research article active vibration control of plate partly...

Research ArticleActive Vibration Control of Plate Partly Treated withACLD Using Hybrid Control

Dongdong Zhang and Ling Zheng

State Key Laboratory of Mechanical Transmission Chongqing University Chongqing 400030 China

Correspondence should be addressed to Ling Zheng zlingcqueducn

Received 26 December 2013 Accepted 23 June 2014 Published 14 July 2014

Academic Editor N Ananthkrishnan

Copyright copy 2014 D Zhang and L Zheng 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

A finite element model of plate partly treated with ACLD treatments is developed based on the constitutive equations of elasticpiezoelectric viscoelastic materials and Hamiltonrsquos principleThe Golla-Hughes-Mctavish (GHM)method is employed to describethe frequency-dependent characteristics of viscoelasticmaterial (VEM) Amodel reduction is completed by using iterative dynamiccondensation and balance model reduction method to design an effective control system The emphasis is concerned on hybrid(combined feedbackfeedforward) control system to attenuate the vibration of plates with ACLD treatments The optimal linearquadratic Gaussian (LQG) controller is considered as a feedback channel and the adaptive filtered-reference LMS (FxLMS)controller is used as a feedforward channel They can be utilized individually or in a hybrid way to suppress the vibration ofplateACLD system The results show that the hybrid controller which combines feedbackfeedforward together can reduce thedisplacement amplitude of plateACLD system subjected to a complicated disturbance substantially without requiringmore controleffort Furthermore the hybrid controller has more rapid and stable convergence rate than the adaptive feedforward FxLMScontroller Meanwhile perfect robustness to phase error of the cancellation path in feedforward controller and the weight matricesin feedback LQG controller is demonstrated in proposed hybrid controller Therefore its application in structural engineering canbe highly appreciated

1 Introduction

Vibration and noise control is of great interest in manyindustrial structures like airplane passenger cabin vehiclebody structure and submarine hull The thin-wall structureis a key part in the bodycabinhull structure The vibratingthin-wall structures like plates and shells which are mainlydisturbed by the harmonic disturbance induced from theengine and other rotating machines as well as stochasticdisturbance make radiation noise into the passenger cabininfluencing the performance of the structure

Active vibration control (AVC) [1] which is well-knownfor its efficiency in low frequency band where traditionalpassive strategies cannot work well has been applied tovarious structures over the past two decades Active vibra-tion control of simple structure such as beams [2] andplates [3] was researched broadly in the open literaturesThakkar and Ganguli [4] investigated the twist vibration

control of helicopter rotor blade by using the piezoelectricactuators Rao et al [5] designed a 119867

infincontroller and

demonstrated experimentally the effectiveness of multimodevibration control in the composite fin-tip of aircraft Kwakand Yang [6] studied the suppression of vibration of ring-stiffened cylindrical shell in contact with external fluid usingpiezoelectric sensor and actuator

Active constrained layer damping (ACLD) is a hybridactive-passive vibration control technique [7] which com-bines the advantages of conventional passive constrainedlayer damping (PCLD) and active vibration control (AVC)Therefore in the same damping treatment the broaderband vibration suppression can be achieved through ACLDtreatments In recent decades extensive efforts had beenmade to reduce the vibration of the thin-wall structureusing ACLD treatments Baz and Ro as the pioneer-ing researchers demonstrated the feasibility of using theACLD in controlling the vibration of rotating beams with

Hindawi Publishing CorporationInternational Journal of Aerospace EngineeringVolume 2014 Article ID 432970 12 pageshttpdxdoiorg1011552014432970

2 International Journal of Aerospace Engineering

the ultimate goal of extending its application of attenuatingthe structural vibration of rotor blades [8]The vibration con-trol of shells [9 10]was also investigated to extend the applica-tion of ACLD in the cabin of aircraft and submarine In thesestudies the PVDF (polyvinylidene fluoride) was served as thesensors and actuators They use the simple proportional andderivative feedback of the transverse deflection to suppressthe vibration of the base structure theoretically and exper-imentally To improve the force transmissibility betweenpiezoelectric actuator and the base structure Liao and Wang[11] proposed an enhanced active constraining layer (EACL)and Gao and Liao [12] extended this configuration Kumaret al [13] added the stand-off layer (SOL) between theviscoelastic layer and the base structure to increase theviscoelastic strain and enhance the effect of the active force Inrecent years piezoelectric fiber-reinforced composite (PFRC)materials [14] and piezoelectric composites (PZC) [15 16]which have a wide range of effective materials propertieswere introduced in the ACLD treatments to serve as activeconstrained layer Shah and Ray [17] focused on investigatingthe effect of the piezoelectric fiber orientation angle in theconstrained layer on the performance of ACLD patchesIn these studies the emphasis was placed on the vibrationcharacteristics analysisenhancement of the performance ofthe ACLD treatments

In the open literatures much attention was paid onactive control methods in the structural activeactive-passivevibration control The optimal feedback control [18] velocityfeedback control [19] fuzzy logic control [3] and the 119867

infin

control [5] were usually employed in active vibration con-trol (AVC) Viswamurthy and Ganguli [20] compared theglobal harmonic control with local blade optimal controlapplied to vibration control of helicopter For active-passiveACLDstructure Baz [21] presented a variational mathemati-calmodel for beams and a globally stable boundary controllerwas investigated to get the high damping characteristics overbroad frequency range Meanwhile Baz [22] developed a1198672robust controller based on the transfer function model

of the ACLD beam The control strategy was stable inthe presence of parameter uncertainty as well as ensuredoptimal disturbance rejection capabilities Furthermore the119867 infinity robust control strategy was utilized by Crassidiset al [23] to maximize disturbance rejection capabilitiesover a desired frequency band Liu et al [24] developed a119867 infinity controller based on the reduced finite elementmodel Rodriguez [25] made a comparison of sliding modecontrol (SMC) with state-feedback control (LQR) appliedto a partially treated ACLD beam In these studies thecontrol strategies can be classified into feedback controltheoriesThese control strategies have been shown to bemoresuitable in applicationswhere the structure is disturbed by thestochastic or impulsive disturbances

Feedforward control which is more effective in the casewhere the deterministic or correlated information about thedisturbance is known is another popular control strategy Itwaswidely used in active vibration and noise control field [2627] However only a few feedforward theories were utilizedin structural vibration control with ACLD treatments Cao[28] employed the feedforward LMS algorithm to control

Plate

Piezoelectric constraining layerViscoelastic

layer

Piezoelectric sensor layer

Figure 1 Schematic drawing of the ACLDplate structure

the vibration of a plate treated with ACLD treatments Thefinite element model was developed and ADF method wasused to identify the performance of VEM Furthermorehybrid control of feedback and feedforward control theorieswere also successfully used to noise and vibration suppression[29 30] However it was rarely investigated for structuralvibration control with ACLD treatments Vasques validatedthe effectiveness of the hybrid control strategies for ACLDbeam [31]

In this paper the focus is placed on combining individualfeedback control based on linear quadratic Gaussian (LQG)and feedforward control based on adaptive filtered-referenceLMS (FxLMS) to develop a hybrid controller for the vibra-tion control of plates treated with ACLD treatments Theperformance of vibration suppression subjected to differentcontrol strategies is compared and the effect of parameterson vibration suppression of the plate with ACLD treatment isfurther investigated

This paper is organized in the following five sections Abrief background is introduced in Section 1 The dynamicmodels including the finite element model of the platetreated with ACLD patches and GHM model of VEM aredeveloped in Section 2 In Section 3 the feedback controllerthe feedforward controller and the hybrid controller aredesigned A numerical example is discussed in Section 4Finally the main results and future work are summarized inSection 5

2 The ACLDPlate System Model Construction

21 Finite Element Model The ACLDplate system is illus-trated in Figure 1 The viscoelastic damping layer is sand-wiched between two piezoelectric layers which serve aspiezosensor and piezoactuator respectively The three layersare bonded to the base plate acting as a smart constraininglayer damping It is assumed that the piezosensor and thebase plate are perfectly bonded together such that they can bereduced to a single equivalent layer Some other assumptionsare made in [32]

The ACLDplate elements considered are four-node two-dimensional elements Each node has seven degrees offreedom to describe the longitudinal displacements 119906

119888and

V119888of the constrained layer the longitudinal displacements

119906119901and V119901of the base plate the transverse deflection 119908 and

the slopes 119909 and 119910 of the deflection line The longitudinaldisplacement functions are expressed as 4-term polynomialsand transverse deflection119908 is expressed as a 12-term polyno-mial Therefore the element deflection vector is given by

Δ(119890)

= Δ1Δ2Δ3Δ4119879

(1)

where Δ119894= 119906119888119894

V119888119894119906119901119894

V119901119894119908119894120579119909119894120579119910119894119879 for 119894 = 1 2 3 4

International Journal of Aerospace Engineering 3

Applying Hamiltonrsquos principle the equations of motionfor an ACLD element can be written as

