Research ArticleAdaptive Model Reference Sliding Mode Control of StructuralNonlinear Vibration

Luyu Li 12 Naibang Wang3 and Han Qin 12

1State Key Laboratory of Coastal and Offshore Engineering Dalian University of Technology Dalian Liaoning 116024 China2School of Civil Engineering Dalian University of Technology Dalian Liaoning 116024 China3East China Architectural Design amp Research Institute Co Ltd Northwest Center Xirsquoan Shanxi 710065 China

Correspondence should be addressed to Luyu Li liluyudluteducn

Received 24 January 2019 Accepted 14 April 2019 Published 28 April 2019

Academic Editor Angelo Marcelo Tusset

Copyright copy 2019 Luyu Li et alampis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper an active mass damper (AMD) with adaptive control design is used to mitigate the vibrations of a multi-degree-of-freedom (MDOF) nonlinear structure under earthquake excitation In the adaptive control design a modified unscented Kalmanfilter (UKF) is developed to identify the unknown states and parameters adaptively Based on the identified states and parametersmodel reference sliding model control (MRSMC) is proposed for structural nonlinear vibration control In the design of MRSMCthe structure with tuned mass damper (TMD) is used as a reference model In the control process the parameters and statesneeded to obtain the control forces are updated adaptively through UKF A numerical example of a three-story shear-type modelwith an active mass damper (AMD) mounted on the top story is used to study the proposed controller ampe interstory shearrestoring forces are simulated by the BoucndashWen model ampis model could simulate the hinge effect of the yielding joints in steelstructures or the performance of the hysteretic energy dissipation devicesampe simulation results demonstrated that with the helpof the modified UKF method and the reference model the vibration of the structure is effectively mitigated under theproposed MRSMC

1 Introduction

Structural control has receivedmuch attention in the researchcommunity during the last few decades [1ndash3] Under strongearthquake excitations a structure can experience nonlineardeformation that may cause damage or even collapse thestructure In steel structures large deformation can induceyielding of the structure In concrete structure yielding of thestructural components can also generate hysteretic forcesNew energy dissipation devices such as steel dampers fric-tion dampers shape memory alloy dampers pounding andimpact dampers and magnetorheological dampers can alsobring nonlinear forces into the structure [4ndash13] ampereforethe study of structural vibration control considering thehysteretic effect is of great importance To model the hys-teretic effect the BoucndashWen model is widely used in civilengineering due to its ability to simulate various hystereticbehaviors [14ndash16]amperefore the nonlinearity of the structure

is simulated using the BoucndashWen model in this paper For anonlinear control design the more the structural character-istics are known the better the structure can be controlledHowever the model parameters of the structural nonlinearityare often unknown and estimations of these parameters arenecessary

Considering the estimation problem of nonlinearstructures the unscented Kalman filter (UKF) method hasbeen used for parameter identification ampe UKF utilizesthe unscented transform (UT) to estimate the propagatedmean and covariance ampe unscented transform which isthe key to UKF uses specially arranged points which arecalled sigma points to go through nonlinear transforms toestimate the updates of the mean and covariance UKF wasdemonstrated to be more effective and accurate than thelinearized counterpart of the Kalman filter the extendedKalman filter ampe two methods were compared in thearticles written in [17ndash20] It is a well-known procedure

HindawiShock and VibrationVolume 2019 Article ID 3612516 13 pageshttpsdoiorg10115520193612516

that has been applied to many real-time control systemsampere are many studies of UKF combined with differentcontrol methods such as fault-tolerant controls modelpredictive controls [21ndash23] LQR controls [24] PID con-trols and its variants [25 26] feedback linearizationcontrols [27] and sliding mode controls [28] Howeverwhen dealing with parameter identification of the systemspossessing latent parameters the results of UKF identifi-cation become less satisfactory Latent parameters are thoseparameters that link indirectly with the observations ampeBoucndashWen model possesses two latent parameters ampestates of the BoucndashWen model are continuous Stateschange very little when using small integration step sizeHow to solve this problem for the BoucndashWen model isimportant for the parameter identification and the con-troller design problems

Based on the estimation results various control algo-rithms can be designed for structural nonlinear vibrationRegarding nonlinear control methods the sliding modecontrol (SMC) shows its prominence in quick responseinsensitivity to disturbances in the structure and ease of use[28ndash33] When specific purposes and performances areexpected model reference controls are often used In vi-bration mitigation a zero reference can induce large controlforces which is often unrealistic for the control actuator torealize A structure with large damping was mostly used asthe reference system [34ndash39]

To realize an active control in civil structures a com-monly used control device is the active mass damper(AMD) An AMD generates control force through activemotors and applies the force to structures by means ofadditional masses Significant progress about using AMDto control structural vibration has been made in civil en-gineering [40ndash45] Some studies are conducted to controlstructural nonlinear vibration using AMD Li et al [46]proposed a fuzzy logic control algorithm for structuralnonlinear vibration control which does not need thestructure model Incorporating the structural model intothe controller design will benefit the control analysis andeffect However fuzzy control is based on the fuzzy rulewhich is specified by the expert experience Without a goodmathematical model fuzzy control may not have a goodcontrol effect especially for structural nonlinearities whichare very complicated In order to overcome this problem amodified UKF is proposed in this paper to estimate un-known parameters and states Based on this information areasonable controller can be proposed to control structuralnonlinear vibration

In this paper MRSMC is combined with UKF to solvethe problem of vibration mitigation of a structure thatcontains nonlinearities ampe UKF is used to identify theparameters and estimate the unknown structural states Toimprove the performance of parameter identification thehysteretic state of the BoucndashWen model is calculated bysubstituting the identified values in the last step into theequilibrium equation ampe obtained state is used as one ofthe observations in the current step to update the identi-fication Using the information estimated by UKF MRSMC

is used to determine the control law from these states andparameters ampe reference model used for MRSMC is thestructure model with TMDampe efficiency of MRSMC-UKFis studied by simulation ampe numerical results demon-strate the effectiveness of this combined MRSMC-UKFmethod

2 Modified UKF

To acquire all the states and parameters an effective iden-tificationmethod is required In this section amodified UKFmethod is developed to effectively identify the unknownstates and parameters in a hysteretic model Traditional UKFused to identify the parameter is reviewed in the followingsubsection

21 State Estimation Using a Traditional UKF UKF ad-dresses the nonlinear system with the state space form asfollows

mk f mkminus1 ukminus1( 1113857 + wkminus1

nk H mkuk( 1113857 + vk(1)

where f(middot) is the nonlinear state function H(middot) is the ob-servation function m is the state vector of the system u isthe input vector of the system and n is the observationvector ampe parameters wkminus1 and vk are respectively theprocess noise and the observation noise vectors and they areassumed to be Gaussian

Generally for parameter identification the parametersare regarded as states and then the parameters are estimatedtogether with the states

x mT θT1113960 1113961T (2)

where θ is the parameter vector of the system and x is theaugmented state vector

22 Modification of the UKF With the uncertainty of theparameters and states the unknown states and relatedparameters are often more difficult to identify using thetraditional UKF Fortunately states in civil engineering arealways continuous and parameters are mostly varyingslowly It is therefore feasible to constrain the freedom ofthe parameter estimation along the time In the modifiedUKF states and parameters estimated in the last step areused to estimate the current states to offer the historicalinformation as a reference by showing the consistency ofthe parameters and continuity of the states

In the modified method the states are divided into twogroups the direct states xd and the latent states xl ampelatent states usually have their own evolving process whichhas no observation variables After each step the latentstates are estimated again as 1113954x++

l by substituting otherestimated states and parameters into the force equilibriumequation ampis estimation is used as the observation of thenext step

2 Shock and Vibration

xk

xlk

xdk

θk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

f1 xlkminus1 xdkminus1 ukminus1( 1113857

f2 xlkminus1 xdkminus1 ukminus1( 1113857

θkminus1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+ wkminus1

yk

nk

1113954x++lk

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

H xlkminus1 xdkminus1 ukminus1( 1113857

g1 xdkminus1 θkminus1ukminus1( 1113857

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ + vk

(3)

where f1 and f2 represent the state update functionscorresponding to xd and xl respectively xk and yk standfor the augmented state and observation vectors re-spectively 1113954x++

lk is calculated by g1 and is the estimation ofthe latent variables used as observation for better esti-mating the state g1 is the expression of xl with respect toxd and u g1 is calculated with the force equilibriumequation xl f1(xl xd u) wkminus1 and vk are the augmentedprocess and observation noise with their covariancematrices being Qkminus1 and Rk respectively

Since the states of the system are continuous in timewhen the time step is small variance of a state is pre-dictably small ampis variance can be counted as the ad-ditive noise In spite of a new uncertain observation beingintroduced the latent variable is confined in a smallerrange which will largely increase the accuracy of theestimation as a whole ampe detailed procedure of themodified UKF identification method is summarized inAppendix A

3 Model Reference Slide ModeControl (MRSMC)

31 Simulation Model and Control Design ampe initialstructural model is a 3-story shear frame structure and thenonlinear behavior exists in the structure In order to studythe effect of control force in the nonlinear field the BoucndashWen model is used to model the nonlinear restoring forcebetween stories and the control goal is to reduce the dis-placement of the third story relative to the ground In thispaper the AMD control system is installed at the top of thestructureampe system schematic diagram is shown in Figure 1

32 Model of the AMD Control System In this work thestiffness and damping elements of the AMD system areobtained by the design method of the optimum TMDparameters equation [47] and the active force of the AMDsystem is designed by the MRSMC control method

ampe displacement states of the controlled system aredefined as x x1 x2 x3 x41113858 1113859

T and y y1 y2 y31113858 1113859T

where x1 x2 and x3 are the displacements against theground x4 is the displacement of the AMD mass relative tothe third story and yi is the interstory displacement ampeparameter zi is a dimensionless hysteretic displacement αA β c n and Dy are the parameters of the BoucndashWenmodel Dy is a parameter controls the magnitude of thehysteretic force of the BoucndashWen model and U is the activeforce generated by the actuator of the AMDampe parametersma ka and ca are the parameters of the AMD system Setting

a value to the ratio of the AMD mass to the main massμ ma3ms the parameter values of the AMD system can besolved from the optimum TMD parameters equation asshown in the following equation [47]

βa ωa

ω0

1minus(μ2)

1113968

1 + μ

ζa

μ(1minus(μ4))

4(1 + μ)(1minus μ2)

1113971

(4)

ampe equilibrium equation for an AMD system excited bythe ground acceleration ag can be expressed as

ma eurox3 + eurox4 + ag1113872 1113873 + kax4 + ca _x4 U (5)

ampe equilibrium equation of the main structure excitedby the ground acceleration is

Meuroxl + C _xl + Kxl + LKnz minusmseag minus fc (6a)

_zi 1

Dyi

Ai _yi minus β _yi

11138681113868111386811138681113868111386811138681113868 zi

11138681113868111386811138681113868111386811138681113868nminus1 minus ci _yi zi

11138681113868111386811138681113868111386811138681113868n

1113872 1113873 i 1 2 3 (6b)

wherexl [x1 x2 x3]

T M msI C

c1 + c2 minusc2minusc2 c2 + c3 minusc3minusc3 c3

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

K

α1k1 + α2k2 minusα2k2minusα2k2 α2k2 + α3k3 minusα3k3

minusα3k3 α3k3

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ L

1 minus11 minus1

1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

Kn

(1minus α1)k1Dy1(1minus α2)k2Dy2

(1minus α3)k3Dy3

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦

z [z1 z2 z3]T e [1 1 1]T and fc [0 0 Uminus kax4

minus ca _x4]T From the equation (6a) we can get the following

equation

ag

ma

ms

ms

ms

BoucndashWen modelActuator UDamping caStiffness ka

k3 c3

k2 c2

k1 c1

Figure 1 Schematic diagram of AMD controlled system

Shock and Vibration 3

ms eurox3 c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3

minus 1minus α3( 1113857k3Dy3z3 minusmsag minusU + kax4 + ca _x4(7)

Substituting this equation into equation (5) the equi-librium equation for the AMD system can be written as

ma eurox4 minusma

ms

1113874c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3 minus 1minus α3( 1113857

middot k3Dy3z31113875 + 1 +ma

ms

1113888 1113889 Uminus kax4 minus ca _x4( 1113857

(8)

ampe governing equations of the AMD control system areexpressed as equations (5) (6a) and (6b) Combining thosetwo equations gives the motion equation of the AMD systemas followsMp eurox + Cp _x + Fp(x z) Meag + HpU

Fp(x z)

0

K 0

minuska

0ma

ms

α3k3 minusma

ms

α3k3 1 +ma

ms

1113888 1113889ka

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

x

+

L

0 0 minusma

ms

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Knz

(9)where Mp isin R4times4 and Cp isin R4times4 are the mass and dampingmatrices respectively while x _x and eurox represent the dis-placement velocity and acceleration vectors of the AMDcontrol system respectively

Specifically

Mp

0

M 0

0

0 0 0 ma

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cp

0

C 0

minusca

0ma

ms

c3 minusma

ms

c3 1 +ma

ms

1113888 1113889ca

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Me minusms minusms minusms 01113858 1113859T

Hp 0 0 minus1 1 +ma

ms1113874 11138751113876 1113877

T

y LTx

(10)

where Hp is the force location vector and y is the interstorydisplacement of the model

33ReferenceModel For vibration control the slidingmodecontrol can make the controlled states track the desiredstates It is the best effect to let the controlled state approachzero however the AMD provides the control force mainlyby the inertia of the massWhen the active force U is equal tozero the above AMD system changes into a TMD systemamperefore in this paper the TMD control system is con-sidered the reference model ampe mathematical model of thereference model is defined as

Mm euroxm + Cm _xm + Fm xm z( 1113857 Mmag

_zi 1

Dyi

Ai _ymi minus β _ymi

11138681113868111386811138681113868111386811138681113868 zi

11138681113868111386811138681113868111386811138681113868nminus1 minus ci _ymi zi

11138681113868111386811138681113868111386811138681113868n

1113872 1113873 i 1 2 3

(11)

where Mm MP Cm Cp Fm Fp Mm Me and theother parameters are defined as above

4 MRSMC-UKF Control Law

In this paper our goal is to reduce the third-story dis-placement and control the displacement of the AMD systemampe signal error is defined as e a(x3 minusxd3) + b(x4 minusxd4) aand b are displacement parameters in which proper valuesare obtained by the optimal parameter module in Simulinkampe control effect is to make the selected signal error e

approach to zero gradually xd3 and xd4 are the states that wewant to track In this simulation we assume that xd3

w1xm3 and xd4 w2xm4 where xm3 and xm4 are the states ofthe reference model and the w1 and w2 are coefficients

ampe sliding mode surface is defined as

S CCe + _e (12)

where Cc gt 0ampe Lyapunov function candidate can be defined as

V 12(SmSS) ampen_V Sms

_S Sms CC _e + a eurox3 + b eurox4( 1113857

S1113858msCc _eminus aw1ms euroxd3 minus bw2ms euroxd4 +(aminus b)

middot c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3 minus 1minus α3( 1113857k3Dy3z31113872 1113873

minus amsag minus aminus 1 +13μ

1113888 1113889b1113888 1113889 Uminus kax4 minus ca _x4( 11138571113859

(13)

amperefore the active force U can be defined as

U 1

aminus(1 +(13μ))b

1113858msCc _eminus aw1ms euroxd3 minus bw2ms euroxd4 +(aminus b)

middot c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3 minus 1minus α3( 1113857k3Dy3z31113872 1113873

+ amsD middot sign(S)1113859 + kax4 + ca _x4

(14)

4 Shock and Vibration

where |ag|leD and D is an approximate rather than a strictupper bound value of an earthquake that is an external inputSubstituting equation (14) into equation (13) the derivativeof V can be rewritten as

_V S minusa middot ms middot ag minus a middot ms middot D middot sign(S)1113872 1113873

minusa middot ms ag middot S + D middot sign(S)1113872 1113873le 0(15)

If equation (15) is satisfied the control law U designed byequation (14) can guarantee the controlled system is stableby the Lyapunov stability theory Only when S 0 can we get_V 0 Since CC gt 0 in equation (12) we can derive that

e⟶ 0 (16)

From LaSallersquos theorem and the work of Xu and Ozguner[48] the active force U can realize the following equation

x3⟶ xd3

x4⟶ xd4(17)

Regarding the control force (14) there is a sign functionin the control force which will cause the chattering phe-nomenon [40 49] In this paper an inverse tangent functionis used to approximate the sign function

5 Numerical Simulation

MATLABSIMULINK is used for carrying out all simula-tions with a sampling frequency of 1000Hz for a period of100 s ampe flow chart of the simulation is shown in Figure 2

Firstly the states of the structure and the parameterswere updated in real time with the UKF Secondly the activeforce can be solved based on the estimated states and theidentified parameters

