## 3 Fault Diagnosis Method  
### VMD Modeling  
The target of VMD is to decompose a real valued input signal \(f\) into a discrete number of sub-signal \(u_{k}\), that have specific sparsity properties while reproducing the input. Each mode \(u_{k}\) is assumed to be mostly compact around a center pulsation \(\omega_{k}\), which is to be determined along with the decomposition [30; 31]. A scheme to assess the bandwidth of a mode is as follows: First, for each mode \(u_{k}\), compute the associated analytic signal by means of the Hilbert transform in order to  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Fault Type Description** & **T6** & **T5** & **T4** & **T3** & **T2** & **T1** & **Coding Number** \\ \hline Normal & 0 & 0 & 0 & 0 & 0 & 1 & \\ \hline  & 0 & 0 & 0 & 0 & 0 & 1 & 2 \\  & 0 & 0 & 0 & 1 & 0 & 3 \\ Single open-circuit & 0 & 0 & 1 & 0 & 0 & 4 \\  & 0 & 0 & 1 & 0 & 0 & 0 & 5 \\  & 0 & 1  
obtain a unilateral frequency spectrum. Second, for each mode, shift the mode's frequency spectrum to \"baseband\", by mixing with an exponential tuned to the respective estimated center frequency. Third, the bandwidth is now estimated through the \(H^{1}\) Gaussian smoothness of the demodulated signal, the squared 2-norm of the gradient. The resulting constrained variational problem is as follows:  
\[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum_{k}\|\partial_{t}\left[\left(\delta(t )+\frac{j}{\pi t}\right)\!u_{k}(t)\right]\!e^{-j\omega_{k}t}\|_{2}^{2}\right\} \mathrm{s.\ t.}\sum_{k}u_{k}=f, \tag{4}\]  
where \(\{u_{k}\}=\left\{ \begin{array}{cc}u_{1},&\cdots,&u_{K}\end{array} \right\}\) is all modes. \(\{\omega_{k}\}=\left\{ \begin{array}{cc}\omega_{1},&\cdots,&\omega_{K}\end{array} \right\}\) is center frequencies of all modes. \(K\) is the number of levels of decomposition. In order to render the problem unconstrained, a quadratic penalty term and Lagrangian multipliers \(\lambda\) are both used. The augmented Lagrangian \(L\) is as follows:  
\[L(\{u_{k}\},\{\omega_{k}\},\lambda)=\alpha\!\sum_{k}\|\partial_{t}\left[\left( \delta(t)+\frac{j}{\pi t}\right)\!u_{k}(t)\right]\!e^{-j\omega_{k}t}\|_{2}^{2} +\left\|f(t)-\sum_{k}u_{k}(t)\right\|_{2}^{2}+\left\langle\lambda(t),f(t)- \sum_{k}u_{k}(t)\right\rangle\!, \tag{5}\]  
Minimization \(u_{k}\) and \(\omega_{k}\), respectively:  
\[\hat{u}_{k}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\sum\limits_{i\
eq k}\hat{u}_ {i}(\omega)+0.5\hat{\lambda}(\omega)}{1+2\alpha(\omega-\omega_{k})^{2}}, \tag{6}\]  
Which is clearly identified as a Wiener filtering of the current residual. The full spectrum of the real mode is obtained by Hermitian symmetric achievement. Conversely, the mode in time domain is procured as the real part of the inverse Fourier transform of this filtered analytic signal. \[\omega_{k}^{n+1}=\frac{\int_{0}^{\infty}\omega\big{|}\hat{u}_{k}(\omega) \big{|}^{2}d\omega}{\int_{0}^{\infty}\big{|}\hat{u}_{k}(\omega)\big{|}^{2}d \omega}, \tag{7}\]  
Which puts the new \(\omega_{k}\) at the center of gravity of the corresponding mode's power spectrum. This mean carrier frequency is the frequency of a least squares linear regression to the instantaneous phase observed in the mode. The decomposition procedure of VMD method is as follows:  
Step 1. Initialize \(\big{\{}a_{k}^{1}\big{\}},\{\omega_{k}^{1}\big{\}},\hat{\lambda}^{1},n \gets 0\). Step 2. Update \(u_{k}\), \(\omega_{k}\) and \(\lambda\), \(n\gets n+1\), \(k=1:K\). \[\hat{u}_{k}^{n+1}(\omega)\leftarrow\frac{\hat{f}(\omega)-\sum_{i<k}\hat{u}_{i }^{n+1}(\omega)-\sum_{i>k}\hat{u}_{i}^{n}(\omega)+0.5\hat{\lambda}^{n}(\omega )}{1+2\alpha(\omega-\omega_{k}^{n})^{2}}\,,\,(\omega\geq 0), \tag{8}\]  
\[\omega_{k}^{n+1}\leftarrow\frac{\int_{0}^{\infty}\omega\big{|}\hat{u}_{k}^{n+ 1}(\omega)\big{|}^{2}d\omega}{\int_{0}^{\infty}\big{|}\hat{u}_{k}^{n+1}(\omega )\big{|}^{2}d\omega}, \tag{9}\]  
\[\hat{\lambda}^{n+1}(\omega)\leftarrow\hat{\lambda}^{n}(\omega)+\tau\Bigg{(}f (\omega)-\sum_{k}\hat{u}_{k}^{n+1}(\omega)\Bigg{)},(\omega\geq 0), \tag{10}\]  
Step 3.