M(119890)Δ(119890) + K(119890)Δ(119890) = F(119890)119889+ F(119890)119888 (2)

where M(119890) = M(119890)119901+ M(119890)119888+ M(119890)V K(119890) = K(119890)

119901+ K(119890)119888+

K(119890)V + 119866VK(119890)

V120573 M(119890)

119901 M(119890)119888 and M(119890)V stand for the element

mass matrices for base plate the constrained layer and theviscoelastic layer K(119890)

119901K(119890)119888 and K(119890)V stand for the element

stiffness matrices for base plate the constrained layer andthe viscoelastic layer K(119890)V120573 is the shear stiffness matrix forviscoelastic layer 119866V is the shear modulus of the viscoelasticmaterials F(119890)

119889is the external force vector and F(119890)

119888is the

control force produced by the piezoelectric actuatorsAssembling the condensed system for all elements yields

the dynamic equation of the plate with ACLD treatments

MΔ + K + 119866VKV120573Δ = F119889+ F119888 (3)

whereK is the global stiffness matrix without considering theshear stiffness of the VEM

22 Viscoelastic Damping Model The frequency- or time-dependent behavior of the viscoelastic material is character-ized by using Golla-Hughes-Mctavish (GHM) method [24]In this approach the shear modulus of VEM is expressed asa series of ldquominioscillatorrdquo terms in the Laplace domain

119904119866 (119904) = 119866infin

[1 +sum

119896

120572119896

1199042

+ 2120589119896120596119896119904

1199042 + 2120589119896120596119896119904 + 1205962

] (4)

where 119866infin corresponds to the equilibrium value of themodulus constants 120572

119896 120589119896 and 120596

119896govern the shape of the

modulus function over the complex 119904-domain A columnmatrix of dissipation coordinates is introduced

1198851015840

(119904) =1205962

1199042 + 2120589120596119904 + 1205962Δ (119904) (5)

Considering a three-term GHM expression (3) can berewritten as follows

M1015840X + C1015840X + K1015840X = F1015840d + F1015840c (6)

where each vector and matrix can be expressed as

M1015840 =

[[[[[[[[

[

M 0 0 0

01205721

1205962

1

Λ 0 0

0 01205722

1205962

2

Λ 0

0 0 01205723

1205962

3

Λ

]]]]]]]]

]

C1015840 =

[[[[[[[[[

[

0 0 0 0

0212057211205771

1205962

1

Λ 0 0

0 0212057221205772

1205962

2

Λ 0

0 0 0212057231205773

1205962

3

Λ

]]]]]]]]]

]

K1015840 =

[[[[[[[

[

119866infinKV120573(1 +

3

sum

119896=1

120572119896) + Kk minus1205721R minus120572

2R minus120572

3R

minus1205721R119879 120572

1Λ 0 0

minus1205722R119879 0 120572

2Λ 0

minus1205723R119879 0 0 120572

3Λ

]]]]]]]

]

X = ΔZ F1015840d = Fd0 F1015840c =

Fc0

(7)

where KV120573 = RVΛVR119879V Λ = 119866infin

ΛV R = RVΛ Z = R119879V Z1015840

ΛV is a diagonal matrix of the positive eigenvalues of matrixKV120573 and RV is the eigenvectors matrix corresponding to theeigenvalues For simplicity the global equation of motion canbe rewritten as

MX + CX + KX = Fd + Fc (8)

23 Model Reduction and State Space Design As well knownthe scale of the finite element model is increased largelydue to the introduction of the dissipation coordinates fromGHM model This requires much computational effort tocalculate the dynamic response of the plates with ACLDtreatments Otherwise not all the state variables in themodelare controllable and observable Therefore model reductionis necessary to design an effective controller Here an iterativedynamic condensation [33] is used to reduce the scale of thefinite element model

Equation (8) can be rewritten as

[M119898119898

M119898119904

M119904119898

M119904119904

]X119898

X119904

+ [C119898119898

C119898119904

C119904119898

C119904119904

]X119898

X119904

+ [K119898119898

K119898119904

K119904119898

K119904119904

]X119898

X119904

= F119898

F119904

(9)


Here amatrix R is defined to relate themaster dofs withthe slave dofs and after 119894 iterations the reduced dynamicequation is

M(119894)119877X + C(119894)

119877X + K(119894)

119877X = F(119894)

119877 (10)

where

M(119894)119877= M119898119898+ R1015840(119894)

119879

M119904119898

+M119898119904R1015840(119894) + R1015840(119894)

119879

M119904119904R1015840(119894)C(119894)

119877

= C119898119898+ R1015840(119894)

119879

C119904119898+ C119898119904R1015840(119894)

+ R1015840(119894)119879

C119904119904R1015840(119894)

K(119894)119877= K119898119898+ R1015840(119894)

119879

K119904119898

+ K119898119904R1015840(119894) + R1015840(119894)

119879

K119904119904R1015840(119894)F(119894)

119877

= F119898+ R1015840(119894)

119879

F119904

R1015840(119894+1) = minusK119904119904

minus1

(M119904119904R1015840(119894)M(119894)

119877

minus1

K(119894)119877minus K119904119898)

R1015840(0) = minusKminus1119904119904K119904119898

(11)

Based on above reduced model the state space form ofthe ACLDplate is derived as follows

x (119905) = Ax (119905) + B119889f119889+ B119888f119888

y (119905) = Cx (119905) + k(12)

where the state space vector x(t) is chosen as X119898

X119898119879

A is system matrix and B119889and B

119888are disturbance input

matrices and the control force distribution matrices C is theoutput matrix and y(t) is the output vector k represents themeasurement noise

A further model reduction is performed with the balancemethod in the state space to guarantee the control systemcontrollable and observable The details are shown in [34]

3 Control System Architecture Design

31 Feedback Control The ultimate aim of the feedbackcontrol is to reduce the motion of the system to the greatestpossible extent in that case the control system is said toact as a regulator Here the linear quadratic regulator (LQR)is designed to control the vibration of plates with ACLDtreatments Cost function or performance index is given by

119869 = int

infin

0

(x(119905)119879Q119909x (119905) + f

119888

119879

(119905)Rf119888(119905)) 119889119905 (13)

or

119869 = int

infin

0

(y(119905)119879Q119910y (119905) + f

119888

119879

(119905)Rf119888(119905)) 119889119905 (14)

where Q119909 Q119910 and R are the state variable the output and

the control input weighting matrices respectivelyThe control input can be calculated using the state

feedback so that

f119888= minusK119888x (119905) (15)

The optimal feedback gain matrix K119888is given by

K119888= Rminus1B119879

119888P (16)

where P is the solution of the famous Riccati equationTherefore considering the control law in (15) and substi-

tuting it into the state space equation of the reduced modelthe closed-loop state equation is given by

x (119905) = (Α minus B119888K119888) x (119905) + B

119889f119889 (17)

In above equation it is assumed that all the state variablescan be measured However that is not always available ina real system an optimal design methodology that onlyutilizes noise partial state information is required Thelinear quadratic Gaussian (LQG) methodology provides ameans of designing such controllers The LQG controller isa combination of a LQR and a Kalman filter a fact known asthe separation principle

The Kalman filter is an optimal state estimator which canminimize the effects of process and measurement noise In(18) the process noise f

119889and measurement noise k

119889are both

assumed to be white and have a Gaussian probability densityfunction Meanwhile they are assumed to be uncorrelatedwith the inputs The correlation properties of the plantand measurement noise vectors are given by the correlationmatrices

119864 (f119889f119879119889) = 119876

119889 119864 (k

119889k119879119889) = 119877119889 (18)

where 119864 denotes the expectation operatorThese matrices areused as weighting matrices of the steady-state Kalman filterproblem [20]

A state estimate x1015840(119905) that minimizes the steady-stateerror covariance is constructed by the Kalman filter

119875 = lim119905rarrinfin

119864([x (119905) minus x1015840 (119905)] [x (119905) minus x1015840 (119905)]119879

) (19)

In this way a new state space model is generated

x1015840 = Ax1015840 + B119888f119891119889119888+ L (yV minus Cx1015840)

[y1015840x1015840] = [

CI] x1015840

(20)

where L is the filter gain it is determined by solving analgebraic Riccati equation yV is the sum of the output and themeasurement noise

32 Feedforward Control The filtered-reference LMS(FxLMS) algorithm is widely studied and has been usedin many active vibration control areas due to its easyimplementation and remarkable performance In a practical


Process noisew(k)

u(k)

+

++

+ ++

+

+

r(k)

Controlinput

Cancellation pathestimate

r998400(k)

Plant

Pu(z)

Disturbance path

Cancellation path

LQG controller

FIR adaptivecontrol filter hi

Filtered-referenceLMS algorithm

yu(k)

y(k)

y(k)

y(k)

(k)

Measurementnoise

y120601(k)

u120601(k)

u120601(k) P120601(z)

P120601(z)

ufb120601 (k)

uff120601 (k)

