International Journal of Innovative Computing, Information and Control Volume 5, Number 5, May 2009
c ICIC International 2008 ISSN 1349-4198 pp. 1285–1295
ROBUST CHANGE DETECTION PROCEDURE WITH APPLICATION IN STRUCTURES MONITORING SUBJECT TO SEISMIC MOTIONS Theodor D. Popescu Research Division National Institute for Research and Development in Informatics 8-10 Averescu Avenue, 011455 Bucharest, Romania
[email protected]
Abstract. This paper puts forward a robust change detection procedure able to analyze the transient seismic motion and the vibration structure response, as well as to determine the changes in the amplitude and frequency content of the structure. This technique makes use of an impulse invariant transformation that will enable us to obtain an equivalent model of the vibration structure, represented by a parallel-form realization of single-degree-of-freedom oscillators that correspond to different vibration modes while a stationary independent and identically distributed signal stands for input. The method was applied to change detection in dynamic characteristics of a structure, a multi-story concrete building subject to a strong seismic motion. Keywords: AR processes, Deconvolution, Detection, Filtering, Frequency response, Parameter estimation, Seismic signal processing.
1. Introduction. The problem of change detection and isolation has received great attention during the last two decades, in the research context and in the applications on real systems, [1]. The change detection schemes, proposed for dynamical systems with various change modes, make use of identification and estimation methods. In this case the problem is to construct an effective detection index, sensitive to the changes in the models, identified for different input-output data sets. Such an index may be Kullback Discrimination Information (KDI), known as a distorsion measure to compare two probability density functions, [2]. Using the parameter estimation approach, the change detection problem can be stated as follows, [3]: Let us consider two data sets In and Ic available. The data set In is a nominal data set obtained under a non change condition, and the data set Ic arises from a data set with a possible change in the system. The data are assumed to originate from a system given by: y(t) = G(θ(t))u(t) + v(t)
(1)
where
θn (t) , for In data set (2) θc (t) , for Ic data set G is a transfer function, discrete or continuous, and v(t) is a term describing the disturbances and the noise. θ(t) =
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The problem can be now stated as deciding which of the following two hypothesis H0 and H1 defined below, is true. H0 : θn = θc H1 : θn 6= θc
(3)
The objective of change detection task is to decide whether the model which was identified on In data set still adequately represents the new data set Ic , obtained during a new inspection of the data. Particular attention is paid to the reduction of the false change instants or false alarms during the data analysis, [4]. In the case of diagnosis or isolation of a change, the problem is to discriminate between the changes in the model parameters, based on the parameter change vector θˆ = θˆn −θˆc and other information (the covariance matrix could be used) obtained during the parameter estimation. Almost all model-based change detection schemes use the assumption that the system itself can be accurately described by a linear finite dimensional model. This leads to the conclusion that arbitrary small changes can be detected for sufficient data records. In practice, there are several problems associated with the usual test procedures: 1. When the system is more complex than the finite dimensional model structure, the parameter estimate θ will still converge, but to a value that now depends upon the experimental conditions. The calculations of the covariance matrix will be much more involved. If the experimental conditions for the experiments n and c are different, the difference parameter vector has no longer zero mean value and the common tests will fail, since the algorithms can not separate changes due to variations in experimental conditions and the real changes in the investigated system. 