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Abstract—This work presents a novel constant
frequency ultrasonic amplitude control (CFUAC) method
based on fuzzy PID (FPID) and amplitude direct feedback.
The frequency shift and amplitude nonlinearity of the
piezoelectric transducer (PT) are measured to determine
the optimal constant control frequency of 19.2 kHz. The
FPID controller is designed to adapt to the nonlinear
changes in different target amplitudes and loads. A directly
PT amplitude feedback method is used to improve signal
anti-interference ability and accuracy. The 5% settling time
and steady-state error of FPID can reach 92.22 ms and
±0.18 μm at the step response under 24 μm. The 5% settling
time and steady-state error of FPID are less than 131.44 ms
and ±0.26 μm at 10 μm under 150 N. The results confirm
that fast and precise control of the vibration amplitude of
an ultrasonic transducer can be realized by the proposed
method. The new CFUAC method lays a foundation for
revealing the ultrasonic welding and metal processing
(UWMP) mechanism and helps to expand the application of
ultrasonic vibration in the fields of precision machining
and high dynamic ultrasonic medical equipment.
Index Terms—ultrasonic transducer, precise amplitude
control, fuzzy PID, direct amplitude feedback, constant
frequency
I. INTRODUCTION
LTRASONIC technology has proven to be indispensable
in many fields such as chemical, medical and
manufacturing industries due to its high efficiency, cleanliness
and environmental friendliness [1-4]. The PT with advantages
of no electromagnetic radiation, compact structure and fast
response [5-8], etc. is widely used in chemical and food
processing [9, 10], medical and medical diagnosis [11, 12],
metal connection and manufacturing industry [13-15]. The
UWMP as important branches of ultrasonic technology are
widely used in welding plastic materials, machining of
brittle-rigid material and surface modification [16], etc.
Ultrasonic welding is extensively used in welding copper,
aluminum and stainless steel, etc. and realizes the advantages of
small welding deformation, high strength and fast speed [17].
Moreover, ultrasonic vibration is used in the precision
This work was supported in part by the National Natural Science
Foundation of China (No. U1913215 and No. 51975144).
The authors are with the State Key Laboratory of Robotics and
System, Harbin Institute of Technology, Harbin 150001, China
(Corresponding author: Yingxiang Liu (liuyingxiang868@hit.edu.cn,
http://homepage.hit.edu.cn/liuyingxiang)).
machining (e.g., drilling, cutting and milling) of difficult-to-cut
materials (e.g., ceramics, glasses and titanium alloys) which
improves the material removal rate, prevents the brittle fracture
of materials and gets better surface finish, etc. [18, 19].
Additionally, in the field of ultrasonic-assisted surface
modification (e.g., ultrasonic peening, ultrasonic surface rolling
and burnishing), the application of ultrasonic vibration has been
proved to be effective in improving the surface quality (e.g.,
surface finish, stress state and microstructure) [20, 21]. These
successful applications show that the application of ultrasonic
vibration has made certain achievements in various machining
fields. However, problems such as surface crack, low weld
strength and low efficiency in the UWMP process are severely
caused by the attenuation and fluctuation of amplitude due to
the change of workpiece shape, hardness, load and ambient
temperature [22, 23]. Additionally, the rapidity in the CFUAC
system is needed in some applications such as ultrasonic
welding which welding time is usually 0.5 s–2 s [17]. More
importantly, the mechanism of UWMP is still unclear, and the
research of the mechanism is inseparable from the constant
frequency and amplitude of ultrasonic vibration. Thus, fast and
precise control of ultrasonic amplitude under constant
frequency has a pivotal role in improving the quality of UWMP,
revealing the mechanism of UWMP and expanding the
application of ultrasonic technology.
Previous studies on the control of ultrasonic amplitude are
mainly focused on resonant frequency tracking (RFT) of the PT,
thereby indirectly keeping the output amplitude maximum. The
RFT is divided into three modes [24]: current maximum control,
power maximum control and phase-locked loop (PLL) control.
