Predictive Torque Control of Permanent Magnet Motor for New-Energy Vehicles Under Low-Carrier-Ratio Conditions

Abstract

The model predictive-torque-control strategy of a permanent magnet synchronous motor (PMSM) has many advantages such as a fast dynamic response and the ease of implementation. However, when the permanent magnet motor has a large number of pole pairs or operates at high-speed, due to constraints such as the inverter switching frequency, sampling time, and algorithm execution time, the motor carrier ratio (the ratio of control frequency to operating frequency) becomes relatively low. The discrete model derived from and based on the forward Euler method has a large model error when the carrier ratio decreases, which leads to voltage vector misjudgment and inaccurate duty cycle calculation, thus leading to the decline of control performance. Meanwhile, the shortcomings of the traditional model predictive-torque-control strategy limit the steady-state performance. In response to the above issues, this paper proposes an improved model predictive-torque-control strategy suitable for low-carrier-ratio conditions. The strategy consists of an improved discrete model that considers rotor-angle-position variations and a model prediction algorithm. It also analyzes the sensitivity of model predictive control to parameter changes and designs an online parameter optimization algorithm. Compared with the traditional forward Euler method, the improved discrete model proposed in this paper has obvious advantages under low-carrier-ratio conditions; at the same time, the parameter optimization process enhances the parameter robustness of the model prediction algorithm. Moreover, the proposed model predictive-torque-control strategy has high torque tracking accuracy. The experimental results verify the feasibility and effectiveness of the proposed strategy.

1. Introduction

In the new-energy vehicle field, permanent magnet motors are required to have low-fluctuation- and high-precision torque output characteristics. Direct torque control of the motor is beneficial for reducing torque fluctuations and improving torque control accuracy [1]. Finite control set model predictive control (FCS-MPC) has gained increasing popularity due to its advantages such as the ease of implementation in multivariable systems and its fast dynamic response [2,3,4]. The model predictive-control strategy applied to permanent magnet synchronous motor systems enables the control of multiple output state variables. This paper aims to maintain the high dynamic response advantages of the FCS-MPC strategy while optimizing the torque quality of the control strategy under steady-state conditions.

In the two-level voltage source inverter, the FCS consists of eight basic voltage vectors with fixed amplitude values and phase angles, and the traditional strategy applies one optimal vector in each control cycle. Therefore, the traditional model predictive torque control has few degrees of freedom and a low control accuracy of torque and flux [5,6]. To enhance the control performance, the duty cycle algorithm is introduced into the model predictive torque control and a duty cycle-based model predictive-torque-control strategy is proposed [7,8]. In this approach, two voltage vectors are applied in one control cycle, and the use of zero vectors adjusts the amplitude of the effective voltage vector, thereby improving the motor control performance. However, the flexibility of its output voltage vector is still insufficient, and the strategy can only adjust the amplitude of the spatial voltage vector in six directions. Consequently, reference [9] proposed a dual-effective-voltage-vector model predictive torque control to improve the trajectory of the electromagnetic torque in one control cycle, thereby achieving the effect of effectively suppressing the torque and flux fluctuations. Compared to the traditional model predictive-torque-control strategy, the duty cycle-based multi-vector model predictive-torque-control strategy significantly enhances control performance, but it still has some defects. Firstly, the relationship between the effective vector selection and the duty cycle calculation is weak. The traditional strategy first selects effective vectors and then calculates the duty cycle based on the selected voltage vectors, which may lead to suboptimal vector action combination [10]. Secondly, the duty cycle calculation is disconnected from the cost function, and the duty cycle calculation based on the deadbeat principle only considers the tracking torque and flux reference value as the targets, ignoring the requirement to reduce torque and flux fluctuation [11]. These defects may cause misjudgments of vector-action combination and inaccurate duty cycle calculation.

When the permanent magnet synchronous motor operates under low-carrier-ratio conditions, the switching frequency within a fundamental current cycle significantly reduces. If the switching frequency equals the control frequency, the corresponding control frequency within one fundamental period also decreases, which will affect the stability and control performance of the system. At the same time, reference [12] points out that when the carrier ratio is low, the rotor position changes significantly within one control period, which will cause errors in the voltage vector synthesis, thereby affecting the dynamic and steady-state performance of the control algorithm. The traditional discrete model assumes that the rotor position angle remains approximately unchanged within one control cycle. Therefore, for low-carrier-ratio conditions, the traditional discrete modeling method is no longer applicable. Relevant scholars have extensively studied this issue. Reference [12] proposed a voltage amplitude- and phase angle-correction process to improve the dynamic and steady-state performance of the system. Reference [13] modeled and analyzed the effects of rotor-position-angle changes and digital controller sampling delay on the stable operation of the model prediction algorithm. Reference [14] analyzed the influence of voltage vector-synthesis error on the model prediction torque control algorithm under low-carrier-ratio conditions for the predictive-torque-control strategy, and improved this phenomenon by combining rotor-position-angle compensation to enhance the control accuracy of torque and flux. Reference [15] further indicates that the proposed control strategy can effectively enhance system stability under low-carrier-ratio conditions while maintaining good decoupling effects and dynamic response capabilities. This finding is highly consistent with the research objective of this paper, which is to verify the effectiveness of the improved strategy through experiments. In Reference [16], Yuan Qingqing et al. established a complex vector model of the motor current loop considering digital and angular delays and conducted a detailed analysis of the performance of different decoupling control methods. In this paper, when establishing the control model under low-carrier-ratio conditions, we can draw on their modeling approach that takes delay factors into account, thereby more accurately describing the operating characteristics of the motor under low-carrier-ratio conditions. In Reference [17], it is noted that under high-speed and low-carrier-ratio conditions, the control system of permanent magnet synchronous motors (PMSM) faces the issue of digital delay, which not only causes time delay but also angular delay, thereby significantly impacting the dynamic and steady-state performance of the system. To address this issue, researchers have accurately modeled the digital delay link and conducted an in-depth analysis of its specific impact on system performance. Reference [18] designed a multi-sampling model predictive-control method to reduce errors caused by sampling delay and improved the accuracy of model predictive control. In addition, some scholars adopted the method of an improved converter topology to solve the problem caused by low carrier ratio. Reference [19] doubled the number of power devices while increasing the number of PWM drive signals, thereby improving the equivalent switching frequency and control frequency, reducing the influence of rotor position changes within one control cycle.

Establishing an accurate discrete model is key to ensuring the performance of model predictive control [20]. In order to simplify the calculation, Reference [21] suggests that a set of linear equations can be obtained using the forward Euler method to calculate the predicted torque and flux. This discrete model can achieve a more accurate prediction of torque and flux when the carrier ratio is relatively high. However, when the carrier ratio decreases, the prediction equation has a large error, which can cause misjudgment of the effective vectors and inaccurate duty cycle calculation, leading to a significant decrease in prediction accuracy and control performance [22,23,24]. This phenomenon is caused by the change in rotor angular position under low-carrier-ratio conditions [13].

To address the above problems, this paper proposes a model predictive-torque-control method for permanent magnet synchronous motors suitable for low-carrier-ratio conditions. Firstly, the shortcomings of the traditional discrete model under low carrier ratio are analyzed, and an improved discrete torque-controlled model that considers rotor position variation is established; secondly, the limitations of the traditional model predictive-torque-control strategy are analyzed and improved, and an improved model predictive-torque-control strategy is proposed. To address problems such as decline in algorithm control performance due to time-varying motor parameters and inverter nonlinear compensation, the parameter optimization algorithm is designed based on the improved model predictive-torque-control strategy to further enhance the torque control performance.

2. Traditional Model Predictive Torque Control

This chapter mainly introduces the principle of the traditional model predictive-torque-control method, which consists of two parts: the discretized mathematical model and the basic principle of the predictive-torque-control method. Firstly, the d-q axis mathematical model of the permanent magnet synchronous motor and the electromagnetic torque equation are expounded, and discretization is carried out using the first-order forward Euler equation on this basis. Then, the principle of predictive torque control is explained, and the differences in control effects at high- and low-speeds and the problems faced are analyzed.

