# A High-Performance Control Method of ConstantV

Authors: Long Chen; Xiaodong Sun; Haobin Jiang; Xing Xu
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

A three-phase induction motor used as a propulsion system for the electric vehicle (EV) is a nonlinear, multi-input multi-output, and strong coupling system. For such a complicated model system with unmeasured and unavoidable disturbances, as well as parameter variations, the conventional vector control method cannot meet the demands of high-performance control. Therefore, a novel control strategy named least squares support vector machines (LSSVM) inverse control is presented in the paper. Invertibility of the induction motor in the constantV
/
f control mode is proved to confirm its feasibility. The LSSVM inverse is composed of an LSSVM approximating the nonlinear mapping of the induction motor and two integrators. The inverse model of the constant V
/
f-controlled induction motor drive is obtained by using LSSVM, and then the optimal parameters of LSSVM are determined automatically by applying a modified particle swarm optimization (MPSO). Cascading the LSSVM inverse with the induction motor drive system, the pseudolinear system can be obtained. Thus, it is easy to design the closed-loop linear regulator. The simulation results verify the effectiveness of the proposed method.


---

## Body

## 1. Introduction

Nowadays, some serious problems such as environment depravation and air pollution are becoming more and more serious, due to the rapid development of the global economy. Electric vehicles (EVs), including fuel cell-powered vehicle and hybrid electric vehicles, are being currently researched and their practicalities are increasingly capturing many countries’ eyes, since they are a way to solve these problems that are tied to exhaust gas-emission and energy-saving issues [1–3].By converting electrical energy into mechanical energy, a motor can propel a vehicle [4, 5]. Compared with the combustion engines, the motors have some main advantages in terms of power density, conversion efficiency, low-speed torque characteristics, and so on [6–9]. In addition, when the motor operates in the braking mode, it can convert the mechanical energy back to electrical energy [10, 11]. Aforementioned characteristics of the motors make the electric drive more energy efficient, more powerful, and more compact. With the rapid development of power electronics, information technology, and the revolution in motor control, the EV technologies are being quickly progressed. Among EV key technologies, selection of a suitable drive, optimum design of the motor topologies, and optimal control strategies are the major factors [12, 13].In general, permanent-magnet synchronous motors (PMSMs) have been popular in EV traction applications, but some problems have recently arisen. One of the key issues is that the cost of rare earth materials, such as neodymium, has sharply increased in the past years [14–16]. Therefore, the induction motors are drawing attention as a promising alternative to PMSMs [17]. The induction motors have many characteristics, such as firm structure, small model, ruggedness, light capacity, low price, and the ability to operate in the extended high speed. Moreover, with the growth of power electronic technology and microprocess technology, high-speed induction motors driven by changeable frequency have many merits on aspects of reliability, low maintenance, low cost, craftwork, and so on [18–20]. So induction motor drive systems play an increasingly important role in driving EV. With the background, our study revisits and renews this old technology, aiming at high-performance control of the induction motor drives in EV applications.The mature control method of induction motor drives is the vector control, which has been widely implemented in electric drives [21, 22]. However, since the rotor flux angle of the induction motor is not directly measurable, vector control of an induction motor is more difficult than that of a PMSM. In addition, another issue in electrical drives is system sensitivity to inaccuracy and changes of motor parameters [23, 24]. Since the vector control systems are very sensitive to such inaccuracies, some parameters of the induction motor drives should be estimated online, and a more robust control structure is required. Therefore, some intelligent control methods, such as fuzzy control [25, 26], sliding mode control [27–29], adaptive control [30, 31], neural network control [32, 33], and other advanced control methods [34–36], are adopted for the induction motor drives. These approaches only improve different aspects of the control performance of induction motor drives. Thus, how to enhance the satisfying dynamic behavior of the induction motor drive for EVs further is very urgent.The aim of this paper is to propose a novel control scheme based on least squares support vector machines (LSSVM) inverse for the constantV
/
f-controlled induction motor drive system for EVs. The mathematic model of the induction motor drive is presented, and a reversible analysis of such system is also performed. Based on the analysis, speed control of the induction motor drive based on LSSVM inverse system method is proposed. The method is combined by LSSVM, which has the abilities of learning and function approximation and the adaptation capacity of system parameter variations, and the inverse system method which can realize the linearization of complex nonlinear system [37, 38]. The LSSVM is used to identify the inverse model of the constant V
/
f-controlled induction motor drive system, and a modified particle swarm optimization (MPSO) algorithm is adopted to optimize the kernel parameter and regularization parameter of the LSSVM. Consequently, a composite pseudolinear system is completed by constructing LSSVM inverse and combining it with the original system. Then the linear control techniques can be applied to design control system to achieve the high performance control of the original nonlinear system. Finally, the simulation testing research is studied using this method, and the control effect is satisfying.


