Sloshing Suppression Control During Liquid Container TransferInvolving Dynamic Tilting using Wigner Distribution Analysis

Yoshiyuki Noda, Ken’ichi Yano, Satoshi Horihata, and Kazuhiko Terashima

Abstract— This paper addresses concerned with the ad-vanced control of the transfer of liquid in a container, with spe-cial consideration given to transfer involving liquid containerwith dynamic tilting as well as the suppression of sloshing(liquid vibration) while maintaining a high transfer speed forthe container. In the transfer with dynamic (continuous) tilting,the sloshing’s natural frequency is varied with the change oftilting angle. In order to suppress the sloshing caused by suchtransfer, a new controller design method is proposed. First,the varying natural frequency is estimated using the WignerDistribution. Then, in order to suppress the sloshing, theposition control is designed using the Hybrid Shape Approachutilizing a time-varying notch filter. The effectiveness of theproposed control system is shown experimentally in a liquidcontainer transfer system.

I. INTRODUCTION

Many industries utilize transfer systems with mechanisms

that induce vibration, for example, crane systems, robotic

manipulators, automatic pouring systems, liquid container

transfer systems, and so on. In these industrial plants, it

is required that vibration is damped, while motion control

is accomplished as the mechanism moves at high speeds.

Recently, in order to meet such requirements, advanced

control systems using various control theories have been

developed. In most of these control systems, vibration

dynamics is treated as a stationary vibration. However, there

exists a transfer system having non-stationary vibration, for

example, a crane system with a changing length of cable, a

liquid container transfer system whose tilting angle changes,

and so on. In the crane system, the vibration’s natural

frequency is varied in terms of the cable length. Therefore,

the natural frequency is estimated theoretically by using the

pendulum model with a varying length of cable, and the

control system is designed with the application of Gain-

Scheduled Control, and so on[1]. However, there exists a

case in which it is difficult to estimate the varying natural

frequency. In the self-transfer-type automatic pouring robot,

the ladle is transferred along the mold line while pouring

the molten metal into the mold by tilting the ladle. In such

a transfer system that involves the tilting of a container, the

natural frequency of sloshing in the container varies with the

tilting; however, it is difficult to theoretically estimate the

varying natural frequency based on the shape of container.

Y. Noda, S. Horihata, and K. Terashima are with Departmentof Production Systems Engineering, Toyohashi University of Tech-nology, 1-1 Hibarigaoka, Tempaku, Toyohashi, Aichi 441-8580 Japannoda@procon.tutpse.tut.ac.jp

K. Yano is with Department of Mechanical and Systems Engineering,Gifu University, 1-1 Yanagido, Gifu, Gifu 501-1193 Japan

Previous studies have described the design of transfer

systems involving tilting motion in consideration of slosh-

ing suppression. For example, sloshing suppression in an

automatic pouring robot has been achieved by using a

Hybrid Shape Approach[2]. Although this robot had the

ability to suppress sloshing during transfer and pouring

motions, the tilting control of ladle was sequentially per-

formed after the transfer of the ladle. As well, sloshing

suppression during the pouring motion and during transfer

was achieved through the use of preview control[3]. These

studies have been performed under the assumption that the

natural frequency of sloshing is constant if the ladle is

slightly tilted, (about 5[deg]). However, in actual pouring

motion, because the tilting angle of ladle has a wide range,

the natural frequency of sloshing varies with the degree of

ladle tilt. No studies have focused on sloshing suppression

in consideration of the change of the natural frequency due

to the tilt of the liquid container.

Therefore, the purpose of this paper is to propose a design

method for sloshing suppression during liquid container

transfer involving the dynamic tilting of the container. First,

the natural frequency of sloshing caused by container trans-

fer involving tilting is estimated by using time-frequency

analysis, which is useful for analyzing non-stationary vi-

bration. In particular, the Wigner Distribution[4] has higher

resolution in time and frequency domain than other time-

frequency analyses (for example, Short Time Fourier Trans-

form and Wavelet Transform). Therefore, in the field of

signal processing, the Wigner Distribution is often applied

to the identification of multi-element structures, fault detec-

tion in rotating machinery, and so on[5]. On the other hand,

this analysis contains interference which does not exist in

real signals with more than two frequency bands. However,

in the present process of liquid container transfer, the first

mode sloshing is dominant, and the natural frequency of

sloshing varies with the change of the container’s liquid

level and tilting angle.

