Research ArticleAdaptive Vibration Control of PiezoactuatedEuler-Bernoulli Beams Using Infinite-Dimensional LyapunovMethod and High-Order Sliding-Mode Differentiation

Teerawat Sangpet1 Suwat Kuntanapreeda1 and Ruumldiger Schmidt2

1Department of Mechanical and Aerospace Engineering King Mongkutrsquos University of Technology North Bangkok1518 Pracharat Sai 1 Bangkok 10800 Thailand2Institute of General Mechanics RWTH Aachen University Templergraben 64 52056 Aachen Germany

Correspondence should be addressed to Teerawat Sangpet teerawatmegmailcom

Received 31 July 2014 Revised 26 November 2014 Accepted 8 December 2014 Published 22 December 2014

Academic Editor Jyoti Sinha

Copyright copy 2014 Teerawat Sangpet et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper presents an adaptive control scheme to suppress vibration of flexible beams using a collocated piezoelectric actuator-sensor configuration A governing equation of the beams is modelled by a partial differential equation based on Euler-BernoullitheoryThus the beams are infinite-dimensional systemsWhereas conventional control design techniques for infinite-dimensionalsystems make use of approximated finite-dimensional models the present adaptive control law is derived based on the infinite-dimensional Lyapunov method without using any approximated finite-dimension model Thus the stability of the control systemis guaranteed for all vibration modes The implementation of the control law requires a derivative of the sensor output forfeedback A high-order sliding mode differentiation technique is used to estimate the derivative The technique features robustexact differentiation with finite-time convergence Numerical simulation and experimental results illustrate the effectiveness of thecontroller

1 Introduction

Flexible structures have attracted interest because of theirlighter weight compared to traditional structures They havebeen widely used in aerospace applications and robotics[1ndash3] However the flexibility leads to vibration problemsTherefore vibration control is needed Over the past fewdecades active vibration control has drawn more interestsfrom researchers since it can effectively suppress the vibration[4ndash7]

Piezoelectric actuators and sensors provide an effectivemeans of vibration suppression of flexible structures [8] Theadvantages of using piezoelectric actuators and piezoelectricsensors include nanometer scale resolution high stiffnessand fast response Many researchers have studied the vibra-tion suppression of flexible structures using piezoelectricactuators and piezoelectric sensors In [9] Tavakolpouret al proposed a self-learning vibration control strategy for

flexible plate structures A control algorithm is based on aP-type iterative learning with displacement feedback Wanget al [10] presented a simple control law for reducing thevibration of the flexible structure Linear feedback controlwas derived using a linear matrix inequality method Qiuet al [11] proposed a neural network controller based onPD control with collocated piezoelectric actuator and sensorThe back-propagation algorithm was utilized for adaptingthe controller parameters In [12] Sangpet et al utilizeda fractional-order control approach to improve the delaymargin in the control systemof a piezoactuated flexible beamThe controller parameters were tuned experimentally Anet al [13] presented a time-delay acceleration feedback con-troller for vibration suppression of cantilever beams Stabilityboundaries of the closed-loop system were determined by aHurwitz stability criterion In [14] Takacs et al presented anadaptive-predictive vibration control system using extendedKalman filter The viability of the control method was

Hindawi Publishing CorporationJournal of EngineeringVolume 2014 Article ID 839128 9 pageshttpdxdoiorg1011552014839128

2 Journal of Engineering

experimentally evaluated In [15] Sangpet et al proposed anobserver-based hysteresis compensation scheme for trackingcontrol of a piezoactuated flexible beam The observer basedon a PI observer and a Kalman-filter algorithmwas employedfor estimation of the hysteresis in the beam The scheme wassuccessfully implemented in experimental tests

The governing equation of vibrating flexible structures isa partial differential equation (PDE) Thus the systems aredistributed-parameter systems and have an infinite numberof vibrationmodesMost of control design techniques exploitan approximated finite-dimensional model by ignoring thehigher frequency modes However this approach can causethe control system to become unstable due to a spillover effect[16 17]

Infinite-dimensional control for the systems modeledby PDEs has been recently investigated Extensions offinite-dimensional techniques to infinite-dimensional sys-tems as well as innovative infinite-dimensional specific con-trol design approaches have been proposed see [18] andreferences therein The Lyapunov-based control method hasplayed amajor role in nonlinear control of finite-dimensionalcomplex systems [19ndash22] Infinite-dimensional Lyapunov-based control is its extension to infinite-dimensional systemswhich has been successfully applied to many engineeringproblems [23ndash28]

In [23] Dadfarnia et al proposed infinite-dimensionalLyapunov-based control for a flexible Cartesian robot witha piezoelectric patch actuator attachment The robot wasmodeled as a flexible cantilever beam with a translationalbase support and a tip mass The controller was designedto exponentially suppress the vibration of the beam andregulate the base motion Coron et al [24] presented a strictLyapunov function for hyperbolic systems of conservationlaws The time derivative of this Lyapunov function can bemade strictly negative definite by an appropriate choice ofthe boundary conditions The method was illustrated by ahydraulic application Cheng et al [25] proposed sliding-mode boundary control of a one-dimensional unstableheat conduction system modeled by parabolic PDE systemswith parameter variations and boundary uncertainties Aninfinite-dimensional sliding manifold was constructed viaLyapunovrsquos direct method Simulation studies were con-ducted to verify the effectiveness of the sliding mode controllaw In [26] Shang and Xu proposed a new control strategyfor a one-dimensional cantilever Euler-Bernoulli beam withan input delay The control system is either exponentially orasymptotically stable depending on the time delay In [27]Guo and Jin presented an active disturbance rejection andsliding mode control approach for a stabilization problem ofEuler-Bernoulli beams with boundary input disturbanceThecontroller design is based on a PDE model In [28] Luem-chamloey and Kuntanapreeda presented an experimentalstudy of active vibration control of flexible beamsTheflexiblebeam was modeled by PDE The controller was designedbased on the infinite-dimensional Lyapunov method Heet al [29] considered vibration control of a flexible stringsystem With Lyapunovrsquos direct method adaptive boundarycontrol was developed to suppress the stringrsquos vibrationand the adaptive law was designed to compensate for the

systemparametric uncertainties Numerical simulationswerecarried out to verify the effectiveness of the controller

In addition high-order sliding-mode differentiation [3031] has been attracting interest It is an alternative differentia-tion that features robust exact differentiation with finite-timeconvergence It has been successfully used in control lawsreplacing the use of derivative measurements [32 33]

This paper presents an adaptive feedback control law tosuppress vibration of Euler-Bernoulli beamswith a collocatedpiezoelectric actuatorsensor pair The actuator and sensorare lead zirconate titanate (PZT) The filtered signal from thesensor is used to provide the feedback signal The governingequation of the beams is a PDE Thus it has an infinitenumber of vibration modes The controller is designed basedon an infinite-dimensional Lyapunov method which doesnot involve any approximated finite-dimensional modelsThus the stability of the control system including the filterdynamic is guaranteed for all vibration modes The high-order differentiation is used to estimate the derivative of thesensor signal and then used in the control law The rest ofthe paper is organized as follows Section 2 provides somepreliminaries Section 3 describes the flexible beam that isused as an experimental test bench Section 4 presents afinite elementmodel of the experimental beamTheproposedcontroller design is given in Section 5 Simulation and exper-imental results are presented in Section 6 The last sectionconcludes the paper

2 Preliminaries

21 System Modeling Consider a flexible cantilever beamwith piezoelectric actuator and sensor patches The patchesare bonded on the top and bottom surfaces of the beam asshown in Figure 1 A governing equation of the beam canbe derived based on an Euler-Bernoulli beam equation andHamiltonrsquos principle [19] as

(119909 119905) =

1

120588 (119909)

[119872119901119900119906 (119905) 119878

10158401015840(119909) minus

120597

2(119864119868 (119909)119908

10158401015840(119909 119905))

120597119909

2

minus 119861 (119909 119905) minus 119862

1015840(119909 119905)]

(1)

with the boundary conditions

119908 (0 119905) = (0 119905) = 119908

1015840(0 119905) = 0

119908

10158401015840(119871 119905) = 119908

101584010158401015840(119871 119905) = 0

(2)

where the dot and prime notations represent derivatives withrespect to time and the variable 119909 respectively 119908(119909 119905) is thetransverse displacement of the beam 119906(119905) is the input voltageapplied to the actuator 119871 is the length of the beam 120588(119909) =

119887119887120588119887119905119887 +119887119901120588119901119905119901119878(119909) 120588119887 120588119901 are the densities of the beam andthe piezoelectric patch actuator respectively 119887119887 119887119901 are thewidths of the beam and the actuator 119905119887 119905119901 are the thicknessesof the beam and the actuator 119878(119909) = 119867(119909 minus 1198971) minus 119867(119909 minus 1198972)

Journal of Engineering 3

L

Piezoelectric actuator

Piezoelectric sensor

x

w(L t)

l2

l1

w(x t)

Figure 1 Beam with a piezoelectric actuator and a piezoelectricsensor

119867(119909) is the Heaviside function 1198971 1198972 are the distances fromthe clamped end to the leading edge and the tailing edge ofthe actuator respectively 119872119901119900 = minus11988711990111986411990111988931(119905119887 + 119905119901) is themoment constant of the actuator119864119868(119909) = 119864119868119887+119864119868119901119878(119909) is thebending stiffness 119864119868119901 = 2119887119901119864119901119905119901(119905

2

1198874+ 1199051198871199051199012+ 119905

3

1199013) 119864119868119887 =

(1198641198871198871198871199053

119887)12 119864119887 119864119901 are the Youngrsquos moduli of the beam and

the actuator and 119861 119862 are the viscous and structural dampingcoefficients respectively

The equation relating the applied voltage 119906 and the axialstrain 120576119909119909 produced by the piezoelectric patch actuator iswritten as [8 11]

120576119909119909 = 11988931

119906

119905119901

(3)

where 11988931 and 119905119901 are the dielectric constant and the thicknessof the actuator respectively The normal stress 120590119909119909 and thebending moment 119872119886 on the beam produced by the actuatorare

120590119909119909 = 11986411990111988931

119906

119905119901

(4)

119872119886 = int

1199051198872+119905119901

1199051198872

120590119909119909119911 119889119860 = 11986411990111988931119887119901119911119906 (5)

where 119864119901 is Youngrsquos modulus of the actuator 119887119901 is the widthof the actuator 119911 = (119905119887 + 119905119901)2 and 119905119887 119905119901 are the thicknessesof the beam and the actuator

By considering the piezoelectric patch sensor as a parallelplate capacitor the voltage V119904(119905) across the two layers pro-duced by the sensor is given as [23]

V119904 (119905) = 120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905)) (6)

where 120595 = 11988711990111990411986411990111990411988931119904(119905119887 + 119905119901119904)2119862119901119904 is the piezoelectricconstant 119887119901119904 is the width of the sensor 119864119901119904 is Youngrsquosmodulus of the sensor 11988931119904 is the dielectric constant of thesensor 119905119901119904 is the thickness of the sensor and 119862119901119904 is thepiezoelectric capacitance

22 Finite Element Equation Consider a finite element (FE)model of a cantilever beam shown in Figure 2 It is assumedthat the beam is divided into 119899 elements Each elementhas two nodes and each node has two degrees of freedomtransverse direction (1199081 1199082) and slope (119908

1015840

1 119908

1015840

2)

The transverse displacement at any 119909 position on theelement can be expressed as [8 30]

[119908 (119909)] = [Φ]

119879[119902] (7)

n minus 1

Piezoelectric actuator

1 2 3 4 n

Piezoelectric sensor

Figure 2 Finite element model

where [119902]

119879= [1199081 119908

1015840

11199082 119908

1015840

2] is the nodal displacement

vector and [Φ] = [1206011 1206012 1206013 1206014] is the mode-shape vectorfunction By using the Lagrangian method the differentialequation of motion to be solved can be expressed as [34 35]

[119872] 119902 + [119862] 119902 + [119870] 119902 = 119865 (8)

where [119872] = 120588119860int

119871

0Φ

119879Φ119889119909 is the mass matrix [119870] =

119864119868 int

119871

0Φ

10158401015840119879Φ

10158401015840119889119909 is the stiffness matrix [119862] is the damping

matrix and 119865 is the forcematrix For Rayleigh proportionaldamping 119862 = 120572119872 + 120573119870 where 120572 120573 are the dampingconstants The force matrix 119865 for the element withoutthe piezoelectric actuator is 119865 = [0 0 0 0]

119879 and isfor the element bonded with the actuator patch 119865 =

[0 11986411990111988931119887119901119911 0 minus 11986411990111988931119887119901119911]119879119906 which is directly derived

from (5) Note that the piezoelectric sensor is not includedin the FE model since the bonded sensor in the experimentalsystem is very thin and light compared to those of the beamand the actuator

23 High-Order Sliding-Mode Differentiation In this sub-section a high-order sliding-mode differentiation methodis shortly described To explain the method let 119891(119905) be afunction from which we want to determine its derivativeConsider the following auxiliary system

= 120578 (9)

The main idea is to find 120578 to keep 119911 minus 119891(119905) = 0 resulting in120578 =

119891(119905) Based on a second-order sliding-mode algorithmone obtains [30]

120578 = 1205781 minus 1205831

1003816

1003816

1003816

1003816

119911 minus 119891(119905)

1003816

1003816

1003816

1003816

12 sign (119911 minus 119891 (119905))

1205781 = minus1205832 sign (119911 minus 119891 (119905))

(10)

where 1205831 1205832 gt 0 and sign (119909) is a signum function A generalform of 119899th-order differentiators can be written as

1205780 = 1205810 1205810 = minus1205830Γ1(119899+1) 1003816

1003816

1003816

1003816

1205780 minus 119891 (119905)

1003816

1003816

1003816

1003816

119899(119899+1)

times sign (1205780 minus 119891 (119905)) + 1205781

4 Journal of Engineering

Piezoelectricactuator

Piezoelectricsensor

Beam

Interface board Power amplifier

Computer

w(L t)

u(t)

(a)

Beam

Laser displacement sensor

Piezoelectric actuator

Power amplifier

Piezoelectric sensor

(b)

Figure 3 Experimental system (a) schematic and (b) photograph

1205781 = 1205811 1205811 = minus1205831Γ1119899 1003816

1003816

1003816

1003816

1205781 minus 1205810

1003816

1003816

1003816

1003816

(119899minus1)119899 sign (1205780 minus 1205810) + 1205782

120578119899minus1 = 120581119899minus1 120581119899minus1 = minus120583119899minus1Γ12 1003816

1003816

1003816

1003816

120578119899minus1 minus 120581119899minus2

1003816

1003816

1003816

1003816

12

times sign (120578119899minus1 minus 120581119899minus2) + 120578119899

120578119899 = minus120583119899Γ sign (120578119899 minus 120581119899minus1)

(11)

where 120578119899 is an approximated value of the 119899th derivative of thefunction 119891(119905) Γ gt 0 is a Lipschitz constant of the function119891(119905) and 120583119894 (119894 = 0 1 119899) are constant parameters Thereader is referred to Levant [30 31] for full details of themethod In this paper thismethodwill be used for estimatingthe first derivative of the signal from the piezoelectric sensor

3 Experimental System

A schematic and a photograph of the experimental systemused in this paper are shown in Figure 3 The system consistsof a flexible cantilever beam a piezoelectric patch actuatora piezoelectric patch sensor a high-voltage power amplifiera 16-bit ADCDAC interface board and a PC The beam ismade of aluminum The dimensions and properties of thebeam are summarized in Table 1 The piezoelectric actuatorand sensor are lead zirconate titanate (PZT)They are bondedon the beam at the distance 119909 = 15mm from the clampedpoint The dimensions and properties of the actuator andsensor are summarized in Table 2 The sensor measures theaxial strain of the beam The system is also equipped witha laser displacement sensor for measuring the beamrsquos tipdeflection however the displacement sensor is not used inthe control loop The signals from the sensors are fed backto the PC through the interface board The control signalfrom the computer is converted to an analogue signal by theinterface board and is transmitted through the high-voltagepower amplifier to the actuator to close the control loopThe gain of the amplifier is 150 Figure 4 displays the pulseresponse and the frequency response functions of the beamwhere the signal from the piezoelectric sensor is considered

Table 1 Beamrsquos dimensions and properties

Dimensionsproperties Symbol Value UnitYoungrsquos modulus 119864119887 70 GPaDensity 120588119887 2700 kgm3

Poisson ratio ] 035 mdashLength 119871 340 mmWidth 119887119887 15 mmThickness 119905119887 08 mm

Table 2 Piezoelectric materialrsquos dimensions and properties

Dimensionsproperties Symbol Value UnitActuatorYoungrsquos modulus 119864119901 6667 GPaDensity 120588119901 7800 kgm3

Length 119897119901 45 mmWidth 119887119901 15 mmThickness 119905119901 05 mmStrain coefficient 11988931 minus171eminus12 mV

Piezoelectric position 1198971 15 mm1198972 60 mm

SensorYoungrsquos modulus 119864119901119904 25 GPaLength 119897119901119904 45 mmWidth 119887119901119904 15 mmThickness 119905119901119904 05 mmPiezoelectric capacitance 119862119901119904 379eminus12 FaradStrain coefficient 11988931119904 23eminus12 mV

as the output The frequency response shows that the firsttwo natural frequencies of the beam are 3928 rads and2247 rads

4 Finite Element Model

A finite element (FE) model of the experimental beam wasdeveloped to be used in the control design process For

Journal of Engineering 5

0 05 1 15 2 25 3 35 4

0

01

02

Time (s)

minus05

minus04

minus03

minus02

minus01

s(t)

(V)

(a)

0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus30

minus20

minus10

103102101

(b)

Figure 4 Experimental systemrsquos responses (a) pulse response and (b) frequency response function

0 05 1 15 2 25 3 35 4

0

005

01

Time (s)

minus025

minus02

minus015

minus01

minus005

s(t)

(V)

(a)

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

103102101

(b)

