# Condition Monitoring of Industrial Equipment Based on Multi-Variables State Estimate Technique
## 3 Multi-Variable State Estimate Technique (MSET)  
By applying MSET, considerable historical data for industrial equipment are used for constructing the normal state envelope space, which is noted as memory matrix \(\mathbf{D}\). While the industrial equipment operating, the measurement parameters are observed real-time, and the related \(n\) parameters construct a vector noted as \(\mathbf{X}_{obs}\). In the time sample \(j\), the observation vector is written as,  
\[\mathbf{X}(t_{j})=[x_{1}(t_{j}),\,x_{2}(t_{j}),\,\ldots,\,x_{n}(t_{j})]^{T} \tag{1}\]  
It should be pointed out that for different equipment condition monitoring, only the parameters and thresholds need to be determined if reusing the proposed method, and the complete preprocess of the case study in our paper is posed in Section 4. \(\mathbf{D}\) is an \(m\times n\) dimension matrix, which means \(m\) normal samples are strictly selected and contained in \(\mathbf{D}\) at last. Comprehensive operating conditions, including high load, low load, change load, and other operating conditions should be contained. Memory matrix \(\mathbf{D}\) could be written as,  
\[D=\left[ \begin{array}{cccc}X(t_{1})&X(t_{2})&\cdots&X(t_{m})\\ \end{array} \right]=\left[ \begin{array}{cccc}x_{1}(t_{1})&x_{1}(t_{2})&\cdots &x_{1}(t_{m})\\ x_{2}(t_{1})&x_{1}(t_{2})&\cdots&x_{2}(t_{m})\\ \vdots&\vdots&\vdots&\vdots\\ x_{n}(t_{1})&x_{n}(t_{2})&\cdots&x_{n}(t_{m})\\ \end{array} \right]_{n\times m} \tag{2}\]  
The vectors included in the memory matrix \(\mathbf{D}\) construct a nonlinear normal state envelope space. After the memory matrix \(\mathbf{D}\) constructed, dynamic state estimation could be accomplished by applying it. The construction of estimation vector for the same time sample \(\mathbf{X}_{est}\) is constructed by the vectors in memory matrix \(\mathbf{D}\), the different of \(\mathbf{X}_{est}\) and \(\mathbf{X}_{obs}\) is applied for condition monitoring. \(\mathbf{X}_{est}\) could be written as,  
\[ \begin{array}{llll}X_{est}&=D\cdot W=D\cdot\left[\begin{array}{cccc}w_{1} &w_{2}&\cdots&w_{m}\\ \end{array}\right]^{T}\\ &=w_{1}X(t_{1})+w_{2}X(t_{2})+\cdots+w_{m}X(t_{m})\end{array}  \tag{3}\]  
The weight matrix \(\mathbf{W}\) stands for similarity between observation vector and components in matrix \(\mathbf{D}\). The optimized weight matrix W could be obtained by minimizing the error between \(\mathbf{X}_{est}\) and \(\mathbf{X}_{obs}\), which is,  
\[e=X_{est}-X_{obs} \tag{4}\]  
According to the least square algorithm, for minimizing \(\|e\|_{2}\) let,  
\[ \begin{array}{ll}S(w)=\sum\limits_{i=i}^{n}e_{i}^{2}=e^{T}e=(X_{obs}-X_{est })^{T}\cdot(X_{obs}-X_{est})\\ =(X_{obs}-D\cdot W)^{T}\cdot(X_{obs}-D\cdot W)=\sum\limits_{i=1}^{n}\left(X_{ obs}(i)-\sum\limits_{j=1}^{m}w_{j}D_{ij}\right)^{2}\end{array}  \tag{5}\]  
where, \(D_{ij}\) is the element for memory matrix in \(i\) line \(j\) column. Solving the partial derivative Equation for \(w_{i}\) in \(S(w)\), which is,  
\[\frac{\partial S}{\partial w_{k}}=-2\sum\limits_{i=1}^{n}\left(X_{obs}(i)- \sum\limits_{j=1}^{m}w_{j}D_{ij}\right)D_{ik}=0 \tag{6}\]  
That is,  
\[\sum\limits_{i=1}^{n}X_{obs}(i)D_{ik}=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^ {m}w_{j}D_{ij}D_{ik}=\sum\limits_{i=1}^{n}\left(\sum\limits_{j=1}^{m}D_{ij}D_{ ik}\right)w_{j} \tag{7}\]The Equation (7) could be expressed as matrix form as,  
\[D^{T}\cdot D\cdot W=D^{T}\cdot X_{obs} \tag{8}\]  
The weight matrix could be solved, that is,  
\[W=\left(D^{T}\cdot D\right)^{-1}\cdot\left(D^{T}\cdot X_{obs}\right) \tag{9}\]  
In particular, if components in memory matrix \(D\) are linear correlative, the matrix \(D^{T}\cdot D\) would be irreversible, which leads the (9) unsolvable. In order to solve the problem properly, there are many alternative operator; Argonne laboratory has also developed a unique algorithm for nonlinear calculation [15].