A High-Precision Torque Control Method for New Energy Vehicle Motors Based on Virtual Signal Injection

Abstract

The operating temperature of new energy vehicles fluctuates significantly, and variations in motor temperature lead to changes in parameters. These changes introduce errors into the motor’s mathematical model, reducing torque accuracy and causing deviations in the Maximum Torque Per Ampere (MTPA). This paper proposes a Gated Recurrent Unit (GRU) neural network-based torque observer that employs virtual signal injection. Specifically, this method innovatively injects a virtual constant signal into the d-q axis current inputs processed by the neural network to derive the partial derivatives of torque concerning the d-axis and q-axis currents. Subsequently, it calculates the derivative of torque concerning the current vector angle (β) using the total differential equation. By leveraging these partial derivatives, the motor parameters are identified online, and the MTPA current reference value is dynamically adjusted based on the identified parameters. Additionally, the GRU’s internal parameters are fine-tuned in real time using the least mean square (LMS) algorithm, which adjusts based on the derivative of torque concerning the current angle and the error between the observed and actual values, thereby enhancing the accuracy of torque observation, and bringing results closer to the true shaft-end torque. Finally, experimental validation confirms the effectiveness and superiority of the proposed method.

1. Introduction

Permanent magnet motors in new energy vehicles have advantages such as compact design, robust operation, and good dynamic response, and are widely used in new energy vehicles [1,2,3]. Efficient and precise torque control is the key to maintaining the excellent characteristics of the motor. Therefore, for the effective utilization of the reluctance torque of the motor and reduce motor losses, in the constant torque region, MTPA control is commonly employed. Under the condition of a constant stator current, the optimal distribution of d-q axis currents is achieved to minimize the stator current and maximize the torque output. However, the motor’s parameters are not fixed. As the motor operates for a longer time and the temperature rises, the parameters of the motor exhibit strong nonlinearity, which greatly affects motor’s torque control accuracy and causes the MTPA trajectory to shift [4,5,6]. Therefore, precise torque control and MTPA point tracking are very difficult in practical applications.

The control strategy for the PMSM employs torque control, typically utilizing the MTPA to regulate the motor. The MTPA trajectory is conventionally calculated using the motor’s nominal parameters [7,8]. However, due to variations in motor parameters caused by temperature increases, the traditional calculation approach yields suboptimal control performance and results in reduced accuracy of the generated motor torque, thereby failing to achieve optimal MTPA operation. Consequently, developing an accurate torque control algorithm that accounts for temperature changes and appropriately distributes the d-q-axis currents is crucial for maintaining the control characteristics of motors. Previous research has proposed various PMSM motor torque control algorithms that consider the impact of temperature changes, which are broadly divided into offline and online algorithms.

The offline algorithm for permanent magnet motors typically employs a lookup table (LUT) generated through experiments or finite element simulations to gain the d-q axis current reference values for MTPA control [9,10,11]. To achieve accurate MTPA control, constructing LUT must account for magnetic saturation and temperature rise effects, necessitating extensive experimentation or simulation, which incurs significant cost and time. Moreover, the use of simple linear interpolation to derive reference current values introduces errors, thereby limiting the applicability of LUT.

Online algorithms for permanent magnet motors encompass real-time parameter estimation [12,13,14,15,16] and signal injection methods [17,18,19,20]. To address the issue of varying motor parameters due to temperature changes, a common approach is to perform real-time online parameter estimation via parameter identification, thereby enhancing torque control precision and enabling MTPA operation. For instance, reference [12] employs a recursive least squares-based strategy to achieve simultaneous real-time estimation of q-axis inductance and flux linkage, thereby resolving potential rank deficiency problems. However, variations in d-axis inductance and stator resistance can introduce interference into the algorithm. Consequently, some researchers have proposed injecting additional signals to new equations for estimating all parameters [13], although this method generates additional losses and is more complex to implement. In recent years, scholars have introduced high-frequency signal injection techniques to achieve MTPA operation. These control strategies are parameter-independent and exhibit good MTPA tracking characteristics. However, the injection of additional signals inevitably leads to increased copper losses and cannot improve torque control accuracy under changing temperatures.

Aiming at the loss problem caused by high-frequency signal injection, the MTPA control of virtual signal injection is proposed. Firstly, inject a virtual signal into the current vector angle β, secondly, dT_e/dβ is extracted to realize MTPA operation. Since no actual physical signals were injected into the motor circuit, additional losses are avoided [17]. However, the virtual signal injection method has certain limitations in the calculation process, on the one hand, the influence of partial derivatives of the second order and above will be ignored in the calculation, on the other hand, the parameters used in the calculation process are mostly nominal values of the motor, so some errors will inevitably occur, and these errors will affect the precision of MTPA and the precision of torque control.

In recent years, intelligent control theory has developed rapidly, and many excellent deep learning algorithms have emerged such as the feedforward neural network (FNN) [21,22,23,24], the convolutional neural network (CNN) [25,26,27,28,29], the generative adversarial network (GAN) [30,31,32,33], and the recurrent neural network (RNN) [34,35,36,37]. Reference [21] achieved high-precision prediction of multiple internal temperatures of the motor through a data-driven approach based on the differential estimation feedforward neural network. Reference [25] applied deep learning technology to classify electroencephalogram (EEG) signals, demonstrating high robustness and accuracy in the field of brain–computer interface (BCI). By using GoogLeNet, a deep convolutional neural network combined with continuous wavelet transform (CWT), the classification accuracy of multi-class motor imagery tasks was significantly improved. Reference [26] realized rapid identification of faults in current sensors based on the multi-channel global maximum pooling convolutional neural network using three-phase current data. In reference [30], high-accuracy detection of early faults in permanent magnet synchronous motor bearings was achieved through an adversarial training mechanism of generators and discriminators. To further enhance the performance of neural networks, scholars have proposed schemes that integrate neural networks with other optimization algorithms. For instance, in reference [38], the fox optimization algorithm is employed instead of the traditional backpropagation scheme of neural networks, significantly improving the robustness and classification accuracy of the model in noisy environments, and reducing the risk of overfitting compared to traditional ANN. In reference [39], Particle Swarm Optimization (PSO) is combined with the traditional RNN to optimize it, significantly improving the accuracy and convergence speed of the model in calculating the iron loss of PMSM and decreasing the likelihood of overfitting compared to traditional RNN. These studies have shown that neural networks have strong nonlinear fitting ability, and the performance of control systems in complex tasks can be significantly improved by different deep learning algorithms. Therefore, the motor torque can be fitted by choosing the network structure and model appropriately. In reference [40], Adaline neural networks are employed for identifying motor parameters and subsequently compensating for the motor torque by utilizing the identified parameters. This method has enhanced the torque control accuracy of the motor and maintained the MTPA operation. However, the structure of this network is simple, and the convergence speed is fast. However, due to the direct contradiction between the convergence step size and the steady-state error, it is difficult to achieve satisfactory convergence results. Reference [41] proposes a torque observer considering motor temperature based on BP neural networks. This method predicts the output torque of the motor by inputting d-q axis current and motor temperature. However, due to its relatively small number of layers, the torque observed by this method has obvious harmonic ripple.