Figure 5 Systemrsquos responses based on FE model (a) pulse response and (b) frequency response function

simplicity 119899 = 11 was chosen The pulse response and thefrequency response functions obtained from the developedFE model are shown in Figure 5 The responses agree withthose of the experimental system

5 Controller Design

The proposed control design is given in this section Thecontrol objective is to suppress the vibration of the beamby using only the signal from the piezoelectric sensor Theschematic for the control loop is shown in Figure 6

Instead of directly using V119904 and V119904 for feedback let V119904 andV119904 pass through a low-pass filter to obtain a new feedbacksignal 119910(119905) as follows

119910 (119905) =

1

120591

(1198861V119904 (119905) + 1198862V119904 (119905) minus 119910 (119905)) (12)

where 120591 gt 0 is the time constant of the filter and 1198861 1198862 ge

0 (1198861 +1198862 = 0) are weighting parameters Note that includinga low-pass filter in the control loop is usually done in practice

r(t)

y(t)

u(t)Controller Power amplifier BeamPiezoelectric

actuator

Piezoelectric sensor

+minus

Figure 6 Controlled system

to attenuate the effect of noise in the feedback signal In thefollowing the dynamic of the filter is also included in thecontroller design

Define a Lyapunov function candidate as

1198811 =1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(13)

6 Journal of Engineering

where 120582 is considered as a controller parameter By taking thetime derivative of (13) and substituting (1) (2) (6) and (12)into the derivative result it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) + 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)))

(14)

By choosing the following control law

119906 (119905) = minus

1205821198862120595119910

120591

minus

1205821198861120595119910

120591

(119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

(

1015840(1198972 119905) minus

1015840(1198971 119905))

(15)

it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(16)

By Barbalatrsquos lemma [36 37] 119910 rarr 0 is achieved Notethat 119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905) can be obtained directly from the

piezoelectric sensor (6) that is V119904(119905) = 120595(119908

1015840(1198972 119905) minus

119908

1015840(1198971 119905)) and

1015840(1198972 119905) minus

1015840(1198971 119905) can be obtained from V119904(119905)

In our experiment we used the high-order sliding-modedifferentiation method described in Section 23 to determineV119904(119905)

Note that control law (15) has a singularity at 1015840(1198972 119905) minus

1015840(1198971 119905) = 0 Thus implementations of the control law must

empirically avoid this singularity For simplicity we chose1198861 = 0 and 1198862 = 1 in the experiment to avoid the singularity

Moreover the parameter 120595 is usually uncertain or evenunknown Thus control law (15) with 1198861 = 0 and 1198862 = 1 ismodified as

119906 = minus

120582119896 (119905) 119910

120591

(17)

where 119896(119905) is an adaptive variable replacing 120595 Define 119890(119905) =

119896(119905) minus 120595 Take

119881 = 1198811 +1

2

119890

2

=

1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(18)

as a new Lyapunov function candidate Taking its derivativewith respect to time and substituting (17) into the result yields

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905)

+ (minus

120582119896 (119905) 119910

120591

) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

+ (119896 (119905) minus 120595)

119896 (119905)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ (

119896 (119905) minus

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

) (119896 (119905) minus 120595)

(19)

By choosing the following adaptive law

119896 (119905) =

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

(20)

it results in

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(21)

Applying Barbalatrsquos lemma [36 37] results in 119910 rarr 0

6 Results

61 Simulation Study A simulation study was conductedusing the FE model described in Section 22 to tune thecontrol parameter 120582 It was observed that the convergencerate of the controlled response increases with 120582 Here thecontrol gain of 120582 = 0052 was chosen Simulation results

Journal of Engineering 7

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 7 FEM simulation result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and(d) tip deflection

are shown in Figure 7 The controller was activated at 119905 =

1 sec The results show that the controller is able to suppressthe vibration effectively Note that 119896(119905) converged to a finitevalue

62 Experimental Results Controller (17) was implementeddigitally on the PC with the sampling period of the con-trol loop of 1msec The high-order sliding differentiationtechnique with Γ = 700 1205830 = 1205831 = 1 was used Theexperimental results are shown in Figure 8 The controllerprovides satisfactory responses similar to those obtained bythe simulation Note that the controller was activated at 119905 =

1 secNote that we set the initial conditions of the FEM sim-

ulation and experiment roughly the same for a comparisonpurpose It was found that there are slight differences in theFEM simulation and experimental results This is mainly dueto themeasurement noise and themismatch in the initial con-ditions The difference between the maximum control inputsis about 1208 The difference of the steady state adaptivegains is about 335 In summary both results are practicallycomparable

7 Conclusions

In this paper we proposed an adaptive controller for vibrationsuppression of flexible structures The structure under con-sideration is a cantilever beam with a collocated piezoelectricactuator and sensor pair The governing equation of thebeam is a partial differential equation Since the beam is aninfinite-dimensional system we derived the control law usingthe infinite-dimensional Lyapunov method which does notrequire any approximated finite-dimension model Thus thestability of the control system is guaranteed for all vibrationmodesWe also employed a high-order slidingmodedifferen-tiation technique to obtain the derivative of the sensorrsquos signalfor being used in the control law The technique featuresrobust exact differentiation with finite-time convergenceEffectiveness of the controller is illustrated both in finite-element model and experiment with results that show goodagreement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Journal of Engineering

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 8 Experimental result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and (d)tip deflection

Acknowledgments

This work was supported by the Royal Golden Jubilee PhDprogram of the Thailand Research Fund The authors alsowould like to thank anonymous referees for their valuablecomments

References

[1] Z Mohamed J M Martins M O Tokhi J Sa da Costa andM A Botto ldquoVibration control of a very flexible manipulatorsystemrdquo Control Engineering Practice vol 13 no 3 pp 267ndash2772005

[2] M Azadi S A Fazelzadeh M Eghtesad and E Azadi ldquoVibra-tion suppression and adaptive-robust control of a smart flexiblesatellite with three axes maneuveringrdquo Acta Astronautica vol69 no 5-6 pp 307ndash322 2011

[3] T P Sales D A Rade and L C G De Souza ldquoPassive vibra-tion control of flexible spacecraft using shunted piezoelectrictransducersrdquo Aerospace Science and Technology vol 29 no 1pp 403ndash412 2013

[4] C C Fuller Active Control of Vibration Academic Press 1996[5] A Preumont Vibration Control of Active Structures Kluwer

Academic 1997

[6] A M Aly ldquoVibration control of buildings using magnetorheo-logical damper a new control algorithmrdquo Journal of Engineer-ing vol 2013 Article ID 596078 10 pages 2013

[7] A Moutsopoulou G E Stavroulakis and A Pouliezos ldquoInno-vation in active vibration control strategy of intelligent struc-turesrdquo Journal of Applied Mathematics vol 2014 Article ID430731 14 pages 2014

[8] B Bandyopadhyay T C Manjunath and M Umapathy Mod-eling Control and Implementation of Smart Structures A FEM-State Space Approach Springer Berlin Germany 2007

[9] A R Tavakolpour M Mailah I Z Mat Darus and O TokhildquoSelf-learning active vibration control of a flexible plate struc-ture with piezoelectric actuatorrdquo Simulation Modelling Practiceand Theory vol 18 no 5 pp 516ndash532 2010

[10] H Wang X-H Shen L-X Zhang and X-J Zhu ldquoActivevibration suppression of flexible structure using a LMI-basedcontrol patchrdquo Journal of Vibration and Control vol 18 no 9pp 1375ndash1379 2012

[11] Z Qiu X Zhang and C Ye ldquoVibration suppression of a flexiblepiezoelectric beam using BP neural network controllerrdquo ActaMechanica Solida Sinica vol 25 no 4 pp 417ndash428 2012

[12] T Sangpet S Kuntanapreeda and R Schmidt ldquoImprovingdelay-margin of noncollocated vibration control of piezo-actuated flexible beams via a fractional-order controllerrdquo Shockand Vibration vol 2014 Article ID 809173 8 pages 2014

Journal of Engineering 9

[13] F An W Chen and M Shao ldquoDynamic behavior of time-delayed acceleration feedback controller for active vibrationcontrol of flexible structuresrdquo Journal of Sound and Vibrationvol 333 no 20 pp 4789ndash4809 2014

[14] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration control of a cantilever beam with real-timeparameter estimationrdquo Shock and Vibration vol 2014 ArticleID 741765 15 pages 2014

[15] T Sangpet S Kuntanapreeda and R Schmidt ldquoHystereticnonlinearity observer design based on Kalman filter for piezo-actuated flexible beamsrdquo International Journal of Automationand Computing vol 11 no 6 pp 627ndash634 2014

[16] X Dong Z Peng W Zhang H Hua and G Meng ldquoResearchon spillover effects for vibration control of piezoelectric smartstructures by ANSYSrdquo Mathematical Problems in Engineeringvol 2014 Article ID 870940 8 pages 2014

[17] A Montazeri J Poshtan and A Yousefi-Koma ldquoDesign andanalysis of robust minimax LQG controller for an experimentalbeam considering spill-over effectrdquo IEEE Transactions on Con-trol Systems Technology vol 19 no 5 pp 1251ndash1259 2011

[18] R Padhi and S F Ali ldquoAn account of chronological devel-opments in control of distributed parameter systemsrdquo AnnualReviews in Control vol 33 no 1 pp 59ndash68 2009

[19] S Zhao H Lin and Z Xue ldquoSwitching control of closedquantum systems via the Lyapunov methodrdquo Automatica vol48 no 8 pp 1833ndash1838 2012

[20] S Kuntanapreeda and P M Marusak ldquoNonlinear extendedoutput feedback control for CSTRs with van de Vusse reactionrdquoComputers amp Chemical Engineering vol 41 pp 10ndash23 2012

[21] B Zhang K Liu and J Xiang ldquoA stabilized optimal nonlinearfeedback control for satellite attitude trackingrdquo Aerospace Sci-ence and Technology vol 27 no 1 pp 17ndash24 2013

[22] A Yang W Naeem G W Irwin and K Li ldquoStability anal-ysis and implementation of a decentralized formation controlstrategy for unmanned vehiclesrdquo IEEE Transactions on ControlSystems Technology vol 22 no 2 pp 706ndash720 2014

[23] M Dadfarnia N Jalili B Xian and D M Dawson ldquoAlyapunov-based piezoelectric controller for flexible cartesianrobot manipulatorsrdquo Journal of Dynamic Systems Measurementand Control vol 126 no 2 pp 347ndash358 2004

[24] J-M Coron B drsquoAndrea-Novel and G Bastin ldquoA strict Lya-punov function for boundary control of hyperbolic systems ofconservation lawsrdquo IEEE Transactions on Automatic Controlvol 52 no 1 pp 2ndash11 2007

[25] M-B Cheng V Radisavljevic and W-C Su ldquoSliding modeboundary control of a parabolic PDE system with parametervariations and boundary uncertaintiesrdquoAutomatica vol 47 no2 pp 381ndash387 2011

[26] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems andControl Letters vol 61 no 11 pp 1069ndash1078 2012

[27] B-Z Guo and F-F Jin ldquoThe active disturbance rejection andsliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbancerdquoAutomatica vol 49 no 9 pp 2911ndash2918 2013

[28] A Luemchamloey and S Kuntanapreeda ldquoActive vibrationcontrol of flexible beambased on infinite-dimensional lyapunovstability theory an experimental studyrdquo Journal of ControlAutomation and Electrical Systems vol 25 no 6 pp 649ndash6562014

[29] W He S Zhang and S S Ge ldquoAdaptive boundary control of anonlinear flexible string systemrdquo IEEE Transactions on ControlSystems Technology vol 22 no 3 pp 1088ndash1093 2014

[30] A Levant ldquoRobust exact differentiation via sliding modetechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998

[31] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[32] XWang and B Shirinzadeh ldquoHigh-order nonlinear differentia-tor and application to aircraft controlrdquoMechanical Systems andSignal Processing vol 46 no 2 pp 227ndash252 2014

[33] I Salgado I Chairez O Camacho and C Yanez ldquoSuper-twisting sliding mode differentiation for improving PD con-trollers performance of second order systemrdquo ISA Transactionsvol 53 no 4 pp 1096ndash1106 2014

[34] S Y Wang ldquoA finite element model for the static and dynamicanalysis of a piezoelectric bimorphrdquo International Journal ofSolids and Structures vol 41 no 15 pp 4075ndash4096 2004

[35] D L Logan A First Course in the Finite Element MethodWadsworth Group 2002

[36] J E Slotine Applied Nonlinear Control Prentice-Hall 1991[37] H K Khalil Nonlinear Systems Prentice-Hall 1996

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

2 Journal of Engineering

experimentally evaluated In [15] Sangpet et al proposed anobserver-based hysteresis compensation scheme for trackingcontrol of a piezoactuated flexible beam The observer basedon a PI observer and a Kalman-filter algorithmwas employedfor estimation of the hysteresis in the beam The scheme wassuccessfully implemented in experimental tests

The governing equation of vibrating flexible structures isa partial differential equation (PDE) Thus the systems aredistributed-parameter systems and have an infinite numberof vibrationmodesMost of control design techniques exploitan approximated finite-dimensional model by ignoring thehigher frequency modes However this approach can causethe control system to become unstable due to a spillover effect[16 17]

Infinite-dimensional control for the systems modeledby PDEs has been recently investigated Extensions offinite-dimensional techniques to infinite-dimensional sys-tems as well as innovative infinite-dimensional specific con-trol design approaches have been proposed see [18] andreferences therein The Lyapunov-based control method hasplayed amajor role in nonlinear control of finite-dimensionalcomplex systems [19ndash22] Infinite-dimensional Lyapunov-based control is its extension to infinite-dimensional systemswhich has been successfully applied to many engineeringproblems [23ndash28]

In [23] Dadfarnia et al proposed infinite-dimensionalLyapunov-based control for a flexible Cartesian robot witha piezoelectric patch actuator attachment The robot wasmodeled as a flexible cantilever beam with a translationalbase support and a tip mass The controller was designedto exponentially suppress the vibration of the beam andregulate the base motion Coron et al [24] presented a strictLyapunov function for hyperbolic systems of conservationlaws The time derivative of this Lyapunov function can bemade strictly negative definite by an appropriate choice ofthe boundary conditions The method was illustrated by ahydraulic application Cheng et al [25] proposed sliding-mode boundary control of a one-dimensional unstableheat conduction system modeled by parabolic PDE systemswith parameter variations and boundary uncertainties Aninfinite-dimensional sliding manifold was constructed viaLyapunovrsquos direct method Simulation studies were con-ducted to verify the effectiveness of the sliding mode controllaw In [26] Shang and Xu proposed a new control strategyfor a one-dimensional cantilever Euler-Bernoulli beam withan input delay The control system is either exponentially orasymptotically stable depending on the time delay In [27]Guo and Jin presented an active disturbance rejection andsliding mode control approach for a stabilization problem ofEuler-Bernoulli beams with boundary input disturbanceThecontroller design is based on a PDE model In [28] Luem-chamloey and Kuntanapreeda presented an experimentalstudy of active vibration control of flexible beamsTheflexiblebeam was modeled by PDE The controller was designedbased on the infinite-dimensional Lyapunov method Heet al [29] considered vibration control of a flexible stringsystem With Lyapunovrsquos direct method adaptive boundarycontrol was developed to suppress the stringrsquos vibrationand the adaptive law was designed to compensate for the

systemparametric uncertainties Numerical simulationswerecarried out to verify the effectiveness of the controller

In addition high-order sliding-mode differentiation [3031] has been attracting interest It is an alternative differentia-tion that features robust exact differentiation with finite-timeconvergence It has been successfully used in control lawsreplacing the use of derivative measurements [32 33]

This paper presents an adaptive feedback control law tosuppress vibration of Euler-Bernoulli beamswith a collocatedpiezoelectric actuatorsensor pair The actuator and sensorare lead zirconate titanate (PZT) The filtered signal from thesensor is used to provide the feedback signal The governingequation of the beams is a PDE Thus it has an infinitenumber of vibration modes The controller is designed basedon an infinite-dimensional Lyapunov method which doesnot involve any approximated finite-dimensional modelsThus the stability of the control system including the filterdynamic is guaranteed for all vibration modes The high-order differentiation is used to estimate the derivative of thesensor signal and then used in the control law The rest ofthe paper is organized as follows Section 2 provides somepreliminaries Section 3 describes the flexible beam that isused as an experimental test bench Section 4 presents afinite elementmodel of the experimental beamTheproposedcontroller design is given in Section 5 Simulation and exper-imental results are presented in Section 6 The last sectionconcludes the paper

2 Preliminaries

21 System Modeling Consider a flexible cantilever beamwith piezoelectric actuator and sensor patches The patchesare bonded on the top and bottom surfaces of the beam asshown in Figure 1 A governing equation of the beam canbe derived based on an Euler-Bernoulli beam equation andHamiltonrsquos principle [19] as

(119909 119905) =

1

120588 (119909)

[119872119901119900119906 (119905) 119878

10158401015840(119909) minus

120597

2(119864119868 (119909)119908

10158401015840(119909 119905))

120597119909

2

minus 119861 (119909 119905) minus 119862

1015840(119909 119905)]

(1)

with the boundary conditions

119908 (0 119905) = (0 119905) = 119908

1015840(0 119905) = 0

119908

10158401015840(119871 119905) = 119908

101584010158401015840(119871 119905) = 0

(2)

where the dot and prime notations represent derivatives withrespect to time and the variable 119909 respectively 119908(119909 119905) is thetransverse displacement of the beam 119906(119905) is the input voltageapplied to the actuator 119871 is the length of the beam 120588(119909) =

119887119887120588119887119905119887 +119887119901120588119901119905119901119878(119909) 120588119887 120588119901 are the densities of the beam andthe piezoelectric patch actuator respectively 119887119887 119887119901 are thewidths of the beam and the actuator 119905119887 119905119901 are the thicknessesof the beam and the actuator 119878(119909) = 119867(119909 minus 1198971) minus 119867(119909 minus 1198972)