The maximum current control [25] changes the frequency of
excitation voltage to ensure the PT current is always at the
maximum state, while the maximum power control [26]
changes the frequency to ensure the product of voltage and
current is maximized. Both methods have achieved good
frequency tracking results, but neither can achieve high
amplitude control accuracy due to the current amplitude
feedback is easily interfered. The PLL control [27, 28] changes
the frequency of the control signal and locks the frequency that
makes the impedance phase to be zero. The PLL method does
not rely on the feedback of the current amplitude but it is still
unable to achieve constant amplitude because the impedance of
PT changes with the resonance frequency and acoustic load
[29]. Thus, Kuang et al. propose to add constant current control
based on PLL to increase the stability of ultrasonic amplitude
[30]. These PLL methods have achieved better amplitude
control effects in some specific systems. However, problems
Pengfei Du, Yingxiang Liu*, Senior Member, IEEE, Weishan Chen, Shijing Zhang, and Jie Deng
Fast and Precise Control for the Vibration
Amplitude of an Ultrasonic Transducer Based on
Fuzzy PID Control
U
such as loss of lock and anti-resonance frequency tracking still
restrict its application [31]. Moreover, The RFT systems
mentioned above limit their rapidity due to the frequency
sweeping process [28, 32], and it could not meet the
requirements of constant amplitude under some high dynamic
loads [17, 33]. More importantly, the RFT method needs to
constantly change the frequency during the machining process
[27, 31], which is not conducive to revealing the mechanism of
UWMP.
In summary, this paper is motivated by two challenges that
are frequently occurred in the field of UWMP. First, many
traditional ultrasonic amplitude control methods use the RFT
method based on electrical parameter feedback. The RFT
algorithms and electrical feedback methods bring in many
drawbacks, such as low amplitude adjustment speed caused by
frequent frequency sweeping, and low amplitude accuracy
caused by electrical parameter feedback that is susceptible to
external interference. The accuracy and speed of amplitude
adjustment limit the accuracy improvement of the UWMP
process and the application of ultrasonic vibration to high
dynamic fields. Second, the traditional RFT method needs
frequent changes in excitation frequency. However, the
research on the mechanism of UWMP needs to be carried out at
a constant frequency, which hinders the revealing of its
mechanism. To address these two challenges, a novel CFUAC
method based on FPID and amplitude direct feedback is
proposed. The proposed method achieves high-precision and
rapid adjustment of ultrasonic amplitude at a constant
frequency, is expected to facilitate the accuracy improvement
and the mechanism revelation of the UWMP process. The
method is illustrated by amplitude control of the ultrasonic
welding process, while it can be extendable to other ultrasonic
machining fields that need high precision and dynamic
performance.
The rest of this work is organized as follows. Section II
introduces the composition of the control system, and the fixed
control frequency is determined according to the frequency
shift and amplitude nonlinearity of the system. The controller is
designed in section III. Experiments of step response and
constant amplitude control under load are executed to verify the
system performance in section IV. Finally, a summary is
provided.
II. SYSTEM CONFIGURATION AND CHARACTERISTIC
ANALYSIS
A. Layout and hardware
The ultrasonic amplitude control system is established based
on the ultrasonic welding platform in order to directly apply the
control scheme, as shown in Fig. 1(a) and (b). A self-designed
sandwich piezoelectric ultrasonic transducer is used in the
ultrasonic welding system [34]. A laser displacement sensor
(LK-H020, Keyence, Japan) with a dynamic measuring
resolution of 5 nm is used to measure the real-time amplitude of
the PT, the sampling rate is set as 392 kHz during the
experiment. The measurement position of the laser
displacement sensor is at the end of the PT, as shown in Fig.
1(c). A 16-bit data acquisition card (NI-9215, National
Instruments, USA) is used to obtain the output signal of the
laser displacement sensor. The control system is built in
LabVIEW (National Instruments, Inc®, Austin, USA) to
receive sensor signals and send out control signals. A 16-bit
digital to analog converter (PCI-1721, ADVANTECH, China)
is used to send excitation signals to the power amplifier. The
analog signal from the D/A converter is amplified by a high
power amplifier (ATA-L8, Aigtek, China) and applied to the PT.
The load application mode of the system is shown in Fig. 1(c),
which is the same as that of ultrasonic welding. Different loads
along the y-axis are applied to PT due to the different air
pressure of the cylinder. A force sensor (JHBM-H1, Jinnuo
Sensor, China) is used to measure the load.
Fig. 1. Experimental system, (a) experimental set-up, (b) control system,
(c) amplitude measurement and load application.
B. Open-loop characteristics of the system
The PT exhibits nonlinearity due to piezoelectric,
mechanical and dielectric nonlinearities of piezoelectric
materials. The nonlinearity of the PT leads to the frequency
shift and amplitude nonlinear change with voltage [35]. Thus,
the nonlinear characteristic is very critical to the design of the
CFUAC controller. The resonant frequency shift and the
amplitude nonlinearity are measured under open-loop condition.
The results are used to design the CFUAC algorithm.
1) Resonant frequency shift
The frequency shift of the PT is measured in the target
voltage range. The vibration velocity versus frequency at
different voltages is measured by a laser Doppler vibrometer
(PSV-400-M2, Polytec, Germany), as shown in Fig. 2. The
resonant frequency decreases from 19.10 kHz to 18.91 kHz
when the excitation voltage increases from 100 VP-P to 700 VP-P.
