Design of optimal coupling gains for synchronization of nonlinear oscillators

This paper develops a structured optimal-control framework to design coupling gains for synchronization of weakly nonlinear oscillator circuits connected in resistive networks with arbitrary topologies. The oscillators are modeled as weakly nonlinear Liénard-type circuits, and the coupling gain amounts to the current gain which scales the output current of the oscillator. The structured optimal-control problem allows us to seek a decentralized control strategy (equivalently, a diagonal feedback matrix) that precludes communications between oscillators. To this end, a sparsity-promoting optimal control algorithm is developed to tune the optimal diagonal feedback-gain matrix with minimal performance sacrifice. This involves solving an ℋ₂ optimal control problem with ℓ₁ regularization by applying the alternating direction method of multipliers (ADMM). Simulation studies with application to voltage regulation in islanded networks composed of power-electronic inverters are provided to validate the approach.