Robust control of a nonlinear electrical oscillator modeled by duffing equation

This paper studies the optimal control problem for a nonlinear electrical circuit exemplified by a Duffing equation. Two cases are considered: first, if the cost function to be optimized is a quadratic one and secondly, if the same quadratic cost function is optimized for a circuit that contains parameter uncertainties in the right-hand side of this nonlinear differential equation. Even though this type of circuit has been object of a variety of control strategies in the past, very few papers have been devoted to the design of optimal control laws with quadratic performance indexes and subject to uncertainties. For that propose, two main techniques are developed in this paper: the so-called state dependent Riccati equation (SDRE) and the Robust Maximum Principle (RMP). Simulation examples are presented to demonstrate that the optimal control strategy works well for a circuit without uncertainties, and that an optimal control of the mini-max type can also be implemented if there are uncertainties in the circuit.