AUTONOMOUS MODAL PARAMETER ESTIMATION BASED ON A STATISTICAL FREQUENCY DOMAIN MAXIMUM LIKELIHOOD APPROACH
Peter Verboven,
Eli Par-loo, Patrick Guillaume Vrije Universiteit Brussel Department of Mechanical Engineering Pleinlaan
2, B-l 050 Brussels,
and Marc Van Overmeire
(WERK)
Belgium
[email protected]
2
ABSTRACT This contribution, presents an autonomous modal parameter estimation procedure based on a statistical frequency-domain maximum likelihood approach. Using improved frequency-domain identification schemes, features such as high accuracy and confidence bounds for the estimated parameters and robustness for different types of testdata make an automation of the modal estimation process possible. Based on a statistical approach, adaptive pole selection criteria are developed. The new approach is illustrated for 2 experimental cases.
1
AUTONOMOUS OVERVIEW
PARAMETER
IDENTIFICATION:
AN
More fundamental research on automating the modal analysis process has mainly dealt with the selection of the correct model order. Most of the classical model order selection tools (Akaike, F-test, MDL,) that are used in time-domain identification were developed in the framework of control theory f3-51. The shortcomings of these methods have been extensively studied in f61. They only allow to verify if the model order used is appropriate or not (detection of under-modeling or over-modeling).
INTRODUCTION
Modal parameter estimation has become a common technique used by engineers to analyze complex mechanical and civil structures such as cars, aircraft, large machinery, bridges, buildings, _._ For these engineers, the modal model is very often not the final goal, but only a means to get condensed experimental data suitable for a variety of mechanical design purposes such as FE model correlation, structural modification design as well as for monitoring and control practices. Although measurement technology and computation power have entered a new era, traditional experimental modal identification can still be user-time intensive and as a result costly. Most important reasons for this are related to the constraints in analyzing large, complex structures which are discussed in [1,21. More specific for the modal identification step, a high modal density, repeated or multiple roots and the lack of error bounds on the estimated parameters, can result in a very difficult model order and mode selection process. The work presented here focuses on this most critical step and proposes a solution for an automated pole selection procedure, based on a statistical approach. An automated approach will result in an important reduction of user-time as well as the possibility for non-expert users to use modal analysis techniques.
1511
Order selection techniques, used for frequency-domain identification, are usually based on a statistical analysis of the residual errors (such as a whiteness test) [?I. This approach can also detect the presence of non-linear distortions. A more recent model order selection approach is based on a statistical analysis of the global minimum of a weighted non-linear leastsquares cost function [*,‘I. Another possible order selection criterion consists in determining the rank of the Jacobian matrix or of the sampled covariance matrix 19310], These techniques can be used to verify if the model order is appropriate or not, or to see if there are non-linear distortions present in the measurements, but they do not allow to separate physical modes from mathematical ones. Moreover, the model orders considered in modal analysis are usually much higher and also the amount of data to be processed is much larger than for control applications (where, most of the time, only single-input, single-output systems are considered). In modal analysis, a stabilization diagram is an important tool that is often used to assist the user in separating physical poles from mathematical ones in order to select the optimal model order. Today, this is still done manually. Selecting the
200
Figure
1: Subframe
structure
with 2 input locations.
300 Freq
(Hz)
Figure 2: FRFs of subframe
structure.
physical poles can be very difficult and time-consuming depending on the type and quality of the measurement, the performance of the estimator and the experience of the user. As a result, attention has been paid recently to the development of autonomous modal parameter estimation techniques [“-‘K and this especially in the view of diagnostic purposes such as for example performing routine modal testing and analysis on the Space Shuttle after each landing. These techniques are based on genetic algorithms. fuzzy logic and neural networks and can be quite computer intensive. Until now, classical time-domain identification techniques (e.g., ERAIDC) were used, and consequently, no information about the quality of the estimates, is taken into account.
tion diagram even for high model orders 1%18*19]. Other advantages of frequency-domain estimators in general are the possibility to estimate a model for data within a certain frequency band as well as to improve the accuracy of the estimated parameters by taking the uncertainty on the measurement data into account. At the same time, this uncertainty information can be transformed into confidence bounds on the estimated parameters. All these attractive features are now used to develop an automated modal identification procedure.
