基于MGC法的非线性随机系统PDF控制

PDF control of nonlinear stochastic systems based on MGC method

对于非线性随机系统,以均值、方差等低阶统计特征作为研究目标往往难以满足实际的控制要求,需要考虑更高阶的统计特征。概率密度函数(Probability density function, PDF)包含了完全统计特征,因此PDF控制能够实现各阶矩的有效控制。针对受高斯白噪声激励的非线性随机系统,将福克-普朗克-柯尔莫哥洛夫(Fokker-PlanckKolmogrov, FPK)方程作为研究工具,提出一种基于多高斯闭合法(MGC)的PDF控制方法。首先,根据目标PDF的形状构造一个由多个高斯型PDF相叠加的PDF;然后,构造一个优化问题,使得该PDF逼近目标PDF;进一步,通过求解FPK方程得到被控系统的状态方程;最后,结合原始状态方程求得控制函数,实现对目标PDF的追踪控制。针对不同类型目标PDF进行的仿真结果表明了所提出方法的可行性和有效性。

For nonlinear stochastic systems, it is difficult to meet the actual control requirements by taking the low-order statistical characteristics such as mean and variance as the research objects, and the higher order statistical characteristics need to be considered. The probability density function(PDF) contains complete statistical characteristics, therefore,PDF control can achieve effective control of all moments. In this paper, aiming at the nonlinear stochastic system excited by Gaussian white noise, the Fokker-Planck-Kolmogrov(FPK) equation is taken as the research tool, and a PDF control method based on the multi-Gaussian closure(MGC) method is proposed. Firstly, according to the shape of the target PDF, a PDF superimposed by multiple Gaussian PDFs is constructed. Then, an optimization problem is built to make the PDF approximate the target PDF. Furthermore, the state equation of the controlled system is obtained by solving the FPK equation. Finally, the control function is calculated according to the original state equation, and the tracking control of the target PDF is implemented. Simulation results of different target PDFs show the feasibility and the effectiveness of the proposed method.