Control of High Speed Linear Induction Motor Using Artificial Neural Networks
Content
HSI 2008
Krakow, Poland, May 25-27, 2008
Control of High Speed Linear Induction Motor Using
Artificial Neural Networks
Mohammadali Abbasian, Jafar Soltani, and Ali Salarvand
Engineering Department, Islamic Azad University- Khorasgan Branch- Isfahan, Iran
[email protected], [email protected]
Abstract- This paper presents a controller for speed and flux
control of a high speed Linear Induction Motor (LIM) Drive
considering end effect. The backstepping control and Artificial
Neural Networks (ANN ) are combined in order to design a robust
controller that is capable of preserving the drive system
robustness subject to all parameter variations and uncertainties.
The overall system stability is proved by Lyaponuv theory. The
effectiveness and validity of the proposed controller is supported
by computer simulation results.
I.
INTRODUCTION
The Linear Induction Motor (LIM) has excellent
performance features such as high-starting thrust force,
alleviation of gear between motor and the motion devices,
reduction of mechanical losses and the size of motion devices,
high speed operation, silence, and so on [1]. Because of above
the advantages, the primary type LIM, shown in Fig. 1, has
been widely used in the field of industrial processes and
transportation applications. The driving principles of the LIM
are similar to the traditional Rotary type Induction Motor
(RIM), but its control characteristics are more complicated
than the RIM, and the motor parameters are time-varying due
to the change of operating conditions, such as speed mover,
temperature, and configuration of rail [2].
In a LIM, the primary winding corresponds to the stator
winding of a rotary induction motor (RIM), while the
secondary corresponds to the rotor. There are some
characteristics differences between the RIM and LIM. The
main difference is that the primary of the LIM has a finite
length, and therefore, there is a fringing field at both ends of
the primary. The infinitely long secondary enters the air-gap
field, carries the magnetic flux along with it, and makes, the
distribution of the electromagnetic quantities nonuniform,
resulting in considerable electric and force losses [3]. The
losses, as well as the flux-profile attenuation, become severer
as the speed increases. Such a phenomena is called ‘end effect’
of LIM.
An accurate equivalent circuit model is indispensable for
high performance control for LIM drive. Most of the existing
models of a LIM depend on field theory [4,5]. Hence, they can
not be directly applied for the nonlinear control and in the most
of the present studies, the RIM model is used in order to
control LIM [6-8]. so, because of the end effect, they can not
be valid in high speed operation of LIM. In [6], an adaptive
Primary(Mover)
Secondary
Fig. 1. Linear Induction Motor.
controller is used in order to control LIM at low speeds. This
controller is robust only against the mechanical parameter
uncertainties. In [7], an indirect field oriented control is used to
control LIM at low speed which is not robust in respect to all
machine parameter variations and uncertainties. In [8], a robust
nonlinear controller has been proposed for a primary type LIM
which is based on combination of SM control and input-output
feedback linearization. Using the RIM model, the drive system
of [8] is robust to parameters variation only at low speed
operation. Using the control method presented in [8], the
main objective of this paper is to introduce a robust nonlinear
controller for LIM which will be robust and stable subject to
the parameter uncertainties and external unknown load force at
low and high speed operation. Reference [9] has described a
secondary field-oriented control scheme for LIM with the end
effect, but the robustness is not considered.
II. LIM MODEL
The LIM fifth order model with end effect, in stationary d:q
axis reference frame with secondary fluxes and primary
currents as state variable can be expressed by[9]:
d λqr
1
L mf
π
i qs + n p V e λqr − λqr
dt
Tr
h
Tr
d λdr L mf
π
1
=
i ds − λdr − n p V e λqr
dt
Tr
Tr
h
=
(1)
(2)
di ds
R
L mf
1− σ
σ&
µ&
= −( s +
+
)i ds + (
−
)λ
σ Ls σT r σ LS
σ Ls L rT r σ LS dr
dt
+
n p L mf π
σ Ls L r h
V e λqr +
1
σ Ls
v ds
(3)
di qs
dt
= −(
Rs
L mf
1− σ
σ&
µ&
+
+
)i qs + (
−
)λqr
σ Ls σT r σ LS
σ L s L rT r σ LS
n p L mf π
1
−
V e λdr +
v qs
σ Ls L r h
σ Ls
Fe =
3π n p µ
2h
(4)
(λdr i qs − λqr i ds ) = M
dV e
+ DV e + Fl
dt
(5)
where i ds , i qs , λdr , λqr , v ds , v qs and V e are the primary
d-q currents, secondary d-q fluxes, primary d-q voltages and
mover speed, respectively. In addition, R s is the primary
resistance, R r is the secondary resistance, n p is the number
of pole pairs, h is the pole pitch, Fl is the external force
disturbance, M is the total mass of mover and D is the
viscous friction coefficient, T r = L r R r is the secondary
time constant,
Ls = L mf + L ls is the primary inductance,
L r = L mf + L lr is the secondary inductance, L ls is the
primary leakage inductance, L lr is the secondary leakage
inductance, σ = 1 − L 2mf Ls L r , µ = L mf L r , δ = σ Ls .
L mf is a time variant parameter as a function of V e given
by [9]:
L mf = L m (1 − f (Q ))
where
(6)
L m = 1.5L mo and L mo is the magnetizing inductance
at zero speed, and:
f (Q ) =
1 − e −Q
,
Q
Q=
lR r
(L m + L lr ) V e
(7)
and l is the primary length.