Next, a sloshing suppression system for container transfer

involving dynamic tilting is designed through the use of

the Hybrid Shape Approach with a time-varying notch

filter. The controller designed using the Hybrid Shape

Approach developed by the author’s group in previous

research allows the fastest possible transfer while meeting

the given control specifications for time and frequency

characteristics[2]. The Hybrid Shape Approach is a design

method aimed at starting control and vibration damping by

using real-time feedback of only liquid container position

data without on-line sloshing feedback. In the present paper,

43rd IEEE Conference on Decision and ControlDecember 14-17, 2004Atlantis, Paradise Island, Bahamas

0-7803-8682-5/04/$20.00 ©2004 IEEE

ThA09.3

3045

the Hybrid Shape Approach is improved, such that the anti-

resonant frequency of a notch-filter is adequately shifted

corresponding to the dynamic change of the sloshing’s

natural frequency as estimated by the Wigner Distribution.

The effectiveness of the proposed system is validated by

applying it to the transfer of liquid by a container involving

pouring motion.

II. EXPERIMENTAL APPARATUS

A schematic diagram of the experimental apparatus of a

liquid container transfer system is shown in Fig.1. In this

study, in order to pour the liquid, a cylindrical container

with nozzle is used. The radius of the three-dimensional

cylindrical container R is 0.12[m] and its height is 0.3[m].

The liquid used in the present experiments is water, whose

static liquid level hs is 0.16[m]. This paper addresses liquid

container transfer on a straight path and the rotation of the

liquid container. The liquid container is transferred using

an AC servo motor equipped with a ball screw, and tilted

by using an AC servo motor directly. The position and

angle are controlled by adjusting the voltage applied to each

motor. The maximum velocity and maximum acceleration

of the motor used for transfer are 0.8[m/s] and 2.0[m/s2],

respectively, and those of the motor used for tilting are

150[deg/s] and 1500[deg/s2]. The position and angle of

the liquid container are detected by encoders fitted to each

motor. In order to evaluate only the experimental results,

level sensors were installed in this apparatus. The position

of the level sensor for measuring sloshing is placed at

position A installed in the transfer direction as shown in

Fig. 1. This sensor is placed to measure the first mode

sloshing in the transfer direction, but not sloshing in the

tilting direction.

AC servo motor

and Encoder for transfer

AC servo motor

and Encoder for tilting

Ball screw

Liquid Container

(R0.12x0.3m)

Level sensor

Nozzule of container

Transfer

Rotation Water

Tray

Transfer

Fig. 1. Schematic diagram of the experimental apparatus

For the AC servo motor using in this apparatus, the

transfer functions PX(s) and PT (s) from the input voltage

u(t) to the position yx(t) and angle yt(t) are respectively

given in the form of the following first-order lag model with

an integrator,

P (s) =Y (s)U(s)

=Km

s(Tms + 1)(1)

, where Tm is the time constant and Km is the gain. These

parameters were identified by adding a step-wise input to

the apparatus. As a result, the parameters of the motor

model for the transfer PX (s) were obtained as Kmx =0.1663[m/sV] and Tmx = 0.0060[s], and those for the

tilting PT (s) were obtained as Kmt = 24.71[deg/sV] and

Tmt = 0.008[s].

In general, if the cylindrical container is not tilted, the

natural frequency fn for a (1,1)-model sloshing of a perfect

fluid is represented by the following equation[2].

fn =12π

√g

Rε1tanh(ε1

hs

R) (2)

, where hs is the static liquid level, R is the radius of the

cylindrical container, g is the gravitational acceleration, and

ε1 is the least positive root for a first-order derivative of

the first kind Bessel function (ε1 = 1.841). In this study,

the target liquid level hs is 0.16[m] and the radius of the

cylindrical container is R = 0.12[m]. Hence, the natural

frequency fn is calculated as 1.9379[Hz] (12.17[rad/s]). As

shown in Fig.1, the process of pouring molten metal in the

casting industry uses the following actions. The container is

tilted forward to pour the molten metal into a mold. In the

self-transfer-type automatic pouring robot, the position of

the container from which the liquid is poured into a mold

is transferred in synchronization with the movement of the

mold. Then, because the quantity of liquid in a container

and the container’s angle are changed, the natural frequency

varies, and these values cannot be obtained from Eq. (2).

So, in the following section, we will show how the natural

frequency of sloshing can be determined for the dynamic

tilting of a container.