Journal of Engineering 3

L

Piezoelectric actuator

Piezoelectric sensor

x

w(L t)

l2

l1

w(x t)

Figure 1 Beam with a piezoelectric actuator and a piezoelectricsensor

119867(119909) is the Heaviside function 1198971 1198972 are the distances fromthe clamped end to the leading edge and the tailing edge ofthe actuator respectively 119872119901119900 = minus11988711990111986411990111988931(119905119887 + 119905119901) is themoment constant of the actuator119864119868(119909) = 119864119868119887+119864119868119901119878(119909) is thebending stiffness 119864119868119901 = 2119887119901119864119901119905119901(119905

2

1198874+ 1199051198871199051199012+ 119905

3

1199013) 119864119868119887 =

(1198641198871198871198871199053

119887)12 119864119887 119864119901 are the Youngrsquos moduli of the beam and

the actuator and 119861 119862 are the viscous and structural dampingcoefficients respectively

The equation relating the applied voltage 119906 and the axialstrain 120576119909119909 produced by the piezoelectric patch actuator iswritten as [8 11]

120576119909119909 = 11988931

119906

119905119901

(3)

where 11988931 and 119905119901 are the dielectric constant and the thicknessof the actuator respectively The normal stress 120590119909119909 and thebending moment 119872119886 on the beam produced by the actuatorare

120590119909119909 = 11986411990111988931

119906

119905119901

(4)

119872119886 = int

1199051198872+119905119901

1199051198872

120590119909119909119911 119889119860 = 11986411990111988931119887119901119911119906 (5)

where 119864119901 is Youngrsquos modulus of the actuator 119887119901 is the widthof the actuator 119911 = (119905119887 + 119905119901)2 and 119905119887 119905119901 are the thicknessesof the beam and the actuator

By considering the piezoelectric patch sensor as a parallelplate capacitor the voltage V119904(119905) across the two layers pro-duced by the sensor is given as [23]

V119904 (119905) = 120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905)) (6)

where 120595 = 11988711990111990411986411990111990411988931119904(119905119887 + 119905119901119904)2119862119901119904 is the piezoelectricconstant 119887119901119904 is the width of the sensor 119864119901119904 is Youngrsquosmodulus of the sensor 11988931119904 is the dielectric constant of thesensor 119905119901119904 is the thickness of the sensor and 119862119901119904 is thepiezoelectric capacitance

22 Finite Element Equation Consider a finite element (FE)model of a cantilever beam shown in Figure 2 It is assumedthat the beam is divided into 119899 elements Each elementhas two nodes and each node has two degrees of freedomtransverse direction (1199081 1199082) and slope (119908

1015840

1 119908

1015840

2)

The transverse displacement at any 119909 position on theelement can be expressed as [8 30]

[119908 (119909)] = [Φ]

119879[119902] (7)

n minus 1

Piezoelectric actuator

1 2 3 4 n

Piezoelectric sensor

Figure 2 Finite element model

where [119902]

119879= [1199081 119908

1015840

11199082 119908

1015840

2] is the nodal displacement

vector and [Φ] = [1206011 1206012 1206013 1206014] is the mode-shape vectorfunction By using the Lagrangian method the differentialequation of motion to be solved can be expressed as [34 35]

[119872] 119902 + [119862] 119902 + [119870] 119902 = 119865 (8)

where [119872] = 120588119860int

119871

0Φ

119879Φ119889119909 is the mass matrix [119870] =

119864119868 int

119871

0Φ

10158401015840119879Φ

10158401015840119889119909 is the stiffness matrix [119862] is the damping

matrix and 119865 is the forcematrix For Rayleigh proportionaldamping 119862 = 120572119872 + 120573119870 where 120572 120573 are the dampingconstants The force matrix 119865 for the element withoutthe piezoelectric actuator is 119865 = [0 0 0 0]

119879 and isfor the element bonded with the actuator patch 119865 =

[0 11986411990111988931119887119901119911 0 minus 11986411990111988931119887119901119911]119879119906 which is directly derived

from (5) Note that the piezoelectric sensor is not includedin the FE model since the bonded sensor in the experimentalsystem is very thin and light compared to those of the beamand the actuator

23 High-Order Sliding-Mode Differentiation In this sub-section a high-order sliding-mode differentiation methodis shortly described To explain the method let 119891(119905) be afunction from which we want to determine its derivativeConsider the following auxiliary system

= 120578 (9)

The main idea is to find 120578 to keep 119911 minus 119891(119905) = 0 resulting in120578 =

119891(119905) Based on a second-order sliding-mode algorithmone obtains [30]

120578 = 1205781 minus 1205831

1003816

1003816

1003816

1003816

119911 minus 119891(119905)

1003816

1003816

1003816

1003816

12 sign (119911 minus 119891 (119905))

1205781 = minus1205832 sign (119911 minus 119891 (119905))

(10)

where 1205831 1205832 gt 0 and sign (119909) is a signum function A generalform of 119899th-order differentiators can be written as

1205780 = 1205810 1205810 = minus1205830Γ1(119899+1) 1003816

1003816

1003816

1003816

1205780 minus 119891 (119905)

1003816

1003816

1003816

1003816

119899(119899+1)

times sign (1205780 minus 119891 (119905)) + 1205781

4 Journal of Engineering

Piezoelectricactuator

Piezoelectricsensor

Beam

Interface board Power amplifier

Computer

w(L t)

u(t)

(a)

Beam

Laser displacement sensor

Piezoelectric actuator

Power amplifier

Piezoelectric sensor

(b)

Figure 3 Experimental system (a) schematic and (b) photograph

1205781 = 1205811 1205811 = minus1205831Γ1119899 1003816

1003816

1003816

1003816

1205781 minus 1205810

1003816

1003816

1003816

1003816

(119899minus1)119899 sign (1205780 minus 1205810) + 1205782

120578119899minus1 = 120581119899minus1 120581119899minus1 = minus120583119899minus1Γ12 1003816

1003816

1003816

1003816

120578119899minus1 minus 120581119899minus2

1003816

1003816

1003816

1003816

12

times sign (120578119899minus1 minus 120581119899minus2) + 120578119899

120578119899 = minus120583119899Γ sign (120578119899 minus 120581119899minus1)

(11)

where 120578119899 is an approximated value of the 119899th derivative of thefunction 119891(119905) Γ gt 0 is a Lipschitz constant of the function119891(119905) and 120583119894 (119894 = 0 1 119899) are constant parameters Thereader is referred to Levant [30 31] for full details of themethod In this paper thismethodwill be used for estimatingthe first derivative of the signal from the piezoelectric sensor

3 Experimental System

A schematic and a photograph of the experimental systemused in this paper are shown in Figure 3 The system consistsof a flexible cantilever beam a piezoelectric patch actuatora piezoelectric patch sensor a high-voltage power amplifiera 16-bit ADCDAC interface board and a PC The beam ismade of aluminum The dimensions and properties of thebeam are summarized in Table 1 The piezoelectric actuatorand sensor are lead zirconate titanate (PZT)They are bondedon the beam at the distance 119909 = 15mm from the clampedpoint The dimensions and properties of the actuator andsensor are summarized in Table 2 The sensor measures theaxial strain of the beam The system is also equipped witha laser displacement sensor for measuring the beamrsquos tipdeflection however the displacement sensor is not used inthe control loop The signals from the sensors are fed backto the PC through the interface board The control signalfrom the computer is converted to an analogue signal by theinterface board and is transmitted through the high-voltagepower amplifier to the actuator to close the control loopThe gain of the amplifier is 150 Figure 4 displays the pulseresponse and the frequency response functions of the beamwhere the signal from the piezoelectric sensor is considered

Table 1 Beamrsquos dimensions and properties

Dimensionsproperties Symbol Value UnitYoungrsquos modulus 119864119887 70 GPaDensity 120588119887 2700 kgm3

Poisson ratio ] 035 mdashLength 119871 340 mmWidth 119887119887 15 mmThickness 119905119887 08 mm

Table 2 Piezoelectric materialrsquos dimensions and properties

Dimensionsproperties Symbol Value UnitActuatorYoungrsquos modulus 119864119901 6667 GPaDensity 120588119901 7800 kgm3

Length 119897119901 45 mmWidth 119887119901 15 mmThickness 119905119901 05 mmStrain coefficient 11988931 minus171eminus12 mV

Piezoelectric position 1198971 15 mm1198972 60 mm

SensorYoungrsquos modulus 119864119901119904 25 GPaLength 119897119901119904 45 mmWidth 119887119901119904 15 mmThickness 119905119901119904 05 mmPiezoelectric capacitance 119862119901119904 379eminus12 FaradStrain coefficient 11988931119904 23eminus12 mV

as the output The frequency response shows that the firsttwo natural frequencies of the beam are 3928 rads and2247 rads

4 Finite Element Model

A finite element (FE) model of the experimental beam wasdeveloped to be used in the control design process For

Journal of Engineering 5

0 05 1 15 2 25 3 35 4

0

01

02

Time (s)

minus05

minus04

minus03

minus02

minus01

s(t)

(V)

(a)

0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus30

minus20

minus10

103102101

(b)

Figure 4 Experimental systemrsquos responses (a) pulse response and (b) frequency response function

0 05 1 15 2 25 3 35 4

0

005

01

Time (s)

minus025

minus02

minus015

minus01

minus005

s(t)

(V)

(a)

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

103102101

(b)

Figure 5 Systemrsquos responses based on FE model (a) pulse response and (b) frequency response function

simplicity 119899 = 11 was chosen The pulse response and thefrequency response functions obtained from the developedFE model are shown in Figure 5 The responses agree withthose of the experimental system

5 Controller Design

The proposed control design is given in this section Thecontrol objective is to suppress the vibration of the beamby using only the signal from the piezoelectric sensor Theschematic for the control loop is shown in Figure 6

Instead of directly using V119904 and V119904 for feedback let V119904 andV119904 pass through a low-pass filter to obtain a new feedbacksignal 119910(119905) as follows

119910 (119905) =

1

120591

(1198861V119904 (119905) + 1198862V119904 (119905) minus 119910 (119905)) (12)

where 120591 gt 0 is the time constant of the filter and 1198861 1198862 ge

0 (1198861 +1198862 = 0) are weighting parameters Note that includinga low-pass filter in the control loop is usually done in practice

r(t)

y(t)

u(t)Controller Power amplifier BeamPiezoelectric

actuator

Piezoelectric sensor

+minus

Figure 6 Controlled system

to attenuate the effect of noise in the feedback signal In thefollowing the dynamic of the filter is also included in thecontroller design

Define a Lyapunov function candidate as

1198811 =1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(13)

6 Journal of Engineering

where 120582 is considered as a controller parameter By taking thetime derivative of (13) and substituting (1) (2) (6) and (12)into the derivative result it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) + 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)))

(14)

By choosing the following control law

119906 (119905) = minus

1205821198862120595119910

120591

minus

1205821198861120595119910

120591

(119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

(

1015840(1198972 119905) minus

1015840(1198971 119905))

(15)

it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(16)

By Barbalatrsquos lemma [36 37] 119910 rarr 0 is achieved Notethat 119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905) can be obtained directly from the

piezoelectric sensor (6) that is V119904(119905) = 120595(119908

1015840(1198972 119905) minus

119908

1015840(1198971 119905)) and

1015840(1198972 119905) minus

1015840(1198971 119905) can be obtained from V119904(119905)

In our experiment we used the high-order sliding-modedifferentiation method described in Section 23 to determineV119904(119905)

Note that control law (15) has a singularity at 1015840(1198972 119905) minus

1015840(1198971 119905) = 0 Thus implementations of the control law must

empirically avoid this singularity For simplicity we chose1198861 = 0 and 1198862 = 1 in the experiment to avoid the singularity

Moreover the parameter 120595 is usually uncertain or evenunknown Thus control law (15) with 1198861 = 0 and 1198862 = 1 ismodified as

119906 = minus

120582119896 (119905) 119910

120591

(17)

where 119896(119905) is an adaptive variable replacing 120595 Define 119890(119905) =

119896(119905) minus 120595 Take

119881 = 1198811 +1

2

119890

2

=

1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(18)

as a new Lyapunov function candidate Taking its derivativewith respect to time and substituting (17) into the result yields

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905)

+ (minus

120582119896 (119905) 119910

120591

) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

+ (119896 (119905) minus 120595)

119896 (119905)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ (

119896 (119905) minus

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

) (119896 (119905) minus 120595)

(19)

By choosing the following adaptive law

119896 (119905) =

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

(20)

it results in

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(21)

Applying Barbalatrsquos lemma [36 37] results in 119910 rarr 0

6 Results

61 Simulation Study A simulation study was conductedusing the FE model described in Section 22 to tune thecontrol parameter 120582 It was observed that the convergencerate of the controlled response increases with 120582 Here thecontrol gain of 120582 = 0052 was chosen Simulation results

Journal of Engineering 7

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 7 FEM simulation result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and(d) tip deflection

are shown in Figure 7 The controller was activated at 119905 =

1 sec The results show that the controller is able to suppressthe vibration effectively Note that 119896(119905) converged to a finitevalue

62 Experimental Results Controller (17) was implementeddigitally on the PC with the sampling period of the con-trol loop of 1msec The high-order sliding differentiationtechnique with Γ = 700 1205830 = 1205831 = 1 was used Theexperimental results are shown in Figure 8 The controllerprovides satisfactory responses similar to those obtained bythe simulation Note that the controller was activated at 119905 =

1 secNote that we set the initial conditions of the FEM sim-

ulation and experiment roughly the same for a comparisonpurpose It was found that there are slight differences in theFEM simulation and experimental results This is mainly dueto themeasurement noise and themismatch in the initial con-ditions The difference between the maximum control inputsis about 1208 The difference of the steady state adaptivegains is about 335 In summary both results are practicallycomparable

7 Conclusions

In this paper we proposed an adaptive controller for vibrationsuppression of flexible structures The structure under con-sideration is a cantilever beam with a collocated piezoelectricactuator and sensor pair The governing equation of thebeam is a partial differential equation Since the beam is aninfinite-dimensional system we derived the control law usingthe infinite-dimensional Lyapunov method which does notrequire any approximated finite-dimension model Thus thestability of the control system is guaranteed for all vibrationmodesWe also employed a high-order slidingmodedifferen-tiation technique to obtain the derivative of the sensorrsquos signalfor being used in the control law The technique featuresrobust exact differentiation with finite-time convergenceEffectiveness of the controller is illustrated both in finite-element model and experiment with results that show goodagreement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Journal of Engineering

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 8 Experimental result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and (d)tip deflection

Acknowledgments

This work was supported by the Royal Golden Jubilee PhDprogram of the Thailand Research Fund The authors alsowould like to thank anonymous referees for their valuablecomments

References

[1] Z Mohamed J M Martins M O Tokhi J Sa da Costa andM A Botto ldquoVibration control of a very flexible manipulatorsystemrdquo Control Engineering Practice vol 13 no 3 pp 267ndash2772005

[2] M Azadi S A Fazelzadeh M Eghtesad and E Azadi ldquoVibra-tion suppression and adaptive-robust control of a smart flexiblesatellite with three axes maneuveringrdquo Acta Astronautica vol69 no 5-6 pp 307ndash322 2011

[3] T P Sales D A Rade and L C G De Souza ldquoPassive vibra-tion control of flexible spacecraft using shunted piezoelectrictransducersrdquo Aerospace Science and Technology vol 29 no 1pp 403ndash412 2013

[4] C C Fuller Active Control of Vibration Academic Press 1996[5] A Preumont Vibration Control of Active Structures Kluwer

Academic 1997

[6] A M Aly ldquoVibration control of buildings using magnetorheo-logical damper a new control algorithmrdquo Journal of Engineer-ing vol 2013 Article ID 596078 10 pages 2013

[7] A Moutsopoulou G E Stavroulakis and A Pouliezos ldquoInno-vation in active vibration control strategy of intelligent struc-turesrdquo Journal of Applied Mathematics vol 2014 Article ID430731 14 pages 2014

[8] B Bandyopadhyay T C Manjunath and M Umapathy Mod-eling Control and Implementation of Smart Structures A FEM-State Space Approach Springer Berlin Germany 2007

[9] A R Tavakolpour M Mailah I Z Mat Darus and O TokhildquoSelf-learning active vibration control of a flexible plate struc-ture with piezoelectric actuatorrdquo Simulation Modelling Practiceand Theory vol 18 no 5 pp 516ndash532 2010

[10] H Wang X-H Shen L-X Zhang and X-J Zhu ldquoActivevibration suppression of flexible structure using a LMI-basedcontrol patchrdquo Journal of Vibration and Control vol 18 no 9pp 1375ndash1379 2012

[11] Z Qiu X Zhang and C Ye ldquoVibration suppression of a flexiblepiezoelectric beam using BP neural network controllerrdquo ActaMechanica Solida Sinica vol 25 no 4 pp 417ndash428 2012

[12] T Sangpet S Kuntanapreeda and R Schmidt ldquoImprovingdelay-margin of noncollocated vibration control of piezo-actuated flexible beams via a fractional-order controllerrdquo Shockand Vibration vol 2014 Article ID 809173 8 pages 2014

Journal of Engineering 9

[13] F An W Chen and M Shao ldquoDynamic behavior of time-delayed acceleration feedback controller for active vibrationcontrol of flexible structuresrdquo Journal of Sound and Vibrationvol 333 no 20 pp 4789ndash4809 2014

[14] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration control of a cantilever beam with real-timeparameter estimationrdquo Shock and Vibration vol 2014 ArticleID 741765 15 pages 2014

[15] T Sangpet S Kuntanapreeda and R Schmidt ldquoHystereticnonlinearity observer design based on Kalman filter for piezo-actuated flexible beamsrdquo International Journal of Automationand Computing vol 11 no 6 pp 627ndash634 2014

[16] X Dong Z Peng W Zhang H Hua and G Meng ldquoResearchon spillover effects for vibration control of piezoelectric smartstructures by ANSYSrdquo Mathematical Problems in Engineeringvol 2014 Article ID 870940 8 pages 2014