In equations 1 to 5, δ& and µ& are functions of other system
states and can be written as follows:
2L L − L
δ& = −L m f& (Q ){1 − mf r2 mf }
Lr
2
µ& = −L m f& (Q )
L lr
L2r
(8)
L mo + Llr
(1 − e −Q − Qe −Q ) ×
MlR r
{
3π n p µ
2h
(λdr i qs − λqr i ds ) − DV e − Fl }
A. ANN Basics
Define W as the controller of ANN weights, then the net
output is [10]:
y =W T φ (x )
(11)
Let S be a compact simply connected set of n , with map
f : S → n , define C m (s ) the functional space such that
f (x ) ∈ C m (s ) , x (t ) ∈ S can be approximated by a neural
network as:
f (x ) = W T φ (x ) + ε (x )
(12)
With ε (x ) a ANN functional reconstruction error vector
and φ (x ) is sigmoid activation function.
B. Robust Backstepping Control of LIM Using ANN
Using the well known fifth order LIM model in a stationary
two axis reference frame where the secondary fluxes and
primary currents are assumed as state variables, the robust
nonlinear controller is designed in the following way:
Dividing the above LIM model into two nonlinear
subsystems, where i ds , i qs are the outputs for the first
subsystem which are simultaneously assumed the fictitious
input of the second sub-system.
Assume that:
1. The reference trajectories V e ∗ and λr* are differentiable
and bounded.
2.the Load force is an unknown constant and resistances,
inductances and moment of inertia are unknown and bounded.
In the first step of the controller design, i ds , i qs are
assumed as fictions controls for the second sub-system. The
main objective is to obtain these controls so that the desired
secondary speed and amplitude signals are perfectly tracked in
spite of machine parameters and external load force
uncertainties. Considering V e ∗ and λr* , the tracking error
equations are obtained as:
e1 = V e −V e ∗
e 2 = (λdr 2 + λqr 2 ) − λr∗2 = λr 2 − λr∗2
(13)
(9)
where f& (Q ) is also a function of the system states as:
f& (Q ) =
III. ROBUST BACKSTEPPING CONTROLER
(10)
Derivating (13) with respect to time (t), and using equations
(1), (2) and (5) yields:
⎧ de
3π n p µ
F
⎪ 1=
(λdr i qs − λqr i ds ) − l −V&e∗
2 hM
M
⎪ dt
⎪
(14)
⎧
⎫
−R
R
⎪ de 2
= 2λdr ⎨ r λdr + r L mf i ds ⎬
⎨
Lr
⎩ Lr
⎭
⎪ dt
⎪
⎧ −R r
⎫
R
⎪
λqr + r L mf i qs ⎬ − 2λr∗λ&r∗
+ 2λqr ⎨
⎪⎩
Lr
⎩ Lr
⎭
416
or:
D1e& = F1 + G1i
2
R r + Lr 2 R s
δ& ⎫⎪ ⎤
⎪⎧ L
− ⎨ mf
+
⎬ i ds ⎥
σ Ls ⎪⎭ ⎥
σ Ls L r 2
⎩⎪
⎥
& ⎫⎪ ⎥
⎧⎪ L 2 R + L 2 R
δ
r
r
s
− ⎨ mf
+
⎬ i qs ⎥
σ Ls ⎪⎭ ⎥
σ Ls L r 2
⎪⎩
⎦
&
+ D& 2G1−1 (Fˆ1 + K 1e ) + D 2G1−1Fˆ1 +
(15)
with:
⎡ −Fl M & ∗
⎤
⎢ µ − µ Ve
⎥
⎥ ,
F1 = ⎢
⎢ 2 2
2 ∗ &∗ ⎥
⎢ − L λr − R µ λr λr ⎥
r
⎣ m
⎦
(
D 2 G1−1K 1D1−1 (F1 − Fˆ1 − K 1e + G1η )
3π n p
⎡ 3π n p
⎤
λqr
λdr ⎥
⎢−
G1 = ⎢ 2 h
2h
⎥,
2λqr ⎥⎦
⎢⎣ 2λdr
⎡M
⎤
0 ⎥
⎢µ
⎡i ⎤
⎡e ⎤
⎥ , i = ⎢ ds ⎥ , e = ⎢ 1 ⎥
D1 = ⎢
i
1 ⎥
⎢
⎣e 2 ⎦
⎣⎢ qs ⎦⎥
⎢ 0 µR ⎥
r
⎣
⎦
(16)
i is
(21) which is necessary to cancel the effect of G1η in (17).
Combining (16) and (18), gives:
D 2η& = F2 − Fˆ2 − K 2η − G1T e
(23)
(17)
−
, η =i−i
D 2η& = F2 + G 2v
Wi , is assumed to be as follows:
(19)
n p π L mf
⎡ L mf R r
µ&
−
)λdr +
V λ
⎢(
2 σL
σ Ls L r h e qr
s
⎢ σ Ls Lr
F2 = D 2 × ⎢
n πL
⎢( L mf R r − µ& )λ − p mf V λ
qr
⎢ σ L L 2 σ Ls
σ Ls Lr h e dr
s r
⎣
F1 = W1T φ1 + ε1
, ε1 < ε1N = cte
F2 = W2T φ2 + ε 2
, ε 2 < ε 2 N = cte
(24)
where φ1 , φ2 provide suitable basis functions. From (23), one
can find that net reconstruction error ε i ( x) is bounded by a
known constant ε iN .
