Time synchronization algorithms for wireless monitoring system

Transcription

1 Source: SPIE s th Annual International Sposiu on Sart Structures and Materials, San Diego, CA, USA, March -6, 3. ie snchronization algoriths for wireless onitoring sste Y. Lei *, A. S. Kireidian, K. K. air, J. P. Lnch and K. H. Law Departent of Civil and Environental Engineering, Stanford Universit, CA, USA 9435 ABSRAC Wireless health onitoring schees are innovative techniques, which effectivel reove the disadvantages associated with current wire-based sensing sstes, i.e., high installation and upkeep costs. However, recorded data sets a have relative tie-delas due to the blockage of sensors or inherent internal clock errors. In this paper, two algoriths are proposed for the snchronization of the recorded asnchronous data easured fro sensing units of a wireless onitoring sste. In the first algorith, the input signal to a structure is easured. ie-dela between an output easureent and the input is identified based on the iniization of errors of the ARX (auto-regressive odel with exogenous input odels for the input-output pair recordings. he second algorith is applicable when a structure is subect to abient excitation and onl output easureents are available. ARMAV (auto-regressive oving average vector odels are constructed fro two output signals and the tie-dela between the is evaluated based on the iniization of errors of the ARMAV odels. he proposed algoriths are verified b siulation data and recorded seisic response data fro ulti-stor buildings. Kewords: Wireless sensors, snchronization, tie series analsis, sste identification, structural health onitoring. IRODUCIO here exists a clear need to onitor the perforance of civil structures over their operational lives. Current wire-based sensing sstes suffer fro high installation and upkeep costs, which liits widespread adoption. In response to the technological and econoic liitations of present coercial onitoring sstes, a novel wireless odule onitoring sste was proposed for the health onitoring of civil structures [-]. It provides a high-perforance et low-cost data acquisition technique for structural health onitoring. When wireless sensing units record easureents independentl, recorded data sets a exist with relative tie-delas due to blockage of sensors or inherent internal clock errors. he relative tie-delas a be saller than the data-sapling interval. hus, it is necessar to perfor tie snchronization aong these recordings for the purpose of accurate structural identification and daage detection. However, little attention has been given to these topics associated with the innovative wireless onitoring sste [3-4]. In this paper, two algoriths are proposed to snchronize easured data with relative tie-delas. In the first algorith, input to a structure is easured. Each output signal is snchronized with the input signal, which is chosen as the reference signal. he second algorith treats asnchronous output easureents fro a structure under abient excitation. In the case of abient excitation, one of the output signals is taken as the reference signal since the input abient vibration cannot be easured. Several nuerical exaples of siulation data and recorded seisic response data fro ulti-stor buildings are used to deonstrate and verif the proposed algoriths.. IME SYCHROIZAIO ALGORIHM FOR IPU-OUPU PAIR RECORDIGS When the excitation to a structure is easured, the excitation signal is selected as a reference signal. Wireless sensing units have tie-delas in recording the output signals relative to the reference signal resulting in asnchronous data. he following asnchronous data of the input and output signals are obtained. x, x,..., x,,...,,,..., o M M, M M,..., M M ( * lei@stanford.edu; phone ( ; fax (

2 ( =,,, M is the output data, M is the nuber of sensing units in the recording output where, x is the excitation, signals, is the nuber of points in the records, relative to the input x. he task is to identif the unknown input. τ is the unknown value of tie-dela in recording the output τ value so that the output data can be snchronized with the. ie snchronization algorith First, the values of the input signal at shifted tie instants can be evaluated b the spline interpolation technique to ield the following set of input data x, x,..., x ( where τ is the value of tie shift. With different values of τ, a set of shifted input signals is obtained. Second, each shifted input signal is paired with one of the output signal na to construct an ARX odel [4-6] as + a kk k = bkx k + ε, τ where is the sapling interval, na and a k are the order and coefficients of the AR odel, respectivel; nb and b k are the order and coefficients of the exogenous input, respectivel; and ε ( t, τ is the prediction error of the odel with a given value of τ. he vectors β and θ are defined as follows, τ = [,..., na, x,..., x nb ] nb k = β ( t (4, a,..., a na, b, b, b,..., bnb ] where superscript denotes a transpose. Eq.(3 can be rewritten as θ = [a (5 = β, τ θ + ε, τ (6 For a given value of τ, the total error V( θ τ, defined as the su of the square of identification errors at all easureent tie instants, is given b where = ε, τ = [ β, τ θ] = nn+ = nn+ V( θ τ (7 ax(na, nb when τ nn = (8 ax(na, nb + fix {-in( τ, τ / } when τ < in which fix{-in( τ, τ / } rounds off the eleent -in( τ, τ / to the nearest integer. he reason that the su in Eq.(8 starts fro nn+ is because of the tie-dela value of τ and the order of the ARX (na, nb odel. he coefficients of the ARX odel θˆ is deterined b iniizing the total error under a given value of τ in Eq.(7, i.e., Eq. (9 gives the following solution for θˆ dv( θ τ / dθ ˆ = (9 θ=θ (3