2.1. PMSM Mathematical Model

The mathematical model of permanent magnet motor in the d-q axis can be expressed as:

$${\displaystyle \frac{\mathrm{d}\psi }{\mathrm{d}t}}=A\psi +u+Ri$$

(1)

where $\psi =\left[\begin{array}{l}{\psi }_{\mathrm{q}}\\ {\psi }_{\mathrm{d}}\end{array}\right]$; $A=\left[\begin{array}{c}0\\ -{\omega }_{\mathrm{e}}\end{array}\right.\left.\begin{array}{c}{\omega }_{\mathrm{e}}\\ 0\end{array}\right]$; $u=\left[\begin{array}{l}{u}_{\mathrm{d}}\\ u_{\mathrm{q}}\end{array}\right]$; $R=\left[\begin{array}{c}-R_{\mathrm{s}}\\ 0\end{array}\right.\left.\begin{array}{c}0\\ -R_{\mathrm{s}}\end{array}\right]$; $i=\left[\begin{array}{l}{i}_{\mathrm{d}}\\ i_{\mathrm{q}}\end{array}\right]$.

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 rotor electrical angular velocity; ψ_d and ψ_q are the d and q-axis components of the stator flux; and R_s is the stator resistance.

The electromagnetic torque equation can be expressed as:

$$T_{\mathrm{e}}={\displaystyle \frac{3}{2}}{P}_{\mathrm{n}}\left[{\psi }_{\mathrm{d}}{i}_{\mathrm{q}}-{\psi }_{\mathrm{q}}{i}_{\mathrm{d}}\right]$$

(2)

where T_e represents the electromagnetic torque and P_n represents the number of pole pairs of the motor.

2.2. Traditional Model Predictive Torque Control Principle

The basic principle and signal-flow diagram of traditional model predictive torque control are shown in Figure 1. Among them, “P” represents the position information acquisition section, “V_dc” represents the DC bus voltage, “T” represents the torque, and “ψ” represents the flux linkage.

According to Equation (2), the torque and flux prediction model is shown in (3):

$$\left\{\begin{array}{l}{T}_{\mathrm{e}}^{k+1}={\displaystyle \frac{3}{2}}{p}_{\mathrm{n}}\left[{\psi }_{\mathrm{d}}^{k+1}{i}_{\mathrm{q}}^{k+1}-{\psi }_{\mathrm{q}}^{k+1}{i}_{\mathrm{d}}^{k+1}\right]\\ {\psi }_{\mathrm{s}}^{k+1}=\sqrt{{({\psi }_{\mathrm{d}}^{k+1})}^2+{({\psi }_{\mathrm{q}}^{k+1})}^2}\end{array}\right.$$

(3)

In the digital control system, the continuous-time domains typically need to be discretized with a fixed step size. In adjacent sampling cycles, the electrical angular velocity ω_e and the control voltage u are usually assumed to be constant. At this point, the first-order forward Euler equation is applied to discretize the permanent magnet motor model, and the discrete model of current and flux in the synchronous rotating coordinate system is obtained as follows:

In the formula, Ld and Lq are the inductance values of the d-axis and q-axis, respectively.

$$\left\{\begin{array}{l}{i}_{\mathrm{d}}^{k+1}=i_{\mathrm{d}}^k+{\displaystyle \frac{T_{\mathrm{s}}}{L_{\mathrm{d}}}}(u_{\mathrm{d}}^k-R_{\mathrm{s}}{i}_{\mathrm{d}}^k+{\omega }_{\mathrm{e}}{L}_{\mathrm{q}}{i}_{\mathrm{q}}^k)\\ i_{\mathrm{q}}^{k+1}=i_{\mathrm{q}}^k+{\displaystyle \frac{T_{\mathrm{s}}}{L_{\mathrm{q}}}}(u_{\mathrm{q}}^k-R_{\mathrm{s}}{i}_{\mathrm{q}}^k-{\omega }_{\mathrm{e}}{L}_{\mathrm{d}}{i}_{\mathrm{d}}^k-{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}})\end{array}\right.$$

(4)

$$\left\{\begin{array}{l}{\psi }_{\mathrm{d}}^{k+1}={\psi }_{\mathrm{d}}^k+u_{\mathrm{d}}^k{T}_{\mathrm{s}}-R_{\mathrm{s}}{i}_{\mathrm{d}}^k{T}_{\mathrm{s}}+{\omega }_{\mathrm{e}}{T}_{\mathrm{s}}{\psi }_{\mathrm{q}}^k\\ {\psi }_{\mathrm{q}}^{k+1}={\psi }_{\mathrm{q}}^k+u_{\mathrm{q}}^k{T}_{\mathrm{s}}-R_{\mathrm{s}}{i}_{\mathrm{q}}^k{T}_{\mathrm{s}}-{\omega }_{\mathrm{e}}{T}_{\mathrm{s}}{\psi }_{\mathrm{d}}^k\end{array}\right.$$

(5)

Traditional predictive torque control is designed based on the above discrete model. When the motor is running at low-speed conditions, the carrier ratio is relatively high, and the rotor position angle does not change significantly within a carrier cycle. Therefore, the error introduced by the discrete model obtained using the forward Euler method is not obvious. However, when the speed increases and the switching frequency remains unchanged, resulting in a decrease in the carrier ratio, the rotor-position-angle changes more within a unit carrier cycle, and the discrete model approximated by the forward Euler method cannot consider the rotor position change within a carrier cycle, resulting in vector screening misjudgment and decreased duty cycle calculation accuracy, causing the motor to become unstable [25,26,27]. Although increasing the switching frequency can alleviate this problem, the inverter’s switching frequency cannot be further improved due to the constraints of power device performance. Therefore, it is necessary to reduce the vector selection and duty cycle calculation errors of the motor system at low-carrier-ratio conditions while maintaining the switching frequency.

3. Model Predictive-Torque-Control Method for Low Carrier Ratio Operating Conditions

This chapter first establishes a discrete model considering the variation in rotor position for the PMSM. Moreover, to address the problem of reduced control frequency within a fundamental current cycle under high-speed conditions, this paper optimizes the relationship between the voltage vector and the duty cycle based on the oversampling control mode. Finally, an improved model predictive-torque-control strategy based on the optimal vector action combination is proposed.

3.1. Predictive Torque Control Model for Low-Carrier-Ratio Conditions

Firstly, ignoring the resistance voltage drop, Equation (1) can be expressed as:

$${\displaystyle \frac{\mathrm{d}\psi }{\mathrm{d}t}}=A\psi +u$$

(6)

By solving the state Formula (6), the solution can be obtained:

$$\psi (t)=e^{\mathrm{A}t}\psi (0)+{\displaystyle {\int }_0^t{e}^{\mathrm{A}(t-\tau )}u(\tau )}\mathrm{d}\tau ,\hspace{1em}t\ge 0$$

(7)

where $e^{\mathrm{A}t}\psi (0)$ is called the zero-input component of the system, which is defined as the solution of the equation under zero control quantity, that is the state variable corresponding to the zero-vector applied to the inverter. ${\displaystyle {\int }_0^t{e}^{\mathrm{A}(t-\tau )}u(\tau )}\mathrm{d}\tau$ is called the zero-state component of the system, which reflects the state variable corresponding to the effective vector applied to the inverter. $e^{\mathrm{A}t}$ is called the state transfer matrix, and τ is the integral variable about time. In order to facilitate the control of digital systems, it is necessary to discretize the continuous time domain. Set the initial state as the sampling value at time kT_s, and discretize Equation (7) with the control period T_s as the step size, that is, by setting t = (k + 1)T_s in Equation (7):

$$\psi (k+1)=e^{\mathrm{A}{T}_{\mathrm{s}}}\psi (k{T}_{\mathrm{s}})+{\displaystyle {\int }_{k{T}_{\mathrm{s}}}^{(k+1)T_{\mathrm{s}}}{e}^{\mathrm{A}((k+1)T_{\mathrm{s}}-\mathsf{\tau })}u(\tau )}d\tau$$

