## 3 NARX Model on the Rotor-Bearing System  
### Identification of the NARX Model  
A broad range of nonlinear systems can be described using NARX models, which can be expressed as [29]:  
\[y(k)=F\big{[}y(k-1)\cdots,y\big{(}k-n_{y}\big{)},u(k-1),\cdots u(k-n_{u})\big{]} \tag{4}\]  
where \(F[\ ]\) is some nonlinear function, \(n_{y}\) and \(n_{u}\) are the maximum time lag for the system output and input, respectively, and \(y(k)\) and \(u(k)\) are the output and input sequence of the system, respectively. The NARX model has several forms, among which the power polynomial representation is most commonly used in nonlinear system identification. Function \(F\) in equation (4) is defined as the following polynomial expression:  
\[y(k)=\theta_{0}+\sum\limits_{i_{1}=1}^{n}\theta_{i_{1}}x_{i_{1}}(k)+\sum \limits_{i_{1}=1}^{n}\sum\limits_{i_{2}=i_{1}}^{n}\theta_{i_{1}}\theta_{i_{2}} x_{i_{1}}(k)x_{i_{2}}(k)+\cdots+\sum\limits_{i_{1}=1}^{n}\cdots\sum\limits_{i_{ 1}=1}^{n}\theta_{i_{1}i_{2}...i_{N}}x_{i_{1}}(k)x_{i_{2}}(k)\cdots x_{i_{l}}(k) \tag{5}\]  
where:  
\[x_{m}(k)=\left\{ \begin{array}{cc}y(k-m)&1\leq m\leq n_{y}\\ u\big{(}k-\big{(}m-n_{y}\big{)}\big{)}&n_{y}+1\leq m\leq n_{y}+n_{u}\end{array} \right. \tag{6}\]  
where \(\theta_{i}\) are model coefficients; \(L\) is the degree of polynomial nonlinearity; \(n=n_{u}+n_{y}\). Consider a generic linear-in-parameters representation of model (5) as:  
\[y(k)=\sum\limits_{m=1}^{M}\theta_{m}p_{m}(k) \tag{7}\]  
where \(\theta_{m}\) is the model coefficient of the \(m\)th term; \(M\) is the number of all possible model terms, \(M=(n+l)!/(n!l! )\); \(p_{m}(k)\) is the regression which is formed by some combinations of the predetermined model variables, which are orderly chosen from the following vector \(x(k)=\big{[}1,y(k-1),L,y\big{(}k-n_{y}\big{)},u(k-1),L,u(k-n_{u})\big{]}\). The NARX model can be obtained by using FROLS (Forward Regression Orthogonal Least Squares) algorithm [30], which is a recursive algorithm based on ERR (Error Reduction Ratio) criterion to select significant model terms. The steps of FROLS are as follows:  
Figure 5: Response of rotor-bearing system under over critical speed condition. * Step 1: Orthogonalization of model: The NARX model (7) can be rewritten into a matrix form: \[y=P\theta\] (8) where \(P=[p_{1},p_{2},L,p_{M}]\) is the regression matrix with the column vectors \(p_{m}=[p_{m}(1),L,p_{m}(N)]^{T},(m=1,L,M)\) which consist of the values \(p_{m}(k),(m=1,K,N)\) over the time history; \(N\) is the number of sampling observations; \(\theta=[\theta_{1},\theta_{2},L,\theta_{M}]^{T}\) is the coefficient vector. Based on the Schmidt's orthogonalization, the model (8) can be orthogonalized into [15]:  
\[y=P\theta=WG \tag{9}\]  
where \(W=[w_{1},w_{2},L,w_{M}]\) is the orthogonalized matrix of \(P,G=[g_{1},g_{2},L,g_{M}]^{T}\) is the corresponding coefficients vector, where:  
\[w_{m}=p_{m}-\sum_{i=1}^{m-1}\frac{<p_{m},w_{i}>}{<w_{i},w_{i}>}w_{i} \tag{10}\] \[g_{m}=\frac{<y,w_{m}>}{<w_{m},w_{m}>} \tag{11}\]
* Step 2: Model term selection: A. Given \(w_{m}^{(1)}=p_{m}\), where the superscript (1) represents the first selecting step. \(g_{m}^{(1)}\) can be calculated based on Equation (11). The ERR value of the \(m\)th model term is given as [31]: \[ERR_{m}^{(1)}=\frac{{(g_{m}^{(1)})}^{2}{<w_{m}^{(1)},w_{m}^{(1)}>}}{<y,y>} \times 100\%\] (12) Then obtain the column \(S_{1}\), which has the largest ERR value: \[s_{1}=argmax\Big{\{}ERR_{m}^{(1)}\Big{\}},1\leq m\leq M\] (13) where \(\text{argmax}\{\ \}\) represents the value of the argument when the function reaches the maximum. Given the corresponding term vector as the first model term vector of the orthogonalized matrix in (9), which means \(w_{1}=w_{S_{1}}^{(1)}\), and given \(\beta_{1}=p_{S_{1}}\). B.