SICE Transaction on Industrial Application
Vol. 9, No. 20, 148/155 (2010)

A Continuous Control Input Generation Method
in Model Predictive Control for an Electromechanical Valve
Addy WAHYUDIE ∗ , Taizo NAKAO ∗∗ , Toshimitsu KAI ∗ , Hideaki SETOGUCHI ∗Masakazu MUKAI ∗ , and Taketoshi KAWABE ∗Abstract : This paper proposes a continuous control input generation method in model predictive control for an electromechanical valve. In previous works, the model predictive control generates a discontinuous control current for thevalve actuation. From a practical point of view, it is difficult for an amplifier to generate a discontinuous current. A plantof the electromechanical valve is augmented with an integrator to generate continuous control input. A traveling timeof the valve armature can be shortened by adjusting parameters of a state weighting matrix in a performance index of
the model predictive control algorithm. In addition, the proposed method has an important feature that it is a deadbeat
controller with a designed terminal condition of the valve armature. Therefore, the armature reaches a target positionand velocity. The controller calculates the continuous optimum control in each sampling instant. A disturbance observer is used for estimating the armature velocity and improving robustness of the controlled system against parameter
uncertainty. The proposed method is validated in computer simulation and experimental results.
Key Words : electromechanical valve, model predictive control, motion control, robust control

1.

Introduction

Significant improvements in internal combustion engine canbe achieved using the electromechanical valve (EMV) system [1]. The EMV system affords the valve timings that are
fully independent of a crank-shanks position. Fuel saving (20%or higher) can be obtained by controlling the intake process atpart load, eliminating intake air throttling and pumping losses.
Pollutant emission reduction (NOx and COx ) is achieved by thepossibility of achieving the high internal Exhaust Gas Recirculation (EGR) rate through both retarding and advancing theexhaust valve closing, resulting NOx reduction up to 90%. Increasing of 10% in the maximum torque and up to 50% at thelow speed torque are produced by optimizing the valve timing at
all operating conditions, avoiding the compromise of mechanically driven camshafts.
A configuration of the EMV system is shown in Fig. 1. The
EMV is driven by two electromagnets, two springs, and an armature. The armature travels between the two electromagnets.
The armature is suspended by the two springs with large spring
constant, enabling it to move at high speed. The springs are attached to make the armature balanced at the neutral position. In“open valve” operation, current at the upper-side electromagnet
is switched off, and the armature moves to the lower electromagnet by the spring force. When the armature moves close to
the lower electromagnet, the current is switched on to this electromagnet, so that pulling control force is available. For “close
valve” operation, the electromagnets system works conversely.
In an actual application, the armature moves within a shortdistance (8–9 [mm]) over a short period of time (approximately

3–4 [ms]) [2, 3]. This short period of time is desirable in order to meet the maximum engine speed (5000 − 6000 [rpm]).
Furthermore, a high speed collision between the armature andthe electromagnet is not desirable. Some authors in [2–4] proposed that the impact velocity should be controlled in less than
0.1 [m/s] for maintaining the acceptable engine acoustic noiselevel, and for avoiding the valve is distorted by mechanicalshock. To overcome the problems, motion of the armature iscontrolled by controlling the current in the pulling electromagnet. However, there are difficulties for controlling motion of thearmature, such as:
1. The control input should be calculated in a very short period of time.
2. The pushing and pulling force are only available from thesprings and the electromagnets, respectively. However, thepushing force cannot be controlled because it generated by
the springs.
3. The magnetic force is effective only when the armature isvery close to the pulling electromagnet.

Shaft
Upper spring
Lower spring
Upper electromagnet
Armature
Lower electromagnet
Cylinder head∗

Graduate School of Information Science and ElectricalEngineering, Kyushu University, Fukuoka, Japan.
∗∗
Nippon Steel Engineering Corp., Kitakyushu, Japan.
(Received June 16, 2010)

