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

Authors: Balamurali Ramakrishnan; Andre Chéagé Chamgoué; Hayder Natiq; Jules Metsebo; Alex Stephane Kemnang Tsafack
Journal: Complexity
 (2022)

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## 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.


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## Body

## 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 Section2. 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]:(1a)di˜ddt˜=u˜d−Rsi˜d+ωeLqi˜qLd,(1b)di˜qdt˜=−Rsi˜q−ωeLdi˜d+kpω˜−ωrefLq,(1c)dω˜dt˜=3PLd−Lqi˜di˜q/4−Bω˜+T˜LJ,where i˜d,i˜q are the d (direct)- and q (quadrature)-axis stator currents, ω˜ is the mechanical rotor speed, ωe is the electrical rotor speed, u˜d is the stator voltage on d axis, Rs is the stator resistance per phase, kp is the feedback coefficient, and ωref is the reference rotor speed, Ld,Lq 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. The normalization of equations (1a)–(1c) leads to the following dimensionless form of the mathematical model of SynRM:(2a)dxdt=ud−bx+yz,(2b)dydt=−y−xz+cz−zref,(2c)dzdt=xy−az+TL,with the following rescaling variables and parameters: τ=Lq/Rs,t˜=τt,a=BLq/JRs,b=Lq/Ld,k=8J/3bP2Ld−LqLq/Rs2,ωref=τzref,x=1/bki˜d,y=1/ki˜q,z=Lq/Rsω˜,c=2kd/kPLq,ud=u˜d/kRs,TL=Pτ2T˜L/2J. The external inputs are removed ud=zref=TL=0, and System (2a)–(2c) becomes(3a)dxdt=−bx+yz,(3b)dydt=−y−xz+cz,(3c)dzdt=xy−az.System (3a)–(3c) is invariant under the transformation: x,y,z⇔x,−y,−z and dissipative if ∇V=∂dx/dt/∂x+∂dy/dt/∂y+∂dz/dt/∂z=−a+b+1<0. It has only one equilibrium point O0,0,0 if Δ=c2−4a<0, three equilibrium points O0,0,0, E1,2c/2,±ab,±cb/a/2 if Δ=0, and five equilibrium points O0,0,0, E11,21c+Δ/2,±ab,±c+Δb/a/2, E11,21c−Δ/2,±ab,±c−Δb/a/2 if Δ>0 [13]. The 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.Figure 1
Two parameters LLE diagrams in (a)a,b space for c=22 and (b)b,c space for a=1.53.
(a)(b)From Figure1, 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
Bifurcation diagrams ofyt (a) and LLE (b) of SynRM versus the parameter a for b=0.2 and c=22. Bifurcation diagrams are obtained by scanning the parameter a upwards (black) and downwards (red). The initial conditions are x0,y0,z0=0.1,0.2,0.2.
(a)(b)Figure2 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 a.Figure 3
Phase planes of SynRM without external inputs for a specific value of parametera: (a) a=1.5, (b) a=1.8, (c) a=2, and (d) a=3. The curves in black line are obtained by using the initial conditions x0,y0,z0=0.1,0.2,0.2, while the curves in red line are obtained by using the initial conditions x0,y0,z0=0.1,−0.2,−0.2. The remaining parameters are b=0.2 and c=22.
(a)(b)(c)(d)The SynRM without external inputs exhibits monostable periodic attractors in Figure3(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 b and c 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 u1=−yz is described by(4a)dxdt=−bx+yz+u1,(4b)dydt=−y−xz+cz,(4c)dzdt=xy−az.The controlled system (4a)–(4c) can be rewritten as(5a)dxdt=−bx,(5b)dydt=−y−xz+cz,(5c)dzdt=xy−az.The solution of equation (5a) is xt=x0e−bt. That is, yield limt⟶∞xt=0. So, system (5a)–(5c) can be reduced as follows:(6a)dydt=−y+cz,(6b)dzdt=−az.The solution of equation (6b) is zt=z0e−at. That is, yield limt⟶∞zt=0. So, system (6a) and (6b) can be rewritten as follows(7)dydt=−y.The solution of equation (7) is yt=y0e−t. That is, yield limt⟶∞yt=0. Therefore, the chaotic behavior found in the SynRM without external inputs can be controlled using the controller u1=−yz. The curves of the state responses and the output of the controller 1 are shown in Figure 4.Figure 4
Time series ofx,y,z, u1 for a=3, b=0.2, and c=22. The initial conditions are (x(0), y(0), z(0)) = (0.1, 0.2, 0.2).The results of Figure4 show the efficiency of the controller u1.