Figure 2 Schematic diagram of the generalized plant for hybrid (combined feedbackfeedforward) controller

vibration control system a cancellation path exists whichhas an important effect on the control results In FxLMSalgorithm the model of the cancellation path is establishedThe primary disturbance 119903(119896) passes through the estimatedmodel of the cancellation path and a filtered-referencesignal 1199031015840(119896) which is correlated with the disturbance isproduced This signal is then adaptively filtered to generatethe necessary control action 119891119891119891

119888to cancel the effect of the

primary disturbanceAn adaptive finite impulse response (FIR) filter whose 119894th

coefficient at the 119896th sample time ℎ119894(119896) is updated using the

LMS algorithm is used in filtering process The coefficientsvector ℎ

119894(119896) and the filter output 119891119891119891

119888(119896) are obtained from

ℎ119894(119896) = ℎ

119894(119896 minus 1) + 2120583119890 (119896 minus 1) 119903

1015840

(119896 minus 1)

119891119891119891

119888(119896) =

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894)

(21)

where 120583 is the step size which has a significant influenceon the convergence rate of the LMS algorithm and 119873 is thenumber of filter coefficients (or the filter length)

The control signal has to pass through a part of thephysical system before the error sensor measures the outputThis physical path 119875

120601(119911) is called the cancelation path this

transfer characteristics also affect the performance of theFxLMS algorithm Therefore the output of the plant due tothe feedforward control input only 119910

120601(119896) is given by

119910120601(119896) =

119872minus1

sum

119895=0

119892119895

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894 minus 119895) (22)

where 119892119894is the discrete impulse response of the cancelation

path 119875120601(119911) which is assumed to be of order119872 Considering

the measurement noise V(119896) the output of the system 119910(119896)can be written as

119910 (119896) = 119910119906(119896) +

119872minus1

sum

119895=0

119892119895

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894 minus 119895) + V (119896) (23)

where 119910119906(119896) is the response due to the effects of the primary

disturbance alone

33 Hybrid Control In general the feedback control is moreappropriate for suppression of stochastic disturbance Mean-while it is susceptible to modeling errors By contrast theadaptive feedforward control is not susceptible to modelingerrors and inherently stable and it is more suitable to reducethe periodic or limited-band disturbanceTherefore it is nec-essary to develop a hybrid (combined feedbackfeedforward)control to attenuate vibration of plates with ACLD treatmentsagainst stochastic and harmonic disturbance simultaneouslyThe schematic diagram of the discrete-time generalized plantfor a hybrid SISO control system is shown in Figure 2 Itis seen that LQG control is considered as feedback channeland an adaptive filtered-reference LMS control is used asfeedforward channel

4 Numerical Simulation and Discussion

To the first bending mode of the plate a cantileverACLDplate as shown in Figure 3 is used as numericalexample to demonstrate the effectiveness of the above con-trollersThe emphasis is on the performance discussion of thecontrollers

41TheNumericalModel of the ACLDPlate System Thebaseplate with the size of 02times01m2 is partly treated with ACLDtreatments The main physical parameters of the base plate


Table 1 Geometrical and physical parameters of ACLDplate system

Aluminum P-5H ZN-1ℎ119901

00008m ℎ119888

00005m ℎV 0001m 1205891

1480119864119901

69GPa 119864119888

745 GPa 120583V 03 1205892

1216120583119901

03 120583119888

032 120588V 7895 kgm31205893

8104120588119901

2800 kgm3120588119888

7450 kgm3119866infin 554200 Pa 120596

1896200

11988931 11988932

186 times 10minus12 CN 120572

13960 120596

2927800

1205722

6569 1205963

7613001205723

1447

Figure 3 The schematic diagram of finite element grid forACLDplate

(aluminum) viscoelastic layer (ZN-1) and piezoelectric layer(P-5H) are listed in Table 1

The cantilever plateACLD is divided into 8 times 4 elementsFour ACLD composite elements are applied in the rootThe locations of the ACLD patches are decided accordingto the modal energy of the first mode [21] which is shownin Figure 4 A model reduction in which the deflection ofthe structure and displacements of the piezoelectric layer areselected to be master dofs and all others are slave dofsis performed according to the model reduction proceduresin Section 3 The bold point in Figure 3 is used as theexcited point and also the response point The comparisonof responses in frequency domain and time domain betweenthe original model and the reduced model is presented inFigure 5 It is clear that the dynamic response of the reducedmodel is nearly coordinate with the dynamic response of theoriginalmodel A SISO control system is considered to designindividual and hybrid controller The input is the controlvoltage applied to the ACLD treatments and the output isselected to be the displacement of the response point of theplate which is assumed to be measurable

42 The Control Results of the Feedback Controller The LQGfeedback controller is tested to mitigate the effects of stochas-tic disturbances A white noise mechanical disturbance ismodeled as process noise (input force disturbance) Theprocess noise is such a signal with 119864(119908119908119879) = 001119873

2 andthe measurement noise is a signal with 119864(119908119908119879) = 10minus101198812The output weighting matrix is set to be 119876 = 1 times 10

6 andthe control weighting matrix 119877 = 1 times 10minus5 In order to avoidthe depolarization the maximum control voltage applied topiezoelectric actuators is set to be 150V The control resultsand the control voltage are presented in Figure 6

13

57

911

510

1520

0

001

002

003

004

Mode number 1

Figure 4 The modal strain energy of the first mode

As can be seen when compared to the PCLDplate sys-tem theACLD treatments can reduce the displacement of theresponse point of the plate significantly The control voltageis limited within 30VThismeans that the proposed feedbackcontrol is effective against the stochastic disturbance

43 The Control Results of the Feedforward Controller Thefeedforward controller based on adaptive FxLMS algorithmis designed to attenuate the effects of the harmonic point-force disturbance on the ACLDplate system in Figure 2 Thedisturbance frequency is close to the first natural frequencyof the ACLDplate system For convenience the estimationmodel of the plant cancellation path is substituted by thecontrol path (control voltage to measured displacement) inthe state space model After that the FxLMS algorithm isimplemented The filter length 119871 is set to be 32 and thestep size 119906 = 001 in the FxLMS controller The results arepresented in Figure 7

It can be seen that the displacement of the plate withACLD treatment is attenuated from 94mm to 008mm atthe end time of 20 s The control voltage is close to 50VThe vibration attenuation is improved substantially by ACLDtreatments when the adaptive FxLMS algorithm is applied Itis clear that the FxLMS algorithm is very useful to attenuatethe vibration of ACLDplate system excited by harmonicdisturbance


Table 2 The convergence situation for hybrid controller and FxLMS controller under different step sizes

Step size 120583001 005 01 05 10 30 35 100

Hybrid controller Y Y Y Y Y Y Y YFxLMS controller Y Y Y Y Y Y N NY convergence N no convergence

20 40 60 80 100 120

Mag

nitu

de (d

B)

Bode diagram

Frequency (Hz)

Full modelReduced model


0 05 1 15 2 25 3

0

001

002Impulse response

Am

plitu

de (m

)

Time (s)

minus40

minus60

minus80

minus100

minus120

minus001

minus002

Figure 5 The frequency and time domain responses of the original and reduced model of the ACLDplate

0 2 4 6 8 10

0

1

2

3

Time (s)

Am

plitu

de (m

)

PCLDACLD

minus1

minus2

minus3

times10minus4

(a)

Time (s)0 2 4 6 8 10

0

10

20

30

40

Con

trol v

olta

ge (V

)

minus10

minus20

minus30

minus40

(b)

Figure 6The displacement and control voltage for a white noise disturbance using the LQG feedback control (a) the displacement response(b) the control voltage

44 The Control Results of the Hybrid Controller Further-more it is particularly true that the system is often subject toboth the harmonic and stochastic disturbances in engineer-ing practice Therefore a hybrid controller which combinesthe LQG feedback control and the FxLMS feedforwardcontrol together is developed Such a hybrid controller is

further designed to attenuate the vibration of the plateACLDsystem subjected to the complicated disturbance In thehybrid controller the FxLMS control is employed as thefeedforward channel which aims to damp out the harmonicdisturbance Meanwhile the LQG feedback control is usedas a compensator The mixed disturbance is the sum of


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

(b)

Figure 7 The displacement control voltage for a single frequency harmonic disturbance using the adaptive filtered-reference LMSfeedforward control (a) the displacement response (b) the control voltage

0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

minus150

(b)

Figure 8 The displacement response for a mixed disturbance using the LQG feedback controller (a) the displacement response (b) thecontrol voltage

the two previous disturbancesThe results of individual LQGfeedback control feedforward control and their combinationare presented in Figures 8 9 and 10 respectively

When the hybrid control system is engaged the feedbackcontroller with the same weights and the feedforward con-troller with the same filter length are used and the step sizeis set to be 05 to accelerate the convergence rate It can beseen from Figures 8(a) and 10(a) that at the end of 20 s thedisplacement amplitude produced by the hybrid controlleris much smaller than that of the LQG feedback controllerIt is also apparent that the fluctuation disappears during theconvergence process and the convergence rate is improvedfrom 10 s to 5 s when the hybrid controller is applied insteadof the feedforward controller Figure 11 shows that the hybridcontrol produces a smallerMSE than the feedforward controlThe maximum control voltage in the feedforward controllerachieves 210V However the control effort in the hybrid

controller is only about 110V a little larger than the stablecontrol voltage of the two single controller

