## 3 Proposed Cross-Working Condition RUL Prediction Method
### RAPP-Based Initial Degradation Detection
#### 3.1.1 DACAEN Model  
DACAEN is an improvement structure based on a deep convolution autoencoder network, which has been introduced as the adversarial mechanism strategy. As shown in Figure 2, DACAEN uses the vibration signal of bearings in normal stages for unsupervised model training. Then, DACAEN with high strength reconstruction ability is used for bearing condition monitoring and determine the starting point of RUL. Unlike traditional DCAE, which takes sample reconstruction loss as the optimization objective, the DACAEN model also combines the discriminant loss of the discriminator. Specifically, a fast Fourier transform (FFT) is first performed, and then the raw vibration signal obtained is fed into DACAEN to reconstruct the sample signal. At the same time, both reconstructed and original samples are input into the discriminator, so that the discriminator can distinguish the real sample and generate the sample, respectively. The structure of the encoder and discriminator in the proposed method is the same, the settings of the hyperparameters are the same, and the corresponding structure of the encoder and decoder is the same, but the corresponding convolution operation is replaced by a deconvolution operation. The specific network structure and model parameters are described in detail in the experimental section.  
The amplitude of the vibration signal becomes progressively larger during the whole process of bearing operation from health to damage. When samples from the normal stage are used as training samples, it is necessary to improve identification between the degraded  
Figure 2: Network structure diagram of the proposed model.  
samples and the normal stage to realize early degradation monitoring and determine the starting point of RUL [25]. Therefore, a convolutional autoencoder network called DACAEN based on an adversarial mechanism strategy was designed in order to improve the ability of normal sample reconstruction and make the health index of the normal stage as small as possible. The input sample is denoted as \(x^{in}\), and the reconstructed sample is denoted as \(x^{gen}\), both of which are generated by \(G\left(x^{in}\right)\). The objective function of DACAEN consists of the following three parts.  
(1) Reconstruction loss \(Loss_{MSE}\): The Euclidean distance between the real sample and the reconstructed sample, since minimizing this loss ensures that the generator produces reconstructed samples that are similar to the real sample.  
\[Loss_{MSE}=\frac{1}{n}\sum_{i=1}^{n}\left|x_{i}^{gen}-x_{i}^{in}\right| \tag{6}\]  
(2) Generator loss \(Loss_{GEN}\): In order to enhance the ability of \(G\) generation and learn domain-invariant representations, the minimal-maximal game between domain discriminator and generator is leveraged. The generated samples obtained from \(G\) need to confuse \(D\) to make a wrong judgment.  
\[Loss_{GEN}=\frac{1}{n}\sum_{i=1}^{n}\left|x_{i}^{gen}-x_{i}^{in}\right|+E_{x^{ in}P_{x}^{in}}\left[\log\left(1-D\left(G(x^{in})\right)\right)\right] \tag{7}\]  
where \(n\) is the sample number.  
(3) Discriminator loss \(Loss_{D}\): Since \(D\) needs to judge the real samples \(x^{in}\) from the generated samples \(x^{gen}\), a fully connected layer is added to output the probability that the generated samples \(x^{gen}\) are derived from the actual space.  
\[Loss_{DIS}=-\left(E_{x^{in}P_{x}^{in}}\left[\log\left(D\left(x^{in}\right) \right)\right]+E_{x^{in}P_{x}^{in}}\left[\log\left(1-D\left(G(x^{in})\right) \right)\right]\right) \tag{8}\]