2017 IEEE 6th Data Driven Control and Learning Systems Conference
May 26-27, 2017, Chongqing, China
Active Vibration Control of Piezoelectricity Cantilever Beam
Using an Adaptive Feedforward Control Method
Jun-Zhou Yue1 , Qiao Zhu2
1. School of Mechanical Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China.
E-mail: yue@my.swjtu.edu.cn
2. School of Mechanical Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China.
E-mail: zhuqiao@home.swjtu.edu.cn
Abstract: This work is focused on the active vibration control of piezoelectric cantilever beam, where an adaptive feeedforward
controller (AFC) is utilized to reject the vibration with unknown multiple frequencies. First, the experiment setup and its mathematical model are introduced. Because the channel between the disturbance and the vibration output is unknown in practice,
a concept of equivalent input disturbance (EID) is used to put a equivalent disturbance into the input channel. In this situation,
the vibration control can be realized by setting the control input be the identified EID. Then, for the disturbance with known
frequencies, the AFC is introduced to reject the disturbance but is sensitive to the frequencies. In order to accurately identify
the unknown frequencies of disturbance in presence of the random disturbances and un-modeled nonlinear dynamics, the timefrequency- analysis method is adopted to precisely identify the unknown frequencies of the disturbance. Finally, experiments
results demonstrate the efficiency of the AFC algorithm.
Key Words: Vibration Control, Adaptive Feedforward Controller, Piezoelectric Cantilever Beam, Disturbance Frequencies.
1
Introduction
tive intelligent Neuro-fuzzy controller was used for achieving a high performance piezoelectric vibration absorber[11].
The sliding mode variable structure control algorithm was
exploited to suppress the vibration of a flexible beam [12].
The genetic algorithm was employed to determine the optimizing delayed feedback for active vibration control of a
cantilever beam [13]. The H∞ and H2 control theories were
also frequently employed to achieve vibration suppression of
smart structures [15, 16].
In the control field, the problem of rejecting sinusoidal
disturbances is a fundamental problem because there are
many practical applications such as vibrating structures [17]
and active noise control [18]. To date, numerous disturbance
rejection techniques have been developed [19, 20]. Among
various kinds of disturbance rejection methods, an adaptive
feedforward control (AFC) approach can perfectly reject the
disturbance consisted of one or more sinusoidal components
by updating the control input to converge to the disturbance
[21, 22]. For the disturbance with unknown frequencies, an
indirect and a direct adaptive algorithms were introduced in
[23] by employing the AFC and a frequency estimator.
In this paper, the AFC algorithm will be used to suppress
the vibration of a cantilever beam due to the fact that the
disturbance causing vibration can be represented as multiple
sinusoidal components.
The aim of this paper is to study the application of AFC
algorithm in cantilever beam vibration control. The structure
of this paper is as follows. Section 2 presents the experimental system of the vibration control and mathematical model.
Section 3 proposes the AFC algorithm to reject the vibration
with unknown frequencies. In Section 4, the effectiveness of
the AFC algorithm is demonstrated by experiment.
In mechanical and civil engineering, efficiency, operation speed, functionality, quality, and low cost are the
main requirements that are manifested in the application of
lightweight and flexible structures. However, mechanical
flexible structures often tend to unwanted vibration that can
significantly lower the performance and even lead to catastrophic system failure. Furthermore, smart materials such as
piezoelectric materials and shape memory alloys are materials that respond with significant change in a property upon
application of an external driving force, which can act as
actuators to apply the controlling force. Therefore, vibration
suppress is well motivated to improve the performance of the
flexible structures by using smart materials. To this end, the
vibration control problem of piezoelectric cantilever beam
as a flexible smart system have drawn much interest of many
researchers in recent years [1, 2]. Notably, the many flexible
engineering structures such as robot arms and aircraft wings
can be modeled as a cantilever beam.
In the devoted literature, the vibration control of flexible smart structures can be split into three main categories.