9 mm

Valve

Fig. 1

Configuration of an electromechanical valve

TRIA 020/10/0920–148 c 2010 SICE

SICE T. on Industrial Application Vol. 9 No. 20 2010
4. Nonlinearity of the electromagnet is large when the armature close to the electromagnets.
5. The valve should be opened/closed within the specified
time in synchronization with the engine revolution.
6. There exist a delay of the electromagnet current.
The control method for the EMV systems have been studied
in recent years. Most of these existed control method (for example papers [2–6]) showed that the armature is asymptoticallyconverge to its desired position and velocity. The papers [7–8]used a iterative learning method for controlling the EMV. However, results showed that many learning processes are needed
before achieving the armature achieving sufficient performance.
This paper apply an online deadbeat controller for controllingthe armature motion using model predictive control (MPC) [9].
In this paper, the proposed method is expected to overcome thefirst five difficulties for controlling the EMV. In this paper, an
improvement concerning a continuous control input generation
for the EMV is proposed in this paper. In previous works [10–11], the MPC (so called the conventional MPC here) generates
a discontinuous optimal control for the EMV. Although satisfactory results in the computer simulation and the experimentusing a laboratory scale testbed are shown using the conventional MPC [12], it is difficult for an amplifier to generate a discontinuous input in an actual application. This because a lower
quality of an amplifier is used in actual application. In order toprovide the continuous control input for the EMV, the input is
integrated using an integrator. The addition of the integrator in
front of the EMV plant tends to yield a longer traveling time of
the armature. However, the paper [13] showed that the travelingtime in the proposed method can be shortened close to the traveling time in the conventional MPC by adjusting parameters ofa state weighting matrix in a performance index of the MPC.
This paper is organized as follows. A model for the augmented plant of the EMV and the proposed controller design
are describe in Section 2. Computer simulation results of the
proposed controller for controlling an actual EMV, are given
in Section 3. The proposed method is implemented on an experimental testbed. Experimental results using the proposed
controller for the testbed are presented in Section 4. Finally,
conclusions are given in Section 5.

2.

Modeling and Controller DesignFirst, the MPC algorithm for a linear plant is proposed. Subsequently, a model of the EMV system is described. The proposed MPC algorithm is applied to the EMV system.
2.1

MPC for a Linear Plant

In this section, the MPC method for a linear plant is described. The linear plant has a state equation in form ofẋ(t) = Ax(t) + Bu(t).

(1)

A and B are n × n system matrix and n × r input matrix, respectively. Here, x(t) ∈ Rn is state vector of the plant and u(t) ∈ Rm
is the plant input. Standard iterative calculation methods are notused for calculation the optimal control in the MPC because it
takes higher computation cost. The following procedure is proposed for calculating the optimal control. Differential equations
are obtained from the procedure.

Let us consider the following performance index
Z
1 tf 
x(τ)T Qx(τ) + u(τ)T Ru(τ) dτ,J(τ) =
2 t

149

(2)

subject to the state equation of linear plant in (1). Here, τ isvirtual time for predictive control. t is the present time of thepredictive horizon and t f is the terminal time of the predictive
horizon. The weighting matrix Q and R are real, constant, symmetric positive semi-definite and positive definite matrix, respectively.
The costate equation and the optimal input for (1) are statedas follows:λ̇(t) = −AT λ(t) − Qx(t),(3)u(t) = −R−1 BT λ(t).

(4)

Costate equation in (3) involves Lagrange multipliers λ that
help us to find the optimal input by converting the constrained minimization problem into an unconstrained minimization problem. The optimal input is obtained by the following procedure.
Using the state equation (1) and the optimal input (4), we getẋ(t) = Ax(t) − BR−1 BT λ(t).
Combining the equations (3) and (5) yield
"
# "
#"
#
ẋ(t)
A −BR−1 BT
x(t)
=
λ̇(t)
−Q
−AT
λ(t)
"
#
x(t)
= H
.
λ(t)

(5)

(6)

The general solution of the state equation the optimal input (6)at τ, given a initial condition at t is
"
#
"
#
x(τ)
x(t)
H(τ−t)
=e
.
(7)
λ(τ)
λ(t)
The transition matrix eH(τ−t) is partitioned into#"
e11 (τ) e12 (τ), e11 (τ), e22 (τ) ∈ Rn×n ,eH(τ−t) =e21 (τ) e22 (τ)
Evaluating (7) at t f given initial condition at t obtainsx(t f ) = e11 (t f )x(t) + e12 (t f )λ(t),
λ(t) = e12 (t f )−1 x(t f ) − e11 (t f )x(t) .
Evaluating (7) at τ by using λ(t) yields
λ(τ) = e21 (t)x(t) + e22 (τ)e12 (t f )−1 x(t f ) − e11 (t f )x(t) .
Using λ(τ) in the last equation, the optimal u(τ) can be obtained by using the following relationshipu(t) = −R−1 BT λ(τ), u(t) = u(τ), for t ≤ τ ≤ t + ∆t,
where ∆t is the sampling period of the MPC. The inverse of R
is exist because R is positive definite matrix. Kalman [14] has
shown that the inverse of e12 is exist for all τ ∈ [t, t f ]. Therefore, the optimal u(τ) is exist.
In the MPC framework for the linear plant, the states variables are updated in each sampling instant. The MPC calculates the optimal u(τ) for the entire predictive horizon in eachsampling instant. The predictive horizon is becoming shorter astime passing. The flowchart in Fig. 2 summarizes the algorithmof the proposed MPC.