III. ESTIMATION OF THE NATURAL

FREQUENCY OF SLOSHING DURING TRNSFER

INCLUDING DYNAMIC TILTING USING WIGNER

DISTRIBUTION

A. Control system and sloshing data for analysis

In order to generate sloshing data, a control system as

shown in Fig. 2 is constructed for a liquid container transfer

system. PX (s) and PT (s) are motor models for transfer and

tilting respectively represented by Eq. (1). SX (s) shows the

sloshing dynamics in the liquid container. Therefore, yx and

yt show the position and tilting angle of liquid container,

and sx is the sloshing caused by the control input ux. This

is measured by level sensor as shown in Fig. 1. The property

of sloshing dynamics SX(s) is varied by the tilting angle yt

as shown in Fig. 2. For transfer control, a position feedback

control system using a feedback controller KX is built for

the motor PX(s). In order to generate sloshing for obtaining

the sloshing data, the feedback controller KX uses only the

proportional gain. In this paper, the transfer reference rx is a

trajectory determined by integrating the maximum velocity

and maximum acceleration, and the starting point is set to

0[m] and the endpoint is 0.24[m]. In order to tilt the liquid

container to 55[deg] and suppress the sloshing caused by

the tilting motion, the control input ut to the tilting motor is

designed using the Hybrid Shape Approach[2]. Moreover, in

3046

order to change the property of sloshing dynamics, forward

tilting begins at 9[s] as shown in Fig. 3, and backward tilting

begins at 15.5[s]. Hence, the natural frequency of sloshing

will be varied at the time interval of 9 ∼ 15.5[s].

P (s)T

P (s)X

S (s)X

+-

Motor for tilting

Motor for transfer

Sloshing

y

y

s

u

u

r x

x

t

x

t

x

Controller

KX

Fig. 2. Control system for sloshing analysis

The experimental results using this control system are

shown in Fig. 3. Diagram (a) shows the position of the

liquid container, diagram (b) is the control input to the

transfer motor, diagram (c) shows the tilting angle, and (d)

is the control input to the tilting motor. In the experimental

results, after transfer of 0.24[m] from the origin , the liquid

container is tilted at up to 55[deg] as mentioned above.

Sloshing in this experiment is shown in Fig. 4. In this Fig.

4, the liquid level is lowered after 11.5[s], since the liquid

is poured from the container. Therefore, the liquid level is

0.16[m] before pouring and 0.087[m] after pouring.

B. Sloshing analysis using the Wigner Distribution

In order to analyze the sloshing caused by the transfer

and the tilting of the liquid container, Wigner Distribution

analysis is utilized. The Wigner Distribution is represented

by the following equation,

WD(t, f) =∫ ∞

−∞sx(t +

τ

2)s∗x(t − τ

2)e−j2πfτdτ (3)

, where t is time, and f is frequency. sx(t) is an analytical

signal, which represents the sloshing data in this paper, and

s∗x(t) is complex conjugate function of the analytical signal.

In Eq. (3), since the Wigner Distribution is the Fourier

0 10 200

0.1

0.2

0.3

Posi

tion

y x [m

]

0 10 20

−2

0

2

Con

trol

inpu

t ux [

V]

0 10 200

20

40

60

Time [s]

Tilt

ing

Ang

le y

t [de

g]

0 10 20−1

−0.5

0

0.5

1

Time [s]

Con

trol

inpu

t ut [

V]

(a) (b)

(c) (d)

Fig. 3. Experimental results of transfer path and tilting angle of container

0 5 10 15 20 25

−0.1

−0.05

0

0.05

Time [s]

Slos

hing

[m

]

Fig. 4. Experimental result of sloshing caused by transfer involve tilting

Transform of a autocorrelation function of the analytical

signal, it is thought that the Wigner Distribution results in

higher resolutions in the time and frequency domain than

does a Short Time Fourier Transform expressed as

Sp(t, f) =∣∣∣∣∫ ∞

−∞sx(t)w(τ )e−j2πfτdτ

∣∣∣∣2

(4)

, where w(t) is a window function.

However, the Wigner Distribution contains an interfer-

ence, which does not exist in real signals, when the analyt-

ical signal has more than two frequency spectrum bands at

the same time. In this study, the effect of interference in this

analysis of sloshing in the present process was insignificant,

because 1st mode sloshing is dominant, and thus sloshing

has an almost single natural frequency. Therefore, the

Wigner Distribution is thought to be available and useful

for this sloshing analysis.

