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一类内部点级联的PDE-ODE系统的边界控制
引用本文:郭春丽,周中成. 一类内部点级联的PDE-ODE系统的边界控制[J]. 控制理论与应用, 2014, 31(6): 779-785
作者姓名:郭春丽  周中成
作者单位:四川文理学院 数学与财经学院,西南大学 数学与统计学院
基金项目:国家自然科学基金资助项目(11301427); 中央高校基本业务费资助项目(XDJK2014B021); 四川省教育厅资助项目(13ZB0101).
摘    要:本文运用backstepping方法研究了一类偏微分方程与常微分方程(PDE-ODE)级联系统的能稳性.常见的级联系统在边界点x=0处级联,而本文所讨论的级联系统在内部点x0∈(0,1)处级联,级联点的改变使得新系统的控制问题更加复杂.针对新系统,首先,我们改进了backstepping方法中的常见变换,改进后的变换与常见变换相比,增加了变换中的核函数,且得到的是带有多个相容性条件的核方程组,给求解带来了困难.文中运用了一系列的技巧解出核函数,从而得到反馈控制器;其次,运用同样的方法找到改进变换的逆变换;最后,选择合适范数,利用变换的有界性证明得到闭环系统的稳定性.

关 键 词:偏微分方程与常微分方程(PDE-ODE)级联系统  边界控制  稳定性  backstepping方法
收稿时间:2013-10-23
修稿时间:2014-02-21

Boundary control for a partial differential equation-ordinary differential equation system cascaded at internal point
GUO Chun-li and ZHOU Zhong-cheng. Boundary control for a partial differential equation-ordinary differential equation system cascaded at internal point[J]. Control Theory & Applications, 2014, 31(6): 779-785
Authors:GUO Chun-li and ZHOU Zhong-cheng
Affiliation:School of Mathematics and Finance, Sichuan University of Arts and Science,School of Mathematics and Statistics, Southwest University
Abstract:Stabilization of a partial differential equation-ordinary differential equation (PDE-ODE) system cascaded at internal point is considered by boundary control. This system is more complicated than the conventional ones because the interconnection point is the internal point x_0in (0, 1). For this new system, a new backstepping transformation is introduced, which contains four kernel functions. Because the number of kernels is increased, the kernel equations with the compatibility conditions are more complicated. Fortunately, the kernel equations can be solved by a series of mathematical tricks to obtain their solutions. Then, the feedback controller is developed by using these kernel functions. The inverse transformation is derived by using the same procedure. Finally, we choose a proper norm and establish the stability of the closed-loop system by the boundedness of the transformation and the inverse transformation.
Keywords:PDE-ODE cascaded system   boundary control   stability   backstepping method
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