An electric scooter with a Vbelt continuously variable transmission (CVT) driven by a permanent magnet synchronous motor (PMSM) has a lot of nonlinear and timevarying characteristics, and accurate dynamic models are difficult to establish for linear controller designs. A PMSM servodrive electric scooter controlled by a novel hybrid modified recurrent Legendre neural network (NN) control system is proposed to solve difficulties of linear controllers under the occurrence of nonlinear load disturbances and parameters variations. Firstly, the system structure of a Vbelt CVT driven electric scooter using a PMSM servo drive is established. Secondly, the novel hybrid modified recurrent Legendre NN control system, which consists of an inspector control, a modified recurrent Legendre NN control with an adaptation law, and a recouped control with an estimation law, is proposed to improve its performance. Moreover, the online parameter tuning method of the modified recurrent Legendre NN is derived according to the Lyapunov stability theorem and the gradient descent method. Furthermore, two optimal learning rates for the modified recurrent Legendre NN are derived to speed up the parameter convergence. Finally, comparative studies are carried out to show the effectiveness of the proposed control scheme through experimental results.
Ⅰ. INTRODUCTION
There are many countries trying to develop electric vehicles in order to reduce both their dependence on petroleum and the effects of air pollution. Scooters are popular for personal transportation. This is partly due to the fact that they are cheap to buy and operate and convenient to park and store. Because scooters are easier and cheaper to own than cars in most parts of the world, the development of control methods is one of the good research areas for electric scooters. Electric scooters are driven by AC motors, so the selection of the AC motors drive systems is important. There are several basic types of AC servo motors such as PMSMs, switched reluctance motors (SRMs) and induction motors (IMs). PMSMs provide higher efficiency, higher power density, and lower power loss for their size when compared to SRMs and IMs. Direct torque control (DTC)
[1]

[4]
and fieldoriented control
[5]

[7]
are the two most popular control techniques used with PMSMs. The classical DTC
[3]
,
[4]
has some drawbacks. One of these drawbacks is the higher torque ripple and high current ripple generated in steady state operation when compared to the fieldoriented control system. On the other hand, the fieldoriented control is much less sensitive to parameter variations and has faster operation in the four quadrants
[5]

[7]
. Hence, fieldoriented control is used to reduce the torque ripple influence in this paper.
Artificial neural networks (ANNs) have become a formidable learning skill for performing many complex tasks such as highly nonlinear approximations
[8]
,
[9]
and the control of dynamical systems
[10]
,
[11]
. One of the major drawbacks of NNs is that they are computationally intensive and need a large number of iterations for their training. In order to reduce the computational complexity, a functionallink NN, which has shown that it is capable of producing performance that is similar to that of NNs but with a much lower computational cost, was introduced in
[12]
,
[13]
. Moreover, a functionallink NNbased nonlinear dynamic system identification with satisfactory results was reported in
[14]
. It is shown that the performance of the functionallink NN is similar to that of a NN but with a faster convergence and less computational complexity. Moreover, a comprehensive survey on various applications for functionallink NNs has been presented in
[15]
.The functionallink NN has emerged as a powerful learning technique for both nonlinear approximations and the dynamic control of systems. A novel Legendre polynomial based linear NN for static function approximation was proposed by Yang
et al
.
[16]
. It was demonstrated that this network has a fast convergence, and provide high accuracy. Recently, Legendre polynomial NNs for channel equalization problems have been proposed in
[17]
,
[18]
. The superior performance of this Legendre NN equalizer over the NNbased and RBFbased equalizers for different nonlinear channel models has been demonstrated. Moreover, the computational complexity of the Legendre NNs is lower than that of the functionallink NN since the evaluation of Legendre polynomials involves less computation than the evaluation of trigonometric functions
[19]
. In addition, the predictive approach is based on a Legendre neural network and a random time strength function via a promising data mining technique used in machine learning and proposed in
[20]
. Furthermore, a computationally efficient Legendre NN for nonlinear active noise cancellation was shown in
[21]
. However, the proposed Legendre NN is only a single layer network structure without a feedback loop. It cannot adequately approximate dynamic behaviors found in PMSM servodriven electric scooter systems with nonlinear and timevarying characteristics.
The recurrent NN has received an increasing amount of attention due to its structural advantage in the modeling of nonlinear systems and the dynamic control of these systems
[22]

[26]
. These networks are capable of effective identification and control of complex process dynamics but at the expense of increased computational complexity. The typical Elman NN, which is a partial recurrent network model that was first proposed by Elman
[27]
, has one hidden layer with a delayed feedback. The Elman NN is capable of providing standard statespace representation for dynamic systems. In order to improve the ability for identifying high order systems, some recurrent modified Elman NNs
[28]

[30]
have been proposed recently. They have been shown to have more advantages than the basic Elman NN, including good performance, high accuracy, dynamic robustness and fast transient performance. However, the parameters of these recurrent modified Elman NNs
[28]

[30]
have a slower convergence speed due to their higher computational complexity and complex network structure. Additionally, the recurrent wavelet NNs proposed by
[31]

[33]
, which combine the properties of the attractor dynamics of the recurrent NN and the good convergence performance of the wavelet NN, can reduce the computational complexity. However, they also have slower convergence speeds due to the adopted fixed learning rates for the parameters in these recurrent wavelet NNs. Therefore, in order to reduce the network complexity and speedup the convergence, the novel simpler NN structure of modified recurrent Legendre NN, which has a selffeedback connection in the hidden layer and a recurrent connection between the output layer and the input layer, is more powerful than the threelayer recurrent NNs for dealing with timevarying and nonlinear dynamic systems. Two optimal learning rates for the parameters in the modified recurrent Legendre NN are proposed to enhance the convergence speed in this paper.
In electric scooters
[34]

[37]
with nonlinear uncertainties, the hybrid recurrent fuzzy NN controller may not provide satisfactory control performance when operated over a wide range of operating conditions. In
[34]
, a hybrid recurrent fuzzy NN controller using a rotor flux estimator was used to control the PMSM without a shaft encoder to drive electric scooter. In
[35]
a hybrid recurrent fuzzy NN control was used for the feedback control of PMSM driven electric scooters with a shaft encoder. A favorable speed tracking responses can only be achieved by using a hybrid recurrent fuzzy NN controller at 1200 rpm in the nominal case and in the parameter variation case. However, poor speed tracking responses are encountered at 2400 rpm due to uncertain perturbations. On the other hand, if the controlled plant has a highly nonlinear uncertainty, the linear controller may not provide satisfactory control performance. Thus, to ensure the control performance and robustness, a Vbelt CVT driven electric scooter using a PMSM servo drive is controlled by the novel hybrid modified recurrent Legendre NN control system developed in this paper. The novel hybrid modified recurrent Legendre NN control system, which is composed of an inspector control, a modified recurrent Legendre NN control with an adaptation law and a recouped control with an estimation law, is applied to a Vbelt CVT driven electric scooter by using the PMSM servo drive system. The novel hybrid modified recurrent Legendre NN control system has a fast convergence ability and good generalization capability. In addition, the adaptation law of the online parameters in the modified recurrent Legendre NN and the estimation law of the recouped controller can be derived according to the Lyapunov stability theorem and the gradient descent method. Furthermore, two optimal learning rates for the modified recurrent Legendre NN are proposed to enhance the convergence speed. The control method, which is not dependent upon the predetermined characteristics of the system, can adapt to any change in the system characteristics. Finally, the control performance of the proposed novel hybrid modified recurrent Legendre NN control system is verified by experimental results.
This paper is organized as follows. The system structure of a Vbelt CVT driven electric scooter using a PMSM servo drive system is reviewed in Section Ⅱ. The design method of the novel hybrid modified recurrent Legendre NN control system is presented in Section Ⅲ. Experimental results are illustrated in Section Ⅳ. Some conclusions are given in Section Ⅴ.
Ⅱ. STRUCTURE OF A VBELT CVT DRIVEN ELECTRIC SCOOTER SYSTEM USING A PMSM SERVO DRIVE
Due to electric scooter systems with unknown nonlinear uncertainties, i.e., load torque, rolling resistance, wind resistance and braking force, the Vbelt CVTs and clutches in scooters can be categorized as functioning in one of two operating modes depending on the speed of the Vbelt CVT output axis: disengaged or completely coupled. At the start of the PMSM drive cycles, the scooter is in an idle state. The clutch is initially disengaged, and subsequent transitions between the modes are controlled by the clutch axis rotational speed. Except for the mechanical losses, the PMSM power is transmitted through the Vbelt CVT and clutch to the wheel in an electric scooter. For the sake of a convenient design, the stator voltage equations of a Vbelt CVT driven electric scooter using a PMSM servo drive in the synchronously rotating reference frame can be described as follows
[5]