### 3.2. Proposed Controller 2

System (3a)–(3c) with the second single controller u2=zx−c is described by(8a)dxdt=−bx+yz,(8b)dydt=−y−xz+cz+u2,(8c)dzdt=xy−az.The controlleru2 into the controlled system (8a)–(8c) can be rewritten as(9a)dxdt=−bx+yz,(9b)dydt=−y,(9c)dzdt=xy−az.The solution of equation (9b) is yt=y0e−t. That is, yield limt⟶∞yt=0. Thus, the system (9a)–(9c) can be reduced as follows:(10a)dxdt=−bx,(10b)dzdt=−az.The solution of system (10a) and (10b) is given by(11a)xt=x0e−bt,(11b)zt=z0e−at.That is, yieldlimt⟶∞yt=0 and limt⟶∞zt=0. Therefore, the chaotic behavior found in the SynRM without external inputs can be controlled using the controller u2=zx−c. The curves of the state responses and the output of the controller are shown in Figure 5.Figure 5
Time series ofx,y,z, u2 for a=3, b=0.2 and c=22. The initial conditions are (x(0), y(0), z(0)) = (0.1, 0.2, 0.2).The results of Figure4 reveal the efficiency of the controller u2.


### 3.3. Proposed Controller 3

System (3a)–(3c) with the third single controller u3=−xy is described by(12a)dxdt=−bx+yz,(12b)dydt=−y−xz+cz,(12c)dzdt=xy−az+u3.Substituting the expression of the controlleru3 into the controlled system (12a)–(12c) becomes(13a)dxdt=−bx+yz,(13b)dydt=−y−xz+cz,(13c)dzdt=−az.The solution of equation (13c) is zt=z0e−at. That, is yield limt⟶∞zt=0. Thus, system (13a)–(13c) can be reduced as follows:(14a)dxdt=−bx,(14b)dydt=−y.The solution of system (14a) and (14b) can be rewritten as follows:(15a)xt=x0e−bt,(15b)yt=y0e−t.That is, yieldlimt⟶∞xt=0 and limt⟶∞yt=0. Therefore, the chaotic behavior found in SynRM without external inputs can be controlled using the controller u3=−xy. The curves of the state responses and the output of the single controller 3 are shown in Figure 6.Figure 6
Time series ofx,y,z, u3 for a=3, b=0.2 and c=22. The initial conditions are (x(0), y(0), z(0)) = (0.1, 0.2, 0.2).The results of Figure6 show the efficiency of the controller u3. 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.


## 3.1. Proposed Controller 1

System (3a)–(3c) with the first single controller u1=−yz is described by(4a)dxdt=−bx+yz+u1,(4b)dydt=−y−xz+cz,(4c)dzdt=xy−az.The controlled system (4a)–(4c) can be rewritten as(5a)dxdt=−bx,(5b)dydt=−y−xz+cz,(5c)dzdt=xy−az.The solution of equation (5a) is xt=x0e−bt. That is, yield limt⟶∞xt=0. So, system (5a)–(5c) can be reduced as follows:(6a)dydt=−y+cz,(6b)dzdt=−az.The solution of equation (6b) is zt=z0e−at. That is, yield limt⟶∞zt=0. So, system (6a) and (6b) can be rewritten as follows(7)dydt=−y.The solution of equation (7) is yt=y0e−t. That is, yield limt⟶∞yt=0. Therefore, the chaotic behavior found in the SynRM without external inputs can be controlled using the controller u1=−yz. The curves of the state responses and the output of the controller 1 are shown in Figure 4.Figure 4
Time series ofx,y,z, u1 for a=3, b=0.2, and c=22. The initial conditions are (x(0), y(0), z(0)) = (0.1, 0.2, 0.2).The results of Figure4 show the efficiency of the controller u1.


