# Synchronization and Antisynchronization ofN-Coupled Complex Permanent Magnet Synchronous Motor Systems with Ring Connection

Authors: Cuimei Jiang; Shutang Liu
Journal: Complexity
 (2017)

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## Abstract

This paper discusses synchronization and antisynchronization ofN-coupled complex permanent magnet synchronous motors systems with ring connection. Based on the direct design method and antisymmetric structure, the appropriate controllers are designed to ensure the occurrence of synchronization and antisynchronization in an array of N-coupled general complex chaotic systems described by a unified mathematical expression with ring connection. The proposed method is flexible and is suitable both for design and for implementation in practice. Numerical results are plotted to show the rapid convergence of errors to zero and further verify the effectiveness and feasibility of the theoretical scheme.


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## Body

## 1. Introduction

Since Fowler et al. [1] were the first to investigate the complex Lorenz equations, complex systems have played an important role in a variety of industrial fields. For example, complex systems can provide an excellent instrument for the description of various physical phenomena, such as detuned laser systems, amplitudes of electromagnetic fields, and thermal convection of liquid flows [2–4]. Another example is when synchronization of complex chaotic systems is used to transmit information in the secure communications, where complex variables (doubling number of real variables) increase the message contents and enhance security of the transmitted information. Hence, synchronization of complex chaotic systems has aroused great interest and been a hot interdisciplinary topic in nonlinear research during the last few decades [5–9]. Some synchronization schemes of real chaotic systems were extended in the complex field, such as complete synchronization [10], antisynchronization [11, 12], projective synchronization [13], and lag synchronization [14]. Additionally, there exist many synchronization schemes of complex chaotic systems, including complex complete synchronization [15], complex projective synchronization [16], complex modified projective synchronization [17, 18], and complex function projective synchronization [7], just to enumerate a few examples.With the mentioned above, most of the works mainly focus on the usual drive-response synchronization within one drive system and one response system; there is very little concern on synchronization of multiple complex chaotic systems. As a matter of fact, synchronization of multiple complex systems can enhance better the security and lead a more bright future in communications than the usual synchronization schemes. Therefore, many researchers have devoted a great deal of time and efforts to study and analyze synchronization of multiple complex chaotic systems. For instance, Zhou et al. [19] investigated the combination synchronization problem in three identical and different hyperchaotic complex systems by means of two real scaling diagonal matrices. As introduced in [20], combination-combination synchronization among different nonlinear complex chaotic systems was realized by designing proper controllers. Sun et al. [21] made an excellent effort for examining combination complex synchronization of three identical hyperchaotic complex systems via two complex scaling diagonal matrices. Noticeably, Jiang and Liu [22] analyzed combination complex synchronization of real chaos and complex chaos with different dimensions, namely, generalized combination complex synchronization. Furthermore, a novel adaptive generalized combination complex synchronization scheme for different real and complex nonlinear systems with unknown parameters was reported in a recent literature [23].To the best of our knowledge, the aforementioned results on synchronization of multiple complex chaotic systems did not consider the case that multiple complex chaotic systems are coupled in a ring structure as shown in Figure1. In this chaotic model, the state variables of the first complex chaotic system couple the Nth system, and the state variables of the second system couple the first,… analogically, until the state variables of the Nth system couple those of the (N-1)th. Thus, the ring structure formed among multiple complex chaotic systems makes them correlative. Consequently, it is much more attractive and challenging for researchers to explore various synchronization schemes of multiple coupled complex chaotic systems with ring connection. Moreover, the previous attention was drawn for the research on synchronization of multiple coupled real chaotic systems with ring connection, such as [24–29] to name a few. To date, the study has rarely been explored to achieve synchronization of multiple coupled complex chaotic systems with ring connection.Figure 1
Diagram of coupled complex chaotic systems with ring connection.Up till now, there are many approaches for realizing chaos synchronization [30–32]. It should be mentioned that activation control is one of the most important approaches for synchronizing chaotic systems. By means of activation method, controllers can be constructed by canceling all nonlinear terms in original systems, and further synchronization of chaotic systems is converted to a linear problem. Meanwhile, the design procedure of sliding mode method is complex and the controller obtained by this method is not simple enough for many systems, while in [27–29, 33] authors presented the direct design approach based on antisymmetric structure to realize synchronization of chaotic systems with real variables. This direct design method has two advantages: (i) it presents an easy procedure for choosing proper controllers in chaos synchronization; (ii) it constructs simple controllers easily to implement. Therefore, in this paper, for N-coupled complex chaotic systems with ring connection, we utilize the direct method based on antisymmetric structure to investigate chaos synchronization.On the other hand, permanent magnet synchronous motor (PMSM) has direct applications in many areas especially for industrial applications in low-medium power range, since it has excellent features such as simple structure, high torque-to-inertia ratio, high torque-to-weight ratio, and low manufacturing cost. Therefore, a considerable amount of research has been devoted to study synchronization and control of the PMSM system [34–38]. However, so far, there are few papers discussing dynamical properties of the PMSM system with complex variables [39, 40]. Meanwhile, complex voltage and current exist widely in motor systems which result in more complex and richer dynamical dynamics in practice. Therefore, it is of practical significance to investigate the complex PMSM system.Motivated by the above discussions, in this paper, we concentrate on synchronization and antisynchronization ofN-coupled complex PMSM systems with ring connection for the first time. By virtue of the direct method, the general controllers are constructed to transform the error system into a nonlinear system with special antisymmetric structure and achieve synchronization and antisynchronization of N-coupled general complex chaotic systems. What is more, this proposed synchronization scheme is simple but efficacious.


