分布参数系统的确定学习理论及其应用
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摘要
流体系统和振动系统广泛存在于自然界和工程界中。随着现代工业的飞速发展,人们对各种装备在轻型化、智能化、精密化程度上的要求和性能不断的提升,使得流体系统和柔性体振动系统被科学界和工业界所重视。由于这些系统本身材料质地的特性,使得其具有无限个自由度,系统的运动状态不再像通常的刚体系统那样可以用有限个参数就能完整的描述,而必须用场才能较完整的描述,即它们的状态既是时间变量的函数也是空间变量的函数,因而这类系统被称为“无限维分布参数系统”。要能对分布参数系统进行较好的设计或作用,首先就需要充分了解该系统的动态行为。但是在未知的动态环境下,要准确的知道分布参数系统的内部动态是一个极富挑战的任务。最近,Wang等人利用自适应控制和系统动力学的概念与方法,提出了确定学习理论。该理论研究非线性系统在未知动态环境下的知识获取、表达、存储和再利用等问题。
本文主要研究含未知内部动态的非线性分布参数系统的确定学习问题,由于分布参数系统的无限维本质特性,使得在未知动态环境下准确建立完整的分布参数系统内部动态异常困难,甚至可以说是在现有条件下不可能完成的任务。我们主要运用确定学习理论提供的框架和方法,在一定的精度内实现对分布参数系统内部动态的准确逼近/辨识。然而,现有的确定学习理论是针对有限维动态系统建立的,而使得其不能直接的应用于无限维分布参数系统。因此,我们首先在合适精度内建立分布参数系统的有限维逼近系统,然后对有限维动态系统应用确定学习理论得到其沿轨迹的准确逼近,进而实现对原分布参数系统的内部动态辨识/逼近。所得到的系统模型不仅在一定精度内反映原分布参数的本质动态特性还易于工程实现,其中主要包括运用何种逼近方法将分布参数系统转化为有限维动态系统、系统的可辨识条件、辨识算法的设计等内容。本文的主要内容概括如下
1.研究一类周向抛物型分布参数系统的确定学习问题。抛物型分布参数系统常常出现在流体、热传导和粒子扩散等物理系统中。系统的“周向”是指分布参数系统描述的物理对象具有环形的几何结构。根据系统的周向结构特点,我们首先利用离散Fourier变换极其逆变换的方法将所考虑的分布参数系统转化为有限维动态系统。其次,我们分析了有限维动态系统的几个重要性质。由于有限统动态系统的系数矩阵是由离散Fourier变换在离散过程中引入的,根据离散Fourier变换与循环矩阵的联系可知系数矩阵是一个循环矩阵;并且由系统的周向几何结构分析可得有限维系统的内部动态是沿系数矩阵的对角线局部占优。对有限维动态系统运用确定学习理论,通过采用径向基函数(Radial Basis Function,RBF)神经网络得到对有限统系统的主要内部动态的沿系统轨迹的精确逼近,进而得到抛物型分布参数系统内部动态沿轨迹的准确逼近。
2.研究涡扇发动机轴流压气机系统的旋转失速初始扰动的建模与快速检测。旋转失速是轴流压气机系统的一种不稳定流动现象,当压气机处于旋转失速状态时,压气机的转子和定子叶片都承受巨大的压力而造成叶片的损伤,失速团的非轴对称流动使得燃烧室和涡轮内局部过热而可能烧坏壁面和叶片。旋转失速会使得压气机的流量和压力突然下降,导致发动机推力的骤然下降而大大限制压气机的性能和降低发动机的效率。由旋转失速造成的叶片槽道内的气流堵塞会引起发动机的喘振(压气机的另一种典型不稳定流动现象/故障)。因为发动机是在失速边界附近效率最大,所以检测旋转失速为提高发动机的性能和预防喘振极为重要。要阻止压气机不进入旋转失速,则需要在旋转失速发生前进行预测。因此,要快速检测旋转失速的初始扰动。快速准确的检测方法是以对初始扰动建立准确建模为基础。因此,我们首先研究旋转失速初始扰动的建模。我们在压气机周向和轴向上布置多个传感器测量压气机的流量和压力信号,运用确定学习理论采用动态RBF神经网络辨识初始扰动的内部动态,得到旋转失速初始扰动的内部动态沿轨迹的准确逼近。因而,建立旋转失速初始扰动的RBF神经网络模型。并将在辨识过程中收敛的RBF神经网络权值在收敛后一段时间内的平均值(常数权值)作为旋转失速初始扰动的模式保存,将多个旋转失速初始扰动建模生成的常数权值保存构成旋转失速初始扰动的模式库。在建模的基础上,我们运用保存的模式库快速检测旋转失速初始扰动。对被检测的压气机系统用模式库中保存的常数权值构造一系列的动态估计器,动态估计器的个数与模式库中保存的模式个数相同,动态估计器的维数与学习辨识过程中构造的RBF神经网络维数相同。由于RFB神经网络模型是旋转失速初始扰动模式内部动态的本质表现,因此动态估计器中嵌入了已辨识的旋转失速初始扰动的内部动态。将被检测的压气机的状态与动态估计器的状态做差生成残差系统,依据确定学习理论的动态模式识别方法,残差度量动态模式的相似性。因此,我们用平均l1范数对残差进行评估,所有残差中平均l1范数最小的那个就是压气机正出现的模式。当旋转失速初始扰动模式所对应的残差的范数最小时,就表明压气机中出现了失速初始扰动,则实现失速初始扰动的快速检测。由于,检测是在失速的初始扰动阶段完成,因此可以预测旋转失速,并且由于旋转失速是喘振的先兆,这样就可以预防喘振和预警发动机。