## 2. Mathematic Model and Reversible Analyses

For current-followed SPWM inverter of the induction motor, when the nonlinear and time delay of the inverter, magnetic saturation, and iron loss of the induction motor are ignored, the state equation is described as a 6-order nonlinear model in stillα
-
β coordinates:
(2)
d
ω
r
d
t
=
n
p
J
(
T
e
-
T
L
)
=
n
p
2
J
(
ψ
s
α
i
s
β
-
ψ
s
β
i
s
α
)
-
n
p
J
T
L
,
d
ψ
s
α
d
t
=
V
cos
⁡
θ
1
-
R
s
i
s
α
,
d
ψ
s
β
d
t
=
V
sin
θ
1
-
R
s
i
s
β
,
d
i
s
α
d
t
=
R
r
L
s
L
r
-
L
m
2
ψ
s
α
+
L
r
L
s
L
r
-
L
m
2
ω
r
ψ
s
β
-
R
r
L
s
+
L
r
R
s
L
s
L
r
-
L
m
2
i
s
α
-
ω
r
i
s
β
+
L
r
L
s
L
r
-
L
m
2
V
cos
⁡
θ
1
,
d
i
s
β
d
t
=
R
r
L
s
L
r
-
L
m
2
ψ
s
β
-
L
r
L
s
L
r
-
L
m
2
ω
r
ψ
s
α
-
R
s
L
r
+
L
s
R
r
L
s
L
r
-
L
m
2
i
s
β
+
ω
r
i
s
α
+
L
r
L
s
L
r
-
L
m
2
V
sin
θ
1
,
d
θ
1
d
t
=
ω
1
,
where ω
1 and ω
r are synchronous angle frequency and rotate speed, respectively. i
s
α and i
s
β are stator currents in α
-
β coordinates, respectively. ψ
s
α and ψ
s
β are stator flux linkage in α
-
β coordinates, respectively. T
r is rotor time constant. L
m is mutual inductance. n
p is the number of pole pairs. L
s and L
r are stator inductance and rotor inductance, respectively. R
s and R
r are stator resistance and rotor resistance, respectively. T
e and T
L are electromagnetic torque and load torque, respectively. θ
1 is angle between the space voltage vector and the α axis. V is the amplitude of the space voltage vector.Given the state variable(2)
x
=
[
ω
r
,
ψ
s
α
,
ψ
s
β
,
i
s
α
,
i
s
β
,
θ
1
]
T
=
[
x
1
,
x
2
,
x
3
,
x
4
,
x
5
,
x
6
]
T
and the input variable
(3)
u
=
[
V
,
ω
1
]
T
=
[
u
1
,
u
2
]
T
,
the output is the ω
r, so
(4)
x
˙
=
f
(
x
,
u
)
=
[
n
p
2
J
(
x
2
x
5
-
x
3
x
4
)
-
n
p
J
T
L
u
1
cos
⁡
x
6
-
R
s
x
4
u
1
sin
x
6
-
R
s
x
5
R
r
L
s
L
r
-
L
m
2
x
2
+
L
r
L
s
L
r
-
L
m
2
x
1
x
3
-
R
r
L
s
+
L
r
R
s
L
s
L
r
-
L
m
2
x
4
-
x
1
x
5
+
L
r
L
s
L
r
-
L
m
2
u
1
cos
⁡
x
6
h
h
h
h
h
h
h
h
h
h
h
R
r
L
s
L
r
-
L
m
2
x
3
-
L
r
L
s
L
r
-
L
m
2
x
1
x
2
-
R
s
L
r
+
L
s
R
r
L
s
L
r
-
L
m
2
x
5
+
x
1
x
4
+
L
r
L
s
L
r
-
L
m
2
u
1
sin
x
6
h
h
h
h
h
h
h
h
h
h
h
u
2
]
,
y
=
h
(
x
)
=
x
1
=
ω
r
.When the induction motor is operating in the constantV
/
f control mode, we can obtain the following expression:
(5)
V
=
k
ω
1
,
where k
=
3
/
2
·
(
220
·
2
)
/
(
2
·
π
·
50
)
=
(
(
3
·
220
)
/
(
2
·
π
·
50
)
) is the proportion coefficient of frequency voltage.So(6)
u
1
=
k
u
2
.In order to analyze the reversibility of the induction motor, the output formula should be differentiated until the input variable is visualized firstly. According to expression (11), the following expressions can be deduced:
(7)
y
(
1
)
=
x
(
1
)
=
n
p
2
J
(
x
2
x
5
-
x
3
x
4
)
-
n
p
J
T
L
,
(8)
y
(
2
)
=
n
p
2
J
(
x
2
x
˙
5
+
x
˙
2
x
5
-
x
3
x
˙
4
-
x
˙
3
x
4
)
=
n
p
2
J
[
x
2
(
R
r
L
s
L
r
-
L
m
2
x
3
-
L
r
L
s
L
r
-
L
m
2
x
1
x
2
hhhhhhhhhh
-
R
s
L
r
+
L
s
R
r
L
s
L
r
-
L
m
2
x
5
+
x
1
x
4
hhhhhhhhhh
+
L
r
L
s
L
r
-
L
m
2
k
u
2
sin
x
6
)
hhhhh
+
(
k
u
2
cos
⁡
x
6
-
R
s
x
4
)
x
5
hhhhh
-
x
3
(
R
r
L
s
L
r
-
L
m
2
x
2
hhhhhhhhhhh
+
L
r
L
s
L
r
-
L
m
2
x
1
x
3
hhhhhhhhhhh
-
R
r
L
s
+
L
r
R
s
L
s
L
r
-
L
m
2
x
4
-
x
1
x
5
hhhhhhhhhhh
+
L
r
L
s
L
r
-
L
m
2
k
u
2
cos
⁡
x
6
)
hhhhh
-
(
k
u
2
sin
x
6
-
R
s
x
5
)
x
4
(
R
r
L
s
L
r
-
L
m
2
x
3
-
L
r
L
s
L
r
-
L
m
2
x
1
x
2
]
,
(9)
∂
y
(
2
)
∂
u
2
=
n
p
2
J
[
x
2
L
r
L
s
L
r
-
L
m
2
k
sin
x
6
+
k
cos
⁡
x
6
x
5
hhhhh
-
x
3
L
r
L
s
L
r
-
L
m
2
k
cos
⁡
x
6
-
k
sin
x
6
x
4
]
=
n
p
2
J
[
L
r
k
L
s
L
r
-
L
m
2
(
x
2
sin
x
6
-
x
3
cos
⁡
x
6
)
hhhhh
+
k
(
cos
⁡
x
6
x
5
-
sin
x
6
x
4
)
L
r
k
L
s
L
r
-
L
m
2
]
.Sincex
∈
Ω
=
{
x
∈
R
6
:
(
L
r
k
/
(
L
s
L
r
-
L
m
2
)
)
(
x
2
sin
x
6
-
x
3
cos
⁡
x
6
)
+
k
(
x
5
cos
⁡
x
6
-
x
4
sin
x
6
)
≠
0
},  the inverse of system is existent. According to implicit function theorem, the inverse system can be written as
(10)
u
=
ξ
(
x
,
y
,
y
˙
,
y
¨
)
.


