System Identification with Noisy Data

Transcription

1 System Identifiction with Noisy Dt U K Dewngn & S V Bri 2 Dept of Civil Engineering, Ntionl Institute of Technology, Ripur, Indi 2 Dept of Civil Engineering, IIT Khrgpur, Khrgpur , Indi E-mil : dewngnumesh25@gmilcom, skbri@civiliitkgpernetin Abstrct - System identifiction from the field obtined dt is not successful till dte becuse of the noise in the sensor or mesured The unknown system prmeter for ny unknown dmge is still long wy to move In this pper finite element technique is pplied on the simulted dmged structures The dmge detection for the noise free dt is extended nd tried for the rndom noisy sensor dt The proposed lgorithm is used to predict the system prmeter on sttic deflection dt with introduction of some rndom noisy dt using simulted structure The developed lgorithm is pplied on bridge truss structure for the identifiction of the dmged system prmeter prediction Key words - Dmge, System identifiction; Noisy dt; Finite element I INTRODUCTION The prediction of the dmged prmeters from the field dt hving noise is still n open chllenge in the field of structurl identifiction The Structurl Helth Monitoring (SHM) of lrge spce structures like erospce structure; stellite vehicles, where once the dmge hd occurred cuse the uncertinty due to unknown dmged prmeter Finite element methods re universlly ccepted s fst computtion tools for structurl behvior prediction Development in the field of computtionl technology nd sensing instruments hs lso progressed lot Field dt re usully noisy The dmge detection, loction nd dmge extent prediction re mong the importnt spect of structurl behvior prediction The min im of reserchers is concentrted on the suitble identifiction lgorithms bsed on uncertin vilble sensor dt which my be liner or nonliner, noisy or noise free sensor dt There re two kinds of prmeter identifiction nmely sttic nd dynmic prmeter identifiction In the dynmic prmeter identifiction there re three unknowns They re mss, stiffness nd dmping The reltionship between the dynmic coefficients (mss, stiffness nd dmping) nd lso its sensitivity effects of one property on nother re still unknowns Such system mkes the nlysis very complicted For uncertin dmge with uncertin noisy dt, no certin techniques re vilble The sttic identifiction method seems to be better thn dynmic identifiction s it is hving only one unknown (stiffness) A brief review on dmge detection minly using the sttic method or sttic combined with dynmic method is presented here The sttic dmge prmeter identifiction pproches by the error term reduction includes minimum devition, sensitivity nlysis, output error optimiztion etc were pproched by [2, [4, [7, [2, [4 nd [5 Dmge detection in composite mterils using system identifiction technique proposed by [4 The output error pproch of system identifiction ws employed to determine the chnges in the nlyticl model in order to minimize the distnce between mesured nd predicted response [2 Used force error estimtor nd displcement error estimtor for sttic prmeter grouping scheme to identify the error by lest squres minimiztion Sttic strin mesurement from multiple loding models for identifiction of the hole nd crcks in liner nisotropy elstic mterils with nonliner optimiztion ws presented by [4 [5 pplied liner constrined nonliner optimiztion problem using the minimiztion of error between the mesured nd computed displcement to find dmge Error sensitivity nlysis is found to be populr method for finding the dmge existence [9, [, [3, [ On nlyzing the sensitivity coefficients for nturl frequency, mode shpes nd modl flexibility, [9 found tht modl flexibility is more sensitive s dmge Interntionl Journl of Advnced Technology in Civil Engineering, ISSN: , Volume-, Issue-2,