3.研究带Dirchilet边界条件的完全共振型波动系统的确定学习问题。根据Lagrangian应力定律、Newton第二定律以及Dirchilet边界条件可知,所讨论的完全共振型波动系统是描述两端固定的柔性振动弦。我们讨论了两大类波动系统:1)系统内部动态部分未知;2)系统内部动态完全未知。我们利用有限差分方法将波动系统在空间域上离散为一个高维的常微分方程组描述的有限维非线性动态系统。然后,利用Gronwall不等式定理和Leary-Schauder不动点定理证明了有限维动态系存在唯一解,并且其解收敛到原波动系统的解。这就保证了这是一个有效的逼近,并且使得有限维动态系统保持了波动系统的本质动态特性。最后,我们运用确定学习理论构造了动态RBF神经网络,其沿轨迹精确辨识了有限维动态系统的内部动态,进而得到有限维截断误差精度内对波动系统内部动态的准确逼近。但是由于第1),2)类波动系统未知内部动态关联的离散点不一样,而使得RBF神经网络的输入维数不同。对于第1)类波动系统,RBF神经网络输入的只是离散点本身的信息,而第2)波动类系统则还要输入相邻两离散点的信息。对输入维数分析可知,输入的离散点是由未知内部动态中包含的最高阶空间微分项来决定。因此,在相同的截断误差精度内,系统中增加对空间变量的未知低阶微分项并不会增加RBF神经网络的输入维数。若要提高逼近精度,则可以通过增加有限维截断误差精度、空间离散点个数和神经元个数来实现。
Fluid systems and vibration systems are widely found in nature and industry. Withthe rapid development of modern industry, the improved performance of light, intelligence,precision of the equipment is requested, which makes ?uid systems and vibration systemsof ?exible body are concerned by the scienti?c and engineering community. Due to thetexture of materials, the system has an in?nite number of degrees of freedom, and thesystem state unlike the rigid-body system can be description by ?nite parameters butmust use a ?eld for complete description, which meas that the system state is not only afunction of time variable but also a function of spatial variables, thus the system is calledas in?nite-dimensional distributed parameter systems (DPS). If one wants to carry outor good design for DPS, a adequate understanding of dynamic behavior of the systemis necessary. However, in uncertain dynamical environments, to accurately acquaint thedynamics of DPS is a challenging task. Recently, Wang etc. propose the deterministiclearning theory (DLT) by utilizing results from concepts and tools of adaptive controland dynamical systems.
