一类非线性系统的参数辨识方法研究
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摘要
受到硬件设备的限制和环境因素的影响，非线性系统广泛存在于过程控制领域。非线性系统的参数辨识是设计控制器的基础。然而，非线性系统结构复杂、存在耦合特性、参数化后维数增大，传统的辨识方法无法应用于非线性系统。因此研究非线性系统的参数辨识具有普遍意义和实用价值。本论文借助辅助模型辨识思想、关键项分离技术以及维尔斯特拉斯逼近理论等技术研究非线性系统的辨识问题，并通过数值仿真验证了方法的性能。本文的具体工作如下。
(1)针对具有已知基的标量非线性系统，借助辅助模型辨识思想以及多新息辨识理论，提出非线性系统基于辅助模型的多新息广义增广随机梯度算法。该方法利用辅助模型输出代替模型中未知中间变量，利用多新息理论提高参数收敛精度。进一步，将提出的方法推广应用到具有已知基的多变量非线性系统中。
(2)针对具有预负载特性的非线性系统，利用关键项分离技术，推导了非线性系统的随机梯度算法，并研究了基于多项式转换技术的双率系统随机梯度算法以及基于损失数据估计的双率系统随机梯度算法。研究表明与基于多项式转换技术的随机梯度算法相比，所提出的损失数据估计随机梯度算法待辨识参数维数少，且能直接辨识出系统参数。
(3)针对结构已知的非线性系统，利用关键项分离技术，给出非线性系统的梯度迭代算法。与在线辨识算法相比，梯度迭代算法对信息向量中含有未知项的模型具有较好的辨识效果。进一步针对结构未知的非线性系统，利用维尔斯特拉斯逼近理论，推导出带遗忘因子的随机梯度算法以及修正的随机梯度算法。
(4)针对一种非线性隶属度函数，给出了基于模型转换的随机梯度算法和梯度迭代算法，由于使用了模型转换技术，所以这两种方法能很好的提高参数辨识效率。进一步，针对维尔斯特拉斯理论中多项式逼近函数，推导出修正的随机梯度算法和带遗忘因子的随机梯度算法，这两种方法都具有较快的参数收敛速度和较小的计算量。
Due to hardware limitations and environmental impacts, nonlinear systems are wide-ly found in process control feld. Nonlinear system identifcation constitutes a crucial partin control designs if the parameters of the nonlinear systems are unknown. Since non-linear systems have complex structures, coupling characteristics and large dimension ofunknown parameters, the traditional identifcation methods for linear systems cannot beapplied for nonlinear systems directly. Therefore, the study of identifcation methods fornonlinear systems is of universal signifcance and has wide application prospects. By usingthe auxiliary model, the key term separation principle and the Weierstrass approxima-tion theorem, this dissertation aims to develop new identifcation methods for nonlinearsystems, and the performances of the methods are illustrated by computer simulations.The main contributions are summarized as follows.
(1) For the nonlinear systems with polynomial nonlinearity, an auxiliary model basedgeneralized extended stochastic gradient (SG) algorithm is developed by using theauxiliary model and the multi-innovation theory. The outputs of the auxiliary modelcan replace the unknown inner variables and the multi-innovation method can im-prove the accuracy of parameter estimation. Furthermore, this method is extendedto identify multivariable nonlinear systems.
(2) Based on the key term separation principle, an SG algorithm is derived for nonlin-ear systems with preload nonlinearity, and a polynomial transformation techniquebased SG algorithm and a missing output estimation model based SG algorithm areproposed for dual-rate nonlinear systems with preload nonlinearity. Compared withthe polynomial transformation technique based SG algorithm, the missing outputestimation model based SG algorithm can directly estimate the parameters of thedual-rate systems and not increase the number of the unknown parameters.
(3) For the nonlinear systems with known hard nonlinearities, the key term separationprinciple based GI algorithm is derived. Compared with the on-line algorithms, theGI algorithm is efective in dealing with nonlinear systems which contain unknowninner variables. Furthermore, based on the Weierstrass approximation theorem, amodifed SG (M-SG) algorithm and a forgetting factor SG (FF-SG) algorithm areproposed for nonlinear systems with unknown hard nonlinearities.
