Synchronous Reluctance Motor: Dynamical Analysis, Chaos Suppression, and Electronic Implementation

Center for Nonlinear Systems, Chennai Institute of Technology, Chennai-600069, Tamilnadu, India Department of Basic Sciences, School of Geology and Mining Engineering, University of Ngaoundere, P. O. BOX 115, Meiganga, Cameroon Information Technology College, Imam Ja’afar Al-Sadiq University, Baghdad 10001, Iraq Department of Hydraulics and Water Management, National Advanced School of Engineering, University of Maroua, P. O. Box 46, Maroua, Cameroon Research Unit of Condensed Matter of Electronics and Signal Processing Department of Physics, Faculty of Sciences, University of Dschang, P. O. Box 67, Dschang, Cameroon


Introduction
An electrical motor converts electrical energy into mechanical energy thank to the discovery by Michael Faraday in the 19th century. He stated that a current carrying coil within a magnetic field will experience a force. Electrical motors can be found in steel rolling mills, drilling machines, railway traction, industrial robots, and in most household items and office equipment [1][2][3][4][5][6]. Today, there are several variants of electric motors including the induction motor [7,8], permanent-magnet brushless motor [9][10][11][12], and variable-reluctance motor. e variable-reluctance motor class takes the advantages of a simple and rugged structure, good compatibility with the power converter, and high recyclability for the core and winding [13]. e variable-reluctance motor is divided into the switched reluctance motor [14,15] and synchronous reluctance motor (SynRM).
e SynRM uses a distributed winding and sinusoidal wave which can essentially eliminate the torque pulsation and acoustic noise problems. It is broadly used in the field of transportation, industrial and agricultural production, commercial and household appliances, medical appliances and equipment, and so on [16][17][18][19][20]. Because of its advantage over other types of electrical motors in simple mechanical construction, there were no slip ring and no permanent magnet and over other servomotors in high efficiency, high power density, and low manufacturing cost [21].
For industrial automation manufacturing, the secure and stable operation of the SynRM is an essential requirement because chaotic behaviors can extremely destabilize the SynRM and even cause the drive system to fail [20]. Hopf Bifurcation and chaos have been found in the SynRM [13]. In this paper, it is demonstrated that the SynRM can exhibit monostable periodic behaviors, bistable periodic behaviors, monostable chaotic behaviors, and bistable chaotic behaviors. e chaotic behaviors found in the SynRM induces instability in this motor and shortens its service time [9].
ereafter, a variety of methods to control chaos have been used to suppress the chaotic behavior in SynRM. A passive adaptive controller [21], a nonlinear feedback controller [22], a controller based on tridiagonal structure matrix stability theory [23], a vector controller [24][25][26], a sliding mode controller [27], and an adaptive sliding mode controller [28] were used for the control of chaotic behavior in SynRM. Most of the existing techniques for the control of chaotic behavior in SynRM use a nonlinear and complicated controller.
To the best of authors' knowledge, no study on the chaos suppression in SynRM without external inputs has been carried out with the single state feedback controller. e single state feedback control method is simple, concise, and easy to implement. erefore, the main contribution of this paper is to investigate the dynamical analysis of SynRM without external inputs and to design three single and simple controllers to suppress chaos in SynRM. e dynamical analysis and chaos suppression via a single controller of SynRM without external inputs are analytically, numerically, and electronically analysed in this paper. e dynamical analysis of SynRM without external inputs is investigated in Section 2. In Section 3, three proposed single controllers are employed to achieve the suppression of chaos in SynRM without external inputs. Section 4 presents the electronic implementation in order to check the existence of dynamical behaviors found in SynRM and the effectiveness of the three proposed single controllers. Finally, conclusions are given in Section 5.

Dynamical Analysis of SynRM without
External Inputs e SynRM can be described by the following rate equations [1,2,13]: where i d , i q are the d (direct)-and q (quadrature)-axis stator currents, ω is the mechanical rotor speed, ω e is the electrical rotor speed, u d is the stator voltage on d axis, R s is the stator resistance per phase, k p is the feedback coefficient, and ω ref is the reference rotor speed, L d , L q are the d-and q-axis stator inductors, P is the number of poles, J, T L , and B are the inertia constant of the motor and load, load torque, and viscous friction coefficient, respectively. e normalization of equations (1a)-(1c) leads to the following dimensionless form of the mathematical model of SynRM: with the following rescaling variables and parameters: . e linear stability analysis of system (2) revealed that the equilibrium points displayed Hopf bifurcation [13]. When the parameters a, b, c are varied, SynRM without external inputs can be expected to exhibit steady state, periodic, and chaotic behaviors. In order to identify the dynamical behaviors of SynRM without external inputs, two parameters LLE diagrams are constructed in Figure 1.
From Figure 1, periodic or steady state regions are characterized as a combination of light blue-light blue-green colors, and chaotic regions are characterized by yellow and red colors. For b � 0.2 and c � 22, the bifurcation diagrams and LLE of SynRM without external inputs as a function of the parameter a are plotted in Figure 2. Figure 2 shows that the SynRM without external inputs exhibits monostable period-3 oscillations, bistable period-3 2 Complexity oscillations followed to period tripling to bistable chaos and monostable chaos interspersed with bistable and monostable periodic regions. e dynamical behaviors shown in Figure 2 are illustrated in Figure 3 for a specific value of a. e SynRM without external inputs exhibits monostable periodic attractors in Figure 3(a), bistable periodic attractors in Figure 3(b), bistable one-scroll chaotic attractors in Figure 3(c), and monostable double-scroll chaotic attractors in Figure 3(d). e bifurcation diagrams of SynRM without external inputs obtained numerically by the parameters b and c reveal monostable chaos and bistable chaos interspersed with monostable and bistable periodic regions

Chaos Suppression in SynRM without External Inputs Using Single Controller
In this section, three single controllers are mathematically designed by using the principle of Lyapunov's method for asymptotic global stability to suppress the chaotic behavior found in SynRM without external inputs [29].

Proposed Controller 1.
System (3a)-(3c) with the first single controller u 1 � − yz is described by dz dt � xy − az. (4c) e controlled system (4a)-(4c) can be rewritten as at is, yield lim t⟶∞ x(t) � 0. So, system (5a)-(5c) can be reduced as follows: e solution of equation (6b) is z(t) � z(0)e − at . at is, yield lim t⟶∞ z(t) � 0. So, system (6a) and (6b) can be rewritten as follows e solution of equation (7) is y(t) � y(0)e − t . at is, yield lim t⟶∞ y(t) � 0. erefore, the chaotic behavior found in the SynRM without external inputs can be controlled using the controller u 1 � − yz. e curves of the state responses and the output of the controller 1 are shown in Figure 4. e results of Figure 4 show the efficiency of the controller u 1 .

Proposed Controller 2. System (3a)-(3c) with the second single controller
e controller u 2 into the controlled system (8a)-(8c) can be rewritten as at is, yield lim t⟶∞ y(t) � 0. us, the system (9a)-(9c) can be reduced as follows:  4 Complexity at is, yield lim t⟶∞ y(t) � 0 and lim t⟶∞ z(t) � 0. erefore, the chaotic behavior found in the SynRM without external inputs can be controlled using the controller e curves of the state responses and the output of the controller are shown in Figure 5. e results of Figure 4 reveal the efficiency of the controller u 2 .

Proposed Controller 3.
System (3a)-(3c) with the third single controller u 3 � − xy is described by Substituting the expression of the controller u 3 into the controlled system (12a)-(12c) becomes at, is yield lim t⟶∞ z(t) � 0.
us, system (13a)-(13c) can be reduced as follows: e solution of system (14a) and (14b) can be rewritten as follows: at is, yield lim t⟶∞ x(t) � 0 and lim t⟶∞ y(t) � 0. erefore, the chaotic behavior found in SynRM without external inputs can be controlled using the controller u 3 � − xy. e curves of the state responses and the output of the single controller 3 are shown in Figure 6. Complexity 5 e results of Figure 6 show the efficiency of the controller u 3 . From practical realization point of view, the single controllers 1 and 3 are preferred because of the inclusion of two states variables (i.e. y and z or x and z) in a single expression signifying a lesser requirement of sensing devices during their fabrication. Hence, this making the system to become cheap.

Circuit Implementation of SynRM without External Inputs and Chaos Suppression in SynRM without External Inputs
e electronic implementation of system (3a)-(3c) is depicted in Figure 7. e electronic circuit of Figure 7 is made of three capacitors, thirteen resistors, six TL081 operational amplifiers, and three analog devices AD633 multipliers. Based on the circuit diagram of Figure 7, the phase portraits of dynamical behaviors found in SynRM without external inputs are illustrated in Figure 8 for specific values of capacitors and resistors. e good qualitative agreement between the Pspice results of Figure 8 and the numerical simulations results of Figure 3 confirms the existence of the dynamical behavior found in SynRM without external inputs. e electronic implementations of the controlled systems (5a)-(5c), (9a)-(9c), and (13a)-(13c) are deduced from the electronic implementation of system (5a)-(5c) in Figure 7 (not shown). e time series of the state responses and the output of the single controller 1 generated from the circuit diagram of the controlled system (5a)-(5c) are shown in Figure 9. e good qualitative agreement between the Pspice results of Figure 9 and the numerical simulations results of Figure 5 confirms the efficiency of proposed single controller 3. e time series of the state responses and the output of the single controller 2 generated from the circuit diagram of the controlled system (9a)-(9c) are shown in Figure 10. e good qualitative agreement between the Pspice results of Figure 10 and the numerical simulations results of Figure 5 confirms the efficiency of proposed single controller 2. e time series of the state responses and the  Complexity output of the single controller 3 generated from the circuit diagram of the controlled system (13a)-(13c) are shown in Figure 11. e good qualitative agreement between the Pspice results of Figure 11 and the numerical simulations results of Figure 6 confirms the efficiency of proposed single controller 3.
Time Time Time Time Figure 9: Time series of chaos suppression in SynRM without external inputs generated from the Pspice oscilloscope for the capacitors and resistors:

Conclusion
is paper is dealt with the dynamical analysis, chaos suppression, and electronic implementation of synchronous reluctance motor without external inputs. e numerical analysis of synchronous reluctance motor without external inputs was revealed as monostable periodic attractors, bistable periodic attractors, monostable double-scroll chaotic attractors, and bistable one-scroll chaotic attractors.
reluctance motor without external inputs, and it was revealed that they were simple and easy to implement. e single controllers 1 and 3 could be a preferable choice because of the use of two states variables (i.e. y and z or x and z) in a single expression. Numerical simulations results were provided to demonstrate the efficiency of three proposed single controllers. To access the physical feasibility of three designed single controllers and the existence of the dynamical behaviors found in synchronous reluctance motor without external inputs, electronic circuits were implemented and validated on OrCAD-PSpice software. In the future works, it will be interesting to study the synchronous reluctance motor with external inputs such as the load torque and the stator voltage.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.