Online Parameter Identification for PMSM Based on Multi-Innovation Extended Kalman Filtering

Abstract

Subject to magnetic saturation, temperature rise, and other factors, the electrical parameters of permanent magnet synchronous motors (PMSMs) in marine electric propulsion systems exhibit time-varying characteristics. Existing parameter identification algorithms fail to fully satisfy the requirements of high-performance PMSM control systems in terms of accuracy, response speed, and robustness. To address these limitations, this paper introduces multi-innovation theory and proposes a novel multi-innovation extended Kalman filter (MIEKF) for the identification of key electrical parameters of PMSMs, including stator resistance, d-axis inductance, q-axis inductance, and permanent magnet flux linkage. Firstly, the extended Kalman filter (EKF) algorithm is applied to linearize the nonlinear system, enhancing the EKF’s applicability for parameter identification in highly nonlinear PMSM systems. Subsequently, multi-innovation theory is incorporated into the EKF framework to construct the MIEKF algorithm, which utilizes historical state data through iterative updates to improve the identification accuracy and dynamic response speed. An MIEKF-based PMSM parameter identification model is then established to achieve online multi-parameter identification. Finally, a StarSim RCP MT1050-based experimental platform for online PMSM parameter identification is implemented to validate the effectiveness and superiority of the proposed MIEKF algorithm under three operational conditions: no-load, speed variation, and load variation. Experimental results demonstrate that (1) across three distinct operating conditions, compared to forget factor recursive least squares (FFRLS) and the EKF, the MIEKF exhibits smaller fluctuation amplitudes, shorter fluctuation durations, mean values closest to calibrated references, and minimal deviation rates and root mean square errors in identification results; (2) under the load increase condition, the EKF shows significantly increased deviation rates while the MIEKF maintains high identification accuracy and demonstrates enhanced anti-interference ability. This research has achieved a comprehensive improvement in parameter identification accuracy, dynamic response speed, convergence effect, and anti-interference performance, providing an electrical parameter identification method characterized by high accuracy, rapid dynamic response, and strong robustness for high-performance control of PMSMs in marine electric propulsion systems.

1. Introduction

The electrification of transportation is advancing rapidly. MEPSs, characterized by their low-carbon and environmentally friendly operation, flexible control, and high energy utilization efficiency, represent the primary development direction for the electrification of marine electric power systems [1]. PMSMs, renowned for their high efficiency, high power density, and superior torque control characteristics [2], serve as the core power equipment within MEPSs. However, under the combined influence of multiple factors such as magnetic saturation and temperature rise, key electrical parameters of the PMSM—including stator resistance, direct-axis (d-axis) inductance, quadrature-axis (q-axis) inductance, and permanent magnet flux linkage—exhibit significant time-varying characteristics [3]. This time-variance can lead to increased torque ripple and degraded control performance of the PMSM, potentially causing severe issues such as loss of control stability under extreme conditions [4]. Consequently, the online, real-time, and accurate identification of these critical electrical parameters is essential for enhancing PMSM control performance and, thereby, ensuring the stable operation of MEPSs.

Currently, online identification algorithms for PMSM’s electrical parameters primarily include the RLS method [5], the MRAS algorithm [6], intelligent algorithms [7], and the EKF algorithm [8]. The RLS algorithm optimizes model parameters by recursively updating weighted least-squares estimates. Its simple structure and high computational efficiency have led to it and its variants being widely used in fields such as real-time identification of motor parameters [5,9] and identification of marine motion parameters and wave peak frequencies [10]. Generally, algorithms based on RLS have difficulty achieving both dynamic response speed and anti-noise performance. The MRAS and its improved algorithm, while inheriting the dynamic parameter update feature of RLS, enhances the system’s anti-noise performance by introducing a reference model and error feedback mechanism, achieving good results in the field of motor parameter identification. For instance, it reduces the current THD and torque ripple caused by motor parameter mismatch [11,12] and enhances the dynamic performance of sensorless drive for SPMSM [13], etc., achieving both control performance optimization and parameter robustness improvement. However, the MRAS requires precise selection of appropriate adaptive law and feedback mechanism parameters, which increases the difficulty and time cost of algorithm implementation. In recent years, intelligent algorithms represented by PSO (such as dynamic self-learning PSO [14], contraction factor anti-predator PSO [15]) and ANN (Adaline neural network [16]), with their strong global search capabilities and unique advantage of not requiring an accurate mathematical model, have been increasingly applied in the field of online parameter identification. Typically, intelligent identification algorithms can accurately track changes in motor parameters, but they suffer from slow convergence speed, and their estimation accuracy is easily affected by the nonlinearity of the inverter and noisy voltage/current. The EKF algorithm [17], by performing local linearization processing on the nonlinear system and fusing observation and process noise covariance information, can effectively suppress noise interference while balancing optimality and real-time performance, and make up for the defects of other algorithms such as insufficient dynamic response and unstable convergence. It provides a new solution for online identification of electrical parameters of strongly coupled and multivariable PMSMs in high-dynamic change operating conditions. Yuan T. et al. [18] propose an extended Kalman particle filter algorithm combining a particle filter based on Bayesian state estimation and the EKF for parameter identification. This method improves the particle degradation problem in particle filters and enhances the ability to handle highly nonlinear systems. The application of the EKF algorithm in PMSM control in Reference [19] effectively improves the control performance of the motor under complex working conditions. However, the EKF still has certain defects in the application of PMSM parameter identification. It only relies on single-step innovation for recursive calculation, making it difficult to improve estimation accuracy; it is easy to fall into local optimal solutions when dealing with complex systems, and when facing complex working conditions with large dynamic changes and strong interference, it cannot quickly and accurately track state changes, affecting the accuracy and stability of parameter identification.

Different from single-step innovation, the multi-innovation theory constructs an innovation vector by integrating continuous multi-step observation information, effectively suppressing measurement noise interference and accelerating the convergence speed of the algorithm, thereby enhancing the robustness and accuracy of the estimation and optimizing the overall dynamic response speed of the system. Although the theory has been preliminarily applied in some fields, there are few reports on its application in the field of PMSM parameter identification. Wang Z. et al. [20] propose a forgetting factor stochastic gradient algorithm based on the multi-innovation theory for parameter estimation of nonlinear exponential autoregressive models in which the multi-innovation theory improves the algorithm performance by increasing data utilization. Ding F et al. [21] demonstrate lemmas and theorems on least squares and multi-innovation least squares parameter estimation algorithms, which can be extended to multivariable stochastic systems and multi-input-output systems. Zhao B. et al. [22] propose a novel identification algorithm for 3-DoF ship maneuvering modeling based on the recursive least squares method combined with multi-innovation of focused innovation processing and nonlinear innovation technology, which shows good accuracy and convergence. Yang D. et al. [23] combine the multi-innovation identification theory with negative gradient search to propose an iterative algorithm based on multi-innovation gradient, which reduces the computation and improves the parameter estimation accuracy.