The frequency reduction of the PT is 110 Hz from 100 VP-P to
300 VP-P and the frequency reduction from 300 VP-P to 700 VP-P
is only 80 Hz. The results show that the frequency shift within
300 VP-P is higher than the frequency shift above 300 VP-P. The
amount of frequency shift gradually decreases as the excitation
voltage increases. In addition, the tested results show that a
good vibration state can be obtained from 18.8 kHz to 19.4 kHz.
It means that the PT can work efficiently and have a larger
amplitude by selecting a fixed frequency point in this range for
constant amplitude control.
Fig. 2. The vibration velocity of the PT varies with frequency under
different voltages.
2) Amplitude nonlinearity
The amplitude at different frequencies is measured in order
to obtain a suitable excitation frequency for fast and precise
control of ultrasonic amplitude. The amplitude of the PT at
frequencies of 19.0 kHz, 19.2 kHz and 19.4 kHz is measured as
shown in Fig. 3.
Fig. 3. The amplitude of the PT at 19.0 kHz, 19.2 kHz and 19.4 kHz.
The amplitude at 19.0 kHz can be divided into three sections
with the voltage increase. The amplitude increases rapidly and
linearly from 6 μm to 15 μm as the voltage increases from 100
VP-P to 175 VP-P in the first stage. In the second stage, the
amplitude sharply increases from 15 μm to 23 μm when the
voltage increases from 175 VP-P to 200 VP-P. In the third stage,
the amplitude increase rate gradually slows down and reaches
the maximum amplitude of 29 μm when the voltage rises from
200 VP-P to 700 VP-P. The amplitude measured at 19.2 kHz can
be divided into two stages. In the first stage, the amplitude
increases from 8 μm to 19 μm when the voltage changes from
100 VP-P to 275 VP-P. In the second stage, the amplitude
increases from 19 μm to 26 μm when the voltage changes from
275 VP-P to 700 VP-P. The amplitude nonlinearity is shown
under 19.2kHz as the amplitude growth rate decreases with the
increase of voltage. Moreover, it is found that the nonlinearity
of the amplitude decreases at 19.4 kHz.
These results indicate that the maximum amplitude of 29 μm
is achieved at 19.0 kHz under 700 VP-P. The amplitude is
reduced by 10.3% and 34.5% respectively under 19.2 kHz and
19.4 kHz at 700 VP-P. It can be recognized that 19.4 kHz is not
suitable for the constant frequency point of ultrasonic
amplitude control as the maxim amplitude at this point is only
19 μm. The amplitude is too small to meet some
ultrasonic-assisted processing that requires large amplitude.
Moreover, the low energy conversion efficiency of the PT at
this frequency leads to severe heat generation.
Although the maximum amplitude can be reached at 19.0
kHz, it is not the best control frequency for CFUAC. These
tested results indicate that the nonlinear trend of the amplitude
is reduced when the excitation frequency is increased from 19.0
kHz to 19.4 kHz with an interval of 0.2 kHz. It should be noted
that a sharp amplitude increase occurs when the voltage
increases from 175 VP-P to 200 VP-P. In other words, the
amplitude increased by 32.53%, while the voltage increased by
only 4.55%. Such phenomenon is not conducive to CFUAC
because very small voltage changes can cause large amplitude
changes. This sudden change of amplitude is mainly caused by
the resonance of the PT when the voltage is increased to 200
VP-P. It can be found clearly from Fig. 2 that the resonant
frequency changes from 19.10 kHz to 19.02 kHz as the voltage
increases from 100 VP-P to 200 VP-P. Strong resonance is excited
as the resonance frequency of the PT coincides with the
excitation frequency. Moreover, the amplitude of the PT at 19.2
kHz is only reduced by 3 μm and the nonlinearity is much
smaller than that of 19.0 kHz. These results indicate that 19.2
kHz is suitable to be used as the frequency for ultrasonic
amplitude control. Thus, 19.2 kHz is chosen as the constant
frequency point for ultrasonic amplitude control.
III. CONTROL STRATEGY
A. Conventional PID (CPID) controller
The PID controller is still the most widely used controller in
modern industrial control in the world due to its simple
structure, convenient design, stable control performance and
low cost. The CPID control system is a typical negative
feedback system. The input of the system can be defined as
follows:
( ) ( )
et A A t=−
(1)
where A is the target amplitude of the system, A(t) is the
real-time feedback amplitude.
The time domain signal of the CPID controller can be
expressed as[36]:
0
()
( ) ( ) ( )
t
PP
I
PD
Kde t
u t K e t e t K T dtT
= + +
(2)
where u(t) is the signal output by the control system, which is
the amplitude of the 16-bit DAC output voltage. KP, TI and TD
are the proportional gain, integral time constant and differential
time constant.