3 ˆ θ = [ β, τ β, τ ] [ β, τ ] ( = nn+ = nn+ he iniu value of V( θ τ is denoted b e ( τ = in V( θ τ θ ( where in gives the iniu value of the function. he variation of e ( τ for a range of τ values is observed. he τ value that gives the iniu value of e ( τ is taken as the estiated value of the tie-dela in recording the output relative to the input signal x, i.e., τ = arg[ ine ( τ ] τ where arg gives the arguent of the function. hen, the corresponding shifted input signal, given b Eq.(, with the value of τ, defined b Eq.(, is snchronous with the output signal. Alternatel, all output signals can be snchronized with the input signal using the above algorith. Snchronous input and output data are obtained.. uerical exaple.. A 3-stor shear building under a sweep sine ground excitation In the first case stud, the 3-stor shear building described b Clough and Penzien [7] is used herein. A sweep sine excitation is applied to the base of the structure. he ground excitation D xd is g ( D x g = sin[.3π( + tt] (3 he excitation has constant aplitude of in/s with a linearl varing frequenc of.3 to 6.3 Hz over 4 seconds. Wireless sensing units at the first, second and third floors have tie-delas of.sec,.4sec and.7sec respectivel, in recording the floor acceleration response relative to the input signal. hese asnchronous data are generated b nuerical siulation. he sapling tie is equal to.sec. Each acceleration response data are paired with the shifted input to appl the proposed algorith. Based on the criteria of optial odel order for ARX odels [4], the orders na and nb are set to 8. Figs. (a-(c illustrate the variations of e (τ, e (τ and e 3 (τ for a range of τ values. Fro these figures, tie-delas in recording the three acceleration response data (relative to ground excitation signal are accuratel evaluated b the iniizing arguents of e (τ ( =,, 3 as described b Eq. (. hese values are found to be identical to the tie-delas introduced in the response signals pointing to the accurac of the snchronization e (τ -6-7 e (τ τ x -3 (sec τ x -3 (sec Fig. (a Variation of error e( τ of the Fig. (b Variation of error ( τ of the st floor response with τ nd floor response with τ

4 e 3 (τ -8-9 e (τ τ x -3 (sec τ x -3 (sec Fig. (c Variation of error e3( τ of the Fig. (a Variation of error ( τ of the 3rd floor response with τ st floor response with τ.. A 3-stor shear building under EL Centro earthquake excitation In the second case stud, the tie snchronization algorith is applied to the sae 3-stor building under the 94 El Centro -S earthquake loading with PGA=.3g. he recorded floor acceleration response at the first, second and third floors are assued to have tie-delas of.4sec,.9sec and.sec respectivel, relative to the input signal. hese are generated nuericall with sapling interval equal to.sec. Figs. (a-(c illustrate the variations of e( τ, e ( τ and e 3 ( τ for a range of τ values. Fro the iniu value of e ( τ ( =,, 3 shown in these figures, it can be shown that the tie-delas in recording the acceleration response data relative to the input signal are evaluated accuratel using Eq.(. e (τ e 3 (τ τ x -3 (sec τ x -3 (sec Fig. (b Variation of error e ( τ of the Fig. (c Variation of error 3 ( τ of the nd floor response with τ 3rd floor response with τ..3 Recorded accelerogras of a 8-stor coercial building subect to Loa Prieta earthquake he tie snchronization algorith is also deonstrated for the strong-otion accelerogras recorded in an 8-stor coercial building in San Francisco subect to the 989 Loa Prieta earthquake. he data were provided b the California Geolog Surve s (CGS Strong Motion Instruentation Progra (SMIP (forerl Division of Mines and Geolog, California Departent of Conservation, ftp://ftp.consrv.ca.gov/pub/dg/csip/. he baseent of the building is excited b one vertical and two horizontal ground otions. Under the condition that the three coponents of excitations recorded at the baseent are snchronized, the above tie-snchronization algorith can be extended to a ulti-input, single-output case. he ARX odel for input-output data in Eq.(3 is rewritten as