(8)

At this time, in order to further obtain the zero-input component of the system, it is necessary to further solve the state matrix e^At. The Laplace transform method is used to solve the state transfer matrix e^At:

$$e^{\mathrm{A}{T}_{\mathrm{s}}}=L^{-1}\left\{{(sI-A)}^{-1}\right\}=L^{-1}\left\{\left[\begin{array}{c}s\\ {\omega }_{\mathrm{e}}\end{array}\right.{\left.\begin{array}{c}-{\omega }_{\mathrm{e}}\\ s\end{array}\right]}^{-1}\right\}=L^{-1}\left[\begin{array}{l}{\displaystyle \frac{s}{s^2+{\omega }_{\mathrm{e}}^2}}\\ -{\displaystyle \frac{{\omega }_{\mathrm{e}}}{s^2+{\omega }_{\mathrm{e}}^2}}\end{array}\right.\left.\begin{array}{l}{\displaystyle \frac{{\omega }_{\mathrm{e}}}{s^2+{\omega }_{\mathrm{e}}^2}}\\ {\displaystyle \frac{s}{s^2+{\omega }_{\mathrm{e}}^2}}\end{array}\right]=\left[\begin{array}{c}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ -\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right.\left.\begin{array}{c}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right]$$

(9)

where L⁻¹ represents the Laplace inverse transform; s represents the Laplace variable.

Then, the zero-input component can be expressed as:

$$e^{\mathrm{A}t}\psi (k{T}_{\mathrm{s}})=\left[\begin{array}{cc}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}& \sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ -\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}& \cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right]\left[\begin{array}{l}{\psi }_{\mathrm{d}}^k\\ {\psi }_{\mathrm{q}}^k\end{array}\right]$$

(10)

Now the zero-state component is derived. Due to the discrete characteristics of the digital controller in practical applications, it is not feasible to directly solve the integration. For the convenience of calculation, combined with the discrete characteristics of the motor system, the action vector within a carrier cycle is analyzed in sections. Taking the vector of the first sector as an example, the action vector is analyzed by dividing it into segments. The component of the first vector on the d-q axis is:

$$\begin{array}{r}{V}_{1\mathrm{d}}(t)=\left\{\begin{array}{l}0,t\le T_1\mathrm{or}t\ge T_2\\ {\displaystyle \frac{2}{3}}{V}_{\mathrm{dc}}\cos ({\theta }_1-{\omega }_e{T}_{\mathrm{s}}),T_1<t<T_2\end{array}\right.\\ V_{1\mathrm{q}}(t)=\left\{\begin{array}{l}0,t\le T_1\mathrm{or}t\ge T_2\\ {\displaystyle \frac{2}{3}}{V}_{\mathrm{dc}}\sin ({\theta }_1-{\omega }_e{T}_{\mathrm{s}}),T_1<t<T_2\end{array}\right.\end{array}$$

(11)

where θ₁ is the phase angle of vector one on the α-β axis; V_dc is the DC voltage; and T₂ − T₁ is the duration of vector one action in the first half of the carrier cycle. Assuming that the voltage applied to the inverter remains constant during each vector action, the integral problem can be transformed into discrete domain analysis. At this time, from Equations (8) and (11), it can be obtained that the zero-state component under the action of vector one in the first half of the carrier cycle is:

$${\displaystyle {\int }_{k{T}_{\mathrm{s}}+T_1}^{k{T}_{\mathrm{s}}+T_2}{e}^{\mathrm{A}(k{T}_{\mathrm{s}}+T_2-\mathsf{\tau })}u(\tau )}d\tau ={\displaystyle \frac{2}{3}}{V}_{\mathrm{dc}}(T_2-T_1)\left[\begin{array}{l}\cos ({\theta }_1-{\omega }_e{T}_{\mathrm{s}})\\ \sin ({\theta }_1-{\omega }_e{T}_{\mathrm{s}})\end{array}\right]$$

(12)

Similarly, the components of other voltage vectors on the d-q axis can be obtained. In the model predictive torque control, two effective vectors and one zero vector are selected from the FCS based on the principle of minimizing the cost function. The seven-segment sequence shown in Figure 2 is usually used, where V_x and V_y are effective vectors (x, y = 1, 2, …, 6). V_0/7 is a zero vector, which is used to fill the remaining duty cycle after the effective vector has acted during the carrier cycle.

From the analysis of the above one vector, it is obtained that after the superposition of each voltage vector action in the whole carrier cycle, there are:

$${\displaystyle {\int }_{k{T}_{\mathrm{s}}}^{(k+1)T_{\mathrm{s}}}{e}^{A((k+1)T_{\mathrm{s}}-\mathsf{\tau })}\mathit{u}(\tau )}d\tau ={\displaystyle \frac{2}{3}}{V}_{\mathrm{dc}}\left[\begin{array}{l}(T_2-T_1)\left[\begin{array}{l}\cos ({\theta }_{\mathrm{x}}-{\omega }_e{T}_{\mathrm{s}})\\ \sin ({\theta }_{\mathrm{x}}-{\omega }_e{T}_{\mathrm{s}})\end{array}\right]\\ +(T_3-T_2)\left[\begin{array}{l}\cos ({\theta }_{\mathrm{y}}-{\omega }_e{T}_{\mathrm{s}})\\ \sin ({\theta }_{\mathrm{y}}-{\omega }_e{T}_{\mathrm{s}})\end{array}\right]\\ +(T_5-T_4)\left[\begin{array}{l}\cos ({\theta }_{\mathrm{y}}-{\omega }_e{T}_{\mathrm{s}})\\ \sin ({\theta }_{\mathrm{y}}-{\omega }_e{T}_{\mathrm{s}})\end{array}\right]\\ +(T_6-T_5)\left[\begin{array}{l}\cos ({\theta }_{\mathrm{x}}-{\omega }_e{T}_{\mathrm{s}})\\ \sin ({\theta }_{\mathrm{x}}-{\omega }_e{T}_{\mathrm{s}})\end{array}\right]\end{array}\right]$$

(13)

Equation (13) assumes that θ_x is the phase angle of the V_x in the α-β axis and θ_y is the phase angle of the V_y in the α-β axis. According to the volt-second balance principle, it is obtained that the vectors and their action times satisfy in the d-q axis:

$$\left\{\begin{array}{r}{\displaystyle \frac{2}{3}}{V}_{\mathrm{dc}}\left[\begin{array}{l}(T_2-T_1)\cos ({\theta }_{\mathrm{x}}-{\omega }_e{T}_{\mathrm{s}})+(T_3-T_2)\cos ({\theta }_{\mathrm{y}}-{\omega }_e{T}_{\mathrm{s}})\\ +(T_5-T_4)\cos ({\theta }_{\mathrm{y}}-{\omega }_e{T}_{\mathrm{s}})+(T_6-T_5)\cos ({\theta }_{\mathrm{x}}-{\omega }_e{T}_{\mathrm{s}})\end{array}\right]=u_{\mathrm{d}}^k{T}_{\mathrm{s}}\\ {\displaystyle \frac{2}{3}}{V}_{\mathrm{dc}}\left[\begin{array}{l}(T_2-T_1)\sin ({\theta }_{\mathrm{x}}-{\omega }_e{T}_{\mathrm{s}})+(T_3-T_2)\sin ({\theta }_{\mathrm{y}}-{\omega }_e{T}_{\mathrm{s}})\\ +(T_5-T_4)\sin ({\theta }_{\mathrm{y}}-{\omega }_e{T}_{\mathrm{s}})+(T_6-T_5)\sin ({\theta }_{\mathrm{x}}-{\omega }_e{T}_{\mathrm{s}})\end{array}\right]=u_{\mathrm{q}}^k{T}_{\mathrm{s}}\end{array}\right.$$

