## 1 Introduction  
The PMSM has the advantages of simple structure, high power density, low failure rate, and reliable operation, and has been widely used in recent years. However, due to the limitation of its own structure, power density, and the various complex motor operating environments, the electromagnetic parameters of a synchronous motor will change with the operation of the equipment, which is likely to lead to the decline of motor vector control performance [1]. Therefore, the online parameter identification of PMSMs has important scientific research and practical application significance for high-performance motor control and fault warning [2]. At present, there are many methods for online parameter identification, such as recursive least squares (RLS) [3, 4], Extended Kalman Filter (EKF) [5, 6], MRAS [7, 8], intelligent optimization algorithm [9, 10], and so on. Among them, RLS is easy to implement but it is unable to obtain the unbiased parameter estimations of output-error systems [4]. The EKF has strong robustness but there is no proven way to design the parameters of EKF, and the gain matrices set by experimental and empirical methods find it hard to achieve good performance [6]. Intelligent optimization algorithms, such as the genetic algorithm and particle swarm optimization, are specialized in parallel searching in solution space and identifying multiple parameters simultaneously; however, this kind of algorithm is difficult to achieve global optimum when dealing with a time-varying multiple-parameter optimization problem [9]. Compared with the above methods, the MRAS exhibits a lot of advantages, such as simple design, small computational complexity, easy implementation in a digital signal processer, and good stability [8]; consequently, MRAS is widely used for online parameter identification and speed control of sensor-less motors. As a part of adaptive control theory, the MRAS has a lot of related research and explorations [7; 8; 11; 12; 13; 14; 15; 16; 17; 18; 19]. Nowadays, three topics are given more attention, including the reference model, MRAS structure, and adaptive law. The traditional reference model generally adopts the \(d\)-\(q\) axis current equation, which has many advantages such as simplicity, practicality, and good stability. In [11; 12], a modified active-power reference model and input impedance angle-based reference model are proposed, respectively. By using those two reference models, sensors and hardware costs can be reduced. However, multiple MRASs are required for parameter identification, which will lead to poor robustness. In addition, the stability of the mentioned methods is difficult to prove. For designing the MRAS structure, the commonly used mode is described in [7; 8; 14; 15; 16; 17; 18], which consists of a reference model, an adjustable model, and the adaptive law. Apart from those, in [13], a new MRAS structure considering the differences in rotor fluxes and electromagnetic torques between the reference model and the adjustable model is proposed, which can effectively improve the performance of speed estimation, however, an appropriate stability proof is also missing. In addition, the adaptive law plays a more significant role in speed estimation and parameter identification [14]. The classical PI adaptive law is the commonly used way because of its simple design and reliable stability; however, the estimation and identification accuracy are seriously affected when high-frequency noise appears [6]. Compared to PI, the switched PI adaptive law is proposed in [18] to improve the anti-disturbance performance of MRAS by flexibly adjusting the proportional and integral gains. Unfortunately, the anti-disturbance ability is not obviously improved due to the anti-disturbance mechanism of PI. Recently, ADRC has been paid much more attention because of its strong anti-disturbance ability.