SYNCHRONIZATION CRITERION OF CHAOTIC PERMANENT MAGNET
SYNCHRONOUS MOTOR VIA OUTPUT FEEDBACK
AND ITS SIMULATION
KALIN SU*, CHUNLAI LI
College of Physics and Electronics, Hunan Institute of Science and Technology,
Yueyang, 414000, China
∗
Corresponding author: sukalinvip@126.com (K. Su)
Received October 3, 2014
Chaos synchronization of permanent magnet synchronous motor (PMSM) has
been central to recent experimental and theoretical investigations. In existing papers,
the implementations of synchronization control require the system states for feedback,
which are effective but unacceptable in practical application. In this paper, an output
feedback controller is proposed for synchronization of PMSM based on the theory of
passive control. Theoretical analysis shows that the control method makes the
synchronization error system between the driving and the response motor systems not
only passive but also asymptotically stable. Numerical simulations are provided to
verify the effectiveness of the proposed scheme.
Key words: permanent magnet synchronous motor; chaos synchronization; output
feedback control.
PACS Nos.: 05.45.-a; 05.45.Xt; 05.45.Gg.
1. INTRODUCTION
Among various electrical machines, permanent magnet synchronous motor
(PMSM) has been studied intensively since it has superior features such as simple
structure, low manufacturing cost, more torque per weight and efficiency [1–4].
With the technological development of control and electronic, the PMSM have
been widely used in direct-drive robotic applications, electric and hybrid vehicle,
especially in industrial applications for low-medium power range.
However, the performance of the PMSM is sensitive to external load
disturbances and system parameter in the plant. The stability of PMSM is a basic
requirement in industrial manufacturing, therefore has received considerable
attention. Investigations show that PMSM displays chaotic behavior when motor
parameters lie in certain value [2, 3]. Chaos in the PMSM, which decreases the
system performance, is highly undesirable in most engineering applications. Many
scientists have devoted themselves to find efficient strategies to control chaos in
PMSM [5–8].
Rom. Journ. Phys., Vol. 60, Nos. 9–10, P. 1409–1419, Bucharest, 2015
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Kalin Su, Chunlai Li
2
More interesting, chaos synchronization of motor, which implies the slave
motor is designed to work following the master motor in the same rhyme of the
angular phase and (or) the amplitude via an appropriate control scheme, has been
attracted to recent theoretical and experimental investigations [9–13]. Although
there are large numbers of control programs for synchronization of chaotic
systems, investigation about synchronization scheme in chaotic motors is few. For
instance, Ge proposed that Brushless DC motors (BLDCM) can be synchronized
by backstepping technique [9]. Liu realized chaos synchronization of Brushless DC
motors by the variable substitution control strategy [10]. Zhao developed a speed
synchronization control strategy for multiple induction motors by employing total
sliding mode control method [12]. Verrelli considered the synchronization problem
for uncertain permanent magnet synchronous motors based on Fourier
approximation theory [13]. All these existing synchronization schemes for chaotic
motors require the system states for feedback, which are unrealistic and
unacceptable in practical. So, it is significant to find an available method in the real
operations for synchronization in motors.
In this paper, a feedback controller which only needs to require the
knowledge of the system output is proposed for synchronization of permanent
magnet synchronous motor (PMSM). The synchronization algorithm is based on
the theory of passive control. Theoretical analysis shows that the synchronization
error system between the driving and the response motor systems is not only
passive but also asymptotically stable by the presented controller. Numerical
simulations are provided to verify the effectiveness of the proposed design.
2. MODEL OF PMSM
The dimensionless mathematical model of PMSM can be described by [2–4]
⎧ did
⎪ dt = −id + ω iq + ud
⎪
⎪ diq
= −iq − ωid + γ ω + uq ,
⎨
⎪ dt
⎪ dω
⎪ dt = σ (iq − ω ) − TL
⎩
(1)
where id and iq denote the stator currents; ω denotes the rotor angular frequency;
ud = KT LU d / bR 2 + KT 2 / bR + γ , uq = KT LU q / bR 2 denote the stator voltages with
L = Ld = Lq ; TL is the external load torque; γ and σ = bL / JR are the operating
parameters. As considered in the study of bifurcations [2], we take the case
ud = uq = TL = 0 in our work.
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Synchronization criterion of chaotic permanent magnet synchronous motor via output feedback 1411
When applying equilibrium condition, one obtain three equilibria of system
(1), as S1 =(0, 0, 0) , S2,3 =(γ − 1, ± γ − 1, ± γ − 1) . The previous investigation
results show that, with the motor parameters σ and γ lying in certain area, all the
three equilibria become unstable, and the PMSM exhibits chaos. The bifurcation
diagram for γ ∈ [14, 24] versus with σ = 5.46 and the typical chaotic attractor with
γ = 20, σ = 5.46 are shown in Figure 1.
Fig. 1 – Bifurcation diagram and chaotic attractor of system (1):
(a) bifurcation diagram; (b) chaotic attractor.