計測自動制御学会産業論文集 第9巻 第20号 2010年

150

Augmented plant

Start

�

MPC u +− u

Set tfżˆ
Disturbanceobserver d̂�Update stateCalculate u(τ )

1
s

EMV

w+

+
d

z
Controllert = t + ∆t
Fig. 3

Configuration of the proposed controlled systemt ≤ u(τ ) ≤ t + ∆t

2.3t = tf ?

No

Yes
End
Fig. 2

Flowchart of the proposed MPC

2.2 Modeling
The EMV consists of an armature, two springs, a shaft, anda valve. The armature, shaft and valve are considered as one
body, and called the moving part. The moving part has amass m. It is suspended by the springs which have the spring
constant k. The damping coefficient of the moving parts is c.
Under those conditions, a motion equation of the moving partcan be expressed asmz̈(t) + cż(t) + kz(t) = w(t) + d(t),(8)

where w and d are the magnetic force and the external disturbance force, respectively. The moving part travels between[−z s , z f ]. Here, z s and z f denote the surface position of theupper and the lower electromagnet, respectively. z = 0 is theneutral position.
We form an augmented plant for the EMV, by appending the
EMV plant (8) with an integrator. The augmented plant for the
EMV can be formulated in the state space equation as follows:ẋ(t) = Ax(t) + B(u(t) + d(t)),
0 w(t) 
 0



0
x(t) =  z(t)  , A =  0


 1
k
−
ż(t)
m
m

(9)



0 
 1 

1  , B =  0  .



− mc
0

The force w is converted to the electromagnetic current i by
the following nonlinear equations
2w(l + k2 )2
.
(10)
i=
k1
Here, l is the gap between the armature and the pulling magnet.
The constants k1 and k2 are obtained through system identification [12]. Equation (10) represents the nonlinear property ofthe electromagnetic force in attracting the armature for various
position of the armature and in various value of electromagnetic current. In this paper, delay of the electromagnet current
in reaching its target value is neglected.

MPC for an EMV

A configuration of the proposed controlled system is shownin Fig. 3. The optimal control force u0 is calculated by the MPC
compensator. The armature position and its velocity are needed
for the calculation. The armature position z is obtained fromdirect measurement. The armature velocity żˆ is estimated using the disturbance observer. The controlled system robustness
against parameter uncertainty is improved using the observerby sending the compensation force dˆ0 . The control force u is
obtained using u0 and dˆ0 . However, the discontinuous signal ofu is tended to generate by the controller. In order to have thecontinuous control force w, the augmented plant of the EMV is
formed by appending the EMV plant with the integrator. The
discontinuous signal u is integrated by this integrator. In thispaper, we assume that there is no external disturbance d.
The controller in Fig. 3 mainly consists of two elements, i.e.,the MPC compensator and the disturbance observer. These elements are described in the following sections.
2.3.1 MPC compensatorIn this section, the MPC algorithm in Section 2.1 is applied to
the EMV system for calculating u0 . In ideal condition (there isno parameter perturbation), the compensation force dˆ0 = 0 andthus u = u0 . The performance index in (2) is used subject to thestate equation of the augmented plant in (9), and positiveness ofthe control force w. The positiveness of w is required to assurethat the armature is pulled to the desired electromagnet by thecontrol force, because the electromagnet can only generate apulling force.
In this paper, we consider an “open valve” operation. The
initial state x(t) and terminal state x(t f ) are defined asx(t) = [ wt zt żt ]T ,hiTx(t f ) = kz f z f 0 ,where wt , zt , żt are present control force, present armature position, and present armature velocity, respectively. z f is the desired position of the armature at terminal time (i.e., the positionof the pulling electromagnet surface). The desired armature velocity is set to 0. The final force is determined so that kz f andpulling force of the springs at surface of the electromagnet isbalance. The weighting matrix Q is formulated as q1 0 0 
Q =  0 q2 0  .