Considering the above, the purpose of this paper is to present an MIEKF method, combining the dual advantages of the EKF and multi-innovation theory, for the identification of key electrical parameters of PMSMs. Linearization of the highly nonlinear PMSM system is performed through the EKF Jacobian matrix; multi-innovation theory is introduced to utilize additional historical data, compensating for EKF linearization errors while effectively suppressing noise, thereby enhancing the accuracy, dynamic response speed, and robustness of parameter identification. This paper is organized as follows: Section 2 expounds on the theory of the MIEKF. The PMSM parameter identification model based on the MIEKF is presented in Section 3. Section 4 verifies the effectiveness of the proposed method through experiments. Conclusions are enclosed in Section 5.

2. Multi-Innovation Extended Kalman Filter

2.1. Extended Kalman Filter

The KF has excellent performance in the field of state estimation and parameter identification [24], but it is usually only suitable for linear systems. Based on inheriting the excellent identification performance of the KF, the EKF linearizes the nonlinear system at each moment, which can be used for parameter identification of highly nonlinear PMSMs.

The block diagram of recursive estimation for a linear system using the KF algorithm is shown in Figure 1.

The linear system can be expressed by the state and measurement equations as follows:

$$\left\{\begin{array}{l}\dot{\mathit{x}}(t)=A(t)\mathit{x}(t)+B(t)\mathit{u}(t)+\mathit{w}(t)\hfill \\ \mathit{y}(t)=C(t)\mathit{x}(t)+\mathit{v}(t)\hfill \end{array}\right.$$

(1)

where $\mathit{x}(t)$ is a state vector containing the system’s internal state information; $\dot{\mathit{x}}(t)$ denotes the derivative of $\mathit{x}(t)$ with respect to time t, reflecting the change rate of the system state over time; $\mathit{u}(t)$ is the input vector composed of control inputs or excitations applied to the system from the outside; $\mathit{y}(t)$ is the measurable system output; $\mathit{w}(t)$ is the process noise vector, representing the impact of uncertain disturbances on state transitions; $\mathit{v}(t)$ is the measurement noise vector, accounting for errors and interference during sensing; and $A(t)$, $B(t)$, and $C(t)$ are the system, input, and output matrixes, respectively.

The state and measurement equations for a continuous nonlinear system can be expressed as follows [25]:

$$\left\{\begin{array}{l}\dot{\mathit{x}}(t)=\mathit{f}(\mathit{x}(t),\mathit{u}(t),t)+\mathit{w}(t)\hfill \\ \mathit{y}=\mathit{h}(\mathit{x}(t),t)+\mathit{v}(t)\hfill \end{array}\right.$$

(2)

where $\mathit{f}(\mathit{x}(t),\mathit{u}(t),t)$ and $\mathit{h}(\mathit{x}(t),t)$ are both nonlinear functions of $\mathit{x}(t)$.

The estimated form of Equation (2) is given by the following equations:

$$\begin{array}{l}\widehat{\dot{\mathit{x}}}=\mathit{f}(\hat{\mathit{x}}(t),\mathit{u}(t),t)\hfill \\ \hat{\mathit{y}}=\mathit{h}(\hat{\mathit{x}}(t),t)\hfill \end{array}$$

(3)

where parameters with the superscript “^” denote identified values, and parameters without “^” represent actual measurement values.

Linearization is subsequently applied to the nonlinear system. Expanding Equation (3) via Taylor series yields the following:

$$\left\{\begin{array}{l}\dot{\mathit{x}}(t)=\mathit{f}(\hat{\mathit{x}}(t),\mathit{u}(t),t)+{\left.{\displaystyle \frac{\partial f(\mathit{x},\mathit{u},t)}{\partial \mathit{x}}}\right|}_{\mathit{x}(t)=\hat{\mathit{x}}(t)}[\mathit{x}(t)-\hat{\mathit{x}}(t)]+{\mathit{R}}_n(\mathit{x})+\mathit{w}(t)\hfill \\ \mathit{y}(t)=\mathit{h}(\hat{\mathit{x}}(t),t)+{\left.{\displaystyle \frac{\partial h(\mathit{x},t)}{\partial \mathit{x}}}\right|}_{\mathit{x}(t)=\hat{\mathit{x}}(t)}[\mathit{x}(t)-\hat{\mathit{x}}(t)]+{\mathit{S}}_n(\mathit{x})+\mathit{v}(t)\hfill \end{array}\right.$$

(4)

where ${\mathit{R}}_n(\mathit{x})$ and ${\mathit{S}}_n(\mathit{x})$ correspond to the high-order terms in the state and measurement equations of the nonlinear system, respectively.

Neglecting the high-order terms ${\mathit{R}}_n(\mathit{x})$ and ${\mathit{S}}_n(\mathit{x})$ in Equation (4), and setting $\Delta \mathit{x}(t)=\mathit{x}(t)-\hat{\mathit{x}}(t)$, the difference between Equations (3) and (4) yields the following linear equation:

$$\left\{\begin{array}{l}\Delta \dot{\mathit{x}}(t)=\mathit{F}(t)\Delta \mathit{x}(t)+\mathit{w}(t)\hfill \\ \Delta \mathit{y}=\mathit{H}(t)\Delta \mathit{x}(t)+\mathit{v}(t)\hfill \end{array}\right.$$

(5)

where $\mathit{F}(t)$ and $\mathit{H}(t)$ are Jacobian matrixes, defined, respectively, as follows:

$$\mathit{F}(t)={\left.{\displaystyle \frac{\partial \mathit{f}(\mathit{x},\mathit{u},t)}{\partial \mathit{x}}}\right|}_{x=\hat{x}}=\left[\begin{array}{cccc}{\displaystyle \frac{\partial {\mathit{f}}_1}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{f}}_1}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{f}}_1}{\partial {\mathit{x}}_n}}\\ {\displaystyle \frac{\partial {\mathit{f}}_2}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{f}}_2}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{f}}_2}{\partial {\mathit{x}}_n}}\\ ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{\partial {\mathit{f}}_n}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{f}}_n}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{f}}_n}{\partial {\mathit{x}}_n}}\end{array}\right]$$