(14)

After substituting Equation (14) into Equation (13) and simplifying, it gives:

$${\displaystyle {\int }_{k{T}_{\mathrm{s}}}^{(k+1)T_{\mathrm{s}}}{e}^{\mathrm{A}((k+1)T_{\mathrm{s}}-\mathsf{\tau })}u(\tau )}d\tau =\left[\begin{array}{cc}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}& \sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ -\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}& \cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right]\left[\begin{array}{l}{u}_{\mathrm{d}}^k\\ u_{\mathrm{q}}^k\end{array}\right]T_{\mathrm{s}}$$

(15)

Similarly, the current discrete model under the state equation can be obtained, which will not be elaborated here. From the derivation process and the results, it can be seen that the zero-state component of the motor state equation is the same regardless of the voltage-vector action sequence used, as long as the volt-second balance principle is satisfied. By combining the above solution process, the improved discrete model of the flux considering the transfer matrix and the rotor position change in one carrier cycle can be obtained as follows:

$${\psi }_{\mathrm{d},\mathrm{q}}^{k+1}=\left[\begin{array}{c}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ -\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\begin{array}{c}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right]{\psi }_{\mathrm{d},\mathrm{q}}^k+\left[\begin{array}{c}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ -\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\begin{array}{c}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right]\left[\begin{array}{l}{u}_{\mathrm{d}}^k\\ u_{\mathrm{q}}^k\end{array}\right]T_{\mathrm{s}}$$

(16)

It shows that the discrete model obtained by solving the state equations generates an additional rotation matrix compared to the traditional discrete model, which considers the rotor position changes within a carrier cycle. When the motor is under high-carrier-ratio conditions, the product of ω_e and T_s is considered to have a small value, and according to the principle of mathematical limit, the rotation matrix can be expressed as:

$$\left[\begin{array}{c}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right.\left.\begin{array}{c}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right]\approx \left[\begin{array}{c}1\\ 0\end{array}\right.\left.\begin{array}{c}0\\ 1\end{array}\right]$$

(17)

Due to the existence of controller delay, the method proposed in this paper adds a delay compensation process to compensate for the controller delay, that is, the predicted value at time k + 1 is used as a one-beat delay compensation for the torque and flux.

In order to reduce the impact of digital control delay under low-carrier-ratio conditions, this paper adopts a duty cycle update strategy based on an oversampling mode, that is, the two parts before and after a carrier are regarded as two independent control cycles [28], and the control algorithm is set to execute twice in one carrier cycle, and sampling and control are performed at the beginning and middle positions of a carrier cycle. In this way, the voltage vector can be screened and calculated twice in each carrier cycle. This further reduces the delay error and prediction algorithm error under model predictive control. As shown in Figure 3, the multi-vector effect under the vector screening strategy based on the oversampling mode is significantly better than other vector combinations.

At this point the equation of the improved prediction model in the oversampling mode at the moment T_k+0.5 can be expressed as:

$$\left\{\begin{array}{l}{T}_{\mathrm{e}}^{k+0.5}={\displaystyle \frac{3}{2}}{p}_{\mathrm{n}}\left[{\psi }_{\mathrm{d}}^{k+0.5}{i}_{\mathrm{q}}^{k+0.5}-{\psi }_{\mathrm{q}}^{k+0.5}{i}_{\mathrm{d}}^{k+0.5}\right]\\ {\psi }_{\mathrm{s}}^{k+0.5}=\sqrt{{({\psi }_{\mathrm{d}}^{k+0.5})}^2+{({\psi }_{\mathrm{q}}^{k+0.5})}^2}\\ \left[\begin{array}{l}{i}_{\mathrm{d}}^{k+0.5}\\ i_{\mathrm{q}}^{k+0.5}\end{array}\right]=\left[\begin{array}{c}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ -{\displaystyle \frac{L_{\mathrm{d}}}{L_{\mathrm{q}}}}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\begin{array}{c}{\displaystyle \frac{L_{\mathrm{q}}}{L_{\mathrm{d}}}}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right]\left[\begin{array}{l}{i}_{\mathrm{d}}^k\\ i_{\mathrm{q}}^k\end{array}\right]+\left[\begin{array}{l}0\\ -{\displaystyle \frac{{\omega }_{\mathrm{e}}{\psi }_{\mathrm{f}}}{L_{\mathrm{q}}}}\end{array}\right]+\left[\begin{array}{c}{\displaystyle \frac{1}{L_{\mathrm{d}}}}\\ 0\end{array}\begin{array}{c}0\\ {\displaystyle \frac{1}{L_{\mathrm{d}}}}\end{array}\right]\left[\begin{array}{c}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\begin{array}{c}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right]\left[\begin{array}{l}{u}_{\mathrm{d}}^k\\ u_{\mathrm{q}}^k\end{array}\right]T_{\mathrm{s}}\\ \left[\begin{array}{l}{\psi }_{\mathrm{d}}^{k+0.5}\\ {\psi }_{\mathrm{q}}^{k+0.5}\end{array}\right]=\left[\begin{array}{c}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ -\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\begin{array}{c}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right]\left[\begin{array}{l}{\psi }_{\mathrm{d}}^k\\ {\psi }_{\mathrm{q}}^k\end{array}\right]+\left[\begin{array}{c}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ -\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\begin{array}{c}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right]\left[\begin{array}{l}{u}_{\mathrm{d}}^k\\ u_{\mathrm{q}}^k\end{array}\right]T_{\mathrm{s}}\end{array}\right.$$

(18)

3.2. Model Predictive-Torque-Control Strategy Based on Optimal Vector-Action Combination

This section proposes a novel vector selection method. It combines the oversampling algorithm and further reduces the delay error and prediction error in the model predictive control. In addition, an online parameter optimization algorithm is proposed to further ensure the good control performance of the model predictive control algorithm.

3.2.1. Analysis of Optimal Vector-Action-Combination Strategies

Firstly, the optimal vector V_opt is determined by finding the effective vectors that yield the minimum value of the cost function G₁:

$$G_1={\left({\displaystyle \frac{{T_{\mathrm{e}}}^{\mathrm{ref}}-{T_{\mathrm{e}}}_{\mathrm{v},\mathrm{m}}^{k+1}}{{T_{\mathrm{e}}}^{\mathrm{ref}}}}\right)}^2+\lambda {\left({\displaystyle \frac{{{\psi }_{\mathrm{s}}}^{\mathrm{ref}}-{{\psi }_{\mathrm{s}}}_{\mathrm{v},\mathrm{m}}^{k+1}}{{{\psi }_{\mathrm{s}}}^{\mathrm{ref}}}}\right)}^2$$

(19)

where ${T_{\mathrm{e}}}^{\mathrm{ref}}$ and ${{\psi }_{\mathrm{s}}}^{\mathrm{ref}}$ represent the torque and stator flux amplitude reference values, λ is the weighting factor of the cost function, and $T_{\mathrm{e}\mathrm{v},\mathrm{m}}^{k+1}$ and ${\psi }_{\mathrm{s}\mathrm{v},\mathrm{m}}^{k+1}$ are the predicted values of torque and magnetic flux when the six effective vectors act individually in the unit control cycle using the improved prediction model on the basis of delay compensation.

To ensure a fixed switching frequency, the suboptimal vector will be selected from the previous effective vector V_opt−1 and the next effective vector V_opt+1 of the optimal vector. The selection process of the suboptimal vector is accompanied by the determination of the optimal duty cycle, and the calculation method of the optimal duty cycle will be introduced in the next section. The flowchart of the optimal vector-action combination is shown in Figure 4.

