一类不确定双曲型PDE-ODE级联系统的自适应边界镇定
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- 英文论文题名：Adaptive boundary stabilization for a class of uncertain hyperbolic PDE–ODE cascade systems
- 论文作者：徐再花 ; 刘允刚 ; 李健
- 英文论文作者：Zaihua Xu ; Yungang Liu ; JIan Li ; School of Control Science and Engineering ; Shandong University ; School of Mathematics and Information Science ; Yantai University
- 年：2017
- 作者机构：山东大学控制科学与工程学院;烟台大学数学与信息科学学院;
- 论文关键词：不确定PDE-ODE级联系统 ; 自适应镇定 ; 非线性发展方程 ; Lyapunov方法
- 英文论文关键词：uncertain PDE–ODE cascade systems ; adaptive stabilization ; nonlinear evolution equation ; Lyapunov method
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（A）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（A）
- 语种：中文
- 分类号：O231
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：6
- 文件大小：522k
- 原文格式：D
- 会议级别：全国
摘要
本文针对一类在非控制边界带有未知参数的不确定双曲型PDE-ODE级联系统,研究其自适应边界镇定问题.与已有密切相关文献相比,本文系统中带有的未知参数使所得闭环系统是一个非线性系统,从而导致已有的分析系统适定性和稳定性的策略不再适用.为了实现本文的控制目标,关键在于为闭环系统和未知参数选取恰当的能量函数和动态补偿器.具体的,通过利用自适应技术和投影算子,引入一个恰当的动态补偿器处理未知参数.在此基础上借助反推技术成功的设计出所期望的自适应边界控制器.然后,利用非线性发展方程理论和通过构造恰当的能量函数,得到整个闭环系统的适定性和稳定性结果.
Adaptive boundary stabilization is investigated for a class of uncertain hyperbolic PDE–ODE cascade systems which allows an unknown parameter at the uncontrolled boundary. Compared with the most closely related literature, the unknown parameter makes the resulting closed-loop system a nonlinear one, which leads to the existing methods of analyzing the wellposedness and stability of a system no longer applicable. To achieve the control objective of this paper, a dynamic compensation for the unknown parameter and an energy function for the considered closed-loop system must be chosen appropriately. Specifically, by using the adaptive technique and projection operator, a proper dynamic compensation is introduced to deal with the unknown parameter. Based on this and by using the backstepping technique, a desired adaptive boundary controller is successfully constructed for the considered system. Then, by the theory of nonlinear evolution equations and constructing an appropriate energy function, the well-posedness and stability results of the entire closed-loop system are obtained.
Adaptive boundary stabilization is investigated for a class of uncertain hyperbolic PDE–ODE cascade systems which allows an unknown parameter at the uncontrolled boundary. Compared with the most closely related literature, the unknown parameter makes the resulting closed-loop system a nonlinear one, which leads to the existing methods of analyzing the wellposedness and stability of a system no longer applicable. To achieve the control objective of this paper, a dynamic compensation for the unknown parameter and an energy function for the considered closed-loop system must be chosen appropriately. Specifically, by using the adaptive technique and projection operator, a proper dynamic compensation is introduced to deal with the unknown parameter. Based on this and by using the backstepping technique, a desired adaptive boundary controller is successfully constructed for the considered system. Then, by the theory of nonlinear evolution equations and constructing an appropriate energy function, the well-posedness and stability results of the entire closed-loop system are obtained.
引文
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[4]B.Z.Guo and F.F.Jin,Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance,IEEE Transactions on Automatic control,60(3):824–830,2015.
[5]Y.Liu,Z.X.Xia,Y.L.Wu,F.Guo and Y.Fu,Vibration suppression of an axially moving system via boundary control and S-curve Ac/deceleration methods,Proceedings of the 35th Chinese Control Conference,Chengdu,China,1329-1333,2016.
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[7]D.Bresch-Pietri and M.Krsti,Output-feedback adaptive control of a wave PDE with boundary anti-damping,Automatica,50(5):1407–1415,2014.
