时变系统辨识方法及其收敛定理

Time-Varying System Identification Methods and Convergence Theorems

介绍了用于辨识方法性能研究的鞅收敛定理和鞅超收敛定理,阐述了其应用范围;讨论了研究辨识算法收敛性的各种激励条件;综述了时变随机系统的各种辨识方法,包括最小二乘类辨识方法(如遗忘因子最小二乘算法、卡尔曼滤波算法、有限数据窗最小二乘算法等)和随机梯度类辨识方法(如遗忘梯度算法、广义投影算法等);同时阐述了时变参数系统辨识领域的一些值得深入研究的课题;最后给出了遗忘梯度算法在不同条件下参数估计误差上界的几个定理,说明数据的平稳性可以改善参数估计精度.

In this paper, we introduce the martingale convergence theorem anu martingale hyperconvergence theorem for analyzing performances of identification methods and states their application ranges. The paper also discusses excitation conditions for studying the convergence of identification algorithms and surveys main identification approaches for time-varying systems, including least squares type algorithms （e. g. , the least squares algorithm with a forgetting factor, Kalman filtering algorithm, finite-data-widow least squares algorithm） and stochastic gradient type algorithms （e. g. , the forgetting gradient algorithm and generalized projection algorithm）. Furthermore, some further research topics for time-varying systems are presented. Finally, we give several convergence theorems of the forgetting gradient algorithm for time- varying systems by using the stochastic process theory and martingale hyperconvergence theorem and indicate that the data stationarity can improve the precision of the parameter estimates.