(6)

$$\mathit{H}(t)={\left.{\displaystyle \frac{\partial \mathit{h}(\mathit{x},t)}{\partial \mathit{x}}}\right|}_{\mathit{x}=\hat{\mathit{x}}}=\left[\begin{array}{cccc}{\displaystyle \frac{\partial {\mathit{h}}_1}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{h}}_1}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{h}}_1}{\partial {\mathit{x}}_{\mathrm{n}}}}\\ {\displaystyle \frac{\partial {\mathit{h}}_2}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{h}}_2}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{h}}_2}{\partial {\mathit{x}}_{\mathrm{n}}}}\\ ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{\partial {\mathit{h}}_{\mathit{m}}}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{h}}_{\mathrm{m}}}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{h}}_{\mathrm{m}}}{\partial {\mathit{x}}_{\mathrm{n}}}}\end{array}\right]$$

(7)

Discretizing the state and measurement equations in Equation (1), we can obtain the following:

$$\left\{\begin{array}{l}{\mathit{x}}_k={\mathit{\varphi }}_{k-1}{\mathit{x}}_{k-1}+{\mathit{w}}_{k-1}\hfill \\ {\mathit{y}}_k={\mathit{H}}_k{\mathit{x}}_{k-1}+{\mathit{v}}_{k-1}\hfill \end{array}\right.$$

(8)

where ${\mathit{\varphi }}_{k-1}$ and ${\mathit{H}}_k$ are the state transition and measurement matrixes, respectively.

When the sampling time T_s is sufficiently small, ${\mathit{\varphi }}_{k-1}$ and ${\mathit{H}}_k$ can be, respectively, approximated as

$${\mathit{\varphi }}_{k-1}\approx \mathit{I}+{\mathit{F}}_{t_{k-1}}{T}_s=\mathit{I}+\left[\begin{array}{cccc}{\displaystyle \frac{\partial {\mathit{f}}_1(\mathit{x},\mathit{u},t_{k-1})}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{f}}_1(\mathit{x},\mathit{u},t_{k-1})}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{f}}_1(\mathit{x},\mathit{u},t_{k-1})}{\partial {\mathit{x}}_{\mathrm{n}}}}\\ {\displaystyle \frac{\partial {\mathit{f}}_2(\mathit{x},\mathit{u},t_{k-1})}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{f}}_2(\mathit{x},\mathit{u},t_{k-1})}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{f}}_2(\mathit{x},\mathit{u},t_{k-1})}{\partial {\mathit{x}}_n}}\\ ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{\partial {\mathit{f}}_n(\mathit{x},\mathit{u},t_{k-1})}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{f}}_n(\mathit{x},\mathit{u},t_{k-1})}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{f}}_n(\mathit{x},\mathit{u},t_{k-1})}{\partial {\mathit{x}}_n}}\end{array}\right]\ast T_{\mathit{s}}$$

(9)

$${\mathit{H}}_k=\left[\begin{array}{cccc}{\displaystyle \frac{\partial {\mathit{h}}_1(\mathit{x},t_k)}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{h}}_1(\mathit{x},t_k)}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{h}}_1(\mathit{x},t_k)}{\partial {\mathit{x}}_n}}\\ {\displaystyle \frac{\partial {\mathit{h}}_2(\mathit{x},t_k)}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial h_2(\mathit{x},t_k)}{\partial x_2}}& \dots & {\displaystyle \frac{\partial {\mathit{h}}_2(\mathit{x},t_k)}{\partial {\mathit{x}}_n}}\\ ⋮& ⋮& ⋮& ⋮\\ {\displaystyle \frac{\partial {\mathit{h}}_m(\mathit{x},t_k)}{\partial {\mathit{x}}_1}}& {\displaystyle \frac{\partial {\mathit{h}}_m(\mathit{x},t_k)}{\partial {\mathit{x}}_2}}& \dots & {\displaystyle \frac{\partial {\mathit{h}}_m(\mathit{x},t_k)}{\partial {\mathit{x}}_n}}\end{array}\right]$$

(10)

In parameter identification, state estimates are categorized into a priori and a posteriori estimate. At the start of identification (k = 0), let $E[\mathit{x}(0)]={\hat{\mathit{x}}}_0$ and $E[({\mathit{x}}_0-E[{\mathit{x}}_0]){({\mathit{x}}_0-E[{\mathit{x}}_0])}^T]={\mathit{P}}_0$. When k = 1, calculate the first measurement. The posteriori estimate ${\hat{\mathit{x}}}_k^{+}$ is obtained by estimating ${\mathit{x}}_k$ using measurements up to and including time k; the priori estimate ${\hat{\mathit{x}}}_k^{-}$ is derived by estimating ${\mathit{x}}_k$ using measurements prior to time k. Though both ${\hat{\mathit{x}}}_k^{+}$ and ${\hat{\mathit{x}}}_k^{-}$ are estimates of ${\mathit{x}}_k$, ${\hat{\mathit{x}}}_k^{+}$ utilizes more measurement data and thus yields greater accuracy than ${\hat{\mathit{x}}}_k^{-}$. Neglecting process noise effects, the priori estimate ${\hat{\mathit{x}}}_k^{-}$ at time k is propagated from the previous state estimate via the nonlinear state transition function:

$${\widehat{\dot{\mathit{x}}}}_{k-1}={\displaystyle \frac{{\hat{\mathit{x}}}_k^{-}-{\hat{\mathit{x}}}_{k-1}^{+}}{T_s}}=\mathit{f}({\hat{\mathit{x}}}_{k-1}^{+},{\mathit{u}}_{k-1})$$

(11)

Thus, the priori estimate ${\hat{\mathit{x}}}_k^{-}$ is expressed as follows:

$${\hat{\mathit{x}}}_k^{-}={\hat{\mathit{x}}}_{k-1}^{+}+\mathit{f}({\hat{\mathit{x}}}_{k-1}^{+},{\mathit{u}}_{k-1})T_s$$

(12)

