小波理论在系统建模与控制中的若干应用研究
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摘要
小波函数及小波变换近乎完美的数学特性使得它日益受到控制科学家和工程人员的青睐,1991年瑞典控制学家Astr(?)m就提出小波函数逼近将成为系统建模、辨识和控制的最新研究方法。其后十几年间,随着小波理论的不断发展和完善,小波在系统辨识和控制中的应用越来越广,部分地解决了系统辨识与控制科学面临的一些问题。本文在前人研究成果的基础上,结合笔者本人的研究工作,给出了小波理论在系统辨识和控制领域中的若干应用算法,主要内容包括:
1.严格证明了白噪声经正交尺度(小波)分解和重构后方差减半的定理,因此可对辨识数据进行多分辨正交尺度分解和重构,尽可能消除噪声对数据的污染,提高辨识精度。在该定理的基础上,笔者将有显式表达的非正交小波作为基函数逼近系统的脉冲响应过程,并经多尺度变换消噪,取得了良好的辨识效果。
2.根据面向控制的辨识理论,推导出一种基于小波分解的线性时变系统在线分频辨识算法,对于工作频段时变的控制系统(如随动系统)可在线调整各子频段的辨识模型加权系数,以期模型在当前工作频段附近尽可能接近真实系统。
3.小波逼近在非线性动态系统辨识中的应用。一是利用小波时频局部化的特点,提出一种尺度因子可调的自适应小波核最小二乘支持向量机,并应用于非线性系统的黑箱建模。二是推导出了等距分布节点的三次B样条函数公式,以此为尺度函数结合B样条高通滤波器系数构造了三次B样条小波,并以B样条小波为基函数,逼近Hammerstein模型中的非线性环节。
4.基于小波逼近的控制策略研究。一是采用离散仿射小波网络逼近一类非匹配不确定性非线性系统,并结合Popov超稳定性理论设计控制参数的自适应律,克服了普通神经网络固有的训练算法计算量大,容易陷入局部极小,收敛性难以保证等缺陷。二是提出以小波函数为基函数应用于预测函数控制,由小波的尺度伸缩特性,根据预测时域内逼近精度要求的高低调整基函数的尺度因子,既保证了预测时域整体优化目标和逼近精度要求,又尽可能减少了基函数个数,实现了优化变量的集结,同时改善了系统的动态特性,增强了系统抑制外部干扰的能力。
5.在正交小波(尺度)分解和重构算法的基础上,得到等效于小波分解和重构
The perfect mathematical characteristics of wavelets and wavelets transform are winning more and more favor from control scientists and engineering personnel. In 1991, Astrom---a Sweden control scientist advanced that wavelets approximation would become the latest researching method in system modeling, identification and control. During the later period over one decade, following with the development of wavelet theory, wavelet has been broadly applied in system identification and control, and partially resolved some problems in system identification and control science. On the base of predecessors's achievements and my research work, some algorithms about the application of wavelets theory in system modeling and control are presented in this paper , the main contents as follows:1. The paper strictly proved the theorem that variance of white noise would decrease to 50% through orthogonal scale (wavelet) decomposition and reconstruction, therefore identification data would be decomposed and reconstructed through orthogonal scaling transform in order to filter noises of data. According to the theorem, non-orthogonal wavelets is taken as basis function of impulse response process and noises is removed through multi-scale decomposition, and good identification result is achieved.2. According to the theory of control-relevant identification, a band-wise identification method online to the linear time-varying system based on wavelets decomposition is presented. According to the method, weighting coefficients of sub-band decomposed by wavelets can be adjusted online if the time-varying system's work band is changed, so the time-varying control request can be implemented by weighting coefficients. The method is fit to some system especially such as servo system.3. The application of wavelets approximation to nonlinear dynamic system. The first is that an adaptive least square wavelets support vector machine whose scale parameter is adjusted and its application to nonlinear system identification. The second is that 3-order B-spline function formula of equidistant points is deduced, and 3-order B-spline wavelet is constructed via B-spline High-Pass filter, then B-spline wavelet is taken as basis function and applied to approximate to the nonlinear part of a Hammerstein model.
