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

Authors: Salahuddin Abdul Rahman; Mohamed Zribi; Nejib Smaoui
Journal: Mathematical Problems in Engineering
 (2019)

---

## 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.


---

## Body

## 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 Section2. 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 Kpωr−ωref, the continuous-time model of the SynRM can be presented as follows:(1)ddtids=1Ldsvds−Rsids+ωeLqsiqs,ddtiqs=1Lqs−Rsiqs−ωeLdsids+Kpωr−ωref,ddtωr=1J3P4Lds−Lqsidsiqs−Bωr−TL.In the above model, the states of the system are thed-axis stator current ids, the q-axis stator current iqs, and the mechanical rotor speed ωr. The parameters of the system are the number of poles P, the electrical rotor speed ωe, the reference speed of the rotor ωref, the proportional gain of the speed controller Kp, the d-axis stator voltage vds, the stator resistance per phase Rs, the d-axis stator inductance Lds, the q-axis stator inductance Lqs, the viscous friction coefficient B, the moment of inertia of the drive system J, and the load torque TL.Gao and Chau [20] and Chau and Wang [9] simplified the above model by adopting a transformation t=τt′ and ids,iqs,ωe to id,iq,ω such that(2)idsiqsωe=bkidkiq1τω,where b=Lqs/Lds, k=8J/3P2Lds−Lqsbτ2, and τ=Lqs/Rs.Then, the system given by equation (1) can be written as follows:(3)ddt′id=−p1id+iqω+vds′,ddt′iq=−iq−idω+p2ω−ωref′,ddt′ω=−p3ω+idiq+TL′,where vd′=vds/kRs, p1=Lqs/Lds, p2=2Kp/kPLqs, p3=BLqs/JRs, wref′=τwref, and TL′=τ2PTL/2J.It was shown in [20] that the above system behaves chaotically for the following choice of the parameters: p1=0.1883, p2=10, and p3=1.5295.A discretized version of the system given by (3), using a first-order approximation (Euler method), is as follows:(4)idk+1=idk+Ts−p1idk+iqkωk+vds′,iqk+1=iqk+Ts−iqk−idkωk+p2ωk−ωref′ωk+1=ωk+Ts−p3ωk+idkiqk+TL′.The states of the above system are defined as follows:(5)x1k=idk,x2k=iqk,x3k=ωk.Therefore, system (4) can be written as(6)x1k+1=x1k+Ts−p1x1k+x2kx3k+vds′,x2k+1=x2k+Ts−x2k−x1kx3k+p2x3k−ωref′,x3k+1=x3k+Ts−p3x3k+x1kx2k+TL′.We simulated system (6) for 100s with parameters p1=0.1883, p2=10, and p3=1.5295 and with time step Ts 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 λ1=0.3593, λ2=−0.0023≈0, and λ3=−3.0701. Since the signs of the Lyapunov exponents are +,0,−, then the system is chaotic. The initial conditions are taken to be id0=8.9, iq0=2.1, and w0=4.6.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.Figure 1
The stateid versus time.Figure 2
The stateiq versus time.Figure 3
The stateω versus time.Figure 4
Plots of the strange attractor of the SynRM.
(a)
(b)
(c)
(d)Therefore, the chaotic behavior of the synchronous reluctance motor is evident in the above figures.


## 3. Problem Formulation

This paper deals with the synchronization of two synchronous reluctance motors. Therefore, we define the master system as follows:(7)x1k+1=x1k+Ts−p1x1k+x2kx3k+vds′,x2k+1=x2k+Ts−x2k−x1kx3k+p2x3k−ωref′,x3k+1=x3k+Ts−p3x3k+x1kx2k+TL′.Also, the slave system is defined as follows:(8)y1k+1=y1k+Ts−p1y1k+y2ky3k+vds′+u1k,y2k+1=y2k+Ts−y2k−y1ky3k+p2y3k−ωref′+u2k,y3k+1=y3k+Ts−p3y3k+y1ky2k+TL′,where u1k and u2k are the controllers to be designed such that the master and the slave systems are synchronized.The error vectorek is defined as follows:(9)k=e1ke2ke3kT=y1k−x1ky2k−x2ky3k−x3kT.Using equations (7)–(9), the error dynamics can be written as follows:(10)e1k+1=1−p1Tse1k+Tsx3ke2k+Tsy2ke3k+u1k,e2k+1=1−Tse2k−Tsy3ke1k−Tsx1ke3k+Tsp2e3k+u2k,e3k+1=1−p3Tse3k+Tsx2ke1k+Tsy1ke2k.The objective of this paper is to synchronize the master and the slave systems by forcing the state vectoryk of the slave system (8) to track the state vector xk of the master system (7), i.e., limk⟶∞ek=limk⟶∞yk−xk=0. This synchronization is done by forcing the error ek to converge to 0,0,0 as k⟶∞. The next section deals with the design of a sliding mode controller to synchronize the master and the slave systems.