5 3
DESCRIPTION
OF TEST
STRUCTURE
AND
A FREQUENCY-DOMAIN
PARAMETER
IDENTIFICA-
DATA
Figure 1 shows the laboratory test structure used for the illustration of the proposed procedure. This frame structure resembles the subframe of a car to be connected to the body at four locations and on which the engine is supposed to be mounted on. A detailed descripfion of the structure and the set-up is given in [17] Using dual random vertical shaker excitation, two uncorrelated white noise signals were applied to the 2 shakers. The 28 responses and 2 input forces were simultaneously recorded for a duration of 32s, sampled at a frequency of 1024Hz. These time signals were then used to calculate all frequency response functions (FRF), using 8 frequency averages and the Hi estimator. Figure 2 shows 2 FRFs in which the 13 physical modes are clearly present.
4
AUTONOMOUS MODAL TION ALGORITHM
Figure 3 is a flowchart of the autonomous approach. There are five principal steps:
modal identification
1. pre-analysis using the LSCF estimator first model order reduction; 2. running the Maximum 3. physical thresholding ematical poles;
Likelihood eliminating
in order to do a
(ML) estimator; the most obvious math-
4. pole evaluation using tools based on the statistical properties of the ML estimator in order to eliminate remaining spurious poles:
APPROACH
Recently, frequency-domain (Total) Least Squares and Maximum Likelihood (ML) identification schemes have been optimized to handle large amounts of data within reasonable computation time ~1 At the same time they have proven to be robust for a high modal density, leading to a very clear stabiliza-
1512
5. running the LSFDX estimator to estimate shapes for the selected set of poles.
The following more detail.
paragraphs
discuss
the mode
each of the five steps
in
so :
40 ++. .+
30 20
+
10
?I!
+ ;+
+
:
+
+’
.*
+
1..
+ ;++
:
I
:
100.
+qz + :
+:
:j;
I
tti ‘::: -+
i
-40
3
-+
+
Q----------, L-2 Thresholder
‘c ;
+t +. *
f
-SO
Pole-Zero
+ ++
-20
MLE
+’
0 -,o
0
++’
,I:
200
.I::
+;;+
300 l=req
400
(HZ)
I
Figure
_ __
4:
LSCF
Pole
Uncertainties
(lines:
physical
poles).
ii
f(stdp)
= sort(db(stdp))
+*
1
parameters -20 -40
+I+-
i-t+
0
10
20
30
40
I 50
diff(f(stdp))
Figure
5.1
3: Flowchart
of Autonomous Estimation Process.
Modal
Parameter
Pre-Analysis
Since no a priori information is available, running the LSCF estimator with an arbitrary model order for 50 modes results in a first set of both physically and typically a large number of mathematical modes due to large over-modeling. The first step of the identification procedure consists of finding an approximation of the appropriate model order.
Figure
indication Pole
Uncertainty
20
30
number
of modes
5: Analysis
of uncertainties.
of the model order.
Analysis
Figure 5 shows the uncertainty of all estimated poles ranked from low to high and the derivative of the function. The maximum of the derivative coincides with 12 modes. Having 13 physical modes, the analysis of the uncertainties on the estimated poles gives, in this case, a good approximation of the number of physical modes. However, if the measurement data is characterized by a low Signal-to-Noise ratio, this approach can fail.
A first approach for model order reduction that was tested, was the analysis of the uncertainties on the poles estimated using the LSCF estimator [Is). Availability of the variances on the FRFs allows, in the case of the LSCF, to do a sensitivity analysis of the parameters for noise on the measurement data resulting in an uncertainty level on the estimated poles (and zeros although the latter calculations are typically much more computer-intensive). As shown in Figure 4, the uncertainties of the poles in case of the subframe data are clearly much smaller for the physical poles. Only the uncertainty of the last physical pole has a similar magnitude as for non-physical poles due to a higher uncertainty level on the measurement data. Nevertheless this behavior of pole uncertainties can be useful to obtain a first
Since this is only a pre-analysis step, the initial model order is reduced to a number of modes (model order) that is 50% more than the number at the maximum of the derivative function. In this case the model order was reduced from 50 to 18 modes.
