th
Source: SPIE’s 10 Annual International Symposium on Smart Structures and Materials, San Diego, CA, USA, March 2-6, 2003.
Time synchronization algorithms for wireless monitoring system Y. Lei*, A. S. Kiremidjian, K. K. Nair, J. P. Lynch and K. H. Law Department of Civil and Environmental Engineering, Stanford University, CA, USA 94305 ABSTRACT Wireless health monitoring schemes are innovative techniques, which effectively remove the disadvantages associated with current wire-based sensing systems, i.e., high installation and upkeep costs. However, recorded data sets may have relative time-delays due to the blockage of sensors or inherent internal clock errors. In this paper, two algorithms are proposed for the synchronization of the recorded asynchronous data measured from sensing units of a wireless monitoring system. In the first algorithm, the input signal to a structure is measured. Time-delay between an output measurement and the input is identified based on the minimization of errors of the ARX (auto-regressive model with exogenous input) models for the input-output pair recordings. The second algorithm is applicable when a structure is subject to ambient excitation and only output measurements are available. ARMAV (auto-regressive moving average vector) models are constructed from two output signals and the time-delay between them is evaluated based on the minimization of errors of the ARMAV models. The proposed algorithms are verified by simulation data and recorded seismic response data from multi-story buildings. Keywords: Wireless sensors, synchronization, time series analysis, system identification, structural health monitoring
1. INTRODUCTION There exists a clear need to monitor the performance of civil structures over their operational lives. Current wire-based sensing systems suffer from high installation and upkeep costs, which limits widespread adoption. In response to the technological and economic limitations of present commercial monitoring systems, a novel wireless module monitoring system was proposed for the health monitoring of civil structures [1-2]. It provides a high-performance yet low-cost data acquisition technique for structural health monitoring. When wireless sensing units record measurements independently, recorded data sets may exist with relative time-delays due to blockage of sensors or inherent internal clock errors. The relative time-delays may be smaller than the data-sampling interval. Thus, it is necessary to perform time synchronization among these recordings for the purpose of accurate structural identification and damage detection. However, little attention has been given to these topics associated with the innovative wireless monitoring system [3-4]. In this paper, two algorithms are proposed to synchronize measured data with relative time-delays. In the first algorithm, input to a structure is measured. Each output signal is synchronized with the input signal, which is chosen as the reference signal. The second algorithm treats asynchronous output measurements from a structure under ambient excitation. In the case of ambient excitation, one of the output signals is taken as the reference signal since the input ambient vibration cannot be measured. Several numerical examples of simulation data and recorded seismic response data from multi-story buildings are used to demonstrate and verify the proposed algorithms.
2. TIME SYNCHRONIZATION ALGORITHM FOR INPUT-OUTPUT PAIR RECORDINGS When the excitation to a structure is measured, the excitation signal is selected as a reference signal. Wireless sensing units have time-delays in recording the output signals relative to the reference signal resulting in asynchronous data. The following asynchronous data of the input and output signals are obtained.
