Neural network based probability density function shape control for unknown stochastic systems

The work presents a numerical solution to the output probability density function (pdf) control for general unknown non-Gaussian stochastic systems. The system is represented by a nonlinear ARMAX model that is subjected to an arbitrary input noise with a known probability density function. At first, a neural network model is proposed to approximate the unknown nonlinear dynamics, where the weight training of the neural network is performed via minimizing the entropy and the mean values of the modelling error. For the trained system model, a secondary recursive pdf model, that relates the conditional output probability density function with the system past input and output, is established via the use of the known pdf of the random noise term. A performance function has therefore been defined upon this secondary model. By minimizing this performance function, a recursive control input formula is derived that aims at making the shape of the conditional output pdf to follow a target shape. A case study has been included in the paper on the closed loop control of a combustion flames distribution system and encouraging simulated results have been initially obtained.