摘要
本文的研究内容主要有两个,即：自抗扰控制在分布参数系统中的应用和非线性波动方程解的性质。自抗绕控制技术对带有未知干扰的非线性系统是非常有效的。它的核心思想是：首先用扩展状态观测器对不确定项进行估计,然后再用该估计来抵消不确定项。自抗扰控制是一个庞大的理论体系和全新的思想,它主要由三部分组成：跟踪微分器,扩展状态观测器,基于扩展状态观测器的误差反馈结构。自抗绕控制不但可以应用于集中参数系统,而且在分布参数控制中也非常有效。波动方程是偏微分方程和分布参数控制理论的一个重要的研究内容,对它的研究必将促进偏微分方程和控制理论的进一步发展。
论文分为四章。
第一章是引言,主要介绍本文的研究背景,国内外研究现状及本文的主要结果。
第二章主要研究跟踪微分器,提出了一种基于泰勒展开的高精度跟踪微分器,即：其中v(t)是输入信号。通过将其与一般的观测形式的高增益微分器进行比较,可以得到我们的微分器具有比一般高增益微分器更高的精度。
第三章主要用自抗绕控制技术稳定两种分布参数系统。在第三章第一节和第三节,分别讨论如下带有不确定干扰的波动方程的稳定性和其中u(x,t)是系统状态,U(t)是控制输入,Y(t),Y1(t),Y2(t)是输出测量,d(t)是未知的干扰项。在第三章第二节,讨论如下带有不确定干扰的Euler-Bernoulli梁方程的稳定性其中w(x,t)是系统状态,U(t)是控制输入,Y(t)是输出测量,d(t)是未知的干扰项。
我们的目标是,在存在不确定干扰d(t)的前提下,利用测量输出Y(t)设计控制U(t)来分别稳定相应的系统。
第四章主要研究一维非线性波动方程全局解的存在性。在第四章第一节和第二节,讨论带有边界阻尼项和内部源项的非线性波动方程的整体解的存在性。在第四章第三节,讨论带有边界源项和内部阻尼项的非线性波动方程的整体解的存在性。在第四章第四节,讨论一种非耗散非线性波动方程的整体解的存在性。
The main contents of this thesis consist of two parts. That is Active Disturbance Rejection Control (ADRC) for the distributed parameters systems and the properties of the solutions for some nonlinear wave equations.
It is very effective to deal with the uncertainty in the nonlinear systems by the ADRC. The main idea of the ADRC is that the uncertainty is first estimated by the extended state observer, and then canceled by its estimation. Theoretical system of the ADRC consist of three parts which are the tracking differentiator, the extended state observer, the output feedback stability based on the extended state observer. The ADRC does well in not only the lumped parameter system but also the distributed parameter system. Wave equations have been one of the important contents of partial differential equation (PDE) and control theory. Studies to wave equations will accelerate the development of PDE and control theory.
This thesis consists of four chapters.
In Chapter1, firstly, a survey on the research background and the research advance of the related work are given. Secondly, the main results obtained in this thesis are listed.
Chapter2is devoted to the study on the tracking differentiator. We proposed the following tracking differentiator by the technique of Taylor expansion. where v(t) is the input signal.
By comparing the presented algorithm with the observable canonical form differen-tiator, it follows that our differentiator obtain the higher accuracy than the traditional high-gain differentiator.
In Chapter3, we apply the ADRC to two distributed parameters systems. In the Section1and Section3of Chapter3, we consider the following wave equation in the presence of disturbance: and where u(x,t) is the state, U(t) is the input, Y(t),Y1(t),Y2(t) are the output, d(t) is the disturbance.
In the Section2of Chapter3, we consider the following Euler-Bernoulli beam equa-tion in the presence of disturbance: where w(x,t) is the state, U(t) is the input, Y(t) is the output, d(t) is the disturbance.
The objective of our paper is to design a continuous controller U(t), which is based on the output Y(t), to stabilize the corresponding system in the presence of disturbance, respectively.
Chapter4is devoted to the existence of the global solution for some one-dimensional wave equations. In the Section1and Section2of Chapter4, we will consider the nonlinear wave equation with boundary damping terms and interior source terms. In Section3, the nonlinear wave equation with boundary source terms and interior damping terms is considered. Section4is devoted to the nonlinear wave equation which is non-dissipative.
引文
[1]W. Wong, W.D. Blair. Steady-state tracking with LFM waveforms. IEEE Transactions on Aerospace and Electronic Systems,2000,36(2),701-709.
[2]H. Lanshammar. On practial evaluation of differentiaction techniques for human gait anal-ysis. Journal of Biomechanics,1982,15(2),99-105.