5 + na a k k = k k = nb bkx nb 3 + b3kx3 k + nb b k + ε k x, τ k (4 where nb, b k, nb, b k and nb 3, b 3k are orders and coefficients of the first, second and third exogenous input, respectivel. Analogousl, it can be shown that the relative tie-dela value of τ can still be evaluated b iniizing e ( τ as described b Eq.(. o get asnchronous acceleration response data sets of the building, recorded accelerogras are artificiall shifted to produce data at shifted tie instants b spline interpolation technique. he proposed tie snchronization algorith is then applied to treat the asnchronous data sets. In this nuerical exaple, one recorded horizontal coponent of accelerogras at the 7th floor and another recorded horizontal coponent of accelerogras at the th floor are shifted so the have tie-delas of.9sec and.4sec relative to the baseent excitations, respectivel. he orders of the ARX odel are set as na =, nb = nb = nb3 =. Figs. 3(a-3(b illustrate the variations of e7( τ and e ( τ for a range of τ values. Fro the iniu value of e ( τ ( =,, 3, the tie-delas in recording acceleration response data relative to the input signal are evaluated as τ 7 =. sec and τ =.5 sec. he error in estiating the tie-dela value is due to the nuerical error in generating the two shifted output and input data b the spline interpolation technique. he original data sapling interval is.sec. If this sapling interval were reduced, the error would also decrease. he order of this error, however, a not be significant for subsequent structural analsis coputations e 7 (τ.7 e (τ τ x -3 (sec τ x -3 (sec Fig. 3(a Variation of error e7 ( τ of the Fig. 3(b Variation of error ( τ of the 7th floor with τ th floor with τ 3. IME SYCHROIZAIO ALGORIHM FOR OUPU RECORDIGS When a structure is subect to abient excitation, the inputs to the structures cannot be easured frequentl. picall onl output signals are recorded b the wireless sensing units instruented at different locations of the structure. One of the easured output signal is chosen as the reference signal. he reaining easured acceleration responses have tie-delas in recording data relative to the reference signal. hus, the following asnchronous output data are easured

6 M,,,..., M, M,..., o M,..., M M (5 where τ i is the unknown tie-dela in recording the output i relative to the reference signal For structures under abient excitation, auto-regressive oving average vector (ARMAV odels have been applied for sste identification of structures [6, 8-]. hese odels onl use tie series of output signals, without the requireent of excitation easureent. he excitation is assued to be a stationar Gaussian white noise. A tie snchronization algorith for output signals based on the ARMAV odels is proposed. 3. ie snchronization algorith he values of the reference signal at shifted tie instants are also evaluated b the spline interpolation to ield the following data,,..., (6 where τ is the value of tie shift. With different values of τ, a set of shifted reference signals is obtained. An ARMAV odel can be constructed fro a shifted reference signal and another output i b p q [n] = ak[n k] + u[n] + bku[n k] ; n (7 k = where p and q are the orders of the AR (auto-regressive and MA (oving average coponents respectivel, a k and b k are atrices of the AR and MA coefficients and is the nuber of points in the records. { [n ], [n + } [ n] = τ and { } i i] processes. he sae ARMAV odel in the state space can be rewritten as u [ n] = u[,n],u[,n] are vectors of stationar zero-ean Gaussian white noise. [n] = A [n ] + B u[n] [n] = C [n] (8 where [n] and u[n] are vectors in the state space of diension p. he are defined as [n] = u[n] = { [n ], [n ],..., [n p + ], [n p + ] } i i { u[,n],u[,n],..., u[,n p + ], u[,n p + ] } A and B are p p diensional atrices containing the coefficients of AR and MA, respectivel, C is the observation atrix [8]. he atrices C and A are expressed b C = [ I... ] ; a I A = o where I is the identit atrix. Paraeters of the ARMAV odels are estiated b the prediction error ethod [8-9]. he vector θ is defined as a i ap I a p (9 (