To sum up, the current torque control algorithm for new energy vehicle motors under temperature changes still has some limitations. Although the existing studies maintain MTPA control while considering torque accuracy, it still has a strong dependence on motor parameters. Based on this, this study takes PMSM as the research object and proposes a high-precision torque control method for new energy vehicles based on virtual signal injection. Different from the existing reference, the proposed method fully considers the influence of motor temperature, identifies motor parameters online using virtual signal injection, and adjusts the GRU torque observer in real time by using the LMS algorithm, which realizes the MTPA optimal current trajectory tracking of high-precision torque control under the change in motor temperature. Article contributions are as follows:

• This paper analyzes the formation mechanism of the reduction in torque control accuracy and MTPA trajectory deviation of PMSM under the influence of temperature and the design Gated Recurrent Unit neural network torque observer considering motor temperature.
• A real-time parameter estimation method based on virtual signal injection was proposed. Through mathematical calculation, the partial derivative of torque with respect to current was obtained, thereby enabling the calculation of motor parameters and dynamic compensation of the maximum torque and (MTPA).
• The LMS algorithm is also employed for online adjustment of the network structure. It takes advantage of the fact that dT_e/dβ is 0 in MTPA to enhance the accuracy of torque measurement.

The rest of this article is as follows: In Section 2, the mathematical formulas for permanent magnet motors and MTPA are introduced, and the influence of motor temperature on the operating characteristics of permanent magnet motors is analyzed. Further, the section presents the relevant contents of the GRU network and the LMS algorithm. In Section 3, the design of the GRU torque observer considering temperature, the online parameter identification strategy based on virtual signal injection, and the dynamic adjustment method of MTPA current reference value are proposed. At the same time, the network parameters are optimized in real time by combining with the LMS algorithm. In Section 4, through building an experimental platform for permanent magnet motors of new energy vehicles, the torque control accuracy, MTPA trajectory tracking performance, and the motor efficiency improvement effect of the proposed method under different temperatures are verified. In Section 5, the proposed method is discussed in relation to some existing torque control strategies and MTPA schemes. In Section 6, the contributions and methods of this paper are summarized.

2. System Model and Network Introduction

2.1. The d-q Axis Model of the Permanent Magnet Motor

In the synchronous coordinate frame, the voltage equation of PMSM is expressed as [4]:

$$\left[\begin{array}{l}{u}_{\mathrm{d}}\\ u_{\mathrm{q}}\end{array}\right]=\left[\begin{array}{l}R+\mathrm{p}{L}_{\mathrm{d}} -{\omega }_{\mathrm{e}}{L}_{\mathrm{q}} \\ {\omega }_{\mathrm{e}}{L}_{\mathrm{d}} R+\mathrm{p}{L}_{\mathrm{q}}\end{array}\right]\left[\begin{array}{l}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right]+\left[\begin{array}{l}0\\ {\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}\end{array}\right]$$

(1)

where u_d and u_q are the d- and q-axis components of the stator voltage; i_d and i_q are the d- and q-axis components of the stator current, ω_e is the electrical angular velocity, ψ_f is the stator flux linkage, R is the stator resistance, L_d and L_q represent d-axis stator inductance and q-axis stator inductance, respectively, and p is the differential symbol.

When the motor is in steady-state operation, Equation (1) can be expressed as:

$$\left\{\begin{array}{l}{u}_{\mathrm{d}}=R{i}_{\mathrm{d}}+L_{\mathrm{d}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{d}}}{dt}}-{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}{i}_{\mathrm{q}}\\ u_{\mathrm{q}}=R{i}_{\mathrm{q}}+L_{\mathrm{q}}{\displaystyle \frac{\mathrm{d}{i}_{\mathrm{q}}}{dt}}+{\omega }_{\mathrm{e}}(L_{\mathrm{d}}{i}_{\mathrm{d}}+{\psi }_{\mathrm{f}})\end{array}\right.$$

(2)

MTPA: The torque of PMSM is composed of the excitation torque and the reluctance torque. To make full use of the reluctance torque to achieve the purpose of minimum current under the same torque, the following expression needs to be met:

$$\left\{\begin{array}{l}{T}_{\mathrm{e}}=1.5p[{\psi }_{\mathrm{f}}{i}_{\mathrm{q}}+(L_{\mathrm{d}}-L_{\mathrm{q}})i_{\mathrm{d}}{i}_{\mathrm{q}}]\\ \min I_s=\sqrt{{i_{\mathrm{d}}}^2+{i_{\mathrm{q}}}^2}\end{array}\right.$$

(3)

where I_s represents the amplitude of the current vector, and p represents the number of polar logarithms. In order to find the extreme value relationship, it is necessary to pass the extreme value equation of Lagrange’s theorem, and the current vector of the motor needs to meet the following expression:

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial (T_{\mathrm{e}}/I_s)}{\partial i_{\mathrm{d}}}}=0\\ {\displaystyle \frac{\partial (T_{\mathrm{e}}/I_s)}{\partial i_{\mathrm{q}}}}=0\end{array}\right.$$

(4)