x ( t 1 ), x ( t 2 ) , ...., x ( t N ) y1 ( t 1 + τ1 ), y1 ( t 2 + τ1 ) , ...., y1 ( t N + τ1 ) y 2 ( t 1 + τ 2 ), y 2 ( t 2 + τ 2 ) , ...., y 2 ( t N + τ 2 ) o y M ( t 1 + τ M ), y M ( t 2 + τ M ) , ...., y M ( t N + τ M )
*
[email protected]; phone (650) 725 0360; fax (650) 725 9755
(1)
where, x is the excitation, y j (j = 1, 2,…, M) is the output data, M is the number of sensing units in the recording output
signals, N is the number of points in the records, τ j is the unknown value of time-delay in recording the output y j relative to the input x. The task is to identify the unknown τ j value so that the output data can be synchronized with the input. 2.1 Time synchronization algorithm
First, the values of the input signal at shifted time instants can be evaluated by the spline interpolation technique to yield the following set of input data x ( t1 + τ 0 ), x ( t 2 + τ 0 ) , ....., x ( t N + τ 0 )
(2)
where τ 0 is the value of time shift. With different values of τ 0 , a set of shifted input signals is obtained. Second, each shifted input signal is paired with one of the output signal y j to construct an ARX model [4-6] as y j (t m + τ j ) +
na ∑ a k y k (t m k =1
+ τ j − k∆ ) =
nb ∑ b k x(t m k =0
+ τ 0 − k∆ ) + ε j ( t m , τ 0 )
(3)
where ∆ is the sampling interval, na and a k are the order and coefficients of the AR model, respectively; nb and b k are the order and coefficients of the exogenous input, respectively; and ε j ( t m , τ 0 ) is the prediction error of the model with a given value of τ 0 . The vectors β and θ are defined as follows
β( t m , τ 0 ) = [ − y j ( t m + τ j − ∆ ),..., − y j ( t m + τ j − na∆ ), x ( t m + τ 0 − ∆ ),..., x ( t m + τ 0 − nb∆ )]T
(4)
θ = [a 1 , a 2 , ..., a na , b 0 , b1 , b 2 , ..., b nb ]T
(5)
where superscript T denotes a transpose. Eq.(3) can be rewritten as y j (t m + τ j ) = βT (t m , τ0 ) θ + ε j (t m , τ0 )
(6)
For a given value of τ 0 , the total error V(θ τ 0 ) , defined as the sum of the square of identification errors at all measurement time instants, is given by V(θ τ 0 ) =
N ε 2j ( t m , τ 0 ) ∑ m = nn +1
=
N ∑ m = nn +1
[ y j ( t m + τ j ) − β T ( t m , τ 0 ) θ] 2
(7)
where max(na, nb) nn = max(na, nb) + fix {-min(τ j , τ 0 ) /∆}
when τ j ≥ 0 when τ j < 0
(8)
in which fix{-min(τ j , τ 0 ) /∆} rounds off the element -min(τ j , τ 0 ) /∆ to the nearest integer. The reason that the sum in Eq.(8) starts from nn+1 is because of the time-delay value of τ 0 and the order of the ARX (na, nb) model. The coefficients of the ARX model θˆ is determined by minimizing the total error under a given value of τ 0 in Eq.(7), i.e., dV( θ τ 0 ) / dθ ˆ = 0 θ=θ Eq. (9) gives the following solution for θˆ
(9)
θˆ =
N N −1 T ∑ [β( t m , τ 0 ) β ( t m , τ 0 ) ] ∑ m = nn +1 m = nn +1
[β( t m , τ 0 ) y j ( t m + τ j )]
(10)
The minimum value of V(θ τ 0 ) is denoted by
e j ( τ 0 ) = min V( θ τ 0 )
(11)
θ
where ‘min’ gives the minimum value of the function. The variation of e j ( τ 0 ) for a range of τ 0 values is observed. The τ 0 value that gives the minimum value of e j ( τ 0 ) is taken as the estimated value of the time-delay in recording the output y j relative to the input signal x, i.e.,
τ j = arg[min e j ( τ 0 )]
(12)
τ0
where ‘arg’ gives the argument of the function. Then, the corresponding shifted input signal, given by Eq.(2), with the value of τ 0 , defined by Eq.(12), is synchronous with the output signal y j . Alternately, all output signals can be synchronized with the input signal using the above algorithm. Synchronous input and output data are obtained.