This thesis studies the DLT of nonlinear DPS with unknown dynamics. Since thein?nite-dimensional essential feature, to establish complete and accurate dynamics is verydi?cult, evenmore it is a impossible task under current conditions. Using the DLT, weprovide a framework and methodology to approximate/identify the dynamics of nonlinearDPS within a certain accuracy. Therefore, we ?rst establish an approximation systemfor DPS in a ?nite-order accuracy; then by using the deterministic learning algorithm,we obtain an accurate radial basis function (RBF) neural network (NN) approximationof the ?nite-dimensional dynamical system (FDDS) along the trajectories; ?nally theidenti?cation of dynamics of the original DPS is achieved in some accuracy. The resultingsystem model can re?ects the nature of the original DPS in the accuracy, and it is easyto be work. The contents contain: ?nite-dimensional approximation method, systemidenti?ability conditions, and identi?cation algorithms, etc. The main contents of thethesis are as follows:
1. We investigate the identi?cation of a class of parabolic DPS. The parabolic DPSusually arises from physical phenomena such as heat conduction, ?ow ?eld and particledi?usion. The considered DPS describe a homogeneous and isotropic object with circulargeometric structure. Based on this feature, we ?rst reduce DPS into a FDDS by thediscrete Fourier transform (DFT) method. Secondly, some important properties of FDDS,including the discrete symmetry and the partial dominance of system dynamics accordingto point-wise observations, are analyzed. Due to the coe?cient matrix is introduced by DFT in discrete process, the coe?cient matrix is a circulant matrix by the connectionbetween DFT and circulant matrix. The system geometry decides the dominant dynamicsof FDDS along the diagonal of coe?cient matrix. Finally, by using the deterministiclearning algorithm, it is show that locally relatively accurate NN approximation of maindynamics of the FDDS is achieved in local region along the recurrent trajectories. Then,a locally identi?cation of the dynamics of the parabolic DPS is accomplished.
2. We investigate the modeling and rapid detection of rotating stall via deterministiclearning. Rotating stall is one distinct aerodynamic instabilities in axial ?ow compressorof turbofan engine. Once a compressor enters fully developed rotating stall, rotor andstator blades are under tremendous stress caused by stalled ?ow to damage the com-pressor, the non-axisymmetric ?ow can result internal overtemperatures in combustionchamber and turbine to burn the wall and blades. Rotating stall makes the mass ?owand press rise through the compressor is decreased which leads to the thrust sudden dropgreatly in engine, and the function is to increase the pressure of the ?ow. Thus, thiscondition severely limit the compressor performance and reduce the e?ciency of the en-gine. Evenmore, the blade channel blockage caused by the rotating stall ?ow can leadsurge which is another typical unsteady ?ow phenomena/fault in compressor. For thesereasons, rapid detection of rotating stall is very important for improving the performanceand preventing surge. To resistant the rotating stall, the key recognizes the rotatingstall precursor (RSP). Rapid and accurate detection method is based on precision modelof RSP. Therefore, the detection process for rotating stall consists of two phases: themodeling/identi?cation phase and the recognition phase. In the identi?cation phase, wearrange multiple sensors at circumferential and axial of compressor to measure ?ow andpressure signals. Then, based on the measured signals, use dynamical RBF NN to i-dentify the dynamics of RSP according to DLT. We obtain the RBF NN model, andaverage of the NN weights in a time segment after convergence process (constant NNweights). The constant NN weights is stored as the pattern for RSP. Using the sameprocedure, a pattern library of compressor can be established for rotating stall. In therecognition phase, based on the established model of RSP, we use pattern library storedin the identi?cation phase to rapid detection RSP. For monitored compressor system, aseries dynamical estimators is constructed by the constant NN weights. Since RFB NNmodel can present the dynamics of RSP, the dynamical estimator embeds the dynamicsof RSP. By comparing the monitored compressor system and dynamical estimators, aseries residual systems is obtained. By the rapid dynamical pattern recognition method,residual can measure the similarity for dynamical patterns. Therefore, we use an averageof l1 norm to evaluate the residuals. The smallest average norm of all residuals is the pattern coming in compressor. If the norm of the RSP is smallest, which is suggest therotating stall is coming; rapid detection of the RSP is achieved. Due to rapid detectioncompletes in rotating stall inception, rotating stall can be predicted, and the surge canbe prevented because rotating stall is a symptom of surge and alert engine.
