Natural gradient algorithm for stochastic distribution systems with output feedback

In this paper, we use natural gradient algorithm to control the shape of the conditional output probability density function for the stochastic distribution systems from the viewpoint of information geometry. The considered system here is of multi-input and single output with an output feedback and a stochastic noise. Based on the assumption that the probability density function of the stochastic noise is known, we obtain the conditional output probability density function whose shape is only determined by the control input vector under the condition that the output feedback is known at any sample time. The set of all the conditional output probability density functions forms a statistical manifold (M), and the control input vector and the output feedback are considered as the coordinate system. The Kullback divergence acts as the distance between the conditional output probability density function and the target probability density function. Thus, an iterative formula for the control input vector is proposed in the sense of information geometry. Meanwhile, we consider the convergence of the presented algorithm. At last, an illustrative example is utilized to demonstrate the effectiveness of the algorithm.