## 4. Sliding Mode Control Design

### 4.1. Design of the Sliding Mode Controller

The error model system given by (10) is considered. The first step in the design of a sliding mode control is to choose the sliding surfaces. Since the system has two control inputs, we need to choose two sliding surfaces. The sliding surfaces S1k and S2k are chosen such that(11)S1k=e1k,(12)S2k=e2k.Also, define the sign function such that(13)sgnσ=1,ifσ>0,0,ifσ=0,−1,ifσ<0.The subsequent development requires that we define the concept of quasisliding mode (QSM) and the concept of quasisliding mode band (QSMB). These concepts were defined in [43]. The quasisliding mode (QSM) is defined as the motion of the system in an ε vicinity of the sliding surface Sk<ε. The width of the band which contains the QSM is called the quasisliding mode band (QSMB); it is defined such that x−Δ<Sx<Δ where 2Δ is the width of the band.Let the positive control parameters bea1, a2, ε1, ε2, q1, q2, and Ts such that 1−q1Ts>0 and 1−q2Ts>0.Also, define the 1 norm of the errorek such that ek1=∑i=1nek.The following theorem gives the first result of the paper.Theorem 1.
The sliding mode control law:(14)u1k=S1k+1a1ek1+1−q1TsS1k−ε1TssgnS1k+a1ek1+1−ε1Tsek1sgnS1k−1−p1Tse1k−x3kTse2k−y2kTse3k,(15)u2k=S2k+1a2ek1+1−q2TsS2k−ε2TssgnS2k+a2ek1+1−ε2Tsek1sgnS2k+y3kTse1k−1−Tse2k+Tsx1ke3k−Tsp2e3k,when applied to the error model given by (10) guarantees the convergence of the errors e1k, e2k, and e3k to zero as k⟶∞.Proof.
Using the sliding surface given by (11), using the dynamical model of the errors given by (10), and using the control laws given by (14), we obtain(16)S1k+1=e1k+1=1−p1Tse1k+Tsx3ke2k+Tsy2ke3k+u1k=S1k+1a1ek1+1−q1TsS1k−ε1TssgnS1k+a1ek1+1−ε1Tsek1sgnS1k.
Also, using the sliding surface given by (12), using the dynamical model of the errors given by (10), and using the control laws given by (15), we obtain(17)S2k+1=e2k+1=1−Tse2k−Tsy3ke1k−Tsx1ke3k+Tsp2e3k+u2k=S2k+1a2ek1+1−q2TsS2k−ε2TssgnS2k+a2ek1+1−ε2Tsek1sgnS2k.
Equations (16) and (17) can be written as follows:(18)Sik+1=Sik+1aiek1+1−qiTsSik−εiTssgnSik+aiek1+1−εiTsek1sgnSik,fori=1,2.
Equation (18) corresponds to the mixed variable speed reaching law defined in [42]. Note that equation (18) satisfies Sarpturk et al. reaching and existence condition given by Sik+1<Sik; this condition can be decomposed into the following two inequalities [59]:(19)Sik+1−SiksgnSik<0,(20)Sik+1+SiksgnSik≥0.
Condition (19) implies that the closed-loop system should be moving in the direction of the sliding surface. On the contrary, condition (20) implies that the closed-loop system is not allowed to go too far in that direction. Thus, conditions (19) and (20) define the lower bound and the upper bound for the control action, respectively.
The above two inequalities can be proven as follows:(21)Sik+1−SiksgnSik=1aiek1+1−qiTsSik−εiTssgnSik+aiek1+1−εiTsek1sgnSiksgnSik=−1aiek1+1qiTsSik+εiTs−aiek1+1εiTsek1<0,(22)Sik+1+SiksgnSik=2Sik+1aiek1+1−qiTsSik−εiTssgnSik+ai/ek1+1−εiTsek1sgnSiksgnSik=2Sik+1aiek1+1−qiTsSik−εiTs+aiek1+1−εiTsek1=2Sik−Ts1aiek1+1qiSik+εi+aiek1+1εiek1,when the sampling time Ts is very small, Sik+1+SiksgnSik>0 outside the QSMB. Using inequalities (21) and (22), it follows that equation (18) satisfies the condition Sik+1<Sik outside the QSMB.
On the sliding surface,S1k=0, we have(23)e1k=0.
Also, on the sliding surface,S2k=0, we have(24)e2k=0.
Sincee1k and e2k converge to zero, then the third difference equation in the error model given by (10) reduces to(25)e3k+1=1−p3Tse3k.
The solution of the above equation ise3k=1−p3Tske30. Since 1−p3Ts<1, the error e3k converges to zero as k⟶∞. Therefore, the errors e1k, e2k, and e3k converge to zero as k⟶∞.The convergence of the errorse1k, e2k, and e3k to zero as k⟶∞ guarantees that the states of the slave systems y1k, y2k, and y3k converge to the states of the master systems x1k, x2k, and x3k, respectively, as k⟶∞. Hence, the states of the master and the slave chaotic systems are synchronized.The proposed sliding mode control scheme contains six control parameters. These parameters area1,a2,ε1,ε2,q1, and q2. A good performance can be achieved by fine tuning the parameters of the controllers u1k and u2k. The process of tuning the parameters of the controllers is not an easy process since multiple parameters must be tuned simultaneously and each tuning must start by repeating the simulations. For the simulation studies, we considered two scenarios. First, we chose the parameters of the controllers manually. Then, we used a genetic algorithm to select the parameters of u1k and u2k. The next subsection briefly describes the algorithm used for tuning the parameters of the controllers.Remark.
The parameters of the controllers area1,a2,ε1,ε2,q1, and q2. For a chosen set of parameters, the controllers u1k and u2k can be easily computed using equations (14) and (15). Hence, it can be concluded that the proposed discrete sliding mode controller involves low computational complexity.