Assumption 3: The ideal weighs are bounded by known
positive values so that:
where:
⎡1 0 ⎤
⎢
⎥
⎣0 1 ⎦
C. F1 , F2 Approximation Using ANN
In this section, functions F1 , F2 are approximated by two
two-layer ANN. In. Using ANNs approximation property,
F1 , F2 as outputs of two two-layer ANN with constant weights,
(18)
In the second step, the control v is obtained in such a way
that η in Equation (17), becomes as small as possible.
Differentiating η with respect to time yields:
⎡v ds ⎤
⎡1 0 ⎤
v = ⎢ ⎥ , G2 = ⎢
⎥ , D 2 = σ Ls
v
⎣0 1 ⎦
⎣⎢ qs ⎦⎥
(22)
second two layer ANN. In addition a term −G1T e is added in
where K1 a design parameter and Fˆ1 the estimate of F1 which
will be estimated in the next section with a two layer ANN.
Substituting (16) into (15) gives:
D1e& = F1 − Fˆ1 − K 1e + G1η
To make η as small as possible, the following control is
chosen :
In (21), Fˆ2 is an estimate of F2 that is also estimated by a
By treating i as a fictitious input, a controller for the ideal
designed as:
, K1 > 0
)
v = G 2−1[−Fˆ2 − K 2η − G1T e ]
From (16) it is clear that G1 is known and invertible matrix.
i = G1−1[− Fˆ1 − K 1e ]
(21)
(20)
W1
≤ W2 M
(25)
≤ Z M , Z = diag {W1 ,W2 }
(26)
F
≤ W1M
, W2
F
or equivalently:
Z
F
The actual inputs to ANN1 are λr ,V e , λ&r∗ ,V&e ∗ and actual
inputs to ANN2 are λr , V e , λ&r∗ , V&e ∗ , λdr , λqr , i ds , i qs , e1 , e 2 .
417
Online ANN approximation of F1 is defined by:
{
Fˆ1 =Wˆ1T φ1
(27)
D1e& =W%1T φ1 − K 1e + G1η + ε1
V& ≤ −λmin ζ
(28)
Fˆ2 =Wˆ2T φ2
(29)
Then error dynamic (22) will be:
D 2η& =W%2T φ2 − K 2η − G1T e + ε 2
D. Updating ANNs Weights
In this part, the stability of proposed controller, is proved
based on Lyapunov stability theory. This analysis shows that
tracking errors and updated weighs are Uniformly Ultimately
Bounded (UUB).
Theory : Let the desired trajectories V e* , λr* be bounded.
Take the control input (21) with weigh updates be provided by:
Γ1 = Γ1T
(31)
⎡ k ( Z%
⎢⎣ ω
}
(33)
, K = diag {K 1 , K 2 }
(34)
ζ = [e ,η ]
2
− Z%
(36)
F
Differentiate (32):
{
}
V& = −ζ T K ζ + k ω e tr Z% T ( Z − Z% ) + ζ T ε
F
(Z M − Z%
F
( Z%
F
F
) + εN ζ
− Z M ) − ε N ⎤⎥
⎦
(37)
( Z%
F
− Z M ) − ε N ⎤⎥ =
⎦
− Z M / 2) 2 − k ω Z M2 / 4 + λmin ζ − ε N ⎤⎥
⎦
(38)
which is guaranteed positive if
ζ > [k ω Z M2 / 4 + ε N ] / λmin
Z%
F
> Z M / 2 + Z M2 / 4 + ε N / k ω
(39)
(40)
Thus, it can be concluded that and are UUB [10] and the
system is stable.
Note 1: Small tracking error bounds may be achieved by
selecting large control gain K . The parameter kω offers a
design tradeoff between the relative eventual magnitudes of
ζ and Z% , a smaller k yields a smaller ζ and a larger
Z%
> 0, Γ 2 = ΓT2
Z% = diag W%1 ,W%2 , Γ = diag {Γ1 , Γ 2 }
T T
F
F
ω
F
> 0 and
scalar positive constant kω . Then the errors η (t ), e(t ) are
UUB. ANN updated weights are bounded. The errors η (t ), e(t )
can be kept as small as desired by increasing gains K 1, K 2 .
Proof: Consider the following Lyapunov candidate:
0 ⎤
⎡D
1
1
% T −1 %
(32)
V = ζT ⎢ 1
⎥ ζ + tr ( Z Γ Z )
D
0
2
2
2⎦
⎣
with any constant matrices
+ k ω ζ Z%
⎡λ
ζ + k ω Z%
⎣⎢ min
(30)
in
Note that there is a term G1η in (27) and a term
(29). This means there are couplings between the error
dynamics (27) and (29).
{
F
F
where λmin is the minimum singular value of K .The righthand side of (37) is negative as long the term inside braces is
positive. Completing the square yields:
−G1T e
&
Wˆ1 = Γ1φ1eT − k ω Γ1 ζ Wˆ1
&
Wˆ2 = Γ 2φ2eT − k ω Γ 2 ζ Wˆ2
2
= − ζ ⎡⎢λmin ζ + k ω Z%
⎣
where W%1 =W 1 −Wˆ1 . Similarly, approximation of F2 is:
T
Z
and having in mind some basics inequalities, from (35) the
following can be obtained:
Linking (18) and (2), gives:
where:
}
tr Z% T ( Z − Z% ) ≤ Z%
F
, and vice versa.