2. If θ is a black-box parameter vector (the coefficients of the model polynomial are the most common choice), it may be difficult to get insight into how variations in the behavior of the system reflect in variations in θ. Very different choices of θ can give good approximations of the same system, which cause obvious problems in change detection schemes. All the problems mentioned above point out the need of some robust change detection schemes to work well and to separate the changes in the experimental conditions from the real changes in the system, especially for systems with arbitrary and non-stationary known or unknown inputs. Examples of such systems are vibration structures: buildings subject to earthquakes, off-shore platforms subject to wave action, bridges subject to wind or earthquakes, mechanical systems subject to fluid interaction. An original approach for change detection and diagnosis in modal characteristics of nonstationary signals for unmeasured natural excitation and application of this technique in vibration monitoring is presented by Basseville and colleagues, [5]. This is based upon the use of instrumental statistics and a statistical approach for detection. A practical application of this approach is given in [6]. Another general approach for measured inputs, suggested by Wahlberg in [7], makes use of some tests developed in frequency domain; this approach is especially recommended for diagnosis purposes. The major part of all change detection schemes seems to be more robust and insightful to work in frequency domain. An application of this approach for a structural system is presented in [8]. Other
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techniques for robust change detection and isolation, using algebraic tools, are given in [9]. The method presented in this paper makes use of an impulse invariant transformation to obtain the equivalent model of the effective vibration structure, represented by a parallel-form realization of single-degree-of-freedom oscillators, corresponding to different vibration modes, having as input a stationary independent and identically distributed sequence. This transformation uses the autoregressive (AR) filters resulted for the quasistationary data blocks of the original non-stationary input, and determined with a segmentation procedure based on Akaike Information Criterion (AIC). In a new step, the same procedure will be used to segment the processed output, in quasi-stationary data blocks. Each transition from one data block to another marks a change in the dynamic properties of the investigated system. For each quasi-stationary data segment, the time and frequency characteristics for diagnosis-making can be evaluated. This method, compared with other techniques operating in the same framework, is robust to the change detection in systems with arbitrary and non-stationary inputs and to the undermodelling errors, [10]. This is proved by Monte-Carlo simulation. Also, it can easily extended to multi-input, multi-output systems. 2. Combined Earthquake - Structure Model. The equation of motion for an N degree-of-freedom system, subject to an earthquake ground motion, outlined as a linear planar, time invariant model with viscous damping can be written as: M x¨(t) + C x(t) ˙ + Kx(t) = −M · [1]¨ xg (t) (4) in which M, C and K are, respectively, the N × N mass, viscous damping and stiffness matrices. Vector x(t) = [x1 (t), . . . , xN (t)]T denotes the relative displacements, the dots mark the derivative with respect to time, [1] the vector [1, 1, . . . , 1]T and x¨g (t) defines the base acceleration. Equation (4) represents a system of N coupled equations which can be uncoupled provided that the viscous damping is proportional (for example Rayleigh assumption C = aM + bK). Let us have x = Φ · y where y is the vector of the modal displacements and Φ is the modal matrix whose column vectors are the individual mode shapes Φr (r = 1, Nm (Nm ≤ N)), i.e. ΦTr = [Φ1r , Φ2r , . . . , ΦN r ]. By multiplying equation (4) by ΦT , we obtain a set of N × Nm uncoupled equations as x¨ri (t) + 2ζr ωr x˙ ri (t) + ωr2 xri (t) = ρri x¨g (t) (5) in which ωr and ζr are the r-th vibration circular frequency and modal damping, respectively. ρri is the participation factor of the r-th mode at response point i, defined as ΦTr · M · [1] (6) ΦTr · M · Φr The displacement at the level i is given by combining all the modal contributions: ρri = −Φir
xi (t) =
Nm X r=1
xri (t)
(7)
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e(t)
-
He (q −1 )
x¨1i (t)
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Hi1 (q −1 )
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x¨g (t)
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.. . - H N m (q −1 ) i
m x¨N (t) i
Figure 1. Excitation-Structure Model The normal modes approach presented above is adopted here. It is assumed that the observed response is the sum of a finite number of modal responses, and each mode is modeled by a second order system. A block diagram of such earthquake excitation structure system is shown in Figure 1. The single-degree-of-freedom oscillator, for all i levels, can be approximated by a transfer function of the form: H r (s) = or:
Kr ωr2 s2 + 2ζr ωr s + ωr2
H r (q −1 ) = q −1
br1 + br2 q −1 1 + ar1 q −1 + ar2 q −2
(8)
(9)
in the discrete case. Our analysis is practically interested in the evaluation of changes in ζr and ωr , using the parameter estimates of the discret model form. The actual earthquake is represented by He (q −1 ) which is assumed to be the transfer function of a finite-order linear, time-variant system , having as input a white noise process e(t). The accelerogram measured at the i level, x¨i (t), can be considered as the realization of a random process containing information about the dynamics of the seismic motion and the dynamics of the structure. This non-stationary signal is treated here as a realization of a discrete-time stochastic process, which is the output of a linear, time-variant system. The dynamic characteristics of a such system are changing with the time and the evolution of the input. 3. Change Detection Procedure. The main hypothesis of the change detection procedure presented here will be that the analyzed signals: the seismic signal and the building response can be considered as non-stationary signals for a long period of time, but as quasi-stationary signals for short time intervals, marked by the change instants. The procedure is as follows: Step 1. Segment the seismic signal x¨g (t) into quasi-stationary data blocks and fit the AR filters for these; here, the acceptance of quasi-stationary reveals that, within a
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AR2 AR0
AR1 L
L+S
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r - detection time Change detection if AIC2 > AIC0 + AIC1
Figure 2. Change Detection Scheme segment, the short-time spectral estimate does not change appreciably with elapse of time. The method, described by Popescu and Demetriu in [11] is based on the non-stationarity analysis procedure suggested by Kitagawa and Akaike, [12], and uses three autoregressive models : a global model AR2 , identified in a growing window, a model AR0 , identified for a data subset supposed stationary and a short term model, AR1 , identified in a sliding window (see Figure 2). Deciding on changing the dynamical properties of the signal involves the evaluation of the Akaike Information Criterion (AIC) for the global model, AIC2 , and for a combination of the latter two models, AIC0,1 = AIC0 + AIC1 . Let x¨g (1), x ¨g (2), . . . , x¨g (L) be a set of initial data considered quasi-stationary, and x¨g (L + 1), . . . , x¨(L + S), an additional data set of S observations, with S a prescribed number. The segmentation procedure is as follows: 1. Fit an autoregressive model AR0 x¨g (t) =
M0 X
am x¨g (t − m) + e(t)
(10)
m=1
with σ02 the innovation variance, where M0 is chosen as giving the minimum of the AIC criterion, for the set of data x¨g (1), x¨g (2), . . . , x¨g (L): AIC0 = L · log σ02 + 2M0
(11)
2. Fit an autoregressive model AR1 x¨g (t) =
M1 X
am x¨g (t − m) + e(t)
(12)
m=1
with σ12 the innovation variance, and M1 chosen as giving the minimum of the AIC criterion, for the set of data x¨g (L + 1), . . . , x¨g (L + S) AIC1 = S · log σ12 + 2M1
(13)
3. Define the first competing model by ”connecting” the autoregressive models AR0 and AR1 . The AIC of this joint model is given by: AIC0,1 = L · log σ02 + S · log σ12 + 2(M1 + M2 )
(14)
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4. Fit an autoregressive model AR2 , x¨g (t) =
M2 X
am x¨g (t − m) + e(t)
(15)
m=1
σ22
with the innovation variance, where M2 is chosen in the same way as above, for the data set x¨g (1), x ¨g (2), . . . , x¨g (L), . . . , x¨g (L + S). 5. Define the model AR2 as the second competing model with the AIC criterion given by: AIC2 = (L + S) · log σ22 + 2M2
(16)
6. If AIC2 is less than AIC0,1 the model AR2 is accepted for initial and additional sets of observations and the two sets of data are considered to be homogeneous (no change in signal characteristics is detected). If not so, we switch to the to new model AR1 (a change in signal characteristics is detected at instant r = L). The procedure is repeated whenever a set of S new observations is obtained. It is so able to adapt to the changes in the signal characteristics. If the characteristics keep unchanged, the model will be improved by the additional observations. An AR model has resulted for each quasi-stationary data segment. It will be used in the next step. The numerical procedure for AR model fitting could be the method of the least squares realized by Householder transformation, [13]. This approach proves a very simple procedure for handling other new observations. It