### 4.2. Sliding Mode Control Parameters Tuned by a Genetic Algorithm

The performance of the closed-loop system is determined by the parameters of the controller:a1,a2,ε1,ε2,q1, and q2. It is important to choose their values using an optimization algorithm. In the literature, different cost functions were used to optimize certain cost function. These cost functions evaluate the performance of the controller. Examples of the commonly used cost functions include the integral square error (ISE) [60], the integral time absolute error (ITAE) [61], the weighted sum of the absolute errors [62], and others [63]. Also, the multiobjective functional optimization was used where several performance measures are optimized [64, 65]. Among the most commonly used cost functions is the quadratic cost function [66]. This cost function is used in this study because it is simple and it guarantees a good performance of the controller. Hence, the GA is adopted to optimize the parameters of the control scheme with respect to the following performance index:(26)Jo=∑j=0NejTQej+ujTRuj,where “N” is the total number of samples which is chosen to be 8000. The matrices Q and R are weighted coefficient matrices that are chosen such as Q=I3 (3 by 3 identity matrix) and R=0.001. The methodology for the adjustment of the parameters of the sliding mode controllers using the genetic algorithm is described in [67].The parameters of the GA are listed in Table1.Table 1
GA parameters.
Population size = 75
Generation number = 25
Crossover rate = 0.8
Mutation rate = 0.01The parameters of the controller obtained by the end of the genetic algorithm tuning process area1=3.6557, ε1=0.0215, q1=0.4438, a2=3.3950, ε2=1.4213, and q2=3.0775.