The first one is the positive position feedback (PPF) that
was firstly proposed in [3]. In [4], the PPF controller was
used to preserve the guaranteed stability margins by suitably
choosing the poles and the damping of its second-order filters (with no zero). Furthermore, a modified PPF was proposed
in [5, 6], which was shown to outperform the conventional
PPF. However, the PPF method is unsuitable for multi-mode
applications. The sencond approach is the resonant control
that is established on the resonant properties of the flexible
structure [7]. Comparison with PPF, the gain selection for
each mode of the resonant controller is independent. An optimal resonant controller with a second-order filter was presented in [8] by displacement or acceleration feedback. The
last one is the advanced linear system control theory. The
PID-based output feedback controller was introduced in [10]
to suppress the vibration of a cantilever beam. The adap-
978-1-5090-5461-9/17/$31.00 ©2017 IEEE
2
Experiment system and modeling
2.1 Experimental setup
In this paper, the control objective is to suppress the vibration of a cantilever beam by using a PZT actuator. Based
on this, an active vibration control system of piezoelectric
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DDCLS'17
Table 1: Physical parameters of the experimental system
cantilever beam is established. To better view the functionality of each component, a schematic of the experimental
setup is shown in Fig. 1. As shown in Fig. 1, a piezo-
beam length, L
piezoelectric patches
position(the left), ra1
piezoelectric patches
position(the right), ra2
actuator voltage constant,
Ca
measuring point of laser sensor,
rb
flexural rigidity constant, Eb I
per unit length mass, ρAb
obtain
∂2
∂r2
Eb I
∂ 2 y(r, t)
∂r2
0.03m
0.06m
-1.2045 ×10−3 N · m/v
0.15m
0.009Nm2
0.0471Kg/m3
+ ρAb
∂ 2 y(r, t)
= 0,
∂t2
(1)
where Eb , I, Ab , and ρ represent Youngaŕs
˛ modulus, the moment of inertia, the cross-section area, and the linear mass
density of the beam, respectively. Furthermore, the elastic
deformation y(r, t) is required to satisfy the following cantilever beam boundary conditions [7]:
Fig. 1: Schematic of experimental setup for active vibration
control of piezoelectric cantilever beam.
electric cantilever beam is used as the object for vibration
control. The beam has a PZT actuator that is paste on its
surface near the cantilever base. The active vibration control
system is fixed on the actuating vibration table (2075E-HT,
The Modal Shop©) which interfaces with a NI DAQ (Data
Acquisition) card (PCIE-6229, NI©) through a power amplifier. The actuating vibration table is used to generate vibration. A laser sensor (HL-C203BE, SUNX©) is utilized to
measure the vertical vibration displacement (mm) at the end
of the beam. Through the Real-Time Workshop Toolbox in
SIMULINK, an industrial computer (IPC-610, Advantech©)
with the NI DAQ card is used for data collection and implementation of control algorithms. The control signal with
−10 ∼ 10 V is amplified to −150 ∼ 150 V by the PZT driving power supply (XE-709, XMT©) and finally sent to the
PZT actuator to suppress the vibration.
2.2
0.2m
∂y(0, t)
= 0,
∂r
∂ 2 y(L, t)
∂ 3 y(L, t)
Eb I
= 0, Eb I
= 0.
2
∂r
∂r3
y(0, t) = 0, Eb I
(2)
By using the assumed modes approach [28], the elastic deformation y(r, t) is expanded as an infinite series in the form
[29]
y(r, t) =
∞
qi (t)φi (r),
(3)
i=1
where qi (t) is the generalized modal coordinate and φi (r) is
the nature mode shape. Furthermore, φi (r) satisfies
φi (r) = C1 cos λi r + C2 sin λi r
+C3 cosh λi r + C4 sinh λi r.
Modeling of piezoelectric cantilever beam
To analyze the dynamics of the experimental beam in Fig.
1, the sketch is displayed in Fig. 2 and the parameters are
given in Table. 1. It should be noted that the subscript a and
b represent beam and piezoelectric actuator respectively in
the following.
