Switched reluctance motors (SRMs) have the simplest and most robust construction among electric motors. Although SRMs have many good characteristics, the doublesalient and high saturated features result in strong nonlinearities. SRMs have never been a popular choice in highaccuracy position control due to the drawback of higher torque ripple compared with conventional motors. The torque ripple can cause large vibrations and acoustic noise [1]. At present, almost all research on SRMs focus on speed or torque control [2]. As an important application aspect, position control of the SRMs still has several problems which must be solved. This paper mainly focuses on solving three problems related to the position control of the SRM: how to enhance the accuracy of position tracking, how to depress torque ripple, and how to reduce the effect of unknown load torque.
Generally speaking, the position control of motors is composed of three control loops: position loop, speed loop, and current loop [3]. As we know, in position control of SRMs, the motor speed is solely controlled by the output of the position loop. That is to say, the speed, which is the differentiation of the rotor position, can not be controlled independently. If one control freedom of motor speed can be added in the position control of the SRM, the positioncontrol flexibility and accuracy of the SRM can be enhanced greatly. Adding this control freedom of speed in the position tracking control of the SRM is a key research problem that is discussed in this paper.
Tracking control is a widely researched problem in modern control theory, which can be divided into trajectorytracking and pathfollowing [4], [5]. Trajectorytracking is concerned with the design of control laws that force the states to reach and follow a time parameterized reference signal. Pathfollowing forces states to follow a specified path without any specific dynamic requirements along the path (such as the constraint of the time or speed). At present, all the position tracking control of the SRMs can be considered trajectory tracking control problems. In position tracking control of the SRMs, if the tracking speed can be controlled deliberately, then the performance of the tracking can be enhanced. The idea in this paper is to add a speed control freedom in the position tracking which originates from the pathfollowing method in [5].
In position tracking control of the SRM, besides the speed control, the torque control performance also need to be considered. At low speed, the torque ripple has strong influence on the position tracking performance. To reduce the torque ripple of the SRM, different torque control techniques have been developed [6][8]. In [6], the direct torque control (DTC) for the SRM was first proposed as the DTC in AC motor. In [7], direct instantaneous torque control (DITC) is used to realize the highdynamic fourquadrant operation of the SRM. In [8], torquesharing function (TSF) is adopted to reduce the torque ripple with torque sharing between different phases. The DTC and DITC can be considered as torque direct control. While the TSF can be considered as indirect torque control. It is of great convenience to use DTC to realize torque control.
There are many elements which affect the position tracking performance of the SRM. Unknown load torque is yet another element which should be considered. If the unknown load torque can be estimated, the position tracking performance can be enhanced further. Backstepping control technique is a systematic and recursive design methodology for nonlinear feedback control [9][11]. The most appealing point of it is to use the virtual control variable to make the original highorder system simple, and the controller can be derived step by step. In this paper, the function of backstepping control is twofold. One is to realize the controller design of speedassigned position tracking, and the other is to estimate the unknown load torque of the SRM.
The disturbance attenuation problem of a nonlinear system has been investigated via a nonlinear model predictive control (MPC) method augmented with a disturbance observer [12]. Disturbance observer has been popularly applied in the design of tracking controllers for motion control systems [13]. Most of the work in the design of disturbance observer is engineeringoriented and lacks sound theoretical justification.In this paper, a disturbance observer is designed based on state equations, which can realize estimation of the unknown load torque. The designed disturbance observer can enhance the estimation accuracy of the load torque further compared with the adaptive estimation in the adaptive backstepping design.
The novelty of the paper can be mainly summarized as following four aspects.
1) The speedassigned method is firstly used in the position tracking control of the SRM to enhance the accuracy of position tracking.
2) The adaptive backstepping control is successfully applied to the speedassignment design of the SRM. The speed control freedom can be easily added in the position tracking control through the backstepping design.
3) The disturbance observer is utilized to realize the load torque estimation of the SRM, which can enhance the estimation accuracy of the load torque compared with the estimation strategy using the adaptive backstepping design.
4) The DTC is used to reduce the torque ripple of the SRM in the position tracking control.
The remainder of this paper is organized as follows. In Section Ⅱ, the model of the SRM is given. In Section Ⅲ, speedassigned position tracking control is designed. In Section Ⅳ, the design of disturbance observer is given. In Section Ⅴ, the simulation procedure is shown, and lots of simulation results are given to certify the rightness and effectiveness of the design. And finally, some conclusions are presented in Section Ⅵ.