### 4.3. Simulation Results

The designed sliding mode control laws in (14) and (15) are applied to the master system given by equation (7) and the slave system given by equation (8). The performance of the controlled error system is simulated using the MATLAB/Simulink software. The parameters of the system are taken as [20] Rs=0.029Ω, vds=0V, wref=0rad/s, Lds=133.3mH, Lqs=25.1mH, J=1.99×10−3Kgm2, B=3.513×10−3Nm/rads−1, P=4, and TL′=0.75. In the simulation, the initial states of the master system are taken to be x10=0.25, x20=0.18, and x30=2. The initial states of the slave system are taken to be y10=0.2, y20=0.2, and y30=2.5.The performance of the system is simulated for 100 seconds. For the first50s, the error system is simulated when u=0. Then, for the next 50s, the controller u is applied.Moreover, chattering is suppressed by replacing the discontinuous sign function in the control law with the saturation function such thatsignS=S/S+δ where δ is an arbitrarily small positive scalar which can be designed to get suitable approximation. The simulations were carried for two sets of the parameters of the controllers. The first simulation was done when the parameters are chosen manually. The second simulation was done when the parameters are determined by the genetic algorithm. The parameters of the controllers are given in Table 2.Table 2
Controller parameters and performance.
Parameter
GA
Manual
q
1
0.4438
1
ε
1
0.0215
0.01
a
1
3.6557
5
q
2
3.0775
5
ε
2
1.4213
2.5
a
2
3.3950
5
Cost function value
6.5
10.85The simulation results are depicted in Figures5–17.Figure 5
The statesx1 and y1 versus time when using the sliding mode controller with manual tuning of the parameters.Figure 6
The statesx2 and y2 versus time when using the sliding mode controller with GA tuning of the parameters.Figure 7
The statesx2 and y2 versus time when using the sliding mode controller with manual tuning of the parameters.Figure 8
The statesx3 and y3 versus time when using the sliding mode controller with GA tuning of the parameters.Figure 9
The statesx3 and y3 versus time when using the sliding mode controller with manual tuning of the parameters.Figure 10
The statesx3 and y3 versus time when using the sliding mode controller with GA tuning of the parameters.Figure 11
The errore1 versus time when using the sliding mode controller.Figure 12
The errore2 versus time when using the sliding mode controller.Figure 13
The errore3 versus time when using the sliding mode controller.Figure 14
The controlleru1 versus time.Figure 15
Zoomed in part of the controlleru1 versus time.Figure 16
The controlleru2 versus time.Figure 17
Zoomed in part of the controlleru2 versus time.The synchronized states versus time when using the manual tuning of the parameters are shown in Figures5, 7, and 9. The synchronized states versus time when using the GA-based tuning of the parameters are shown in Figures 6, 8, and 10. The errors e1k, e2k, and e3k versus time are depicted in Figures 11–13. It is clear from these figures that the errors converge to zero. The simulation results indicate that the proposed control scheme works well. However, it is clear that better results were obtained when using parameters which are tuned using a genetic algorithm. This is especially true when one considers the simulation results of e2k versus time.The controllers versus time are presented in Figures14–17. Again, better results were obtained when using parameters of the controller which are tuned using a genetic algorithm.Therefore, it can be concluded that the proposed controller works well for both sets of parameters. However, a better performance is obtained when using parameters of the control scheme which are tuned using a genetic algorithm.


