一类 2$\times$2 双曲偏微分系统的 PDP 边界控制

Abstract

Abstract:

This paper studies the exponential stability of a single open-channel system with constant slope and bottom friction, which is described by a $2\times2$ hyperbolic partial differential equation. The position feedback and delayed position feedback (PDP for short) boundary controller is designed to solve the problem of feedback stabilization. Firstly, the well-posedness of the system is proved by using operator semigroup theory. Then, the exponential stability of the closed-loop system is analyzed by constructing an appropriate Lyapunov function, and sufficient conditions for feedback parameters and time-delay are obtained. In addition, the asymptotic expressions of the eigenvalues and the eigenfunctions of the system operator are given by spectral analysis method. Finally, a numerical example is used to evaluate the performance of the PDP controller.

Key words: Hyperbolic PDE system, Lyapunov function, PDP boundary controller, Exponential stability

CLC Number:

O231.4

Pang Yuting,Zhao Dongxia,Zhao Xin,Gao Caixia. The PDP Boundary Control for a Class of 2$\times$2 Hyperbolic Partial Differential System[J].Acta mathematica scientia,Series A, 2023, 43(5): 1471-1482.

Trendmd

References

[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
[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
[3]	Bastin G, Coron J M, Tamasoiu S O. Stability of linear density-flow hyperbolic systems under PI boundary control. Automatica, 2015, 53: 37-42
[4]	Zhang L G, Prieur C, Qiao J F. PI boundary control of linear hyperbolic balance laws with stabilization of ARZ traffic flow models. Systems & Control Letters, 2019, 123: 85-91
[5]	Bastin G, Coron J M. Stability and Boundary Stabilization of 1-D Hyperbolic Systems. Switzerland: Birkh?user, 2016
[6]	Vazquez R, Krstic M, Coron J M. Backstepping boundary stabilization and state estimation of a $2\times2$ linear hyperbolic system. Orlando: IEEE Conference on Decision and Control and European Control Conference, 2011
[7]	Litrico X, Fromion V. Boundary control of hyperbolic conservation laws using a frequency domain approach. Automatica, 2009, 45(3): 647-656
[8]	Bastin G, Coron J M, D'Andrea-Novel B, et al. Boundary control for exact cancellation of boundary disturbances in hyperbolic systems of conservation laws. Seville: IEEE Conference on Decision and Control, and the European Control Conference, 2005
[9]	Bastin G, Coron J M. Exponential stability of PI control for Saint-Venant equations with a friction term. Methods and Applications of Analysis, 2019, 26(2): 101-112
[10]	Coron J M, D'Andrea-Novel B, Bastin G. A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws. IEEE Transactions on Automatic Control, 2007, 52(1): 2-11
[11]	Pazy A. Semigroup of Linear Operators and Applications to Partial Differential Equations. New York: Springer-Verlag, 1983
[12]	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
[13]	Coron J M, Tamasoiu S O. Feedback stabilization for a scalar conservation law with PID boundary control. Chinese Annals of Mathematics, 2015, 36B(5): 763-776
[14]	Chentouf B, Wang J M. Boundary feedback stabilization and Riesz basis property of a 1-d first order hyperbolic linear system with $L^\infty$-coefficients. Journal of Differential Equations, 2009, 246(3): 1119-1138
[15]	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
[16]	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
[17]	Astr?m K J, H?gglund T. PID Controllers: Theory, Design, and Tunning. Research Triangle Park, NC: ISA-The Instrumentation, Systems and Automation Society, 1995
[18]	Litrico X, Fromion V, Scorletti G. Robust feedforward boundary control of hyperbolic conservation laws. Networks and Heterogeneous Media, 2017, 2(4): 717-731
[19]	Bastin G, Coron J M, Hayat A. Feedforward boundary control of $2\times2$ nonlinear hyperbolic systems with application to Saint-Venant equations. European Journal of Control, 2021, 57: 41-53
[20]	Tallman G, Smith O. Analog study of dead-beat posicast control. Ire Transactions on Automatic Control, 2003, 4(1): 14-21
[21]	Silva G J, Datta A, Bhattacharyya S P. PID Controllers for Time-Delay Systems. Boston: Birkh?user, 2005
[22]	Cai G, Chen L. Some problems of delayed feedback control. Advances in Mechanics, 2013, 43(1): 21-28
[23]	Suh I H, Bien Z. Proportional minus delay controller. IEEE Transactions on Automatic Control, 1979, 24(2): 370-372
[24]	Suh I H, Bien Z. Use of time-delay actions in the controller design. IEEE Transactions on Automatic Control, 1980, 25: 600-603
[25]	Atay F M. Balancing the inverted pendulum using position feedback. Applied Mathematics Letters, 1999, 12(5): 51-56
[26]	Hu H Y. Using delayed state feedback to stabilize periodic motions of an oscillator. Journal of Sound and Vibration, 2004, 275(3-5): 1009-1025
[27]	Liu B, Hu H. Stabilization of linear undamped systems via position and delayed position feedbacks. Journal of Sound and Vibration, 2008, 312(3): 509-525
[28]	Krstic M. Control of an unstable reaction-diffusion PDE with long input delay. Systems & Control Letters, 2009, 58(10/11): 773-782
[29]	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
[29]	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
[30]	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
[31]	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