0 0 q3In the MPC framework for the EMV, the variables wt , zt , z˙t ,and t are updated in each sampling instant, while w f , z f and t f

SICE T. on Industrial Application Vol. 9 No. 20 2010ẋˆ(t)0 = (A0 − LC 0 ) x̂(t)0 + Ly + B0 u(t).
"
#
"
#
A B
B
0
0
A =
, B =
, C0 = [ 0 1 0 0 ] ,
0 0
0


x0 (t) = w(t) z(t) ż(t) d0 (t) T , y(t) = C 0 x(t)0 .
The observer gain L is designed by using a pole placement
procedure. An appropriate L is chosen so that (A0 − LC 0 ) has
stable eigenvalues, and the respond of the observer is faster than
the response of the MPC.
1200

Table 1

3.

Parameter

Value

Unit

Sampling time of the MPC
Mass (m)Spring constant, kViscous constant, cUpper magnet position, −z sTarget position, z f
z when the MPC is started
z when the MPC is terminated

4 × 10−5
0.215
20 × 10412−4.3754.375
3.375
4.3745sec
Kg
N/m
N.sec/m
mm
mm
mm
mm

In this section, computer simulations are conducted to control the EMV with its actual parameters (true industrial specifications) using the proposed method. The computer simulations show the effectiveness of the proposed method to overcome the first five problems in the difficulties for controlling
motion of the armature. The parameters of actual EMV [2] andthe simulation setup, are listed in Table 1. In the table, we use
5 × 10−5 [s] for the MPC sampling time . This sampling time issmall enough for controlling the armature and have been successfully used in [5] for implementation of its online control
law using a sophisticated control board. Therefore, the sampling time is considered to be available. Motion of the armature
is controlled by controlling the current in the pulling electromagnet because we cannot controlled the spring force. The armature motion can only be controlled while it is closer than one
millimeter from the pulling electromagnet, due to limitation of
maximum current in the electromagnet. The nonlinearity of theelectromagnet can be compensated using the nonlinear equation in (10). When the armature reaches a very close position
from the pulling magnet, the MPC control action is terminatedand the holding force is applied to keep armature on the magnet
surface. The observer gain L is designed so that the eigenvalues of (A0 − LC 0 ) equal −2πp × √90, −2πp × 90.1, −2πp × 90.2,
and −2πp × 90.3, where p = k/m. Computer simulations
are conducted using the proposed method and the conventional
!3

5

x 10

4

3
2.6

Control action

1000

Parameters of an actual EMV and its simulation setup

Simulation Results Using an Actual EMV Parametersz [m]are fixed in all sampling instant. Furthermore, t f is a designed
traveling time of the armature.
Now, the positiveness of w is considered. From the resultof numerical simulation in Fig. 4, important facts are observed
that: a short traveling time makes the w negative between t0
and t f , and a long traveling time makes the w positive between
t0 and t f . t0 is the time when MPC start to work. According
to the relationship, the following physical interpretation is also
obtained. A short traveling time requires relatively a large force
to slow down the armature velocity and place it to the desired
final position, comparing to the case of long traveling time.
A shorter t f needs a sudden acceleration and a sudden deceleration of the armature. This sudden deceleration leads to anegative w for the armature. A longer t f admits a more moderate acceleration and a more moderate deceleration of the armature. This moderate deceleration force allows w still positive. Although w always positive for a sufficiently large t f , thesmallest t f is selected to meet the maximum engine speed. The
choice of t f also consider limitation of the force that can beproduced by the pulling electromagnet. For implementation,
the admissible value of t f that meets those requirements, can beobtained through computer simulation.
2.3.2 Disturbance observer
The disturbance observer in Fig. 3 is used to estimate thearmature velocity żˆ. The disturbance observer is also used tocompensate the perturbation of m, c, k and influence of the
disturbance d in low frequency band. A structure of the disturbance observer for the EMV system is simply constructed byappending the compensation force dˆ0 to the state vector in (9)and augmenting the corresponding row in the state matrix withzeroes:

151

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6
!3

x 10
3

żˆ [m/s]

800

w[N]

600

Longer tf

400

2
1
0
2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6
!3

x 10
1000

w [N]