In general the acceleration of the structure is easy tomeasure therefore in the simulation we assumed that onlythe acceleration state of the structure is known To study thestructure with uncertainties we assumed that the parameterski α A β and c are unknown ampe parameters n and Dyi inthe BoucndashWen model have relatively little effect on thenonlinear behavior and are assumed to be knownampe valuesof the structural parameters are shown in Table 1

ampe El Centro earthquake with amplitude 490Gal isemployed as the seismic excitation ampe initial states of thestructure are set to zero and the mass ratio μ of the AMDcontrol system is set to 005 All the unknown parameters areassumed to be 06 times the actual value respectively

In the simulation the coefficients of the sliding modesurface are defined as Cc 2 a 3 b 01 w1 05 andw2 5 For the x3 the AMD control system based onMRSMC is better than the TMD system because of w1 05To improve the control of x3 at the expense of magnifyingthe displacement of the AMD mass we set w2 to 5 and themagnified displacement is in our acceptable range

51 State Estimation and Parameter IdentificationEquations (6a) and (6b) shows that there is a little correlationbetween the acceleration states eurox and the unknown BoucndashWen parameters that are closely related to the values of

state z If the observations that are input to the UKF havelittle correlation with the unknown parameters the iden-tification effect will be poor ampe state z is also needed to begiven to the UKF as the observation however the z state isdifficult to measure In this paper we propose a novelmethod First at time k using the estimated values of thestate and parameters the calculated value of the activecontrol force and the measured values of the external inputsand acceleration to calculate z by the dynamic equation ofthe AMD control system are shown in equation (6a) Secondthe measured value of acceleration at time k and state z isinput to the UKF as the observed value at time k to estimatethe state and parameters at time k+ 1 ampe calculationformula of the z state is expressed as

ze1

11minus α1( 1113857k1Dy1

1113858minusc1 _x1 + ca _x4 minus α1k1x1 + kax4

minusms eurox1 + eurox2 + eurox3 + 3ag1113872 1113873minusU1113859

ze2

11minus α2( 1113857k2Dy2

1113858c2 _x1 minus c2 _x2 + ca _x4 + α2k2x1

minus α2k2x2 + kax4 minusms eurox2 + eurox3 + 2ag1113872 1113873minusU1113859

ze3

11minus α3( 1113857k3Dy3

1113858c3 _x2 minus c3 _x3 + ca _x4 + α3k3x2

minus α3k3x3 + kax4 minusms eurox3 + ag1113872 1113873minusU1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

It should be noted that all the unknown variables in theabove equation used the value estimated in real time Figures 3and 4 show the parameter identification effect using only theacceleration as the observation input and using both accel-eration and the state z as the observation input respectivelyampe identified results are shown in Table 2 ampe state esti-mation results are shown in Figures 5ndash7 and the estimated

agPlant

Model

MRSMC UKFForce Identified parameter

Estimated

Figure 2 Flow chart of the simulation

Table 1 Structural parameters for simulation

Story ms (t) ki (104 kNm) ci (kNlowast sm) Dyi (cm) n

13456

9315 545 1952 7605 445 17

3 6165 359 15

Shock and Vibration 5

200 40 60 80 100020406

Ture valueIdentified value

051

15A

020406

Time (sec)

020406

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

γ

α

β

(a)

Ture valueIdentified value

Time (sec)

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

020406

0608

1

020406

020406

A

γ

α

β

(b)

Figure 3 Results of the BoucndashWen parameter identification under different observations (a) Only acceleration is input to UKF (b) Bothacceleration and state z are input to UKF

20 40 60 80 10005

115

times108

times107

times107

True valueIdentified value

05

10

Time (sec)

05

10

k1

k2

k3

0

20 40 60 80 1000

20 40 60 80 1000

(a)

times108

times107

times107

True valueIdentified value

051

15k1

k2

k3

05

10

Time (sec)

05

10

20 40 60 80 1000

20 40 60 80 1000

20 40 60 80 1000

(b)

Figure 4 Results of the stiffness identification under different observations (a) Only acceleration is input to UKF (b) Both acceleration andstate z are input to UKF

Table 2 Identified value and error of the unknown parameters

Parameters Actual value Initial valueIdentified by augmented

observation Identified by acceleration

Value Error () Value Error ()k1 (107) 9315 55890 93537 042 92308 09k2 (107) 7605 45630 76404 047 75410 084k3 (107) 6165 36990 62121 076 61211 071α 05 03 05070 141 05169 337A 1 06 09906 094 10176 176β 05 03 05190 380 03917 2166c 05 03 05187 373 03893 2213

6 Shock and Vibration

6420 108Time (sec)

ndash01

ndash005

0

005

01

True valueEstimated value

Rel x

1 (m

)

(a)

6420 108Time (sec)

ndash015

ndash01

ndash005

0

005

01

015

True valueEstimated value

Rel x

2 (m

)

(b)

True valueEstimated value

6420 108Time (sec)

ndash01

ndash005

0

005

01

015

02

Rel x

3 (m

)

(c)

Figure 5 Comparison of structural displacement trajectory under MRSMC (a) Displacement of the first story (b) Displacement of thesecond story (c) Displacement of the third story

6420 108Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

True valueEstimated value

(a)

Int x

2 (m

)

True valueEstimated value

6420 108Time (sec)

ndash005

0

005

(b)

Figure 6 Continued

Shock and Vibration 7

states effectively converge the actual states It is beneficial toobtain the control force when the UKF can identify theunknown parameters in a short time with a small error

52 Control Results For chattering reduction the sign(middot) isreplaced by arctan(middot) in equation (15) since the arctan(middot)

function generates smooth control actions [37ndash39] ampe

control effect of AMD is compared with the response of thestructure without control and the TMD control systemand the result of the control are shown in Figure 8 ampecontrol results of the interstory displacement are shown inFigure 9 ampe states of AMD mass are shown in Figure 10ampe active force generated by the actuator is shown inFigure 11 and the value of the sliding surface is shown inFigure 12

Int x

3 (m

)

True valueEstimated value

0 1 2 3 4 5 6 7 8 9 10Time (sec)

ndash01

ndash005

0

005

01

(c)

Figure 6 Comparison of structural interstory displacement trajectory under MRSMC (a) Interstory displacement of the first story (b)Interstory displacement of the second story (c) Interstory displacement of the third story

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 1 (m

)

True valueEstimatied value

(a)

z 2 (m

)

True valueEstimatied value

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

(b)

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 3 (m

)

True valueEstimatied value

(c)

Figure 7 Dimensionless hysteretic displacement trajectory (a) z1 (b) z2 (c) z3

8 Shock and Vibration

200 40 60Time (sec)

ndash01

ndash005

0

005

01

Rel x

1 (m

)

No controlTMD controlAMD control

(a)

200 40 60Time (sec)

Rel x

2 (m

)

No controlTMD controlAMD control

ndash02ndash015

ndash01ndash005

0005

01015

02

(b)

200 40 60Time (sec)

Rel x

3 (m

)

No controlTMD controlAMD control

ndash03

ndash02

ndash01

0

01

02

03

(c)

Figure 8 Comparison of each story response for different control strategies (a) ampe 1st story response (b) ampe 2nd story response (c) ampe3rd story response

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

No controlTMD controlAMD control

(a)

Int x

2 (m

)

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(b)

Figure 9 Continued

Shock and Vibration 9

Int x

3 (m

)100 20 30 40 50 60

Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(c)

Figure 9 Comparison of each interstory displacement for different control strategies (a) ampe 1st interstory displacement (b) ampe 2ndinterstory displacement (c) ampe 3rd interstory displacement

100 20 30 40 50 60

100 20 30 40 50 60

100 20 30 40 50 60

ndash3

0

3

x 4 (m

)

TMD controlAMD control

ndash20

0

20

v 4 (m

)

Time (sec)

ndash100

0

100

a4

(m)

Figure 10 Responses of AMD and TMD device

Time (sec)

ndash6

ndash4

ndash2

0

2

4

Forc

e app

lied

on th

e str

uctu

re

times106

100 20 30 40 50 60

Figure 11 Applied active force

10 Shock and Vibration

In Table 3 different criterions are used to demonstratethe control effect of utilizing the method introduced in thispaper In terms of the maximum interstory drift the controleffect of the introduced AMD method is mostly far betterthan the TMD control effect in the control of the first twostories though the control effect of these two methods in thethird story is comparable When the input potential ofstructure which is expressed as the 2-norm of the interstorydrift is studied the AMD control shows overwhelmingadvantage compared with the TMD control effect

6 Conclusions

ampe vibration mitigation of a nonlinear structure using anactive mass damper with an adaptive control design isstudied in this paper and the adaptive force is provided bythe model reference sliding model control (MRSMC)method Due to the unknown parameters in the systemthis paper proposed a novel control method by combiningthe unscented Kalman filter (UKF) with the MRSMC toform the integrated MRSMC-UKF ampe UKF is used toidentify the unknown parameters and estimate thestructural states and the MRSMC is used to determine thecontrol law by these states and parameters based on theinformation estimated by UKF ampe numerical model is anonlinear 3-story frame structure with the AMD device onthe top floor ampe BoucndashWen model is used to model thenonlinear restoring force of the structure ampe stiffness ofthe simulated structure and the parameters of the BoucndashWen model are assumed to be unknown and these pa-rameters are estimated in real time by using the proposedmethod based on the measured acceleration states ampecontrol effect of AMD is compared with the responses ofthe structure without control and with the TMD controlsystem It turns out that the proposed MRSMC-UKF notonly efficiently estimates the states and identifies the pa-rameters but also effectively controls the structural non-linear vibration Based on the proposed method a betteractive control device could be developed to suppressstructural nonlinear vibration of the high-rise buildingsunder earthquake excitation Moreover other innovativeactive or semiactive control methods for structural non-linear vibration may also be proposed since a good non-linear mathematical model can be obtained by themodified UKF

Appendix

In this appendix the procedure of the modified UKF methodis presented as follows

First initialize the algorithm with1113954x+0 E x0( 1113857

P+0 E x0 minus 1113954x+

0( 1113857 x0 minus 1113954x+0( 1113857

T1113876 1113877

(A1)

where 1113954 represents the estimation of the correspondingvariable x0 is the initial value of the states 1113954x+

0 is the firstestimation of x0 P0 is the estimated covariance of x0 by thesame principle P is the estimated covariance of x andE() calculates the expectation of a random variable

ampen the algorithm startsAt the k-th stepUse 1113954x+

kminus1 which is the estimation of xkminus1 at the kminus 1-thstep to formulate the sigma point vectors 1113954x(i)

kminus1

1113954x(i)kminus1 1113954x+

kminus1 + 1113957x(i) i 1 2n (A2)

where 1113957x(i) (nP+

kminus11113968

)T

i i 1 n and 1113957x(i+n)

minus(nP+

kminus11113968

)T

i i 1 n

Calculate the estimated latent state 1113954x++lk with

1113954x++lk g1 xdkminus1 θkminus1 ukminus1( 1113857 (A3)

Substitute the sigma points into the system model toobtain the updated sigma points 1113954x(i)

k

1113954x(i)k f 1113954x(i)

kminus1ukminus11113872 1113873 i 1 2n (A4)

With these updated sigma points the first estimations ofthe mean and covariance of the states in step k are

1113954xminusk 12n

1113944

2n

i11113954x(i)

k

Pminusk 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954x(i)k minus 1113954xminusk1113872 1113873 + Qkminus1

(A5)

Substituting the sigma points into the observationfunction gives

1113954n(i)k H 1113954xi

kuk1113872 1113873 (A6)

From the sigma points of the observation vector

1113954y(i)k

1113954n(i)k

1113954x(i)lk

1113890 1113891 1113954x(i)lk is the value in 1113954x(i)

k corresponding to the

latent statesampe estimation of the observation and its covariance

together with the cross covariance of x and y are computedas follows

Table 3 Control effect under different criteria

Story max|di(t)||AMDcontrolmax|di(t)||TMDcontrol

max|di(t)||AMDcontrolmax|di(t)||No control

di(t)2|AMDcontroldi(t)2|TMDcontrol

di(t)2|AMDcontroldi(t)2|No control

1 7805 6878 7507 62562 6167 5013 5371 44363 10267 8328 8495 7038

100 20 30 40 50 60Time (sec)

ndash12ndash1

ndash08ndash06ndash04ndash02

00204

Slid

ing

surfa

ce

Figure 12 Convergence of the sliding surface

Shock and Vibration 11

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

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Submit your manuscripts atwwwhindawicom

xk

xlk

xdk

θk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

f1 xlkminus1 xdkminus1 ukminus1( 1113857

f2 xlkminus1 xdkminus1 ukminus1( 1113857

θkminus1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦+ wkminus1

yk

nk

1113954x++lk

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦

H xlkminus1 xdkminus1 ukminus1( 1113857

g1 xdkminus1 θkminus1ukminus1( 1113857

⎡⎢⎢⎢⎢⎣ ⎤⎥⎥⎥⎥⎦ + vk

(3)

where f1 and f2 represent the state update functionscorresponding to xd and xl respectively xk and yk standfor the augmented state and observation vectors re-spectively 1113954x++

lk is calculated by g1 and is the estimation ofthe latent variables used as observation for better esti-mating the state g1 is the expression of xl with respect toxd and u g1 is calculated with the force equilibriumequation xl f1(xl xd u) wkminus1 and vk are the augmentedprocess and observation noise with their covariancematrices being Qkminus1 and Rk respectively

Since the states of the system are continuous in timewhen the time step is small variance of a state is pre-dictably small ampis variance can be counted as the ad-ditive noise In spite of a new uncertain observation beingintroduced the latent variable is confined in a smallerrange which will largely increase the accuracy of theestimation as a whole ampe detailed procedure of themodified UKF identification method is summarized inAppendix A

3 Model Reference Slide ModeControl (MRSMC)

31 Simulation Model and Control Design ampe initialstructural model is a 3-story shear frame structure and thenonlinear behavior exists in the structure In order to studythe effect of control force in the nonlinear field the BoucndashWen model is used to model the nonlinear restoring forcebetween stories and the control goal is to reduce the dis-placement of the third story relative to the ground In thispaper the AMD control system is installed at the top of thestructureampe system schematic diagram is shown in Figure 1

32 Model of the AMD Control System In this work thestiffness and damping elements of the AMD system areobtained by the design method of the optimum TMDparameters equation [47] and the active force of the AMDsystem is designed by the MRSMC control method

ampe displacement states of the controlled system aredefined as x x1 x2 x3 x41113858 1113859

T and y y1 y2 y31113858 1113859T

where x1 x2 and x3 are the displacements against theground x4 is the displacement of the AMD mass relative tothe third story and yi is the interstory displacement ampeparameter zi is a dimensionless hysteretic displacement αA β c n and Dy are the parameters of the BoucndashWenmodel Dy is a parameter controls the magnitude of thehysteretic force of the BoucndashWen model and U is the activeforce generated by the actuator of the AMDampe parametersma ka and ca are the parameters of the AMD system Setting

a value to the ratio of the AMD mass to the main massμ ma3ms the parameter values of the AMD system can besolved from the optimum TMD parameters equation asshown in the following equation [47]

βa ωa

ω0

1minus(μ2)

1113968

1 + μ

ζa

μ(1minus(μ4))

4(1 + μ)(1minus μ2)

1113971

(4)

ampe equilibrium equation for an AMD system excited bythe ground acceleration ag can be expressed as

ma eurox3 + eurox4 + ag1113872 1113873 + kax4 + ca _x4 U (5)

ampe equilibrium equation of the main structure excitedby the ground acceleration is

Meuroxl + C _xl + Kxl + LKnz minusmseag minus fc (6a)

_zi 1

Dyi

Ai _yi minus β _yi

11138681113868111386811138681113868111386811138681113868 zi

11138681113868111386811138681113868111386811138681113868nminus1 minus ci _yi zi

11138681113868111386811138681113868111386811138681113868n

1113872 1113873 i 1 2 3 (6b)

wherexl [x1 x2 x3]

T M msI C

c1 + c2 minusc2minusc2 c2 + c3 minusc3minusc3 c3

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

K

α1k1 + α2k2 minusα2k2minusα2k2 α2k2 + α3k3 minusα3k3

minusα3k3 α3k3

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦ L

1 minus11 minus1

1

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

Kn

(1minus α1)k1Dy1(1minus α2)k2Dy2

(1minus α3)k3Dy3

⎡⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎦

z [z1 z2 z3]T e [1 1 1]T and fc [0 0 Uminus kax4

minus ca _x4]T From the equation (6a) we can get the following

equation

ag

ma

ms

ms

ms

BoucndashWen modelActuator UDamping caStiffness ka

k3 c3

k2 c2

k1 c1

Figure 1 Schematic diagram of AMD controlled system

Shock and Vibration 3

ms eurox3 c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3

minus 1minus α3( 1113857k3Dy3z3 minusmsag minusU + kax4 + ca _x4(7)