[7]
:
where
v_{qr}
and
v_{dr}
are the
d
axis and
q
axis stator voltages,
i_{qr}
and
i_{dr}
are the
d
axis and
q
axis stator currents,
L_{qr}
and
L_{dr}
are the
d
axis and
q
axis stator inductances,
λ_{fd}
is the
d
axis permanent magnet flux linkage,
R_{r}
is the stator resistance, and
ω_{f}
is the electrical angular speed.
The electromagnetic torque
T_{e}
of a Vbelt CVT driven electric scooter using a PMSM servo drive can be described as:
where
P_{r}
is the number of poles. Then, the dynamic equation of a Vbelt CVT driven electric scooter using a PMSM servo drive
[34

37]
can be represented as:
where
T_{l}
(
T_{a},
Δ
T_{p}
,
F_{l}
(
B_{g}
),
υ_{a}
(
ν_{r}
,
B_{g}
),τ
_{a}
(
ν_{r}
),
ω_{r}
^{2}
) =
T_{a}
+ Δ
T_{p}
+
T_{un}
[36]
,
[37]
represents the lumped nonlinear external disturbances and parameter variations, (e.g. the electric scooter system including the fixed load torque
T_{a}
, the variation of the parameters Δ
T_{p}
, the rolling resistance
υ_{a}
(
v_{r}
,
B_{g}
), the wind resistance τ
_{a}
(
v_{r}
) and the braking force
F_{l}
(
B_{g}
) ),
represents the variation of the parameters,
T_{un}
=
F_{l}
(
B_{g}
)+
υ_{a}
(
v_{r}
,
B_{g}
)+
τ_{a}
(
v_{r}
)
ω_{r}
^{2}
represents the unknown nonlinear load torque,
v_{r}
is the total wind velocity,
B_{g}
is the total frictional coefficient of the ground surface,
B_{r}
is the total viscous frictional coefficient, and
J_{r}
is the total moment of inertia including the electric scooter itself.
ω_{r}
= 2
ω_{f}
/
P_{r}
is the rotor speed. Due to
L_{dr}
=
L_{qr}
, for a surfacemounted PMSM, the second term of (3) is zero. Moreover,
λ_{fd}
is a constant for the surfacemounted PMSM. The rotor flux is produced in the
d
axis only, while the current vector is generated in the
q
axis for the fieldoriented control. When the
d
axis rotor flux is a constant and the torque angle is
π
/ 2
[6

7]
, the maximum torque per ampere can be reached for the fieldoriented control. The electromagnetic torque
T_{e}
is linearly proportional to the
q
axis current
i_{qr}
, which is determined by the closedloop control. The control principle of the PMSM servodrive system is based on the field orientation. A Vbelt CVT driven electric scooter using a PMSM servo drive system with the implementation of fieldoriented control can be reduced as:
in which
k_{r}
= 3
P_{r}λ_{fd}
/4 is the torque constant. A block diagram of a Vbelt CVT driven electric scooter using a PMSM servo drive system is shown in
Fig. 1
.
Block diagram of a Vbelt CVT driven electric scooter using a PMSM servo drive.
The whole system of a Vbelt CVT driven electric scooter using a PMSM servo drive can be indicated as follows: a fieldoriented institution, a PI current loop control, one set of sinusoidal pulsewidthmodulation (PWM) control circuits, one set of interlock and isolated circuits, a voltage source inverter with three sets of insulatedgate bipolar transistor (IGBT) power modules, and a speed controller
[33]

[35]
. The PI current loop controller is a current loop tracking controller. In order to attain a good dynamic response, all of the gains for a wellknown PI current loop controller are listed as follows:
k_{pc}
= 9.5 and
k_{ic}
=
k_{pc}
/
T_{ic}
= 2.8 through some heuristic knowledge
[38]

[40]
of the tuning of the PI controller. The fieldoriented institution consists of a coordinate transformation, sin
θ_{f}
/cos
θ_{f}
generation and lookup table generation. A TMS320C32 digital signal processor (DSP) control system is used to implement the fieldoriented institution control and speed control. A Vbelt CVT driven electric scooter using a PMSM servo drive is manipulated under the lumped nonlinear external disturbances
Ⅲ. DESIGN OF THE CONTROL SYSTEM
The nonlinear uncertainties of a Vbelt CVT driven electric scooter using a PMSM servo drive such as the nonlinear friction force of the transmission belt and clutch, rolling resistance, wind resistance and braking force, lead to degenerate tracking responses in the command current and speed. These nonlinear uncertainties cause variations in the rotor inertia and friction. For a convenient novel hybrid modified recurrent Legendre NN design, the dynamic equation of the Vbelt CVT driven electric scooter using the PMSM servo drive from (4) can be rewritten as:
where
u_{n} = i^{*}_{qr}
is the command current of the stator of the PMSM.
A_{a} = B_{r}/J_{r}
,
B_{a} = k_{r}/J_{r}
and
C_{a} = 1/J_{r}
are three known constants. When uncertainties including the lumped nonlinear external disturbances occur, the parameters are assumed to be bounded, i.e., │
A_{a}ω_{r}
│ ≤
D
_{1}
(
ω_{r}
) , │
C_{a}T_{l} (T_{a},ΔT_{p} ,F_{l} (B_{g} ), ν_{a} (ν_{r}
,
B_{g}
),
τ_{a} (ν_{r} ),ω_{r}^{2}
) │≤
D
_{2}
, and
D
_{3}
≤
B_{a}
, where
D
_{1}
(
ω_{r}
) is a known continuous function, and
D
_{2}
and
D
_{3}
are two known constants. Then, the tracking error can be defined as:
where
ω
^{*}
represents the desired command rotor speed, and
e
is the tracking error between the desired rotor speed and the actual rotor speed. If all of the parameters of the Vbelt CVT driven electric scooter using the PMSM servo drive system including the lumped nonlinear external disturbances and parameter variations are well known, the ideal control law can be designed as:
in which
k
_{1}
is a positive constant. Substituting (8) into (6), the error dynamic equation can be obtained as:
The system state can track the desired trajectory gradually if
e
(
t
)→0 as
t
→∞in (9). However, a novel hybrid modified recurrent Legendre NN control system is proposed to control the Vbelt CVT driven electric scooter using the PMSM servo drive under uncertain perturbations.
 A. Novel Hybrid Modified Recurrent Legendre NN Control
The configuration of the proposed novel hybrid modified recurrent Legendre NN control system, which is composed of an inspector control system, a modified recurrent Legendre NN controller and a recouped controller, is described in
Fig. 2
.
Block diagram of the novel hybrid modified recurrent Legendre NN control system.
where
u_{in}
is the proposed inspector control that is capable of stabilizing around a predetermined bound area in the states of the controlled system. The modified recurrent Legendre NN control
u_{rc}
is a major tracking controller, which is used to imitate an ideal control law. The recouped control
u_{re}
is designed to recoup the difference between the ideal control law and the modified recurrent Legendre NN control. Since the inspector control caused an overdone and chattering effort, the modified recurrent Legendre NN control and the recouped control are proposed to reduce and smooth the control effort when the system states are inside a predetermined bound area. When the modified recurrent Legendre NN approximation properties cannot be ensured, the inspector control is able to take action. For the condition of the divergence of states, the design of the novel hybrid modified recurrent Legendre NN control system is essential to stretch the divergent states back to the predestinated bound area. The novel hybrid modified recurrent Legendre NN control system can uniformly approximate the ideal control law inside the bound area. Then the stability of the system can be warranted. Error dynamic equations from (6) to (10) can be acquired as:
Firstly, the inspected control
u_{in}
can be designed as:
in which sgn(·) is a sign function. When the modified recurrent Legendre NN approximation properties cannot be ensured, the inspector control law is able to take action, i.e.,
I_{in}
= 1. Due to inadequate bound values, e.g.,
D
_{1}
(
ω_{r}
),
D
_{2}
,
D
_{3}
and the sign function, the inspected control can produce an overdone and chattering effort. However, the modified recurrent Legendre NN control and the recouped control can be devised to eliminate this flaw. The modified recurrent Legendre NN control is raised to imitate an ideal control
u_{n}
^{*}
. Then the recouped control is posed to recoup the difference between the ideal control
u_{n}
^{*}
and the modified recurrent Legendre NN control
u_{rc}
.
 B. Modified Recurrent Legendre NN
The architecture of the proposed threelayer modified recurrent Legendre NN is depicted in
Fig. 3
. It is composed of an input layer, a hidden layer and an output layer. The activation functions and signal actions of the nodes in each layer of the modified recurrent Legendre NN can be described as follows:
Structure of the threelayer modified recurrent Legendre NN.
1) First layer – input layer:
Each node
i
in this layer is indicated by using ∏, which multiplies all of the input signals by one another. The output signals are the results of the product. The inputs and outputs for each node
i
in this layer can be expressed as:
x
_{1}
^{1}
=
ω
^{*}

