Robust Flux-Weakening Control Strategy Against Multiple Parameter Variations for Interior Permanent Magnet Synchronous Motors

Abstract

Interior permanent magnet synchronous motors (IPMSMs) are widely adopted in electric vehicles due to their high torque density and efficiency, and they require flux-weakening operation to achieve high-speed performance under certain driving conditions. However, the traditional current vector control (CVC)-based flux-weakening strategies suffer from performance degradation when motor parameters, such as inductances and flux linkage, vary with temperature and operating conditions. To address this issue, this paper proposes a robust flux-weakening control strategy against multiple parameter variations. First, three sequential sliding-mode observers (SMOs) that form a sliding-mode observer suite (SMOS), whose stability is analyzed using Lyapunov theory, are designed to estimate the flux linkage, q-axis inductance, and d-axis inductance, respectively. Second, an error-analysis extraction (EAE) is developed to refine the parameter estimation accuracy by analytically solving a set of linear equations derived from observer results. Third, the accurately estimated parameters are applied to the CVC framework to generate adaptive reference currents, achieving robust and stable flux-weakening control performance. Finally, simulation and experiment are conducted to demonstrate that the proposed strategy effectively enhances control performance under multiple parameter variations.

1. Introduction

For the sake of environmental friendliness and sustainability, the electric vehicle (EV) market has undergone rapid expansion, driven by global efforts to decarbonize transportation and the accelerating adoption of battery-electric powertrains [1,2,3]. At present, traction motors that serve as the heart of EV propulsion systems are increasingly dominated by interior permanent-magnet synchronous machines (IPMSMs) owing to their superior torque density, high efficiency, and wide constant-power speed range [4,5]. For instance, Tesla Model 3 and Model Y employ interior permanent-magnet drive units, while enterprises such as Toyota, BYD, and Nidec have also developed IPMSM-based propulsion systems for their latest EV platforms [6,7]. Considering that IPMSMs have become the mainstream solution for EV traction, it is essential to achieve high-performance motor control strategies, ensuring vehicle dynamic performance, driving safety, and energy efficiency, etc.

During low- and medium-speed driving conditions, such as vehicle start-up, urban traffic, and hill climbing, high torque output and smooth dynamic response are particularly important. To achieve these objectives, field-oriented control (FOC) and direct torque control (DTC) are widely adopted for IPMSMs due to their capability of decoupling torque and flux components, which enables fast torque response and precise current regulation [8]. In these regions, maximum torque per ampere (MTPA) control is typically incorporated to utilize both magnetic and reluctance torque components, thereby improving torque efficiency and reducing current amplitude. In high-speed cruising scenarios on highways, the back electromotive force (EMF) of the motor approaches the inverter voltage limit, making it difficult to maintain sufficient torque using conventional control methods [9]. Therefore, flux-weakening control is applied to reduce the effective air-gap flux, enabling the motor to operate beyond its base speed and ensuring stable and efficient high-speed performance [10]. It needs to be mentioned that, for the flux-weakening control process, constant-power control principle should be considered to ensure that the EV powertrains work under safe conditions [11]. In recent years, the study of flux-weakening control strategies for high-speed operation of IPMSMs has attracted significant research attention, since it is crucial for EV applications yet difficult to implement effectively.

Nowadays, typical flux-weakening approaches include current vector control (CVC), voltage closed-loop control, and predictive- or optimization-based control strategies [12,13,14,15,16,17,18]. The CVC method injects a negative d-axis current to counteract the back EMF, achieving high-speed operation with simple implementation and reliable performance [12,13,14]. In terms of the voltage closed-loop approach, it directly regulates the stator voltage amplitude within the inverter limit through feedback, improving adaptability to parameter variations and voltage utilization [15,16,17]. Moreover, model-predictive and maximum torque per voltage-based flux-weakening methods have been proposed to further enhance dynamic response and torque capability in deep flux-weakening regions [18]. Although each method has unique advantages, the CVC approach remains the most widely used in EV traction systems due to its simple structure, ease of implementation, and satisfactory performance under most driving conditions.

Although the CVC method is widely adopted in practical applications, its effectiveness strongly depends on the accuracy of the motor model [19]. Specifically, CVC determines the reference d-axis current based on machine parameters such as the stator inductances and permanent-magnet flux linkage. Any deviation in these parameters directly affects the accuracy of the reference current and consequently degrades the torque output and voltage utilization. In real EV applications, however, maintaining precise parameter values is extremely difficult due to the complex and time-varying operating environment. Factors including temperature rise, magnetic saturation, and manufacturing tolerances can cause significant deviations in the stator resistance, d/q-axis inductances, and magnet flux linkage from their nominal values [20,21]. Such parameter variations may lead to inaccurate current commands, reduced flux-weakening capability, torque ripple, and even system instability during high-speed operation [22]. Therefore, enhancing the robustness of flux-weakening control against parameter variations has become a crucial research focus for ensuring reliable high-speed performance of IPMSM traction systems in EVs.

To cope with the performance degradation caused by parameter mismatches in flux-weakening control, various solutions have been proposed, which can generally be grouped into two categories: compensation-based and parameter identification-based methods. The compensation-based method mitigates the impact of parameter variations indirectly by estimating or compensating for the lumped effect of parameter uncertainties and external disturbances within the control loop [23]. Techniques such as disturbance observers, sliding-mode compensators, or extended state observers are typically employed to construct correction terms that enhance current and torque stability [24]. While this approach can improve robustness without requiring precise knowledge of each parameter, it does not directly address the root cause of parameter mismatch [25]. The compensation accuracy depends heavily on observer bandwidth and gain tuning, and residual torque errors or slow dynamic responses may still occur under rapidly changing conditions. In contrast, the parameter identification-based method aims to deal with parameter mismatch more directly by estimating the stator resistance, inductance, and flux linkage in real time [26,27,28]. Usually, the estimation of one parameter usually assumes that the others are known and fixed, which rarely holds true in practical EV applications. Because these parameters vary simultaneously with temperature, magnetic saturation, and operating conditions, this interdependence leads to cumulative estimation errors and unreliable identified values. Specifically, a single sliding-mode observer (SMO) is established to estimate the flux linkage of an IPMSM, while the inductance and resistance are assumed to remain constant [29]. However, the literature [30] reports that such a traditional SMO yields inaccurate estimation results because it relies on a priori and fixed parameter values. Overall, if a parameter identification method could accurately and simultaneously estimate multiple key parameters, it would represent one of the most effective solutions for mitigating parameter mismatch in flux-weakening control, which is highly required in EV applications.