(4)
where λi are the roots of the following equation 1 +
cos λi L cosh λi L = 0 [29]. Then, from [7], Chapter 3 in
[28], and Chapter 5 in [29], the ith mode equation of the
piezoelectric cantilever beam can be obtained by
ρAb L3 (q̈i (t) + θi2 qi (t)) = Ca (φi (ra2 ) − φi (ra1 ))u(t),
(5)
Fig. 2: Piezoelectric Laminate Beam
where u(t) is the control voltage of the PZT actuator and θi
is the resonant frequency.
For the purpose of simplifying the mathematical model,
we only consider the first two modes of the system vibration
(i = 1, 2). Then, from (3) and (5), we can get the following
state space model
˙
ξ(t)
= Aξ(t) + B1 u(t),
(6)
y(t) = Cξ(t),
Let y(r, t) be the elastic deformation of the beam. Then,
based on the classic Bernoulli–Euler beam equation [14], we
where y(t) = y(rb , t) denotes the vertical vibration displacement at the end of the beam, ξ(t) =
[q1 (t), q̇1 (t), q2 (t), q̇2 (t)]T , and
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DDCLS'17
⎡
0
⎢ −θ12
A=⎢
⎣ 0
0
1
0
0
0
⎡
0
0
0
−θ22
⎤
0
0 ⎥
⎥,
1 ⎦
0
⎤
0
2
1 ⎥
Ca ⎢
⎢ φ1 (ra )−φ1 (ra ) ⎥ , C=[φ1 (rb ), 0, φ2 (rb ), 0].
B1=
⎦
0
ρAb L3 ⎣
2
1
φ2 (ra )−φ2 (ra )
Obviously, substituting the values in Table 1 into (6) yields the dynamic model of the experimental beam in Fig. 1.
2.3
Fig. 3: Frequency responses of the experimental data and the
identified model.
System identification
The parameters in Table 1 is difficult to be exactly measured for a given experimental beam. So, it is inevitable the
dynamic model (6) exists a significant modeling error. As
such, the experimental modeling method is employed here.
Seventeen sinusoidal signal test points with frequency
changed frow 0 to 100 Hz and amplitude 1 V were applied to
the piezoelectric patches as input data and the displacement
of the bending vibration was measured as output data for the
identification of a transfer function model. The blue lines in
the Fig. 3 is the frequency response of the experimental data. Combine the experiment data and the dynamic model of
the cantilever beam, we obtain the following transfer function by using the System Identification Toolbox (MATLAB
R2014a). It should be note that this article only needs to use
the transfer function model, so the specific value of the state space coefficient matrix A, B, C is not given. And the
mathematical model of the plant
Y (s) = G(s)U (s),
Next, let us give the precise definition of EID.
Definition 2.1 Let the control input u(t) be zero, the output
of the system (9) caused by the disturbance d(t) be y(t), and
the output of the system (10) caused by the disturbance de (t)
be ye (t). Then, the disturbance de (t) is called an EID of the
disturbance d(t) if y(t) = ye (t) for any t ≥ 0.
Let
Ω = {pi (t) sin(ωi t + φi )}, i = 0, . . . , n, n < ∞,
where ωi (≥ 0) and φi are constants, and pi (t) denotes any
polynomials in time t. Now, we need to introduce the existence of the EID de (t).
Lemma 2.1 (see Lemma 1 in [25]) Assume that the transfer function G(s) exists a controllable and observable statespace realization and has no poles on the imaginary axis.
If the output y(t) caused by the disturbance d(t) belongs to
the set Ω, then there always exists an EID de (t) ∈ Ω on the
control input channel.
(7)
where Y (s) and U (s) are the Laplace transform of y(t) and
u(t), respectively, and
5.078 × 10−7 s2 + 6.602 × 10−6 s + 0.05012
. (8)
G(s) = 1.055 × 10−9 s4 + 2.109 × 10−8 s3
+1.82 × 10−4 s2 + 9.7 × 10−4 s + 0.9
Notably, the transfer function G(s) in (8) exists a controllable and observable state-space realization and has no poles
on the imaginary axis and it is reasonable to assume that
the vibration output y(t) always belongs to set Ω both in the
laboratory and practice. Therefore, the EID de (t) has always
existed and the model (10) will be considered in this paper.
