Exceedance probability criterion based stochastic optimal polynomial control of Duffing oscillators

An optimal polynomial control strategy is developed in the context of the physical stochastic optimal control scheme of structures that is well-adapted to randomly-driven non-linear dynamical systems. A class of Duffing oscillators with polynomial active tendons subjected to random ground motions is investigated for illustrative purposes. Numerical studies reveal that using an exceedance probability criterion with the minimum of the failure probability of system quantities in energy trade-off sense, a linear control with the 1st-order controller suffices even for strongly non-linear systems. This bypasses the need to utilize non-linear controls with the higher-order controller which may be associated with dynamical instabilities due to time delay and computational dynamics. The statistical variability, meanwhile, of system responses gains an obvious reduction, and the system performance is significantly improved. The 1st-order controller, however, does not have the same control effect to the higher-order controller when control criteria currently in used are employed, e.g. system second-order statistics evaluation and Lyapunov asymptotic stability condition, as indicated in the comparative studies of the exceedance probability criterion against the two control criteria. Besides, the proposed optimal polynomial control is insensitive to the non-linearity strength of the class of base-excited non-linear oscillators whereby a robust control of systems can be implemented, while the LQG control in conjunction with the statistical linearization technique, using a band-limited white noise input, does not have this advantage.