###### Abstract
Model-based approaches for fault diagnosis utilize signal estimation techniques, system identification approaches, and output observation schemes. [PERSON] et al. [13] obtained the residual signals that were calculated based on the difference between original and estimated ones for fault diagnosis using a model-based approach. Although this approach is stable and reliable, there are significant challenges related to signal modeling. Mathematical modeling and identification techniques are the most important methods for signal/system modeling. Although mathematical-based modeling has advantages such as reliability and accuracy, it is difficult to use this method to model nonlinear and complex signals/systems. On the other hand, an identification-based approach is able to model the dynamic behavior of the system by analyzing data extracted from the sensors. Regression-based approaches, such as the autoregressive technique [14] and autoregressive with external input [15], are the most important schemes for identification-based modeling. To improve the robustness of regression-based approaches, the Laguerre filter technique was suggested in [16]. The application of a fuzzy technique to a regression-based identification technique was introduced in [17]. One of the important schemes for model-based fault diagnosis is observation-based approaches. These techniques can be designed by various algorithms, which are categorized into two groups: linear-based observation algorithms and nonlinear-based observation techniques. The proportional-integral observer, proportional-multi-integral observer, and proportional-integral-derivative observer each fall into the group of linear observers. For example, [PERSON] and [PERSON] [18] introduced a proportional-integral (PI) observer for bearing fault diagnosis. To develop the performance of the PI observer, the proportional-multi-integral (PMI) observer for fault diagnosis of bearings was introduced in [19]. Apart from the advantages of linear observers, such as their simplicity of implementation, these methods suffer from a lack of robustness and accuracy in fault modeling. Nonlinear observation approaches have been proposed to overcome the problems of linear observation techniques. Various observers can be listed in the group of nonlinear observers, such as linearized feedback methods, backstepping approaches, Lyapunov algorithms, variable structure schemes, fuzzy methods, and neural network algorithms. For instance, linearized feedback has been applied for accurate fault modeling in [11]. In [20], the vibration signals were modeled by mathematical techniques, and the fault signal was estimated by a feedback linearization observer. This technique has two significant challenges: stability and system modeling using a mathematical approach. Variable structure algorithms and Lyapunov methods are two influential, robust, nonlinear algorithms [21]. These two algorithms have been used in diverse applications for fault diagnosis in bearings, robotics, and pipelines. High-frequency fluctuations in unknown conditions are a significant challenge of variable structure observers. To reduce the effect of high-frequency fluctuations, various techniques have been suggested, including the super-twisting technique [22], saturation algorithm [23], and high-order variable structure [24]. In this research article, a robust smooth sliding digital twin is recommended for bearing anomaly identification. This method consists of three main parts. First, the AE bearing signals are resampled and the RMS feature is extracted from them. Next, the robust smooth sliding digital twin is designed for modeling and to estimate the RMS feature. After that, the SVM technique is used for feature classification and crack size identification. The main part of this research is the digital twin design. So, to design the proposed digital twin, the indirect (feature) signals are modeled using the proposed autoregressive fuzzy Gauss-Laguerre algorithm.