The error between real-time average amplitude and target
amplitude are collected to calculate the output of the control
system in each cycle. The output of the proportional,
integration and differential can be obtained by using the
discrete calculation method as follows:
( )
()
PP
U n K E n=
(3)
( ) ( ) ( ) ( )
-1
12
P
II
I
E n E n
K
U n U n T
T
+
=
−
+
(4)
( ) ( ) ( )
( )
-1
D
DP
T
U n K A n A n
T
= − −
(5)
where UP(n) is the output of proportional operation in the nth
cycle, UI(n) is the output of integration operation in the nth
cycle (the trapezoidal discrete integration is used for
calculation to avoid drastic changes in the integration results
caused by sudden changes in real-time or target amplitude),
UD(n) is the output of differential operation in the nth cycle (the
real-time amplitude A(n) is differentiated but not the error to
avoid drastic changes in the differential result caused by sudden
changes in the target amplitude), E(n) is the error of the nth
cycle, A(n) is the feedback amplitude of the nth cycle, ΔT is the
sampling time of the controller.
The proportional, integral and differential operation are
added to get the total output U(n) of the system as follows:
( ) ( ) ( ) ( )
P I D
U n U n U n U n++=
(6)
B. FPID controller
The fast and precise control of ultrasonic amplitude in large
amplitude range and variable load is difficult to achieve by
traditional control methods due to the nonlinearity of the PT
[35]. The nonlinearity of the PT is caused by the dielectric,
mechanical and piezoelectric nonlinearity of the piezoelectric
material, which lead to serious heating, a saturation of the
amplitude and system stability reduce as the voltage rises [37].
This makes the precise modeling and amplitude control of the
PT very difficult [38]. Thus, an FPID controller is selected for
adapting to the nonlinearity of the PT. The FPID control
strategy is widely used in the nonlinear system due to the
establishing of fuzzy logic by human experience and
non-dependent of the precise mathematical model [39].
Additionally, the robust and fast control of nonlinear systems
can be achieved through real-time tuning of PID parameters by
a fuzzy controller [40, 41].
A basic configuration of the fuzzy control system is shown in
Fig. 4. It can be recognized that the fuzzification, fuzzy
inference mechanism and defuzzification modules are
intelligentized by the knowledge base. The knowledge base is
the key to the system, which contains a fuzzy control rule base
and a database. The inference module infers according to the
fuzzy control rules and given conditions. The fuzzification
module implements a mapping from real input space to fuzzy
space, while the defuzzification module establishes a mapping
from the fuzzy space to the real output space[42].
Fig. 4. Basic configuration of a fuzzy control system.
1) The structure of FPID control system
The block diagram of the FPID control system for fast and
precise control of ultrasonic amplitude is shown in Fig. 5. The
inputs of the fuzzy logic controller are ultrasonic amplitude
error E and error change rate ER. The error change rate ER can
be defined as follows:
( ) ( ) ( )
1E n E n
ER n T
−−
=
(7)
Fig. 5. Block diagram of FPID control system for CFUAC.
The outputs of the fuzzy logic controller are the variation of
proportional gain ΔKP, integral gain ΔTI and differential gain
ΔTD. The initial values of the PID gains are Kp0, TI0 and TD0, and
the PID gains adjusted by the fuzzy controller are as follows:
0
0
0
P P P
I I I
D D D
K K K
T T T
T T T
= +
= +
= +
(8)
2) Fuzzy design of input and output variables
The input and output parameters of the fuzzy controller are E,
ER, ΔKp, ΔTI, ΔTD. The fuzzy domains of those parameters are
{FE}, {FER}, {FKP}, {FTI}, and {FTD}. In addition, the basic
domains are {BE}, {BER}, {BKP}, {BTI}, and {BTD}. The
conversion of the basic domain and the fuzzy domain is carried
out by using scaling factor which can be defined as follows:
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
e
er
p p p
I i I
D d D
k
k
K n k FK n
T n k FT n
FE n E n
FER n E
T n k FT n
Rn
=
=
=
=
=
(9)
where ke, ker, kp, ki and kd, are the scaling factors of E, ER, ΔKp,
ΔTI and ΔTD. The values of those domains and scaling factors
for the fuzzy controller are shown in Table I.
