变量带误差动态系统的广义极大似然辨识方法

Towards the accurate modeling of dynamic systems from noisy input-output observations, the proposal is aimed to develop a generalized maximum likelihood estimator for dynamic errors-in-variables systems, which can handle arbitrary random inputs, colored disturbances and high-order dynamic systems. Regarding the challenge that the order of the dynamic errors-in-variables system is unknown, the nonparametric modeling strategy is originally proposed for dynamic errors-in-variables systems. The kernel modeling is exploited to describe the inherent information of the dynamic system and of the noise disturbance, based on which a regularized nonparametric method is developed for errors-in-variables modeling and identification. The generalized maximum likelihood identification method, which has been studied in the presence of white noises, is further extended for parametric errors-in-variables modeling, and the sufficient conditions are derived, which obtain the system identifiability. By virtue of combining the nonparametric and parametric modeling strategies, the identification process of dynamic errors-in-variables models is performed without user’s interaction. The proposed generalized maximum likelihood estimator is utilized to identify nonlinear dynamic systems from noisy input-output data. A linearized framework is formally defined, which allows to obtain the best linear approximation of the nonlinear system in a least squares sense. The ways of detecting and quantifying the induced nonlinear distortions are studied for random excitations, and efforts are placed on clarifying the influence of the nonlinear distortions on the properties of the proposed estimator, which is constructed using linear system theory. Finally, the proposed identification method is experimentally validated. By carrying out the proposed research, a big step will be made towards the errors-in-variables modeling and identification of industrial applications.

本课题围绕均含有噪声干扰的输入输出通道之间的精确建模问题，针对任意平稳随机输入、有色量测噪声、高阶动态系统等工程实际情形，发展一类变量带误差系统的广义极大似然辨识方法。针对变量带误差系统阶次未知的问题，提出变量带误差非参数建模的新思路，采用核函数理论刻画动态系统及量测噪声模型的先验信息，发展基于正则化的变量带误差非参数辨识新方法。进一步拓展前期建立的变量带误差参数模型的广义极大似然辨识理论，阐明系统满足可辨识性的充分条件，集成非参数辨识，实现基于非参数-参数建模的变量带误差模型辨识过程的自动化。延伸广义极大似然辨识方法至非线性系统情形，建立变量带误差非线性系统的最优线性近似方法，发展系统非线性扰动的探测及量化方法，阐明非线性扰动对广义极大似然参数估计量性质的影响，最后开展所提出辨识方法的实验评价研究。通过本课题的开展，将为变量带误差系统建模与辨识的工程应用奠定理论基础。

变量带误差系统辨识旨在从均含有干扰噪声的输入输出数据中建立真实输入与真实输出之间的关联。有色干扰噪声情形下的变量带误差建模符合实际工程情形和具有重要应用价值。然而，有色噪声模型的引入急剧增加变量带误差建模与辨识的难度。本项目围绕非参数辨识、参数辨识与方法应用三方面开展了研究。(1) 在参数辨识方面，建立了两类变量带误差系统的辨识方法：（i）针对有色噪声情形下，真实输入为非参数任意序列，采用参数化建模动态系统和输入输出干扰噪声的动态特性，阐明了可辨识性的充分条件，建立了变量带误差广义极大似然辨识方法，并证明了其在Godambe信息准则下的最优性；（ii）针对ARMAX系统模型和输入白噪声情形，采用有理模型参数化建模系统、真实输入和输出干扰噪声，通过多变量ARMA模型把变量带误差模型分解为两个子辨识问题，阐明了多变量ARMA模型可辨识的充分条件，建立了所有参数联合估计的极大似然辨识方法。（2）在非参数辨识方面，开展了特定输入信号（冲击激励、参考信号已知等）下的非参数辨识研究:（i）针对参考信号已知情形，建立了低阻尼多变量变量带误差系统的辨识方法，阻尼的估计精度高，用户友好；（ii）针对冲击激励，采用高斯过程刻画频率响应函数的稳定和光滑性质，建立了适用于数字图像相关测试的频率响应函数的非参数估计方法。（3）在方法应用方面，针对声学测试和信号处理，建立了高阶声学相对传递函数的极大似然辨识方法，应用于麦克风校准和声学定位；针对数字图像相关的光学模态测试，提出了频率响应函数的空间-频率模型，发展了特定位置上的频率响应函数高精度估计方法，应用于光学模态测试。通过本课题的实施，建立的有色噪声下变量带误差辨识新方法丰富了系统辨识理论，为声源定位、模态分析、故障诊断与自动控制等领域应用研究提供理论与方法支撑。

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