By solving the Equation (4), it can be obtained that the d-q axis current expression in MTPA control is:

$$\left\{\begin{array}{l}{i}_{\mathrm{q}}={\displaystyle \frac{\frac{8T_{\mathrm{e}}{\psi }_{\mathrm{f}}}{3p}+\sqrt{{(\frac{8T_{\mathrm{e}}{\psi }_{\mathrm{f}}}{3p})}^2-4[{{\psi }_{\mathrm{f}}}^2-4{(L_{\mathrm{d}}-L_{\mathrm{q}})}^2][{(\frac{4T_{\mathrm{e}}}{3p})}^2-{{\psi }_{\mathrm{f}}}^2]}}{2[{{\psi }_{\mathrm{f}}}^2-4{(L_{\mathrm{d}}-L_{\mathrm{q}})}^2]}}\\ i_{\mathrm{d}}={\displaystyle \frac{-{\psi }_{\mathrm{f}}+\sqrt{{{\psi }_{\mathrm{f}}}^2+4{(L_{\mathrm{d}}-L_{\mathrm{q}})}^2{i_{\mathrm{q}}}^2}}{2(L_{\mathrm{d}}-L_{\mathrm{q}})}}\end{array}\right.$$

(5)

Figure 1 shows the curve of MTPA. MTPA curve is a track curve formed by a series of MTPA operating points, from which the d-q axis current distribution scheme under different stator current vectors can be seen. Within the MTPA control strategy, the optimal operating point corresponds to the tangential intersection of the current vector locus and the constant-torque characteristic curve in the parametric plane. The current vector circle is the set of different stator current vectors with the same stator current vector, and the constant torque curve is the set of stator current vectors with the same torque output. The point of tangency between the two represents the stator current vector that requires the smallest amplitude to output the same torque. The stator current vector corresponding to angle β₂ in Figure 1 and the stator current vector corresponding to β₁ and β₃ output the same size electromagnetic torque, but the required stator current vector amplitude is the smallest.

2.2. Analysis of the Influence of Motor Temperature on the Operating Characteristics of the Permanent Magnet Motor

From Equations (3) and (5), it can be found that accurate motor parameters are needed to calculate motor torque and MTPA. However, because the temperature rise will cause the motor parameters to change, the direct use of the calculation method will lead to a deviation of the calculation results, resulting in decreased motor torque control precision and increased current loss. Figure 2 shows the MTPA trajectory curve under two different motor temperatures and the 80 N·m constant torque curve under two different motor temperatures. It can be found that the MTPA trajectory with a higher motor temperature rotates anti-clockwise relative to the MTPA trajectory with lower motor temperature. This means that if the MTPA trajectory at room temperature continues to be used after the temperature has risen, it will cause the motor operating current to increase, which will result in lower motor efficiency and higher current losses. Moreover, under the same torque, the higher the motor temperature, the farther the constant torque curve is from the origin of the d-q axis current coordinate system, which means that the current required by the motor to output the same torque at high temperature is larger.

2.3. The Gated Recurrent Unit Neural Network

In the operation of the motor, the parameters will change with the working condition, and the traditional neural network only uses the current data to estimate the torque, ignoring the historical information, resulting in poor robustness and large training errors. Although the recurrent neural network (RNN) can capture time-series information, they are susceptible to issues of vanishing or exploding gradients. For this purpose, this paper uses an optimized structural variant of RNN, namely GRU. By updating the gate and resetting the gate mechanism, GRU can effectively retain and forget historical information, solve the gradient problem, simplify the structure, improve the computational efficiency and real-time performance, and is suitable for the motor torque prediction task. Its internal structure is shown in Figure 3, and its expression is:

$$\left\{\begin{array}{l}{Z}_t=\sigma (W_z\cdot [H_{t-1,}{X}_t])\\ R_t=\sigma (W_r\cdot [H_{t-1,}{X}_t])\\ \widetilde{H_t}=\tanh (W\cdot [R_t*H_{t-1,}{X}_t])\\ H_t=(1-Z_t)*H_{t-1,}+Z_t*\widetilde{H_t}\end{array}\right.$$

(6)

where $\sigma$(·) represents the sigmoid function, tanh(·) represents the hyperbolic tangent function, X_t represents the input information of the current moment, H_t−1 represents the hidden state of the previous moment, which acts as memory information of the neural network including the data seen by the previous node, and H_t represents the output of the current unit. R_t and Z_t represent the status of the reset door and the update door, respectively, and $\widetilde{H_t}$ represent the candidate hiding state; W_z, W_r and W represent the weight value of the reset door, the update door and candidate hiding state, respectively.

2.4. The LMS Algorithm

As a classical adaptive filtering algorithm, LMS is often used in adaptive control due to its high computational efficiency. The main goal of the LMS algorithm is to minimize the error value through the gradient descent method, to optimize the system structure and to minimize the error between the system output and the expected output. The basic structure of the LMS algorithm consists of the input signal, the expected signal, and the error signal, and its basic structure diagram is shown in Figure 4, where H(k) represents the input of the system at moment k, y(k) is the system output signal after the adjustment of the input signal at moment k, and d(k) is the expected signal of the system. $\epsilon (k)$ is the error signal at time k of the system, which is obtained by the difference between the output signal of the system and the expected signal and is used to adjust the weight coefficient of the algorithm. Finally, the output signal is approximated to the expected signal through iterative learning. The convergence expression of the LMS algorithm is:

$$\left\{\begin{array}{l}y(k)=W(k)H(k)+b\\ \epsilon (k)=d(k)-y(k)\\ W(k+1)=W(k)+2\eta H(k)\epsilon (k)\\ 0<2\eta H{(k)}^2<1\end{array}\right.$$

(7)

where W(k) is the weight parameter of the system, b is the bias parameter of the system, and η is the step size of the weight calculation.

3. Proposed Methods

To improve the torque observation accuracy of the GRU torque observer and realize MTPA operation to compensate the influence of temperature change, this paper proposes a GRU torque observer and MTPA current reference value adjustment strategy based on virtual signal injection.

3.1. The GRU Network Considering Motor Temperature

From Equations (2) and (3), it is found that motor torque is related to i_d, i_q, ω_e, u_d and u_q. Therefore, the network input designed in this paper includes the above variables, and the motor torque is also affected by temperature changes. Therefore, the motor temperature T_d is added as an additional input to make the prediction of the neural network torque more accurate and closer to the true value of torque. The GRU torque observer considering temperature is shown in Figure 5. The key technical points of the proposed GRU torque observer considering temperature are shown in

Table 1. The key technical points of the proposed GRU torque observer.

Category	Design Proposal	Advantages
Neural network	GRU	Considering timing information, high computational efficiency and simple structure
Weight initialization	Xavier	Avoid drastic changes in gradient, high stability, high training efficiency
Weight update	Adam	High universality and fast convergence

.

