Pade Approximants as a Modal Identification Technique

Luiz Antonio Barbosa Coelho and Luiz Antonio Baccala Escola Politecnica, University of Sao Paulo Av. Prof. Luciano Gualberto, Trav. 3, #158 Sao Paulo, SP, Brazil 05508-900 The output-only identification of modal parameters of large structures with low damping and under adverse signalto-noise condition is a practical problem of interest that still poses an enormous challenge since usual modal identification techniques generally prove inaccurate, particularly when damping estimation is concerned. This work addresses this problem by presenting a novel estimation technique based on Pade approximants, derived from quantum mechanics resonance problems. Via Monte Carlo simulation techniques we show the superior performance of the new approach over a high-resolution subspace method we previously described in [1]. Finally we illustrate some of its statistical behavior by applying it to both synthetic and real data. 1-INTRODUCTION Modal analysis of large civil structures has well known relevance for structural damage identification and diagnosis. To achieve high identification accuracy, one must often resort to the use of expensive shakers as the problem involves the identification of slowly damped modes that are difficult to characterize in the sole presence of the natural excitation provided by wind or traffic. The difficulty stems from the poor performance of most modal identification methods when the signal-to-noise ratio (SNR) is small. In [1] we showed that a state-of-the-art method that employs subspace identification can accurately pinpoint modal frequencies but is less accurate in regard to damping coefficient estimation. Furthermore, the latter method was also shown to degrade abruptly around an SNR of 5 dB, which is not uncommon in practical settings. To overcome these limitations, we have looked into alternative approaches and the use of Pade approximants, a fairly classical theme, has proved useful in the conditions of natural excitation based identification, which is usually characterized by large amounts available data under low SNR. The motivation for the present approach comes from the analogous problem in nuclear magnetic resonance (NMR), where frequency and damping rate allow chemical compound identification. NMR identification via Pade approximants was pioneered by Belkic [2, 3] under the comparatively benign SNR = 50dB conditions. Pade approximants were also employed by O'Sullivan and Cowan [4] to obtain the transient acoustic impulse response of theaters. In this latter case, the fast damping range of the transients facilitated the separation between the actual observed dynamics and the effects of observation noise as discussed in Sec. 2. The problem of modal analysis, via a semi-empirical application of Pade approximants, has also appeared in the well-founded work of Barone [5, 6, 7]. The rest of the present paper is organized as follows. We begin with a brief revision the fundamentals of Pade approximation and discuss the fundamental role of the idea of Froissart doublets, followed by the statement of our proposed algorithm for the natural excitation setting, in which several independent observations are available. Proceedings of the IMAC-XXVII February 9-12, 2009 Orlando, Florida USA ©2009 Society for Experimental Mechanics Inc. Finally numerical Monte Carlo experiments illustrate the accuracy and the gentle algorithm degradation with decreasing SNR. In Sec. 4 we show the results of applying the algorithm to the modal identification of the Z24 bridge [8,9]. 2BACKGROUND The general modal response signal x(t) can be written as a sum of W damped sinusoids x(l) ~ Re[ta,e" ' U, ' ]""' ; (1 ) where ak >0 represent signal amplitudes, OJk E[-Jr ,Jr] the modal frequencies, dk the damping factors and where {rpk} are uniformly distributed phases (in [Jr , Jr]). Also, let j = ~ and t = n.M represent discrete time samples obtained at an f s sampling rate. In practice one must also consider the effect of additive random noise ret) leading to an observed signal model of the form c(t) = x(t) + ret) where the noise is further assumed zero mean Gaussian of (J" 2 variance. Modal identification is based on using an observed sequence fen) to determine the parameters in (1). One may represent the time observations with help of its z-transform written as N, fez) = LCnz n (2) n=1 whose description may be further compacted with help of the so-called Pade approximants which consist of rational functions [M I N] j (z) = PM (z) , where M and N are the degrees of polynomials that approximate the QN(Z) power series expansion (Taylor series) of f(z) , an analytical function on z = 0, up to degree M + N. Take the power series i=O The Pade approximant must satisfy: when z ~ 0 and QN(O) oj:. 0, where 0(.) is Landau's big-Oh symbol, meaning that the right side is a power series over Z k , beginning with degree k = M + N + 1, up to k = +00 . As a consequence YCiZi _ PM (z) = 0 i=O QN(Z) (3) We usually impose a normalization condition, QN(O)=l, as the ratio [M I N] j (z) can be multiplied by any number and we must have QN (0) oj:. 0 . So, let and PM(z) = Po + PI Z+ . + PMZM QN(Z) = 1 + qIZ+ . +qNZN M+N Multiplying (3) byQN(z) results QN(Z) ~>iZ i _PM (Z)=O i=O From this relation we can extract the following equations: Co = Po cl +COql = PI c2 +C1ql +COq2 = P 2 Or, in matrix form: Co 0 0 1 Po cl Co 0 ql PI cM CM-l Co q M P M CM CM-l CM-N+l ql CM+l CM+1 CM CM-N+2 q 2 CM+2 CM+N-l CM+N-2 CM q N CM+N For higher values of M and N, solving for polynomial coefficients is often difficult unless more numerically accurate methods based on singular value decomposition (SVD) are used [10]. When one computes [0 / 2W] the resulting rational approximant reduces to the well known Prony method for fitting exponentials to data [11]. In the Prony method, the modal parameters Z k = e dk + jOJk are obtained as the roots of Q2W (Z). It is also well known that the Prony approach suffers from two serious drawbacks. First of all the 'order' 2W, i.e. the number of sinusoids, must be known a priori. But perhaps, more importantly, the method is very sensitive to additive noise (no less than 80 dB of SNR is adequate), making it useless for the problem at hand. It may seems surprising that insisting on Pade approximation would lead anywhere. However, more general approximants of the [N / N] type do have a peculiar behavior with respect to the effect of noise. This behavior was first noted by Froissart and reported by one of his associates [12]. In the noise-free case, i.e. when estimation errors are due solely to numerical rounding effects in both computing the approximant and the factorization of the resulting polynomial, the roots of QN(Z) can be split into two subsets: (a) one containing barely perturbed estimates of Z k (true poles) and (b) another subset of non-modal roots that are circularly distributed at a constant radius from the origin of the complex plane (spurious poles). Furthermore a subset of the roots of PN(z) are in close correspondence with the non-modal roots by forming the so called Froissart doublets, i.e. close pole-zero root pairs from QN(Z) and PN(z) (respectively), whose distance is of the order of the numerical machine precision used in the computations (see Fig. 1 and Fig. 2). The proximity of the non-modal poles and zeros is easy to understand as the poles need numerator zeros to approximately cancel their effect on f (z) . 1 . 5,----~-~-~-~-~-__,