Hybrid synchronization and parameter estimation of a complex...

1Bull. Pol. Acad. Sci. Tech. Sci. 69(3) 2021, e137056

BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 69(3), 2021, Article number: e137056DOI: 10.24425/bpasts.2021.137056

CONTROL AND INFORMATICS

© 2021 The Author(s). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

Abstract. The synchronisation of a complex chaotic network of permanent magnet synchronous motor systems has increasing practical impor-tance in the field of electrical engineering. This article presents the control design method for the hybrid synchronization and parameter esti-mation of ring-connected complex chaotic network of permanent magnet synchronous motor systems. The design of the desired control law is a challenging task for control engineers due to parametric uncertainties and chaotic responses to some specific parameter values. Controllers are designed based on the adaptive integral sliding mode control to ensure hybrid synchronization and estimation of uncertain terms. To apply the adaptive ISMC, firstly the error system is converted to a unique system consisting of a nominal part along with the unknown terms which are computed adaptively. The stabilizing controller incorporating nominal control and compensator control is designed for the error system. The compensator controller, as well as the adopted laws, are designed to get the first derivative of the Lyapunov equation strictly negative. To give an illustration, the proposed technique is applied to 4-coupled motor systems yielding the convergence of error dynamics to zero, estimation of uncertain parameters, and hybrid synchronization of system states. The usefulness of the proposed method has also been tested through computer simulations and found to be valid.

Key words: chaotic system; hybrid synchronization (HS); complex chaotic permanent magnet synchronous motor; adaptive integral sliding mode control (ISMC); Lyapunov function.

Hybrid synchronization and parameter estimation of a complex chaotic network of permanent magnet synchronous motors

using adaptive integral sliding mode control

Nazam SIDDIQUE * and Fazal U. REHMANCapital University of Science and Technology, Islamabad Expressway, Kahuta Road, Zone-V Islamabad, Pakistan

*e-mail: [email protected]

Manuscript submitted 2020-09-29, revised 2021-02-12, initially accepted for publication 2021-03-01, published in June 2021

BULLETIN OF THE POLISH ACADEMY OF SCIENCESTECHNICAL SCIENCES, Vol. 69(3), 2021, Article number: e137056DOI: 10.24425/bpasts.2021.137056

Hybrid synchronization and parameter estimation of a complexchaotic network of permanent magnet synchronous motors


Nazam SIDDIQUE∗, and Fazal U. REHMANCapital University of Science and Technology, Pakistan

Abstract. The synchronization of a complex chaotic network of permanent magnet synchronous motor systems has increasing practical im-portance in the field of electrical engineering. This article presents the control design method for the hybrid synchronization and parameterestimation of ring-connected complex chaotic network of permanent magnet synchronous motor systems. The design of the desired control lawis a challenging task for control engineers due to parametric uncertainties and chaotic responses to some specific parameter values. Controllersare designed based on the adaptive integral sliding mode control to ensure hybrid synchronization and estimation of uncertain terms. To applythe adaptive ISMC, firstly the error system is converted to a unique system consisting of a nominal part along with the unknown terms whichare computed adaptively. The stabilizing controller incorporating nominal control and compensator control is designed for the error system. Thecompensator controller, as well as the adopted laws, are designed to get the first derivative of the Lyapunov equation strictly negative. To givean illustration, the proposed technique is applied to 4-coupled motor systems yielding the convergence of error dynamics to zero, estimation ofuncertain parameters, and hybrid synchronization of system states. The usefulness of the proposed method has also been tested through computersimulations and found to be valid.

Key words: chaotic system, hybrid synchronization (HS), complex chaotic permanent magnet synchronous motor, adaptive integral slidingmode control (ISMC), Lyapunov function.

1. Introduction

Since the pioneering work by A.C. Fowler et al. [1], com-plex chaotic systems have become an interesting field of re-search over the last few decades, especially synchronizationof complex natured chaotic systems have attracted the atten-tion of many researchers. Complex systems have a broad rangeof applications in industrial areas and it is very important tounderstand numerous physical systems like a chaotic com-plex system. Synchronization of complex chaotic systems is afascinating subject, especially in communications [2–5]. Syn-chronization methods used for simple chaotic systems are ex-tended for complex systems like lag synchronization [6], anti-synchronization [7], adaptive anti-synchronization for unknownparameters [8], hybrid synchronization (HS) and parameteridentification of chaotic systems coupled in ring topology [9]and projective synchronization [10]. In addition, there are someother techniques reported in the literature [11–17], which weredesigned particularly for complex chaotic systems.

All the techniques mentioned above are mainly designed forsynchronization of two or more [18] complex systems, whereone is usually the drive system and the other is the responsesystem. The purpose of these kinds of techniques was that thestates of the response system follow the trajectories of drive

∗e-mail: [email protected]

Manuscript submitted 20XX-XX-XX, initially accepted for publication20XX-XX-XX, published in June 2021.

systems. Instead of using simple techniques, the synchroniza-tion technique for multiple coupled complex chaotic systemscan improve the protection of message signals in secure com-munication and it also has a bright future in the communicationfield. Consequently, many researchers are attracted to this fieldof research and are making their efforts to analyze multiple cou-pled systems. Wang et al. [19] evaluated adaptive combinationsynchronization of complex and real dynamical systems. Zhouet al. [20] investigated adaptive synchronization for uncertaincomplex networks.

The hybrid synchronization (HS) of multiple coupledcomplex dynamical systems was reported in [21], wheresynchronization/anti-synchronization are achieved for complexsystems connected in the ring topology and with known param-eters. A ring topology is shown in Fig. 1, where the states offirst system track the states of N-th system, states of the sec-ond follow the states of the first complex chaotic systems andso on, finally the states of the last system follow the states of(N−1)-th complex chaotic system. Synchronization and anti-synchronization for real chaotic systems coupled in the ring

Fig. 1. Ring topology

Bull. Pol. Acad. Sci. Tech. Sci. 69(3) 2021, e137056 1







1. Introduction















1. Introduction















1. Introduction









http://creativecommons.org/licenses/by/4.0/

http://orcid.org/ 0000-0003-4592-8104

mailto:[email protected]

2

N. Siddique and F.U. Rehman

Bull. Pol. Acad. Sci. Tech. Sci. 69(3) 2021, e137056


topology are achieved in [21], but complex systems connectedin the ring topology is a less focused area of research.

Several different methods were reported in the literature[21, 22] for chaotic synchronization. The active control tech-nique is a significant and easy control method because it pro-vides a convenient way to select and implement the controllers.In an active control technique, the controllers are selected tonullify the nonlinear terms, which are present in the system,therefore, chaotic synchronization becomes a linear problem.The direct design control method is investigated in [23]. Mean-while, the sliding mode control (SMC) [24] is a bit difficultapproach. However, it has a lot of advantages, like the fast re-sponse and robustness in opposition to the parameter deviationand external disturbances.

A modification of SMC technique is the integral sliding modecontrol (ISMC) [25] which combines the discontinuous controland nominal control. The main advantage of applying the ISMCis that it eliminates the reaching phase. So, robustness is guar-anteed throughout the system response.

Medium power permanent magnet synchronous motors(PMSM) are very useful in industrial applications due to theirsignificant features like a small size, low cost, and high torque.Especially, the simple structure of PMSM, in which there isno field winding present in the motor, makes it first choice forthe industrial applications. Therefore, a lot of research has beenconducted to investigate control and synchronization of realpermanent magnet synchronous motors [26–28], whereas muchless work is being done for complex variable permanent magnetsynchronous motors [29]. Recently, some new results related tothe control of PMSM were documented in [30–32]. Practically,in PMSM, complex currents and complex voltages are presentin the dynamical model and there is a possibility that one or allthe parameters of the systems are disturbed due to noise. So, itis practically sound to estimate the unknown parameters for HSof PMSM systems. In this research, an effort has been made toreach HS and identification of uncertain parameters for a com-plex chaotic network of PMSM systems connected in the ringtopology.

2. System description

In [33] the mathematical model of a field-oriented PMSM rotorsystem is given as:

diddt

=(−R1id +wiqLq +ud)

Ld,

diqdt

=(−R1iq +widLd −wψr +uq)

Lq,

dwdt

=(npψriq +npidiq(Ld −Lq)−wβ −TL)

J.

(1)

According to this mathematical model (1) the dynamic statevariables iq, id represent currents and w is angular frequency;ud is the direct axis and uq is the quadrature-axis component ofinput stator voltages; TL is the applied load torque; J is the arcticmoment of inertia; R1 represents resistance of stator wingding;

β is the adhesive damping constant; Ld represents direct axis in-ductance and Lq represents quadrature-axis inductance; ψr rep-resents rotor flux and np are the total number of rotor poles.When there is an even air gap, uniform flux distribution, and themotor operates at no-load, the mathematical model of PMSMcan simply be presented as:

x1 = (x2 − x1)(a) ,

x2 = bx1 − x2 − x1x3 ,

x3 = x1x2 − x3 .

(2)

This system has two complex variables x1,x2 and two constantparameters a,b. In [22] another complex model of permanentmagnet synchronous motor is presented as:

x1 = (x2 − x1)(a),

x2 = bx1 − x1x3 − x2 ,

x3 = 0.5(x2x1 + x2x1),

(3)

where x1, x2 are in complex conjugate form with j =√−1.

This system shows chaotic behavior when its constant parame-ters are selected as b = 20,a = 11. Figures 2(a)–2(b), depict thechaotic behavior of this system. There are many properties ofPMSM systems studied in [34]. In this research paper, we areexamining parameter identification and HS of PMSM systemsusing adaptive ISMC.

