Mathematical Problems in Engineering

Volume 2018, Article ID 7253210, 10 pages

https://doi.org/10.1155/2018/7253210

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

^{1}College of Electrical and Information Engineering, Hunan University, Changsha, Hunan 410082, China^{2}College of Electrical and Information, Hunan Institute of Engineering, Xiangtan, Hunan 411100, China^{3}College of Information Engineering, Xiangtan University, Xiangtan 411105, China

Correspondence should be addressed to Jiande Yan; moc.qq@23743304

Received 11 December 2017; Revised 28 March 2018; Accepted 17 April 2018; Published 15 May 2018

Academic Editor: Xue-Jun Xie

Copyright © 2018 Jiande Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. 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 [2–4].

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 [3–6]. 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 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 observer-backstepping 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.

#### 2. 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, 13–16]:where , 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 , are external disturbance. Without loss of generality, we assume that the disturbances are slowly time varying; that is, .

Define ; according to (1), the mathematical model of PMSM can be written as follows: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.

#### 3. Controller Design

##### 3.1. Design of LMI-Based Disturbance Observer

In this section, for nonlinear system (2), assume that the nonlinear function is Lipschitz; that is, for all ,where is Lipschitz constant.

Based on the above assumption, the observer of nonlinear system (2) is designed aswhere and is observer gain matrix to be determined.

The nonlinear observer (5) can be written in the following form:

Defining the observer error , we have

Setting , we havewhere .

Thus, the design problem of observer is transformed into the stability problem of error system (9). To obtain an LMI-based observer design method, the following Lemmas are necessary.

Lemma 1 (see [19]). *Given real matrices and of appropriate dimensions,for all satisfying , if and only if there exists a constant , such that*

Lemma 2 (Schur complement [20]). *For a real matrix , the following conclusions are equivalent:*(1)*;*(2)*, and ;*(3)*, 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 , such thatwhere , then the error dynamics (10) is asymptotically stable. Furthermore, the observer gain can be chosen as .*

*Proof. *Define monochromatic Lyapunov function as ; taking the derivative along system (10), we havewhere .

Using Lemma 1 and condition (4), we can obtainCombined with the above formula, inequality (14) is equivalent toIf we definethenBy the stability theory of Lyapunov, the observer dynamic error system (10) is asymptotically stable. Besides, seeing and applying the Schur complement, inequality is equivalent to (13). The proof is completed.

*3.2. 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 dynamicsIn 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 . Define the speed tracking error ; thuswhere .

Define the first Lyapunov function aswhere , is the integral of the velocity error,

Taking the derivative of and using inequality (18), we haveDefine the virtual control inputwhere .

Therefore,Using the classical inequality yieldswhere *.* If the parameters , >0 and are properly selected such that , thenwhich 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 asDefine the axis current tracking error Choose the second Lyapunov function to stabilize axis current tracking error dynamics asFrom (7), (8), and (27), the following result can be easily obtained:Substituting (20) into (29), we haveDefineThenThe derivative of time is given byIn order to keep the *-*axis current tracking error asymptotically stable, the control law can be selected asFurther, consider the following inequalities:where are positive real numbers*. *Therefore, inequality (33) can be further simplified aswhere .

If choosing the right parameter , satisfies the following conditions:then . Thus, the dynamic error of -axis current is asymptotically stable.

*Step 3. *The expected value of -axis current is . Define the tracking error as follows:The derivative of isSelect the third Lyapunov function aswhich results inIf we select the control law asthen inequality (41) can be reduced toBased on inequality , ,inequality (43) can be rewritten asIf the parameters , , and , are properly selected such thatthen , which indicates that the -axis current dynamic error is also asymptotically stable. The objective of tracking control of PMSM is completed.

*4. 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.*