Response minimization of MDOF nonlinear system driven by fGn

Abstract

Stochastic optimal control studies the Markov control of Markov diffusion process. For a system driven by the fractional Gaussian noise (fGn), the system response is non-Markov process. Thus, stochastic optimal control methods cannot be directly applied to stochastic optimal control of multi-degree-of-freedom (MDOF) nonlinear system driven by fGn. In the present paper, it is pointed out that fGn can be approximated to wideband noise under certain condition and the stochastic optimal control method based on the stochastic averaging method (SAM) of quasi-Hamiltonian system driven by wideband noise and dynamical programming can then be applied. The study in the present paper focuses on minimizing the response of MDOF nonlinear systems driven by fGn. Firstly, a controlled and fGn-driven quasi integrable Hamiltonian system is considered. Under certain conditions, the SAM of quasi-Hamiltonian systems driven by wideband noise is applied to obtain the partially-averaged and controlled Itô stochastic differential equations (SDEs). Then, applying the dynamical programming principle to the partially-averaged and controlled Itô SDEs yields the Hamilton-Jacobi-Bellman (HJB) equation. Solving HJB equation yields the optimal control force. Substitute the obtained optimal control force into the partially-averaged and controlled Itô SDEs and complete averaging to obtain completely averaged and optimal controlled Itô SDEs. Finally, solve the associated reduced Fokker-Planck-Kolmogorov (FPK) equation to obtain the response of the optimal controlled system. The proposed method is verified by comparing the theoretical results with those from Monte Carlo simulation.