Figure 5 shows the relationship between the input and output of the neural network as follows:

$$T_{GRU}=(\sum _{i=1}^6{W}_i{H}_i+b)$$

(8)

where W_i is the weight parameter of the network, b is the network bias parameter, and H_i is the input of the network into the full connection.

In the training process of neural networks, due to the differences in dimensions of different input features, if a unified learning rate and gradients based on different dimensions are used to update parameters of each dimension, the training process may be slow, or even unable to converge to the optimal solution. Therefore, it is particularly important to preprocess the data properly. In this paper, the normalization method is used as a means of data preprocessing, its expression is as follows:

$$x_{norm}={\displaystyle \frac{x-x_{\min }}{x_{\max }-x_{\min }}}$$

(9)

where x_max and x_min are the maximum and minimum values of an input dimension, and x is the original value of the input dimension. The equation can map the input of the network to the range [0, 1].

To obtain the expected output torque, the network output needs to be subjected to inverse normalization after the network training. The equation is as follows:

$$T_{GRU}=T_{\mathrm{norm}}(T_{\max }-T_{\min })+T_{\min }$$

(10)

where T_max and T_min represent the maximum torque and the minimum torque in the training data, respectively. And T_norm is the torque before the inverse normalization of torque. This equation can restore the model and reverse the output data back to the original data range.

3.2. Principle of Motor Parameter Calculation

We inject signal A into the d-axis current and q-axis current of the neural network inputs, that is, i_d + A, i_q, u_d, u_q, ω_e and T_d are used as network inputs, and i_d, i_q + A, u_d, u_q, ω_e and T_d are used as network inputs, and the network outputs are T_GRU(i_d + A, i_q) and T_GRU (i_d, i_q + A), respectively.

Take T_GRU (i_d + A, i_q) and T_GRU (i_d, i_q + A) and perform Taylor series expansion:

$$\left\{\begin{array}{l}{T}_{GRU}(i_{\mathrm{d}}+A,i_{\mathrm{q}})=T_{GRU}(i_{\mathrm{d}},i_{\mathrm{q}})+A*{\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{d}}}}+{\displaystyle \frac{1}{2}}{A}^2{\displaystyle \frac{\partial }{\partial i_{\mathrm{d}}}}({\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{d}}}})+\dots \\ T_{GRU}(i_{\mathrm{d}},i_{\mathrm{q}}+A)=T_{GRU}(i_{\mathrm{d}},i_{\mathrm{q}})+A*{\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{q}}}}+{\displaystyle \frac{1}{2}}{A}^2{\displaystyle \frac{\partial }{\partial i_{\mathrm{q}}}}({\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{q}}}})+\dots \end{array}\right.$$

(11)

where the expressions of partial derivatives ∂T_e/∂i_d and ∂T_e/∂i_q can be obtained from Equation (3) as follows:

$${\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{d}}}}=1.5p(L_{\mathrm{d}}-L_{\mathrm{q}})i_{\mathrm{q}}$$

(12)

$${\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{q}}}}=1.5p({\psi }_{\mathrm{f}}+(L_{\mathrm{d}}-L_{\mathrm{q}})i_{\mathrm{d}})$$

(13)

Since ∂T_GRU/∂i_d does not contain i_d and ∂T_GRU/∂i_q does not contain i_q, the second-order partial derivatives of T_GRU with respect to i_d and i_q can be expressed as:

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial }{\partial i_{\mathrm{d}}}}({\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{d}}}})={\displaystyle \frac{\partial }{\partial i_{\mathrm{d}}}}\left[1.5p(L_{\mathrm{d}}-L_{\mathrm{q}})i_{\mathrm{q}})\right]=0\\ {\displaystyle \frac{\partial }{\partial i_{\mathrm{q}}}}({\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{q}}}})={\displaystyle \frac{\partial }{\partial i_{\mathrm{q}}}}\left[1.5p({\psi }_{\mathrm{f}}+(L_{\mathrm{d}}-L_{\mathrm{q}})i_{\mathrm{d}})\right]=0\end{array}\right.$$

(14)

As can be seen from Equation (14), it can be seen that the second-order partial derivatives of T_GRU with respect to i_d and i_q are both 0, while the higher-order partial derivatives involve the second-order partial derivatives being differentiated again, so the results are also 0. Equation (11) is equivalent to the following equation:

$$\left\{\begin{array}{l}{T}_{GRU}(i_{\mathrm{d}}+A,i_{\mathrm{q}})=T_{GRU}(i_{\mathrm{d}},i_{\mathrm{q}})+A{\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{d}}}}\\ T_{GRU}(i_{\mathrm{d}},i_{\mathrm{q}}+A)=T_{GRU}(i_{\mathrm{d}},i_{\mathrm{q}})+A{\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{q}}}}\end{array}\right.$$

(15)

Based on the output of the neural network, we have obtained T_GRU(i_d + A,i_q), T_GRU(i_d,i_q + A) and T_GRU(i_d,i_q). At this point, since the injected signal A is set by us and is a known number, we can directly calculate ∂T_GRU/∂i_d and ∂T_GRU/∂i_q by taking the difference. By substituting Equations (12) and (13) into Equation (15), the inductance value can be obtained by addition, subtraction and elimination method, and then the flux value under the current working condition can be calculated by the inductance value obtained. The specific equation is shown as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{d}}}}=1.5p(L_{\mathrm{d}}-L_{\mathrm{q}})i_{\mathrm{q}}={\displaystyle \frac{T_{GRU}(i_{\mathrm{d}}+A,i_{\mathrm{q}})-T_{GRU}(i_{\mathrm{d}},i_{\mathrm{q}})}{A}}\\ {\displaystyle \frac{\partial T_{GRU}}{\partial i_{\mathrm{q}}}}=1.5p({\psi }_{\mathrm{f}}+(L_{\mathrm{d}}-L_{\mathrm{q}})i_{\mathrm{d}})={\displaystyle \frac{T_{GRU}(i_{\mathrm{d}},i_{\mathrm{q}}+A)-T_{GRU}(i_{\mathrm{d}},i_{\mathrm{q}})}{A}}\\ (L_{\mathrm{d}}-L_{\mathrm{q}})={\displaystyle \frac{\partial T_{GRU}/\partial i_{\mathrm{d}}}{1.5p{i}_{\mathrm{q}}}}\\ {\psi }_{\mathrm{f}}={\displaystyle \frac{\partial T_{GRU}/\partial i_{\mathrm{q}}}{1.5p}}-{\displaystyle \frac{\partial T_{GRU}/\partial i_{\mathrm{d}}}{1.5p{i}_{\mathrm{q}}}}{i}_{\mathrm{d}}\end{array}\right.$$

