title : feasible performance evaluations of digitally...
TRANSCRIPT
Title : Feasible Performance Evaluations of Digitally-Controlled Auxiliary Resonant Commutation Snubber-Assisted Three Phase Vo
326 Journal of Power Electronics, Vol. 12, No. 6, November 2012
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m
r
r
ds
ss
r
r
qs
ss
r
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i
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d
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i
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http://dx.doi.org/10.6113/JPE.2012.12.2.326
330 Journal of Power Electronics, Vol. 12, No. 2, March 2012
Robust Recurrent Wavelet Interval Type-2 Fuzzy-Neural-Network Control for a DSP-Based 327
Robust Recurrent Wavelet Interval Type-2 Fuzzy-Neural-Network Control for a DSP-Based PMSM Servo Drive System
Fayez F. M. El-Sousy
College of Engineering, Department of Electrical Engineering, Salman bin Abdulaziz University, Al-Kharj, Saudi Arabia and on leave from the Department of Power Electronics and Energy Conversion, Electronics Research Institute, Cairo, Egypt
Abstract
In this paper, an intelligent robust control system (IRCS) for precision tracking control of permanent-magnet synchronous motor (PMSM) servo drive is proposed. The IRCS comprises a recurrent wavelet-based interval type-2 fuzzy-neural-network controller (RWIT2FNNC), an RWIT2FNN estimator (RWIT2FNNE) and a compensated controller. The RWIT2FNNC combines the merits of the self-constructing interval type-2 fuzzy logic system, recurrent neural network and wavelet neural network. Moreover, it performs the structure and parameter-learning concurrently. The RWIT2FNNC is used as the main tracking controller to mimic the ideal control law (ICL) while the RWIT2FNNE is developed to approximate an unknown dynamic function including the lumped parameter uncertainty. Furthermore, the compensated controller is designed to achieve L2 tracking performance with desired attenuation level and to deal with the uncertainties including the approximation error, optimal parameter vectors and higher order terms in Taylor series. Moreover, the adaptive learning algorithms for the compensated controller and the RWIT2FNNE are derived using the Lyapunov stability theorem to train the parameters of the RWIT2FNNE online. A computer simulation and an experimental system are developed to validate the effectiveness of the proposed IRCS. All control algorithms are implemented in a TMS320C31 DSP-based control computer. The simulation and experimental results confirm that the IRCS grants robust performance and precise response regardless of load disturbances and PMSM parameters uncertainties.
Key words: L2 tracking performance, Lyapunov satiability theorem, Permanent-magnet synchronous motor (PMSM) servo drive, Recurrent interval type-2 fuzzy-neural-network (RIT2FNN), Wavelet neural networks
I. Introduction
Recently, advancements in magnetic materials, semiconductor power devices and control theories have made the permanentmagnet synchronous motor (PMSM) drives play a vitally important role in motion-control applications. PMSMs are widely used in high-performance applications such as industrial robots and machine tools because of their compact size, high-power density, high air-gap flux density, high-torque/inertia ratio, high torque capability, high efficiency and free maintenance. The overall performance of a speed and/or position control of PMSM drives depend not only on the quickness and the precision of the system response, but also on the robustness of the control strategy which has been carried out to assure the same performances if exogenous disturbances and variations of the system parameters occur. In fact, the control of PMSM drives often necessitates the determination of the machine parameters. The online variations of parameters, which essentially depend on temperature variation, saturation and skin effects, external load disturbance and unmodeled dynamics in practical applications can affect the PMSM servo drive performances [1][6]. On the other hand, the computed torque controller (CTC) is utilized to linearize the nonlinear equation by cancellation of some, or all, nonlinear terms such that a linear feedback controller is designed to achieve the desired closed-loop performance. However, an objection to the real-time use of such control scheme is the lack of knowledge of uncertainties [7][9]. Therefore, to compensate for various uncertainties and nonlinearities, sophisticated control strategy is very important in PMSM servo drives.
Nowadays, intelligent control techniques in much research have been developed to improve the performance of the PMSM servo drives and to deal with the nonlinearities and uncertainties using fuzzy logic, neural network, wavelet neural networks (WNN) and/or the hybrid of them [10][15]. The concept of incorporating fuzzy logic into a neural network (NN) to constitute fuzzy-neural-network (FNN) has grown into a popular research topic [1625]. However, all of these analyses and implementation are focused on type-1 FNN. On the other hand, a type-2 fuzzy neural network (T2FNN) consists of a type-2 fuzzy linguistic process as the antecedent part and an interval neural network as the consequent part. The interval T2FNN (IT2FNN) is a multi-layer network for the realization of type-2 fuzzy inference system, and it can be constructed from a set of type-2 fuzzy rules. Furthermore, the IT2FNN possess the merits of both the type-2 fuzzy systems and the neural network. Therefore, it does not require mathematical models and have the ability to approximate nonlinear systems. In addition, the IT2FNN is superior to the type-1 FNN in the control of complicated and highly nonlinear system such as PMSM servo drive system. On the other hand, there were only a few works to analyze and simulate the type-2 FNN or IT2FNN [2636]. In [52, 53], at nominal parameters of the PMSM, a two-degrees-of-freedom integral plus proportional and rate feedback (2DOF I-PD) position controller is designed and analyzed. Although the desired tracking and regulation position control can be realized using the 2DOF I-PD position controller at the nominal PMSM parameters, the performance of the servo drive is still sensitive to parameter variations. To solve this problem, an IRCS is proposed.
In this paper, an IRCS is proposed for identification and control the rotor position of the PMSM servo drive. First, based on the principle of L2 tracking performance, a position tracking controller is designed and analyzed. The IRCS comprises an RWIT2FNN controller (RWIT2FNNC), RWIT2FNN estimator (RWIT2FNNE) and a compensated controller. In the proposed control scheme, the RWIT2FNNC, which combines the merits of the self-constructing interval type-2 fuzzy logic system, recurrent neural network and wavelet neural network, is used as the main tracking controller to mimic the ICL. Additionally, to relax the requirement of the lumped uncertainty, an RWIT2FNNE is developed to approximate an unknown dynamic function. In addition, a compensated controller is designed to achieve L2 tracking performance with desired attenuation level and to deal with the uncertainties including the approximation error, optimal parameter vectors, and higher order terms in Taylor series. Moreover, the adaptive learning algorithms for the compensated controller and the RWIT2FNNE are derived using the Lyapunov stability theorem to train the parameters of the RWIT2FNNE online, so that the stability of the PMSM servo drive can be guaranteed. A computer simulation is developed and an experimental system is established for demonstration and to verify the effectiveness of the proposed IRCS for PMSM servo drive. All control algorithms has been implemented in a control computer based on a TMS320C31 and TMS320P14 DSPs control board. The dynamic performance of the PMSM servo drive has been studied under load changes and parameters uncertainties. The numerical simulations and experimental results are given to demonstrate the effectiveness of the proposed IRCS.
