Dynamic Fuzzy Wavelet Neuroemulator for Non-linear Control of Irregular Buildins Structures

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2008; 74:10451066Published online 6 December 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2195

Dynamic fuzzy wavelet neuroemulator for non-linear control ofirregular building structures

Xiaomo Jiang1 and Hojjat Adeli2,,1Department of Civil and Environmental Engineering, Vanderbilt University, Nashville, TN 37235, U.S.A.

2Department of Civil and Environmental Engineering and Geodetic Science, The Ohio State University,470 Hitchcock Hall, 2070 Neil Ave, Columbus, OH 43210, U.S.A.

SUMMARY

A new non-linear control model is presented for active control of three-dimensional (3D) buildingstructures. Both geometrical and material non-linearities are included in the structural control formulation.A dynamic fuzzy wavelet neuroemulator is presented for predicting the structural response in future timesteps. Two dynamic coupling actions are taken into account simultaneously in the control model: (a)coupling between lateral and torsional motions of the structure and (b) coupling between the actuator andthe structure. The new neuroemulator is validated using two irregular 3D steel building structures, a 12-story structure with vertical setbacks and an 8-story structure with plan irregularity. Numerical validationsin both time and frequency domains demonstrate that the new neuroemulator provides accurate predictionof structural displacement responses, which is required in neural network models for active control ofstructures. In the companion paper, a floating-point genetic algorithm is presented for finding the optimalcontrol forces needed for active non-linear control of building structures using the dynamic fuzzy waveletneuroemulator presented in this paper. Copyright q 2007 John Wiley & Sons, Ltd.

Received 5 November 2006; Revised 27 July 2007; Accepted 22 August 2007

KEY WORDS: non-linear control; wavelet neural network; neuroemulator; active control

INTRODUCTION

Over the past two decades, a large amount of research has been conducted on the development andimplementation of active, semi-active, and hybrid control of structures (e.g. [123]), and variouscontrol strategies have been proposed, such as sliding mode control [22] and optimal polynomialcontrol [23]. However, most of the studies have focused on the application of classical linearcontrol theories, such as linear quadratic regulator (LQR) feedback control algorithm [6] and linear

Correspondence to: Hojjat Adeli, Department of Civil and Environmental Engineering and Geodetic Science,The Ohio State University, 470 Hitchcock Hall, 2070 Neil Ave, Columbus, OH 43210, U.S.A.

E-mail: [email protected]

Copyright q 2007 John Wiley & Sons, Ltd.

1046 X. JIANG AND H. ADELI

quadratic Gaussian (LQG) control algorithm [2]. These algorithms control structural responseseffectively only when the structure is small and assumed to behave linearly.

There are several motivations for considering active non-linear control of structures. The linearcontrol algorithms are not effective for active non-linear control of highrise building structuressubjected to extreme loadings. Highrise building structures may experience yielding and non-linearbehavior (geometrical or material non-linearity or both) under severe earthquakes or strong winds.Any structural damage changes the structural stiffnesses during the extreme dynamic event. Insuch cases, a computational model based on the assumption that the controlled structure behaveslinearly would be inadequate for representing the actual behavior of the structure. Linear controlalgorithms cannot be used to control the response of structures in the non-linear range effectively[17]. On the other hand, maintaining a linear behavior for a large controlled structure such asa highrise building during an extreme dynamic event would require actuators with impracticallylarge capacities.

Neural networks are known for their ability to model complex and non-linear phenomena whereno explicit mathematical model exists [2426] and for their adaptability for handling incompleteinformation. In the past decade or so, a number of articles have been published on the developmentand application of neural network-based or other adaptive/intelligent control algorithms for activelinear/non-linear control of mostly small structural systems [8, 1618, 2732]. Ghaboussi andJoghataie [28] first use a neural network-based emulator to identify the response of a 3-storytwo-dimensional (2D) frame structure. Then, a neural network-based controller is trained usingthe emulator for linear control of the structure. Structural displacement and acceleration responsesduring the previous two time steps and actuator electric signals during the previous three time stepsare used as inputs to the neural network model. Bani-Hani and Ghaboussi [16, 17] extended theneural network-based algorithm for non-linear control of a benchmark 3-story 2D frame structureconsidering material non-linearity. The authors show that the neural network-based approachesprovide adequate accuracy for active control of such small 2D frame structures.

Recently, a few journal articles have been published on active control of three-dimensional(3D) highrise building structures subjected to earthquake loads (e.g. [3337]). Fur et al. [33] usean active tuned mass damper (ATMD) system for an 8-story building structure considering thelateraltorsional coupling. The control gain, an important parameter required in feedback controlalgorithms such as LQR, is obtained through the complete feedback of position and velocity.Al-Dawod et al. [34] present a fuzzy logic-based controller for active control of 3- and 20-story3D symmetric building structures. Ohtori et al. [35] define 3-, 9-, and 20-story symmetric steelstructures for benchmark active control studies without presenting any active non-linear controlalgorithm. The 9-story structure has also been investigated in a semi-active control study usingmagnetorheological dampers [36] and an active control study using H2/LQG controller [37].

Recently, Adeli and Kim [20] presented a novel wavelet-hybrid feedback-linear mean square(LMS) algorithm for robust control of civil structures. It is shown that the wavelet transform can beused to enhance the performance of feedback control algorithms. An active mass driver benchmarkstructure is used to validate the effectiveness of the proposed control model. Simulation resultsdemonstrate that the proposed model is effective for control of both steady and transient vibrationswithout any significant additional computational burden. Kim and Adeli [38] present a hybridcontrol system through judicious combination of a passive supplementary damping system witha semi-active tuned liquid column damper (TLCD) system. The wavelet-hybrid feedback LMScontrol algorithm [20] is used to find optimal values of the control parameters for an 8-story frameusing three different simulated earthquake ground accelerations. The wavelet-hybrid feedback LMS

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2008; 74:10451066DOI: 10.1002/nme

WAVELET NEUROEMULATOR FOR NON-LINEAR CONTROL OF IRREGULAR STRUCTURES 1047

algorithm has also been used for vibration control of cable-stayed bridges under various seismicexcitations [39].

Recent literature indicates that adaptive/intelligent control algorithms, such as those developedby Adeli and Kim [20], are advantageous over classical feedback control algorithms for active orhybrid control of structures for several reasons. First, these algorithms can tolerate the imprecisionin the sensed data. Second, they require less prior knowledge of the structural system to becontrolled. Third, they can be used to handle non-linearity. Finally, they usually converge quicklyand are therefore practical for online active control of large-scale structures.