Hilbert

transformConjugation

Butterfly

operation

Fourier

Transform

Window

Absolute

of real part

sx(t)

WD(t,f)analytical signal

Fig. 5. Computation process of Wigner Distribution

The computation process of the Wigner Distribution is

shown in Fig. 5. In the first step, a signal is sectioned by a

window. Next, in order to obtain the analytical signal sx(t)in Eq. (3), the Hilbert Transform is introduced for the signal

sectioned by the window. After that, a butterfly operation is

implemented for the analytical signal sx(t) and a conjugated

signal s∗x(t). In this process, the autocorrelation function of

the signal sectioned by the window is obtained, and energy

at the natural frequency of sloshing is emphasized. Then,

the signal after the butterfly operation is transformed by

5 10 15 20 25

−0.04

−0.02

0

0.02

0.04

Slos

hing

[m

]

Time [s]

Fig. 6. Sloshing data for analysis

3047

Fig. 7. Result of sloshing analysis by Wigner Distribution

6 8 10 12 14 16 18 20 221.4

1.6

1.8

2

2.2

2.4

Freq

uenc

y [H

z]

Time [s]

Fig. 8. Energy contour of sloshing by Wigner Distribution

Fourier Transform into the frequency domain. In the final

step, only the absolute the value of the real part of data in

the frequency domain is used to clarify the result, since the

analytical result is unclear because it contains the internal

interference. This process is implemented for each shift in

time. Therefore, the energy for the signal in the time and

frequency domain is obtained.

In this study, the sloshing data shown in Fig. 4 is analyzed

using the Wigner Distribution. However, the data includes

the step-wise signal, since liquid in the container is poured.

The step-wise signal increases in the low frequency domain.

Therefore, in order to eliminate the step-wise signal from

the sloshing data, the sloshing data is pre-compensated by

the following equation,

sx(k) = sx(k) − 12N + 1

N∑m=−N

sx(k + m) (5)

, where t = k∆T , and ∆T is sampling time, which is

0.01[s] in this study. The second term in Eq. (5) represents

a moving average. N in the moving average is selected to

be 50 in order to obtain only sloshing which is different

from the liquid level. The sloshing data compensated by

Eq. (5) is shown in Fig. 6. In the analysis using the Wigner

Distribution, sloshing data from 3[s] to 26[s] is used, since

the sloshing to 3[s] includes the fluctuation of the liquid

surface due to the acceleration of transfer.

The result obtained by Wigner Distribution analysis of the

sloshing data is shown in Fig. 7. This analysis is performed

at a sampling time of 0.01[s] and a window size of 526,

Fig. 9. Result of sloshing analysis by Short Time Fourier Transform

6 8 10 12 14 16 18 20 221.4

1.6

1.8

2

2.2

2.4

Freq

uenc

y [H

z]Time [s]

Fig. 10. Energy contour of sloshing by Short Time Fourier Transform

which shifts from 6[s] to 23[s] in the sloshing data. In Fig.

7, the 1st mode sloshing appears near 2[Hz]. Moreover,

3rd mode sloshing and interference appear near 3[Hz] and

2.5[Hz] respectively. However, the natural frequency can be

easily estimated, since the 1st mode sloshing is the largest

at each time. If larger interference appears in the analysis,

Reduced Interference Distribution should be applied[6].

Furthermore, the Wigner Distribution for the sloshing

data is shown in Fig. 8, as an energy contour in the time and

frequency domain. Fig. 8 shows that the region covering the

greatest number of contour cycles has higher energy than

the other region. Therefore, the energy in the region from

6[s] to 8[s] and near 1.9[Hz] is the highest in the time-

frequency domain. It is seen that the energy peak during the

time span from 9[s] to 20[s] shifts toward a low frequency

when the liquid container is tilted.

As a comparison of the sloshing analysis using the

Wigner Distribution with other analytical methods, a slosh-

ing analysis using the well-known Short Time Fourier

Transform is shown in Fig. 9, and the energy contour in

time and frequency is shown in Fig. 10. In this analysis,

the window size and the shifting time are as the same

as those of the Wigner Distribution. In Fig. 9, 3rd mode

sloshing scarcely appears. Therefore, using Short Time

Fourier Transform, it is seen clearly that tilting varies

the sloshing property of liquid transfer. In this section,

Short Time Fourier Transform is apparently more useful

than the Wigner Distribution. However, with respect to the

representation of natural frequency, we remark that the

Short Time Fourier Transform is less accurate than the

3048

6 8 10 12 14 16 18 20 221.4

1.6

1.8

2

2.2

2.4Pe

ak F

requ

ency

[H

z]

Time [s]

Fig. 11. Peak frequency obtained from Wigner Distribution

6 8 10 12 14 16 18 20 221.4

1.6

1.8

2

2.2

2.4

Peak

Fre

quen

cy [

Hz]

Time [s]

Fig. 12. Peak frequency obtained from Short Time Fourier Transform

Wigner Distribution. The reason for this is described in the

next section.