4. Some research of control strategy based on wavelet approximation. The first is that discrete affine wavelet network is applied to approximation of nonlinear system with mismatched uncertainties, and adaptive control law is designed according to the theory of hyperstability, so the drawbacks of BP network such as large computation online, local minimum aiid uncertain convergency are overcomed. The second is that a predictive functional control using wavelets as basis function is presented. Because wavelets has the properties of compact support and multi-scale analysis, the number of wavelets functions and their location are set flexibly according to the system's controlling precision and the different requirements of approximation precision at different intervals of the horizon. So the target function of the whole horizon are optimized and some important points are emphasized, meanwhile the optimal parameters have been aggregated. 5. On the base of orthogonal wavelet (scaling function) decomposition and reconstruction algorithm, the paper presents the equivalent wavelet (scaling function) FIR filter, and provides two theorems about filter coefficients. The filters are applied to solve the next two problems: The first, aiming at that the relay is sensitive to noise and extracting characteristic parameters from output values polluted by noises is hard, applying equivalent wavelet filter to eliminate noise online from feedback signal, and denoising of output wave-forms offline using wavelet thresholding to extract characteristic parameters easily . The second, in order to eliminate measurement noises in system's outputs for dynamical matrix control (DMC), output measurements are filtered by wavelets denoising online, and the reduction coefficient of reference locus in DMC is automatically adjusted according to wavelets correlation values, so system's anti-disturb (such as load disturb) ability is enhanced.6. Based on operation matrix via orthogonal wavelets, two simplified control algorithm based on orthogonal wavelets transform are presented: The first, the integral operation matrix of Harr wavelets is applied to approximation of hierarchical control of linear large scale systems, the differential equations are converted into a set of linear algebraic equations, so complicated calculous operation has been transformed to simple matrix operation, and computation has been reduced effectively. The second, an iterative learning control algorithm based on Harr wavelets is presented to address the terminal control
problem of a linear time-varying systems. By employing the orthogonality and boundary values of Harr wavelets, the differential equations are converted into a set of linear algebraic equations, and the control problem is simplified to finding Harr wavelets coefficients of control variables, so the method avoids computing the state transfer matrix, and the coefficients are determined iteratively by steepest descent learning algorithm.
1.严格证明了白噪声经正交尺度(小波)分解和重构后方差减半的定理,因此可对辨识数据进行多分辨正交尺度分解和重构,尽可能消除噪声对数据的污染,提高辨识精度。在该定理的基础上,笔者将有显式表达的非正交小波作为基函数逼近系统的脉冲响应过程,并经多尺度变换消噪,取得了良好的辨识效果。
2.根据面向控制的辨识理论,推导出一种基于小波分解的线性时变系统在线分频辨识算法,对于工作频段时变的控制系统(如随动系统)可在线调整各子频段的辨识模型加权系数,以期模型在当前工作频段附近尽可能接近真实系统。
3.小波逼近在非线性动态系统辨识中的应用。一是利用小波时频局部化的特点,提出一种尺度因子可调的自适应小波核最小二乘支持向量机,并应用于非线性系统的黑箱建模。二是推导出了等距分布节点的三次B样条函数公式,以此为尺度函数结合B样条高通滤波器系数构造了三次B样条小波,并以B样条小波为基函数,逼近Hammerstein模型中的非线性环节。
4.基于小波逼近的控制策略研究。一是采用离散仿射小波网络逼近一类非匹配不确定性非线性系统,并结合Popov超稳定性理论设计控制参数的自适应律,克服了普通神经网络固有的训练算法计算量大,容易陷入局部极小,收敛性难以保证等缺陷。二是提出以小波函数为基函数应用于预测函数控制,由小波的尺度伸缩特性,根据预测时域内逼近精度要求的高低调整基函数的尺度因子,既保证了预测时域整体优化目标和逼近精度要求,又尽可能减少了基函数个数,实现了优化变量的集结,同时改善了系统的动态特性,增强了系统抑制外部干扰的能力。
5.在正交小波(尺度)分解和重构算法的基础上,得到等效于小波分解和重构
The perfect mathematical characteristics of wavelets and wavelets transform are winning more and more favor from control scientists and engineering personnel. In 1991, Astrom---a Sweden control scientist advanced that wavelets approximation would become the latest researching method in system modeling, identification and control. During the later period over one decade, following with the development of wavelet theory, wavelet has been broadly applied in system identification and control, and partially resolved some problems in system identification and control science. On the base of predecessors's achievements and my research work, some algorithms about the application of wavelets theory in system modeling and control are presented in this paper , the main contents as follows:1. The paper strictly proved the theorem that