2.2 Numerical example 2.2.1 A 3-story shear building under a sweep sine ground excitation
In the first case study, the 3-story shear building described by Clough and Penzien [7] is used herein. A sweep sine excitation is applied to the base of the structure. The ground excitation DxD g is
DxD g ( t ) = sin[0.3π(2 + t)t]
(13)
The excitation has constant amplitude of 1 in/s2 with a linearly varying frequency of 0.3 to 6.3 Hz over 40 seconds. Wireless sensing units at the first, second and third floors have time-delays of 0.002sec, 0.004sec and 0.007sec respectively, in recording the floor acceleration response relative to the input signal. These asynchronous data are generated by numerical simulation. The sampling time is equal to 0.01sec. Each acceleration response data are paired with the shifted input to apply the proposed algorithm. Based on the criteria of optimal model order for ARX models [4], the orders na and nb are set to 8. Figs. 1(a)-1(c) illustrate the variations of e1(τ0), e2(τ0) and e3(τ0) for a range of τ0 values. From these figures, time-delays in recording the three acceleration response data (relative to ground excitation signal) are accurately evaluated by the minimizing arguments of ej(τ0) (j = 1, 2, 3) as described by Eq. (12). These values are found to be identical to the time-delays introduced in the response signals pointing to the accuracy of the synchronization. 10-5 10-6 10-7
e2(τ 0)
e1(τ 0)
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τ0
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x10-3
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(sec)
Fig. 1(a) Variation of error e1 ( τ 0 ) of the 1st floor response with τ 0
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τ 0x10-3 (sec)
Fig. 1(b) Variation of error e 2 ( τ 0 ) of the 2nd floor response with τ 0
100
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e1(τ 0)
e3(τ 0)
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Fig. 1(c) Variation of error e 3 ( τ 0 ) of the 3rd floor response with τ 0
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x10-3 (sec)
Fig. 2(a) Variation of error e1 ( τ 0 ) of the 1st floor response with τ 0
2.2.2 A 3-story shear building under EL Centro earthquake excitation
In the second case study, the time synchronization algorithm is applied to the same 3-story building under the 1940 El Centro N-S earthquake loading with PGA=0.3g. The recorded floor acceleration response at the first, second and third floors are assumed to have time-delays of 0.004sec, 0.009sec and 0.012sec respectively, relative to the input signal. These are generated numerically with sampling interval equal to 0.02sec. Figs. 2(a)-2(c) illustrate the variations of e1 ( τ 0 ) , e 2 ( τ 0 ) and e 3 ( τ 0 ) for a range of τ0 values. From the minimum value of e j ( τ 0 ) (j = 1, 2, 3) shown in these figures, it can be shown that the time-delays in recording the acceleration response data relative to the input signal are evaluated accurately using Eq.(12).
100
100
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10-2
e3(τ 0)
e2(τ 0)
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Fig. 2(b) Variation of error e 2 ( τ 0 ) of the 2nd floor response with τ 0
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Fig. 2(c) Variation of error e 3 ( τ 0 ) of the 3rd floor response with τ 0
2.2.3 Recorded accelerograms of a 18-story commercial building subject to Loma Prieta earthquake
The time synchronization algorithm is also demonstrated for the strong-motion accelerograms recorded in an 18-story commercial building in San Francisco subject to the 1989 Loma Prieta earthquake. The data were provided by the California Geology Survey’s (CGS) Strong Motion Instrumentation Program (SMIP) (formerly Division of Mines and Geology, California Department of Conservation, ftp://ftp.consrv.ca.gov/pub/dmg/csmip/). The basement of the building is excited by one vertical and two horizontal ground motions. Under the condition that the three components of excitations recorded at the basement are synchronized, the above time-synchronization algorithm can be extended to a multi-input, single-output case. The ARX model for input-output data in Eq.(3) is rewritten as
na
nb1
nb 2
k=0 nb 3 + ∑ b 3k x 3 ( t m k=0
k=0
y j ( t m + τ j ) + ∑ a k y k ( t m + τ j − k∆ ) = ∑ b1k x1 ( t m + τ 0 − k∆ ) + ∑ b 2 k x 2 ( t m + τ 0 − k∆ ) k =1
(14)
+ τ 0 − k∆ ) + ε j ( t m , τ 0 )
where nb1 , b1k , nb 2 , b 2 k and nb 3 , b 3k are orders and coefficients of the first, second and third exogenous input, respectively. Analogously, it can be shown that the relative time-delay value of τ j can still be evaluated by minimizing e j ( τ 0 ) as described by Eq.(12). To get asynchronous acceleration response data sets of the building, recorded accelerograms are artificially shifted to produce data at shifted time instants by spline interpolation technique. The proposed time synchronization algorithm is then applied to treat the asynchronous data sets. In this numerical example, one recorded horizontal component of accelerograms at the 7th floor and another recorded horizontal component of accelerograms at the 12th floor are shifted so they have time-delays of 0.009sec and 0.014sec relative to the basement excitations, respectively. The orders of the ARX model are set as na = 12, nb1 = nb 2 = nb 3 = 12 . Figs. 3(a)-3(b)
0.730
0.722
0.725
0.718
e12 (τ 0)
e7(τ 0)
illustrate the variations of e 7 ( τ 0 ) and e12 ( τ 0 ) for a range of τ 0 values. From the minimum value of e j ( τ 0 ) (j = 1, 2, 3), the time-delays in recording acceleration response data relative to the input signal are evaluated as τ 7 = 0.010 sec and τ12 = 0.015 sec . The error in estimating the time-delay value is due to the numerical error in generating the two shifted output and input data by the spline interpolation technique. The original data sampling interval is 0.02sec. If this sampling interval were reduced, the error would also decrease. The order of this error, however, may not be significant for subsequent structural analysis computations.