## 2. System Descriptions

Consider a PMSM system in field-oriented rotor which takes the following form:(1)diddt=-R1id+ωLqiq+udLd,diqdt=-R1iq+ωLdid+uq-ωψrLq,dωdt=npψriq+npLd-Lqidiq-TL-βωJ,where id, iq, and ω are the state variables which represent currents and motor angular frequency, respectively; ud and uq are the direct-axis stator and quadrature-axis stator voltage components, respectively; J is the polar moment of inertia; TL is the external load torque; β is the viscous damping coefficient; R1 is the stator resistance; Ld and Lq are the direct-axis and quadrature-axis inductances, respectively; ψr is rotor flux; and np is pole-pairs.When the air gap is even and the motor operates with no load and power outage, the dimensionless equations of a PMSM system can be expressed as follows:(2)x˙1=ax2-x1,x˙2=bx1-x2-x1x3,x˙3=x1x2-x3,where a and b are parameters. Considering that current in system (1) is plural and variables x1 and x2 in system (2) are complex numbers, Wang and Zhang [39] introduced a complex PMSM system as follows:(3)x˙1=ax2-x1,x˙2=bx1-x2-x1x3,x˙3=12x¯1x2+x1x¯2-x3,where x1 and x2 are complex variables and x¯1 and x¯2 are conjugate complex numbers of x1 and x2, j=-1. When a=11 and b=20, system (3) is chaotic and the phase portraits are shown in Figure 2.Figure 2
Chaotic behavior and attractors of the complex PMSM system (3) in different phase planes and projections.
(a) (b)Authors studied some basic dynamical properties of the complex PMSM system (3) in [39, 40]. But there are many properties of the complex PMSM system remaining to be uncovered. In the sequel, we will discuss synchronization and antisynchronization of N-coupled complex PMSM systems with ring connection.The main construct of the rest of the paper is organized as follows. Section3 investigates synchronization and antisynchronization of N-coupled general complex chaotic systems described by a unified mathematical expression with ring connection. In Section 4, the complex controllers are provided to realize synchronization and antisynchronization of 4-coupled complex PMSM systems with ring connection. Finally, Section 5 gives some concluding remarks.


