Time Synchronization Algorithms for Wireless Monitoring System

7/27/2019 Time Synchronization Algorithms for Wireless Monitoring System

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* [email protected]; phone (650) 725 0360; fax (650) 725 9755

Source: SPIEs 10t

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 recordedseismic 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 performtime 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.

)t(y....,,)t(y),t(y

)t(y....,,)t(y),t(y

)t(y....,,)t(y),t(y

)t(x....,,)t(x),t(x

MNMM2MM1M

2N2222212

1N1121111

N21

+++

+++

+++

o

(1)


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where, x is the excitation, jy (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, j is the unknown value of time-delay in recording the output jy

relative to the input x. The task is to identify the unknown 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 yieldthe following set of input data

)t(x.....,,)t(x),t(x 0N0201 +++ (2)

where 0 is the value of time shift. With different values of 0 , a set of shifted input signals is obtained.

Second, each shifted input signal is paired with one of the output signal jy to construct an ARX model [4-6] as

),t()kt(xb)kt(ya)t(y 0mjnb

0k0mk

na

1kjmkkjmj ++=+++

=

=(3)

where is the sampling interval, na and ka are the order and coefficients of the AR model, respectively; nb and kb

are the order and coefficients of the exogenous input, respectively; and ),t( 0mj is the prediction error of the model

with a given value of 0 . The vectors and are defined as follows

T0m0mjmjjmj0m )]nbt(x),...,t(x),nat(y),...,t(y[),t( ++++= (4)

Tnb210na21 ]b...,,b,b,b,a...,,a,a[= (5)

where superscript T denotes a transpose. Eq.(3) can be rewritten as

),t(),t()t(y 0mj0mT

jmj +=+ (6)

For a given value of 0 , the total error )(V 0 , defined as the sum of the square of identification errors at all

measurement time instants, is given by

2N

1nnm0m

Tjmj

N

1nnm0m

2j0 ]),t()t(y[),t()(V

+=

+=+== (7)

where


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])t(y),t([]),t(),t([N

1nnmjmj0m

1N

1nnm0m

T0m

+=

+=+= (10)

The minimum value of )(V 0 is denoted by

)(Vmin)(e 00j = (11)

where min gives the minimum value of the function. The variation of )(e 0j for a range of 0 values is observed.

The 0 value that gives the minimum value of )(e 0j is taken as the estimated value of the time-delay in recording the

output jy relative to the input signal x, i.e.,

)](eminarg[ 0j0

j =

(12)

where arg gives the argument of the function. Then, the corresponding shifted input signal, given by Eq.(2), with the

value of 0 , defined by Eq.(12), is synchronous with the output signal jy . 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 gxDD is

t)t](23.0sin[)t(xg +=DD (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 criteriaof 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(0), e2(0) and e3(0) for a range of 0 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(0) (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.

Fig. 1(a) Variation of error )(e 01 of the Fig. 1(b) Variation of error )(e 02 of the

1st floor response with 0 2nd floor response with 0

0 1 2 3 4 5 6 7 8 9 10

0x10

-3 (sec)

10-7

10-6

10-5

e

1(0

)

0 1 2 3 4 5 6 7 8 9 10

0x10

-3 (sec)

10-9

10-8

10-7

10-6

e

2(0

)


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Fig. 1(c) Variation of error )(e 03 of the Fig. 2(a) Variation of error )(e 01 of the

3rd floor response with 0 1st floor response with 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 ElCentro 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

)(e 01 , )(e 02 and )(e 03 for a range of 0 values. From the minimum value of )(e 0j (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).

Fig. 2(b) Variation of error )(e 02 of the Fig. 2(c) Variation of error )(e 03 of the

2nd floor response with 0 3rd floor response with 0

2.2.3 Recorded accelerograms of a 18-story commercial building subject to Loma Prieta earthquakeThe 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 Surveys (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 thebuilding 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

0 1 2 3 4 5 6 7 8 9 10

0x10-3 (sec)

10-10

10-9

10-8

10-7

10-6

e3

(0

)

0 2 4 6 8 10 12 14 16 18 20

0x10-3 (sec)

10-4

10-3

10-2

10-1

100

e1

(0)

0 2 4 6 8 10 12 14 16 18 20

0x10-3 (sec)

10-5

10-4

10-310-2

10-1

100

e2

(0

)

0 2 4 6 8 10 12 14 16 18 20

0x10

-3 (sec)

10-4

10-3

10-2

10-1

100

e3

(0

)


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),t()kt(xb

)kt(xb)kt(xb)kt(ya)t(y

0mj

3nb

0k0m3k3

2nb

0k0m2k2

1nb

0k0m1k1

na

1kjmkkjmj

+++

+++=+++

=

=

=

=(14)

where k11 b,nb , k22 b,nb and k33 b,nb 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 j can still be evaluated by minimizing

)(e 0j 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 ofaccelerograms 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 12nbnbnb,12na 321 ==== . Figs. 3(a)-3(b)

illustrate the variations of )(e 07 and )(e 012 for a range of 0 values. From the minimum value of )(e 0j (j = 1, 2,

3), the time-delays in recording acceleration response data relative to the input signal are evaluated as sec010.07 =

and sec015.012

= . 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.

