Abstract
Dynamical analysis, chaos suppression and electronic implementation of the synchronous reluctance motor (SynRM) without external inputs are investigated in this paper. The different dynamical behaviors (including monostable periodic behaviors, bistable periodic behaviors, monostable chaotic behaviors, and bistable chaotic behaviors) found in the SynRM without external inputs are illustrated in the two parameters largest Lyapunov exponent (LLE) diagrams, one parameter bifurcation diagram, and phase portraits. The three single controllers are designed to suppress the chaotic behaviors found in SynRM without external inputs. The three proposed single controllers are simple and easy to implement. Numerical simulation results show that the three proposed single controllers are effective. Finally, the dynamical behaviors found in the SynRM without external inputs and the physical feasibility of the three proposed single controllers are validated through circuit implementation on OrCAD-PSpice software.
1. 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 [1–6]. Today, there are several variants of electric motors including the induction motor [7,8], permanent-magnet brushless motor [9–12], and variable-reluctance motor. The 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]. The variable-reluctance motor is divided into the switched reluctance motor [14,15] and synchronous reluctance motor (SynRM). The 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 [16–20]. 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. The chaotic behaviors found in the SynRM induces instability in this motor and shortens its service time [9]. Thereafter, 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 [24–26], 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. The single state feedback control method is simple, concise, and easy to implement. Therefore, 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. The dynamical analysis and chaos suppression via a single controller of SynRM without external inputs are analytically, numerically, and electronically analysed in this paper. The 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.
2. Dynamical Analysis of SynRM without External Inputs
The SynRM can be described by the following rate equations [1, 2, 13]:where are the d (direct)- and q (quadrature)-axis stator currents, is the mechanical rotor speed, is the electrical rotor speed, is the stator voltage on d axis, is the stator resistance per phase, is the feedback coefficient, and is the reference rotor speed, are the d- and q-axis stator inductors, is the number of poles, , , and are the inertia constant of the motor and load, load torque, and viscous friction coefficient, respectively. The normalization of equations (1a)–(1c) leads to the following dimensionless form of the mathematical model of SynRM:with the following rescaling variables and parameters: The external inputs are removed , and System (2a)–(2c) becomes
System (3a)–(3c) is invariant under the transformation: and dissipative if . It has only one equilibrium point if , three equilibrium points , if , and five equilibrium points , , if [13]. The linear stability analysis of system (2) revealed that the equilibrium points displayed Hopf bifurcation [13]. When the parameters 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.
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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 and , the bifurcation diagrams and LLE of SynRM without external inputs as a function of the parameter are plotted in Figure 2.
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Figure 2 shows that the SynRM without external inputs exhibits monostable period-3 oscillations, bistable period-3 oscillations followed to period tripling to bistable chaos and monostable chaos interspersed with bistable and monostable periodic regions. The dynamical behaviors shown in Figure 2 are illustrated in Figure 3 for a specific value of .
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The 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). The bifurcation diagrams of SynRM without external inputs obtained numerically by the parameters and reveal monostable chaos and bistable chaos interspersed with monostable and bistable periodic regions followed by monostable period-3-oscillations, but the results have not presented here for brevity.
3. 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].
3.1. Proposed Controller 1
System (3a)–(3c) with the first single controller is described by
The controlled system (4a)–(4c) can be rewritten as
The solution of equation (5a) is . That is, yield . So, system (5a)–(5c) can be reduced as follows:
The solution of equation (6b) is . That is, yield . So, system (6a) and (6b) can be rewritten as follows
The solution of equation (7) is . That is, yield . Therefore, the chaotic behavior found in the SynRM without external inputs can be controlled using the controller . The curves of the state responses and the output of the controller 1 are shown in Figure 4.
The results of Figure 4 show the efficiency of the controller .
3.2. Proposed Controller 2
System (3a)–(3c) with the second single controller is described by
The controller into the controlled system (8a)–(8c) can be rewritten as
The solution of equation (9b) is . That is, yield . Thus, the system (9a)–(9c) can be reduced as follows:
The solution of system (10a) and (10b) is given by
That is, yield and . Therefore, the chaotic behavior found in the SynRM without external inputs can be controlled using the controller . The curves of the state responses and the output of the controller are shown in Figure 5.
The results of Figure 4 reveal the efficiency of the controller .
3.3. Proposed Controller 3
System (3a)–(3c) with the third single controller is described by
Substituting the expression of the controller into the controlled system (12a)–(12c) becomes
The solution of equation (13c) is . That, is yield . Thus, system (13a)–(13c) can be reduced as follows:
The solution of system (14a) and (14b) can be rewritten as follows:
That is, yield and . Therefore, the chaotic behavior found in SynRM without external inputs can be controlled using the controller . The curves of the state responses and the output of the single controller 3 are shown in Figure 6.
The results of Figure 6 show the efficiency of the controller . 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.
4. Circuit Implementation of SynRM without External Inputs and Chaos Suppression in SynRM without External Inputs
The electronic implementation of system (3a)–(3c) is depicted in Figure 7.
The 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.
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The 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. The 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). The 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.
The 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. The 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.
The 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. The time series of the state responses and the output of the single controller 3 generated from the circuit diagram of the controlled system (13a)–(13c) are shown in Figure 11.
The 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.
5. Conclusion
This paper is dealt with the dynamical analysis, chaos suppression, and electronic implementation of synchronous reluctance motor without external inputs. The 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. Thanks to the principle of Lyapunov’s method for asymptotic global stability, three single controllers were designed to suppress chaotic behavior found in synchronous reluctance motor without external inputs, and it was revealed that they were simple and easy to implement. The 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
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was partially funded by the Center for Nonlinear Systems, Chennai Institute of Technology, India via funding number CIT/CNS/2021/RD/064.