Constrained PI tracking control for output probability distributions based on two-step neural networks

In this paper, a new method for the control of the shape of the conditional output probability density function (pdf) for general nonlinear dynamic stochastic systems is presented using two-step neural networks (NNs). Following the square-root B-spline NN approximation to the measured output pdf, the problem is transferred into the tracking of dynamic weights. Different from the previous related works, time-delay dynamic NNs with undetermined parameters are employed to identify the nonlinear relationships between the control input and the weighting vectors. In order to achieve the required control objective and satisfy the state constraints due to the property of output pdfs, a constrained PI tracking controller is designed by solving a class of linear matrix inequalities and algebraic equations. With the proposed tracking controller and adaptive projection algorithms, both identification and tracking errors can be made to converge to zero, and the state constraints can also be simultaneously guaranteed. Finally, two simulated examples are given, which effectively demonstrate the use of the proposed control algorithm.