ω_{r}
=
e
is the tracking error between the desired speed
ω
^{*}
and the rotor speed
ω_{r}
.
x
_{2}
^{1}
=
e
(1z
^{1}
)=Δ
e
. is the tracking error change.
is the recurrent weight between the output layer and the input layer.
N
denotes the number of iterations.
is the output value of the output layer in the modified recurrent Legendre NN.
2) Second layer – hidden layer:
The single node
j
th in this layer is labeled with Σ. The net input and the net output for node
j
th of this layer are expressed as:
Legendre polynomials
[17]

[19]
are selected for the activation function of the hidden layer. The Legendre polynomials are denoted by
L_{n}
(
x
), where
n
is the order of expansion, 1<
x
<1 is the argument of the polynomial,
m
is the number of nodes, and
γ
is the selfconnecting feedback gain of the hidden layer which is selected to be between 0 and 1. The zero, the first and the second order Legendre polynomials are given by
L
_{0}
(
x
) = 1,
L
_{1}
(
x
) =
x
and
L
_{2}
(
x
) = (3
x
^{2}
1)/2, respectively. The higher order polynomials are given by
L
_{3}
(
x
) = (5
x
^{3}
3
x
)/2 and
L
_{4}
(
x
) = (35
x
^{4}
30
x
^{2}
+3)/8. The higher order Legendre polynomials are generated by the recursive formula given by:
L
_{n+1}
(
x
) = [(2
n
+1)
xL_{n}
(
x
) 
nL_{n1}
(
x
)]/(
n
+1).
3) Third layer – output layer:
The single node
k
th in this layer is labeled with Σ. It computes the overall output as a summation of all of the input signals. The net input and the net output for node
k
th in this layer are expressed as:
where
is the connective weight between the hidden layer and the output layer;
is the activation function, which is selected as a linear function; and
x
^{3}
_{j}
(
N
) =
y
^{2}
_{j}
(
N
) represents the
j
th input to the node of the output layer. The output value of the modified recurrent Legendre NN can be represented as
y
^{3}
_{k}
(
N
) =
u_{rc}
. Then the output value of the modified recurrent Legendre NN,
u_{rc}
, can be denoted as:
in which
is the adjustable weight parameter vector between the hidden layer and the output layer of the recurrent Legendre NN.
is the input vector in the output layer of the modified recurrent Legendre NN, in which
is determined by the selected Legendre polynomials.
 C. Recouped Control with Estimation Law
By assuming that the sampling frequency is sufficiently high and that the backEMF is constant during a sampling period, the recouped control with estimation law of a PMSM are given as follows: In order to evolve the recouped control
u_{re}
, the minimum approximation error
δ
is defined as:
in which
is an ideal weight vector to reach the minimum approximation error. It is assumed that the absolute value of
δ
is less than a small positive value
λ
, i.e., │
δ
│ <
λ
. Then, the error dynamic equation from (11) can be rewritten as:
Then, the Lyapunov function is selected as:
in
is the bound estimated error,
μ
_{a}
>0 is the learning rate, and
η
is the adaptation gain. By differentiating the Lyapunov function with respect to
t
and using (18), (19) can be rewritten as:
To satisfy
the adaptation law
and the recouped controller
u_{re}
with the estimation law
can be designed as follows:
By substituting (12), (21) and (22) into (20) and using (12) with
I
_{in}
= 0, (20) can be represented as:
By substituting (23) into (24), (24) can be obtained as:
From (25),
is a negative semidefinite value, i.e.
V
_{1}
(
t
)≤
V
_{1}
(0). This implies that
e
and
are bounded. For proof that the hybrid modified recurrent Legendre NN control system can gradually become stable, the function is defined as:
Integrating (26) with respect to
t
results in:
Since
V
_{1}
(0) is bounded, and
V
_{1}
(
t
) is nonincreasing and bounded:
Differentiating (26) with respect to
t
gives:
All of the variables in the right side of (18) are bounded. This implies that
is also bounded. Then,
ε
(
t
) is a uniformly continuous function
[41

