11.5 System Identification

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Wavelet analysis lends itself to system-identification applications. For instance, frequency localization

properties allow detection and decoupling of individual vibration modes of multi-degree-of-freedom

(multi-DoF) linear systems. The wavelet representation of the system response can be truncated to an

appropriate scale parameter in order to filter measurement noise. Also, the wavelet transform coefficients

can be related directly to the system parameters, as long as specific bases are used.

Early investigations trace back to the work by Robertson et al. [77], who have used the DWT for the

estimation of the impulse-response function of multi-DoF systems. Compared with alternative timedomain

techniques, the DWT-based extraction procedure offers significant advantages. It is robust since

singularities in the procedure-related matrices can generally be avoided by selecting orthonormal wavelet

functions. Further, the reconstructed impulse-response function captures the low-frequency components,

referred to as static modes and mode shape errors, which ordinarily are difficult to estimate.

An important application of wavelet analysis to structural identification is due to Staszewski [78], who

has used complex Morlet wavelets for modal damping estimation. Specifically, Staszewski has interpreted

in terms of the wavelet transform some concepts already used in well-established methods, where the

Hilbert transform has been applied to a free-vibration linear response [79]. In the case of light damping,

the free response in each mode xjðtÞ may be approximated in the complex plane by an analytical signal,

given by Equation 11.37. The modulus of the Morlet wavelet transform of xjðtÞ can be expressed as

􀀈 􀀈

Wxj ða; bÞ

􀀈 􀀈

< Aj e2zjvj b

􀀈 􀀈

C^

–

ð^iajvj

ffiffiffiffiffiffiffiffi

1 2 z2j

q

Þ

􀀈 􀀈

ð11:72Þ

where Aj is the residue magnitude, and vj and zj are the mode natural frequency and damping ratio,

respectively. In Equation 11.72, the symbol aj denotes the specific scale value, related to the mode natural

frequency vj by the closed-form relation 11.51, typical of Morlet wavelets. Assuming that the natural

frequency vj has been previously computed, the damping ratio zj can then be estimated as the slope of a

straight line, representing the cross section wavelet modulus (Equation 11.72) plotted in a

semilogarithmic scale. That is,

ln

􀀈 􀀈

ðWxj ðaj; bÞÞ

􀀈 􀀈

< 2zjvjb þ ln

􀀍

Aj

􀀈 􀀈

C^

–

ð^iajvj

ffiffiffiffiffiffiffiffi

1 2 z2j

q

Þ

􀀈 􀀈

􀀎

ð11:73Þ

Staszewski has also proposed an alternative damping estimation method based on the ridge and skeletons

of the wavelet transform. A ridge is a curve of local maxima in the mean-square wavelet map, and the

Wavelets — Concepts and Applications 11-15

© 2005 by Taylor & Francis Group, LLC

corresponding skeleton is given by the values of the wavelet transform restricted to the ridge. As a result

of the localization properties of the wavelet transform, the ridges and skeletons of the wavelet transform

can be detected separately for each mode. Specifically, the real part of the skeleton of the wavelet

transform gives the impulse-response function for each single mode from which a straightforward

estimate of the damping ratio is obtained from a logarithmic equation analogous to Equation 11.73. A

generalization of the method for nonlinear systems can also be formulated [80].

Ruzzene et al. [81] have also presented a damping estimation algorithm based on the same concepts

and leading to analogous results. Certain issues have been addressed in detail concerning the frequency

resolution of the adopted wavelet basis, crucial for detecting coupled modes, and appropriate algorithms

for ridges extraction [50]. Lardies and Gouttebroze [82] have estimated modal parameters via ambient

records without input measurements. To this end, the random decrement method (see Ref. [83] and the

references therein) has been used to convert ambient vibration response into a free vibration response.

Also, a modified Morlet wavelet basis has been developed with enhanced properties for modal parameters

estimation. The method devised by Staszewski and by Ruzzene et al. has also been implemented by Slavic

et al. [84] by replacing the Morlet wavelets by Gabor wavelets, whose time and frequency resolution may

be adjusted by an appropriate parameter. Explicit conditions have been given on the frequency

bandwidths of the Gabor wavelet transform, in order to estimate the instantaneous frequencies of two

adjacent modes.

Damping coefficients have been estimated using a logarithmic decrement formula, where the ratio

of the wavelet transform at two subsequent extremes of the pseudo-period Tj ¼ 2p=vj of the response

in each mode is involved for a selected wavelet transform scale [85,86]. For the procedure to estimate

the damping coefficient associated with the fundamental mode, it is sufficient to adapt the analyzing

scale so that the higher frequency modes are filtered. For an arbitrary mode j; low-pass filtering is

used to cancel the fundamental and the first j 2 1 modes. Ghanem and Romeo [87] have formulated

a wavelet-Galerkin method for time-varying systems, where both damping and stiffness parameters are

computed by solving a matrix equation. The latter is built by a standard Galerkin method by

projecting the solution of the differential equation of motion onto a subspace described by the wavelet

scaling functions of a compactly supported Daubechies wavelet basis. The method is accurate for both

free and forced vibration responses. A formulation for nonlinear systems has also been proposed [88].

Another application is due to Yu and Xiao [89], who have used wavelet transform to identify the

parameters of a Preisach model of hysteresis (see Refs. [69,90] and the references therein). The output

function of the Preisach model is expanded in terms of the scaling functions of a given wavelet basis.

Then, the coefficients of such an expansion are determined by fitting a number of experimental data

points with a minimum energy method. From the output function, the so-called Preisach function

can be determined in a closed form.

A comprehensive application of wavelet-analysis concepts to system-identification problems has been

given by Le and Argoul [91]. They have developed closed-form expressions to compute the damping

ratio, the natural frequency and the shape of each mode, based on ridges and skeletons of the wavelet

transformed free vibration response. As an alternative, Yin et al. [92] have proposed to apply the wavelet

transform to the frequency response function (FRF) of the system. Specifically, given the FRF of an

N-DoF system in the form

HðvÞ ¼

XN

r¼1

Ar

iv 2 lr þ

Ar

iv 2 lr

" #

ð11:74Þ

where lr is the rth complex pole and Ar the rth residue, a complex fractional function

cy ðxÞ ¼

1

ð1 þ ixÞyþ1 ¼ e2ðyþ1Þlogð1þixÞ; y [ Rþ ð11:75Þ

11-16 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

is selected as a wavelet basis. Based on Equation 11.75, a closed-form expression can be established for the

CWT of Equation 11.74 multiplied by ð

ffiffi

paÞ2y : Specifically,

Hy ða; bÞ ¼ a2ðyþ1Þ=2

ð1

21

HðvÞc

–

y

v 2 b

a

􀀏 􀀐

dv

¼ 2paðyþ1Þ=2

XN

r¼1

Ar

ða þ ib 2 lr Þyþ1 þ

Ar

ða þ ib 2 lr Þyþ1

􀁻 !

ð11:76Þ

Natural frequencies and damping ratios can be estimated by locating the maxima of Equation 11.76 in

the ða; bÞ plane.

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