# Fault-Diagnosis Method for Rotating Machinery Based on SVMD Entropy and Machine Learning
## 2 Signal-Decomposition Methods  
### Variational-Mode Decomposition (VMD)  
In the VMD algorithm, the intrinsic-mode function (IMF) is redefined as an AM-FM signal with Equation (1):  
\[u_{k}(t)=A_{k}(t)cos(\Phi_{k}(t)) \tag{1}\]  
where \(A_{k}(t)\) is the instantaneous amplitude of \(u_{k}(t)\); \(\Phi_{k}(t)\) is the instantaneous phase of \(u_{k}(t)\).  
When the VMD algorithm obtains the IMF component, it gets rid of the circular-sieving and stripping-signal-processing method used by the EMD algorithm and transfers the signal-decomposition process to the variational framework, realizing the adaptive decomposition of the signal by searching for the optimal solution of the constrained variational model. The frequency center and bandwidth of each IMF component are constantly updated in the process of iteratively solving the variational model. Finally, the adaptive division of the signal-frequency band can be completed according to the frequency-domain characteristics of the actual signal, and several narrowband IMF components can be obtained. Assuming that the original signal is decomposed into \(K\) IMF components, the corresponding expression of the constrained variational model is as follows:  
\[\min_{u_{k},\omega_{k}}\left\{\sum_{k}\left\|\partial_{t}\left[\left(\delta(t )+\frac{i}{nt}\right)u_{k}(t)\right]e^{-i\omega_{k}t}\right\|_{2}^{2}\right\} \tag{2}\]  
where \(u_{k}\) represents \(K\) IMF components obtained by the decomposition, \(\omega_{k}\) denotes the central frequency of each component, and \(\delta(t)\) is the Dirac function.  
To find the optimal solution of the above constrained variational problem, an incremental Lagrange function of the following form is introduced:  
\[\begin{split}& L(u_{k},\omega_{k},\lambda)=\alpha\sum_{k}\left\| \partial_{t}\left[\left(\delta(t)+\frac{i}{nt}\right)u_{k}(t)\right]e^{-i \omega_{k}t}\right\|_{2}^{2}\\ &\quad+\left\|f(t)-\sum_{k}u_{k}(t)\right\|_{2}^{2}+\left\langle \lambda(t),f(t)-\sum_{k}u_{k}(t)\right\rangle\end{split} \tag{3}\]  
where \(\alpha\) is the penalty factor, and \(\lambda\) is the Lagrange multiplier.  
The alternating direction-multiplier algorithm is used to find the saddle point of the above incremental Lagrange function, which is the optimal solution of the constrained variational model of Equation (3), so as to decompose the original signal into \(K\) narrowband IMF components. The specific implementation process is as follows.  
1. Initialize \(u_{k}^{1}\), \(\omega_{k}^{1}\), \(\lambda^{1}\), and \(n\) to 0.
2. \(n=n+1\); execute the entire loop.
3. Execute the first loop of the inner level based on \(u_{k}^{n+1}=\arg_{u_{k}}\text{min}L\left(u_{i<k}^{n+1},\ \lambda^{n},\ u_{i\geq k}^{n},\ \omega_{i}^{n}\right)\) and update \(u_{k}\).
4. \(k=k+1\) and repeat step (3) until \(k=K\); then end the first loop of the inner layer.
5. Execute the second loop of the inner level based on \(\omega_{k}^{n+1}=\arg_{\omega_{k}}\text{min}L(u_{i}^{n+1},\ \omega_{i<k}^{n+1},\ \omega_{i \geq k}^{n+1},\ \lambda^{n})\) and update \(\omega_{k}\).
6. \(k=k+1\) and repeat step (5) until \(k=K\), then end the second loop in the inner layer.
7. Based on \(\lambda^{n+1}=\lambda^{n}+\tau\left(f-\sum_{k}u_{k}^{n+1}\right)\), update \(\lambda\).
8. Repeat steps (2)\(\sim\)(7) until the iteration-stop condition \(\sum_{k}\left(\left\|u_{k}^{n+1}-u_{k}^{n}\right\|_{2}^{2}/\left\|u_{k}^{n} \right\|_{2}^{2}\right)<\varepsilon\) is satisfied, end the whole loop, and output the result to get \(K\) narrowband IMF components.