## 2 Methodology
### Fault Signal Detector Design  
Figure 3 shows the block diagram of proposed faults signals detector based on federal Kalman filter and chi-square detector. Every Kalman filter used in our work is standard; we take the sub Kalman_Filter_1 for GPS signal channel as example. The state vector and measurement vector can be written as:  
\[\left\{ \begin{array}{l}X_{GPS}=\left[\Delta\dot{x},\Delta\dot{y},\Delta \varphi,\Delta\dot{\varphi},\Delta X,\Delta Y\right]\\ Y_{GPS}=\Delta\varphi\end{array} \right. \tag{24}\]  
The state space and measurement space equations of Kalman Filter from moment \(k-1\) to moment \(k\) can be written as:  
\[\left\{ \begin{array}{l}X_{GPS,k}=F_{GPS,k}X_{GPS,k-1}+w_{GPS,k}\\ Y_{GPS,k}=H_{GPS,k}X_{GPS,k}+v_{GPS,k}\end{array} \right. \tag{25}\]  
where \(F_{GPS,k}\) is the state transition matrix, \(w_{GPS,k}\) is the process noise, \(H_{GPS,k}\) is the measurement transition matrix, and \(v_{GPS,k}\) is the measurement noise. The equations of Kalman Filter can be written as:  
\[\left\{ \begin{array}{l}\hat{X}_{GPS,k|k-1}=F_{GPS,k}\hat{X}_{GPS,k-1|k-1}\\ P_{GPS,k|k-1}=F_{GPS,k}P_{GPS,k-1|k-1}F_{GPS,k}^{T}+Q_{GPS,k}\\ K_{GPS,k}=P_{GPS,k|k-1}H_{GPS,k}^{T}\Big{(}H_{GPS,k}P_{GPS,k|k-1}+R_{GPS,k}^{ T}\Big{)}^{-1}\\ \hat{X}_{GPS,k|k}=\hat{X}_{GPS,k|k-1}+K_{GPS,k}\Big{(}Y_{GPS,k}-H_{GPS,k}\hat {X}_{GPS,k|k-1}\Big{)}\\ P_{GPS,k|k}=\Big{(}I-K_{GPS,k}H_{GPS,k}\Big{)}P_{GPS,k|k-1}\end{array} \right. \tag{26}\]  
where \(P_{GPS,k|k}\) is systematic covariance matrix, \(K_{GPS,k}\) is the gain matrix, \(Q_{GPS,k}\) is the process noise covariance matrix, and \(R_{GPS,k}\) is the measurement noise covariance matrix. The state propagator can be written as:  
\[\left\{ \begin{array}{l}\hat{X}_{i,k|k-1}=F_{i,k}\hat{X}_{i,k-1|k-1}\\ P_{i,k|k-1}=F_{i,k}P_{i,k-1|k-1}F_{i,k}^{T}+Q_{i,k}\end{array} \right.,\;\;i=1,2 \tag{27}\]  
The \(\chi^{2}\) chi-square detector is widely used to detect malfunctions in stochastic dynamic systems [26]. This method can be divided into three types: (1) \(\chi^{2}\) detector for residual error, which can be used for sensor fault detection; (2) \(\chi^{2}\) detector state with a single state propagator, which can be used for the state transfer error detection and sensor fault detecting with the disadvantage of error accumulation;  
Figure 3: The block diagram of proposed faults signals detector. and (3) \(\chi^{2}\) detector with a double state propagators, which can be used for the state transfer error detection without error accumulation. In this paper, we use the \(\chi^{2}\) detector for residual error and the \(\chi^{2}\) detector with a double state propagators to form a fault signal detector with a parallel structure, as shown in Figure 4.  
To reduce the number of propagators, we designed two double state propagators, the outputs of which are used for each state \(\chi^{2}\) detector. The working principle can be described as following: during \(t_{k}\) there is a malfunction and a switch \(K_{1}\) is in position \"\(L_{1}\)\", switch \(K_{2}\) is in position \"\(L_{2}\)\", and State_Propagator_1 is \"dirty\" due to this malfunction, but the output of State_Propagator_2 was obtained using the previous correct state, that is, this output is correct and can be used to correct malfunctions. During \(t_{k+1}\), errors in prediction 1 are corrected using the output of Kalman filter. After time \(\Delta t\), the \(K_{1}\) switch is in position \"\(L_{2}\)\", the switch \(K_{2}\) is in position \"\(L_{1}\)\", and the State_Propagator_2 is used to correct the malfunction of the state. The error state vectors of state propagators can be defined as:  
\[\left\{ \begin{array}{l}e_{i,k}=X_{k}-\hat{X}_{i,k}\\ e_{GPS,k}=X_{k}-\hat{X}_{GPS,k}\end{array} \right. \tag{28}\]  
where \(X_{k}\) is true state vector, \(\hat{X}_{GPS,k}\) is estimation error from Sub_Kalman_Filter_1, and \(\hat{X}_{i,k}\) is estimation error from state propagator \(i\).