Stochastic optimal semi-active control of nonlinear systems by using MR dampers

A stochastic optimal semi-active control strategy for strongly nonlinear oscillator subjected to external and/or parametric excitations of Gaussian white noises using magneto-rheological (MR) damper is proposed. The dynamic behavior of an MR damper is characterized by using the Bouc-Wen hysteretic model. The control force produced by the MR damper is split into a passive part and a semi-active part. The passive part is incorporated with the uncontrolled system to form a passive control system. Then the system is converted into an equivalent nonlinear non-hysteretic stochastic control system, from which an W stochastic differential equation for total energy is derived by using the stochastic averaging method of energy envelope. For the ergodic control problem, a dynamical programming equation for the controlled total energy process is established based on the stochastic dynamical programming principle. The optimal control law is obtained by minimizing the dynamical programming equation and can be implemented by the MR damper without clipping. Then the fully averaged Ito equation for the controlled total energy process is obtained by replacing the control force with the optimal control force and completing the averaging. Finally, the response of semi-actively controlled system is obtained from solving the final dynamical programming equation and the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged Ito equation. The efficacy of the proposed stochastic optimal semi-active control strategy is illustrated by using the numerical results for two examples.