45 The Performance Discussions to the Controllers Table 2lists the convergence situations for hybrid controller andFxLMS controller under different step sizes It can be seenthat the hybrid controller can converge over a larger range ofstep size This implies that the hybrid controller has a betterrobustness than the FxLMS controller

The effect of different step sizes on the performanceof ACLDplate with the FxLMS and hybrid controller isdiscussed It is seen in Figure 12 that when the step size isincreasedMSE decreases rapidly meanwhile the fluctuationincreasesThismeans that convergence process is not smoothanymore which results in the fluctuation of the controlvoltage shown in Figure 9(b) On the other hand at the end


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

200

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus200

(b)

Figure 9 The displacement response for a mixed disturbance using the FxLMS feedforward controller (a) the displacement response (b)the control voltage

0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

150

50

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus150

minus50

(b)

Figure 10 The displacement response for a mixed disturbance using the hybrid controller (a) the displacement response (b) the controlvoltage

of the iteration process the MSE approaches to the samevalue Figure 13 shows that MSE varies with iteration numberin hybrid control under different step sizes It can be seenclearly that the fluctuation is reduced substantially in theconvergence process and the convergence rate is improvedgreatly when the step size increases This results in a stablecontrol voltage in Figure 10(b)

In general the cancellation pathmodel of the feedforwardcontroller is obtained by the system identification algorithmTherefore the errors including amplitude error and phaseerror in the cancellation path has a significant effect on thecontroller performance Figures 14 and 15 show that the effectof phase errors of the cancellation path on the performanceof ACLDplate is considerable It is seen that when thephase error achieves 30∘ the control process in feedforwardcontrol will not converge anymore However when the phaseerror achieves 60∘ the control process is still convergent

in the hybrid control It can be concluded that the hybridcontrol is not sensitive to the phase error compared with thefeedforward control

Figure 16 shows the MSE using LQG feedback controlunder different119876matrix Figure 17 demonstrates the MSE byusing hybrid control under different119876matrix It can be foundthat the performance of the feedback control has a distinctfluctuation when different 119876 matrix is applied However theperformance of the hybrid control changes indistinctivelyThat implies that the hybrid control is more stable thanindividual LQG control

Based on the above discussions it can be inferred thatthe hybrid control will be a potential for vibration controlof aerospace and other structures The hybrid controllerdemonstrates a better performance than that of individualfeedbackfeedforward controller with different parametersMeanwhile even though one of individual controllers is out


005

004

003

002

001

00

5000 10000 15000 20000

MSE

Iteration number (n)

HybridFeedforwardFeedback

3

2

1

010000 15000 20000

times10minus3

Figure 11TheMSEusing the feedforward control feedback controland hybrid control

012

01

05

15

004

008

002

006

0

1

2

05000 10000

10000

15000

15000

20000

20000

MSE


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

Figure 12 The MSE using the feedforward control under differentstep sizes

of work the other will still be effective on suppressing thevibration of structure However it is reasonable that themodeling accuracy of the cancellation path will make aninfluence on performance for vibration control in practicalapplications

5 Conclusions

In this paper a finite element model of ACLDplate systemis developed The behavior of the viscoelastic material ismodeled by the Golla-Hughes-Mctavish (GHM) methodThe model reduction is carried out by means of dynamiccondensation and balance reduction methods respectivelyto design an effective control system The emphasis has been

0

1

2

3

4

5

6

MSE

0 5000 10000 15000 20000


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

u = 10

Figure 13 The MSE using the hybrid control under different stepsizes

004

008

002

006

0

MSE

0 5000 10000 15000 20000


Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Figure 14 The MSE using the feedforward control under differentphase errors of the cancellation path

placed on vibration control of ACLDplate using the hybridcontroller which is composed of the LQG feedback controllerand the adaptive FxLMS feedforward controller The per-formance of these controllers with different parameters isfurther discussed in detail

The results show that the reduction process is effective inlow frequency range The LQG feedback controller is veryuseful to attenuate the vibration of the ACLDplate systemagainst stochastic disturbance while the adaptive FxLMSfeedforward controller is more appropriate to attenuate thevibration of the ACLDplate system induced by harmonicdisturbance Furthermore the LQG feedback controller and


0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘

Figure 15 The MSE using the hybrid control under different phaseerrors of the cancellation path

2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106

Figure 16TheMSE using the LQG feedback control under different119876matrix

the adaptive FxLMS feedforward controller are combinedas the hybrid controller which can reduce substantially thedisplacement amplitude of plateACLD system subjected toa complicated disturbance without requiring more controleffort and is more stable and smooth On the other handthe hybrid controller demonstrates a rapider and more stableconvergence rate than the adaptive feedforward controllerMeanwhile the hybrid controller demonstrates much betterrobustness than individual LQG feedback control or adaptivefeedforward control

0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106

Figure 17 The MSE using the hybrid control under different 119876matrix

It is worthmentioning that although the hybrid controllerpresented is shown theoretically with better performance fora singlemode vibration control the research for SISOMIMOvibration control of several modes using hybrid controlstrategies is more applicable in practical flexible structures ofaerospace and other engineering fields An attempt to addressthese issues is recommended to future studies

Conflict of Interests

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

Acknowledgments

This work was supported by Nature Science Foundation ofChina (no 50775225) the State Key Laboratory of Mechani-cal Transmission ChongqingUniversity (no 0301002109165)These financial supports are gratefully acknowledged

References

[1] R C Fuller J S Elliott and A P Nelson Active Control ofVibration Academic Press London UK 1997

[2] C M A Vasques and J D Rodrigues ldquoActive vibration controlof smart piezoelectric beams comparison of classical andoptimal feedback control strategiesrdquo Computers amp Structuresvol 84 no 22-23 pp 1402ndash1414 2006

[3] A H N Shirazi H R Owji and M Rafeeyan ldquoActivevibration control of an FGM rectangular plate using fuzzy logiccontrollersrdquo Procedia Engineering vol 14 pp 3019ndash3026 2011

[4] D Thakkar and R Ganguli ldquoInduced shear actuation ofhelicopter rotor blade for active twist controlrdquo Thin-WalledStructures vol 45 no 1 pp 111ndash121 2007

[5] A K Rao K Natesan M S Bhat and R Ganguli ldquoExperimen-tal demonstration of 119867

infincontrol based active vibration sup-

pression in composite fin-tip of aircraft using optimally placed


piezoelectric patch actuatorsrdquo Journal of Intelligent MaterialSystems and Structures vol 19 no 6 pp 651ndash669 2008

[6] M K Kwak and D H Yang ldquoActive vibration control of a ring-stiffened cylindrical shell in contact with unbounded externalfluid and subjected to harmonic disturbance by piezoelectricsensor and actuatorrdquo Journal of Sound and Vibration vol 332no 20 pp 4775ndash4797 2013

[7] A Benjeddou ldquoAdvances in hybrid active-passive vibration andnoise control via piezoelectric and viscoelastic constrained layertreatmentsrdquo Journal of Vibration and Control vol 7 no 4 pp565ndash602 2001

[8] A Baz and J Ro ldquoVibration control of rotating beams withactive constrained layer dampingrdquo Smart Materials and Struc-tures vol 10 no 1 pp 112ndash120 2001

[9] M C Ray J Oh and A Baz ldquoActive constrained layer dampingof thin cylindrical shellsrdquo Journal of Sound and Vibration vol240 no 5 pp 921ndash935 2001

[10] T H Chen Vibration control of cylindrical shells with activeconstrained layer damping [PhD thesis] Catholic UniversityWashington DC USA 1996

[11] W H Liao and K W Wang ldquoOn the active-passive hybridcontrol actions of structures with active constrained layertreatmentsrdquo Journal of Vibration and Acoustics Transactions ofthe ASME vol 119 no 4 pp 563ndash571 1997

[12] J X Gao and W H Liao ldquoVibration analysis of simplysupported beamswith enhanced self-sensing active constrainedlayer damping treatmentsrdquo Journal of Sound and Vibration vol280 no 1-2 pp 329ndash357 2005

[13] S Kumar R Kumar andR Sehgal ldquoEnhancedACLD treatmentusing stand-off-layer FEM based design and experimentalvibration analysisrdquo Applied Acoustics vol 72 no 11 pp 856ndash872 2011

[14] J Shivakumar M H Ashok and M C Ray ldquoActive control ofgeometrically nonlinear transient vibrations of laminated com-posite cylindrical panels using piezoelectric fiber reinforcedcompositerdquo Acta Mechanica vol 224 no 1 pp 1ndash15 2013

[15] P H Shah and M C Ray ldquoActive control of laminated com-posite truncated conical shells using vertically and obliquelyreinforced 1-3 piezoelectric compositesrdquo European Journal ofMechanics ASolids vol 32 pp 1ndash12 2012