42]
. It is denoted that
by using Barbalat’s lemma
[41

42]
. Therefore
e
(
t
)→0 as
t
→∞. From the above proof, the novel hybrid modified recurrent Legendre NN control system becomes gradually stable. Moreover, the tracking error
e
(
t
) of the system will converge to zero. Furthermore, in order to avoid the chattering phenomenon of recouped controller
u_{re}
, the sign the function sgn(
B_{a}e
) can be replaced by the equation
where
in which
ρ
_{0}
and
τ
are positive constants.
 D. Parameter Tuning Methodology and Convergence Analysis
According to the Lyapunov stability theorem and the gradient descent method, an online parameters training methodology for the modified recurrent Legendre NN can be derived and trained effectively. Then the parameter of adaptation law
shown in (21) can be computed by the gradient descent method for the proper choice of two learning rates. The parameter convergence can be guaranteed but the convergent speed is reduced due to a smaller value for the learning rate. On the other hand, the parameter convergence may have an oscillation due to a larger value for the learning rate. In order to train the modified recurrent Legendre NN efficiently, two optimal learning rates will be derived to achieve a fast convergence of the output tracking error. The adaptation law
shown in (21) can be rewritten as:
To train the parameters of the modified recurrent Legendre NN effectively, the method to obtain a recursively gradient vector is very important. Each component can be defined as a derivative of the cost function in the training algorithm. The gradient vector is calculated in the direction opposite the flow of the output of each node. This is done by means of the chain rule. In order to describe the online training algorithm of the modified recurrent Legendre NN, a cost function is defined as
[43]
:
The adaptation law of the connective weight by means of the gradient descent method can be represented as:
where
μ_{α}
is the learning rate. Comparing (30) with (32), yields ∂
V
_{2}
/∂
y
^{3}
_{k}
= 
eB_{a}
. The convergence analysis in the following theorem is used to derive a specific learning rate to assure the convergence of the output tracking error. The adaptation law of the recurrent weight
by using the gradient descent method can be calculated as:
in which
μ_{b}
is the learning rate. Then, two optimal learning rates are derived to assure the convergence of the output tracking error. The convergence analysis is provided in the following two theorems.
Theorem 1:
Assume that
μ_{α}
is the learning rate of the modified recurrent Legendre NN connective weight between the hidden layer and the output layer. Meanwhile, define
P
_{1wmax}
as
P
_{1wmax}
≡
max_{N}
∥
P
_{1w}
(
N
)∥, where
P
_{1w}
(
N
) =
∂
y
^{3}
_{k}
∂
w
^{2}
_{kj}
, and ∥·∥ is the Euclidean norm in ℜ
^{n}
.
μ_{α}
is chosen as
[30
,
43]
:
Then, the convergence of the output tracking error is guaranteed. Furthermore, the optimal learning rate, which achieves a fast convergence, can be obtained as:
Proof:
Since:
The discretetype Lyapunov function is selected as:
The change in the Lyapunov function is obtained by:
Next, the error difference can be represented by:
in which Δ
e
(
N
) is the output error change, and Δ
represents change in the weight. (39) can be obtained by means of (30), (31), (32) and (36):
Thereby:
By means of (38) to (42),the Δ
V
_{2}
(
N
) can be rewritten as:
If
μ_{α}
is chosen as 0 <
μ_{α}
< 2 /{(
P
_{1wmax}
)
^{2}
[
eB_{a}
/
e
(
N
)]
^{2}
}, the Lyapunov stability of
V
_{2}
(
N
) > 0 and Δ
V
_{2}
< 0 is guaranteed. Then, the output tracking error will converge to zero as
t
→ 0. This completes the proof of the theorem. Furthermore, the optimal learning rate, which achieves a fast convergence, corresponds to
[30
,
43]
:
i.e.:
which comes from the derivative of (43) with respect to
μ_{α}
, and is equal to zero. This produces interesting results for the optimal learning rate which can be online tuned at each instant.
Theorem 2:
Assume that
μ_{b}
is the learning rate of the modified recurrent Legendre NN recurrent weight between the output layer and the input layer. Meanwhile,
P
_{2wmax}
should be defined as
P
_{2wmax}
≡
max_{N}
∥
P
_{2w}
(
N
)∥ , where
P
_{2w}
(
N
) =∂
y_{k}
^{3}
/∂
w
^{1}
_{ik}
, and ∥·∥ is the Euclidean norm in ℜ
^{n}
.
μ_{b}
is chosen as
[30
,
43]
:
Then, the convergence of the output tracking error is guaranteed. Furthermore, the optimal learning rate which achieves a fast convergence can be obtained as:
Proof:
Since:
The discretetype Lyapunov function can be selected as (36) and the change in the Lyapunov function is obtained by (37). Then, the error difference can be represented by:
where Δ
e
(
N
) is the output error change, and Δ
represents change in the weight. (49) by using (31), (33) and (48), can be represented as:
Thereby:
By using (38), (49) to (52), Δ
V
_{2}
(
N
)can be rewritten as:
If
μ_{b}
is chosen as 0 <
μ_{b}
< 2 /{(
P
_{2wmax}
)
^{2}
[
B_{a}e
/
e
(
N
)]
^{2}
}, the Lyapunov stability of
V
_{2}
(
N
) > 0 and Δ
V
_{2}
(
N
) < 0 is guaranteed. Then, the output tracking error will converge to zero as
t
→0. This completes the proof of the theorem. Moreover, the optimal learning rate, which achieves fast convergence, corresponds to
[30
,
43]
:
i.e.:
which comes from the derivative of (53) with respect to
μ_{b}
, and is equal to zero. This produces interesting results for the optimal learning rate which can be online tuned at each instant. In summary, the online tuning algorithm for the modified recurrent Legendre NN controller is based on the adaptation laws (32) and (33) for the connective weight adjustment and the recurrent weights adjustment with the two optimal learning rates in (35) and (47), respectively. Moreover, the modified recurrent Legendre NN weight estimation errors are fundamentally bounded
[44]
. The modified recurrent Legendre NN weight estimation errors are bounded, which is required to ensure that the control signal is bounded.
Ⅳ. EXPERIMENTAL RESULTS
The whole system of the DSPbased control system for a Vbelt CVT driven electric scooter using a PMSM servo drive system is shown in
Fig. 1
. A photo of the experimental setup is shown in
Fig. 4
. The control algorithm was executed by a TMS320C32 DSP control system including multichannels of D/A, eight channels programmable PWM and an encoder interface circuit. The methodologies proposed for the realtime control implementation in the DSP control system are composed of the main program and the interrupt service routine (ISR), as shown in
Fig. 5
. In the main program, the parameters and input/output(I/O) initializations are processed first. Then, the interrupt interval for the ISR is set. After enabling the interrupt, the main program is used to monitor the control data. The ISR with a 2
ms
sampling interval is used for reading the rotor position of the PMSM servodriven electric scooter from the encoder, and the threephase currents from the A/D converter. This is done by calculating the rotor position and speed, executing the lookup table and the coordinate translation, executing the PI current control, executing the novel hybrid modified recurrent Legendre NN control system, and outputting the threephase current commands to the sinusoidal PWM circuit for switching the threesets of IGBT power module inverters via the interlock and isolated circuits. The IGBT power module voltage source inverter is executed by a currentcontrolled sinusoidal PWM with a switching frequency of 15
k
Hz by using a triangular carrier wave. In addition, the measured bandwidth of the speed control loop is about 24Hz, and the measured bandwidth of the current control loop by no load is about 240Hz for a PMSM servodrive system without mounted Vbet CVT and wheel systems. The specifications of the surfacemounted PMSM are a threephase 48V, 750W,16.5A,3600
rpm
.The parameters of the PMSM are given as follows:
k_{r}
= 0.86
Nm/A
,
R
_{1}
= 2.5Ω, and
L
_{d1}
=
L
_{q1}
= 6.53
mH
by using an open circuit test, short test, rotor block test, and loading test. To show the control performance of the proposed novel hybrid modified recurrent Legendre NN control system, three cases are provided in the experimentation. These cases are the 125.6
rad/s
(1200
rpm
) case, the 251.2
rad/s
(2400
rpm
) case and 376.8
rad/s
(3600
rpm
) case. With this application it is necessary to decrease the bandwidth of the speed due to high stiffness between the motor and the nonlinear load. This is especially true with an electric scooter mounted with Vbet CVT and wheel systems. Therefore, the adopted three bandwidths of the speed control loop with the three reference models in the 125.6
rad/s
(1200
rpm
) case, the 251.2
rad/s
(2400
rpm
) case, and the 376.8
rad/s
(3600
rpm
) case are about 6.25Hz, 2.8Hz, and 2.0Hz, respectively.
Photo of the experimental setup.
Flowchart of the executing program by using the DSP control system.
Since an electric scooter is a nonlinear and timevarying system, all of the gains of the wellknown PI controller are
k_{ps}
=13.5,
k_{is}
=
k_{ps}/T_{is}
=1.8 through some heuristic knowledge
[38