[17] A Montazeri J Poshtan and A Yousefi-Koma ldquoDesign andanalysis of robust minimax LQG controller for an experimentalbeam considering spill-over effectrdquo IEEE Transactions on Con-trol Systems Technology vol 19 no 5 pp 1251ndash1259 2011

[18] R Padhi and S F Ali ldquoAn account of chronological devel-opments in control of distributed parameter systemsrdquo AnnualReviews in Control vol 33 no 1 pp 59ndash68 2009

[19] S Zhao H Lin and Z Xue ldquoSwitching control of closedquantum systems via the Lyapunov methodrdquo Automatica vol48 no 8 pp 1833ndash1838 2012

[20] S Kuntanapreeda and P M Marusak ldquoNonlinear extendedoutput feedback control for CSTRs with van de Vusse reactionrdquoComputers amp Chemical Engineering vol 41 pp 10ndash23 2012

[21] B Zhang K Liu and J Xiang ldquoA stabilized optimal nonlinearfeedback control for satellite attitude trackingrdquo Aerospace Sci-ence and Technology vol 27 no 1 pp 17ndash24 2013

[22] A Yang W Naeem G W Irwin and K Li ldquoStability anal-ysis and implementation of a decentralized formation controlstrategy for unmanned vehiclesrdquo IEEE Transactions on ControlSystems Technology vol 22 no 2 pp 706ndash720 2014

[23] M Dadfarnia N Jalili B Xian and D M Dawson ldquoAlyapunov-based piezoelectric controller for flexible cartesianrobot manipulatorsrdquo Journal of Dynamic Systems Measurementand Control vol 126 no 2 pp 347ndash358 2004

[24] J-M Coron B drsquoAndrea-Novel and G Bastin ldquoA strict Lya-punov function for boundary control of hyperbolic systems ofconservation lawsrdquo IEEE Transactions on Automatic Controlvol 52 no 1 pp 2ndash11 2007

[25] M-B Cheng V Radisavljevic and W-C Su ldquoSliding modeboundary control of a parabolic PDE system with parametervariations and boundary uncertaintiesrdquoAutomatica vol 47 no2 pp 381ndash387 2011

[26] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems andControl Letters vol 61 no 11 pp 1069ndash1078 2012

[27] B-Z Guo and F-F Jin ldquoThe active disturbance rejection andsliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbancerdquoAutomatica vol 49 no 9 pp 2911ndash2918 2013

[28] A Luemchamloey and S Kuntanapreeda ldquoActive vibrationcontrol of flexible beambased on infinite-dimensional lyapunovstability theory an experimental studyrdquo Journal of ControlAutomation and Electrical Systems vol 25 no 6 pp 649ndash6562014

[29] W He S Zhang and S S Ge ldquoAdaptive boundary control of anonlinear flexible string systemrdquo IEEE Transactions on ControlSystems Technology vol 22 no 3 pp 1088ndash1093 2014

[30] A Levant ldquoRobust exact differentiation via sliding modetechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998

[31] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[32] XWang and B Shirinzadeh ldquoHigh-order nonlinear differentia-tor and application to aircraft controlrdquoMechanical Systems andSignal Processing vol 46 no 2 pp 227ndash252 2014

[33] I Salgado I Chairez O Camacho and C Yanez ldquoSuper-twisting sliding mode differentiation for improving PD con-trollers performance of second order systemrdquo ISA Transactionsvol 53 no 4 pp 1096ndash1106 2014

[34] S Y Wang ldquoA finite element model for the static and dynamicanalysis of a piezoelectric bimorphrdquo International Journal ofSolids and Structures vol 41 no 15 pp 4075ndash4096 2004

[35] D L Logan A First Course in the Finite Element MethodWadsworth Group 2002

[36] J E Slotine Applied Nonlinear Control Prentice-Hall 1991[37] H K Khalil Nonlinear Systems Prentice-Hall 1996

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

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Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

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SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 3

L

Piezoelectric actuator

Piezoelectric sensor

x

w(L t)

l2

l1

w(x t)

Figure 1 Beam with a piezoelectric actuator and a piezoelectricsensor

119867(119909) is the Heaviside function 1198971 1198972 are the distances fromthe clamped end to the leading edge and the tailing edge ofthe actuator respectively 119872119901119900 = minus11988711990111986411990111988931(119905119887 + 119905119901) is themoment constant of the actuator119864119868(119909) = 119864119868119887+119864119868119901119878(119909) is thebending stiffness 119864119868119901 = 2119887119901119864119901119905119901(119905

2

1198874+ 1199051198871199051199012+ 119905

3

1199013) 119864119868119887 =

(1198641198871198871198871199053

119887)12 119864119887 119864119901 are the Youngrsquos moduli of the beam and

the actuator and 119861 119862 are the viscous and structural dampingcoefficients respectively

The equation relating the applied voltage 119906 and the axialstrain 120576119909119909 produced by the piezoelectric patch actuator iswritten as [8 11]

120576119909119909 = 11988931

119906

119905119901

(3)

where 11988931 and 119905119901 are the dielectric constant and the thicknessof the actuator respectively The normal stress 120590119909119909 and thebending moment 119872119886 on the beam produced by the actuatorare

120590119909119909 = 11986411990111988931

119906

119905119901

(4)

119872119886 = int

1199051198872+119905119901

1199051198872

120590119909119909119911 119889119860 = 11986411990111988931119887119901119911119906 (5)

where 119864119901 is Youngrsquos modulus of the actuator 119887119901 is the widthof the actuator 119911 = (119905119887 + 119905119901)2 and 119905119887 119905119901 are the thicknessesof the beam and the actuator

By considering the piezoelectric patch sensor as a parallelplate capacitor the voltage V119904(119905) across the two layers pro-duced by the sensor is given as [23]

V119904 (119905) = 120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905)) (6)

where 120595 = 11988711990111990411986411990111990411988931119904(119905119887 + 119905119901119904)2119862119901119904 is the piezoelectricconstant 119887119901119904 is the width of the sensor 119864119901119904 is Youngrsquosmodulus of the sensor 11988931119904 is the dielectric constant of thesensor 119905119901119904 is the thickness of the sensor and 119862119901119904 is thepiezoelectric capacitance

22 Finite Element Equation Consider a finite element (FE)model of a cantilever beam shown in Figure 2 It is assumedthat the beam is divided into 119899 elements Each elementhas two nodes and each node has two degrees of freedomtransverse direction (1199081 1199082) and slope (119908

1015840

1 119908

1015840

2)

The transverse displacement at any 119909 position on theelement can be expressed as [8 30]

[119908 (119909)] = [Φ]

119879[119902] (7)

n minus 1

Piezoelectric actuator

1 2 3 4 n

Piezoelectric sensor

Figure 2 Finite element model

where [119902]

119879= [1199081 119908

1015840

11199082 119908

1015840

2] is the nodal displacement

vector and [Φ] = [1206011 1206012 1206013 1206014] is the mode-shape vectorfunction By using the Lagrangian method the differentialequation of motion to be solved can be expressed as [34 35]

[119872] 119902 + [119862] 119902 + [119870] 119902 = 119865 (8)

where [119872] = 120588119860int

119871

0Φ

119879Φ119889119909 is the mass matrix [119870] =

119864119868 int

119871

0Φ

10158401015840119879Φ

10158401015840119889119909 is the stiffness matrix [119862] is the damping

matrix and 119865 is the forcematrix For Rayleigh proportionaldamping 119862 = 120572119872 + 120573119870 where 120572 120573 are the dampingconstants The force matrix 119865 for the element withoutthe piezoelectric actuator is 119865 = [0 0 0 0]

119879 and isfor the element bonded with the actuator patch 119865 =

[0 11986411990111988931119887119901119911 0 minus 11986411990111988931119887119901119911]119879119906 which is directly derived

from (5) Note that the piezoelectric sensor is not includedin the FE model since the bonded sensor in the experimentalsystem is very thin and light compared to those of the beamand the actuator

23 High-Order Sliding-Mode Differentiation In this sub-section a high-order sliding-mode differentiation methodis shortly described To explain the method let 119891(119905) be afunction from which we want to determine its derivativeConsider the following auxiliary system

= 120578 (9)

The main idea is to find 120578 to keep 119911 minus 119891(119905) = 0 resulting in120578 =

119891(119905) Based on a second-order sliding-mode algorithmone obtains [30]

120578 = 1205781 minus 1205831

1003816

1003816

1003816

1003816

119911 minus 119891(119905)

1003816

1003816

1003816

1003816

12 sign (119911 minus 119891 (119905))

1205781 = minus1205832 sign (119911 minus 119891 (119905))

(10)

where 1205831 1205832 gt 0 and sign (119909) is a signum function A generalform of 119899th-order differentiators can be written as

1205780 = 1205810 1205810 = minus1205830Γ1(119899+1) 1003816

1003816

1003816

1003816

1205780 minus 119891 (119905)

1003816

1003816

1003816

1003816

119899(119899+1)

times sign (1205780 minus 119891 (119905)) + 1205781

4 Journal of Engineering

Piezoelectricactuator

Piezoelectricsensor

Beam

Interface board Power amplifier

Computer

w(L t)

u(t)

(a)

Beam

Laser displacement sensor

Piezoelectric actuator

Power amplifier

Piezoelectric sensor

(b)

Figure 3 Experimental system (a) schematic and (b) photograph

1205781 = 1205811 1205811 = minus1205831Γ1119899 1003816

1003816

1003816

1003816

1205781 minus 1205810

1003816

1003816

1003816

1003816

(119899minus1)119899 sign (1205780 minus 1205810) + 1205782

120578119899minus1 = 120581119899minus1 120581119899minus1 = minus120583119899minus1Γ12 1003816

1003816

1003816

1003816

120578119899minus1 minus 120581119899minus2

1003816

1003816

1003816

1003816

12

times sign (120578119899minus1 minus 120581119899minus2) + 120578119899

120578119899 = minus120583119899Γ sign (120578119899 minus 120581119899minus1)

(11)

where 120578119899 is an approximated value of the 119899th derivative of thefunction 119891(119905) Γ gt 0 is a Lipschitz constant of the function119891(119905) and 120583119894 (119894 = 0 1 119899) are constant parameters Thereader is referred to Levant [30 31] for full details of themethod In this paper thismethodwill be used for estimatingthe first derivative of the signal from the piezoelectric sensor

3 Experimental System

A schematic and a photograph of the experimental systemused in this paper are shown in Figure 3 The system consistsof a flexible cantilever beam a piezoelectric patch actuatora piezoelectric patch sensor a high-voltage power amplifiera 16-bit ADCDAC interface board and a PC The beam ismade of aluminum The dimensions and properties of thebeam are summarized in Table 1 The piezoelectric actuatorand sensor are lead zirconate titanate (PZT)They are bondedon the beam at the distance 119909 = 15mm from the clampedpoint The dimensions and properties of the actuator andsensor are summarized in Table 2 The sensor measures theaxial strain of the beam The system is also equipped witha laser displacement sensor for measuring the beamrsquos tipdeflection however the displacement sensor is not used inthe control loop The signals from the sensors are fed backto the PC through the interface board The control signalfrom the computer is converted to an analogue signal by theinterface board and is transmitted through the high-voltagepower amplifier to the actuator to close the control loopThe gain of the amplifier is 150 Figure 4 displays the pulseresponse and the frequency response functions of the beamwhere the signal from the piezoelectric sensor is considered

Table 1 Beamrsquos dimensions and properties

Dimensionsproperties Symbol Value UnitYoungrsquos modulus 119864119887 70 GPaDensity 120588119887 2700 kgm3

Poisson ratio ] 035 mdashLength 119871 340 mmWidth 119887119887 15 mmThickness 119905119887 08 mm

Table 2 Piezoelectric materialrsquos dimensions and properties

Dimensionsproperties Symbol Value UnitActuatorYoungrsquos modulus 119864119901 6667 GPaDensity 120588119901 7800 kgm3

Length 119897119901 45 mmWidth 119887119901 15 mmThickness 119905119901 05 mmStrain coefficient 11988931 minus171eminus12 mV

Piezoelectric position 1198971 15 mm1198972 60 mm

SensorYoungrsquos modulus 119864119901119904 25 GPaLength 119897119901119904 45 mmWidth 119887119901119904 15 mmThickness 119905119901119904 05 mmPiezoelectric capacitance 119862119901119904 379eminus12 FaradStrain coefficient 11988931119904 23eminus12 mV

as the output The frequency response shows that the firsttwo natural frequencies of the beam are 3928 rads and2247 rads

4 Finite Element Model

A finite element (FE) model of the experimental beam wasdeveloped to be used in the control design process For

Journal of Engineering 5

0 05 1 15 2 25 3 35 4

0

01

02

Time (s)

minus05

minus04

minus03

minus02

minus01

s(t)

(V)

(a)

0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus30

minus20

minus10

103102101

(b)

Figure 4 Experimental systemrsquos responses (a) pulse response and (b) frequency response function

0 05 1 15 2 25 3 35 4

0

005

01

Time (s)

minus025

minus02

minus015

minus01

minus005

s(t)

(V)

(a)

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

103102101

(b)

Figure 5 Systemrsquos responses based on FE model (a) pulse response and (b) frequency response function

simplicity 119899 = 11 was chosen The pulse response and thefrequency response functions obtained from the developedFE model are shown in Figure 5 The responses agree withthose of the experimental system

5 Controller Design

The proposed control design is given in this section Thecontrol objective is to suppress the vibration of the beamby using only the signal from the piezoelectric sensor Theschematic for the control loop is shown in Figure 6

Instead of directly using V119904 and V119904 for feedback let V119904 andV119904 pass through a low-pass filter to obtain a new feedbacksignal 119910(119905) as follows

119910 (119905) =

1

120591

(1198861V119904 (119905) + 1198862V119904 (119905) minus 119910 (119905)) (12)

where 120591 gt 0 is the time constant of the filter and 1198861 1198862 ge

0 (1198861 +1198862 = 0) are weighting parameters Note that includinga low-pass filter in the control loop is usually done in practice

r(t)

y(t)

u(t)Controller Power amplifier BeamPiezoelectric

actuator

Piezoelectric sensor

+minus

Figure 6 Controlled system

to attenuate the effect of noise in the feedback signal In thefollowing the dynamic of the filter is also included in thecontroller design

Define a Lyapunov function candidate as

1198811 =1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(13)

6 Journal of Engineering

where 120582 is considered as a controller parameter By taking thetime derivative of (13) and substituting (1) (2) (6) and (12)into the derivative result it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) + 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)))

(14)

By choosing the following control law

119906 (119905) = minus

1205821198862120595119910

120591

minus

1205821198861120595119910

120591

(119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

(

1015840(1198972 119905) minus

1015840(1198971 119905))

(15)

it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(16)

By Barbalatrsquos lemma [36 37] 119910 rarr 0 is achieved Notethat 119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905) can be obtained directly from the

piezoelectric sensor (6) that is V119904(119905) = 120595(119908

1015840(1198972 119905) minus

119908

1015840(1198971 119905)) and

1015840(1198972 119905) minus

1015840(1198971 119905) can be obtained from V119904(119905)

In our experiment we used the high-order sliding-modedifferentiation method described in Section 23 to determineV119904(119905)

Note that control law (15) has a singularity at 1015840(1198972 119905) minus

1015840(1198971 119905) = 0 Thus implementations of the control law must

empirically avoid this singularity For simplicity we chose1198861 = 0 and 1198862 = 1 in the experiment to avoid the singularity

Moreover the parameter 120595 is usually uncertain or evenunknown Thus control law (15) with 1198861 = 0 and 1198862 = 1 ismodified as

119906 = minus

120582119896 (119905) 119910

120591

(17)

where 119896(119905) is an adaptive variable replacing 120595 Define 119890(119905) =

119896(119905) minus 120595 Take

119881 = 1198811 +1

2

119890

2

=

1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(18)

as a new Lyapunov function candidate Taking its derivativewith respect to time and substituting (17) into the result yields

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905)

+ (minus

120582119896 (119905) 119910

120591

) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

+ (119896 (119905) minus 120595)

119896 (119905)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ (

119896 (119905) minus

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

) (119896 (119905) minus 120595)

(19)

By choosing the following adaptive law

119896 (119905) =

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

(20)

it results in

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(21)

Applying Barbalatrsquos lemma [36 37] results in 119910 rarr 0

6 Results

61 Simulation Study A simulation study was conductedusing the FE model described in Section 22 to tune thecontrol parameter 120582 It was observed that the convergencerate of the controlled response increases with 120582 Here thecontrol gain of 120582 = 0052 was chosen Simulation results

Journal of Engineering 7

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 7 FEM simulation result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and(d) tip deflection

are shown in Figure 7 The controller was activated at 119905 =

1 sec The results show that the controller is able to suppressthe vibration effectively Note that 119896(119905) converged to a finitevalue

62 Experimental Results Controller (17) was implementeddigitally on the PC with the sampling period of the con-trol loop of 1msec The high-order sliding differentiationtechnique with Γ = 700 1205830 = 1205831 = 1 was used Theexperimental results are shown in Figure 8 The controllerprovides satisfactory responses similar to those obtained bythe simulation Note that the controller was activated at 119905 =

1 secNote that we set the initial conditions of the FEM sim-

ulation and experiment roughly the same for a comparisonpurpose It was found that there are slight differences in theFEM simulation and experimental results This is mainly dueto themeasurement noise and themismatch in the initial con-ditions The difference between the maximum control inputsis about 1208 The difference of the steady state adaptivegains is about 335 In summary both results are practicallycomparable

7 Conclusions

In this paper we proposed an adaptive controller for vibrationsuppression of flexible structures The structure under con-sideration is a cantilever beam with a collocated piezoelectricactuator and sensor pair The governing equation of thebeam is a partial differential equation Since the beam is aninfinite-dimensional system we derived the control law usingthe infinite-dimensional Lyapunov method which does notrequire any approximated finite-dimension model Thus thestability of the control system is guaranteed for all vibrationmodesWe also employed a high-order slidingmodedifferen-tiation technique to obtain the derivative of the sensorrsquos signalfor being used in the control law The technique featuresrobust exact differentiation with finite-time convergenceEffectiveness of the controller is illustrated both in finite-element model and experiment with results that show goodagreement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Journal of Engineering