Substituting this equation into equation (5) the equi-librium equation for the AMD system can be written as

ma eurox4 minusma

ms

1113874c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3 minus 1minus α3( 1113857

middot k3Dy3z31113875 + 1 +ma

ms

1113888 1113889 Uminus kax4 minus ca _x4( 1113857

(8)

ampe governing equations of the AMD control system areexpressed as equations (5) (6a) and (6b) Combining thosetwo equations gives the motion equation of the AMD systemas followsMp eurox + Cp _x + Fp(x z) Meag + HpU

Fp(x z)

0

K 0

minuska

0ma

ms

α3k3 minusma

ms

α3k3 1 +ma

ms

1113888 1113889ka

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

x

+

L

0 0 minusma

ms

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Knz

(9)where Mp isin R4times4 and Cp isin R4times4 are the mass and dampingmatrices respectively while x _x and eurox represent the dis-placement velocity and acceleration vectors of the AMDcontrol system respectively

Specifically

Mp

0

M 0

0

0 0 0 ma

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cp

0

C 0

minusca

0ma

ms

c3 minusma

ms

c3 1 +ma

ms

1113888 1113889ca

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Me minusms minusms minusms 01113858 1113859T

Hp 0 0 minus1 1 +ma

ms1113874 11138751113876 1113877

T

y LTx

(10)

where Hp is the force location vector and y is the interstorydisplacement of the model

33ReferenceModel For vibration control the slidingmodecontrol can make the controlled states track the desiredstates It is the best effect to let the controlled state approachzero however the AMD provides the control force mainlyby the inertia of the massWhen the active force U is equal tozero the above AMD system changes into a TMD systemamperefore in this paper the TMD control system is con-sidered the reference model ampe mathematical model of thereference model is defined as

Mm euroxm + Cm _xm + Fm xm z( 1113857 Mmag

_zi 1

Dyi

Ai _ymi minus β _ymi

11138681113868111386811138681113868111386811138681113868 zi

11138681113868111386811138681113868111386811138681113868nminus1 minus ci _ymi zi

11138681113868111386811138681113868111386811138681113868n

1113872 1113873 i 1 2 3

(11)

where Mm MP Cm Cp Fm Fp Mm Me and theother parameters are defined as above

4 MRSMC-UKF Control Law

In this paper our goal is to reduce the third-story dis-placement and control the displacement of the AMD systemampe signal error is defined as e a(x3 minusxd3) + b(x4 minusxd4) aand b are displacement parameters in which proper valuesare obtained by the optimal parameter module in Simulinkampe control effect is to make the selected signal error e

approach to zero gradually xd3 and xd4 are the states that wewant to track In this simulation we assume that xd3

w1xm3 and xd4 w2xm4 where xm3 and xm4 are the states ofthe reference model and the w1 and w2 are coefficients

ampe sliding mode surface is defined as

S CCe + _e (12)

where Cc gt 0ampe Lyapunov function candidate can be defined as

V 12(SmSS) ampen_V Sms

_S Sms CC _e + a eurox3 + b eurox4( 1113857

S1113858msCc _eminus aw1ms euroxd3 minus bw2ms euroxd4 +(aminus b)

middot c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3 minus 1minus α3( 1113857k3Dy3z31113872 1113873

minus amsag minus aminus 1 +13μ

1113888 1113889b1113888 1113889 Uminus kax4 minus ca _x4( 11138571113859

(13)

amperefore the active force U can be defined as

U 1

aminus(1 +(13μ))b

1113858msCc _eminus aw1ms euroxd3 minus bw2ms euroxd4 +(aminus b)

middot c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3 minus 1minus α3( 1113857k3Dy3z31113872 1113873

+ amsD middot sign(S)1113859 + kax4 + ca _x4

(14)

4 Shock and Vibration

where |ag|leD and D is an approximate rather than a strictupper bound value of an earthquake that is an external inputSubstituting equation (14) into equation (13) the derivativeof V can be rewritten as

_V S minusa middot ms middot ag minus a middot ms middot D middot sign(S)1113872 1113873

minusa middot ms ag middot S + D middot sign(S)1113872 1113873le 0(15)

If equation (15) is satisfied the control law U designed byequation (14) can guarantee the controlled system is stableby the Lyapunov stability theory Only when S 0 can we get_V 0 Since CC gt 0 in equation (12) we can derive that

e⟶ 0 (16)

From LaSallersquos theorem and the work of Xu and Ozguner[48] the active force U can realize the following equation

x3⟶ xd3

x4⟶ xd4(17)

Regarding the control force (14) there is a sign functionin the control force which will cause the chattering phe-nomenon [40 49] In this paper an inverse tangent functionis used to approximate the sign function

5 Numerical Simulation

MATLABSIMULINK is used for carrying out all simula-tions with a sampling frequency of 1000Hz for a period of100 s ampe flow chart of the simulation is shown in Figure 2

Firstly the states of the structure and the parameterswere updated in real time with the UKF Secondly the activeforce can be solved based on the estimated states and theidentified parameters

In general the acceleration of the structure is easy tomeasure therefore in the simulation we assumed that onlythe acceleration state of the structure is known To study thestructure with uncertainties we assumed that the parameterski α A β and c are unknown ampe parameters n and Dyi inthe BoucndashWen model have relatively little effect on thenonlinear behavior and are assumed to be knownampe valuesof the structural parameters are shown in Table 1

ampe El Centro earthquake with amplitude 490Gal isemployed as the seismic excitation ampe initial states of thestructure are set to zero and the mass ratio μ of the AMDcontrol system is set to 005 All the unknown parameters areassumed to be 06 times the actual value respectively

In the simulation the coefficients of the sliding modesurface are defined as Cc 2 a 3 b 01 w1 05 andw2 5 For the x3 the AMD control system based onMRSMC is better than the TMD system because of w1 05To improve the control of x3 at the expense of magnifyingthe displacement of the AMD mass we set w2 to 5 and themagnified displacement is in our acceptable range

51 State Estimation and Parameter IdentificationEquations (6a) and (6b) shows that there is a little correlationbetween the acceleration states eurox and the unknown BoucndashWen parameters that are closely related to the values of

state z If the observations that are input to the UKF havelittle correlation with the unknown parameters the iden-tification effect will be poor ampe state z is also needed to begiven to the UKF as the observation however the z state isdifficult to measure In this paper we propose a novelmethod First at time k using the estimated values of thestate and parameters the calculated value of the activecontrol force and the measured values of the external inputsand acceleration to calculate z by the dynamic equation ofthe AMD control system are shown in equation (6a) Secondthe measured value of acceleration at time k and state z isinput to the UKF as the observed value at time k to estimatethe state and parameters at time k+ 1 ampe calculationformula of the z state is expressed as

ze1

11minus α1( 1113857k1Dy1

1113858minusc1 _x1 + ca _x4 minus α1k1x1 + kax4

minusms eurox1 + eurox2 + eurox3 + 3ag1113872 1113873minusU1113859

ze2

11minus α2( 1113857k2Dy2

1113858c2 _x1 minus c2 _x2 + ca _x4 + α2k2x1

minus α2k2x2 + kax4 minusms eurox2 + eurox3 + 2ag1113872 1113873minusU1113859

ze3

11minus α3( 1113857k3Dy3

1113858c3 _x2 minus c3 _x3 + ca _x4 + α3k3x2

minus α3k3x3 + kax4 minusms eurox3 + ag1113872 1113873minusU1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

It should be noted that all the unknown variables in theabove equation used the value estimated in real time Figures 3and 4 show the parameter identification effect using only theacceleration as the observation input and using both accel-eration and the state z as the observation input respectivelyampe identified results are shown in Table 2 ampe state esti-mation results are shown in Figures 5ndash7 and the estimated

agPlant

Model

MRSMC UKFForce Identified parameter

Estimated

Figure 2 Flow chart of the simulation

Table 1 Structural parameters for simulation

Story ms (t) ki (104 kNm) ci (kNlowast sm) Dyi (cm) n

13456

9315 545 1952 7605 445 17

3 6165 359 15

Shock and Vibration 5

200 40 60 80 100020406

Ture valueIdentified value

051

15A

020406

Time (sec)

020406

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

γ

α

β

(a)

Ture valueIdentified value

Time (sec)

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

020406

0608

1

020406

020406

A

γ

α

β

(b)

Figure 3 Results of the BoucndashWen parameter identification under different observations (a) Only acceleration is input to UKF (b) Bothacceleration and state z are input to UKF

20 40 60 80 10005

115

times108

times107

times107

True valueIdentified value

05

10

Time (sec)

05

10

k1

k2

k3

0

20 40 60 80 1000

20 40 60 80 1000

(a)

times108

times107

times107

True valueIdentified value

051

15k1

k2

k3

05

10

Time (sec)

05

10

20 40 60 80 1000

20 40 60 80 1000

20 40 60 80 1000

(b)

Figure 4 Results of the stiffness identification under different observations (a) Only acceleration is input to UKF (b) Both acceleration andstate z are input to UKF

Table 2 Identified value and error of the unknown parameters

Parameters Actual value Initial valueIdentified by augmented

observation Identified by acceleration

Value Error () Value Error ()k1 (107) 9315 55890 93537 042 92308 09k2 (107) 7605 45630 76404 047 75410 084k3 (107) 6165 36990 62121 076 61211 071α 05 03 05070 141 05169 337A 1 06 09906 094 10176 176β 05 03 05190 380 03917 2166c 05 03 05187 373 03893 2213

6 Shock and Vibration

6420 108Time (sec)

ndash01

ndash005

0

005

01

True valueEstimated value

Rel x

1 (m

)

(a)

6420 108Time (sec)

ndash015

ndash01

ndash005

0

005

01

015

True valueEstimated value

Rel x

2 (m

)

(b)

True valueEstimated value

6420 108Time (sec)

ndash01

ndash005

0

005

01

015

02

Rel x

3 (m

)

(c)

Figure 5 Comparison of structural displacement trajectory under MRSMC (a) Displacement of the first story (b) Displacement of thesecond story (c) Displacement of the third story

6420 108Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

True valueEstimated value

(a)

Int x

2 (m

)

True valueEstimated value

6420 108Time (sec)

ndash005

0

005

(b)

Figure 6 Continued

Shock and Vibration 7

states effectively converge the actual states It is beneficial toobtain the control force when the UKF can identify theunknown parameters in a short time with a small error

52 Control Results For chattering reduction the sign(middot) isreplaced by arctan(middot) in equation (15) since the arctan(middot)

function generates smooth control actions [37ndash39] ampe

control effect of AMD is compared with the response of thestructure without control and the TMD control systemand the result of the control are shown in Figure 8 ampecontrol results of the interstory displacement are shown inFigure 9 ampe states of AMD mass are shown in Figure 10ampe active force generated by the actuator is shown inFigure 11 and the value of the sliding surface is shown inFigure 12

Int x

3 (m

)

True valueEstimated value

0 1 2 3 4 5 6 7 8 9 10Time (sec)

ndash01

ndash005

0

005

01

(c)

Figure 6 Comparison of structural interstory displacement trajectory under MRSMC (a) Interstory displacement of the first story (b)Interstory displacement of the second story (c) Interstory displacement of the third story

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 1 (m

)

True valueEstimatied value

(a)

z 2 (m

)

True valueEstimatied value

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

(b)

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 3 (m

)

True valueEstimatied value

(c)

Figure 7 Dimensionless hysteretic displacement trajectory (a) z1 (b) z2 (c) z3

8 Shock and Vibration

200 40 60Time (sec)

ndash01

ndash005

0

005

01

Rel x

1 (m

)

No controlTMD controlAMD control

(a)

200 40 60Time (sec)

Rel x

2 (m

)

No controlTMD controlAMD control

ndash02ndash015

ndash01ndash005

0005

01015

02

(b)

200 40 60Time (sec)

Rel x

3 (m

)

No controlTMD controlAMD control

ndash03

ndash02

ndash01

0

01

02

03

(c)

Figure 8 Comparison of each story response for different control strategies (a) ampe 1st story response (b) ampe 2nd story response (c) ampe3rd story response

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

No controlTMD controlAMD control

(a)

Int x

2 (m

)

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(b)

Figure 9 Continued

Shock and Vibration 9

Int x

3 (m

)100 20 30 40 50 60

Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(c)

Figure 9 Comparison of each interstory displacement for different control strategies (a) ampe 1st interstory displacement (b) ampe 2ndinterstory displacement (c) ampe 3rd interstory displacement

100 20 30 40 50 60

100 20 30 40 50 60

100 20 30 40 50 60

ndash3

0

3

x 4 (m

)

TMD controlAMD control

ndash20

0

20

v 4 (m

)

Time (sec)

ndash100

0

100

a4

(m)

Figure 10 Responses of AMD and TMD device

Time (sec)

ndash6

ndash4

ndash2

0

2

4

Forc

e app

lied

on th

e str

uctu

re

times106

100 20 30 40 50 60

Figure 11 Applied active force

10 Shock and Vibration

In Table 3 different criterions are used to demonstratethe control effect of utilizing the method introduced in thispaper In terms of the maximum interstory drift the controleffect of the introduced AMD method is mostly far betterthan the TMD control effect in the control of the first twostories though the control effect of these two methods in thethird story is comparable When the input potential ofstructure which is expressed as the 2-norm of the interstorydrift is studied the AMD control shows overwhelmingadvantage compared with the TMD control effect

6 Conclusions

ampe vibration mitigation of a nonlinear structure using anactive mass damper with an adaptive control design isstudied in this paper and the adaptive force is provided bythe model reference sliding model control (MRSMC)method Due to the unknown parameters in the systemthis paper proposed a novel control method by combiningthe unscented Kalman filter (UKF) with the MRSMC toform the integrated MRSMC-UKF ampe UKF is used toidentify the unknown parameters and estimate thestructural states and the MRSMC is used to determine thecontrol law by these states and parameters based on theinformation estimated by UKF ampe numerical model is anonlinear 3-story frame structure with the AMD device onthe top floor ampe BoucndashWen model is used to model thenonlinear restoring force of the structure ampe stiffness ofthe simulated structure and the parameters of the BoucndashWen model are assumed to be unknown and these pa-rameters are estimated in real time by using the proposedmethod based on the measured acceleration states ampecontrol effect of AMD is compared with the responses ofthe structure without control and with the TMD controlsystem It turns out that the proposed MRSMC-UKF notonly efficiently estimates the states and identifies the pa-rameters but also effectively controls the structural non-linear vibration Based on the proposed method a betteractive control device could be developed to suppressstructural nonlinear vibration of the high-rise buildingsunder earthquake excitation Moreover other innovativeactive or semiactive control methods for structural non-linear vibration may also be proposed since a good non-linear mathematical model can be obtained by themodified UKF

Appendix

In this appendix the procedure of the modified UKF methodis presented as follows

First initialize the algorithm with1113954x+0 E x0( 1113857

P+0 E x0 minus 1113954x+

0( 1113857 x0 minus 1113954x+0( 1113857

T1113876 1113877

(A1)

where 1113954 represents the estimation of the correspondingvariable x0 is the initial value of the states 1113954x+

0 is the firstestimation of x0 P0 is the estimated covariance of x0 by thesame principle P is the estimated covariance of x andE() calculates the expectation of a random variable

ampen the algorithm startsAt the k-th stepUse 1113954x+

kminus1 which is the estimation of xkminus1 at the kminus 1-thstep to formulate the sigma point vectors 1113954x(i)

kminus1

1113954x(i)kminus1 1113954x+

kminus1 + 1113957x(i) i 1 2n (A2)

where 1113957x(i) (nP+

kminus11113968

)T

i i 1 n and 1113957x(i+n)

minus(nP+

kminus11113968

)T

i i 1 n

Calculate the estimated latent state 1113954x++lk with

1113954x++lk g1 xdkminus1 θkminus1 ukminus1( 1113857 (A3)

Substitute the sigma points into the system model toobtain the updated sigma points 1113954x(i)

k

1113954x(i)k f 1113954x(i)

kminus1ukminus11113872 1113873 i 1 2n (A4)

With these updated sigma points the first estimations ofthe mean and covariance of the states in step k are

1113954xminusk 12n

1113944

2n

i11113954x(i)

k

Pminusk 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954x(i)k minus 1113954xminusk1113872 1113873 + Qkminus1

(A5)

Substituting the sigma points into the observationfunction gives

1113954n(i)k H 1113954xi

kuk1113872 1113873 (A6)