This algorithm is used to optimize the command combinations for motor control. First, the cost function G₁ is used to determine the optimal vector, and then the suboptimal vector candidates are identified. Next, the cost function values of the suboptimal vectors under specific duty cycles are calculated. Finally, the function values are compared, and the optimal command combination is selected to enable the motor to operate stably and efficiently.

$$G_2={\left({\displaystyle \frac{{T_{\mathrm{e}}}^{\mathrm{ref}}-{T_{\mathrm{e}}}^{k+1}}{{T_{\mathrm{e}}}^{\mathrm{ref}}}}\right)}^2+\lambda {\left({\displaystyle \frac{{{\psi }_{\mathrm{s}}}^{\mathrm{ref}}-{{\psi }_{\mathrm{s}}}^{k+1}}{{{\psi }_{\mathrm{s}}}^{\mathrm{ref}}}}\right)}^2$$

(20)

where ${T_{\mathrm{e}}}^{k+1}$, ${{\psi }_{\mathrm{s}}}^{k+1}$ represent the prediction of torque and flux at the next moment when the optimal vector and the alternative suboptimal vector combine with the optimal duty cycle to act on the control system. As Figure 4 shows, after the optimal vector is determined, the alternative suboptimal vectors are confirmed, and the optimal duty ratio is calculated for each of the two suboptimal vectors. This process generates two sets of different alternative-optimal-vector combinations, which are then evaluated by the cost function G₂ to determine the optimal vector combination. The entire process involves strong cascading between vector selection and duty cycle calculation, resulting in a more accurate vector combination.

3.2.2. Optimal Duty Cycle Calculation Method

This section proposes an optimal duty cycle calculation method. As previously mentioned, the vector combination selected based on the duty cycle update strategy in the oversampling mode is significantly better than other vector combinations. The optimal vector action combination algorithm has been proposed in the previous section and now introduces the optimal duty cycle calculation method.

Assuming the optimal vector action time is μ_σμ_θT_s, the suboptimal vector action time is (1 − μ_σ)μ_θT_s, and the zero vector action time is (1 − μ_θ)T_s. Then, the following relationship:

$$\left\{\begin{array}{c}{T_{\mathrm{e}}}^{k+1}={T_{\mathrm{e}}}^{k+0.5}+s_{\mathrm{opt}}^{T_{\mathrm{e}}}{\mu }_{\mathsf{\sigma }}{\mu }_{\mathsf{\theta }}{T}_{\mathrm{s}}+s_{\mathrm{sub}}^{T_{\mathrm{e}}}(1-{\mu }_{\mathsf{\sigma }}){\mu }_{\mathsf{\theta }}{T}_{\mathrm{s}}+s_0^{T_{\mathrm{e}}}(1-{\mu }_{\mathsf{\theta }})T_{\mathrm{s}}\\ {{\psi }_{\mathrm{s}}}^{k+1}={{\psi }_{\mathrm{s}}}^{k+0.5}+s_{\mathrm{opt}}^{{\psi }_{\mathrm{s}}}{\mu }_{\mathsf{\sigma }}{\mu }_{\mathsf{\theta }}{T}_{\mathrm{s}}+s_{\mathrm{sub}}^{{\psi }_{\mathrm{s}}}(1-{\mu }_{\mathsf{\sigma }}){\mu }_{\mathsf{\theta }}{T}_{\mathrm{s}}+s_0^{{\psi }_{\mathrm{s}}}(1-{\mu }_{\mathsf{\theta }})T_{\mathrm{s}}\end{array}\right.$$

(21)

where, as Figure 3 shows, $s_{\mathrm{opt}}^{T_{\mathrm{e}}}$, $s_{\mathrm{sub}}^{T_{\mathrm{e}}}$, and $s_0^{T_{\mathrm{e}}}$, respectively, represent the slopes of torque under the action of the optimal vector, the suboptimal vector, and the zero vector, and $s_{\mathrm{opt}}^{{\psi }_s}$, $s_{\mathrm{sub}}^{{\psi }_{\mathrm{s}}}$, and $s_0^{{\psi }_{\mathrm{s}}}$, respectively, represent the slopes of the stator flux under the action of the optimal vector, the suboptimal vector, and the zero vector.

The problem of finding the vector action time that minimizes the cost function G₂ under various action combinations is a constrained quadratic programming problem. Due to the complexity of solving for the vector action time, a step-by-step approach was adopted in this paper. First, neglecting the effect of the zero vector, μ_θ = 1, substituting Equation (21) into the cost function G₂ we get as follows:

$$G_2\left({\mu }_{\sigma }\right)=\left(a^2+b^2\right){\mu }_{\sigma }^2-2\left(ma+nb\right){\mu }_{\sigma }+\left(m^2+n^2\right)$$

(22)

From the above equation, it can be seen that the cost function G₂ can be represented as a quadratic function problem of the duty cycle. The quadratic function is an upward-opening curve that does not intersect the μ_σ-axis. Based on its properties, it is known that there exists an optimal solution μ_σ that minimizes the cost function subject to the condition $\mathrm{d}{G}_2({\mu }_{\mathsf{\sigma }})/\mathrm{d}{\mu }_{\mathsf{\sigma }}=0$. The solution can be obtained by solving:

$${\mu }_{\sigma }={\displaystyle \frac{ma{\left({\psi }_{\mathrm{s}}^{\mathrm{ref}}\right)}^2+\lambda nb{\left(T_{\mathrm{e}}^{\mathrm{ref}}\right)}^2}{a^2{\left({\psi }_{\mathrm{s}}^{\mathrm{ref}}\right)}^2+\lambda b^2{\left(T_{\mathrm{e}}^{\mathrm{ref}}\right)}^2}}$$

(23)

where

$$\left\{\begin{array}{l}a={T_{\mathrm{e}}}^{\mathrm{v},\mathrm{opt}}-{T_{\mathrm{e}}}^{\mathrm{v},\mathrm{sub}},b={{\psi }_{\mathrm{s}}}^{\mathrm{v},\mathrm{opt}}-{{\psi }_{\mathrm{s}}}^{\mathrm{v},\mathrm{sub}}\\ m={T_{\mathrm{e}}}^{\mathrm{ref}}-{T_{\mathrm{e}}}^{\mathrm{v},\mathrm{sub}},n={{\psi }_{\mathrm{s}}}^{\mathrm{ref}}-{{\psi }_{\mathrm{s}}}^{\mathrm{v},\mathrm{sub}}\end{array}\right.$$

(24)

${T_{\mathrm{e}}}^{\mathrm{v},\mathrm{opt}}$, ${{\psi }_{\mathrm{s}}}^{\mathrm{v},\mathrm{opt}}$, ${T_{\mathrm{e}}}^{\mathrm{v},\mathrm{sub}}$, and ${{\psi }_{\mathrm{s}}}^{\mathrm{v},\mathrm{sub}}$ represent the predicted torque and flux of the optimal vector and suboptimal vector acting on a control cycle. After obtaining the μ_σ that minimizes G₂(μ_σ), the effect of the zero vector is taken into account, and the value of μ_θ that minimizes the cost function G₂ can be obtained:

$${\mu }_{\sigma }={\displaystyle \frac{m_0{a}_0{\left({\psi }_{\mathrm{s}}^{\mathrm{ref}}\right)}^2+\lambda n_0{b}_0{\left(T_{\mathrm{e}}^{\mathrm{ref}}\right)}^2}{a_0^2{\left({\psi }_{\mathrm{s}}^{\mathrm{ref}}\right)}^2+\lambda b_0^2{\left(T_{\mathrm{e}}^{\mathrm{ref}}\right)}^2}}$$

(25)

where

$$\left\{\begin{array}{l}{a}_0=({T_{\mathrm{e}}}^{\mathrm{v},\mathrm{opt}}-{T_{\mathrm{e}}}^{\mathrm{v},\mathrm{sub}}){\mu }_{\mathsf{\sigma }}+{T_{\mathrm{e}}}^{\mathrm{v},\mathrm{sub}}-{T_{\mathrm{e}}}^{\mathrm{v},0}\\ b_0=({{\psi }_{\mathrm{s}}}^{\mathrm{v},\mathrm{opt}}-{{\psi }_{\mathrm{s}}}^{\mathrm{v},\mathrm{sub}}){\mu }_{\mathsf{\sigma }}+{{\psi }_{\mathrm{s}}}^{\mathrm{v},\mathrm{sub}}-{{\psi }_{\mathrm{s}}}^{\mathrm{v},0}\\ m_0={T_{\mathrm{e}}}^{\mathrm{ref}}-{T_{\mathrm{e}}}^{\mathrm{v},0},n_0={{\psi }_{\mathrm{s}}}^{\mathrm{ref}}-{{\psi }_{\mathrm{s}}}^{\mathrm{v},0}\end{array}\right.$$

(26)

This part adopts the improved prediction model proposed in this paper. By combining the optimal duty cycle calculation method and the selection of the optimal vector combination, it allows us to obtain the optimal vector-action combination within the unit control cycle. Compared with the traditional model predictive-torque-control strategy, the vector selection and duty cycle calculation are more reasonable.