1513
Pole-Zero
Checker
first step significantly
improves
the starting values which are during the pre-
taken from the last run of !Ce LSCF estimator The second approach tested for model order reduction, is based on the detection of “canceling pole-zero pairs”, a typical characteristic for the mathematical or ‘noise’ modes L9t. In order to eliminate these pole-zero pairs, a “Pole-Zero Checker’ was developed. This was possible since the model structure, used for the implementation of the proposed frequencydomain estimators (LSCF, ML), is a (common denominator) transfer function model with polynomials in both numerator and denominator i’a’. The polynomial coefficients which are estimated, are related to the poles and the zeros of the model. As discussed in [I’, 191, the frequency-domain formulation of the LSCF (and also the ML) estimator is robust to large model orders. This automated procedure, detects a “canceling pole-zero paii’ in the case that one or more zeros fall within the “threshold circle” around the pole for 90% or more of the estimated transfer functions (defined as the “pole-zero threshold ratio”). The radius R of the “threshold circle” is calculated using the uncertainty level on each estimated pole using following expression: R = K
.upoie
The most important algorithm parameters of the ML estimator are the model order, the number of iterations (and the frequency band of interest). A maximum number of 50 iterations is chosen since this, for structural dynamics analysis, typically results in the convergence of the ML estimator, In the case that convergence is reached faster, the ML estimator is stopped earlier. For each iteration, a new set of modal parameters is obtained. In addition to generating modal parameters, it is possible to calculate the standard deviations for all modal parameters based on the uncertainty on the measurement data and this for each iteration. Using the uncertainty information allows to weight the equation error in ML sense yielding very accurate estimates of the parameters and their standard deviation,
6
- pb)
the multiplier of the pole uncertainty apore and pb the probability limit for the event that the true pole falls inside the circle. During this pre-analysis step, this probability is chosen pb = 68%. This multiplier M, is used to convert the pole uncertainty to a 2 dimensional boundary. This “canceling pole-zero” elimination process has proven to be more robust to noise on the data compared to “uncertainty analysis” method, and as a result is incorporated in the procedure. The initial model order is now reduced using a recursive approach as shown in Figure 3. In the case of the subframe data, the model order was reduced from 50 to 20 modes during three LSCF runs.
. Damping l
Maximum
Likelihood
THRESHOLDER
< 0% (unstable
pole) or > 10% or
Frequency within 1% of edges of analysis
bandwidth
The damping cutoff of 10% is a valid selection for typical mechanical structures having modal damping ratios of the magnitude of 2% or less up to 5% for highly damped structures. The frequency cutoff is appiied to eliminate the mathematical poles on the edges of the analysis bandwidth that result from the fact that the ML estimator uses a model defined in the discrete frequency-domain.
7 5.2
PHYSICAL
The physical thresholding step consists of a very straightforward elimination of unfeasible modes generated by the ML estimator due to the remaining overmodeling after pre-analysis. Using the following simple and obvious criteria, most nonphysical modes will be eliminated:
(1)
with MC = J-2.&(1
analysis step.
POLE SELECTOR
Estimation
The frequency-domain Maximum Likelihood (ML) estimatoras presented in [IS], is used as “data analysis engine”. This ML estimator has been successfully applied in a large number of cases for estimating structural modal parameters from both forced response (FRFs) and ambient response data. The ML estimator estimates the model parameters through an iterative optimization process (such as Gauss-Newton) and as a result starting values must by available. The quality of these starting values is an important factor for the convergence behavior of the ML estimator. The model order reduction in the
1514
The next step of the identification procedure is passing the “Pole Selector”. This fourth step currently consists of 2 tools which will be now discussed in more detail.
7.1
Pole-Zero
Checker
Although the model order reduction using the pre-analysis the physical thresholding step will have eliminated most physical modes, the subset of poles can still contain poles correspond to “noise modes” as it is also the case for the frame data. Figure 6 shows the poles (after thresholding)
and nonthat subfor
300
200 Freq
Figure
6:
ML
Pole
diagram
Freq
(Hz)
before
Pole
Figure
Selector.
all iterations, clearly showing the lines of poles at the natural frequencies of the structure as well as the averaged sum of all FRFs. Looking closer also reveals some additional lines of poles around 62Hz and 376Hz. Especially the second line around 62Hz could make a manual pole selection from this diagram very difficult since both lines seem to have a ‘stable’ behavior. However, these additional lines can be explained again by the presence of “canceling pole-zero pairs” as is shown in Figure 7 by zooming in on the pole diagram in those 2 frequency areas. These pole-zero pairs, are a purely mathematical phenomenon, which nevertheless can seriously hamper automating the procedure. In order to check for “canceling pole-zero pairs”, the “PoleZero Checker”, explained in section 5.1, was slightly modified for this “Pole Selector”. The ML estimator returns a set of poles and zeros for each iteration and subsequently a polezero check is done for each result. The ML estimator calculates the uncertainties on the poles for each iteration. Again, the pole uncertainties are used for a similar expression as equation (1). however the probability used to calculate the radius of the “threshold circle” is pb = 98%, while the “pole-zero threshold ratio” of 90% is applied again. However in the case of a high modal density or a high uncertainty on the measurement data, this “Pole-Zero Checkei’ can fail since only the number zeros within the “threshold circle” of a certain pole is evaluated for the “pole-zero threshold ratio” of 90%. This can be avoided by performing an additional check on the correlation between the pole and all zeros.
1515
7.2
7: Zoom
Correlation
(Hz)
Freq
on pole
diagram
(*:
pole,
(Hz)
x: zero).
Analysis
Based on the stochastic properties of the measurement data, the ML estimator can also calculate, the correlation coefficient between each pole and zero. However, this calculation also requires the uncertainties on the zeros, which represents a significant computational load compared to the rest of the ML estimation process. The correlation matrix between poles and zeros gives more insight for the detection of canceling pole-zero pairs since it also tells the level of coherence or correlation that exists between each pole and zero. Figure 6, shows the correlation matrix between the poles and zeros for all modes (after physical thresholding) calculated during the last ML iteration. The modes that have a high correlation between the pole and all zeros within the respective “threshold circle”, are the ones that correspond to the spurious modes around 63Hz and 350Hz. The “Pole Selector uses this correlation information to double check the poles that are indicated as being non-physical by the “Pole-Zero Checker”. A candidate “canceling pole-zero pair” is eliminated when 90% or more of the correlation values is above 95%. Passing the “Pole Selectoi’ results in Figure 9, which clearly shows that all the poles (lines) corresponding with spurious poles are eliminated from the solution sets. In this case, the number of degrees of freedom was small (56) and the correlation were calculated for each iteration. The final set of poles is selected on the basis of lines having minimal 5 subsequent poles. However, in case of large problems, the”Pole Selector’ only uses results of the last ML iteration, and the final set of poles is now based only on the results if this last iteration.
mode nr Figure
7.3
LSFDX
8: Pole-zero
correlation
VALIDATION
Figure 9: ML Pole diagram
- modeshapes
OF AUTONOMOUS
300
Freq
matrix.
9
Once the final set of poles is accurately identified, the spatial information is currently estimated in a fourth step using the LSFDX estimator, which estimates mode shape values and the modal participation factor values at the same time. This is done by estimating first the complete residue matrix for each mode. Next this matrix is split into one mode shapevector and one participation factor vector using a Singular Value Decomposition technique. Since this is again a frequency-domain estimator, results can be further improved, by using the weighted version, which takes the uncertainty information on the data Into account.
8
200 (Hz)
after Pole Selector.
CONCLUSIONS
In this paper an automated modal parameter identification procedure was proposed. The attractive features of the frequency-domain maximum likelihood (ML) estimator (high accuracy, confidence bounds, robustness to high model order) make it a very suitable data analysis tool for an automated procedure. Since the selection criteria are developed in a stochastic framework, this procedure is robust to high noise levels. At this stage, although only the poles were supervised during the identification process, it was already possible to identify the structural modes with a very high accuracy. Spurious modes were all eliminated from the modal model. In the future, this procedure will be also tested for operational measurements, typically containing more measurement noise.
PROCEDURE
In order to validate the proposed procedure, it was applied to a composite plate structure with higher damping characteristics. Forced scanning laser-vibrometer measurements using a single shaker applying a multi-sine signal resulted in 224 FRFs with their variances. After the pre-analysis step, the model order was reduced from 50 to 34 after 2 LSCF runs. Next the ML estimator estimated the model parameters of 34 modes and this for 50 iterations (convergence was reached). The poles that are left after the “physical thresholding” are plotted for all iterations in Figure 10. The two pole lines at 150Hz and 330Hz do not correspond with system poles. After passing the ‘Pole Selector’ step, these lines disappear as shown in Figure 11 and again a successful identification of the system poles was done in this case.
1516
ACKNOWLEDGEMENTS This research has been supported by the Institute for the Promotion of Innovation by Science and Technology in Flanders (IWT) and by the Research Council (OZR) of the Vrije Universiteit Brussel (VUB). The authors also would like to thank LMS International for providing the measurement data on the subframe.
REFERENCES [l] Allemang, R. J., Modal Analysis - Where Do We Go From Here?, Proc. of 1 1th International Modal Analysis Conference, p. keynote paper, February 1993. [Z] Van der Auweraer, H., Leurs, W., Mas, P. and Hermans, L., Modal Parameter Estimation From lnconsisfenf Data Sets, Proc. of 19th International Modal Analysis Conference, pp. 763-771,
50
50 -
45
45
40
40
35 $ 2 30 2 s 25 .x=
35 p 30. 2 g 257
E 20
E 20-
15
15
10
10
5
5-
01
I 50
100
150 Freq
Figure
February
10:
ML
Pole
"
300
50
before
Pole
Selector.