## 3. Basic Conception of LSSVM

As an interesting variant of the standard support vector machines (SVM), least squares support vector machines (LSSVM) have been proposed by Suykens and Vandewalle for solving pattern recognition and nonlinear function estimation problems [39, 40]. Here, we simply present the basic principle of LSSVM. Considering a given training set {
x
i
,
y
i
}
i
=
1
l, with input data x
i
∈
R
n and output data y
i
∈
R, in feature space, the regression model takes the form
(11)
f
(
x
)
=
w
T
Φ
(
x
)
+
b
,
where the nonlinear mapping Φ
(
·
)
:
R
n
→
R
H maps the input data into a higher dimensional feature space, w
∈
R
H is a weight vector of the same dimension as the feature space, and b is a threshold. Then the following optimization problem for the LSSVM is formulated:
(12)
min
⁡
ect
to
J
(
w
,
ξ
)
=
1
2
∥
w
∥
2
+
1
2
γ
∑
i
=
1
l
ξ
i
2
subject
to
y
i
=
w
T
Φ
(
x
i
)
+
b
+
ξ
i
,
where ξ is approximation error at the instant i, J is a loss function, and γ is an adjustable constant which determines penalties for estimation errors.The corresponding Lagrange function is described as(13)
L
(
w
,
b
,
ξ
,
α
)
=
J
(
w
,
ξ
)
-
∑
i
=
1
l
α
i
[
w
T
Φ
(
x
i
)
+
b
+
ξ
i
-
y
i
]
,
where α
=
(
α
1
,
α
2
,
α
3
,
…
,
α
l
)
T is a vector with the Lagrange multipliers.According to the optimization conditions by Karush-Kuhn-Tucker, the optimal values can be found by setting the derivatives of the Lagrange function equal to zero:(14)
∂
L
∂
w
=
0
⟶
w
=
∑
i
=
1
l
α
i
Φ
(
x
i
)
,
∂
L
∂
b
=
0
⟶
∑
i
=
1
l
α
i
=
0
,
∂
L
∂
ξ
i
=
0
⟶
α
i
=
γ
ξ
i
,
i
=
1,2
,
…
,
l
,
∂
L
∂
α
i
=
0
⟶
w
T
Φ
(
x
i
)
+
b
+
ξ
i
-
y
i
=
0
.Then the optimization problem can be rewritten as(15)
[
0
1
l
×
1
T
1
l
×
1
Ω
+
γ
-
1
I
]
[
b
α
]
=
[
0
y
]
,
where y
=
[
y
1
,
y
2
,
y
3
,
…
,
y
l
]
T, 1
l
×
1
=
[
1,1
,
…
,
1
]
T, I
=
diag
⁡
[
1,1
,
…
,
1
], Ω
=
{
Ω
i
j
}
l
×
l
,
Ω
i
j
=
Φ
T
(
x
i
)
·
Φ
(
x
j
), i
=
1
,
2
,
…
,
l.According to Mercer’s condition, there is kernel functionK
(
x
i
,
x
)
=
Φ
T
(
x
i
)
·
Φ
(
x
), which causes such LSSVM model
(16)
f
(
x
)
=
∑
i
=
1
l
α
i
K
(
x
i
,
x
)
+
b
,
where α and b are obtained by solving (7). In this paper, we will focus on RBF kernel function which corresponds to
(17)
K
(
x
i
,
x
)
=
exp
⁡
(
-
|
x
i
-
x
|
2
2
σ
2
)
,
where σ is the kernel parameter. The structure of the LSSVM is shown in Figure 1.Figure 1
Structure of least squares support vector machine.