As shown in Figure 5, under the multi-vector interaction mechanism of optimal torque control predicted by the improved model proposed in this paper, compared with the traditional sampling and duty cycle updating strategy, the improved duty cycle strategy does not increase the switching frequency, and the torque fluctuation is smaller, and the torque accuracy is higher. By combining the improved duty cycle update strategy with the optimal vector-action-combination strategy, it can further enhance the strategy’s accuracy without increasing the carrier ratio.

3.2.3. Online Parameter Optimization Strategy

From the previous analysis, it can be seen that both the traditional discrete model and the improved model have a certain dependence on the motor parameters, but the motor in the actual operation process, due to temperature rise, magnetic saturation, and other phenomena, will cause nonlinear changes in the parameters. Firstly, changes in the motor parameters can cause deviations in the delay compensation section; secondly, the errors in the delay compensation section can further affect the vector selection and optimal duty cycle calculation in the prediction section; finally, the vector selection and optimal duty cycle calculation in the prediction section are also affected by changes in motor parameters [29,30]. The theoretical analysis of the impact caused by these problems on the delay compensation section is discussed below.

$$\left\{\begin{array}{l}\stackrel{\wedge }{i_{\mathrm{d}}^{k+0.5}}=i_{\mathrm{d}}^{k+0.5}+\mathsf{\Delta }{i}_{\mathrm{d}}^{k+0.5}\\ \stackrel{\wedge }{i_{\mathrm{q}}^{k+0.5}}=i_{\mathrm{q}}^{k+0.5}+\mathsf{\Delta }{i}_{\mathrm{q}}^{k+0.5}\\ \stackrel{\wedge }{{\psi }_{\mathrm{d}}^{k+0.5}}={\psi }_{\mathrm{d}}^{k+0.5}+\mathsf{\Delta }{\psi }_{\mathrm{d}}^{k+0.5}\\ \stackrel{\wedge }{{\psi }_{\mathrm{q}}^{k+0.5}}={\psi }_{\mathrm{q}}^{k+0.5}+\mathsf{\Delta }{\psi }_{\mathrm{q}}^{k+0.5}\end{array}\right.$$

(27)

The first term on the right side of Equation (27) represents the delay compensation value of the state quantity at k + 0.5 time, which is calculated using nominal parameters. The Δ represents the difference between the actual value of the state variable and the delay compensation value, which is caused by the parameter mismatch and the nonlinear factors of the inverter. The resulting errors in delay compensation torque and flux are given by the following:

$$\left\{\begin{array}{l}\mathsf{\Delta }{T}_{\mathrm{e}}^{k+0.5}=\mathsf{\Delta }{i}_{\mathrm{q}}^{k+0.5}{\psi }_{\mathrm{d}}^{k+0.5}+\mathsf{\Delta }{\psi }_{\mathrm{d}}^{k+0.5}{i}_{\mathrm{q}0}^{k+0.5}-\mathsf{\Delta }{i}_{\mathrm{d}}^{k+0.5}{\psi }_{\mathrm{q}}^{k+0.5}-\mathsf{\Delta }{\psi }_{\mathrm{q}}^{k+0.5}{i}_{\mathrm{d}0}^{k+0.5}\\ \mathsf{\Delta }{\psi }_{\mathrm{s}}^{k+0.5}=\sqrt{{(\mathsf{\Delta }{\psi }_{\mathrm{d}}^{k+0.5})}^2+{(\mathsf{\Delta }{\psi }_{\mathrm{q}}^{k+0.5})}^2+2({\psi }_{\mathrm{d}0}^{k+0.5}\mathsf{\Delta }{\psi }_{\mathrm{d}}^{k+0.5}+{\psi }_{\mathrm{q}0}^{k+0.5}\mathsf{\Delta }{\psi }_{\mathrm{q}}^{k+0.5})}\\ \mathsf{\Delta }{i}_{\mathrm{d}}^{k+0.5}=({\displaystyle \frac{L_{\mathrm{q}}}{L_{\mathrm{d}}}}-{\displaystyle \frac{L_{\mathrm{q}0}}{L_{\mathrm{d}0}}})i_{\mathrm{q}}^k\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+T_{\mathrm{s}}(u_{\mathrm{d}}^k-u_{\mathrm{d}}^{\mathrm{dead}})({\displaystyle \frac{1}{L_{\mathrm{d}}}}-{\displaystyle \frac{1}{L_{\mathrm{d}0}}})+({\displaystyle \frac{{\psi }_{\mathrm{f}}}{L_{\mathrm{d}}}}-{\displaystyle \frac{{\psi }_{\mathrm{f}0}}{L_{\mathrm{d}0}}})(\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}-1)\\ \mathsf{\Delta }{i}_{\mathrm{q}}^{k+0.5}=({\displaystyle \frac{L_{\mathrm{d}}}{L_{\mathrm{q}}}}-{\displaystyle \frac{L_{\mathrm{d}0}}{L_{\mathrm{q}0}}})i_{\mathrm{d}}^k\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+T_{\mathrm{s}}(u_{\mathrm{q}}^k-u_{\mathrm{q}}^{\mathrm{dead}})({\displaystyle \frac{1}{L_{\mathrm{q}}}}-{\displaystyle \frac{1}{L_{\mathrm{q}0}}})-({\displaystyle \frac{{\psi }_{\mathrm{f}}}{L_{\mathrm{q}}}}-{\displaystyle \frac{{\psi }_{\mathrm{f}0}}{L_{\mathrm{q}0}}})\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \mathsf{\Delta }{\psi }_{\mathrm{d}}^{k+0.5}=\mathsf{\Delta }{\psi }_{\mathrm{d}}^k\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+\mathsf{\Delta }{\psi }_{\mathrm{q}}^k\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+u_{\mathrm{d}}^{\mathrm{dead}}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+u_{\mathrm{q}}^{\mathrm{dead}}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \mathsf{\Delta }{\psi }_{\mathrm{q}}^{k+0.5}=-\mathsf{\Delta }{\psi }_{\mathrm{d}}^k\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+\mathsf{\Delta }{\psi }_{\mathrm{q}}^k\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}-u_{\mathrm{d}}^{\mathrm{dead}}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+u_{\mathrm{q}}^{\mathrm{dead}}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right.$$