Finally, the recursive equations of the EKF for the discrete, nonlinear system are obtained as follows:

$$\left\{\begin{array}{l}{\hat{\mathit{x}}}_k^{-}={\mathit{x}}_{k-1}^{+}+\mathit{f}({\hat{\mathit{x}}}_{k-1}^{+},{\mathit{u}}_{k-1})T_s\hfill \\ {\mathit{P}}_k^{-}={\mathit{\varphi }}_{k-1}{\mathit{P}}_{k-1}^{+}{\mathit{\varphi }}_{k-1}^T+{\mathit{Q}}_{k-1}\hfill \\ {\mathit{K}}_k={\mathit{P}}_k^{-}{\mathit{H}}_k^T{({\mathit{H}}_k{\mathit{P}}_k^{-}{\mathit{H}}_k^T+{\mathit{R}}_k)}^{-1}\hfill \\ {\hat{\mathit{x}}}_k^{+}={\hat{\mathit{x}}}_k^{-}+{\mathit{K}}_k({\mathit{y}}_k-{\mathit{H}}_k{\hat{\mathit{x}}}_k^{-})\hfill \\ {\mathit{P}}_k^{+}={\mathit{P}}_k^{-}-{\mathit{K}}_k{\mathit{H}}_k{\mathit{P}}_k^{-}\hfill \end{array}\right.$$

(13)

where ${\mathit{P}}_k^{-}$ and ${\mathit{P}}_k^{+}$ are covariance matrixes of the priori estimate error and the posteriori estimate error, respectively, and Q_k and R_k are covariance matrixes of the process noise and the observation noise, respectively.

2.2. Multi-Innovation Theory

The conventional EKF exhibits limitations in identification accuracy and dynamic response speed when applied to nonlinear PMSM systems. In this section, the multi-innovation theory is introduced. By using the data information updated iteratively at multiple historical moments, the residual scalar in the EKF is expanded into an innovation matrix, enabling reuse and online updating of historical state data. This enhancement improves both identification accuracy and dynamic response performance.

Let $\mathit{\theta }$ denote the parameter vector to be identified, and its stochastic gradient identification algorithm is expressed as

$$\hat{\mathit{\theta }}(k)=\hat{\mathit{\theta }}(k-1)+{\mathit{K}}_k\mathit{e}(k)$$

(14)

where ${\mathit{K}}_k$ is the Kalman gain vector, and $\mathit{e}(k)$ is the error vector (termed scalar innovation), defined as

$${\mathit{e}}_k=\mathit{Z}(k)-\hat{\mathit{Z}}(k-1)$$

(15)

where $\mathit{Z}(k)$ is the measurement at time k, and $\hat{\mathit{Z}}(k-1)$ is the estimate of $\mathit{Z}(k)$ at time k − 1.

The innovation length significantly impacts identification accuracy [26]. Expanding the scalar innovation $\mathit{e}(k)$ by innovation length p yields the extended innovation matrix ${\mathit{E}}_p(k)$:

$${\mathit{E}}_p(k)=\left[\begin{array}{c}\mathit{e}(k)\\ \mathit{e}(k-1)\\ \dots \\ \mathit{e}(k-p)\end{array}\right]=\left[\begin{array}{c}\mathit{Z}(k)-\hat{\mathit{Z}}(k-1)\\ \mathit{Z}(k-1)-\hat{\mathit{Z}}(k-2)\\ \dots \\ \mathit{Z}(k-p)-\hat{\mathit{Z}}(k-p-1)\end{array}\right]$$

(16)

Concurrently, ${\mathit{K}}_k$ is expanded into the gain matrix ${\mathit{K}}_{\mathit{p}}(k)$:

$${\mathit{K}}_{\mathit{p}}(k)=\left[\begin{array}{c}\mathit{K}(k)\\ \mathit{K}(k-1)\\ \dots \\ \mathit{K}(k-p)\end{array}\right]=\left[\begin{array}{c}{\mathit{P}}_k^{-}{\mathit{H}}_k^T{({\mathit{H}}_k{\mathit{P}}_k^{-}{\mathit{H}}_k^T+{\mathit{R}}_k)}^{-1}\\ {\mathit{P}}_{k-1}^{-}{\mathit{H}}_{k-1}^T{({\mathit{H}}_{k-1}{\mathit{P}}_{k-1}^{-}{\mathit{H}}_{k-1}^T+{\mathit{R}}_{k-1})}^{-1}\\ \dots \\ {\mathit{P}}_{k-p}^{-}{\mathit{H}}_{k-p}^T{({\mathit{H}}_{k-p}{\mathit{P}}_{k-p}^{-}{\mathit{H}}_{k-p}^T+{\mathit{R}}_{k-p})}^{-1}\end{array}\right]$$

(17)

With ${\mathit{E}}_p(k)$ and ${\mathit{K}}_{\mathrm{p}}(k)$ obtained through extension, Equation (14) is reformulated as follows:

$$\hat{\mathit{\theta }}(k)=\hat{\mathit{\theta }}(k-1)+{\mathit{K}}_p(k){\mathit{E}}_p(k)$$

(18)

Consequently, term ${\hat{\mathit{x}}}_{\mathit{k}}^{+}$ in the EKF recursion (Equation (13)) is replaced by

$${\hat{\mathit{x}}}_{\mathit{k}}^{+}={\hat{\mathit{x}}}_{\mathit{k}}^{-}+{\mathit{K}}_{\mathit{p}}(k){\mathit{E}}_{\mathit{p}}(k)$$

(19)

This process enables reuse and online updating of data from p past time steps.

3. PMSM Parameter Identification Model Based on MIEKF

3.1. Construction of PMSM Parameter Identification Model Based on MIEKF

The PMSM constitutes a multivariable, strongly coupled nonlinear system. To facilitate analysis, disturbance factors including magnetic saturation effects, core eddy currents, hysteresis losses, and higher-order harmonics are neglected. Under the d-q synchronous rotating reference frame, the stator voltage state equations of the PMSM simplify to

$$\left\{\begin{array}{l}{u}_d=R_s{i}_d+L_d{\displaystyle \frac{d{i}_d}{dt}}-{\omega }_e{L}_q{i}_q\\ u_q=R_s{i}_q+L_q{\displaystyle \frac{d{i}_q}{dt}}+{\omega }_e{L}_d{i}_d+{\omega }_e{\psi }_f\end{array}\right.$$

(20)

where $u_d$ and $u_q$ are the d-axis and q-axis voltages, respectively; $i_d$ and $i_q$ are the d-axis and q-axis currents, respectively; $L_d$ and $L_q$ are the d-axis and q-axis inductances, respectively; $R_s$ is the stator resistance; ${\psi }_f$ is the permanent magnet flux linkage; and ${\omega }_e$ is the electrical angular velocity.