## 3. General Method for Synchronization and Antisynchronization ofN-Coupled Complex Chaotic Systems

A general form ofN-coupled complex chaotic systems with ring connection can be configured as(4)X˙1=A1X1+G1X1+Q1XN-X1,X˙2=A2X2+G2X2+Q2X1-X2,⋮X˙N=ANXN+GNXN+QNXN-1-XN,where X1,X2,…,XN∈Cn (N>2) are complex state vectors of chaotic systems and Xk=Xkr+jXki; A1,A2,…,AN∈Rn×n are constant matrices (Ak≠Al,k≠l); Gk(Xk) are the continuous complex nonlinear functions (Gk≠Gl,k≠l); Qk=diag⁡(qk1,qk2,…,qkn) are n-dimensional real diagonal matrices; and qkl≥0 are the ideal gains which represent the coupled parameters (k=1,2,…,N;l=1,2,…,n).Suppose that the first system is chosen as the aim system, and add the controllers to the remaining systems. Thus system (4) can be rewritten as(5)X˙1=A1X1+G1X1+Q1XN-X1,X˙2=A2X2+G2X2+Q2X1-X2+U1,X˙3=A3X3+G3X3+Q3X2-X3+U2,⋮X˙N=ANXN+GNXN+QNXN-1-XN+UN-1.The synchronization error vector is defined as ek=Xk+1-δXk(k=1,2,…,N-1;δ=1,-1). Apparently, if δ=1, then the error system ek can degenerate into the complete synchronization error system. And if δ=-1, then the error system ek can degenerate into the antisynchronization error system. Taking into account system (5), we have the following error dynamical system as(6)e˙t=e˙rt+je˙it=Σe1e2⋮eN-1+δA2-A1+1-δQ2-δN-δQ1X1+G2X2-δG1X1+U1δA3-A2+δ-1Q2-Q3X2+G3X3-δG2X2+U2-δU1⋮δAN-AN-1+δ-1QN-1-QNXN-1+GNXN-δGN-1XN-1+UN-1-δUN-2,where(7)Σ=A2-δN-1Q1-Q2-δN-2Q1-δN-3Q1⋯-δ2Q1-δQ1Q2A3-Q30⋯00⋮⋮⋮⋱⋮⋮000⋯QN-1AN-QN.Furthermore, it is not difficult to obtain that(8)e˙1re˙2r⋮e˙N-1r=Σe1re2r⋮eN-1r+δA2-A1+1-δQ2-δN-δQ1X1r+G2rX2-δG1rX1+U1rδA3-A2+δ-1Q2-Q3X2r+G3rX3-δG2rX2+U2r-δU1r⋮δAN-AN-1+δ-1QN-1-QNXN-1r+GNrXN-δGN-1rXN-1+UN-1r-δUN-2r,e˙1ie˙2i⋮e˙N-1i=Σe1ie2i⋮eN-1i+δA2-A1+1-δQ2-δN-δQ1X1i+G2iX2-δG1iX1+U1iδA3-A2+δ-1Q2-Q3X2i+G3iX3-δG2iX2+U2i-δU1i⋮δAN-AN-1+δ-1QN-1-QNXN-1i+GNiXN-δGN-1iXN-1+UN-1i-δUN-2i.Thus the essential goal of this paper is to design complex controllersUk=Ukr+jUki (k=1,2,…,N-1) such that the error dynamical error e(t) tends to zero as t→∞; namely, (9)limt→∞ert=0,limt→∞eit=0, and then (10)limt→∞et=0, where · is the Euclidean norm. This implies that synchronization and antisynchronization of N-coupled complex chaotic systems are realized.In what follows, applying the direct method based on antisymmetric structure, we investigate synchronization and antisynchronization ofN-coupled chaotic systems with complex variables.First, the controllersUkr and Uki (k=1,2,…,N-1) are chosen as(11)U1r=V1r-δA2-A1+1-δQ2-δN-δQ1X1r-G2rX2+δG1rX1,U2r=V2r-δA3-A2+δ-1Q2-Q3X2r-G3rX3+δG2rX2+δU1r,⋮UN-1r=VN-1r-δAN-AN-1+δ-1QN-1-QNXN-1r-GNrXN+δGN-1rXN-1+δUN-2r,U1i=V1i-δA2-A1+1-δQ2-δN-δQ1X1i-G2iX2+δG1iX1,U2i=V2i-δA3-A2+δ-1Q2-Q3X2i-G3iX3+δG2iX2+δU1i,⋮UN-1i=VN-1i-δAN-AN-1+δ-1QN-1-QNXN-1i-GNiXN+δGN-1iXN-1+δUN-2i,where(12)V1rV2r⋮VN-1r=Θe1re2r⋮eN-1r,V1iV2i⋮VN-1i=Θe1ie2i⋮eN-1i,and Θ is a real coefficient matrix. Then the error dynamical systems (8) can be represented as(13)e˙rt=Φtert,e˙it=Φteit,where Φ(t)=Σ+Θ. Obviously, there is no unique choice for such a matrix Θ to ensure that the error dynamical system (13) is asymptotically stable at the origin. In general, Θ is defined as a state-dependent real coefficient matrix. Then we can obtain the sufficient stability conditions of system (13) by transforming it into a stable system with antisymmetric structure. Before we give our main result, the following lemma should be introduced first.Lemma 1 (see [41]).