This paper is organized as follows. Section 2 presents the fieldoriented control (FOC) and dynamic analysis of PMSM servo drive. Both the problem formulation and the description of the IRCS of the PMSM servo drive are introduced. The design methodology for the compensated controller and the IRCS are given in Section 3. Also, The design procedures and adaptive learning algorithms of the proposed IRCS and the compensated controller are described in details in Section 3. The validity of the design procedure and the robustness of the proposed controller is verified by means of computer simulation and experimental analysis. Control algorithms have been developed in a control computer that is based on a TMS320C31 and TMS320P14 DSP DS1102 board. The dynamic performance of the PMSM drive system has been studied under load changes and parameter uncertainties. Numerical simulations and experimental results are provided to validate the effectiveness of the proposed control system in Section 4. Conclusions are introduced in Section 5.
II. Modeling of PMSM and Dynamic Analysis
The voltage equations of the stator windings in the rotating reference frame can be expressed in (1) and (2). Then, using FOC and setting d-axis current as zero, the electromagnetic torque is obtained as given in (3) and (4) [1]. The parameters of the surface-mounted PMSM are listed in Table (1).
(1)
r
qs
ss
r
r
ds
ss
r
ds
s
r
ds
i
L
i
dt
d
L
i
R
V
w
-
+
=
(2)
The electromagnetic torque can be expressed as:
r
qs
t
r
qs
m
e
i
K
i
P
T
=
=
l
).
2
/
).(
2
/
3
(
(3)
L
r
m
r
m
e
T
dt
d
P
dt
d
P
J
T
+
+
=
q
b
q
2
2
2
2
(4)
From (3) and (4), the motion dynamics can be simplified as:
L
m
r
qs
m
t
r
m
m
r
T
J
P
i
P
J
K
P
J
.
1
.
2
.
)
2
/
(
2
.
*
-
+
-
=
q
b
q
&
&
&
(5)
L
m
m
r
m
r
T
D
t
U
B
A
.
)
(
.
+
+
=
q
q
&
&
&
(6)
Now, assume that the parameters of the PMSM are well known and the external load disturbance is absent, rewriting (6) can represent the model of the PMSM servo drive system.
)
(
.
)
(
)
(
t
U
B
t
A
t
m
r
m
r
+
=
q
q
&
&
&
(7)
By considering the dynamics in (6) with parameter variations, disturbance load and unpredictable uncertainties will give:
L
m
mn
m
mn
r
m
mn
r
T
D
D
t
U
B
B
t
A
A
t
).
(
)
(
).
(
)
(
)
(
)
(
D
+
+
D
+
+
D
+
=
q
q
&
&
&
(8)
)
(
)
(
.
)
(
)
(
t
t
U
B
t
A
t
mn
r
mn
r
G
+
+
=
q
q
&
&
&
(9)
where Amn, Bmn and Dmn are the nominal parameters of Am, Bm and Dm respectively. (Am, (Bm, (Dm and TL are the uncertainties due to mechanical parameters Jm and (m, and ((t) is the lumped parameter uncertainty and is defined as:
L
m
mn
m
r
m
T
D
D
t
U
B
t
A
t
)
(
)
(
.
)
(
)
(
D
+
+
D
+
D
=
G
q
&
(10)
The bound of the lumped parameter uncertainty (PU) is assumed to be given, that is,
q
K
t
G
)
(
(11)
where K( is a given positive constants.
TABLE I
Parameters of PMSM Used in Simulation and Experimentation
Quantity
Symbol
Value
Nominal power
Pn
1 hp (3-phase)
Stator self inductance
Lss
0.05 H
Stator resistance
Rs
1.5 (
Voltage constant
(m
0.314 V.s/rad
Number of poles
P
4
Rotor inertia
Jm
0.003 kg.m2
Friction coefficient
(m
0.0009 N.m/rad/sec
Nominal speed (electrical)
(r
377 rad/sec
Rated torque
Te
3.6 N.m
Rated current
I
4 A
Rated voltage
VL-L
208 V
Rated frequency
f
60 Hz
Torque constant
Kt
0.95 N.m/A
Resolution of the encoder
n
4(10000 p/r
III. Intelligent Robust Control System (IRCS)
In this section, an IRCS is designed for identification and control the rotor position of the PMSM servo drive. The IRCS comprises an RWIT2FNNC, RWIT2FNNE and a compensated controller. In the proposed control scheme, the RWIT2FNNC is used as the main tracking controller to mimic the ICL. Additionally, to relax the requirement of the lumped uncertainty, an RWIT2FNNE is developed to approximate an unknown dynamic function. In addition, a compensated controller is designed to achieve L2 tracking performance with desired attenuation level and to deal with the uncertainties including the approximation error, optimal parameter vectors, and higher order terms in Taylor series. Moreover, the adaptive learning algorithms for the compensated controller and the RWIT2FNNE are derived using the Lyapunov stability theorem to train the parameters of the RWIT2FNNE online, so that the stability of the PMSM servo drive can be guaranteed.
The control problem is to find a control law so that the rotor position
)
(
t
r
q
can track the desired position
)
(
t
m
r
q
. To achieve this control objective, we define a tracking error vector as
T
m
m
e
e
E
]
[
q
q
&
=
. where
)
(
t
m
r
q
and
)
(
t
m
r
q
&
are the desired position and speed of the PMSM servo drive system;
)]
(
)
(
[
t
t
e
r
m
r
m
q
q
q
-
=
,
)]
(
)
(
[
t
t
e
r
m
r
m
q
q
q
&
&
&
-
=
, and
)]
(
)
(
[
t
t
e
r
m
r
m
q
q
q
&
&
&
&
&
&
-
=
denote the position, speed and acceleration errors of the PMSM servo drive system. The PMSM parameters are assumed to be precisely known and the external load torque is assumed to be measurable. The ideal control law (ICL) is designed as [51]
]
)
(
)
(
)
(
[
)
(
)
(
1
*
KE
t
t
A
t
B
t
i
t
U
r
mn
m
r
mn
r
qs
ICL
+
G
-
-
=
=
-
q
q
&
&
&
(12)
where
]
[
2
1
k
k
K
=
, in which k1 and k2 are positive constants. Substituting (12) into (9) results the error dynamics,
0
)
(
)
(
)
(
1
2
=
+
+
t
e
k
t
e
k
t
e
m
m
m
q
q
q
&
&
&
.