[3]P.R. Mahaparea. Mixed coordinate tracking of generalized maneuvering targets using ac-celeraction and jerk models. IEEE Transactionson Aerospace and Electronic Systems,2000, 36(3),992-1000.
[4]J. Cullum. Numerical differentiation and regularization. SIAM Journal of Numerical Anal-ysis,1971,8(2),254-265.
[5]H. Khalil. Robust servomechanism output feedback controller for feedback linearizable sys-tems. Automatica,1994,30,1587-1599.
[6]S. Ibrir. Linear time-derivative trackers. Automatica,2004,40,397-405.
[7]B.Z. Guo, Z.L. Zhao. On convergence of nonlinear tracking differentiator. International Jour-nal of Control,2011,84,693-701.
[8]X. Wang, B. Shirinzadeh. Rapid-convergent nonlinear differentiator. Mechanical Systems and Signal Processing,2012,28,414-431.
[9]J.Q. Han, W. Wang. Nonlinear tracking-differentiator. Journal of Systems Science and Math-ematical Sciences,1994,14,177-183.
[10]J. Davila, L. Fridman, A. Levant. Second-order sliding-modes observer for mechanical sys-tems. IEEE Transactions on Automatic Control,2005,50,1785-1789.
[11]A.N. Atassi, H.K. Khalil. Separation results for the stabilization of nonlinear systems using different high-gain observer designs. Systems Control Letter,2000,39,183-191.
[12]B.Z. Guo, J.Q. Han, F.B. Xi. Linear tracking-differentiator and application to online esti-mation of the frequency of a sinusoidal signal with random noise perturbation. International Journal of Systems,2002,33,351-358.
[13]A. Levant. Robust exact differentiation via sliding mode technique. Automatica,1998,34, 379-384.
[14]A. Levant. High-order sliding modes, differentiation and output-feedback control. Interna-tional Journal of Control,2003,76,924-941.
[15]J.H. Ahrens, H. K. Khalil. High-gain observers in the presence of measurement noise:A switched-gain approach. Automatica,2009,45,936-943.
[16]A. M. Dabroom, H. K. Khalil. Discrete-time implementation of high-gain observers for nu-merical differentiation. International Journal of Control,1997,12,1523-15371.
[17]X. Wang, Z. Chen, G. Yang. Finite-time-convergent differentiator based on singular pertur-bation technique. IEEE Trans. Automat. Control,2007,52(9),1731-1737.
[18]H. K. Khalil. Nonlinear systems, New York, Prentice-Hall,1996.
[19]B.Z. Guo, F.F. Jin. 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,99 10.1109/TAC.2012.2218669
[20]Bao-Zhu Guo, Jun-Jun Liu. Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schroinger equation subject to boundary control matched disturbance.2013, Int. J. Robust. Nonlinear Control, DOI:10.1002/rnc.2977.
[21]M. Tsutsum. On solutions of semilinear differention equations in Hillbert space. Math Japon., 1972,17,173-193.
[22]O. A. Oleinik, E. V. Radkevich. Method of introducing of parameterevolution equations. Russian Math. Surver,1978,33,7-84.
[23]S. Alinhac. Blow-up for nonlinear hyperbolic equations. Birkhauser, Boston, Berlin,1995.
[24]H. A. Levine. The roal of critical exponents in blowup theorems. SIAM Review,1974,32(2), 262-288.
[25]H. A. Levine. A note on a nonexistence theorem for nonlinear wave equations. SIAM J. Math. Anal.,1974,5(5),644-648.
[26]H. A. Levine, L. E. Payne. Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations. J. Math. Anal. Appl.1976,55,329-334.
[27]M. Ohta. Blow-up of solutions of dissipative nonlinear wave equations. Hokkaido Math. J., 1997,26,115-124.
[28]K. Ono. Global existence, decay and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings. J. Diff. Equa.,1997,137,273-301.
[29]O. A. Ladyzhenskaya, V. K. Kalantarov. Blow up theorems for quasilinear parabolic and hyperbolic equations. Zap. nauckn. SLOMI Steklov,1977,69,77-102.
[30]Zhijian Yang. Global existence, asympotic behavior and blow up of solutions for a class of nonlinear wave equations with dissipative term. J. Diff. Equa.,2003,187,520-540.
[31]J. A. Esquivel-Avila. Qualitative analysis of a nonlinear wave equation. Disc. Cont. Dyna. Sys.,10 (3) 787-804.
[32]D. H. Sattinger. On global solution of nonlinear hyperbolic equations. Arch. Rational Mech. Anal.1968,30,148-172.
[33]L. Payne, D. Sattinger. Saddle points and instability of nonlinear hyperbolic equations. Israel Math. J.1981,22,273-303.
[34]H. A. Levine. Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J. Math. Anal.1974,5(4),138-146.
