Applying constrained nonlinear generalized PI strategy to PDF tracking control through square root B-spline models

For stochastic systems with non-Gaussian variables, the classical control approaches where only expectation and variance are concerned cannot cover the control requirement of the closed loop in some practical processes. In this paper, the tracking control problem for output probability density functions (PDFs) is studied using square root B-spline expansions and non-linear weight dynamical models. After the measurable output PDFs are approximated by the B-spline expansions, a non-linear dynamical model can be established between the control input and the weights related to the PDFs. The tracking control problem for the output PDFs can be reduced to a constrained tracking problem for the non-linear weight dynamics. For this non-linear weight model, a generalized proportional-integral (PI) control strategy is proposed in discrete time context. The objective of the control is to make sure that the output PDFs of the system can follow a given target function, and the closed-loop system is exponentially stable and satisfies the constraint imposed on the state vector. The LMI-based convex optimization approach is adopted to design the parameters of the proposed PI controllers. This result also generalizes some previous works for classical constrained PI tracking control of non-linear discrete-time systems. Simulations are given to demonstrate the efficiency of the proposed approach. Acknowledgments This work is jointly supported by The Leverhulme Trust, UK under Grant No. F00038/D and National Science Foundation of China under Grant No. 60128303 and 60474050. These are gratefully acknowledged.