TABLE I
THE DOMAINS AND SCALING FACTORS FOR FUZZY CONTROLLER
Variable
E
ER
ΔKP
ΔTI
ΔTD
Basic
domain
[-26,26]
[-4,4]
[-0.005,0.005]
[-4E-5,
4E-5]
[-5E-5,
5E-5]
Fuzzy
domain
[-3,3]
[-3,3]
[-3,3]
[-3,3]
[-3,3]
Scaling
factor
0.12
0.75
0.0019
1.4E-5
1.7E-5
3) Design of fuzzy levels
The more refined the fuzzy set is divided, the more detailed
language description is needed, and the higher the system
precision. On the other hand, the rapidity and stability of the
system will be reduced. The input and output fuzzy control
subsets both adopt a level of seven by considering the system
rapidity and accuracy. The fuzzy subset is {NB, NM, NS, ZO,
PS, PM, PB}, and the NB, NM, NS, ZO, PS, PM, PB stand for
negative big, negative middle, negative small, zero, positive
small, positive middle and positive big. Thus, the fuzzy set of
variables E, ER, ΔKp, ΔTI, ΔTD is {NB, NM, NS, ZO, PS, PM,
PB}.
TABLE II
THE FPID CONTROLLER RULE BASE FOR ΔKP, ΔTI AND ΔTD
ΔKp/ΔTI/ΔTD
E
NB
NM
NS
ZO
PS
PM
PB
ER
NB
NB/PB/ZO
NB/PB/PM
NM/PB/PB
NM/PM/PB
NS/PM/PB
NS/ZO/PM
ZO/ZO/ZO
NM
NB/PB/ZO
NM/PB/PM
NS/PM/PM
NS/PM/PB
NS/PS/PM
ZO/ZO/PM
PS/ZO/ZO
NS
NM/PM/ZO
NS/PS/PS
NS/PS/PM
ZO/PS/PM
ZO/ZO/PM
PS/NS/PS
PM/NS/ZO
ZO
NM/PM/ZO
NS/PS/ZO
ZO/PS/ZO
ZO/ZO/ZO
ZO/NS/ZO
PS/NS/ZO
PM/NM/ZO
PS
NS/PS/ZO
NS/PS/PS
ZO/ZO/PM
ZO/NS/ PM
PS/NS/PM
PS/NS/PS
PM/NM/ZO
PM
ZO/ZO/ZO
ZO/ZO/PM
PS/NS/PM
PS/NM/PB
PS/NM/PM
PM/NB/PM
PB/NM/ZO
PB
ZO/ZO/ZO
PS/ZO/PM
PS/NS/PB
PM/NM/PB
PM/NB/PB
PB/NB/PM
PB/NB/ZO
4) Design of membership function
The membership functions are divided into Gauss, trapezoid
and triangle, etc. The Gaussian function has a wide range of
applications and high stability, but it will increase the amount
of system calculation and reduce the system rapidity. The
stability of trapezoidal and triangular functions is lower than
the Gauss function, but the controller has higher running
rapidity. Moreover, the high resolution and high sensitivity of
the system are realized through the sharp membership function
like triangle [43].
The Gaussian and triangular membership functions are
selected by considering system rapidity and stability. The
Gaussian function is selected to make the control stable when
large amplitude error E and error change rate ER happens. The
triangular function is selected to improve the rapidity of the
system when E and ER are small. The narrow triangular
membership function is used near zero to obtain better control
sensitivity when the amplitude fluctuation is small, as shown in
Fig. 6(a). The membership functions of the CFUAC system are
established based on the above criteria and multiple amplitude
control experiments as shown in Fig. 6.
Fig. 6. Membership function of (a) E, ER, ΔTI and ΔTD, (b) ΔKp.
5) Design of fuzzy rule base
The fuzzy rules between amplitude error E, amplitude error
change rate ER and output variables are established according
to the following principles:
Principle (1): The KP is increased to improve the rapidity and
reduce the deviation value rapidly when the amplitude
deviation is large. In addition, the KP is reduced to avoid
unstable oscillation of the system when the amplitude deviation
is small.
Principle (2): The TI is designed to be large to avoid
overshoot caused by integral saturation when the amplitude
deviation is large. Moreover, the TI is gradually decreased to
reduce the steady-state error of the system as the amplitude
deviation decreases.
Principle (3): The TD is set as small to increase the rapidity of
the system when the amplitude deviation is large. Moreover,
the TD is increased to enhance stability when the deviation is
small. In addition, the TD is set as large to avoid overshoot when
ER is large. The fuzzy rule base of the CFUAC system is
designed based on the above principles, as shown in Table II.
The first rule in the fuzzy rule base can be expressed as:
If E is NB and ER is NB, then ΔKp is NB, ΔTI is PB, ΔTD is
ZO.
A total of 49 fuzzy rules are generated through Table II for
the CFUAC system. Moreover, the centroid method is used to
defuzzify output variables.