[16] M C Ray and J N Reddy ldquoActive damping of laminated cylin-drical shells conveying fluid using 1ndash3 piezoelectric compositesrdquoComposite Structures vol 98 pp 261ndash271 2013

[17] P H Shah and M C Ray ldquoActive structural-acoustic controlof laminated composite truncated conical shells using smartdamping treatmentrdquo Journal of Vibration and Acoustics Trans-actions of the ASME vol 135 no 2 Article ID 021001 2013

[18] M V Kumar P Sampath S Suresh S N Omkar and RGanguli ldquoDesign of a stability augmentation system for ahelicopter using LQR control and ADS-33 handling qualitiesspecificationsrdquo Aircraft Engineering and Aerospace Technologyvol 80 no 2 pp 111ndash123 2008

[19] J Rohlfing P Gardonio and D J Thompson ldquoComparison ofdecentralized velocity feedback control for thin homogeneousand stiff sandwich panels using electrodynamic proof-massactuatorsrdquo Journal of Sound and Vibration vol 330 no 5 pp843ndash867 2011

[20] S R Viswamurthy and R Ganguli ldquoPerformance sensitivityof helicopter global and local optimal harmonic vibrationcontrollerrdquo Computers and Mathematics with Applications vol56 no 10 pp 2468ndash2480 2008

[21] A Baz ldquoBoundary control of beams using active constrainedlayer dampingrdquo Journal of Vibration and Acoustics vol 119 no2 pp 166ndash172 1997

[22] A Baz ldquoRobust control of active constrained layer dampingrdquoJournal of Sound andVibration vol 211 no 3 pp 467ndash480 1998

[23] J L Crassidis A Baz and N Wereley ldquo119867infin

control of activeconstrained layer dampingrdquo Journal of Vibration and Controlvol 6 no 1 pp 113ndash136 2000

[24] T Liu H Hua and Z Zhang ldquoRobust control of plate vibrationvia active constrained layer dampingrdquo Thin-Walled Structuresvol 42 no 3 pp 427ndash448 2004

[25] J J Rodriguez Comparison of sliding mode and state feedbackcontrol applied to a partilly treated actively damped beam [MSthesis] University of Texas Austin Tex USA 2006

[26] P Belanger A Berry Y Pasco O Robin Y St-Amant andS Rajan ldquoMulti-harmonic active structural acoustic controlof a helicopter main transmission noise using the principalcomponent analysisrdquo Applied Acoustics vol 70 no 1 pp 153ndash164 2009

[27] L Lei Z Gu and M Lu ldquoMIMO hybrid control of structuralresponses for helicopterrdquoChinese Journal of Aeronautics vol 16no 3 pp 151ndash156 2003

[28] Y Q Cao Study on vibration and noise control for car bodystructure based on smart constrained layer damping [PhDthesis] Chongqing University Chongqing China 2011

[29] WR SaundersHH Robertshaw andRA Burdisso ldquoAhybridstructural control approach for narrow-band and impulsivedisturbance rejectionrdquo Noise Control Engineering Journal vol44 no 1 pp 11ndash21 1996

[30] D P Man and A Preumont ldquoHybrid feedback-feedforwardcontrol for vibration suppressionrdquo Journal of Structure Controlvol 3 no 1-2 pp 33ndash44 1996

[31] H Kim andH Adeli ldquoHybrid feedback-least mean square algo-rithm for structural controlrdquo Journal of Structural Engineeringvol 130 no 1 pp 120ndash127 2004

[32] Z Ling L Zheng and Y Wang ldquoVibration and dampingcharacteristics of cylindrical shells with active constrained layerdamping treatmentsrdquo Smart Materials and Structures vol 20no 2 pp 1ndash9 2011

[33] Z Q Qu ldquoA multi-step method for matrix condensation offinite element modelsrdquo Journal of Sound and Vibration vol 214no 5 pp 965ndash971 1998

[34] M J Lam D J Inman and W R Saunders ldquoHybrid dampingmodels using the Golla-Hughes-McTavish method with inter-nally balanced model reduction and output feedbackrdquo SmartMaterials and Structures vol 9 no 3 pp 362ndash373 2000

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



the ultimate goal of extending its application of attenuatingthe structural vibration of rotor blades [8]The vibration con-trol of shells [9 10]was also investigated to extend the applica-tion of ACLD in the cabin of aircraft and submarine In thesestudies the PVDF (polyvinylidene fluoride) was served as thesensors and actuators They use the simple proportional andderivative feedback of the transverse deflection to suppressthe vibration of the base structure theoretically and exper-imentally To improve the force transmissibility betweenpiezoelectric actuator and the base structure Liao and Wang[11] proposed an enhanced active constraining layer (EACL)and Gao and Liao [12] extended this configuration Kumaret al [13] added the stand-off layer (SOL) between theviscoelastic layer and the base structure to increase theviscoelastic strain and enhance the effect of the active force Inrecent years piezoelectric fiber-reinforced composite (PFRC)materials [14] and piezoelectric composites (PZC) [15 16]which have a wide range of effective materials propertieswere introduced in the ACLD treatments to serve as activeconstrained layer Shah and Ray [17] focused on investigatingthe effect of the piezoelectric fiber orientation angle in theconstrained layer on the performance of ACLD patchesIn these studies the emphasis was placed on the vibrationcharacteristics analysisenhancement of the performance ofthe ACLD treatments

In the open literatures much attention was paid onactive control methods in the structural activeactive-passivevibration control The optimal feedback control [18] velocityfeedback control [19] fuzzy logic control [3] and the 119867

infin

control [5] were usually employed in active vibration con-trol (AVC) Viswamurthy and Ganguli [20] compared theglobal harmonic control with local blade optimal controlapplied to vibration control of helicopter For active-passiveACLDstructure Baz [21] presented a variational mathemati-calmodel for beams and a globally stable boundary controllerwas investigated to get the high damping characteristics overbroad frequency range Meanwhile Baz [22] developed a1198672robust controller based on the transfer function model

of the ACLD beam The control strategy was stable inthe presence of parameter uncertainty as well as ensuredoptimal disturbance rejection capabilities Furthermore the119867 infinity robust control strategy was utilized by Crassidiset al [23] to maximize disturbance rejection capabilitiesover a desired frequency band Liu et al [24] developed a119867 infinity controller based on the reduced finite elementmodel Rodriguez [25] made a comparison of sliding modecontrol (SMC) with state-feedback control (LQR) appliedto a partially treated ACLD beam In these studies thecontrol strategies can be classified into feedback controltheoriesThese control strategies have been shown to bemoresuitable in applicationswhere the structure is disturbed by thestochastic or impulsive disturbances

Feedforward control which is more effective in the casewhere the deterministic or correlated information about thedisturbance is known is another popular control strategy Itwaswidely used in active vibration and noise control field [2627] However only a few feedforward theories were utilizedin structural vibration control with ACLD treatments Cao[28] employed the feedforward LMS algorithm to control

Plate

Piezoelectric constraining layerViscoelastic

layer

Piezoelectric sensor layer

Figure 1 Schematic drawing of the ACLDplate structure

the vibration of a plate treated with ACLD treatments Thefinite element model was developed and ADF method wasused to identify the performance of VEM Furthermorehybrid control of feedback and feedforward control theorieswere also successfully used to noise and vibration suppression[29 30] However it was rarely investigated for structuralvibration control with ACLD treatments Vasques validatedthe effectiveness of the hybrid control strategies for ACLDbeam [31]

In this paper the focus is placed on combining individualfeedback control based on linear quadratic Gaussian (LQG)and feedforward control based on adaptive filtered-referenceLMS (FxLMS) to develop a hybrid controller for the vibra-tion control of plates treated with ACLD treatments Theperformance of vibration suppression subjected to differentcontrol strategies is compared and the effect of parameterson vibration suppression of the plate with ACLD treatment isfurther investigated

This paper is organized in the following five sections Abrief background is introduced in Section 1 The dynamicmodels including the finite element model of the platetreated with ACLD patches and GHM model of VEM aredeveloped in Section 2 In Section 3 the feedback controllerthe feedforward controller and the hybrid controller aredesigned A numerical example is discussed in Section 4Finally the main results and future work are summarized inSection 5

2 The ACLDPlate System Model Construction

21 Finite Element Model The ACLDplate system is illus-trated in Figure 1 The viscoelastic damping layer is sand-wiched between two piezoelectric layers which serve aspiezosensor and piezoactuator respectively The three layersare bonded to the base plate acting as a smart constraininglayer damping It is assumed that the piezosensor and thebase plate are perfectly bonded together such that they can bereduced to a single equivalent layer Some other assumptionsare made in [32]

The ACLDplate elements considered are four-node two-dimensional elements Each node has seven degrees offreedom to describe the longitudinal displacements 119906

119888and

V119888of the constrained layer the longitudinal displacements

119906119901and V119901of the base plate the transverse deflection 119908 and

the slopes 119909 and 119910 of the deflection line The longitudinaldisplacement functions are expressed as 4-term polynomialsand transverse deflection119908 is expressed as a 12-term polyno-mial Therefore the element deflection vector is given by