This study requires simultaneous identification of four key electrical parameters: $R_s$, $L_d$, $L_q$, and ${\psi }_f$. However, the stator voltage state equation (Equation (20)) has a rank of only two, rendering it rank-deficient. This condition may cause convergence difficulties and erroneous identification results. To resolve the rank deficiency, a stepwise identification approach using dq-axis voltage equations is adopted. One EKF is used to identify $R_s$ and ${\psi }_f$, and another EKF is used to identify $L_d$ and $L_q$. The identification principle proceeds as follows:

First, assuming $L_d$ and $L_q$ remain constant during the sampling period T_s, and taking parameters $a=R_s$ and $b={\psi }_f$ as system state variables for identification, selected state variables and output variables can be expressed as follows:

$$\left\{\begin{array}{l}\mathit{x}={\left[\begin{array}{cccc}{i}_{\mathrm{d}}& i_{\mathrm{q}}& a& b\end{array}\right]}^T\\ \mathit{y}={\left[\begin{array}{cc}{i}_{\mathrm{d}}& i_{\mathrm{q}}\end{array}\right]}^T\end{array}\right.$$

(21)

Building upon the linearization method for discrete nonlinear systems established earlier, an EKF-based parameter identification approach for PMSMs is formulated. The mathematical model of the PMSM is expressed in state-space form as follows:

$$\left\{\begin{array}{l}\dot{\mathit{x}}={\mathit{f}}^{\mathit{R}\mathit{\Psi }}(\mathit{x},\mathit{u},\mathit{t})+\mathit{w}\\ \mathit{y}=\mathit{h}\left(\mathit{x},t\right)+\mathit{v}\end{array}\right.$$

(22)

where

$$\left\{\begin{array}{l}\mathit{y}={[i_d,i_q]}^T\\ \mathit{x}={[i_d,i_q,a,b]}^T\\ \dot{\mathit{x}}={\displaystyle \frac{d{[i_d,i_q,\widehat{a},\widehat{b}]}^T}{dt}}\\ {\mathit{f}}^{\mathit{R}\mathit{\Psi }}(\mathit{x},\mathit{u},t)={[{\displaystyle \frac{-a{i}_d+{\omega }_e{i}_q{L}_q+u_d}{L_d}},{\displaystyle \frac{-{\omega }_e{i}_d{L}_d-a{i}_q+u_q-{\omega }_{e}b}{L_q}},0,0]}^T\\ \mathit{h}\left(\mathit{x},t\right)=H_k={\left.{\displaystyle \frac{\partial h}{\partial \mathit{x}}}\right|}_{\mathit{x}={\hat{\mathit{x}}}_k^{-}}=\left[\begin{array}{l}\begin{array}{cccc}1& 0& 0& 0\end{array}\\ \begin{array}{cccc}0& 1& 0& 0\end{array}\end{array}\right]\end{array}\right.$$

(23)

The discrete state transition matrix for this identification system is derived by computing the partial derivatives of each element in matrix ${\mathit{f}}^{\mathit{R}\mathit{\Psi }}(\mathit{x},\mathit{u},t)$ above with respect to the four state variables:

$${\mathit{\varphi }}_{\mathit{k}-1}^{\mathit{R}\mathit{\Psi }}=\mathit{I}+{\left.{\displaystyle \frac{\partial \mathit{h}}{\partial \mathit{x}}}\right|}_{\mathit{x}={\hat{\mathit{x}}}_{k-1}^{-}}·T_s=\left[\begin{array}{cccc}1-{\displaystyle \frac{a}{L_d(k)}}& {\displaystyle \frac{{\omega }_e(k)L_q(k)}{L_d(k)}}& -{\displaystyle \frac{i_d(k)}{L_d(k)}}& 0\\ -{\displaystyle \frac{{\omega }_e(k)L_d(k)}{L_q(k)}}& 1-{\displaystyle \frac{a}{L_q(k)}}& 0& -{\displaystyle \frac{{\omega }_e(k)}{L_q(k)}}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]·T_s$$

(24)

Subsequently, assuming $R_s$ and ${\psi }_f$ remain constant during sampling period $T_s$, and letting $a=1/L_d$ and $b=1/L_q$, we obtain the following:

$${\mathit{f}}^{\mathit{L}}(\mathit{x},\mathit{u},t)=\left[\begin{array}{c}-a{R}_s{i}_d+{\omega }_e{i}_q{\displaystyle \frac{a}{b}}+a{u}_d\\ -{\omega }_e{i}_d{\displaystyle \frac{b}{a}}-b{R}_s{i}_q+b{u}_q-b{\omega }_e{\psi }_f\\ 0\\ 0\end{array}\right]$$

(25)

Furthermore, the discrete state transition matrix is thus given by

$${\mathit{\varphi }}_{\mathit{k}-1}^{\mathit{L}}=\mathit{I}+{\left.{\displaystyle \frac{\partial \mathit{h}}{\partial \mathit{x}}}\right|}_{\mathit{x}={\hat{\mathit{x}}}_{k-1}^{-}}·T_s=\left[\begin{array}{cccc}1-a{R}_s(k)& {\displaystyle \frac{a{\omega }_e(k)}{b}}& \mathrm{A}& {\displaystyle \frac{-a{\omega }_e(k)i_q(k)}{b^2}}\\ -{\displaystyle \frac{{\omega }_e(k)b}{a}}& 1-b{R}_s(k)& {\displaystyle \frac{b{\omega }_e(k)i_d(k)}{a^2}}& {\rm B}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]·T_s$$

(26)

where

$$\begin{array}{l}\mathrm{A}=u_d(k)-R_s(k)i_d(k)+{\displaystyle \frac{{\omega }_e(k)i_q(k)}{b}}\\ \mathrm{B}=u_q(k)-R_s(k)i_q(k)-{\omega }_e(k){\psi }_f(k)-{\displaystyle \frac{{\omega }_e(k)i_d(k)}{a}}\end{array}$$

(27)

3.2. Process of PMSM Parameter Identification Based on MIEKF

Building upon the preceding method, the workflow of the MIEKF parameter identification algorithm is illustrated in Figure 2, with detailed procedural steps as follows:

(1) Initialize the mean ${\hat{\mathit{x}}}_0^{+}$ and the covariance ${\mathit{P}}_0^{+}$. The noise matrixes ${\mathit{Q}}_k$ and ${\mathit{R}}_k$ are assigned via trial and error method.