(16)

3.3. MTPA Current Reference Value Online Adjustment Policy

The calculated parameters, namely Equation (16) and network output torque T_GRU, are substituted into Equation (5) to obtain the q-axis current compensation value.

$$\left\{\begin{array}{l}{T}_x=T^{*}-T_{GRU}\\ \Delta i_{\mathrm{q}}={\displaystyle \frac{\frac{8T_x{\psi }_{\mathrm{f}}}{3p}+\sqrt{{(\frac{8T_x{\psi }_{\mathrm{f}}}{3p})}^2-4[{{\psi }_{\mathrm{f}}}^2-4{(L_{\mathrm{d}}-L_{\mathrm{q}})}^2][{(\frac{4T_x}{3p})}^2-{{\psi }_{\mathrm{f}}}^2]}}{2[{{\psi }_{\mathrm{f}}}^2-4{(L_{\mathrm{d}}-L_{\mathrm{q}})}^2]}}\end{array}\right.$$

(17)

where T_x is the torque error. T^* is the given torque reference value.

The d-axis current reference value of MTPA is related to the q-axis current and the motor parameters. When calculating inductance parameters and flux linkage of the motor based on virtual signal injection, Equations (16) and (17) are substituted into MTPA Equation (5), and the equation is as follows:

$$\Delta i_{\mathrm{d}}={\displaystyle \frac{-{\psi }_{\mathrm{f}}+\sqrt{{{\psi }_{\mathrm{f}}}^2+4{(L_{\mathrm{d}}-L_{\mathrm{q}})}^2{(i_{\mathrm{q}}+\Delta i_{\mathrm{q}})}^2}}{2(L_{\mathrm{d}}-L_{\mathrm{q}})}}-i_{\mathrm{d}}$$

(18)

3.4. The Network Correction Principle

The traditional neural network is generated by offline training, so the generated neural network is fixed, and its internal parameters are also fixed, but the fixed neural network cannot be dynamically adjusted and cannot adapt to dynamic conditions. This paper proposes a method combining LMS algorithm to process the GRU network as a whole, and dynamically adjust network parameters by judging whether the network output is correct, so as to improve the accuracy and robustness of the network. In this paper, the output of the current unit of the GRU network, H_i, is specifically utilized as the input to the LMS algorithm. Meanwhile, the weight parameters W_i in the fully connected layer are designated as the weights to be adjusted by the LMS algorithm. The internal parameters of the GRU, such as the weights of the update gate and the reset gate, are kept constant after offline pre-training to avoid disrupting its capability of capturing time-series features. By adjusting the weights in the fully connected layer, the mapping relationship from the hidden state to the torque output can be directly corrected, thereby enhancing the accuracy of observation while maintaining the stability of the GRU’s gating mechanism.

One of the characteristics of MTPA control is that dT_e/dβ is equal to 0, when the MTPA current reference value is adjusted by Equations (17) and (18), theoretically speaking, dT_e/dβ is 0 at this time. The following equation needs to be met:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{T}_{\mathrm{e}}}{\mathrm{d}\beta }}={\displaystyle \frac{\partial T_{\mathrm{e}}}{\partial i_d}}(-I_s\cos \beta )+{\displaystyle \frac{\partial T_{\mathrm{e}}}{\partial i_q}}(-I_s\sin \beta )=0\\ -I_s\cos \beta =-(i_{\mathrm{q}}+\Delta i_{\mathrm{q}})\\ -I_s\sin \beta =i_{\mathrm{d}}+\Delta i_{\mathrm{d}}\\ -{\displaystyle \frac{\partial T_{\mathrm{e}}}{\partial i_{\mathrm{d}}}}(i_{\mathrm{q}}+\Delta i_{\mathrm{q}})+{\displaystyle \frac{\partial T_{\mathrm{e}}}{\partial i_{\mathrm{q}}}}(i_{\mathrm{d}}+\Delta i_{\mathrm{d}})=0\end{array}\right.$$

(19)

where −I_scosβ is the q-axis current after adjustment, and −I_ssinβ is the d-axis current after adjustment. At this time, due to MTPA current adjustment, the value of dT_e/dβ is theoretically 0. If the calculation result is not 0, it is proved that the output of the Gated Recurrent Unit generates errors. At this time, the LMS algorithm is used to adjust the internal parameters of the GRU, as shown in Figure 6. The equation is:

$$\left\{\begin{array}{l}\epsilon (k)=0-{\displaystyle \frac{\mathrm{d}{T}_{\mathrm{e}}}{\mathrm{d}\beta }}\\ W(k+1)=W(k)+2\eta H(k)\epsilon (k)\end{array}\right.$$

(20)

where η is the step of weight calculation. ε(k) is the error signal at time k. H(k) is the input into the fully connected network at time k. And W(k) is the internal weight parameter of the network at time k, which is the weight parameter in the fully connected part of the GRU network. In this paper, based on the gradient descent method, the error between the actual dT_e/dβ and 0 is utilized to adjust the weight parameters of the fully connected layer of GRU in real time. However, it does not further affect the internal gating parameters of GRU. This design avoids complex time-series gradient calculations, reduces the computational burden of real-time adjustment, and at the same time utilizes the inherent gating mechanism of GRU to maintain the effective utilization of historical information.

To guarantee the algorithm’s convergence, the equation should be:

$$0<2\eta H{(k)}^2<1$$

(21)

The stability of the algorithm is influenced by η. As pointed out in reference [42], for the normalized data, the selection of η within the range of [0.001, 0.1] is a verified safe range. The advantages and disadvantages of the selection of the step size parameter are analyzed as follows. When η = 0.001, the convergence is slow, although the steady-state error is small, it is difficult to adapt to dynamic conditions and has insufficient real-time performance. When η = 0.1, it may cause weight oscillation and increase the steady-state error. When η = 0.01, this can balance the relationship between the convergence speed and the steady-state error; therefore, 0.01 is chosen as the convergence step size in this paper.