Suppose the control gain K is chosen such that all roots of the characteristic polynomial of error dynamics lie strictly in the open left half of the complex plane. This implies that the position tracking error will converge to zero when time tends to infinity. The PMSM servo drive system is asymptotically stable when the control effort UICL is applied. However, the equation of the ideal control law is not feasible in practice because the PMSM parameters vary and the variation cannot be measured or predicted and the external load torque in a PMSM system is not known. Therefore the control effort (12) cannot be made. Hence, an IRCS is proposed herein to approach the ICL. The configuration of the proposed IRCS for PMSM servo drive is shown in Fig. 1. The proposed control law is assumed to take the following form:
)
(
)
(
)
(
)
(
2
2
2
*
t
U
t
U
t
U
t
U
L
qs
FNNE
RWIT
qs
FNNC
RWIT
qs
qs
+
+
=
(13)
where
)
(
)
(
*
2
t
i
t
U
e
qs
FNNC
RWIT
qs
=
is the RWIT2FNNC,
)
(
)
(
2
t
t
U
FNNE
RWIT
qs
W
=
is the RWIT2FNNE and
)
(
2
t
U
L
qs
is the compensated controller.A. Description of Wavelet Bases and Wavelet Neural Network (WNN)
The architecture of a WNN is shown in Fig. 2, which is a three-layer neural network including input, hidden, and output layer. Each output (i.e.
L
,
,
2
1
H
H
etc.) formulated by a sub-WNN is defined as a WNN base. In the RWIT2FNN, the WNN bases do not exist in the initial state, and they are generated on line concurrently with the fuzzy rules using the structure learning algorithm. The WNNs are characterized by weights and wavelet bases. Each linear synaptic weight of wavelet basis is adjustable by learning. Notably, the ordinary wavelet neural network model applications are often useful for normalizing the input vectors into the interval (0; 1) [37]-[38]. The
)
(
.
i
b
a
x
f
functions which are used to input vectors to fire up the wavelet interval are then calculated. Obviously, a value
b
a
.
f
is obtained as follows:
Error
Function
++
FNNCRWIT
qs
U
2
Reference Model
-
+
*
r
m
r
r
ICL
qs
U
Intelligent Robust Control
System (IRCS)
Ideal Control
Law
Operator
Index
I
Error
Function
E
-
+
E
E
m
e
e
6
Adaptive
Estimation Law
Compensated
Controller
E
RWIT2FNNE
CC
qs
U
+
+
yyyyy
W
,,,,
,
,
,
,
W
Adaptive Laws
54321
E
E
r
K
Error
Dynamics
m
r
On-Line Structure
Learning Algorithm
new
k
new
k
new
k
new
k
W,,,
ij
ij
Lk
Rk
jk
W
k
ij
ij
4
Lk
4
Rk
jk
W
k
,,,,
W
On-Line Parameter
Learning Algorithm
Recurrent Wavelet-Based
Interval Type-2 Fuzzy-
Neural-Network Controller
(RWIT2FNNC)
E
+
PMSM
+
+
)/(
mm
m
Jss
K
r
K
t
cr
qs
i
r
qs
i0
cr
ds
i
s
ss
s
R
/1
/1
s
ss
s
R
/1
/1
G
c
-q
G
c
-d
*r
qs
e
*r
ds
e
*r
qs
V
*r
ds
V
r
ds
i
r
qs
e
r
ds
e
+
-
++
++
+
+-
-
-
+
ssr
L
ssr
L
ssr
L
ssr
L
0L
m
/1
L
T
e
T
Fig. 1. Structure of the proposed intelligent robust control system (IRCS) using RWIT2FNN for PMSM servo drive.
)
cos(
,
)
(
0
5
.
0
5
.
0
)
cos(
)
(
.
b
ax
otherwise
x
x
x
i
b
a
i
i
i
-
=
-
=
f
f
(14)
where b = 1; . . . ; a and a = 1; . . .;m.
=
=
=
m
a
a
b
M
1
1
1
(15)
The above equation formulates the nonorthogonal wavelets in a finite range, where b denotes a shifting parameter, the maximum value of which equals the corresponding scaling parameter a. In the RWIT2FNN model, the wavelet bases do not exist in the initial state, and the amount generated by the online learning algorithm is consistent between wavelet bases and fuzzy rules. Obviously, a crisp value
b
a
.
f
can be obtained as follows:
X
x
n
i
i
b
a
b
a
=
=
1
.
.
)
(
f
j
(16)
where
X
means the number of input dimension. The final output of the wavelet neural networks is
m
m
s
jM
s
j
s
j
s
j
b
a
M
k
s
jk
s
j
W
W
W
W
W
H
.
1
.
1
3
0
.
1
2
0
.
0
1
.
1
j
j
j
j
j
+
+
+
+
=
=
=
L
(17)
where
s
j
H
denotes the local output of the WNN for output Hs and the jth rule, the link weight
s
jk
W
is the output action strength associated with the sth output, jth rule and kth
b
a
.
f
, and M denotes the number of wavelet bases, which equals the number of existing fuzzy rules in the RWIT2FNN model.
B. Structure of the Recurrent Wavelet-Based IT2FNN
This subsection introduces the structure of the RWIT2FNN model. The RWIT2FNN integrates the interval type-2 fuzzy logic system, recurrent neural network and WNN. The goal of integrating the RWIT2FNN model with WNN model is to improve the accuracy of function approximation. The interval type-2 Gaussian MF is constructed by a type-1 Gaussian MF with an adjustable uncertain mean and an adjustable standard deviation [27-29]. Figure 3 shows a 2-D type-2 Gaussian MF with an adjustable uncertain mean in
]
,
[
m
m
and an adjustable standard deviation (. It can be described as
Input
Layer
Hidden
Layer
Output
Layer
1
1
x
1
H
3
H
2
H
Normalization
1
2
x
W
11
W
12
W
13
W
21
W
22
W
23
W
33
W
31
W
32
Fig. 2. Wavelet network basis.