This paper proposes a robust flux-weakening control strategy against multiple parameter variations for IPMSMs in EVs. The proposed method can simultaneously estimate multiple motor parameters, avoiding the assumption that other parameters remain accurate when identifying one. Based on the estimated parameters, adaptive current references are generated to realize robust and stable flux-weakening control under varying operating conditions. The main contributions and novelties of this study are summarized as follows:

• Instead of estimating one parameter at a time while assuming the others to be known and fixed, three sequential sliding-mode observers are jointly designed to form a sliding-mode observer suite (SMOS) that simultaneously targets the flux linkage, q-axis inductance, and d-axis inductance. The three observers operate in a sequential manner under identical operating conditions, making the estimation of even a single parameter inherently dependent on the entire suite rather than on an isolated observer. Lyapunov-based analysis is further employed to derive explicit stability constraints for the SMOS, ensuring reliable operation under EV-relevant flux-weakening scenarios.
• Based on the SMOS, an error-analysis extraction (EAE) procedure is developed to combine the observer outputs with offline-measured values and formulate a system of three linear equations that is analytically solved to obtain corrected flux-linkage and inductance values. This process explicitly removes the mutual coupling among parameters, enabling accurate and simultaneous identification of flux linkage and inductance even when the original offline measurements exhibit large deviations. As a result, the overall robustness and consistency of the parameter-identification process are significantly enhanced compared with traditional single-observer-based schemes. This method is suitable for EV powertrains whose parameters vary frequently.
• Unlike traditional flux-weakening methods that use fixed parameters to calculate the reference current in the CVC framework, the accurately estimated flux linkage and inductance are utilized for reference current generation in the proposed method. Together with a numerically derived decoupling strategy, a robust flux-weakening control strategy is achieved. This method is developed by explicitly incorporating the operating characteristics of electric vehicle powertrains, including frequent parameter variations induced by varying working conditions, thereby endowing the proposed strategy with inherent innovation and practical value.

The structure of the rest of the paper is as follows. Section 2 presents the proposed sliding-mode multiple parameter identification strategy based on EAE. Section 3 introduces the robust CVC-based flux-weakening control strategy. Section 4 provides simulation and experimental results to validate the effectiveness of the proposed method. Finally, Section 5 concludes the paper.

2. Sliding-Mode Multiple Parameter Identification Based on EAE

This section first introduces the modeling of IPMSMs used in EVs. Then, the sliding-mode flux linkage observer (SMFLO), sliding-mode q-axis inductance observer (SMQIO), and sliding-mode d-axis inductance observer (SMDIO) are established to constitute the SMOS, and their stability is analyzed. Finally, the error-analysis extraction (EAE) is presented to obtain accurate parameter values.

2.1. Modeling of IPMSMs

In general, constructing a mathematical model from the physical motor enables clear identification of the parameters that must be measured or estimated for control, while also forming the basis of model-based parameter identification. For IPMSMs employed in EV applications, the electrical and mechanical behaviors in the direct–quadrature (dq) rotating frame can be formulated as [31]:

$${\displaystyle \frac{\mathrm{d}{i}_q}{\mathrm{d}t}}=-{\displaystyle \frac{L_d}{L_q}}p{\omega }_m{i}_d-{\displaystyle \frac{R_s}{L_q}}{i}_q+{\displaystyle \frac{u_q}{L_q}}-{\displaystyle \frac{{\psi }_f}{L_q}}p{\omega }_m$$

(1)

$${\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}=-{\displaystyle \frac{R_s}{L_q}}{i}_d+{\displaystyle \frac{L_q}{L_d}}p{\omega }_m{i}_q+{\displaystyle \frac{u_q}{L_q}}$$

(2)

$$T_e=1.5p ({\psi }_f{i}_q+(L_d-L_q)i_d{i}_q)$$

(3)

$${\displaystyle \frac{\mathrm{d}{\omega }_m}{\mathrm{d}t}}={\displaystyle \frac{1}{J}}(T_e-T_l-B{\omega }_m)$$

(4)

where ω_m is the mechanical angular speed, T_e and T_l are the electromagnetic and load torque, respectively. J and B denote the rotor moment of inertia and damping coefficient, respectively. i_d and i_q represent the stator d-axis and q-axis currents, while u_d and u_q are the corresponding control voltages. L_d and L_q are the real-time d-axis and q-axis inductances, and R_s and p denote the stator winding resistance and the number of pole pairs. ψ_f represents the real-time flux linkage of the permanent magnet. In this study, it is reasonably assumed that R_s, J, and B can be accurately obtained by offline or online methods, as they vary only slightly during operation and are not directly involved in the CVC-based flux-weakening control process.

2.2. Stable Sliding-Mode Observer Suite

(a) Design of SMDIO, SMQIO, and SMFLO constituting SMOS

When constructing sliding-mode parameter observers based on the motor model, the key is to identify which equations contain the measurable or estimable information of the target parameters. Specifically, (1) simultaneously contains the information related to the d-axis and q-axis inductances, as well as the flux linkage. From the perspective of sliding-mode variable-structure theory, this equation could theoretically be used to design any observer. However, it has been found that when the variable to be observed appears in the numerator of the differential equation, the convergence of the constructed observer becomes difficult to guarantee [22]. Therefore, in practice, (1) is only suitable for constructing the d-axis inductance observer and the flux linkage observer. In contrast, in (2), the q-axis inductance appears in the denominator, which makes it appropriate for designing the SMQIO. As for (4), although it is not a differential equation, it contains both inductance and flux-linkage information. By substituting (3) into (4), a new differential equation, (5), can be obtained, which can then be used to construct any observers based on the sliding-mode variable-structure theory [32,33].

$${\displaystyle \frac{\mathrm{d}{\omega }_m}{\mathrm{d}t}}={\displaystyle \frac{1}{J}}[1.5p ({\psi }_f{i}_q+(L_d-L_q)i_d{i}_q)-T_l-B{\omega }_m]$$

(5)

Based on the above analysis, (1), (2), and (5) are selected to construct the SMFLO, SMQIO, and SMDIO, respectively. The construction process of the sliding-mode observers is shown in Figure 1.