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 8 Experimental result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and (d)tip deflection

Acknowledgments

This work was supported by the Royal Golden Jubilee PhDprogram of the Thailand Research Fund The authors alsowould like to thank anonymous referees for their valuablecomments

References

[1] Z Mohamed J M Martins M O Tokhi J Sa da Costa andM A Botto ldquoVibration control of a very flexible manipulatorsystemrdquo Control Engineering Practice vol 13 no 3 pp 267ndash2772005

[2] M Azadi S A Fazelzadeh M Eghtesad and E Azadi ldquoVibra-tion suppression and adaptive-robust control of a smart flexiblesatellite with three axes maneuveringrdquo Acta Astronautica vol69 no 5-6 pp 307ndash322 2011

[3] T P Sales D A Rade and L C G De Souza ldquoPassive vibra-tion control of flexible spacecraft using shunted piezoelectrictransducersrdquo Aerospace Science and Technology vol 29 no 1pp 403ndash412 2013

[4] C C Fuller Active Control of Vibration Academic Press 1996[5] A Preumont Vibration Control of Active Structures Kluwer

Academic 1997

[6] A M Aly ldquoVibration control of buildings using magnetorheo-logical damper a new control algorithmrdquo Journal of Engineer-ing vol 2013 Article ID 596078 10 pages 2013

[7] A Moutsopoulou G E Stavroulakis and A Pouliezos ldquoInno-vation in active vibration control strategy of intelligent struc-turesrdquo Journal of Applied Mathematics vol 2014 Article ID430731 14 pages 2014

[8] B Bandyopadhyay T C Manjunath and M Umapathy Mod-eling Control and Implementation of Smart Structures A FEM-State Space Approach Springer Berlin Germany 2007

[9] A R Tavakolpour M Mailah I Z Mat Darus and O TokhildquoSelf-learning active vibration control of a flexible plate struc-ture with piezoelectric actuatorrdquo Simulation Modelling Practiceand Theory vol 18 no 5 pp 516ndash532 2010

[10] H Wang X-H Shen L-X Zhang and X-J Zhu ldquoActivevibration suppression of flexible structure using a LMI-basedcontrol patchrdquo Journal of Vibration and Control vol 18 no 9pp 1375ndash1379 2012

[11] Z Qiu X Zhang and C Ye ldquoVibration suppression of a flexiblepiezoelectric beam using BP neural network controllerrdquo ActaMechanica Solida Sinica vol 25 no 4 pp 417ndash428 2012

[12] T Sangpet S Kuntanapreeda and R Schmidt ldquoImprovingdelay-margin of noncollocated vibration control of piezo-actuated flexible beams via a fractional-order controllerrdquo Shockand Vibration vol 2014 Article ID 809173 8 pages 2014

Journal of Engineering 9

[13] F An W Chen and M Shao ldquoDynamic behavior of time-delayed acceleration feedback controller for active vibrationcontrol of flexible structuresrdquo Journal of Sound and Vibrationvol 333 no 20 pp 4789ndash4809 2014

[14] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration control of a cantilever beam with real-timeparameter estimationrdquo Shock and Vibration vol 2014 ArticleID 741765 15 pages 2014

[15] T Sangpet S Kuntanapreeda and R Schmidt ldquoHystereticnonlinearity observer design based on Kalman filter for piezo-actuated flexible beamsrdquo International Journal of Automationand Computing vol 11 no 6 pp 627ndash634 2014

[16] X Dong Z Peng W Zhang H Hua and G Meng ldquoResearchon spillover effects for vibration control of piezoelectric smartstructures by ANSYSrdquo Mathematical Problems in Engineeringvol 2014 Article ID 870940 8 pages 2014

[17] A Montazeri J Poshtan and A Yousefi-Koma ldquoDesign andanalysis of robust minimax LQG controller for an experimentalbeam considering spill-over effectrdquo IEEE Transactions on Con-trol Systems Technology vol 19 no 5 pp 1251ndash1259 2011

[18] R Padhi and S F Ali ldquoAn account of chronological devel-opments in control of distributed parameter systemsrdquo AnnualReviews in Control vol 33 no 1 pp 59ndash68 2009

[19] S Zhao H Lin and Z Xue ldquoSwitching control of closedquantum systems via the Lyapunov methodrdquo Automatica vol48 no 8 pp 1833ndash1838 2012

[20] S Kuntanapreeda and P M Marusak ldquoNonlinear extendedoutput feedback control for CSTRs with van de Vusse reactionrdquoComputers amp Chemical Engineering vol 41 pp 10ndash23 2012

[21] B Zhang K Liu and J Xiang ldquoA stabilized optimal nonlinearfeedback control for satellite attitude trackingrdquo Aerospace Sci-ence and Technology vol 27 no 1 pp 17ndash24 2013

[22] A Yang W Naeem G W Irwin and K Li ldquoStability anal-ysis and implementation of a decentralized formation controlstrategy for unmanned vehiclesrdquo IEEE Transactions on ControlSystems Technology vol 22 no 2 pp 706ndash720 2014

[23] M Dadfarnia N Jalili B Xian and D M Dawson ldquoAlyapunov-based piezoelectric controller for flexible cartesianrobot manipulatorsrdquo Journal of Dynamic Systems Measurementand Control vol 126 no 2 pp 347ndash358 2004

[24] J-M Coron B drsquoAndrea-Novel and G Bastin ldquoA strict Lya-punov function for boundary control of hyperbolic systems ofconservation lawsrdquo IEEE Transactions on Automatic Controlvol 52 no 1 pp 2ndash11 2007

[25] M-B Cheng V Radisavljevic and W-C Su ldquoSliding modeboundary control of a parabolic PDE system with parametervariations and boundary uncertaintiesrdquoAutomatica vol 47 no2 pp 381ndash387 2011

[26] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems andControl Letters vol 61 no 11 pp 1069ndash1078 2012

[27] B-Z Guo and F-F Jin ldquoThe active disturbance rejection andsliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbancerdquoAutomatica vol 49 no 9 pp 2911ndash2918 2013

[28] A Luemchamloey and S Kuntanapreeda ldquoActive vibrationcontrol of flexible beambased on infinite-dimensional lyapunovstability theory an experimental studyrdquo Journal of ControlAutomation and Electrical Systems vol 25 no 6 pp 649ndash6562014

[29] W He S Zhang and S S Ge ldquoAdaptive boundary control of anonlinear flexible string systemrdquo IEEE Transactions on ControlSystems Technology vol 22 no 3 pp 1088ndash1093 2014

[30] A Levant ldquoRobust exact differentiation via sliding modetechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998

[31] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[32] XWang and B Shirinzadeh ldquoHigh-order nonlinear differentia-tor and application to aircraft controlrdquoMechanical Systems andSignal Processing vol 46 no 2 pp 227ndash252 2014

[33] I Salgado I Chairez O Camacho and C Yanez ldquoSuper-twisting sliding mode differentiation for improving PD con-trollers performance of second order systemrdquo ISA Transactionsvol 53 no 4 pp 1096ndash1106 2014

[34] S Y Wang ldquoA finite element model for the static and dynamicanalysis of a piezoelectric bimorphrdquo International Journal ofSolids and Structures vol 41 no 15 pp 4075ndash4096 2004

[35] D L Logan A First Course in the Finite Element MethodWadsworth Group 2002

[36] J E Slotine Applied Nonlinear Control Prentice-Hall 1991[37] H K Khalil Nonlinear Systems Prentice-Hall 1996

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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Volume 2014

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SensorsJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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Navigation and Observation

International Journal of

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DistributedSensor Networks

International Journal of

4 Journal of Engineering

Piezoelectricactuator

Piezoelectricsensor

Beam

Interface board Power amplifier

Computer

w(L t)

u(t)

(a)

Beam

Laser displacement sensor

Piezoelectric actuator

Power amplifier

Piezoelectric sensor

(b)

Figure 3 Experimental system (a) schematic and (b) photograph

1205781 = 1205811 1205811 = minus1205831Γ1119899 1003816

1003816

1003816

1003816

1205781 minus 1205810

1003816

1003816

1003816

1003816

(119899minus1)119899 sign (1205780 minus 1205810) + 1205782

120578119899minus1 = 120581119899minus1 120581119899minus1 = minus120583119899minus1Γ12 1003816

1003816

1003816

1003816

120578119899minus1 minus 120581119899minus2

1003816

1003816

1003816

1003816

12

times sign (120578119899minus1 minus 120581119899minus2) + 120578119899

120578119899 = minus120583119899Γ sign (120578119899 minus 120581119899minus1)

(11)

where 120578119899 is an approximated value of the 119899th derivative of thefunction 119891(119905) Γ gt 0 is a Lipschitz constant of the function119891(119905) and 120583119894 (119894 = 0 1 119899) are constant parameters Thereader is referred to Levant [30 31] for full details of themethod In this paper thismethodwill be used for estimatingthe first derivative of the signal from the piezoelectric sensor

3 Experimental System

A schematic and a photograph of the experimental systemused in this paper are shown in Figure 3 The system consistsof a flexible cantilever beam a piezoelectric patch actuatora piezoelectric patch sensor a high-voltage power amplifiera 16-bit ADCDAC interface board and a PC The beam ismade of aluminum The dimensions and properties of thebeam are summarized in Table 1 The piezoelectric actuatorand sensor are lead zirconate titanate (PZT)They are bondedon the beam at the distance 119909 = 15mm from the clampedpoint The dimensions and properties of the actuator andsensor are summarized in Table 2 The sensor measures theaxial strain of the beam The system is also equipped witha laser displacement sensor for measuring the beamrsquos tipdeflection however the displacement sensor is not used inthe control loop The signals from the sensors are fed backto the PC through the interface board The control signalfrom the computer is converted to an analogue signal by theinterface board and is transmitted through the high-voltagepower amplifier to the actuator to close the control loopThe gain of the amplifier is 150 Figure 4 displays the pulseresponse and the frequency response functions of the beamwhere the signal from the piezoelectric sensor is considered

Table 1 Beamrsquos dimensions and properties

Dimensionsproperties Symbol Value UnitYoungrsquos modulus 119864119887 70 GPaDensity 120588119887 2700 kgm3

Poisson ratio ] 035 mdashLength 119871 340 mmWidth 119887119887 15 mmThickness 119905119887 08 mm

Table 2 Piezoelectric materialrsquos dimensions and properties

Dimensionsproperties Symbol Value UnitActuatorYoungrsquos modulus 119864119901 6667 GPaDensity 120588119901 7800 kgm3

Length 119897119901 45 mmWidth 119887119901 15 mmThickness 119905119901 05 mmStrain coefficient 11988931 minus171eminus12 mV

Piezoelectric position 1198971 15 mm1198972 60 mm

SensorYoungrsquos modulus 119864119901119904 25 GPaLength 119897119901119904 45 mmWidth 119887119901119904 15 mmThickness 119905119901119904 05 mmPiezoelectric capacitance 119862119901119904 379eminus12 FaradStrain coefficient 11988931119904 23eminus12 mV

as the output The frequency response shows that the firsttwo natural frequencies of the beam are 3928 rads and2247 rads

4 Finite Element Model

A finite element (FE) model of the experimental beam wasdeveloped to be used in the control design process For

Journal of Engineering 5

0 05 1 15 2 25 3 35 4

0

01

02

Time (s)

minus05

minus04

minus03

minus02

minus01

s(t)

(V)

(a)

0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus30

minus20

minus10

103102101

(b)

Figure 4 Experimental systemrsquos responses (a) pulse response and (b) frequency response function

0 05 1 15 2 25 3 35 4

0

005

01

Time (s)

minus025

minus02

minus015

minus01

minus005

s(t)

(V)

(a)

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

103102101

(b)

Figure 5 Systemrsquos responses based on FE model (a) pulse response and (b) frequency response function

simplicity 119899 = 11 was chosen The pulse response and thefrequency response functions obtained from the developedFE model are shown in Figure 5 The responses agree withthose of the experimental system

5 Controller Design

The proposed control design is given in this section Thecontrol objective is to suppress the vibration of the beamby using only the signal from the piezoelectric sensor Theschematic for the control loop is shown in Figure 6

Instead of directly using V119904 and V119904 for feedback let V119904 andV119904 pass through a low-pass filter to obtain a new feedbacksignal 119910(119905) as follows

119910 (119905) =

1

120591

(1198861V119904 (119905) + 1198862V119904 (119905) minus 119910 (119905)) (12)

where 120591 gt 0 is the time constant of the filter and 1198861 1198862 ge

0 (1198861 +1198862 = 0) are weighting parameters Note that includinga low-pass filter in the control loop is usually done in practice

r(t)

y(t)

u(t)Controller Power amplifier BeamPiezoelectric

actuator

Piezoelectric sensor

+minus

Figure 6 Controlled system

to attenuate the effect of noise in the feedback signal In thefollowing the dynamic of the filter is also included in thecontroller design

Define a Lyapunov function candidate as

1198811 =1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(13)

6 Journal of Engineering

where 120582 is considered as a controller parameter By taking thetime derivative of (13) and substituting (1) (2) (6) and (12)into the derivative result it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) + 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)))

(14)

By choosing the following control law

119906 (119905) = minus

1205821198862120595119910

120591

minus

1205821198861120595119910

120591

(119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

(

1015840(1198972 119905) minus

1015840(1198971 119905))

(15)

it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(16)

By Barbalatrsquos lemma [36 37] 119910 rarr 0 is achieved Notethat 119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905) can be obtained directly from the

piezoelectric sensor (6) that is V119904(119905) = 120595(119908

1015840(1198972 119905) minus

119908

1015840(1198971 119905)) and

1015840(1198972 119905) minus

1015840(1198971 119905) can be obtained from V119904(119905)

In our experiment we used the high-order sliding-modedifferentiation method described in Section 23 to determineV119904(119905)

Note that control law (15) has a singularity at 1015840(1198972 119905) minus

1015840(1198971 119905) = 0 Thus implementations of the control law must

empirically avoid this singularity For simplicity we chose1198861 = 0 and 1198862 = 1 in the experiment to avoid the singularity

Moreover the parameter 120595 is usually uncertain or evenunknown Thus control law (15) with 1198861 = 0 and 1198862 = 1 ismodified as

119906 = minus

120582119896 (119905) 119910

120591

(17)

where 119896(119905) is an adaptive variable replacing 120595 Define 119890(119905) =

119896(119905) minus 120595 Take

119881 = 1198811 +1

2

119890

2

=

1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(18)

as a new Lyapunov function candidate Taking its derivativewith respect to time and substituting (17) into the result yields

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905)

+ (minus

120582119896 (119905) 119910

120591

) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

+ (119896 (119905) minus 120595)

119896 (119905)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ (

119896 (119905) minus

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

) (119896 (119905) minus 120595)

(19)

By choosing the following adaptive law

119896 (119905) =

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

(20)

it results in

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(21)

Applying Barbalatrsquos lemma [36 37] results in 119910 rarr 0

6 Results

61 Simulation Study A simulation study was conductedusing the FE model described in Section 22 to tune thecontrol parameter 120582 It was observed that the convergencerate of the controlled response increases with 120582 Here thecontrol gain of 120582 = 0052 was chosen Simulation results

Journal of Engineering 7

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 7 FEM simulation result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and(d) tip deflection

are shown in Figure 7 The controller was activated at 119905 =

1 sec The results show that the controller is able to suppressthe vibration effectively Note that 119896(119905) converged to a finitevalue

62 Experimental Results Controller (17) was implementeddigitally on the PC with the sampling period of the con-trol loop of 1msec The high-order sliding differentiationtechnique with Γ = 700 1205830 = 1205831 = 1 was used Theexperimental results are shown in Figure 8 The controllerprovides satisfactory responses similar to those obtained bythe simulation Note that the controller was activated at 119905 =

1 secNote that we set the initial conditions of the FEM sim-

ulation and experiment roughly the same for a comparisonpurpose It was found that there are slight differences in theFEM simulation and experimental results This is mainly dueto themeasurement noise and themismatch in the initial con-ditions The difference between the maximum control inputsis about 1208 The difference of the steady state adaptivegains is about 335 In summary both results are practicallycomparable

7 Conclusions

In this paper we proposed an adaptive controller for vibrationsuppression of flexible structures The structure under con-sideration is a cantilever beam with a collocated piezoelectricactuator and sensor pair The governing equation of thebeam is a partial differential equation Since the beam is aninfinite-dimensional system we derived the control law usingthe infinite-dimensional Lyapunov method which does notrequire any approximated finite-dimension model Thus thestability of the control system is guaranteed for all vibrationmodesWe also employed a high-order slidingmodedifferen-tiation technique to obtain the derivative of the sensorrsquos signalfor being used in the control law The technique featuresrobust exact differentiation with finite-time convergenceEffectiveness of the controller is illustrated both in finite-element model and experiment with results that show goodagreement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Journal of Engineering

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 8 Experimental result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and (d)tip deflection

Acknowledgments

This work was supported by the Royal Golden Jubilee PhDprogram of the Thailand Research Fund The authors alsowould like to thank anonymous referees for their valuablecomments

References

[1] Z Mohamed J M Martins M O Tokhi J Sa da Costa andM A Botto ldquoVibration control of a very flexible manipulatorsystemrdquo Control Engineering Practice vol 13 no 3 pp 267ndash2772005

[2] M Azadi S A Fazelzadeh M Eghtesad and E Azadi ldquoVibra-tion suppression and adaptive-robust control of a smart flexiblesatellite with three axes maneuveringrdquo Acta Astronautica vol69 no 5-6 pp 307ndash322 2011

[3] T P Sales D A Rade and L C G De Souza ldquoPassive vibra-tion control of flexible spacecraft using shunted piezoelectrictransducersrdquo Aerospace Science and Technology vol 29 no 1pp 403ndash412 2013