## 4. MPSO-Based Parameters Optimization for LSSVM

### 4.1. MPSO

Particle swarm optimization (PSO), a novel evolutionary computation technique, was proposed by Kennedy and Eberhart in 1995 [41–43]. It is inspired by social behavior of bird flocking and fish schooling and has been found to be robust in solving nonlinear optimization problems. Compared with other stochastic approaches, PSO can generate solutions of high quality with relative shorter calculation time and has more stable convergence features. In the original PSO algorithm, particles flying through the search space are affected by two factors: one is the best position ever found of the individual, and another is the best position of the group.The position and velocity ofith individual (called particle) in d-dimensional search space can be represented as u
i
=
[
u
i
1
,
u
i
2
,
…
,
u
i
d
] and v
i
=
[
v
i
1
,
v
i
2
,
…
,
v
i
d
], respectively. The local best of the ith particle can be denoted as p
i
=
[
p
i
1
,
p
i
2
,
…
,
p
i
d
] and the global best found so far can be denoted as p
g
=
[
p
g
1
,
p
g
2
,
…
,
p
g
d
]. Then, at each iteration, the new positions and velocities of each particle can be calculated as shown in the following formulas:
(18)
v
i
(
k
+
1
)
=
ω
v
i
(
k
)
+
c
1
r
1
(
p
i
(
k
)
-
u
i
(
k
)
)
+
c
2
r
2
(
p
g
(
k
)
-
u
i
(
k
)
)
,
(19)
u
i
(
k
+
1
)
=
u
i
(
k
)
+
v
i
(
k
)
,
where i
=
1,2
,
…
,
m. m is the number of particles. k is the number of current iteration. u
i
(
k
) is the position of ith particle at iteration k. p
i
(
k
) is the best local position of ith particle at iteration k. p
g
(
k
) is the best global position of all particles at iteration k. v
i
(
k
) is the velocity of ith particle at iteration k. r
1 and r
2 are random variables drawn from a uniform distribution in the range [
0
,
1
] to provide a stochastic weight of the different components participating in the particle velocity definition. c
1 and c
2 are two acceleration constants regulating the relative velocities with respect to the best global and local positions, respectively. ω is the inertia weight used as a tradeoff between global and local exploration capabilities of the swarm. Let the inertia weight be a high value ω
max
⁡ in the early evolution and linearly decrease to ω
min
⁡ at the maximal number of iterations. Its mathematical representation can be described as
(20)
ω
=
ω
min
⁡
+
n
max
⁡
-
n
n
max
⁡
(
ω
max
⁡
-
ω
min
⁡
)
,
where n
max
⁡ is the maximal number of iterations and n is the current number of iterations.In order to overcome the limitation that basic PSO was not convergent because of the maximum speed parameters and big acceleration constants(
c
1
,
c
2
), Clerc introduced the shrinkage factor η. This approach can ensure that the PSO algorithm convergent during the search process. So, in this paper, we apply the MPSO method with shrinkage factor η to optimize the LSSVM’s parameters. Then the mathematical representations of PSO algorithm given in (18) can be changed to
(21)
v
i
(
k
+
1
)
=
η
(
+
c
2
r
2
(
p
g
(
k
)
-
u
i
(
k
)
)
ω
v
i
(
k
)
+
c
1
r
1
(
p
i
(
k
)
-
u
i
(
k
)
)
h
h
h
h
+
c
2
r
2
(
p
g
(
k
)
-
u
i
(
k
)
)
)
,
where
(22)
η
=
2
|
2
-
φ
-
φ
2
-
4
φ
|
,
φ
=
c
1
+
c
2
>
4
.Aiming at the regularization parameterγ and kernel parameter σ which are needed to be optimized, choose mean square root error (RMSE) of the LSSVM as shown in (14) as the fitness function f
(
·
) of the MPSO:
(23)
f
(
u
)
=
f
(
γ
,
σ
)
=
E
RMSE
=
1
l
∑
i
=
1
l
(
y
i
-
y
^
i
)
2
,
where y
i and y
^
i are real value and model output one, respectively. The minimization of the fitness function can be seen as a mechanism to guarantee a reasonable choice of the optimized parameters (
γ
,
σ
).


### 4.2. Procedure of LSSVM’s Parameters Optimization Using MPSO

The steps of the MPSO-LSSVM estimation are summarized as follows.Step 1.
Generate training and test samples sets, and normalize the data.Step 2.
Initialize the parameters of MPSO. The values of the parameters are as follows:m
=
60, d
=
2,  n
max
⁡
=
500,  c
1
=
c
2
=
2.05, ω
max
⁡
=
0.9, and ω
min
⁡
=
0.4.Step 3.
Generate an initial swarm of sizem. Set the best position of each particle with its initial position; that is, p
i
=
u
i
(
i
=
1
,
2
,
…
,
m
).Step 4.
Set to zero the velocity vectorsv
i
(
i
=
1
,
2
,
…
,
m
) that are associated with the m particles.Step 5.
For each candidate particleu
i
(
i
=
1
,
2
,
…
,
m
), train an LSSVM estimator on the corresponding training set and with the estimation of γ and σ that are conveyed by u
i. Then, compute its fitness function f
(
u
i
).Step 6.
Update the velocity of each particle using (21). To perform the update, the best global position p
g is selected. p
g is chosen as the position exhibiting the minimal value of the considered fitness function over all explored trajectories. In detail, compare each particle’s current fitness value f
(
u
i
) with the fitness value of its best local position f
(
p
i
). If f
(
u
i
)
<
f
(
p
i
), let p
i
=
u
i. And compare the current fitness value f
(
u
i
) with the fitness value of the best globe position f
(
p
g). If f
(
u
i
)
<
f
(
p
g
), let p
g
=
u
i.Step 7.
Update the position of each particle by means of (19). In the event of a particle flying beyond the predefined boundary of the search space, set the position of the particle at the space boundary and reverse its search direction by means of multiplying its velocity vector by −1.Step 8.
Ifn
max
⁡
≥
500 or RMSE < 1 × 10−3, output the optimal parameter values; else return to Step 5.


