AN OPERATIONAL ANALYSIS TECHNIQUE FOR THE MIMO SITUATION BASED ON CEPSTRAL TECHNIQUES

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1 AN OPERATIONAL ANALYSIS TECNIQUE FOR TE MIMO SITUATION BASE ON CEPSTRAL TECNIQUES Wee-Lee Chia, R. B. Randall, J. Antoni, School of Mechanical and Manufacturing Engineering The University of New South Wales, Sydney 05, School of Mechanical and Manufacturing Engineering The University of New South Wales, Sydney 05, Lab. Roberval, The University of Technology of Compiègne, Compiègne, Australia Australia France Abstract This paper loos at blind system identification leading to operational modal analysis using a MIMO (multiple input, multiple output version of the mean differential cepstrum (MC for a series of random transient inputs (eg burst random in the presence of other noise. If the inputs X are white, the system from the generic system equation: Y X can theoretically be identified by utilizing its output vector Y only. By maing certain assumptions, it is possible to derive the matri differential equation: ' 0, where is a matri analogous to the frequency domain version of the mean differential cepstrum of Y. The use of Singular Value ecomposition: U V where U and V are unitary (rotation matrices and is a diagonal (scaling matri, and applying this mathematical form in the calculation of the system s cross spectral matri S YY yields U U (where the superscript means ermitian transpose. Only the unitary matri V remains to be identified, and this can theoretically be derived from a different matri differential equation V' zz V 0 (where Z U Y V X, which should be easier solve as V should be better conditioned than. There is no closed form solution to these matri differential equations but numerical solutions are being sought using finite difference methods. Eperimental results have been obtained from a simple two-input-two-output test rig. The paper describes the results obtained from processing these signals. 1 Bacground on Mean ifferential Cepstrum (MC The ey advantage of the cepstrum is its ability to separate source and transmission path effects [1] (i.e. its deconvolutive or homomorphic property due to the logarithmic nature of its formulation. Its definition varies according to formulation but we are only concerned with the definition of the Comple Cepstrum. Page 1 of 8

2 1.1 Comple Cepstrum The comple cepstrum is the inverse Fourier Transform of the logarithmic spectrum: 1 C( τ I [lny ( ] Y ( I[ y( t] A( e t : time lny ( ln A( + jφ( ω : frequency τ : quefrency (same dimensions as time A ( : Amplitude I : Fourier transform φ ( : Phase jφ ( The homomorphic nature results from the logarithmic operation, since forcing and transfer functions that are convolutive in the time domain and multiplicative in the frequency domain, are additive in the log spectrum and cepstrum (SIMO only. One drawbac is the requirement for the phase function f(w to be a continuous function of frequency rather than the principal value modulo p as is normally measured [1] and hence the need for phase unwrapping. In minimum phase systems, the phase is simply the inverse ilbert transform of the log amplitude, and thus does not have to be measured or unwrapped. ( ifferential Cepstrum (C The differential cepstrum is the inverse Fourier transform of the derivative of the logarithmic spectrum for a deterministic signal: 1 d 1 Y ( d y ( τ I lny ( I (1. dω Y ( The homomorphic property of the C is coupled with the added advantage that it can be directly calculated from the time signal[], so that no phase unwrapping is required. This is not applicable to random/stochastic signals but is the basis for the formulation of the MC for the purpose of this application. 1.3 Mean ifferential Cepstrum (MC The Mean ifferential Cepstrum is the inverse Fourier transform of the partial derivative of the logarithmic spectral correlation density (SC [3] which can be shown to give: d (1.3 1 ( τ I ( ( * ( Y ( Y ( * ( Y( Y ( where ( is the epected value which is identical to Eq. (1. if the signal is deterministic. The MC is developed for stochastic/random signals, allowing for ensemble averaging to remove noise. Lie the C, it retains the homomorphic feature. see Appendi A Page of 8

3 1.4 MIMO version of MC X( 1 ( y Y( y 1 Input assumptions: (i mutually uncorrelated Y ( ( X( (ii broad band ecitation and (iii by convention has unitary power (scale indeterminacy such that the Cross Spectral Matri of the input, S XX (XX I (identity matri In a MIMO system where Y is the Fourier components vector of the measurement at frequencyω, the transfer matri can be separated from input X using the MC property. For simplification, that is being solved for, is a time shifted version of the actual system. By analogy to the previous section, the MC in a MIMO system can be defined such that (frequency domain: Y Y YY (1.4 Theoretical evelopment ( ( ([ X X ] X ( [ X] X ( ( ( YY YY + XX + XX XX [ jc] [ jc] [ jc] [ jc] + [ ] ( ( I where superscript denotes hermitian transpose The MIMO version of the MC in Eq. (1.4 can be simplified using the input assumptions to formulate the matri differential equation which theoretically gives access to the system matri ; as mentioned earlier it is a time shifted version of the actual system matri. In an attempt to mae the matri differential equation better conditioned (resonance peas, singular value decomposition (SV is used to transform the problem into two secondary equations..1 Matri differential equation of In the quefrency domain, the pure imaginary diagonal matri [ jc] corresponds to time displacements, which can be set to zero. Note that input X is assumed to be mutually uncorrelated, broad band and by convention have unit power, meaning that differences in scaling are assigned to the transfer functions. ence in its cross spectral matri, the cross terms are zero and lie terms are unity i.e. an identity matri. I (.1 see Appendi A Page 3 of 8