Ifε:[0,∞]→R satisfies limt→∞∫0tε2(t)dt<∞, ε˙(t) exists, and it is bounded, then limt→∞ε(t)=0.Theorem 2.
Synchronization and antisynchronization ofN-coupled complex chaotic systems (5) will occur by using the control laws (11), if the state-dependent real coefficient matrix Φ(t)=Φ1(t)+Φ2 satisfies the following assumptions:(14)Φ1Tt=-Φ1t,Φ2=diag⁡-φ1,-φ2,…,-φN-1,where φk=diag⁡(φk1,φk2,…,φkn) (k=1,2,…,N-1) and φkl>0 (l=1,2,…,n).Proof.
Choose a positive definite Lyapunov candidate function as(15)Vt=12ertTert+12eitTeit.Then the time derivative of V(t) along the trajectory of the error system (13) is as follows:(16)V˙t=12ertTΦTt+Φtert+12eitTΦTt+Φteit.Substitution of (14) into (16) yields (17)V˙t=12ertTΦ2T+Φ2ert+12eitTΦ2T+Φ2eit=ertTΦ2ert+eitTΦ2eit=e1rtT,e2rtT,…,eN-1rtT-φ100⋯000-φ20⋯00⋮⋮⋮⋱⋮⋮000⋯0-φN-1e1rte2rt⋮eN-1rt+e1itT,e2itT,…,eN-1itT-φ100⋯000-φ20⋯00⋮⋮⋮⋱⋮⋮000⋯0-φN-1e1ite2it⋮eN-1it=-e1rtTφ1e1rt-e2rtTφ2e2rt-⋯-eN-1rtTφN-1eN-1rt-e1itTφ1e1it-e2itTφ2e2it-⋯-eN-1itTφN-1eN-1it≤-φmine1rtTe1rt+e2rtTe2rt+⋯+eN-1rtTeN-1rt+e1itTe1it+⋯+eN-1itTeN-1it=-φminet2,where φmin=min1≤k≤N-1,1≤l≤n{φkl}. Thus taking the integral of both sides, we have (18)∫0tet2dt≤V0-Vtφmin≤V0φmin<∞.As e˙r(t) and e˙i(t) exist and they are bounded, then (19)detdt=12et-12e˙rtTert+2e˙itTeit=e˙rtTert+e˙itTeitet. Hence, we get that the derivative of e(t) exists and it is bounded. According to Lemma 1, we have limt→∞e(t)=0. That implies that the closed-loop system (13) is globally asymptotically stable. Therefore, N-coupled complex chaotic systems (5) realize synchronization and antisynchronization. This completes the proof.The following corollary can be easily obtained from Theorem2.Corollary 3.
If the structures of general complex chaotic systems are identical, that is,Ak=Al=A and Gk(·)=Gl(·)=G(·), then complex controllers Uk=Ukr+jUki can be constructed as (20)U1=V1-1-δQ2-δN-δQ1X1-GX2+δGX1,U2=V2-δ-1Q2-Q3X2-GX3+δGX2+δU1,⋮UN-1=VN-1-δ-1QN-1-QNXN-1-GXN+δGXN-1+δUN-2, where Vk=Vkr+jVki (k=1,2,…,N-1) are defined as in (12) and Θ is a real coefficient matrix. Similar to the aforementioned discussion, we can achieve synchronization and antisynchronization of N-coupled identical complex chaotic systems with ring connection based on antisymmetric structure.As above-mentioned analysis, we easily know that the error dynamical systems (8) can be transformed into system (13) by utilizing complex control laws Uk. Moreover, the suitable real coefficient matrix Θ is chosen to translate system (13) into a stable system with an antisymmetric structure. It is known that the antisymmetric structure in Theorem 2 is the generalization of the tridiagonal structure [42, 43]. The error system constructed with the antisymmetric structure is more convenient than the one with tridiagonal structure when the original system has some zero elements at the tridiagonal position and nonzero elements at other positions. And we find that selecting the coefficient matrix plays an important role in achieving synchronization and antisynchronization of N-coupled complex chaotic systems with ring connection.