40]
on the tuning of the PI controller in the 125.6
rad/s
(1200
rpm
) case for speed tracking in order to achieve good transient and steadystate control performances. The experimental results of the wellknown PI controller for the Vbelt CVT driven electric scooter using the PMSM servo drive system in the125.6
rad/s
(1200
rpm
)case and in the 251.2
rad/s
(2400
rpm
) case are shown in
Fig. 6
and
Fig. 7
, respectively.
Experimental result of the wellknown PI controller controlled for the Vbelt CVT driven electric scooter by using the PMSM servo drive system at 125.6rad/s (1200rpm): (a) tracking response of rotor speed command ω_{r}^{*}, desired rotor speed command ω^{*} and measured rotor speed ω_{r}; (b) response of tracking error e amplification; (c) current command i_{a}^{*} and measured current i_{a} in phase a; (d) electromagnetic torque T_{e}.
Experimental result of the wellknown PI controller controlled for the Vbelt CVT driven electric scooter by using the PMSM servo drive system at 251.2rad/s (2400rpm): (a) tracking response of rotor speed command ω_{r}^{*}, desired rotor speed command ω^{*} and measured rotor speed ω_{r}; (b) response of tracking error e amplification ; (c) current command i_{a}^{*} and measured current i_{a} in phase a; (d) electromagnetic torque T_{e}.
The responses of the rotor speed command
ω_{r}
^{*}
, the desired rotor speed command
ω
^{*}
and the measured rotor speed
ω_{r}
are shown in
Fig. 6
(a) and
Fig. 7
(a). The response of the tracking error
e
amplification is shown in
Fig. 6
(b) and
Fig. 7
(b). The tracking response of the current command
i_{a}
^{*}
and the measured current
i_{a}
in phase
a
is shown in
Fig. 6
(c) and
Fig. 7
(c). The dynamic response of the electromagnetic torque
T_{e}
is shown in
Fig. 6
(d) and
Fig. 7
(d).
In addition, the experimental result of the wellknown PI controller for the Vbelt CVT driven electric scooter using the PMSM servo drive system in the 376.8
rad/s
(3600
rpm
)case under high speed perturbation is shown in
Fig. 8
, where responses of the rotor speed command
ω_{r}
^{*}
, the desired otor speed command
ω
^{*}
and the measured rotor speed
ω_{r}
are shown in
Fig. 8
(a). The response of the tracking error
e
is shown in
Fig. 8
(b). The response of the tracking error
e
amplification is shown in
Fig. 8
(c). The dynamic response of the electromagnetic torque
T_{e}
is shown in
Fig. 8
(d). Since the low speed operation is the same as the nominal case due to smaller lumped nonlinear external disturbances, the response of the speed shown in
Fig. 6
(a) has better tracking performance. In addition, the degenerate tracking response of the speed shown in
Fig. 7
(a) and
Fig. 8
(a) is obvious due to bigger lumped nonlinear disturbances (e.g. rolling resistance and parameter variations)in the 251.2
rad/s
(2400
rpm
) case and in the 376.8
rad/s
(3600
rpm
) case. Moreover, the electromagnetic torque brings in a greater torque ripple, as shown in
Fig. 6
(d),
Fig. 7
(d) and
Fig. 8
(d), due to a Vbelt CVT system with highly nonlinear external disturbances, such as Vbelt shaking friction, and the action friction between the primary pulley and the second pulley. Moreover, the sluggish tracking response of the current shown
Fig. 7
(c) is obtained for the wellknown PI controlled Vbelt CVT driven electric scooter using the PMSM servo drive system. The linear controller has weak robustness under larger lumped nonlinear external disturbances in high speed operations due to the lack of appropriately gains tuning or degenerate nonlinear effect.
Experimental results of the wellknown PI controlled for the Vbelt CVT driven electric scooter by using the PMSM servo drive system at 376.8 rad/s (3600rpm): (a) tracking response of rotor speed command ω_{r}^{*}, desired rotor speed command ω^{*} and measured rotor speed ω_{r}; (b) response of tracking error e; (c)response of tracking error e amplification ; (d) electromagnetic torque T_{e}.
For comparison, the control performance with the proposed novel hybrid modified recurrent Legendre NN control system, for the threelayer recurrent NN control system is adopted in this paper. The threelayer recurrent NN has the recurrent loop between the output layer and the input layer and it has the sigmoid activation function in the hidden layer. It has two, three and one neurons in the input layer, the hidden layer and the output layer, respectively. All of the control gains of the threelayer recurrent NN control system are chosen to achieve the best transient control performance while considering the requirement of stability. For simplicity, the recurrent weight between the output layer and the input layer in the threelayer recurrent NN is to set 1. Moreover, the connective weight between the input layer and the hidden layer, and the connective weight between the hidden layer and the output layer in the threelayer recurrent NN are initialized with random numbers. In addition, all of the learning rates are set as fixed constants. Furthermore, the normalized inputs and references have zero and unity, respectively. Also the network outputs should be converted back to the original units of reference. The parameter adjustment process remains continually active for the duration of the experiment. The experimental results of the threelayer recurrent NN control system for the Vbelt CVT driven electric scooter using the PMSM servo drive system in the 125.6
rad/s
(1200
rpm
) case and the 251.2
rad/s
(2400
rpm
) case are shown in
Fig. 9
and
Fig. 10
. The response of the rotor speed command
ω_{r}
^{*}
, the desired rotor speed
ω
^{*}
and the measured rotor speed
ω_{r}
are shown in
Figs. 9
(a) and
Fig. 10
(a). The response of the tracking error
e
amplification is shown in
Fig. 9
(b) and
Fig. 10
(b). The tracking response of the current command
i_{a}
^{*}
and the measured current
i_{a}
in phase
a
are shown in
Fig. 9
(c) and
Fig. 10
(c). The dynamic response of the electromagnetic torque
T_{e}
is shown in
Fig. 9
(d) and
Fig. 10
(d). In addition, the experimental results of the threelayer recurrent NN control system for the Vbelt CVT driven electric scooter using the PMSM servo drive system in the 376.8
rad/s
(3600
rpm
) case under a high speed perturbation is shown in
Fig. 11
, where response of the rotor speed command
ω_{r}
^{*}
, the desired rotor speed command
ω
^{*}
and the measured rotor speed
ω_{r}
are shown in
Fig. 11
(a). The response of the tracking error
e
is shown in
Fig. 11
(b). The response of the tracking error
e
amplification is shown in
Fig. 11
(c). The dynamic response of the electromagnetic torque
T_{e}
is shown in
Fig. 11
(d).
Experimental results of the threelayer recurrent NN control system controlled for the Vbelt CVT driven electric scooter by using the PMSM servo drive system at 125.6rad/s (1200rpm): (a) tracking response of rotor speed command ω_{r}^{*}, desired rotor speed command ω^{*} and measured rotor speed ω_{r}; (b) response of tracking error e amplification ; (c) current command i_{α}^{*} and measured current i_{α} in phase α; (d) electromagnetic torque T_{e}.
Experimental results of the threelayer recurrent NN control system controlled for the Vbelt CVT driven electric scooter by using the PMSM servo drive system at 251.2rad/s (2400rpm): (a) tracking response of rotor speed command ω_{r}^{*}, desired rotor speed command ω^{*} and measured rotor speed ω_{r}; (b) response of tracking error e amplification ; (c) current command i_{α}^{*} and measured current i_{α} in phase α; (d) electromagnetic torque T_{e}.
Experimental results of the threelayer recurrent NN control system controlled for the Vbelt CVT driven electric scooter by using the PMSM servo drive system at 376.8 rad/s (3600rpm): (a) tracking response of rotor speed command ω_{r}^{*}, desired rotor speed command ω^{*} and measured rotor speed ω_{r}; (b) response of tracking error e; (c)response of tracking error e amplification ; (d) electromagnetic torque T_{e}.
However, due to the online adaptive mechanism of the threelayer recurrent NN and the recouped controller, accurate tracking responses of the speed and current can be obtained. These experimental results show that the threelayer recurrent NN control system has better control performance than the wellknown PI controller for the Vbelt CVT driven electric scooter using the PMSM servo drive system in the 251.2