3. We investigate the DLT of completely resonant wave systems with Dirchiletboundary conditions. According to Lagrangian stress theorem, Newton’s second lawand the Dirchilet boundary conditions, considered completely resonant wave system isdescribed the ?exible vibrating string with ?xed endpoints. Two types of wave systemare discussed: 1) the unknown part of dynamics within the system; 2) the unknown allof dynamics within the system. We use ?nite di?erence method to discretize the wavesystem into a higher-dimensional nonlinear dynamic system in the space domain. Then,Gronwall inequality theorem and Leary-Schauder ?xed point theorem are used to provethe existence and uniqueness of solution of FDDS, and the solution of FDDS convergeto the solution of wave system. Therefore, these ensure that the approximation is valid,and makes the FDDS keeps the essence dynamics of the wave system. Finally, accordingto DLT, we employ the dynamical RBF NN to accurately identify the FDDS along thetrajectory. Then the accurate approximation of dynamics of the wave system is obtainedin some accuracy. Since it is di?erence that unknown dynamics of the ?rst and secondtype systems associated with the discrete points, the input dimension of RBF NN isdi?erent. For the ?rst type system, the input of RBF NN is only the discrete pointitself; while the second type system, the input of RBF NN is not only the discrete pointitself but also adjacent two discrete points. Analysis the input dimension, the input ofRBF NN is decide by the most order of spatial di?erential term. Therefore, if truncationerror accuracy is same, increasing the low-order di?erential item will not increase theinput dimension of RBF NN. To improve the approximation accuracy can be succeed byincreasing the ?nite-dimensional truncation error accuracy, the number of space discretepoints and the number of neurons.
本文主要研究含未知内部动态的非线性分布参数系统的确定学习问题,由于分布参数系统的无限维本质特性,使得在未知动态环境下准确建立完整的分布参数系统内部动态异常困难,甚至可以说是在现有条件下不可能完成的任务。我们主要运用确定学习理论提供的框架和方法,在一定的精度内实现对分布参数系统内部动态的准确逼近/辨识。然而,现有的确定学习理论是针对有限维动态系统建立的,而使得其不能直接的应用于无限维分布参数系统。因此,我们首先在合适精度内建立分布参数系统的有限维逼近系统,然后对有限维动态系统应用确定学习理论得到其沿轨迹的准确逼近,进而实现对原分布参数系统的内部动态辨识/逼近。所得到的系统模型不仅在一定精度内反映原分布参数的本质动态特性还易于工程实现,其中主要包括运用何种逼近方法将分布参数系统转化为有限维动态系统、系统的可辨识条件、辨识算法的设计等内容。本文的主要内容概括如下
1.研究一类周向抛物型分布参数系统的确定学习问题。抛物型分布参数系统常常出现在流体、热传导和粒子扩散等物理系统中。系统的“周向”是指分布参数系统描述的物理对象具有环形的几何结构。根据系统的周向结构特点,我们首先利用离散Fourier变换极其逆变换的方法将所考虑的分布参数系统转化为有限维动态系统。其次,我们分析了有限维动态系统的几个重要性质。由于有限统动态系统的系数矩阵是由离散Fourier变换在离散过程中引入的,根据离散Fourier变换与循环矩阵的联系可知系数矩阵是一个循环矩阵;并且由系统的周向几何结构分析可得有限维系统的内部动态是沿系数矩阵的对角线局部占优。对有限维动态系统运用确定学习理论,通过采用径向基函数(Radial Basis Function,RBF)神经网络得到对有限统系统的主要内部动态的沿系统轨迹的精确逼近,进而得到抛物型分布参数系统内部动态沿轨迹的准确逼近。
