Identification procedure by total least-squares

Transcription

1 Identification procedure by total least-squares Andrea Malengo and Francesca Pennecchi Istituto Nazionale di Ricerca Metrologica, Italy 7th Workshop on Analysis of Dynamic Measurements Paris - October 15-16, 1

2 Accelerometer model Second-order differential equation with unknown parameters ω, δ, ρ, modelling the output of an accelerometer. The transfer function Gω is G j S e j, where Sω and φω are the measured magnitude and the phase delay of the system, obtained during sinusoidal excitation calibration. : resonant frequency :damping : transformation constant of the input acceleration

3 Parameter identification Link & all, Modelling accelerometers for transient signals using calibration measurement upon sinusoidal excitation [Measurement 4 7, pp ]: weighted linear least-squares applied to Gω -1. Our proposal: total least-squares TLS directly applied to rationalized Gω. Note: Inversion of Gω and, hence, reverse determination of the original parameters are avoided; Uncertainties associated not only with measured Sω and φω, but also with the independent variable ω can be taken into account; Possible information on someone of parameters ρ, δ, ω expressed, for example, in form of an estimate and the associated uncertainty, can be taken into account.

4 Total least-squares TLS Let us consider the general fitting model y = fx, p, with p = p 1,, p k the vector of parameters to be estimated. The sum to be minimized is χ = dx V x -1 dx T + dy V y -1 dy T, where x = x 1,, x n and y = y 1,, y n are the observation vectors having covariance matrices V x and V y, respectively; dx = x 1 - X 1,, x n - X n and dy = y 1 - fx 1, p,, y n - fx n, p are the residual vectors; X = X 1,, X n is the parameter vector to be estimated together with p.

5 TLS cont. In general, the χ function is non-linear in its parameters p, X and a numerical solution is necessary for its minimization. We implemented an algorithm based on the MATLAB function fminunc.m, available within the MATLAB Optimization Toolbox. The algorithm can deal with any model function fx, p and any form of the covariance matrices V x and V y. A relevant paper on the algorithm is under development and will be summitted to Metrologia.

6 Application to accelerometer calibration x = ω 1,, ω n and V x = diagu ω S = Sω 1,, Sω n and V S = diagu S φ = φω 1,, φω n and V φ = diagu φ G = Sω 1 expjφω 1,, Sω n expjφω n y = ReG, ImG V y = cov[rey MC Imy MC ], where y MC are values obtained with Monte Carlo method applied on S and φ p = Ω 1,, Ω n, ω, δ, ρ 4 4,, ; j j j j G Rationalized transfer function j e S j G

7 Application to accelerometer calibration cont. The χ function is written in a m-file chisquare.m which is passed to fminunc.m as an input, together with a starting point of parameter estimates x [par_opt, fval] = fminunc@chisquare, x, optimset ; par_opt: parameter estimates; fval: minimum of χ function Function chisquare.m minimizes χ = dx V x -1 dx T + dy V y -1 dy T, where dx = ω 1 - Ω 1,, ω n - Ω n dy 1,,n = y 1 -ReGΩ 1, ω, δ, ρ,, y n ReGΩ n, ω, δ, ρ, with Re G ;,, 4 dy n+1,,n = y n+1 -ImGΩ 1, ω, δ, ρ,, y n ImGΩ n, ω, δ, ρ, with Im G ;,, 4

8 Example 1 n = 1, simulated data k = 3, number of parameters u ω = π.5 rad/s Expected χ = n k = 17 Results f 5.79 khz u f.49 khz u S u S obs pc/ms pc/ms - - From Table 1 in [Measurement 4 7, pp ] Frequency u rel S 1-3 uφ/deg range/khz <= ,1] ,15] , ] 5.5

9 Uncertainty evaluation by MC method For each ω i, calculate Sω i and φω i satisfying the estimated Gω i For each ω i, generate M values: ω ik, k = 1,, M, drawn from normal distributions having parameters ω i and u i S ik, k = 1,, M, drawn from normal distributions having parameters Sω i and u Si φ ik, k = 1,, M, drawn them from normal distributions having parameters φω i and u i For each k = 1,, M, apply TLS to the data set, obtaining ω k, δ k, ρ k Take the uncertainty for parameters ω, δ, ρ as the standard deviation of the numerical distributions

10 TLS when information on a parameter is available Let us assume that an estimate of a model parameter, with an associated uncertainty, is available before the measurement. Let us consider, for example, we know ω * with uω *. Hence, ω * is considered among the other uncertain observations: x = ω 1,, ω n, ω * and V x = diagu ω 1,, u ω n,u ω * dx = ω 1 - Ω 1,, ω n Ω n, ω * ω dy unchanged V y unchanged Expected χ = n k +1 The method is generalizable to more than one parameter for which information is available.

11 Example Same data and uncertainties as in Example 1 Expected χ = 18 Several values simulated for f * and uf * Example 1 f 5.79 khz,.63, u.63 S obs.5471 pc/ms u f.49 khz -, u S.67pC/ms - Case 1 Case Case 3 Case 4 f * uf * f * uf * f * uf * f * uf * khz est. unc. est. unc. est. unc. est. unc. f / khz δ S / pc/ms χ

12 Conclusions Main advantages of the proposed method are: It is applicable to any function even nonlinear in the model parameters. In the considered case of accelerometer calibration, the generality of the approach may also allow to work with more complicated expressions for Gω. Uncertainty associated not only with the dependent variable, but also with the independent variable can be taken into account. Possible information on someone of the model parameters can be straightforward taken into account, numbering such available estimates with the associated uncertainty along with the experimental uncertain measurements.

13 Future developments Comparison of the performance of WLS and TLS on the same data set. Application of TLS method to accelerometer calibration results upon shock excitation. Application of TLS method to more complicated expressions for transfer function Gω, modelling, for example, the influence of the frequency response of the charge amplifier.

14 The research leading to these results has received funding from the European Union on the basis of Decision No 91/9/EC. Thank you for the attention