(2) Update the priori estimate ${\hat{\mathit{x}}}_k^{-}$ and the covariance of the state estimate error ${\mathit{P}}_k^{-}$:

$${\hat{\mathit{x}}}_k^{-}={\mathit{x}}_{k-1}^{+}+\mathit{f}({\hat{\mathit{x}}}_{k-1}^{+},{\mathit{u}}_{k-1})T_s$$

(28)

$${\mathit{P}}_k^{-}={\mathbf{\varphi }}_{k-1}{\mathit{P}}_{k-1}^{+}{\mathbf{\varphi }}_{k-1}^T+{\mathit{Q}}_{k-1}$$

(29)

(3) Define the innovation matrix ${\mathit{E}}_p(k)$:

$${\mathit{E}}_p(k)={\left[\mathit{e}(k),(k-1),\cdot \cdot \cdot ,\mathit{e}(k-p)\right]}^T$$

(30)

(4) Calculate the Kalman gain matrix ${\mathit{K}}_p(k)$:

$${\mathit{K}}_p(k)=[{\mathit{K}}_k,{\mathit{K}}_{k-1},\cdot \cdot \cdot ,{\mathit{K}}_{k-p}]$$

(31)

(5) Update the posteriori estimate ${ \mathit{x} ^}_k^{+}$ and the state error covariance ${\mathit{P}}_k^{+}$:

$${\hat{\mathit{x}}}_k^{+}={\hat{\mathit{x}}}_k^{-}+K_p(k)[\mathit{Z}(k)-\hat{\mathit{Z}}(k-1)]$$

(32)

$${\mathit{P}}_k^{+}={\mathit{P}}_k^{-}-K_p(k){\mathit{P}}_k^{-}{K}^T(k)$$

(33)

(6) Update the output value.

Repeat steps (2)–(5), and then the PMSM parameter identification results can be obtained.

4. Experimental Verification

4.1. Experimental Platform

To verify the effectiveness of the proposed MIEKF online parameter identification method for PMSMs, an experimental platform was established based on the RCP (MT1050) from ModelingTech (Shanghai, China). It primarily consists of a test motor, a load motor, drivers of the test and load motors, an RCP unit, and a host computer, as illustrated in Figure 3. A PMSM vector control system was implemented within the StarSim RCP software (version 5.2.0), depicted in Figure 4. The inputs to the MIEKF identification algorithm include the dq-axis voltage signals output from the vector control system, the dq-axis current signals collected by the current sensor and transformed via coordinate transformation, and the electrical angular velocity signal measured by the position encoder. The parameters of the tested PMSM are listed in

Table 1. The main parameters of the test PMSM.

Motor Parameter	Value	Motor Parameter	Value
Rated power (kW)	5.5	Stator resistance (Ω)	1.08
Rated speed (r/min)	1500	d-axis inductance (mH)	8.38
Rated current (A)	10	q-axis inductance (mH)	25.6
Rated torque (N·m)	36	Permanent magnet flux linkage (Wb)	0.416
Rated frequency (Hz)	50	Number of pole pairs	4

.

4.2. Algorithm Evaluation Metrics and Parameter Settings

4.2.1. Evaluation Metrics

To assess the identification performance of the algorithms, this study employed a set of evaluation metrics, including the mean value μ, deviation rate $\epsilon$, and RMSE. The calculation formulae for these metrics are as follows:

$$\mu ={\displaystyle \frac{{\displaystyle {\sum }_{k=1}^N{\widehat{x}}_k}}{N}}$$

(34)

$$\epsilon ={\displaystyle \frac{{\displaystyle {\sum }_{k=1}^N\left|{\widehat{x}}_k-x_k\right|}}{N}}\times 100\%$$

(35)

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{k=1}^N{({\widehat{x}}_k-x_k)}^2}}$$

(36)

4.2.2. Selection of Innovation Length

As discussed previously, the innovation length p significantly influences the identification accuracy of the MIEKF algorithm. The identification deviation rates of key PMSM electrical parameters (such as $R_s$, $L_d$, $L_q$, and ${\psi }_f$) using the MIEKF under different innovation lengths are presented in

Table 2. The identification deviation rate $\epsilon$ of the MIEKF under different innovation lengths.

Innovation Lengths	Rs (%)	Ld (%)	Lq (%)	${\mathit{\psi }}_{\mathit{f}}$ (%)
p = 1	0.5113	0.7217	0.3922	0.2243
p = 3	0.4923	0.5751	0.0856	0.2108
p = 5	0.4746	0.1897	0.1554	0.2297
p = 7	0.4397	0.1046	0.1005	0.2026
p = 10	0.4081	0.0928	0.0726	0.1929

. The simulation conditions were PMSM operating under no-load condition at a speed of 1000 r/min. Note: when the innovation length p = 1, the MIEKF degenerates into the standard EKF.

As indicated in Table 2, the identification deviation rates $\epsilon$ of the MIEKF for all parameters generally decrease with the increase of innovation length p. For instance, the $\epsilon$ for parameter L_d is 0.7217% at p = 1 but decreases substantially to 0.0928% at p = 10. Although a larger p enhances parameter identification accuracy, it also leads to increased computational load and longer processing times, potentially causing data saturation under limited memory conditions. Therefore, practical applications require selecting an appropriate p that balances controller hardware capabilities with identification accuracy requirements. Further analysis of Table 2 reveals that as p increases, the reduction in $\epsilon$ exhibits a decelerating trend. Take L_d for example, the $\epsilon$ decreases 0.0851 when p changes from 5 to 7, which is only 0.0118 when p changes from 7 to 10. The same phenomenon also exists for other parameters. To balance the identification accuracy and the computation, p = 7 is selected as the best innovation length for the MIEKF in subsequent studies.

4.2.3. Parameter Initialization Settings of EKF

The main parameters requiring initialization in the EKF comprise the covariance matrixes of state estimation error P, process noise Q, and observation noise R.

