Lyapunov-based controller design for bounded dynamic stochastic distribution control

Following developments in the modelling and control of the output probability density function of dynamic stochastic systems, an 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