is well-known that the AIC criterion tends to overestimate the true model order (even for N → ∞), while the other methods provide consistent order estimates. However, for a finite number of data points Akaike’s criterion is expected to take a smaller risk in underestimating the model order. This is the main reason for which we used this criterion rather than other ones. Step 2. The variance of e(t), corresponding to different time intervals, is stabilized by dividing it by the envelope function given by standard deviations of residuals of each model, obtained for quasi-stationary data blocks. The resulted residuals can be assumed to be a stationary independent and identically distributed process with mean zero and variance 1. At this stage the autocorrelation function of the residuals must be estimated, to confirm the lack of correlation. Step 3. Divide the building response by the envelope function determined above. Step 4. Filter the resulted building response, containing information about the dynamics of seismic signal as well as the dynamics of the structure, with the AR filters resulted at the Step 1 above, corresponding to the quasi-stationary time intervals of the seismic signal. If the series of the model residuals obtained at Step 2 can be assimilated to a white noise process, the filtered response will only contain information on the dynamic properties of the analyzed structure. This signal can be considered as the output of a linear, timevariant filter having as input a white noise signal. Step 5. Segment the filtered response in quasi-stationary data blocks, using the procedure described at Step 1 above. Each transition from a data block to another will represent a change in the dynamic properties of the system. For each quasi- stationary
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data segment, the characteristics of the system in time and frequency domain can be evaluated for diagnosis-making. Starting from the resulted AR models is possible to evaluate the structural vibration modes for each quasi-stationary data segment. 4. Case Study. The previously described procedure was applied to strong motion records obtained in a 12-storey reinforced concrete building, during the August 30/31, 1986 Romanian earthquake. The transversal (N-S), longitudinal (E-W) and vertical components of the acceleration recorded at the basement and at the roof level of the structure have been analyzed. The seismic acceleration and the response acceleration for all components are sampled at 0.02 seconds and are represented in Figure 3, Figure 4 and Figure 5, respectively. 1 0.5 0 -0.5 -1 0
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4 2 0 -2 -4 0
Figure 3. Input-output data: N-S direction For regular symmetric structures, such as the investigated building, the identification results obtained assuming a multi-input, single-output system are practically similar to those obtained for a multi-input, multi-output system. The identification results obtained in this case [14] point out the fact that the transfer function corresponding to the input in the direction of the output is more dominant, suggesting that the building could have been identified using only the input in the direction of the output. This is not surprising, since the structure is symmetric and there is very little torsion. In this case, the change detection procedure has been applied for 3 single-input singleoutput systems, corresponding to investigated directions of seismic wave propagation: N-S, E-W and vertical.
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1 0.5 0 -0.5 -1 0
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Figure 4. Input-output data: E-W direction 0.4 0.2 0 -0.2 -0.4 -0.6 0
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Figure 5. Input-output data: Vertical direction
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The Fourier amplitude spectra of the records point out that their frequency content is mainly in the range 0-5 Hz for transversal and longitudinal components and in the range 0-12 Hz for vertical component. Therefore, the transversal and longitudinal data were low-pass filtered using a zero-phase, second-order Butterworth filter, with the cut-off frequency of 5 Hz; the vertical components were low-pass filtered with a similar filter with cut-off of 12 Hz. After a preliminary analysis of the seismic components, for different length of the sliding window, we find the length of 200 data (4 sec.) as the best for this application. Using the segmentation procedure, described in the previous section, the excitation components were divided into quasi-stationary data segments after