In addition, the frequency responses of the experiment and
the identified model (8) are shown in Fig. 3. It is displayed
in Fig. 3 that the identified model is consistent with the experimental data in the frequency interval [1, 100]Hz. Furthermore, it is shown in Fig. 3 that there are two notable
resonant frequencies θ1 = 12Hz and θ2 = 65Hz.
2.4
Remark 2.1 The concept of EID plays an important role in
the design of vibration controller. For the unknown disturbance d(t), the main work of this paper is to identify the EID
de (t) and then to set the control input u(t) be de (t).
Equivalent input disturbance (EID)
3
Considering the disturbance causing the vibration, the
model (7) should be rewritten as
Y (s) = G(s)U (s) − Gd (s)D(s),
TFA-based AFC algorithm
In this section, a well-known disturbance rejection
method, AFC algorithm, will be carefully employed in the
vibration control of the piezoelectricity cantilever beam.
Here, we assume that the unknown EID de (t) in (10) consists of multiple sinusoidal signals, i.e.,
(9)
where D(s) (or d(t)) is the unknown disturbance generated
by the actuating vibration table here and Gd (s) is the transfer
function between the disturbance D(s) and the output Y (s).
In practice, the transfer function Gd (s) is difficult (even impossible) to be identified. This motivates us to introduce the
EID. That is, a disturbance De (s) (or de (t)) on the input
channel is employed to equal the impact of the disturbance
D(s), which makes the system (9) become
Y (s) = G(s)(U (s) − De (s)).
(11)
de,i (t) = θc,i cos(ωi t) + θs,i sin(ωi t),
de (t) =
N
de,i (t),
(12)
i=1
where de,i (t) is the sinusoidal component of the EID de (t),
θc,i , θs,i ∈ R determine the amplitude and phase, and ωi is
(10)
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DDCLS'17
Table 2: Steady-state vibration amplitudes of u(t) = 0 and
the AFC algorithm (13) with frequencies 11.48, 11.495 and
11.5Hz for the disturbance with frequency 11.5Hz
the frequency.
Then, the simple case with unknown parameters θc,i , θs,i
but known frequency ωi is firstly discussed. From the woks
in [24], the AFC algorithm can be summarized as follows to
perfectly reject the EID de (t) in (10):
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
˙
θ̂c,i
˙
θ̂s,i
⎪
⎪
⎪
⎪
ui (t)
⎪
⎪
⎩
u(t)
GR (ωi ) −GI (ωi )
θ̂c,i ,
GI (ωi ) GR (ωi ) GR (ωi ) −GI (ωi )
θ̂s,i ,
−gy(t)
GI (ωi ) GR (ωi )
θ̂c,i cos(ωi t) + θ̂s,i sin(ωi t),
N
i=1 ui (t),
Control
input
u(t) = 0
AFC with 11.48Hz
AFC with 11.495Hz
AFC with 11.5Hz
=
=
=
=
−gy(t)
Steady-state
amplitude
0.55mm
0.248mm
0.024mm
0.0005mm
Suppression
ratios
100%
45.09%
4.36%
0.09%
(13)
accurately estimate the frequencies of the disturbance in the
presence of the un-modeled nonlinear dynamics and random
disturbances. This motivate us to find an effective frequency estimation method, for instance, time frequency analysis,
that is the short-time Fourier transform (STFT), is available
to extract the unknown frequencies of the external disturbances from the vibration data. It should be noted the TFA
is a well established signal processing approach, which can
easily deal with un-modeled nonlinear dynamics and random
disturbances in real time for multiple frequencies.
where ui (t) is the sub-input with respect to the sinusoidal
component of (12) with frequency ωi , g > 0 is the adaptive
gain, and GR (ωi ) and GI (ωi ) are the real part and the imaginary part of G(jωi ), respectively. Furthermore, the parameters θ̂c,i and θ̂s,i are the estimates of θc,i , θs,i , respectively.
Let
cos(ωi t)
, i = 1, . . . , N.