Compared with FPID, The CPID is difficult to compensate
for the amplitude nonlinear change caused by voltage and load,
because of the certainty of the calculation method and the
invariance of gains. This leads to problems such as increased
overshoot, severe oscillation, and prolonged stabilization time,
which reduces the stability and rapidity of the system. However,
the gains of FPID can be adjusted online according to the
amplitude error and error change rate, to compensate for the
nonlinear characteristics of the system for achieving a better
control effect under a large amplitude and load range. In order
to obtain a fair comparison, we have tuned the CPID gains. We
choose those gains that yield the best performance in the sense
of minimum rise time, without overshoot and avoiding the
possibility of transducer amplitude adjustment instability. The
best tuning result of CPID gains of KP, TI and TD are 0.006,
0.00018 and 0.00006.
IV. RESULTS AND DISCUSSION
Fast realization of different target amplitudes and fast
stabilization of amplitudes under different loads are two
important requirements in the UWMP process. For these two
requirements, step response of different target amplitudes and
load disturbance experiments with fixed amplitudes are carried
out to investigate the precise and rapidity of the CFUC method.
A. CFUAC under different amplitudes
For testing the rapidity and accuracy of the proposed control
method to achieve different target amplitudes, step response
experiments are carried out. The experimental results of step
response with target amplitudes of 8 μm, 16 μm and 24 μm are
plotted in Fig. 7(a), (c) and (e), while Fig. 7(b), (d) and (f)
represent the gains of FPID over time. The response parameters
corresponding to Fig. 7(a), (c) and (e) are shown in Table III.
It can be found clearly from Table III that the rising time
using FPID is reduced by 16.67%, 63.82% and 52.98%, while
the settling time is reduced by 10.65%, 61.91% and 59.91%. It
can be recognized that the rapidity of the FPID controller is
improved more obviously when the target amplitude is greater
than 16 μm. Such phenomenon is mainly caused by the large
increase of proportional gain KP in FPID control, when the
target amplitude increases.
In addition, the steady-state error of FPID is ±0.18 μm when
the target amplitude is 24 μm, while that of CPID is ±0.32 μm,
as shown in Fig. 7(e). The steady-state errors of FPID control
are reduced by 30%, 54.17% and 43.75%, in the step response
of 8 μm, 16 μm and 24 μm. Such phenomenon can be explained
as: the differential time constant TD is increased by the fuzzy
controller when the error is small and the error rate is large, and
this suppresses the system error from continuing to increase and
makes the output amplitude more stable. The results
demonstrate that FPID controller has obvious advantages in
improving accuracy and rapidity in a large amplitude range,
which is helpful for the application of ultrasonic vibration in
high dynamic demand.
Fig. 7. Different amplitude responses at excitation frequency of 19.2 kHz
and its FPID parameters change:(a), (c) and (e) step responses at 8 μm,
16 μm and 24 μm; (b), (d) and (f) FPID parameters change under 8 μm,
16 μm and 24 μm. TABLE III
THE RESPONSE PARAMETERS UNDER DIFFERENT AMPLITUDES
Target
8 μm
16μm
24μm
Controller
PID
FPID
PID
FPID
PID
FPID
Tr (ms)
38.38
31.99
123.71
44.75
153.52
72.18
Ts (ms)
46.28
41.35
177.80
67.72
230.03
92.22
Mp (%)
3.48
4.74
0
0
0
0
Es (μm)
±0.10
±0.07
±0.24
±0.11
±0.32
±0.18
It can be found clearly from Fig. 7(b), (d) and (f) that KP, TI
and TD in the FPID controller are tuned over time according to
the change of the system error and the error rate. The system
error and error rate are large when step response occurs. Thus,
the KP increased rapidly, and the TI is fast decreased by the
fuzzy controller. The system rapidity is improved by the tuning
of the PID gains. Moreover, the KP and TI are almost unchanged
due to the error reducing when the system reaches steady-state.
In addition, the amplitude stability is improved due to the
sensitivity of the TD change caused by the ER change.
B. Amplitude stability of different load and amplitude
For investigating the stability and rapidity of the proposed
method in different amplitude ranges, amplitude control
experiments under different targets and loads are performed.
The load range of 50 N to 150 N is set to meet the
corresponding load requirements of ultrasonic welding. The
following parameters are defined to evaluate the rapidity and
accuracy of the amplitude control system under load
disturbance, as shown in Fig. 8(a). The amplitude fluctuation
(AF) is defined as the difference between the maximum
amplitude and the minimum amplitude under load condition.
The steady-state error (Es) is defined as the maximum
amplitude deviation from 1.5 s to 2 s. The settling time (Ts) is
defined as the time from the application of the load to the
amplitude recovery to 95% of the target amplitude.
The experimental results of amplitude stability at 10 μm
under 50 N, 100 N and 150 N are plotted in Fig. 8(a), (c) and (e),
while Fig. 8(b), (d) and (f) represent the amplitude stability at
20 μm. The response parameters of Fig. 8 are shown in Table
IV.