2.研究涡扇发动机轴流压气机系统的旋转失速初始扰动的建模与快速检测。旋转失速是轴流压气机系统的一种不稳定流动现象,当压气机处于旋转失速状态时,压气机的转子和定子叶片都承受巨大的压力而造成叶片的损伤,失速团的非轴对称流动使得燃烧室和涡轮内局部过热而可能烧坏壁面和叶片。旋转失速会使得压气机的流量和压力突然下降,导致发动机推力的骤然下降而大大限制压气机的性能和降低发动机的效率。由旋转失速造成的叶片槽道内的气流堵塞会引起发动机的喘振(压气机的另一种典型不稳定流动现象/故障)。因为发动机是在失速边界附近效率最大,所以检测旋转失速为提高发动机的性能和预防喘振极为重要。要阻止压气机不进入旋转失速,则需要在旋转失速发生前进行预测。因此,要快速检测旋转失速的初始扰动。快速准确的检测方法是以对初始扰动建立准确建模为基础。因此,我们首先研究旋转失速初始扰动的建模。我们在压气机周向和轴向上布置多个传感器测量压气机的流量和压力信号,运用确定学习理论采用动态RBF神经网络辨识初始扰动的内部动态,得到旋转失速初始扰动的内部动态沿轨迹的准确逼近。因而,建立旋转失速初始扰动的RBF神经网络模型。并将在辨识过程中收敛的RBF神经网络权值在收敛后一段时间内的平均值(常数权值)作为旋转失速初始扰动的模式保存,将多个旋转失速初始扰动建模生成的常数权值保存构成旋转失速初始扰动的模式库。在建模的基础上,我们运用保存的模式库快速检测旋转失速初始扰动。对被检测的压气机系统用模式库中保存的常数权值构造一系列的动态估计器,动态估计器的个数与模式库中保存的模式个数相同,动态估计器的维数与学习辨识过程中构造的RBF神经网络维数相同。由于RFB神经网络模型是旋转失速初始扰动模式内部动态的本质表现,因此动态估计器中嵌入了已辨识的旋转失速初始扰动的内部动态。将被检测的压气机的状态与动态估计器的状态做差生成残差系统,依据确定学习理论的动态模式识别方法,残差度量动态模式的相似性。因此,我们用平均l1范数对残差进行评估,所有残差中平均l1范数最小的那个就是压气机正出现的模式。当旋转失速初始扰动模式所对应的残差的范数最小时,就表明压气机中出现了失速初始扰动,则实现失速初始扰动的快速检测。由于,检测是在失速的初始扰动阶段完成,因此可以预测旋转失速,并且由于旋转失速是喘振的先兆,这样就可以预防喘振和预警发动机。
3.研究带Dirchilet边界条件的完全共振型波动系统的确定学习问题。根据Lagrangian应力定律、Newton第二定律以及Dirchilet边界条件可知,所讨论的完全共振型波动系统是描述两端固定的柔性振动弦。我们讨论了两大类波动系统:1)系统内部动态部分未知;2)系统内部动态完全未知。我们利用有限差分方法将波动系统在空间域上离散为一个高维的常微分方程组描述的有限维非线性动态系统。然后,利用Gronwall不等式定理和Leary-Schauder不动点定理证明了有限维动态系存在唯一解,并且其解收敛到原波动系统的解。这就保证了这是一个有效的逼近,并且使得有限维动态系统保持了波动系统的本质动态特性。最后,我们运用确定学习理论构造了动态RBF神经网络,其沿轨迹精确辨识了有限维动态系统的内部动态,进而得到有限维截断误差精度内对波动系统内部动态的准确逼近。但是由于第1),2)类波动系统未知内部动态关联的离散点不一样,而使得RBF神经网络的输入维数不同。对于第1)类波动系统,RBF神经网络输入的只是离散点本身的信息,而第2)波动类系统则还要输入相邻两离散点的信息。对输入维数分析可知,输入的离散点是由未知内部动态中包含的最高阶空间微分项来决定。因此,在相同的截断误差精度内,系统中增加对空间变量的未知低阶微分项并不会增加RBF神经网络的输入维数。若要提高逼近精度,则可以通过增加有限维截断误差精度、空间离散点个数和神经元个数来实现。
Fluid systems and vibration systems are widely found in nature and industry. Withthe rapid development of modern industry, the improved performance of light, intelligence,precision of the equipment is requested, which makes ?uid systems and vibration systemsof ?exible body are concerned by the scienti?c and engineering community. Due to thetexture of materials, the system has an in?nite number of degrees of freedom, and thesystem state unlike the rigid-body system can be description by ?nite parameters butmust use a ?eld for complete description, which meas that the system state is not only afunction of time variable but also a function of spatial variables, thus the system is calledas in?nite-dimensional distributed parameter systems (DPS). If one