P reflects the degree of uncertainty in the initial state estimates. For the PMSM parameter identification scenario in this study, the initial estimates of parameters in the state vector (such as $R_s$, $L_d$, $L_q$, and ${\psi }_f$) are derived from calibrated values of the test PMSM. Given the high confidence in these initial estimates, the initial covariance can be set to a small value to avoid significant identification fluctuations caused by excessive initial uncertainty. Accordingly, the initial value of P in this work is set to ${\mathit{P}}_0^{rf+}=diag\left(\left[\begin{array}{cccc}0.1& 0.1& 1& 0.5\end{array}\right]\right)$ and ${\mathit{P}}_0^{l+}=diag\left(\left[\begin{array}{cccc}0.1& 0.1& 0.01& 0.01\end{array}\right]\right)$.

Q characterizes the statistical properties of noise during system state transitions, representing the uncertainty in model predictions. For PMSM systems, it quantifies the noise intensity from processes such as parameter variations due to magnetic saturation and temperature rise. For initial value determination, this study employs a trial-and-error approach by comparing the MIEKF identification performance under multiple Q_k. Results indicate minimal $\epsilon$ and RMSE when ${\mathit{Q}}_k=diag\left(\left[\begin{array}{cccc}1& 1& 50& 50\end{array}\right]\right)$, achieving optimal balance between dynamic response speed during parameter variations and avoidance of noise over-fitting. Furthermore, as the time-varying processes of $R_s$, $L_d$, $L_q$, and ${\psi }_f$ are mutually independent with uncoupled process noise, all non-diagonal elements are set to 0. Thus, Q_k is configured as ${\mathit{Q}}_k=diag\left(\left[\begin{array}{cccc}1& 1& 50& 50\end{array}\right]\right)$.

R characterizes the statistical properties of sensor measurement noise, representing the uncertainty in sensor readings. This study first determines the initial value range of this matrix by referencing the accuracy of the current and voltage sensors used in the experimental platform. Subsequently, through a trial-and-error approach comparing the MIEKF identification performance under multiple R_k, it is found that when ${\mathit{R}}_k=diag\left(\left[\begin{array}{cc}1& 1\end{array}\right]\right)$, the $\epsilon$ and RMSE of identification results are minimized, achieving optimal measurement noise suppression while ensuring effective updating of measurement information. Consequently, R_k is set to ${\mathit{R}}_k=diag\left(\left[\begin{array}{cc}1& 1\end{array}\right]\right)$.

4.3. Validation of the Effectiveness of MIEKF

This section verifies the parameter identification effectiveness of the MIEKF under three operational conditions: no-load, speed variation, and load variation. Comparative analysis with classical FFRLS and EKF algorithms further demonstrates the MIEKF’s superiority.

4.3.1. No-Load Condition

The PMSM was started under no-load conditions and then stabilized at 1000 r/min. The identification experimental curves of key electrical parameters are illustrated in Figure 5, while evaluation metrics for assessing identification accuracy are listed in

Table 3. Evaluation metrics for key electrical parameters under no-load conditions.

Parameter	Calibrated Value	Method	μ	ε (%)	RMSE
Rs (mΩ)	1080	FFRLS	1113.51	3.1028	33.4963
		EKF	1080.03	0.0028	0.1016
		MIEKF	1080.01	0.0009	0.0419
Ld (μH)	8380	FFRLS	8379.78	0.0026	12.9354
		EKF	8379.26	0.0088	0.7973
		MIEKF	8379.91	0.0011	0.0606
Lq (μH)	25,600	FFRLS	25,586.41	0.0532	4.0162
		EKF	25,601.79	0.0069	3.3493
		MIEKF	25,600.09	0.0004	2.4607
${\psi }_f$ ($\mathrm{mWb}$)	416	FFRLS	428.888	3.0981	12.8875
		EKF	416.003	0.0007	0.0030
		MIEKF	416.001	0.0002	0.0008

.

It can be seen from Figure 5 that under no-load operational condition, FFRLS exhibits non-convergent identification results for $R_s$ and ${\psi }_f$, revealing inadequate adaptability to parameter variations. The EKF displays substantial outcome fluctuations that compromise precision, while the MIEKF achieves convergent identification with minimal oscillations, especially for $R_s$ and $L_d$, delivering optimal performance.

Table 3 further indicates that all three algorithms (FFRLS, EKF, and MIEKF) yield PMSM parameter identification results close to calibrated values. Specifically, FFRLS exhibits the highest $\epsilon$ and RMSE. For example, its $R_s$ identification achieves 3.1028% $\epsilon$ and 33.4963 RMSE, significantly exceeding the EKF (0.0028% $\epsilon$ and 0.1016 RMSE) and MIEKF (0.0009% $\epsilon$ and 0.0419 RMSE). Both the EKF and MIEKF demonstrate favorable results, with the MIEKF attaining mean values closer to calibrated references alongside lower $\epsilon$ and RMSE. For instance, the $\epsilon$ of L_q decreases from 0.0069% (EKF) to 0.0004% (MIEKF), while RMSE reduces from 3.3493 to 2.4607—a 26.5% reduction. Thus, the MIEKF achieves the highest identification accuracy under no-load condition, validating its effectiveness.

4.3.2. Speed Variation Condition

The PMSM was started under no-load condition and then stabilized at 1000 r/min. At t = 5 s, the speed began increasing and reached 1100 r/min after 5 s, with the rate of increase in speed constrained to 20 r/s. The identification experimental curves of key electrical parameters throughout this process are presented in Figure 6, while evaluation metrics for identification accuracy are listed in

Table 4. Evaluation metrics for key electrical parameters under speed variation conditions.

Parameter	Calibrated Value	Method	μ	ε (%)	RMSE
Rs (mΩ)	1080	FFRLS	1113.45	3.1009	33.4218
		EKF	1080.07	0.0065	0.0208
		MIEKF	1080.003	0.0002	0.0093
Ld (μH)	8380	FFRLS	8381.47	0.0175	7.8521
		EKF	8379.96	0.0005	0.0347
		MIEKF	8379.99	0.0001	0.0156
Lq (μH)	25,600	FFRLS	25,598.83	0.0046	2.0153
		EKF	25,600.32	0.0013	2.0825
		MIEKF	25,600.17	0.0007	1.0234
${\psi }_f$ ($\mathrm{mWb}$)	416	FFRLS	428.875	3.0949	12.8876
		EKF	416.0006	0.0001	0.0004
		MIEKF	416.0008	0.0002	0.0005

.