C. Estimation of sloshing’s natural frequency

In order to estimate the natural frequency of sloshing

using the Wigner Distribution, the energy peak on each time

is plotted, as shown in Fig. 11. The peak frequency before

tilting the container is 1.95[Hz], and the peak frequency

after tilting is 1.85[Hz]. The static liquid level before the

container is tilted is 0.16[m], then the theoretical natural fre-

quency of sloshing as represented in Eq.(2) is 1.9379[Hz].

As well, the static liquid level after tilting is 0.087[m],

and thus the sloshing’s theoretical natural frequency is

1.8207[Hz]. Therefore, it is seen that the peak frequency in

the Wigner Distribution has estimated the natural frequency

of sloshing as well. In this paper, the peak frequency in

the Wigner Distribution is called the sloshing’s estimated

natural frequency. Furthermore, in this analysis, the useful

result is obtained such that the estimated natural frequency

during tilting is in a lower frequency domain than that after

tilting.

In order to compare the sloshing’s estimated natural

frequency as obtained by the Wigner Distribution, the

estimated result obtained by Short Time Fourier Transform

is shown in Fig. 12. By comparison of Fig. 11 and Fig.

12, the Wigner Distribution shows an impressively high

resolution. If the window size of the Short Time Fourier

Transform is increased in order to achieve higher resolution

in the frequency domain, the accuracy in the time domain

become lessens. Therefore, for the estimation of natural

frequency, it is seen that the Wigner Distribution is effective.

D. Simulation of sloshing by sloshing’s estimated naturalfrequency

The following model is used to verify the sloshing’s

estimated natural frequency. Initially, the sloshing model,

m

c

l

θ

L

hs

h

x

Fig. 13. Pendulum model for sloshing in the liquid container

in which the liquid container is not tilted, is introduced.

We assume that the sloshing in the liquid container is

approximated by the pendulum model[7] shown in Fig. 13.

In Fig. 13, x-direction is the transfer direction of the liquid

container and L is the diameter of the liquid container. his the level of the static liquid and hs is a fluctuation of

the static liquid level, while m indicates the weight of the

liquid in the container and c and l indicate the coefficient

of the fluid’s viscosity and the length of the pendulum for

sloshing. Further, θ indicates the angle of the slope of the

sloshing.

In Fig. 13, the differential equation regarding θ is shown

in the following equation[7].

ml2 θ − mlx cos θ + mgl sin θ + cl2θcos2θ = 0 (6)

Then, the liquid level hs is represented by the equation

hs =L

2tan θ (7)

The linearization of Eqs.(6) and (7) produce the follow-

ing: θ = − c

m θ − gl θ + 1

l xhs = L

2 θ(8)

, where θ 0. Therefore, the transfer function of Eq.(8) is

represented as follows:

Sx(s) =Hs(s)A(s)

=Kω2

n

s2 + 2ζωns + ω2n

(9)

, where a(t) = x(t), K = L

2g

ωn =√

gl , ζ = c

2m

√lg

(10)

In the sloshing model involving tilting, the sloshing’s

natural frequency ωn in Eq. (9) varies. Therefore, the

transfer function of the sloshing model caused by transfer

involving tilting is represented as follows:

Sx(s, τ) =KΩ2

s2 + 2ζΩs + Ω2

∣∣∣∣Ω=ωn(τ)

(11)

, where ωn(τ ) is the sloshing’s estimated natural frequency

in Fig. 11.

3049

10 12 14 16 18 20 22

-0.02

-0.01

0

0.01

0.02S

losh

ing [

m]

SimulationExperiment

10 12 14 16 18 20 22

-0.02

-0.01

0

0.01

0.02

Slo

shin

g [

m]

Time [s]

(a) Fixed natural frequency

(c) Wigner Distribution

10 12 14 16 18 20 22

-0.02

-0.01

0

0.01

0.02

Slo

shin

g [

m]

(b) Short Time Fourier Transform

Fig. 14. Comparison with sloshing simulations and experiments

Simulation results using Eq. (11) are shown in Fig. 14.

In order to compare the proposed sloshing model in Eq.

(11) with the natural frequency obtained by the Wigner

Distribution, the sloshing model including a constant natural

frequency fixed at 1.9379[Hz] and that with a natural

frequency are estimated by Short Time Fourier Transform.

A bold line in diagram (a) is the simulation result ob-

tained by using the sloshing model with the fixed natural

frequency, and the bold lines in diagrams (b) and (c) are

the simulation results obtained by the proposed sloshing

model in Eq. (11) using Short Time Fourier Transform and

Wigner Distribution, respectively. In both (a), (b), and (c),

the thin lines indicate the experimental result of sloshing

caused by the transfer involving the tilting as shown in Fig.

