基于高阶统计理论的线性与非线性系统辨识的研究
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摘要
应用高阶统计理论和人工神经网络理论,本文对线性离散系统ARMA模型及非线性离散系统二阶Volterra模型的系统辨识方法进行了较深入的探讨,获得了一些理论成果。
本文首先提出了一种基于模型输入输出三阶累积量对非高斯非最小相位ARMA模型进行辨识的最小平方递推算法-CRLS算法。该算法利用ARMA模型输出信号的三阶累积量和输入输出信号的三阶互累积量构造代价函数,在三阶累积量域实现对AR及MA子模型的RLS(Recursive Least Squares)辨识。该算法假设AR及MA子模型的阶p,q事先已知,在估计MA子模型的参数时,需要计算残留时间序列(RTS)。由于在三阶累积量域中高斯随机信号能被完全消除,当所测得的模型输出信号染有高斯噪声时,该算法因此仍能获得ARMA模型参数的一致估计值。理论和仿真结果均表明,该算法具有良好的收敛性和准确性。
当ARMA模型的输入输出信号同时染有高斯噪声时,上述CRLS算法便不再适用。对于一类更一般ARMA模型的辨识,模型的阶p,q及相应的参数均完全未知。本文于是提出了一种基于模型含噪声的输入输出信号的三阶累积量对非高斯非最小相位ARMA模型的阶p,q及其相应参数完全辨识的递推算法。该算法按照模型的阶次递推,当递推阶次等于或大于AR子模型的阶p时,输出信号的平方误差达到最小,此时即获得ARMA模型的阶以及相应参数的估计值。理论和仿真结果均表明,该算法具有良好的收敛性。
在系统辨识的实际工程应用中,很多时候仅能测得模型的输出信号,而模型的输入信号是不可测的。这时只能利用所测得的输出信号对系统的阶次和参数进行辨识,即所谓盲辨识(Blind Identification)。本文首先提出了一种基于模型输出信号三阶累积量对非高斯非最小相位ARMA模型进行辨识的算法。该算法分别按照AR及MA子模型的阶次p,q递推,AR部分的阶次和参数辨识仅仅基于所测输出信号的三阶累积量,因而称之为盲辨识;而MA部分的阶次和参数辨识不仅与输出信号的三阶累积量有关,而且与输出信号与假设的系统输入信号的三阶互累积量相关,因而整个ARMA模型的辨识称之为半盲辨识(SemiblindIdentification)。在MA子模型参数估计中,由于没有直接计算残留时间系列(RTS),只是通过AR的参数估计值与所测输出信号的三阶累积量来计算RTS的三阶累积量,从而节省了存储空间,减小了计算复杂程度,同时也减少了AR参数估计误差对MA参数估计的影响。该算法是从一阶AR和MA子模型开始递推,逐步增加递推阶次而达到逼近所观测输出信号的目的。理论分析表明,当递推阶次增加到等于或大于其相应真实阶次时,算法中所构造的代价函数达到最小并保持稳定。因此,可以根据代价函数特征曲线来估计AR及MA的阶次和相应的参数。由于该算法基于所测输出信号的三阶累积量,即使输出信号染有高斯噪声或其它具有对称分布的非高斯噪声,该算法也可完全消除这些噪声对系统辨识的影响。理论和仿真结果表明,该算法具有良好的一致收敛性。
鉴于上述MA子模型的非盲辨识特性,本文又提出了一种完全基于模型输出信号三阶累积量的MA子模型的盲辨识算法。类似上述AR子模型的盲辨识算法,通过依MA阶次递推来估计出子模型的阶次和参数。从而整个ARMA模型的辨识成为完全的盲辨识。
由于非线性离散Volterra系统的广泛应用,本文对二阶Volterra模型的盲辨识机理进行了较深入的研究。
首先,本文提出了一种基于自相关分析和受约束前馈神经网络理论的对二阶Volterra模型进行盲辨识的算法。假设该模型由不可测的独立同分布(i.i.d.)随机信号驱动。利用所测得的模型输出信号,构造一个三层前馈神经网络,其激励函数为恒定值,神经元的连接权为需估计的二阶Volterra模型的核,即模型的参数集。当网络输入一组与所辨识的Volterra模型阶次相关的伪二进制信号时,网络的输出即为该模型输出信号的自相关函数。不断训练该网络至权收敛,便可获得二阶Volterra模型的相关参数。
本文将上述ARMA线性模型与二阶Volterra非线性模型的盲辨识算法初步应用到实际火车振动信号的建模中。
在此基础上,本文对在三阶累积量域中二阶Volterra模型的可盲辨识性作了进一步的研究。通过理论推导和计算机数字仿真,初步揭示了模型输出信号的三阶累积量与模型参数之间的非线性解析关系,为进一步探讨三阶或更高阶Volterra模型的可盲辨识性提供了理论基础。本文最后将基于三阶累积量的前馈神经网络推广到一般意义的非线性系统辨识。神经网络的输入空间由系统的输入输出过去观测值构成,输人神经元个数直接与所辨识系统的阶次相关,即输入神经元个数至少为系统阶次的两倍。因而可以通过检测网络的辨识误差特征曲线的变化来确定系统的阶次。由于网络训练的代价函数引入了三阶累积量,因而可消除系统输出信号中的高斯噪声或其它具有对称分布的非高斯噪声,从而增强神经网络的稳定性。仿真结果表明该算法具有较强的可辨识能力。
This dissertation considers the identification of linear ARMA models as well as truncated nonlinear Volterra systems- quadratic nonlinear systems based on higher-order statistics and artificial neural networks. Very promising results are drawn as follows.
