COMPARATIVE STUDY OF 5 RECURSIVE SYSTEM IDENTIFICATION METHODS IN EARTHQUAKE ENGINEERING

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1 COMPARATIVE STUDY OF 5 RECURSIVE SYSTEM IDENTIFICATION METHODS IN EARTHQUAKE ENGINEERING FanLang Kong Dept. of Civil Engineering, State University of New York at Buffalo Amherst, NY 14260, U.S.A ABSTRACT Comparison of 5 most popular recursive identification algorithms is conducted for modal analysis purpose. The identification techniques are critically assessed and compared by considering the noise level effect. Some results and conclusions are obtained. 1. INTRODUCTION During the past two decades, many different identification and parameter estimation methods for dynamic process have been described in the literature. The comparative evaluations of different recursive identification algorithms were made by several researchers for control purpose (Isermann et al, 1974 and Saridis, 1974), the comparative studies, however, are very scarce in terms of modal parameters, such as resonant frequency and damping ratio, which are very important to the earthquake engineering. This paper attempts to introduce some comparison of 5 most popular identification algorithmsthe algorithms selected for this paper a, the recursive least squares method, the recursive instrumental variable method, the recursive generalized least square method, the prediction error method, and the recursive maximum likelihood method. In order to verify and check the accuracy of theses algorithms, the identification techniques are critically assessed by using data produced by numerical simulation. Further, the comparison is conducted by considering the noise effect to the identification results. Two cases, 5% and 10% noise level in the meaning of RMS are studied separately. Some results and conclusions are obtained. 2. MATHEMATICAL MODELS Consider a linear dynamic system represented by the differential equation [Ml i (t) + [Cl i (t) + [K] z (t) = [Dl f(t) (1) in which [Ml, [Cl, and K] are n x n mass, damping and stiffness matrices, respectively, z(t) is the n-dimensional response vector, and D] is an n x 1 input matrix. 1230

2 Assume that me displacement in some points of the system are measured and the measurements are subjected to observation noise vector v(t). That is y(t) = WI z(t) + e(t) (2) The noise vector E (t) is assumed to be a white Gaussian process independent of f(t). The symbol H denotes a n x 1 observation matrix. It can be shown (Goodwin and Sin, 1984) that under mildly restrictive conditions, such as controllability and observability of the system, the state space model can be converted into an compact ARMA model. 2n C ajyk-j = g Pjfk-j +&ki j-0 j=l a, = 1 (3) Eq. (3) can be written in the compact form where 6. and 0. denote the percent of critical da&ping ritio and the natural frequency, respectively, associated with the jth mode of vibration of the structure, and i = Ji. 3. RECURSICE IDENTIFICATION METHODS 3.1 Recursive Least Square Method The main concern in the least square algorithms is, how to choose the parameter 0 in such a manner as to minimize the sum of the squares of the residual series, { Ek}.The corresponding algorithm can be summarized as (Goodwin and Payne, 1977), b.+t = &.+K,+,(Y,+,-:+,h) (10) K k+i = pkxk+, 4 a+x:+,p,x,+, (11) ) yk = x:8+& k (4) 4 = -yk... -Yk-,, fk+l fk... f,-. ) 6: = [%... %- Pk+l B,... IL,] (Q According to Pandit and Mehta (1985) the roots of the polynomial h +a,h a. = 0 (7) are related to the eigenvalue of the original system by the equation pj = -I,w,+iw,a (9) The least squares estimations are biased if the residue series Ed are correlated. 3.2 Recursive Inshwnental Variable Method To maximize the correlation with the output while minimizing the correlation with the residues, the instrumental variables h, is used in the from P 1;+, = (l- I)Pk+rer+l (14) where fl, denotes the parameters of the auxiliary system, and for earthquake data, the values of y may vaty between 0.03 and 1231