200

0

Shorter tf

!200

!400
2.6

500

0
2.6

2.7

2.8

2.9

3

3.1

3.2

Time [s]
2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

Time [s]Fig. 4

The relationship between w and t f

3.5

3.6
!3

x 10

Fig. 5

3.3

3.4

3.5

3.6
!3

x 10

Computer simulation results of the EMV with actual parameterusing the conventional MPC method (green) and the proposed
method with q1 = 0 (black), q1 = 1 × 108 (red), and q1 = 1 × 109(blue)

計測自動制御学会産業論文集 第9巻 第20号 2010年

152

Table 2

!3

4.38

x 10

Methodq1 = 0q1 = 1 × 108
q1 = 1 × 109
Conventional MPC

z [m]

4.375
4.37
4.365
3.42

3.44

3.46

3.48

3.5

3.52

3.54

Comparison of the traveling time in Fig. 5

3.56
!3

Traveling time [s]3.540 × 10−3
3.517 × 10−3
3.492 × 10−3
3.473 × 10−3x 10żˆ [m/s]

0.2

Table 3

0.1

0
3.42

3.44

3.46

3.48

3.5

3.52

3.54

3.56

The impact velocities in Fig. 7 using the proposed method
Simulation caseżˆ [m/s]

Nominal case
5% perturbation
10% perturbation

0.04
0.03
0.05

!3

x 10

w [N]

1000

500

0
3.42

Fig. 6

3.44

3.46

3.48

3.5

Time [s]

3.52

3.54

3.56
!3

x 10

Zoomed computer simulation results of Fig. 5 at its final travelingtime

MPC method [12]. The parameter q2 and q3 in the matrix Q
are set to 0, while q1 is varied. The weighting matrix R is set
to 1. The simulation result is shown in Fig. 5 and its zoomed
computer simulation results at its final traveling time is shownin Fig. 6. The conventional MPC gives the minimum travelingtime of the armature, compared to the one using the proposed
method. The shorter t f of the proposed method can be obtainedby adjusting the value of q1 . The corresponding traveling timein Fig. 5 is listed in Table 2.
Computer simulations are conducted for perturbed EMV
model using the proposed method with q1 = 1 × 109 to evaluate robustness of the controlled system. In practical, there are
perturbations in spring constant because of inaccuracy in springmachining. Perturbations in mass are unlikely to be happened.
Influence of the viscous constant perturbations is small, owing
!3

5

x 10

z [m]

4
3
2
2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5
!3

x 10

żˆ [m/s]

3
2
1
0
2.5

2.6

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5
!3

x 10

its relatively small value. Here, perturbations of spring constant
are set at 5% and 10% larger than the nominal value. The simulation results are shown in Fig. 7. The control forces are kept
continuously positive during the control action for the nominalcase and perturbed ones by the proposed method. The armature
traveling time is kept within the designed t f , for the nominalcase and its perturbations. The shape of w in perturbed cases
are similar with the nominal case by using the disturbance observer. The impact velocities in Fig. 7 are given in Table 3. All
the impact velocities are less than 0.1 [m/s]. Since the armature
can travel within its designed traveling time, this enables us tomake synchronization with the timing of engine revolution andmeets the required of maximum engine speed.
Therefore, the computer simulation results showed that the
proposed method can overcome the first five difficulties for controlling motion of the armature. The proposed method enables
low impact velocity for the armature and robustness of the controlled system is preserved

4.

Experimental Results

A laboratory scale testbed is used for implementing the proposed method. The testbed is composed to study the principal properties of the proposed method such as the effectiveness of the proposed method for controlling the traveling timeof the armature, providing low impact velocity, and low impact sound level. This testbed has different specifications compared with the actual EMV system that described in Section 3.
Lower specifications or inexpensive parts/devices (i.e., electromagnets, power supply, control board, sensors) are used for the
testbed. Due to the limitation of power supply that we can provide for controlling the testbed of EMV, a longer t f is used in
the experiments, compared with the t f in Section 3. This longert f is implemented using a lighter mass of the armature and thesprings with lower spring constant.
First, the experimental device setup for implementation of
the proposed method on a testbed is described. Sequently, theexperimental results using the testbed are showed.

w [N]