[4] C C Fuller Active Control of Vibration Academic Press 1996[5] A Preumont Vibration Control of Active Structures Kluwer

Academic 1997

[6] A M Aly ldquoVibration control of buildings using magnetorheo-logical damper a new control algorithmrdquo Journal of Engineer-ing vol 2013 Article ID 596078 10 pages 2013

[7] A Moutsopoulou G E Stavroulakis and A Pouliezos ldquoInno-vation in active vibration control strategy of intelligent struc-turesrdquo Journal of Applied Mathematics vol 2014 Article ID430731 14 pages 2014

[8] B Bandyopadhyay T C Manjunath and M Umapathy Mod-eling Control and Implementation of Smart Structures A FEM-State Space Approach Springer Berlin Germany 2007

[9] A R Tavakolpour M Mailah I Z Mat Darus and O TokhildquoSelf-learning active vibration control of a flexible plate struc-ture with piezoelectric actuatorrdquo Simulation Modelling Practiceand Theory vol 18 no 5 pp 516ndash532 2010

[10] H Wang X-H Shen L-X Zhang and X-J Zhu ldquoActivevibration suppression of flexible structure using a LMI-basedcontrol patchrdquo Journal of Vibration and Control vol 18 no 9pp 1375ndash1379 2012

[11] Z Qiu X Zhang and C Ye ldquoVibration suppression of a flexiblepiezoelectric beam using BP neural network controllerrdquo ActaMechanica Solida Sinica vol 25 no 4 pp 417ndash428 2012

[12] T Sangpet S Kuntanapreeda and R Schmidt ldquoImprovingdelay-margin of noncollocated vibration control of piezo-actuated flexible beams via a fractional-order controllerrdquo Shockand Vibration vol 2014 Article ID 809173 8 pages 2014

Journal of Engineering 9

[13] F An W Chen and M Shao ldquoDynamic behavior of time-delayed acceleration feedback controller for active vibrationcontrol of flexible structuresrdquo Journal of Sound and Vibrationvol 333 no 20 pp 4789ndash4809 2014

[14] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration control of a cantilever beam with real-timeparameter estimationrdquo Shock and Vibration vol 2014 ArticleID 741765 15 pages 2014

[15] T Sangpet S Kuntanapreeda and R Schmidt ldquoHystereticnonlinearity observer design based on Kalman filter for piezo-actuated flexible beamsrdquo International Journal of Automationand Computing vol 11 no 6 pp 627ndash634 2014

[16] X Dong Z Peng W Zhang H Hua and G Meng ldquoResearchon spillover effects for vibration control of piezoelectric smartstructures by ANSYSrdquo Mathematical Problems in Engineeringvol 2014 Article ID 870940 8 pages 2014

[17] A Montazeri J Poshtan and A Yousefi-Koma ldquoDesign andanalysis of robust minimax LQG controller for an experimentalbeam considering spill-over effectrdquo IEEE Transactions on Con-trol Systems Technology vol 19 no 5 pp 1251ndash1259 2011

[18] R Padhi and S F Ali ldquoAn account of chronological devel-opments in control of distributed parameter systemsrdquo AnnualReviews in Control vol 33 no 1 pp 59ndash68 2009

[19] S Zhao H Lin and Z Xue ldquoSwitching control of closedquantum systems via the Lyapunov methodrdquo Automatica vol48 no 8 pp 1833ndash1838 2012

[20] S Kuntanapreeda and P M Marusak ldquoNonlinear extendedoutput feedback control for CSTRs with van de Vusse reactionrdquoComputers amp Chemical Engineering vol 41 pp 10ndash23 2012

[21] B Zhang K Liu and J Xiang ldquoA stabilized optimal nonlinearfeedback control for satellite attitude trackingrdquo Aerospace Sci-ence and Technology vol 27 no 1 pp 17ndash24 2013

[22] A Yang W Naeem G W Irwin and K Li ldquoStability anal-ysis and implementation of a decentralized formation controlstrategy for unmanned vehiclesrdquo IEEE Transactions on ControlSystems Technology vol 22 no 2 pp 706ndash720 2014

[23] M Dadfarnia N Jalili B Xian and D M Dawson ldquoAlyapunov-based piezoelectric controller for flexible cartesianrobot manipulatorsrdquo Journal of Dynamic Systems Measurementand Control vol 126 no 2 pp 347ndash358 2004

[24] J-M Coron B drsquoAndrea-Novel and G Bastin ldquoA strict Lya-punov function for boundary control of hyperbolic systems ofconservation lawsrdquo IEEE Transactions on Automatic Controlvol 52 no 1 pp 2ndash11 2007

[25] M-B Cheng V Radisavljevic and W-C Su ldquoSliding modeboundary control of a parabolic PDE system with parametervariations and boundary uncertaintiesrdquoAutomatica vol 47 no2 pp 381ndash387 2011

[26] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems andControl Letters vol 61 no 11 pp 1069ndash1078 2012

[27] B-Z Guo and F-F Jin ldquoThe active disturbance rejection andsliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbancerdquoAutomatica vol 49 no 9 pp 2911ndash2918 2013

[28] A Luemchamloey and S Kuntanapreeda ldquoActive vibrationcontrol of flexible beambased on infinite-dimensional lyapunovstability theory an experimental studyrdquo Journal of ControlAutomation and Electrical Systems vol 25 no 6 pp 649ndash6562014

[29] W He S Zhang and S S Ge ldquoAdaptive boundary control of anonlinear flexible string systemrdquo IEEE Transactions on ControlSystems Technology vol 22 no 3 pp 1088ndash1093 2014

[30] A Levant ldquoRobust exact differentiation via sliding modetechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998

[31] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[32] XWang and B Shirinzadeh ldquoHigh-order nonlinear differentia-tor and application to aircraft controlrdquoMechanical Systems andSignal Processing vol 46 no 2 pp 227ndash252 2014

[33] I Salgado I Chairez O Camacho and C Yanez ldquoSuper-twisting sliding mode differentiation for improving PD con-trollers performance of second order systemrdquo ISA Transactionsvol 53 no 4 pp 1096ndash1106 2014

[34] S Y Wang ldquoA finite element model for the static and dynamicanalysis of a piezoelectric bimorphrdquo International Journal ofSolids and Structures vol 41 no 15 pp 4075ndash4096 2004

[35] D L Logan A First Course in the Finite Element MethodWadsworth Group 2002

[36] J E Slotine Applied Nonlinear Control Prentice-Hall 1991[37] H K Khalil Nonlinear Systems Prentice-Hall 1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 5

0 05 1 15 2 25 3 35 4

0

01

02

Time (s)

minus05

minus04

minus03

minus02

minus01

s(t)

(V)

(a)

0

10

20

30

Frequency (rads)

Mag

nitu

de (d

B)

minus30

minus20

minus10

103102101

(b)

Figure 4 Experimental systemrsquos responses (a) pulse response and (b) frequency response function

0 05 1 15 2 25 3 35 4

0

005

01

Time (s)

minus025

minus02

minus015

minus01

minus005

s(t)

(V)

(a)

0

10

20

Frequency (rads)

Mag

nitu

de (d

B)

minus40

minus30

minus20

minus10

103102101

(b)

Figure 5 Systemrsquos responses based on FE model (a) pulse response and (b) frequency response function

simplicity 119899 = 11 was chosen The pulse response and thefrequency response functions obtained from the developedFE model are shown in Figure 5 The responses agree withthose of the experimental system

5 Controller Design

The proposed control design is given in this section Thecontrol objective is to suppress the vibration of the beamby using only the signal from the piezoelectric sensor Theschematic for the control loop is shown in Figure 6

Instead of directly using V119904 and V119904 for feedback let V119904 andV119904 pass through a low-pass filter to obtain a new feedbacksignal 119910(119905) as follows

119910 (119905) =

1

120591

(1198861V119904 (119905) + 1198862V119904 (119905) minus 119910 (119905)) (12)

where 120591 gt 0 is the time constant of the filter and 1198861 1198862 ge

0 (1198861 +1198862 = 0) are weighting parameters Note that includinga low-pass filter in the control loop is usually done in practice

r(t)

y(t)

u(t)Controller Power amplifier BeamPiezoelectric

actuator

Piezoelectric sensor

+minus

Figure 6 Controlled system

to attenuate the effect of noise in the feedback signal In thefollowing the dynamic of the filter is also included in thecontroller design

Define a Lyapunov function candidate as

1198811 =1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(13)

6 Journal of Engineering

where 120582 is considered as a controller parameter By taking thetime derivative of (13) and substituting (1) (2) (6) and (12)into the derivative result it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) + 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)))

(14)

By choosing the following control law

119906 (119905) = minus

1205821198862120595119910

120591

minus

1205821198861120595119910

120591

(119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

(

1015840(1198972 119905) minus

1015840(1198971 119905))

(15)

it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(16)

By Barbalatrsquos lemma [36 37] 119910 rarr 0 is achieved Notethat 119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905) can be obtained directly from the

piezoelectric sensor (6) that is V119904(119905) = 120595(119908

1015840(1198972 119905) minus

119908

1015840(1198971 119905)) and

1015840(1198972 119905) minus

1015840(1198971 119905) can be obtained from V119904(119905)

In our experiment we used the high-order sliding-modedifferentiation method described in Section 23 to determineV119904(119905)

Note that control law (15) has a singularity at 1015840(1198972 119905) minus

1015840(1198971 119905) = 0 Thus implementations of the control law must

empirically avoid this singularity For simplicity we chose1198861 = 0 and 1198862 = 1 in the experiment to avoid the singularity

Moreover the parameter 120595 is usually uncertain or evenunknown Thus control law (15) with 1198861 = 0 and 1198862 = 1 ismodified as

119906 = minus

120582119896 (119905) 119910

120591

(17)

where 119896(119905) is an adaptive variable replacing 120595 Define 119890(119905) =

119896(119905) minus 120595 Take

119881 = 1198811 +1

2

119890

2

=

1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(18)

as a new Lyapunov function candidate Taking its derivativewith respect to time and substituting (17) into the result yields

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905)

+ (minus

120582119896 (119905) 119910

120591

) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

+ (119896 (119905) minus 120595)

119896 (119905)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ (

119896 (119905) minus

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

) (119896 (119905) minus 120595)

(19)

By choosing the following adaptive law

119896 (119905) =

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

(20)

it results in

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(21)

Applying Barbalatrsquos lemma [36 37] results in 119910 rarr 0

6 Results

61 Simulation Study A simulation study was conductedusing the FE model described in Section 22 to tune thecontrol parameter 120582 It was observed that the convergencerate of the controlled response increases with 120582 Here thecontrol gain of 120582 = 0052 was chosen Simulation results

Journal of Engineering 7

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 7 FEM simulation result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and(d) tip deflection

are shown in Figure 7 The controller was activated at 119905 =

1 sec The results show that the controller is able to suppressthe vibration effectively Note that 119896(119905) converged to a finitevalue

62 Experimental Results Controller (17) was implementeddigitally on the PC with the sampling period of the con-trol loop of 1msec The high-order sliding differentiationtechnique with Γ = 700 1205830 = 1205831 = 1 was used Theexperimental results are shown in Figure 8 The controllerprovides satisfactory responses similar to those obtained bythe simulation Note that the controller was activated at 119905 =

1 secNote that we set the initial conditions of the FEM sim-

ulation and experiment roughly the same for a comparisonpurpose It was found that there are slight differences in theFEM simulation and experimental results This is mainly dueto themeasurement noise and themismatch in the initial con-ditions The difference between the maximum control inputsis about 1208 The difference of the steady state adaptivegains is about 335 In summary both results are practicallycomparable

7 Conclusions

In this paper we proposed an adaptive controller for vibrationsuppression of flexible structures The structure under con-sideration is a cantilever beam with a collocated piezoelectricactuator and sensor pair The governing equation of thebeam is a partial differential equation Since the beam is aninfinite-dimensional system we derived the control law usingthe infinite-dimensional Lyapunov method which does notrequire any approximated finite-dimension model Thus thestability of the control system is guaranteed for all vibrationmodesWe also employed a high-order slidingmodedifferen-tiation technique to obtain the derivative of the sensorrsquos signalfor being used in the control law The technique featuresrobust exact differentiation with finite-time convergenceEffectiveness of the controller is illustrated both in finite-element model and experiment with results that show goodagreement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Journal of Engineering

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 8 Experimental result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and (d)tip deflection

Acknowledgments

This work was supported by the Royal Golden Jubilee PhDprogram of the Thailand Research Fund The authors alsowould like to thank anonymous referees for their valuablecomments

References

[1] Z Mohamed J M Martins M O Tokhi J Sa da Costa andM A Botto ldquoVibration control of a very flexible manipulatorsystemrdquo Control Engineering Practice vol 13 no 3 pp 267ndash2772005

[2] M Azadi S A Fazelzadeh M Eghtesad and E Azadi ldquoVibra-tion suppression and adaptive-robust control of a smart flexiblesatellite with three axes maneuveringrdquo Acta Astronautica vol69 no 5-6 pp 307ndash322 2011

[3] T P Sales D A Rade and L C G De Souza ldquoPassive vibra-tion control of flexible spacecraft using shunted piezoelectrictransducersrdquo Aerospace Science and Technology vol 29 no 1pp 403ndash412 2013

[4] C C Fuller Active Control of Vibration Academic Press 1996[5] A Preumont Vibration Control of Active Structures Kluwer

Academic 1997

[6] A M Aly ldquoVibration control of buildings using magnetorheo-logical damper a new control algorithmrdquo Journal of Engineer-ing vol 2013 Article ID 596078 10 pages 2013

[7] A Moutsopoulou G E Stavroulakis and A Pouliezos ldquoInno-vation in active vibration control strategy of intelligent struc-turesrdquo Journal of Applied Mathematics vol 2014 Article ID430731 14 pages 2014

[8] B Bandyopadhyay T C Manjunath and M Umapathy Mod-eling Control and Implementation of Smart Structures A FEM-State Space Approach Springer Berlin Germany 2007

[9] A R Tavakolpour M Mailah I Z Mat Darus and O TokhildquoSelf-learning active vibration control of a flexible plate struc-ture with piezoelectric actuatorrdquo Simulation Modelling Practiceand Theory vol 18 no 5 pp 516ndash532 2010

[10] H Wang X-H Shen L-X Zhang and X-J Zhu ldquoActivevibration suppression of flexible structure using a LMI-basedcontrol patchrdquo Journal of Vibration and Control vol 18 no 9pp 1375ndash1379 2012

[11] Z Qiu X Zhang and C Ye ldquoVibration suppression of a flexiblepiezoelectric beam using BP neural network controllerrdquo ActaMechanica Solida Sinica vol 25 no 4 pp 417ndash428 2012

[12] T Sangpet S Kuntanapreeda and R Schmidt ldquoImprovingdelay-margin of noncollocated vibration control of piezo-actuated flexible beams via a fractional-order controllerrdquo Shockand Vibration vol 2014 Article ID 809173 8 pages 2014

Journal of Engineering 9

[13] F An W Chen and M Shao ldquoDynamic behavior of time-delayed acceleration feedback controller for active vibrationcontrol of flexible structuresrdquo Journal of Sound and Vibrationvol 333 no 20 pp 4789ndash4809 2014

[14] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration control of a cantilever beam with real-timeparameter estimationrdquo Shock and Vibration vol 2014 ArticleID 741765 15 pages 2014

[15] T Sangpet S Kuntanapreeda and R Schmidt ldquoHystereticnonlinearity observer design based on Kalman filter for piezo-actuated flexible beamsrdquo International Journal of Automationand Computing vol 11 no 6 pp 627ndash634 2014

[16] X Dong Z Peng W Zhang H Hua and G Meng ldquoResearchon spillover effects for vibration control of piezoelectric smartstructures by ANSYSrdquo Mathematical Problems in Engineeringvol 2014 Article ID 870940 8 pages 2014

[17] A Montazeri J Poshtan and A Yousefi-Koma ldquoDesign andanalysis of robust minimax LQG controller for an experimentalbeam considering spill-over effectrdquo IEEE Transactions on Con-trol Systems Technology vol 19 no 5 pp 1251ndash1259 2011

[18] R Padhi and S F Ali ldquoAn account of chronological devel-opments in control of distributed parameter systemsrdquo AnnualReviews in Control vol 33 no 1 pp 59ndash68 2009

[19] S Zhao H Lin and Z Xue ldquoSwitching control of closedquantum systems via the Lyapunov methodrdquo Automatica vol48 no 8 pp 1833ndash1838 2012

[20] S Kuntanapreeda and P M Marusak ldquoNonlinear extendedoutput feedback control for CSTRs with van de Vusse reactionrdquoComputers amp Chemical Engineering vol 41 pp 10ndash23 2012

[21] B Zhang K Liu and J Xiang ldquoA stabilized optimal nonlinearfeedback control for satellite attitude trackingrdquo Aerospace Sci-ence and Technology vol 27 no 1 pp 17ndash24 2013

[22] A Yang W Naeem G W Irwin and K Li ldquoStability anal-ysis and implementation of a decentralized formation controlstrategy for unmanned vehiclesrdquo IEEE Transactions on ControlSystems Technology vol 22 no 2 pp 706ndash720 2014

[23] M Dadfarnia N Jalili B Xian and D M Dawson ldquoAlyapunov-based piezoelectric controller for flexible cartesianrobot manipulatorsrdquo Journal of Dynamic Systems Measurementand Control vol 126 no 2 pp 347ndash358 2004

[24] J-M Coron B drsquoAndrea-Novel and G Bastin ldquoA strict Lya-punov function for boundary control of hyperbolic systems ofconservation lawsrdquo IEEE Transactions on Automatic Controlvol 52 no 1 pp 2ndash11 2007

[25] M-B Cheng V Radisavljevic and W-C Su ldquoSliding modeboundary control of a parabolic PDE system with parametervariations and boundary uncertaintiesrdquoAutomatica vol 47 no2 pp 381ndash387 2011