3. BASIC THEORY OF PASSIVE CONTROL
In this section, the basic conception of passivity of nonlinear affine system
and passivity-based control method are summarized. Here, the following nonlinear
differential equation is considered:
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Kalin Su, Chunlai Li
⎧ x = f ( x) + g ( x )u
,
⎨
⎩ y = h( x )
4
(2)
where x ∈ R n is the state variable; y ∈ R m is the output; u ∈ R m is the external
input; f ( x ) and g ( x ) are smooth vector fields; h( x ) is a smooth mapping.
The conception of passivity can be described as below [14].
Definition 1. If there exists a continuously differentiable positive semidefinite storage function V(x) satisfying
u T y ≥ V ( x)
then system (2) is said to be passive.
Moreover, if
u T y ≥ V ( x) + y T ρ ( y ) and yT ρ ( y ) > 0
(3)
(4)
for any y ≠ 0 , we called the output is strictly passive. And the output is strictly
passive if
(5)
uT y ≥ V ( x) + ( y )
for some positive definite function ( y ) .
If system (2) is strictly passive or it’s output is strictly passive and zero-state
observable, then the origin of system (2) is asymptotically stable with u = 0.
Furthermore, if the storage function V(x) is radially unbounded, the origin is
globally asymptotically stable.
Definition 2. If x = 0 is an asymptotically stable equilibrium
of f ( x ) and Lg h(0) is nonsingular, then system (2) is a minimum phase system.
When system (2) is minimum phase and the distribution spanned by the
vector fields g1 ( x ), g 2 ( x )
g m ( x) is involutive, system (2) can be represented
as the following generalized form
⎧ z = f ( z ) + q ( z , w) w
,
⎨
⎩ w = α ( z , w ) + β ( z , w )u
(6)
where z = ϑ ( x) , β ( z , w) is nonsingular for any ( z , w) .
The physical meaning of passive system is that the energy of the nonlinear
affine system can’t increase unless an external source is supplied. In other words, a
passive system cannot store more energy than that supplied externally. The passive
system is stable naturally. We can utilize the input-output relationship based on
energy-related considerations to analyze stability properties.
The problem for passivity-based control is to design a feedback control
scheme for achieving the passivity. It’s necessary that system (2) is a minimum
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Synchronization criterion of chaotic permanent magnet synchronous motor via output feedback 1413
phase system when utilizing feedback passivity. Therefore, the system (6) cannot
be made passive by feedback if its zero-dynamics is unstable.
4. SYNCHRONIZATION OF PMSM VIA OUTPUT FEEDBACK
In order to observe the synchronization behavior of the two identical chaotic
motor systems, we assume that the drive system is given as (1) with
ud = uq = TL = 0 .
And the response system with controller is expressed by
⎧ did′
⎪ dt = −id′ + ω ′iq′
⎪
⎪ diq′
= −iq′ − ω ′id′ + γ ω ′ + u ,
⎨
⎪ dt
⎪ dω ′
⎪ dt = σ (iq′ − ω ′)
⎩
(7)
where u is the control function to be designed.
Defining the synchronization error e1 = id′ − id , e2 = iq′ − iq , e3 = ω ′ − ω , then
one can obtain the error dynamical system
⎧ de1
⎪ dt = −e1 + iq′ e3 + ω e2
⎪
⎪ de2
= −e2 + γ e3 − id′ e3 − ω e1 + u .
⎨
⎪ dt
⎪ de3
⎪ dt = σ (e2 − e3 )
⎩
(8)
It follows that if controlled system (8) is stabilized, then synchronization
error e will converge to a zero equilibrium point, which means that the trajectories
of response system with the controller u asymptotically synchronize the trajectories
of the driving system.
In order to achieve the goal, we introduce a dynamical variable e3 , which is
an estimator of e3 .
Define z = [ z1 , z2 , z3 ] = [ e1 , e3 , e3 − e3 ] , y = e2 , then system (8) can be
T
represented as
T
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Kalin Su, Chunlai Li
⎧ dz1
⎪ dt = − z1 + iq′ z2 + ω y
⎪
⎪ dz 2 = σ ( y − z )
2
⎪ dt
.
⎨
d
z
⎪ 3 = −σ z + yh( y )
3
⎪ dt
⎪ dy
⎪ = − y + γ z2 − id′ z2 − ω z1 + u
⎪⎩ dt
When write system (9) in the normal form (6), we have
⎧ z = f ( z ) + q ( z , w) w
,
⎨
w
=
z
w
+
z
w
u
α
β
(
,
)
(
,
)
⎩
6
(9)
(10)
where
f 0 ( z ) = [ − z1 + iq′ z 2 , − σ z2 , − σ z3 ]T , q0 ( z , y ) = [ω , σ , h( y )]T
α ( z , y ) = − y + γ z2 − id′ z2 − ω z1 , β ( z , y ) = 1
Suppose that the bound of iq′ is B , namely, iq′ < B . We define a Lyapunov
function candidate
V0 ( z ) =
η
2
z12 +
1 2 1 2
z 2 + z3
2
2
for the zero dynamics z = f 0 ( z ) , where η ≤ 4σ / B 2 . Taking the derivative for
V0 ( z ) with respect to time yields
∂V0 ( z )
f0 ( z )
∂z
= η z1 ( − z1 + iq′ z2 ) − σ z2 2 − σ z3 2
V0 ( z ) =
= −η z12 + iq′η z1 z2 − σ z2 2 − σ z32
≤ −η z12 + η B i z1 z2 − σ z2 2 − σ z3 2
≤ −η z12 + 2 ησ i z1 z2 − σ z2 2 − σ z3 2
= −( η i z1 − σ i z2 ) 2 − σ z3 2
<0
Therefore, the origin of the zero dynamics is stable. Thus, system (9) is a
minimum phase system.