Δ(119890)

= Δ1Δ2Δ3Δ4119879

(1)

where Δ119894= 119906119888119894

V119888119894119906119901119894

V119901119894119908119894120579119909119894120579119910119894119879 for 119894 = 1 2 3 4



M(119890)Δ(119890) + K(119890)Δ(119890) = F(119890)119889+ F(119890)119888 (2)

where M(119890) = M(119890)119901+ M(119890)119888+ M(119890)V K(119890) = K(119890)

119901+ K(119890)119888+

K(119890)V + 119866VK(119890)

V120573 M(119890)






119888is the



MΔ + K + 119866VKV120573Δ = F119889+ F119888 (3)



119904119866 (119904) = 119866infin

[1 +sum

119896

120572119896

1199042

+ 2120589119896120596119896119904

1199042 + 2120589119896120596119896119904 + 1205962

] (4)


119896 120589119896 and 120596



1198851015840

(119904) =1205962

1199042 + 2120589120596119904 + 1205962Δ (119904) (5)


M1015840X + C1015840X + K1015840X = F1015840d + F1015840c (6)


M1015840 =

[[[[[[[[

[

M 0 0 0

01205721

1205962

1

Λ 0 0

0 01205722

1205962

2

Λ 0

0 0 01205723

1205962

3

Λ

]]]]]]]]

]

C1015840 =

[[[[[[[[[

[

0 0 0 0

0212057211205771

1205962

1

Λ 0 0

0 0212057221205772

1205962

2

Λ 0

0 0 0212057231205773

1205962

3

Λ

]]]]]]]]]

]

K1015840 =

[[[[[[[

[

119866infinKV120573(1 +

3

sum

119896=1

120572119896) + Kk minus1205721R minus120572

2R minus120572

3R

minus1205721R119879 120572

1Λ 0 0

minus1205722R119879 0 120572

2Λ 0

minus1205723R119879 0 0 120572

3Λ

]]]]]]]

]

X = ΔZ F1015840d = Fd0 F1015840c =

Fc0

(7)


ΛV R = RVΛ Z = R119879V Z1015840


MX + CX + KX = Fd + Fc (8)



[M119898119898

M119898119904

M119904119898

M119904119904

]X119898

X119904

+ [C119898119898

C119898119904

C119904119898

C119904119904

]X119898

X119904

+ [K119898119898

K119898119904

K119904119898

K119904119904

]X119898

X119904

= F119898

F119904

(9)



M(119894)119877X + C(119894)

119877X + K(119894)

119877X = F(119894)

119877 (10)

where

M(119894)119877= M119898119898+ R1015840(119894)

119879

M119904119898

+M119898119904R1015840(119894) + R1015840(119894)

119879

M119904119904R1015840(119894)C(119894)

119877

= C119898119898+ R1015840(119894)

119879

C119904119898+ C119898119904R1015840(119894)

+ R1015840(119894)119879

C119904119904R1015840(119894)

K(119894)119877= K119898119898+ R1015840(119894)

119879

K119904119898

+ K119898119904R1015840(119894) + R1015840(119894)

119879

K119904119904R1015840(119894)F(119894)

119877

= F119898+ R1015840(119894)

119879

F119904

R1015840(119894+1) = minusK119904119904

minus1

(M119904119904R1015840(119894)M(119894)

119877

minus1

K(119894)119877minus K119904119898)

R1015840(0) = minusKminus1119904119904K119904119898

(11)


x (119905) = Ax (119905) + B119889f119889+ B119888f119888

y (119905) = Cx (119905) + k(12)


X119898119879







119869 = int

infin

0

(x(119905)119879Q119909x (119905) + f

119888

119879

(119905)Rf119888(119905)) 119889119905 (13)

or

119869 = int

infin

0

(y(119905)119879Q119910y (119905) + f

119888

119879

(119905)Rf119888(119905)) 119889119905 (14)



feedback so that

f119888= minusK119888x (119905) (15)


K119888= Rminus1B119879

119888P (16)



x (119905) = (Α minus B119888K119888) x (119905) + B

119889f119889 (17)




119889are both


119864 (f119889f119879119889) = 119876

119889 119864 (k

119889k119879119889) = 119877119889 (18)




119864([x (119905) minus x1015840 (119905)] [x (119905) minus x1015840 (119905)]119879

) (19)



[y1015840x1015840] = [

CI] x1015840

(20)




Process noisew(k)

u(k)

+

++

+ ++

+

+

r(k)

Controlinput


r998400(k)

Plant

Pu(z)

Disturbance path

Cancellation path

LQG controller



yu(k)

y(k)

y(k)

y(k)

(k)

Measurementnoise

y120601(k)

u120601(k)

u120601(k) P120601(z)

P120601(z)

ufb120601 (k)

uff120601 (k)









ℎ119894(119896) = ℎ

119894(119896 minus 1) + 2120583119890 (119896 minus 1) 119903

1015840

(119896 minus 1)

119891119891119891

119888(119896) =

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894)

(21)





120601(119896) is given by

119910120601(119896) =

119872minus1

sum

119895=0

119892119895

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894 minus 119895) (22)




119910 (119896) = 119910119906(119896) +

119872minus1

sum

119895=0

119892119895

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894 minus 119895) + V (119896) (23)


disturbance alone








00008m ℎ119888

00005m ℎV 0001m 1205891

1480119864119901

69GPa 119864119888

745 GPa 120583V 03 1205892

1216120583119901

03 120583119888

032 120588V 7895 kgm31205893

8104120588119901

2800 kgm3120588119888

7450 kgm3119866infin 554200 Pa 120596

1896200

11988931 11988932


13960 120596

2927800

1205722

6569 1205963

7613001205723

1447







13

57

911

510

1520

0

001

002

003

004

Mode number 1







Step size 120583001 005 01 05 10 30 35 100


20 40 60 80 100 120

Mag

nitu

de (d

B)

Bode diagram

Frequency (Hz)



0 05 1 15 2 25 3

0

001

002Impulse response

Am

plitu

de (m

)

Time (s)

minus40

minus60

minus80

minus100

minus120

minus001

minus002


0 2 4 6 8 10

0

1

2

3

Time (s)

Am

plitu

de (m

)

PCLDACLD

minus1

minus2

minus3

times10minus4

(a)

Time (s)0 2 4 6 8 10

0

10

20

30

40

Con

trol v

olta

ge (V

)

minus10

minus20

minus30

minus40

(b)





0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

minus150

(b)








0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

200

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus200

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

150

50

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus150

minus50

(b)








005

004

003

002

001

00

5000 10000 15000 20000

MSE



3

2

1

010000 15000 20000

times10minus3


012

01

05

15

004

008

002

006

0

1

2

05000 10000

10000

15000

15000

20000

20000

MSE


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1



5 Conclusions


0

1

2

3

4

5

6

MSE

0 5000 10000 15000 20000


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

u = 10


004

008

002

006

0

MSE

0 5000 10000 15000 20000


Phase error = 0∘

Phase error = 15∘

Phase error = 30∘





0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘


2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106



0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106





Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











M(119890)Δ(119890) + K(119890)Δ(119890) = F(119890)119889+ F(119890)119888 (2)

where M(119890) = M(119890)119901+ M(119890)119888+ M(119890)V K(119890) = K(119890)

119901+ K(119890)119888+

K(119890)V + 119866VK(119890)

V120573 M(119890)






119888is the



MΔ + K + 119866VKV120573Δ = F119889+ F119888 (3)



119904119866 (119904) = 119866infin

[1 +sum

119896

120572119896

1199042

+ 2120589119896120596119896119904

1199042 + 2120589119896120596119896119904 + 1205962

] (4)


119896 120589119896 and 120596



1198851015840

(119904) =1205962

1199042 + 2120589120596119904 + 1205962Δ (119904) (5)


M1015840X + C1015840X + K1015840X = F1015840d + F1015840c (6)


M1015840 =

[[[[[[[[

[

M 0 0 0

01205721

1205962

1

Λ 0 0

0 01205722

1205962

2

Λ 0

0 0 01205723

1205962

3

Λ

]]]]]]]]

]

C1015840 =

[[[[[[[[[

[

0 0 0 0

0212057211205771

1205962

1

Λ 0 0

0 0212057221205772

1205962

2

Λ 0

0 0 0212057231205773

1205962

3

Λ

]]]]]]]]]

]

K1015840 =

[[[[[[[

[

119866infinKV120573(1 +

3

sum

119896=1

120572119896) + Kk minus1205721R minus120572

2R minus120572

3R

minus1205721R119879 120572

1Λ 0 0

minus1205722R119879 0 120572

2Λ 0

minus1205723R119879 0 0 120572

3Λ

]]]]]]]

]

X = ΔZ F1015840d = Fd0 F1015840c =

Fc0

(7)


ΛV R = RVΛ Z = R119879V Z1015840


MX + CX + KX = Fd + Fc (8)



