卡爾曼濾波器（Kalman Filter, KF）及其衍生演算法，如擴增卡爾曼濾波器（Extended Kalman Filter, EKF）與無跡卡爾曼濾波器（Unscented Kalman Filter, UKF），已廣泛應用於狀態估測、信號追蹤與控制系統等。然而，這些方法均依賴於系統狀態與觀測輸出的緊密相關性，當模型是不可偵測系統（undetectable）時，其估測性能將顯著下降。為了克服此限制，近年來結合神經網路與卡爾曼濾波之架構逐漸受到重視，其中神經網路擴增卡爾曼濾波器（Neural Network Extended Kalman Filter, NNEKF）基於自適性動態修正的能力,成為一個重要的發展方式。
本論文探討一個人造的非線性動態系，其模型描述如下:
rac{operatorname{dx}}{dt}= rac{operatorname{d}}{dt}ξ1ξ2=0-ξ13t+ ϖ1tϖ2t,
y=[0, 1]x(t)=ξ2(t),
其中 omega= ϖ1ϖ2 分別為狀態向量x=ξ1ξ2的相關擾動雜訊。
這個人造系統具有以下特性：
(1) 固系統之雅可比矩陣(Jacobian Matrix)具奇異性，非線性模型無法在局部通過線性系統逼近
(2) 退化之演化函數與觀測輸出同時僅描繪狀態之局部部分量，致使系統不可觀測。
本論文著重於各式卡爾曼濾波器得數位實現的比較。通過比較，可確定神經網路擴增卡爾曼濾波器的優越性。

Kalman filtering (KF) and its extended algorithms, e.g., Extended Kalman filter (EKF) and the Unscented Kalman Filter (UKF), etc., are ubiquitously applied to signal tracking, system state estimation, optimal control problems. However, the efficiency of these proposed algorithms is highly related to the detectability of the target system. Whenever the target system is undetectability, the performance of these Kalman-like filters is obviously reduced. To overcome the drawback causing from undetectability, recently, a scheme based on the combination of neural networks, neural network extended Kalman filter (NNEKF) is developed.
In the thesis, an artificial nonlinear system
rac{operatorname{dx}}{dt}= rac{operatorname{d}}{dt}ξ1ξ2=0-ξ13t+ ϖ1tϖ2t,
y=[0, 1]x(t)=ξ2(t),
is investigated, where ϖ= ϖ1ϖ2 is the noise perturbation of the state x =ξ1,ξ2 which is usually related to a normal distribution with zero mean, i.e., ϖ~N(0,omega).
The target system has the following properties :
(1) The Jacobian is singular, a local linearized approximation of the target system may not exist.
(2) The undetectability of the target system implies that the true state cannot be tracked by using the estimated state of Kalman-like filter. Finally, based on numerical implement on several variant examples, the efficiency is realized.

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