4 Hybrid Substructure Assembly Techniques for Efficient and Robust. . . 37 where vk represents noise which is correlated with the output. Based on this perception of a disturbed linear in- and output relation in time domain, several non-parametric approaches to estimate the FRF exist. Note that the non-parametric approaches will yield a non-symbolic transfer functions of the formG∈Cp×o×nf for an evaluation at nf frequencies. Empirical Transfer Function Estimate (ETFE) The empirical transfer function estimate is based on the Discrete Fourier TransformationU(ωk) ∈ Co andY(ωk) ∈ Cp of the measured input and output signal, which are evaluated at nf frequencies ωk withk ∈ {1, . . . ,nf}. The estimate is given by division of the transformed quantities: GETFE(ωk) =[gij(ωk)], gij(ω) = Yi(ωk) Uj(ωk) , i ∈ {1, ..,p}, j ∈ {1, . . . ,o}. (4.2) For arbitrary inputs and in the absence of noise, this expression is still an estimate of the actual FRF. The ETFE is only accurate for periodic inputs. In addition, the ETFE is not consistent—the variance of GETFE does not decrease as the number of samples rises [17, p. 556]. This leads to approaches based on smoothing. Smoothing, Non-parametric Approaches Several approaches exist that are based on smoothing operations. Two major ideas can be found: one idea is to smooth the ETFE directly. This can be achieved by weighted averages at each frequency or by Welch’s averaging process, which divides the available time data into segments, see [17, p. 561] and [20]. A different approach is based on spectral analysis (SPA), where smoothed estimates of the power spectral density of the input ¯Suu(ωk) and the cross spectral density of the output ¯Syu(ωk) are used. This leads to GSPA(ωk) = ¯Syu(ωk)¯S−1 uu (ωk). (4.3) Like the smoothing of the ETFE, the estimates of the spectral quantities can be calculated by weighted averages or Welch averaging. The latter one leads to the so-called H1 estimator. Another approach is to use the Blackman-Tukey method [2, 3]. This approach will be used for the subsequent comparison. 4.2.1.2 State-Space Identification Compared to the non-parametric approaches, the objective of state-space identification is to find a state-space model of order n that can be used to explain the observed in- and output samples. In this case, the parameters are the state-space matrices A∈Rn×n, B ∈Rn×o, C∈Rp×n, D∈Rp×o andK∈Rn×p of the time-discrete state-space innovations form [17, pp. 659] xk+1 =Axk +Buk +Kek, yk =Cxk +Duk +ek, (4.4) where xk ∈ Rn is an unknown state and ek ∈ Rp is a stationary disturbance. Additional to estimating the matrices A,B,C,D,K, the noise variance E(eke T k ) is identified. For the identification of state space models, two main strategies are available. On the one hand, the matrices can be determined via subspace methods, which are based on non-iterative projections, which makes them computationally efficient [4]. On the other hand, predictor-error methods can be applied, which results in the minimization of an objective function. This ultimately leads to the application of iterative solution methods which is likely to increase the computational effort, yet can yield more accurate results. Both approaches are sketched in more detail below. Subspace Identification (SID) The basic process of SID is summarized in [10, p. 341]. Based on the given in- and output data, the extended observability matrix of the system can be estimated by projections. This matrix leads toA, C, and the noise parameters. With this knowledge, the matrices Band Dcan be determined. In this framework, several different algorithms, which use different weighting matrices in the estimation process to determine Aand C were developed. In this context, the algorithms MOESP [19], CVA [9] and SSARX [5] are mentioned, since these alternatives are available off-the-shelf in MATLAB. Prediction-Error Methods (PEM) Prediction-error methods are based on determining a set of parameters such that =argmin N ! k=0 l(yk − ˆyk( )), (4.5) where l is a scalar function that measures the prediction error y − ˆy( ) [10, pp. 199]. The function l can be chosen such that the result is estimated for example in a least squares or maximum likelihood sense. For state spaces, the number of
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