It can be seen from Figure 6 that under speed variation condition, FFRLS identification results exhibit large fluctuations, instability, and significant glitches. This method excessively relies on historical data, resulting in sluggish response to electrical parameter changes induced by speed variations. Through linearization of nonlinear systems, the EKF is capable of promptly tracking and responding to the speed variations. However, its identification results for L_d (Figure 6c) and L_q (Figure 6d) show considerable oscillations, indicating a need for enhanced noise immunity. Compared with the EKF, the MIEKF exhibits reduced fluctuations in the identification results of all electrical parameters and demonstrates superior dynamic response performance, especially for ${\psi }_f$ (Figure 6b) and L_d (Figure 6c). It shows significantly enhanced robustness.

Table 4 further indicates that FFRLS exhibits the highest $\epsilon$ and RMSE, while the MIEKF achieves mean values closest to calibrated references. Its $\epsilon$ and RMSE are either lower than or comparable to those of the EKF, yielding the highest identification accuracy. Compared to the accuracy metrics before the speed increases (Table 3), all three algorithms show marginal changes in $\epsilon$ and RMSE for $R_s$ and ${\psi }_f$ identification. However, significant reductions are observed for evaluation metrics of $L_d$ and $L_q$. For instance, the $\epsilon$ of $L_d$ identified by the EKF decreases from 0.0088% to 0.005% with RMSE reduction from 0.7973 to 0.0349, while the $\epsilon$ of $L_d$ identified by the MIEKF improves from 0.0011% to 0.0001% and RMSE decreases from 0.0606 to 0.0156.

4.3.3. Load Variation Conditions

The PMSM was started with a no-load condition and then stabilized at 1000 r/min. At t = 5 s, the load torque was gradually increased from no-load to 10 N·m. The identification experimental curves for key electrical parameters during this loading process are presented in Figure 7, with evaluation metrics for identification accuracy listed in

Table 5. Evaluation metrics for key electrical parameters under load variation conditions.

Parameter	Calibrated Value	Method	μ	ε (%)	RMSE
Rs (mΩ)	108	FFRLS	1113.89	3.1379	33.4488
		EKF	1080.03	0.0027	0.0124
		MIEKF	1079.95	0.0046	0.0285
Ld (μH)	8380	FFRLS	8381.75	0.0209	1.2459
		EKF	8380.94	0.0112	0.0323
		MIEKF	8380.13	0.0015	0.0296
Lq (μH)	25,600	FFRLS	25,597.98	0.0079	2.0135
		EKF	25,595.33	0.0183	0.0005
		MIEKF	25,593.83	0.0241	0.0005
${\psi }_f$ ($\mathrm{mWb}$)	416	FFRLS	426.851	2.6084	12.8876
		EKF	417.933	0.4647	2.8827
		MIEKF	416.056	0.0135	1.1989

.

As illustrated in Figure 7 under the load variation condition, FFRLS exhibits the most pronounced fluctuations in identification curves, particularly for $R_s$ and ${\psi }_f$. Following severe oscillations, its results demonstrate a continuing downward trend, indicating low identification accuracy and weak anti-interference capability. While both the EKF and MIEKF stabilize after significant initial transients, the MIEKF achieves smaller amplitude deviations and faster settling during loading transitions. For instance, for parameter ${\psi }_f$, amplitude deviation 0.07 Wb (EKF) vs. 0.008 Wb (MIEKF), transition time 2.5 s (EKF) vs. 1.2 s (MIEKF). This confirms that the MIEKF has superior robustness and dynamic response performance.

Further analysis of Table 5 reveals that under load variation conditions, FFRLS maintains the poorest identification performance. While the EKF shows marginally lower $\epsilon$ and RMSEs for $R_s$ and $L_q$ compared to the MIEKF, its metrics for $L_d$ and ${\psi }_f$ substantially exceed those of the MIEKF. Overall, even if the EKF performs comparably for specific parameters, the MIEKF provides the best overall performance, demonstrating exceptional robustness under load changes, particularly for ${\psi }_f$, $\epsilon$ decreases from 0.4647% (EKF) to 0.0135% (MIEKF), while RMSE drops from 2.8827 (EKF) to 1.1989 (MIEKF).

Furthermore, compared to the no-load condition (Table 3), the $\epsilon$ of both the EKF and MIEKF increases overall after load application, with this trend being significantly more pronounced for the EKF. For instance, the $\epsilon$ for ${\psi }_f$ is identified by the EKF rising from 0.0007% (no-load condition) to 0.4647% (10 N·m load condition), whereas the MIEKF increases only marginally from 0.0002% (no-load condition) to 0.0135% (10 N·m load condition). This phenomenon primarily stems from the multi-innovation mechanism’s effective suppression of disturbances during PMSM operation, enabling the MIEKF to maintain higher identification accuracy and enhanced robustness. In contrast to speed variation conditions (Table 4), all three algorithms exhibit substantially greater $\epsilon$ under load variation, confirming that load changes exert a significantly larger impact on parameter identification accuracy.

5. Conclusions

Aiming at the problems of low identification accuracy and weak anti-interference ability caused by the time-varying characteristics of PMSM parameters affected by factors such as magnetic saturation and temperature rise, this paper proposed an MIEKF parameter identification method for PMSMs. It was based on the EKF framework and linearizes nonlinear systems using Jacobian matrices. It introduced the multi-innovation theory and iteratively updates historical state data to compensate for EKF linearization errors, effectively suppresses measurement noise and interference, and thereby improves identification accuracy, dynamic response speed, and robustness. An online parameter identification experimental platform for PMSMs was established based on RCP (MT1050) to experimentally validate the proposed method. Identification results and accuracy evaluation metrics demonstrate that under three operating conditions—no-load, speed variation, and load variation—compared with FFRLS and the EKF, the MIEKF algorithm exhibits the smallest fluctuation amplitude, the shortest fluctuation duration, mean values closest to calibrated values, minimal $\epsilon$, and minimal RMSE, indicating the highest accuracy and dynamic response performance. After the load increases, the $\epsilon$ of identification results for both the EKF and MIEKF algorithms increase overall, but the MIEKF shows significant suppression effect against this disturbance, maintaining high identification accuracy and demonstrating stronger robustness. The proposed MIEKF satisfies the requirements of MEPSs for PMSMs regarding rapid dynamic response, small steady-state error, and high control precision under different operating conditions. This method incorporates multiple sources of historical innovation and exhibits strong capabilities in nonlinear processing. It holds significant potential for application and extension in nonlinear domains that demand high-precision state or parameter estimation, such as target tracking, navigation and positioning, battery management systems, and industrial robotics.