0.720 0.715
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0.706 0
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Fig. 3(a) Variation of error e 7 ( τ 0 ) of the 7th floor with τ0
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τ 0x10-3 (sec)
Fig. 3(b) Variation of error e12 ( τ 0 ) of the 12th floor with τ0
3. TIME SYNCHRONIZATION ALGORITHM FOR OUTPUT RECORDINGS When a structure is subject to ambient excitation, the inputs to the structures cannot be measured frequently. Typically only output signals are recorded by the wireless sensing units instrumented at different locations of the structure. One of the measured output signal y j is chosen as the reference signal. The remaining measured acceleration responses have time-delays in recording data relative to the reference signal. Thus, the following asynchronous output data are measured
y1 ( t1 + τ1 j ), y1 ( t 2 + τ1 j ) , ...., y1 ( t N + τ1 j ) y j ( t 1 ),
y 2 ( t 2 ) , ...., y j ( t N )
(15)
o y M ( t 1 + τ Mj ), y M ( t 2 + τ Mj ) , ...., y M ( t N + τ Mj ) where τ ij is the unknown time-delay in recording the output y i relative to the reference signal y j .
For structures under ambient excitation, auto-regressive moving average vector (ARMAV) models have been applied for system identification of structures [6, 8-10]. These models only use time series of output signals, without the requirement of excitation measurement. The excitation is assumed to be a stationary Gaussian white noise. A time synchronization algorithm for output signals based on the ARMAV models is proposed. 3.1 Time synchronization algorithm
The values of the reference signal at shifted time instants are also evaluated by the spline interpolation to yield the following data y j ( t 1 + τ 0 ), y j ( t 2 + τ 0 ) , ....., y j ( t N + τ 0 )
(16)
where τ 0 is the value of time shift. With different values of τ 0 , a set of shifted reference signals is obtained. An ARMAV model can be constructed from a shifted reference signal and another output y i by p
q
k =1
k =0
y[ n ] = ∑ a k y[ n − k ] + u[ n ] + ∑ b k u[ n − k ] ; 1 ≤ n ≤ N
(17)
where p and q are the orders of the AR (auto-regressive) and MA (moving average) components respectively, ak and bk are 2 × 2
{
matrices of the AR and MA coefficients and N is the number of points in the records.
}
y[ n ] = y j [n + τ 0 ], y i [ n + τ ij ] T and u[ n ] = {u[1, n ], u[2, n ]}T are vectors of stationary zero-mean Gaussian white noise
processes. The same ARMAV model in the state space can be rewritten as y[ n ] = A y[ n − 1] + B u[ n ]
(18)
y[ n ] = C y[ n ]
where y[n ] and u[n ] are vectors in the state space of dimension 2p. They are defined as
{
}
y[ n ] = y j [ n + τ 0 ], y i [ n + τ ij ],....., y j [n − p + 1], y i [n − p + 1] T
(19)
u[ n ] = {u[1, n ], u[2, n ],....., u[1, n − p + 1], u[2, n − p + 1]}T
A and B are 2p × 2p dimensional matrices containing the coefficients of AR and MA, respectively, C is the observation matrix [8]. The matrices C and A are expressed by
C = [I
0
...