3 0.1 (Kong, 1994). This method suggests the following recursive algorithm, ik+, = &+K,+,(Y,+,-x:+&r) (13 K k+l = pk=k+i I[ 1+ x:+~p~z~:, Ij (16) estimation and convergence. Ii: = [-yk Yk-n.Uk-1P 1 L.... Ilk-., &k-l,.... &-. I (23) bt = [a,,..., a,, b,,..., b,, c,,.... CJ (24) P L+f = [I-pk(*))pkw 3.3 Recursive Generalized Least Square Method The method of the generalized least squares tries to overcome the problem of bias by introducing filters to get uncorrelated residuals, and the filter may take the form T &k = -fk&k-q+?k (19) The corresponding algorithm can be summarized as L ek+i = ~k+kk+1(~k+1-~k+$k) (20) tk = yk - n&e 1 (25) Similar algorithm as Eq.(23) - Eq(25) can be used here. This method has been found to have good convergence properties in almost all case studied, although seldom a counter-example to general convergence may be got (Ljung, 1975). 3.5 Recursive Maximum Likelihood Method To overcome the convergence problem in predictions error method, the recursive maximum likelihood estimator is defined as: hk+, = kik+k k+l(yktg:~i) (26) K k+l = k@k+ I (i k@k+ lj(27) Kk+1 = P,f,+,/(l fxk+~pkjtk+,) c21) P k+, = k( -(a]) c2 ) P k+, = ( I -p, ( 1 +;;;;;;;k+ J )pk (22) Where jrk and ii, are the filtered values of Yk and +c. 3.4 Prediction Error Method In prediction error method, the innovations sequence {$} is used to improve the 4. NUMERICAL RESULTS A conventional single-degree-of-freedom structure shown in Fig.1 is considered, its natural frequency is 20 rad/s, and the damping ratio is 1%. Starting from zero initial condition, the response of the structure can be generated by direct integration, as shown in Fig.2, assuming El-Centro 1940 NS earthquake as input. Typical estimation of 1232

4 ARMA parameters is shown in Fig.3, and the estimation results of modal parameters are shown in table 1. In order to investigate the sensitivity of results to noise level on the observations, two cases ate studies herein, one is the response under noise level=5% in RMS, another under noise level=lo%. The comparison of frequency estimations is shown in Fig.4, table 2 and 3 summarize the comparison of modal parameters under two cases, respectively. 5. CONCLUSIONS The verification and comparison of S recursive identification method is discussed in this paper. It is found that all methods give nearly exact estimation under noise-free condition, and give fairly good estimation under higher noise level, say, 10% in RMS. As suspected, damping ratio is much easier to be effected by the noise level than natural frequency, which is very important for earthquake engineering and field testing. 6. ACKNOWLEDGEMENTS This paper is supervised by Dr. Roger Ghanem. His instmction is well acknowledged. 7. REFERENCES Ghanem, R. G., H. Gavin and M. Shinozuka, (1991), Experimental Verifiution of a number of Structural System Ident& jcution Algorithms, Technical Report NCEER Goodwin, G. C. and R. L. Payne, Dynamic System Identtjication: Experiment Design and Data Analysis, (1977), Academic Press, Inc. Goodwin, G.C. and K.S. Sin, (1984), Adaptive Filtering Prediction and Control, Prentice-Hall, Inc., Englewood Cliffs, NJ. Isemrann, R., Baur, U., Bamberger, W., Kneppo, P., and Siebert, H., (1974), Comparison of Six On-line Identification and Parameter Estimation Methods, Automatica, Vol.10, , Pergamon Press. Jaxwinski, (1970), Stochastic Process and Filtering Theory, Academic Press, Inc. Kong, EL., (1994), Application of System Identijcation Techniques to the Analysis of Soil-Pile-Structure Interaction, M.S thesis, State University of New York at Buffalo, Buffalo, NY. Ljung, L., (1975), Counterexamples to General Convergence of a Commonly Used Recursive Identijcation Method, IEEE Transactions on Automatic Control, Vol. AC-20, N0.5, Pandit, S. W. and N.P. Mehta, (1985), Data Dependent Systems Approach to Modal Analysis Via State Space, ASME Paper No. 85-WA/DSC-1. Saridis, G.N., (1974), Comparison of Six On-line Identification Algorithms, Automatica, Vol.10, 69-79, Pergaman Press. M -fj * Fig.1 Single-degree-of-freedom structure 1233

5 Fig.2 Response of the shuctue Fig.3 Typical ARMA esrimation 1234

6 Fig.4 Estimated natural frequency under 10% noise level 1235

7 Table 1: Estimation Results Under Noise Free Parameters Exact Values Least Squares Method Instrumental Variable Method ( Generalized Least Squares Method Prediction Error Method I Maximum Likelihood Method Table 2: Estimation Results Under Noise Level=5 % Parameters Exact Values Least Squares Method Instrumental Variable Method Generalized Least Squares Method Prediction Error Method I Maximum Likelihood Method Table 3: Estimation Results Under Noise Level = 10% Parameters Natural Frequency Damping Ratio J%act Values Least Squares Method Instrumental Variable Method Generalized Least Squares Method Prediction Error Method Maximum Likelihood Method I