3. Fig. 14 includes the results obtained by 9[s] to 23[s],

since these clearly show the sloshing’s properties varied due

to the tilting. The results show that, the sloshing model

having a fixed natural frequency and using Short Time

Fourier Transform does not closely represent the sloshing

we are attending to here. On the other hand, the results

of the proposed sloshing model correlated closely with

the experimental result, demonstrating that the proposed

method using the Wigner Distribution accurately explains

the sloshing caused by transfer involving tilting.

IV. SLOSHING SUPPRESSION CONTROL USING

THE HYBRID SHAPE APPROACH

In order to suppress sloshing, a control system is designed

using the Hybrid Shape Approach. This approach, described

in detail in [2], is a method for starting control and vibration

damping, and satisfies hybrid specifications in both the

frequency domain (gain margin, phase margin, vibration

characteristics, and so on) and the time domain (transient

response, settling time, overshoot, restriction of control

input, and so on) by using real-time feedback of only the

container’s position data.

In the approach described in the previous paper, sloshing

can be damped by furnishing the controller with a notch

filter at the sloshing’s natural frequency. In this paper, a

time-varying notch filter corresponding to the sloshing’s

estimated natural frequency is introduced into the original

Hybrid Shape Approach. The design method using this

approach is described in the following.

A. Control system

For position control and sloshing suppression, the system

shown in Fig. 2 is used. In order to suppress sloshing, a

feedback controller KX is designed using Hybrid Shape

Approach and is equipped with a time-varying notch filter.

B. Selection of control element

In order to satisfy the desired specifications, the controller

is formulated as follows.

Kx(s, τ) =n∏

i=1

Ki(s, τ) (12)

Eq. (1) descibes a servo system equipped with an integrator.

Thus, according to the Internal Model principle, a propor-

tional control (P control) system is sufficient to avoid the

offset. Therefore, the proportional gain (K1) is selected as

the first element of the controller Kx(s, τ) given by Eq.

(12).

K1 = KP (13)

In order to reduce the influences of higher-mode sloshing

and noise, a low-pass filter (K2), which makes the controller

express low gain at a high frequency domain, is selected as

the second element of the controller.

K2 =1

Tls + 1(14)

A time-varying notch filter (K3) is selected as the final

element of the controller in order to suppress sloshing.

K3 =s2 + 2ζΩs + Ω2

s2 + Ωs + Ω2

∣∣∣∣Ω=ω(τ)

(15)

The parameter ζ in Eq.(15) is given as ζ = 0.001, and

ωn(τ ) is given as the sloshing’s estimated natural frequency

in Fig. 11. The notch filter makes it possible to suppress

residual vibration without directly measuring the liquid

vibration. Finally, the transfer function of the controller are

given as Eq.(16).

Kx(s, τ) =3∏

i=1

Ki

=KP (s2 + 2ζΩs + Ω2)

(Tls + 1)(s2 + Ωs + Ω2)

∣∣∣∣Ω=ωn(τ)

(16)

In Eq.(16), KP and Tl are unknown parameters. These

parameters are reasonably determined by solving an opti-

mization problem.

3050

C. Formulation of design specifications

In the Hybrid Shape Approach, various control specifica-

tions in the time and frequency domains can be given. The

specifications of the controller in both domains are formu-

lated by making use of penalty function, and the controller

Kx(s, τ) is then calculated to satisfy those specifications.

In the control design, Spec.(I) ∼ Spec.(III) were given for

the controller.

• Spec.(I): The controller gain should be less than 0[dB]

at ω = 314[rad/s] in order to decrease the influences

of the higher-order sloshing mode and noise. A penalty

is given if the following constraint does not hold:

|K(τ, ω)| < 0[dB] (17)

• Spec.(II): Due to the restriction of the input magni-

tudes, the input voltage u should not exceed a mag-

nitude of ±10[V]. A penalty is given if the following

constraint does not hold:

max |u| < 10[V] (18)

• Spec.(III): Maximum overshoot should not exceed a

magnitude of 10−3[m]. A penalty is given if the

following constraint does not hold:

max(Os) < 10−3[m] (19)

D. Formulation of an optimization problem

The following optimization problem using penalty terms

is formulated with Eq. (17) ∼ Eq. (19)

minK(s)

J = Ts + JP (20)

where,

JP = w1 + w2 + · · ·+ wi + · · · ,

Ts = mint | |yf − y(t + σ)| < ye, σ > 0 , (21)

and t is the time, y(t) is the position of the container at

time t, yf is the target position, Ts is the settling time to the

reference trajectory rx in the control system shown in Fig.

