the measurement delays. The convergence analysis is developed for local objectives of quadratic type; this restriction, which is required by the stochastic-averaging argument, is stated explicitly and discussed. Using stochastic averaging theory and Lyapunov–Krasovskii functionals for delay systems, we prove that, for every finite initial history, the estimates converge almost surely to an arbitrarily small neighborhood of the optimum for every fixed tolerance, and we derive an explicit admissible range for the constant communication delays together with a sequential procedure for selecting the design parameters. Two case studies, namely a three-generator microgrid and a six-generator microgrid, are used to illustrate the theoretical results.