[26] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems andControl Letters vol 61 no 11 pp 1069ndash1078 2012

[27] B-Z Guo and F-F Jin ldquoThe active disturbance rejection andsliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbancerdquoAutomatica vol 49 no 9 pp 2911ndash2918 2013

[28] A Luemchamloey and S Kuntanapreeda ldquoActive vibrationcontrol of flexible beambased on infinite-dimensional lyapunovstability theory an experimental studyrdquo Journal of ControlAutomation and Electrical Systems vol 25 no 6 pp 649ndash6562014

[29] W He S Zhang and S S Ge ldquoAdaptive boundary control of anonlinear flexible string systemrdquo IEEE Transactions on ControlSystems Technology vol 22 no 3 pp 1088ndash1093 2014

[30] A Levant ldquoRobust exact differentiation via sliding modetechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998

[31] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[32] XWang and B Shirinzadeh ldquoHigh-order nonlinear differentia-tor and application to aircraft controlrdquoMechanical Systems andSignal Processing vol 46 no 2 pp 227ndash252 2014

[33] I Salgado I Chairez O Camacho and C Yanez ldquoSuper-twisting sliding mode differentiation for improving PD con-trollers performance of second order systemrdquo ISA Transactionsvol 53 no 4 pp 1096ndash1106 2014

[34] S Y Wang ldquoA finite element model for the static and dynamicanalysis of a piezoelectric bimorphrdquo International Journal ofSolids and Structures vol 41 no 15 pp 4075ndash4096 2004

[35] D L Logan A First Course in the Finite Element MethodWadsworth Group 2002

[36] J E Slotine Applied Nonlinear Control Prentice-Hall 1991[37] H K Khalil Nonlinear Systems Prentice-Hall 1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Engineering

where 120582 is considered as a controller parameter By taking thetime derivative of (13) and substituting (1) (2) (6) and (12)into the derivative result it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) + 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909

minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ 119906 (119905) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(1198861120595 (119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

+ 1198862120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)))

(14)

By choosing the following control law

119906 (119905) = minus

1205821198862120595119910

120591

minus

1205821198861120595119910

120591

(119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905))

(

1015840(1198972 119905) minus

1015840(1198971 119905))

(15)

it yields

1198811 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(16)

By Barbalatrsquos lemma [36 37] 119910 rarr 0 is achieved Notethat 119908

1015840(1198972 119905) minus 119908

1015840(1198971 119905) can be obtained directly from the

piezoelectric sensor (6) that is V119904(119905) = 120595(119908

1015840(1198972 119905) minus

119908

1015840(1198971 119905)) and

1015840(1198972 119905) minus

1015840(1198971 119905) can be obtained from V119904(119905)

In our experiment we used the high-order sliding-modedifferentiation method described in Section 23 to determineV119904(119905)

Note that control law (15) has a singularity at 1015840(1198972 119905) minus

1015840(1198971 119905) = 0 Thus implementations of the control law must

empirically avoid this singularity For simplicity we chose1198861 = 0 and 1198862 = 1 in the experiment to avoid the singularity

Moreover the parameter 120595 is usually uncertain or evenunknown Thus control law (15) with 1198861 = 0 and 1198862 = 1 ismodified as

119906 = minus

120582119896 (119905) 119910

120591

(17)

where 119896(119905) is an adaptive variable replacing 120595 Define 119890(119905) =

119896(119905) minus 120595 Take

119881 = 1198811 +1

2

119890

2

=

1

2119872119901119900

int

119871

0

120588 (119909)

2(119909 119905) 119889119909

+

1

2119872119901119900

int

119871

0

119864119868 (119909)119908

101584010158402(119909 119905) 119889119909 +

120582

2

119910

2

(18)

as a new Lyapunov function candidate Taking its derivativewith respect to time and substituting (17) into the result yields

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905)

+ (minus

120582119896 (119905) 119910

120591

) (

1015840(1198972 119905) minus

1015840(1198971 119905))

+

120582119910

120591

(120595 (

1015840(1198972 119905) minus

1015840(1198971 119905)) minus 119910)

+ (119896 (119905) minus 120595)

119896 (119905)

= minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

+ (

119896 (119905) minus

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

) (119896 (119905) minus 120595)

(19)

By choosing the following adaptive law

119896 (119905) =

120582119910 (

1015840(1198972 119905) minus

1015840(1198971 119905))

120591

(20)

it results in

119881 = minus

119861

119872119901119900

int

119871

0

2(119909 119905) 119889119909 minus

119862

2119872119901119900

2(119871 119905) minus

120582119910

2

120591

le minus

120582119910

2

120591

le 0

(21)

Applying Barbalatrsquos lemma [36 37] results in 119910 rarr 0

6 Results

61 Simulation Study A simulation study was conductedusing the FE model described in Section 22 to tune thecontrol parameter 120582 It was observed that the convergencerate of the controlled response increases with 120582 Here thecontrol gain of 120582 = 0052 was chosen Simulation results

Journal of Engineering 7

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 7 FEM simulation result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and(d) tip deflection

are shown in Figure 7 The controller was activated at 119905 =

1 sec The results show that the controller is able to suppressthe vibration effectively Note that 119896(119905) converged to a finitevalue

62 Experimental Results Controller (17) was implementeddigitally on the PC with the sampling period of the con-trol loop of 1msec The high-order sliding differentiationtechnique with Γ = 700 1205830 = 1205831 = 1 was used Theexperimental results are shown in Figure 8 The controllerprovides satisfactory responses similar to those obtained bythe simulation Note that the controller was activated at 119905 =

1 secNote that we set the initial conditions of the FEM sim-

ulation and experiment roughly the same for a comparisonpurpose It was found that there are slight differences in theFEM simulation and experimental results This is mainly dueto themeasurement noise and themismatch in the initial con-ditions The difference between the maximum control inputsis about 1208 The difference of the steady state adaptivegains is about 335 In summary both results are practicallycomparable

7 Conclusions

In this paper we proposed an adaptive controller for vibrationsuppression of flexible structures The structure under con-sideration is a cantilever beam with a collocated piezoelectricactuator and sensor pair The governing equation of thebeam is a partial differential equation Since the beam is aninfinite-dimensional system we derived the control law usingthe infinite-dimensional Lyapunov method which does notrequire any approximated finite-dimension model Thus thestability of the control system is guaranteed for all vibrationmodesWe also employed a high-order slidingmodedifferen-tiation technique to obtain the derivative of the sensorrsquos signalfor being used in the control law The technique featuresrobust exact differentiation with finite-time convergenceEffectiveness of the controller is illustrated both in finite-element model and experiment with results that show goodagreement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Journal of Engineering

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 8 Experimental result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and (d)tip deflection

Acknowledgments

This work was supported by the Royal Golden Jubilee PhDprogram of the Thailand Research Fund The authors alsowould like to thank anonymous referees for their valuablecomments

References

[1] Z Mohamed J M Martins M O Tokhi J Sa da Costa andM A Botto ldquoVibration control of a very flexible manipulatorsystemrdquo Control Engineering Practice vol 13 no 3 pp 267ndash2772005

[2] M Azadi S A Fazelzadeh M Eghtesad and E Azadi ldquoVibra-tion suppression and adaptive-robust control of a smart flexiblesatellite with three axes maneuveringrdquo Acta Astronautica vol69 no 5-6 pp 307ndash322 2011

[3] T P Sales D A Rade and L C G De Souza ldquoPassive vibra-tion control of flexible spacecraft using shunted piezoelectrictransducersrdquo Aerospace Science and Technology vol 29 no 1pp 403ndash412 2013

[4] C C Fuller Active Control of Vibration Academic Press 1996[5] A Preumont Vibration Control of Active Structures Kluwer

Academic 1997

[6] A M Aly ldquoVibration control of buildings using magnetorheo-logical damper a new control algorithmrdquo Journal of Engineer-ing vol 2013 Article ID 596078 10 pages 2013

[7] A Moutsopoulou G E Stavroulakis and A Pouliezos ldquoInno-vation in active vibration control strategy of intelligent struc-turesrdquo Journal of Applied Mathematics vol 2014 Article ID430731 14 pages 2014

[8] B Bandyopadhyay T C Manjunath and M Umapathy Mod-eling Control and Implementation of Smart Structures A FEM-State Space Approach Springer Berlin Germany 2007

[9] A R Tavakolpour M Mailah I Z Mat Darus and O TokhildquoSelf-learning active vibration control of a flexible plate struc-ture with piezoelectric actuatorrdquo Simulation Modelling Practiceand Theory vol 18 no 5 pp 516ndash532 2010

[10] H Wang X-H Shen L-X Zhang and X-J Zhu ldquoActivevibration suppression of flexible structure using a LMI-basedcontrol patchrdquo Journal of Vibration and Control vol 18 no 9pp 1375ndash1379 2012

[11] Z Qiu X Zhang and C Ye ldquoVibration suppression of a flexiblepiezoelectric beam using BP neural network controllerrdquo ActaMechanica Solida Sinica vol 25 no 4 pp 417ndash428 2012

[12] T Sangpet S Kuntanapreeda and R Schmidt ldquoImprovingdelay-margin of noncollocated vibration control of piezo-actuated flexible beams via a fractional-order controllerrdquo Shockand Vibration vol 2014 Article ID 809173 8 pages 2014

Journal of Engineering 9

[13] F An W Chen and M Shao ldquoDynamic behavior of time-delayed acceleration feedback controller for active vibrationcontrol of flexible structuresrdquo Journal of Sound and Vibrationvol 333 no 20 pp 4789ndash4809 2014

[14] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration control of a cantilever beam with real-timeparameter estimationrdquo Shock and Vibration vol 2014 ArticleID 741765 15 pages 2014

[15] T Sangpet S Kuntanapreeda and R Schmidt ldquoHystereticnonlinearity observer design based on Kalman filter for piezo-actuated flexible beamsrdquo International Journal of Automationand Computing vol 11 no 6 pp 627ndash634 2014

[16] X Dong Z Peng W Zhang H Hua and G Meng ldquoResearchon spillover effects for vibration control of piezoelectric smartstructures by ANSYSrdquo Mathematical Problems in Engineeringvol 2014 Article ID 870940 8 pages 2014

[17] A Montazeri J Poshtan and A Yousefi-Koma ldquoDesign andanalysis of robust minimax LQG controller for an experimentalbeam considering spill-over effectrdquo IEEE Transactions on Con-trol Systems Technology vol 19 no 5 pp 1251ndash1259 2011

[18] R Padhi and S F Ali ldquoAn account of chronological devel-opments in control of distributed parameter systemsrdquo AnnualReviews in Control vol 33 no 1 pp 59ndash68 2009

[19] S Zhao H Lin and Z Xue ldquoSwitching control of closedquantum systems via the Lyapunov methodrdquo Automatica vol48 no 8 pp 1833ndash1838 2012

[20] S Kuntanapreeda and P M Marusak ldquoNonlinear extendedoutput feedback control for CSTRs with van de Vusse reactionrdquoComputers amp Chemical Engineering vol 41 pp 10ndash23 2012

[21] B Zhang K Liu and J Xiang ldquoA stabilized optimal nonlinearfeedback control for satellite attitude trackingrdquo Aerospace Sci-ence and Technology vol 27 no 1 pp 17ndash24 2013

[22] A Yang W Naeem G W Irwin and K Li ldquoStability anal-ysis and implementation of a decentralized formation controlstrategy for unmanned vehiclesrdquo IEEE Transactions on ControlSystems Technology vol 22 no 2 pp 706ndash720 2014

[23] M Dadfarnia N Jalili B Xian and D M Dawson ldquoAlyapunov-based piezoelectric controller for flexible cartesianrobot manipulatorsrdquo Journal of Dynamic Systems Measurementand Control vol 126 no 2 pp 347ndash358 2004

[24] J-M Coron B drsquoAndrea-Novel and G Bastin ldquoA strict Lya-punov function for boundary control of hyperbolic systems ofconservation lawsrdquo IEEE Transactions on Automatic Controlvol 52 no 1 pp 2ndash11 2007

[25] M-B Cheng V Radisavljevic and W-C Su ldquoSliding modeboundary control of a parabolic PDE system with parametervariations and boundary uncertaintiesrdquoAutomatica vol 47 no2 pp 381ndash387 2011

[26] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems andControl Letters vol 61 no 11 pp 1069ndash1078 2012

[27] B-Z Guo and F-F Jin ldquoThe active disturbance rejection andsliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbancerdquoAutomatica vol 49 no 9 pp 2911ndash2918 2013

[28] A Luemchamloey and S Kuntanapreeda ldquoActive vibrationcontrol of flexible beambased on infinite-dimensional lyapunovstability theory an experimental studyrdquo Journal of ControlAutomation and Electrical Systems vol 25 no 6 pp 649ndash6562014

[29] W He S Zhang and S S Ge ldquoAdaptive boundary control of anonlinear flexible string systemrdquo IEEE Transactions on ControlSystems Technology vol 22 no 3 pp 1088ndash1093 2014

[30] A Levant ldquoRobust exact differentiation via sliding modetechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998

[31] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[32] XWang and B Shirinzadeh ldquoHigh-order nonlinear differentia-tor and application to aircraft controlrdquoMechanical Systems andSignal Processing vol 46 no 2 pp 227ndash252 2014

[33] I Salgado I Chairez O Camacho and C Yanez ldquoSuper-twisting sliding mode differentiation for improving PD con-trollers performance of second order systemrdquo ISA Transactionsvol 53 no 4 pp 1096ndash1106 2014

[34] S Y Wang ldquoA finite element model for the static and dynamicanalysis of a piezoelectric bimorphrdquo International Journal ofSolids and Structures vol 41 no 15 pp 4075ndash4096 2004

[35] D L Logan A First Course in the Finite Element MethodWadsworth Group 2002

[36] J E Slotine Applied Nonlinear Control Prentice-Hall 1991[37] H K Khalil Nonlinear Systems Prentice-Hall 1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 7

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 7 FEM simulation result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and(d) tip deflection

are shown in Figure 7 The controller was activated at 119905 =

1 sec The results show that the controller is able to suppressthe vibration effectively Note that 119896(119905) converged to a finitevalue

62 Experimental Results Controller (17) was implementeddigitally on the PC with the sampling period of the con-trol loop of 1msec The high-order sliding differentiationtechnique with Γ = 700 1205830 = 1205831 = 1 was used Theexperimental results are shown in Figure 8 The controllerprovides satisfactory responses similar to those obtained bythe simulation Note that the controller was activated at 119905 =

1 secNote that we set the initial conditions of the FEM sim-

ulation and experiment roughly the same for a comparisonpurpose It was found that there are slight differences in theFEM simulation and experimental results This is mainly dueto themeasurement noise and themismatch in the initial con-ditions The difference between the maximum control inputsis about 1208 The difference of the steady state adaptivegains is about 335 In summary both results are practicallycomparable

7 Conclusions

In this paper we proposed an adaptive controller for vibrationsuppression of flexible structures The structure under con-sideration is a cantilever beam with a collocated piezoelectricactuator and sensor pair The governing equation of thebeam is a partial differential equation Since the beam is aninfinite-dimensional system we derived the control law usingthe infinite-dimensional Lyapunov method which does notrequire any approximated finite-dimension model Thus thestability of the control system is guaranteed for all vibrationmodesWe also employed a high-order slidingmodedifferen-tiation technique to obtain the derivative of the sensorrsquos signalfor being used in the control law The technique featuresrobust exact differentiation with finite-time convergenceEffectiveness of the controller is illustrated both in finite-element model and experiment with results that show goodagreement

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

8 Journal of Engineering

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 8 Experimental result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and (d)tip deflection

Acknowledgments

This work was supported by the Royal Golden Jubilee PhDprogram of the Thailand Research Fund The authors alsowould like to thank anonymous referees for their valuablecomments

References

[1] Z Mohamed J M Martins M O Tokhi J Sa da Costa andM A Botto ldquoVibration control of a very flexible manipulatorsystemrdquo Control Engineering Practice vol 13 no 3 pp 267ndash2772005

[2] M Azadi S A Fazelzadeh M Eghtesad and E Azadi ldquoVibra-tion suppression and adaptive-robust control of a smart flexiblesatellite with three axes maneuveringrdquo Acta Astronautica vol69 no 5-6 pp 307ndash322 2011

[3] T P Sales D A Rade and L C G De Souza ldquoPassive vibra-tion control of flexible spacecraft using shunted piezoelectrictransducersrdquo Aerospace Science and Technology vol 29 no 1pp 403ndash412 2013

[4] C C Fuller Active Control of Vibration Academic Press 1996[5] A Preumont Vibration Control of Active Structures Kluwer

Academic 1997

[6] A M Aly ldquoVibration control of buildings using magnetorheo-logical damper a new control algorithmrdquo Journal of Engineer-ing vol 2013 Article ID 596078 10 pages 2013

[7] A Moutsopoulou G E Stavroulakis and A Pouliezos ldquoInno-vation in active vibration control strategy of intelligent struc-turesrdquo Journal of Applied Mathematics vol 2014 Article ID430731 14 pages 2014

[8] B Bandyopadhyay T C Manjunath and M Umapathy Mod-eling Control and Implementation of Smart Structures A FEM-State Space Approach Springer Berlin Germany 2007

[9] A R Tavakolpour M Mailah I Z Mat Darus and O TokhildquoSelf-learning active vibration control of a flexible plate struc-ture with piezoelectric actuatorrdquo Simulation Modelling Practiceand Theory vol 18 no 5 pp 516ndash532 2010

[10] H Wang X-H Shen L-X Zhang and X-J Zhu ldquoActivevibration suppression of flexible structure using a LMI-basedcontrol patchrdquo Journal of Vibration and Control vol 18 no 9pp 1375ndash1379 2012

[11] Z Qiu X Zhang and C Ye ldquoVibration suppression of a flexiblepiezoelectric beam using BP neural network controllerrdquo ActaMechanica Solida Sinica vol 25 no 4 pp 417ndash428 2012

[12] T Sangpet S Kuntanapreeda and R Schmidt ldquoImprovingdelay-margin of noncollocated vibration control of piezo-actuated flexible beams via a fractional-order controllerrdquo Shockand Vibration vol 2014 Article ID 809173 8 pages 2014

Journal of Engineering 9

[13] F An W Chen and M Shao ldquoDynamic behavior of time-delayed acceleration feedback controller for active vibrationcontrol of flexible structuresrdquo Journal of Sound and Vibrationvol 333 no 20 pp 4789ndash4809 2014

[14] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration control of a cantilever beam with real-timeparameter estimationrdquo Shock and Vibration vol 2014 ArticleID 741765 15 pages 2014

[15] T Sangpet S Kuntanapreeda and R Schmidt ldquoHystereticnonlinearity observer design based on Kalman filter for piezo-actuated flexible beamsrdquo International Journal of Automationand Computing vol 11 no 6 pp 627ndash634 2014

[16] X Dong Z Peng W Zhang H Hua and G Meng ldquoResearchon spillover effects for vibration control of piezoelectric smartstructures by ANSYSrdquo Mathematical Problems in Engineeringvol 2014 Article ID 870940 8 pages 2014

[17] A Montazeri J Poshtan and A Yousefi-Koma ldquoDesign andanalysis of robust minimax LQG controller for an experimentalbeam considering spill-over effectrdquo IEEE Transactions on Con-trol Systems Technology vol 19 no 5 pp 1251ndash1259 2011

[18] R Padhi and S F Ali ldquoAn account of chronological devel-opments in control of distributed parameter systemsrdquo AnnualReviews in Control vol 33 no 1 pp 59ndash68 2009

[19] S Zhao H Lin and Z Xue ldquoSwitching control of closedquantum systems via the Lyapunov methodrdquo Automatica vol48 no 8 pp 1833ndash1838 2012

[20] S Kuntanapreeda and P M Marusak ldquoNonlinear extendedoutput feedback control for CSTRs with van de Vusse reactionrdquoComputers amp Chemical Engineering vol 41 pp 10ndash23 2012

[21] B Zhang K Liu and J Xiang ldquoA stabilized optimal nonlinearfeedback control for satellite attitude trackingrdquo Aerospace Sci-ence and Technology vol 27 no 1 pp 17ndash24 2013

[22] A Yang W Naeem G W Irwin and K Li ldquoStability anal-ysis and implementation of a decentralized formation controlstrategy for unmanned vehiclesrdquo IEEE Transactions on ControlSystems Technology vol 22 no 2 pp 706ndash720 2014

[23] M Dadfarnia N Jalili B Xian and D M Dawson ldquoAlyapunov-based piezoelectric controller for flexible cartesianrobot manipulatorsrdquo Journal of Dynamic Systems Measurementand Control vol 126 no 2 pp 347ndash358 2004

[24] J-M Coron B drsquoAndrea-Novel and G Bastin ldquoA strict Lya-punov function for boundary control of hyperbolic systems ofconservation lawsrdquo IEEE Transactions on Automatic Controlvol 52 no 1 pp 2ndash11 2007

[25] M-B Cheng V Radisavljevic and W-C Su ldquoSliding modeboundary control of a parabolic PDE system with parametervariations and boundary uncertaintiesrdquoAutomatica vol 47 no2 pp 381ndash387 2011

[26] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems andControl Letters vol 61 no 11 pp 1069ndash1078 2012

[27] B-Z Guo and F-F Jin ldquoThe active disturbance rejection andsliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbancerdquoAutomatica vol 49 no 9 pp 2911ndash2918 2013

[28] A Luemchamloey and S Kuntanapreeda ldquoActive vibrationcontrol of flexible beambased on infinite-dimensional lyapunovstability theory an experimental studyrdquo Journal of ControlAutomation and Electrical Systems vol 25 no 6 pp 649ndash6562014

[29] W He S Zhang and S S Ge ldquoAdaptive boundary control of anonlinear flexible string systemrdquo IEEE Transactions on ControlSystems Technology vol 22 no 3 pp 1088ndash1093 2014

[30] A Levant ldquoRobust exact differentiation via sliding modetechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998

[31] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[32] XWang and B Shirinzadeh ldquoHigh-order nonlinear differentia-tor and application to aircraft controlrdquoMechanical Systems andSignal Processing vol 46 no 2 pp 227ndash252 2014

[33] I Salgado I Chairez O Camacho and C Yanez ldquoSuper-twisting sliding mode differentiation for improving PD con-trollers performance of second order systemrdquo ISA Transactionsvol 53 no 4 pp 1096ndash1106 2014

[34] S Y Wang ldquoA finite element model for the static and dynamicanalysis of a piezoelectric bimorphrdquo International Journal ofSolids and Structures vol 41 no 15 pp 4075ndash4096 2004

[35] D L Logan A First Course in the Finite Element MethodWadsworth Group 2002

[36] J E Slotine Applied Nonlinear Control Prentice-Hall 1991[37] H K Khalil Nonlinear Systems Prentice-Hall 1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Engineering

0 05 1 15 2 25 3 35 4

0

5

10

15

20

Time (s)

minus20

minus15

minus10

minus5y(t)

(Vs

)

(a)

0 05 1 15 2 25 3 35 4

0

05

1

Time (s)

minus1

minus05

u(t)

(V)

(b)

0 05 1 15 2 25 3 35 40

001

002

003

004

005

006

Time (s)

k(t)

(V)

(c)

0 05 1 15 2 25 3 35 4

0

05

1

15

2

Time (s)

minus2

minus15

minus1

minus05w(Lt)

(mm

)

(d)

Figure 8 Experimental result (the controller was activated at 119905 = 1 sec) (a) feedback signal (b) control signal (c) adaptive variable and (d)tip deflection

Acknowledgments

This work was supported by the Royal Golden Jubilee PhDprogram of the Thailand Research Fund The authors alsowould like to thank anonymous referees for their valuablecomments

References

[1] Z Mohamed J M Martins M O Tokhi J Sa da Costa andM A Botto ldquoVibration control of a very flexible manipulatorsystemrdquo Control Engineering Practice vol 13 no 3 pp 267ndash2772005

[2] M Azadi S A Fazelzadeh M Eghtesad and E Azadi ldquoVibra-tion suppression and adaptive-robust control of a smart flexiblesatellite with three axes maneuveringrdquo Acta Astronautica vol69 no 5-6 pp 307ndash322 2011

[3] T P Sales D A Rade and L C G De Souza ldquoPassive vibra-tion control of flexible spacecraft using shunted piezoelectrictransducersrdquo Aerospace Science and Technology vol 29 no 1pp 403ndash412 2013

[4] C C Fuller Active Control of Vibration Academic Press 1996[5] A Preumont Vibration Control of Active Structures Kluwer

Academic 1997

[6] A M Aly ldquoVibration control of buildings using magnetorheo-logical damper a new control algorithmrdquo Journal of Engineer-ing vol 2013 Article ID 596078 10 pages 2013

[7] A Moutsopoulou G E Stavroulakis and A Pouliezos ldquoInno-vation in active vibration control strategy of intelligent struc-turesrdquo Journal of Applied Mathematics vol 2014 Article ID430731 14 pages 2014

[8] B Bandyopadhyay T C Manjunath and M Umapathy Mod-eling Control and Implementation of Smart Structures A FEM-State Space Approach Springer Berlin Germany 2007

[9] A R Tavakolpour M Mailah I Z Mat Darus and O TokhildquoSelf-learning active vibration control of a flexible plate struc-ture with piezoelectric actuatorrdquo Simulation Modelling Practiceand Theory vol 18 no 5 pp 516ndash532 2010

[10] H Wang X-H Shen L-X Zhang and X-J Zhu ldquoActivevibration suppression of flexible structure using a LMI-basedcontrol patchrdquo Journal of Vibration and Control vol 18 no 9pp 1375ndash1379 2012

[11] Z Qiu X Zhang and C Ye ldquoVibration suppression of a flexiblepiezoelectric beam using BP neural network controllerrdquo ActaMechanica Solida Sinica vol 25 no 4 pp 417ndash428 2012

[12] T Sangpet S Kuntanapreeda and R Schmidt ldquoImprovingdelay-margin of noncollocated vibration control of piezo-actuated flexible beams via a fractional-order controllerrdquo Shockand Vibration vol 2014 Article ID 809173 8 pages 2014

Journal of Engineering 9

[13] F An W Chen and M Shao ldquoDynamic behavior of time-delayed acceleration feedback controller for active vibrationcontrol of flexible structuresrdquo Journal of Sound and Vibrationvol 333 no 20 pp 4789ndash4809 2014

[14] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration control of a cantilever beam with real-timeparameter estimationrdquo Shock and Vibration vol 2014 ArticleID 741765 15 pages 2014

[15] T Sangpet S Kuntanapreeda and R Schmidt ldquoHystereticnonlinearity observer design based on Kalman filter for piezo-actuated flexible beamsrdquo International Journal of Automationand Computing vol 11 no 6 pp 627ndash634 2014

[16] X Dong Z Peng W Zhang H Hua and G Meng ldquoResearchon spillover effects for vibration control of piezoelectric smartstructures by ANSYSrdquo Mathematical Problems in Engineeringvol 2014 Article ID 870940 8 pages 2014

[17] A Montazeri J Poshtan and A Yousefi-Koma ldquoDesign andanalysis of robust minimax LQG controller for an experimentalbeam considering spill-over effectrdquo IEEE Transactions on Con-trol Systems Technology vol 19 no 5 pp 1251ndash1259 2011

[18] R Padhi and S F Ali ldquoAn account of chronological devel-opments in control of distributed parameter systemsrdquo AnnualReviews in Control vol 33 no 1 pp 59ndash68 2009

[19] S Zhao H Lin and Z Xue ldquoSwitching control of closedquantum systems via the Lyapunov methodrdquo Automatica vol48 no 8 pp 1833ndash1838 2012

[20] S Kuntanapreeda and P M Marusak ldquoNonlinear extendedoutput feedback control for CSTRs with van de Vusse reactionrdquoComputers amp Chemical Engineering vol 41 pp 10ndash23 2012

[21] B Zhang K Liu and J Xiang ldquoA stabilized optimal nonlinearfeedback control for satellite attitude trackingrdquo Aerospace Sci-ence and Technology vol 27 no 1 pp 17ndash24 2013

[22] A Yang W Naeem G W Irwin and K Li ldquoStability anal-ysis and implementation of a decentralized formation controlstrategy for unmanned vehiclesrdquo IEEE Transactions on ControlSystems Technology vol 22 no 2 pp 706ndash720 2014

[23] M Dadfarnia N Jalili B Xian and D M Dawson ldquoAlyapunov-based piezoelectric controller for flexible cartesianrobot manipulatorsrdquo Journal of Dynamic Systems Measurementand Control vol 126 no 2 pp 347ndash358 2004

[24] J-M Coron B drsquoAndrea-Novel and G Bastin ldquoA strict Lya-punov function for boundary control of hyperbolic systems ofconservation lawsrdquo IEEE Transactions on Automatic Controlvol 52 no 1 pp 2ndash11 2007

[25] M-B Cheng V Radisavljevic and W-C Su ldquoSliding modeboundary control of a parabolic PDE system with parametervariations and boundary uncertaintiesrdquoAutomatica vol 47 no2 pp 381ndash387 2011

[26] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems andControl Letters vol 61 no 11 pp 1069ndash1078 2012

[27] B-Z Guo and F-F Jin ldquoThe active disturbance rejection andsliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbancerdquoAutomatica vol 49 no 9 pp 2911ndash2918 2013

[28] A Luemchamloey and S Kuntanapreeda ldquoActive vibrationcontrol of flexible beambased on infinite-dimensional lyapunovstability theory an experimental studyrdquo Journal of ControlAutomation and Electrical Systems vol 25 no 6 pp 649ndash6562014

[29] W He S Zhang and S S Ge ldquoAdaptive boundary control of anonlinear flexible string systemrdquo IEEE Transactions on ControlSystems Technology vol 22 no 3 pp 1088ndash1093 2014

[30] A Levant ldquoRobust exact differentiation via sliding modetechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998

[31] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[32] XWang and B Shirinzadeh ldquoHigh-order nonlinear differentia-tor and application to aircraft controlrdquoMechanical Systems andSignal Processing vol 46 no 2 pp 227ndash252 2014

[33] I Salgado I Chairez O Camacho and C Yanez ldquoSuper-twisting sliding mode differentiation for improving PD con-trollers performance of second order systemrdquo ISA Transactionsvol 53 no 4 pp 1096ndash1106 2014

[34] S Y Wang ldquoA finite element model for the static and dynamicanalysis of a piezoelectric bimorphrdquo International Journal ofSolids and Structures vol 41 no 15 pp 4075ndash4096 2004

[35] D L Logan A First Course in the Finite Element MethodWadsworth Group 2002

[36] J E Slotine Applied Nonlinear Control Prentice-Hall 1991[37] H K Khalil Nonlinear Systems Prentice-Hall 1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Engineering 9

[13] F An W Chen and M Shao ldquoDynamic behavior of time-delayed acceleration feedback controller for active vibrationcontrol of flexible structuresrdquo Journal of Sound and Vibrationvol 333 no 20 pp 4789ndash4809 2014

[14] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration control of a cantilever beam with real-timeparameter estimationrdquo Shock and Vibration vol 2014 ArticleID 741765 15 pages 2014

[15] T Sangpet S Kuntanapreeda and R Schmidt ldquoHystereticnonlinearity observer design based on Kalman filter for piezo-actuated flexible beamsrdquo International Journal of Automationand Computing vol 11 no 6 pp 627ndash634 2014

[16] X Dong Z Peng W Zhang H Hua and G Meng ldquoResearchon spillover effects for vibration control of piezoelectric smartstructures by ANSYSrdquo Mathematical Problems in Engineeringvol 2014 Article ID 870940 8 pages 2014

[17] A Montazeri J Poshtan and A Yousefi-Koma ldquoDesign andanalysis of robust minimax LQG controller for an experimentalbeam considering spill-over effectrdquo IEEE Transactions on Con-trol Systems Technology vol 19 no 5 pp 1251ndash1259 2011

[18] R Padhi and S F Ali ldquoAn account of chronological devel-opments in control of distributed parameter systemsrdquo AnnualReviews in Control vol 33 no 1 pp 59ndash68 2009

[19] S Zhao H Lin and Z Xue ldquoSwitching control of closedquantum systems via the Lyapunov methodrdquo Automatica vol48 no 8 pp 1833ndash1838 2012

[20] S Kuntanapreeda and P M Marusak ldquoNonlinear extendedoutput feedback control for CSTRs with van de Vusse reactionrdquoComputers amp Chemical Engineering vol 41 pp 10ndash23 2012

[21] B Zhang K Liu and J Xiang ldquoA stabilized optimal nonlinearfeedback control for satellite attitude trackingrdquo Aerospace Sci-ence and Technology vol 27 no 1 pp 17ndash24 2013

[22] A Yang W Naeem G W Irwin and K Li ldquoStability anal-ysis and implementation of a decentralized formation controlstrategy for unmanned vehiclesrdquo IEEE Transactions on ControlSystems Technology vol 22 no 2 pp 706ndash720 2014

[23] M Dadfarnia N Jalili B Xian and D M Dawson ldquoAlyapunov-based piezoelectric controller for flexible cartesianrobot manipulatorsrdquo Journal of Dynamic Systems Measurementand Control vol 126 no 2 pp 347ndash358 2004

[24] J-M Coron B drsquoAndrea-Novel and G Bastin ldquoA strict Lya-punov function for boundary control of hyperbolic systems ofconservation lawsrdquo IEEE Transactions on Automatic Controlvol 52 no 1 pp 2ndash11 2007

[25] M-B Cheng V Radisavljevic and W-C Su ldquoSliding modeboundary control of a parabolic PDE system with parametervariations and boundary uncertaintiesrdquoAutomatica vol 47 no2 pp 381ndash387 2011

[26] Y F Shang and G Q Xu ldquoStabilization of an Euler-Bernoullibeam with input delay in the boundary controlrdquo Systems andControl Letters vol 61 no 11 pp 1069ndash1078 2012

[27] B-Z Guo and F-F Jin ldquoThe active disturbance rejection andsliding mode control approach to the stabilization of the Euler-Bernoulli beam equation with boundary input disturbancerdquoAutomatica vol 49 no 9 pp 2911ndash2918 2013

[28] A Luemchamloey and S Kuntanapreeda ldquoActive vibrationcontrol of flexible beambased on infinite-dimensional lyapunovstability theory an experimental studyrdquo Journal of ControlAutomation and Electrical Systems vol 25 no 6 pp 649ndash6562014

[29] W He S Zhang and S S Ge ldquoAdaptive boundary control of anonlinear flexible string systemrdquo IEEE Transactions on ControlSystems Technology vol 22 no 3 pp 1088ndash1093 2014

[30] A Levant ldquoRobust exact differentiation via sliding modetechniquerdquo Automatica vol 34 no 3 pp 379ndash384 1998

[31] A Levant ldquoHigher-order sliding modes differentiation andoutput-feedback controlrdquo International Journal of Control vol76 no 9-10 pp 924ndash941 2003

[32] XWang and B Shirinzadeh ldquoHigh-order nonlinear differentia-tor and application to aircraft controlrdquoMechanical Systems andSignal Processing vol 46 no 2 pp 227ndash252 2014

[33] I Salgado I Chairez O Camacho and C Yanez ldquoSuper-twisting sliding mode differentiation for improving PD con-trollers performance of second order systemrdquo ISA Transactionsvol 53 no 4 pp 1096ndash1106 2014

[34] S Y Wang ldquoA finite element model for the static and dynamicanalysis of a piezoelectric bimorphrdquo International Journal ofSolids and Structures vol 41 no 15 pp 4075ndash4096 2004

[35] D L Logan A First Course in the Finite Element MethodWadsworth Group 2002

[36] J E Slotine Applied Nonlinear Control Prentice-Hall 1991[37] H K Khalil Nonlinear Systems Prentice-Hall 1996

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of