Mathematical Problems in Engineering

Volume 2019, Article ID 5765390, 13 pages

https://doi.org/10.1155/2019/5765390

## Synchronization of Synchronous Reluctance Motors Using the Discrete Sliding Mode Control Technique

Correspondence should be addressed to Mohamed Zribi; moc.liamg@01ibirz.demahom

Received 30 July 2019; Revised 30 September 2019; Accepted 21 October 2019; Published 4 December 2019

Academic Editor: Haiyan Lu

Copyright © 2019 Salahuddin Abdul Rahman 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

The synchronous reluctance motor (SynRM) drive system is known to exhibit chaotic behavior under specified conditions. In this paper, the discrete-time sliding mode control (DSMC) technique is used to synchronize two SynRMs starting from different sets of initial conditions. The mixed variable speed reaching law is adopted in the design of the controller scheme. The parameters of the designed control scheme are tuned using a genetic algorithm (GA). Simulation results are presented to demonstrate the effectiveness of the proposed controller. In addition, the performance of the proposed control scheme is studied through simulations when bounded disturbances and mismatches between the parameters of the systems and those of the control scheme exist. The simulation results show that the designed control scheme is robust to bounded external disturbances and to mismatches in the parameters of the systems.

#### 1. Introduction

Chaos theory is related to the study of the behavior of dynamical systems that exhibit sensitive dependence on initial conditions. The early discovery of chaotic systems goes back to the work of Poincare [1] during his attempt to prove the stability of the solar system through his work on the restricted three-body problem. A major contribution of chaos theory goes to Lorenz in his work on weather prediction [2]. These days, chaos can be found in different branches of science including mechanics [3], biology [4], chemistry [5], electronics [6], optics [7], and secure communication [8].

The importance of investigating chaos in electric motors comes from its direct applications in many areas of real life. For example, electric motors play important roles in industrial machinery, electrical locomotives, and electrical submersibles thruster drives [9]. The work of Kuroe and Hayashi [10] in the late 1980’s addressed the occurrence of chaos in motor drives. After that time, many researchers worked on chaos, and its control and its synchronization in several types of motor drive systems.

Chaos was investigated in the brushless DC motor (BLDCM) [11, 12], permanent magnet synchronous motor (PMSM) [13–15], stepper motor (SM) [16], induction motor (IM) [17, 18], switched reluctance motor (SRM) drive, and synchronous reluctance motor (SynRM) [19–21].

Chaotification and chaos synchronization were utilized in some industrial applications. Chaos synchronization is important when two or more electric motor drives need to work in synchrony. Chaotification of electric motor drives was applied in several industrial applications. For example, chaotification of DC motors was applied for the purpose of industrial mixing by Ye and Chau in [22, 23]. The inherent nature of stretching and folding of chaos helped in improving the efficiency of mixing. Also, it was shown that an abrasive machine using chaotic rotation is superior to a machine using periodic waves in raising abrasive efficiency [24]. Other industrial applications of chaotic motor drives can be found in vibratory compaction [25], dishwashers [26], air-conditioners [27], etc.

The variable-reluctance motor (VRM) is divided into two types, namely, the SRM and the SynRM. While the SRM uses rectangular waves and concentrated windings, the SynRM adopts sinusoidal waves and distributed windings. Therefore, the SynRM is capable of eliminating torque pulsation and acoustic noise problems that are intractable in the SRM. Also, the SynRM can offer the capability of electrical fault tolerance, such as one-phase open circuit, one-phase short circuit, and phase-to-phase short circuit [20].

SRM drives and SynRMs were shown to exhibit chaotic behaviors by Chau et al. [19] and Gao and Chau [20], respectively. A modeling, analysis, and experimentation of chaos in a switched reluctance drive system are presented in the work of Chau and Chen [28].

Controlling chaotic systems is a challenging problem because of the sensitivity of these systems to perturbations. Many control techniques were proposed to control chaos in SRM drives and SynRMs. For example, a passive adaptive control of chaos in a SynRM is presented in the work of Wei and Luo [29]. Tang et al. [30] designed a controller for the purpose of stabilization of a SynRM. Also, chaos is controlled in a SynRM using a nonlinear feedback control technique via control of the *d*-axis and the *q*-axis components of the stator voltage in [31]. However, less work has been done on the chaos synchronization of SynRMs. Because of its wide industrial applications, chaos synchronization of electric motors problem has been extensively studied in the literature. Chaos synchronization has been investigated in IM [32], BLDCM [33], PMSM [34], and others. This motivates us to deal with the chaos synchronization problem of SynRMs in this paper since SynRM takes advantage of structure simplicity, power efficiency, and the core and winding high recyclability [20].