From the sigma points of the observation vector

1113954y(i)k

1113954n(i)k

1113954x(i)lk

1113890 1113891 1113954x(i)lk is the value in 1113954x(i)

k corresponding to the

latent statesampe estimation of the observation and its covariance

together with the cross covariance of x and y are computedas follows

Table 3 Control effect under different criteria

Story max|di(t)||AMDcontrolmax|di(t)||TMDcontrol

max|di(t)||AMDcontrolmax|di(t)||No control

di(t)2|AMDcontroldi(t)2|TMDcontrol

di(t)2|AMDcontroldi(t)2|No control

1 7805 6878 7507 62562 6167 5013 5371 44363 10267 8328 8495 7038

100 20 30 40 50 60Time (sec)

ndash12ndash1

ndash08ndash06ndash04ndash02

00204

Slid

ing

surfa

ce

Figure 12 Convergence of the sliding surface

Shock and Vibration 11

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

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ms eurox3 c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3

minus 1minus α3( 1113857k3Dy3z3 minusmsag minusU + kax4 + ca _x4(7)

Substituting this equation into equation (5) the equi-librium equation for the AMD system can be written as

ma eurox4 minusma

ms

1113874c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3 minus 1minus α3( 1113857

middot k3Dy3z31113875 + 1 +ma

ms

1113888 1113889 Uminus kax4 minus ca _x4( 1113857

(8)

ampe governing equations of the AMD control system areexpressed as equations (5) (6a) and (6b) Combining thosetwo equations gives the motion equation of the AMD systemas followsMp eurox + Cp _x + Fp(x z) Meag + HpU

Fp(x z)

0

K 0

minuska

0ma

ms

α3k3 minusma

ms

α3k3 1 +ma

ms

1113888 1113889ka

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

x

+

L

0 0 minusma

ms

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Knz

(9)where Mp isin R4times4 and Cp isin R4times4 are the mass and dampingmatrices respectively while x _x and eurox represent the dis-placement velocity and acceleration vectors of the AMDcontrol system respectively

Specifically

Mp

0

M 0

0

0 0 0 ma

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Cp

0

C 0

minusca

0ma

ms

c3 minusma

ms

c3 1 +ma

ms

1113888 1113889ca

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Me minusms minusms minusms 01113858 1113859T

Hp 0 0 minus1 1 +ma

ms1113874 11138751113876 1113877

T

y LTx

(10)

where Hp is the force location vector and y is the interstorydisplacement of the model

33ReferenceModel For vibration control the slidingmodecontrol can make the controlled states track the desiredstates It is the best effect to let the controlled state approachzero however the AMD provides the control force mainlyby the inertia of the massWhen the active force U is equal tozero the above AMD system changes into a TMD systemamperefore in this paper the TMD control system is con-sidered the reference model ampe mathematical model of thereference model is defined as

Mm euroxm + Cm _xm + Fm xm z( 1113857 Mmag

_zi 1

Dyi

Ai _ymi minus β _ymi

11138681113868111386811138681113868111386811138681113868 zi

11138681113868111386811138681113868111386811138681113868nminus1 minus ci _ymi zi

11138681113868111386811138681113868111386811138681113868n

1113872 1113873 i 1 2 3

(11)

where Mm MP Cm Cp Fm Fp Mm Me and theother parameters are defined as above

4 MRSMC-UKF Control Law

In this paper our goal is to reduce the third-story dis-placement and control the displacement of the AMD systemampe signal error is defined as e a(x3 minusxd3) + b(x4 minusxd4) aand b are displacement parameters in which proper valuesare obtained by the optimal parameter module in Simulinkampe control effect is to make the selected signal error e

approach to zero gradually xd3 and xd4 are the states that wewant to track In this simulation we assume that xd3

w1xm3 and xd4 w2xm4 where xm3 and xm4 are the states ofthe reference model and the w1 and w2 are coefficients

ampe sliding mode surface is defined as

S CCe + _e (12)

where Cc gt 0ampe Lyapunov function candidate can be defined as

V 12(SmSS) ampen_V Sms

_S Sms CC _e + a eurox3 + b eurox4( 1113857

S1113858msCc _eminus aw1ms euroxd3 minus bw2ms euroxd4 +(aminus b)

middot c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3 minus 1minus α3( 1113857k3Dy3z31113872 1113873

minus amsag minus aminus 1 +13μ

1113888 1113889b1113888 1113889 Uminus kax4 minus ca _x4( 11138571113859

(13)

amperefore the active force U can be defined as

U 1

aminus(1 +(13μ))b

1113858msCc _eminus aw1ms euroxd3 minus bw2ms euroxd4 +(aminus b)

middot c3 _x2 minus c3 _x3 + α3k3x2 minus α3k3x3 minus 1minus α3( 1113857k3Dy3z31113872 1113873

+ amsD middot sign(S)1113859 + kax4 + ca _x4

(14)

4 Shock and Vibration

where |ag|leD and D is an approximate rather than a strictupper bound value of an earthquake that is an external inputSubstituting equation (14) into equation (13) the derivativeof V can be rewritten as

_V S minusa middot ms middot ag minus a middot ms middot D middot sign(S)1113872 1113873

minusa middot ms ag middot S + D middot sign(S)1113872 1113873le 0(15)

If equation (15) is satisfied the control law U designed byequation (14) can guarantee the controlled system is stableby the Lyapunov stability theory Only when S 0 can we get_V 0 Since CC gt 0 in equation (12) we can derive that

e⟶ 0 (16)

From LaSallersquos theorem and the work of Xu and Ozguner[48] the active force U can realize the following equation

x3⟶ xd3

x4⟶ xd4(17)

Regarding the control force (14) there is a sign functionin the control force which will cause the chattering phe-nomenon [40 49] In this paper an inverse tangent functionis used to approximate the sign function

5 Numerical Simulation

MATLABSIMULINK is used for carrying out all simula-tions with a sampling frequency of 1000Hz for a period of100 s ampe flow chart of the simulation is shown in Figure 2

Firstly the states of the structure and the parameterswere updated in real time with the UKF Secondly the activeforce can be solved based on the estimated states and theidentified parameters

In general the acceleration of the structure is easy tomeasure therefore in the simulation we assumed that onlythe acceleration state of the structure is known To study thestructure with uncertainties we assumed that the parameterski α A β and c are unknown ampe parameters n and Dyi inthe BoucndashWen model have relatively little effect on thenonlinear behavior and are assumed to be knownampe valuesof the structural parameters are shown in Table 1

ampe El Centro earthquake with amplitude 490Gal isemployed as the seismic excitation ampe initial states of thestructure are set to zero and the mass ratio μ of the AMDcontrol system is set to 005 All the unknown parameters areassumed to be 06 times the actual value respectively

In the simulation the coefficients of the sliding modesurface are defined as Cc 2 a 3 b 01 w1 05 andw2 5 For the x3 the AMD control system based onMRSMC is better than the TMD system because of w1 05To improve the control of x3 at the expense of magnifyingthe displacement of the AMD mass we set w2 to 5 and themagnified displacement is in our acceptable range

51 State Estimation and Parameter IdentificationEquations (6a) and (6b) shows that there is a little correlationbetween the acceleration states eurox and the unknown BoucndashWen parameters that are closely related to the values of

state z If the observations that are input to the UKF havelittle correlation with the unknown parameters the iden-tification effect will be poor ampe state z is also needed to begiven to the UKF as the observation however the z state isdifficult to measure In this paper we propose a novelmethod First at time k using the estimated values of thestate and parameters the calculated value of the activecontrol force and the measured values of the external inputsand acceleration to calculate z by the dynamic equation ofthe AMD control system are shown in equation (6a) Secondthe measured value of acceleration at time k and state z isinput to the UKF as the observed value at time k to estimatethe state and parameters at time k+ 1 ampe calculationformula of the z state is expressed as

ze1

11minus α1( 1113857k1Dy1

1113858minusc1 _x1 + ca _x4 minus α1k1x1 + kax4

minusms eurox1 + eurox2 + eurox3 + 3ag1113872 1113873minusU1113859

ze2

11minus α2( 1113857k2Dy2

1113858c2 _x1 minus c2 _x2 + ca _x4 + α2k2x1

minus α2k2x2 + kax4 minusms eurox2 + eurox3 + 2ag1113872 1113873minusU1113859

ze3

11minus α3( 1113857k3Dy3

1113858c3 _x2 minus c3 _x3 + ca _x4 + α3k3x2

minus α3k3x3 + kax4 minusms eurox3 + ag1113872 1113873minusU1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

It should be noted that all the unknown variables in theabove equation used the value estimated in real time Figures 3and 4 show the parameter identification effect using only theacceleration as the observation input and using both accel-eration and the state z as the observation input respectivelyampe identified results are shown in Table 2 ampe state esti-mation results are shown in Figures 5ndash7 and the estimated

agPlant

Model

MRSMC UKFForce Identified parameter

Estimated

Figure 2 Flow chart of the simulation

Table 1 Structural parameters for simulation

Story ms (t) ki (104 kNm) ci (kNlowast sm) Dyi (cm) n

13456

9315 545 1952 7605 445 17

3 6165 359 15

Shock and Vibration 5

200 40 60 80 100020406

Ture valueIdentified value

051

15A

020406

Time (sec)

020406

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

γ

α

β

(a)

Ture valueIdentified value

Time (sec)

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

020406

0608

1

020406

020406

A

γ

α

β

(b)

Figure 3 Results of the BoucndashWen parameter identification under different observations (a) Only acceleration is input to UKF (b) Bothacceleration and state z are input to UKF

20 40 60 80 10005

115

times108

times107

times107

True valueIdentified value

05

10

Time (sec)

05

10

k1

k2

k3

0

20 40 60 80 1000

20 40 60 80 1000

(a)

times108

times107

times107

True valueIdentified value

051

15k1

k2

k3

05

10

Time (sec)

05

10

20 40 60 80 1000

20 40 60 80 1000

20 40 60 80 1000

(b)

Figure 4 Results of the stiffness identification under different observations (a) Only acceleration is input to UKF (b) Both acceleration andstate z are input to UKF

Table 2 Identified value and error of the unknown parameters

Parameters Actual value Initial valueIdentified by augmented

observation Identified by acceleration

Value Error () Value Error ()k1 (107) 9315 55890 93537 042 92308 09k2 (107) 7605 45630 76404 047 75410 084k3 (107) 6165 36990 62121 076 61211 071α 05 03 05070 141 05169 337A 1 06 09906 094 10176 176β 05 03 05190 380 03917 2166c 05 03 05187 373 03893 2213

6 Shock and Vibration

6420 108Time (sec)

ndash01

ndash005

0

005

01

True valueEstimated value

Rel x

1 (m

)

(a)

6420 108Time (sec)

ndash015

ndash01

ndash005

0

005

01

015

True valueEstimated value

Rel x

2 (m

)

(b)

True valueEstimated value

6420 108Time (sec)

ndash01

ndash005

0

005

01

015

02

Rel x

3 (m

)

(c)

Figure 5 Comparison of structural displacement trajectory under MRSMC (a) Displacement of the first story (b) Displacement of thesecond story (c) Displacement of the third story

6420 108Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

True valueEstimated value

(a)

Int x

2 (m

)

True valueEstimated value

6420 108Time (sec)

ndash005

0

005

(b)

Figure 6 Continued

Shock and Vibration 7

states effectively converge the actual states It is beneficial toobtain the control force when the UKF can identify theunknown parameters in a short time with a small error

52 Control Results For chattering reduction the sign(middot) isreplaced by arctan(middot) in equation (15) since the arctan(middot)

function generates smooth control actions [37ndash39] ampe

control effect of AMD is compared with the response of thestructure without control and the TMD control systemand the result of the control are shown in Figure 8 ampecontrol results of the interstory displacement are shown inFigure 9 ampe states of AMD mass are shown in Figure 10ampe active force generated by the actuator is shown inFigure 11 and the value of the sliding surface is shown inFigure 12

Int x

3 (m

)

True valueEstimated value

0 1 2 3 4 5 6 7 8 9 10Time (sec)

ndash01

ndash005

0

005

01

(c)

Figure 6 Comparison of structural interstory displacement trajectory under MRSMC (a) Interstory displacement of the first story (b)Interstory displacement of the second story (c) Interstory displacement of the third story

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 1 (m

)

True valueEstimatied value

(a)

z 2 (m

)

True valueEstimatied value

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

(b)

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 3 (m

)

True valueEstimatied value

(c)

Figure 7 Dimensionless hysteretic displacement trajectory (a) z1 (b) z2 (c) z3

8 Shock and Vibration

200 40 60Time (sec)

ndash01

ndash005

0

005

01

Rel x

1 (m

)

No controlTMD controlAMD control

(a)

200 40 60Time (sec)

Rel x

2 (m

)

No controlTMD controlAMD control

ndash02ndash015

ndash01ndash005

0005

01015

02

(b)

200 40 60Time (sec)

Rel x

3 (m

)

No controlTMD controlAMD control

ndash03

ndash02

ndash01

0

01

02

03

(c)

Figure 8 Comparison of each story response for different control strategies (a) ampe 1st story response (b) ampe 2nd story response (c) ampe3rd story response

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

No controlTMD controlAMD control

(a)

Int x

2 (m

)

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(b)

Figure 9 Continued

Shock and Vibration 9

Int x

3 (m

)100 20 30 40 50 60

Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(c)

Figure 9 Comparison of each interstory displacement for different control strategies (a) ampe 1st interstory displacement (b) ampe 2ndinterstory displacement (c) ampe 3rd interstory displacement

100 20 30 40 50 60

100 20 30 40 50 60

100 20 30 40 50 60

ndash3

0

3

x 4 (m

)

TMD controlAMD control

ndash20

0

20

v 4 (m

)

Time (sec)

ndash100

0

100

a4

(m)

Figure 10 Responses of AMD and TMD device

Time (sec)

ndash6

ndash4

ndash2

0

2

4

Forc

e app

lied

on th

e str

uctu

re

times106

100 20 30 40 50 60

Figure 11 Applied active force

10 Shock and Vibration

In Table 3 different criterions are used to demonstratethe control effect of utilizing the method introduced in thispaper In terms of the maximum interstory drift the controleffect of the introduced AMD method is mostly far betterthan the TMD control effect in the control of the first twostories though the control effect of these two methods in thethird story is comparable When the input potential ofstructure which is expressed as the 2-norm of the interstorydrift is studied the AMD control shows overwhelmingadvantage compared with the TMD control effect

6 Conclusions

ampe vibration mitigation of a nonlinear structure using anactive mass damper with an adaptive control design isstudied in this paper and the adaptive force is provided bythe model reference sliding model control (MRSMC)method Due to the unknown parameters in the systemthis paper proposed a novel control method by combiningthe unscented Kalman filter (UKF) with the MRSMC toform the integrated MRSMC-UKF ampe UKF is used toidentify the unknown parameters and estimate thestructural states and the MRSMC is used to determine thecontrol law by these states and parameters based on theinformation estimated by UKF ampe numerical model is anonlinear 3-story frame structure with the AMD device onthe top floor ampe BoucndashWen model is used to model thenonlinear restoring force of the structure ampe stiffness ofthe simulated structure and the parameters of the BoucndashWen model are assumed to be unknown and these pa-rameters are estimated in real time by using the proposedmethod based on the measured acceleration states ampecontrol effect of AMD is compared with the responses ofthe structure without control and with the TMD controlsystem It turns out that the proposed MRSMC-UKF notonly efficiently estimates the states and identifies the pa-rameters but also effectively controls the structural non-linear vibration Based on the proposed method a betteractive control device could be developed to suppressstructural nonlinear vibration of the high-rise buildingsunder earthquake excitation Moreover other innovativeactive or semiactive control methods for structural non-linear vibration may also be proposed since a good non-linear mathematical model can be obtained by themodified UKF

Appendix

In this appendix the procedure of the modified UKF methodis presented as follows

First initialize the algorithm with1113954x+0 E x0( 1113857

P+0 E x0 minus 1113954x+

0( 1113857 x0 minus 1113954x+0( 1113857

T1113876 1113877

(A1)

where 1113954 represents the estimation of the correspondingvariable x0 is the initial value of the states 1113954x+

0 is the firstestimation of x0 P0 is the estimated covariance of x0 by thesame principle P is the estimated covariance of x andE() calculates the expectation of a random variable

ampen the algorithm startsAt the k-th stepUse 1113954x+

kminus1 which is the estimation of xkminus1 at the kminus 1-thstep to formulate the sigma point vectors 1113954x(i)

kminus1

1113954x(i)kminus1 1113954x+

kminus1 + 1113957x(i) i 1 2n (A2)

where 1113957x(i) (nP+

kminus11113968

)T

i i 1 n and 1113957x(i+n)

minus(nP+

kminus11113968

)T

i i 1 n

Calculate the estimated latent state 1113954x++lk with

1113954x++lk g1 xdkminus1 θkminus1 ukminus1( 1113857 (A3)

Substitute the sigma points into the system model toobtain the updated sigma points 1113954x(i)

k

1113954x(i)k f 1113954x(i)

kminus1ukminus11113872 1113873 i 1 2n (A4)