## 4.1. MPSO

Particle swarm optimization (PSO), a novel evolutionary computation technique, was proposed by Kennedy and Eberhart in 1995 [41–43]. It is inspired by social behavior of bird flocking and fish schooling and has been found to be robust in solving nonlinear optimization problems. Compared with other stochastic approaches, PSO can generate solutions of high quality with relative shorter calculation time and has more stable convergence features. In the original PSO algorithm, particles flying through the search space are affected by two factors: one is the best position ever found of the individual, and another is the best position of the group.The position and velocity ofith individual (called particle) in d-dimensional search space can be represented as u
i
=
[
u
i
1
,
u
i
2
,
…
,
u
i
d
] and v
i
=
[
v
i
1
,
v
i
2
,
…
,
v
i
d
], respectively. The local best of the ith particle can be denoted as p
i
=
[
p
i
1
,
p
i
2
,
…
,
p
i
d
] and the global best found so far can be denoted as p
g
=
[
p
g
1
,
p
g
2
,
…
,
p
g
d
]. Then, at each iteration, the new positions and velocities of each particle can be calculated as shown in the following formulas:
(18)
v
i
(
k
+
1
)
=
ω
v
i
(
k
)
+
c
1
r
1
(
p
i
(
k
)
-
u
i
(
k
)
)
+
c
2
r
2
(
p
g
(
k
)
-
u
i
(
k
)
)
,
(19)
u
i
(
k
+
1
)
=
u
i
(
k
)
+
v
i
(
k
)
,
where i
=
1,2
,
…
,
m. m is the number of particles. k is the number of current iteration. u
i
(
k
) is the position of ith particle at iteration k. p
i
(
k
) is the best local position of ith particle at iteration k. p
g
(
k
) is the best global position of all particles at iteration k. v
i
(
k
) is the velocity of ith particle at iteration k. r
1 and r
2 are random variables drawn from a uniform distribution in the range [
0
,
1
] to provide a stochastic weight of the different components participating in the particle velocity definition. c
1 and c
2 are two acceleration constants regulating the relative velocities with respect to the best global and local positions, respectively. ω is the inertia weight used as a tradeoff between global and local exploration capabilities of the swarm. Let the inertia weight be a high value ω
max
⁡ in the early evolution and linearly decrease to ω
min
⁡ at the maximal number of iterations. Its mathematical representation can be described as
(20)
ω
=
ω
min
⁡
+
n
max
⁡
-
n
n
max
⁡
(
ω
max
⁡
-
ω
min
⁡
)
,
where n
max
⁡ is the maximal number of iterations and n is the current number of iterations.In order to overcome the limitation that basic PSO was not convergent because of the maximum speed parameters and big acceleration constants(
c
1
,
c
2
), Clerc introduced the shrinkage factor η. This approach can ensure that the PSO algorithm convergent during the search process. So, in this paper, we apply the MPSO method with shrinkage factor η to optimize the LSSVM’s parameters. Then the mathematical representations of PSO algorithm given in (18) can be changed to
(21)
v
i
(
k
+
1
)
=
η
(
+
c
2
r
2
(
p
g
(
k
)
-
u
i
(
k
)
)
ω
v
i
(
k
)
+
c
1
r
1
(
p
i
(
k
)
-
u
i
(
k
)
)
h
h
h
h
+
c
2
r
2
(
p
g
(
k
)
-
u
i
(
k
)
)
)
,
where
(22)
η
=
2
|
2
-
φ
-
φ
2
-
4
φ
|
,
φ
=
c
1
+
c
2
>
4
.Aiming at the regularization parameterγ and kernel parameter σ which are needed to be optimized, choose mean square root error (RMSE) of the LSSVM as shown in (14) as the fitness function f
(
·
) of the MPSO:
(23)
f
(
u
)
=
f
(
γ
,
σ
)
=
E
RMSE
=
1
l
∑
i
=
1
l
(
y
i
-
y
^
i
)
2
,
where y
i and y
^
i are real value and model output one, respectively. The minimization of the fitness function can be seen as a mechanism to guarantee a reasonable choice of the optimized parameters (
γ
,
σ
).


## 4.2. Procedure of LSSVM’s Parameters Optimization Using MPSO

The steps of the MPSO-LSSVM estimation are summarized as follows.Step 1.
Generate training and test samples sets, and normalize the data.Step 2.
Initialize the parameters of MPSO. The values of the parameters are as follows:m
=
60, d
=
2,  n
max
⁡
=
500,  c
1
=
c
2
=
2.05, ω
max
⁡
=
0.9, and ω
min
⁡
=
0.4.Step 3.
Generate an initial swarm of sizem. Set the best position of each particle with its initial position; that is, p
i
=
u
i
(
i
=
1
,
2
,
…
,
m
).Step 4.
Set to zero the velocity vectorsv
i
(
i
=
1
,
2
,
…
,
m
) that are associated with the m particles.Step 5.
For each candidate particleu
i
(
i
=
1
,
2
,
…
,
m
), train an LSSVM estimator on the corresponding training set and with the estimation of γ and σ that are conveyed by u
i. Then, compute its fitness function f
(
u
i
).Step 6.
Update the velocity of each particle using (21). To perform the update, the best global position p
g is selected. p
g is chosen as the position exhibiting the minimal value of the considered fitness function over all explored trajectories. In detail, compare each particle’s current fitness value f
(
u
i
) with the fitness value of its best local position f
(
p
i
). If f
(
u
i
)
<
f
(
p
i
), let p
i
=
u
i. And compare the current fitness value f
(
u
i
) with the fitness value of the best globe position f
(
p
g). If f
(
u
i
)
<
f
(
p
g
), let p
g
=
u
i.Step 7.
Update the position of each particle by means of (19). In the event of a particle flying beyond the predefined boundary of the search space, set the position of the particle at the space boundary and reverse its search direction by means of multiplying its velocity vector by −1.Step 8.
Ifn
max
⁡
≥
500 or RMSE < 1 × 10−3, output the optimal parameter values; else return to Step 5.