Ⅱ. MODEL OF THE SRMThe model of the SRM can be referenced to [14] and given by the following state equations:
$ \dot{\theta}=\omega $  (1) 
$ \dot{\omega}=\frac{B}{J}\omega+\frac{1}{J}\left(\sum\limits_{k=1}^{N_{ph}}T_kT_L\right) $  (2) 
$ \dot{i}_k=\left(\frac{\partial \psi_k}{\partial i_k}\right)^{1}\left[Ri_k\frac{\partial \psi_k}{\partial \theta}\omega+u_k\right] $  (3) 
where
To simplify the position tracking control of the SRM, the mechanical part is selected as the controlled part. The electromagnetic part is enclosed in the inner loop, which will be given in Section Ⅴ. The simplified mechanical part are given as the following state equations:
$ \dot{\theta}=\omega $  (4) 
$ \dot{\omega}=a\omega +bu+d $  (5) 
where
Before the design of the position tracking control, we first give three assumptions.
Assumption 1: The load torque
Remark 1: The load torque is considered as the only uncertainty to the controller design in this paper. If the uncertainties of
Assumption 2: Because the mechanical dynamics of the motor is slower than the electrical dynamics, the load torque of the SRM can be assumed to satisfy that
Assumption 3:
Assumption 4: We assume that
The general position tracking control can be called timeassigned position tracking control. Timeassigned position tracking control means to be at specific position along the reference signal at specific time instances. The timeassigned position tracking can be presented as following equation:
$ \begin{align} \lim\limits_{t\rightarrow +\infty}\eta(t)=\lim\limits_{t\rightarrow +\infty}\theta_d(t)\theta(t)=0 \end{align} $  (6) 
where
In contrast to timeassigned control, speedassigned control means to design a desired speed at specific position. If the reference position
$ \begin{align} \lim\limits_{t\rightarrow +\infty}\eta(t)=\lim\limits_{t\rightarrow +\infty}v_d(\theta_d(t), t)\dot{\gamma}(t)=0 \end{align} $  (7) 
where
$ \begin{align} \dot{\gamma}(t)=v_d(\theta_d(t), t)\eta(t). \end{align} $  (8) 
To simplify the function presentation, we will use
The design of the adaptive backstepping controller for speedassigned position tracking of the SRM can be achieved through two steps.
Step 1: The position tracking error
$ \begin{align} x_1=\theta\theta_d(\gamma). \end{align} $  (9) 
The time derivative of
$ \begin{align} \dot{x}_1=\omega\theta_d^\gamma(v_d\eta). \end{align} $  (10) 
According (10), the virtual variable
$ \begin{align} x_2 =\omega + k_1 x_1 \theta _d^\gamma v_d \end{align} $  (11) 
where
$ \begin{align} \dot {x}_1 =k_1 x_1 +x_2 +\theta _d^\gamma \eta. \end{align} $  (12) 
The first Lyapunov function
$ \begin{align} V_1=\frac{1}{2}x_1^2. \end{align} $  (13) 
The time derivative of Lyapunov function
$ \begin{align} \dot {V}_1 =k_1 x_1^2 +x_1 x_2 +x_1 \theta _d^\gamma \eta. \end{align} $  (14) 
Step 2: The second Lyapunov function
$ \begin{align} V_2 =\frac{1}{2}x_1^2 +\frac{1}{2}x_2^2+\frac{1}{2}\tilde{d}^2+\frac{1}{2}\eta^2. \end{align} $  (15) 
The time derivative of the second Lyapunov function can be written as
$ \begin{align} \dot {V}_2 =k_1 x_1^2 +x_1 x_2 +x_1 \theta _d^\gamma \eta +x_2 \dot {x}_2+\tilde{d}\dot{\tilde{d}}+\eta\dot{\eta} \end{align} $  (16) 
where the time derivative of
$ \begin{align} \dot {x}_2 =&\ a\omega +bu+d+k_1 (k_1 x_1 +x_2 +\theta _d^\gamma \eta )\nonumber\\&\theta _d^\gamma \dot {v}_d \theta _d^{\gamma 2} (v_d \eta )v_d. \end{align} $  (17) 
Until now, the control
$ \begin{align} u=&\frac{1}{b}\left[a\omega+\hat{d}+(1k_1^2)x_1\right.\notag\\ &\, +\left.(k_1+k_2)x_2 \theta_d^\gamma\dot{v}_d\theta_d^{\gamma2}v_d^2\right]\end{align} $  (18) 
where
$ \dot{\tilde{d}}=k_3\tilde{d}x_2 $  (19) 
$ \dot{\eta}=k_4\eta\theta_d^\gamma x_1(k_1\theta_d^\gamma+\theta_d^{\gamma2}v_d)x_2 $  (20) 
where
$ \begin{align} \dot {V}_2=k_1 x_1^2k_2 x_2^2 k_3\tilde {d}^2k_4 \eta ^2. \end{align} $  (21) 
According to Barbalat's Lemma [17], we can obtain
The proposed control structure of SRM with the disturbance observer is given in Fig. 1. As seen from the figure, the composite controller consists of two parts: a controller and a disturbance observer. In the position tracking control of SRM, the unknown load torque
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Fig. 1 The control structure with the disturbance observer. 
From Assumption 2, we can know
$ \begin{align} \dot {z}=kz+k(k \omega +a \omega +bu) \end{align} $  (22) 
and
$ \begin{align} \hat {d}=z+k \omega. \end{align} $  (23) 
From the disturbance observer, we can obtain the estimated load torque
Theorem 1: Consider the model (4) and (5) of SRM with unknown load disturbance
Proof: Combining the model (4) and (5) of SRM and the disturbance observer (22) and (23), the differentiation of
$ \begin{align} \dot {\tilde {d}}=&\, kz+k(k\omega +a \omega +bu)k(a \omega +bu+d) \nonumber\\=&\, k(\hat {d}+d)=k\tilde {d}. \end{align} $  (24) 
Equation (24) can be simplified as
$ \begin{align} \dot{\tilde{d}}=k\tilde {d}. \end{align} $  (25) 
This implies that
The above disturbance observer has the following three advantages.
1) The design of the disturbance observer is very simple. It can enhance the disturbance attenuation ability of any control system with the same structure.
2) The disturbance observer is exponentially stable, and its stability is regardless of the design of the controller. This can enhance the flexibility of the controller design.
3) The disturbance observer can be designed before or after the controller design. It is very meaningful to simplify the control system design. This can be used for other control systems with the form described as (4) and (5).
Ⅴ. SIMULATION PROCEDURE AND ANALYSISThe structure of speedassigned position tracking control of the SRM is given in Fig. 2. In Fig. 2, the speedassigned position tracking controller is abbreviated as SAPTC. The output of the SAPTC block
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Fig. 2 The control structure of the SRM system. 
1) Measurement of
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Fig. 3 The measured electromagnetic torque and flux linkage. 
2) Computation of
The coenergy can be computed from the measured torque data at a constant current
$ \begin{align} W_c(\theta_0, I)=\frac{1}{2}L_uI^2+\int_{0}^{\theta_0}T(\theta, I)d\theta \end{align} $  (26) 
where
$ \begin{align} \psi(\theta, i)=\frac{\partial W_c(\theta, i)}{\partial i} \end{align} $  (27) 
where
To acquire the fluxlinkage characteristics of the SRM, two steps of computation are needed.
1) The data of coenergy
2) The data of fluxlinkage is acquired from the data of the coenergy
Similar to induction motors, equivalent space voltage vectors may be defined for the SRM. The voltage space vectors for each phase are defined as lying on the center axis of the stator pole. The definition of the voltage space vectors for a four phase 8/6 SRM are shown in Fig. 4(a). The voltage state definition of one phase of SRM are given in Fig. 5.
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Fig. 4 The definition of the voltage and fluxlinkage vector. 
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Fig. 5 The voltage state definition of one phase. 
According to the eight effective voltage space vectors in Fig. 4(a), the formation of fluxlinkage directions are given in Fig. 4(b). As in Fig. 4(c), the relation between composite vector of fluxlinkage and the phase fluxlinkage vector are given as following equations:
$ \psi_\alpha =\psi_A\psi_C $  (28) 
$ \psi_\beta =\psi_B\psi_D $  (29) 
$ \psi =\sqrt{\psi_\alpha^2+\psi_\beta^2}. $  (30) 
One of the eight possible states is chosen at a time in order to keep the stator flux linkage and the motor torque within hysteresis bands. The space distribution of eight effective voltage space vectors are given in Fig. 6(a), where
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Fig. 6 Effective voltage phase vectors. 
In the simulation of the speedassigned position tracking control of the SRM, there are five control variables that need to be regulated. The control variables are given as
To test the effectiveness of the speedassigned position tracking control of the SRM, four cases of simulation are given. The desired position reference signal is set as
In the first case of simulation, the assignedspeed of the SRM is set to be
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Fig. 7 The simulation results of the first case. 
In the second case of simulation, the assignedspeed of the SRM is reduced to be
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Fig. 8 The simulation results of the second case. 
In the third case of simulation, the assigned speed of the SRM is set to be
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Fig. 9 The simulation results of the third case. 
In the fourth case of simulation, the assigned speed of the SRM is reduced to be
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Fig. 10 The simulation results of the fourth case. 
The comparison between adaptive estimation and disturbance observer of the speedassigned position tracking under different conditions are given in Table Ⅱ and Table Ⅲ, where the position error, speed error, and torque estimated error represent steady errors. And AE and DO present adaptive estimation and disturbance observer respectively.
To demonstrate the superiority of the proposed method, the speedassigned position tracking control is compared with the traditional PID controller. The structure of the PID control for the SRM is given in Fig. 11. Because the PID control does not need the load torque estimation and cannot realize the speed assignment, the load torque estimation and adaptive estimation units are omitted.
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Fig. 11 The structure of position control with PID control. 
The parameters of the PID controller for the position control of the SRM are given as following:
$ \begin{align*}P=5, \quad I=18, \quad D=0.00045.\end{align*} $ 
In the simulation, the speedassigned position tracking control of the SRM with disturbance observer is compared with the PID controllers. Two cases of the simulation results are given. The reference position and load torque are the same as the conditions in above two cases. The comparisons of the PID control and the speedassigned position tracking control of the SRM with disturbance observer are given in Fig. 12. Fig. 12 shows the reference position, actual position, the tracking error with PID control, and speedassigned strategy based on disturbance observer under different assigned speed. From the comparison, we can conclude that the speedassigned position tracking with disturbance observer has higher tracking accuracy and better tracking dynamics than the traditional PID controller.
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Fig. 12 The comparison between PID controller and speedassigned control with disturbance observer. 
From Figs. 710, Fig. 12 and Tables Ⅱ and Ⅲ given above for the speedassigned position tracking, we can reach the following six conclusions.
1) The speedassigned method can add a speed control freedom to the position tracking of the SRM. When a low speed is assigned to the position tracking, the position tracking error can be reduced. This is very meaningful for position control of SRM.
2) The disturbance observer can estimate the unknown load torque of the SRM. The convergent speed of the disturbance observer can be designed fast enough to suit the controller design.
3) The backstepping controller is effective for the speedassigned position tracking control of the SRM, and the controller has robustness for varying load torque.
4) The DTC has very good direct torque and fluxlinkage control performance for the SRM. There is no need for the control of turnon and turnoff angles. This is the virtue of the DTC to realize the fourquadrant operation of SRM. The transition procedure of SRM in four operating quadrants is very fluent.
5) The torque ripple of the SRM is reduced with combination of the DTC and disturbance observer. In the position tracking procedure, we can see that the torque ripple is small enough to realize the high accuracy of position tracking.
6) The speedassigned strategy is superior to the traditional PID control for the SRM. The speedassigned position tracking control method has large potential for the position control of the SRM in the future.
Ⅵ. CONCLUSIONIn this paper, a novel speedassigned position tracking control of the SRM is designed which achieves excellent position control performance. The main contribution of this paper can be summarized as following three aspects.
1) A speed control freedom is successfully added to the position tracking control of the SRM. This can enhance the flexibility of position tracking control of the SRM.
2) The speedassigned position tracking control of the SRM is realized with adaptive backstepping design. This design makes the speedassigned error and load torque estimation an easy job.
3) The disturbance observer can enhance the accuracy of the load torque estimation. It is very meaningful for the position control of SRM.
The speedassignment method proposed in this paper has a general applicability. The design method can be easily extended to the position control of other motors.
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