With these updated sigma points the first estimations ofthe mean and covariance of the states in step k are

1113954xminusk 12n

1113944

2n

i11113954x(i)

k

Pminusk 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954x(i)k minus 1113954xminusk1113872 1113873 + Qkminus1

(A5)

Substituting the sigma points into the observationfunction gives

1113954n(i)k H 1113954xi

kuk1113872 1113873 (A6)

From the sigma points of the observation vector

1113954y(i)k

1113954n(i)k

1113954x(i)lk

1113890 1113891 1113954x(i)lk is the value in 1113954x(i)

k corresponding to the

latent statesampe estimation of the observation and its covariance

together with the cross covariance of x and y are computedas follows

Table 3 Control effect under different criteria

Story max|di(t)||AMDcontrolmax|di(t)||TMDcontrol

max|di(t)||AMDcontrolmax|di(t)||No control

di(t)2|AMDcontroldi(t)2|TMDcontrol

di(t)2|AMDcontroldi(t)2|No control

1 7805 6878 7507 62562 6167 5013 5371 44363 10267 8328 8495 7038

100 20 30 40 50 60Time (sec)

ndash12ndash1

ndash08ndash06ndash04ndash02

00204

Slid

ing

surfa

ce

Figure 12 Convergence of the sliding surface

Shock and Vibration 11

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

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Submit your manuscripts atwwwhindawicom

where |ag|leD and D is an approximate rather than a strictupper bound value of an earthquake that is an external inputSubstituting equation (14) into equation (13) the derivativeof V can be rewritten as

_V S minusa middot ms middot ag minus a middot ms middot D middot sign(S)1113872 1113873

minusa middot ms ag middot S + D middot sign(S)1113872 1113873le 0(15)

If equation (15) is satisfied the control law U designed byequation (14) can guarantee the controlled system is stableby the Lyapunov stability theory Only when S 0 can we get_V 0 Since CC gt 0 in equation (12) we can derive that

e⟶ 0 (16)

From LaSallersquos theorem and the work of Xu and Ozguner[48] the active force U can realize the following equation

x3⟶ xd3

x4⟶ xd4(17)

Regarding the control force (14) there is a sign functionin the control force which will cause the chattering phe-nomenon [40 49] In this paper an inverse tangent functionis used to approximate the sign function

5 Numerical Simulation

MATLABSIMULINK is used for carrying out all simula-tions with a sampling frequency of 1000Hz for a period of100 s ampe flow chart of the simulation is shown in Figure 2

Firstly the states of the structure and the parameterswere updated in real time with the UKF Secondly the activeforce can be solved based on the estimated states and theidentified parameters

In general the acceleration of the structure is easy tomeasure therefore in the simulation we assumed that onlythe acceleration state of the structure is known To study thestructure with uncertainties we assumed that the parameterski α A β and c are unknown ampe parameters n and Dyi inthe BoucndashWen model have relatively little effect on thenonlinear behavior and are assumed to be knownampe valuesof the structural parameters are shown in Table 1

ampe El Centro earthquake with amplitude 490Gal isemployed as the seismic excitation ampe initial states of thestructure are set to zero and the mass ratio μ of the AMDcontrol system is set to 005 All the unknown parameters areassumed to be 06 times the actual value respectively

In the simulation the coefficients of the sliding modesurface are defined as Cc 2 a 3 b 01 w1 05 andw2 5 For the x3 the AMD control system based onMRSMC is better than the TMD system because of w1 05To improve the control of x3 at the expense of magnifyingthe displacement of the AMD mass we set w2 to 5 and themagnified displacement is in our acceptable range

51 State Estimation and Parameter IdentificationEquations (6a) and (6b) shows that there is a little correlationbetween the acceleration states eurox and the unknown BoucndashWen parameters that are closely related to the values of

state z If the observations that are input to the UKF havelittle correlation with the unknown parameters the iden-tification effect will be poor ampe state z is also needed to begiven to the UKF as the observation however the z state isdifficult to measure In this paper we propose a novelmethod First at time k using the estimated values of thestate and parameters the calculated value of the activecontrol force and the measured values of the external inputsand acceleration to calculate z by the dynamic equation ofthe AMD control system are shown in equation (6a) Secondthe measured value of acceleration at time k and state z isinput to the UKF as the observed value at time k to estimatethe state and parameters at time k+ 1 ampe calculationformula of the z state is expressed as

ze1

11minus α1( 1113857k1Dy1

1113858minusc1 _x1 + ca _x4 minus α1k1x1 + kax4

minusms eurox1 + eurox2 + eurox3 + 3ag1113872 1113873minusU1113859

ze2

11minus α2( 1113857k2Dy2

1113858c2 _x1 minus c2 _x2 + ca _x4 + α2k2x1

minus α2k2x2 + kax4 minusms eurox2 + eurox3 + 2ag1113872 1113873minusU1113859

ze3

11minus α3( 1113857k3Dy3

1113858c3 _x2 minus c3 _x3 + ca _x4 + α3k3x2

minus α3k3x3 + kax4 minusms eurox3 + ag1113872 1113873minusU1113859

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(18)

It should be noted that all the unknown variables in theabove equation used the value estimated in real time Figures 3and 4 show the parameter identification effect using only theacceleration as the observation input and using both accel-eration and the state z as the observation input respectivelyampe identified results are shown in Table 2 ampe state esti-mation results are shown in Figures 5ndash7 and the estimated

agPlant

Model

MRSMC UKFForce Identified parameter

Estimated

Figure 2 Flow chart of the simulation

Table 1 Structural parameters for simulation

Story ms (t) ki (104 kNm) ci (kNlowast sm) Dyi (cm) n

13456

9315 545 1952 7605 445 17

3 6165 359 15

Shock and Vibration 5

200 40 60 80 100020406

Ture valueIdentified value

051

15A

020406

Time (sec)

020406

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

γ

α

β

(a)

Ture valueIdentified value

Time (sec)

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

020406

0608

1

020406

020406

A

γ

α

β

(b)

Figure 3 Results of the BoucndashWen parameter identification under different observations (a) Only acceleration is input to UKF (b) Bothacceleration and state z are input to UKF

20 40 60 80 10005

115

times108

times107

times107

True valueIdentified value

05

10

Time (sec)

05

10

k1

k2

k3

0

20 40 60 80 1000

20 40 60 80 1000

(a)

times108

times107

times107

True valueIdentified value

051

15k1

k2

k3

05

10

Time (sec)

05

10

20 40 60 80 1000

20 40 60 80 1000

20 40 60 80 1000

(b)

Figure 4 Results of the stiffness identification under different observations (a) Only acceleration is input to UKF (b) Both acceleration andstate z are input to UKF

Table 2 Identified value and error of the unknown parameters

Parameters Actual value Initial valueIdentified by augmented

observation Identified by acceleration

Value Error () Value Error ()k1 (107) 9315 55890 93537 042 92308 09k2 (107) 7605 45630 76404 047 75410 084k3 (107) 6165 36990 62121 076 61211 071α 05 03 05070 141 05169 337A 1 06 09906 094 10176 176β 05 03 05190 380 03917 2166c 05 03 05187 373 03893 2213

6 Shock and Vibration

6420 108Time (sec)

ndash01

ndash005

0

005

01

True valueEstimated value

Rel x

1 (m

)

(a)

6420 108Time (sec)

ndash015

ndash01

ndash005

0

005

01

015

True valueEstimated value

Rel x

2 (m

)

(b)

True valueEstimated value

6420 108Time (sec)

ndash01

ndash005

0

005

01

015

02

Rel x

3 (m

)

(c)

Figure 5 Comparison of structural displacement trajectory under MRSMC (a) Displacement of the first story (b) Displacement of thesecond story (c) Displacement of the third story

6420 108Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

True valueEstimated value

(a)

Int x

2 (m

)

True valueEstimated value

6420 108Time (sec)

ndash005

0

005

(b)

Figure 6 Continued

Shock and Vibration 7

states effectively converge the actual states It is beneficial toobtain the control force when the UKF can identify theunknown parameters in a short time with a small error

52 Control Results For chattering reduction the sign(middot) isreplaced by arctan(middot) in equation (15) since the arctan(middot)

function generates smooth control actions [37ndash39] ampe

control effect of AMD is compared with the response of thestructure without control and the TMD control systemand the result of the control are shown in Figure 8 ampecontrol results of the interstory displacement are shown inFigure 9 ampe states of AMD mass are shown in Figure 10ampe active force generated by the actuator is shown inFigure 11 and the value of the sliding surface is shown inFigure 12

Int x

3 (m

)

True valueEstimated value

0 1 2 3 4 5 6 7 8 9 10Time (sec)

ndash01

ndash005

0

005

01

(c)

Figure 6 Comparison of structural interstory displacement trajectory under MRSMC (a) Interstory displacement of the first story (b)Interstory displacement of the second story (c) Interstory displacement of the third story

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 1 (m

)

True valueEstimatied value

(a)

z 2 (m

)

True valueEstimatied value

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

(b)

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 3 (m

)

True valueEstimatied value

(c)

Figure 7 Dimensionless hysteretic displacement trajectory (a) z1 (b) z2 (c) z3

8 Shock and Vibration

200 40 60Time (sec)

ndash01

ndash005

0

005

01

Rel x

1 (m

)

No controlTMD controlAMD control

(a)

200 40 60Time (sec)

Rel x

2 (m

)

No controlTMD controlAMD control

ndash02ndash015

ndash01ndash005

0005

01015

02

(b)

200 40 60Time (sec)

Rel x

3 (m

)

No controlTMD controlAMD control

ndash03

ndash02

ndash01

0

01

02

03

(c)

Figure 8 Comparison of each story response for different control strategies (a) ampe 1st story response (b) ampe 2nd story response (c) ampe3rd story response

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

No controlTMD controlAMD control

(a)

Int x

2 (m

)

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(b)

Figure 9 Continued

Shock and Vibration 9

Int x

3 (m

)100 20 30 40 50 60

Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(c)

Figure 9 Comparison of each interstory displacement for different control strategies (a) ampe 1st interstory displacement (b) ampe 2ndinterstory displacement (c) ampe 3rd interstory displacement

100 20 30 40 50 60

100 20 30 40 50 60

100 20 30 40 50 60

ndash3

0

3

x 4 (m

)

TMD controlAMD control

ndash20

0

20

v 4 (m

)

Time (sec)

ndash100

0

100

a4

(m)

Figure 10 Responses of AMD and TMD device

Time (sec)

ndash6

ndash4

ndash2

0

2

4

Forc

e app

lied

on th

e str

uctu

re

times106

100 20 30 40 50 60

Figure 11 Applied active force

10 Shock and Vibration

In Table 3 different criterions are used to demonstratethe control effect of utilizing the method introduced in thispaper In terms of the maximum interstory drift the controleffect of the introduced AMD method is mostly far betterthan the TMD control effect in the control of the first twostories though the control effect of these two methods in thethird story is comparable When the input potential ofstructure which is expressed as the 2-norm of the interstorydrift is studied the AMD control shows overwhelmingadvantage compared with the TMD control effect

6 Conclusions

ampe vibration mitigation of a nonlinear structure using anactive mass damper with an adaptive control design isstudied in this paper and the adaptive force is provided bythe model reference sliding model control (MRSMC)method Due to the unknown parameters in the systemthis paper proposed a novel control method by combiningthe unscented Kalman filter (UKF) with the MRSMC toform the integrated MRSMC-UKF ampe UKF is used toidentify the unknown parameters and estimate thestructural states and the MRSMC is used to determine thecontrol law by these states and parameters based on theinformation estimated by UKF ampe numerical model is anonlinear 3-story frame structure with the AMD device onthe top floor ampe BoucndashWen model is used to model thenonlinear restoring force of the structure ampe stiffness ofthe simulated structure and the parameters of the BoucndashWen model are assumed to be unknown and these pa-rameters are estimated in real time by using the proposedmethod based on the measured acceleration states ampecontrol effect of AMD is compared with the responses ofthe structure without control and with the TMD controlsystem It turns out that the proposed MRSMC-UKF notonly efficiently estimates the states and identifies the pa-rameters but also effectively controls the structural non-linear vibration Based on the proposed method a betteractive control device could be developed to suppressstructural nonlinear vibration of the high-rise buildingsunder earthquake excitation Moreover other innovativeactive or semiactive control methods for structural non-linear vibration may also be proposed since a good non-linear mathematical model can be obtained by themodified UKF

Appendix

In this appendix the procedure of the modified UKF methodis presented as follows

First initialize the algorithm with1113954x+0 E x0( 1113857

P+0 E x0 minus 1113954x+

0( 1113857 x0 minus 1113954x+0( 1113857

T1113876 1113877

(A1)

where 1113954 represents the estimation of the correspondingvariable x0 is the initial value of the states 1113954x+

0 is the firstestimation of x0 P0 is the estimated covariance of x0 by thesame principle P is the estimated covariance of x andE() calculates the expectation of a random variable

ampen the algorithm startsAt the k-th stepUse 1113954x+

kminus1 which is the estimation of xkminus1 at the kminus 1-thstep to formulate the sigma point vectors 1113954x(i)

kminus1

1113954x(i)kminus1 1113954x+

kminus1 + 1113957x(i) i 1 2n (A2)

where 1113957x(i) (nP+

kminus11113968

)T

i i 1 n and 1113957x(i+n)

minus(nP+

kminus11113968

)T

i i 1 n

Calculate the estimated latent state 1113954x++lk with

1113954x++lk g1 xdkminus1 θkminus1 ukminus1( 1113857 (A3)

Substitute the sigma points into the system model toobtain the updated sigma points 1113954x(i)

k

1113954x(i)k f 1113954x(i)

kminus1ukminus11113872 1113873 i 1 2n (A4)

With these updated sigma points the first estimations ofthe mean and covariance of the states in step k are

1113954xminusk 12n

1113944

2n

i11113954x(i)

k

Pminusk 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954x(i)k minus 1113954xminusk1113872 1113873 + Qkminus1

(A5)

Substituting the sigma points into the observationfunction gives

1113954n(i)k H 1113954xi

kuk1113872 1113873 (A6)

From the sigma points of the observation vector

1113954y(i)k

1113954n(i)k

1113954x(i)lk

1113890 1113891 1113954x(i)lk is the value in 1113954x(i)

k corresponding to the

latent statesampe estimation of the observation and its covariance

together with the cross covariance of x and y are computedas follows

Table 3 Control effect under different criteria

Story max|di(t)||AMDcontrolmax|di(t)||TMDcontrol

max|di(t)||AMDcontrolmax|di(t)||No control

di(t)2|AMDcontroldi(t)2|TMDcontrol

di(t)2|AMDcontroldi(t)2|No control

1 7805 6878 7507 62562 6167 5013 5371 44363 10267 8328 8495 7038

100 20 30 40 50 60Time (sec)

ndash12ndash1

ndash08ndash06ndash04ndash02

00204

Slid

ing

surfa

ce

Figure 12 Convergence of the sliding surface

Shock and Vibration 11

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

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Submit your manuscripts atwwwhindawicom

200 40 60 80 100020406

Ture valueIdentified value

051

15A

020406

Time (sec)

020406

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

γ

α

β

(a)

Ture valueIdentified value

Time (sec)

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

200 40 60 80 100

020406

0608

1

020406

020406

A

γ

α

β

(b)

Figure 3 Results of the BoucndashWen parameter identification under different observations (a) Only acceleration is input to UKF (b) Bothacceleration and state z are input to UKF

20 40 60 80 10005

115

times108

times107

times107

True valueIdentified value

05

10

Time (sec)

05

10

k1

k2

k3

0

20 40 60 80 1000

20 40 60 80 1000

(a)

times108

times107

times107

True valueIdentified value

051

15k1

k2

k3

05

10

Time (sec)

05

10

20 40 60 80 1000

20 40 60 80 1000

20 40 60 80 1000

(b)

Figure 4 Results of the stiffness identification under different observations (a) Only acceleration is input to UKF (b) Both acceleration andstate z are input to UKF

Table 2 Identified value and error of the unknown parameters

Parameters Actual value Initial valueIdentified by augmented

observation Identified by acceleration

Value Error () Value Error ()k1 (107) 9315 55890 93537 042 92308 09k2 (107) 7605 45630 76404 047 75410 084k3 (107) 6165 36990 62121 076 61211 071α 05 03 05070 141 05169 337A 1 06 09906 094 10176 176β 05 03 05190 380 03917 2166c 05 03 05187 373 03893 2213

6 Shock and Vibration

6420 108Time (sec)

ndash01

ndash005

0

005

01

True valueEstimated value

Rel x

1 (m

)

(a)

6420 108Time (sec)

ndash015

ndash01

ndash005

0

005

01

015

True valueEstimated value

Rel x

2 (m

)

(b)

