# Inductance Estimation Based on Wavelet-GMDH for Sensorless Control of PMSM
###### Abstract
Therefore, even with appropriate back electromotive force constants, exposure to periodic load torque fluctuations can degrade the system stability and speed estimation performance due to the occurrence of noise, phase delays, and other factors. Unlike model-based methods, soft computing methods utilizing data-driven approaches allow for the selection of input variables and define partial expressions of input-output data to objectively predict system equations. This enables more accurate and faster parameter learning and estimation, especially in cases of nonlinear relationships where input-output relationships or model functions are not specific. Particularly, the group method of data handling (GMDH), a form of a self-organizing neural network (SONN), estimates output parameters in the form of polynomial expressions in multilayered structures. GMDH offers fast computation speed, good real-time performance, and excellent accuracy, making it advantageous for embedded systems compared to neural networks, fuzzy algorithms, and others. Due to its flexible structure that evolves through learning, GMDH does not have fixed layer numbers and node counts; instead, it iteratively creates and disbands layers. Furthermore, GMDH effectively reduces parameter estimation errors by combining multiple input-output datasets and dynamically generating and modeling the required set of coefficients [15]. Compared to conventional model-based techniques for estimating the inductance parameters of PMSM sensorless drives, GMDH based estimation offers several key advantages. Firstly, GMDH based learning methods are well suited for complex nonlinear models such as motor systems. By learning a multilayered structure of subexpressions represented in polynomial form, it minimizes the time and cost required for modeling complex nonlinear systems. Secondly, GMDH based learning accurately estimates unknown parameter variables and compensates for delays or distortions, resulting in superior performance compared to other methods in terms of comparative verification and RMS (root mean square) error. Thirdly, the learning process enables easy adaptation to various operational load conditions such as step input loads, periodic loads, random profiles, and more [16; 17; 18; 19]. While recent research has highlighted the application of machine learning in motor control systems, attempts to apply machine learning for sensorless control parameter prediction and compensation control are rare due to implementation challenges, optimization issues, and others. Particularly, applying GMDH based learning techniques to nonlinear systems without appropriately selecting the design parameters can degrade the control characteristics of the system. Moreover, in the presence of unknown disturbances or significant noise in the inputs for machine learning, the estimated results cannot be trusted, and have a negative impact on the system. Therefore, it is necessary not only to design appropriate training parameters suitable for nonlinear sensorless systems but also to perform optimized learning through noise reduction and the refinement of the input data for machine learning. The combination of wavelet transform for signal analysis and artificial neural networks is increasingly being applied in research. Primarily, studies are conducted where wavelet transform is used for signal analysis to extract feature vectors, followed by the application of artificial neural networks for diagnostics. Wavelet transform is widely used in signal processing and image processing as it removes noise from high-frequency components and enables simultaneous analysis in the time-frequency domain. However, such research is limited to cases applying to physical faults or winding faults in electric motors [20; 21]. In addition, there are studies that have integrated wavelets and machine learning such as GMDH to perform load forecasting or failure prediction of power systems [22; 23], but no cases have been applied to the parameter estimation of PMSM.