## 4. Synchronization and Antisynchronization ofN-Coupled Complex PMSM Systems with Ring Connection

In the following, we investigate synchronization and antisynchronization ofN-coupled complex PMSM systems with ring connection. Now, 4-coupled complex PMSM systems with ring connection and designed controllers can be described as (21)x˙11=ax12-x11+q11x41-x11,x˙12=bx11-x12-x11x13+q12x42-x12,x˙13=12x¯11x12+x11x¯12-x13+q13x43-x13,x˙21=ax22-x21+q21x11-x21+u11,x˙22=bx21-x22-x21x23+q22x12-x22+u12,x˙23=12x¯21x22+x21x¯22-x23+q23x13-x23+u13,x˙31=ax32-x31+q31x21-x31+u21,x˙32=bx31-x32-x31x33+q32x22-x32+u22,x˙33=12x¯31x32+x31x¯32-x33+q33x23-x33+u23,x˙41=ax42-x41+q41x31-x41+u31,x˙42=bx41-x42-x41x43+q42x32-x42+u32,x˙43=12x¯41x42+x41x¯42-x43+q43x33-x43+u33,where (22)Ak=A=-aa0b-1000-1,GkXk=GXk=0-xk1xk312x¯k1xk2+xk1x¯k2,Qk=diag(qk1,qk2,qk3) are the coupled matrices (k=1,2,3,4), and Uk=(uk1,uk2,uk3)T(k=1,2,3) are the complex control inputs.The synchronization error can be presented asek=Xk+1-δXk (k=1,2,3). Then the error dynamical system can be written in the form (23)e˙t=A-δ3Q1-Q2-δ2Q1-δQ1Q2A-Q300Q3A-Q4e1te2te3t+1-δQ2-δ4-δQ1X1+GX2-δGX1+U1δ-1Q2-Q3X2+GX3-δGX2+U2-δU1δ-1Q3-Q4X3+GX4-δGX3+U3-δU2.


### 4.1. Synchronization of 4-Coupled Complex PMSM Systems

Whenδ=1, the error system is degraded into the following complete synchronization error system:(24)e˙rt=A-Q1-Q2-Q1-Q1Q2A-Q300Q3A-Q4e1rte2rte3rt+GrX2-GrX1+U1rGrX3-GrX2+U2r-U1rGrX4-GrX3+U3r-U2r,e˙it=A-Q1-Q2-Q1-Q1Q2A-Q300Q3A-Q4e1ite2ite3it+GiX2-GiX1+U1iGiX3-GiX2+U2i-U1iGiX4-GiX3+U3i-U2i.Choose the real coefficient matrix Θ=Θ1Θ2Θ3 as follows: (25)Θ1=0000x11r0000-a-b00000000000000000,Θ2=q11-q2100000-q3100-x11rq12-q220-a-b000-q32000q13-q2300000-q33,Θ3=q11000000000q120000-a-b0000q13000000.Thus, complex controllers Uk (k=1,2,3) are provided as (26)U1=U1r+jU1i=x11re22r-a+be11r+x21rx23-x11rx13-x21rx22r-x21ix22i+x11rx12r+x11ix12i+jx11re22i-a+be11i+x21ix23-x11ix130,U2=U2r+jU2i=q11-q21e11r-q31e31r+x11re22rq12-q22e12r-a+be21r+e11r-q32e32r-x11re11r+x31rx33-x11rx13q13-q23e13-q33e33-x31rx32r-x31ix32i+x11rx12r+x11ix12i+jq11-q21e11i-q31e31i+x11re22iq12-q22e12i-a+be21i+e11i-q32e32i-x11re11i+x31ix33-x11ix130,U3=U3r+jU3i=2q11-q21e11r-q31e31r+x11re22r2q12-q22e12r-a+be11r+e21r+e31r-q32e32r-x11re11r+x41rx43-x11rx132q13-q23e13-q33e33-x41rx42r-x41ix42i+x11rx12r+x11ix12i+j2q11-q21e11i-q31e31i+x11re22i2q12-q22e12i-a+be11i+e21i+e31i-q32e32i-x11re11i+x41ix43-x11ix130.So we have e˙r(t)=(Φ1(t)+Φ2)er(t) and e˙i(t)=(Φ1(t)+Φ2)ei(t), where Φ2=diag(-a-q11-q21, -1-q12-q22, -1-q13-q23, -a-q31, -1-q32, -1-q33, -a-q41, -1-q42, -1-q43) and (27)Φ1t=0a0-q11x11r0-q1100-a000-q1200-q12000000-q1300-q13q11000a0-q3100-x11rq120-a000-q32000q1300000-q33q1100q31000a00q1200q320-a0000q1300q33000. If the conditions (28)-a-q11-q21<0,-1-q12-q22<0,-1-q13-q23<0,-a-q31<0,-1-q32<0,-1-q33<0,-a-q41<0,-1-q42<0,-1-q43<0hold, then the error system is asymptotically stable due to Theorem 2. Hence, complete synchronization of 4-coupled complex PMSM systems is realized.The simulation results are illustrated with the initial conditionX1(0)=(1+2j,3+4j,5)T, X2(0)=(6+16j,15+14j,30)T, X3(0)=(10+4j,10+21j,8)T, and X4(0)=(5+9j,11+4j,10)T. Furthermore, selecting qk1=qk2=qk3=1 (k=1,2,3,4), we obtain simulation results as shown in Figures 3 and 4. Figure 3 displays the state variables for complete synchronization of the 4-coupled complex PMSM systems. As we can see from Figure 4 the errors of synchronization converge asymptotically to zero in a quite short period. Thus, we achieve complete synchronization of 4-coupled complex PMSM systems with ring connection.Figure 3
The state trajectoriesxk1, xk2, and xk3 (k=1,2,3,4) for synchronization of 4-coupled complex PMSM systems.
(a) (b) (c) (d) (e)Figure 4
The error dynamics of synchronization among4-coupled complex PMSM systems.
(a) (b) (c)


