带有未知内部扰动的星形Euler-Bernoulli梁网络的指数跟踪控制

数学物理学报 ›› 2019, Vol. 39 ›› Issue (3): 596-610.

带有未知内部扰动的星形Euler-Bernoulli梁网络的指数跟踪控制

张雅轩^1,*(),许跟起²,郭燕妮¹

¹ 中国民航大学理学院数学系天津 300300
² 天津大学数学学院天津 300350

收稿日期:2017-05-27 出版日期:2019-06-26 发布日期:2019-06-27
通讯作者: 张雅轩 E-mail:bunnyxuan@tju.edu.cn
基金资助:
国家自然科学基金(61503385);国家自然科学基金(11705279);中央高校基本科研业务费(3122018L004);天津市教委科研计划项目(2018KJ253)

Exponential Tracking Control for a Star-Shaped Network of Euler-Bernoulli Beams with Unknown Internal Disturbance

Yaxuan Zhang^1,*(),Genqi Xu²,Yanni Guo¹

¹ Department of Mathematics, College of Science, Civil Aviation University of China, Tianjin 300300
² College of Mathematics, Tianjin University, Tianjin 300350

Received:2017-05-27 Online:2019-06-26 Published:2019-06-27
Contact: Yaxuan Zhang E-mail:bunnyxuan@tju.edu.cn
Supported by:
the NSFC(61503385);the NSFC(11705279);the Fundamental Research Funds for the Central Universities(3122018L004);the Scientific Research Project of Tianjin Municipal Education Commission(2018KJ253)

摘要/Abstract

摘要：

该文研究了带有未知内部扰动的星形Euler-Bernoulli梁网络的指数跟踪控制问题.首先将该问题等价转化为跟踪网络与被跟踪网络的误差网络的镇定问题.利用滑模控制思想，对误差网络设计了非线性反馈控制方案.通过对状态空间选取适当的范数，运用单调算子理论得到了误差网络的适定性.通过构造适当的Lyapunov函数，证明误差网络按任一收敛率指数稳定.这表明跟踪网络能够按任一给定速率以指数速度跟踪到目标网络.

关键词: 跟踪控制, 未知内部扰动, 星形Euler-Bernoulli梁网络, 滑模控制, 指数镇定

Abstract:

In this paper, the exponential tracking control for a star-shaped network of Eulerbernoulli beams with unknown internal disturbance is studied. The problem is transformed into the stabilization of the error system between the objective network and the active network. The idea of sliding-mode control is used to design a nonlinear feedback control law. The solvability of the error system is obtained via monotone operator theory under an appropriately chosen space norm. The error system is proved to be exponentially stabilized at any decay rate by a suitable Lyapunov functional. So the objective network can track the active network exponentially at any designated rate.

Key words: Tracking control, Unknown internal disturbance, Star-shaped network of Eulerbernoulli beams, Sliding-mode control, Exponential stabilization

中图分类号:

O231.9

张雅轩,许跟起,郭燕妮. 带有未知内部扰动的星形Euler-Bernoulli梁网络的指数跟踪控制[J]. 数学物理学报, 2019, 39(3): 596-610.

Yaxuan Zhang,Genqi Xu,Yanni Guo. Exponential Tracking Control for a Star-Shaped Network of Euler-Bernoulli Beams with Unknown Internal Disturbance[J]. Acta mathematica scientia,Series A, 2019, 39(3): 596-610.