0

20

10

10 20

20

x3r

10

30

x2r

0

x1r

40

0-10

-10

-20 -20

(a)

-15 -10 -5 0 5 10 15

x1i

-15

-10

-5

0

5

10

15

20

x2i

(b)

Fig. 2. (a) Chaotic behaviour of PMSM system on x1r,x2r,x3r space,(b) Chaotic behaviour of PMSM system on x1i,x2i space

The remaining paper is organized as: in Section 3, HS controlproblem formulation. In Section 4, the proposed control algo-rithm is discussed where as, in Section 5, HS for PMSM sys-tems is discussed. In Section 6, simulation results are discussedand in Section 7, the paper is concluded.

3. HS control problem formulation

In general, N complex chaotic systems connected in a ringtopology can be configured as:

x1 = f1(x1)+F1(x1)θ1 +D1(xN − x1),

x2 = f2(x2)+F2(x2)θ2 +D2(x1 − x2),

...xN = fN(xN)+FN(xN)θN +DN(xN−1 − xN),

(4)

2 Bull. Pol. Acad. Sci. Tech. Sci. 69(3) 2021, e137056








diddt


Ld,

diqdt


Lq,

dwdt


J.

(1)



x1 = (x2 − x1)(a) ,

x2 = bx1 − x2 − x1x3 ,

x3 = x1x2 − x3 .

(2)


x1 = (x2 − x1)(a),

x2 = bx1 − x1x3 − x2 ,

x3 = 0.5(x2x1 + x2x1),

(3)



0

20

10

10 20

20

x3r

10

30

x2r

0

x1r

40

0-10

-10

-20 -20

(a)

-15 -10 -5 0 5 10 15

x1i

-15

-10

-5

0

5

10

15

20

x2i

(b)





x1 = f1(x1)+F1(x1)θ1 +D1(xN − x1),

x2 = f2(x2)+F2(x2)θ2 +D2(x1 − x2),


(4)









diddt


Ld,

diqdt


Lq,

dwdt


J.

(1)



x1 = (x2 − x1)(a) ,

x2 = bx1 − x2 − x1x3 ,

x3 = x1x2 − x3 .

(2)


x1 = (x2 − x1)(a),

x2 = bx1 − x1x3 − x2 ,

x3 = 0.5(x2x1 + x2x1),

(3)



0

20

10

10 20

20

x3r

10

30

x2r

0

x1r

40

0-10

-10

-20 -20

(a)

-15 -10 -5 0 5 10 15

x1i

-15

-10

-5

0

5

10

15

20

x2i

(b)





x1 = f1(x1)+F1(x1)θ1 +D1(xN − x1),

x2 = f2(x2)+F2(x2)θ2 +D2(x1 − x2),


(4)


Hybrid synchronization and parameter estimation of a complex chaotic network . . .

where x1,x2, ...,xN ∈ Cn, are defined as the complex state vec-tors, xi = (xi1,xi2,xi3, ...,xin)

T , xk = xkr + jxki,k = 1,2,3, ...,N,j =

√−1, both subscripts r and i represent real as well as imag-

inary components from the beginning to the end of this paper,fi : Cn → Cn are the continuous nonlinear function, θi ∈ ℜp

are unknown parameters, Fi(xi) ∈ C(n× p) are matrices, Di =diag{di1,di2,di3...,diN}, i = 1,2,3, ...,N are N-dimensional di-agonal matrices, as well as di j ≥ 0 represents connected termsof Di. In Fig. 1, the complex chaotic dynamic systems are con-nected in a ring, in which the dynamic states of the 1st systemcouples the Nth, the 2nd system couples the 1st, so on, and fi-nally, the N-th complex chaotic system couples the (N−1)-th.

The network model presented in (4) is very practical andunique in the sense that it contains unknown constant terms θi.The constant terms of (4) are assumed to be uncertain due tonoise or some other unwanted external disturbances. The un-certain terms will be estimated by the proposed control algo-rithm. In this research, we have utilized this coupling schemeto investigate HS and it can be mathematically represented as:

x1 = f1(x1)+F1(x1)θ1 +D1(xN − x1),

x2 = f2(x2)+F2(x2)θ2 +D2(x1 − x2)+u1 ,

...xN = f(xN)+FN(xN)θN +DN(xN−1 − xN)+uN−1 ,

(5)

where uk = ukr + juki,k = 1,2, ...,N−1 are the complex inputs.The hybrid synchronization for multiple connected system isdefined as:

Definition 1. The chaotic dynamic system (5), we say, thereexists hybrid synchronization conceding that the controllersui, i = 1,2,3, ...,N−1 are selected in such a manner that allthe trajectories x1(t),x2(t), ...,xN(t)) in (5) with either initialconditions (x1(0),x2(0), ...,xN(0)) satisfy: For the errors ei =(ei1,ei2, ...,eiN)

T , we have

limt→∞

ei = limt→∞

xi(t)+qxi+1(t) = 0, i = 1,2,3, ...,N−1.

For the anti-synchronization we choose q = 1 and for completesynchronization q = −1. The problem of hybrid synchroniza-tion can be resolved by designing appropriate controllers ui toget ei = (ei1,ei2, ...,eiN)

T → 0 asymptotically.

4. The proposed control algorithm

For the hybrid synchronization, we define the error vectors as:

e1 = x2 +qx1 ,

e2 = x3 +qx2 ,

...eN−1 = xN +qxN−1 .

(6)

Let θi be the estimate of θi and let θi = θi − θi be the errors inestimating the parameters θi, i = 1,2, · · · ,N, respectively. Thefirst derivative of (6) yields the following:

e1

e2

e3...

eN−1

=

f2(x2)+q f1(x1)+F2(x2)θ2 +qF1(x1)θ1

+D2(x1 − x2)+qD1(xN − x1)

f3(x3)+q f2(x2)+F3(x3)θ3 +qF2(x2)θ2

+D3(x2 − x3)+qD2(x1 − x2)

f4(x4)+q f3(x3)+F4(x4)θ4 +qF3(x3)θ3

+D4(x3 − x4)+qD3(x2 − x3)...

fN(xN)+q fN−1(xN−1)+FN(xN)θN

+qFN−1(xN−1)θN−1 +DN(xN−1 − xN)

+qDN−1(xN−2 − xN−1)

+

F2(x2)θ2 +qF1(x1θ1)

F3(x3)θ3 +qF2(x2θ2)

F4(x4)θ4 +qF3(x3θ3)...

FN(xN)θN

+qFN−1(xN−1θN−1)

+

1 0 0 · · · 0q 1 0 · · · 00 q 1 · · · 0...

......

......

0 0 · · · q 1

u1

u2...

uN−1

. (7)

By choosing:

u1

u2...

uN−1

=

1 0 0 · · · 0q 1 0 · · · 00 q 1 · · · 0...

......

......

0 0 · · · q 1

−1

+

e2

e3...

eN−1

v

−A

(8)

where, v is the new input vector and

A =

f2(x2)+q f1(x1)+F2(x2)θ2 +qF1(x1)θ1

+D2(x1 − x2)+qD1(xN − x1)

f3(x3)+q f2(x2)+F3(x3)θ3 +qF2(x2)θ2

+D3(x2 − x3)+qD2(x1 − x2)

f4(x4)+q f3(x3)+F4(x4)θ4 +qF3(x3)θ3

+D4(x3 − x4)+qD3(x2 − x3)...



+qDN−1(xN−2 − xN−1)

. (9)

and replacing (8) in (7) the error dynamic system presented in(7) becomes as:

e1 = e2 +F2(x2)θ2 +qF1(x1)θ1 ,

e2 = e3 +F3(x3)θ3 +qF2(x2)θ2 ,

e3 = e4 +F4(x4)θ4 +qF3(x3)θ3 ,

...

eN−2 = eN−1+FN−1(xN−1)θN−1+qFN−2(xN−2)θN−2,

eN−1 = v+FN(xN)θN +qFN−1(xN−1)θN−1 .

(10)


3

Hybrid synchronization and parameter estimation of a complex chaotic network of permanent magnet synchronous motors using adaptive...




T , xk = xkr + jxki,k = 1,2,3, ...,N,j =





x1 = f1(x1)+F1(x1)θ1 +D1(xN − x1),

x2 = f2(x2)+F2(x2)θ2 +D2(x1 − x2)+u1 ,


(5)



T , we have

limt→∞

ei = limt→∞

xi(t)+qxi+1(t) = 0, i = 1,2,3, ...,N−1.





e1 = x2 +qx1 ,

e2 = x3 +qx2 ,

...eN−1 = xN +qxN−1 .

(6)


e1

e2

e3...

eN−1

=

f2(x2)+q f1(x1)+F2(x2)θ2 +qF1(x1)θ1

+D2(x1 − x2)+qD1(xN − x1)

f3(x3)+q f2(x2)+F3(x3)θ3 +qF2(x2)θ2

+D3(x2 − x3)+qD2(x1 − x2)

f4(x4)+q f3(x3)+F4(x4)θ4 +qF3(x3)θ3

+D4(x3 − x4)+qD3(x2 − x3)...



+qDN−1(xN−2 − xN−1)

+

F2(x2)θ2 +qF1(x1θ1)

F3(x3)θ3 +qF2(x2θ2)

F4(x4)θ4 +qF3(x3θ3)...

FN(xN)θN


+

1 0 0 · · · 0q 1 0 · · · 00 q 1 · · · 0...

......

......