Firstly, third-order cumulant RLS algorithm for nonGaussian nonminimum phase ARMA models (CRLS) is suggested in this thesis. A cost function based on the third-order cumulants and cross-cumulants is defined for the development of the CRLS identification algorithm. While dealing with ARMA models, the orders of AR and MA sub-models p, q are assumed known a priori. The construction of a residual time series (RLS) is required for the MA parameters estimation. Due to the third-order cumulant properties, the CRLS algorithm can suppress Gaussian noise and is capable of providing an consistent estimate under a noisy environment. Theoretical analyses and numerical simulations show that the proposed approach has strong consistency and convergency.
When both inputs and outputs of ARMA models are contaminated by additive Gaussian noises of unknown power spectral density, the above CRLS method is not suitable any more. A novel order-recursive methodology for identifying ARMA models is then introduced herein. The major objective of the proposed algorithm, in which a priori knowledge of the model orders are not required, is to determine the model orders together with the corresponding parameters simultaneously. It is performed order-recursively until the updated order is equal to or greater than the true AR submodel's order, where the norm of output error squares (NES) reaches its minimum. Both ARMA orders and parameters are then estimated correctly. Numerical simulations testify the convergency of the proposed approach.
In many practical cases, however, the inputs of the identified systems are not measurable and the identification must be performed blindly, that simply based on some statistical properties of the output signals. This thesis develops a novel identification methodology for nonminimum phase ARMA models of which the models' orders are not given. It is based on the third-order statistics of given noisy output observations and assumed input random sequences. Semiblind identification approach is thereby named. At each updated order the MA parameters are estimated without computing the residual time series with the result of decreasing the computational complexity and memory consumption. Effects of the AR estimation error on the MA parameters estimation are reduced also. The procedure begins with a first-order AR and MA submodel and updates the model order with an increment of one. At each test order, the corresponding parameters are estimated by using the least squares algorithm to minimize the well-defined cost function. When the test order reaches its true one, the magnitude of the cost function is minimum and may hold stable even when the test order further increases. we can conclude that the estimates of ARMA orders are just at the last turning point of the cost function characteristic curves, where the corresponding parameters can be estimated. Theoretical statements and simulation results illustrate that the method provides accurate estimates of unknown ARMA models despite the output measurements are corrupted by arbitrary Gaussian noises of unknown p.d.f..