First, for the SMFLO, the offline-measured parameters are utilized. Based on the sliding-mode variable-structure theory, the observer shown in Figure 2 is designed as follows:

$${\displaystyle \frac{\mathrm{d}{i_q}^{*}}{\mathrm{d}t}}=-{\displaystyle \frac{L_{d\_m}}{L_{q\_m}}}p{\omega }_m{i}_d-{\displaystyle \frac{R_s}{L_{q\_m}}}{i_q}^{*}+{\displaystyle \frac{u_q}{L_{q\_m}}}-{\displaystyle \frac{{\lambda }_{1}F(\overline{i_q})}{L_{q\_m}}}p{\omega }_m$$

(6)

where L_{d_m} and L_{q_m} are the offline-measured d-axis and q-axis inductances. i_q* is the estimated q-axis current of the observer. λ₁ is the gain coefficient. $\overline{i_q}$ is the error between the estimated current and the real one, that is, $\overline{i_q}={i_q}^{*}-i_q$. $F(\overline{i_q})$ is the switching function, which is described as:

$$F(\overline{i_q})=\left\{\begin{array}{l} 1, if \overline{i_q}\ge 0 \\ -1, if \overline{i_q}<0 \end{array}\right.$$

(7)

When the system gets stable, the estimated flux linkage ψ_f^* can be obtained by:

$${\psi }_f^{*}={\lambda }_{1}F(\overline{i_q})$$

(8)

Then, the offline-measured inductance L_{d_m} is used to construct the SMQIO, which is shown in Figure 3:

$${\displaystyle \frac{\mathrm{d}{i_d}^{*}}{\mathrm{d}t}}=-{\displaystyle \frac{R_s}{L_{d\_m}}}{i_d}^{*}+{\displaystyle \frac{{\lambda }_{2}F(\overline{i_d})}{L_{d\_m}}}p{\omega }_m{i}_q+{\displaystyle \frac{u_d}{L_{d\_m}}}$$

(9)

where i_d^* is the estimated d-axis current of the observer. λ₂ is the gain coefficient. $\overline{i_d}$ is the error between the estimated d-axis current and the real one. In terms of the switching function $F(\overline{i_d})$, it can be expressed in a form similar to (7). When the system reaches the stable state, the estimated inductance L_q* is:

$${L_q}^{*}={\lambda }_{2}F(\overline{i_d})$$

(10)

After obtaining ψ_f*, it needs to be substituted into (5) to construct the SMDIO (shown in Figure 4) on the basis of the variable-structure theory:

$${\displaystyle \frac{\mathrm{d}{{\omega }_m}^{*}}{\mathrm{d}t}}={\displaystyle \frac{1}{J}} [1.5p ({\psi }_{f\_m}{i}_q+({\lambda }_{3}F(\overline{{\omega }_m})-L_{q\_m})i_d{i}_q)-T_l-B{{\omega }_m}^{*}]$$

(11)

where ω_m* is the estimated speed of the observer. ψ_{f_m} represents the offline-measured flux linkage. λ₃ is the gain coefficient. $\overline{{\omega }_m}$ is the error between the estimated speed and the real speed. $F(\overline{{\omega }_m})$ is the switching function. When the system arrives at the equilibrium state, the estimated d-axis inductance L_d* is:

$${L_d}^{*}={\lambda }_{3}F(\overline{{\omega }_m})$$

(12)

(b) Stability Analysis of SMOS

The reliable acquisition of flux linkage and inductance information using the three proposed SMOS requires each observer to operate stably. As the observers are executed sequentially in the digital processor, they are inherently independent of one another. Accordingly, three individual Lyapunov functions are constructed to analyze their respective stability. Based on the Lyapunov stability criterion, the q-axis current, d-axis current, and speed sliding surfaces S for the SMFLO, SMQIO, and SMDIO are defined as follows:

$$S={\left[S_q,S_d, S_{\omega }\right]}^T={\left[\overline{i_q}, \overline{i_d}, \overline{{\omega }_m}\right]}^T$$

(13)

Based on (13), the Lyapunov functions V are established as:

$$V={\displaystyle \frac{1}{2}}{\left[{S_q}^2,{S_d}^2,{S_{\omega }}^2\right]}^T={\displaystyle \frac{1}{2}}{\left[{\overline{i_q}}^2, {\overline{i_d}}^2, {\overline{{\omega }_m}}^2\right]}^T$$

(14)

Taking the derivative of (14), it can be obtained that:

$${\displaystyle \frac{\mathrm{d}V}{\mathrm{d}t}}={\left[\overline{i_q}{\displaystyle \frac{\mathrm{d}\overline{i_q}}{\mathrm{d}t}}, \overline{i_d}{\displaystyle \frac{\mathrm{d}\overline{i_d}}{\mathrm{d}t}}, \overline{{\omega }_m} {\displaystyle \frac{\mathrm{d}\overline{{\omega }_m} }{\mathrm{d}t}}\right]}^T$$

(15)

Considering the offline-measured parameters and subtracting (1), (2), and (5) from (6), (9), and (11), we can obtain:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}\overline{i_q}}{\mathrm{d}t}}=-{\displaystyle \frac{R_s}{L_{q\_m}}}\overline{i_q}-{\displaystyle \frac{{\lambda }_{1}F(\overline{i_q})-{\psi }_{f\_m}}{L_{q\_m}}}p{\omega }_m\\ {\displaystyle \frac{\mathrm{d}\overline{i_d}}{\mathrm{d}t}}=-{\displaystyle \frac{R_s}{L_{d\_m}}}\overline{i_d}+{\displaystyle \frac{{\lambda }_{2}F(\overline{i_d})-L_{q\_m}}{L_{d\_m}}}p{\omega }_m{i}_q\\ {\displaystyle \frac{\mathrm{d}\overline{{\omega }_m}}{\mathrm{d}t}}={\displaystyle \frac{1}{J}} [1.5p ({\lambda }_{3}F(\overline{{\omega }_m})-L_{d\_m})i_d{i}_q)-T_l-B\overline{{\omega }_m}]\end{array}\right.$$

(16)