0 0 · · · q 1

u1

u2...

uN−1

. (7)

By choosing:

u1

u2...

uN−1

=

1 0 0 · · · 0q 1 0 · · · 00 q 1 · · · 0...

......

......

0 0 · · · q 1

−1

+

e2

e3...

eN−1

v

−A

(8)


A =

f2(x2)+q f1(x1)+F2(x2)θ2 +qF1(x1)θ1

+D2(x1 − x2)+qD1(xN − x1)

f3(x3)+q f2(x2)+F3(x3)θ3 +qF2(x2)θ2

+D3(x2 − x3)+qD2(x1 − x2)

f4(x4)+q f3(x3)+F4(x4)θ4 +qF3(x3)θ3

+D4(x3 − x4)+qD3(x2 − x3)...



+qDN−1(xN−2 − xN−1)

. (9)


e1 = e2 +F2(x2)θ2 +qF1(x1)θ1 ,

e2 = e3 +F3(x3)θ3 +qF2(x2)θ2 ,

e3 = e4 +F4(x4)θ4 +qF3(x3)θ3 ,

...



(10)




T , xk = xkr + jxki,k = 1,2,3, ...,N,j =





x1 = f1(x1)+F1(x1)θ1 +D1(xN − x1),

x2 = f2(x2)+F2(x2)θ2 +D2(x1 − x2)+u1 ,


(5)



T , we have

limt→∞

ei = limt→∞

xi(t)+qxi+1(t) = 0, i = 1,2,3, ...,N−1.





e1 = x2 +qx1 ,

e2 = x3 +qx2 ,

...eN−1 = xN +qxN−1 .

(6)


e1

e2

e3...

eN−1

=

f2(x2)+q f1(x1)+F2(x2)θ2 +qF1(x1)θ1

+D2(x1 − x2)+qD1(xN − x1)

f3(x3)+q f2(x2)+F3(x3)θ3 +qF2(x2)θ2

+D3(x2 − x3)+qD2(x1 − x2)

f4(x4)+q f3(x3)+F4(x4)θ4 +qF3(x3)θ3

+D4(x3 − x4)+qD3(x2 − x3)...



+qDN−1(xN−2 − xN−1)

+

F2(x2)θ2 +qF1(x1θ1)

F3(x3)θ3 +qF2(x2θ2)

F4(x4)θ4 +qF3(x3θ3)...

FN(xN)θN


+

1 0 0 · · · 0q 1 0 · · · 00 q 1 · · · 0...

......

......

0 0 · · · q 1

u1

u2...

uN−1

. (7)

By choosing:

u1

u2...

uN−1

=

1 0 0 · · · 0q 1 0 · · · 00 q 1 · · · 0...

......

......

0 0 · · · q 1

−1

+

e2

e3...

eN−1

v

−A

(8)


A =

f2(x2)+q f1(x1)+F2(x2)θ2 +qF1(x1)θ1

+D2(x1 − x2)+qD1(xN − x1)

f3(x3)+q f2(x2)+F3(x3)θ3 +qF2(x2)θ2

+D3(x2 − x3)+qD2(x1 − x2)

f4(x4)+q f3(x3)+F4(x4)θ4 +qF3(x3)θ3

+D4(x3 − x4)+qD3(x2 − x3)...



+qDN−1(xN−2 − xN−1)

. (9)


e1 = e2 +F2(x2)θ2 +qF1(x1)θ1 ,

e2 = e3 +F3(x3)θ3 +qF2(x2)θ2 ,

e3 = e4 +F4(x4)θ4 +qF3(x3)θ3 ,

...



(10)


4




To apply the ISMC, firstly we have to define the nominal systemfor (10):

e1 = e2 ,

e2 = e3 ,

e3 = e4 ,

...eN−2 = eN−1 ,

eN−1 = vo .

(11)

To stabilize the error system in (11), the Hurwitz sliding sur-

face is designed as: σo = (1 +ddt)N−2e1 = e1 + c1e2 + ...+

cN−3eN−1, where the coefficients ci are chosen in such a waythat σo becomes Hurwitz polynomial. The time derivative of theabove sliding surface will be like this σo = e2 + c1e3 + c2e4 +...+ cN−3eN−1 + vo. By choosing vo = −e2 − c1e3 − c2e4 −...− cN−3eN−1 − kσo,k > 0, we have σo =−kσo, consequentlyσo → 0, which gives e1,e2, ...,eN−1 → 0. In consequence sys-tem (11) is asymptotically stable. Moreover, by designing slid-ing surfaces for the above system (10) as: σ = σo + z where zin this equation is an integral parameter which shall be com-puted subsequently. To avert reaching phase set z(0) such thatσ(0) = 0. The time derivative of this sliding manifold will beas following:

σ = σo + z

= e1 + c1e2 + ...+ cN−3eN−2 + eN−1 + v+ z

= e2 +N−1

∑i=3

ci−2ei + v+ z+qF1(x1)θ1

+((1+qc1)F2(x2)θ2

)

+N−4

∑i=1

(ci +qci+1)Fi+2(xi+2)θi+2

+((cN−3 +q)FN−1(xN−1)θN−1

)+FN(xN)θN . (12)

The new input term v in (12) is defined as v = vs +v0, where v0is the nominal input vector and the other term vs is a compen-sator input vector which will be computed later. The Lyapunovstability function for (12) is defined as:

V =12

{σT σ + θ T

1 θ1 + θ T2 θ2

}+

N−4

∑i=1

θ Ti+2θi+2

+ θ TN−1θN−1 + θ T

N θN (13)

by properly defining the adaptive laws θi, θi, i = 1,2,3, ...,N,and computing the compensator input vector vs such that thefist derivative of (13) can be achieved as V < 0.

Theorem 1. For the Lyapunov equation as described in (13) itis possible to get V < 0 conceding that θi, θi, i = 1,2,3, ...,Nand vs are chosen as:

z =−e2 −N−1

∑i=3

ci−2ei − vo,

vs =−kσ − ksign(σ),

˙θ1 =−qFT1 (x1)σ − k1θ1,

˙θ1 =− ˙θ1 ,

˙θ2 =−(1+qc1)FT2 (x2)σ − k2θ2,

˙θ2 =− ˙θ2 ,

˙θi+2 =−(ci +qci+1)FTi+1(xi+2)σ − kN−1 − ki+2θi+2,

˙θi+2 =− ˙θi+2, i = 1, ...,N−4,˙θN−1 =−(cN−3 +q)F6TN−1σ − kN−1θN−1,

˙θN−1 =− ˙θN−1,

˙θN =−FTN (xN)σ − kN θN ,

˙θNc =− ˙θN .

(14)

Proof. Since:

V = σ T σ + θ T1

˙θ1 + θ T2

˙θ2

+N−4

∑i=1

θ Ti+2

˙θi+2 + θ TN−1

˙θN−1 + θ TN

˙θN

= σT

{e2 +

N−1

∑i=3

ci−2ei + vo + vs + z

}+ θ T

1

{˙θ1 +qFFT

1 (x1)σ}

+ θ T2

{˙θ2 +(1+qc1)FT

2 (x2)σ}

+N−4

∑i=1

θ Ti+2

{˙θi+2 +(ci +qci+1)FT

i+2(xi+2)σ}

+ θ TN−1

{˙θN−1 +(cN−3 +q)FT

N−1(xN−1)σ}

+ θ TN

{˙θN +FT

N (xN)σ}. (15)

By replacing (14) in (15) we get:

V =−kσ2 −N

∑i=1

kiθ Ti θi − kσT sign(σ), k > 0. (16)

This shows σ and θi → 0 consequentlyei → 0, i = 1,2,3, ..,N−1.

5. HS of ring-connected complex PMSM systems

In this section, we investigate HS of N-coupled complex PMSMsystems connected in ring topology. Assuming N = 4, the cou-pled complex PMSM systems in a ring topology can be repre-sented as:




e1 = e2 ,

e2 = e3 ,

e3 = e4 ,

...eN−2 = eN−1 ,

eN−1 = vo .

(11)




σ = σo + z

= e1 + c1e2 + ...+ cN−3eN−2 + eN−1 + v+ z

= e2 +N−1

∑i=3


+((1+qc1)F2(x2)θ2

)

+N−4

∑i=1


+((cN−3 +q)FN−1(xN−1)θN−1

)+FN(xN)θN . (12)


V =12

{σT σ + θ T

1 θ1 + θ T2 θ2

}+

N−4

∑i=1

θ Ti+2θi+2

+ θ TN−1θN−1 + θ T

N θN (13)



z =−e2 −N−1

∑i=3

ci−2ei − vo,


˙θ1 =−qFT1 (x1)σ − k1θ1,

˙θ1 =− ˙θ1 ,

˙θ2 =−(1+qc1)FT2 (x2)σ − k2θ2,

˙θ2 =− ˙θ2 ,



˙θN−1 =− ˙θN−1,


˙θNc =− ˙θN .

(14)

Proof. Since:

V = σ T σ + θ T1

˙θ1 + θ T2

˙θ2

+N−4

∑i=1

θ Ti+2

˙θi+2 + θ TN−1

˙θN−1 + θ TN

˙θN

= σT

{e2 +

N−1

∑i=3


}+ θ T

1

{˙θ1 +qFFT

1 (x1)σ}

+ θ T2

{˙θ2 +(1+qc1)FT

2 (x2)σ}

+N−4

∑i=1

θ Ti+2


i+2(xi+2)σ}

+ θ TN−1

{˙θN−1 +(cN−3 +q)FT

N−1(xN−1)σ}

+ θ TN

{˙θN +FT

N (xN)σ}. (15)


V =−kσ2 −N

∑i=1








e1 = e2 ,

e2 = e3 ,

e3 = e4 ,

...eN−2 = eN−1 ,

eN−1 = vo .