rad/s
(2400
rpm
) case and in the 376.8
rad/s
(3600
rpm
) case under the occurrence of large lumped nonlinear disturbances. Meanwhile, the electromagnetic torque
T_{e}
exhibits a lower torque ripple, as shown in
Fig. 9
(d),
Fig. 10
(d) and
Fig. 11
(d).
The control gains of the proposed novel hybrid modified recurrent Legendre NN control system are
η
= 0.1,
λ
= 0.5 according to the estimated uncertainty bound for the recouped controller compensating perturbations in order to achieve the best transient control performance. All of the control gains of the novel hybrid modified recurrent Legendre NN control system are chosen to achieve the best transient control performance, while considering the requirement of stability. Usually, some heuristics can be used to roughly initialize the parameters of the novel hybrid modified recurrent Legendre NN control system for practical applications. The effects of the inaccurate selection of the initialized parameters can be retrieved by the online parameter training methodology. For simplicity, all of the recurrent weights between the output layer and the input layer in the modified recurrent Legendre NN are to set to 1. Moreover, the connective weights between the hidden layer and the output layer in the modified recurrent Legendre NN are initialized with a random number. Furthermore, the normalized inputs and references are zero and unity, respectively. In addition, the network outputs should be converted back to the original units of the references. The parameter adjustment process remains continually active for the duration of the experiment. The structure of the modified recurrent Legendre NN controller has 2 nodes, 3 nodes and 1 node in the input layer, the hidden layer and the output layer, respectively. The experimental results of the novel hybrid modified recurrent Legendre NN control system for the Vbelt CVT driven electric scooter using the PMSM servo drive system in the125.6
rad/s
(1200
rpm
) case and in the 251.2
rad/s
(2400
rpm
) case are shown in
Fig. 12
and
Fig. 13
. The response of the rotor speed command
ω_{r}
^{*}
, the desired rotor speed command
ω
^{*}
and the measured rotor speed
ω_{r}
are shown in
Fig. 12
(a) and
Fig. 13
(a). The response of the tracking error
e
amplification is shown in
Fig. 12
(b) and
Fig. 13
(b). The tracking response of the current command
i_{a}
^{*}
and the measured current
i_{a}
in phase
a
are shown in
Fig. 12
(c) and
Fig. 13
(c). The dynamic response of the electromagnetic torque
T_{e}
is shown in
Fig. 12
(d) and
Fig. 13
(d). In addition, the experimental results of novel hybrid modified recurrent Legendre NN control system for the Vbelt CVT driven electric scooter using the PMSM servo drive system in the 376.8
rad/s
(3600
rpm
) case under high speed perturbations is shown in
Fig. 14
. The response of the rotor speed command
ω_{r}
^{*}
, the desired rotor speed command
ω
^{*}
and the measured rotor speed
ω_{r}
are shown in
Fig. 14
(a). The response of the tracking error
e
is shown in
Fig. 14
(b). The response of the tracking error
e
amplification is shown in
Fig. 14
(c).The dynamic response of the electromagnetic torque
T_{e}
is shown in
Fig. 14
(d).
Experimental results of the novel hybrid modified recurrent Legendre NN control system controlled for the Vbelt CVT driven electric scooter by using the PMSM servo drive system at 125.6rad/s (1200rpm): (a) tracking response of rotor speed command ω_{r}^{*}, desired rotor speed command ω^{*} and measured rotor speed ω_{r} ; (b) response of tracking error e amplification ; (c) current command i_{α}^{*} and measured current i_{α} in phase α; (d) electromagnetic torque T_{e}.
Experimental results of the novel hybrid modified recurrent Legendre NN control system controlled for the Vbelt CVT driven electric scooter by using the PMSM servo drive system at 251.2rad/s (2400rpm): (a) tracking response of rotor speed command ω_{r}^{*}, desired rotor speed command ω^{*} and measured rotor speed ω_{r} ; (b) response of tracking error e amplification ; (c) current command i_{a}^{*} and measured current i_{α} in phase α; (d) electromagnetic torque T_{e}.
Experimental results of the novel hybrid modified recurrent Legendre NN control system controlled for the Vbelt CVT driven electric scooter by using the PMSM servo drive system at 376.8 rad/s (3600rpm): (a) tracking response of rotor speed command ω_{r}^{*}, desired rotor speed command ω^{*} and measured rotor speed ω_{r} ; (b) response of tracking error e ; (c)response of tracking error e amplification ; (d) electromagnetic torque T_{e}.
From the experimental results, accurate tracking performance is obtained for the novel hybrid modified recurrent Legendre NN controller for the Vbelt CVT driven electric scooter using the PMSM servo drive owing to the online adaptive mechanism of the modified recurrent Legendre NN and action of the recouped controller. Therefore, these results show that the novel hybrid modified recurrent Legendre NN control system has better control performance than the wellknown PI controller for speed perturbations for the Vbelt CVT driven electric scooter using the PMSM servo drive system.
Additionally, the small chattering phenomena of the currents in phase α, shown in
Fig. 9
(c),
Fig. 10
(c) and
Fig. 12
(c) and
Fig. 13
(c), are induced by online adjustments of the threelayer recurrent NN and the novel hybrid modified recurrent Legendre NN, respectively, to cope with the highly nonlinear dynamics of the system.
Furthermore, the speed response, shown in
Fig. 13
(a) and
Fig. 14
(a) from using the novel hybrid modified recurrent Legendre NN control system due to its lower computational complexity, has a faster convergence than the speed response shown in
Fig. 10
(a) and
Fig. 11
(a) by using the threelayer recurrent NN control system in the 251.2
rad/s
(2400
rpm
) case and in the 376.8
rad/s
(3600
rpm
) case. Moreover, the dynamic response of electromagnetic torque
T_{e}
, shown in
Fig. 12
(d),
Fig. 13
(d) and
Fig. 14
(d),from using the novel hybrid modified recurrent Legendre NN control system has a lower torque ripple than that obtained using the wellknown PI control and the threelayer recurrent NN control system shown in
Fig. 5
(d),
Fig. 6
(d),
Fig. 7
(d) and in
Fig. 8
(d),
Fig. 9
(d),
Fig. 10
(d), respectively.
Moreover, the experimental results of the learning rate variation in the modified recurrent Legendre NN are shown in
Fig. 15
, where the response of the learning rate
μ_{α}
of the connection weights in the 125.6
rad/s
(1200rpm) case is shown in
Fig. 15
(a). The response of the learning rate
μ_{b}
of the recurrent weights in the 125.6
rad/s
(1200rpm) case is shown in
Fig. 15
(b). The response of the learning rate
μ_{α}
of the connection weights in the 251.2
rad/s
(2400rpm) case is shown in
Fig. 15
(c). The response of the learning rate
μ_{b}
of the recurrent weights in the 251.2
rad/s
(2400rpm) case is shown in
Fig. 15
(d). The learning rate
μ_{α}
variation of the connection weights and the learning rate
μ_{b}
variation of the recurrent weights in the 251.2
rad/s
(1200rpm)case and in the 251.2
rad/s
(2400rpm)case have fast convergences to be bounded from the experimental results.
Experimental result of the learning rate variations in the modified recurrent Legendre NN: (a) response of the learning rate μ_{α} the connection weights at 125.6 rad/s (1200 rpm) case; (b) response of the learning rate μ_{b} of the recurrent weights at 125.6 rad/s (1200 rpm) case; (c) response of the learning rate μ_{α} of the connection weights at 251.2 rad/s (2400 rpm) case; (d) response of the learning rate μ_{b} of the recurrent weights at 251.2 rad/s (2400 rpm).
Finally, the measured rotor speed response under step load torque disturbances is given. The condition under a
T_{l}
= 2
Nm
(
T_{a}
) +
T_{un}
load torque disturbance with adding a load and shedding a load is tested while using the wellknown PI controller, the threelayer recurrent NN control system and the novel hybrid modified recurrent Legendre NN control system. The experimental result under a
T_{l}
= 2
Nm
(
T_{a}
) +
T_{un}
load torque disturbance with adding a load and shedding a load at the command rotor speed 251.2
rad/s
(2400
rpm
) is shown in
Figs. 16