## 5. LSSVM Inverse Control

From the mathematical model of the constantV
/
f-controlled induction motor described as (2), it can be seen that the induction motor drive system is a nonlinear and strong coupling system. Therefore, it is very difficult to get the accurate analytic expression described in (10). To effectively solve this thorny problem, an LSSVM is employed to identify the inverse model of the constant V
/
f-controlled induction motor drive system, since the LSSVM has the ability of approaching an arbitrary nonlinear function with satisfactory accuracy. The proposed LSSVM inverse consists of an LSSVM approximating the nonlinear mapping (10) and two integrators charactering its dynamic behaviors. In view of the principle of the LSSVM regression and the control principle of the inverse system scheme, we can easily obtain the whole implementation steps of the LSSVM inverse control scheme for the induction motor drive system for EVs.Step 1.
A superposition of the random signals and constant is chosen as the input excitation signal. The original induction motor drive system is adequately excited, and then we can obtain its static and dynamic performance. We not only need the static data of the constantV
/
f-controlled induction motor drive system but also need the dynamic ones for the LSSVM learning the inverse model of the constant V
/
f-controlled induction motor. Consequently, the complete inverse model of induction motor drive system can just be gotten. During the measuring process, because there are various random noises jamming and the error of measurement devices themselves, there are generally some errors between measured values and real ones. In order to effectively overcome these disadvantages, the 2-order filters are used for adopting data. Note that the variation period of the given speed signal must be chosen properly; otherwise the practical system cannot follow the given signal. In this paper, the variation period of the all kinds of given signal is set at 10 s and the sampling period of speed is set at 0.1 s. The whole operation time is 200 s and 2000 sampled data are obtained.Step 2.
By using precise seven-point algorithm, the 1-order and 2-order offline derivative of speed output response can be obtained. Therefore, the training sample sets{
x
i
,
u
}
(
i
=
1,2
,
…
,
l
,
l
=
1000
) can be formed by chosing from original sampling period equidistantly, where, x
=
[
y
(
2
)
,
y
(
1
)
,
y
]
=
[
ω
r
(
2
)
,
ω
r
(
1
)
,
ω
r
] and u
=
ω
1 are, respectively, the input data and the expected output data of the LSSVM which learns the inverse model of the constant V
/
f-controlled induction motor drive system. Figure 2 shows the collected training data.Collected training data.
(a)
ω
1
(b)
ω
rStep 3.
According to the MPSO optimization procedure mentioned aforementioned, the best values of LSSVM parameters areγ
=
950 and σ
=
2.6, and its RMSE (i.e., the fitness function) is equal to 3.421 × 10
-
3. Through learning the LSSVM with training sample sets, the corresponding input vector coefficient α
i (where the zero coefficients are also included in the formula) and threshold value b can be obtained. Therefore, in view of the current input x, the output of the αth-order inversion can be identified as
(24)
u
(
x
)
=
∑
i
=
1
d
α
i
(
Φ
T
(
x
i
)
·
Φ
(
x
)
)
+
b
=
∑
i
=
1
d
α
i
K
(
x
i
,
x
)
+
b
.Step 4.
By combining the LSSVM inverse with the induction motor drive system, a 2-order compound system, named pseudolinear system, can be obtained as shown in Figure3. The input and output linearization of the original constant V
/
f-controlled induction motor drive system is achieved. Although the relationship between input and output of the compound system is linear, there are still some nonlinear factors in the system. Therefore, the compound system is not an ideal linear system and is called the pseudolinear system. Since the compound system is an open-loop unstable system and various uncertainties are existing in practical applications, the additional closed-loop controller should be designed. In this paper, the proportional integral (PI) controller is designed for the closed-loop controller. Figure 4 shows the whole control diagram of LSSVM inverse for the induction motor drive system.Figure 3
Diagram of pseudolinear system.Figure 4
Control diagram of LSSVM inverse control.


## 6. Simulation Test Research

The parameters of the induction motor areP
e
=
5 KW,   R
s
=
5.35 Ω,   R
r
=
4.8
5 Ω,   L
s
=
0
.
41
H, L
r
=
0.46 H,  L
m
=
0
.
4
7 H, and J
=
0.00
18 kg·m2. Rated speed is 1600 r/min.