1000
500
0
2.5

2.6

2.7

2.8

2.9

3

Time [s]Fig. 7

3.1

3.2

3.3

3.4

3.5
!3

x 10

Computer simulation results of the EMV with the actual parameters for nominal case (black), 5% perturbation of the k value (red),
10% perturbation of the k value (blue) using the proposed method,
and 10% perturbation of the k value without using the compensation force (green)

4.1 Experimental Device Setup
The experimental device setup is shown in Fig. 8. Realtime implementation of the control law is coded in C using adSPACE DS1104 control board. In each sampling instant, thecontrol board reads the armature position as analog input, estimates the armature velocity, and calculates the control input.
The control input from the control board is amplified using aLM3886 audio amplifier from National Semiconductor. A DC
power supply is used for the audio amplifier that can generate ± 50 [V] / 5 [A]. The pulling force is provided by two par-

SICE T. on Industrial Application Vol. 9 No. 20 2010

153

!3

Position sensor
EMV

DS1104

DC power supply

5

z [m]

Sound levelsensorżˆ [m/s]

Current
sensor

x 10

4
3
0.044
0.2

w [N]

0.05

0.052

0.054

0.056

0.046

0.048

0.05

0.052

0.054

0.056

0.046

0.048

0.05

0.052

0.054

0.056

0.046

0.048

0.05

0.052

0.054

0.056

3.5
0

i [A]

The experimental device setup

14 cm

0.048

0.1
0
0.044
7

Fig. 8

0.046

0.044
4
2
0
0.044

Time [s]Position sensor

Fig. 10
Shaft

Comparison of the measured (black) and simulated (blue) results
using the proposed method

0.16Upper spring

0.14

30 cm
Lower spring

0.12

Upper magnet
Armature
Lower magnet

0.1żˆ [m/s]

8 mm

0.08
0.06
0.04
0.02

Fig. 9The testbed of an electromechanical valve0
3.2

Table 4

Experiment
Simulation
On-off
3.4

3.6

Parameters of the testbed and its experimental setup
Parameter

Value

Unit

Sampling time of the MPC
Mass mSpring constant, kViscous constant, c
k1
k2QUpper magnet position, −z sTarget position, z f
z when MPC is started
z when MPC is terminatedTraveling time t f

1.8 × 10−40.285
1333.31.62393.59
1.33790−4.300
4.300
3.300
4.29
5.58 × 10−2

sec
Kg
N/m
N.sec/m
N.mm2 /A2
N.mm2 /A2
−
mm
mm
mm
mm
sec

allel electromagnets. The armature displacement is measured
by an inductive displacement sensor (Keyence EX-422V). A
U.R.D. HCS-20-AP device is used as a current sensor. Impact
sound level is measured using a precision sound level sensor(Rion N-15). The sound sensor is set to “A” weighting option, to give actual impression of loudness. Detail of the EMV
testbed is shown in Fig. 9.

Fig. 11

4.2

3.8

z [m]

4

4.2

4.4
!3

x 10

Comparison of the experimental and computer simulation results
in a phase plane

Experimental Results

In this section, effectiveness of the proposed method in controlling the armature of the EMV testbed is showed. The parameters of the testbed and its experimental setup are listed inTable 4. The system identification process for obtaining parameters m, c, k, k1 , k2 are described in [12]. Similar with Section 3, the armature is controlled within the one millimeter fromthe pulling electromagnet. The experimental and simulated results for the testbed using the proposed method, are shown in
Fig. 10. The experimental results are similar with its simulated
results. The dSPACE DS1104 control board can calculate thecontrol input of the MPC in each sampling time. A ripple control signal is appeared in w because of noise on z and żˆ due tothe noise of position sensor. Uncertainties of the model and delay of the current in reaching its target value are also become
another source of the ripple control signal. In the experimental
results, the target position is reached slightly faster than the de-計測自動制御学会産業論文集 第9巻 第20号 2010年

154

SPL [dB]

90
80
70
60
50
2

2.5

3

3.5

Time [s]

4

4.5

5

SPL [dB]

90

5.

80
70
60
50
2

Fig. 12

using the proposed method and its resulted impact sounds aremeasured. A low average of impact sound level (77.97 [dB])
and a small standard deviation (1.9 [dB]) are obtained, as shownin Fig. 13. The results show ability of the proposed method torepeatedly enable the low impact sound level. In average, a difference of loudness (7.63 [dB]) is obtained using the proposed
method and the on-off control. Therefore, the proposed method
can suppress the maximum value of the impact sound level by
7.63 [dB].

