Multivariable extremum-seeking by periodic switching functions with application to Raman optical amplifiers

A periodic switching function based extremum-seeking sliding-mode control strategy is addressed for uncertain nonlinear systems. We generalize previous results to a more challenging multivariable scenario, where the diagonal dominance condition is relaxed and can now be understood as a triangular dominance condition, which is significantly less restrictive. As an essential advantage over earlier results in the literature, the complete global convergence proof (with respect to the closed-loop initial conditions) of the overall multi input multi-output (MIMO) closed-loop system is provided, rather than local stability results. The practical application of the proposed multivariable controller is exemplified through the optimization problem of the output signal power spectrum of Raman optical amplifiers. From a practical point of view, the extremum-seeking scheme enables the automatic adjustment of the control pump power signals in the presence of plant uncertainties, guaranteeing a flat power spectrum of the downstream data signals. Using a multi-objective approach, we develop the optimization algorithm to achieve this goal by simultaneously minimizing the output ripple and the deviation from some desired power level. The convergence analysis and control design are carried out in the presence of uncertainties, and the algorithm performance is tested through numerical examples, where a particular optical communication system with 32 channels is considered. This numerical example uses model parameters acquired experimentally and a representative fiber model, such that the results are close to what is expected to find in a practical implementation.