(11)




σ = σo + z

= e1 + c1e2 + ...+ cN−3eN−2 + eN−1 + v+ z

= e2 +N−1

∑i=3


+((1+qc1)F2(x2)θ2

)

+N−4

∑i=1


+((cN−3 +q)FN−1(xN−1)θN−1

)+FN(xN)θN . (12)


V =12

{σT σ + θ T

1 θ1 + θ T2 θ2

}+

N−4

∑i=1

θ Ti+2θi+2

+ θ TN−1θN−1 + θ T

N θN (13)



z =−e2 −N−1

∑i=3

ci−2ei − vo,


˙θ1 =−qFT1 (x1)σ − k1θ1,

˙θ1 =− ˙θ1 ,

˙θ2 =−(1+qc1)FT2 (x2)σ − k2θ2,

˙θ2 =− ˙θ2 ,



˙θN−1 =− ˙θN−1,


˙θNc =− ˙θN .

(14)

Proof. Since:

V = σ T σ + θ T1

˙θ1 + θ T2

˙θ2

+N−4

∑i=1

θ Ti+2

˙θi+2 + θ TN−1

˙θN−1 + θ TN

˙θN

= σT

{e2 +

N−1

∑i=3


}+ θ T

1

{˙θ1 +qFFT

1 (x1)σ}

+ θ T2

{˙θ2 +(1+qc1)FT

2 (x2)σ}

+N−4

∑i=1

θ Ti+2


i+2(xi+2)σ}

+ θ TN−1

{˙θN−1 +(cN−3 +q)FT

N−1(xN−1)σ}

+ θ TN

{˙θN +FT

N (xN)σ}. (15)


V =−kσ2 −N

∑i=1








e1 = e2 ,

e2 = e3 ,

e3 = e4 ,

...eN−2 = eN−1 ,

eN−1 = vo .

(11)




σ = σo + z

= e1 + c1e2 + ...+ cN−3eN−2 + eN−1 + v+ z

= e2 +N−1

∑i=3


+((1+qc1)F2(x2)θ2

)

+N−4

∑i=1


+((cN−3 +q)FN−1(xN−1)θN−1

)+FN(xN)θN . (12)


V =12

{σT σ + θ T

1 θ1 + θ T2 θ2

}+

N−4

∑i=1

θ Ti+2θi+2

+ θ TN−1θN−1 + θ T

N θN (13)



z =−e2 −N−1

∑i=3

ci−2ei − vo,


˙θ1 =−qFT1 (x1)σ − k1θ1,

˙θ1 =− ˙θ1 ,

˙θ2 =−(1+qc1)FT2 (x2)σ − k2θ2,

˙θ2 =− ˙θ2 ,



˙θN−1 =− ˙θN−1,


˙θNc =− ˙θN .

(14)

Proof. Since:

V = σ T σ + θ T1

˙θ1 + θ T2

˙θ2

+N−4

∑i=1

θ Ti+2

˙θi+2 + θ TN−1

˙θN−1 + θ TN

˙θN

= σT

{e2 +

N−1

∑i=3


}+ θ T

1

{˙θ1 +qFFT

1 (x1)σ}

+ θ T2

{˙θ2 +(1+qc1)FT

2 (x2)σ}

+N−4

∑i=1

θ Ti+2


i+2(xi+2)σ}

+ θ TN−1

{˙θN−1 +(cN−3 +q)FT

N−1(xN−1)σ}

+ θ TN

{˙θN +FT

N (xN)σ}. (15)


V =−kσ2 −N

∑i=1








e1 = e2 ,

e2 = e3 ,

e3 = e4 ,

...eN−2 = eN−1 ,

eN−1 = vo .

(11)




σ = σo + z

= e1 + c1e2 + ...+ cN−3eN−2 + eN−1 + v+ z

= e2 +N−1

∑i=3


+((1+qc1)F2(x2)θ2

)

+N−4

∑i=1


+((cN−3 +q)FN−1(xN−1)θN−1

)+FN(xN)θN . (12)


V =12

{σT σ + θ T

1 θ1 + θ T2 θ2

}+

N−4

∑i=1

θ Ti+2θi+2

+ θ TN−1θN−1 + θ T

N θN (13)



z =−e2 −N−1

∑i=3

ci−2ei − vo,


˙θ1 =−qFT1 (x1)σ − k1θ1,

˙θ1 =− ˙θ1 ,

˙θ2 =−(1+qc1)FT2 (x2)σ − k2θ2,

˙θ2 =− ˙θ2 ,



˙θN−1 =− ˙θN−1,


˙θNc =− ˙θN .

(14)

Proof. Since:

V = σ T σ + θ T1

˙θ1 + θ T2

˙θ2

+N−4

∑i=1

θ Ti+2

˙θi+2 + θ TN−1

˙θN−1 + θ TN

˙θN

= σT

{e2 +

N−1

∑i=3


}+ θ T

1

{˙θ1 +qFFT

1 (x1)σ}

+ θ T2

{˙θ2 +(1+qc1)FT

2 (x2)σ}

+N−4

∑i=1

θ Ti+2


i+2(xi+2)σ}

+ θ TN−1

{˙θN−1 +(cN−3 +q)FT

N−1(xN−1)σ}

+ θ TN

{˙θN +FT

N (xN)σ}. (15)


V =−kσ2 −N

∑i=1








e1 = e2 ,

e2 = e3 ,

e3 = e4 ,

...eN−2 = eN−1 ,

eN−1 = vo .

(11)




σ = σo + z

= e1 + c1e2 + ...+ cN−3eN−2 + eN−1 + v+ z

= e2 +N−1

∑i=3


+((1+qc1)F2(x2)θ2

)

+N−4

∑i=1


+((cN−3 +q)FN−1(xN−1)θN−1

)+FN(xN)θN . (12)


V =12

{σT σ + θ T

1 θ1 + θ T2 θ2

}+

N−4

∑i=1

θ Ti+2θi+2

+ θ TN−1θN−1 + θ T

N θN (13)



z =−e2 −N−1

∑i=3

ci−2ei − vo,


˙θ1 =−qFT1 (x1)σ − k1θ1,

˙θ1 =− ˙θ1 ,

˙θ2 =−(1+qc1)FT2 (x2)σ − k2θ2,

˙θ2 =− ˙θ2 ,



˙θN−1 =− ˙θN−1,


˙θNc =− ˙θN .

(14)

Proof. Since:

V = σ T σ + θ T1

˙θ1 + θ T2

˙θ2

+N−4

∑i=1

θ Ti+2

˙θi+2 + θ TN−1

˙θN−1 + θ TN

˙θN

= σT

{e2 +

N−1

∑i=3


}+ θ T

1

{˙θ1 +qFFT

1 (x1)σ}

+ θ T2

{˙θ2 +(1+qc1)FT

2 (x2)σ}

+N−4

∑i=1

θ Ti+2


i+2(xi+2)σ}

+ θ TN−1

{˙θN−1 +(cN−3 +q)FT

N−1(xN−1)σ}

+ θ TN

{˙θN +FT

N (xN)σ}. (15)


V =−kσ2 −N

∑i=1








e1 = e2 ,

e2 = e3 ,

e3 = e4 ,

...eN−2 = eN−1 ,

eN−1 = vo .

(11)




σ = σo + z

= e1 + c1e2 + ...+ cN−3eN−2 + eN−1 + v+ z

= e2 +N−1

∑i=3


+((1+qc1)F2(x2)θ2

)

+N−4

∑i=1


+((cN−3 +q)FN−1(xN−1)θN−1

)+FN(xN)θN . (12)


V =12

{σT σ + θ T

1 θ1 + θ T2 θ2

}+

N−4

∑i=1

θ Ti+2θi+2

+ θ TN−1θN−1 + θ T

N θN (13)



z =−e2 −N−1

∑i=3

ci−2ei − vo,


˙θ1 =−qFT1 (x1)σ − k1θ1,

˙θ1 =− ˙θ1 ,

˙θ2 =−(1+qc1)FT2 (x2)σ − k2θ2,

˙θ2 =− ˙θ2 ,



˙θN−1 =− ˙θN−1,


˙θNc =− ˙θN .

(14)

Proof. Since:

V = σ T σ + θ T1

˙θ1 + θ T2

˙θ2

+N−4

∑i=1

θ Ti+2

˙θi+2 + θ TN−1

˙θN−1 + θ TN

˙θN

= σT

{e2 +

N−1

∑i=3


}+ θ T

1

{˙θ1 +qFFT

1 (x1)σ}

+ θ T2

{˙θ2 +(1+qc1)FT

2 (x2)σ}

+N−4

∑i=1

θ Ti+2


i+2(xi+2)σ}

+ θ TN−1

{˙θN−1 +(cN−3 +q)FT

N−1(xN−1)σ}

+ θ TN

{˙θN +FT

N (xN)σ}. (15)


V =−kσ2 −N

∑i=1







x11 = a(x12 − x11)+d11(x41 − x11),

x12 = bx11 − x12 − x11x13 +d12(x42 − x12),

x13 = 0.5(x11x12 + x11x12)− x13 +d13(x43 − x13),

x21 = a(x22 − x21)+d21(x11 − x21)+µ11 ,

x22 = bx21 − x22 − x21x23 +d22(x12 − x22)+µ12 ,

x23 = 0.5(x21x22 + x21x22)− x23 +d23(x13 − x23)+µ13 ,

x31 = a(x32 − x31)+d31(x21 − x31)+µ21 ,

x32 = bx31 − x32 − x31x33 +d32(x22 − x32)+µ22 ,

x33 = 0.5(x31x32 + x31x32)− x33 +d33(x23 − x33)+µ23 ,

x41 = a(x42 − x41)+d41(x31 − x41)+µ31 ,

x42 = bx41 − x42 − x41x43 +d42(x32 − x42)+µ32 ,

x43 = 0.5(x41x42 + x41x42)− x43 +d33(x33 − x43)+µ33 ,

(17)

where, xk1 = xk1r + jxk1i, xk2 = xk2r + jxk2i are complex andxk3 = xk3r are real, xk1, xk2 denote the complex conjugate vari-ables of xk1, xk2, k = 1,2,3,4 and a, b are unknown real terms.

