Quantitative Observability of PMSM Inductance Estimation: A Unified Multiscale Framework

Abstract

The convergence stability and accuracy of permanent magnet synchronous motor (PMSM) parameter estimation are fundamentally governed by model observability. While fundamental-frequency approaches are established, the theoretical observability of emerging models relying on high-frequency current-derivative information remains largely unexplored. This article proposes a real-time quantitative framework for evaluating the observability of PMSM parameter-estimation basis models within a unified multiscale perspective. The proposed method introduces a condition-number-based metric derived from the information matrix, enabling real-time assessment of model excitation and conditioning. Most existing parameter-estimation methods are summarized into three basis models: the fundamental-frequency time-scale model (FFTM), the switching-time-scale full model (STFM), and the virtual-voltage-vector model (VVVM). These models are comparatively analyzed within this framework. Analytical results reveal the excitation-dependent conditioning characteristics of each model and identify their weakly observable conditions. An adaptive observer guided by the proposed metric is utilized specifically to demonstrate the effectiveness of the observability assessment. Both numerical and experimental results confirm that the proposed criterion provides a reliable and real-time index for model observability evaluation and parameter-identification stability.