The focus of this study is active non-linear control of 3D building structures based on thewavelet neural network (WNN) model developed by authors. Both material and geometrical non-linearities are considered in modeling the structural response under strong earthquake loadings.Furthermore, the structural modeling takes into account two coupling actions: the coupling actionbetween the actuators and the structure and the coupling between the lateral and torsional motionsof 3D irregular structures. A dynamic fuzzy WNN is developed as a fuzzy wavelet neuroemulatorto predict structural responses from the immediate past structural responses and actuator dynamics.In the companion paper, a floating-point genetic algorithm is presented for finding the optimalcontrol forces for active non-linear control of building structures using the dynamic fuzzy waveletneuroemulator presented in this paper [40].

COUPLED NON-LINEAR DYNAMICS OF 3D BUILDING STRUCTURES

Non-linear dynamics of irregular 3D buildings with an active control systemA 3D highrise building structure can in general have both plan and elevation irregularities. Ingeneral, the center of mass, CM, does not coincide with the center of resistance, CR, in eachfloor (Figure 1(a)), and the centers of mass and resistance of floors do not lie on the samevertical axes (their locations can vary from floor to floor) due to different story stiffnesses alongthe two principal directions. This difference may exist even for a structure with a symmetricplan because different members with different stiffnesses may be used in a symmetric floorarrangement. In such situations there is a coupling between lateral and torsional motions. Thecoupling may result in the maximum lateral displacement in a direction other than the direc-tion of minimum stiffness. In this case, neglecting the coupling between lateral and torsionalmotions usually underestimates the maximum responses of the structure under earthquakeloadings [41].

In this study, floor diaphragms are assumed to be rigid and axial deformations of the columns areneglected. As such, the irregular building structure is modeled using three displacement degreesof freedom (DOFs) for each floor: translations in x- and y-directions and a rotation about thevertical axis z passing through the center of resistance. The total number of DOFs is thereforem =3L , where L is the number of stories. The structural displacement vector at time t is expressedas

u(t)=[u1(t) v1(t) 1(t) u2(t) v2(t) 2(t) uL(t) vL(t) L(t)]T (1)where ui (t),vi (t), and i (t) are the translations in x- and y-directions and the rotation about thevertical axis z of the i th floor, respectively. The superscript T represents the transpose of thematrix. The equation of motions of 3D building structures with an active control system subjected



x

yCenter ofMass

z

CM

x

y

CR

zCenter of

Resistance

Seismic excitation xg(t)

Actuator

CRCM

x

x

y

y

(a)

(b)

Figure 1. Structural model of a 3D building: (a) 3D building and (b) typical story with four actuators.

to seismic excitations is expressed as

Mu(t)+Cu(t)+R(x, t)=IcF(t)M0Ig xg(t) (2)where M and C represent mm mass and damping coefficient matrices of the structure, respec-tively; R(x, t) is the m1 restoring force vector; F(t) is the r 1 control force vector whoseelements F(t) are in the form of time series; Ic is an mr location matrix representing the locationof the actuators; and xg(t) is the input horizontal earthquake acceleration which can have anyarbitrary orientation. In this study, we assume that two pairs of actuators are used in every floorof the building along two perpendicular axes x and y (Figure 1(b)). Each pair consists of twoidentical actuators, representing the control force F(t) in the corresponding direction. In practicalapplications, the moment caused by actuators in two directions can be minimized by properlyassigning the control force to the two actuators in each direction. For the sake of simplifyingthe computations, we assume that the distance between the center of mass (CM) and the centerof resistance (CR, ) in each floor is very small. Thus, the moment caused by actuators in twodirections is ignored. The focus of the current paper is on novel computational techniques, notactuator placement. Thus, the proposed methodologies are not tied to any particular configuration.As such, r =2L . The angle of orientation of the actuators is =0 for one pair and =90 foranother pair. The total number of actuators used in the building is therefore 4L .



In Equation (2), M0 is an mm diagonal mass matrix whose diagonal terms are the same as thediagonal terms of the full matrix M, and Ig is an m1 orientation matrix denoting the orientationof the external earthquake excitation:

Ig =[cos sin 0 cos sin 0 cos sin 0]T (3)where is the direction angle of the earthquake motion measured from the x-axis (Figure 1). Thelocation matrix Ic in Equation (2) is expressed as follows:

Ic =diagcos1 sin1sin2 cos2

0 0

cos3 sin3sin4 cos4

0 0

cosr1 sinr1sinr cosr

0 0

mr(4)

Figure 2 shows a building structure with an active control system. The earthquake excitation isrepresented by the horizontal ground acceleration in the form of a time series, xg(t). The activecontrol system consists of a computer, hydraulic actuators, and sensors. The computer is used toreceive, process, and send signals between the actuators and the structure via sensors and cable lines.When the structure is subjected to an earthquake loading, the structural response at the top floor,for example u(t) in the x-direction, is sent to the controller along with the earthquake excitations.Optimal control forces from actuators are needed to minimize the structural displacement at thetop of the structure in every time step. Each optimal control force is converted to a correspondingelectric control signal Es(t) based on the properties of the actuator to be discussed later. Eachactuator is then excited by the control signal to generate a required control force.

Non-linear hysteretic effectIn order to represent the material non-linear and hysteretic behavior of a highrise buildingstructure, the restoring force of a structural element at time t , R(x, t), is defined by the followingequation proposed by Wen [42]:

R(x, t)=akx +(1a)kdy (5)where x is the displacement of the element, RL(x)=kx and RNL(v)=(1)kdy represent thelinear and non-linear parts of the restoring force, respectively, in which v represents the non-lineardisplacement. The parameters k and k (


Seismic excitation )(txg

Sensor

Active actuator

system

u(t)

)(txg

(t)

Sensor

Sensor

Actuator

Es(t)

Es(t)

Controller

Figure 2. Building with an active control system in one direction. xg(t)=seismic accel-eration excitation at time t , x(t)=structural displacement response, Es(t)=electric signal

to the actuator, fc(t)=control force from the actuator.