参考文献 28

1	Valencia C, Vellasco M, Figueiredo K. Trajectory tracking control using echo state networks for the corobot's arm//Kin J-H, et al. Robot Intelligence Technology and Applications 2, Advances in Intelligent Systems and Computing. Switzerland: Springer Inter Publish, 2014, 274: 433-447
2	Kumar N , Panwar V , Sukavanam N , et al. Neural network-based nonlinear tracking control of kinematically redundant robot manipulators. Math Comput Modelling, 2011, 53 (9/10): 1889- 1901
3	Mehrabian A, Tafazoli S, Khorasani K. Cooperative tracking control of Euler-Lagrange systems with switching communication network topologies//2010 IEEE/ASME International Conference on Advanced Intelligent Mechatronics. Montréal, Canada:IEEE, 2010:756-761
4	Xiong J , Li J . Non-fragile consensus algorithms for a network of diffusion PDEs with boundary local interaction. Internat J Systems Sci, 2017, 48 (9): 1- 7
5	Demetriou M . Synchronization and consensus controllers for a class of parabolic distributed parameter systems. Syst Control Lett, 2013, 62 (1): 70- 76
6	Pilloni A , Pisano A , Orlov Y , et al. Consensus-based control for a network of diffusion PDEs with boundary local interaction. IEEE Trans Autom Control, 2016, 61 (9): 2708- 2713 doi: 10.1109/TAC.2015.2506990
7	Jadachowski L , Meurer T , Kugi A . Backstepping observers for linear PDEs on higher-dimensional spatial domains. Automatica, 2016, 51: 85- 97
8	Freudenthaler G , Meurer T . PDE-based tracking control for multi-agent deployment. IFAC-PapersOnLine, 2016, 49 (18): 582- 587 doi: 10.1016/j.ifacol.2016.10.228
9	Qi J , Vazquez R , Krstic M . Multi-agent deployment in 3-d via PDE control. IEEE Trans Autom Control, 2015, 60 (4): 891- 906 doi: 10.1109/TAC.2014.2361197
10	Shifmann J. A tracking controller for the Euler-Bernoulli beam//1990 IEEE International Conference on Robotics and Automation. Cincinnati, OH:IEEE, 1990, 928-933
11	Shtessel T, Edwards C, Fridman L. Sliding Mode Control and Observation. Basel: Birkhäuser, 2014
12	Fu Q , Xu G . Exponential stabilization of 1-d wave equation with distributed disturbance. WSEAS Trans Math, 2015, 14: 192- 201
13	Tang S, Krstic M. Sliding mode control to the stabilization of a linear 2×2 hyperbolic system with boundary input disturbance//Proceedings of 2014 American Control Conference. Portland, America:IEEE, 2014:1027-1032
14	Cheng M, Su W. Boundary stabilization and matched disturbance rejection of hyperbolic PDE systems:a sliding-mode approach//Proceedings of IEEE American Control Conference. Montréal:IEEE, 2012:5360-5365
15	Orlov Y , Pilloni A , Pisano A , et al. Consensus-based leader-follower tracking for a network of perturbed diffusion PDEs via local boundary interaction. IFAC-PapersOnLine, 2016, 49 (8): 228- 233 doi: 10.1016/j.ifacol.2016.07.446
16	张丽萍, 刘东毅, 张国山. 带有内部扰动的Timoshenko梁系统的指数稳定性. 数学物理学报, 2017, 37A (1): 185- 198 doi: 10.3969/j.issn.1003-3998.2017.01.017
	Zhang L , Liu D , Zhang G . Exponential stabilization of a Timoshenko beam system with internal disturbances. Acta Math Sci, 2017, 37A (1): 185- 198 doi: 10.3969/j.issn.1003-3998.2017.01.017
17	Liu J , Wang J . Boundary stabilization of a cascade of ODE-wave systems subjected to boundary control matched disturbance. Internat. J Robust Linear Control, 2017, 27: 252- 280 doi: 10.1002/rnc.v27.2
18	Wang J , Liu J , Ren B , et al. Sliding mode control to stabilization of cascaded heat PDE-ODE systems subject to boundary control matched disturbance. Automatica, 2015, 52: 23- 34 doi: 10.1016/j.automatica.2014.10.117
19	Han J . From PID to active disturbance rejection control. IEEE Trans. Industrial Electronics, 2009, 56 (3): 900- 906 doi: 10.1109/TIE.2008.2011621
20	Guo B , Liu J , AL-Fhaid A , et al. The active disturbance rejection control approach to stabilisation of coupled heat and ODE system subject to boundary control matched disturbance. Internat J Control, 2015, 88 (8): 1554- 1564 doi: 10.1080/00207179.2015.1010179
21	Guo B , Jin F . Sliding mode and active disturbance rejection control to stabilization of one-dimensional anti-stable wave equations subject to disturbance in boundary input. IEEE Trans Automat Control, 2013, 58 (5): 1269- 1274 doi: 10.1109/TAC.2012.2218669
22	Guo B , Jin F . The active disturbance rejection and sliding mode control approach to stabilization of the Euler-bernoulli beam equations with boundary input disturbance. Automatica, 2013, 49 (9): 2911- 2918 doi: 10.1016/j.automatica.2013.06.018
23	Guo B , Liu J . Sliding mode control and active disturbance rejection control to the stabilization of onedimensional Schrödinger equation subject to boundary control matched disturbance. Internat J Robust Nonlinear Control, 2014, 24 (16): 2194- 2212 doi: 10.1002/rnc.v24.16
24	Wang J , Guo B . Riesz basis and stabilization for the flexible structure of a symmetric tree-shaped beam network. Math Meth Appl Sci, 2008, 31: 289- 314 doi: 10.1002/(ISSN)1099-1476
25	Barbu V . Nonlinear Differential Equations of Monotone Types in Banach Spaces. New York: SpringerVerlag, 2010
26	Polyak B . Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Soviet Math Dokl, 1966, 7 (2): 72- 75
27	Merentes N , Nikodem K . Remarks on strongly convex functions. Aequat Math, 2010, 80: 193- 199 doi: 10.1007/s00010-010-0043-0
28	Bauschke H , Combettes L . Convex Analysis and Monotone Operator Theory in Hilbert Spaces. New York: Springer-Verlag, 2017