Fig. 3(a) Variation of error )(e 07 of the Fig. 3(b) Variation of error )(e 012 of the

7th floor with 0 12th floor with 0

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 jy is chosen as the reference signal. The remaining measured acceleration responses havetime-delays in recording data relative to the reference signal. Thus, the following asynchronous output data aremeasured

0 2 4 6 8 10 12 14 16 18 200x10

-3 (sec)

0.710

0.715

0.720

0.725

0.730

e7

(0

)

0 2 4 6 8 10 12 14 16 18 20

0x10-3 (sec)

0.706

0.710

0.714

0.718

0.722

e12

(0

)


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)t(y....,,)t(y),t(y

)t(y....,,)t(y),t(y

)t(y....,,)t(y),t(y

MjNMMj2MMj1M

Nj221j

j1N1j121j111

+++

+++

o(15)

where ij is the unknown time-delay in recording the output iy relative to the reference signal jy .

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 timesynchronization 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

)t(y.....,,)t(y),t(y 0Nj02j01j +++ (16)

where 0 is the value of time shift. With different values of 0 , a set of shifted reference signals is obtained.

An ARMAV model can be constructed from a shifted reference signal and another output iy by

Nn1;]kn[]n[]kn[]n[p

1k

q

0kkk ++=

=

=ubuyay (17)

where p and q are the orders of the AR (auto-regressive) and MA (moving average) components respectively, akand bk

are 22 matrices of the AR and MA coefficients and N is the number of points in the records.

{ }Tiji0j ]n[y],n[y]n[ ++=y and { }T]n,2[u],n,1[u]n[ =u are vectors of stationary zero-mean Gaussian white noise

processes. The same ARMAV model in the state space can be rewritten as

]n[]n[

]n[]1n[]n[

yCy

uByAy

=

+=(18)

where ]n[y and ]n[u are vectors in the state space of dimension 2p. They are defined as

{ }

{ }T

Tijiji0j

]1pn,2[u],1pn,1[u],.....,n,2[u],n,1[u]n[

]1pn[y],1pn[y],.....,n[y],n[y]n[

++=

++++=

u

y(19)

A and B are 2p2p dimensional matrices containing the coefficients of AR and MA, respectively, C is the observation

matrix [8]. The matrices C and A are expressed by

==

000

000;]00...0[

p1p21

I

I

aaaa

AIC

m

mo

m

m

(20)

where I is the identity matrix. Parameters of the ARMAV models are estimated by the prediction error method [8-9].

The vectoris definedas


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Tq210p21 ],,...,,,,...,[ bbbbaaa= (21)

The prediction error vector ],n[ 0 of the ARMAV model under a given value of 0 can be expressed as

]n[]n[],n[ 0 yy = (22)

where ]n[y is the vector of actual measured output values and ]n[y denotes the predicted value by the ARMAV model

[10]. With a given value of 0 , can be obtained as the minimum point of a criterion function )(V 0 , i.e.,

)](Vmin[arg 0=

(23)

where the criterion function )(V 0 is given as [8]

= =

T0

N

1n00 ],n[],n[

N

1det)(V (24)

The minimum value of the criterion function under a given value of0 )(w 0ij is defined as

)(Vmin)(w 00ij =

(25)

The variation of )(w 0ij for a range of 0 values is observed. The value of 0 , which gives the minimum value of

)(w 0ij , is taken as the estimated value of the time-delay in recording the selected output iy relative to the reference

signal jy , i.e.,

)](wmin[arg 0ij0

ij =

(26)

Finally, the shifted reference signal, given by Eq.(16), with 0 defined by Eq.(26), is synchronous with the output iy .

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 todemonstrate 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.

Fig. 4 The benchmark building [11] Fig. 5 (a) Variation of error )(w 014 of the

1st floor with 0

0 1 2 3 4 5 6 7 8 9 10

0x10-3 (sec)

4.8

5.0

5.2

5.4

5.6

5.8

w14(

0)x10-8


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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 theoutput 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 )(w 014 , )(w 024 and

)(w 034 for a range of 0 values. From the values of 0 , which produce the minimum values of )(w 04i (i = 1, 2,3), the time-delays in recording acceleration response data relative to the reference signal are evaluated accurately.

Fig. 5(b) Variation of error )(w 024 of the Fig. 5 (c) Variation of error )(w 034 of the

2nd floor with 0 the 3rd floor with 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 modelsfor 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 outputmeasurements 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 structuralhealth 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 appreciatetheir continued support.

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0 1 2 3 4 5 6 7 8 9 10

0x10-3 (sec)

7.0

7.3

7.6

7.9

8.2

w24

(0

)x10-8

0 1 2 3 4 5 6 7 8 9 10

0x10

-3 (sec)

6.0

6.6

7.2

7.8

8.4

9.0

w34

(0

)x10-8


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Time Synchronization Algorithms for Wireless Monitoring System

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