## 1. Introduction

In recent years, the study on Electric Vehicles/Hybrid Electric Vehicles (EVs/HEVs) has warmed up due to the serious air pollution and the stringent regulations on emissions as well as constraints on energy resources. For the electric propulsion systems in EVs/HEVs, permanent magnet synchronous motors (PMSM) are extensively applied owing to the high efficiency, high power density and low maintenance cost [

1,

2].

There are two common strategies for PMSM, Field-Oriented Control (FOC) and Direct Torque Control (DTC), which can realize the motor’s high-performance operation. Different from the decoupled current control in FOC, the conventional DTC uses two hysteresis regulators and a switching table to control electromagnetic torque and stator flux linkage directly. So DTC strategy possesses the merits of simple implementation, fast dynamic response, and good robustness against motor parameters’ variation and external disturbances. Hence, the DTC is better than FOC for the application of EVs which require high torque response [

3]. However, this scheme has several major disadvantages, namely large torque and flux ripples, variable switching frequency and high sampling requirement for digital implementation [

4]. To address these problems, numerous modified DTC methods have been proposed from various perspectives. These methods are classified into several categories.

The basic idea is to add more available voltage vectors. The typical approach uses new hardware topologies. In [

5], a three-level inverter is employed instead of the classical two-level inverter, hence generating 27 voltage vectors, and 19 of them are non-repetitive. In [

6], a three-phase to three-phase matrix converter is employed, and 21 voltage vectors are available in the DTC algorithm. However, in order to use more voltage vectors, the hysteresis controllers must have more output levels. The stator flux vector space is subdivided into more sections and the switching table is expanded. Therefore, it inevitably increases the complexity and hardware cost of the system.

In the classical DTC method, the voltage vector is utilized in the entire cycle of motor control. If we can control the applied time of the active voltage vectors, the torque control will be more accurate, and the torque and flux ripples can also be suppressed. Based on this thought, duty cycle control is introduced in the DTC, which adjusts the duty ratio of the active voltage vectors. The key point is how to obtain the duty ratio. Varieties of methods are proposed and differ in the optimization objectives for duty cycle determination [

7,

8,

9]. The method in [

7] determines the duty ratio on the basis of torque ripple RMS (Root-Mean-Square) minimization over one cycle. In [

8], direct mean torque control forces the mean torque equal to the reference value in one cycle. The author in [

9] simplifies the algorithm of duty cycle determination. He applies two adjacent active voltage vectors to the motor control in each cycle, which can regulate the torque and flux precisely. In these strategies, many motor parameters are used to calculate the torque slopes. Hence, the robustness of the system is deteriorated and computation burden is increased. Furthermore, according to the condition where one unique duty determination is used under different speed and torque, there always exists significant steady error of torque.

Another category of the enhanced DTC schemes is incorporated of space vector modulation (SVM). SVM-based DTC can generate voltage vectors with adjustable amplitude and phase, so it can control the electromagnetic torque and stator flux more precisely. As a result, two main advantages can be realized, namely, torque and flux ripples reduction and the fixed switching frequency. The core issue of this strategy is how to obtain the reference voltage vector. There are several different methods in this respect, such as deadbeat controller [

10], torque predictive controller [

11] and proportional-integral controller [

12].

The aforementioned strategies are proposed to achieve high-performance torque control, but speed regulation is also important in traction applications. For instance, the speed control performance of the electric motor has a great influence on the entire gear shifting process in the clutchless automatic mechanical transmission systems [

13]. In general, the PI controller is adopted in the speed loop for improved DTC methods due to its simplicity and clear functionality [

14]. However, a big drawback of the PI controller is that the optimal PI control parameter is depend on the system itself and sensitive to the system uncertainties. As a result, the system variations degrade the control performance dramatically. To overcome this disadvantage, the sliding mode controller is used instead of PI controller in electric drives. It features fast response, disturbances rejection and simple implementation.

This paper not only optimize the torque control in the DTC method, but also focuses on the speed regulation. In order to reduce the torque ripples and fix the switching frequency, the SVM technique is adopted, which an arbitrary reference voltage vector can be synthesized with two adjacent primary voltage vectors and one zero voltage vector. Moreover, a SMC-based torque controller is used to regulate the torque angle increment, and a novel SMC-based speed controller is designed instead of a classical PI controller. Consequently, the proposed scheme improves the dynamic response time and enhances the robustness performance against parameter variations and external loading disturbances.

The rest of this paper is organized as follows:

Section 2 presents the principle of the conventional DTC for PMSM. In

Section 3, the improved DTC method is proposed and analyzed in details. The basic control performance of the conventional and proposed schemes are studied and compared through simulations in

Section 4. This paper is concluded with a summary in

Section 5.

## 3. Proposed DTC Method

#### 3.1. Generation of Reference Voltage Vector

The conventional DTC uses a limited number of voltage vectors with specific amplitude and fixed direction in each control period, while the SVM-based DTC can synthesize an arbitrary reference voltage vector with multiple vectors in each sampling interval. How to obtain the reference voltage vector is core to the process. In [

15,

16], the commanding voltage vector can be obtained through the reference stator flux vector in stationary reference frame. The method is named as reference flux vector calculator (RFVC). It calculates the expected voltage vector according to the load angle increment, the estimated position of the stator, and the amplitude of the reference flux linkage.

The stator voltage and its discrete form can be transformed from Equation (9) into the following equation:

where

${T}_{s}$ is the sampling interval.

${u}_{s}\left(k+1\right)$ is the required voltage vector that forces flux linkage to reach the reference flux linkage vector

${\mathsf{\psi}}_{s}^{\ast}$ at the end of the next sampling interval. Their relationship is shown in

Figure 6. Equation (17) can be decomposed in

$\alpha \beta $ reference frame and are rewritten as:

#### 3.2. Design of SMC-Based Torque Controller

From the equation, the relation between the error of torque and the increment of load angel

$\Delta \delta $ is nonlinear. In [

15,

16], the PI controller is used to generate the load angel increment required to minimize the error between reference

${T}_{e}{}^{*}$ and actual torque. However, owing to the characteristic of the PI controller, the performance varies according to the gain and can also be degraded under system uncertainties.

In this paper, a novel SMC-based torque controller is used to regulate the torque angle increment. Its main objective is to control torque of the motor fast and accurately. In the state space, a switching algorithm is used to move the state trajectory onto a selected surface. The sliding surface is chosen to realize the first-order dynamics, the sliding mode operation is followed:

where

${x}_{T}={T}_{e}^{\ast}-{T}_{e}$ is the torque error, and

${K}_{T}$ is the positive control gain.

The operation of sliding mode restricts the controlled state onto corresponding sliding surface, namely ${S}_{T}=\stackrel{\u2022}{{S}_{T}}=0$. The equivalent dynamics can be defined in this condition.

In order to restrain the shock, a boundary layer was set up for the sliding surface and inserted a linear region.

The control law for the load angle increment is selected as:

where

${K}_{1}$ is the proportional gain in the boundary layer and

${T}_{s}$ is the sampling period.

$u$ is the control rate of the SMC.

${\varphi}_{t}$ is the width of the SMC boundary layer.

$\Delta {\theta}_{s}$ is restrained by the maximum voltage vector of the electric power capacity.

$\Delta {\theta}_{s\mathrm{max}}$ is the maximum flux linkage angle increment. It can be expressed as follows:

Compared to the conventional SMC, the proposed method combined a proportional section in the boundary layer. Proportional gain ${K}_{1}$ can converge the system state to the sliding mode surface faster when the error is large. The system state can be restricted onto the surface ${S}_{T}$ by the control law in the sliding mode.

The PMSM’s DTC control system cannot use a typical symmetry maximum control input, because the control input

$u$ has an extra item

${T}_{s}{\omega}_{r}$. As it shown in the Equation (22), the rising edge control input

${u}^{+}$ is different from the falling edge control input

${u}^{-}$, due to this extra item which is related to the current motor speed. When the speed is high, the response of torque falling is easily overdone on the falling edge because of

$\left|{u}^{-}\right|>\left|{u}^{+}\right|$. In order to solve this problem, a unique dynamic asymmetric boundary layer is designed for this asymmetric SMC control input to improve robustness. The thickness of the boundary layer on both sides of the sliding surface depends on the speed and

${K}_{2}$. It can be expressed as follows:

This method adds a proportional section inside the SMC boundary and design a dynamic asymmetric boundary. It can converge the system state to the sliding mode surface faster with low overshoot and high robustness.

#### 3.3. Design of SMC-Based Speed Controller

To design the SMC-based speed controller, the dynamic torque Equation (6) can be rewritten as:

The speed error ${x}_{r}$ is chosen as the state variable, namely ${x}_{r}={\omega}_{r}^{*}-{\omega}_{r}$. Where ${\omega}_{r}^{*}$ is the reference speed.

The derivative of

${x}_{r}$ is expressed as:

In order to achieve fast convergence and strong robustness, the switching surface is designed as:

where

${K}_{r}$ is a linear feedback gain. When the state trajectory of system Equation (26) is trapped on the switching surface, namely

${S}_{r}={\dot{S}}_{r}=0$, the equivalent dynamics of the system is governed by:

According to the sliding surface, a control law is designed, which guarantees the existence of the sliding mode.

The speed controller can be designed as:

where

${K}_{3}$ is the switching gain and

${G}_{sw}\left(\right)$ is a switch function.

In order to restrain the shock, a boundary layer was set up for the sliding surface and inserted a linear region.

${\mathrm{G}}_{\mathrm{sw}}\left(\right)$ is designed as:

where ${\delta}_{r}$ is thickness of the boundary layer and ${K}_{p}$ is the proportional gain in the boundary layer. In order to eliminate the steady state error, inserted an integral of ${x}_{r}$ in the boundary layer.

The control structure of proposed DTC method is presented in

Figure 7.