Control of Nonlinear Distributed Parameter Systems Based on Global Approximation

Changqing Du ,Guansheng Xing

doi:10.1155/2014/845370

Abstract

We extend an iterative approximation method to nonlinear, distributed parameter systems given by partial differential and functional equations. The nonlinear system is approached by a sequence of linear time-varying systems, which globally converges in the limit to the original nonlinear systems considered. This allows many linear control techniques to be applied to nonlinear systems. Here we design a sliding mode controller for a nonlinear wave equation to demonstrate the effectiveness of this method.

Highlights

The control of finite-dimensional nonlinear system of the form ẋ (t) = A (x, N (x, θ)) x (t) + Bu (1)has recently been studied via a sequence of linear timevarying (LTV) approximation of the form ẋ[i] (t) = A (x[i−1], N (x[i−1], θ)) x[i] (t) + B (x[i−1] (t)) u, i = 1, 2, 3, 4, . . . , (2)where N(x, θ) is some nonlinear function defined over the interval [t − θ, t].This iterative linear approximation method makes it convenient to control nonlinear systems via linear feedback control technique
[6] Wu and Li design a fuzzy observer-based controller based on T-S model of the ordinary differential equations (ODE), while in [7, 8] Wu and Li design linear model feedback controllers based on neural network approximation of the ODE
[9] Deng et al develop a spectral approximation method to distributed thermal processing, and a hybrid general regression NN is trained to be a nonlinear model of the original PDE, which allows many control methods to be applied to this kind of nonlinear PDEs

Summary

Introduction

In [9] Deng et al develop a spectral approximation method to distributed thermal processing, and a hybrid general regression NN is trained to be a nonlinear model of the original PDE, which allows many control methods to be applied to this kind of nonlinear PDEs. In this paper we will extend the above-mentioned iterative linear approximation theory to the nonlinear wave equation, which is a typical infinite-dimensional nonlinear PDE of the form. The nonlinear wave equation is transformed into a sequence of LTV approximations, each of which has an infinite-dimensional sliding surface which is time-varying.

Sliding Control for Finite-Dimensional Nonlinear Systems

Linear Approximation of Nonlinear Wave Equation

Conclusion

Sliding Control of Nonlinear Wave Equation

Simulation Result