Equations (5) and (6) represent the non-linear stiffness hysteretic model of a structural elementwhere its restoring force is a function of a displacement variable x and a hysteretic variable v.The restoring force vector for the entire structure is expressed in matrix form as

R(x, t)=KGX(t)+KMV(t) (7)where KG and KM are structural stiffness matrices. The stiffness matrix KG includes geometricnon-linearity terms, whereas stiffness matrix KM includes both geometric and material non-linearityterms; X(t) and V(t) are the total displacement and non-linear displacement variable vectors attime t (representing the second term in the right-hand side of Equation (5)), respectively.

Electrohydraulic actuator

The importance of the actuatorstructure coupling in active control of structures has been recognizedin the recent literature [8, 17, 28, 32, 43]. The linear hydraulic actuator is employed in this study foractive control of a building structure because of its ability to generate relatively large forces with arelatively small response time in the order of a few milli-seconds and maintain the force with verylittle power [8]. The level of force produced by a hydraulic actuator can be in the order of 1000 kN(220 kilo-pounds) (http://www.nees.buffalo.edu/pdfs/244 Actuators.pdf). The magnitude of the



k1

1 kk

n

fy

x

f

dy

fy = kdy = yield force

Displacement

Forc

e

k 1

Figure 3. Loaddisplacement curve for an element considering materialnon-linearity and hysteretic behavior.

control force, F(t), is found by solving the following differential equation numerically in eachtime step of the dynamic analysis [44]:

q(t)= Ap xp+ ClAp F(t)+Vc

2ApF(t) (8)

where Ap is the effective cross-sectional area of the piston; xp is the piston velocity; Cl is theleakage coefficient; Vc is the volume of the cylinder; and is the compressibility coefficient ofthe fluid. The displacement of the actuator/piston is assumed to be the same as the displacementof the floor it is attached to.

State-space model

Considering material non-linearity and hysteretic behavior represented by Equation (5), geometricalnon-linear behavior of the structure, and the dynamics of the actuator represented by Equation (8),the non-linear equation of the motion for a structure with an active control system subjected toearthquake loading is expressed in the state-space form as follows:

Z(t)=AZ(t)+BQc(t)+BFg(t) (9)where Z(t)=[u(t),V(t),F(t), u(t)]T is a column vector of 4m state-space variables, in whichF(t)=IcF(t). The 4m4m matrix A is expressed by

A=

0 0 0 I0 0 0 D0 0 G1 G2

M1KG M1KM M1Ic M1C

(10)



where D,KG, and KM are mm matrices incorporating the structural stiffness, geometrical andmaterial non-linearities. The terms in these matrices are defined as follows:

Di, j =keffi, jf y

[1 1+sgn(xi, ji, j )

2|i, j |n

](11)

(KG)i, j = keffi, j (12)(KM)i, j = (1)keffi, j dyi, j (13)

where the superscript eff on k denotes the effective lateral stiffness of the column element whichtakes into account the geometrical non-linearity.

In Equation (10), 0 and I are the mm zero and identity matrices, respectively, and G1 and G2are two mm diagonal matrices representing the characteristics of hydraulic actuators as follows:

G1 = diag

2x ClxVCx

0 0

02yCly

VCy0

0 0 0

, . . . ,

2x ClxVCx

0 0

02yCly

VCy0

0 0 0

L1

(14)

G2 = diag

2x A2PxVCx

0 0

02y A2Py

VCy0

0 0 0

, . . . ,

2x A2PxVCx

0 0

02y A2Py

VCy0

0 0 0

L1

(15)

The matrices Qc(t) and Fg(t) are 4m1 actuator flow rate and external excitation vectors, respec-tively, expressed as follows:

Qc(t)=

0101

Icq(t)

01

, Fg(t)=

010101

M0Ig xg(t)

(16)

where q(t) is a 2L 1 vector representing the flow rate of hydraulic actuators [44] andB=01 01 01 M1I1 (17)

is a 4m1 matrix, in which 01 =[0 0 0]T and I1 =[1 1 1]T are m1 zero and unitcolumn vectors, respectively.

The non-linear state-space equation represented by Equation (9) is solved numerically using thefourth-order RungeKutta method with a proper integration time step (e.g. t =0.01s, a smallertime step may be required for a larger structure). In order to obtain the mass, stiffness, and damping



coefficient matrices of the structure, building structures are modeled with finite elements in twosteps. In the first step, columns and beams are modeled as 3D frame elements with two endnodes, each node having six DOFs (three displacements and three rotations). In the second step,dynamic condensation is applied to reduce the structural model to one with rigid floor diaphragmsand no axial column deformations. The resulting structural model has three DOFs for each floor(translations in x- and y-directions and a rotation about the vertical axis z passing through thecenter of resistance, shown in Figure 1).

DYNAMIC FUZZY WNN EMULATOR

Recently, the authors developed a novel multi-paradigm dynamic time-delay fuzzy WNN modelfor nonparametric identification of structures using the non-linear autoregression moving averagewith exogenous inputs (NARMAX) approach [45]. The model integrates two intelligent computingtechniques (dynamic neural networks and fuzzy logic), a signal processing technique (wavelettransform), and the chaos theory with the goal of improving the accuracy and adaptability ofnonparametric system identification. Furthermore, an adaptive LevenbergMarquardt-least-squares(LM-LS) algorithm with a backtracking inexact linear search scheme has been developed fortraining of the dynamic fuzzy WNN model [46]. The approach avoids the second-order differentia-tion required in the GaussNewton algorithm and overcomes the numerical instabilities encounteredin the steepest descent algorithm with improved learning convergence rate and high-computationalefficiency.

In order to control the response of a given structure effectively the structural response infuture time steps have to be estimated. This is necessary in order to determine the magni-tude of the required control forces. In this study, the dynamic fuzzy WNN model developedby the authors has been used as the neuroemulator to predict the non-linear structural responsein future time steps from the immediate past structural response and actuator dynamics. Thegeneral discrete dynamic inputoutput mapping in the dynamic fuzzy WNN approach is expressedas follows [45]:

f (Xk)=M

i=1wi

Dj=1

(Xk j ci j

ai j

)+

Dj

b j Xk j +d0, k =1, . . . ,Na, a , (.) L2() (18)

where f (Xk) is the predicted structural response; (.) is the nonorthogonal Mexican hat waveletfunction used in this work [47]; Xkj is the j th value in the kth input vector (or data point) Xk ,and ci j is the j th value in the i th cluster of the multidimensional input vector obtained using thefuzzy C-means clustering approach [48]. The parameter D is the dimension or the size of the inputvector in the NARMAX approach. The parameter M is the number of wavelets obtained using themodified GramSchmidt algorithm [49] and the Akaikes final prediction error method (AFPE)[50], which is also equal to the number of the fuzzy clusters as well as the number of wavelet nodesused in the WNN model. The parameters ai j =0 denote the frequency (or scale) correspondingto the multidimensional input vector; wi represents the i th wavelet coefficient linking the hiddennode to the output; b j is the weight of the link of the j th input to the output (for the linear partof the neural network, input is connected directly to the output); d0 is a bias term, and is theset of real numbers. The notation L2() represents the square summable constructed state-spacevectors. The parameter Na is the number of input vectors.



