## 2 The Detection and Isolation Method of the Parity Space Method
### The Improvement of the SVD-Based Method  
The method above has the following disadvantages, which are analyzed and proved in [31]:  
1. The detection function \(F_{D}\) has the probability to disobey the \(\chi^{2}\) distribution, because the \(\chi^{2}\) distribution requires that each variable is independent. The matrix \(\mathbf{V}\) is singular, so elements of the vector \(\mathbf{p}=\mathbf{V}\mathbf{Z}\) are not independent. As a result, the threshold determined by \(\chi^{2}\) distribution is inappropriate. 2. The isolation function \(F_{1}^{i}\) given in Equation (9) may cause the inequality of isolation probability among the sensors. The isolation function is designed to indicate the similarity between the parity vector and the fault reference vector, and the sensor with the highest similarity is most likely to get faulty. From Equation (9), the values of isolation function are subjected to the modules of \(\mathbf{V}_{i}\), but there is no guarantee that the modules of each fault reference vector \(\mathbf{V}_{i}\) are equal. The similarity between two vectors is supposed to only depend on the projection or the angular between them. In other words, the modules of fault reference vectors are not supposed to influence the isolation function. A more appropriate method is to use the angular between the parity vector and the fault reference vector \(\mathbf{V}_{1}\). For example, consider a simple situation: \(\mathbf{V}_{1}=[0,1]\), \(\mathbf{V}_{2}=[4,0]\) and \(\mathbf{p}=[1,\sqrt{3}]\). Calculate the isolation function values with Equation (9), and there are \(F_{1}^{1}=\sqrt{3}\) and \(F_{1}^{2}=4\). So the result of the original method is that the first senor is faulty. However, the angular between \(\mathbf{V}_{1}\) and \(\mathbf{p}\) is \(30^{\circ}\), and the angle between \(\mathbf{V}_{2}\) and \(\mathbf{p}\) is \(60^{\circ}\). It is obvious that the second sensor is most likely to have faults, and the original method produced an incorrect result. 3. The isolation strategy is flawed as it cannot isolate the negative faults. For example, we simulate a set of measurement signals based on 6-gyros-dodecahedron redundant system. A positive step fault and a negative fault are injected in the fourth gyro in turn. The signal, detection function, and isolation function are shown in Figure 2. Analyzing the figure, the detection function is able to detect the faults once the faults are injected. But the isolation strategy chooses the sensor to isolate, which corresponds to the largest isolation function value; however, the negative fault will bring down the isolation function values of the faulty sensor. So the isolation function fails to isolate negative faults. Figure 2: The fault detection and isolation function in two cases: (**a**) The detection functions of positive fault and negative fault; (**b**) The isolation functions of positive fault and negative fault. To solve the problems above, the method is improved as follows: To solve the first problem, the decoupling matrix \(\mathbf{V}\) is redefined and the parity vector is standardized as follows:  
\[\mathbf{V}=\mathbf{U}_{2}^{T} \tag{11}\]  
\[\mathbf{p}=\mathbf{V}\mathbf{Z}/\mathbf{\sigma} \tag{12}\]  
Since this improvement, the \(\mathbf{p}^{T}\mathbf{p}\) obeys \(\chi^{2}(n-3)\) distribution, and the threshold can be determined with a given false alarm rate \(\alpha\). To solve the last two problems, the isolation function is redefined as follows:  
\[\mathbf{F}_{1}^{i}=\left(\frac{\mathbf{p}^{T}\mathbf{V}_{i}}{\left|\mathbf{p}\|\mathbf{V}_{i} \right|}\right)^{2}=\frac{\left(\mathbf{p}^{T}\mathbf{V}_{i}\right)^{2}}{\mathbf{V}_{i}^{ T}\mathbf{V}_{i}\mathbf{p}^{T}\mathbf{p}} \tag{13}\]  
The improved isolation function indicates the angular between the parity vector \(\mathbf{p}\) and the \(i\)-th fault reference vector \(\mathbf{V_{i}}\), and the effect of the module of \(\mathbf{V_{i}}\) is eliminated. Therefore, the impurity caused by the difference of modules of fault reference vectors will not influence the isolation results.