### 4.2. Antisynchronization of 4-Coupled Complex PMSM Systems

Whenδ=-1, the error system is degraded into the following antisynchronization error system: (29)e˙t=A+Q1-Q2-Q1Q1Q2A-Q300Q3A-Q4e1te2te3t+2Q2-Q1X1+GX2+GX1+U12Q3-Q2X2+GX3+GX2+U2+U12Q4-Q3X3+GX4+GX3+U3+U2.By choosing the real coefficient matrixΘ=Θ1Θ2Θ3, (30)Θ1=0-a-b0q11-q2100-q11000000q12-q2200-q12000000q13-q2300-q13,Θ2=0000-a-b0-q31000000000-q32000000000-q33,Θ3=0000000-a-b-x32r0000000x32r00000000000, we design the complex controllers Uk=(uk1,uk2,uk3)T (k=1,2,3) as (31)u11=-a+be12+q11-q21e21-q11e31-2q21-q11x11,u12=q12-q22e22-q12e32-2q22-q12x12+x21x23+x11x13,u13=q13-q23e23-q13e33-2q23-q13x13-x21rx22r-x21ix22i-x11rx12r-x11ix12i,u21=a+be12-e22+q11-q31e31-q11-q21e21+2x11+2q21-q31x21,u22=q12-q32e32-q12-q22e22+2x12+2q22-q32x22+x31x33-x11x13,u23=q13-q33e33-q13-q23e23+2x13+2q23-q33x23-x31rx32r-x31ix32i+x11rx12r+x11ix12i,u31=a+be22-e32-e12+q31-q11e31-x32re32+q11-q21e21+2x11+2q31-q41x31-2q21-q31x21,u32=q32-q12e32+q12-q22e22+2x12+x32re31+2q32-q42x32-2q22-q32x22+x41x43+x11x13,u33=q33-q13e33+q13-q23e23+2x13+2q33-q43x33-2q23-q33x23-x41rx42r-x41ix42i-x11rx12r-x11ix12i.Then, we obtain the following error system e˙(t)=Φ(t)e(t)=(Φ1(t)+Φ2)e(t), where Φ2=diag(-a+q11-q21, -1+q12-q22, -1+q13-q23, -a-q31, -1-q32, -1-q33, -a-q41, -1-q42, -1-q43) and (32)Φ1t=0-b0-q2100000b000-q22000000000-q23000q21000-b0-q31000q220b000-q32000q2300000-q33000q31000-b-x32r00000q320b+x32r0000000q33000. If the conditions (33)-a+q11-q21<0,-1+q12-q22<0,-1+q13-q23<0,-a-q31<0,-1-q32<0,-1-q33<0,-a-q41<0,-1-q42<0,-1-q43<0hold, then the error system is asymptotically stable under complex controllers Uk (k=1,2,3) according to Theorem 2. Hence, antisynchronization of 4-coupled complex PMSM systems is realized.The simulation results are illustrated with the initial conditionX1(0)=(1+2j,3+6j,5)T, X2(0)=(4-2j,1+2j,8)T, X3(0)=(-3+4j,-2+5j,-1)T, and X4(0)=(5-5j,4-2j,3)T. Furthermore, selecting q11=q12=q13=q31=q32=q33=1, q21=q22=q23=2, and q41=q42=q43=20, we have simulation results as displayed in Figures 5 and 6. Figure 5 shows the state variables for antisynchronization of the 4-coupled complex PMSM systems. It follows from Figure 6 that the errors of synchronization converge asymptotically to zero in a quite short period. As expected, antisynchronization of 4-coupled complex PMSM systems can occur.Figure 5
The state trajectoriesxk1, xk2, and xk3 (k=1,2,3,4) for antisynchronization of 4-coupled complex PMSM systems.
(a) (b) (c) (d) (e)Figure 6
The error dynamics of antisynchronization among4-coupled complex PMSM systems.
(a) (b) (c)