Since the earthquake can occur in any horizontal direction two fuzzy WNN-based emulators,x- and y-neuroemulators, are created to predict the corresponding structural displacement responsesin the x- and y-directions. Details of how to construct the dynamic fuzzy WNN model usinga given time series are presented in Adeli and Jiang [45]. The procedures to construct thetwo neuroemulators are similar. The response data in the x-direction are used to construct thex-neuroemulator and those in the y-direction are used for the y-neuroemulator. The resultingnumbers of the hidden or wavelet nodes in the hidden layer of the two neuroemulators are ingeneral different.

TRAINING OF THE FUZZY WNN NEUROEMULATOR

Bani-Hani and Ghaboussi [17] investigate two neuroemulators using the BP neural network algo-rithm: linearly trained neuroemulator and non-linearly trained neuroemulator. They show that theemulator trained using structural non-linear response data can predict structural responses moreaccurately than that trained using structural linear response data, especially when the structure issubjected to strong earthquake excitations. The neuroemulator presented in this paper is a non-physically based model (or black box), which does not explicitly involve any physical parametersof the structure, in the context of both training and testing. The neuroemulator is trained to mapnon-linear relationships between input and output data to represent the physical model of thecorresponding structure; then the trained neuroemulator is used to predict the structural responseunder various input excitations.

In the current study, the dynamic fuzzy WNN neuroemulator is trained using the structuraldisplacement response data generated from the non-linear structural analysis taking into accountboth geometrical and material non-linearities as described previously. The training data are gener-ated from the non-linear analyses of the structure subjected to different earthquake loadings.Figure 4 shows the training diagram of the dynamic fuzzy WNN model using the adaptive LM-LSlearning algorithm for active non-linear control of structures. The training is done in three steps,identified by numbers (1)(3) in Figure 4 as follows:

(1) The inputoutput data sets, Xk and yk (k =1,2, . . . ,Na), are formed from structuralresponses, actuator forces, and earthquake excitations, where the kth input state-spacevector (time t during the earthquake excitation), Xk , is constructed as follows:Xk =[xk, fk,ak]T =xk, xk+1, . . . , xk+x ,Fk,Fk+1, . . . ,Fk+c , xgk , xgk+1, . . . , xgk+g T (19)where xk , Fk , and xgk are the kth structural displacement response, actuator/control force,and earthquake excitation, respectively. The parameters x , c, and g are time delays forthe three variables representing dimensions of the inputs (number of previous discrete timesteps used for any input time series). Their optimal dimensions are obtained by the falsenearest neighbor (FNN) method, as discussed in Jiang and Adeli [51]. The number of datasets Na is obtained by

Na = N max(x ,c,g) (20)where N is the total number of data points used to train the system.

(2) The dynamic fuzzy WNN model with a recurrent feedback topology is created as describedin Adeli and Jiang [45].



)( kk fy X=

1

ky

Seismic excitation )(t

Structuralsystem

x

yz

Sensor

C-mean

2) Dynamicfuzzy WNN

model

1) ConstructState space

X

E

Feed

back

ky

Adaptive LM-LSalgorithm Converge?

Stop

YesNo

3) Training model

Updateparameters

x(t)

)(tF

)(txg..

Controller

Figure 4. Training diagram of the dynamic fuzzy WNN model using the adaptive LM-LS learning algo-rithm for active non-linear control of structures. F(t)=actuator/control force at time t , xg(t)=groundacceleration, y(t)=measured structural response at time t , yk =kth measured structural response(k represents time step t), X=[X1 X2 Xk]T = input state space vector for all structural response data,f (Xk)= fuzzy WNN mapping function represented by Equation (18), yk = f (Xk)=kth computed output,

yk1 = feedback structural response, E =error function.

(3) The dynamic fuzzy WNN model is trained using the adaptive LM-LS algorithm and theconstructed inputoutput data sets [46].

The trained neuroemulator is tested using the data generated from both non-linear and linearstructural analyses. The ability of the trained model to predict both linear and non-linear responsesof the structure is investigated.

FINDING OPTIMAL CONTROL FORCES

A floating-point genetic algorithm is developed for finding the optimal control forces for activelinear or non-linear control of highrise building structures in tandem with the dynamic fuzzywavelet neuroemulator presented in this paper. The corresponding control signals are sent to theactuators that generate the control forces to be applied on the structure. Details of the geneticcontrol algorithm and how to implement the control methodology are presented in the companionpaper [40].



NUMERICAL VALIDATION OF THE DYNAMIC FUZZY WAVELET NEUROEMULATOR

In order to demonstrate that the new dynamic fuzzy wavelet neuroemulator model can accuratelypredict the dynamic responses of structures with irregularities, two 3D multistory steel buildingstructures, a 12-story structure with vertical setbacks (Example 1), and an 8-story structure withplan irregularity (Example 2) are studied. Kim and Adeli [41] used the same examples in theirstudy of hybrid control of irregular highrise building structures assuming linear behavior. Theyreport that a 2D dynamic analysis neglecting the effect of coupling between lateral and torsionalmotions underestimates the maximum response of the structure up to 4% for Example 1 and 7%for Example 2.

The non-linear dynamic responses of the two example structures subjected to earthquake exci-tations are obtained by solving their dynamic equation of motion represented by Equation (6).Both geometric and material non-linearities are considered in the structural analysis. A steel yieldstress of fy =36ksi (248.3N/mm2 in Equation (5)), a yield ratio of =0.1 and a yield exponentof n =2 (Equation (5)), and a damping ratio of 2% are used for both structures.