In (17), if the paramaters a and b are uncertain and their esti-mates are a and b respectively, the error in estimation of uncer-tain parameters can be described as: a = a− a, b = b− b, then(17) can be written in vector form as:

x1r = f1r +F1rθ +F1rθ +D1G1r ,

x1i = f1i +F1iθ +F1iθ +D1G1i ,

x2r = f2r +F2rθ +F2rθ +D2G2r +µ1r ,

x2i = f2i +F2iθ +F2iθ +D2G2i +µ1i ,





(18)

where

fkr =

0−xk2r − xk1rxk3

0.5(xk1rxk2r + xk1ixk2i)− xk3

,

fki =


0

, k = 1,2,3,4;

Fkr =

xk2r − xk1r 00 xk1r

0 0

,

Fki =

xk2i − xk1i 00 xk1i

0 0

,

(19)

G1r =

x41r − x11r

x42r − x12r

x4r − x13

, G1i =

x41i − x11i

x42i − x12i

0

,

G2r =

x11r − x21r

x12r − x22r

x13 − x43

, G2i =

x11i − x21i

x12i − x22i

0

,

G3r =

x21r − x31r

x22r − x32r

x23 − x33

, G3i =

x21i − x31i

x22i − x32i

0

,

G4r =

x31r − x41r

x32r − x42r

x33 − x43

, G4i =

x31i − x41i

x32i − x42i

x33 − x43

,

ulr =

ul1r

ul2r

ulr

, uli =

ul1i

ul2i

0

, l = 1,2,3,

θ =

[ab

], θ =

[ab

].

(19)

Defining the error as: ek = ekr + jeki = xk+1 + qxk =x(k+1)r +qxkr + jx(k+1)i +qxki, this gives ekr = x(k+1)r + qxkrand, eki = x(k+1)i +qxki, k = 1,2,3. The error dynamics of thissystem becomes as:

e1r

e2r

e3r

=

( f2r +q f1r)+(F2r +qF1rθ)+D2G2r +qD1G1r)



+

F2r +qF1r

F3r +qF2r

F4r +qF3r

+

I 0 0−qI I 0

0 −qI I

µ1r

µ2r

µ3r

,

e1i

e2i

e3i

=

( f2i +q f1i)+(F2i +qF1iθ)+D2G2i +qD1G1i)



+

F2i +qF1i

F3i +qF2i

F4i +qF3i

+

I 0 0−qI I 0

0 −qI I

µ1i

µ2i

µ3i

(20)

by choosing,

µ1r

µ2r

µ3r

=

I 0 0−qI I 0

0 −qI I

−1

(Fqr1)(Fqr2)(Fqr3)

+

e2r

e3r

Vr

,

µ1i

µ2i

µ3i

=

I 0 0−qI I 0

0 −qI I

−1

(Fqi1)(Fqi2)(Fqi3)

+

e2i

e3i

Vi

,

(21)

whereFqr1 = ( f2r +q f1r)+(F2r +qF1rθ)+D2G2r +qD1G1r Fqr2 =


5




x11 = a(x12 − x11)+d11(x41 − x11),

x12 = bx11 − x12 − x11x13 +d12(x42 − x12),

x13 = 0.5(x11x12 + x11x12)− x13 +d13(x43 − x13),

x21 = a(x22 − x21)+d21(x11 − x21)+µ11 ,

x22 = bx21 − x22 − x21x23 +d22(x12 − x22)+µ12 ,

x23 = 0.5(x21x22 + x21x22)− x23 +d23(x13 − x23)+µ13 ,

x31 = a(x32 − x31)+d31(x21 − x31)+µ21 ,

x32 = bx31 − x32 − x31x33 +d32(x22 − x32)+µ22 ,

x33 = 0.5(x31x32 + x31x32)− x33 +d33(x23 − x33)+µ23 ,

x41 = a(x42 − x41)+d41(x31 − x41)+µ31 ,

x42 = bx41 − x42 − x41x43 +d42(x32 − x42)+µ32 ,

x43 = 0.5(x41x42 + x41x42)− x43 +d33(x33 − x43)+µ33 ,

(17)

where, xk1 = xk1r + jxk1i, xk2 = xk2r + jxk2i are complex andxk3 = xk3r are real, xk1, xk2 denote the complex conjugate vari-ables of xk1, xk2, k = 1,2,3,4 and a, b are unknown real terms.

In (17), if the paramaters a and b are uncertain and their esti-mates are a and b respectively, the error in estimation of uncer-tain parameters can be described as: a = a− a, b = b− b, then(17) can be written in vector form as:

x1r = f1r +F1rθ +F1rθ +D1G1r ,

x1i = f1i +F1iθ +F1iθ +D1G1i ,







(18)

where

fkr =


0.5(xk1rxk2r + xk1ixk2i)− xk3

,

fki =


0

, k = 1,2,3,4;

Fkr =

xk2r − xk1r 00 xk1r

0 0

,

Fki =

xk2i − xk1i 00 xk1i

0 0

,

(19)

G1r =

x41r − x11r

x42r − x12r

x4r − x13

, G1i =

x41i − x11i

x42i − x12i

0

,

G2r =

x11r − x21r

x12r − x22r

x13 − x43

, G2i =

x11i − x21i

x12i − x22i

0

,

G3r =

x21r − x31r

x22r − x32r

x23 − x33

, G3i =

x21i − x31i

x22i − x32i

0

,

G4r =

x31r − x41r

x32r − x42r

x33 − x43

, G4i =

x31i − x41i

x32i − x42i

x33 − x43

,

ulr =

ul1r

ul2r

ulr

, uli =

ul1i

ul2i

0

, l = 1,2,3,

θ =

[ab

], θ =

[ab

].

(19)

Defining the error as: ek = ekr + jeki = xk+1 + qxk =x(k+1)r +qxkr + jx(k+1)i +qxki, this gives ekr = x(k+1)r + qxkrand, eki = x(k+1)i +qxki, k = 1,2,3. The error dynamics of thissystem becomes as:

e1r

e2r

e3r

=




+

F2r +qF1r

F3r +qF2r

F4r +qF3r

+

I 0 0−qI I 0

0 −qI I

µ1r

µ2r

µ3r

,

e1i

e2i

e3i

=




+

F2i +qF1i

F3i +qF2i

F4i +qF3i

+

I 0 0−qI I 0

0 −qI I

µ1i

µ2i

µ3i

(20)

by choosing,

µ1r

µ2r

µ3r

=

I 0 0−qI I 0

0 −qI I

−1

(Fqr1)(Fqr2)(Fqr3)

+

e2r

e3r

Vr

,

µ1i

µ2i

µ3i

=

I 0 0−qI I 0

0 −qI I

−1

(Fqi1)(Fqi2)(Fqi3)

+

e2i

e3i

Vi

,

(21)

whereFqr1 = ( f2r +q f1r)+(F2r +qF1rθ)+D2G2r +qD1G1r Fqr2 =


6




( f3r + q f2r)+ (F3r + qF2rθ)+D3G3r + qD2G2r Fqr3 = ( f4r +

q f3r)+(F4r +qF3rθ)+D4G4r +qD3G3rFqi1 = ( f2i + q f1i) + (F2i + qF1iθ) + D2G2i + qD1G1i Fqi2 =

( f3i + q f2i) + (F3i + qF2iθ) + D3G3i + qD2G2i Fqi3 = ( f4i +

q f3i)+(F4i −+F3iθ)+D4G4i +qD3G3i.The system (20) becomes:

e1r

e2r

e3r

=

e2r

e3r

Vr

+

F2r +qF1r

F3r +qF2r

F4r +qF3r

θ ,

e1i

e2i

e3i

=

e2i

e3i

Vi

+

F2i +qF1i

F3i +qF2i

F4i +qF3i

θ

(22)

taking the nominal system for (22) as:

e1r

e2r

e3r

=

e2r

e3r

Vr0

,

e1i

e2i

e3i

=

e2i

e3i

Vi0

(23)

and defining the sliding surface for nominal system (23) as:

σ0r = e1r +2e2r + e3r ,

σ0i = e1i +2e2i + e3i(24)

then the first derivative of (24) becomes as:

σ0r = e1r +2e2r + e3r = e2r +2e3r + v0r ,

σ0i = e1i +2e2i + e3i = e2i +2e3i + v0i(25)

by choosing,

v0r =−e2r −2e3r − k1sign(σ0r), k1 > 0,

v0i =−e2i −2e3i − k2sign(σ0i), k2 > 0(26)

we have,

σ0r =−k1σ0r − k1sign(σ0r),

σ0i =−k2σ0i − k2sign(σ0i).(27)

In consequence, the nominal system (24) is asymptotically sta-ble. The sliding surface for error dynamics presented in (22) isdefined as:

σr = σ0r + zr = e1r +2e2r + e3r + zr ,

σi = σ0i + zi = e1i +2e2i + e3i + zi ,(28)

where zr, zi are some itegral terms computed later. Now to avertthe reaching phase, take up zr(0), zi(0) such that σr(0) = 0,σi(0) = 0. Choosing vr = v0r + vsr, vi = voi + vi, where, v0r,v0iare the nominal inputs and vsr,vsi are compensator terms which

will be computed later. The first derivative of (28) can be de-rived as:

σr = σ0r + zr = e1r +2e2r + e3r + zr

= e2r +(F2r +qF1r)θ +2e3r +2(F3r +qF2r)θ +(F4r

+qF3r)θ + v0r + vsr + zr

σi = σ0i + zi = e1i +2e2i + e3i + zi

= e2i +(F2i +qF1i)θ +2e3i +2(F3i +qF2i)θ +(F4i

+qF3i)θ + v0i + vsi + zi

(29)

The Lyapunov stability function for (29) can be defined as:

V =12

σTr σr +

12

σTi σi +

12

θ T θ . By properly defining the adap-

tive laws θ , θ and computing vsr, vsi, it is possible to get firstderivative of Lyapunov stability function as V < 0.

