Learning in Wireless Control Systems over Non-Stationary Channels

This paper considers a set of multiple independent control systems that are each connected over a non-stationary wireless channel. The goal is to maximize control performance over all the systems through the allocation of transmitting power within a fixed budget. This can be formulated as a constrained optimization problem examined using Lagrangian duality. By taking samples of the unknown wireless channel at every time instance, the resulting problem takes on the form of empirical risk minimization, a well-studied problem in machine learning. Due to the non-stationarity of wireless channels, optimal allocations must be continuously learned and updated as the channel evolves. The quadratic convergence property of Newton's method motivates its use in learning approximately optimal power allocation policies over the sampled dual function as the channel evolves over time. Conditions are established under which Newton's method learns approximate solutions with a single update, and the subsequent sub-optimality of the control problem is further characterized. Numerical simulations illustrate the near-optimal performance of the method and resulting stability on a wireless control problem.