To further illustrate the role of the proposed virtual constant signal injection in the derivation of the partial derivative of torque, sinusoidal signal Asin (ω_ht) is injected into the current vector angle β by the traditional virtual signal injection method. At this time, the current vector angle is:

$${\beta }_1=\beta +A\sin ({\omega }_{\mathrm{h}}\mathrm{t})$$

(22)

The torque after signal injection can be expressed by Taylor expansion as follows:

$$\begin{array}{l}{T}_{\mathrm{e}}({\beta }_1)=T_{\mathrm{e}}(\beta +A\sin ({\omega }_{\mathrm{h}}\mathrm{t})) \\ ==T_{\mathrm{e}}(\beta )+{\displaystyle \frac{\partial T_{\mathrm{e}}}{\partial \beta }}A\sin ({\omega }_{\mathrm{h}}\mathrm{t})+{\displaystyle \frac{\partial }{\partial \beta }}({\displaystyle \frac{\partial T_{\mathrm{e}}}{\partial \beta }})A^2{\sin }^2({\omega }_{\mathrm{h}}\mathrm{t})+\dots \end{array}$$

(23)

To obtain the dT_e/dβ. The virtual signal injection method neglects the second-order and higher-order partial derivatives in Equation (23), while there is a functional relationship between torque and β as follows:

$$T_{\mathrm{e}}=1.5p[{\psi }_{\mathrm{f}}{I}_{\mathrm{s}}\cos \beta -{\displaystyle \frac{1}{2}}(L_{\mathrm{d}}-L_{\mathrm{q}}) I_{\mathrm{s}}^2\sin 2\beta ]$$

(24)

Therefore, if the higher-order partial derivatives are simply ignored, it will to some extent affect the extraction of partial derivative information. Therefore, this paper proposes the method of injecting virtual constant signals to obtain accurate partial derivative information.

In summary, Figure 7 shows the overall control block diagram of the proposed method.

4. Experimental Results and Analysis

To verify the effectiveness of the proposed strategy, this paper builds a set of new energy vehicle PMSM experiment platform, as shown in Figure 8. The torque sensor used is HBM-T12. The sampling frequency and carrier frequency of the control system are both 10 kHz.

Table 2. Main parameters of the PMSM.

Parameter	Sign	Value	Unit
Rated current	IN	437	A
Rated voltage	UN	380	V
Rated load	TN	215	N·m
Rated speed	nN	4000	r/min
Stator resistance	Rs	0.031	Ω
d-axis inductance	Ld	0.0118	mH
q-axis inductance	Lq	0.02375	mH
Poles pairs	P	4	-
Permanent magnet flux	ψf	0.0576	Wb

shows the parameters of the motor. The torque loss of the experimental system under 4000 r/min is approximately 0.9 N·m, and shaft-end torque is obtained by the torque sensor.

To measure the temperature of the motor, a thermistor PT100 was embedded at the end of the winding. PT100 is a platinum thermistor whose resistance varies linearly with temperature. The device used was a TP9000 multiplexed temperature logger (Toprie, Shenzhen, China) (accuracy ± 0.02% rdg.) + 0.3 °C) (measuring temperature range −200 °C to + 660 °C).

Table 3. Resistance values of PT100 at different temperatures.

Temperature (°C)	PT100 Resistance Value (Ω)
0	100
20	107.8
40	115.60
60	123.34
80	130.90
100	138.51
120	146.07
140	153.38

gives the resistance values of PT100 at different temperatures.

4.1. Experimental Results of the Effect of Temperature Rise on the Motor

To prove the influence of temperature change on torque control accuracy and MTPA trajectory, the motor was run for 20 min at a motor speed of 4000 r/min and a given torque value of 140, 160, and 200 N·m, respectively. As shown in

Table 4. Temperature rise data of the PMSM.

Given Torque (N·m)	Mean Value (N·m)	Minimum Value (N·m)	Temperature (°C)
140	137.5	133.14	16.5–108.1
160	157.1	151.47	16.5–125.8
200	192.9	187	16.5–149.5

, the motor temperature rose from normal temperature to 108.1 °C, 125.8 °C, and 149.5 °C, respectively. Table 4 shows the temperature rise data of PMSM, when the torque is given to 140 N·m, the motor torque decreases by 6.86 N·m at most. When the torque is set to 160 N·m, the motor torque decreases by 8.53 N·m at most. When the torque is given to 200 N·m, the motor torque decreases by 13 N·m at most. This is consistent with the conclusion that the motor torque control accuracy decreases under the temperature change analyzed above. The greater the given torque value, the greater the motor temperature change, and the higher the motor temperature, the lower the torque control accuracy of the motor.

Figure 9 shows the MTPA trajectory diagram under three different motor temperatures when the given current amplitude changes from 40–450 A with step size of 40 A to 420 A and finally rises to 450 A at a rotational speed of 4000 r/min. Obviously, when the motor temperature rises, the MTPA trajectory has obvious counterclockwise shift. This is consistent with the conclusion that MTPA trajectory shifts under temperature change analyzed above, and the higher the temperature, the greater the shift.

4.2. High-Precision Torque Control Experiment Results

To compare the observation performance of the GRU torque observer before and after correction, the Mean Absolute Error (MAE) is adopted to compare the GRU torque observer before and after correction. The MAE serves as one of the metrics for evaluating the accuracy of prediction models. It calculates the average of the absolute values of the differences between the predicted values and the actual values. The MAE is expressed as:

$$\mathrm{MAE}={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N\left|T_{m,i}-T_{GRU,i}\right|}$$

(25)

where N represents the number of data points, T_m,i is the true value of the end-axial torque at the i-th position, and T_GRU,i is the observed torque of the i-th network. If the MAE value is smaller, it indicates that the prediction is better.

When the motor is operated at a low temperature of 20–30 °C with a torque reference value of 140-160-200 N·m and a speed of 4000 r/min, the comparison of torque observation between the GRU torque observer considering motor temperature and the GRU torque observer corrected by the LMS algorithm is shown in Figure 10. The ordinary GRU torque observer has a larger error compared with the corrected GRU torque observer when tracking the true shaft-end torque. The maximum error between the ordinary GRU torque observer and the true shaft-end torque is approximately 8 N·m. In contrast, the GRU torque observer corrected by the LMS algorithm can track the true shaft-end torque more accurately, with a smaller error that is maintained within a range of 5 N·m.

