In this paper,a stochastic minimax optimal control strategy for uncertain quasi-Hamilionian systems is proposed based on stochastic averaging method,stochastic maximum principle and stochastic differential game theory.Firstly,the partially averaged It(o) stochastic differential equations are derived using the stochastic averaging method for quasi-Hamiltonian systems,while the system state transits from rapid variable of velocity and displacement into the slow variable of energy.Secondly,the stochastic optimal control of Hamiltonian system with a given performance index is converted into a minimax control problem based on the stochastic differential game theory.Thirdly,forward-backward stochastic differential equations of the system and the adjoint process were established according to stochasticmaximum principle.The worst disturbances are generated by minimizing the Hamiltonian function,while maximizing the minimal Hamiltonian function results in the worst-case optimal controls.The worst disturbances and the worst-case optimal controls are then substituting into the partially averaged It(o) equation in order to obtain the fully averaged It(o) equation.The responses of controlled system are predicted by solving the Fokker-Planck-Kolmogorov (FPK) equation associated with the fully averaged It(o) equation.Meanwhile,the control effectiveness can also be computed.Finally,the proposed stochastic optimal control of uncertain quasi-Hamiltonian system is applied into a two-DOF nonlinear system.The effectiveness of the minimax control strategy is validated by numerical results.%本文提出了不确定拟哈密顿系统、基于随机平均法、随机极大值原理和随机微分对策理论的一种随机极大极小最优控制策略.首先,运用拟哈密顿系统的随机平均法,将系统状态从速度和位移的快变量形式转化为能量的慢变量形式,得到部分平均的It8随机微分方程;其次,给定控制性能指标,对于不确定拟哈密顿系统的随机最优控制,根据随机微分对策理论,将其转化为一个极小极大控制问题;再根据随机极大值原理,建立关于系统与伴随过程的前向-后向随机微分方程,随机最优控制表达为哈密顿控制函数的极大极小条件,由此得到最坏情形下的扰动参数与极大极小最优控制;然后,将最坏扰动参数与最优控制代入部分平均的It8随机微分方程并完成平均,求解与完全平均的It(o)随机微分方程相应的Fokker-Planck-Kolmogorov (FPK)方程,可得受控系统的响应量并计算控制效果;最后,将上述不确定拟哈密顿系统的随机最优控制策略应用于一个两自由度非线性系统,通过数值结果说明该随机极大极小控制策略的控制效果.
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