[8]B.d’Andréa-Novel and J.M.Coron,Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach,Automatica,36(4):587–593,2000.
[9]Z.H.Xu and Y.G.Liu,Adaptive stabilization for a class of PDE–ODE cascade systems with uncertain harmonic disturbances,ESAIM:Control,Optimisation and Calculus of Variations,23(2):497–515,2017.
[10]S.Zhang,W.He and D.Q.Huang,Active vibration control for a flexible string system with input backlash,IET Control Theory&Applications,10(7):800–805,2016.
[11]H.C.Zhou,B.Z.Guo and Z.H.Wu,Output feedback stabilisation for a cascaded wave PDE–ODE system subject to boundary control matched disturbance,International Journal of Control,89(12):2396–2405,2016.
[12]J.J.Liu and J.M.Wang,Boundary stabilization of a cascade of ODE–wave systems subject to boundary control matched disturbance,International Journal of Robust and Nonlinear Control,27(2):252–280,2017.
[13]M.Krsti and A.Smyshlyaev,Adaptive boundary control for unstable parabolic PDEs-part I:Lyapunov design,IEEE Transactions on Automatic Control,53(7):1575–1591,2008.
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[15]E.Zeidler,Nonlinear Functional Analysis and its Applications III:Variational Methods and Optimization,Springer,New York,1985.
[2]W.Guo and B.Z.Guo,Adaptive output feedback stabilization for one-dimensional wave equation with corrupted observation by harmonic disturbance,SIAM Journal on Control and Optimization,51(2):1679–1706,2013.
[3]H.Y.P.Feng and S.J.Li,The stability for a onedimensional wave equation with nonlinear uncertainty on the boundary,Nonlinear Analysis:Theory,Methods&Applications,89:202–207,2013.
[4]B.Z.Guo and F.F.Jin,Output feedback stabilization for one-dimensional wave equation subject to boundary disturbance,IEEE Transactions on Automatic control,60(3):824–830,2015.
[5]Y.Liu,Z.X.Xia,Y.L.Wu,F.Guo and Y.Fu,Vibration suppression of an axially moving system via boundary control and S-curve Ac/deceleration methods,Proceedings of the 35th Chinese Control Conference,Chengdu,China,1329-1333,2016.
[6]M.Krsti,Adaptive control of an anti-stable wave PDE,Dynamics of Continuous,Discrete and Impulsive Systems Series A:Mathematical Analysis,17:853–882,2010.
[7]D.Bresch-Pietri and M.Krsti,Output-feedback adaptive control of a wave PDE with boundary anti-damping,Automatica,50(5):1407–1415,2014.
[8]B.d’Andréa-Novel and J.M.Coron,Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach,Automatica,36(4):587–593,2000.
[9]Z.H.Xu and Y.G.Liu,Adaptive stabilization for a class of PDE–ODE cascade systems with uncertain harmonic disturbances,ESAIM:Control,Optimisation and Calculus of Variations,23(2):497–515,2017.
[10]S.Zhang,W.He and D.Q.Huang,Active vibration control for a flexible string system with input backlash,IET Control Theory&Applications,10(7):800–805,2016.
[11]H.C.Zhou,B.Z.Guo and Z.H.Wu,Output feedback stabilisation for a cascaded wave PDE–ODE system subject to boundary control matched disturbance,International Journal of Control,89(12):2396–2405,2016.
[12]J.J.Liu and J.M.Wang,Boundary stabilization of a cascade of ODE–wave systems subject to boundary control matched disturbance,International Journal of Robust and Nonlinear Control,27(2):252–280,2017.
[13]M.Krsti and A.Smyshlyaev,Adaptive boundary control for unstable parabolic PDEs-part I:Lyapunov design,IEEE Transactions on Automatic Control,53(7):1575–1591,2008.
[14]A.Pazy,Semigroups of Linear Operators and Applications to Partial Differential Equations,Springer,New York,1983.
[15]E.Zeidler,Nonlinear Functional Analysis and its Applications III:Variational Methods and Optimization,Springer,New York,1985.