When the motor is operated at a medium temperature of 50–60 °C with a torque reference value of 140-160-200 N·m and a speed of 4000 r/min, the comparison of torque observation between the GRU torque observer considering motor temperature and the GRU torque observer corrected by the LMS algorithm is shown in Figure 11. It can be seen that the ordinary GRU torque observer has a larger error compared with the corrected GRU torque observer when tracking the true shaft-end torque. The maximum error between the ordinary GRU torque observer and the true shaft-end torque is approximately 12 N·m. In contrast, the GRU torque observer corrected by the LMS algorithm can track the true shaft-end torque more accurately, with a smaller error that is maintained within a range of 5 N·m.

When the motor is operated at a high temperature of 100–110 °C with a torque reference value of 140-160-200 N·m and a speed of 4000 r/min, the comparison of torque observation between the GRU torque observer considering motor temperature and the GRU torque observer corrected by the LMS algorithm is shown in Figure 12. It can be seen that the ordinary GRU torque observer has a larger error compared with the corrected GRU torque observer when tracking the true shaft-end torque. The maximum error between the ordinary GRU torque observer and the true shaft-end torque is approximately 17 N·m. In contrast, the GRU torque observer corrected by the LMS algorithm can track the true shaft-end torque more accurately, with a smaller error that is maintained within a range of 5 N·m.

To evaluate the performance of the GRU torque observer improved by the LMS algorithm, this paper adds the MAE as an evaluation criterion. The results are shown in

Table 5. The MAE values of the GRU torque observers before and after correction under different temperatures.

Temperature (°C)	GRU (MAE)	The Modified GRU (MAE)
140	4.556	2.146
160	7.572	2.347
200	10.397	2.797

. It can be found that compared with the uncorrected GRU torque observer, the corrected GRU torque observer has a significantly smaller MAE value and better performance. Moreover, the MAE value of the corrected GRU torque observer at low temperatures (20–30 °C) is reduced by 52.8% compared with that of the uncorrected GRU torque observer, at medium temperatures (50–60 °C) it is reduced by 69%, and at high temperatures (100–110 °C) it is reduced by 73.1%. In conclusion, as the temperature rises, the advantages of the modified GRU torque observer become more and more prominent.

Figure 10, Figure 11 and Figure 12 show that with the increase in motor temperature, the performance of the GRU torque observer gradually decreases, and the observation error becomes more and more obvious. The modified GRU torque observer shows excellent performance at low, medium, and high temperatures, and its observation error is within the range of 5 N·m. The torque observation accuracy of the modified GRU torque observer is better and has less error than that of the unmodified GRU torque observer. Therefore, the torque observation methods adopted in the following sections are all modified GRU torque observers.

To verify the effectiveness of the high-precision torque control algorithm proposed in this paper, when the speed is 4000 r/min, the given torque reference is 140-160-200 N·m step change, the motor temperature is 20–30 °C at low temperature, 50–60 °C at medium temperature, and 100–110 °C at high temperature, respectively, the proposed method is used to control the motor torque, and compared with the traditional calculation method of MTPA. Figure 13 is a comparison diagram of the tracking effect of the electromagnetic torque of the motor on the given value of the torque of the proposed method and the calculation method, wherein Figure 13a represents the calculation method and Figure 13b represents the proposed method. Figure 13 shows the comparison of the tracking effect of the electromagnetic torque of the proposed method and the calculated method on the given torque when the reference of the given torque is 140-160-200 N·m step and the rotational speed is 4000 r/min running at a low temperature of 20–30 °C. Figure 13 shows that the electromagnetic torque calculated by the calculation method can track the reference value of the torque at low temperatures. This is because the influence of low temperature on the motor parameters is relatively small. But because the motor parameters are affected by other factors such as magnetic saturation in addition to temperature, the tracking effect of the calculation method is significantly worse under the condition of high torque 200 N·m. At low temperature, the method presented in this paper shows excellent performance in electromagnetic torque tracking under any torque condition, and the torque control precision is higher. In Figure 13, Figure 14 and Figure 15, T_e* represents the given torque reference value.

Figure 14 shows the comparison of the tracking effect of the electromagnetic torque of the proposed method and the calculated method on the given torque when the reference of the given torque is 140-160-200 N·m step and the rotational speed is 4000 r/min, and the operation is carried out at a medium temperature of 50–60 °C. Figure 14 shows that at the medium temperature, the torque tracking effect and torque control accuracy of the calculation method become worse and worse with the gradual increase in torque, and the torque obviously cannot track the torque reference value under 200 N·m and 160 N·m working conditions. The proposed method shows excellent performance of electromagnetic torque tracking under any torque condition at medium temperature, and the torque control precision is higher.

Figure 15 shows the comparison of the tracking effect of the electromagnetic torque of the proposed method and the calculation method on the given torque when the reference of the given torque is 140-160-200 N·m step and the rotational speed is 4000 r/min running at a high temperature of 100–110 °C. Figure 15 shows that at high temperatures, the torque tracking effect and torque control accuracy of the calculation method become worse and worse with the gradual increase in torque, and the torque cannot track the reference value of the torque under the three working conditions of 140-160-200 N·m. At high temperature, the electromagnetic torque tracking effect of the proposed method shows excellent performance under any torque working condition—a higher torque control accuracy.

It can be seen from Figure 13, Figure 14 and Figure 15 that the torque control accuracy of the traditional calculation method gradually decreases with the increase in motor temperature, while the proposed method shows superior performance at low, medium, and high temperatures, and the advantages of the proposed method are more obvious compared with the traditional calculation method with the increase in temperature.

4.3. Optimal Current Trajectory Tracking

Figure 16 shows the results of MTPA tracking performance when the motor temperature is between 100 and 110 °C. Under a constant torque condition, the calculation method requires a larger current to output the same torque, and it seriously deviates from the optimal trajectory. Moreover, as the torque increases, the deviation also gradually becomes larger. Compared with the calculation method, the proposed method has higher tracking accuracy for the MTPA trajectory and better performance.

In addition, the accuracy of the MTPA affects the efficiency of the permanent magnet motor. To verify the superiority of the proposed MTPA method in terms of motor efficiency, By using the Yokogawa-WT3000 power analyzer (Yokogawa Test & Measurement Corporation, Tokyo, Japan), the efficiency curves of the proposed method and the calculation method for the motor were observed when the given torque was 160 N·m. Figure 17a shows the motor efficiency plots of the calculation method at motor temperatures of 50 °C and 100 °C for a given torque of 160 N·m, and Figure 17b shows the motor efficiency plots of the proposed method at motor temperatures of 50 °C and 100 °C for a given torque of 160 N·m. From Figure 17a,b, the motor efficiency of the proposed method is improved by 1.007% and 1.145%, respectively, compared to the computational method.