The original Chichi earthquake and 200% Chichi earthquake record of Richter magnitude7.3, which occurred at Foothills, Western Taiwan, on 21 September 1999, are applied to thestructures. The structure is assumed to behave linearly when the difference of the maximumdisplacement obtained from the non-linear and linear analyses is within 1% of the linear maximumdisplacement. On the basis of this definition, under the Chichi earthquake loading, the 12-storystructure (Example 1) behaves non-linearly while the 8-story structure (Example 2) behaves linearly.When the 200% Chichi earthquake loading is applied, both structures behave non-linearly.

The displacement responses of the uncontrolled structure obtained from 200% Chichi earth-quake representing the structural non-linear behavior are used to train the neuroemulator, whilethe displacement responses from the original Chichi earthquake excitation are used to test theneuroemulator. Displacement responses are obtained at each floor in the x and y horizontal direc-tions at increments of 0.01 s over a period of 40 s, equal to the duration of the Chichi earthquakeexcitation. Thus, the number of sample data, N in Equation (20), is 4000.

The non-linear control model is trained using the results obtained when the structure is subjectedto (a) earthquake excitation (uncontrolled structure) and (b) both earthquake excitation and actuatorforces simultaneously. This will enhance the adaptability of the fuzzy wavelet neuroemulatorbecause for some cases no control forces may be necessary, for example, when earthquake excitationis small.

Example 1: Twelve-story irregular steel building

Example structure and its response. This example is an irregular 12-story 3D steel buildingstructure with vertical setbacks as shown in Figure 5. This structure was created by Adeli andSaleh [13, 14] for the study of structural control and used by Jiang and Adeli [46] for non-linearsystem identification of structures and Kim and Adeli [41] for hybrid control of the irregularstructure. It consists of 148 members and 77 nodes. In earlier papers [14, 46], the structure wasmodeled as a 3D space frame with 462 DOFs. In this article, floor diaphragms are assumedto be rigid and axial column deformations are neglected. The rigid-floor structural model has36 DOFs.

The members of the structure are W shapes as indicated in Figure 5. The structure is subjectedto the combination of uniformly distributed floor dead and live loads of 60 psf (2.88 kPa) and 50 psf(2.38 kPa), respectively, and the Chichi earthquake records. The Chichi earthquake records were



2 5.5m

W21 57

W21 50

W21

50

y

x

12

4.5m

W14

90

W14

68

W14

61

A

Groundexcitation

yx

z

=-130

Sect

ion

I Se

ctio

nII W21 50

W21 50

W21 57

Beam/column Actuator location

W21

50

W21 50

A

(a) (c)

(b)

Figure 5. Twelve-story steel building with vertical irregularity (Example 1): (a) perspectiveview; (b) plan of section I; and (c) plan of section II.

used as the representative of one of the most destructive earthquakes of the past few decades forillustration. The modal analyses of this structure conducted by Kim and Adeli [41] demonstratethat the coupling effect of lateral and torsional vibrations is most significant when the earthquakeexcitation is applied in the direction with =13 (Figure 5). Therefore, this is the angle usedin this research to perform non-linear dynamic analyses of the structure for training and testingthe fuzzy wavelet neuroemulator. Figure 6 shows the x and y displacements of joint A at the topfloor of the structure (identified in Figure 5) when the structure is subjected to the original Chichiand 200% of Chichi earthquakes in the direction with =13.

Actuators. Four double-acting actuators with the same properties are installed in every floor alongthe exterior envelope of the structure as indicated with dashed lines in Figures 5(b) and (c). Theproperties of an actuator used in this study are based on data provided by one manufacturer(http://www.mts.com/vehicles/testline/pdfs/100-016-993.pdf) and are summarized in Table I.



0 5 10 15 20 25 30 35 40-1

-0.5

0

0.5

t (s)

t (s)

Disp

lace

men

t (m)

Disp

lace

men

t (m)

Original earthquake 200% of original earthquake

-1.5-1

-0.50

0.51

1.52

Acc

eler

atio

n (g)

0 5 10 15 20 25 30 35 40

-0.1

0

0.1

0.2

0.3

-0.2

(a)

(b)

(c)

Figure 6. Displacements of joint A at the top floor of the structure (identified in Figure 7) when thestructure is subjected to the original Chichi and 200% Chichi earthquakes in the direction with =13:(a) 200% Chichi earthquake, Taiwan, 21 September 1999; (b) displacements of joint A in the x-direction;

and (c) displacements of joint A in the y-direction.

Table I. Properties of the actuator.

Variables Description Value

Ap Effective piston area 3.368103 m2Vc Chamber volume 1.01103 m3Cl Leakage coefficient 0.11010 m5/(Ns) Compressibility coefficient 2.11010 N/m2qmax Maximum flow rate 2.0103 m3/sFmax Maximum actuator force 68 kN Servovalve time constant 0.15 sg Servovalve constant 2.1104 m3/s/volt



-6

-3

0

3

6x104

F(t)

(N)

0 5 10 15 20 25 30 35 40t(s)

Figure 7. Randomly generated control forces in the x-direction over the same earthquake duration of 40 sfor the actuator with properties shown in Table I.

For training the non-linear control model, two sets of control forces are randomly generated in thex- and y-directions with a maximum actuator force of 68 kN (Table I) and a uniform distributionin the interval of (68, 68 kN). Figure 7 shows the randomly generated control forces in thex-direction over the same earthquake duration of 40 s.

Constructing fuzzy wavelet neuroemulators. Three sets of time series data are needed to constructeach one of the two fuzzy WNN neuroemulators in the x- and y-directions: the structuraldisplacement response at joint A (Figure 5(a)), earthquake excitation, and control force. Eachset of time series consists of 4000 data points. Optimum time-delay steps of x =2, g =5, andu =8 are obtained for the displacement response at joint A, the earthquake excitation, and thecontrol force using the FNN method, respectively. The same numbers of time steps are obtainedfor the data sets in both x- and y-directions. The number of input vectors created is thus equal toNa =4000max{2,5,8}=3992 (Equation (20)).

A five-level wavelet decomposition was performed on the 3992 sets of input vectors with the sizeof x +u +g =15 (Equation (18)) using the Mexican hat wavelet function. The empty waveletswhose supports do not contain any data are eliminated, resulting in 109 and 105 nonempty waveletsfor x- and y-direction vectors, respectively. On the basis of the AFPE criterion and using themodified GramSchmidt algorithm [52], it is concluded that three out of the 109 and four outof the 105 nonempty wavelets are sufficient to construct x- and y-fuzzy WNN neuroemulators,respectively. As such, there are three WNN nodes in the hidden layer of the x-neuroemulator forpredicting the x-directional displacement response and four WNN nodes in the hidden layer of they-neuroemulator for predicting the y-directional displacement response. For a discussion on theconcept of empty wavelets and how the necessary number of nonempty wavelets is chosen, referto Jiang and Adeli [52].