### 6.1. Predicted Result Comparison of the Inverse Model

The remaining 1000 groups of data from the whole sample data sets are chosen and then adopted for testing sample to compare the predicted performance of the inverse model. To effectively test the performance of the inversion, the mean square root of error(
E
RMSE
) and maximal absolute error (
E
MAXE
) are considered as
(25)
E
RMSE
=
1
l
∑
i
=
1
l
(
y
i
-
y
^
i
)
2
,
E
MAXE
=
max
⁡
i
=
1
l
|
y
i
-
y
^
i
|
,
where i
=
1,2
,
…
,
l and y
i and y
^
i are real and predicting values.In order to verify the predicted performance of the LSSVM inversion with the MPSO algorithm (MPSO-LSSVM), the standard SVM and LSSVM are utilized for the training sample to develop the inversion of the constantV
/
f-controlled induction motor drive system. The CPU runtime (t
/
s) and the corresponding key performance indicators (E
RMSE and E
MAXE) of the models are listed in Table 1. From Table 1, it can be seen that the generalization ability and predicted precision of MPSO-LSSVM inversion is superior to the standard SVM and LSSVM inversion.Table 1
Comparison of predicting results ofE
RMSE, E
MAXE, and t.
E
RMSE
E
MAXE
t
/s
Standard SVM
0.0091
0.0073
60. 472
Standard LSSVM
0.0053
0.0042
23. 193
MPSO-LSSVM
1.306
e
-
04
7.616
e
-
04
11.178


### 6.2. Simulation Results Analysis

The constantV
/
f-controlled induction motor drive system under the control schemes of the proposed method and vector control are simulated using Matlab/Simulink.Case 1 (performance comparison of speed startup response without load disturbances).
In this case, Figures5 and 6 are, respectively, the simulation results of speed startup response curves in the case of rated load with the proposed method and vector control. The dashed and solid lines are, respectively, the reference value and practical one. Compared with the speed response with vector control, the speed response under LSSVM inverse control has a shorter settling time and much smaller overshoot and steady-state error. The corresponding performance index of the speed response is shown in Table 2. From Figures 5 and 6 and Table 2, we can see that the constant V
/
f-controlled induction motor drive system has a better speed-adjusting performance by using the proposed method.Table 2
Speed performance comparison in the case of rated load.
Settling  time
Overshoot
Steady-state error
Proposed method
1.4 s
7.33%
0.56
Vector control method
2 s
32.62%
6.83Figure 5
Speed startup response curves in the case of rated load with vector control method.Figure 6
Speed startup response curves in the case of rated load with the proposed control method.Case 2 (performance comparison of tracking triangular wave without load disturbances).
In this case, Figures7 and 8 show, respectively, the simulation results of tracking triangular wave in the case of rated load with LSSVM inverse control and vector control methods. In Figures 7 and 8, the amplitudes of the triangular waves are all between 1200 r/min and 1600 r/min, and the dashed and the solid lines are, respectively, the given motor input signals and response curves. From Figures 7 and 8, we can see that, by adopting LSSVM inverse control method, the setting time of the constant V
/
f-controlled induction motor drive system is shorter, and its dynamic performance is also better. Additionally, the response curve can more accurately and quickly track the given motor input signal, and the control accuracy can be improved form 86% to 96%.Figure 7
Response of tracking triangle curves in the case of rated load with vector control method.Figure 8
Response of tracking triangle curves in the case of rated load with proposed control method.Case 3 (performance comparison with rotor resistance variation).
Of all the motor parameters, the rotor resistance has the greatest impact on the control performance of the motor drive system. In this case, a random variable is added to the rated value of the motor rotor resistance, and its amplitude is not more than 10% of the rated resistance value. The comparison results of the proposed LSSVM inverse control and the vector control methods in the case of rotor resistance variation and without load disturbances are shown in Figures9 and 10, respectively. From Figures 9 and 10, it can be seen that, compared with the vector control method, the LSSVM inverse scheme has a smoother speed curve with almost no overshoot. In addition, by adopting the proposed LSSVM inverse method, the relative steady error can be reduced from 12% to 4%.Figure 9
Speed response curves in the case of rotor resistance variation with vector control method.Figure 10
Speed response curves in the case of rotor resistance variation with the proposed control method.Case 4 (performance comparison with load disturbances).
To compare the capacity of being insensitive to the load torque variation of the proposed LSSVM inverse control and the vector control schemes, the simulation comparison was also carried out under sudden load disturbance impact. When the constantV
/
f-controlled induction motor drive system is running at steady state of 1600 r/min, a sudden load torque disturbance was applied at 100 s and removed immediately, and the simulation results of speed response curves during the sudden application and removal of the load disturbance are shown in Figures 11 and 12. From Figures 11 and 12, it can be seen that the speed response by using the proposed method recovers faster when the load torque disturbance was added and removed. We can see that the proposed method can effectively attenuate the speed deviation caused by load disturbances and has a shorter recovery time. Moreover, from Figures 11 and 12, we can see that the control accuracy can be improved form 90% to 97% by using the proposed method. Therefore, it obviously suggests that the constant V
/
f-controlled induction motor drive system has better robustness performance compared with the vector control method and that the LSSVM inverse method possesses a good adaptation and a much better dynamic performance against load disturbance.Figure 11
Speed response curves in the case of load disturbance with vector control method.Figure 12
Speed response curves in the case of load disturbance with the proposed control method.