]
,
[
,
)
(
2
1
exp
)
(
2
2
~
m
m
m
s
m
l
-
-
=
x
x
A
(18)
The type-2 fuzzy set has a region called a footprint of uncertainty and bounded by an upper MF and a lower MF [34], which are denoted as
)
(
~
x
A
l
and
)
(
~
x
A
l
, respectively. The suggested recurrent WNN-based RIT2FNN is presented as follows:
Rj : IF
1
1
x
is
j
A
1
~
and ... and
1
n
x
is
j
n
A
~
and
)
(
N
h
j
is
j
F
THEN
j
H
is
b
a
M
k
jk
W
.
1
j
=
,
)
1
(
+
N
h
j
is
j
Q
and y1 is
]
,
[
4
4
Lk
Rk
v
v
(19)
)(
~
x
A
)(
~
x
A
Fig. 3. Interval type-2 fuzzy set with adjustable uncertain mean and adjustable standard deviation.
where Rj is the jth rule; hj is the internal variable;
j
H
is the jth WNN base which is the jth output of the local model for rule Rj,
j
A
1
~
is the interval type-2 fuzzy set of the antecedent part, Fj is the output of the recurrent layer;
]
,
[
4
4
Lk
Rk
v
v
is a centroid set with a membership grade of the secondary MF setting to unity, which can be called the weighting interval set, derived from interval type-2 fuzzy sets in the consequent part [31];
3
k
y
is the output of the layer 3;
j
Q
, and
jk
W
are the consequent part parameters for the outputs hj, and
b
a
.
f
, respectively;
5
o
y
is the output of the RWIT2FNN.
The architecture of the RWIT2FNN is a five-layer IT2FNN embedded with dynamic feedback connections and an WNN as shown in Fig. 4. The basic functions and signal propagation of each layer are described as follows.
1) Layer 1: input layer: Each node i in this layer is an input node, which corresponds to one input variable. These nodes only pass the input signal to the next layer and the input variables of the recurrent IT2FNN and WNN are the same. In this layer, the node input and output are represented as
1
1
)
(
i
i
x
N
net
=
(20)
2
,
1
)
(
))
(
(
)
(
1
1
1
1
=
=
=
i
N
net
N
net
f
N
y
i
i
i
i
(21)
)
(
1
1
t
e
x
m
q
=
and
)
(
1
2
t
e
x
m
q
&
=
(22)
where
1
i
x
represents the ith input to the node of layer 1, N denotes the number of iterations. The input variables are
)
(
1
1
r
m
r
m
e
x
q
q
q
-
=
=
, the tracking position error between the desired position command
)
(
t
m
r
q
, and the rotor position
)
(
t
m
r
q
and
)
(
1
2
r
m
r
m
e
x
q
q
q
&
&
&
-
=
=
, tracking position error change of the rotor.
2) Layer 2: membership layer: In this layer, each node performs an interval type-2 fuzzy MF, as shown in Fig. 2. For the jth node the input and output of the membership node can be described as follows:
2
2
2
2
/
)
)(
2
/
1
(
)
(
ij
ij
i
j
x
N
net
s
m
-
-
=
(23)
(
)
(
)
s
j
as
N
y
as
N
y
x
N
net
N
net
f
x
A
N
y
ij
ij
j
ij
ij
j
ij
ij
i
j
j
j
i
j
i
j
,..,
1
,
)
(
)
(
2
1
exp
))
(
exp(
))
(
(
)
(
~
)
(
2
2
2
2
2
2
2
2
2
2
=
=
=
=
-
-
=
=
=
=
m
m
m
m
s
m
(24)
where (ij and (ij are the mean and standard deviation of the Gaussian function in the jth term of the ith input linguistic variable
2
i
x
to the node of layer 2, respectively, and s the number of linguistic values with respect to each input node. As shown in Fig. 2-b, type-2 MFs can be represented as an interval bound by the upper MF
)
(
~
x
A
l
and the lower MF
)
(
~
x
A
l
. Therefore, the output of layer 2
)
(
2
N
y
j
is also represented as
)]
(
),
(
[
2
2
N
y
N
y
j
j
.
3) Layer 3: rule layer: This layer includes rule layer and recurrent layer of the RWIT2FNN and the output layer of the WNN in which each output is defined as a WNN base. For the internal variable hk in the recurrent layer, the following sigmoid membership function is used:
}
exp{
1
1
k
k
h
F
-
+
=
, k=1, 2,.,m (25)
Q
=
=
m
k
k
k
k
u
N
h
1
3
)
(
(26)
where hk is the recurrent unit acting as the memory element, and (k is the recurrent weight. Moreover, the neurons in the rule layer represent the preconditioning part of one interval type-2 fuzzy logic rule. Thus, the neuron in this layer is denoted by (, which multiplies the incoming signals from layer 2 and the recurrent layer, and outputs the product result, i.e., the firing strength of a rule. For the kth rule node:
)
(
)
(
3
3
3
3
N
u
x
F
N
net
k
j
j
jk
k
k
=
v
(27)
)
(
))
(
(
)
(
3
3
3
3
N
net
N
net
f
N
u
k
k
k
k
=
=
(28)
Substituting (25) and (27) into (28) will yield,
}
exp{
1
1
)
(
)
(
1
2
3
3
1
3
3
3
-
+
=
=
=
=
n
j
j
jk
k
k
n
j
j
jk
k
k
y
h
N
u
x
F
N
u
v
v
(29)
,......,
1
,
}
exp{
1
1
)
(
}
exp{
1
1
)
(
)
(
1
2
3
3
1
2
3
3
3
n
k
y
h
N
u
y
h
N
u
N
u
n
j
j
jk
k
k
n
j
j
jk
k
k
k
=
-
+
=
-
+
=
=
=
=
v
v
(30)
Substituting (24) into (30) will give the output of this layer
]
,
[
3
3
k
k
u
u
as follows:
(
)
(
)
(
)
(
)
n
k
x
h
N
u
x
h
N
u
N
u
n
j
ij
ij
i
jk
k
k
n
j
ij
ij
i
jk
k
k
k
,......,
1
,
2
1
exp
}
exp{
1
1
)
(
2
1
exp
}
exp{
1
1
)
(
)
(
1
2
2
2
3
3
1
2
2
2
3
3
3
=
-
-
-
+
=
-
-
-
+
=
=
=
=
s
m
v
s
m
v
(31)
where
3
j
x
represents the jth input to the node of layer 3;
3
jk
v
are the weights between the membership layer and the rule layer and are set to be equal to unity to simplify the implementation for the real-time control; and n is the number of rules. Similar to layer 2, the output of this part in layer 3 is represented as
]
,
[
3
3
k
k
u
u
. In the second part in layer 3, the neuron in this layer multiplies the incoming signals, which are
k
H
from the output of the WNN and
3
k
u
from the output of layer 3 in the recurrent WIT2FNN part. The mathematical function of each node k is derived with (17) as:
3
.