Substituting (16) into (15), it can be obtained that:

$${\displaystyle \frac{\mathrm{d}V}{\mathrm{d}t}}=\left[\begin{array}{l}-{\displaystyle \frac{R_s}{L_{q\_m}}}{\overline{i_q}}^2-{\displaystyle \frac{{\lambda }_{1}F(\overline{i_q})-{\psi }_{f\_m}}{L_{q\_m}}}p{\omega }_m\overline{i_q}, \\ -{\displaystyle \frac{R_s}{L_{d\_m}}}{\overline{i_d}}^2+{\displaystyle \frac{{\lambda }_{2}F(\overline{i_d})-L_{q\_m}}{L_{d\_m}}}p{\omega }_m{i}_q\overline{i_d}, \\ {\displaystyle \frac{1}{J}} [1.5p ({\lambda }_{3}F(\overline{{\omega }_m})-L_{d\_m})i_d{i}_q\overline{{\omega }_m})-B{\overline{{\omega }_m}}^2]\end{array}\right]$$

(17)

To make the proposed observers stable, the following inequality conditions should be satisfied:

$$V>0, {\displaystyle \frac{\mathrm{d}V}{\mathrm{d}t}}<0$$

(18)

Apparently, the first condition in (18) is inherently satisfied. As long as the second condition is satisfied, the observers remain stable. In this case, the following equations can be obtained:

$$\left\{\begin{array}{l}-{\displaystyle \frac{R_s}{L_{q\_m}}}{\overline{i_q}}^2-{\displaystyle \frac{{\lambda }_{1}F(\overline{i_q})-{\psi }_{f\_m}}{L_{q\_m}}}p{\omega }_m\overline{i_q}<0\\ -{\displaystyle \frac{R_s}{L_{d\_m}}}{\overline{i_d}}^2+{\displaystyle \frac{{\lambda }_{2}F(\overline{i_d})-L_{q\_m}}{L_{d\_m}}}p{\omega }_m{i}_q\overline{i_d}<0 \\ {\displaystyle \frac{1}{J}} [1.5p ({\lambda }_{3}F(\overline{{\omega }_m})-L_{d\_m})i_d{i}_q\overline{{\omega }_m})-B{\overline{{\omega }_m}}^2]<0\end{array}\right.$$

(19)

By neglecting those terms that are constantly smaller than zero in (19), it can be further derived that:

$$\left\{\begin{array}{l}-{\displaystyle \frac{{\lambda }_{1}F(\overline{i_q})-{\psi }_{f\_m}}{L_{q\_m}}}p{\omega }_m\overline{i_q}<0\\ {\displaystyle \frac{{\lambda }_{2}F(\overline{i_d})-L_{q\_m}}{L_{d\_m}}}p{\omega }_m{i}_q\overline{i_d}<0 \\ {\displaystyle \frac{1.5p}{J}}({\lambda }_{3}F(\overline{{\omega }_m})-L_{d\_m})i_d{i}_q\overline{{\omega }_m})<0\end{array}\right.$$

(20)

Interestingly, (20) shows that, apart from the estimation errors $\overline{i_q}$, $\overline{i_d}$ and $\overline{{\omega }_m}$, the motor states, including motor speed ω_m and currents i_d and i_q, influence the stability of the observers. First, under highway cruising conditions, EVs do not undergo reverse operation, and the motor continuously produces positive torque in a single rotational direction. Second, this study focuses on the flux-weakening control process, where the d-axis current should be smaller than zero. Overall, it can be obtained that:

$${\omega }_m>0, i_q>0, i_d<0$$

(21)

With consideration of (21), (20) can be further derived as:

$$\left\{\begin{array}{l}({\lambda }_{1}F(\overline{i_q})-{\psi }_{f\_m})\overline{i_q}>0\\ ({\lambda }_{2}F(\overline{i_d})-L_{q\_m})\overline{i_d}<0 \\ ({\lambda }_{3}F(\overline{{\omega }_m})-L_{d\_m})\overline{{\omega }_m})>0\end{array}\right.$$

(22)

Considering the signs of the estimation errors, (22) can be described as:

$$\begin{array}{l}\left\{\begin{array}{l}{\lambda }_1>{\psi }_{f\_m}, \mathrm{if} \overline{i_q}>0 \\ {\lambda }_1>-{\psi }_{f\_m}, \mathrm{if} \overline{i_q}<0 \end{array}\right.\to {\lambda }_1>{\psi }_{f\_m}\\ \left\{\begin{array}{l}{\lambda }_2<L_{q\_m}, \mathrm{if} \overline{i_d}>0\\ {\lambda }_2<-L_{q\_m}, \mathrm{if} \overline{i_d}<0\end{array}\right.\to {\lambda }_2<-L_{q\_m}\\ \left\{\begin{array}{l}{\lambda }_3>L_{d\_m}, \mathrm{if} \overline{{\omega }_m}>0\\ {\lambda }_3>-L_{d\_m}, \mathrm{if} \overline{{\omega }_m}<0\end{array}\right.\to {\lambda }_3>L_{d\_m}\end{array}$$

(23)

The above equations illustrate that there exist certain conditions to enure that the proposed SMDIO, SMQIO, and SMFLO are stable. In practice, as long as the gain coefficients are selected following the principles in (23), the flux linkage and inductance information can be estimated. It can be observed that the admissible ranges of the observer gains defined by (23) are relatively wide. Consequently, further tuning of these gains to achieve an appropriate compromise between convergence speed and estimation fluctuation in practical applications becomes an important issue.

(c) Parameter Tuning Based on the Torque Variation Test for SMOS

After deriving the stability conditions in (23), it can be observed that the admissible ranges of the observer gains are relatively wide. Although any gain selection within these ranges guarantees stability, different gain values lead to different trade-offs between dynamic response speed and steady-state estimation fluctuation. Therefore, a systematic tuning procedure is required to select practically optimal gains for the SMOS.

For sliding-mode observers, the gain magnitude introduces an intrinsic trade-off [34]. Increasing the gain generally shortens the reaching and convergence time, but it also amplifies switching activity and sensitivity to measurement noise, which enlarges the steady-state fluctuation of the estimated quantities. Conversely, decreasing the gain reduces the steady-state fluctuation but slows down the convergence. As a consequence, within the stability-guaranteed region, there must exist an intermediate gain value that provides a balanced compromise between response speed and estimation accuracy.

Based on the above theoretical analysis, the optimal observer gain can be obtained through an experimental testing procedure. Accordingly, a parameter tuning method Based on the torque-variation test is proposed. Taking the SMFLO as an example, the tuning procedure is described as follows.