wants to carry outor good design for DPS, a adequate understanding of dynamic behavior of the systemis necessary. However, in uncertain dynamical environments, to accurately acquaint thedynamics of DPS is a challenging task. Recently, Wang etc. propose the deterministiclearning theory (DLT) by utilizing results from concepts and tools of adaptive controland dynamical systems.
This thesis studies the DLT of nonlinear DPS with unknown dynamics. Since thein?nite-dimensional essential feature, to establish complete and accurate dynamics is verydi?cult, evenmore it is a impossible task under current conditions. Using the DLT, weprovide a framework and methodology to approximate/identify the dynamics of nonlinearDPS within a certain accuracy. Therefore, we ?rst establish an approximation systemfor DPS in a ?nite-order accuracy; then by using the deterministic learning algorithm,we obtain an accurate radial basis function (RBF) neural network (NN) approximationof the ?nite-dimensional dynamical system (FDDS) along the trajectories; ?nally theidenti?cation of dynamics of the original DPS is achieved in some accuracy. The resultingsystem model can re?ects the nature of the original DPS in the accuracy, and it is easyto be work. The contents contain: ?nite-dimensional approximation method, systemidenti?ability conditions, and identi?cation algorithms, etc. The main contents of thethesis are as follows:
1. We investigate the identi?cation of a class of parabolic DPS. The parabolic DPSusually arises from physical phenomena such as heat conduction, ?ow ?eld and particledi?usion. The considered DPS describe a homogeneous and isotropic object with circulargeometric structure. Based on this feature, we ?rst reduce DPS into a FDDS by thediscrete Fourier transform (DFT) method. Secondly, some important properties of FDDS,including the discrete symmetry and the partial dominance of system dynamics accordingto point-wise observations, are analyzed. Due to the coe?cient matrix is introduced by DFT in discrete process, the coe?cient matrix is a circulant matrix by the connectionbetween DFT and circulant matrix. The system geometry decides the dominant dynamicsof FDDS along the diagonal of coe?cient matrix. Finally, by using the deterministiclearning algorithm, it is show that locally relatively accurate NN approximation of maindynamics of the FDDS is achieved in local region along the recurrent trajectories. Then,a locally identi?cation of the dynamics of the parabolic DPS is accomplished.