[M119898119898

M119898119904

M119904119898

M119904119904

]X119898

X119904

+ [C119898119898

C119898119904

C119904119898

C119904119904

]X119898

X119904

+ [K119898119898

K119898119904

K119904119898

K119904119904

]X119898

X119904

= F119898

F119904

(9)



M(119894)119877X + C(119894)

119877X + K(119894)

119877X = F(119894)

119877 (10)

where

M(119894)119877= M119898119898+ R1015840(119894)

119879

M119904119898

+M119898119904R1015840(119894) + R1015840(119894)

119879

M119904119904R1015840(119894)C(119894)

119877

= C119898119898+ R1015840(119894)

119879

C119904119898+ C119898119904R1015840(119894)

+ R1015840(119894)119879

C119904119904R1015840(119894)

K(119894)119877= K119898119898+ R1015840(119894)

119879

K119904119898

+ K119898119904R1015840(119894) + R1015840(119894)

119879

K119904119904R1015840(119894)F(119894)

119877

= F119898+ R1015840(119894)

119879

F119904

R1015840(119894+1) = minusK119904119904

minus1

(M119904119904R1015840(119894)M(119894)

119877

minus1

K(119894)119877minus K119904119898)

R1015840(0) = minusKminus1119904119904K119904119898

(11)


x (119905) = Ax (119905) + B119889f119889+ B119888f119888

y (119905) = Cx (119905) + k(12)


X119898119879







119869 = int

infin

0

(x(119905)119879Q119909x (119905) + f

119888

119879

(119905)Rf119888(119905)) 119889119905 (13)

or

119869 = int

infin

0

(y(119905)119879Q119910y (119905) + f

119888

119879

(119905)Rf119888(119905)) 119889119905 (14)



feedback so that

f119888= minusK119888x (119905) (15)


K119888= Rminus1B119879

119888P (16)



x (119905) = (Α minus B119888K119888) x (119905) + B

119889f119889 (17)




119889are both


119864 (f119889f119879119889) = 119876

119889 119864 (k

119889k119879119889) = 119877119889 (18)




119864([x (119905) minus x1015840 (119905)] [x (119905) minus x1015840 (119905)]119879

) (19)



[y1015840x1015840] = [

CI] x1015840

(20)




Process noisew(k)

u(k)

+

++

+ ++

+

+

r(k)

Controlinput


r998400(k)

Plant

Pu(z)

Disturbance path

Cancellation path

LQG controller



yu(k)

y(k)

y(k)

y(k)

(k)

Measurementnoise

y120601(k)

u120601(k)

u120601(k) P120601(z)

P120601(z)

ufb120601 (k)

uff120601 (k)









ℎ119894(119896) = ℎ

119894(119896 minus 1) + 2120583119890 (119896 minus 1) 119903

1015840

(119896 minus 1)

119891119891119891

119888(119896) =

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894)

(21)





120601(119896) is given by

119910120601(119896) =

119872minus1

sum

119895=0

119892119895

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894 minus 119895) (22)




119910 (119896) = 119910119906(119896) +

119872minus1

sum

119895=0

119892119895

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894 minus 119895) + V (119896) (23)


disturbance alone








00008m ℎ119888

00005m ℎV 0001m 1205891

1480119864119901

69GPa 119864119888

745 GPa 120583V 03 1205892

1216120583119901

03 120583119888

032 120588V 7895 kgm31205893

8104120588119901

2800 kgm3120588119888

7450 kgm3119866infin 554200 Pa 120596

1896200

11988931 11988932


13960 120596

2927800

1205722

6569 1205963

7613001205723

1447







13

57

911

510

1520

0

001

002

003

004

Mode number 1







Step size 120583001 005 01 05 10 30 35 100


20 40 60 80 100 120

Mag

nitu

de (d

B)

Bode diagram

Frequency (Hz)



0 05 1 15 2 25 3

0

001

002Impulse response

Am

plitu

de (m

)

Time (s)

minus40

minus60

minus80

minus100

minus120

minus001

minus002


0 2 4 6 8 10

0

1

2

3

Time (s)

Am

plitu

de (m

)

PCLDACLD

minus1

minus2

minus3

times10minus4

(a)

Time (s)0 2 4 6 8 10

0

10

20

30

40

Con

trol v

olta

ge (V

)

minus10

minus20

minus30

minus40

(b)





0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

minus150

(b)








0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

200

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus200

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

150

50

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus150

minus50

(b)








005

004

003

002

001

00

5000 10000 15000 20000

MSE



3

2

1

010000 15000 20000

times10minus3


012

01

05

15

004

008

002

006

0

1

2

05000 10000

10000

15000

15000

20000

20000

MSE


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1



5 Conclusions


0

1

2

3

4

5

6

MSE

0 5000 10000 15000 20000


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

u = 10


004

008

002

006

0

MSE

0 5000 10000 15000 20000


Phase error = 0∘

Phase error = 15∘

Phase error = 30∘





0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘


2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106



0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106





Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











M(119894)119877X + C(119894)

119877X + K(119894)

119877X = F(119894)

119877 (10)

where

M(119894)119877= M119898119898+ R1015840(119894)

119879

M119904119898

+M119898119904R1015840(119894) + R1015840(119894)

119879

M119904119904R1015840(119894)C(119894)

119877

= C119898119898+ R1015840(119894)

119879

C119904119898+ C119898119904R1015840(119894)

+ R1015840(119894)119879

C119904119904R1015840(119894)

K(119894)119877= K119898119898+ R1015840(119894)

119879

K119904119898

+ K119898119904R1015840(119894) + R1015840(119894)

119879

K119904119904R1015840(119894)F(119894)

119877

= F119898+ R1015840(119894)

119879

F119904

R1015840(119894+1) = minusK119904119904

minus1

(M119904119904R1015840(119894)M(119894)

119877

minus1

K(119894)119877minus K119904119898)

R1015840(0) = minusKminus1119904119904K119904119898

(11)


x (119905) = Ax (119905) + B119889f119889+ B119888f119888

y (119905) = Cx (119905) + k(12)


X119898119879







119869 = int

infin

0

(x(119905)119879Q119909x (119905) + f

119888

119879

(119905)Rf119888(119905)) 119889119905 (13)

or

119869 = int

infin

0

(y(119905)119879Q119910y (119905) + f

119888

119879

(119905)Rf119888(119905)) 119889119905 (14)



feedback so that

f119888= minusK119888x (119905) (15)


K119888= Rminus1B119879

119888P (16)



x (119905) = (Α minus B119888K119888) x (119905) + B

119889f119889 (17)




119889are both


119864 (f119889f119879119889) = 119876

119889 119864 (k

119889k119879119889) = 119877119889 (18)




119864([x (119905) minus x1015840 (119905)] [x (119905) minus x1015840 (119905)]119879

) (19)



[y1015840x1015840] = [

CI] x1015840

(20)




Process noisew(k)

u(k)

+

++

+ ++

+

+

r(k)

Controlinput


r998400(k)

Plant

Pu(z)

Disturbance path

Cancellation path

LQG controller



yu(k)

y(k)

y(k)

y(k)

(k)

Measurementnoise

y120601(k)

u120601(k)

u120601(k) P120601(z)

P120601(z)

ufb120601 (k)

uff120601 (k)









ℎ119894(119896) = ℎ

119894(119896 minus 1) + 2120583119890 (119896 minus 1) 119903

1015840

(119896 minus 1)

119891119891119891

119888(119896) =

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894)

(21)





120601(119896) is given by

119910120601(119896) =

119872minus1

sum

119895=0

119892119895

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894 minus 119895) (22)




119910 (119896) = 119910119906(119896) +

119872minus1

sum

119895=0

119892119895

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894 minus 119895) + V (119896) (23)


disturbance alone








00008m ℎ119888

00005m ℎV 0001m 1205891

1480119864119901

69GPa 119864119888

745 GPa 120583V 03 1205892

1216120583119901

03 120583119888

032 120588V 7895 kgm31205893

8104120588119901

2800 kgm3120588119888

7450 kgm3119866infin 554200 Pa 120596

1896200

11988931 11988932


13960 120596

2927800

1205722

6569 1205963

7613001205723

1447







13

57

911

510

1520

0

001

002

003

004

Mode number 1







Step size 120583001 005 01 05 10 30 35 100


20 40 60 80 100 120

Mag

nitu

de (d

B)

Bode diagram

Frequency (Hz)



0 05 1 15 2 25 3

0

001

002Impulse response

Am

plitu

de (m

)

Time (s)

minus40

minus60

minus80

minus100

minus120

minus001

minus002


0 2 4 6 8 10

0

1

2

3

Time (s)

Am

plitu

de (m

)

PCLDACLD

minus1

minus2

minus3

times10minus4

(a)

Time (s)0 2 4 6 8 10

0

10

20

30

40

Con

trol v

olta

ge (V

)

minus10

minus20

minus30

minus40

(b)





0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

minus150

(b)








0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

200

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus200

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

150

50

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus150

minus50

(b)








