# Blended Filter-Based Detection for Thruster Valve Failure and Control Recovery Evaluation for RLV
## 2 Blending Filter-Based Thruster Fault Detection
### Blending the Physical Filter and Kalman Filter for Fault Detection
If there is an approximate control signal noise in the input of the physical system, it may lead to the following: one is that the input of the physical system will deviate from the input of the model; the other is that the input of physical system has a great influence on the sensor data generated by the actual physical system, which is then introduced into the Kalman filter for state estimation. These challenges may result in the failure of the Kalman filter to estimate the state, as shown in Figure 2.  
In the RLV flight, several thruster missions, including on-orbit, de-orbit, and initial reentry, are performed in the near-earth space environment. However, electromagnetic radiation in the near-earth space environment is very complex [22]. Furthermore, the noise caused by electromagnetic induction or charging effect of the vehicle can easily interfere with the control signal [23]. However, the vehicle control input usually works within a certain frequency band. For example, the flight attitude control command period is 20 ms, the guidance command period is 100 ms, and the no-command state still sends the reference pulse based on the command period. Therefore, in this study, a bandpass physical filter was introduced and applied before the control signal input in order to suppress the approximate control signal noise. The input position of the physical filter is shown in Figure 1. The approximate control signal noise outside the frequency band, which is caused by electromagnetic radiation in space, will be suppressed after passing through the physical filter. This ensures that the state estimation performed using the Kalman filter is more accurate. A sample circuit diagram of the physical filter is shown below in Figure 3.  
Figure 3: Example of a bandpass filter: (**a**) bandpass filter circuit diagram; (**b**) frequency response diagram. Figure 2: Kalman filter state estimation under the interference of approximate control signal noise. The filter is designed using the FilterLab 2.0 software; the passband frequency is from 10 Hz to 50 Hz, which meets the control input period. The introduction of the physical filter improved the input consistency between the model and the physical system as shown in the simulation results given in Figure 4. Blending the physical filter and Kalman filter results in a smaller state estimation bias in comparison to the Kalman filter. In the attitude loop, the onboard calculation period is typically in the order of several tens of milliseconds. In this paper, the residual deviation between the predicted output of the blending filter (\(\hat{\mathbf{z}}_{k}\)) and the sensor measurement feedback (\(\mathbf{z}_{k}\)) was used as the observation output in each period for three types of faults, that is:  
\[\mathbf{r}_{k}=\mathbf{z}_{k}-\hat{\mathbf{z}}_{k} \tag{10}\] \[\mathbf{r}_{k}=\mathbf{H}\mathbf{x}_{k}+\mathbf{v}_{k}-\mathbf{H} \hat{\mathbf{x}}_{k}\]  
Fault-free condition: \(E[\mathbf{r}_{k}]=\)0, their probability density obeys the normal distribution of expectation 0, variance \(R\). Fault condition: \(E[\mathbf{r}_{k}]\
eq 0\)  
A variance test is used for the residual evaluation for a selected time frame, and analysis; see Section 4.1. Establishing the binary hypothesis:  
H0: fault-free  
H1: fault  
The designed fault detection function (weighted sum of squared residuals) was:  
\[\lambda_{k}=\mathbf{r}_{k}^{T}\mathbf{A}_{k}^{-1}\mathbf{r}_{k}, \tag{11}\]  
where \(\mathbf{A}_{k}=diag(\mathbf{\sigma}_{k}^{2})\), \(\mathbf{\sigma}_{k}\) are the \(\mathbf{r}_{k}\) group standard deviation, then \(\lambda_{k}\sim\chi^{2}(m)\), \(m\) equals the dimension of \(\mathbf{z}_{k}\). The fault-free was determined by:  
Figure 2.0  
The fault was detected by:  
Figure 2.1  
To determine the \(T_{D}\) according to the requirement of the system false alarm rate \(P_{F}\):  
Figure 2.2  
Figure 2.3 Blending of the physical filter and Kalman filter state estimation under the interference of approximate control signal noise. then,  
Figure 2.4  
where \(\alpha_{T}\) is the expected value.