To further demonstrate the performance of the proposed method, a three-dimensional surface graph was constructed to compare the motor efficiencies of the two methods under different torque conditions and operating temperatures. Two colors are used in the graph to distinguish the two methods: orange represents the proposed method, and blue represents the computational method. From Figure 18, the motor efficiency of the proposed method is generally higher than that of the computational method under all test conditions. Specifically, the efficiency surface of the proposed method (orange) is always above the efficiency surface of the computational method (blue) within the entire temperature and torque range.

Figure 16 verifies the superior performance of the proposed method under MTPA optimal current tracking, Figure 17 and Figure 18 verifies the advantages of the proposed method in terms of motor efficiency.

In summary, the proposed method can improve the torque control accuracy of the motor under temperature change, and the proposed method has excellent performance in torque control performance under different temperature and load conditions, and the method has improved the motor efficiency and MTPA optimal current tracking accuracy compared with the calculation method.

5. Discussion

In MTPA control, the existing methods are generally faced with theoretical limitations and engineering practice challenges. Although the signal injection method realizes the control through specific mathematical conditions, it essentially relies on the open-loop architecture and lacks the real-time torque observation and closed-loop compensation mechanism, so the control accuracy is easily affected by unmodeled dynamics. Among them, the high-frequency signal injection method will increase the loss due to the introduction of the actual disturbed signal, and it is sensitive to parameter change, and the signal processing process is easily disturbed by noise, which reduces the demodulation accuracy. Virtual signal injection rule easily produces torque deviation under complex working conditions because it ignores high-order nonlinear terms and parameter mismatch. The model reference adaptive algorithm identifies L_d, L_q and ψ_f through the voltage equation. Due to the fact that the number of equations are less than that of unknowns, the rank deficiency problem is likely to occur, which in turn leads to the failure of convergence. Moreover, sampling noise also interferes with the model equation, further causing the results to diverge. The recursive least squares method (RLS) estimates system parameters by minimizing the sum of squared errors. However, RLS has strong linear assumptions for the system, and when the motor parameters exhibit nonlinear characteristics with temperature variations, its estimation accuracy will significantly decrease. Moreover, the complexity of RLS calculation is high, which may affect the real-time performance of system control. Noise will be regarded as linear time-varying parameters, leading to cumulative errors, and ultimately the noise algorithm results will diverge.

The proposed approach can not only accurately monitor the motor torque but can also identify the motor parameters online through the virtual signal injection method, thereby achieving MTPA operation. Moreover, by using the GRU update gate and the reset gate to retain the features of the signals that are useful at the current moment, the proposed method effectively filters out noise interference. Additionally, the proposed method combines GRU and LMS algorithms. Since noise is usually random, the LMS algorithm ignores these random fluctuations during the optimization process, thereby reducing the impact of noise on the weight update. The LMS algorithm can adapt to the dynamic input signals by adjusting the weight parameters in real time. Even if the input signal contains noise, the algorithm can minimize the error signal by continuously adjusting the weight parameters, thereby improving the robustness of the system.

In references [43,44], two classic optimization algorithms are presented, namely the gradient-based optimization method and the centralized optimization method. The gradient-based optimization method directly controls the motor through the gradient descent method, which strongly relies on the acquisition of gradient information. However, the motor is a strongly coupled nonlinear system, making it difficult to obtain gradient information, especially when the temperature rise has a nonlinear impact on the motor parameters. As a result, the performance of the gradient-based optimization algorithm will significantly deteriorate. The centralized optimization method treats the motor system as a whole, collects relevant information and parameters of the motor, and then formulates strategies through a central controller. This method requires centralized processing of all data, has high computational complexity, and is relatively dependent on precise mathematical models. However, during actual operation, the motor model parameters will change, making the centralized optimization method difficult to precisely control the motor under conditions of significant temperature variations. In contrast, the proposed method directly calculates the partial derivatives through virtual signal injection and neural networks, avoiding the reliance on gradient information. Moreover, the proposed method can identify motor parameters in real time and dynamically adjust the MTPA current reference value according to temperature changes, thereby maintaining high-precision torque control under different temperature conditions.

The traditional network model (such as the BP neural network) is limited by the one-way data-driven mechanism, and its fixed parameter characteristics are difficult to respond to the time-varying characteristics of the motor operating parameters in real time, and it lacks the modeling ability of the time correlation of dynamic processes. Although sequential memory units are introduced into RNN, gradient anomalies are still encountered in long sequence processing. The gated structure network developed for this purpose shows improvement advantages: the LSTM constructs a dynamic memory unit through three gated systems to effectively balance the storage and updating of historical information; GRU uses only two gates to design, which improves the computational efficiency while maintaining the ability of timing feature extraction and provides a more feasible solution for real-time control system. In addition, the GRU combined with the LMS algorithm adjusts the weight parameters and uses the difference between 0 and dT_e/dβ as the error signal, which not only further optimizes the prediction accuracy under dynamic conditions, but also adapts to the time-varying motor working conditions.

6. Conclusions

To solve the problem that the torque control accuracy of PMSM decreases and the MTPA trajectory shifts when the temperature changes, in this paper, a GRU torque observer considering motor temperature is proposed, and the parameters of the motor are calculated through virtual signal injection to compensate the d-q axis current of MTPA. Finally, LMS is used to adjust the internal weight parameters of the neural network in real time. We summarize the advantages of the proposed method as follows:

• The temperature factor is taken into account in the torque observer to improve the accuracy of torque observation;
• The partial derivative information of torque to d-q axis current calculated by virtual signal injection is used to identify the motor parameters online, so as to realize the correction of the MTPA current reference value.
• The partial derivative information of torque to current vector angle calculated by the LMS algorithm and virtual signal injection is used to adjust the internal parameters of the neural network in real time, which solves the problem of fixed network parameters after traditional offline training and enhances the robustness and prediction accuracy of the network.

In summary, the proposed method not only improves the torque observation accuracy but also achieves MTPA operation under the environment of temperature change, thus improving the torque control accuracy of the motor control system.