005

004

003

002

001

00

5000 10000 15000 20000

MSE



3

2

1

010000 15000 20000

times10minus3


012

01

05

15

004

008

002

006

0

1

2

05000 10000

10000

15000

15000

20000

20000

MSE


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1



5 Conclusions


0

1

2

3

4

5

6

MSE

0 5000 10000 15000 20000


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

u = 10


004

008

002

006

0

MSE

0 5000 10000 15000 20000


Phase error = 0∘

Phase error = 15∘

Phase error = 30∘





0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘


2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106



0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106





Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Process noisew(k)

u(k)

+

++

+ ++

+

+

r(k)

Controlinput


r998400(k)

Plant

Pu(z)

Disturbance path

Cancellation path

LQG controller



yu(k)

y(k)

y(k)

y(k)

(k)

Measurementnoise

y120601(k)

u120601(k)

u120601(k) P120601(z)

P120601(z)

ufb120601 (k)

uff120601 (k)









ℎ119894(119896) = ℎ

119894(119896 minus 1) + 2120583119890 (119896 minus 1) 119903

1015840

(119896 minus 1)

119891119891119891

119888(119896) =

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894)

(21)





120601(119896) is given by

119910120601(119896) =

119872minus1

sum

119895=0

119892119895

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894 minus 119895) (22)




119910 (119896) = 119910119906(119896) +

119872minus1

sum

119895=0

119892119895

119873minus1

sum

119894=0

ℎ119894(119896) 1199031015840

(119896 minus 119894 minus 119895) + V (119896) (23)


disturbance alone








00008m ℎ119888

00005m ℎV 0001m 1205891

1480119864119901

69GPa 119864119888

745 GPa 120583V 03 1205892

1216120583119901

03 120583119888

032 120588V 7895 kgm31205893

8104120588119901

2800 kgm3120588119888

7450 kgm3119866infin 554200 Pa 120596

1896200

11988931 11988932


13960 120596

2927800

1205722

6569 1205963

7613001205723

1447







13

57

911

510

1520

0

001

002

003

004

Mode number 1







Step size 120583001 005 01 05 10 30 35 100


20 40 60 80 100 120

Mag

nitu

de (d

B)

Bode diagram

Frequency (Hz)



0 05 1 15 2 25 3

0

001

002Impulse response

Am

plitu

de (m

)

Time (s)

minus40

minus60

minus80

minus100

minus120

minus001

minus002


0 2 4 6 8 10

0

1

2

3

Time (s)

Am

plitu

de (m

)

PCLDACLD

minus1

minus2

minus3

times10minus4

(a)

Time (s)0 2 4 6 8 10

0

10

20

30

40

Con

trol v

olta

ge (V

)

minus10

minus20

minus30

minus40

(b)





0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

minus150

(b)








0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

200

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus200

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

150

50

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus150

minus50

(b)








005

004

003

002

001

00

5000 10000 15000 20000

MSE



3

2

1

010000 15000 20000

times10minus3


012

01

05

15

004

008

002

006

0

1

2

05000 10000

10000

15000

15000

20000

20000

MSE


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1



5 Conclusions


0

1

2

3

4

5

6

MSE

0 5000 10000 15000 20000


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

u = 10


004

008

002

006

0

MSE

0 5000 10000 15000 20000


Phase error = 0∘

Phase error = 15∘

Phase error = 30∘





0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘


2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106



0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106





Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












00008m ℎ119888

00005m ℎV 0001m 1205891

1480119864119901

69GPa 119864119888

745 GPa 120583V 03 1205892

1216120583119901

03 120583119888

032 120588V 7895 kgm31205893

8104120588119901

2800 kgm3120588119888

7450 kgm3119866infin 554200 Pa 120596

1896200

11988931 11988932


13960 120596

2927800

1205722

6569 1205963

7613001205723

1447







13

57

911

510

1520

0

001

002

003

004

Mode number 1







Step size 120583001 005 01 05 10 30 35 100


20 40 60 80 100 120

Mag

nitu

de (d

B)

Bode diagram

Frequency (Hz)



0 05 1 15 2 25 3

0

001

002Impulse response

Am

plitu

de (m

)

Time (s)

minus40

minus60

minus80

minus100

minus120

minus001

minus002


0 2 4 6 8 10

0

1

2

3

Time (s)

Am

plitu

de (m

)

PCLDACLD

minus1

minus2

minus3

times10minus4

(a)

Time (s)0 2 4 6 8 10

0

10

20

30

40

Con

trol v

olta

ge (V

)

minus10

minus20

minus30

minus40

(b)





0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

minus150

(b)








0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

200

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus200

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

150

50

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus150

minus50

(b)








005

004

003

002

001

00

5000 10000 15000 20000

MSE



3

2

1

010000 15000 20000

times10minus3


012

01

05

15

004

008

002

006

0

1

2

05000 10000

10000

15000

15000

20000

20000

MSE


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1



5 Conclusions


0

1

2

3

4

5

6

MSE

0 5000 10000 15000 20000


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

u = 10


004

008

002

006

0

MSE

0 5000 10000 15000 20000


Phase error = 0∘

Phase error = 15∘

Phase error = 30∘





0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘


2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106



0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106





Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Step size 120583001 005 01 05 10 30 35 100


20 40 60 80 100 120

Mag

nitu

de (d

B)

Bode diagram

Frequency (Hz)



0 05 1 15 2 25 3

0

001

002Impulse response

Am

plitu

de (m

)

Time (s)

minus40

minus60

minus80

minus100

minus120

minus001

minus002


0 2 4 6 8 10

0

1

2

3

Time (s)

Am

plitu

de (m

)

PCLDACLD

minus1

minus2

minus3

times10minus4

(a)

Time (s)0 2 4 6 8 10

0

10

20

30

40

Con

trol v

olta

ge (V

)

minus10

minus20

minus30

minus40

(b)





0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

minus150

(b)








0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

200

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus200

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

150

50

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus150

minus50

(b)








005

004

003

002

001

00

5000 10000 15000 20000

MSE



3

2

1

010000 15000 20000

times10minus3


012

01

05

15

004

008

002

006

0

1

2

05000 10000

10000

15000

15000

20000

20000

MSE


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1



5 Conclusions


0

1

2

3

4

5

6

MSE

0 5000 10000 15000 20000


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

u = 10


004

008

002

006

0

MSE

0 5000 10000 15000 20000


Phase error = 0∘

Phase error = 15∘

Phase error = 30∘





0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘


2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106



0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106





Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

50

100

150

Time (s)

Con

trol v

olta

ge (V

)

minus50

minus100

minus150

(b)








0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

200

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus200

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

150

50

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus150

minus50

(b)








005

004

003

002

001

00

5000 10000 15000 20000

MSE



3

2

1

010000 15000 20000

times10minus3


012

01

05

15

004

008

002

006

0

1

2

05000 10000

10000

15000

15000

20000

20000

MSE


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1



5 Conclusions


0

1

2

3

4

5

6

MSE

0 5000 10000 15000 20000


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

u = 10


004

008

002

006

0

MSE

0 5000 10000 15000 20000


Phase error = 0∘

Phase error = 15∘

Phase error = 30∘





0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘


2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106



0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106





Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

200

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus200

(b)


0 5 10 15 20

0

0005

001

Time (s)

minus001

minus0005

Am

plitu

de (m

)

PCLDACLD

(a)

0 5 10 15 20

0

100

150

50

Time (s)

Con

trol v

olta

ge (V

)

minus100

minus150

minus50

(b)








005

004

003

002

001

00

5000 10000 15000 20000

MSE



3

2

1

010000 15000 20000

times10minus3


012

01

05

15

004

008

002

006

0

1

2

05000 10000

10000

15000

15000

20000

20000

MSE


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1



5 Conclusions


0

1

2

3

4

5

6

MSE

0 5000 10000 15000 20000


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

u = 10


004

008

002

006

0

MSE

0 5000 10000 15000 20000


Phase error = 0∘

Phase error = 15∘

Phase error = 30∘





0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘


2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106



0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106





Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










005

004

003

002

001

00

5000 10000 15000 20000

MSE



3

2

1

010000 15000 20000

times10minus3


012

01

05

15

004

008

002

006

0

1

2

05000 10000

10000

15000

15000

20000

20000

MSE


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1



5 Conclusions


0

1

2

3

4

5

6

MSE

0 5000 10000 15000 20000


times10minus3

u = 001

u = 005

u = 01

u = 05

u = 1

u = 10


004

008

002

006

0

MSE

0 5000 10000 15000 20000


Phase error = 0∘

Phase error = 15∘

Phase error = 30∘





0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘


2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106



0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106





Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1

2

3

4

0 5000 10000 15000 20000


times10minus3

MSE

Phase error = 0∘

Phase error = 15∘

Phase error = 30∘

Phase error = 60∘


2

4

6

8

10

12

0 5000 10000 15000 20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106



0

0002

0004

0006

0008

001

05

1

15

0 5000 10000 15000

15000 17500

20000

20000


times10minus3

MSE

Q = 2 times 105

Q = 5 times 105

Q = 1 times 106

Q = 2 times 106

Q = 4 times 106





Acknowledgments


References










































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article active vibration control of plate partly...

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