True valueEstimated value

6420 108Time (sec)

ndash01

ndash005

0

005

01

015

02

Rel x

3 (m

)

(c)

Figure 5 Comparison of structural displacement trajectory under MRSMC (a) Displacement of the first story (b) Displacement of thesecond story (c) Displacement of the third story

6420 108Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

True valueEstimated value

(a)

Int x

2 (m

)

True valueEstimated value

6420 108Time (sec)

ndash005

0

005

(b)

Figure 6 Continued

Shock and Vibration 7

states effectively converge the actual states It is beneficial toobtain the control force when the UKF can identify theunknown parameters in a short time with a small error

52 Control Results For chattering reduction the sign(middot) isreplaced by arctan(middot) in equation (15) since the arctan(middot)

function generates smooth control actions [37ndash39] ampe

control effect of AMD is compared with the response of thestructure without control and the TMD control systemand the result of the control are shown in Figure 8 ampecontrol results of the interstory displacement are shown inFigure 9 ampe states of AMD mass are shown in Figure 10ampe active force generated by the actuator is shown inFigure 11 and the value of the sliding surface is shown inFigure 12

Int x

3 (m

)

True valueEstimated value

0 1 2 3 4 5 6 7 8 9 10Time (sec)

ndash01

ndash005

0

005

01

(c)

Figure 6 Comparison of structural interstory displacement trajectory under MRSMC (a) Interstory displacement of the first story (b)Interstory displacement of the second story (c) Interstory displacement of the third story

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 1 (m

)

True valueEstimatied value

(a)

z 2 (m

)

True valueEstimatied value

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

(b)

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 3 (m

)

True valueEstimatied value

(c)

Figure 7 Dimensionless hysteretic displacement trajectory (a) z1 (b) z2 (c) z3

8 Shock and Vibration

200 40 60Time (sec)

ndash01

ndash005

0

005

01

Rel x

1 (m

)

No controlTMD controlAMD control

(a)

200 40 60Time (sec)

Rel x

2 (m

)

No controlTMD controlAMD control

ndash02ndash015

ndash01ndash005

0005

01015

02

(b)

200 40 60Time (sec)

Rel x

3 (m

)

No controlTMD controlAMD control

ndash03

ndash02

ndash01

0

01

02

03

(c)

Figure 8 Comparison of each story response for different control strategies (a) ampe 1st story response (b) ampe 2nd story response (c) ampe3rd story response

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

No controlTMD controlAMD control

(a)

Int x

2 (m

)

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(b)

Figure 9 Continued

Shock and Vibration 9

Int x

3 (m

)100 20 30 40 50 60

Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(c)

Figure 9 Comparison of each interstory displacement for different control strategies (a) ampe 1st interstory displacement (b) ampe 2ndinterstory displacement (c) ampe 3rd interstory displacement

100 20 30 40 50 60

100 20 30 40 50 60

100 20 30 40 50 60

ndash3

0

3

x 4 (m

)

TMD controlAMD control

ndash20

0

20

v 4 (m

)

Time (sec)

ndash100

0

100

a4

(m)

Figure 10 Responses of AMD and TMD device

Time (sec)

ndash6

ndash4

ndash2

0

2

4

Forc

e app

lied

on th

e str

uctu

re

times106

100 20 30 40 50 60

Figure 11 Applied active force

10 Shock and Vibration

In Table 3 different criterions are used to demonstratethe control effect of utilizing the method introduced in thispaper In terms of the maximum interstory drift the controleffect of the introduced AMD method is mostly far betterthan the TMD control effect in the control of the first twostories though the control effect of these two methods in thethird story is comparable When the input potential ofstructure which is expressed as the 2-norm of the interstorydrift is studied the AMD control shows overwhelmingadvantage compared with the TMD control effect

6 Conclusions

ampe vibration mitigation of a nonlinear structure using anactive mass damper with an adaptive control design isstudied in this paper and the adaptive force is provided bythe model reference sliding model control (MRSMC)method Due to the unknown parameters in the systemthis paper proposed a novel control method by combiningthe unscented Kalman filter (UKF) with the MRSMC toform the integrated MRSMC-UKF ampe UKF is used toidentify the unknown parameters and estimate thestructural states and the MRSMC is used to determine thecontrol law by these states and parameters based on theinformation estimated by UKF ampe numerical model is anonlinear 3-story frame structure with the AMD device onthe top floor ampe BoucndashWen model is used to model thenonlinear restoring force of the structure ampe stiffness ofthe simulated structure and the parameters of the BoucndashWen model are assumed to be unknown and these pa-rameters are estimated in real time by using the proposedmethod based on the measured acceleration states ampecontrol effect of AMD is compared with the responses ofthe structure without control and with the TMD controlsystem It turns out that the proposed MRSMC-UKF notonly efficiently estimates the states and identifies the pa-rameters but also effectively controls the structural non-linear vibration Based on the proposed method a betteractive control device could be developed to suppressstructural nonlinear vibration of the high-rise buildingsunder earthquake excitation Moreover other innovativeactive or semiactive control methods for structural non-linear vibration may also be proposed since a good non-linear mathematical model can be obtained by themodified UKF

Appendix

In this appendix the procedure of the modified UKF methodis presented as follows

First initialize the algorithm with1113954x+0 E x0( 1113857

P+0 E x0 minus 1113954x+

0( 1113857 x0 minus 1113954x+0( 1113857

T1113876 1113877

(A1)

where 1113954 represents the estimation of the correspondingvariable x0 is the initial value of the states 1113954x+

0 is the firstestimation of x0 P0 is the estimated covariance of x0 by thesame principle P is the estimated covariance of x andE() calculates the expectation of a random variable

ampen the algorithm startsAt the k-th stepUse 1113954x+

kminus1 which is the estimation of xkminus1 at the kminus 1-thstep to formulate the sigma point vectors 1113954x(i)

kminus1

1113954x(i)kminus1 1113954x+

kminus1 + 1113957x(i) i 1 2n (A2)

where 1113957x(i) (nP+

kminus11113968

)T

i i 1 n and 1113957x(i+n)

minus(nP+

kminus11113968

)T

i i 1 n

Calculate the estimated latent state 1113954x++lk with

1113954x++lk g1 xdkminus1 θkminus1 ukminus1( 1113857 (A3)

Substitute the sigma points into the system model toobtain the updated sigma points 1113954x(i)

k

1113954x(i)k f 1113954x(i)

kminus1ukminus11113872 1113873 i 1 2n (A4)

With these updated sigma points the first estimations ofthe mean and covariance of the states in step k are

1113954xminusk 12n

1113944

2n

i11113954x(i)

k

Pminusk 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954x(i)k minus 1113954xminusk1113872 1113873 + Qkminus1

(A5)

Substituting the sigma points into the observationfunction gives

1113954n(i)k H 1113954xi

kuk1113872 1113873 (A6)

From the sigma points of the observation vector

1113954y(i)k

1113954n(i)k

1113954x(i)lk

1113890 1113891 1113954x(i)lk is the value in 1113954x(i)

k corresponding to the

latent statesampe estimation of the observation and its covariance

together with the cross covariance of x and y are computedas follows

Table 3 Control effect under different criteria

Story max|di(t)||AMDcontrolmax|di(t)||TMDcontrol

max|di(t)||AMDcontrolmax|di(t)||No control

di(t)2|AMDcontroldi(t)2|TMDcontrol

di(t)2|AMDcontroldi(t)2|No control

1 7805 6878 7507 62562 6167 5013 5371 44363 10267 8328 8495 7038

100 20 30 40 50 60Time (sec)

ndash12ndash1

ndash08ndash06ndash04ndash02

00204

Slid

ing

surfa

ce

Figure 12 Convergence of the sliding surface

Shock and Vibration 11

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

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Submit your manuscripts atwwwhindawicom

6420 108Time (sec)

ndash01

ndash005

0

005

01

True valueEstimated value

Rel x

1 (m

)

(a)

6420 108Time (sec)

ndash015

ndash01

ndash005

0

005

01

015

True valueEstimated value

Rel x

2 (m

)

(b)

True valueEstimated value

6420 108Time (sec)

ndash01

ndash005

0

005

01

015

02

Rel x

3 (m

)

(c)

Figure 5 Comparison of structural displacement trajectory under MRSMC (a) Displacement of the first story (b) Displacement of thesecond story (c) Displacement of the third story

6420 108Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

True valueEstimated value

(a)

Int x

2 (m

)

True valueEstimated value

6420 108Time (sec)

ndash005

0

005

(b)

Figure 6 Continued

Shock and Vibration 7

states effectively converge the actual states It is beneficial toobtain the control force when the UKF can identify theunknown parameters in a short time with a small error

52 Control Results For chattering reduction the sign(middot) isreplaced by arctan(middot) in equation (15) since the arctan(middot)

function generates smooth control actions [37ndash39] ampe

control effect of AMD is compared with the response of thestructure without control and the TMD control systemand the result of the control are shown in Figure 8 ampecontrol results of the interstory displacement are shown inFigure 9 ampe states of AMD mass are shown in Figure 10ampe active force generated by the actuator is shown inFigure 11 and the value of the sliding surface is shown inFigure 12

Int x

3 (m

)

True valueEstimated value

0 1 2 3 4 5 6 7 8 9 10Time (sec)

ndash01

ndash005

0

005

01

(c)

Figure 6 Comparison of structural interstory displacement trajectory under MRSMC (a) Interstory displacement of the first story (b)Interstory displacement of the second story (c) Interstory displacement of the third story

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 1 (m

)

True valueEstimatied value

(a)

z 2 (m

)

True valueEstimatied value

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

(b)

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 3 (m

)

True valueEstimatied value

(c)

Figure 7 Dimensionless hysteretic displacement trajectory (a) z1 (b) z2 (c) z3

8 Shock and Vibration

200 40 60Time (sec)

ndash01

ndash005

0

005

01

Rel x

1 (m

)

No controlTMD controlAMD control

(a)

200 40 60Time (sec)

Rel x

2 (m

)

No controlTMD controlAMD control

ndash02ndash015

ndash01ndash005

0005

01015

02

(b)

200 40 60Time (sec)

Rel x

3 (m

)

No controlTMD controlAMD control

ndash03

ndash02

ndash01

0

01

02

03

(c)

Figure 8 Comparison of each story response for different control strategies (a) ampe 1st story response (b) ampe 2nd story response (c) ampe3rd story response

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

No controlTMD controlAMD control

(a)

Int x

2 (m

)

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(b)

Figure 9 Continued

Shock and Vibration 9

Int x

3 (m

)100 20 30 40 50 60

Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(c)

Figure 9 Comparison of each interstory displacement for different control strategies (a) ampe 1st interstory displacement (b) ampe 2ndinterstory displacement (c) ampe 3rd interstory displacement

100 20 30 40 50 60

100 20 30 40 50 60

100 20 30 40 50 60

ndash3

0

3

x 4 (m

)

TMD controlAMD control

ndash20

0

20

v 4 (m

)

Time (sec)

ndash100

0

100

a4

(m)

Figure 10 Responses of AMD and TMD device

Time (sec)

ndash6

ndash4

ndash2

0

2

4

Forc

e app

lied

on th

e str

uctu

re

times106

100 20 30 40 50 60

Figure 11 Applied active force

10 Shock and Vibration

In Table 3 different criterions are used to demonstratethe control effect of utilizing the method introduced in thispaper In terms of the maximum interstory drift the controleffect of the introduced AMD method is mostly far betterthan the TMD control effect in the control of the first twostories though the control effect of these two methods in thethird story is comparable When the input potential ofstructure which is expressed as the 2-norm of the interstorydrift is studied the AMD control shows overwhelmingadvantage compared with the TMD control effect

6 Conclusions

ampe vibration mitigation of a nonlinear structure using anactive mass damper with an adaptive control design isstudied in this paper and the adaptive force is provided bythe model reference sliding model control (MRSMC)method Due to the unknown parameters in the systemthis paper proposed a novel control method by combiningthe unscented Kalman filter (UKF) with the MRSMC toform the integrated MRSMC-UKF ampe UKF is used toidentify the unknown parameters and estimate thestructural states and the MRSMC is used to determine thecontrol law by these states and parameters based on theinformation estimated by UKF ampe numerical model is anonlinear 3-story frame structure with the AMD device onthe top floor ampe BoucndashWen model is used to model thenonlinear restoring force of the structure ampe stiffness ofthe simulated structure and the parameters of the BoucndashWen model are assumed to be unknown and these pa-rameters are estimated in real time by using the proposedmethod based on the measured acceleration states ampecontrol effect of AMD is compared with the responses ofthe structure without control and with the TMD controlsystem It turns out that the proposed MRSMC-UKF notonly efficiently estimates the states and identifies the pa-rameters but also effectively controls the structural non-linear vibration Based on the proposed method a betteractive control device could be developed to suppressstructural nonlinear vibration of the high-rise buildingsunder earthquake excitation Moreover other innovativeactive or semiactive control methods for structural non-linear vibration may also be proposed since a good non-linear mathematical model can be obtained by themodified UKF

Appendix

In this appendix the procedure of the modified UKF methodis presented as follows

First initialize the algorithm with1113954x+0 E x0( 1113857

P+0 E x0 minus 1113954x+

0( 1113857 x0 minus 1113954x+0( 1113857

T1113876 1113877

(A1)

where 1113954 represents the estimation of the correspondingvariable x0 is the initial value of the states 1113954x+

0 is the firstestimation of x0 P0 is the estimated covariance of x0 by thesame principle P is the estimated covariance of x andE() calculates the expectation of a random variable

ampen the algorithm startsAt the k-th stepUse 1113954x+

kminus1 which is the estimation of xkminus1 at the kminus 1-thstep to formulate the sigma point vectors 1113954x(i)

kminus1

1113954x(i)kminus1 1113954x+

kminus1 + 1113957x(i) i 1 2n (A2)

where 1113957x(i) (nP+

kminus11113968

)T

i i 1 n and 1113957x(i+n)

minus(nP+

kminus11113968

)T

i i 1 n

Calculate the estimated latent state 1113954x++lk with

1113954x++lk g1 xdkminus1 θkminus1 ukminus1( 1113857 (A3)

Substitute the sigma points into the system model toobtain the updated sigma points 1113954x(i)

k

1113954x(i)k f 1113954x(i)

kminus1ukminus11113872 1113873 i 1 2n (A4)

With these updated sigma points the first estimations ofthe mean and covariance of the states in step k are

1113954xminusk 12n

1113944

2n

i11113954x(i)

k

Pminusk 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954x(i)k minus 1113954xminusk1113872 1113873 + Qkminus1

(A5)

Substituting the sigma points into the observationfunction gives

1113954n(i)k H 1113954xi

kuk1113872 1113873 (A6)

From the sigma points of the observation vector

1113954y(i)k

1113954n(i)k

1113954x(i)lk

1113890 1113891 1113954x(i)lk is the value in 1113954x(i)

k corresponding to the

latent statesampe estimation of the observation and its covariance

together with the cross covariance of x and y are computedas follows

Table 3 Control effect under different criteria

Story max|di(t)||AMDcontrolmax|di(t)||TMDcontrol

max|di(t)||AMDcontrolmax|di(t)||No control

di(t)2|AMDcontroldi(t)2|TMDcontrol

di(t)2|AMDcontroldi(t)2|No control

1 7805 6878 7507 62562 6167 5013 5371 44363 10267 8328 8495 7038

100 20 30 40 50 60Time (sec)

ndash12ndash1

ndash08ndash06ndash04ndash02

00204

Slid

ing

surfa

ce

Figure 12 Convergence of the sliding surface

Shock and Vibration 11

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

states effectively converge the actual states It is beneficial toobtain the control force when the UKF can identify theunknown parameters in a short time with a small error

52 Control Results For chattering reduction the sign(middot) isreplaced by arctan(middot) in equation (15) since the arctan(middot)

function generates smooth control actions [37ndash39] ampe

control effect of AMD is compared with the response of thestructure without control and the TMD control systemand the result of the control are shown in Figure 8 ampecontrol results of the interstory displacement are shown inFigure 9 ampe states of AMD mass are shown in Figure 10ampe active force generated by the actuator is shown inFigure 11 and the value of the sliding surface is shown inFigure 12

Int x

3 (m

)

True valueEstimated value

0 1 2 3 4 5 6 7 8 9 10Time (sec)

ndash01

ndash005

0

005

01

(c)

Figure 6 Comparison of structural interstory displacement trajectory under MRSMC (a) Interstory displacement of the first story (b)Interstory displacement of the second story (c) Interstory displacement of the third story

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 1 (m

)

True valueEstimatied value

(a)

z 2 (m

)

True valueEstimatied value

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

(b)

6420 108Time (sec)

ndash15

ndash1

ndash05

0

05

1

15

z 3 (m

)

True valueEstimatied value

(c)