0
a 1 I 0] ; A = o 0
a2 0
m m
a p −1 0
m 0
m
I
ap 0 0
(20)
where I is the identity matrix. Parameters of the ARMAV models are estimated by the prediction error method [8-9]. The vector θ is defined as
θ = [a1 , a 2 ,...a p , b 0 , b1 , b 2 ,..., b q ]T
(21)
The prediction error vector ε[n θ,τ 0 ] of the ARMAV model under a given value of τ 0 can be expressed as ε[ n θ,τ 0 ] = y[n ] − yˆ [n ]
(22)
where y[n ] is the vector of actual measured output values and yˆ [n ] denotes the predicted value by the ARMAV model [10]. With a given value of τ , θˆ can be obtained as the minimum point of a criterion function V( θ τ ) , i.e., 0
0
θˆ = arg [min V(θ τ 0 )]
(23)
θ
where the criterion function V(θ τ 0 ) is given as [8]
1 N V(θ τ 0 ) = det ∑ ε[ n θ,τ 0 ] ε[ n θ,τ 0 ]T N n =1
(24)
The minimum value of the criterion function under a given value of τ0 w ij ( τ 0 ) is defined as w ij ( τ 0 ) = min V(θ τ 0 )
(25)
θ
The variation of w ij ( τ 0 ) for a range of τ0 values is observed. The value of τ 0 , which gives the minimum value of w ij ( τ 0 ) , is taken as the estimated value of the time-delay in recording the selected output y i relative to the reference signal y j , i.e., τ ij = arg [min w ij ( τ 0 )]
(26)
τ0
Finally, the shifted reference signal, given by Eq.(16), with τ 0 defined by Eq.(26), is synchronous with the output y i . Alternately, other output measurements can be synchronized with the reference signal. 3.2 Numerical example
A 4-story 2-bay by 2-bay shear building under ambient wind loading at each floor in the y-direction is considered to demonstrate the application of the proposed algorithm. This is one of the cases in the benchmark problem proposed by the ASCE Task Group on structural health monitoring [11] and is illustrated in Fig. 4. More information on the benchmark problem can be obtained from the web site: http://wusceel.cive.wustl.edu/asce.shm/benchmarks.htm. 5.8
w14(τ 0)x10-8
5.6 5.4 5.2 5.0 4.8 0
Fig. 4 The benchmark building [11]
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Fig. 5 (a) Variation of error w14 ( τ 0 ) of the 1st floor with τ 0
It is assumed that wireless sensing units at the first, second and third floors have time-delays of 0.006sec, 0.004sec and 0.003sec respectively, in recording the floor acceleration response relative to the acceleration response of the fourth floor. These are generated by the MATLAB program provided by the ASCE Task Group. The sampling interval of the output data is 0.001sec. The above algorithm is applied to these asynchronous data. Acceleration response signal of the fourth floor is chosen as the reference signal. Figs. 5(a)-5(c) illustrate the variations of w14 ( τ 0 ) , w 24 ( τ 0 ) and w 34 ( τ 0 ) for a range of τ 0 values. From the values of τ 0 , which produce the minimum values of w i4 ( τ 0 ) (i = 1, 2, 3), the time-delays in recording acceleration response data relative to the reference signal are evaluated accurately. 9.0 8.4
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w34 (τ 0)x10-8
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Fig. 5(b) Variation of error w 24 ( τ 0 ) of the 2nd floor with τ 0
Fig. 5 (c) Variation of error w 34 ( τ 0 ) of the the 3rd floor with τ 0
4. CONCLUSIONS In this paper, two time synchronization algorithms are proposed to treat asynchronous data recorded by wireless sensing units for the purpose of accurate structural parameter identification and damage detection. The first algorithm can be used when the input to a structure is measured. Output data are synchronized with the input based on the ARX models for the input-output pairs. The algorithm is simple and its validity has been test by several numerical examples of simulation data and practically recorded seismic response data of buildings. Time-delays in recording output measurements relative to the measured ground input can be accurately evaluated as long as the numerical error due to interpolation of signal is small. The second algorithm can synchronize recorded outputs from structures under ambient excitation. It is based on the ARMAV model for a pair of output data, which requires more numerical effort compared to the first algorithm. Simulation data of the benchmark building proposed by the ASCE Task Group on structural health monitoring show that the second algorithm can accurately synchronize output measurements.
ACKNOWLEDGEMENT This research is funded by the National Science Foundation through Grant No. CMS-0121841. We greatly appreciate their continued support.
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