## 2 The Detection and Isolation Method of the Parity Space Method  
### The SVD-Based Method and Improvement  
The parity space method is a diagnosis method based on analytic redundancy. Its basic idea is to construct a parity matrix based on the hardware redundancy or analytical redundancy equation of the system, and use the actual observation to check the parity (consistency) of the system mathematical model (analytic redundancy relationship). The method needs to find the parity equation decoupled from the faults, and then the parity vector and the fault detection function are constructed based on the parity equation. With the detection threshold determined by hypothesis testing theory, the faults can be detected once the detection function values exceed the threshold. In this paper, the SVD-based method [22] is used as an example to deduce the relation between detection functions and data anomalies. This relation is the basis to analyze anomalies. In normal working, the observation equation of a redundant system composed of \(n\) gyroscopes can be expressed as follows:  
\[\textbf{Z=}\textbf{H}\textbf{x+}\boldsymbol{\varepsilon} \tag{1}\]  
In the equation, \(\textbf{Z}\) is the measurement data vector of \(n\)-dimensional; \(\textbf{H}_{n\times 3}\) is the geometric configuration matrix; \(\boldsymbol{\varepsilon}\) is the \(n\)-dimensional white Gaussian noise vector which statistical characteristics meet:  
\[E[\boldsymbol{\varepsilon}]=0,E[\boldsymbol{\varepsilon}\boldsymbol{ \varepsilon}^{T}]=\sigma^{2}\textbf{I}_{\text{n}} \tag{2}\]When the anomalies appear, the observation equation can be expressed as:  
\[\mathbf{Z}\mathbf{=}\mathbf{H}\mathbf{x}\mathbf{+}\mathbf{b}\mathbf{+}\mathbf{\varepsilon} \tag{3}\]  
In Equation (3), \(\mathbf{b}\) is the anomaly vector. Several typical anomaly vectors are described in Table 1.  
Note that, \"Duration\" refers to the number of sampling points from the anomaly appearing to the disappearing; \"N\" is the number of sampling points in a single diagnostic period; \"\(\backslash\)\" indicates that the anomalies vectors are not fixed or too difficult to express by mathematical equation; \(\dot{d}_{io}\) is the outlier vector; \(\mathbf{e}\) is the white noise vector with zero mean value; \(\mathbf{d}\) is the constant drift vector with fixed values of each element; \(\mathbf{K}_{e}\) is the anomaly coefficient matrix with fixed values of each element. \(\mathbf{K}_{e}\) is also a diagonal matrix, and the elements of normal sensors at the diagonal line equal to zero. To detect anomalies, in [22], an SVD-based method was presented and a detailed derivation and verification was conducted. The first step is to decompose the geometric configuration matrix \(\mathbf{H}\) through singular value decomposition. \[\mathbf{H}=\mathbf{U}_{\rm H}\left( \begin{array}{c}\mathbf{S}_{\rm H}\\ 0\end{array} \right)\mathbf{I}_{\rm H}^{T} \tag{4}\]  
In Equation (4), the matrix \(\mathbf{S}_{\rm H}\) is a diagonal matrix. The decoupling matrix \(\mathbf{V}\) is constructed from the matrix \(\mathbf{U}_{\rm H}\) as follows:  
\[\mathbf{U}_{\rm H}=[\mathbf{U}_{1}\vdots\mathbf{U}_{2}] \tag{5}\]  
\[\mathbf{V}=\mathbf{U}_{2}\mathbf{U}_{2}^{T} \tag{6}\]  
where \(\mathbf{U}_{1}\in\mathbb{R}^{m\times 3}\) and \(\mathbf{U}_{2}\in\mathbb{R}^{m\times(n-3)}\). The parity vector \(\mathbf{p}\) is calculated with the matrix \(\mathbf{V}\) and the measurement vector \(\mathbf{Z}\) as follows:  
\[\mathbf{p}=\mathbf{V}\mathbf{Z} \tag{7}\]  
In [22] the fault detection function and isolation function are constructed as follows:  
\[F_{\rm D}=\mathbf{p}^{T}\mathbf{p} \tag{8}\]  
\[F_{\rm i}^{\rm i}=\mathbf{p}^{T}\mathbf{V}_{\rm i} \tag{9}\]  
where the \(\mathbf{V}_{\rm i}\) is the fault reference vector, which refers to the \(i\)-th column of the matrix \(\mathbf{V}\), and the \(F_{\rm i}^{\rm i}\) corresponds the \(i\)-th sensor. The decision rule to detect fault is given as follows:  
\[\left\{ \begin{array}{l}H_{0}:F_{D}<T_{D}\\ H_{1}:F_{D}>T_{D}\end{array} \right. \tag{10}\]