2 System Identifiction with Noisy Dt indictor Sensitivity with some other term pproch like orthogonl sensitivity, non-linerity, modl updting were pproched by[3, [6, [20 etc The orthogonl condition sensitivities ws developed by [6 for the dmged nd undmged structure mode shpes using FEM in lminted rectngulr plte of composite structures [20 presented sensitivity-bsed finite element (FE) model updting for dmge detection The modl flexibility residul formultion nd its grdient were used for formultion The dmge detection procedure ws illustrted on simulted exmple with noisy dt nd on reinforced concrete bem model Dmge detection of cble-styed bridges by chnges in cble forces, ws optimized on cble force error between mesurement results nd nlyticl model by [7 [0 developed method used continuous strin dt from fiber optic sensor nd neurl network model [8 used sensitive chrcteristic of strin to identify dmge in structures for strin-bsed dmge identifiction [8 Used conventionl single-objective optimiztion pproch defines the objective function by combining multiple error terms into single one, for weker constrint in solving the identifiction problem The bove literture review indictes tht both sttic s well dynmic methods were used in the system prmeter prediction of structure using simulted, experimentl nd from the field dt Sttic methods used either strin or displcement s mesured dt, while in cse of dynmic methods frequency nd mode shpes dt were used populrly Sensitivity method, nlyses the effect of prmeter chnges on the other prmeters When the structurl prmeters re unknown, the sensitivity nlysis hs no mening In ddition, noise in the mesured dt my completely chnge the mtrix property using mtrix inversion The structurl prmeter identifiction from the field dt hs no well-estblished solution, until now The pper is ttributed to the sttic prmeter identifiction process with the noisy sensor dt The objective of this pper is to develop new modified pproch for the noisy sensor dt with few mesurement The initil prmeter identifiction lgorithm hs been tken from [3 but some noise were introduced in the sensor dt This will led to ner relistic field sitution Finite element method for dmge detection using sttic test dt for smller subgroups of mtrix ws pplied [3 for dmge existence prediction with only few mesurements The noise vlues were vried between ± 4 % errors in the sensor dt A finite element model of bridge truss structure presented for the demonstrtion II INITIAL AND DEDELOPED APPROACH An lgorithm to find the prmeter extent for noise free dt ws developed by [3 Assuming the structure behvior of structure is liner throughout the test, the force displcement reltionship in the sttic cse for undmged structure is given by [ F = [ K [ U () nd for dmged structure by [ F = [ K [ U (2) d Prtitioning into mesured nd unmesured displcements f Kd K db U = fb Kdb Kd Ub (3) [ f = [[ K [ K b -- [ K -- [ K b [ U + [ K b [ K -- [ f b (4) [ f, [ f nd [ b U re mesured from test The difference between the mesured nd theoreticlly clculted vlue will be the error term If the stiffness prmeters re correct, then error mtrix [ E ( p) will be zero, otherwise non-zero [ E ( p) = [[ K [ K b [ K [ K b [ U + [ K b [ K [ f b [ f (5) [ E( p) { E ( p) } + { S( δ p) } [ Δ p (6) The error sensitivity expression ws clculted using first order Tylor series expnsion of error mtrix [ E ( p) The stiffness prmeters were obtined by minimiztion of error function with respect to unknown prmeter ( p) using the lest squre optimiztion The error sensitivity expression hs been modified for noisy sensor dt s the displcement gets modified due to sensor noise nd unknown dmged The unknown prmeters become the function of both Interntionl Journl of Advnced Technology in Civil Engineering, ISSN: , Volume-, Issue-2,

3 System Identifiction with Noisy Dt displcements The sensitivity mtrix ws reclculted with this new modified expression, [ [ [ k [ k [ k [ [ [ [ b b [ S( pi) = [ k kb kb k pi pi pi + k k k k U + k k k k [ k [ U [ [ [ [ [ [ [ b b b b b p i [ kb [ k [ [ [ k k k [ k + b * pi pi [ f [ K [ K b [ K [ K b [ U * [ K b [ K [ K b [ k [ fb [ fb [ k [ k + [ f [ [ k + K [ U * [ K b [ K pi pi [ k [ k [ [ [ [ U + f k + K * [ K b [ K (7) pi pi pi p i The present method is ble to predict the dmge existence in structure with only few mesurements Using lest squre optimiztion technique, ll the unknown dmged prmeters cn be identified fter some itertion successfully with the noisy sensor dt The finite element method revisited nd row-echelon form of mtrix developed for dmge detection using sttic displcement nd force dt, Dewngn U K (200) [4 The row-echelon form hs n dvntge of prtitioning mtrix into smller subgroupsthe noise vlues were vried between ± 4 % errors in the sensor dt A finite element model of bridge truss structure presented for the demonstrtion The modified lgorithm is implemented using MATLAB [2 III EXAMPLES Bsed on the lgorithm discussed in the bove section, the computer progrm ws written in MATLAB nd tested Bridge Truss Structure Bridge Truss Structure Structurl Detils: All elements re hving the modulus of elsticity E = 20 GP nd initil undmged 5 2 cross sectionl re A = 6X0 mm The structurl configurtion is shown in Fig 38 A single concentrted lod is pplied t ech joint The deflections t ech joint re mesured The force mtrix [ F is found out nd corresponding displcement mtrix [ U is mesured For bridge truss structure, [ s shown in Figure, the modulus of elsticity of ll elements ws 2068 GP nd initil undmged cross sectionl re of ll members ws 500 mm 2 The noise ws introduced up to ± 5 % error in the sensor dt for sensor numbers 5, 6, 7 nd 8 displcement dof Different lod combintions re considered nd they re tbulted in the Tble 2 with sensor noise vlue Previously discussed lgorithms were pplied to this problem nd results re given in Tble 2 nd Figure 4 for typicl cses For the noisy sensor dt set combintion on tower truss the computed prmeter vlues were compred with the ctul prmeter vlue From the plotted grph s shown in Figure 2, it is cler tht with the noisy sensor dt, [3 lgorithm vlues re fr wy from the ctul vlue of the prmeter The modified proposed lgorithm vlues re nerly close to the ctul vlue of the prmeters The lgorithm could identify the dmge extent for members wy from the supports Interntionl Journl of Advnced Technology in Civil Engineering, ISSN: , Volume-, Issue-2,