detecting change instants; also, the AR models, of maximum order 3, were determined for these quasi-stationary data segments: 12 for N-S component, 11 for E-W component and 12 for vertical component. A lack of correlation in the autocorrelation functions of the normalized residuals of the excitation components could be noted. The building response components, after dividing them by the envelope of the standard deviation of the seismic components and filtering with the AR models, resulted for the seismic quasi-stationary components, have been segmented in quasi-stationary data segments, by the same procedure, using AR models of maximum order 10. The analysis procedure applied in this case determined the change instants in the dynamic characteristics of the system. The evolution of frequency response of the system for the investigated directions, and for the quasi-stationary data segments resulted is presented in Figure 6, Figure 7 and Figure 8, under the form of the amplitude-frequency characteristics. It resulted 9, 10 and 10 change instants in dynamics of system for the N-S, E-W and vertical directions, respectively. The AR models obtained at this phase can be used to estimate the modal characteristics of the structure for each direction. The software support in this case study was provided by CHANGE program package, [15]. 5. Conclusions. A robust method for change detection in the dynamic characteristics of a structural system, subject to a strong seismic motion, was presented and investigated in a case study. On the basis of the results reported here, as well as on the basis of the results obtained in Monte-Carlo simulation, it is tentatively concluded that this approach enables us to adequately detect changing instants in dynamic characteristics of vibration structures subject to transient natural excitations. REFERENCES [1] Gustafsson, F., Adaptive Filtering and Change Detection, Wiley, 2001. [2] Basseville, M. and I. Nikiforov, Detection of Abrupt Changes. Theory and Applications. Prentice Hall, Information and System Sciences Series, NY, 1993. [3] Popescu, Th., Change detection and data segmentation methods, Proc. of the 11th WSEAS International Conference on Systems, Agios Nikolaos, Crete Island, Greece, pp. 136-141, 2007. [4] Chowdhury, F. N., B. Jiang and C. M. Belcastro, Reduction of false alarms in fault detection problems, International Journal of Innovative Computing, Information and Control, vol. 2, pp. 481490, 2006. [5] Basseville, M., A. Benveniste, G. Moustakides and A. Rougee, Detection and diagnosis of changes in eigenstructure of nonstationary multivariable systems, Automatica, vol. 23, pp. 479-489, 1987.
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Figure 6. Frequency response evolution for N-S direction
Figure 7. Frequency response evolution for E-W direction
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Figure 8. Frequency response evolution for Vertical direction [6] Popescu, Th., Some experiences with change detection in dynamical systems, Proc. of the 8th International Conference, KES 2004, Wellington, New Zealand, pp. 1220-1227, 2004. [7] Wahlberg, B., Robust Frequency Domain Fault Detection/Diagnosis, Technical Report EE/CIC8902, University of Newcastle, 1989. [8] Popescu, Th. and S. Demetriu, Robust change detection in dynamic systems with model uncertainties, Prepr. of the 4th IFAC Conference on System Structure and Control, Bucharest, Romania, pp. 290-296, 1997. o [9] Berdjag, D., C. Christophe and V. Cocquempot, Nonlinear model decomposition for robust fault detection and isolation using algebraic tools, International Journal of Innovative Computing, Information and Control, vol. 2, pp. 1337-1354, 2006. [10] Popescu, Th., Robust change detection method using Akaike information criterion for signal segmentation, Soft Methodology and Random Information Systems, Springer Verlag, Berlin, pp. 749-756, 2004. [11] Popescu, Th. and S. Demetriu, Analysis and simulation of strong ground motions using ARMA models, Automatica, vol. 24, pp. 721-737, 1990. [12] Kitagawa, G. and H. Akaike, A procedure for modelling of nonstationary time series, Ann. Inst. Statist. Math., vol. 30, B, 19, 1978. [13] S¨oderstrom, S. and P. Stoica, System Identification, Prentice Hall, London, 1989. [14] Popescu, Th. and S. Demetriu, Identification of a multi-story structure from earthquake records, Prepr. of the 10th Symposium on System Identification, SYSID’94, Copenhagen, Danemark, pp. 411-416, 1994. [15] Popescu, Th., CHANGE - Program package for change detection in signals and systems, Prepr. of the 11th IFAC Symposium on System Identification, SYSID’97, Fukuoka, Japan, pp. 1619-1625, 1997.