Fig. 8. Amplitude stability under load of 50 N, 100 N and 150N: (a), (c)
and (e) under amplitude of 10 μm; (b), (d) and (f) under amplitude of 20
μm.
TABLE IV
·IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS
COMPARISON OF PARAMETERS BETWEEN CPID AND FPID AMPLITUDE CONTROL AT 10/20 MICRON UNDER DIFFERENT LOADS
Load
50 N
100 N
150 N
Controller
Open-loop
PID
FPID
Open-loop
PID
FPID
Open-loop
PID
FPID
AF (μm)
1.28/2.68
0.94/1.71
0.79/1.21
2.45/3.41
1.73/2.34
1.22/1.63
3.13/4.63
2.16/2.78
1.68/1.96
Es (μm)
/
±0.25/±0.31
±0.16/±0.25
/
±0.29/±0.39
±0.23/±0.28
/
±0.37/±0.42
±0.26/±0.30
Ts (ms)
/
72.35/116.58
66.18/80.74
/
187.47/240.69
87.22/128.10
/
258.70/348.21
131.44/145.85
It can be found clearly from Table IV that the maximum
reduction of open-loop amplitude at 10 μm and 20 μm are
31.3% and 24.2% under 150 N, and the steady-state error of the
FPID under this condition are ±0.26 μm and ±0.30 μm. Results
show that the CFUAC has an obvious inhibitory effect on the
amplitude attenuation caused by the load, and the high
precision constant amplitude control is realized.
In addition, the maximum AFs of the open-loop at 10 μm and
20 μm are 3.13 μm and 4.63 μm, while that of the close-loop is
reduced to 1.68 μm and 1.96 μm under 150 N. The results
indicate that the CFUAC strategy can restrain the amplitude
fluctuation under different loads that is beneficial to suppress
the amplitude fluctuation caused by load fluctuation in UWMP.
Fig. 8(a) shows that the settling time of CPID and FPID is
72.35 ms and 66.18 ms. It means that the system rapidity is
slightly affected by the two controllers under light load and
small target amplitude. Such phenomenon can be explained as:
the PID gain is slightly tuned for small amplitude attenuation
due to light load, which has little effect on the PID controller.
Fig. 8(e) shows that the settling time of the CPID is 258.70 ms,
while that of the FPID is 131.44 ms which was reduced by
49.19%. It means that the FPID control has obvious advantages
in improving the system rapidity under a large load. The main
reason is that the ΔTD and ΔKP increase greatly due to the great
amplitude decrease and decrease rate under heavy load. In
addition, Fig. 8(b), (d) and (f) shows that the settling time of
CPID is 116.58 ms, 240.69 ms and 348.21 ms, while that of the
FPID is 80.74 ms, 128.10 ms and 145.85 ms which reduced by
30.74%, 46.78% and 58.11%. It means that the FPID control
has obvious advantages in improving the rapidity of the system
at high amplitude under different loads. The main reason is that
the disturbance of the load under large amplitude causes the
sharp change of amplitude and amplitude change rate, resulting
in a large increase of ΔTD and ΔKP.
Fig. 8(e) and (f) show that the AFs of the CPID under 10 μm
and 20 μm are 2.16 μm and 2.78 μm, while that of the FPID are
reduced to 1.68 μm and 1.96 μm which decreased by 21.82%
and 29.5% under 150 N. The results show that the FPID can
suppress the amplitude fluctuation due to the real-time tuning
of the PID gains. Furthermore, the suppressions of amplitude
fluctuation are large at the target amplitude of 20 μm. This is
mainly due to a significant gain tuning of the fuzzy controller
caused by the large amplitude deviation and deviation rate
under high amplitude.
Moreover, compared with the CPID the maximum
steady-state errors decrease of the FPID under 10 μm is 34.7%
which decrease from ±0.25 μm to ±0.16 μm, as shown in Fig.
8(a). Meanwhile, the maximum steady-state errors decrease of
the FPID under 20 μm is 28.6%, as shown in Fig. 8(f). It means
that the FPID controller can greatly improve the accuracy of
steady-state amplitude under load conditions. Such
phenomenon can be explained as: the TD of the FPID controller
can be tuned by the fuzzy controller by sensing the small
amplitude fluctuation of the system, which is helpful to
improve the stability and accuracy in UWMP.
In conclusion, the settling time of no-load step response
under 10 μm using the RFT method is about 1 s to 2 s [30],
while the maximum settling time under 24 μm in this study is
92.22 ms, which indicates the advantage of the proposed
CFUAC method in rapidity. In addition, Even though the
traditional RFT method keeps the PT in resonance, the
amplitude is still not constant due to the change of the PT
impedance with the resonance frequency and load [29].