## 3.2. Proposed Controller 2

System (3a)–(3c) with the second single controller u2=zx−c is described by(8a)dxdt=−bx+yz,(8b)dydt=−y−xz+cz+u2,(8c)dzdt=xy−az.The controlleru2 into the controlled system (8a)–(8c) can be rewritten as(9a)dxdt=−bx+yz,(9b)dydt=−y,(9c)dzdt=xy−az.The solution of equation (9b) is yt=y0e−t. That is, yield limt⟶∞yt=0. Thus, the system (9a)–(9c) can be reduced as follows:(10a)dxdt=−bx,(10b)dzdt=−az.The solution of system (10a) and (10b) is given by(11a)xt=x0e−bt,(11b)zt=z0e−at.That is, yieldlimt⟶∞yt=0 and limt⟶∞zt=0. Therefore, the chaotic behavior found in the SynRM without external inputs can be controlled using the controller u2=zx−c. The curves of the state responses and the output of the controller are shown in Figure 5.Figure 5
Time series ofx,y,z, u2 for a=3, b=0.2 and c=22. The initial conditions are (x(0), y(0), z(0)) = (0.1, 0.2, 0.2).The results of Figure4 reveal the efficiency of the controller u2.


## 3.3. Proposed Controller 3

System (3a)–(3c) with the third single controller u3=−xy is described by(12a)dxdt=−bx+yz,(12b)dydt=−y−xz+cz,(12c)dzdt=xy−az+u3.Substituting the expression of the controlleru3 into the controlled system (12a)–(12c) becomes(13a)dxdt=−bx+yz,(13b)dydt=−y−xz+cz,(13c)dzdt=−az.The solution of equation (13c) is zt=z0e−at. That, is yield limt⟶∞zt=0. Thus, system (13a)–(13c) can be reduced as follows:(14a)dxdt=−bx,(14b)dydt=−y.The solution of system (14a) and (14b) can be rewritten as follows:(15a)xt=x0e−bt,(15b)yt=y0e−t.That is, yieldlimt⟶∞xt=0 and limt⟶∞yt=0. Therefore, the chaotic behavior found in SynRM without external inputs can be controlled using the controller u3=−xy. The curves of the state responses and the output of the single controller 3 are shown in Figure 6.Figure 6
Time series ofx,y,z, u3 for a=3, b=0.2 and c=22. The initial conditions are (x(0), y(0), z(0)) = (0.1, 0.2, 0.2).The results of Figure6 show the efficiency of the controller u3. 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.Figure 7
Circuit diagram of system (3a)–(3c) describing SynRM without external inputs.The electronic circuit of Figure7 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.Figure 8
Phase planes of the dynamical behaviors of circuit in Figure7 observed on the Pspice oscilloscope for specific values of R6: (a) R6=6.67kΩ, (b) R6=5.56kΩ, (c)R6=5kΩ, and (d) R6=3.33kΩ. The others values are C1=C2=C3=10nF, R3=R8=R9=R10=R11=R12=R13=10kΩ, R5=454.55kΩ, R1=50kΩ, R2=R4=R7=1kΩ.
(a)(b)(c)(d)The good qualitative agreement between the Pspice results of Figure8 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.Figure 9
Time series of chaos suppression in SynRM without external inputs generated from the Pspice oscilloscope for the capacitors and resistors:C1=C2=C3=10nF, R3=R8=R9=R10=R11=R12=R13=10kΩ, R5=454.55kΩ, R1=50kΩ, R2=R4=R7=1kΩ, R6=3.33kΩ.The good qualitative agreement between the Pspice results of Figure9 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.Figure 10
Time series of chaos suppression in SynRM without external inputs generated from the Pspice oscilloscope for the capacitors and resistors:C1=C2=C3=10nF, R3=R8=R9=R10=R11=R12=R13=10kΩ, R5=454.55kΩ, R1=50kΩ, R2=R4=R7=1kΩ, R6=3.33kΩ.The good qualitative agreement between the Pspice results of Figure10 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.Figure 11
Time series of chaos suppression in SynRM without external inputs generated from the Pspice oscilloscope for the capacitors and resistors:C1=C2=C3=10nF, R3=R8=R9=R10=R11=R12=R13=10kΩ, R5=454.55kΩ, R1=50kΩ, R2=R4=R7=1kΩ, R6=3.33kΩ.The good qualitative agreement between the Pspice results of Figure11 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.



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*Source: 8080048-2022-03-17.xml*