## 4.1. Design of the Sliding Mode Controller

The error model system given by (10) is considered. The first step in the design of a sliding mode control is to choose the sliding surfaces. Since the system has two control inputs, we need to choose two sliding surfaces. The sliding surfaces S1k and S2k are chosen such that(11)S1k=e1k,(12)S2k=e2k.Also, define the sign function such that(13)sgnσ=1,ifσ>0,0,ifσ=0,−1,ifσ<0.The subsequent development requires that we define the concept of quasisliding mode (QSM) and the concept of quasisliding mode band (QSMB). These concepts were defined in [43]. The quasisliding mode (QSM) is defined as the motion of the system in an ε vicinity of the sliding surface Sk<ε. The width of the band which contains the QSM is called the quasisliding mode band (QSMB); it is defined such that x−Δ<Sx<Δ where 2Δ is the width of the band.Let the positive control parameters bea1, a2, ε1, ε2, q1, q2, and Ts such that 1−q1Ts>0 and 1−q2Ts>0.Also, define the 1 norm of the errorek such that ek1=∑i=1nek.The following theorem gives the first result of the paper.Theorem 1.
The sliding mode control law:(14)u1k=S1k+1a1ek1+1−q1TsS1k−ε1TssgnS1k+a1ek1+1−ε1Tsek1sgnS1k−1−p1Tse1k−x3kTse2k−y2kTse3k,(15)u2k=S2k+1a2ek1+1−q2TsS2k−ε2TssgnS2k+a2ek1+1−ε2Tsek1sgnS2k+y3kTse1k−1−Tse2k+Tsx1ke3k−Tsp2e3k,when applied to the error model given by (10) guarantees the convergence of the errors e1k, e2k, and e3k to zero as k⟶∞.Proof.
Using the sliding surface given by (11), using the dynamical model of the errors given by (10), and using the control laws given by (14), we obtain(16)S1k+1=e1k+1=1−p1Tse1k+Tsx3ke2k+Tsy2ke3k+u1k=S1k+1a1ek1+1−q1TsS1k−ε1TssgnS1k+a1ek1+1−ε1Tsek1sgnS1k.
Also, using the sliding surface given by (12), using the dynamical model of the errors given by (10), and using the control laws given by (15), we obtain(17)S2k+1=e2k+1=1−Tse2k−Tsy3ke1k−Tsx1ke3k+Tsp2e3k+u2k=S2k+1a2ek1+1−q2TsS2k−ε2TssgnS2k+a2ek1+1−ε2Tsek1sgnS2k.
Equations (16) and (17) can be written as follows:(18)Sik+1=Sik+1aiek1+1−qiTsSik−εiTssgnSik+aiek1+1−εiTsek1sgnSik,fori=1,2.
Equation (18) corresponds to the mixed variable speed reaching law defined in [42]. Note that equation (18) satisfies Sarpturk et al. reaching and existence condition given by Sik+1<Sik; this condition can be decomposed into the following two inequalities [59]:(19)Sik+1−SiksgnSik<0,(20)Sik+1+SiksgnSik≥0.
Condition (19) implies that the closed-loop system should be moving in the direction of the sliding surface. On the contrary, condition (20) implies that the closed-loop system is not allowed to go too far in that direction. Thus, conditions (19) and (20) define the lower bound and the upper bound for the control action, respectively.
The above two inequalities can be proven as follows:(21)Sik+1−SiksgnSik=1aiek1+1−qiTsSik−εiTssgnSik+aiek1+1−εiTsek1sgnSiksgnSik=−1aiek1+1qiTsSik+εiTs−aiek1+1εiTsek1<0,(22)Sik+1+SiksgnSik=2Sik+1aiek1+1−qiTsSik−εiTssgnSik+ai/ek1+1−εiTsek1sgnSiksgnSik=2Sik+1aiek1+1−qiTsSik−εiTs+aiek1+1−εiTsek1=2Sik−Ts1aiek1+1qiSik+εi+aiek1+1εiek1,when the sampling time Ts is very small, Sik+1+SiksgnSik>0 outside the QSMB. Using inequalities (21) and (22), it follows that equation (18) satisfies the condition Sik+1<Sik outside the QSMB.
On the sliding surface,S1k=0, we have(23)e1k=0.
Also, on the sliding surface,S2k=0, we have(24)e2k=0.
Sincee1k and e2k converge to zero, then the third difference equation in the error model given by (10) reduces to(25)e3k+1=1−p3Tse3k.
The solution of the above equation ise3k=1−p3Tske30. Since 1−p3Ts<1, the error e3k converges to zero as k⟶∞. Therefore, the errors e1k, e2k, and e3k converge to zero as k⟶∞.The convergence of the errorse1k, e2k, and e3k to zero as k⟶∞ guarantees that the states of the slave systems y1k, y2k, and y3k converge to the states of the master systems x1k, x2k, and x3k, respectively, as k⟶∞. Hence, the states of the master and the slave chaotic systems are synchronized.The proposed sliding mode control scheme contains six control parameters. These parameters area1,a2,ε1,ε2,q1, and q2. A good performance can be achieved by fine tuning the parameters of the controllers u1k and u2k. The process of tuning the parameters of the controllers is not an easy process since multiple parameters must be tuned simultaneously and each tuning must start by repeating the simulations. For the simulation studies, we considered two scenarios. First, we chose the parameters of the controllers manually. Then, we used a genetic algorithm to select the parameters of u1k and u2k. The next subsection briefly describes the algorithm used for tuning the parameters of the controllers.Remark.
The parameters of the controllers area1,a2,ε1,ε2,q1, and q2. For a chosen set of parameters, the controllers u1k and u2k can be easily computed using equations (14) and (15). Hence, it can be concluded that the proposed discrete sliding mode controller involves low computational complexity.