2. We investigate the modeling and rapid detection of rotating stall via deterministiclearning. Rotating stall is one distinct aerodynamic instabilities in axial ?ow compressorof turbofan engine. Once a compressor enters fully developed rotating stall, rotor andstator blades are under tremendous stress caused by stalled ?ow to damage the com-pressor, the non-axisymmetric ?ow can result internal overtemperatures in combustionchamber and turbine to burn the wall and blades. Rotating stall makes the mass ?owand press rise through the compressor is decreased which leads to the thrust sudden dropgreatly in engine, and the function is to increase the pressure of the ?ow. Thus, thiscondition severely limit the compressor performance and reduce the e?ciency of the en-gine. Evenmore, the blade channel blockage caused by the rotating stall ?ow can leadsurge which is another typical unsteady ?ow phenomena/fault in compressor. For thesereasons, rapid detection of rotating stall is very important for improving the performanceand preventing surge. To resistant the rotating stall, the key recognizes the rotatingstall precursor (RSP). Rapid and accurate detection method is based on precision modelof RSP. Therefore, the detection process for rotating stall consists of two phases: themodeling/identi?cation phase and the recognition phase. In the identi?cation phase, wearrange multiple sensors at circumferential and axial of compressor to measure ?ow andpressure signals. Then, based on the measured signals, use dynamical RBF NN to i-dentify the dynamics of RSP according to DLT. We obtain the RBF NN model, andaverage of the NN weights in a time segment after convergence process (constant NNweights). The constant NN weights is stored as the pattern for RSP. Using the sameprocedure, a pattern library of compressor can be established for rotating stall. In therecognition phase, based on the established model of RSP, we use pattern library storedin the identi?cation phase to rapid detection RSP. For monitored compressor system, aseries dynamical estimators is constructed by the constant NN weights. Since RFB NNmodel can present the dynamics of RSP, the dynamical estimator embeds the dynamicsof RSP. By comparing the monitored compressor system and dynamical estimators, aseries residual systems is obtained. By the rapid dynamical pattern recognition method,residual can measure the similarity for dynamical patterns. Therefore, we use an averageof l1 norm to evaluate the residuals. The smallest average norm of all residuals is the pattern coming in compressor. If the norm of the RSP is smallest, which is suggest therotating stall is coming; rapid detection of the RSP is achieved. Due to rapid detectioncompletes in rotating stall inception, rotating stall can be predicted, and the surge canbe prevented because rotating stall is a symptom of surge and alert engine.
3. We investigate the DLT of completely resonant wave systems with Dirchiletboundary conditions. According to Lagrangian stress theorem, Newton’s second lawand the Dirchilet boundary conditions, considered completely resonant wave system isdescribed the ?exible vibrating string with ?xed endpoints. Two types of wave systemare discussed: 1) the unknown part of dynamics within the system; 2) the unknown allof dynamics within the system. We use ?nite di?erence method to discretize the wavesystem into a higher-dimensional nonlinear dynamic system in the space domain. Then,Gronwall inequality theorem and Leary-Schauder ?xed point theorem are used to provethe existence and uniqueness of solution of FDDS, and the solution of FDDS convergeto the solution of wave system. Therefore, these ensure that the approximation is valid,and makes the FDDS keeps the essence dynamics of the wave system. Finally, accordingto DLT, we employ the dynamical RBF NN to accurately identify the FDDS along thetrajectory. Then the accurate approximation of dynamics of the wave system is obtainedin some accuracy. Since it is di?erence that unknown dynamics of the ?rst and secondtype systems associated with the discrete points, the input dimension of RBF NN isdi?erent. For the ?rst type system, the input of RBF NN is only the discrete pointitself; while the second type system, the input of RBF NN is not only the discrete pointitself but also adjacent two discrete points. Analysis the input dimension, the input ofRBF NN is decide by the most order of spatial di?erential term. Therefore, if truncationerror accuracy is same, increasing the low-order di?erential item will not increase theinput dimension of RBF NN. To improve the approximation accuracy can be succeed byincreasing the ?nite-dimensional truncation error accuracy, the number of space discretepoints and the number of neurons.
引文
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