## 4.1. Synchronization of 4-Coupled Complex PMSM Systems

Whenδ=1, the error system is degraded into the following complete synchronization error system:(24)e˙rt=A-Q1-Q2-Q1-Q1Q2A-Q300Q3A-Q4e1rte2rte3rt+GrX2-GrX1+U1rGrX3-GrX2+U2r-U1rGrX4-GrX3+U3r-U2r,e˙it=A-Q1-Q2-Q1-Q1Q2A-Q300Q3A-Q4e1ite2ite3it+GiX2-GiX1+U1iGiX3-GiX2+U2i-U1iGiX4-GiX3+U3i-U2i.Choose the real coefficient matrix Θ=Θ1Θ2Θ3 as follows: (25)Θ1=0000x11r0000-a-b00000000000000000,Θ2=q11-q2100000-q3100-x11rq12-q220-a-b000-q32000q13-q2300000-q33,Θ3=q11000000000q120000-a-b0000q13000000.Thus, complex controllers Uk (k=1,2,3) are provided as (26)U1=U1r+jU1i=x11re22r-a+be11r+x21rx23-x11rx13-x21rx22r-x21ix22i+x11rx12r+x11ix12i+jx11re22i-a+be11i+x21ix23-x11ix130,U2=U2r+jU2i=q11-q21e11r-q31e31r+x11re22rq12-q22e12r-a+be21r+e11r-q32e32r-x11re11r+x31rx33-x11rx13q13-q23e13-q33e33-x31rx32r-x31ix32i+x11rx12r+x11ix12i+jq11-q21e11i-q31e31i+x11re22iq12-q22e12i-a+be21i+e11i-q32e32i-x11re11i+x31ix33-x11ix130,U3=U3r+jU3i=2q11-q21e11r-q31e31r+x11re22r2q12-q22e12r-a+be11r+e21r+e31r-q32e32r-x11re11r+x41rx43-x11rx132q13-q23e13-q33e33-x41rx42r-x41ix42i+x11rx12r+x11ix12i+j2q11-q21e11i-q31e31i+x11re22i2q12-q22e12i-a+be11i+e21i+e31i-q32e32i-x11re11i+x41ix43-x11ix130.So we have e˙r(t)=(Φ1(t)+Φ2)er(t) and e˙i(t)=(Φ1(t)+Φ2)ei(t), where Φ2=diag(-a-q11-q21, -1-q12-q22, -1-q13-q23, -a-q31, -1-q32, -1-q33, -a-q41, -1-q42, -1-q43) and (27)Φ1t=0a0-q11x11r0-q1100-a000-q1200-q12000000-q1300-q13q11000a0-q3100-x11rq120-a000-q32000q1300000-q33q1100q31000a00q1200q320-a0000q1300q33000. If the conditions (28)-a-q11-q21<0,-1-q12-q22<0,-1-q13-q23<0,-a-q31<0,-1-q32<0,-1-q33<0,-a-q41<0,-1-q42<0,-1-q43<0hold, then the error system is asymptotically stable due to Theorem 2. Hence, complete synchronization of 4-coupled complex PMSM systems is realized.The simulation results are illustrated with the initial conditionX1(0)=(1+2j,3+4j,5)T, X2(0)=(6+16j,15+14j,30)T, X3(0)=(10+4j,10+21j,8)T, and X4(0)=(5+9j,11+4j,10)T. Furthermore, selecting qk1=qk2=qk3=1 (k=1,2,3,4), we obtain simulation results as shown in Figures 3 and 4. Figure 3 displays the state variables for complete synchronization of the 4-coupled complex PMSM systems. As we can see from Figure 4 the errors of synchronization converge asymptotically to zero in a quite short period. Thus, we achieve complete synchronization of 4-coupled complex PMSM systems with ring connection.Figure 3
The state trajectoriesxk1, xk2, and xk3 (k=1,2,3,4) for synchronization of 4-coupled complex PMSM systems.
(a) (b) (c) (d) (e)Figure 4
The error dynamics of synchronization among4-coupled complex PMSM systems.
(a) (b) (c)