## 4.2. Sliding Mode Control Parameters Tuned by a Genetic Algorithm

The performance of the closed-loop system is determined by the parameters of the controller:a1,a2,ε1,ε2,q1, and q2. It is important to choose their values using an optimization algorithm. In the literature, different cost functions were used to optimize certain cost function. These cost functions evaluate the performance of the controller. Examples of the commonly used cost functions include the integral square error (ISE) [60], the integral time absolute error (ITAE) [61], the weighted sum of the absolute errors [62], and others [63]. Also, the multiobjective functional optimization was used where several performance measures are optimized [64, 65]. Among the most commonly used cost functions is the quadratic cost function [66]. This cost function is used in this study because it is simple and it guarantees a good performance of the controller. Hence, the GA is adopted to optimize the parameters of the control scheme with respect to the following performance index:(26)Jo=∑j=0NejTQej+ujTRuj,where “N” is the total number of samples which is chosen to be 8000. The matrices Q and R are weighted coefficient matrices that are chosen such as Q=I3 (3 by 3 identity matrix) and R=0.001. The methodology for the adjustment of the parameters of the sliding mode controllers using the genetic algorithm is described in [67].The parameters of the GA are listed in Table1.Table 1
GA parameters.
Population size = 75
Generation number = 25
Crossover rate = 0.8
Mutation rate = 0.01The parameters of the controller obtained by the end of the genetic algorithm tuning process area1=3.6557, ε1=0.0215, q1=0.4438, a2=3.3950, ε2=1.4213, and q2=3.0775.


## 4.3. Simulation Results

The designed sliding mode control laws in (14) and (15) are applied to the master system given by equation (7) and the slave system given by equation (8). The performance of the controlled error system is simulated using the MATLAB/Simulink software. The parameters of the system are taken as [20] Rs=0.029Ω, vds=0V, wref=0rad/s, Lds=133.3mH, Lqs=25.1mH, J=1.99×10−3Kgm2, B=3.513×10−3Nm/rads−1, P=4, and TL′=0.75. In the simulation, the initial states of the master system are taken to be x10=0.25, x20=0.18, and x30=2. The initial states of the slave system are taken to be y10=0.2, y20=0.2, and y30=2.5.The performance of the system is simulated for 100 seconds. For the first50s, the error system is simulated when u=0. Then, for the next 50s, the controller u is applied.Moreover, chattering is suppressed by replacing the discontinuous sign function in the control law with the saturation function such thatsignS=S/S+δ where δ is an arbitrarily small positive scalar which can be designed to get suitable approximation. The simulations were carried for two sets of the parameters of the controllers. The first simulation was done when the parameters are chosen manually. The second simulation was done when the parameters are determined by the genetic algorithm. The parameters of the controllers are given in Table 2.Table 2
Controller parameters and performance.
Parameter
GA
Manual
q
1
0.4438
1
ε
1
0.0215
0.01
a
1
3.6557
5
q
2
3.0775
5
ε
2
1.4213
2.5
a
2
3.3950
5
Cost function value
6.5
10.85The simulation results are depicted in Figures5–17.Figure 5
The statesx1 and y1 versus time when using the sliding mode controller with manual tuning of the parameters.Figure 6
The statesx2 and y2 versus time when using the sliding mode controller with GA tuning of the parameters.Figure 7
The statesx2 and y2 versus time when using the sliding mode controller with manual tuning of the parameters.Figure 8
The statesx3 and y3 versus time when using the sliding mode controller with GA tuning of the parameters.Figure 9
The statesx3 and y3 versus time when using the sliding mode controller with manual tuning of the parameters.Figure 10
The statesx3 and y3 versus time when using the sliding mode controller with GA tuning of the parameters.Figure 11
The errore1 versus time when using the sliding mode controller.Figure 12
The errore2 versus time when using the sliding mode controller.Figure 13
The errore3 versus time when using the sliding mode controller.Figure 14
The controlleru1 versus time.Figure 15
Zoomed in part of the controlleru1 versus time.Figure 16
The controlleru2 versus time.Figure 17
Zoomed in part of the controlleru2 versus time.The synchronized states versus time when using the manual tuning of the parameters are shown in Figures5, 7, and 9. The synchronized states versus time when using the GA-based tuning of the parameters are shown in Figures 6, 8, and 10. The errors e1k, e2k, and e3k versus time are depicted in Figures 11–13. It is clear from these figures that the errors converge to zero. The simulation results indicate that the proposed control scheme works well. However, it is clear that better results were obtained when using parameters which are tuned using a genetic algorithm. This is especially true when one considers the simulation results of e2k versus time.The controllers versus time are presented in Figures14–17. Again, better results were obtained when using parameters of the controller which are tuned using a genetic algorithm.Therefore, it can be concluded that the proposed controller works well for both sets of parameters. However, a better performance is obtained when using parameters of the control scheme which are tuned using a genetic algorithm.