1
3
3
.
.
)
(
k
b
a
M
k
s
jk
k
k
k
u
W
u
H
N
y
=
=
=
j
(32)
From (31) and (32)
n
k
N
u
W
N
u
W
N
y
n
j
k
b
a
M
k
s
jk
n
j
k
b
a
M
k
s
jk
k
,......,
1
,
))
(
.(
))
(
.(
)
(
1
3
.
1
1
3
.
1
3
=
=
=
=
=
=
j
j
(33)
(
)
(
)
(
)
(
)
n
k
x
W
h
x
W
h
N
y
n
j
ij
ij
i
jk
b
a
M
k
s
jk
k
ij
ij
i
jk
n
j
b
a
M
k
s
jk
k
k
,......,
1
,
2
1
exp
.
}
exp{
1
1
2
1
exp
.
}
exp{
1
1
)
(
1
2
2
2
3
.
1
2
2
2
3
1
.
1
3
=
-
-
-
+
-
-
-
+
=
=
=
=
=
s
m
v
j
s
m
v
j
(34)
4) Layer 4: type-reduction layer: This layer is used to implement the type-reduction. Type-reduction is very intensive, and there exist many kinds of type-reduction, such as centroid, height, centre-of-sets and modified height. In a type-1 FLS, the height defuzzification is computationally inexpensive and gives satisfactory results. However, in a type-2 FLS, the height type-reduction does not perform as well. The centre-of-sets type-reduction does a better job. Therefore the centre-of-set type-reduction algorithm [32] is adopted in this paper. Furthermore, the process of this layer is described as follows:
=
=
=
n
k
k
n
k
k
k
l
N
y
N
y
N
net
1
3
1
3
4
4
)
(
)
(
)
(
v
(35)
1
,
)
(
)
(
)
(
)
(
)
(
))
(
(
)
(
1
3
1
3
4
4
1
3
1
3
4
4
4
4
4
4
=
=
=
=
=
=
=
=
=
=
=
=
l
y
N
y
N
y
y
y
N
y
N
y
y
N
net
N
net
f
N
y
L
T
L
n
k
Lk
n
k
Lk
Lk
Ll
R
T
R
n
k
Rk
n
k
Rk
Rk
Rl
l
l
l
l
v
v
v
v
(36)
where
]
,
[
4
4
4
Lk
Rk
k
v
v
v
is the centroid of the type-2 interval consequent set,
T
Rn
R
R
R
]
[
4
4
2
4
1
v
v
v
v
L
=
,
T
Ln
L
L
L
]
[
4
4
2
4
1
v
v
v
v
L
=
,
T
n
k
Rk
Rn
n
k
Rk
R
n
k
Rk
R
R
N
y
N
y
N
y
N
y
N
y
N
y
y
=
=
=
=
1
3
3
1
3
3
2
1
3
3
1
)
(
)
(
)
(
)
(
)
(
)
(
L
(37)
T
n
k
Lk
Ln
n
k
Lk
L
n
k
Lk
L
L
N
y
N
y
N
y
N
y
N
y
N
y
y
=
=
=
=
1
3
3
1
3
3
2
1
3
3
1
)
(
)
(
)
(
)
(
)
(
)
(
L
(38)
In order to compute
)
(
4
N
y
Rl
and
)
(
4
N
y
Ll
, we need to compute
]
,
[
4
4
Lk
Rk
v
v
. This can be done using exact computational procedure given in [32]. Here briefly provide the computation procedure for
)
(
4
N
y
Rl
and
)
(
4
N
y
Ll
. First, the weighting interval set
]
,
[
4
4
Lk
Rk
v
v
)
,....,
1
(
n
k
=
should be set first before the computation of
)
(
4
N
y
l
. Moreover, Algorithms 1 and 2 listed below are the type-reduction algorithms to compute
)
(
4
N
y
Rl
and
)
(
4
N
y
Ll
[32], [34], respectively.
Algorithm 1: Without loss of generality, assume that the weighting interval
4
Rk
v
and
4
Rl
y
are arranged in an ascending order, i.e.,
4
4
2
4
1
Rn
R
R
v
v
v
L
and
4
4
2
4
1
Rn
R
R
y
y
y
L
.
(1)Compute
4
Rl
y
in (36) by
)
(
(
)
(
3
3
N
y
N
y
k
Rk
=
2
/
))
(
3
N
y
k
+
for
)
,....,
1
(
n
k
=
, and let
4
4
~
Rl
Rl
y
y
@
.
(2)Find
)
1
~
1
(
-
n
R
R
such that
4
)
1
~
(
4
4
~
~
+
R
R
Rl
R
R
y
v
v
.
(3)Compute
4
Rl
y
in (36) with
)
(
)
(
3
3
N
y
N
y
k
Rk
=
for
R
k
~
and
)
(
)
(
3
3
N
y
N
y
k
Rk
=
for
R
k
~
>
, and set
4
4
~
~
Rl
Rl
y
y
@
.
(4)If
4
4
~
~
~
Rl
Rl
y
y
@
, then go to step (5). If
4
4
~
~
~
Rl
Rl
y
y
=
, then set
4
4
~
~
Rl
Rl
y
y
=
and stop.
(5)Set
4
4
~
~
~
Rl
Rl
y
y
=
and return to step (2).
Algorithm 2: Without loss of generality, assume that the weighting interval
4
Lk
v
and
4
Ll
y
are arranged in an ascending order, i.e.,
4
4
2
4
1
Ln
L
L
v
v
v
L
and
4
4
2
4
1
Ln
L
L
y
y
y
L
.
(1)Compute
4
Ll
y
in (36) by
)
(
(
)
(
3
3
N
y
N
y
k
Lk
=
2
/
))
(
3
N
y
k
+
for
)
,....,
1
(
n
k
=
, and let
4
4
~
Ll
Ll
y
y
@
.
(2)Find
)
1
~
1
(
-
n
L
L
such that
4
)
1
~
(
4
4
~
~
+
L
L
Ll
L
L
y
v
v
.
