星形明渠网络系统的 PDP 反馈控制和指数镇定

摘要/Abstract

摘要：

该文研究基于 Saint-Venant 方程组的具有底部坡度和底部摩擦的星形明渠网络, 该系统包含 $n$ 个子系统: $n-1$ 个入口通道和 $1$ 个出口通道. 设在线输出量测和输入控制均位于边界上, 结合闸门前后流量关系等约束, 建立了具有瞬时位置和时滞位置线性组合的反馈控制律. 考虑到时滞项可由一阶双曲偏微分方程刻画, 并结合黎曼坐标变换和线性化手段将闭环系统改写为 PDE-PDE 无穷维耦合系统的形式. 通过构造加权 Lyapunov 函数讨论了系统在 $L^2$ 范数下的指数稳定性, 给出了控制参数和时滞值的充分条件, 进而证明了在线输出量测被调控到指定的参考信号. 最后, 利用 Matlab 数学软件进行数值仿真, 佐证参数条件的合理性和结论的有效性.

关键词: Saint-Venant 方程组, PDP 反馈控制器, Lyapunov 方法, 指数稳定性

Abstract:

Based on the Saint-Venant equations, the star-shaped open channel network with bottom slope and bottom friction is studied in this paper. The system consists of $n$ subsystems: $n-1$ inlet channels and one outlet channel. It is assumed that both online output measurement and input control are located on the boundary. The feedback control law with a linear combination of position and delayed position is established based on the restriction of flow relation before and after the gate. Considering that the delay term can be characterized by the first-order hyperbolic partial differential equation, the closed-loop system is rewritten into the form of PDE-PDE infinite-dimensional coupling system by means of Riemannian coordinate transformation and linearization. By constructing the weighted Lyapunov function, the exponential stability of the system under the $L^2$-norm is discussed, and the sufficient conditions for the control parameters and time-delay value are given. Furthermore, it is proved that the online output measurement is regulated to the specified reference signal. Finally, numerical simulation is carried out by using Matlab to prove the rationality of the parameters conditions and the validity of the conclusion.

Key words: Saint-Venant equations, PDP feedback controller, Lyapunov method, Exponential stability

中图分类号:

O231.4

庞玉婷, 赵东霞. 星形明渠网络系统的 PDP 反馈控制和指数镇定[J]. 数学物理学报, 2023, 43(6): 1803-1813.

Pang Yuting, Zhao Dongxia. The PDP Feedback Control and Exponential Stabilization of a Star-Shaped Open Channels Network System[J]. Acta mathematica scientia,Series A, 2023, 43(6): 1803-1813.

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参考文献 19

[1]	Litrico X, Fromion V, Baume J P, et al. Experimental validation of a methodology to control irrigation canals based on Saint-Venant equations. Control Engineering Practice, 2005, 13(11): 1425-1437 doi: 10.1016/j.conengprac.2004.12.010
[2]	Santos V D, Prieur C. Boundary control of open channels with numerical and experimental validations. IEEE Transactions on Control Systems Technology, 2008, 16(6): 1252-1264 doi: 10.1109/TCST.2008.919418
[3]	Bastin G, Coron J M, d'Andrea-Novel B. On Lyapunov stability of linearised Saint-Venant equations for a sloping channel. Networks and Heterogeneous Media, 2009, 4(2): 177-187 doi: 10.3934/nhm.2009.4.177
[4]	Halleux J D, Prieur C, Coron J M, et al. Boundary feedback control in networks of open channels. Automatica, 2003, 39(8): 1365-1376 doi: 10.1016/S0005-1098(03)00109-2
[5]	Halleux J D, Bastin G, d'Andréa-Novel B, et al. A Lyapunov approach for the control of multi reach channels modelled by Saint-Venant equations. IFAC Nonlinear Control Systems, 2001, 34(6): 1429-1434
[6]	Trinh N T, Andrieu V, Xu C Z. Boundary PI controllers for a star-shaped network of $2\times2$ systems governed by hyperbolic partial differential equations. International Federation of Automatic Control, 2017, 50(1): 7070-7075
[7]	Trinh N T, Andrieu V, Xu C Z. Output regulation for a cascaded network of $2\times2$ hyperbolic systems with PI controller. Automatica, 2018, 91: 270-278 doi: 10.1016/j.automatica.2018.01.010
[8]	Zhao D X, Fan D X, Guo Y P. The spectral analysis and exponential stability of a 1-d $2\times2$ hyperbolic system with proportional feedback control. International Journal of Control, Automation and Systems, 2022, 20(8): 2633-2640
[9]	Hayat A, Shang P P. A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope. Automatica, 2019, 100: 52-60 doi: 10.1016/j.automatica.2018.10.035
[10]	Bastin G, Coron J M. Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Switzerland: Birkhäuser, 2016
[11]	Tallman G, Smith O. Analog study of dead-beat posicast control. Ire Transactions on Automatic Control, 2003, 4(1): 14-21 doi: 10.1109/TAC.1958.1104844
[12]	蔡国平, 陈龙祥. 时滞反馈控制的若干问题. 力学进展, 2013, 43(1): 21-28
	Cai G P, Chen L X. Some problems of delayed feedback control. Advances in Mechanics, 2013, 43(1): 21-28
[13]	Liu B, Hu H Y. Stabilization of linear undamped systems via position and delayed position feedbacks. Journal of Sound and Vibration, 2008, 312(3): 509-525 doi: 10.1016/j.jsv.2007.11.001
[14]	Wang J M, Lv X W, Zhao D X. Exponential stability and spectral analysis of the pendulum system under position and delayed position feedbacks. International Journal of Control, 2011, 84(5): 904-915 doi: 10.1080/00207179.2011.582886
[15]	Zhao D X, Wang J M. Exponential stability and spectral analysis of the inverted pendulum system under two delayed position feedbacks. Journal of Dynamical and Control Systems, 2012, 18(2): 269-295 doi: 10.1007/s10883-012-9143-6
[16]	Atay F M. Balancing the inverted pendulum using position feedback. Applied Mathematics Letters, 1999, 12(5): 51-56
[17]	Chentouf B, Smaoui N. Time-Delayed feedback control of a hydraulic model governed by a diffusive wave system. Complexity, 2020: Article ID 4986026
[18]	范东霞, 赵东霞, 史娜, 等. 一类扩散波方程的PDP反馈控制和稳定性分析. 数学物理学报, 2021, 41A(4): 1088-1096
	Fan D X, Zhao D X, Shi N, et al. The PDP feedback control and stability analysis of a diffusive wave equation. Acta Mathematica Scientia, 2021, 41A(4): 1088-1096
[19]	庞玉婷, 赵东霞, 赵鑫, 等. 一类 $2\times2$ 双曲偏微分系统的 PDP 边界控制. 数学物理学报, 2023, 43A(5): 1471-1482
	Pang Y T, Zhao D X, Zhao X, et al. The PDP boundary control for a class of $2\times2$ hyperbolic partial differential system. Acta Mathematica Scientia, 2023, 43A(5): 1471-1482