2, and ye is the admissible error set to 10−3[m]. Minimizing

Ts indicates that the trajectory of the container quickly

reaches the target point. If none of the above constraint

conditions are held, the penalty function wi in Eq. (20) is

added as wi = 108 to the cost function J . The index i of

wi corresponds to the above specification number.

E. Computation of a controller

Feedback controller Kx(s, τ ) in Eq. (16) is derived to

minimize the cost function given in Eq. (20). The simplex

method [8] is adopted as the optimization method, where the

reflection coefficient is α = 1.0, the expansion coefficient

is β = 0.5, and the contraction coefficient is γ = 2.0. The

solution obtained by the simplex method is not always the

globally optimal one. Hence, the initial simplex values and

weights of the cost function are modified in order to obtain

the desired characteristics of the controller. The initial

simplex values were assigned to be KP = (17.2, 50, 1)

and Tl = (0.01, 0.005, 0.1). The reference trajectory of the

transfer system using optimization was determined by inte-

grating the maximum velocity and maximum acceleration

of the present experimental apparatus. In the design, the

starting point was set to be yx = 0[m], relay point was

set to be yx = 0.24[m], and the endpoint was set to be yx

= 0.48[m]. A starting time when shifting from the relay

point to the endpoint was 14[s], and thus the tilting angle

is 55[deg]. This situation is the transfer at the middle time

point of tilting as shown in Fig. 16.

As a result, around 20 iterations were required for

convergence, and it took about 36 seconds to compute

an optimization problem when using a personal computer

(Pentium IV 2.4GHz CPU). In the computation, Kp = 9.10and Tl = 0.055 were obtained as cost function J = 3.21.

The frequency response of the controller is shown in

Fig.15. In Fig. 15, the gains of proposed controller at

τ = 0[s], τ = 15[s], and τ = 25[s] are shown. The

proposed controller at τ = 0[s] represents the tilting angle

at 0[deg] before tilting, τ = 15[s] indicates the tilting

angle at 55[deg] as the maximum forward tilting, and τ =25[s] indicates the tilting angle at 0[deg] as after following

backward tilting. From this figure, it can be seen that the

controller gain decreases at the sloshing’s estimated natural

frequency. Also, it can be confirmed that it is less than

0[dB] at ωl=314[rad/s] (50[Hz]). Since the gains decrease

monotonously decreasing in the high-frequency domain, it

is also clear that the controller can decrease the influences

of high-mode sloshing and noise. Furthermore, a high-speed

transfer of the container is achieved because the controller

is high-gain in the low-frequency domain.

100

101

102

−40

−20

0

20

Angular frequency [rad/s]

Gai

n [d

B]

τ =0[s] τ =15[s]

τ =25[s]

Fig. 15. Bode diagram of the proposed controller

The simulation results using the developed controller

confirmed that the container quickly transferred to the target

point, and the restriction of the magnitude of control input

was satisfied. These results cannot be shown here due to the

limitations of this paper; however, the effectiveness of this

system can be evaluated by the experimental results shown

in the following section, because the experimental results

are almost the same as those of the simulation.

V. EXPERIMENTAL RESULTS

Using the liquid container transfer system as shown in

Fig. 1, the effectiveness of the proposed control system will

be verified in this section. The tilting condition is as the

same as that shown in Fig. 3, and the transfer reference

trajectory rx is as the same as that shown in Fig. ??.

3051

A controller with a notch filter, whose natural frequency

is fixed at 1.9379[Hz], is described for comparison with

the proposed system. The controller is designed using the

original Hybrid Shape Approach.

Fig. 16 shows the experimental results regarding the

position of the container, the control input to the motor

for transfer, and the tilting angle of the proposed control

system. Fig. 17 shows the experimental results of sloshing

when using controllers equipped with the fixed notch filter

and the time-varying notch filter. In Fig. 16, diagram (a)

shows position yx of the liquid container and reference

trajectory rx. The dashed line is the reference trajectory,

and the solid line shows that of the position of the liquid

container. Diagram (b) shows control inputs to the motor for

the transfer. Diagram (c) shows tilting angle of the liquid

container. In Fig. 17, diagram (a) shows the experimental

result from 0[s] to 8[s] when using the controller equipped

with the fixed notch filter, while diagram (b) shows that by

the proposed controller using time-varying notch filter. Dia-

grams (c) and (d) show the experimental results from 14[s]

to 22[s] when using controllers equipped with the fixed

notch filter and the time-varying notch filter, respectively. In

these diagrams, the static liquid level is set at 0.0[m]. The

tilting angle in both experiments using the fixed and time-

varying notch filter is the same. The experimental result

regarding the position of the container and the control input

to the fixed notch filter were almost same as those using

the time-varying notch filter. (These experimental results

were not shown due to the limitations of this paper.) The

settling times to controllers when using the fixed notch filter

and the time-varying notch filter were 3.17[s] and 3.21[s],

respectively, the controller using the fixed notch filter being

slightly faster than the proposed control system by 1.25[%].