## 4.2. Antisynchronization of 4-Coupled Complex PMSM Systems

Whenδ=-1, the error system is degraded into the following antisynchronization error system: (29)e˙t=A+Q1-Q2-Q1Q1Q2A-Q300Q3A-Q4e1te2te3t+2Q2-Q1X1+GX2+GX1+U12Q3-Q2X2+GX3+GX2+U2+U12Q4-Q3X3+GX4+GX3+U3+U2.By choosing the real coefficient matrixΘ=Θ1Θ2Θ3, (30)Θ1=0-a-b0q11-q2100-q11000000q12-q2200-q12000000q13-q2300-q13,Θ2=0000-a-b0-q31000000000-q32000000000-q33,Θ3=0000000-a-b-x32r0000000x32r00000000000, we design the complex controllers Uk=(uk1,uk2,uk3)T (k=1,2,3) as (31)u11=-a+be12+q11-q21e21-q11e31-2q21-q11x11,u12=q12-q22e22-q12e32-2q22-q12x12+x21x23+x11x13,u13=q13-q23e23-q13e33-2q23-q13x13-x21rx22r-x21ix22i-x11rx12r-x11ix12i,u21=a+be12-e22+q11-q31e31-q11-q21e21+2x11+2q21-q31x21,u22=q12-q32e32-q12-q22e22+2x12+2q22-q32x22+x31x33-x11x13,u23=q13-q33e33-q13-q23e23+2x13+2q23-q33x23-x31rx32r-x31ix32i+x11rx12r+x11ix12i,u31=a+be22-e32-e12+q31-q11e31-x32re32+q11-q21e21+2x11+2q31-q41x31-2q21-q31x21,u32=q32-q12e32+q12-q22e22+2x12+x32re31+2q32-q42x32-2q22-q32x22+x41x43+x11x13,u33=q33-q13e33+q13-q23e23+2x13+2q33-q43x33-2q23-q33x23-x41rx42r-x41ix42i-x11rx12r-x11ix12i.Then, we obtain the following error system e˙(t)=Φ(t)e(t)=(Φ1(t)+Φ2)e(t), where Φ2=diag(-a+q11-q21, -1+q12-q22, -1+q13-q23, -a-q31, -1-q32, -1-q33, -a-q41, -1-q42, -1-q43) and (32)Φ1t=0-b0-q2100000b000-q22000000000-q23000q21000-b0-q31000q220b000-q32000q2300000-q33000q31000-b-x32r00000q320b+x32r0000000q33000. If the conditions (33)-a+q11-q21<0,-1+q12-q22<0,-1+q13-q23<0,-a-q31<0,-1-q32<0,-1-q33<0,-a-q41<0,-1-q42<0,-1-q43<0hold, then the error system is asymptotically stable under complex controllers Uk (k=1,2,3) according to Theorem 2. Hence, antisynchronization of 4-coupled complex PMSM systems is realized.The simulation results are illustrated with the initial conditionX1(0)=(1+2j,3+6j,5)T, X2(0)=(4-2j,1+2j,8)T, X3(0)=(-3+4j,-2+5j,-1)T, and X4(0)=(5-5j,4-2j,3)T. Furthermore, selecting q11=q12=q13=q31=q32=q33=1, q21=q22=q23=2, and q41=q42=q43=20, we have simulation results as displayed in Figures 5 and 6. Figure 5 shows the state variables for antisynchronization of the 4-coupled complex PMSM systems. It follows from Figure 6 that the errors of synchronization converge asymptotically to zero in a quite short period. As expected, antisynchronization of 4-coupled complex PMSM systems can occur.Figure 5
The state trajectoriesxk1, xk2, and xk3 (k=1,2,3,4) for antisynchronization of 4-coupled complex PMSM systems.
(a) (b) (c) (d) (e)Figure 6
The error dynamics of antisynchronization among4-coupled complex PMSM systems.
(a) (b) (c)


## 5. Conclusions

A lot of progress has been made in the research of dynamical properties and synchronization for PMSM system with real variables, while the study on complex PMSM system is seldom reported in the previous literatures. So this paper analyzes and validates synchronization and antisynchronization ofN-coupled complex PMSM systems with ring connection. First of all, for N-coupled general complex chaotic systems described by a unified mathematical expression with ring connection, we realize synchronization and antisynchronization based on the direct design method which is flexible and is suitable both for design and for implementation in practice. Thus, the proposed method is successfully applied to achieve synchronization and antisynchronization of N-coupled complex PMSM systems with ring connection on the basis of antisymmetric structure. Moreover, the synchronization scheme is so universal and simple that can be applicable to all complex chaotic systems, including those that can exhibit hyperchaotic behavior. Therefore, it is believed that the proposed scheme will play an important role in engineering practice.



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*Source: 6743184-2017-01-26.xml*