Output PDFs control for linear stochastic systems with arbitrarily bounded random parameters: A new application of the Laplace transform

Following the recent developments on the shape control of the output probability density functions for non-Gaussian and dynamic stochastic systems with fixed parameters ([12]-[16]), this paper presents a controller design for the shape control of output probability density functions for non-Gaussian dynamic stochastic systems represented by an ARMAX model. The coefficients in the ARMAX model are random and are represented by their known probability density functions and the system is also subjected to a random noise. All these random parameters and noise are assumed bounded and non-Gaussian. To formulate a simple relationship among the probability density function of the system output and those of random parameters and the noise term, the Laplace transform is applied to all the probability density functions. As a result, a simple mathematical relationship amongst all the transferred probability density functions of the system output and random parameters has been established. Since controlling the shape of the output probability density function is equivalent to controlling the shape of its Laplace transform, a new performance function is introduced, whose minimization is performed so as to design an optimal control input sequence that makes the shape of the output probability density function follow a target distribution. A gradient search technique is applied in the optimization.