(14)
i (t) =
sin(ωi t)
4
Then, the following lemma is given to show the stability of
the closed-loop system (10) with the controller (13).
Experimental implementation
In order to illustrate the efficiency of the AFC algorithm,
the following three cases are considered. Case 1 demonstrates the advantage of the AFC algorithm (13) by comparing with the PID controller in suppressing a vibration with
the resonant frequency 12Hz. In Case 2, the control effects
for a sinusoidal vibration with unknown time-invariant frequency 11Hz is given to show the advantage of the TFAbased frequency identification method by comparing with an
adaptive frequency estimator. Finally, the efficiency of TFAbased AFC algorithm is illustrated in Case 3 by rejecting a
vibration with unknown and time-varying frequencies.
Case 1. The disturbance is sinusoidal with known timeinvariant frequency. Let the EID de (t) (12) be
Lemma 3.1 (see Theorem 2.6.5 in [26]) If the transfer function G(s) is strictly positive real, i (t) is persistently exciting, and i (t), ˙ i (t) ∈ L∞ (i = 1, . . . , N ), then the closedloop system (10) with controller (13) is globally exponentially stable.
Next, let us discuss the sensitivity of the AFC algorithm
(13) for the frequency by experiment. For the situation that
the actuating vibration table is used to generate the vibration
with the frequency 11.5Hz, the experimental results of the
AFC (13) with different frequency estimates are given in the
following Fig. 4, and the Tab. 2 shows steady state amplitude
of the beam with different estimates of frequency.
de (t) = θc cos(2πωt) + θs sin(2πωt)
(15)
that is generated by the actuating vibration table, where
θc , θs are unknown parameters but ω = 12Hz is known.
In order to show the superiority of the disturbance rejection method, AFC algorithm (13), the classic PID controller
is also utilized. Then, executing the PID controller and the
AFC algorithm, vibration control effects are shown in Fig. 5,
and the Tab. 3 shows the steady state amplitude of the beam
correspondence in the Fig. 5.
Table 3: Steady-state vibration amplitudes of u(t) = 0, the
PID controller, and the AFC algorithm (13) for the disturbance (15) with known frequency 12Hz.
Fig. 4: Vibration control profiles of u(t) = 0 and the AFC
algorithm (13) with frequencies 11.48, 11.495 and 11.5Hz
for the disturbance with frequency 11.5Hz.
Control
Input
u(t) = 0
PID controller
AFC algorithm (13)
It is shown in Fig. 4 that the AFC algorithm (13) is sensitive to the frequency. This indicates the accurate frequency identification method is critical to successfully utilize the
AFC algorithm (13) to achieve the vibration control for the
disturbance with unknown frequencies. In [24], an indirect
and a direct AFC algorithm was proposed to deal with the EID de (t) (12) with unknown frequencies. However, the adaptive frequency estimators employed in [24] are difficult to
Steady-state
amplitude
0.55mm
0.213mm
0.005mm
Suppression
ratios
100%
38.73%
0.91%
Fig. 5 shows that both the PID controller and the AFC algorithm (13) have a strong ability to suppress the disturbance
(15). Furthermore, we also see that the vibration control performance of the AFC algorithm is better than that of the PID
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DDCLS'17
the frequency estimate 10.74Hz. Obviously, the frequency
estimate of the STFT is more precise than that of the frequency estimator but at the cost of time. After the frequency
estimate is obtained, both the indirect and TFA-based AFC
algorithms are switched from the initial input u = 0 to the
AFC algorithm (13).
Case 3. The disturbance consists of multiple sinusoidal
components with unknown and time-varying frequencies.
The EID de (t) (12) satisfies
⎧
θc,1 cos(2πω1 t)+
⎪
⎪
⎪ θ sin(2πω t),
⎪
t ∈ [0, 80],
⎪
s,1
1
⎪
⎨ 2
(θ
cos(2πω
t)+
c,i
i
i=1
(16)
de (t) =
t ∈ [80, 160],
θs,i sin(2πωi t)),
⎪
⎪
⎪
⎪
θ cos(2πω2 t)+
⎪
⎪
⎩ c,2
t ≥ 160,
θs,2 sin(2πω2 t),
Fig. 5: Vibration control profiles of u(t) = 0, the PID controller and the AFC algorithm (13) for the disturbance (15)
with known frequency 12Hz.
where θc,1 , θs,1 , θc,2 , θs,2 are the unknown parameters and
the unknown frequencies ω1 = 11Hz and ω2 = 15Hz. The
control effect of the TFA-based AFC algorithm is given in
the following Fig. 7. The Tab. 5 shows the steady state
amplitude of the beam in different time. Furthermore, the
AFC-based sub-inputs u1 and u2 are shown in Fig. 8.
Fig. 6: Vibration control profiles of u(t) = 0, the indirect
AFC algorithm, and the TFA-based AFC algorithm for the
disturbance (15) unknown frequency 12Hz. The circle on the
time axis represents the switching point of the corresponding
control algorithm.
controller. The main reason is that the AFC algorithm is designed to perfectly reject sinusoidal vibrations, but the PID
controller aims at suppressing any vibrations.
Case 2. The disturbance is sinusoidal with unknown timeinvariant frequency. Let the EID de (t) (12) be that in (15).
But the frequency ω equals 11Hz and is unknown. In order
to show the advantages of the proposed TFA-based AFC algorithm , the indirect AFC algorithm given in [24] is also
employed. The vibration control profiles of u(t) = 0, indirect AFC, and TFA-based AFC are displayed in the following Fig. 6, and the Tab. 5 shows the steady state amplitude
of the beam under different control methods.
Fig. 7: Vibration control profile the TFA-based AFC algorithm for the disturbance (16).
Table 5: Steady-state vibration amplitudes of multifrequency disturbance with unknown frequency 11 and
15Hz.
time
70-80s
150-160s
240-250s
Table 4: Steady-state vibration amplitudes of u(t) = 0, the
indirect AFC algorithm, and the TFA-based AFC algorithm
for the disturbance (15) unknown frequency 11Hz.
Control
Input
u(t) = 0
Indirect AFC algorithm
TFA-based AFC algorithm
Steady-state
amplitude
0.26mm
0.08mm
0.010mm
Steady-state amplitude
0.011mm
0.08mm
0.010mm
Fig. 7 shows that the TFA-based AFC algorithm is effective to the EID (16) with unknown and time-varying frequencies. At times t = 4.108s and t = 84.096s, the frequency
estimates 10.9863Hz and 14.8926Hz are obtained, respectively. In addition, the cantilever beam starts to oscillate
when the sinusoidal component de,1 = θc,1 cos(2πω1 t) +
θs,1 sin(2πω1 t) is disappeared at time t = 160s. The reason
of this phenomenon is that the corresponding sub-input u1 is
in the transient process of convergence.
Suppression
ratios
100%
30.77%
3.85%
It is shown in Fig. 6 that the AFC algorithm is effective to reject the sinusoidal disturbance with unknown timeinvariant frequency. At time t = 2s, the adaptive frequency
estimator employed in the indirect AFC algorithm obtains
5
Conclusions
In order to suppress the vibration of the piezoelectric cantilever beam with unknown multiple sinusoidal components,
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DDCLS'17
[11]
[12]
[13]
Fig. 8: The control profiles of the AFC-based sub-inputs u1
and u2 .
[14]
a new application-oriented AFC algorithm is presented by
using the TFA approach to precisely and rapidly identify the
unknown frequencies of the vibration. First, the vibration rejection can be achieved by updating the input to be the EID
that is the equivalent disturbance of the vibration in the input
channel. Then, the AFC method is introduced to perfectly
reject the vibration with known frequencies but is shown to
be sensitive to the frequency estimation. Consequently, the
TFA approach, e.g., the STFT, is utilized to accurately identify the unknown frequencies of the vibration. Based on the
accurate identified frequencies, the AFC can be successfully
used to suppress the vibration of the piezoelectric cantilever
beam.
[15]
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