(3)Compute
4
Ll
y
in (36) with
)
(
)
(
3
3
N
y
N
y
k
Lk
=
for
L
k
~
and
)
(
)
(
3
3
N
y
N
y
k
Rk
=
for
L
k
~
>
, and set
4
4
~
~
Ll
Ll
y
y
@
.
(4)If
4
4
~
~
~
Ll
Ll
y
y
, then go to step (5). If
4
4
~
~
~
Ll
Ll
y
y
@
, then set
4
4
~
~
Ll
Ll
y
y
=
and stop.
(5)Set
4
4
~
~
~
Ll
Ll
y
y
=
and return to step (2).
The Algorithms 1 and 2 provide the method to separate
)
(
3
N
y
Rk
and
)
(
3
N
y
Lk
into two sides by the points
R
~
and
L
~
, respectively. Moreover, one side uses lower firing strengths
)
(
3
N
y
k
, and the other side uses upper firing strengths
)
(
3
N
y
k
. Therefore, (36) can be rewritten as
1
,
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
~
1
1
~
3
3
~
1
1
~
3
4
3
4
1
3
1
3
4
4
~
1
1
~
3
3
~
1
1
~
3
4
3
4
1
3
1
3
4
4
4
=
+
+
=
=
+
+
=
=
=
=
+
=
=
+
=
=
=
=
+
=
=
+
=
=
=
l
N
y
N
y
N
y
N
y
N
y
N
y
y
N
y
N
y
N
y
N
y
N
y
N
y
y
N
y
L
k
n
L
k
Lk
Lk
L
k
n
L
k
Lk
Lk
Lk
Lk
n
k
Lk
n
k
Lk
Lk
Ll
R
k
n
R
k
Rk
Rk
R
k
n
R
k
Rk
Rk
Rk
Rk
n
k
Rk
n
k
Rk
Rk
Rl
l
v
v
v
v
v
v
(39)
5) Layer 5: output layer: This layer performs the linear combination of
)
(
4
N
y
Rl
and
)
(
4
N
y
Ll
, i.e.,
)
(
2
2
*
4
4
5
t
U
y
y
y
FNNC
RWIT
qs
Ll
Rl
o
=
+
=
(40)
)
,
,
,
,
(
2
1
)
(
2
1
1
5
Q
=
+
=
W
x
y
y
y
y
T
L
T
L
R
T
R
o
s
m
v
v
v
(41)
where
5
o
y
is the output of RWIT2FNN and
FNNC
RWIT
U
2
*
qs
is the control effort of the servo drive system.
T
T
L
T
R
]
[
v
v
v
=
,
T
L
R
y
y
y
]
[
=
,
]
[
1
2
1
1
1
x
x
x
=
,
]
[
2
21
1
11
S
S
m
m
m
m
m
L
L
=
,
]
[
2
21
1
11
S
S
s
s
s
s
s
L
L
=
]
[
1
Mj
j
W
W
W
L
=
, and
]
[
1
j
Q
Q
=
Q
L
.
C. On-Line Learning Algorithm of the RWIT2FNNC
1) Structure Learning Algorithm
The structure learning algorithm is responsible for on-line rule generation. The first task in structure learning is to determine when to generate a new rule. The way the input space is partitioned determines the number of rules extracted from the training data, as well as the number of fuzzy sets in the universe of discourse for each input variable. Geometrically, a rule corresponds to a cluster in the input space, and a rule firing strength can be regarded as the degree to which input data belong to the cluster. Based on this concept, a previous study [39-41] used the rule firing strength as a criterion for type-1 fuzzy rule generation. This idea is extended to type-2 fuzzy rule generation criteria in an RWIT2FNNC.
2) Parameter Learning Algorithm
Membership
Layer j
Rule
Layer k
Type-Reduction
Layer l
Output
Layer o
FNNERWIT
qso
e
qs
FNNCRWIT
qso
UyoriUy
25*2*54
Rk
1
3
jk
4
Lk
1
F
Input
Layer
Hidden
Layer
Output
Layer
1
1
x
1
H
3
H
2
H
Normalization
1
2
x
W
11
W
12
W
13
W
21
W
22
W
23
W
33
W
31
W
32
1
1
x
1
2
x
z
-1
z
-1
z
-1
Input
Layer i
Consequents
Antecedents
4
Rl
4
Ll
],[
3
3
k
k
yy],[
2
2
j
j
yy],[
3
3
k
k
uu
j
A
1
~
j
A
2
~
2
F
3
F
)(
3
Nh)(
1
Nh)(
2
Nh )1(
3
Nh )1(
1
Nh )1(
2
Nh
1
3
2
Fig. 4. Structure of the recurrent wavelet-based interval type-2 fuzzy-neural-network controller (RWIT2FNNC).
The central part of the parameter-learning algorithm for the RWIT2FNNC concerns how to recursively obtain a gradient vector in which each element in the learning algorithm is defined as the derivative of an energy function with respect to a parameter of the network. This is done by means of the chain rule, and the method is generally referred to as the backpropagation learning rule, because the gradient vector is calculated in the direction opposite to the flow of the output of each node. To describe the online parameter learning algorithm of the RWIT2FNNC using the supervised gradient descent method, first the energy function is assumed
2
2
)
(
2
1
)
(
2
1
m
r
m
r
e
E
q
q
q
q
=
-
=
(42)
where
m
e
q
is the error signal between the desired rotor position and the actual position.
Then, the update laws for the parameters in the RWIT2FNNC are described as follows:
Layer 5: During the learning process of the RWIT2FNNC, the error term to be propagated is calculated as:
-
=
-
=
-
=
5
5
5
5
.
.
.
o
r
r
m
m
o
m
m
o
o
y
e
e
E
y
e
e
E
y
E
q
q
d
q
q
q
q
q
q
q
(43)
Layer 4: In this layer the error term needs to be calculated and propagated:
5
4
5
5
4
4
2
1
.
o
l
o
o
l
l
net
y
y
E
net
E
d
d
q
q
=
-
=
-
=
(44)
and the weighting interval factors are updated by the amount
=
-
=
-
=
D
=
n
k
k
k
l
Lk
Ll
Ll
l
l
Lk
Lk
y
y
y
y
net
net
E
E
1
3
3
4
4
4
4
4
4
4
4
d
h
v
h
v
h
v
v
q
v
q
v
(45)
=
-
=
-
=
D
=
n
k
k
k
l
Rk
Rl
Rl
l
l
Rk
Rk
y
y
y
y
net
net
E
E
1
3
3
4
4
4
4
4
4
4
4
d
h
v
h
v
h
v
v
q
v
q
v
(46)
Layer 3: The error term is computed as follows
3
1
3
1
3
3
3
4
1
3
1
3
4
4
3
3
3
3
3
4
4
5
5
3
3
)
(
)
(
k
n
k
k
n
k
k
k
k
RLk
n
k
k
n
k
k
RLk
l
k
k
k
k
k
l
l
o
o
k
k
H
y
y
y
y
y
y
net
u
u
y
y
net
net
y
y
E
net
E
+
+
-
+
=
=
-
=
=
=
=
=
v
v
d
d
q
q
(47)
where
4
RLk
v
can be
4
Rk
v
and
4
Lk
v
.