Note 2 : If Wˆi (0) are taken as zeroes the linear proportional
control term − K ζ stabilizes the system on an interim basis.
IV. SYSTEM SIMULATION
Based on the proposed control strategy in this paper, the
block diagram of LIM drive control is shown in Fig. 2.
A C ++ computer program was developed for system
simulation. In this program, the nonlinear equations are solved
based on static forth order Range-Kutta method. The proposed
control method, is tested by simulation for a three-phase LIM
with parameters shown in Table (1).
In this simulation, the controller gains are obtained by trial
and error and are given as
K 1 = diag {1000,1000} , K 2 = diag {2000,1000}
(35)
k ω = 1 , Γi = 10I
Now, using Schwarz inequality [10]:
418
(41)
Table 1
λr (Wb )
: LIM PARAMETERS
0.12
Primary resistance
R s = 5.36Ω
0.1
Secondary resistance
R r = 3.53Ω
0.08
D = 36.08Kg / sec
0.06
np =1
0.04
Primary leakage inductance
Ls = 0.0029H
0.02
Secondary leakage inductance
L r = 0.0029H
Magnetizing Inductance at zero speed
L m 0 = 0.0681H
Viscous friction coefficient
Number of pole pairs
Pole pitch
h = 0.027m
Total Mass of moving element
M = 2.78kg
λr∗
0
0
λr
0.5
1
1.5
2
t (sec)
Fig. 3a. Secondary flux
V e (m / s )
1.2
V e∗
1
Simulation results shown in Fig. 3 (3a to 3g) , are obtained
in the case of an exponential reference flux rising up from zero
to 0.1 W b − t at t = 0s with a time constant of τ = 0.1sec , an
exponential reference speed from zero to 1m/sec at t = 0sec ,
a step load disturbance from zero to 5Nm at t = 0.5sec , step
down to 3Nm at t = 1sec and motor electromechanical
parameters assumed to be twice their nominal value.
Fig. 4 (4a to 4g) shows the simulation results obtained for an
exponential reference flux rising up from zero to 0.1 W b − t at
t = 0sec and a sinusoidal reference speed. In flux and speed
control performance, motor electromechanical parameters
assumed to be twice their nominal values.
Referring to Fig. 3f and 4.f, it is clear that the maximum of
f (Q ) function is 0.16. This means that the motor magnetizing
inductance reduces to 84 percent its original value, when motor
moves at the spped of 1m/s.
Ve
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
t (sec)
Fig. 3b. Mover speed
i ds (A )
40
30
20
10
0
-10
-20
-30
CONTROLLER
Fˆ1
-40
0
0.5
1
1.5
2
t (sec)
ANN1
Fig. 3c. d-axis current
i qs (A )
Fˆ2
40
ANN2
30
20
λr∗ ,V e∗
10
− G1T
CONTROLLER
PWM
INVERTER
0
LIM
-10
-20
-30
-40
0
Fig. 2. The overall block diagram of LIM drive system control
0.5
Fig. 3d. q-axis current
419
1
1.5
2
t (sec)
v as (V )
V e (m / s )
800
V e∗
1
600
400
Ve
0.5
200
0
0
-200
-0.5
-400
-1
-600
-800
0
0.5
1
1.5
0
2
t (sec)
0.5
1
Figure 3e. Inverter's output voltage
1.5
t (sec)
Fig. 4b. Mover speed
i ds ( A )
f (Q )
40
0.16
30
0.14
20
0.12
10
0.1
0
0.08
-10
0.06
-20
0.04
-30
0.02
0
0
0.5
1
1.5
-40
0
2
0.5
1
t (sec)
Figure 3f. f(Q) function
1.5
t (sec)
Fig. 4c. d-axis current
Fl (N .m )
i qs (A )
6
30
5
20
4
10
7
40
0
3
-10
2
-20
1
-30
0
0
0.5
1
1.5
t (sec)
2
-40
0
0.5
1
Figure 3g. External force disturbance
1.5
t (sec)
Fig. 4d. q-axis current
λr (W b − t )
v as (V )
0.12
600
λr*
0.1
0.08
400
λr
200
0.06
0
0.04
-200
0.02
-400
0
0
0.5
Figure 4a. Secondary flux
1
-600
0
1.5
t (sec)
0.5
Fig. 4e. Inverter's output voltage
420
1
1.5
t (sec)
f (Q )
Computer simulation results obtained, confirm the
effectiveness and validity of the proposed controller. These
results also confirm that the drive system control is robust and
stable against the parameters uncertainties and unknown load
force disturbance.
0.2
0.15
0.1
0.05
REFERENCES
0
0
0.5
1
[1]
1.5
t (sec)
Fig. 4f. f(Q) function
[2]
Fl (N .m )
[3]
7
6
[4]
5
4
[5]
3
2
[6]
1
0
0
0.5
1
1.5
2
t (sec)
[7]
Fig. 4g. External force disturbance
[8]
CONCLUSIONS
In this paper, a composite nonlinear controller has been
proposed for the LIM secondary flux and speed tracking
control. The nonlinear controller is designed based on the LIM
fifth order model in a fixed two axis reference frame,
combining the backstepping control and ANN. The overall
stability of this controller is proved by Lyapunov theory.