Primarily, candidate gain values are generated within the stability-guaranteed region defined by (23). Starting from a conservative stable setting, the gain is increased gradually using a step size of one, thereby forming a set of trial gain values. For instance, if the conservative initial gain is selected as 1.0, the subsequent gain values used for assessment are 2.0, 3.0, 4.0, … For each trial gain value, the following torque-variation test is conducted:

(1)

No-load rated-speed operation: the motor is operated at no load and accelerated to the nominal rated speed. After reaching steady state, the estimated parameter produced by the SMOS is monitored, and the maximum steady-state fluctuation amplitude Δψ_f* is recorded (the peak-to-peak variation within a fixed observation window).

(2)

Step-load excitation: a step load torque of unit load (1 Nm) is abruptly applied. The time required for the estimated parameter to transition from the disturbed state to a new steady state is measured and defined as the response time T_r.

(3)

Cost evaluation and selection: substitute the maximum steady-state fluctuation amplitude Δψ_f* and response time T_r into the cost function (24). The gain value that minimizes J is selected as the optimal gain within the stability-guaranteed region.

$$J={({\displaystyle \frac{T_r}{\mu T_s}})}^2+{({\displaystyle \frac{\Delta {{\psi }_f}^{*}}{{\psi }_{f\_m}}})}^2$$

(24)

where J represents the value of the cost function. T_s denotes the control period, and μ represents the maximum acceptable number of convergence periods. In this study, μ is set to 12.

2.3. EAE-Based SMOS for Parameter Calculation

For the proposed three SMOs, their performance relies on both offline-measured and estimated parameter values. When the measured parameters deviate from their actual values, the overall estimation accuracy deteriorates. To address this issue, an EAE is introduced to enhance the reliability of parameter estimation and observer precision.

By retaining the original parameters in the corresponding equations and subtracting (1), (2), and (5) from (6), (9), and (11), respectively, the estimation errors of the flux linkage and the q- and d-axis inductances can be derived as follows:

$$\left\{\begin{array}{l}{{\psi }_f}^{*}-{\psi }_f=-(L_{d\_m}-L_d)i_d -{\displaystyle \frac{R_s}{p{\omega }_m}}\overline{i_q}-{\displaystyle \frac{1}{p{\omega }_m}}(L_{q\_m}{\displaystyle \frac{\mathrm{d}{i_q}^{*}}{\mathrm{d}t}}-L_q{\displaystyle \frac{\mathrm{d}{i}_q}{\mathrm{d}t}})\\ {L_q}^{*}-L_q={\displaystyle \frac{1}{p{\omega }_m{i}_q}}(L_{d\_m}{\displaystyle \frac{\mathrm{d}{i_d}^{*}}{\mathrm{d}t}}-L_d{\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}})+{\displaystyle \frac{R_s}{p{\omega }_m{i}_q}}\overline{i_d} \\ {L_d}^{*}-L_d={\displaystyle \frac{J}{1.5p{i}_d{i}_q}}{\displaystyle \frac{\mathrm{d}\overline{{\omega }_m}}{\mathrm{d}t}}+{\displaystyle \frac{B}{1.5p{i}_d{i}_q}}\overline{{\omega }_m}+L_{q\_m}-L_q-{\displaystyle \frac{1}{i_d}}({\psi }_{f\_m}-{\psi }_f)\end{array}\right.$$

(25)

As for the SMOs, because the estimated states equal the real ones (i_q = i_q*, i_d = i_d*, and ω_m = ω_m*) when they work stably, (25) can be simplified as:

$$\left\{\begin{array}{l}{{\psi }_f}^{*}-{\psi }_f=-(L_{d\_m}-L_d)i_d -{\displaystyle \frac{(L_{q\_m}-L_q)}{p{\omega }_m}}{\displaystyle \frac{\mathrm{d}{i}_q}{\mathrm{d}t}}\\ {L_q}^{*}-L_q={\displaystyle \frac{(L_{d\_m}-L_d)}{p{\omega }_m{i}_q}}{\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}} \\ {L_d}^{*}-L_d=L_{q\_m}-L_q-{\displaystyle \frac{({\psi }_{f\_m}-{\psi }_f)}{i_d}}\end{array}\right.$$

(26)

In (26), ψ_f, L_d and L_q represent the accurate flux linkage and inductance values which are unknown, while the other variables including ψ_f*, L_d*, L_q*, L_{q_m}, L_{d_m}, and ψ_{f_m} are known. Hence, it is a ternary system of first-order equations, and the solution concerning the flux linkage and inductance of the system (denoted as ψ_{f_ca}, L_{d_cal}, L_{q_cal}) can be derived by the use of Maple:

$$\left\{\begin{array}{l}{\psi }_{f\_cal}={\displaystyle \frac{\left(\begin{array}{l}{p}^2{{\omega }_m}^2{i_d}^2{i}_q(L_{d\_m}-{L_d}^{\mathrm{*}}+L_{q\_m}-{L_q}^{\mathrm{*}})+(L_{q\_m}-{L_q}^{\mathrm{*}})p{\omega }_m{i}_d{i}_q\frac{\mathrm{d}{i}_q}{\mathrm{d}t}\\ -p{\omega }_m{i}_d{{\psi }_f}^{\mathrm{*}}\frac{\mathrm{d}{i}_d}{\mathrm{d}t}+(L_{d\_m} -{L_d}^{\mathrm{*}})i_d\frac{\mathrm{d}{i}_q}{\mathrm{d}t} \cdot \frac{\mathrm{d}{i}_d}{\mathrm{d}t}-{{\psi }_f}^{\mathrm{*}}\frac{\mathrm{d}{i}_q}{\mathrm{d}t} \cdot \frac{\mathrm{d}{i}_d}{\mathrm{d}t}\end{array}\right)}{p{\omega }_m{i}_d +\frac{\mathrm{d}{i}_q}{\mathrm{d}t} }}\\ L_{q\_cal}={\displaystyle \frac{(L_{d\_m}+L_{q\_m}-{L_d}^{\mathrm{*}})p{\omega }_m{i}_d+({{\psi }_f}^{\mathrm{*}}-{\psi }_{f\_m})p{\omega }_m+L_{q\_m}\frac{\mathrm{d}{i}_q}{\mathrm{d}t}}{p{\omega }_m{i}_d+\frac{\mathrm{d}{i}_q}{\mathrm{d}t}}} \\ L_{d\_cal}={\displaystyle \frac{\left(\begin{array}{l}(L_{d\_m}-{L_d}^{\mathrm{*}}+L_{q\_m}-{L_q}^{\mathrm{*}})p^2{{\omega }_m}^2{i}_d{i}_q+({{\omega }_f}^{\mathrm{*}}-{\psi }_{f\_m})p^2{{\omega }_m}^2{i}_q\\ +(L_{q\_m}-{L_q}^{\mathrm{*}})\frac{\mathrm{d}{i}_q}{\mathrm{d}t}p{\omega }_m{i}_q+L_{d\_m}\frac{\mathrm{d}{i}_d}{\mathrm{d}t}(p{\omega }_m{i}_d+\frac{\mathrm{d}{i}_q}{\mathrm{d}t})\end{array}\right)}{p{\omega }_m{i}_d+\frac{\mathrm{d}{i}_q}{\mathrm{d}t}}}\end{array}\right.$$

(27)

By using (27), it can be noted that the accurate parameter values can be calculated.

3. Robust CVC-Based Flux-Weakening Control Strategy

The structure of the proposed flux-weakening control strategy is illustrated in Figure 5, where DCR represents the d-axis current regulator, QCR is the q-axis current regulator and SR is the speed regulator. In addition to employing accurately estimated parameters to calculate the d-axis reference current for enhanced robustness, the proposed flux-weakening strategy exhibits another key feature: the integration of deviation decoupling into the control process. The detailed explanations of the proposed strategy are provided below.

3.1. Deviation Decoupling

In the field-weakening region, the interaction between the d-axis and q-axis currents becomes more pronounced due to the increased role of the demagnetizing current component i_d. As the d-axis current is intentionally driven negative to reduce the air-gap flux and extend the speed range, it causes a substantial cross-coupling effect in the stator voltage equations. The pω_mi_d and pω_mi_q terms lead to mutual interference between the voltage and current loops, thereby degrading current regulation accuracy. Without effective decoupling, the q-axis torque current control becomes sluggish, and the d-axis flux-weakening current may oscillate or deviate from its reference, resulting in torque ripple and instability at high speed. Moreover, as the inverter operates closer to its voltage limit in the flux-weakening region, even slight coupling-induced disturbances can push the system into saturation, further reducing control bandwidth. Therefore, introducing an appropriate decoupling strategy is essential to achieve accurate current regulation, maintain dynamic stability, and ensure smooth torque production across the entire speed range [35]. This part introduces a deviation decoupling technique. For clarity of explanation, Figure 6 illustrates the motor’s s-domain model that includes the current regulation dynamics following the electrical motor properties (1) and (2), where G_d(s) and G_q(s) denote the d-axis current regulator (DCR) and q-axis current regulator (QCR), respectively, and s represents the Laplace operator. It is evident that the d-axis and q-axis currents are mutually dependent rather than isolated, and hence, variations in one axis inevitably affect the other. Such cross-interaction, commonly referred to as coupling, prevents the voltage components u_d and u_q from independently governing their respective currents.

Based on the above analysis, it can be concluded that implementing an appropriate decoupling mechanism to isolate the two current channels constitutes the core logic of the decoupling design. As illustrated in Figure 7, a deviation-based decoupling scheme comprising three functional modules, G₁, G₂, and G₃, is employed to mitigate the mutual coupling between the d-axis and q-axis channels. The design of this strategy involves two key considerations. First, the G₃ module is embedded in the q-axis control path to counteract the influence of the motional EMF, which introduces coupling terms that hinder precise current regulation. In practice, it can be expressed as:

$$G_3=p{\omega }_m{\psi }_f$$

(28)

Then, the goal of decoupling is to utilize G₁ and G₂ to convert the original machine models (1) and (2) into the following form:

$$\left\{\begin{array}{l}{u}_{d1}={\displaystyle \frac{1}{L_{d}s+R_s}}{i}_d \\ u_{q1}={\displaystyle \frac{1}{L_{q}s+R_s}}{i}_q \end{array}\right.$$

(29)

where u_d1 and u_q1 are the manipulated voltages after decoupling.

After transforming (29) into a model in the time domain, it can be obtained that:

$$\left\{\begin{array}{l}{u}_{d1}=L_d{\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}+i_d{R}_s\\ u_{q1}=L_q{\displaystyle \frac{\mathrm{d}{i}_q}{\mathrm{d}t}}+i_q{R}_s\end{array}\right.$$

(30)

The principal challenge of the proposed decoupling approach lies in the design of G₁ and G₂, which must be properly formulated to achieve complete decoupling between the d-axis and q-axis currents. To solve the issue, firstly, u_d and u_q in Figure 7 need to be described as:

$$\left\{\begin{array}{l}{u}_d=(i_{dr}-i_d)\cdot G_d+(i_{qr}-i_q)\cdot G_2\\ u_q=(i_{qr}-i_q)\cdot G_q+(i_{dr}-i_d)\cdot G_1+G_3\end{array}\right.$$

(31)

where i_dr and i_qr are the d-axis and q-axis reference currents, respectively. Substituting (31) into (1) and (2), and it can be obtained that:

$$\left\{\begin{array}{l}{u}_{d1}+G_2(i_{qr}-i_q)+p{\omega }_m{L}_q{i}_q=L_d{\displaystyle \frac{\mathrm{d}{i}_d}{\mathrm{d}t}}+i_d{R}_s\\ u_q+G_1(i_{dr}-i_d)-p{\omega }_m{L}_d{i}_d=L_q{\displaystyle \frac{\mathrm{d}{i}_q}{\mathrm{d}t}}+i_q{R}_s \end{array}\right.$$

(32)

Then, by comparing (32) with (30), the following conditions should be satisfied to get a decoupled system:

$$\left\{\begin{array}{l}{G}_2(i_{qr}-i_q)+p{\omega }_m{L}_q{i}_q=0\\ G_1(i_{dr}-i_d)-p{\omega }_m{L}_d{i}_d=0 \end{array}\right.$$

(33)

Further, G₁ and G₂ are derived as:

$$\left\{\begin{array}{l}{G}_1={\displaystyle \frac{p{\omega }_m{L}_d{i}_d}{i_{dr}-i_d}}\\ G_2=-{\displaystyle \frac{p{\omega }_m{L}_q{i}_q}{i_{qr}-i_q}}\end{array}\right.$$

(34)

By substituting ψ_{f_ca}, L_{d_cal}, and L_{q_cal} obtained in Section 2 into (34), robust decoupling of the IPMSM in the dq reference frame is achieved. The manipulated voltages generated by the d-axis and q-axis controllers can regulate the corresponding currents independently.

3.2. Implementation of Robust CVC-Based Flux-Weakening Strategy

Combining the analysis in Section 2, the main implementation procedures of the proposed robust CVC-based flux-weakening strategy incorporating the deviation decoupling scheme can be summarized as follows:

(a) Parameter pre-estimation: Based on the offline-measured parameters, activate the SMOS to estimate the flux linkage and d/q-axis inductances based on the motor’s real-time operating states.

(b) EAE and parameter correction: Apply the EAE method to refine the initially estimated parameters by solving the derived linear equations. The corrected flux linkage and inductance values are regarded as accurate short-term parameters and updated periodically during operation.

(c) Robust reference current generation: Using the updated parameter values, as shown in (34), compute the reference d-axis reference current and the maximum q-axis reference current that is used for current limitation [36]. This ensures that the flux-weakening reference currents are always consistent with the actual motor parameters.

$$i_{dr}=-{\displaystyle \frac{{\psi }_{f\_cal}}{2L_{d\_cal}}}+{\displaystyle \frac{1}{L_{d\_cal}}}\sqrt{{\displaystyle \frac{{{\psi }_{f\_cal}}^2}{4}}+{\displaystyle \frac{{V_{dc}}^2}{3{{\omega }_{mr}}^2}}-{\displaystyle \frac{{L_{q\_cal}}^2}{{L_{d\_cal}}^2}}{i_{qr}}^2}$$

(35)

where i_dr and i_qr are the d-axis and q-axis reference currents, respectively. V_dc is the DC-link voltage. ω_mr is the reference speed.

(d) Deviation decoupling and voltage control: Integrate the deviation decoupling modules into the voltage regulation loop. By applying the decoupled manipulated voltages u_d1 and u_q1, the d-axis and q-axis current controllers independently regulate their respective current components, effectively suppressing cross-coupling.

(e) Flux-weakening control execution: Implement the overall flux-weakening control in the high-speed region by commanding the adaptive reference currents to the current controllers. The system maintains a constant power output characteristic while ensuring voltage utilization and torque stability under multiple parameter variations.

(f) Periodic update and steady operation: Repeat the parameter estimation and correction cycle at scheduled intervals, assuming the estimated parameters remain unchanged within each short control period. This mechanism guarantees robust flux-weakening operation with high stability and adaptability over varying temperature and load conditions.

4. Verification Results

To verify the effectiveness of the proposed flux-weakening control strategy under multiple parameter variations, experiments were performed on a three-phase IPMSM whose parameters are listed in

Table 1. Parameters of IPMSM used for verification.

Parameter	Value	Unit
winding resistance Rs	0.605	Ω
d-axis inductance Ld	12.650	mH
q-axis inductance Lq	13.500	mH
the number of pole pairs p	2	-
rated speed ωmrated	560	rpm
flux linkage ψf	0.687	Wb

. Neglecting effects such as magnetic saturation and thermal variation, the parameters in Table 1, which are measured by using authentic offline-measured techniques, can be regarded as the accurate parameter values. The experimental setup is shown in Figure 8. Insulated gate bipolar transistor (IGBT) modules FS100R12PT4 and drive modules PSPC-420E-EP4F constitute the voltage inverter with a control frequency of 10 kHz (T_s = 0.1 ms), with a DC-bus voltage of 150 V. The rotor position and speed are detected by a 1024-line rotary encoder. Hall sensors, LA25-NP and LV25-P are used to measure the phase currents and DC-link voltage, respectively. An induction motor driven by Automation Drive PM240-2 (controlled by CU250S-2) in torque control mode is coupled to the test machine, providing the required load torque. The proposed control algorithms are implemented on a dSPACE control board, and data acquisition is carried out through the dSPACE ControlDesk interface. The measured current and voltage signals are transmitted through the patch board to a host computer for data logging and subsequent analysis.

4.1. Parameter Identification Results

The core of the proposed flux-weakening control algorithm with robustness against multiple parameter variations lies in the adoption of a multi-parameter identification technique. Therefore, the first step is to verify the superiority of the proposed identification method compared with conventional fixed-parameter approaches. In this part, it is assumed that the measured parameters deviate from those listed in Table 1, satisfying the following conditions: L_{d_m} = 3L_d, L_{q_m} = 2L_q, and ψ_{f_m} =0.5ψ_f. Then, the parameter identification results obtained using the proposed SMOS under different operating conditions are presented and compared with those derived from the traditional SMO-based identification method. It should be noted that the term “traditional SMO” refers to a single observer constructed directly based on sliding-mode variable-structure theory to estimate the target parameters. In other words, each standalone observer presented in (6), (9), and (11), respectively, is treated as one traditional SMO. Their gains satisfy the conditions shown in (23) and are optimized by using the torque variation test technique, with λ₁, λ₂, and λ₃ being 6.5, 1.5, and 1.5, respectively. The outputs of the traditional SMOs are directly used for the proposed EAE-based parameter identification. This setup can ensure that the traditional observers work under the same conditions as those used to construct the SMOS.

(a)

Steady-State performance

Figure 9 shows the system performance and parameter identification results when the d-axis current is set as −1.5 A. The experimental setups are as follows. The motor speed is constantly set as 500 rpm under a load of 3 Nm.

From Figure 9, it can be seen that first, the motor operates stably under the pre-set speed and current references, where the q-axis current remains approximately 2.13 A under a 3 Nm load. Second, when the observer gains λ₁, λ₂, and λ₃ are selected according to the analytical results in (23), the three SMOs maintain stable operation, as evidenced by the fact that the estimation errors of all states remain nearly zero. Third, using the traditional SMOs, the identified flux linkage, q-axis inductance, and d-axis inductance are 0.82 Wb, 0.011 H, and 0.0079 H, deviating from the real values by 19.3%, 18.5%, and 37.5%, respectively. Fourth, after applying the proposed parameter calculation strategy, the estimated flux linkage, q-axis inductance, and d-axis inductance are 0.67 Wb, 0.0134 H, and 0.0125 H, with deviations of only 2.5%, 0.7%, and 1.2%, respectively. Comparatively, the proposed parameter identification method relying on the EAE-based SMOS achieves substantially higher accuracy and thus exhibits stronger robustness against parameter variations.

Further, when the d-axis current is set as −3 A, the system performance and parameter identification results are depicted in Figure 10. Similar to the observations in Figure 9, the motor continues to operate stably. For an IPMSM, the presence of reluctance torque causes the required q-axis current to decrease as the negative d-axis current increases. In addition, consistent with the results in Figure 9, the proposed parameter identification strategy maintains significantly higher accuracy than the traditional methods. Specifically, the proposed method estimates the flux linkage, q-axis inductance, and d-axis inductance as 0.68 Wb, 0.0132 H, and 0.0123 H, respectively, which are much closer to the real parameter values than those obtained using the traditional approach, as illustrated in Figure 10c.

(b)

Dynamic performance

Finally, in addition to the steady-state conditions discussed above, the transient experimental results are shown in Figure 11. In this test, the d-axis current is fixed at −3 A, and the load torque is abruptly changed from 3 Nm to 1 Nm at 1.5 s. First, when the load decreases, the motor speed increases and the q-axis current drops, which is consistent with the motor characteristics. Second, for the traditional parameter identification method, the estimated parameters vary significantly with the operating condition. Specifically, the estimated flux linkage, q-axis inductance, and d-axis inductance shift from 0.87 Wb, 0.019 H, and 0.082 H to 0.62 Wb, 0.022 H, and 0.01 H, respectively. Although some values become closer to the true parameters, others deviate further, reflecting the intrinsic instability and limited accuracy of traditional approaches. Third, with the proposed parameter identification method, the estimated parameters exhibit no abrupt changes before and after the load variation, demonstrating high robustness and accuracy. Fourth, even during the dynamic transition, the proposed SMOs remain stable, indicating reliable observer convergence and suitability for real-time control.

4.2. Comparative Results of Flux-Weakening Strategy

(a)

Without parameter mismatch

When the measured inductance and flux linkage are consistent with the real values listed in Table 1, the performances of the traditional and the proposed flux-weakening strategies are compared. Figure 12a,b illustrate the system responses when the motor accelerates from standstill to 600 rpm. It can be observed that, in the absence of parameter mismatch, both the traditional and the proposed flux-weakening strategies operate effectively. Specifically, the motor speed successfully reaches and stabilizes at 600 rpm under both methods. Moreover, the resulting d-axis current is approximately −0.6 A in both cases, regardless of whether the motor parameters are obtained from offline measurements or identified using the proposed SMOS-based method. This result indirectly confirms the accuracy of the proposed parameter identification approach, indicating that it yields parameter values consistent with the actual motor characteristics even when no parameter mismatch is present.

(b)

With parameter mismatch

In this part, when the measured inductance and flux linkage satisfy the following conditions: L_{d_m} = 2L_d, L_{q_m} = 3L_q, and ψ_{f_m} =0.5ψ_f, the current performance of the traditional constant-parameter flux-weakening method and the proposed updated-parameter flux-weakening method for computing the d-axis current reference is compared.

Figure 13 and Figure 14 illustrate the system performance at a target speed of 600 rpm when the traditional and proposed flux-weakening strategies are applied. Both strategies are able to drive the motor to 600 rpm and maintain stable operation, demonstrating that flux-weakening is effective under this operating condition. Nevertheless, a noticeable difference appears once the motor reaches steady state. Specifically, the traditional method produces a final d-axis current of −0.9 A, whereas the proposed method yields −0.7 A. Since the d-axis current directly influences the flux level and copper losses, this discrepancy leads to different energy-conversion efficiencies. With the proposed method, the overall efficiency reaches 88.5%, which is slightly higher than that achieved by the traditional approach. This improvement confirms that the proposed flux-weakening strategy not only ensures speed regulation but also enhances energy utilization, indicating clear performance advantages over the traditional method.

Figure 15 and Figure 16 compare the system performance at a speed of 700 rpm and a load of 1 Nm when the traditional and proposed flux-weakening strategies are implemented. Similar to the results shown in Figure 13 and Figure 14, both flux-weakening methods are capable of accelerating the motor to 700 rpm. However, a clear difference emerges once steady-state operation is reached. Specifically, the conventional method results in a steady-state d-axis current of −1.4 A, whereas the proposed method reduces this value to nearly −1.1 A (8%). Since the d-axis current directly affects the magnetic flux level and copper losses, this difference leads to distinct energy-conversion efficiencies. As a result, the proposed method achieves an overall efficiency of 89%, outperforming the conventional approach, which attains an efficiency of 86%.

5. Conclusions

This paper presents a robust flux-weakening control strategy for IPMSMs in EV applications to address performance degradation caused by multiple parameter variations. By integrating a multi-parameter identification framework with an enhanced CVC-based flux-weakening strategy, reliable high-speed operation is achieved even when inductance and flux linkage deviate significantly from their nominal values.

Specifically, an SMOS constituting three sequential sliding-mode observers (SMOs) is developed to estimate the flux linkage, q-axis inductance, and d-axis inductance, whose stability is guaranteed by Lyapunov analysis. Combined with the proposed EAE method, the identification errors of flux linkage, q-axis inductance, and d-axis inductance are reduced from 19.3%, 18.5%, and 37.5% using traditional SMO-based methods to 2.5%, 0.7%, and 1.2%, respectively, under steady-state conditions, while maintaining stable convergence during load transients.

Based on the accurately identified parameters, the proposed flux-weakening strategy effectively mitigates the impact of parameter mismatch. Experimental results show that, under parameter mismatch, the steady-state d-axis current is reduced from −0.9 A to −0.7 A at 600 rpm and from −1.4 A to −1.1 A at 700 rpm, accompanied by an efficiency improvement from 86% to 89% compared with the conventional constant-parameter CVC method. These results confirm that the proposed approach enhances the robustness and energy efficiency of IPMSM flux-weakening control.

Although the proposed method effectively addresses inductance and flux-linkage variations, it does not explicitly consider magnetic saturation and temperature-dependent resistance effects in the current formulation. Hence, future work will focus on extending the proposed SMOS-EAE framework to account for these factors, thereby further improving the robustness of the flux-weakening control strategy under practical operating conditions.