4 . Singular value decomposition (SV Y X using SV: U V where [U] and [V] are unitary matrices U Y VX [] is a diagonal matri with real values S Y Y YY ( ( UV X X V U ( S I U VV U I Let UV XX V U Z U Y V X Matrices U and can be solved using eigenvalue decomposition (EV: (. S YY U U (.3 Application of MC to Eq. (. (frequency domain, ( ( ( [ VX VX ] XV + I ( ( V XX V + V XX V SXX I jc[ ] V V + 0 V - V 0 (.4 Similar to section.1, the pure imaginary time shift constant[ jc] can be set to zero. The resulting matri differential equation in terms of unitary matri V should be easier to solve than Eq. (.1. Its resolution can theoretically be obtained through a finite difference operation performed frequency by frequency. This together with U and from Eq. (.3 will allow the determination of. Page 4 of 8

5 .3 Transformation of V - V 0 for numerical application Bacward difference: V V [ V( ω - V( ω ] ( ω. V ( ω ( d dω. ( ω V( ω. V ( ω - V( ω. V ( ω I - V( ω. V ( ω d V( ω. V ( ω I - ω. ( ω owever, this numerical scheme does not enforce V( ω to be unitary Finite difference approimation; if dω is small, V( ω e. V( ω (.5 It can be proven that j ( ω j( j is hermitian and therefore e is unitary. Solution based on V( being unitary will ensure that V( is always unitary and also stable. Note that the first ω ω jq term on the RS of the Eq. (.5 taes the form e (where Q is a hermitian matri and ( d is simply a scaling factor. ereinafter Eq. (.5 will be nown as the Chain Equation method. 3 Eperiment Set-Up V ( ω V ( ω - dω. V ( ω. ( ω ( ω ( ω - dω. ( ω. ( ω V V V ( d I - ω. ( ω. V( ω - dω. ( ω ( e j( - d. j ( ω ( V( ω. V( ω The eperimental rig used is a cantilever beam with two lumped masses as shown in Figure 1, representing a simple Two-Input-Two-Output (TITO system. The input shaers (s1, s introduce ecitation to the system through the force transducers (f1, f. The output signals at the lumped masses are measured with the accelerometers (a1, a. 485 s1 s f1 f a1 30 a 15 Figure 1 Eperimental Set-up 70 Material: Steel (All dimension in mm see Appendi A Page 5 of 8

6 4 Results/Conclusion Figure (a - Results from Joint iagonalisation Method [3] Figure (b - Results from Chain Equation Method Figure (a shows the results obtained via the Joint iagonalisation method described in the paper [3] by J. Antoni while Figure (b shows that from the Chain Equation method proposed in the previous sections. The former uses a process which solves the FRF for each frequency independently but at the same time suffers an unnown permutation for each frequency solved. The latter relies on the set of values solved for the previous frequency and consequently has no permutation problem after to the first set of estimates (i.e. zero frequency value which was obtained from the Joint iagonalisation method in this case. In both methods, estimation of the poles loos reasonable in terms of relative amplitude and position but estimation on the zeros is poor. 5 Acnowledgement This research is supported by STO, Australia. Page 6 of 8

7 6 References [1] R. B. Randall, Frequency Analysis, Brüel & Kjær, enmar, 1987 [] Y. Gao, R. B. Randall, etermination of Frequency Response Functions from Response Measurements I. Etraction of Poles and Zeros from Response Cepstra, Mechanical Systems and Signal Processing, 1996 [3] J. Antoni, F. Guillet, M. El Badaoui, F. Bonnardot, Blind separation of convolved cyclostationary processes, Signal Processing, 85 (1, Jan. 005 p51-66 Appendi A A1 Comple scaling indeterminacy and assumption of system [] in BSI problems In blind system identification (BSI problems, the resolution can only be up to a certain unnown scale factor, a. It is required to mae assumptions about the input X to achieve identification: Y X ax a a is the comple scaling indeterminacy with unnown magnitude and phase. For BSI in a MIMO scenario, mn is assumed to be a full column ran matri with equal or more response measurements (outputs than ecitations (inputs i.e. m n A Stochastic input X assumptions Autocorrelation of a stationary white signal (t as a function of time lag ( τ : R τ σ δ τ ( ( Similarly for a non-stationary white stochastic signal (t: R ( τ σ ( t δ( τ where σ ( t is the instantaneous power as a function of time, δ ( τ is a delta function as a function of time lag (In both cases, they only have non zero values forτ 0 The spectral correlation density (SC of stochastic (t is the double Fourier Transform of { τ } SXX I R (, t S ( α, 1 t ατ ω I{ σ (} { ( t I δ τ }, α 0 ω α Evaluating ( XX ( + S ( αω, S ( α ω α 0, 0 (power spectrum even S S S + j S jc α α 0 dα α 0 dα α 0 dα α 0 d d d ( αω, ( α Re { ( α } Im { ( α } ence ( XX jc[ ] 0 jc [ ] [ ] R : Page 7 of 8

8 A3 Solution for a time shifted version of [] Y( ( X( - jc ( ω Y X X e where is a time shifted version of (i ( ( - ( ( ( e ω ω ω jc ω e e e jc jc jc jc ( YY ( X + (- ( X + X ( X (ii 1 ( YY ( YY e ( XX e - jc( e ( XX e + e ( XX e jc ( jc ( - jc ( jc ( jc ( jc ( S I S I jc XX XX - jc ( ω jc - j C + j ( ω e Ce jc ( YY ( XX e ( XX e S I XX - jc( jc( ( i ( ii o e I ence which is the same form as Eq. (.1. ( ( The use of allow the simplification of the matri differential equation. Moreover, the actual can only be solved to a certain unnown scale factor as shown in A1. A4 Ensuring unitary property of V ( ω For any unitary matri V, there eists hermitian Q s.t. VV I VV 0 V V + V V 0 j V V j j V V j ( j [ j ] jq V e, n n Q Q n n n n To ensure unitary property of V ( ω, each term on the RS of the Eq. (.5 must be unitary. is SKEW ERMITIAN j is ERMITIAN Page 8 of 8