Theorem 2. For the Lyapunov equation V =12

σTr σr +

12

σTi σi +

12

θ T θ it is possible to get V < 0 conceding that the

adaptive laws ˙θ , ˙θ and the values of vsr, vi are chosen as:

zr =−e2r −2e3r − v0r,vsr =−k−3σr − k3sign(σr)

zi =−e2i −2e3i − v0r,vsi =−k−4σi − k4sign(σi)

˙θ =−σ Tr {(F2r +qF1r)

T +2(F3r +qF2r)T +(F4r +qF3r)

T}−σT

i {(F2i +qF1i)T +2(F3i +qF2i)

T +(F4i +qF3i)T}−K5θ

˙θ =− ˙θ ,ki > 0, i = 1, · · · ,5.(30)

Proof. Since:

V = σ Tr σr +σT

i σi + θ T ˙θ ,

V = σ Tr {e2r +(F2r +qF1r)θ +2e3r +2(F3r)+qF2r)θ

+(F4r +qF3r)θ + v0r + vsr + zr}+σTi {e2i +(F2i +qF1i)θ

+2e3i +2(F3i +qF2i)θ +(F4i +qF3i)θ + v0i + vsi + z}

+ θ T ˙θ (31)

V = σTr {e2r +2e3r + v0r + vsr + zr}+σT

i {e2i +2e3i + v0i + vsi

+ zi}+ θ T [ ˙θ +σTr {(F2r +qF1r)

T +2(F3r +qF2r)T +(F4r

+qF3r)T}+σT

i {(F2i +qF1i)T +2(F3i +qF2i)

T

+(F4i +qFT3i )}].

By replacing the values of adaptive laws ˙θ , ˙θ and the valuesof vsr, vi proposed in (30), the above system (31) becomes:

V =−k1σTr σr − k2σT

i σi − k2θ T θ − k3σTr sign(σr)

− k4σTi sign(σi)< 0. (32)

This shows that the designed sliding surfaces σr, σi and adap-tive laws θ → 0; therefore, ekr,eki → 0, k = 1,2,3,4. The con-vergence of error system to zero ensures HS of coupled com-plex chaotic permanent magnet motors systems connected inring topology.



6. Simulation results and discussion

Simulation results were presented by taking the following ini-tial conditions, x1(0) = (1+2 j,3+6 j,5)T , x2(0) = (4−2 j,1+2 j,8)T , x3(0) = (−3+4 j,−2+5 j,−1)T , x4(0) = (5−5 j,4−2 j,3)T . Figure 3(a) displays the concurrence of both the realand imaginary parts of synchronization error dynamics e11r,e12r, e13r, e11i, e12i to zero, Fig. 3(b) displays the concurrenceof real as well as imaginary parts of synchronization error dy-namics e21r, e22r, e23r, e21i, e22i to zero, Fig. 3(c) displays theconcurrence of real as well as imaginary parts of synchroniza-tion error dynamics e31r, e32r, e33r, e31i, e32i to zero. Figure 4depicts that dynamic states of all the systems are synchronized.Figure 5(a) shows the convergence of anti-synchronization er-ror dynamics e11r, e12r, e13r, e11i, e12i to zero, Fig. 5(b) showsthe convergence of anti-synchronization error dynamics e21r,e22r, e23r, e21i, e22i to zero, Fig. 5(c) shows the convergence ofanti-synchronization error dynamics e31r, e32r, e33r, e31i, e32i tozero. Anti-synchronization phenomena of all the systems con-nected in the ring connection is depicted in Fig. 6(a–e). Fromthis, it is very clear that the dynamic states of second systemsare anti-synchronized with the dynamic states of the first sys-tem. The states of the third system are anti-synchronized withthe second system but these are synchronized with the first sys-tem due to the connection arrangement. Similarly, the states ofthe fourth system are anti-synchronized with the third systembut are synchronized with the second system. Figure 6(f) showsthat the estimated parameters a, b converge to their true valuesa, b respectively.

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-10

-5

0

5

10

15

20

25

30

e11r,e

12r,e

13r,e

11i,e

12i

e11r

e11i

e12r

e12i

e13

(a)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-30

-25

-20

-15

-10

-5

0

5

10

e21r,e

21i,e

22r,e

22i,e

23

e21r

e21i

e22r

e22i

e23

(b)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-20

-15

-10

-5

0

5

e31r,e

31i,e

32r,e

32i,e

33

e31r

e31i

e32r

e32i

e33

(c)

Fig. 3. Convergence of real as well as imaginary parts of syn-chronization error dynamics e11r,e12r,e13r,e11i,e12i, (b) Conver-gence of real as well as imaginary parts of synchronization errorse21r,e22r,e23r,e21i,e22i, (c) Convergence of real as well as imaginary

parts of synchronization errors e31r,e32r,e33r,e31i,e32i

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-10

-5

0

5

10

15

20

x1

1r,x

21

r,x3

1r,x

41

r

x11r

x21r

x31r

x41r

(a)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-10

-5

0

5

10

15

20

25

30

x1

1i,x

21

i,x3

1i,x

41

i

x11i

x21i

x31i

x41i

(b)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-20

-10

0

10

20

30

40

x1

2r,x

22

r,x3

2r,x

42

r

x12r

x22r

x32r

x42r

(c)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-10

-5

0

5

10

15

20

25

30

35

40

x1

2i,x

22

i,x3

2i,x

42

i

x12i

x22i

x32i

x42i

(d)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

0

5

10

15

20

25

30

35

40

45

50

x1

3,x

23,x

33,x

43

x13

x23

x33

x43

(e)

Fig. 4. (a) Synchronization of real parts of system statesx11r,x21r,x31r,x41r, (b) Synchronization of imaginary parts of systemstates x11i,x21i,x31i,x41i, (c) Synchronization of real parts of systemstates x12r,x22r,x32r,x42r,(d) Synchronization of imaginary parts ofsystem states x12i,x22i,x32i,x42i,(e) Synchronization of system states

x13,x23,x33,x43

0 5 10 15 20 25

Time(s)

0

10

20

30

40

50

60

e1

1r,e

12

r,e1

3r,e

11

i,e1

2i

e11r

e12r

e13r

e11i

e12i

(a)

0 5 10 15 20 25

Time(s)

-40

-30

-20

-10

0

10

20

30

40

50

60

e2

1r,e

22

r,e2

3r,e

21

i,e2

2i

e21r

e22r

e23r

e21i

e22i

(b)

0 5 10 15 20 25

Time(s)

-80

-60

-40

-20

0

20

40

e3

1r,e

32

r,e3

3r,e

31

i,e3

2i

e31r

e32r

e33r

e31i

e32i

(c)

Fig. 5. (a) Convergence of real and imaginary parts of anti-synchronization errors e11r,e12r,e13r,e11i,e12i, (b) Convergenceof real and imaginary parts of anti-synchronization errorse21r,e22r,e23r,e21i,e22i, (c) Convergence of real and imaginary

parts of anti-synchronization errors e31r,e32r,e33r,e31i,e32i


7




6. Simulation results and discussion

Simulation results were presented by taking the following ini-tial conditions, x1(0) = (1+2 j,3+6 j,5)T , x2(0) = (4−2 j,1+2 j,8)T , x3(0) = (−3+4 j,−2+5 j,−1)T , x4(0) = (5−5 j,4−2 j,3)T . Figure 3(a) displays the concurrence of both the realand imaginary parts of synchronization error dynamics e11r,e12r, e13r, e11i, e12i to zero, Fig. 3(b) displays the concurrenceof real as well as imaginary parts of synchronization error dy-namics e21r, e22r, e23r, e21i, e22i to zero, Fig. 3(c) displays theconcurrence of real as well as imaginary parts of synchroniza-tion error dynamics e31r, e32r, e33r, e31i, e32i to zero. Figure 4depicts that dynamic states of all the systems are synchronized.Figure 5(a) shows the convergence of anti-synchronization er-ror dynamics e11r, e12r, e13r, e11i, e12i to zero, Fig. 5(b) showsthe convergence of anti-synchronization error dynamics e21r,e22r, e23r, e21i, e22i to zero, Fig. 5(c) shows the convergence ofanti-synchronization error dynamics e31r, e32r, e33r, e31i, e32i tozero. Anti-synchronization phenomena of all the systems con-nected in the ring connection is depicted in Fig. 6(a–e). Fromthis, it is very clear that the dynamic states of second systemsare anti-synchronized with the dynamic states of the first sys-tem. The states of the third system are anti-synchronized withthe second system but these are synchronized with the first sys-tem due to the connection arrangement. Similarly, the states ofthe fourth system are anti-synchronized with the third systembut are synchronized with the second system. Figure 6(f) showsthat the estimated parameters a, b converge to their true valuesa, b respectively.