[9]
[10]
421
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Krakow, Poland, May 25-27, 2008
Control of High Speed Linear Induction Motor Using
Artificial Neural Networks
Mohammadali Abbasian, Jafar Soltani, and Ali Salarvand
Engineering Department, Islamic Azad University- Khorasgan Branch- Isfahan, Iran
[email protected], [email protected]
Abstract- This paper presents a controller for speed and flux
control of a high speed Linear Induction Motor (LIM) Drive
considering end effect. The backstepping control and Artificial
Neural Networks (ANN ) are combined in order to design a robust
controller that is capable of preserving the drive system
robustness subject to all parameter variations and uncertainties.
The overall system stability is proved by Lyaponuv theory. The
effectiveness and validity of the proposed controller is supported
by computer simulation results.
I.
INTRODUCTION
The Linear Induction Motor (LIM) has excellent
performance features such as high-starting thrust force,
alleviation of gear between motor and the motion devices,
reduction of mechanical losses and the size of motion devices,
high speed operation, silence, and so on [1]. Because of above
the advantages, the primary type LIM, shown in Fig. 1, has
been widely used in the field of industrial processes and
transportation applications. The driving principles of the LIM
are similar to the traditional Rotary type Induction Motor
(RIM), but its control characteristics are more complicated
than the RIM, and the motor parameters are time-varying due
to the change of operating conditions, such as speed mover,
temperature, and configuration of rail [2].
In a LIM, the primary winding corresponds to the stator
winding of a rotary induction motor (RIM), while the
secondary corresponds to the rotor. There are some
characteristics differences between the RIM and LIM. The
main difference is that the primary of the LIM has a finite
length, and therefore, there is a fringing field at both ends of
the primary. The infinitely long secondary enters the air-gap
field, carries the magnetic flux along with it, and makes, the
distribution of the electromagnetic quantities nonuniform,
resulting in considerable electric and force losses [3]. The
losses, as well as the flux-profile attenuation, become severer
as the speed increases. Such a phenomena is called ‘end effect’
of LIM.
An accurate equivalent circuit model is indispensable for
high performance control for LIM drive. Most of the existing
models of a LIM depend on field theory [4,5]. Hence, they can
not be directly applied for the nonlinear control and in the most
of the present studies, the RIM model is used in order to
control LIM [6-8]. so, because of the end effect, they can not
be valid in high speed operation of LIM. In [6], an adaptive
Primary(Mover)
Secondary
Fig. 1. Linear Induction Motor.
controller is used in order to control LIM at low speeds. This
controller is robust only against the mechanical parameter
uncertainties. In [7], an indirect field oriented control is used to
control LIM at low speed which is not robust in respect to all
machine parameter variations and uncertainties. In [8], a robust
nonlinear controller has been proposed for a primary type LIM
which is based on combination of SM control and input-output
feedback linearization. Using the RIM model, the drive system
of [8] is robust to parameters variation only at low speed
operation. Using the control method presented in [8], the
main objective of this paper is to introduce a robust nonlinear
controller for LIM which will be robust and stable subject to
the parameter uncertainties and external unknown load force at
low and high speed operation. Reference [9] has described a
secondary field-oriented control scheme for LIM with the end
effect, but the robustness is not considered.
II. LIM MODEL
The LIM fifth order model with end effect, in stationary d:q
axis reference frame with secondary fluxes and primary
currents as state variable can be expressed by[9]:
d λqr
1
L mf
π
i qs + n p V e λqr − λqr
dt
Tr
h
Tr
d λdr L mf
π
1
=
i ds − λdr − n p V e λqr
dt
Tr
Tr
h
=
(1)
(2)
di ds
R
L mf
1− σ
σ&
µ&
= −( s +
+
)i ds + (
−
)λ
σ Ls σT r σ LS
σ Ls L rT r σ LS dr
dt
+
n p L mf π
σ Ls L r h
V e λqr +
1
σ Ls
v ds
(3)
di qs
dt
= −(
Rs
L mf
1− σ
σ&
µ&
+
+
)i qs + (
−
)λqr
σ Ls σT r σ LS
σ L s L rT r σ LS
n p L mf π
1
−
V e λdr +
v qs
σ Ls L r h
σ Ls
Fe =
3π n p µ
2h
(4)
(λdr i qs − λqr i ds ) = M
dV e
+ DV e + Fl
dt
(5)
where i ds , i qs , λdr , λqr , v ds , v qs and V e are the primary
d-q currents, secondary d-q fluxes, primary d-q voltages and
mover speed, respectively. In addition, R s is the primary
resistance, R r is the secondary resistance, n p is the number
of pole pairs, h is the pole pitch, Fl is the external force
disturbance, M is the total mass of mover and D is the
viscous friction coefficient, T r = L r R r is the secondary
time constant,
Ls = L mf + L ls is the primary inductance,
L r = L mf + L lr is the secondary inductance, L ls is the
primary leakage inductance, L lr is the secondary leakage
inductance, σ = 1 − L 2mf Ls L r , µ = L mf L r , δ = σ Ls .
L mf is a time variant parameter as a function of V e given
by [9]:
L mf = L m (1 − f (Q ))
where
(6)
L m = 1.5L mo and L mo is the magnetizing inductance
at zero speed, and:
f (Q ) =
1 − e −Q
,
Q
Q=
lR r
(L m + L lr ) V e
(7)
and l is the primary length.
