## 3 Aircraft Elastic Deformation Estimation and Inertial Network Algorithm Design
### Distributed Data Fusion Algorithm
#### 3.2.2 Distributed State Fusion Algorithm  
The local dynamic model on each node of the inertial network system can be described as  
\[ \begin{array}{l}x_{I}(t_{k})=\Phi_{I}(t_{k},t_{k-1})x_{I}(t_{k-1})+G_{I}(t_{ k-1})w_{I}(t_{k-1})\\ z_{I}(t_{k})=D_{I}(t_{k})x_{I}(t_{k})+v_{I}(t_{k})\end{array}  \tag{27}\]  
where \(J=1,2,3\cdots n\) represents each node; the local state vector \(x_{I}\) can be divided into local system state \(x_{1,J}\) and local sensor error state \(x_{2,J}\). \[x_{J}=\left[ \begin{array}{cc}x_{1J}^{\rm T}&x_{2J}^{\rm T}\end{array} \right] ^{\rm T} \tag{28}\]  
The local system states at the nodes are called similar states, and the transformation between these similar states is given by the dynamic transformation matrix. The measurement vector \(z_{I}\) represents all the observations (including the observations of the node) after receiving the inertial measurement information of other nodes, which can be decomposed into two sub-vectors. \[z_{J}=\left[ \begin{array}{cc}z_{JL}^{\rm T}&z_{JA}^{\rm T}\end{array} \right] ^{\rm T} \tag{29}\]  
where \(z_{JL}\) is the measurement vector provided by the sensor of this node, and \(z_{JA}\) is the combination of inertial measurements absorbed from other nodes. Since the two measurement vectors are independent of each other, the following decomposition can be obtained as follows:  
\[ \begin{array}{l}D_{I}=\left[\begin{array}{cc}D_{IL}^{\rm T}&D_{JA}^{\rm T }\end{array}\right]^{\rm T}\\ v_{I}=\left[\begin{array}{cc}v_{JL}^{\rm T}&v_{JA}^{\rm T}\end{array}\right] ^{\rm T}\\ R_{J}=\mbox{blockdiag}\big{[}\begin{array}{cc}R_{JL}&R_{JA}\end{array}\end{array}  \end{array} \tag{30}\]The local Kalman filter uses the assimilated sensor measurements to estimate the local state, and the local state fusion filter combines the local estimation with the estimation after assimilation of other nodes to update the local estimation. The Kalman filter algorithm is applied to the local dynamic model on each node, and the local state estimation \(\hat{x}\) and covariance matrix \(P_{J}\) can be obtained. \[ \begin{array}{l}P_{J}^{-1}\big{(}t_{k}^{+}\big{)}=P_{J}^{-1}\big{(}t_{k}^{-} \big{)}+D_{J}^{\mathsf{T}}(t_{k})R_{J}^{-1}(t_{k})D_{J}(t_{k})\\ =P_{J}^{-1}\big{(}t_{k}^{-}\big{)}+\sum\limits_{k=I,J}D_{I}^{\mathsf{T}}(t_{k} )R_{I}^{-1}(t_{k})D_{I}(t_{k})\\ P_{J}^{-1}\big{(}t_{k}^{+}\big{)}\hat{x}_{I}\big{(}t_{k}^{+}\big{)}=P_{J}^{-1} \big{(}t_{k}^{-}\big{)}\hat{x}_{I}\big{(}t_{k}^{-}\big{)}+D_{J}^{\mathsf{T}}(t _{k})R_{J}^{-1}(t_{k})z_{J}(t_{k})\\ =P_{J}^{-1}\big{(}t_{k}^{-}\big{)}\hat{x}_{I}\big{(}t_{k}^{-}\big{)}+\sum \limits_{l=I,J,J}D_{I}^{\mathsf{T}}(t_{k})R_{I}^{-1}(t_{k})z_{I}(t_{k})\end{array}  \tag{31}\]  
The mean square error matrix of state one-step prediction and state one-step prediction in Formula (31) is  
\[ \begin{array}{l}\hat{x}_{J}\big{(}t_{k}^{-}\big{)}=\Phi_{J}(t_{k},t_{k-1}) \overline{x}_{J}(t_{k-1})\\ P_{J}\big{(}t_{k}^{-}\big{)}=\Phi_{J}(t_{k},t_{k-1})\overline{P}_{J}(t_{k-1}^{ +})\Phi_{J}^{\mathsf{T}}(t_{k},t_{k-1})\\ +G_{J}(t_{k-1})Q_{I}(t_{k-1})G_{J}^{\mathsf{T}}(t_{k-1})\end{array}  \tag{32}\]  
where \(\overline{x}\) and \(\overline{P}\) are the local state updates obtained by the local state fusion filter. Therefore, it is only necessary to calculate the values of \(\overline{x}\) and \(\overline{P}\) at each moment, so that the filter can work normally and output the final result. The calculation method will be explained below. Where we introduce the nonlinear functional model. \[J=\int_{a}^{b}f^{2}(t)dt \tag{33}\]  
In the optimal control system, the above equation can be expressed as  
\[J=\int_{a}^{b}\big{(}x^{T}Px\big{)}dt \tag{34}\]  
After discretization of the Equation (34), it can be written as  
\[J=\sum x^{T}Px \tag{35}\]  
A state fusion filter is established on each node.