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-10

-5

0

5

10

15

20

25

30

e11r,e

12r,e

13r,e

11i,e

12i

e11r

e11i

e12r

e12i

e13

(a)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-30

-25

-20

-15

-10

-5

0

5

10

e21r,e

21i,e

22r,e

22i,e

23

e21r

e21i

e22r

e22i

e23

(b)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-20

-15

-10

-5

0

5

e31r,e

31i,e

32r,e

32i,e

33

e31r

e31i

e32r

e32i

e33

(c)

Fig. 3. Convergence of real as well as imaginary parts of syn-chronization error dynamics e11r,e12r,e13r,e11i,e12i, (b) Conver-gence of real as well as imaginary parts of synchronization errorse21r,e22r,e23r,e21i,e22i, (c) Convergence of real as well as imaginary

parts of synchronization errors e31r,e32r,e33r,e31i,e32i

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-10

-5

0

5

10

15

20

x1

1r,x

21

r,x3

1r,x

41

r

x11r

x21r

x31r

x41r

(a)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-10

-5

0

5

10

15

20

25

30

x1

1i,x

21

i,x3

1i,x

41

i

x11i

x21i

x31i

x41i

(b)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-20

-10

0

10

20

30

40

x1

2r,x

22

r,x3

2r,x

42

r

x12r

x22r

x32r

x42r

(c)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

-10

-5

0

5

10

15

20

25

30

35

40

x1

2i,x

22

i,x3

2i,x

42

i

x12i

x22i

x32i

x42i

(d)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

0

5

10

15

20

25

30

35

40

45

50

x1

3,x

23,x

33,x

43

x13

x23

x33

x43

(e)

Fig. 4. (a) Synchronization of real parts of system statesx11r,x21r,x31r,x41r, (b) Synchronization of imaginary parts of systemstates x11i,x21i,x31i,x41i, (c) Synchronization of real parts of systemstates x12r,x22r,x32r,x42r,(d) Synchronization of imaginary parts ofsystem states x12i,x22i,x32i,x42i,(e) Synchronization of system states

x13,x23,x33,x43

0 5 10 15 20 25

Time(s)

0

10

20

30

40

50

60

e1

1r,e

12

r,e1

3r,e

11

i,e1

2i

e11r

e12r

e13r

e11i

e12i

(a)

0 5 10 15 20 25

Time(s)

-40

-30

-20

-10

0

10

20

30

40

50

60

e2

1r,e

22

r,e2

3r,e

21

i,e2

2i

e21r

e22r

e23r

e21i

e22i

(b)

0 5 10 15 20 25

Time(s)

-80

-60

-40

-20

0

20

40

e3

1r,e

32

r,e3

3r,e

31

i,e3

2i

e31r

e32r

e33r

e31i

e32i

(c)

Fig. 5. (a) Convergence of real and imaginary parts of anti-synchronization errors e11r,e12r,e13r,e11i,e12i, (b) Convergenceof real and imaginary parts of anti-synchronization errorse21r,e22r,e23r,e21i,e22i, (c) Convergence of real and imaginary

parts of anti-synchronization errors e31r,e32r,e33r,e31i,e32i


8




0 5 10 15 20 25

Time(s)

-40

-30

-20

-10

0

10

20

30

40

x1

1r,x

21

r,x3

1r,x

41

r

x11r

x21r

x31r

x41r

(a)

0 5 10 15 20 25

Time(s)

-40

-30

-20

-10

0

10

20

30

40

50

60

x1

1i,x

21

i,x3

1i,x

41

i

x11i

x21i

x31i

x41i

(b)

0 5 10 15 20 25

Time(s)

-60

-40

-20

0

20

40

60

x1

2r,x

22

r,x3

2r,x

42

r

x12r

x22r

x32r

x42r

(c)

0 5 10 15 20 25

Time(s)

-100

-80

-60

-40

-20

0

20

40

60

80

100

x1

2i,x

22

i,x3

2i,x

42

i

x12i

x22i

x32i

x42i

(d)

0 5 10 15 20 25

Time(s)

-100

-80

-60

-40

-20

0

20

40

60

80

100

x1

3,x

23,x

33,x

43

x13

x23

x33

x43

(e)

0 2 4 6 8 10 12 14 16 18 20

Time(s)

0

5

10

15

20

25

30

a,b

a

b

(f)

Fig. 6. (a) Anti-synchronization of real parts of system statesx11r,x21r,x31r,x41r, (b) Anti-synchronization of real parts of systemstates x12r,x22r,x32r,x42r, (c) Anti-synchronization of real parts ofsystem states x13,x23,x33,x43, (d) Anti-synchronization of imaginaryparts of system states x11i,x21i,x31i,x41i, (e) Anti-synchronization ofimaginary parts of system states x12i,x22i,x32i,x42i, (f) Convergence

of estimated parameters a, b to their true values a,b

7. Conclusion

The synchronization of complex chaotic PMSM systems is ofincreased practical importance in the field of electrical engi-neering. This article presents the control design method for HSand parameter identification of ring-connected complex per-manent magnet synchronous motor (PMSM) systems. Designof the desired control law is a challenging task for controlengineering applications due to parametric uncertainties andchaotic response to some specific parameter values. In order toachieve HS and to estimate the unknown parameters for com-plex PMSM systems, an adaptive integral sliding mode con-trol is proposed. To apply adaptive ISMC, the error system isfirstly converted to a unique system, consisting of nominal parttogether with some unknown terms which are computed adap-tively. The stabilizing controller incorporating nominal controland compensator control is designed for the error system. Onthe other hand, the compensation controller and the adaptedlaws are designed to get the first derivative of the Lyapunov

equation strictly negative. Simulation results verify the appro-priatness of the proposed technique by showing the conver-gence of error dynamics to zero and HS of system states. More-over, this work can be extended further by incorporating cou-pling delays.

REFERENCES

[1] A.C. Fowler, J.D. Gibbon, and M.J. McGuinness, “The complexLorenz equations”, Physica D 4, 139–163 (1982).

[2] P. Liu, H. Song and X. Li, “Observe-based projective synchro-nization of chaotic complex modified Van Der Pol-Duffing os-cillator with application to secure communication”, J. Comput.Nonlinear Dyn. 10, 051015 (2015).

[3] G.M. Mahmoud, and A.A. Shaban, “On periodic solutions ofparametrically excited complex non-linear dynamical systems”,Physica A 278(3-4), 390–404 (2000).

[4] G.M. Mahmoud and A.A. Shaban, “Periodic attractors of com-plex damped non-linear systems”, Int. J. Non-Linear Mech.35(2), 309–323 (2000).

[5] G.M. Mahmoud, “Periodic solutions of strongly non-linearMathieu oscillators”, Int. J. Non-Linear Mech. 32(6), 1177–1185(1997).

[6] G.M. Mahmoud and E.E. Mahmoud, “Lag synchronization ofhyperchaotic complex nonlinear systems”, Nonlinear Dynamics67, 1613–1622 (2012).

[7] P. Liu, and S. Liu, “Anti-synchronization between differentchaotic complex systems”, Phys. Scr. 83, 065006 (2011).

[8] S. Liu and P. Liu, “Adaptive anti-synchronization of chaoticcomplex nonlinear systems with unknown parameters”, Nonlin-ear Anal.-Real World Appl. 12, 3046–3055(2011).

[9] N. Siddique and F.U. Rehman, “Parameter Identification andHybrid Synchronization in an Array of Coupled Chaotic Sys-tems with Ring Connection: An Adaptive Integral Sliding ModeApproach”, Math. Probl. Eng. 2018, 6581493 (2018).

[10] G.M. Mahmoud, E.E. Mahmoud, and A.A. Arafa, “Projectivesynchronization for coupled partially linear complex-variablesystems with known parameters”, Math. Meth. Appl. Sci. 40(4),1214–1222 (2017).

[11] D.W. Qian, Y.F Xi, and S.W. Tong, “Chaos synchronization ofuncertain coronary artery systems through sliding mode”, Bull.Pol. Acad. Sci. Tech. Sci. 67(3), 455–462 (2019).

[12] G.M. Mahmoud, E.E. Mahmoud, and A.A. Arafa, “On modifiedtime delay hyperchaotic complex Lü system”, Nonlinear Dyn.80(1-2), 855–869 (2015).

[13] G.M. Mahmoud, T. Bountis, M.A. Al-Kashif, and A.A. Shaban,“Dynamical properties and synchronization of complex non-linear equations for detuned lasers”, Dynam. Syst. 24(1), 63–79(2009).

[14] J.-B. Hu, H. Wei, Y.-F. Feng, and X.-B. Yang, “Synchronizationof fractional chaotic complex networks with delay”, Kybernetika55, 203–215 (2019).

[15] N.A. Almohammadi, E.O. Alzahrani, and M.M. El-Dessoky,“Combined modified function projective synchronization of dif-ferent systems through adaptive control”, Arch. Control Sci. 29,133–146 (2019).

