Lyapunov-based controller design for bounded dynamic stochastic distribution control

Following recent development in the modelling and control of the output probability density function of dynamic stochastic systems, a new approach for the controller design is presented using square root approximation, where a set of B-spline functions are used to approximate the square root of the measured output probability density function to guarantee its positiveness. A performance function is defined which measures the tracking error of the output probability density function with respect to a given distribution. Instead of finding an optimal control which minimises this performance function and then analysing the stability of the closed-loop system, the new approach directly uses the performance function as a Lyapunov function to design the required controller. As a result, the controller obtained not only guarantees the decreasing of the performance function with respect to time, but also stabilises the closed-loop system, realising an asymptotically tracking performance of the output probability density function with respect to its target distribution. The algorithm described has been tested on a simulated example and desired results have been achieved.