Both controllers were able to damp sloshing at the tilting

angle 0[deg] shown in Fig. 17(a) and (b). However, in Fig.

17(c), the sloshing is caused by the transfer during tilting.

On the other hand, in Fig. 17(d), the sloshing is dampened

by the proposed control system equipped with the time-

varying notch filter. Therefore, during the transfer of a liquid

container involving tilting, the proposed control system can

achieve perfect sloshing suppression.

VI. CONCLUSIONS

In this paper, we proposed a design method for suppress-

ing sloshing in a liquid container during transfer involving

dynamic tilting. In order to estimate the varying natural

frequency of sloshing, the Wigner Distribution was applied.

A comparison of simulations of sloshing demonstrated

that the Wigner Distribution estimated the change of the

natural frequency more effectively than did conventional

Short Time Fourier Transform. A control system for liquid

container transfer involving tilting was then designed using

the Hybrid Shape Approach, utilizing a time-varying notch

filter. Experimental results showed that the proposed control

system effectively achieved sloshing suppression.

0 5 10 15 20 250

0.2

0.4

0.6

Posi

tion

[m]

0 5 10 15 20 250

0.5

1

1.5

Con

trol

inpu

t [V

]

0 5 10 15 20 250

20

40

60

Tilt

ing

angl

e [d

eg]

Time [s]

Reference trajectory

Transfer trajectory

Fig. 16. Experimental results by using the proposed controller

0 2 4 6 8−0.02

−0.01

0

0.01

0.02

Slos

hing

[m

]

0 2 4 6 8−0.02

−0.01

0

0.01

0.02

14 16 18 20 22−0.02

−0.01

0

0.01

0.02

Slos

hing

[m

]

Time [s]14 16 18 20 22

−0.02

−0.01

0

0.01

0.02

Time [s]

(a)Fixed notch filter from 0[s]

(b)Time−varying notch filter from 0[s]

(c)Fixed notch filter from 14[s]

(d)Time−varying notch filter from 14[s]

Fig. 17. Experimental results of sloshing caused by the control systemby using the fixed and the time varying notch filter

REFERENCES

[1] K. Takagi and H. Nishimura, Gain-Scheduled Control of a TowerCrane Considering Varying Load-Rope Length, Journal of the JapanSociety of Mechanical Engineers International Journal. Ser C, 1999,Vol.42, No.4, pp 914-921.

[2] K. Yano, T. Toda, and K. Terashima, Sloshing Suppression Control ofAutomatic Pouring Robot by Hybrid Shape Approach, Proceedingsof the 40th IEEE Conference on Decision and Control, 2001, pp1328-1333.

[3] H. Katsube and M. Nagai, Study of Liquid Surface Wave Control ofan Incline-Type Automatic Pouring Machine (in japanese), Journalof the Japan Society of Mechanical Engineers. Ser C, 2001, Vol. 67,No. 658, pp 1792-1799.

[4] J. Ville, Theorie et applications de la notion de signal analytique,Cables and Transm., 1948, Vol. 2, No. 1, pp.61-74.

[5] J. Wahl T and S. Bolton J, The Use of the Wigner Distributionto Identify Wave-Types in Multi-Element Structures, Noise andVibration Conference, 1993, pp.205-214.

[6] S. Ishimitsu and H. Kitagawa, Interference-Term Elimination forWigner Distribution using the Frequency Domain Adaptive Filter,Journal of the Japan Society of Mechanical Engineers InternationalJournal. Ser C, 1996,Vol.39, No.4, pp.808-814.

[7] K. Yano and K. Terashima, Robust Liquid Container Transfer Controlfor Complete Sloshing Suppression, IEEE Trans. on Control SystemsTechnology, 2001, vol. 9, no. 3, pp.483-493.

[8] L. Collatz and W. Wetterling, ”Optimization Problem”, AppliedMathematical Sciences (Springer-Verlag New York Inc.), 1997, vol.17.

3052