## 5. Robustness Studies

In this section, we will study the robustness of the proposed control scheme. The robustness will be studied for two different cases. The first case will test the robustness to bounded external disturbances. The second case will test the robustness to model uncertainties.


### 5.1. Case 1: Robustness to External Disturbances

In this case, the effects of bounded external disturbances are added to the slave system. So the equations of the slave system after the addition of the disturbances are as follows:(27)y1k+1=y1k+Ts−p1y1k+y2ky3k+vds′+u1k+d1k,y2k+1=y2k+Ts−y2k−y1ky3k+p2y3k−ωref′+u2k+d2k,y3k+1=y3k+Ts−p3y3k+y1ky2k+TL′+d3k,where d1k, d2k, and d3k represent the disturbances. Moreover, it is assumed that these disturbances are bounded with their bounds being known. If a bound on the disturbances is not available, then a disturbance estimator can be used.In the simulations, the disturbances are taken to be(28)d1k=0.15,d2k=0.2cosk,d3k=0.15sink.Figure18 depicts the errors e1k, e2k, and e3k versus time when using the proposed sliding mode control scheme and when disturbances are present in the model of the slave system. The simulation results show that the errors converge to zero, and hence, it can be concluded that the proposed control scheme works well and it is able to suppress the effect of bounded external disturbances.Figure 18
The errorse1, e2, and e3 versus time when using the sliding mode controller with disturbances.


### 5.2. Case 2: Robustness to Mismatches in the Parameters of the Systems and Those of the Controllers

For the second case, we will assume that there is a mismatch between the parametersp1 and p2 of the motors and the parameters p1 and p2 of the controllers. The parameters p1 and p2 of the controllers will be denoted as p¯1 and p¯2 such that p¯1=p1+Δp1 and p¯2=p2+Δp2. The sliding mode controllers (14) and (15) with the parameters p¯1 and p¯2 become(29)u1k=S1k+1a1ek1+1−q1TsS1k−ε1TssgnS1k+a1ek1+1−ε1Tsek1sgnS1k−1−p¯1Tse1k+x3kTse2k+y2kTse3k,(30)u2k=S2k+1a2ek1+1−q2TsS2k−ε2TssgnS2k+a2ek1+1−ε2Tsek1sgnS2k+y3kTse1k−1−Tse2k−Tsp¯2−Tsx1ke3k.The sliding mode controllers given by (29) and (30) are applied to the chaotic synchronous motors given by (7) and (8) for Δp1=0.15 and Δp2=2. The initial conditions and the other parameters of the controllers are the same as in the previous subsection.Figure19 depicts the errors e1k, e2k, and e3k versus time when using the proposed sliding mode control scheme and when there is a mismatch between the parameters of the systems and those of the controllers. The simulation results show that the errors converge to zero, and hence, it can be concluded that the proposed control scheme works well and it is able to suppress the effect of mismatches in the parameters of the systems with those of the controllers.Figure 19
The errorse1, e2, and e3 versus time when using the sliding mode controller with mismatch in the parameters of the systems and those of the controller.Hence, it can be concluded that Figures18 and 19 indicate that the proposed control scheme works well when there are bounded disturbances acting on the slave system and when there are mismatches between the parameters of the controller and those of the motors systems. Thus, the simulation studies indicate that the developed sliding mode control scheme is robust with respect to disturbances and with respect to mismatches between the parameters of the controllers and those of the motor systems.