Theorem 1. For the controlled error dynamical system (9), if the control
scheme is designed as
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Synchronization criterion of chaotic permanent magnet synchronous motor via output feedback 1415
u = −(σ + γ )e3 + id′ z2 + v
h( y ) = σ + γ
,
(11)
where v is the external input signal, then the controlled system (9) is a passive
system and globally asymptotically stabilized at the zero equilibrium point.
Namely, the trajectories of response system with the controller u asymptotically
synchronize the trajectories of the driving system.
Proof. Choose the storage function as
V ( z ) = V0 ( z ) +
1 2
y
2
The function V ( z ) is positive definite and radially unbounded. Taking
derivative of V ( z ) with respect to time along the trajectory of the controlled error
system (9), we can obtain
∂V0 ( z )
∂V ( z )
f0 ( z) + 0
q0 ( z , y ) y + yy
∂z
∂z
∂V ( z )
≤[ 0
q 0 ( z , y ) + α ( z , y ) + β ( z , y )u ] y
∂z
= [ω z1 + σ z2 + h( y ) z3 − y + γ z2 − id′ z2 − ω z1 + u ] y
= [(σ + γ ) z2 + h( y ) z3 − y − id′ z2 + u ] y
V ( z) =
When consider the control scheme, one have
V ( z ) ≤ ( − y + v ) y = − y 2 + vy .
(12)
Namely, vy ≥ V ( z ) + ( y ) , where ( y ) = y 2 . So, the output of system (9)
with the control scheme (11) is strictly passive. Thus, system (8) will be
asymptotically stabilized at the zero equilibrium point by using the control scheme
(11).
Remark. It’s known that a dynamic system is more stable with lower energy.
On the other hand, we know that a passive system cannot store more energy than
that supplied externally. Therefore, we have the conclusion from (12) that with the
increasing of control parameter v, the stability of the controlled system (8) will
reduce, although the synchronization is achieved, and that the initial values don’t
produce effect on the stability of synchronization.
5. NUMERICAL SIMULATION
To demonstrate and verify the validity of the proposed synchronization
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Kalin Su, Chunlai Li
8
scheme, some numerical simulations are presented in this section. For comparing
conveniently, in all the process of numerical simulation, the ODE45 method in
Matlab is adopted to solve the nonlinear systems. Since the initial values don’t
affect the stability of synchronization, so in all the numerical process, we set the
initial values of systems (1) and (7)
i d (0), i q (0), ω (0) = (3, 1, 12) ,
( i' (0), i' (0), ω '(0) ) = (1, 3, 3 ) .
d
(
)
q
First, the control parameter is set to be v = 0.2. The corresponding
simulation results are shown in Fig. 2. Fig. 2 (a) shows the state trajectory; Fig. 2
(b) shows the synchronization error; and Fig. 2 (c) shows the time graphic of the
control function u. As we know that the synchronization errors converge
asymptotically to zero, and the two motor systems are indeed synchronized.
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Synchronization criterion of chaotic permanent magnet synchronous motor via output feedback 1417
Fig. 2 – Synchronization with v = 0.2: (a) state trajectory; (b) synchronization error;
(c) control function u.
Then, we select a larger control parameter as v = 10. The corresponding
simulation results are shown in Figs. 3 (a), (b) and (c). As we know that the two
motor systems are synchronized with a larger control parameter. However, both
the synchronization errors and energy consumption of control function increase in
comparison with a less parameter v.
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Kalin Su, Chunlai Li
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Fig. 3 – Synchronization with v = 10: (a) state trajectory; (b) synchronization error;
(c) control function u.
6. CONCLUSION
In this paper, a feedback controller which only needs to require the knowledge
of the system output is proposed for synchronization of PMSM based on the theory
of passive control. Theoretical analysis shows that the synchronization error system
between the driving and the response motor systems is not only passive but also
asymptotically stable by the presented controller. Numerical simulations are provided
to verify the effectiveness of the proposed design.
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Synchronization criterion of chaotic permanent magnet synchronous motor via output feedback 1419
Acknowledgments. This work was supported by the Research Foundation of Education Bureau
of Hunan Province, China (Grant No. 13C372) and the Hunan Provincial Natural Science Foundation
of China (Grant No. 10JJ30052).
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