辫识时变随机系统遗忘因子算法的收敛性分析

文「１」给出了辨识时变随机系统遗忘因子算法的收敛性分子，本文指出了文「１」中的一些证明和结论上的错误，并给出了相应的改正和完善。     

多变量系统的辅助模型辨识方法的收敛性分析*

丁锋，谢新民[1]假设协方差阵P0-1（t）（s）的最大特征值与最小特征值之比（即条件数）有界，和系统噪声{w（t）}方差有界，证明了辅助模型辨识其法参数估计的一致收致性．本文将分别放松这两个条件，即ⅰ）条件数无界，ⅱ）系统噪声为非平稳噪声，且方差无界，探讨了辅助模型算法的收敛性．分析表明在这种较弱的假设下，辅助模型算法保持很强的鲁棒性，参数估计是一致收敛的．     

鞅超收敛定理与遗忘因子最小二乘算法的收敛法分析

本文扩展了用于分析时不变系统辨识算法收敛性的鞅收敛定理，建立了鞅超收敛定理。它可以作为工具来分析时变系统的各种辨识算法的收敛性，为地变系统收敛性和稳定性分析这一困难课题提供了新方法，开辟了新路。     

时变系统遗忘因子最小二乘法的有界收敛性

利用随机过程理论研究了遗忘因子最小二乘法 (FFLS)的有界收敛性, 给出了参数估计误差的上界. 分析表明: i)对于时不变确定性系统, FFLS算法产生的参数估计以指数速度收敛于真参数; ii)对于时不变随机系统, FFLS算法给出有界均方估计误差; iii)对于时变随机系统, FFLS算法可以跟踪时变参数, 且跟踪误差有界.     

时变系统遗忘因子最小二乘法的有界性收敛性

利用随机过程理论研究了遗忘因子最小二乘法（FFLS）的有界收敛性，给出了参数估计误差的上界，分析表明：i)对于时不变确定性系统；FFLS算法产生的参数估计以指数速度收敛于真参数；ii)对于时不变随机系统，FFLS算法给出界均方估计误差，iii)对于时变随机系统，FFLS算法可以跟踪时变参数，且跟踪误差有界。     

通用进化算法的收敛性分析

传统进化算法的收敛性专注于具体算法,对应的研究成果也仅仅适用于具体算法。为了研究所有进化算法的收敛性问题,提出了一种包含所有操作类型算子的通用进化算法,建立了一套概率空间用于研究算法的收敛性,所有有关算法的术语都用严格的数学语言加以定义。在概率空间中,有七个算法收敛性定理被完整地证明,其中之一找到了算法依概率收敛的充分必要条件。更为重要的是,这些定理适用所有进化算法。它建立了一个体系,用来指导进化算法的设计,从理论上判断进化算法的收敛性。     

时变系统最小均方算法的性能分析

在无过程数据平稳性假设和各态遍历等条件下,运用随机过程理论研究了最小方算法（LMS）的有界收敛性,给出了估计误差的上界,论述了LMS算法收敛因子或步长的选择方法,以使参数估计误差上界最小。这对于提高LMS算法的实际应用效果有着重要意义。LMS算法的收敛性分析表明：（1）对于确定性时不变系统,LMS算法是指数速度收敛的;（2）对于确定性时变系统,收敛因子等于1,LMS算法的参数估计误差上界最小;（3）对于时变或不变随机系统,LMS算法的参数估计误差一致有上界。     

时变多变量系统辨识的一种方法

本文提出时变多变量系统辨识的一种新方法，并分析了算法的收敛性和稳定性。仿真例子表明所提出算法是有效的。     

多变量输出误差系统的辅助模型随机梯度辨识算法

考虑了多变量输出误差系统的辨识问题. 使用系统可得到的输入输出数据构造一个辅助模型, 用辅助模型的输出代替信息向量中的未知变量, 提出了一个基于辅助模型的随机梯度辨识算法. 使用鞅收敛定理的收敛性分析表明: 提出的算法给出的参数估计收敛于它们的真值. 给出了带遗忘因子的辅助模型随机梯度算法来改进参数估计精度, 仿真结果证实了提出的结论.     

时变系统有限数据窗最小二乘辨识的有界收敛性

利用随机过程理论证明了有限数据窗最小二乘法的有界收敛性,给出了参数估计误差 上界的计算公式,阐述了获得最小均方参数估计误差上界时数据窗长度的选择方法.分析表明, 对于时不变随机系统,数据窗长度越大,均方参数估计误差上界越小;对于确定性时变系统,数 据窗长度越小,均方参数估计误差上界越小.因此,对于时变随机系统,一个折中方案是寻求一 个最佳数据窗长度,以使均方参数估计误差最小.该文的研究成果对于提高辨识算法的实际应 用效果有重要意义.     

一种快速对角回归神经网络控制算法

文[1]定理1给出了一个基于Lyapunov函数的三层对角回归神经网络(DRNN)任意权参数学习速率的自适应调整算法, 而推导各层权自适应学习速率时没有严格满足定理1成立的必要条件, 故没能找到各学习速率的准确范围. 依据文[1]定理1,精确给出了各权向量及权矩阵学习速率的调整算法, 结果表明DRNN应具有更大的学习速率, 对应更加快速的收敛算法. 给出了相应的仿真结果.     