Figure
[13] af the Statistical Mode/ Identification, Control, Vol. AC-19, No. 6, pp. 716-
by Shortest 1978.
Leontaritis, I. and Billings, lion Methods for Non-Linear No. 1, pp. 31 l-341, 1967.
150
Data Description,
Automat-
Pintelon, R., Schoukens, J. and Vandersteen, G., Model Seiecbon Through a Sfatisfical Analysis of the Global Minimum of a Weighted Non-Linear Least Squares Cost Function, Proc. of 35th Conference on Decision and Control, pp. 3100-3106. December 1996. EsfimaDomain Control,
[lo]
Davila, C. and Chiang, H.. An Aigorithm for Pole-Zero Mode/ Order Estimation, IEEE Transactions on Signal !ng,Vol.43, No.4, pp. 1013-1017,April 1995.
[Ill
R., James Ill, G. and Zimmerman, 0.. AuPappa, tonomous Modal ldentificafion of the Space Shuttle Tail Rudder, ASME Design Engineering Technical ConferenceslDETC, , NO. DETC97NIB.4250, September 14-17 1997.
[12]
Pappa, R.. Woodard, S. and Juang, J., A Benchmark Problem for Development of Aufonomous Structural Modal Idenilficafion, Proc. of 15th International Modal Analysis Conference, pp. 1071-1077, February 3-6 1997.
ML
Pole
diagram
after
250
Pole
300
Selector.
James Ill, G. H., Zimmerman, D. C. and Chhipwadia, K. S., Application ofAutonomous Modal Idenfificafion to Traditionaland Ambient Data Sets. Proc. of 17th International Modal Analysis Conference, pp. 840-645, 1999.
[15]
Woodard, S. and Pappa, R.. Development ficafion Accuracy lndicafors Using Fuzzy Engineering Technical ConferencesiDETC, 4258, September 14-17 1997.
[16]
Lim, T., Cabell, R. and Silcox, R., On-Line fdenfificafion of Modal Parameters Using Artificial Neural Networks, J. of Vibration and Acoustics, Vol. 116, No. 4, pp. 649-656, 1996.
[17]
Hermans, L., Van der Auweraer, H. and Abdelghani, M., A Critical Evaluation of Modal Parameter Extraction Schemes for Output-Only Data, 15th Modal Analysis Conference Japan, pp. 662668, September 1997.
[18]
Guillaume, P., Verboven, P. and Vanlanduit, S., FrequencyDomain Maximum Likelihood Identification of Modal Paramsfers wifh Confidence Intervals, Proc. of 23rd International Seminar on Modal Analysis, pp. 359-366, September 1996.
[19]
Verboven, P., Guillaume, P., Van Overmeire, M. and Vanlanduit, S., Improved Frequency-Domain Total Least Squares Identification for Modal Analysis Applications, Proc. of 5th National Congress on Theoretical and Applied Mechanics, pp. 211-214, May 2000.
S., Mode/ Se/ecfion and ValidaSystems, Int. J. of Control, Vol. 45.
[9] R&in, Y., Schoukens, J. and Pintelon, R., Order fion for Linear Time-lnvarianf Systems Using Frequency Idenfificahon Methods, IEEE Transactions on Automatic Vol. 12, No. 10. pp. 1406-1416, October 1997.
11:
200 (Hz)
1141 Chhipwadia, K. S., Zimmerman, D. C. and James Ill, G. H., Evolving Autonomous Modal Paramefer Esfimafion, Proc. of 17th International Modal Analysis Conference, pp. 619-625, 1999.
Idenfifi-
[7] Schoukens, J., Dobrowiecki, T. and Pintelon, R., Identificalion of Linear Sysfems m fhe Presence of Nonlinear Distortions. A Frequency Domain Approach. Part I: Non-Parametric Identification, Proc. of 34th Conference on Decision and Control, pp. 1216-1221. Dec. 13-15 1995. [6]
100
Freq
Soderstrom, T.. On Mode/ Structure Testing in System cation, Int. J. of Control, Vol. 26. No. 2, pp. l-18, 1977.
[5] Rissanen, J., Modeiing ica, Vol. 14, pp. 465-471, [6]
diagram
250
2000.
[3] Akaike, H., A New Look IEEE Trans. on Automatic 723. Dee 1974. [4]
200 (Hz)
System Process-
1517
of Structural IdenfiLogic, ASME Design , No. DETC97NIB-