variance of white noise would decrease to 50% through orthogonal scale (wavelet) decomposition and reconstruction, therefore identification data would be decomposed and reconstructed through orthogonal scaling transform in order to filter noises of data. According to the theorem, non-orthogonal wavelets is taken as basis function of impulse response process and noises is removed through multi-scale decomposition, and good identification result is achieved.2. According to the theory of control-relevant identification, a band-wise identification method online to the linear time-varying system based on wavelets decomposition is presented. According to the method, weighting coefficients of sub-band decomposed by wavelets can be adjusted online if the time-varying system's work band is changed, so the time-varying control request can be implemented by weighting coefficients. The method is fit to some system especially such as servo system.3. The application of wavelets approximation to nonlinear dynamic system. The first is that an adaptive least square wavelets support vector machine whose scale parameter is adjusted and its application to nonlinear system identification. The second is that 3-order B-spline function formula of equidistant points is deduced, and 3-order B-spline wavelet is constructed via B-spline High-Pass filter, then B-spline wavelet is taken as basis function and applied to approximate to the nonlinear part of a Hammerstein model.
4. Some research of control strategy based on wavelet approximation. The first is that discrete affine wavelet network is applied to approximation of nonlinear system with mismatched uncertainties, and adaptive control law is designed according to the theory of hyperstability, so the drawbacks of BP network such as large computation online, local minimum aiid uncertain convergency are overcomed. The second is that a predictive functional control using wavelets as basis function is presented. Because wavelets has the properties of compact support and multi-scale analysis, the number of wavelets functions and their location are set flexibly according to the system's controlling precision and the different requirements of approximation precision at different intervals of the horizon. So the target function of the whole horizon are optimized and some important points are emphasized, meanwhile the optimal parameters have been aggregated. 5. On the base of orthogonal wavelet (scaling function) decomposition and reconstruction algorithm, the paper presents the equivalent wavelet (scaling function) FIR filter, and provides two theorems about filter coefficients. The filters are applied to solve the next two problems: The first, aiming at that the relay is sensitive to noise and extracting characteristic parameters from output values polluted by noises is hard, applying equivalent wavelet filter to eliminate noise online from feedback signal, and denoising of output wave-forms offline using wavelet thresholding to extract characteristic parameters easily . The second, in order to eliminate measurement noises in system's outputs for dynamical matrix control (DMC), output measurements are filtered by wavelets denoising online, and the reduction coefficient of reference locus in DMC is automatically adjusted according to wavelets correlation values, so system's anti-disturb (such as load disturb) ability is enhanced.6. Based on operation matrix via orthogonal wavelets, two simplified control algorithm based on orthogonal wavelets transform are presented: The first, the integral operation matrix of Harr wavelets is applied to approximation of hierarchical control of linear large scale systems, the differential equations are converted into a set of linear algebraic equations, so complicated calculous operation has been transformed to simple matrix operation, and computation has been reduced effectively. The second, an iterative learning control algorithm based on Harr wavelets is presented to address the terminal control
problem of a linear time-varying systems. By employing the orthogonality and boundary values of Harr wavelets, the differential equations are converted into a set of linear algebraic equations, and the control problem is simplified to finding Harr wavelets coefficients of control variables, so the method avoids computing the state transfer matrix, and the coefficients are determined iteratively by steepest descent learning algorithm.
引文
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