## 5.1. Case 1: Robustness to External Disturbances

In this case, the effects of bounded external disturbances are added to the slave system. So the equations of the slave system after the addition of the disturbances are as follows:(27)y1k+1=y1k+Ts−p1y1k+y2ky3k+vds′+u1k+d1k,y2k+1=y2k+Ts−y2k−y1ky3k+p2y3k−ωref′+u2k+d2k,y3k+1=y3k+Ts−p3y3k+y1ky2k+TL′+d3k,where d1k, d2k, and d3k represent the disturbances. Moreover, it is assumed that these disturbances are bounded with their bounds being known. If a bound on the disturbances is not available, then a disturbance estimator can be used.In the simulations, the disturbances are taken to be(28)d1k=0.15,d2k=0.2cosk,d3k=0.15sink.Figure18 depicts the errors e1k, e2k, and e3k versus time when using the proposed sliding mode control scheme and when disturbances are present in the model of the slave system. The simulation results show that the errors converge to zero, and hence, it can be concluded that the proposed control scheme works well and it is able to suppress the effect of bounded external disturbances.Figure 18
The errorse1, e2, and e3 versus time when using the sliding mode controller with disturbances.


## 5.2. Case 2: Robustness to Mismatches in the Parameters of the Systems and Those of the Controllers

For the second case, we will assume that there is a mismatch between the parametersp1 and p2 of the motors and the parameters p1 and p2 of the controllers. The parameters p1 and p2 of the controllers will be denoted as p¯1 and p¯2 such that p¯1=p1+Δp1 and p¯2=p2+Δp2. The sliding mode controllers (14) and (15) with the parameters p¯1 and p¯2 become(29)u1k=S1k+1a1ek1+1−q1TsS1k−ε1TssgnS1k+a1ek1+1−ε1Tsek1sgnS1k−1−p¯1Tse1k+x3kTse2k+y2kTse3k,(30)u2k=S2k+1a2ek1+1−q2TsS2k−ε2TssgnS2k+a2ek1+1−ε2Tsek1sgnS2k+y3kTse1k−1−Tse2k−Tsp¯2−Tsx1ke3k.The sliding mode controllers given by (29) and (30) are applied to the chaotic synchronous motors given by (7) and (8) for Δp1=0.15 and Δp2=2. The initial conditions and the other parameters of the controllers are the same as in the previous subsection.Figure19 depicts the errors e1k, e2k, and e3k versus time when using the proposed sliding mode control scheme and when there is a mismatch between the parameters of the systems and those of the controllers. The simulation results show that the errors converge to zero, and hence, it can be concluded that the proposed control scheme works well and it is able to suppress the effect of mismatches in the parameters of the systems with those of the controllers.Figure 19
The errorse1, e2, and e3 versus time when using the sliding mode controller with mismatch in the parameters of the systems and those of the controller.Hence, it can be concluded that Figures18 and 19 indicate that the proposed control scheme works well when there are bounded disturbances acting on the slave system and when there are mismatches between the parameters of the controller and those of the motors systems. Thus, the simulation studies indicate that the developed sliding mode control scheme is robust with respect to disturbances and with respect to mismatches between the parameters of the controllers and those of the motor systems.


## 6. Conclusion

In this paper, the discrete sliding mode control technique is used to design a robust synchronization control scheme for two chaotic synchronous reluctance motors. It is shown that the proposed control scheme guarantees the convergence of the state of the slave system to the states of the master system. The proposed control scheme is simulated using MATLAB using manually chosen control parameters and control parameters which are determined by using a genetic algorithm. The simulation results clearly show the convergence of the errors to zero which validate the proposed control design. Moreover, the simulation results indicate that better results were obtained when using control gains determined by a genetic algorithm. In addition, the simulation studies indicate that the proposed controller is robust to bounded disturbances and to mismatches in the parameters of the systems with those used in the control scheme.Future work will address the design of observer-based controllers to synchronize chaotic synchronous reluctance motors. Other avenues for further research include the design of second-order sliding mode controllers as well as fuzzy controllers for chaotic synchronous reluctance motors.



---

*Source: 5765390-2019-12-04.xml*