In equations 1 to 5, δ& and µ& are functions of other system
states and can be written as follows:
2L L − L
δ& = −L m f& (Q ){1 − mf r2 mf }
Lr
2
µ& = −L m f& (Q )
L lr
L2r
(8)
L mo + Llr
(1 − e −Q − Qe −Q ) ×
MlR r
{
3π n p µ
2h
(λdr i qs − λqr i ds ) − DV e − Fl }
A. ANN Basics
Define W as the controller of ANN weights, then the net
output is [10]:
y =W T φ (x )
(11)
Let S be a compact simply connected set of n , with map
f : S → n , define C m (s ) the functional space such that
f (x ) ∈ C m (s ) , x (t ) ∈ S can be approximated by a neural
network as:
f (x ) = W T φ (x ) + ε (x )
(12)
With ε (x ) a ANN functional reconstruction error vector
and φ (x ) is sigmoid activation function.
B. Robust Backstepping Control of LIM Using ANN
Using the well known fifth order LIM model in a stationary
two axis reference frame where the secondary fluxes and
primary currents are assumed as state variables, the robust
nonlinear controller is designed in the following way:
Dividing the above LIM model into two nonlinear
subsystems, where i ds , i qs are the outputs for the first
subsystem which are simultaneously assumed the fictitious
input of the second sub-system.
Assume that:
1. The reference trajectories V e ∗ and λr* are differentiable
and bounded.
2.the Load force is an unknown constant and resistances,
inductances and moment of inertia are unknown and bounded.
In the first step of the controller design, i ds , i qs are
assumed as fictions controls for the second sub-system. The
main objective is to obtain these controls so that the desired
secondary speed and amplitude signals are perfectly tracked in
spite of machine parameters and external load force
uncertainties. Considering V e ∗ and λr* , the tracking error
equations are obtained as:
e1 = V e −V e ∗
e 2 = (λdr 2 + λqr 2 ) − λr∗2 = λr 2 − λr∗2
(13)
(9)
where f& (Q ) is also a function of the system states as:
f& (Q ) =
III. ROBUST BACKSTEPPING CONTROLER
(10)
Derivating (13) with respect to time (t), and using equations
(1), (2) and (5) yields:
⎧ de
3π n p µ
F
⎪ 1=
(λdr i qs − λqr i ds ) − l −V&e∗
2 hM
M
⎪ dt
⎪
(14)
⎧
⎫
−R
R
⎪ de 2
= 2λdr ⎨ r λdr + r L mf i ds ⎬
⎨
Lr
⎩ Lr
⎭
⎪ dt
⎪
⎧ −R r
⎫
R
⎪
λqr + r L mf i qs ⎬ − 2λr∗λ&r∗
+ 2λqr ⎨
⎪⎩
Lr
⎩ Lr
⎭
416
or:
D1e& = F1 + G1i
2
R r + Lr 2 R s
δ& ⎫⎪ ⎤
⎪⎧ L
− ⎨ mf
+
⎬ i ds ⎥
σ Ls ⎪⎭ ⎥
σ Ls L r 2
⎩⎪
⎥
& ⎫⎪ ⎥
⎧⎪ L 2 R + L 2 R
δ
r
r
s
− ⎨ mf
+
⎬ i qs ⎥
σ Ls ⎪⎭ ⎥
σ Ls L r 2
⎪⎩
⎦
&
+ D& 2G1−1 (Fˆ1 + K 1e ) + D 2G1−1Fˆ1 +
(15)
with:
⎡ −Fl M & ∗
⎤
⎢ µ − µ Ve
⎥
⎥ ,
F1 = ⎢
⎢ 2 2
2 ∗ &∗ ⎥
⎢ − L λr − R µ λr λr ⎥
r
⎣ m
⎦
(
D 2 G1−1K 1D1−1 (F1 − Fˆ1 − K 1e + G1η )
3π n p
⎡ 3π n p
⎤
λqr
λdr ⎥
⎢−
G1 = ⎢ 2 h
2h
⎥,
2λqr ⎥⎦
⎢⎣ 2λdr
⎡M
⎤
0 ⎥
⎢µ
⎡i ⎤
⎡e ⎤
⎥ , i = ⎢ ds ⎥ , e = ⎢ 1 ⎥
D1 = ⎢
i
1 ⎥
⎢
⎣e 2 ⎦
⎣⎢ qs ⎦⎥
⎢ 0 µR ⎥
r
⎣
⎦
(16)
i is
(21) which is necessary to cancel the effect of G1η in (17).
Combining (16) and (18), gives:
D 2η& = F2 − Fˆ2 − K 2η − G1T e
(23)
(17)
−
, η =i−i
D 2η& = F2 + G 2v
Wi , is assumed to be as follows:
(19)
n p π L mf
⎡ L mf R r
µ&
−
)λdr +
V λ
⎢(
2 σL
σ Ls L r h e qr
s
⎢ σ Ls Lr
F2 = D 2 × ⎢
n πL
⎢( L mf R r − µ& )λ − p mf V λ
qr
⎢ σ L L 2 σ Ls
σ Ls Lr h e dr
s r
⎣
F1 = W1T φ1 + ε1
, ε1 < ε1N = cte
F2 = W2T φ2 + ε 2
, ε 2 < ε 2 N = cte
(24)
where φ1 , φ2 provide suitable basis functions. From (23), one
can find that net reconstruction error ε i ( x) is bounded by a
known constant ε iN .