The update law of
jk
W
is
b
a
k
l
W
jk
k
k
o
o
jk
W
jk
u
W
y
y
y
y
E
W
E
W
.
3
4
3
3
5
5
j
d
h
h
h
q
v
q
=
-
=
-
=
D
(48)
The update of (k is:
-
-
=
Q
-
=
Q
-
=
DQ
=
Q
Q
Q
n
i
j
k
k
k
k
k
k
k
k
k
k
k
k
k
N
y
N
u
F
F
h
h
F
F
u
u
E
E
1
2
3
3
3
3
)
(
)
1
(
)
1
(
d
h
h
h
q
q
(49)
Layer 2: The error term is computed as follows:
k
k
j
j
j
k
k
k
k
k
k
l
l
o
o
j
j
F
net
y
y
net
net
u
u
y
y
net
net
y
y
E
net
E
3
2
2
2
3
3
3
3
3
3
4
4
5
5
2
2
d
d
q
q
=
=
-
=
(50)
The update laws of means are:
2
2
3
2
2
2
2
2
3
3
)
(
)
(
ij
ij
i
jk
j
ij
j
j
j
j
k
k
ij
ij
x
net
net
y
y
net
net
E
E
s
m
v
d
h
m
h
m
h
m
m
q
m
q
m
-
=
-
=
-
=
D
(51)
2
2
3
2
2
2
2
2
3
3
)
(
)
(
ij
ij
i
jk
j
ij
j
j
j
j
k
k
ij
ij
x
net
net
y
y
net
net
E
E
s
m
v
d
h
m
h
m
h
m
m
q
m
q
m
-
=
-
=
-
=
D
(52)
Moreover, the update law of the standard deviation is:
2
2
3
2
2
2
2
2
3
3
)
(
)
(
ij
ij
i
jk
j
ij
j
j
j
j
k
k
ij
ij
x
net
net
y
y
net
net
E
E
s
m
v
d
h
s
h
s
h
s
s
q
s
q
s
-
=
-
=
-
=
D
(53)
where
v
h
,
W
h
,
Q
h
,
m
h
and
s
h
are the learning-rate parameters of the weighting interval factors, the link weights of both the RIT2FNN and WNN, feedback weights, and means and standard deviations, respectively. To accelerate training, the weighting interval factors, the link weights, feedback weights, means and standard deviations of the membership functions are updated by including a momentum term as follows:
)
1
(
)
(
)
(
)
1
(
4
4
4
4
-
D
+
D
+
=
+
N
N
N
N
Lk
Lk
Lk
Lk
v
a
v
v
v
(54)
)
1
(
)
(
)
(
)
1
(
4
4
4
4
-
D
+
D
+
=
+
N
N
N
N
Rk
Rk
Rk
Rk
v
a
v
v
v
(55)
)
1
(
)
(
)
(
)
1
(
-
D
+
D
+
=
+
N
W
N
W
N
W
N
W
jk
jk
jk
jk
a
(56)
)
1
(
)
(
)
(
)
1
(
-
DQ
+
DQ
+
Q
=
+
Q
N
N
N
N
k
k
k
k
a
(57)
)
1
(
)
(
)
(
)
1
(
-
D
+
D
+
=
+
N
N
N
N
ij
ij
ij
ij
m
a
m
m
m
(58)
)
1
(
)
(
)
(
)
1
(
-
D
+
D
+
=
+
N
N
N
N
ij
ij
ij
ij
m
a
m
m
m
(59)
)
1
(
)
(
)
(
)
1
(
-
D
+
D
+
=
+
N
N
N
N
ij
ij
ij
ij
s
a
s
s
s
(60)
where N denotes the iteration number.
The exact calculation of the Jacobian of the system
)
/
(
5
o
r
y
q
which is contained in (43),
)
/
(
5
o
y
E
q
, cannot be determined due to the uncertainties of the servo drive system dynamic, such as parameter variations and external load disturbances. To overcome this problem and to increase the online learning rate of the network parameters, the delta adaptation law is adopted as follows:
m
m
m
o
e
k
e
q
q
q
d
&
+
=
5
(61)
where
m
k
q
is a positive constant.
D. RWIT2FNN Estimator and Compensated Controller
In this section, the RWIT2FNNE is developed to approximate an unknown dynamic function including the lumped parameter uncertainty. Furthermore, a compensated controller is designed to achieve L2 tracking performance with desired attenuation level. Moreover, the adaptive learning algorithms for the compensated controller and the RWIT2FNNE are derived using the Lyapunov stability theorem to train the parameters of the RWIT2FNNE online.
To achieve the control objective, the tracking errors are defined as
)]
(
)
(
[
t
t
e
r
m
r
m
q
q
q
-
=
,
)]
(
)
(
[
t
t
e
r
m
r
m
q
q
q
&
&
&
-
=
, and
)]
(
)
(
[
t
t
e
r
m
r
m
q
q
q
&
&
&
&
&
&
-
=
. We define a tracking error function as follows:
m
m
e
t
e
t
E
q
q
g
+
=
)
(
)
(
&
(62)
where
)
(
t
m
r
q
,
)
(
t
m
r
q
&
and
)
(
t
m
r
q
&
&
are the desired position, speed and acceleration of the PMSM servo drive;
)
(
t
e
m
q
,
)
(
t
e
m
q
&
and
)
(
t
e
m
q
&
&
denote the position, speed and acceleration errors of the PMSM servo drive. The weighting factor, (, is used to normalize the contribution of
)
(
t
e
m
q
and
)
(
t
e
m
q
&
in the error function E(t). By differentiating (62), the error function becomes:
)
(
)
(
)
(
)
(
1
1
t
t
U
t
E
A
B
t
E
B
mn
mn
mn
W
+
+
=
-
-
&
(63)
where the nonlinear function ((t) is defined as:
)}
)
(
(
)
(
)
(
)
(
.