In view of unblind feature of the above MA submodels identification, a blind identification approach is then presented in this study. As per the above AR scheme, the developed method performs order-recursively to estimate MA orders and parameters.
Because of wide application of nonlinear Volterra systems, the blind identifiability of quadratic nonlinear systems is considered in this thesis.
Blind identification of quadratic nonlinear systems based on constrained neural networks and autocorrelations of the system outputs is addressed firstly. The driven sequence of the system is assumed an unobservable i.i.d. random variable. Different from the conventional neural network, the proposed network is of a three-layer constrained feedforward topology, in which activation functions hold constant and weights of summing junctions represent the system kernels. The inputs of the network consist of pseudo-random binary signals which are generated in terms of the monitored system's order. Conventional back-propagation policy is used to train the constrained network such that the outputs of the network arrive at the desired second-order statistics of the system output when the weights equal their corresponding exact kernels. Very promising results on identifying nonlinear systems are obtained and discussed through numerical simulations.
The blind identification methods for ARMA models and quadratic nonlinear systems mentioned above are also utilized to model the real vibration signals measured from a running railway carriage.
Based on the above development, blind identifiability of quadratic nonlinear systems in higher-order statistics domain is presented in this study. Some nonlinear analytic relations between the third-order cumulants of the outputs and the quadratic kernels are characterized through theoretical statements and simulations. It provides a useful starting point for implementation of truncated Volterra nonlinear system identification using conventional techniques or neural networks methodologies.
Lastly, an input-output model based on feedforward neural networks for a generic nonlinear dynamic system identification is considered. In the developed neural networks based nonlinear system identification model, the input space of the network consists of input-output past observations of the identified system and the size of the input space is therefore directly related to the system order, i.e., the number of the input space is at least double the system order. The neural network uses a higher-order cumulants based cost function together with an effective training algorithm, the cumulant-based Weights Decoupled Extended Kalman Filter (CWDEKF) strategy. By monitoring the identification error characteristic curve, the system order and subsequently an appropriate network structure for systems identification can be determined. The obtained results are promising which indicate that generic nonlinear systems can be identified by the proposed neural network models.