Training fuzzy wavelet neuroemulators. Two sets of data are used to train each neuroemulator,one set at a time, denoted as Set1 and Set2. In Set1, original Chichi earthquake excitationsduring the previous g =5 time intervals and the resulting structural displacement responses ofjoints A (Figure 5(a)) during the previous x =2 time intervals are used as inputs. The structuralresponses in the x-direction (Figure 6(b)) are used for training the x-neuroemulator and those in the



y-direction (Figure 6(c)) for the y-neuroemulator. In Set2, 200% of Chichi earthquake excitations(scaled) during the previous g =5 time intervals, the randomly generated control forces duringthe previous u =8 time intervals, and the structural displacement responses of joints A during theprevious x =2 time intervals subjected to combined earthquake and control loadings are used asinputs. The control forces in the x-direction (Figure 7) and their corresponding responses are usedfor training the x-neuroemulator and those in the y-direction for training the y-neuroemulator. Inboth sets of training, the current displacement response of joint A is the output of the dynamicfuzzy wavelet neuroemulators. Set1 data are used to illustrate the effectiveness of the proposedneuroemulator in predicting linear response of the structure, while Set2 data are used to illustrateits effectiveness in predicting non-linear response of the structure.

The training of the model using the adaptive LM-LS learning algorithm converges very fast afteronly six iterations for training the x-neuroemulator and eight iterations for the y-neuroemulatorusing each of the two sets of data. The model is implemented in a combination of C++ program-ming language and MATLAB 6.1 [53] on a Windows XP Professional platform and a 1.5 GHzIntel Pentium 4 processor. The CPU time for training the x-neuroemulator using each set of datais only 53 s and for training the y-neuroemulator, only 55 s.

Testing fuzzy wavelet neuroemulators. The performance of the trained neuroemulators is evaluatedby comparing their predicted displacement response with the finite element simulation resultsobtained directly by Equation (9). The evaluation is conducted in both time and frequency domainsfor non-linear response of the structure subjected to 100% of Chichi earthquake. Each of thetrained x- and y-neuroemulators is tested using two sets of data defined previously for 100% ofChichi earthquake. Figure 8 shows the predicted and simulated structural displacement responsesof joint A of the 12-story structure in the x- and y-directions under the combined Chichi earthquakeexcitation and actuator forces. The relative root mean square (RRMS) errors between the simulatedand predicted results for the four test cases (two in the x and two in the y-direction) are summarizedin Table II. The RRMS errors of the displacement responses for all four sets of test data are lessthan 0.3, which is quite small.

The pseudospectra method, a power density spectrum method, proposed by authors for damagedetection of highrise buildings [54] is employed in this study to evaluate the accuracy of the trainedneuroemulators in the frequency domain. The multiple signal classification (MUSIC) method isemployed to compute the pseudospectrum from the structural response time series. Figure 9 showsa comparison of pseudospectra of the predicted and simulated displacement responses of joint Aof the 12-story structure in the x- and y-directions under the combined earthquake and actuatorloadings. The pseudospectra of the predicted and simulated structural responses match quite wellfor both x- and y-directions, thus demonstrating that the fuzzy WNN neuroemulators provideaccurate prediction of structural displacement responses.

Example 2: Eight-story irregular steel building

Example structure and its response. This example structure is an 8-story 3D steel building structurewith a plan irregularity and a height of 36 m shown in Figure 10. This structure was created byKim and Adeli [39] for the study of hybrid control of 3D irregular buildings. It consists of 208members and 99 nodes. The members of the structures are various W shapes as described in Kimand Adeli [39]. Floor diaphragms are assumed to be rigid and axial column deformations areneglected. The resulting structural model has 24 DOFs.



0 5 10 15 20 25 30 35 40-1

-0.5

0

0.5

1

Disp

lace

men

t (m)

t(s)

Simulated data Predicted by fuzzywavelet neuroemulator

-1.5

-1

-0.5

0

0.5

1

1.5

(a)

0 5 10 15 20 25 30 35 40t(s)(b)

Figure 8. Predicted and simulated displacement responses of joint A of a 12-story structure (Example 1)subjected to combined Chichi earthquake excitation and actuator forces: (a) displacements of joint A in

the x-direction and (b) displacements of joint A in the y-direction.

Table II. Relative root mean square (RRMS) error of four test cases in Example 1.Earthquake Combined earthquake excitation

Loading type excitation and actuator forces

x-direction 0.29 0.15y-direction 0.23 0.24

The structure is subjected to the combination of uniformly distributed floor dead and live loadsof 100 psf (4.78 kPa) and 70 psf (3.35 kPa), respectively, and the Chichi earthquake records. Themodal analyses of this structure conducted by Kim and Adeli [39] demonstrate that the couplingeffect of lateral and torsional vibrations is most significant when the earthquake excitation isapplied in the direction with =83.4 (Figure 10). Therefore, this is the angle used in this researchto perform non-linear dynamic analyses of the structure for training and testing the new fuzzywavelet neuroemulator.

Actuators. Four double-acting actuators with the same properties are installed in every floor asindicated with dashed lines in Figure 10(b). Their properties are the same as those employedin Example 1. The control forces are randomly generated and used similar to that explained forExample 1.



-50

0

50

100

150

Frequency (Hz)

Mag

nitu

de (d

B)

Simulated data

0 /10 /5 3/10 2/5 /2 3/5 7/10 4/5 9/10

Frequency (Hz)0 /10 /5 3/10 2/5 /2 3/5 7/10 4/5 9/10

Predicted by fuzzywavelet neuroemulator

(a)

-50

0

50

100

150

Mag

nitu

de (d

B)

(b)

Figure 9. Comparison of pseudospectra of predicted and simulated displacement responses of joint Aof 12-story structure (Example 1) under combined earthquake and actuator loadings: (a) comparison of

displacements in the x-direction and (b) comparison of displacements in the y-direction.

