Disturbance Observer-Based Backstepping Control of PMSM for the Mine Traction Electric Locomotive

For the Permanent Magnet Synchronous Motor (PMSM) control system of the Mine Traction Electric Locomotive (MTEL), the fluctuation of the load will lead to the resonance of the velocity of the MTEL. In addition, the speed sensor is easy to be damaged due to the moisture, dust, and vibration. To solve the above problems, a disturbance observer-based (DOB) backstepping control of PMSM for the MTEL is proposed in this paper. First, a full-dimensional Luenberger observer for PMSM is designed and the asymptotically stability of the observer is proved. Next, through the designing of the virtual control input that includes the reconstruction disturbances and using backstepping control strategy, the DOB controller is proposed. The obtained controller can achieve high precision speed tracking and disturbance rejection. Finally, the effectiveness and feasibility of the designed system are verified by Matlab simulation and experiment results.


Introduction
With the development of power electronics technology and control technology, PMSM has been widely applied in various industrial sectors due to its compact size, high torque/inertia ratio, high torque/weight ratio, and absence of rotor loss [1].However, PMSM is a complicated high-order, nonlinear system with multiple variables and strong coupling characteristics as well as external disturbances.Over the last decades, various design methods have been developed [2][3][4].
To gain the information on rotor speed and position of the motor, a commonly used control strategy is to install an encoder or other kinds of sensors on the rotor shaft, but it would increase the system cost and reduce the system reliability.In recent years, to enhance the system performance and reduce the adverse effects of sensors on the system, much attention has been given to achieve sensorless operation [3][4][5][6].In [7], a model reference adaptive system (MRAS) technique has been used for speed estimation in sensorless speed control of PMSM.To obtain the rotor speed of the motor, a reduced-order linear Luenberger observer was proposed in [8].However, the rate of convergence for the Luenberger observer was determined through pole assignment.Furthermore, a passive full-order observer was designed to estimate rotor speed.In order to improve the robustness and accuracy of position and speed estimations, the sliding-mode observers were widely used in very recent years.However, the chattering phenomenon in sliding-mode observer is the major drawback [9].
The disturbance observer does not need to establish accurate mathematical model for the disturbance signal [10].Recently, disturbance observer-based (DOB) control methods have been applied to PMSM system for better robustness against system disturbance [11].On the basis of disturbance observer, sliding model controller was adopted to realize Permanent Magnet Synchronous Motor control proposed in [12], but the control design of low pass filter is sensitive to the noise.In [13], an integral state observer-based controller was designed to improve disturbance rejection performance of PMSM.In [14], a DOB state feedback controller was designed for PMSM system.By using the same disturbance observer, a sensorless control method for PMSM drive was developed 2 Mathematical Problems in Engineering in [15].The proposed DOB controller involved the use of a back electromotive force observer and a torque observer to estimate rotor position and compensate for load torque disturbance, respectively.For the mismatched disturbance, in [16], a DOB integral sliding-mode control approach for linear systems with mismatched disturbances was presented.The disturbance observer is proposed to generate the disturbance estimate, which can be incorporated in the controller to counteract the disturbance.In [17], the load factor of friction was considered and the sliding-mode variable structure controller was designed.Using nonlinear disturbance observer to approximate system uncertainty, a disturbance observerbackstepping control was proposed in [18].However, the observer and controller are designed separately.
Motivated by the discussions above, in this paper, we mainly investigate backstepping speed control for PMSM based on disturbance observer.The contributions include the following: (1) A nonlinear disturbance observer is first constructed to estimate the external slowly time varying disturbance by using system state variables.(2) Based on Lyapunov stability theory, the linear matrix inequality-(LMI-) based design method of DOB is obtained.(3) Based on backstepping control theory, the PMSM rotor speed and current tracking controllers are designed.Meanwhile, global asymptotic stability is guaranteed by Lyapunov stability analysis.
The rest of this paper is organized as follows.In Section 2, the mathematic model of PMSM and problem formulation are presented.The LMI-based nonlinear disturbance observer design and stability analysis as well as the DOB backstepping controller design method are obtained in Section 3. The system simulation and experimental results are presented in Section 4. Some conclusions are drawn in Section 5.

Mathematical Model of Permanent Magnet Synchronous Motor
Assuming that the magnetic circuit of PMSM is unsaturated, magnetic hysteresis and eddy current loss are ignored; the traditional mathematical model of the PMSM can be given by the following equations under the - coordinate framework [3,[13][14][15][16]: where where In this paper, the main control objective is to design a DOB backstepping controller to keep all the signals in the closed loop system bounded and ensure global asymptotic convergence of the desired speed and current tracking errors to zero eventually.

Design of LMI-Based Disturbance Observer.
In this section, for nonlinear system (2), assume that the nonlinear function where  is Lipschitz constant.Based on the above assumption, the observer of nonlinear system (2) is designed as where  1 = [ 11  12  13 ] and  2 is observer gain matrix to be determined.
The nonlinear observer (5) can be written in the following form: î Defining the observer error   = () − x(),   =  − d, we have Setting  = [  ,   ], we have where ]. Thus, the design problem of observer is transformed into the stability problem of error system (9).To obtain an LMIbased observer design method, the following Lemmas are necessary.
Lemma 1 (see [19]).Given real matrices  and  of appropriate dimensions,  ()  +     ()   < 0 (11) for all () satisfying   ()() ≤ , if and only if there exists a constant  > 0, such that Lemma 2 (Schur complement [20]).For a real matrix Ω = Ω  , the following conclusions are equivalent: (3) Ω 22 > 0, and Based on the above Lemmas and applying Lyapunov stability theory, the design method of LMI-based observer can be obtained by the following result.Theorem 3.For nonlinear systems (2), suppose that the observer holds the form (5); if there exist symmetrical positive definite matrix  and matrix  of appropriate dimensions together with real scalar  > 0, such that where Π =  +    −  −     +  2 , then the error dynamics ( 10) is asymptotically stable.Furthermore, the observer gain can be chosen as  =  −1 .
Using Lemma 1 and condition (4), we can obtain Combined with the above formula, inequality ( 14) is equivalent to If we define then By the stability theory of Lyapunov, the observer dynamic error system (10) is asymptotically stable.Besides, seeing  =  and applying the Schur complement, inequality Ψ < 0 is equivalent to (13).The proof is completed.

DOB Backstepping Controller Design.
Backstepping control is an efficient method for nonlinear system.In this paper, the disturbance observer-based backstepping (DBS) control design can be established by the following three steps.
Step 1.Consider the motor rotor mechanical angular velocity dynamics In the first step of the design of backstepping control, a virtual control input of the motor speed  has to be determined.Let  * be the desired trajectory and  * = 0. Define the speed tracking error   =  * − ; thus where w =  − ŵand ĩ =   − î .
Define the first Lyapunov function as where  = ()  (),  is the integral of the velocity error,  > 0,  = ∫  0   .Taking the derivative of  1 and using inequality (18), we have Define the virtual control input where  1 > 0. Therefore, Using the classical inequality ± ⩽  2 + (1/4) 2 ( > 0) yields where and  1 are properly selected such that  1 > 0,  2 > 0, then which ensures that the speed tracking error will converge asymptotically to zero.
Step 2. According to (23), the virtual input current of the  axis can be chosen as Define the  axis current tracking error   = î *  − î .Choose the second Lyapunov function to stabilize  axis current tracking error dynamics as From ( 7), (8), and (27), the following result can be easily obtained: Substituting ( 20) into (29), we have Define Then The derivative of  2 time  is given by In order to keep the -axis current tracking error asymptotically stable, the control law   can be selected as Further, consider the following inequalities: where  3 ,  4 are positive real numbers.Therefore, inequality (33) can be further simplified as where If choosing the right parameter  3 ,  4 satisfies the following conditions: then V2 < 0. Thus, the dynamic error of -axis current is asymptotically stable.
Step 3. The expected value of -axis current is î *  = 0. Define the tracking error as follows: The derivative of   is Select the third Lyapunov function as which results in If we select the control law   as then inequality (41) can be reduced to Based on inequality − 3   w ⩽  5  3  2  +  3 w2 /4 5 ,  5 > 0,inequality (43) can be rewritten as If the parameters  1 ,  2 ,  3 and  4 ,  5 are properly selected such that then V3 < 0, which indicates that the -axis current dynamic error is also asymptotically stable.The objective of tracking control of PMSM is completed.

Numerical Simulation and Experimental Results
In this section, the numerical example and experimental results are presented to demonstrate the validity of the proposed DBS control scheme.The MATLAB/Simulink model of the proposed DBS control system is shown in Figure 1.
The experimental platform is a MTEL as shown in Figure 2. The structure of the MTEL is shown in Figure 3. MTEL have two wheel sets, and each wheel set is equipped with a PMSM, while the PMSM is driven by motor driver.
Stringing provides 550 V direct current, which is power for the MTEL, MTEL takes electricity power from the stringing through pantograph.The experimental platform is a Mine Traction Electric Locomotive (MTEL) as shown in Figure 2. The structure of the MTEL is shown in Figure 3. MTEL has two wheel sets, and each wheel set is equipped with a PMSM, while the PMSM is driven by motor driver.Stringing provides 550 V direct current, which is power for the MTEL; MTEL takes electricity power from the stringing through pantograph.
By some calculations, it can be found that all the conditions of the DBS controller are satisfied.

Simulation Results.
The initial torque of the motor is 0 N⋅m, and the rotation speed is 1000 r/min.At 0.4 s the external load suddenly becomes 140 N⋅m.Then the external load suddenly return back to 0 N⋅m at 0.9 s.A comparison is made between the proposed DBS control scheme and the traditional backstepping (TBS) control scheme.shows that when TBS is adopted, the speed fluctuation range is ±40 r/min and the stabilization time was 0.05 s.
However, when DBS control scheme is applied, the speed fluctuation range is ±20 r/min and the stabilization time was 0.02 s. Figure 6 shows that the torque fluctuation range is ±40 N⋅m of TBS, while the torque fluctuation range of DBS is ±20 N⋅m.
The three-phase currents of the TBS and DBS control scheme are shown in Figures 7 and 8, respectively.It can be seen that the DBS control method can track the reference rotation speed with smaller stability error, smaller overshoots, and less load torque fluctuations than that of the TBS method.Figures 9 and 10 show actual and estimated speed when the LMI-based disturbance observer and traditional observer scheme [16,18] are applied.It can be seen that the proposed observer can estimate the actual speed accurately, and it is more accurate, more efficient, and more stable than the traditional observer scheme.As can be seen in Figure 10, the torque fluctuation range of the TBS scheme is ±40 N⋅m when the load torque is up to 140 N⋅m in 0.72 second and the stabilization time is 0.08 second; the torque fluctuation range of the DBS scheme is ±20 N⋅m when the load torque is up to 140 N⋅m in 0.72 second and the stabilization time is 0.04 second.Three-phase current waveform of TBS scheme and DBS scheme are, respectively, shown in Figures 13 and 14, which illustrate the low ripple.The actual and estimated speed responses are shown in Figures 15  and 16, which agree with the simulation result well.

Conclusions
In this paper, a disturbance observer-based (DOB) backstepping speed tracking control method has been presented for the speed tracking control of PMSM for MTEL.Through disturbance estimation, the DOB backstepping control strategy can achieve high precision speed tracking and disturbance

Figure 4 :
Figure 4: Electrical schematic diagram of the MTEL driving system.

4. 2 .
Experimental Results.The results of the experiment are shown in Figures11 and 12
,   are - axis stator voltages;   ,   are - axis stator currents;  is the stator resistor;  is the stator inductance; and   is load torque. is the rotation inertia,  is the viscosity friction coefficient,  is the pole pair,  is the rotor mechanical angular velocity, and  1 ,  2 are external disturbance.Without loss of generality, we assume that the disturbances are slowly time varying; that is, ḋ 1 = 0, ḋ 2 = 0.