Stochastic minimax semi-active control for MDOF nonlinear uncertain systems under combined harmonic and wide-band noise excitations using MR dampers

• The control strategy is very effective and has a quite good performance even for large parametric uncertainty.
• The application and effectiveness are illustrated to be quite practical.
• By using the stochastic averaging method, the dimension of the control system is reduced and the system state control is transformed into the system energy control.
• The combined harmonic and wide-band noise excitations have been investigated deeply.

A stochastic minimax semi-active control strategy for multi-degrees-of-freedom (MDOF) strongly nonlinear systems under combined harmonic and wide-band noise excitations is proposed. First, a stochastic averaging procedure is introduced for controlled uncertain strongly nonlinear systems using generalized harmonic functions and the control forces produced by Magneto-rheological (MR) dampers are split into the passive part and the active part. Then, a worst-case optimal control strategy is derived by solving a stochastic differential game problem. The worst-case disturbances and the optimal semi-active controls are obtained by solving the Hamilton–Jacobi–Isaacs (HJI) equations with the constraints of disturbance bounds and MR damper dynamics. Finally, the responses of optimally controlled MDOF nonlinear systems are predicted by solving the Fokker–Planck–Kolmogorov (FPK) equation associated with the fully averaged Itô equations. Two examples are worked out in detail to illustrate the proposed control strategy. The effectiveness of the proposed control strategy is verified by using the results from Monte Carlo simulation.