The PDP Boundary Control for a Class of 2$\times$2 Hyperbolic Partial Differential System

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Wu Wenbin, Ren Xue, Zhang Ran. Traveling Waves for a Discrete Diffusive Vaccination Model with Delay [J]. Acta mathematica scientia,Series A, 2025, 45(3): 858-874.
[2]	Zhang Chunguo, Sun Baonan, Fu Yuzhi, Yu Xin. Stabilization of 2-D Mindlin Timoshenko Plate Systems with Local Damping [J]. Acta mathematica scientia,Series A, 2024, 44(4): 946-959.
[3]	Gao Caixia, Zhao Dongxia. The Delayed Control and Input-to-State Stability of ARZ Traffic Flow Model with Disturbances [J]. Acta mathematica scientia,Series A, 2024, 44(4): 960-977.
[4]	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.
[5]	He Xuyang,Mao Mingzhi,Zhang Tengfei. Existence and Stability of a Class of Impulsive Neutral Stochastic Functional Differential Equations with Poisson Jump [J]. Acta mathematica scientia,Series A, 2023, 43(4): 1221-1243.
[6]	Zou Yonghui,Xu Xin. Existence of Back-Flow Point for the Two-Dimensional Compressible Prandtl Equation [J]. Acta mathematica scientia,Series A, 2023, 43(3): 691-701.
[7]	Yu Yang. Global Attractivity of a Nonlocal Delayed and Diffusive SVIR Model [J]. Acta mathematica scientia,Series A, 2021, 41(6): 1864-1870.
[8]	Dongxia Fan,Dongxia Zhao,Na Shi,Tingting Wang. The PDP Feedback Control and Stability Analysis of a Diffusive Wave Equation [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1088-1096.
[9]	Lixiang Feng,Defen Wang. Global Stability of an Epidemic Model with Quarantine and Incomplete Treatment [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1235-1248.
[10]	Xingshou Huang,Ricai Luo,Wusheng Wang. Stability Analysis for a Class Neural Network with Proportional Delay Based on the Gronwall Integral Inequality [J]. Acta mathematica scientia,Series A, 2020, 40(3): 824-832.
[11]	Zhongwei Cao,Xiangdan Wen,Wei Feng,Li Zu. Dynamics of a Nonautonomous SIRI Epidemic Model with Random Perturbations [J]. Acta mathematica scientia,Series A, 2020, 40(1): 221-233.
[12]	Xinzhe Zhang,Guofeng He,Gang Huang. Dynamical Properties of a Delayed Epidemic Model with Vaccination and Saturation Incidence [J]. Acta mathematica scientia,Series A, 2019, 39(5): 1247-1259.
[13]	Chao Yang,Runjie Li. Existence and Stability of Periodic Solution for a Lasota-Wazewska Model with Discontinuous Harvesting [J]. Acta mathematica scientia,Series A, 2019, 39(4): 785-796.
[14]	Xiaojie Jing, Aimin Zhao, Guirong Liu. Global Stability of a Measles Epidemic Model with Partial Immunity and Environmental Transmission [J]. Acta mathematica scientia,Series A, 2019, 39(4): 909-917.
[15]	Wei Fengying, Lin Qingteng. Extinction and Distribution for an SIQS Epidemic Model with Quarantined-Adjusted Incidence [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1148-1161.