[35]H.A. Levine. Instability and nonexistence of global solutions to nonlinear wave equations of the form. Trans. Amer. Math. Soc.1974,192,1-21.
[36]J. Ball. Remark on blow-up and nonexistence theorems for nonlinear evolution equations. Quart. J. math. Oxford 1977,28,437-486.
[37]R. T. Glassey. Blow-up theorems for nonlinear wave equation. Math. Z.1973,132,183-203.
[38]V. Georgiev, G. Todorova. Existence of solution of the wave equation with nonlinear damping and source term. J. Differential Equation,1994,109,295-308.
[39]Salim A. Messaoudi. Blow-up in a nonlinearly damped wave equation. Math. Nachy.2001, 231,1-7.
[40]E. Vitillaro. Global nonexistence theorem for a class of evolution equations with dissipation. Arch. Ration. Mech. Anal,1999149,155-182.
[41]A. Haraux, E. Zuazua. Decay estimates for some semilinear damping hyperbolic problems. Arch. Ration. Mech. Anal.,1988,150,191-206.
[42]Salim A. Messaoudi. Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic eqution. J. Math. Anal. Appl.,2006,320,902-915.
[43]E. Vitillaro. Global nonexistence for the wave equations with nonlinear boundary damping and source terms. J. Differential Equations,2002,186,259-298.
[44]R. Ikehata, T. Suzuki. Stable and unstable sets for evolution equations of parabolic and hyperbolic type. Hiroshima Math. J.,1996,26,475-491.
[45]R. Ikehata. Some remark on the wave equations with nonlinear damping and source terms. Nonlinear Analysis,1996,27,1165-175.
[46]M. Aassila, M.M. Cavalcanti, V.N. Domingos Cavalcanti. Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term. Calc. Var. Partial Differential Equations,2002,15,155-183.
[47]L. Bociu. I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping. Discrete Contin. Dyn. Syst.2008,22, 835-860.
[48]M.M. Cavalcanti, V.N. Domingos Cavalcanti, I. Lasiecka. Well-posedness and optimal de-cay rates for the wave equation with nonlinear boundary damping-source interaction. J. Differential Equations,2007,236,407-459.
[49]Wenjun Liu, Jun Yu. On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms. Nonlinear Analysis,2011,74,2175-2190.
[50]L.Q. Lu, S.J. Li, S.G. Chai. On a viscoelastic equation with nonlinear boundary damping and source terms:global existence and decay of the solution. Nonlinear Anal. RWA,2011, 12 (1),295-303.
[51]E. Vitillaro. A potential well theory for the wave equation with nonlinear source and bound-ary damping terms. Glasgow Math. J.,2002,44,375-395.
[52]G. Chen, a note on the boundary stabilization of the wave equation. Siam J. Control and Optimization,1981,19,106-113.
[53]G. Chen. Energy decay estimates and exact boundary value controllability for the wave equatiuon in a bounded domain. J. Math. Pures.,1979,58(9),249-273.
[54]A. Pazy. Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York,1983.
[55]B.Z. Guo, K.Y. Yang. Dynamic stabilization of an Euler-Bernoulli beam equation with time delay in boundary observation. Automatica,2009,45,1468-1475.
[56]G. Weiss, Admissible Observation Operators for Linear Semigroups,65 (1989), Israel Journal of Mathematics,17-43.
[57]B.Z. Guo and C.Z. Xu, The Stabilization of a One-Dimensional Wave Equation by Boundary Feedback With Noncollocated Observation, IEEE Trans. Automat. Control,52 (2007),371-377
[58]H.W. Zhang, Q.Y. Hu. Asymptotic behavior and nonexistence of wave equation with non-linear boundary condition. Commun. Pure Appl. Anal.,2005,4 (4),861-869.
[59]M.M. Cavalcanti, V.N.D. Cavalcanti, P. Martinez. Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term. J. Differential Equations, 2004,203 (1),119-158.
[60]E. Vitillaro. Some new results on global nonexistence and blow-up for evolution problems with positive initial energy. Rend. Istit. Mat. Univ. Trieste,2000,31,245-275.
[61]L.X. Truong, L.T.P. Ngoc, A.P.N. Dinh, N.T. Long. The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions. Nonlinear Anal. RWA,2010,11,1289-1303.
[62]L.X. Truong, L.T.P. Ngoc, A.P.N. Dinh, N.T. Long. Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type. Nonlinear Anal.,2011, dx.doi.org/10.1016/j.na.2011.07015.
[63]H. A. Levine, L.E. Payne. Nonexistence theorems for the heat equation with nonlinear bound-ary conditions and for the porous medium equation backward in time. J. Differential Equa-tions,1974,16,319-334.