4 System Identifiction with Noisy Dt U 2 U 4 U U U 6 U 5 3 U 8 U U 0 U U U 2 U U 4 U 3 9 U 6 U 5 U 8 U mm = 3926 mm Fig : Rilwy bridge truss 2 U 2 U U 4 U U 2 U 6 7 U 2 U 2 U 4 U 3 U Force degree of freedom U U 2 U 4 U U Figure 2 Rilwy bridge truss (Cse 2) Displcement degree of freedom 2 5 Sensor 7,8,9,0,7,8,9,20 ll with 4% Noise Actul Prmeter Predicted Element Number Snyei nd Onipede (99) Modified Proposed Algorithm Fig 3 : Comprtive study of dmge extent prediction (Bridge truss Structure) TABLE I: RESULTS OF NOISY SENSOR DATA WITH MODIFIED ALGORITHM ON BRIDGE TRUSS Cse Member dmged Applied force dof Mesured displcement dof Noise in Sensor Members Converged Members Diverged All up to 5% -4, 6-4, ,6,5 nd 2 2 All 7-0, , up to 5% -4, 6-4, ,6,5 nd 2 3 All 7-0, , -4 4 up to 5% -4, 6-4, ,6,5 nd 2 4 All ,6,7 nd, 8 up to 4% -4, 6-4, ,6,5 nd 2 Interntionl Journl of Advnced Technology in Civil Engineering, ISSN: , Volume-, Issue-2,

5 System Identifiction with Noisy Dt IV RESULTS Considering the vrious multiple dmged member combintions, the existence of dmge is detected Different combintions of force DOF nd displcement DOF used re given in Tble for vrious cses From the plotted grph s shown in Figure 4, it is cler tht with the noisy sensor dt, the modified proposed lgorithm vlues re nerly close to the ctul vlue of the prmeters The lgorithm could identify the dmge extent for members wy from the supports V DISCUSSION In this study, it ws clerly observed from the demonstrted exmples tht the modified lgorithm could predict dmge extent in the presence of noise in structurl response Further it ws found tht the lgorithms filed to predict dmge extent in the members, which were ner to the support Hence, the present work is very well justified for the lrge spce structures where except the support connected members, ll other members, dmge extent could be predicted The filure of the lgorithm in identifying the dmge extent in members ner support is mtter of further investigtion VI CONCLUSION The dmge prmeter identifiction with the noisy sensor dt ws crried out for different rndom noise vlue Algorithm developed by [3 hs been modified for the noise in sensor vlue Finite element method for dmge detection using sttic test dt for smller subgroups of mtrix ws pplied [3 for dmge existence prediction with only few mesurements The noise vlues were vried between ± 4 % errors in the sensor dt The lgorithm works well for lower noise level up to vlue ± 4 % errors in the sensor dt nd the unknown dmged prmeter could be extrcted even from the noisy dt set of structurl response using bove technique for the members with only few mesurements Hence the lgorithm is useful for dmge prediction with noisy sensor dt on lrge structures for the members, which re wy from the supports NOTATION dof = Degree of freedom ddof = Displcement degree of freedom fdof = Force degree of freedom {E(p)} = Error function vector [ F = Applied force mtrix t force dof t mesured dof [Fb = Unpplied force mtrix t force dof for unmesured displcement dof FEM = Finite element method [K = Undmged globl stiffness mtrix [Kd = Sub mtrix of [K corresponding to mesured dof nd pplied force dof [Kdb = Sub mtrix of [K corresponding to mesured dof nd unpplied force dof [Kd = Sub mtrix of [K corresponding to unmesured dof nd unpplied force dof [Kdb = Sub mtrix of [K corresponding to unmesured dof nd pplied force dof [Kd = Dmged globl stiffness mtrix n = Number of elements p = Unknown prmeter s vlues r = Row number [Sd = Dmged globl stiffness mtrix [S = Undmged globl stiffness mtrix [S = Sub mtrix of [S corresponding to mesured dof nd pplied force dof [Sb = Sub mtrix of [S corresponding to mesured dof nd unpplied force dof [S = Sub mtrix of [S corresponding to unmesured dof nd unpplied force dof [Sb = Sub mtrix of [S corresponding to unmesured dof nd pplied force dof {S(p)} = Sensitivity mtrix [ U = Trnsformtion mtrix [U = Mesured displcements mtrix [Ub = Unmesured displcements mtrix REFERENCES [ S R Agrwl, Steel Bridges In Indin Rilwys Advnces in Steel Structures, (Ed) V Klynrmn, Proc of the Ntionl Symposium on Advnces in Steel Structures, Tt McGrw Hill, Publishing Compny Ltd, New Delhi,,990, pp [2 M R Bnn, M R Bnn nd K D Hjelmstd, Prmeter Estimtion of Structures From Sttic Response II: Numericl simultion studies, Journl of Structurl Engineering, vol 20, Nov,994, pp [3 U K Dewngn, Structurl Dmge Existence Prediction with Few Mesurements Interntionl Interntionl Journl of Advnced Technology in Civil Engineering, ISSN: , Volume-, Issue-2,

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