18
. The experimental result of the measured rotor speed response and measured current in phase
a
while using the wellknown PI controller under a
T_{l}
= 2
Nm
(
T_{a}
) +
T_{un}
load disturbance with adding a load and shedding a load at 251.2
rad/s
(2400
rpm
) is shown in
Fig. 16
(a), and the amplified speed error
e_{c}
response is shown in
Fig. 16
(b). The experimental results of the measured rotor speed response and the measured current in phase
a
while using the threelayer recurrent NN control system under a
T_{l}
= 2
Nm
(
T_{a}
) +
T_{un}
load disturbance with adding a load and shedding a load at 251.2
rad/s
(2400
rpm
) is shown in
Fig. 17
(a), and the amplified speed error
e_{c}
response is shown in
Fig. 17
(b). The experimental result of the measured rotor speed response and the measured current in phase a while using the novel hybrid modified recurrent Legendre NN control system under a
T_{l}
= 2
Nm
(
T_{a}
) +
T_{un}
load disturbance with adding a load and shedding a load at 251.2
rad/s
(2400
rpm
) is shown in
Fig. 18
(a), and the amplified speed error
e_{c}
response is shown in
Fig. 18
(b). From the experimental results, the degenerated responses due to a
T_{l}
= 2
Nm
(
T_{a}
) +
T_{un}
load disturbance are greatly improved when using the novel hybrid modified recurrent Legendre NN control system. From the experimental results, it can be seen that the transient response of the novel hybrid modified recurrent Legendre NN control system is better than that of the wellknown PI controller and the threelayer recurrent NN control system in terms of load regulation.
Experimental results under T_{l} = 2Nm (T_{α}) +T_{un} load disturbance with adding load and shedding load at 251.2 rad/s (2400 rpm) case by using the wellknown PI controller: (a) command rotor angular speed ω_{r}^{*}, measured rotor angular speed ω_{r} and measured current i_{a} in phase α; (b) amplified speed error e .
Experimental results under T_{l} = 2Nm (T_{α}) +T_{un} load disturbance with adding load and shedding load at 251.2 rad/s (2400 rpm) case by using the threelayer recurrent NN control system: (a) command rotor angular speed ω_{r}^{*}, measured rotor angular speed ω_{r} and measured current i_{a} in phase α; (b) amplified speed error e .
Experimental results under T_{l} = 2Nm (T_{α}) +T_{un} load disturbance with adding load and shedding load at 251.2 rad/s (2400 rpm) case by using the novel hybrid modified recurrent Legendre NN control system: (a) command rotor angular speed ω_{r}^{*}, measured rotor angular speed ω_{r} and measured current i_{a} in phase α; (b) amplified speed error e .
Moreover, the novel hybrid modified recurrent Legendre NN control system has a faster convergence and better load regulation than the threelayer recurrent NN control system under a
T_{l}
= 2
Nm
(
T_{α}
) +
T_{un}
load disturbance.
In addition, control performance comparisons of the PI controller, the threelayer recurrent NN control system and then novel hybrid modified recurrent Legendre NN control system are summarized in
Table I
. They are shown for the experimental results under the four test cases. In
Table I
, the novel hybrid modified recurrent Legendre NN control system results in a smaller tracking speed error with respect to the PI controller and the threelayer recurrent NN control system. According to the tabulated measurements, the proposed novel hybrid modified recurrent Legendre NN control system yields superior control performance when compared with the PI controller and the threelayer recurrent NN control system. Furthermore, characteristic performance comparisons of the PI controller, the threelayer recurrent NN control system and the novel hybrid modified recurrent Legendre NN control system are summarized in
Table II
for the experimental results. In
Table II
, it can be seen that the various performances of the novel hybrid modified recurrent Legendre NN control system are superior to those of the PI controller and the threelayer recurrent NN control system.
PERFORMANCE COMPARISONS OF CONTROL SYSTEMS
PERFORMANCE COMPARISONS OF CONTROL SYSTEMS
CHARACTERISTIC PERFORMANCE COMPARISONS OF CONTROL SYSTEMS
CHARACTERISTIC PERFORMANCE COMPARISONS OF CONTROL SYSTEMS
Ⅴ. CONCLUSIONS
A novel hybrid modified recurrent Legendre NN control system has been successfully developed in this paper to control Vbelt CVT driven electric scooters using a PMSM servo drive system with robust control characteristics.The adopted modified recurrent Legendre NN has a selffeedback connection in the hidden layer and a recurrent connection between the output layer and the input layer. It is more powerful than the threelayer recurrent NN for dealing with timevarying and nonlinear dynamical systems. Moreover, in order to improve the ability in terms of identifying high order systems and the convergence speed of the parameters, a modified recurrent Legendre NN with two optimal learning rates is also proposed in this paper. It has been proved through experimental results that the modified recurrent Legendre NN has more advantages than the threelayer recurrent NN with the sigmoid activation function, including better performance, higher accuracy, increased dynamic robustness and faster transient performance.
The main contributions of this paper are as follows as: (1) a dynamic models of a Vbelt CVT driven electric scooter with unknown nonlinear and timevarying characteristics using a PMSM servo drive system was successful derived; (2) a novel hybrid modified recurrent Legendre NN control system for Vbelt CVT driven electric scooters using a PMSM servo drive system under the occurrence of the variations in the rotor inertia and load torque disturbance was successful applied to enhance robustness; (3) the online parameters tuning methodology of the modified recurrent Legendre NN and the estimation law of the remunerated controller using the Lyapunov stability theorem were successful derived; (4) two optimal learning rates for the connective weight and recurrent weight of the modified recurrent Legendre NN according to the discretetype Lyapunov function were successful derived in order to speedup convergence; (5) a novel hybrid modified recurrent Legendre NN control system, which has a fast convergence ability and fast capture in terms of the system’s nonlinear and timevarying behaviors, is successful developed; (6) the proposed novel hybrid modified recurrent Legendre NN control system has a lower torque ripple than the wellknown PI controller and the threelayer recurrent NN control systems. Therefore, the control performance of the proposed hybrid modified recurrent Legendre NN control system is more suitable than the wellknown PI controller and the threelayer recurrent NN control systems for Vbelt CVT driven electric scooters using a PMSM servo drive system.
BIO
ChihHong Lin was born in Taichung, Taiwan, R.O.C. He received his B.S. and M.S. degrees in Electrical Engineering from the National Taiwan University of Science and Technology, Taipei, Taiwan, R.O.C., in 1989 and 1991, respectively. He received his Ph.D. degree in Electrical Engineering from Chung Yuan Christian University, Chung Li, Taiwan, R.O.C.,in 2001.He is currently an Associate Professor in the Department of Electrical Engineering, National United University, Miao Li, Taiwan, R.O.C. His current research interests include power electronics, motor servo drives and intelligent control.