Assumption 3: The ideal weighs are bounded by known
positive values so that:
where:
⎡1 0 ⎤
⎢
⎥
⎣0 1 ⎦
C. F1 , F2 Approximation Using ANN
In this section, functions F1 , F2 are approximated by two
two-layer ANN. In. Using ANNs approximation property,
F1 , F2 as outputs of two two-layer ANN with constant weights,
(18)
In the second step, the control v is obtained in such a way
that η in Equation (17), becomes as small as possible.
Differentiating η with respect to time yields:
⎡v ds ⎤
⎡1 0 ⎤
v = ⎢ ⎥ , G2 = ⎢
⎥ , D 2 = σ Ls
v
⎣0 1 ⎦
⎣⎢ qs ⎦⎥
(22)
second two layer ANN. In addition a term −G1T e is added in
where K1 a design parameter and Fˆ1 the estimate of F1 which
will be estimated in the next section with a two layer ANN.
Substituting (16) into (15) gives:
D1e& = F1 − Fˆ1 − K 1e + G1η
To make η as small as possible, the following control is
chosen :
In (21), Fˆ2 is an estimate of F2 that is also estimated by a
By treating i as a fictitious input, a controller for the ideal
designed as:
, K1 > 0
)
v = G 2−1[−Fˆ2 − K 2η − G1T e ]
From (16) it is clear that G1 is known and invertible matrix.
i = G1−1[− Fˆ1 − K 1e ]
(21)
(20)
W1
≤ W2 M
(25)
≤ Z M , Z = diag {W1 ,W2 }
(26)
F
≤ W1M
, W2
F
or equivalently:
Z
F
The actual inputs to ANN1 are λr ,V e , λ&r∗ ,V&e ∗ and actual
inputs to ANN2 are λr , V e , λ&r∗ , V&e ∗ , λdr , λqr , i ds , i qs , e1 , e 2 .
417
Online ANN approximation of F1 is defined by:
{
Fˆ1 =Wˆ1T φ1
(27)
D1e& =W%1T φ1 − K 1e + G1η + ε1
V& ≤ −λmin ζ
(28)
Fˆ2 =Wˆ2T φ2
(29)
Then error dynamic (22) will be:
D 2η& =W%2T φ2 − K 2η − G1T e + ε 2
D. Updating ANNs Weights
In this part, the stability of proposed controller, is proved
based on Lyapunov stability theory. This analysis shows that
tracking errors and updated weighs are Uniformly Ultimately
Bounded (UUB).
Theory : Let the desired trajectories V e* , λr* be bounded.
Take the control input (21) with weigh updates be provided by:
Γ1 = Γ1T
(31)
⎡ k ( Z%
⎢⎣ ω
}
(33)
, K = diag {K 1 , K 2 }
(34)
ζ = [e ,η ]
2
− Z%
(36)
F
Differentiate (32):
{
}
V& = −ζ T K ζ + k ω e tr Z% T ( Z − Z% ) + ζ T ε
F
(Z M − Z%
F
( Z%
F
F
) + εN ζ
− Z M ) − ε N ⎤⎥
⎦
(37)
( Z%
F
− Z M ) − ε N ⎤⎥ =
⎦
− Z M / 2) 2 − k ω Z M2 / 4 + λmin ζ − ε N ⎤⎥
⎦
(38)
which is guaranteed positive if
ζ > [k ω Z M2 / 4 + ε N ] / λmin
Z%
F
> Z M / 2 + Z M2 / 4 + ε N / k ω
(39)
(40)
Thus, it can be concluded that and are UUB [10] and the
system is stable.
Note 1: Small tracking error bounds may be achieved by
selecting large control gain K . The parameter kω offers a
design tradeoff between the relative eventual magnitudes of
ζ and Z% , a smaller k yields a smaller ζ and a larger
Z%
> 0, Γ 2 = ΓT2
Z% = diag W%1 ,W%2 , Γ = diag {Γ1 , Γ 2 }
T T
F
F
ω
F
> 0 and
scalar positive constant kω . Then the errors η (t ), e(t ) are
UUB. ANN updated weights are bounded. The errors η (t ), e(t )
can be kept as small as desired by increasing gains K 1, K 2 .
Proof: Consider the following Lyapunov candidate:
0 ⎤
⎡D
1
1
% T −1 %
(32)
V = ζT ⎢ 1
⎥ ζ + tr ( Z Γ Z )
D
0
2
2
2⎦
⎣
with any constant matrices
+ k ω ζ Z%
⎡λ
ζ + k ω Z%
⎣⎢ min
(30)
in
Note that there is a term G1η in (27) and a term
(29). This means there are couplings between the error
dynamics (27) and (29).
{
F
F
where λmin is the minimum singular value of K .The righthand side of (37) is negative as long the term inside braces is
positive. Completing the square yields:
−G1T e
&
Wˆ1 = Γ1φ1eT − k ω Γ1 ζ Wˆ1
&
Wˆ2 = Γ 2φ2eT − k ω Γ 2 ζ Wˆ2
2
= − ζ ⎡⎢λmin ζ + k ω Z%
⎣
where W%1 =W 1 −Wˆ1 . Similarly, approximation of F2 is:
T
Z
and having in mind some basics inequalities, from (35) the
following can be obtained:
Linking (18) and (2), gives:
where:
}
tr Z% T ( Z − Z% ) ≤ Z%
F
, and vice versa.