8

4.5m

=36

m

y

x

Ground excitationy

x

=83.40

z

A

3 6m = 18m

2

6m

=12

m

Beam

Actuatorlocation

(a)

(b)

Figure 10. Eight-story steel building with plan irregularity (Example 2): (a) perspective view and (b) plan.



Constructing fuzzy wavelet neuroemulators. Following the procedure for constructing the neuroem-ulators explained for Example 1, optimum time-delay steps of x =2, g =5, and u =14 areobtained for the displacement response at joint A, the earthquake excitation, and the controlforce using the FNN method, respectively. The number of input vectors created is equal toNa =4000max{2,5,14}=3986 (Equation (19)). The 3986 sets of input vectors are used toconstruct the neuroemulators. The resulting x-neuroemulator has 21 input nodes in the input layer(x +u +g =21 in Equation (18)) and two hidden/wavelet nodes in the hidden layer, whereas theresulting y-neuroemulator has 21 input nodes in the input layer and three hidden/wavelet nodesin the hidden layer.

Training fuzzy wavelet neuroemulators. Two sets of data, denoted by Set1 and Set2, similar tothose explained for Example 1, are used to train each neuroemulator, one set at a time. The trainingof the model using the adaptive LM-LS learning algorithm converges after only four iterations fortraining the x-neuroemulator and six iterations for the y-neuroemulator using each of the two setsof data.

Testing fuzzy wavelet neuroemulators. Testing is performed similar to that for Example 1 under100% of Chichi earthquake. This structure, however, behaves linearly under this earthquake loadingas mentioned earlier. Figure 11 shows the predicted and simulated displacement responses of

-0.6-0.4-0.2

00.20.40.6

Disp

lace

men

t (m)

0 5(a)

Disp

lace

men

t (m)

(b)

10 15 20 25 30 35 40-1

-0.5

0

0.5

1

t(s)

0 5 10 15 20 25 30 35 40t(s)

Simulated data Predicted by fuzzywavelet neuroemulator

Figure 11. Predicted and simulated displacement responses of joint A of an 8-story building (Example 2)subjected to combined Chichi earthquake excitation and actuator forces: (a) displacements of joint A in

the x-direction and (b) displacements of joint A in the y-direction.



Table III. Relative root mean square (RRMS) error of four test cases in Example 2.Earthquake Combined earthquake excitation

Loading type excitation and actuator forces

x-direction 0.22 0.14y-direction 0.20 0.19

joint A of the structure in the x- and y-directions under the combined Chichi earthquake excitationand actuator forces. The RRMS errors between the simulated and predicted results for the four testcases (two in the x and two in the y-direction) are summarized in Table III. The RRMS errors ofthe displacement responses for all four sets of test data are less than 0.25 which is quite small. Theperformance of the trained neuroemulators is also evaluated using the pseudospectra method in thefrequency domain for the Chichi earthquake. The pseudospectra of the predicted and simulatedstructural responses match quite well for both x- and y-directions.

CONCLUDING REMARKS

An effective model has been developed for active non-linear control of large 3D structures. Bothgeometric and material non-linearities are included in the control formulation. Coupling betweenlateral and torsional motions of the structure as well as the actuator dynamics are taken into accountin the control model.

A dynamic fuzzy wavelet neuroemulator was presented for predicting the structural responsein future time steps. It is demonstrated that the dynamic fuzzy wavelet neuroemulator providesaccurate prediction of structural displacement responses in both linear and non-linear ranges. Inthe companion paper [40], a floating-point genetic algorithm is presented for finding the optimalcontrol forces needed for active non-linear control of building structures.

REFERENCES

1. Masri SF, Bekey GA, Caughey TK. On-line control of nonlinear flexible structures. Journal of Applied Mechanics1982; 49(12):871884.

2. Stein G, Athans M. The LQG/LTR procedure for multivariable feedback control design. IEEE Transactions onAutomatic Control 1987; AC32(2):105114.

3. Miller RK, Masri SF, Dehghanyar T, Caughey TK. Active vibration control of large civil structures. Journal ofEngineering Mechanics (ASCE) 1988; 114(9):15421570.

4. Yang JN, Long FX, Wong D. Optimal control of nonlinear structures. Journal of Applied Mechanics 1988;55(4):931938.

5. Yang JN, Danielians A, Liu SC. Hybrid control of nonlinear and hysteretic systems I&II. Journal of EngineeringMechanics (ASCE) 1992; 118:14231439, 14411457.

6. Soong TT. Active Structural Control: Theory and Practice. Longmans: New York, 1990.7. Chang CC, Yang HTY. Control of building using active tuned mass dampers. Journal of Engineering Mechanics

(ASCE) 1995; 121(3):355366.8. Nikzad K, Ghaboussi J, Paul SL. Actuator dynamics and delay compensation using neurocontrollers. Journal of

Engineering Mechanics (ASCE) 1996; 122(10):966975.9. Tomasula DP, Spencer BF, Sain MK. Nonlinear control strategies for limiting dynamic response extremes. Journal

of Engineering Mechanics (ASCE) 1996; 122(3):218229.10. Thomson M, Schooling SP, Soufian M. The practical application of a nonlinear identification methodology.

Control Engineering Practice 1996; 4(3):295306.



11. Housner GW, Bergman LA, Caughey TK, Chassiakos AG, Claus RO, Masri SF, Skelton RE, Soong TT, SpencerBF, Yao JTP. Structural control: past, present, and future. Journal of Engineering Mechanics (ASCE) 1997;123(9):897971.

12. Adeli H, Saleh A. Optimal control of adaptive/smart bridge structures. Journal of Structural Engineering (ASCE)1997; 123(2):218226.

13. Adeli H, Saleh A. Control, Optimization, and Smart StructuresHigh-performance Bridges and Buildings of theFuture. Wiley: New York, 1999.

14. Saleh A, Adeli H. Optimal control of adaptive/smart multistory building structures. Computer-Aided Civil andInfrastructure Engineering 1998; 13(6):389403.

15. Lu LT, Chiang WL, Tang JP. LQG/LTR control methodology in active structural control. Journal of EngineeringMechanics (ASCE) 1998; 124(4):446454.

16. Bani-Hani K, Ghaboussi J. Neural networks for structural control of a benchmark problem, active tendon system.Earthquake Engineering and Structural Dynamics 1998; 27(11):12251245.

17. Bani-Hani K, Ghaboussi J. Nonlinear structural control using neural networks. Journal of Engineering Mechanics(ASCE) 1998; 124(3):319327.