Takahashi I.
,
Noguchi T.
1986
“A new quickresponse and high efficiency control strategy of an induction motor,”
IEEE Trans. Ind. Appl.
22
(5)
820 
827
DOI : 10.1109/TIA.1986.4504799
Habetler T. G.
,
Profumo F.
,
Pastorelli M.
,
Tolbert L. M.
1992
“Direct torque control of induction machines using space vector modulation,”
IEEE Trans. Ind. Appl.
28
(5)
1045 
1053
DOI : 10.1109/28.158828
Zhong L.
,
Rahman M. F.
,
Hu W. Y.
,
Lim K. W.
1997
“Analysis of direct torque control in permanent magnet synchronous motor drives,”
IEEE Trans. Power Electron.
12
(3)
528 
536
DOI : 10.1109/63.575680
Zhong L.
,
Rahman M. F.
,
Hu W. Y.
,
Lim K. W.
,
Rahman M. A.
1999
“A direct torque controller for permanent magnet synchronous motor drives,”
IEEE Trans. Energy Convers.
14
(3)
637 
642
DOI : 10.1109/60.790928
Novotny D. W.
,
Lipo T. A.
1996
Vector Control and Dynamics of AC Drives
Oxford University Press
Krishnan R.
2001
Electric Motor Drives: Modeling, Analysis, and Control
Prentice Hall
Lin F. J.
1997
“Realtime IP position controller design with torque feedforward control for PM synchronous motor,”
IEEE Trans. Ind. Electron.
4
(3)
398 
407
Haykin S.
1994
Neural Networks
Maxwell Macmillan
124 
128
Eskander M. N.
2002
“Minimization of losses in permanent magnet synchronous motors using neural network,”
Journal of Power Electronics
2
(3)
220 
229
Sastry P. S.
,
Santharam G.
,
Unnikrishnan K. P.
1994
“Memory neural networks for identification and control of dynamical systems,”
IEEE Trans. Neural Netw.
5
(2)
306 
319
DOI : 10.1109/72.279193
Ko J. S.
,
Choi J. S.
,
Chung D. H.
2012
“Maximum torque control of an IPMSM drive using an adaptive learning fuzzyneural network,”
Journal of Power Electronics
12
(3)
468 
476
DOI : 10.6113/JPE.2012.12.3.468
Pao Y. H.
1989
Adaptive Pattern Recognition and Neural Networks
AddisonWesley
Patra J. C.
,
Pal R. N.
,
Chatterji B. N.
,
Panda G.
1999
“Identification of nonlinear dynamic systems using functional link artificial neural networks,”
IEEE Trans. Syst., Man, Cybern. B, Cybern.
29
(2)
254 
262
DOI : 10.1109/3477.752797
Dehuriand S.
,
Cho S. B.
2010
“A comprehensive survey on functional link neural networks and an adaptive PSOBP learning for CFLNN,”
Neural Computing and Applications
19
(2)
187 
205
DOI : 10.1007/s0052100902885
Yang S. S.
,
Tseng C. S.
1996
“An orthogonal neural network for function approximation,”
IEEE Trans. Syst., Man, Cybern. B, Cybern.
26
(5)
779 
785
DOI : 10.1109/3477.537319
Patra J. C.
,
Chin W. C.
,
Meher P. K.
,
Chakraborty G.
2008
“LegendreFLANNbased nonlinear channel equalization in wireless communication systems,”
in Proc. IEEE Int. Conf. Systems, Man, Cybernetics
1826 
1831
Patra J. C.
,
Meher P. K.
,
Chakraborty G.
2009
“Nonlinear channel equalization for wireless communication systems using Legendre neural networks,”
Signal Processing
89
(12)
2251 
2262
DOI : 10.1016/j.sigpro.2009.05.004
Patra J. C.
,
Bornand C.
2010
“Nonlinear dynamic system identification using Legendre neural network,”
in Proc. Int. Joint Conf. Neural Networks
1 
7
Liu F.
,
Wang J.
2012
“Fluctuation prediction of stock market index by Legendre neural network with random time strength function,”
Neurocomputing
83
12 
21
DOI : 10.1016/j.neucom.2011.09.033
Das K. K.
,
Satapathy J. K.
2012
“Novel algorithms based on Legendre neural network for nonlinear active noise control with nonlinear secondary path,”
Int. J. Computer Science and Information Technology
3
(5)
5036 
5039
Madyastha R. K.
,
Aazhang B.
1994
“An algorithm for Training multilayer perceptrons for data classification and function interpolation,”
IEEE Trans. Circuits SystemsI
41
(12)
866 
875
DOI : 10.1109/81.340848
Chow T. W. S.
,
Fang Y.
1992
“A recurrent neuralnetworkbased realtime learning control strategy applying to nonlinear systems with unknown dynamics,”
IEEE Trans. Ind. Electron.
45
(1)
151 
161
DOI : 10.1109/41.661316
Li X. D.
,
Ho J. K. L.
,
Chow T. W. S.
2005
“Approximation of dynamical timevariant systems by continuoustime recurrent neural networks,”
IEEE Trans. Circuits Syst. II, Exp. Briefs
52
(10)
656 
660
DOI : 10.1109/TCSII.2005.852006
Lu C. H.
,
Tsai C. C.
2008
“Adaptive predictive control with recurrent neural network for industrial processes: an application to temperature control of a variablefrequency oilcooling machine,”
IEEE Trans. Ind. Electron.
55
(3)
1366 
1375
DOI : 10.1109/TIE.2007.896492
Payam A. F.
,
Hashemnia M. N.
,
Faiz J.
2011
“Robust DTC control of doublyfed induction machines based on inputoutput feedback linearization using recurrent neural networks,”
Journal of Power Electronics
11
(5)
719 
725
DOI : 10.6113/JPE.2011.11.5.719
Yang X. Y.
,
Xu D. P.
,
Han X. J.
,
Zhou H. N.
2005
“Predictive functional control with modified Elman neural network for reheated steam temperature,”
in IEEE Int. Conf. Machine Learning Cybernetics
4699 
4703
Lin C. H.
2013
“Novel modified Elman neural network control for PMSG system based on wind turbine emulator,”
Mathematical Problems in Engineering
Article ID 753756, 15 pages
2013
Lin C. H.
2013
“Recurrent modified Elman neural network control of PM synchronous generator system using wind turbine emulator of PM synchronous servo motor drive,”
Int. J. Electrical Power and Energy Systems
52
143 
160
DOI : 10.1016/j.ijepes.2013.03.021
Yoo S. J.
,
Park J. B.
,
Choi Y. H.
2005
“Stable predictive control of Chaotic systems using selfrecurrent wavelet neural network,”
Int. J. Automatic Control Systems
3
(1)
43 
55
Lu C. H.
2009
“Design and application of stable predictive controller using recurrent wavelet neural networks,”
IEEE Trans. Ind. Electron.
56
(9)
3733 
3742
DOI : 10.1109/TIE.2009.2025714
Lin C. H.
2014
“Hybrid recurrent wavelet neural network control of PMSM servodrive system for electric scooter,”
Int. J. Automatic Control Systems
12
(1)
177 
187
DOI : 10.1007/s1255501201902
Lin C. H.
,
Lin C. P.
2012
“The hybrid RFNN control for a PMSM drive system using rotor flux estimator,”
Int. J. Power Electronics
4
(1)
33 
48
DOI : 10.1504/IJPELEC.2012.044150
Lin C. H.
,
Chiang P. H.
,
Tseng C. S.
,
Lin Y. L.
,
Lee M. Y.
2010
“Hybrid recurrent fuzzy neural network control for permanent magnet synchronous motor applied in electric scooter,”
in 6th Int. Power Electronics Conf.
1371 
1376
Tseng C. Y.
,
Chen L. W.
,
Lin Y. T.
,
Li J. Y.
2008
“A hybrid dynamic simulation model for urban scooters with a mechanicaltype CVT,”
in IEEE Int. Conf. Automation and Logistics
519 
519
Tseng C. Y.
,
Lue Y. F.
,
Lin Y. T.
,
Siao J. C.
,
Tsai C. H.
,
Fu L. M.
2009
“Dynamic simulation model for hybrid electric scooters,”
in IEEE Int. Symp. Industrial Electronics
1464 
1469
Astrom K. J.
,
Hagglund T.
1995
PID Controller: Theory, Design, and Tuning
Instrument Society of America
Hagglund T.
,
Astrom K. J.
2004
“Revisiting the ZieglerNichols tuning rules for PI control – part II: the frequency response method,”
Asian J. Control
6
(4)
469 
482
DOI : 10.1111/j.19346093.2004.tb00368.x
Slotine J. J. E.
,
Li W.
1991
Applied Nonlinear Control
Prentice Hall
Astrom K. J.
,
Wittenmark B.
1995
Adaptive Control
AddisonWesley
Ku C. C.
,
Lee K. Y.
1995
“Diagonal recurrent neural networks for dynamic system control,”
IEEE Trans. Neural Netw.
6
(1)
144 
156
DOI : 10.1109/72.363441
Lewis F. L.
,
Campos J.
,
Selmic R.
2002
NeuroFuzzy Control of Industrial Systems with Actuator Nonlinearities
SIAM Frontiers in Applied Mathematics