With the development of nonlinear control theory, the sliding mode control technique became an effective method to control nonlinear systems and to achieve the synchronization of chaotic and hyperchaotic systems [35]. The sliding mode control technique represents a powerful and robust technique [36–39]. Also, since discrete-time systems are easier to implement and using computers and DSP chips to implement controllers of discrete-time systems has become very common and simple [40, 41], a discrete sliding mode control technique will be adopted in this work. Moreover, the mixed variable speed reaching law [42] which can overcome the chattering drawback of the well-known exponential reaching law [43] is used to further enhance the performance of the designed controllers.

Another problem that control designers are faced with is how to tune the parameters of a controller so that a good performance is obtained. Some researchers have used optimization techniques to tune the control parameters. Nature-inspired optimization algorithms have become increasingly common in solving optimization problems in recent years [44]. These algorithms include two types of methods: evolutionary algorithms (EAs) [45] and swarm intelligence (SI) algorithms [46]. EAs are inspired by the principles of population genetics and natural selection. Examples of evolutionary algorithms include genetic algorithms (GAs) [47], genetic programming [48], and others [49, 50]. Swarm Intelligence algorithms are founded on the principles of collective behavior observed in biological systems. Examples of SI algorithms are the cuckoo search (CS) [51], particle swarm optimization (PSO) [52], and others [53, 54]. In the literature, EAs and SI algorithms were successfully applied to the offline design of controllers. They have been used to tune the parameters of controllers [55, 56] or to obtain the structure of the controller [57], or both [58].

In this paper, the chaotic behavior in the SynRM is investigated. The DSMC technique is used to design controllers for the purpose of synchronization. The contributions of this paper can be summarized as follows: (i) chaos is investigated in the SynRM and chaotic behavior is verified by calculating the Lyapunov exponents of the system, (ii) the problem of chaos synchronization is formulated, and the DSMC technique is used to design the controllers to synchronize two SynRMs starting from different sets of initial conditions, (iii) the mixed variable speed reaching law is adopted to overcome the drawbacks of the other reaching laws, and (iv) a genetic algorithm is used to tune the parameters of the designed controller to guarantee the optimum choice of parameters so that a predefined index is minimized. Simulation results are presented to validate the designed controllers. In addition, the robustness of the designed controllers against external disturbances and parameters mismatch is verified using simulations.

The remainder of the paper is organized as follows: the model of the SynRM is described in Section 2. Section 3 formulates the synchronization problem of two SynRM chaotic systems. The design of a discrete sliding mode control scheme to synchronize the two SynRM systems is presented in Section 4. In Section 5, a robustness study of the designed controller scheme is presented. Finally, the conclusion is given in Section 6.

#### 2. Model of the Synchronous Reluctance Motor

A detailed study of chaos in the SynRM was performed by Gao and Chau in [20]. By using field-oriented control and considering the feedback effect , the continuous-time model of the SynRM can be presented as follows:

In the above model, the states of the system are the *d*-axis stator current , the *q*-axis stator current , and the mechanical rotor speed . The parameters of the system are the number of poles *P*, the electrical rotor speed , the reference speed of the rotor , the proportional gain of the speed controller , the *d*-axis stator voltage , the stator resistance per phase , the *d*-axis stator inductance , the *q*-axis stator inductance , the viscous friction coefficient *B*, the moment of inertia of the drive system *J*, and the load torque .

Gao and Chau [20] and Chau and Wang [9] simplified the above model by adopting a transformation and to such thatwhere , , and .

Then, the system given by equation (1) can be written as follows:where , , , , , and .

It was shown in [20] that the above system behaves chaotically for the following choice of the parameters: , , and .

A discretized version of the system given by (3), using a first-order approximation (Euler method), is as follows:

The states of the above system are defined as follows:

Therefore, system (4) can be written as

We simulated system (6) for with parameters , , and and with time step set to 0.001 seconds. These parameters result in a chaotic behavior of the system. The Lyapunov exponents of the system were calculated and we found them to be , , and . Since the signs of the Lyapunov exponents are , then the system is chaotic. The initial conditions are taken to be , , and .

The simulation results of the chaotic system (6) are presented in Figures 1–4. Figures 1–3 show the states of the system. Figure 4 shows the plots of the strange attractor of the SynRM.