(4) For a nonlinear membership function, two model transformation based estimationmethods are proposed. The proposed methods can greatly save the computational cost by using the model transformation technique. Furthermore, an FF-SG algorith-m and an M-SG algorithm are proposed for the polynomial function of the Weier-strass approximation theorem, and the proposed methods have quick convergencerate and low computational load.
(1)针对具有已知基的标量非线性系统，借助辅助模型辨识思想以及多新息辨识理论，提出非线性系统基于辅助模型的多新息广义增广随机梯度算法。该方法利用辅助模型输出代替模型中未知中间变量，利用多新息理论提高参数收敛精度。进一步，将提出的方法推广应用到具有已知基的多变量非线性系统中。
(2)针对具有预负载特性的非线性系统，利用关键项分离技术，推导了非线性系统的随机梯度算法，并研究了基于多项式转换技术的双率系统随机梯度算法以及基于损失数据估计的双率系统随机梯度算法。研究表明与基于多项式转换技术的随机梯度算法相比，所提出的损失数据估计随机梯度算法待辨识参数维数少，且能直接辨识出系统参数。
(3)针对结构已知的非线性系统，利用关键项分离技术，给出非线性系统的梯度迭代算法。与在线辨识算法相比，梯度迭代算法对信息向量中含有未知项的模型具有较好的辨识效果。进一步针对结构未知的非线性系统，利用维尔斯特拉斯逼近理论，推导出带遗忘因子的随机梯度算法以及修正的随机梯度算法。
(4)针对一种非线性隶属度函数，给出了基于模型转换的随机梯度算法和梯度迭代算法，由于使用了模型转换技术，所以这两种方法能很好的提高参数辨识效率。进一步，针对维尔斯特拉斯理论中多项式逼近函数，推导出修正的随机梯度算法和带遗忘因子的随机梯度算法，这两种方法都具有较快的参数收敛速度和较小的计算量。
Due to hardware limitations and environmental impacts, nonlinear systems are wide-ly found in process control feld. Nonlinear system identifcation constitutes a crucial partin control designs if the parameters of the nonlinear systems are unknown. Since non-linear systems have complex structures, coupling characteristics and large dimension ofunknown parameters, the traditional identifcation methods for linear systems cannot beapplied for nonlinear systems directly. Therefore, the study of identifcation methods fornonlinear systems is of universal signifcance and has wide application prospects. By usingthe auxiliary model, the key term separation principle and the Weierstrass approxima-tion theorem, this dissertation aims to develop new identifcation methods for nonlinearsystems, and the performances of the methods are illustrated by computer simulations.The main contributions are summarized as follows.
(1) For the nonlinear systems with polynomial nonlinearity, an auxiliary model basedgeneralized extended stochastic gradient (SG) algorithm is developed by using theauxiliary model and the multi-innovation theory. The outputs of the auxiliary modelcan replace the unknown inner variables and the multi-innovation method can im-prove the accuracy of parameter estimation. Furthermore, this method is extendedto identify multivariable nonlinear systems.
(2) Based on the key term separation principle, an SG algorithm is derived for nonlin-ear systems with preload nonlinearity, and a polynomial transformation techniquebased SG algorithm and a missing output estimation model based SG algorithm areproposed for dual-rate nonlinear systems with preload nonlinearity. Compared withthe polynomial transformation technique based SG algorithm, the missing outputestimation model based SG algorithm can directly estimate the parameters of thedual-rate systems and not increase the number of the unknown parameters.
(3) For the nonlinear systems with known hard nonlinearities, the key term separationprinciple based GI algorithm is derived. Compared with the on-line algorithms, theGI algorithm is efective in dealing with nonlinear systems which contain unknowninner variables. Furthermore, based on the Weierstrass approximation theorem, amodifed SG (M-SG) algorithm and a forgetting factor SG (FF-SG) algorithm areproposed for nonlinear systems with unknown hard nonlinearities.
(4) For a nonlinear membership function, two model transformation based estimationmethods are proposed. The proposed methods can greatly save the computational cost by using the model transformation technique. Furthermore, an FF-SG algorith-m and an M-SG algorithm are proposed for the polynomial function of the Weier-strass approximation theorem, and the proposed methods have quick convergencerate and low computational load.
引文
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