[16] H. Su, Z. Rong, M.Z.Q. Chen, X. Wang, G. Chen and H. Wang,“Decentralized adaptive pinning control for cluster synchroniza-tion of complex dynamical networks”, IEEE Trans. Cybern. 43,2182–2195 (2013).


compensation controller and the adapted laws are designed to get the first derivative of the Lyapunov equation strictly neg-ative. Simulation results verify the appropriatness of the pro-posed technique by showing the convergence of error dynamics to zero and HS of system states. Moreover, this work can be extended further by incorporating coupling delays.

References [1] A.C. Fowler, J.D. Gibbon, and M.J. McGuinness, “The complex

Lorenz equations”, Physica D 4, 139–163 (1982). [2] P. Liu, H. Song, and X. Li, “Observe-based projective synchro-

nization of chaotic complex modified Van Der Pol-Duffing os-cillator with application to secure communication”, J. Comput. Nonlinear Dyn. 10, 051015 (2015).

[3] G.M. Mahmoud and A.A. Shaban, “On periodic solutions of parametrically excited complex non-linear dynamical systems”, Physica A 278(3‒4), 390–404 (2000).

[4] G.M. Mahmoud and A.A. Shaban, “Periodic attractors of com-plex damped non-linear systems”, Int. J. Non-Linear Mech. 35(2), 309–323 (2000).

[5] G.M. Mahmoud, “Periodic solutions of strongly non-linear Mathieu oscillators”, Int. J. Non-Linear Mech. 32(6), 1177–1185 (1997).

[6] G.M. Mahmoud and E.E. Mahmoud, “Lag synchronization of hyperchaotic complex nonlinear systems”, Nonlinear Dynamics 67, 1613–1622 (2012).

[7] P. Liu and S. Liu, “Anti-synchronization between different cha-otic complex systems”, Phys. Scr. 83, 065006 (2011).

[8] S. Liu and P. Liu, “Adaptive anti-synchronization of chaotic complex nonlinear systems with unknown parameters”, Nonlin-ear Anal.-Real World Appl. 12, 3046–3055 (2011).

[9] N. Siddique and F.U. Rehman, “Parameter Identification and Hybrid Synchronization in an Array of Coupled Chaotic Sys-tems with Ring Connection: An Adaptive Integral Sliding Mode Approach”, Math. Probl. Eng. 2018, 6581493 (2018).

[10] G.M. Mahmoud, E.E. Mahmoud, and A.A. Arafa, “Projective synchronization for coupled partially linear complex-variable systems with known parameters”, Math. Meth. Appl. Sci. 40(4), 1214–1222 (2017).

[11] D.W. Qian, Y.F Xi, and S.W. Tong, “Chaos synchronization of uncertain coronary artery systems through sliding mode”, Bull. Pol. Acad. Sci. Tech. Sci. 67(3), 455–462 (2019).

[12] G.M. Mahmoud, E.E. Mahmoud, and A.A. Arafa, “On modified time delay hyperchaotic complex Lü system”, Nonlinear Dynam-ics 80(1‒2), 855–869 (2015).

[13] G.M. Mahmoud, T. Bountis, M.A. Al-Kashif, and A.A. Shaban, “Dynamical properties and synchronization of complex nonlin-ear equations for detuned lasers”, Dynam. Syst. 24(1), 63–79 (2009).

[14] J.-B. Hu, H. Wei, Y.-F. Feng, and X.-B. Yang, “Synchronization of fractional chaotic complex networks with delay”, Kybernetika 55, 203–215 (2019).

[15] N.A. Almohammadi, E.O. Alzahrani, and M.M. El-Dessoky, “Combined modified function projective synchronization of different systems through adaptive control”, Arch. Control Sci. 29, 133–146 (2019).

[16] H. Su, Z. Rong, M.Z.Q. Chen, X. Wang, G. Chen and H. Wang, “Decentralized adaptive pinning control for cluster synchroni-zation of complex dynamical networks”, IEEE Trans. Cybern. 43, 2182–2195 (2013).

7. Conclusion

The synchronization of complex chaotic PMSM systems is of increased practical importance in the field of electrical engi-neering. This article presents the control design method for HS and parameter identification of ring-connected complex perma-nent magnet synchronous motor (PMSM) systems. Design of the desired control law is a challenging task for control engi-neering applications due to parametric uncertainties and chaotic response to some specific parameter values. In order to achieve HS and to estimate the unknown parameters for complex PMSM systems, an adaptive integral sliding mode control is proposed. To apply adaptive ISMC, the error system is firstly converted to a unique system, consisting of nominal part together with some unknown terms which are computed adaptively. The stabiliz-ing controller incorporating nominal control and compensator control is designed for the error system. On the other hand, the

Fig. 6. (a) Anti-synchronization of real parts of system states x11r, x21r, x31r, x41r, (b) Anti-synchronization of real parts of system states x12r, x22r, x32r, x42r, (c) Anti-synchronization of real parts of system states x13, x23, x33, x43, (d) Anti-synchronization of imaginary parts of system states x11i, x21i, x31i, x41i, (e) Anti-synchronization of imag-inary parts of system states x12i , x22i , x32i , x42i , (f ) Convergence of

estimated parameters a , b to their true values a, b

9



[17] A. Khan and U. Nigar, “Sliding mode disturbance observer con-trol based on adaptive hybrid projective compound combination synchronization in fractional-order chaotic systems”, Int. J. Con-trol Autom. Syst. 31, 885–899 (2020).

[18] G.M. Mahmoud, E.A. Mansour, and T.M. Abed-Elhameed, “On fractional-order hyperchaotic complex systems and their gener-alized function projective combination synchronization”, Optik 130, 398–406 (2017).

[19] S. Wang, X. Wang, X. Wang, and Y. Zhou, “Adaptive general-ized combination complex synchronization of uncertain real and complex nonlinear systems”, AIP Adv. 6, 045011 (2016).

[20] J. Zhou, A. Oteafy, and N. Smaoui, “Adaptive synchronization of an uncertain complex dynamical network”, IEEE Trans. Autom. Control 51, 652–656 (2006).

[21] X. Chen, J. Qiu, J. Cao, and H. He, “Hybrid synchronization behavior in an array of coupled chaotic systems with ring con-nection”, Neurocomputing 173, 1299–1309 (2016).

[22] F. Zhang, C. Mu, X. Wang, I. Ahmed, and Y. Shu, “Solution bounds of a new complex PMSM system”, Nonlinear Dynamics 74, 1041–1051 (2013).

[23] Y.Wang, Y. Fan, Q.Wang, and Y. Zhang, “Stabilization and synchronization of complex dynamical networks with different dynamics of nodes via decentralized controllers”, IEEE Trans. Circuits Syst. I-Regul. Pap. 59, 1786–1795 (2012).

[24] L. Zarour, K. Abed, M. Hacil, and A Borni, “Control and opti-misation of photovoltaic water pumping system using sliding mode”, Bull. Pol. Acad. Sci. Tech. Sci. 67(3), 605–611 (2019).

[25] N. Siddique, F.U. Rehman, M. Wasif, W. Abbasi, and Q. Khan, “Parameter Estimation and Synchronization of Vaidyanathan Hyperjerk Hyper-Chaotic System via Integral Sliding Mode Control”, 2018 AEIT Conference IEEE, 1–5 (2018).

[26] K. Urbanski, “A new sensorless speed control structure for PMSM using reference model”, Bull. Pol. Acad. Sci. Tech. Sci. 65(4), 489–496 (2017).

[27] X. Sun, Z. Shi, Y. Zhou, W. Zebin, S. Wang,B. Su, L. Chen, and K. Li, “Digital control system design for bearingless permanent magnet synchronous motors”, Bull. Pol. Acad. Sci. Tech. Sci. 66(5), 687–698 (2018).

[28] T. Tarczewski, M. Skiwski, L.J. Niewiara, and L.M. Grzesiak, “High-performance PMSM servo-drive with constrained state feedback position controller”, Bull. Pol. Acad. Sci. Tech. Sci. 66(1), 49–58 (2018).

[29] W. Xing-Yuan and Z. Hao, “Backstepping-based lag synchro-nization of a complex permanent magnet synchronous motor system”, Chin. Phys. B 22, 048902 (2013).

[30] Z. Zhang, Z. Li, M.P. Kazmierkowski, J. Rodríguez, and R. Ken-nel, “Robust Predictive Control of Three-Level NPC Back-to-Back Power Converter PMSG Wind Turbine Systems With Revised Predictions”, IEEE Trans. Power Electron. 33(11), 9588– 9598 (2018).

[31] N. Hoffmann, F.W. Fuchs, M.P. Kazmierkowski, and D. Schröder, “Digital current control in a rotating reference frame – Part I: System modeling and the discrete time-domain current control-ler with improved decoupling capabilities”, IEEE Trans. Power Electron. 31(7), 5290–5305 (2016).

[32] H. Won, Y.-K. Hong, M. Choi, H.-s. Yoon, S. Li and T. Haskew, “Novel Efficiency-shifting Radial-Axial Hybrid Interior Perma-nent Magnet Sychronous Motor for Electric Vehicle”, 2020 IEEE Energy Conversion Congress and Exposition (ECCE), Detroit, USA, 2020, pp. 47–52.

[33] C. Jiang and S. Liu, “Synchronization and Antisynchroniza-tion of-Coupled Complex Permanent Magnet Synchronous Motor Systems with Ring Connection”, Complexity 4, 1–15 (2017).

[34] M. Karabacak and H.I. Eskikurt, “Speed and current regula-tion of a permanent magnet synchronous motor via nonlinear and adaptive backstepping control”, Math. Comput. Model. 53, 2015–2030 (2011).

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