本文首先提出了一种基于模型输入输出三阶累积量对非高斯非最小相位ARMA模型进行辨识的最小平方递推算法-CRLS算法。该算法利用ARMA模型输出信号的三阶累积量和输入输出信号的三阶互累积量构造代价函数,在三阶累积量域实现对AR及MA子模型的RLS(Recursive Least Squares)辨识。该算法假设AR及MA子模型的阶p,q事先已知,在估计MA子模型的参数时,需要计算残留时间序列(RTS)。由于在三阶累积量域中高斯随机信号能被完全消除,当所测得的模型输出信号染有高斯噪声时,该算法因此仍能获得ARMA模型参数的一致估计值。理论和仿真结果均表明,该算法具有良好的收敛性和准确性。
当ARMA模型的输入输出信号同时染有高斯噪声时,上述CRLS算法便不再适用。对于一类更一般ARMA模型的辨识,模型的阶p,q及相应的参数均完全未知。本文于是提出了一种基于模型含噪声的输入输出信号的三阶累积量对非高斯非最小相位ARMA模型的阶p,q及其相应参数完全辨识的递推算法。该算法按照模型的阶次递推,当递推阶次等于或大于AR子模型的阶p时,输出信号的平方误差达到最小,此时即获得ARMA模型的阶以及相应参数的估计值。理论和仿真结果均表明,该算法具有良好的收敛性。
在系统辨识的实际工程应用中,很多时候仅能测得模型的输出信号,而模型的输入信号是不可测的。这时只能利用所测得的输出信号对系统的阶次和参数进行辨识,即所谓盲辨识(Blind Identification)。本文首先提出了一种基于模型输出信号三阶累积量对非高斯非最小相位ARMA模型进行辨识的算法。该算法分别按照AR及MA子模型的阶次p,q递推,AR部分的阶次和参数辨识仅仅基于所测输出信号的三阶累积量,因而称之为盲辨识;而MA部分的阶次和参数辨识不仅与输出信号的三阶累积量有关,而且与输出信号与假设的系统输入信号的三阶互累积量相关,因而整个ARMA模型的辨识称之为半盲辨识(SemiblindIdentification)。在MA子模型参数估计中,由于没有直接计算残留时间系列(RTS),只是通过AR的参数估计值与所测输出信号的三阶累积量来计算RTS的三阶累积量,从而节省了存储空间,减小了计算复杂程度,同时也减少了AR参数估计误差对MA参数估计的影响。该算法是从一阶AR和MA子模型开始递推,逐步增加递推阶次而达到逼近所观测输出信号的目的。理论分析表明,当递推阶次增加到等于或大于其相应真实阶次时,算法中所构造的代价函数达到最小并保持稳定。因此,可以根据代价函数特征曲线来估计AR及MA的阶次和相应的参数。由于该算法基于所测输出信号的三阶累积量,即使输出信号染有高斯噪声或其它具有对称分布的非高斯噪声,该算法也可完全消除这些噪声对系统辨识的影响。理论和仿真结果表明,该算法具有良好的一致收敛性。
鉴于上述MA子模型的非盲辨识特性,本文又提出了一种完全基于模型输出信号三阶累积量的MA子模型的盲辨识算法。类似上述AR子模型的盲辨识算法,通过依MA阶次递推来估计出子模型的阶次和参数。从而整个ARMA模型的辨识成为完全的盲辨识。
由于非线性离散Volterra系统的广泛应用,本文对二阶Volterra模型的盲辨识机理进行了较深入的研究。
首先,本文提出了一种基于自相关分析和受约束前馈神经网络理论的对二阶Volterra模型进行盲辨识的算法。假设该模型由不可测的独立同分布(i.i.d.)随机信号驱动。利用所测得的模型输出信号,构造一个三层前馈神经网络,其激励函数为恒定值,神经元的连接权为需估计的二阶Volterra模型的核,即模型的参数集。当网络输入一组与所辨识的Volterra模型阶次相关的伪二进制信号时,网络的输出即为该模型输出信号的自相关函数。不断训练该网络至权收敛,便可获得二阶Volterra模型的相关参数。
本文将上述ARMA线性模型与二阶Volterra非线性模型的盲辨识算法初步应用到实际火车振动信号的建模中。
在此基础上,本文对在三阶累积量域中二阶Volterra模型的可盲辨识性作了进一步的研究。通过理论推导和计算机数字仿真,初步揭示了模型输出信号的三阶累积量与模型参数之间的非线性解析关系,为进一步探讨三阶或更高阶Volterra模型的可盲辨识性提供了理论基础。本文最后将基于三阶累积量的前馈神经网络推广到一般意义的非线性系统辨识。神经网络的输入空间由系统的输入输出过去观测值构成,输人神经元个数直接与所辨识系统的阶次相关,即输入神经元个数至少为系统阶次的两倍。因而可以通过检测网络的辨识误差特征曲线的变化来确定系统的阶次。由于网络训练的代价函数引入了三阶累积量,因而可消除系统输出信号中的高斯噪声或其它具有对称分布的非高斯噪声,从而增强神经网络的稳定性。仿真结果表明该算法具有较强的可辨识能力。
This dissertation considers the identification of linear ARMA models as well as truncated nonlinear Volterra systems- quadratic nonlinear systems based on higher-order statistics and artificial neural networks. Very promising results are drawn as follows.
Firstly, third-order cumulant RLS algorithm for nonGaussian nonminimum phase ARMA models (CRLS) is suggested in this thesis. A cost function based on the third-order cumulants and cross-cumulants is defined for the development of the CRLS identification algorithm. While dealing with ARMA models, the orders of AR and MA sub-models p, q are assumed known a priori. The construction of a residual time series (RLS) is required for the MA parameters estimation. Due to the third-order cumulant properties, the CRLS algorithm can suppress Gaussian noise and is capable of providing an consistent estimate under a noisy environment. Theoretical analyses and numerical simulations show that the proposed approach has strong consistency and convergency.
When both inputs and outputs of ARMA models are contaminated by additive Gaussian noises of unknown power spectral density, the above CRLS method is not suitable any more. A novel order-recursive methodology for identifying ARMA models is then introduced herein. The major objective of the proposed algorithm, in which a priori knowledge of the model orders are not required, is to determine the model orders together with the corresponding parameters simultaneously. It is performed order-recursively until the updated order is equal to or greater than the true AR submodel's order, where the norm of output error squares (NES) reaches its minimum. Both ARMA orders and parameters are then estimated correctly. Numerical simulations testify the convergency of the proposed approach.
In many practical cases, however, the inputs of the identified systems are not measurable and the identification must be performed blindly, that simply based on some statistical properties of the output signals. This thesis develops a novel identification methodology for nonminimum phase ARMA models of which the models' orders are not given. It is based on the third-order statistics of given noisy output observations and assumed input random sequences. Semiblind identification approach is thereby named. At each updated order the MA parameters are estimated without computing the residual time series with the result of decreasing the computational complexity and memory consumption. Effects of the AR estimation error on the MA parameters estimation are reduced also. The procedure begins with a first-order AR and MA submodel and updates the model order with an increment of one. At each test order, the corresponding parameters are estimated by using the least squares algorithm to minimize the well-defined cost function. When the test order reaches its true one, the magnitude of the cost function is minimum and may hold stable even when the test order further increases. we can conclude that the estimates of ARMA orders are just at the last turning point of the cost function characteristic curves, where the corresponding parameters can be estimated. Theoretical statements and simulation results illustrate that the method provides accurate estimates of unknown ARMA models despite the output measurements are corrupted by arbitrary Gaussian noises of unknown p.d.f..
In view of unblind feature of the above MA submodels identification, a blind identification approach is then presented in this study. As per the above AR scheme, the developed method performs order-recursively to estimate MA orders and parameters.
Because of wide application of nonlinear Volterra systems, the blind identifiability of quadratic nonlinear systems is considered in this thesis.
Blind identification of quadratic nonlinear systems based on constrained neural networks and autocorrelations of the system outputs is addressed firstly. The driven sequence of the system is assumed an unobservable i.i.d. random variable. Different from the conventional neural network, the proposed network is of a three-layer constrained feedforward topology, in which activation functions hold constant and weights of summing junctions represent the system kernels. The inputs of the network consist of pseudo-random binary signals which are generated in terms of the monitored system's order. Conventional back-propagation policy is used to train the constrained network such that the outputs of the network arrive at the desired second-order statistics of the system output when the weights equal their corresponding exact kernels. Very promising results on identifying nonlinear systems are obtained and discussed through numerical simulations.
The blind identification methods for ARMA models and quadratic nonlinear systems mentioned above are also utilized to model the real vibration signals measured from a running railway carriage.
Based on the above development, blind identifiability of quadratic nonlinear systems in higher-order statistics domain is presented in this study. Some nonlinear analytic relations between the third-order cumulants of the outputs and the quadratic kernels are characterized through theoretical statements and simulations. It provides a useful starting point for implementation of truncated Volterra nonlinear system identification using conventional techniques or neural networks methodologies.
Lastly, an input-output model based on feedforward neural networks for a generic nonlinear dynamic system identification is considered. In the developed neural networks based nonlinear system identification model, the input space of the network consists of input-output past observations of the identified system and the size of the input space is therefore directly related to the system order, i.e., the number of the input space is at least double the system order. The neural network uses a higher-order cumulants based cost function together with an effective training algorithm, the cumulant-based Weights Decoupled Extended Kalman Filter (CWDEKF) strategy. By monitoring the identification error characteristic curve, the system order and subsequently an appropriate network structure for systems identification can be determined. The obtained results are promising which indicate that generic nonlinear systems can be identified by the proposed neural network models.
引文
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