Figure 7 Dimensionless hysteretic displacement trajectory (a) z1 (b) z2 (c) z3

8 Shock and Vibration

200 40 60Time (sec)

ndash01

ndash005

0

005

01

Rel x

1 (m

)

No controlTMD controlAMD control

(a)

200 40 60Time (sec)

Rel x

2 (m

)

No controlTMD controlAMD control

ndash02ndash015

ndash01ndash005

0005

01015

02

(b)

200 40 60Time (sec)

Rel x

3 (m

)

No controlTMD controlAMD control

ndash03

ndash02

ndash01

0

01

02

03

(c)

Figure 8 Comparison of each story response for different control strategies (a) ampe 1st story response (b) ampe 2nd story response (c) ampe3rd story response

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

No controlTMD controlAMD control

(a)

Int x

2 (m

)

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(b)

Figure 9 Continued

Shock and Vibration 9

Int x

3 (m

)100 20 30 40 50 60

Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(c)

Figure 9 Comparison of each interstory displacement for different control strategies (a) ampe 1st interstory displacement (b) ampe 2ndinterstory displacement (c) ampe 3rd interstory displacement

100 20 30 40 50 60

100 20 30 40 50 60

100 20 30 40 50 60

ndash3

0

3

x 4 (m

)

TMD controlAMD control

ndash20

0

20

v 4 (m

)

Time (sec)

ndash100

0

100

a4

(m)

Figure 10 Responses of AMD and TMD device

Time (sec)

ndash6

ndash4

ndash2

0

2

4

Forc

e app

lied

on th

e str

uctu

re

times106

100 20 30 40 50 60

Figure 11 Applied active force

10 Shock and Vibration

In Table 3 different criterions are used to demonstratethe control effect of utilizing the method introduced in thispaper In terms of the maximum interstory drift the controleffect of the introduced AMD method is mostly far betterthan the TMD control effect in the control of the first twostories though the control effect of these two methods in thethird story is comparable When the input potential ofstructure which is expressed as the 2-norm of the interstorydrift is studied the AMD control shows overwhelmingadvantage compared with the TMD control effect

6 Conclusions

ampe vibration mitigation of a nonlinear structure using anactive mass damper with an adaptive control design isstudied in this paper and the adaptive force is provided bythe model reference sliding model control (MRSMC)method Due to the unknown parameters in the systemthis paper proposed a novel control method by combiningthe unscented Kalman filter (UKF) with the MRSMC toform the integrated MRSMC-UKF ampe UKF is used toidentify the unknown parameters and estimate thestructural states and the MRSMC is used to determine thecontrol law by these states and parameters based on theinformation estimated by UKF ampe numerical model is anonlinear 3-story frame structure with the AMD device onthe top floor ampe BoucndashWen model is used to model thenonlinear restoring force of the structure ampe stiffness ofthe simulated structure and the parameters of the BoucndashWen model are assumed to be unknown and these pa-rameters are estimated in real time by using the proposedmethod based on the measured acceleration states ampecontrol effect of AMD is compared with the responses ofthe structure without control and with the TMD controlsystem It turns out that the proposed MRSMC-UKF notonly efficiently estimates the states and identifies the pa-rameters but also effectively controls the structural non-linear vibration Based on the proposed method a betteractive control device could be developed to suppressstructural nonlinear vibration of the high-rise buildingsunder earthquake excitation Moreover other innovativeactive or semiactive control methods for structural non-linear vibration may also be proposed since a good non-linear mathematical model can be obtained by themodified UKF

Appendix

In this appendix the procedure of the modified UKF methodis presented as follows

First initialize the algorithm with1113954x+0 E x0( 1113857

P+0 E x0 minus 1113954x+

0( 1113857 x0 minus 1113954x+0( 1113857

T1113876 1113877

(A1)

where 1113954 represents the estimation of the correspondingvariable x0 is the initial value of the states 1113954x+

0 is the firstestimation of x0 P0 is the estimated covariance of x0 by thesame principle P is the estimated covariance of x andE() calculates the expectation of a random variable

ampen the algorithm startsAt the k-th stepUse 1113954x+

kminus1 which is the estimation of xkminus1 at the kminus 1-thstep to formulate the sigma point vectors 1113954x(i)

kminus1

1113954x(i)kminus1 1113954x+

kminus1 + 1113957x(i) i 1 2n (A2)

where 1113957x(i) (nP+

kminus11113968

)T

i i 1 n and 1113957x(i+n)

minus(nP+

kminus11113968

)T

i i 1 n

Calculate the estimated latent state 1113954x++lk with

1113954x++lk g1 xdkminus1 θkminus1 ukminus1( 1113857 (A3)

Substitute the sigma points into the system model toobtain the updated sigma points 1113954x(i)

k

1113954x(i)k f 1113954x(i)

kminus1ukminus11113872 1113873 i 1 2n (A4)

With these updated sigma points the first estimations ofthe mean and covariance of the states in step k are

1113954xminusk 12n

1113944

2n

i11113954x(i)

k

Pminusk 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954x(i)k minus 1113954xminusk1113872 1113873 + Qkminus1

(A5)

Substituting the sigma points into the observationfunction gives

1113954n(i)k H 1113954xi

kuk1113872 1113873 (A6)

From the sigma points of the observation vector

1113954y(i)k

1113954n(i)k

1113954x(i)lk

1113890 1113891 1113954x(i)lk is the value in 1113954x(i)

k corresponding to the

latent statesampe estimation of the observation and its covariance

together with the cross covariance of x and y are computedas follows

Table 3 Control effect under different criteria

Story max|di(t)||AMDcontrolmax|di(t)||TMDcontrol

max|di(t)||AMDcontrolmax|di(t)||No control

di(t)2|AMDcontroldi(t)2|TMDcontrol

di(t)2|AMDcontroldi(t)2|No control

1 7805 6878 7507 62562 6167 5013 5371 44363 10267 8328 8495 7038

100 20 30 40 50 60Time (sec)

ndash12ndash1

ndash08ndash06ndash04ndash02

00204

Slid

ing

surfa

ce

Figure 12 Convergence of the sliding surface

Shock and Vibration 11

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

200 40 60Time (sec)

ndash01

ndash005

0

005

01

Rel x

1 (m

)

No controlTMD controlAMD control

(a)

200 40 60Time (sec)

Rel x

2 (m

)

No controlTMD controlAMD control

ndash02ndash015

ndash01ndash005

0005

01015

02

(b)

200 40 60Time (sec)

Rel x

3 (m

)

No controlTMD controlAMD control

ndash03

ndash02

ndash01

0

01

02

03

(c)

Figure 8 Comparison of each story response for different control strategies (a) ampe 1st story response (b) ampe 2nd story response (c) ampe3rd story response

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

Int x

1 (m

)

No controlTMD controlAMD control

(a)

Int x

2 (m

)

100 20 30 40 50 60Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(b)

Figure 9 Continued

Shock and Vibration 9

Int x

3 (m

)100 20 30 40 50 60

Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(c)

Figure 9 Comparison of each interstory displacement for different control strategies (a) ampe 1st interstory displacement (b) ampe 2ndinterstory displacement (c) ampe 3rd interstory displacement

100 20 30 40 50 60

100 20 30 40 50 60

100 20 30 40 50 60

ndash3

0

3

x 4 (m

)

TMD controlAMD control

ndash20

0

20

v 4 (m

)

Time (sec)

ndash100

0

100

a4

(m)

Figure 10 Responses of AMD and TMD device

Time (sec)

ndash6

ndash4

ndash2

0

2

4

Forc

e app

lied

on th

e str

uctu

re

times106

100 20 30 40 50 60

Figure 11 Applied active force

10 Shock and Vibration

In Table 3 different criterions are used to demonstratethe control effect of utilizing the method introduced in thispaper In terms of the maximum interstory drift the controleffect of the introduced AMD method is mostly far betterthan the TMD control effect in the control of the first twostories though the control effect of these two methods in thethird story is comparable When the input potential ofstructure which is expressed as the 2-norm of the interstorydrift is studied the AMD control shows overwhelmingadvantage compared with the TMD control effect

6 Conclusions

ampe vibration mitigation of a nonlinear structure using anactive mass damper with an adaptive control design isstudied in this paper and the adaptive force is provided bythe model reference sliding model control (MRSMC)method Due to the unknown parameters in the systemthis paper proposed a novel control method by combiningthe unscented Kalman filter (UKF) with the MRSMC toform the integrated MRSMC-UKF ampe UKF is used toidentify the unknown parameters and estimate thestructural states and the MRSMC is used to determine thecontrol law by these states and parameters based on theinformation estimated by UKF ampe numerical model is anonlinear 3-story frame structure with the AMD device onthe top floor ampe BoucndashWen model is used to model thenonlinear restoring force of the structure ampe stiffness ofthe simulated structure and the parameters of the BoucndashWen model are assumed to be unknown and these pa-rameters are estimated in real time by using the proposedmethod based on the measured acceleration states ampecontrol effect of AMD is compared with the responses ofthe structure without control and with the TMD controlsystem It turns out that the proposed MRSMC-UKF notonly efficiently estimates the states and identifies the pa-rameters but also effectively controls the structural non-linear vibration Based on the proposed method a betteractive control device could be developed to suppressstructural nonlinear vibration of the high-rise buildingsunder earthquake excitation Moreover other innovativeactive or semiactive control methods for structural non-linear vibration may also be proposed since a good non-linear mathematical model can be obtained by themodified UKF

Appendix

In this appendix the procedure of the modified UKF methodis presented as follows

First initialize the algorithm with1113954x+0 E x0( 1113857

P+0 E x0 minus 1113954x+

0( 1113857 x0 minus 1113954x+0( 1113857

T1113876 1113877

(A1)

where 1113954 represents the estimation of the correspondingvariable x0 is the initial value of the states 1113954x+

0 is the firstestimation of x0 P0 is the estimated covariance of x0 by thesame principle P is the estimated covariance of x andE() calculates the expectation of a random variable

ampen the algorithm startsAt the k-th stepUse 1113954x+

kminus1 which is the estimation of xkminus1 at the kminus 1-thstep to formulate the sigma point vectors 1113954x(i)

kminus1

1113954x(i)kminus1 1113954x+

kminus1 + 1113957x(i) i 1 2n (A2)

where 1113957x(i) (nP+

kminus11113968

)T

i i 1 n and 1113957x(i+n)

minus(nP+

kminus11113968

)T

i i 1 n

Calculate the estimated latent state 1113954x++lk with

1113954x++lk g1 xdkminus1 θkminus1 ukminus1( 1113857 (A3)

Substitute the sigma points into the system model toobtain the updated sigma points 1113954x(i)

k

1113954x(i)k f 1113954x(i)

kminus1ukminus11113872 1113873 i 1 2n (A4)

With these updated sigma points the first estimations ofthe mean and covariance of the states in step k are

1113954xminusk 12n

1113944

2n

i11113954x(i)

k

Pminusk 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954x(i)k minus 1113954xminusk1113872 1113873 + Qkminus1

(A5)

Substituting the sigma points into the observationfunction gives

1113954n(i)k H 1113954xi

kuk1113872 1113873 (A6)

From the sigma points of the observation vector

1113954y(i)k

1113954n(i)k

1113954x(i)lk

1113890 1113891 1113954x(i)lk is the value in 1113954x(i)

k corresponding to the

latent statesampe estimation of the observation and its covariance

together with the cross covariance of x and y are computedas follows

Table 3 Control effect under different criteria

Story max|di(t)||AMDcontrolmax|di(t)||TMDcontrol

max|di(t)||AMDcontrolmax|di(t)||No control

di(t)2|AMDcontroldi(t)2|TMDcontrol

di(t)2|AMDcontroldi(t)2|No control

1 7805 6878 7507 62562 6167 5013 5371 44363 10267 8328 8495 7038

100 20 30 40 50 60Time (sec)

ndash12ndash1

ndash08ndash06ndash04ndash02

00204

Slid

ing

surfa

ce

Figure 12 Convergence of the sliding surface

Shock and Vibration 11

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

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Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

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Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

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Navigation and Observation

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wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Int x

3 (m

)100 20 30 40 50 60

Time (sec)

ndash01

ndash005

0

005

01

No controlTMD controlAMD control

(c)

Figure 9 Comparison of each interstory displacement for different control strategies (a) ampe 1st interstory displacement (b) ampe 2ndinterstory displacement (c) ampe 3rd interstory displacement

100 20 30 40 50 60

100 20 30 40 50 60

100 20 30 40 50 60

ndash3

0

3

x 4 (m

)

TMD controlAMD control

ndash20

0

20

v 4 (m

)

Time (sec)

ndash100

0

100

a4

(m)

Figure 10 Responses of AMD and TMD device

Time (sec)

ndash6

ndash4

ndash2

0

2

4

Forc

e app

lied

on th

e str

uctu

re

times106

100 20 30 40 50 60

Figure 11 Applied active force

10 Shock and Vibration

In Table 3 different criterions are used to demonstratethe control effect of utilizing the method introduced in thispaper In terms of the maximum interstory drift the controleffect of the introduced AMD method is mostly far betterthan the TMD control effect in the control of the first twostories though the control effect of these two methods in thethird story is comparable When the input potential ofstructure which is expressed as the 2-norm of the interstorydrift is studied the AMD control shows overwhelmingadvantage compared with the TMD control effect

6 Conclusions

ampe vibration mitigation of a nonlinear structure using anactive mass damper with an adaptive control design isstudied in this paper and the adaptive force is provided bythe model reference sliding model control (MRSMC)method Due to the unknown parameters in the systemthis paper proposed a novel control method by combiningthe unscented Kalman filter (UKF) with the MRSMC toform the integrated MRSMC-UKF ampe UKF is used toidentify the unknown parameters and estimate thestructural states and the MRSMC is used to determine thecontrol law by these states and parameters based on theinformation estimated by UKF ampe numerical model is anonlinear 3-story frame structure with the AMD device onthe top floor ampe BoucndashWen model is used to model thenonlinear restoring force of the structure ampe stiffness ofthe simulated structure and the parameters of the BoucndashWen model are assumed to be unknown and these pa-rameters are estimated in real time by using the proposedmethod based on the measured acceleration states ampecontrol effect of AMD is compared with the responses ofthe structure without control and with the TMD controlsystem It turns out that the proposed MRSMC-UKF notonly efficiently estimates the states and identifies the pa-rameters but also effectively controls the structural non-linear vibration Based on the proposed method a betteractive control device could be developed to suppressstructural nonlinear vibration of the high-rise buildingsunder earthquake excitation Moreover other innovativeactive or semiactive control methods for structural non-linear vibration may also be proposed since a good non-linear mathematical model can be obtained by themodified UKF

Appendix

In this appendix the procedure of the modified UKF methodis presented as follows

First initialize the algorithm with1113954x+0 E x0( 1113857

P+0 E x0 minus 1113954x+

0( 1113857 x0 minus 1113954x+0( 1113857

T1113876 1113877

(A1)

where 1113954 represents the estimation of the correspondingvariable x0 is the initial value of the states 1113954x+

0 is the firstestimation of x0 P0 is the estimated covariance of x0 by thesame principle P is the estimated covariance of x andE() calculates the expectation of a random variable

ampen the algorithm startsAt the k-th stepUse 1113954x+

kminus1 which is the estimation of xkminus1 at the kminus 1-thstep to formulate the sigma point vectors 1113954x(i)

kminus1

1113954x(i)kminus1 1113954x+

kminus1 + 1113957x(i) i 1 2n (A2)

where 1113957x(i) (nP+

kminus11113968

)T

i i 1 n and 1113957x(i+n)

minus(nP+

kminus11113968

)T

i i 1 n

Calculate the estimated latent state 1113954x++lk with

1113954x++lk g1 xdkminus1 θkminus1 ukminus1( 1113857 (A3)

Substitute the sigma points into the system model toobtain the updated sigma points 1113954x(i)

k

1113954x(i)k f 1113954x(i)

kminus1ukminus11113872 1113873 i 1 2n (A4)

With these updated sigma points the first estimations ofthe mean and covariance of the states in step k are

1113954xminusk 12n

1113944

2n

i11113954x(i)

k

Pminusk 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954x(i)k minus 1113954xminusk1113872 1113873 + Qkminus1

(A5)

Substituting the sigma points into the observationfunction gives

1113954n(i)k H 1113954xi

kuk1113872 1113873 (A6)