(28)

where

$$\left\{\begin{array}{l}\mathsf{\Delta }{i}_{\mathrm{d}}^{k+0.5}=({\displaystyle \frac{L_{\mathrm{q}}}{L_{\mathrm{d}}}}-{\displaystyle \frac{L_{\mathrm{q}0}}{L_{\mathrm{d}0}}})i_{\mathrm{q}}^k\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+T_{\mathrm{s}}(u_{\mathrm{d}}^k-u_{\mathrm{d}}^{\mathrm{dead}})({\displaystyle \frac{1}{L_{\mathrm{d}}}}-{\displaystyle \frac{1}{L_{\mathrm{d}0}}})+({\displaystyle \frac{{\psi }_{\mathrm{f}}}{L_{\mathrm{d}}}}-{\displaystyle \frac{{\psi }_{\mathrm{f}0}}{L_{\mathrm{d}0}}})(\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}-1)\\ \mathsf{\Delta }{i}_{\mathrm{q}}^{k+0.5}=({\displaystyle \frac{L_{\mathrm{d}}}{L_{\mathrm{q}}}}-{\displaystyle \frac{L_{\mathrm{d}0}}{L_{\mathrm{q}0}}})i_{\mathrm{d}}^k\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+T_{\mathrm{s}}(u_{\mathrm{q}}^k-u_{\mathrm{q}}^{\mathrm{dead}})({\displaystyle \frac{1}{L_{\mathrm{q}}}}-{\displaystyle \frac{1}{L_{\mathrm{q}0}}})-({\displaystyle \frac{{\psi }_{\mathrm{f}}}{L_{\mathrm{q}}}}-{\displaystyle \frac{{\psi }_{\mathrm{f}0}}{L_{\mathrm{q}0}}})\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right.$$

(29)

$$\left\{\begin{array}{l}\mathsf{\Delta }{\psi }_{\mathrm{d}}^{k+0.5}=\mathsf{\Delta }{\psi }_{\mathrm{d}}^k\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+\mathsf{\Delta }{\psi }_{\mathrm{q}}^k\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+u_{\mathrm{d}}^{\mathrm{dead}}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+u_{\mathrm{q}}^{\mathrm{dead}}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\\ \mathsf{\Delta }{\psi }_{\mathrm{q}}^{k+0.5}=-\mathsf{\Delta }{\psi }_{\mathrm{d}}^k\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+\mathsf{\Delta }{\psi }_{\mathrm{q}}^k\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}-u_{\mathrm{d}}^{\mathrm{dead}}\sin {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}+u_{\mathrm{q}}^{\mathrm{dead}}\cos {\omega }_{\mathrm{e}}{T}_{\mathrm{s}}\end{array}\right.$$

(30)

where L_d is the actual parameter of the d-axis inductance, L_q is the actual parameter of the q-axis inductance, ψ_f is the actual parameter of the permanent flux, L_d0 is the nominal value of the d-axis inductance, L_q0 is the nominal value of the q-axis inductance, ψ_f0 is the nominal value of the permanent flux, $i_{\mathrm{d}0}^{k+0.5}$, $i_{\mathrm{q}0}^{k+0.5}$, ${\psi }_{\mathrm{d}0}^{k+0.5}$, and ${\psi }_{\mathrm{q}0}^{k+0.5}$ are the delay compensation state variables calculated using the nominal parameters values, and $u_{\mathrm{d}}^{\mathrm{dead}}$ and $u_{\mathrm{q}}^{\mathrm{dead}}$ are the voltage drops caused by the nonlinearity of the inverter on the d-q axis.

The online optimization of parameters [31] proposed in this paper are shown in Figure 6.

Step 1: First, sample the current to obtain i_dq. After delay compensation, the model—predicted current $i_{\mathrm{d}\mathrm{q}}^{\mathrm{p}\mathrm{r}\mathrm{e}}$ is obtained. Compare the model predicted current$i_{\mathrm{d}\mathrm{q}}^{\mathrm{p}\mathrm{r}\mathrm{e}}$ with the system input and calculate the error E. Determine whether the error E is greater than or equal to the set threshold b. If E ≥ b, proceed to step 2; if E < b, the process ends.

Step 2: The input of the reference model is i_dq, and the input of the adjustable model is$i_{\mathrm{d}\mathrm{q}}^{\mathrm{p}\mathrm{r}\mathrm{e}}$. After comparing the outputs of the two, the error E is calculated. Determine whether the error E is less than or equal to the set threshold a. If E ≤ a, the process ends; if E > a, then use, ψ_r, L_s, and L_m to adjust the parameters in small steps, and then continue the error judgment until the condition is met and the process ends.

In the experimental system, the real value of the shaft end torque is difficult to feed back into the control algorithm, but the stator current can be sampled in real time and fed back into the algorithm. Therefore, the sampled three-phase current transformed by the coordinate transformation into $i_{\mathrm{d}\mathrm{q}}^{\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{l}}$ are used as the reference model. As introduced in the previous section on delay compensation, delay compensation is proposed to solve the impact of one-beat delay in digital control systems. Therefore, the compensated current value $i_{\mathrm{d}\mathrm{q}}^{k+0.5}$ can be used as an adjustable model, and the absolute difference between the two can be defined as the generalized state deviation E.

Considering that the system difference is a random value, but its numerical distribution is approximately Gaussian, the upper limit b and lower limit a of the difference E during stable motor operation are recorded. When the updated upper limit value stably exceeds the value of b during the operation, the motor parameters are considered to be in a mismatched state, and the parameter optimization algorithm is initiated. The motor parameters are adjusted in small steps based on the continuously changing E so that the difference between the output of the reference model and the adjustable model is continuously reduced. When E again meets the judgment conditions, the parameter optimization is completed, and the motor parameter values at this time are recorded and updated in the model predictive-torque-control strategy for the next parameter mismatch judgment.

3.3. Algorithmic Implementation

Figure 7 shows the overall block diagram of the model predictive torque control proposed in this paper. The control system includes the torque- and flux delay-compensation section, the selection vector section and duty cycle-calculation section, the improved sampling- and duty cycle update strategy, the parameter optimization section, and the motor inverter system. In addition, the voltage drop caused by the nonlinearity of the inverter on the d-q axis is calculated and compensated.

The reference torque and speed target values are input, and relevant reference quantities are obtained through look-up tables. The cost function G₁ is used to select the optimal voltage vector and calculate the optimal duty cycle. Multiple calculation and processing steps are carried out, such as online parameter optimization, torque, and flux-linkage delay compensation. Then, the cost function G₂ is utilized to calculate the optimal sub-duty cycle based on the oversampling mode. The duty cycle is updated according to the results and fed back to the system, enabling precise closed-loop control of the motor’s torque, speed, and flux-linkage.

4. Results

In order to verify the correctness of the theoretical analysis and the advantages of the proposed method in terms of torque control accuracy and fluctuation suppression performance, this paper conducts experimental verification with a 40 kw PMSM system.

Table 1. The rated parameters of the PMSM.

Parameters	Symbolic Representation	Value
Rated voltage/V	Vdc	320
Rated torque/(N·m)	Te	72
Rated speed/(r/min)	n	3000
Pole pairs	Pn	4
d-axis inductance Ld/mH	Ld	0.1099
q-axis inductance Lq/mH	Lq	0.3453
Flux linkage/Wb	ψf	0.038749
Stator resistance/Ω	Rs	0.03

lists the motor parameters. In the experimental system, the carrier frequency of the control system is 5 kHz. The experimental system’s friction torque is about 0.3 N·m at low-speed conditions, about 0.4 N·m at medium-speed conditions, and about 0.55 N·m at high-speed conditions. The load torque is measured from the torque sensor.

The waveforms of A-phase current, flux, and torque when the motor is running at low-speed 600 RPM with 60 N·m condition are shown in Figure 8. It should be noted that only the improved prediction model was employed here, and no improved method was utilized. Comparing the waveforms of the traditional prediction model and the improved prediction model, it can be observed that there is little difference in the steady-state performance at low-speed conditions, which is consistent with the previous analysis. This confirms the validity of using the traditional prediction model in many research papers that focused on low-speed conditions.

The waveforms of A-phase current, flux, and torque when the motor is running at medium-speed 3000 RPM with 60 N·m condition are shown in Figure 9. It should be noted that only the improved prediction model was employed here, and no improved method was utilized. Under the medium-speed conditions, torque control experiments using the traditional prediction model result in significant current distortions and tracking errors of about 13 N·m and large torque fluctuations. In contrast, the improved prediction model has significantly improved torque and flux tracking performance, and the current waveform is more sinusoidal with an obvious reduction in torque and flux fluctuations.