2.5

3

3.5

Time [s]

4

4.5

5

Sound pressure level of the armature impact using the proposed
method (upper), and the on-off control (lower)

15

Frequency

105Conclusion

An continuous control input generation method in MPC for
an EMV has been proposed. A set of differential equations
have been derived for calculating the optimal control for the
EMV. The simulation results for EMV using actual parameters
have showed that the proposed method can overcome the first
five difficulties for controlling the armature of EMV and has
met the requirements in the actual application. In the computer
simulation and the experimental results, the proposed method
has been implemented successfully using relatively small sampling times of the MPC compared with their traveling times of
the armature. Based on the experimental results, the proposed
method has been proved successfully in handling the nonlinearity of the electromagnet force for controlling the testbed.
The armature has reached desired states within the certain designed traveling time in the simulation and the experimental results. The proposed method has enabled the low impact velocity, while robustness against the uncertain parameter model has
been preserved. The proposed method has effectively reduced
the impact sound level. The ability of the proposed method formaintaining the low impact sound level in many cycles has been
showed. Hence, it can be concluded that the proposed method
gives satisfactory performances, despite the difficulties for controlling the EMV.
References

072

Fig. 13

74

76

78

SPL [dB]

80

82

84

Distribution of the sound pressure level for fifty experimentalcycles, using the proposed method

signed t f . The t f discrepancy because stopping condition of the
MPC is influenced by noise of the position sensor.
Phase plane plots of the armature positions and its velocities
are given in Fig. 11. The figure also shows experimental result when on-off control (i.e., the holding force is given to thelower electromagnets, when the armature reaches 1 [mm] fromthe desired position) is applied to the testbed. The phase plant
plots show that the states in experimental result using the proposed method are converged to its simulated states within thedesigned t f .
Effectiveness of the proposed method to reduce an impact
sound level is shown in Fig. 12. A maximum Sound Pressure
Level (SPL) of 77.92 [dB] and 85.6 [dB] are resulted by the proposed method and the on-off control, respectively. This result
also indicates effectiveness of the proposed method to provide
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Addy WAHYUDIE (Student Member)
He finished his B. Eng. in 2002 from Gadjah Mada
University, and M. Eng. in 2005 from Chulalongkorn
University. Currently, he is a Doctoral student in the
Graduate School of Information Science and ElectricalEngineering, Kyushu University. His research interests
include robust optimal control theory and its applications.

Taizo NAKAO
He received his B. E. and M.E. from the Department
of Electrical Engineering, Kyushu University in 2008 and2010, respectively. His research interests include receding horizon control and its applications in automotive
control technology. He is currently with Nippon SteelEngineering Corp.

Toshimitsu KAI
He received his B. E. from the Department of Electrical Engineering, Kyushu University in 2009. Currently,
he is a student in the master’s in the Graduate School ofInformation Science and Electrical Engineering, Kyushu
University. His research interests include receding horizon control and its applications.

Hideaki SETOGUCHIHe received his B. E., from the Department of Electrical Engineering, Kyushu University in 2010. Currently,
He is a student in the master’s degree in the Automotive
Science Department, Kyushu University. His research interests include receding horizon control and its application on automotive control technology.

Masakazu MUKAI (Member)He received the B.E., M.E. and Dr. of Engineering degrees in Electrical Engineering from Kanazawa University, in 2000, 2002, and 2005, respectively. He is currently with the Graduate School of Information Scienceand Electrical Engineering, Kyushu University. His research interests include receding horizon control and its
applications. He is a member of the ISCIE, IEEJ, and
IEEE.

155

Taketoshi KAWABE (Member)He received the B. Sc. degree from the Department
of Applied Physics, School of Science and Engineering,
Waseda University in 1981, and M. Sc. from the Major in Pure and Applied Physics, Waseda University in
1984. He had been working with Nissan Research Center, Nissan Motor Co., Ltd. since 1984 to 2005. He was a
research student of the Department of Mathematical Engineering and Information Physics, the University of Tokyo in 1992 and1993, and received the Ph. D degree from the University of Tokyo in 1994.
He is currently a professor in the Faculty of Information Science and Electrical Engineering, Kyushu University since April 2005. His research interests include motion control, vibration control and related automotive control technology. He is a member of the JSME, IEEJ, Society of Automotive
Engineering of Japan and IEEE.