## 6.1. Predicted Result Comparison of the Inverse Model

The remaining 1000 groups of data from the whole sample data sets are chosen and then adopted for testing sample to compare the predicted performance of the inverse model. To effectively test the performance of the inversion, the mean square root of error(
E
RMSE
) and maximal absolute error (
E
MAXE
) are considered as
(25)
E
RMSE
=
1
l
∑
i
=
1
l
(
y
i
-
y
^
i
)
2
,
E
MAXE
=
max
⁡
i
=
1
l
|
y
i
-
y
^
i
|
,
where i
=
1,2
,
…
,
l and y
i and y
^
i are real and predicting values.In order to verify the predicted performance of the LSSVM inversion with the MPSO algorithm (MPSO-LSSVM), the standard SVM and LSSVM are utilized for the training sample to develop the inversion of the constantV
/
f-controlled induction motor drive system. The CPU runtime (t
/
s) and the corresponding key performance indicators (E
RMSE and E
MAXE) of the models are listed in Table 1. From Table 1, it can be seen that the generalization ability and predicted precision of MPSO-LSSVM inversion is superior to the standard SVM and LSSVM inversion.Table 1
Comparison of predicting results ofE
RMSE, E
MAXE, and t.
E
RMSE
E
MAXE
t
/s
Standard SVM
0.0091
0.0073
60. 472
Standard LSSVM
0.0053
0.0042
23. 193
MPSO-LSSVM
1.306
e
-
04
7.616
e
-
04
11.178


## 6.2. Simulation Results Analysis

The constantV
/
f-controlled induction motor drive system under the control schemes of the proposed method and vector control are simulated using Matlab/Simulink.Case 1 (performance comparison of speed startup response without load disturbances).
In this case, Figures5 and 6 are, respectively, the simulation results of speed startup response curves in the case of rated load with the proposed method and vector control. The dashed and solid lines are, respectively, the reference value and practical one. Compared with the speed response with vector control, the speed response under LSSVM inverse control has a shorter settling time and much smaller overshoot and steady-state error. The corresponding performance index of the speed response is shown in Table 2. From Figures 5 and 6 and Table 2, we can see that the constant V
/
f-controlled induction motor drive system has a better speed-adjusting performance by using the proposed method.Table 2
Speed performance comparison in the case of rated load.
Settling  time
Overshoot
Steady-state error
Proposed method
1.4 s
7.33%
0.56
Vector control method
2 s
32.62%
6.83Figure 5
Speed startup response curves in the case of rated load with vector control method.Figure 6
Speed startup response curves in the case of rated load with the proposed control method.Case 2 (performance comparison of tracking triangular wave without load disturbances).
In this case, Figures7 and 8 show, respectively, the simulation results of tracking triangular wave in the case of rated load with LSSVM inverse control and vector control methods. In Figures 7 and 8, the amplitudes of the triangular waves are all between 1200 r/min and 1600 r/min, and the dashed and the solid lines are, respectively, the given motor input signals and response curves. From Figures 7 and 8, we can see that, by adopting LSSVM inverse control method, the setting time of the constant V
/
f-controlled induction motor drive system is shorter, and its dynamic performance is also better. Additionally, the response curve can more accurately and quickly track the given motor input signal, and the control accuracy can be improved form 86% to 96%.Figure 7
Response of tracking triangle curves in the case of rated load with vector control method.Figure 8
Response of tracking triangle curves in the case of rated load with proposed control method.Case 3 (performance comparison with rotor resistance variation).
Of all the motor parameters, the rotor resistance has the greatest impact on the control performance of the motor drive system. In this case, a random variable is added to the rated value of the motor rotor resistance, and its amplitude is not more than 10% of the rated resistance value. The comparison results of the proposed LSSVM inverse control and the vector control methods in the case of rotor resistance variation and without load disturbances are shown in Figures9 and 10, respectively. From Figures 9 and 10, it can be seen that, compared with the vector control method, the LSSVM inverse scheme has a smoother speed curve with almost no overshoot. In addition, by adopting the proposed LSSVM inverse method, the relative steady error can be reduced from 12% to 4%.Figure 9
Speed response curves in the case of rotor resistance variation with vector control method.Figure 10
Speed response curves in the case of rotor resistance variation with the proposed control method.Case 4 (performance comparison with load disturbances).
To compare the capacity of being insensitive to the load torque variation of the proposed LSSVM inverse control and the vector control schemes, the simulation comparison was also carried out under sudden load disturbance impact. When the constantV
/
f-controlled induction motor drive system is running at steady state of 1600 r/min, a sudden load torque disturbance was applied at 100 s and removed immediately, and the simulation results of speed response curves during the sudden application and removal of the load disturbance are shown in Figures 11 and 12. From Figures 11 and 12, it can be seen that the speed response by using the proposed method recovers faster when the load torque disturbance was added and removed. We can see that the proposed method can effectively attenuate the speed deviation caused by load disturbances and has a shorter recovery time. Moreover, from Figures 11 and 12, we can see that the control accuracy can be improved form 90% to 97% by using the proposed method. Therefore, it obviously suggests that the constant V
/
f-controlled induction motor drive system has better robustness performance compared with the vector control method and that the LSSVM inverse method possesses a good adaptation and a much better dynamic performance against load disturbance.Figure 11
Speed response curves in the case of load disturbance with vector control method.Figure 12
Speed response curves in the case of load disturbance with the proposed control method.


## 7. Conclusion

This paper aims at how to further improve the performance of the induction motor drive system used in EVs widely. The design and implementation procedure of a control scheme named LSSVM inverse control for a high-performance induction motor drive system for the traction purpose of EVs has been investigated. According to the characteristic of the constantV
/
f control mode, the mathematic model of the induction motor drive system is deduced and an inverse system model suitable for the constant V
/
f control mode is also obtained. Based on these, the LSSVM inverse control method is applied to control the induction motor drive system. The simulation test results testify that the proposed method is feasible and it can realize the high performance control of the induction motor drive system. It also offers a new method for the control of the induction motor drive system.



---

*Source: 386174-2014-01-21.xml*