From the sigma points of the observation vector

1113954y(i)k

1113954n(i)k

1113954x(i)lk

1113890 1113891 1113954x(i)lk is the value in 1113954x(i)

k corresponding to the

latent statesampe estimation of the observation and its covariance

together with the cross covariance of x and y are computedas follows

Table 3 Control effect under different criteria

Story max|di(t)||AMDcontrolmax|di(t)||TMDcontrol

max|di(t)||AMDcontrolmax|di(t)||No control

di(t)2|AMDcontroldi(t)2|TMDcontrol

di(t)2|AMDcontroldi(t)2|No control

1 7805 6878 7507 62562 6167 5013 5371 44363 10267 8328 8495 7038

100 20 30 40 50 60Time (sec)

ndash12ndash1

ndash08ndash06ndash04ndash02

00204

Slid

ing

surfa

ce

Figure 12 Convergence of the sliding surface

Shock and Vibration 11

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

In Table 3 different criterions are used to demonstratethe control effect of utilizing the method introduced in thispaper In terms of the maximum interstory drift the controleffect of the introduced AMD method is mostly far betterthan the TMD control effect in the control of the first twostories though the control effect of these two methods in thethird story is comparable When the input potential ofstructure which is expressed as the 2-norm of the interstorydrift is studied the AMD control shows overwhelmingadvantage compared with the TMD control effect

6 Conclusions

ampe vibration mitigation of a nonlinear structure using anactive mass damper with an adaptive control design isstudied in this paper and the adaptive force is provided bythe model reference sliding model control (MRSMC)method Due to the unknown parameters in the systemthis paper proposed a novel control method by combiningthe unscented Kalman filter (UKF) with the MRSMC toform the integrated MRSMC-UKF ampe UKF is used toidentify the unknown parameters and estimate thestructural states and the MRSMC is used to determine thecontrol law by these states and parameters based on theinformation estimated by UKF ampe numerical model is anonlinear 3-story frame structure with the AMD device onthe top floor ampe BoucndashWen model is used to model thenonlinear restoring force of the structure ampe stiffness ofthe simulated structure and the parameters of the BoucndashWen model are assumed to be unknown and these pa-rameters are estimated in real time by using the proposedmethod based on the measured acceleration states ampecontrol effect of AMD is compared with the responses ofthe structure without control and with the TMD controlsystem It turns out that the proposed MRSMC-UKF notonly efficiently estimates the states and identifies the pa-rameters but also effectively controls the structural non-linear vibration Based on the proposed method a betteractive control device could be developed to suppressstructural nonlinear vibration of the high-rise buildingsunder earthquake excitation Moreover other innovativeactive or semiactive control methods for structural non-linear vibration may also be proposed since a good non-linear mathematical model can be obtained by themodified UKF

Appendix

In this appendix the procedure of the modified UKF methodis presented as follows

First initialize the algorithm with1113954x+0 E x0( 1113857

P+0 E x0 minus 1113954x+

0( 1113857 x0 minus 1113954x+0( 1113857

T1113876 1113877

(A1)

where 1113954 represents the estimation of the correspondingvariable x0 is the initial value of the states 1113954x+

0 is the firstestimation of x0 P0 is the estimated covariance of x0 by thesame principle P is the estimated covariance of x andE() calculates the expectation of a random variable

ampen the algorithm startsAt the k-th stepUse 1113954x+

kminus1 which is the estimation of xkminus1 at the kminus 1-thstep to formulate the sigma point vectors 1113954x(i)

kminus1

1113954x(i)kminus1 1113954x+

kminus1 + 1113957x(i) i 1 2n (A2)

where 1113957x(i) (nP+

kminus11113968

)T

i i 1 n and 1113957x(i+n)

minus(nP+

kminus11113968

)T

i i 1 n

Calculate the estimated latent state 1113954x++lk with

1113954x++lk g1 xdkminus1 θkminus1 ukminus1( 1113857 (A3)

Substitute the sigma points into the system model toobtain the updated sigma points 1113954x(i)

k

1113954x(i)k f 1113954x(i)

kminus1ukminus11113872 1113873 i 1 2n (A4)

With these updated sigma points the first estimations ofthe mean and covariance of the states in step k are

1113954xminusk 12n

1113944

2n

i11113954x(i)

k

Pminusk 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954x(i)k minus 1113954xminusk1113872 1113873 + Qkminus1

(A5)

Substituting the sigma points into the observationfunction gives

1113954n(i)k H 1113954xi

kuk1113872 1113873 (A6)

From the sigma points of the observation vector

1113954y(i)k

1113954n(i)k

1113954x(i)lk

1113890 1113891 1113954x(i)lk is the value in 1113954x(i)

k corresponding to the

latent statesampe estimation of the observation and its covariance

together with the cross covariance of x and y are computedas follows

Table 3 Control effect under different criteria

Story max|di(t)||AMDcontrolmax|di(t)||TMDcontrol

max|di(t)||AMDcontrolmax|di(t)||No control

di(t)2|AMDcontroldi(t)2|TMDcontrol

di(t)2|AMDcontroldi(t)2|No control

1 7805 6878 7507 62562 6167 5013 5371 44363 10267 8328 8495 7038

100 20 30 40 50 60Time (sec)

ndash12ndash1

ndash08ndash06ndash04ndash02

00204

Slid

ing

surfa

ce

Figure 12 Convergence of the sliding surface

Shock and Vibration 11

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

1113954yk 12n

1113944

2n

i11113954y(i)

k

Pminusy 12n

1113944

2n

i11113954y(i)

k minus 1113954yk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T+ Rk

Pxy 12n

1113944

2n

i11113954x(i)

k minus 1113954xminusk1113872 1113873 1113954y(i)k minus 1113954yk1113872 1113873

T

(A7)

Compute the Kalman gain matrix1113954x+

k 1113954xminusk + Kk yk minus 1113954yk( 1113857

P+k Pminusk minusKkPyK

Tk

(A8)

Continue to the k + 1-th step1113954x+

k is the states and parameters estimation in the k-thstep and P+

k is its covariance

Data Availability

No data were used to support this study

Conflicts of Interest

ampe authors declare that they have no conflicts of interest

Acknowledgments

ampis work was supported in part by the National ScienceFoundation of China under Award Nos 51678116 and51378093

References

[1] B F Spencer Jr and S Nagarajaiah ldquoState of the art ofstructural controlrdquo Journal of Structural Engineering vol 129no 7 pp 845ndash856 2003

[2] B Basu O S Bursi F Casciati et al ldquoA european associationfor the control of structures joint perspective recent studies incivil structural control across Europerdquo Structural Control andHealth Monitoring vol 21 no 12 pp 1414ndash1436 2014

[3] F Casciati J Rodellar and U Yildirim ldquoActive and semi-active control of structuresmdashtheory and applications a reviewof recent advancesrdquo Journal of Intelligent Material Systemsand Structures vol 23 no 11 pp 1181ndash1195 2012

[4] G Wang S Veeramani and N M Wereley ldquoAnalysis ofsandwich plates with isotropic face plates and a viscoelasticcorerdquo Journal of Vibration and Acoustics vol 122 no 3pp 305ndash312 2000

[5] Y Zhou X Lu D Weng and R Zhang ldquoA practical designmethod for reinforced concrete structures with viscousdampersrdquo Engineering Structures vol 39 pp 187ndash198 2012

[6] Q Xue J Zhang J He and C Zhang ldquoControl performanceand robustness of pounding tuned mass damper for vibrationreduction in SDOF structurerdquo Shock and Vibration vol 2016Article ID 8021690 15 pages 2016

[7] W Wang X Hua X Wang Z Chen and G Song ldquoNu-merical modeling and experimental study on a novelpounding tuned mass damperrdquo Journal of Vibration andControl vol 24 no 17 pp 4023ndash4036 2017

[8] L Tian K Rong P Zhang and Y Liu ldquoVibration control of apower transmission tower with pounding tuned mass damperunder multi-component seismic excitationsrdquo Applied Sci-ences vol 7 no 5 p 477 2017

[9] W Fu C Zhang L Sun et al ldquoExperimental investigation of abase isolation system incorporating MR dampers with thehigh-order single step control algorithmrdquo Applied Sciencesvol 7 no 4 p 344 2017

[10] Z Lu DWang and Y Zhou ldquoExperimental parametric studyon wind-induced vibration control of particle tuned massdamper on a benchmark high-rise buildingrdquo Ae StructuralDesign of Tall and Special Buildings vol 26 no 8 articlee1359 2017

[11] Z Huang X G Hua Z Q Chen and H W Niu ldquoModelingtesting and validation of an eddy current damper forstructural vibration controlrdquo Journal of Aerospace Engineer-ing vol 31 no 5 article 04018063 2018

[12] Z Wang Z Chen H Gao and H Wang ldquoDevelopment of aself-powered magnetorheological damper system for cablevibration controlrdquo Applied Sciences vol 8 no 1 p 118 2018

[13] J Ye and L Xu ldquoMember discrete element method for staticand dynamic responses analysis of steel frames with semi-rigid jointsrdquo Applied Sciences vol 7 no 7 p 714 2017

[14] S J Dyke B F Spencer M K Sain and J D CarlsonldquoSeismic response reduction using magnetorheologicaldampersrdquo IFAC Proceedings Volumes vol 29 no 1pp 5530ndash5535 1996

[15] B F Spencer Jr S J Dyke M K Sain and J D CarlsonldquoPhenomenological model for magnetorheological dampersrdquoJournal of Engineering Mechanics vol 123 no 3 pp 230ndash2381997

[16] N M Kwok Q P Ha M T Nguyen J Li and B SamalildquoBouc-Wen model parameter identification for a MR fluiddamper using computationally efficient GArdquo ISA Trans-actions vol 46 no 2 pp 167ndash179 2007

[17] A Al-Hussein and A Haldar ldquoStructural health assessment ata local level using minimum informationrdquo EngineeringStructures vol 88 pp 100ndash110 2015

[18] L DrsquoAlfonso W Lucia P Muraca and P Pugliese ldquoMobilerobot localization via EKF and UKF a comparison based onreal datardquo Robotics and Autonomous Systems vol 74pp 122ndash127 2015

[19] H Gurung and A Banerjee ldquoSelf-Sensing shapememory alloywire actuator based on unscented kalman filterrdquo Sensors andActuators A Physical vol 251 pp 258ndash265 2016

[20] J S Lee I Y Choi S Kim and D S Moon ldquoKinematicmodeling of a track geometry using an unscented Kalmanfilterrdquo Measurement vol 94 pp 707ndash716 2016

[21] H A Izadi Y Zhang and B W Gordon ldquoFault tolerantmodel predictive control of quad-rotor helicopters with ac-tuator fault estimationrdquo IFAC Proceedings Volumes vol 44no 1 pp 6343ndash6348 2011

[22] A Rahideh andM H Shaheed ldquoConstrained output feedbackmodel predictive control for nonlinear systemsrdquo ControlEngineering Practice vol 20 no 4 pp 431ndash443 2012

[23] A Mirzaee and K Salahshoor ldquoFault diagnosis and accom-modation of nonlinear systems based on multiple-modeladaptive unscented Kalman filter and switched MPC andH-infinity loop-shaping controllerrdquo Journal of Process Con-trol vol 22 no 3 pp 626ndash634 2012

[24] M S Miah E N Chatzi and F Weber ldquoSemi-active controlfor vibration mitigation of structural systems incorporatinguncertaintiesrdquo Smart Materials and Structures vol 24 no 5article 055016 2015

12 Shock and Vibration

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

[25] B Allotta A Caiti L Chisci et al ldquoAn unscented Kalmanfilter based navigation algorithm for autonomous underwatervehiclesrdquo Mechatronics vol 39 pp 185ndash195 2016

[26] R J Caverly and J R Forbes ldquoState estimator design for asingle degree of freedom cable-actuated systemrdquo Journal ofthe Franklin Institute vol 353 no 18 pp 4845ndash4869 2016

[27] S Ramadurai S N Kosari H H King H J Chizeck andB Hannaford ldquoApplication of unscented kalman filter to acable driven surgical robot a simulation studyrdquo in Pro-ceedings of the 2012 IEEE International Conference on Roboticsand Automation (ICRA) IEEE St Paul MN USA May 2012

[28] H Aschemann and T Meinlschmidt ldquoCascaded nonlinearcontrol of a duocopter with disturbance compensation by anunscented kalman filterrdquo IFAC-PapersOnLine vol 48 no 1pp 904ndash909 2015

[29] L Li G Song and J Ou ldquoAdaptive fuzzy sliding mode basedactive vibration control of a smart beam with mass un-certaintyrdquo Structural Control and Health Monitoring vol 18no 1 pp 40ndash52 2011

[30] M Singla L-S Shieh G Song L Xie and Y Zhang ldquoA newoptimal sliding mode controller design using scalar signfunctionrdquo ISA Transactions vol 53 no 2 pp 267ndash279 2014

[31] M Heydari H Salarieh and M Behzad ldquoStochastic chaossynchronization using unscented Kalman-Bucy filter andsliding mode controlrdquo Mathematics and Computers in Sim-ulation vol 81 no 9 pp 1770ndash1784 2011

[32] M Parsapour S RayatDoost and H Taghirad ldquoPositionbased sliding mode control for visual servoing systemrdquo inProceedings of the 2013 First RSIISM International Conferenceon Robotics and Mechatronics (ICRoM) IEEE Tehran IranFebruary 2013

[33] L Li and H Liang ldquoSemiactive control of structural nonlinearvibration considering the MR damper modelrdquo Journal ofAerospace Engineering vol 31 no 6 article 04018095 2018

[34] R L Wang S C Ho and N Ma ldquoActive model referencevibration control of a flexible beam with surface-bonded PZTsensor and actuatorrdquo Journal of Vibroengineering vol 18no 1 2016

[35] R Agarwala S Ozcelik and M Faruqi ldquoActive vibrationcontrol of a multi-degree-of-freedom structure by the use ofdirect model reference adaptive controlrdquo in Proceedings of the2000 American Control Conference IEEE Chicago IL USAJune 2000

[36] H Gu and G Song ldquoActive vibration suppression of a flexiblebeamwith piezoceramic patches using robust model referencecontrolrdquo Smart Materials and Structures vol 16 no 4pp 1453ndash1459 2007

[37] G Song and R Mukherjee ldquoA comparative study of con-ventional nonsmooth time-invariant and smooth time-varying robust compensatorsrdquo IEEE Transactions on Con-trol Systems Technology vol 6 no 4 pp 571ndash576 1998

[38] G Song R Mukherjee G Song and R W Longman ldquoIn-tegrated sliding-mode adaptive-robust controlrdquo IEEE Pro-ceedingsmdashControl Aeory and Applications vol 146 no 4pp 341ndash347 1999

[39] H Gu G Song and H Malki ldquoChattering-free fuzzy adaptiverobust sliding-mode vibration control of a smart flexiblebeamrdquo Smart Materials and Structures vol 17 no 3 article035007 2008

[40] C Li and B Cao ldquoHybrid active tuned mass dampers forstructures under the ground accelerationrdquo Structural Controland Health Monitoring vol 22 no 4 pp 757ndash773 2015

[41] M Soleymani and M Khodadadi ldquoAdaptive fuzzy controllerfor active tuned mass damper of a benchmark tall building

subjected to seismic and wind loadsrdquoAe Structural Design ofTall and Special Buildings vol 23 no 10 pp 781ndash800 2014

[42] Y C He and Q Li ldquoDynamic responses of a 492-m-high tallbuilding with active tuned mass damping system during atyphoonrdquo Structural Control and Health Monitoring vol 21no 5 pp 705ndash720 2013

[43] N D Anh H-L Bui N-L Vu and D-T Tran ldquoApplicationof hedge algebra-based fuzzy controller to active control of astructure against earthquakerdquo Structural Control and HealthMonitoring vol 20 no 4 pp 483ndash495 2013

[44] Y Ikeda ldquoAn active mass damper designed using ARXmodelsof a building structurerdquo Earthquake Engineering amp StructuralDynamics vol 45 no 13 pp 2185ndash2205 2016

[45] J Tu X Lin B Tu J Xu and D Tan ldquoSimulation and ex-perimental tests on active mass damper control system basedon model reference adaptive control algorithmrdquo Journal ofSound and Vibration vol 333 no 20 pp 4826ndash4842 2014

[46] L Li G Song and J Ou ldquoHybrid active mass damper (AMD)vibration suppression of nonlinear high-rise structure usingfuzzy logic control algorithm under earthquake excitationsrdquoStructural Control and Health Monitoring vol 18 no 6pp 698ndash709 2011

[47] H-C Tsai and G-C Lin ldquoOptimum tuned-mass dampers forminimizing steady-state response of support-excited anddamped systemsrdquo Earthquake Engineering amp Structural Dy-namics vol 22 no 11 pp 957ndash973 1993

[48] R Xu and U Ozguner ldquoSliding mode control of a class ofunderactuated systemsrdquo Automatica vol 44 no 1 pp 233ndash241 2008

[49] L Li G Song and J Ou ldquoNonlinear structural vibrationsuppression using dynamic neural network observer andadaptive fuzzy slidingmode controlrdquo Journal of Vibration andControl vol 16 no 10 pp 1503ndash1526 2010

Shock and Vibration 13

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom