球对称<i>n</i>维反应扩散方程的边界观测与输出反馈控制

齐洁; 王川; 潘峰

doi:10.16383/j.aas.2015.c140741

球对称n维反应扩散方程的边界观测与输出反馈控制

doi: 10.16383/j.aas.2015.c140741

齐洁^1,2,
王川^1,2,
潘峰^1,2

1.
东华大学信息科学与技术学院上海 201620;
2.
东华大学数字化纺织服装技术教育部工程研究中心上海 201620

基金项目:

国家自然科学基金(61134009, 61104154), 中央高校基本科研业务费专项资金(2232015D3-24, 2232012D3-19)资助

详细信息

作者简介:
王川东华大学信息科学与技术学院硕士研究生. 主要研究方向为偏微分方程边界控制, 多智能体系统协同控制.E-mail: 785407617@qq.com

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出版历程
- 收稿日期: 2014-10-24
- 修回日期: 2015-02-27
- 刊出日期: 2015-07-20

Boundary Observer and Output-feedback Control for a Class of n-dimensional Symmetric Advection-diffusion Equations

QI Jie^1,2,
WANG Chuan^1,2,
PAN Feng^1,2

1.
School of Information Science and Technology, Donghua Univer-sity, Shanghai 201620;
2.
Engineering Research Center of Digitized Textile and Fashion Technology, Ministry of Education, Donghua University, Shanghai 201620

Funds:

Supported by National Natural Science Foundation of China (61134009, 61104154) and Fundamental Research Funds for the Central Universities (2232015D3-24, 2232012D3-19)

摘要

摘要: 许多实际系统可用n 维超球坐标系来描述, 并且系统有球对称的性质, 因而可通过研究半径方向的状态变化, 得到系统的全局动态过程. 通过将高维的对称系统转化为等价的径向一维方程, 本文采用边界Backstepping 方法设计了球对称反应扩散方程的输出反馈控制器. 使用容易测量的边界状态值, 设计了状态观测器来估计系统在空间域的所有状态, 从而实现输出反馈控制. 本文扩展了连续Backstepping 方法,提出了n维球坐标的Volterra 积分映射, 从而求出了显式表达的控制器和状态观测器. 论文用Lyapunov 函数法证明了输出反馈系统在H1范数下指数稳定, 表明状态对初值的连续依赖, 确保控制系统具有较好的性质, 不会在空间某点发散. 最后进行了数值仿真, 仿真结果表明系统在输出反馈控制律的作用下收敛到稳态值.
- 边界观测器 /
- 状态估计 /
- 反步法 /
- 扩散反应方程 /
- 分布式参数系统
Abstract: Many systems in application are modelled in n-sphere coordinates and also have symmetric properties. For symmetric system, we can obtain the dynamics of n-D states by investigating 1-D states along the radial direction. This pa-per uses the boundary backstepping method to design an output feedback control law for a symmetrical n-D reaction-diffusion equation by studying an equivalent 1-D radial system. We de-sign an observer by boundary sensor to estimate the whole states in space which are needed in the feedback control. We extend the continuous backstepping method by proposing an n-D sphere Volterra integration mapping, which makes it possible to get ex-plicit controller and observer. We have proved the exponential stability in the H1 norm by using Lyapunov function, which means that the states are continuous in terms of initial condi-tions and that the system has good properties in that it will not diverge in some special point. Finally, we do some simulations to illustrate that the system can converge to the equilibrium actuated by the output feedback control.
- Boundary observer /
- state estimation /
- backstep-ping /
- advection-diffusion equation /
- distributed parameter system

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参考文献(31)

[1]	Padhi R, Ali S F. An account of chronological developments in control of distributed parameter systems. Annual Reviews in Control, 2009, 33(1): 59-68
[2]	Erturk A, Inman D J. A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. Journal of Vibration and Acoustics, 2008, 130(4): 041002
[3]	Butkovskii A G. The maximum principle for optimum systems with distributed parameters. Automation and Remote Control, 1961, 22(10): 1156-1169
[4]	Triggiani R. Boundary feedback stabilizability of parabolic equations. Applied Mathematics and Optimization, 1980, 6(1): 201-220
[5]	Fard M P, Sagatun S I. Exponential stabilization of a transversely vibrating beam via boundary control. Journal of Sound and Vibration, 2001, 240(4): 613-622
[6]	Li Jian, Liu Yun-Gang. Adaptive boundary control for a class of uncertain heat equations. Acta Automatica Sinica, 2012, 38(3): 469-473 (李健, 刘允刚. 一类不确定热方程自适应边界控制. 自动化学报, 2012, 38(3): 469-473)
[7]	Guo B Z, Kang W. The Lyapunov approach to boundary stabilization of an anti-stable one-dimensional wave equation with boundary disturbance. International Journal of Robust and Nonlinear Control, 2014, 24(1): 54-69
[8]	Rebarber R. Conditions for the equivalence of internal and external stability for distributed parameter systems. IEEE Transactions on Automatic Control, 1993, 38(6): 994-998
[9]	Ge Z Q, Zhu G T, Feng D X. Degenerate semi-group methods for the exponential stability of the first order singular distributed parameter systems. Journal of Systems Science & Complexity, 2008, 21(2): 260-266
[10]	Cheng M B, Radisavljevic V, Su W C. Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties. Automatica, 2011, 47(2): 381-387
[11]	Guo B Z, Liu J J. Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional schrödinger equation subject to boundary control matched disturbance. International Journal of Robust and Nonlinear Control, 2014, 24(16): 2194-2212
[12]	Guo B Z, Jin F F. Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance. IEEE Transactions on Automatic Control, 2015, 60(3): 824-830
[13]	Li Shao-Yuan. Model Predictive Control for Global Operating Condition System and Its Implementation. Beijing: Science Press, 2008. (李少远. 全局工况系统预测控制及其应用. 北京: 科学出版社, 2008.)
[14]	Dubljevic S, El-Farra N H, Mhaskar P, Christofides P D. Predictive control of parabolic PDEs with state and control constraints. International Journal of Robust and Nonlinear Control, 2006, 16(16): 749-772
[15]	Luo Yi-Ping, Xia Wen-Hua, Liu Guo-Rong, Deng Fei-Qi. LMI approach to exponential stabilization of distributed parameter control systems with delay. Acta Automatica Sinica, 2009, 35(3): 299-304 (罗毅平, 夏文华, 刘国荣, 邓飞其. 时滞分布参数控制系统指数镇定的LMI方法. 自动化学报, 2009, 35(3): 299-304)
[16]	Krstic M. Systematization of approaches to adaptive boundary stabilization of PDEs. International Journal of Robust and Nonlinear Control, 2006, 16(16): 801-818
[17]	Vazquez R, Krstic M. Boundary observer for output-feedback stabilization of thermaluid convection loop. IEEE Transactions on Control Systems Technology, 2010, 18(4): 789-797
[18]	Li Xiao-Guang, Liu Jin-Kun. Continuum backstepping control algorithms in partial differential equation orientation: a review. Control Theory & Applications, 2012, 29(7): 825-832 (李晓光, 刘金琨. 面向偏微分方程的连续反演控制算法综述. 控制理论与应用, 2012, 29(7): 825-832)
[19]	Smyshev A, Krstic M. Closed-form boundary state feedbacks for a class of 1-D partial integro--differential equations. IEEE Transactions on Automatic Control, 2004, 49(12): 2185-2202
[20]	Krstic M, Smyshev A. Boundary Control of PDEs: A Course on Backstepping Designs. Philadelphia: SIAM, 2008.
[21]	Qi J, Vazquez R, Krstic M. Multi-agent deployment in 3-D via PDE control. IEEE Transactions on Automatic Control, 2015, 60(4): 891-906
[22]	Vazquez R, Krstic M. Control of 1-D parabolic PDEs with Volterra nonlinearities ---Part I: Design. Automatica, 2008, 44(11): 2778-2790
[23]	Xu C, Schuster E, Vazquez E, Krstic M. Stabilization of linearized 2D magnetohydrodynamic channel flow by backstepping boundary control. Systems & Control Letters, 2008, 57 (10): 805-812
[24]	Vazquez R, Schuster E, Krstic M. A Closed-Form Full-State feedback controller for stabilization of 3D magnetohydrodynamic channel flow. Journal of Dynamic Systems, Measurement, and Control, 2009 131: 041001
[25]	Vazquez R, Krstic M. Boundary observer for output-feedback stabilization of thermal-fluid convection loop. IEEE Transactions on Control Systems Technology, 2010, 18(4): 789-797
[26]	Bai Y, Xie C. Temperature control of reaction-diffusion process in ball. In: Proceedings of the 23rd Chinese Control and Decision Conference (CCDC). Mianyang, China, 2011. 2291-2295
[27]	Vazquez R, Krstic M. Explicit boundary control of a reaction-diffusion equation on a disk. In: Proceedings of the 19th World Congress The International Federation of Automatic Control. Cape Town, South Africa: IFAC, 2014. 1562-1567
[28]	Tang K T. Mathematical Methods for Engineers and Scientists. Berlin: Springer-Verlag, 2007. 142-186
[29]	Qi Jie, Qi Jin-Peng. Boundary stabilization for a 2-D reaction-diffusion equation with symmetrical initial data. Acta Automatica Sinica, 2015, 41(1): 209-214 (齐洁, 齐金鹏. 具有对称初始数据的二维反应扩散方程的边界镇定. 自动化学报, 2015, 41(1): 209-214)
[30]	Brezis H. Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York: Springer, 2011. 326-328
[31]	Lai M C. A note on finite difference discretizations for Poisson equation on a disk. Numerical Methods for Partial Differential Equations, 2001, 17(3): 199-203

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球对称n维反应扩散方程的边界观测与输出反馈控制

doi: 10.16383/j.aas.2015.c140741

作者简介:
王川东华大学信息科学与技术学院硕士研究生. 主要研究方向为偏微分方程边界控制, 多智能体系统协同控制.E-mail: 785407617@qq.com

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Boundary Observer and Output-feedback Control for a Class of n-dimensional Symmetric Advection-diffusion Equations

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球对称n维反应扩散方程的边界观测与输出反馈控制

doi: 10.16383/j.aas.2015.c140741

作者简介: 王川东华大学信息科学与技术学院硕士研究生. 主要研究方向为偏微分方程边界控制, 多智能体系统协同控制.E-mail: 785407617@qq.com

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出版历程

Boundary Observer and Output-feedback Control for a Class of n-dimensional Symmetric Advection-diffusion Equations

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出版历程

目录

作者简介:
王川东华大学信息科学与技术学院硕士研究生. 主要研究方向为偏微分方程边界控制, 多智能体系统协同控制.E-mail: 785407617@qq.com