Analysis of the Kalman filter based estimation algorithm: an orthogonal decomposition approach

In this paper we shall provide new analysis on some fundamental properties of the Kalman filter based parameter estimation algorithms using an orthogonal decomposition approach based on the excited subspace. A theoretical analytical framework is established based on the decomposition of the covariance matrix, which appears to be very useful and effective in the analysis of a parameter estimation algorithm with the existence of an unexcited subspace. The sufficient and necessary condition for the boundedness of the covariance matrix in the Kalman filter is established. The idea of directional tracking is proposed to develop a new class of algorithms to overcome the windup problem. Based on the orthogonal decomposition approach two kinds of directional tracking algorithms are proposed. These algorithms utilize a time-varying covariance matrix and can keep stable even in the case of unsufficient and/or unbounded excitation.     

广义离散随机线性系统的最优递推滤波方法(Ⅱ)

本文对文献[1]给出的广义离散随机线性系统最优估计误差协方差阵进行了分析,在一定 条件下得到了误差协方差阵的上界和下界,继而讨论了由文献[1]给出的滤波器的稳定性.     

State-saturated resilient filtering for nonlinear complex networks under event-triggering protocol

This article is concerned with the resilient state-saturated filtering issue for nonlinear complex networks via the event-triggering protocol. The nonlinear inner coupling is taken into account, thereby better reflecting the nature of the complex networks. A set of Bernoulli-distributed sequences are introduced to model the randomly occurring nonlinearities with a given probability. The signum function is utilized to characterize the state saturation owing to the physical limits on the system. For the purpose of energy saving, an event-triggering protocol is adopted to govern the regulation of the transmission. The objective of this article is to develop an event-triggering resilient filtering for nonlinear complex networks subject to state saturations as well as randomly occurring nonlinearities. By using matrix analysis techniques, we first guarantee the upper bound on the filtering error covariance by means of recursions and subsequently minimize such an upper bound by looking for the proper gain matrix relying on the solutions to two difference equations. Moreover, the performance evaluation of the designed filtering scheme is conducted by analyzing the boundedness of the estimation error in the mean square sense. Finally, an experimental example is exploited to validate the usefulness of the state-saturated resilient filtering algorithm.     

Optimal selection of the forgetting matrix into an iterative learning control algorithm

A recursive optimal algorithm, based on minimizing the input error covariance matrix, is derived to generate the optimal forgetting matrix and the learning gain matrix of a P-type iterative learning control (ILC) for linear discrete-time varying systems with arbitrary relative degree. This note shows that a forgetting matrix is neither needed for boundedness of trajectories nor for output tracking. In particular, it is shown that, in the presence of random disturbances, the optimal forgetting matrix is zero for all learning iterations. In addition, the resultant optimal learning gain guarantees boundedness of trajectories as well as uniform output tracking in presence of measurement noise for arbitrary relative degree.     

Quantized innovations Kalman filter: stability and modification with scaling quantization

The stability of quantized innovations Kalman filtering (QIKF) is analyzed. In the analysis, the correlation between quantization errors and measurement noises is considered. By taking the quantization errors as a random perturbation in the observation system, the QIKF for the original system is equivalent to a Kalman-like filtering for the equivalent state-observation system. Thus, the estimate error covariance matrix of QIKF can be more exactly analyzed. The boundedness of the estimate error covariance matrix of QIKF is obtained under some weak conditions. The design of the number of quantized levels is discussed to guarantee the stability of QIKF. To overcome the instability and divergence of QIKF when the number of quantization levels is small, we propose a Kalman filter using scaling quantized innovations. Numerical simulations show the validity of the theorems and algorithms.     

Comments on "Realization and reduction of Markovian models from nonstationary data"

In the above paper, Baram determines a basis in which to construct realizations from measurement data and for model reduction of these realizations. The basis selected by Baram is similar to the "cost-decoupled" basis introduced earlier [1], [2], with one exception. Both have diagonal state covariances, but the second matrix diagonalized is different. Of course, this makes the two algorithms different and the nature of these differences is pointed out in this correspondence. The principal difference is that Baram's algorithm for model reduction approximates (in a least squares sense) all of the covariance sequences, whereas the model reduction of [1], [2] matches the first two covariance sequences exactly. The cost decoupled basis guarantees several different properties: 1) output covariance matching, and 2) an equivalent quadratic performance metric (i.e., "cost-equivalent" realizations).     

噪声方差不确定约束系统的滚动时域估计

针对噪声方差不确定的约束系统,讨论了一种鲁棒滚动时域估计(MHE)方法.首先,根据噪声方差不确定模型,找到满足所有不确定性的最小方差上界,在线性矩阵不等式(LMI)框架下求解优化问题,得到近似到达代价的表达形式;然后再融合预测控制的滚动优化原理,把系统的硬约束直接表述在优化问题中,在线优化性能指标,估计出当前时刻系统的状态.仿真时与鲁棒卡尔曼滤波方法进行比较,结果表明了该方法的有效性.