7 θ = [ a, a,... ap, b, b, b,..., bq ] ( he prediction error vector ε [ n θ, τ] of the ARMAV odel under a given value of τ can be expressed as ε [ n θ, τ] = [n] ˆ[n] ( where [n] is the vector of actual easured output values and ŷ [n] denotes the predicted value b the ARMAV odel []. With a given value of τ, θˆ can be obtained as the iniu point of a criterion function V( θ τ, i.e., θ ˆ = arg [in V( θ τ ] θ (3 where the criterion function V( θ τ is given as [8] V( θ τ = det ε[n θ, τ] ε[n θ, τ] (4 n= he iniu value of the criterion function under a given value of τ ( is defined as wi τ wi( τ = inv( θ τ θ (5 he variation of wi( τ for a range of τ values is observed. he value of τ, which gives the iniu value of wi( τ, is taken as the estiated value of the tie-dela in recording the selected output i relative to the reference signal, i.e., τ i = arg [in wi( τ ] τ (6 Finall, the shifted reference signal, given b Eq.(6, with τ defined b Eq.(6, is snchronous with the output. i Alternatel, other output easureents can be snchronized with the reference signal. 3. uerical exaple A 4-stor -ba b -ba shear building under abient wind loading at each floor in the -direction is considered to deonstrate the application of the proposed algorith. his is one of the cases in the benchark proble proposed b the ASCE ask Group on structural health onitoring [] and is illustrated in Fig. 4. More inforation on the benchark proble can be obtained fro the web site: w 4 (τ x τ x -3 (sec Fig. 4 he benchark building [] Fig. 5 (a Variation of error w4( τ of the st floor with τ 4.8

8 It is assued that wireless sensing units at the first, second and third floors have tie-delas of.6sec,.4sec and.3sec respectivel, in recording the floor acceleration response relative to the acceleration response of the fourth floor. hese are generated b the MALAB progra provided b the ASCE ask Group. he sapling interval of the output data is.sec. he above algorith is applied to these asnchronous data. Acceleration response signal of the fourth floor is chosen as the reference signal. Figs. 5(a-5(c illustrate the variations of w4 ( τ, w 4 ( τ and w34 ( τ for a range of τ values. Fro the values of τ, which produce the iniu values of wi4 ( τ (i =,, 3, the tie-delas in recording acceleration response data relative to the reference signal are evaluated accuratel w 4 (τ x w 34 (τ x τ x -3 (sec τ x -3 (sec Fig. 5(b Variation of error ( τ of the Fig. 5 (c Variation of error ( τ of the w4 nd floor with τ the 3rd floor with τ w COCLUSIOS In this paper, two tie snchronization algoriths are proposed to treat asnchronous data recorded b wireless sensing units for the purpose of accurate structural paraeter identification and daage detection. he first algorith can be used when the input to a structure is easured. Output data are snchronized with the input based on the ARX odels for the input-output pairs. he algorith is siple and its validit has been test b several nuerical exaples of siulation data and practicall recorded seisic response data of buildings. ie-delas in recording output easureents relative to the easured ground input can be accuratel evaluated as long as the nuerical error due to interpolation of signal is sall. he second algorith can snchronize recorded outputs fro structures under abient excitation. It is based on the ARMAV odel for a pair of output data, which requires ore nuerical effort copared to the first algorith. Siulation data of the benchark building proposed b the ASCE ask Group on structural health onitoring show that the second algorith can accuratel snchronize output easureents. ACKOWLEDGEME his research is funded b the ational Science Foundation through Grant o. CMS-84. We greatl appreciate their continued support. REFERECES. E.G. Straser and A. S. Kireidian, Modular, Wireless Daage Monitoring Sste for Structures. Report o. 8, John A. Blue Earthquake Engineering Center, Departent of Civil and Environental Engineering, Stanford Universit, Stanford, CA J. P. Lnch, K. H. Law, A. S. Kireidian, E. Carrer,. W. Kenn, A. Partridge and A. Sundararaan, Validation Of A Wireless Modular Monitoring Sste for Structures. Proceedings of Sart Structures and Materials, SPIE, San Diego CA,.

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