)
(
{
)
(
1
m
r
mn
m
m
r
L
m
mn
m
r
m
mn
e
t
A
e
t
T
D
D
t
U
B
t
A
B
t
q
q
g
q
g
q
q
+
-
+
+
D
+
+
D
+
D
=
W
-
&
&
&
&
&
(64)
Taking into consideration the parameter variations of PMSM servo drive system, ((t) is not only nonlinear but is also a time-varying function, consisting of commands, PMSM servo system parameters and load torque disturbance. Since the unknown function ((t) is very difficult to obtain in advance in practical application, the RWIT2FNNE is employed to estimate ((t) online. Furthermore, since the output of RWIT2FNNE, ((t), is not able to approximate
)
(
t
W
accurately, a compensated controller
)
(
2
t
U
L
qs
is used to attenuate the approximation error. Thus, the control law is defined as
)
(
)
(
)
(
2
t
U
t
t
U
L
qs
E
qs
+
W
=
(65)
where
)
(
)
(
2
t
U
t
FNNE
RWIT
qs
=
W
is the output of the RWIT2FNNE.
Applying the control law (63)(65), the closed-loop dynamics of the PMSM servo drive system can be expressed as follows:
)
(
)
(
~
)
(
)
(
)
(
)
(
2
2
1
1
t
U
t
t
U
t
U
t
E
A
B
t
E
B
L
qs
FNNC
RWIT
qs
mn
mn
mn
-
W
+
-
+
=
-
-
&
(66)
where the approximation error
)
(
~
t
W
is denoted as
W
+
Q
-
Q
=
W
-
W
=
W
e
s
m
v
s
m
v
)
,
,
,
,
(
)
,
,
,
,
(
2
1
)
(
~
1
*
*
*
*
1
*
W
x
y
W
x
y
T
T
(67)
where
W
e
is a minimum reconstructed error due to the insufficient number of rules;
*
v
,
*
m
,
*
s
,
*
W
and
*
Q
are the optimal parameters of
v
,
m
,
s
,
W
and
Q
;
v
,
m
,
s
,
W
and
Q
are the estimated values of the optimal parameters (
*
v
,
*
m
,
*
s
,
*
W
and
*
Q
) as provided by the tuning algorithms that will be introduced. The approximation error
)
(
~
t
W
in (67) is rewritten as
W
+
+
=
W
e
v
v
y
y
T
T
~
2
1
~
2
1
~
*
(68)
where
)
(
~
*
v
v
v
-
=
and
)
(
~
*
y
y
y
-
=
.
The weights of the RWIT2FNNE are updated online to make its output approximate the unknown nonlinear function ((t) accurately. To achieve this, the linearization technique is used to transform the nonlinear output of RWIT2FNNE into partially linear form so that the Lyapunov theorem extension can be applied. The expansion of
y
~
in Taylor series is obtained as follows:
Z
y
W
y
y
y
Z
y
y
W
W
W
y
W
y
y
y
y
y
y
y
y
T
T
W
T
T
L
R
W
W
L
R
L
R
L
R
L
R
+
Q
+
+
+
+
Q
-
Q
Q
Q
+
-
+
-
+
-
=
=
Q
Q
=
Q
=
=
=
~
~
~
~
)
(
)
(
)
(
)
(
~
~
~
*
*
*
*
s
m
s
s
s
s
m
m
m
m
s
m
s
s
m
m
(69)
where
m
m
m
m
m
=
=
L
R
y
y
y
,
s
s
s
s
s
=
=
L
R
y
y
y
,
W
W
L
R
W
W
y
W
y
y
=
=
,
Q
=
Q
Q
Q
Q
=
L
R
y
y
y
,
)
(
~
*
m
m
m
-
=
,
)
(
~
*
s
s
s
-
=
,
)
(
~
*
W
W
W
-
=
,
)
(
~
*
Q
-
Q
=
Q
and Z is a vector of higher order terms and assumed to be pounded by a positive constant. Substituting (69) into (68) will yield
G
+
Q
+
+
+
+
Q
-
-
-
-
=
+
+
+
=
W
Q
Q
W
)
~
~
~
~
(
2
1
)
(
~
2
1
~
2
1
~
2
1
~
~
2
1
~
T
T
W
T
T
T
T
T
W
T
T
T
T
T
T
y
W
y
y
y
y
W
y
y
y
y
y
y
y
s
m
v
s
m
v
e
v
v
v
s
m
s
m
(70)
W
Q
Q
+
Q
+
+
+
-
+
Q
+
+
+
=
G
e
s
m
v
s
m
v
s
m
s
m
)
(
2
1
)
(
2
1
*
*
*
*
*
*
*
*
*
T
T
W
T
T
T
T
T
W
T
T
T
y
W
y
y
y
Z
y
W
y
y
y
(71)
where the uncertain term
G
is assumed to be bounded by a small positive constant
r
G
. In order to develop the L2 compensated controller, from (66) and (70), the error equation in can be rewritten as follows.
G
+
Q
+
+
+
+
Q
-
-
-
-
+
-
=
Q
Q
-
-
)
~
~
~
~
(
2
1
)
(
~
2
1
)
(
)
(
)
(
2
1
1
T
T
W
T
T
T
T
T
W
T
T
T
L
qs
mn
mn
mn
y
W
y
y
y
y
W
y
y
y
y
t
U
t
E
A
B
t
E
B
s
m
v
s
m
v
s
m
s
m
&
(72)
In case of the existence (, consider a specified L2 tracking performance
]
,
0
[
],
,
0
[
)
(
~
2
1
)
0
(
~
)
0
(
~
2
1
)
0
(
~
)
0
(
~
2
1
)
0
(
~
)
0
(
~
2
1
)
0
(
~
)
0
(
~
2
1
)
0
(
~
)
0
(
~
2
1
)
0
(
2
1
)
(
(
2
0
2
2
6
5
4
3
2
1
2
1
0
2
T
L
T
d
W
W
E
B
d
E
T
T
T
T
T
T
mn
T
G
"
G
+
Q
Q
+
+
+
+
+
-
t
t
r
h
h
h
s
s
h
m
m
h
v
v
h
t
t
(73)
where (1, (2, (3, (4 (5 and (6 are strictly positive learning rates;
r
is a prescribed attenuation level. If ( is squared integrable, that