Note 2 : If Wˆi (0) are taken as zeroes the linear proportional
control term − K ζ stabilizes the system on an interim basis.
IV. SYSTEM SIMULATION
Based on the proposed control strategy in this paper, the
block diagram of LIM drive control is shown in Fig. 2.
A C ++ computer program was developed for system
simulation. In this program, the nonlinear equations are solved
based on static forth order Range-Kutta method. The proposed
control method, is tested by simulation for a three-phase LIM
with parameters shown in Table (1).
In this simulation, the controller gains are obtained by trial
and error and are given as
K 1 = diag {1000,1000} , K 2 = diag {2000,1000}
(35)
k ω = 1 , Γi = 10I
Now, using Schwarz inequality [10]:
418
(41)
Table 1
λr (Wb )
: LIM PARAMETERS
0.12
Primary resistance
R s = 5.36Ω
0.1
Secondary resistance
R r = 3.53Ω
0.08
D = 36.08Kg / sec
0.06
np =1
0.04
Primary leakage inductance
Ls = 0.0029H
0.02
Secondary leakage inductance
L r = 0.0029H
Magnetizing Inductance at zero speed
L m 0 = 0.0681H
Viscous friction coefficient
Number of pole pairs
Pole pitch
h = 0.027m
Total Mass of moving element
M = 2.78kg
λr∗
0
0
λr
0.5
1
1.5
2
t (sec)
Fig. 3a. Secondary flux
V e (m / s )
1.2
V e∗
1
Simulation results shown in Fig. 3 (3a to 3g) , are obtained
in the case of an exponential reference flux rising up from zero
to 0.1 W b − t at t = 0s with a time constant of τ = 0.1sec , an
exponential reference speed from zero to 1m/sec at t = 0sec ,
a step load disturbance from zero to 5Nm at t = 0.5sec , step
down to 3Nm at t = 1sec and motor electromechanical
parameters assumed to be twice their nominal value.
Fig. 4 (4a to 4g) shows the simulation results obtained for an
exponential reference flux rising up from zero to 0.1 W b − t at
t = 0sec and a sinusoidal reference speed. In flux and speed
control performance, motor electromechanical parameters
assumed to be twice their nominal values.
Referring to Fig. 3f and 4.f, it is clear that the maximum of
f (Q ) function is 0.16. This means that the motor magnetizing
inductance reduces to 84 percent its original value, when motor
moves at the spped of 1m/s.
Ve
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
t (sec)
Fig. 3b. Mover speed
i ds (A )
40
30
20
10
0
-10
-20
-30
CONTROLLER
Fˆ1
-40
0
0.5
1
1.5
2
t (sec)
ANN1
Fig. 3c. d-axis current
i qs (A )
Fˆ2
40
ANN2
30
20
λr∗ ,V e∗
10
− G1T
CONTROLLER
PWM
INVERTER
0
LIM
-10
-20
-30
-40
0
Fig. 2. The overall block diagram of LIM drive system control
0.5
Fig. 3d. q-axis current
419
1
1.5
2
t (sec)
v as (V )
V e (m / s )
800
V e∗
1
600
400
Ve
0.5
200
0
0
-200
-0.5
-400
-1
-600
-800
0
0.5
1
1.5
0
2
t (sec)
0.5
1
Figure 3e. Inverter's output voltage
1.5
t (sec)
Fig. 4b. Mover speed
i ds ( A )
f (Q )
40
0.16
30
0.14
20
0.12
10
0.1
0
0.08
-10
0.06
-20
0.04
-30
0.02
0
0
0.5
1
1.5
-40
0
2
0.5
1
t (sec)
Figure 3f. f(Q) function
1.5
t (sec)
Fig. 4c. d-axis current
Fl (N .m )
i qs (A )
6
30
5
20
4
10
7
40
0
3
-10
2
-20
1
-30
0
0
0.5
1
1.5
t (sec)
2
-40
0
0.5
1
Figure 3g. External force disturbance
1.5
t (sec)
Fig. 4d. q-axis current
λr (W b − t )
v as (V )
0.12
600
λr*
0.1
0.08
400
λr
200
0.06
0
0.04
-200
0.02
-400
0
0
0.5
Figure 4a. Secondary flux
1
-600
0
1.5
t (sec)
0.5
Fig. 4e. Inverter's output voltage
420
1
1.5
t (sec)
f (Q )
Computer simulation results obtained, confirm the
effectiveness and validity of the proposed controller. These
results also confirm that the drive system control is robust and
stable against the parameters uncertainties and unknown load
force disturbance.
0.2
0.15
0.1
0.05
REFERENCES
0
0
0.5
1
[1]
1.5
t (sec)
Fig. 4f. f(Q) function
[2]
Fl (N .m )
[3]
7
6
[4]
5
4
[5]
3
2
[6]
1
0
0
0.5
1
1.5
2
t (sec)
[7]
Fig. 4g. External force disturbance
[8]
CONCLUSIONS
In this paper, a composite nonlinear controller has been
proposed for the LIM secondary flux and speed tracking
control. The nonlinear controller is designed based on the LIM
fifth order model in a fixed two axis reference frame,
combining the backstepping control and ANN. The overall
stability of this controller is proved by Lyapunov theory.
[9]
[10]
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