18. Hung SL, Kao CY, Lee JC. Active pulse structural control using artificial neural networks. Journal of EngineeringMechanics (ASCE) 2000; 126(8):839849.

19. Connor JJ. Introduction to Structural Motion Control. Prentice-Hall: Upper Saddle River, NJ, 2003.20. Adeli H, Kim H. Wavelet-hybrid feedback-least mean square algorithm for robust control of structures. Journal

of Structural Engineering (ASCE) 2004; 130(2):128137.21. Kim H, Adeli H. Hybrid feedback-least mean square algorithm for structural control. Journal of Structural

Engineering (ASCE) 2004; 130(2):120127.22. Edwards C, Spurgeon S. Sliding Mode Control: Theory and Applications. Taylor & Francis, CRC Press: London,

Boca Raton, FL, 1998.23. Agrawal AK, Yang JN. Optimal polynomial control of seismically excited linear structures. Journal of Engineering

Mechanics (ASCE) 1996; 122(8):753761.24. Adeli H, Hung SL. Machine LearningNeural Networks, Genetic Algorithms, and Fuzzy Sets. Wiley: New York,

1995.25. Adeli H, Park HS. Neurocomputing for Design Automation. CRC Press: Boca Raton, FL, 1998.26. Adeli H. Neural networks in civil engineering: 19892000. Computer-Aided Civil and Infrastructure Engineering

2001; 16(2):126142.27. Chen HM, Tsai KH, Qi GZ, Yang JCS, Amini F. Neural network for structure control. Journal of Computing

in Civil Engineering (ASCE) 1995; 9(2):168176.28. Ghaboussi J, Joghataie A. Active control of structures using neural networks. Journal of Engineering Mechanics

(ASCE) 1995; 121(4):555567.29. Kim JT, Jung HJ, Lee IW. Optimal structural control using neural networks. Journal of Engineering Mechanics

(ASCE) 2000; 126(2):201205.30. Kim DH, Seo SH, Lee IW. Optimal neurocontroller for nonlinear benchmark structure. Journal of Engineering

Mechanics (ASCE) 2004; 130(4):424429.31. Brown AS, Yang HTY. Neural networks for multiobjective adaptive structural control. Journal of Structural

Engineering (ASCE) 2001; 127(2):203210.32. Kim DH, Lee IW. Neurocontrol of seismically excited steel structures through sensitivity evaluation scheme.

Earthquake Engineering and Structural Dynamics 2001; 30(9):13611377.33. Fur LS, Yang HTY, Ankireddi S. Vibration control of tall building buildings under seismic and wind loads.

Journal of Structural Engineering (ASCE) 1996; 122(8):948957.34. Al-Dawod M, Smali B, Kwok K, Naghdy F. Fuzzy controller for seismically excited nonlinear buildings. Journal

of Engineering Mechanics (ASCE) 2004; 130(4):407415.35. Ohtori Y, Christenson RE, Spencer BF, Dyke SJ. Benchmark control problems for seismically excited nonlinear

buildings. Journal of Engineering Mechanics (ASCE) 2004; 130(4):366385.36. Yoshida O, Dyke SJ. Response control of full-scale irregular buildings using magnetorheological dampers. Journal

of Structural Engineering (ASCE) 2005; 131(5):734742.37. Tan P, Dyke SJ. Integrated device placement and control design in civil structures using genetic algorithms.

Journal of Structural Engineering (ASCE) 2005; 131(10):14891496.38. Kim H, Adeli H. Hybrid control of smart structures using a novel wavelet-based algorithm. Computer-Aided

Civil and Infrastructure Engineering 2005; 20(1):722.



39. Kim H, Adeli H. Wavelet hybrid feedback-LMS algorithm for robust control of cable-stayed bridges. Journal ofBridge Engineering (ASCE) 2005; 10(2):116123.

40. Jiang XM, Adeli H. Neuro-genetic algorithm for nonlinear active control of highrise buildings. InternationalJournal for Numerical Methods in Engineering 2007; DOI: 10.1002/nme.2274.

41. Kim H, Adeli H. Hybrid control of irregular steel highrise building structures under seismic excitations.International Journal for Numerical Methods in Engineering 2005; 63(12):17571774.

42. Wen YK. Method for random vibration for hysteretic structures. Journal of the Engineering Mechanics Division(ASCE) 1976; 102(EM2):249263.

43. Dyke SJ, Spencer BF, Sain MK. Role of controlstructure interaction in protective system design. Journal ofEngineering Mechanics (ASCE) 1995; 121(2):322338.

44. Desilva CW. Control Sensors and Actuators. Prentice-Hall: Englewood Cliffs, NJ, 1989.45. Adeli H, Jiang XM. Dynamic fuzzy wavelet neural network model for structural system identification. Journal

of Structural Engineering (ASCE) 2006; 132(1):102111.46. Jiang XM, Adeli H. Dynamic wavelet neural network for nonlinear identification of highrise buildings. Computer-

Aided Civil and Infrastructure Engineering 2005; 20(5):316330.47. Zhou Z, Adeli H. Time-frequency signal analysis of earthquake records using Mexican hat wavelets. Computer-

Aided Civil and Infrastructure Engineering 2003; 18(5):379389.48. Bezdek JC. Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum: New York, 1981.49. Zhang Q. Using wavelet network in nonparametric estimate. IEEE Transactions on Neural Networks 1997;

8(2):227236.50. Ljung L. System IdentificationTheory for the User (2nd edn). Prentice-Hall: Upper Saddle River, NJ, 1999.51. Jiang XM, Adeli H. Fuzzy clustering approach for accurate embedding dimension identification in chaotic time

series. Integrated Computer-Aided Engineering 2003; 10(3):287302.52. Jiang XM, Adeli H. Dynamic wavelet neural network model for traffic flow forecasting. Journal of Transportation

Engineering (ASCE) 2005; 131(10):771779.53. Matlab. The Language of Technical Computing. Mathworks Inc.: Natick, MA, 2001.54. Adeli H, Jiang XM. Psuedospectra, MUSIC, and dynamic fuzzy wavelet neural network for damage detection

of highrise building using sensor data. International Journal for Numerical Methods in Engineering 2007;71(5):606629.


Dynamic Fuzzy Wavelet Neuroemulator for Non-linear Control of Irregular Buildins Structures

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