Stochastic Optimal Control For MDOF Nonlinear Systems Under Combined Harmonic And Wide-Band Noise Excitations

The stochastic optimal control strategy for multi-degrees-of-freedom (MDOF) strongly nonlinear systems under combined harmonic and wide-band noise excitations is investigated in this paper, including unbounded optimal control, bounded optimal control and minimax optimal semi-active control of uncertain systems using Magneto-rheological (MR) dampers.For the case of optimal unbounded control, first,a stochastic averaging procedure is introduced for controlled strongly nonlinear systems under combined harmonic and wide-band noise excitations using generalized harmonic functions. Then, the dynamical programming equation is derived from the partially averaged Ito equations and solved to yield the optimal control law.For the case of optimalbounded control, the optimal bounded control obtainedconsistsof optimal unbounded control and bang-bangcontrol.When the desired control is greater than the actuator limit, the control is the form of Bang-Bang control, which can be obtained without solving the dynamic programming equation. When the desired control is less than the actuator limits,the control is the form ofoptimal unbounded control.For the case of minimax optimal semi-active control, the control forces produced by Magneto-rheological (MR) dampers are split into the passive part and the semi-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.For the above cases, the responses of optimally controlled MDOF nonlinear systems are predicted by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged Ito equations.The results show that the proposed control strategy has higher control effectiveness and control efficiency.