Additionally, the CFUAC control method can achieve a
steady-state error of ±0.30 μm under 150 N. Therefore, the
proposed CFUAC method can achieve fast and precise control
of ultrasonic amplitude. The developed CFUAC method will
have broad application prospects in ultrasonic-assisted
precision machining and will help reveal the mechanisms of
UWMP.
V. CONCLUSION
In this work, a novel CFUAC method was proposed,
designed and tested. A closed-loop control system was built.
The frequency shift and amplitude nonlinearity of the system
was measured, and according to that, the constant control
frequency point of 19.2 kHz was determined. The FPID
controller was designed based on the nonlinearity of the PT. A
series of experiments were accomplished, and compared with
CPID conclusions were drawn as follows. The settling time and
steady-state error of step response within 24 um were less than
92.22 ms and ±0.18 μm, the maximum rapidity and accuracy
can be improved by 61.91% and 54.17%. In addition, the
settling time and steady-state error under 150 N were better
than ±0.30 μm and 145.85 ms in constant amplitude control
within 20 μm, the maximum rapidity and accuracy can be
improved by 58.11% and 34.70%. Compared with the
traditional RFT method, the control method proposed in this
paper realized the rapid and precise control of the amplitude
under constant frequency. The realization of the CFUAC
method provides a new feasible idea for the amplitude control
of ultrasonic transducer. Furthermore, the proposed method had
good rapidity and sub-micron accuracy in a large amplitude and
load range due to the use of FPID and direct amplitude
feedback. These merits make the proposed method more
conducive to improve the quality and reveal the mechanism of
the UWMP, and it has great potentials for applications in
precision machining and high dynamic ultrasonic medical
equipment. The extension of the application of the proposed
method will be the research focus in future.
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Biographical notes Pengfei Du was born in Sichuan Province, China,
in 1989. He received his B.E. degree and M.E.
degree from the School of Mechanical
Engineering and Automation, Northeastern
University, China, in 2013 and 2015, respectively.
He is currently a Ph.D. candidate at the Harbin
Institute of Technology, China. His research
interests include piezoelectric actuating and
vibration-assisted machining.
Yingxiang Liu (M’12, SM’16) was born in Hebei
Province, China, in 1982. He received his B.E.
degree in mechanical engineering, M.E. and
Ph.D. degrees in mechatronics engineering from
the School of Mechatronics Engineering at
Harbin Institute of Technology, China, in 2005,
2007 and 2011, respectively. He joined the
School of Mechatronics Engineering at the
Harbin Institute of Technology in 2011, where he
has been a professor since December 2013, and
is also a member of the State Key Laboratory of
Robotics and System. He is the vice director of the Department of
Mechatronic Control and Automation. He was a Visiting Scholar at the
Mechanical Engineering Department, University of California, Berkeley,
from August 2013 to August 2014. His research interests include
piezoelectric actuators, ultrasonic motors, ultrasonic transducers,
micro-nano manipulations, nano positioning, piezoelectric micro jets,
vibration control, bionic robots, fish robots, soft robots, micro robots and
artificial muscles. He has served as an Associate Editor of IEEE
Transactions on Industrial Electronics, an Associate Editor of IEEE
Access and a Topic Editor of Materials.
Weishan Chen was born in Hebei, China, in
1965. He received his B.E. and the M.E. degrees
in precision instrumentation engineering, and the
Ph.D. degree in Mechatronics engineering from
Harbin Institute of Technology, China, in 1986,
1989, and 1997, respectively. Since 1999, he
has been a professor with the School of
Mechatronics Engineering at the Harbin Institute
of Technology. His research interests include
ultrasonic driving, smart materials and structures,
bio-robotics.
Shijing Zhang was born in Guizhou Province,
China, in 1994. He received his B.E. degree from
the School of Mechanical Engineering at the
Guizhou University, China, in 2017. He received
his M.E. degrees in mechatronics engineering
from the School of Mechatronics Engineering,
Harbin Institute of Technology, China, in 2019.
He is currently a Ph.D. candidate in
mechatronics engineering at the Harbin Institute
of Technology, China. His research interests
include precision piezoelectric actuating with
multi-DOF and micro-nano manipulating.
Jie Deng was born in Shaanxi Province, China,
in 1992. He received his B.E. degree in
mechanical engineering from the School of
Mechanical Engineering, Northwestern
Polytechnical University, China in 2014. He
received the M.E. and Ph.D. degrees in
mechatronics engineering from the School of
Mechatronics Engineering at Harbin Institute of
Technology, China, in 2016 and 2020,
respectively. He is currently a lecturer at the
Harbin Institute of Technology, China. His
research interests include piezoelectric actuators, piezoelectric robots,
micro-nano manipulations and nano positioning.