The experimental results at a high-speed 6000 RPM with a carrier ratio of about 12 are shown in Figure 10. It should be noted that the traditional model cannot be implemented in high-speed conditions, while the improved model can operate at high speeds. Therefore, only the experimental results of the improved model under the traditional method are listed here. The carrier ratio is calculated based on the ratio between the traditional fixed switching frequency and the motor operating frequency. The improved prediction model has a torque tracking error of approximately 3 N·m, and the current waveform is weak with large flux and torque fluctuations. As can be seen from Figure 10, although the improved prediction model allows the motor system to operate stably at high-speeds with a low switching frequency, its steady-state performance and tracking performance are weak. To address this problem, this paper proposes an improved model prediction-torque-control strategy in Section 3.2, and the specific experimental results will be analyzed subsequently.

Next, the online parametric algorithm strategy proposed in this paper is experimentally verified. To evaluate the effectiveness, a parameter mismatch experiment is conducted by setting the inductance and permanent magnet flux parameters used in the control strategy to 1.2 times their nominal values. Figure 11a shows the waveforms of the A-phase current, flux, and torque under 6000 RPM with 60 N·m condition. According to the parameter optimization judgment condition, the parameter optimization algorithm was initiated and continued until completion, and the torque waveform quality returns to the level before parameter mismatch, as demonstrated by the A-phase current, flux, and torque waveforms in Figure 11b.

When parameter mismatching occurs, both the torque waveform tracking performance and steady-state performance are affected, with the parameter mismatch being the cause of this deviation. Parameter mismatch also causes the flux reference value to deviate. In Figure 11a, the flux reference value is 0.073 Wb after parameter mismatch. In Figure 11b, the stator flux linkage reference value is restored to normal. It can be seen that when the parameter mismatch phenomenon occurs, both the torque waveform tracking performance and steady-state performance are affected. At this time, the delayed compensation i_d,q values deviate from the actual i_d,q values obtained through the coordinate transformation of the sampling circuit and the delay compensation cannot be accurately compensate. According to the parameter optimization judgment conditions, the parameter optimization algorithm is activated. The parameter optimization process continuously reduces the generalized system error E until it recovers to the normal level, as Figure 12 shows.

The analysis of current THD under normal parameter conditions is shown in Figure 13a. The analysis of current THD under parameter mismatch conditions is shown in Figure 13b. The analysis of current THD after the parameter optimization completion is shown in Figure 13c. This comparison reveals that the parameter optimization algorithm not only ensures the recovery of torque accuracy but also does not reduce the current quality.

Next the improved model predictive-torque-control strategy proposed in this paper is verified. The waveforms of A-phase current, flux, and torque when the motor is running at 600 RPM with 60 N·m condition are shown in Figure 14. It is worth noting that the advantages of the improved prediction model have been demonstrated in the previous section and will be used in subsequent experiments. Comparing the results, the improved model predictive control strategy reduces flux and torque fluctuations, produces a more sinusoidal current waveform, and torque can track the torque reference value well. The current harmonic distortion rate is analyzed through FFT analysis, and the improved model predictive-torque-control strategy shows a significantly lower distortion rate. The effect comparison of the two models in a low-speed state is shown in

Table 2. The effect comparison of the two models in a low-speed state.

	Average Torque Tracking Error	Total Harmonic Distortion (THD)/%
Traditional strategy	2.33%	6.60
Improved strategy	1.67%	3.98

.

The waveforms of A-phase current, flux, and torque when the motor is running at 3000 RPM with 60 N·m condition are shown in Figure 15. Comparing the results, the improved model predictive control strategy reduces flux and torque fluctuations, produces a more sinusoidal current waveform, and torque can track the torque reference value well. The current harmonic distortion rate is analyzed through FFT analysis, and the improved model predictive-torque-control strategy shows better performance. The effect comparison of the two models in a medium-speed state is shown in

Table 3. The effect comparison of the two models in a medium-speed state.

	Average Torque Tracking Error	Total Harmonic Distortion (THD)/%
Traditional strategy	4.17%	13.36
Improved strategy	1.17%	7.86

.

The waveforms of A-phase current, flux, and torque when the motor is running at 6000 RPM with the carrier ratio of approximately 12 are shown in Figure 16. By comparing the waveforms, it can be seen that the improved model predictive-torque-control strategy not only achieves stable operation at high-speed conditions but also achieves a significantly improved tracking performance. Moreover, it demonstrates superior performance in flux and torque fluctuations while maintaining a high level of tracking accuracy. The THD of the current is analyzed by FFT and compared as shown in Figure 16. It can be seen that the improved model predictive-torque-control strategy is obviously superior. The effect comparison of the two models in high-speed state is shown in

Table 4. The effect comparison of the two models in high-speed state.

	Average Torque Tracking Error	Total Harmonic Distortion (THD)/%
Traditional strategy	6.00%	21.66
Improved strategy	0.67%	10.07

.

5. Conclusions

This paper proposes a model predictive-torque-control strategy for permanent magnet synchronous motors (PMSM) applicable to low-carrier-ratio operating conditions. The main contributions of this study are summarized as follows:

(1)

This paper analyzes the reasons why the predictive model based on the forward Euler method struggles to achieve accurate torque control under low-carrier-ratio conditions from the perspective of model errors and establishes a motor discrete model considering the rotor position variation. This model is applied to the delay-compensation section and the selection of the optimal-vector-action-combination section.

(2)

The cascade relationship between the voltage vector and the duty cycle is improved, and the optimal duty cycle calculation method is proposed, which is combined with the improved duty cycle updating method to further improve the algorithm accuracy. Using a unit carrier as the analysis period, the oversampling algorithm ensures that the selected voltage vector and interaction time form an optimal combination throughout the entire carrier cycle.

(3)

The parameter sensitivity of the model predictive-torque-control strategy is analyzed, and an online parameter optimization algorithm is proposed. The parameter optimization algorithm not only ensures the accuracy of the algorithm during motor operation but also resolves the modeling errors caused by neglecting the resistance voltage drop to some extent. Experimental results demonstrate that the addition of this section is beneficial for solving the problem of decreased torque control accuracy caused by parameter mismatch occurs.

In conclusion, the proposed model predictive-torque-control strategy is a highly effective algorithm, which has a wide range of application prospects in the permanent magnet motor drive system of new-energy vehicles. Future research can be carried out on multiple aspects. Regarding model optimization, artificial intelligence algorithms such as deep learning and neural networks can be considered for more accurate modeling of the complex operating characteristics of motors. This can enhance the adaptability and accuracy of the model and improve the control precision. To address the issue of computational burden, more efficient optimization algorithms or hardware acceleration technologies can be explored. For example, application-specific integrated circuits (ASICs) or field-programmable gate arrays (FPGAs) can be used for the hardware implementation of algorithms. This allows for reducing the computation time while ensuring control performance, thereby enhancing the system’s real-time performance. To improve the generality of the strategy, it is necessary to expand the scope of experimental motors, covering motors with different power ratings, pole pairs, and other parameters. In-depth research on the application effects of the strategy on different motors can help establish more general control models and parameter adjustment methods. Additionally, the integration of this strategy with other advanced control technologies, such as adaptive control and sliding mode control, can be explored to further enhance the comprehensive control performance of permanent magnet synchronous motors under low-carrier-ratio conditions.

The method is primarily designed for motors operating at high speeds with a low carrier-to-signal ratio. By establishing a discrete model that takes into account the variation in electrical angles and optimizing the duty cycle calculation using an oversampling mode, this method not only can be applied to the drive of high-speed centralized motors in the field of new-energy vehicles but is also suitable for the drive of direct-drive hub motors with multiple pole pairs. Moreover, with appropriate modifications, this method can still be utilized in scenarios where the motor speed is high, or the number of motor pole pairs is large.