Robust Power Control of Optical Networks with Time-delays

We study the stability of power control algorithms applied to optical networks in the presence of both time-delays and uncertainties. The objective of power control algorithms acting on optical networks is to ensure each signal channel attains an optimal optical signal-to-noise ratio (OSNR) value such that transmission errors are minimized. The inputs to the optical network are the transmitter powers and the outputs of the optical network are the OSNR values. The primal control algorithms adjust the channel powers at the transmitters using the channel OSNR values as feedbacks to attain OSNR optimality. We also present the dual control algorithm located at the links which transmits a channel price as an additional feedback to the primal control algorithms. Together, these are called primal-dual control algorithms. We present robust OSNR models for optical networks with multiple time-delays. Specifically, we consider additive system uncertainties, input multiplicative uncertainties on the signal powers, and transmitter noise uncertainties, all within a norm-bounded uncertainty framework. We analyze and modify both central cost based algorithms and game-theoretic based algorithms, with an emphasis on the latter, to ensure the stability of the closed-loop system. We apply time-delay stability analyses to exploit the structures of the closed-loop systems for each type of control algorithm. These techniques include frequency analyses, Lyapunov-Razumikhin techniques, and Lyapunov-Krasovskii techniques. Due to nonlinearities in the closed-loop system models, and their time-scale separated dynamics, we apply singular perturbation theory modified to handle either Lyapunov-Razumikhin theory or Lyapunov-Krasovskii theory. Singular perturbation theory, modified for time-delays, allows us to decouple complicated closed-loop systems into two simpler subsystems, one on a "slow" time-scale, and the other on a "fast" time-scale. We develop stability conditions for primal algorithms applied to arbitrary networks with delays. We also develop stability conditions for primal-dual algorithms applied to single-links, single-sink networks, two channel networks, and multi-link networks with both time-delays and uncertainties. The main results are presented as either LMI conditions and algebraic criteria. Simulations verify the stability of the closed-loop systems in the presence of time-delays. In addition, the simulations show the stabilization of perturbed systems at the expense of transient convergence time.Ph