A Novel PDF Shape Control Approach for Nonlinear Stochastic Systems

In this work, a novel shape control approach of the probability density function (PDF) for nonlinear stochastic systems is presented. First, we provide the formula for the PDF shape controller without devising the control law of the controller. Then, based on the exact analytical solution of the Fokker-Planck-Kolmogorov (FPK) equation, the product function of the polynomial and the exponential polynomial is regarded as the stationary PDF of the state response. To validate the performance of the proposed control approach, we compared it with the exponential polynomial method and the multi-Gaussian closure method by implementing comparative simulation experiments. The results show that the novel PDF shape control approach is effective and feasible. Using an equal number of parameters, our method can achieve a similar or better control effect as the exponential polynomial method. By comparison with the multi-Gaussian closure method, our method has clear advantages in PDF shape control performance. For all cases, the integral of squared error and the errors of first four moments of our proposed method were very small, indicating superior performance and promising good overall control effects of our method. The approach presented in this study provides an alternative for PDF shape control in nonlinear stochastic systems.