Validation on a Simulated Structure

For particularly large problems, typically requiring a segmentation number in excess of 1000, evaluating the generating model at some frequency is extremely costly in computational effort. It therefore supports maximum inference of error characteristics from successive comparisons and approximations of fitting models. Consequently, the global mismatch error function of (7.3.2) is used.

7.3.2 An Iterative Strategy

FDPM as defined in section 6.3.2 is purely capable of filling a transfer function matrix based on a number of samples and solving for the numerator and denominator coefficients. This solution gives a rational function analytical approximation to the system defined by those P points. Since P = 2N defines N poles and associated residues, this assumes that there are

N poles required to correctly characterise the frequency response of the given structure (c.f.

section 7.2.5). By using the continuous error measure of section 7.3.1, a relative error in the two FM approximations is given; specific problems related to this global approximation are given in section 7.3.4. Each continuous mismatch error function ∆M M_ij has 1 error maxi- mum; it is thus deduced that the GM is least accurately defined at that frequency. The GM therefore needs to be evaluated at that frequency and new, more accurate FMs defined. Each new GM sample regenerates at least 2 FMs. The iterative successive-approximation method, otherwise known as adaptive sampling is hence defined. For the implementation discussed in this document, the frequency corresponding to the largest error across all mismatch functions defines the new sample to be obtained from the GM.

A minimum of 4 GM samples is required to allow 2 FMs to be generated. The maximum number of samples permitted for any FM is limited by the condition number of its matrix; for the target system used in this investigation, the largest condition number is roughly 9 × 1013_.

At the GM samples used for developing the FMs, the mismatch error will be approximately zero (c.f. section 7.1.2). Due to the large condition numbers of the matrices used to calculate the rational function coefficients, the coefficients are limited in accuracy by computational precision. The cumulative error in the entire rational function polynomial expansion may result in numerical errors at the original GM samples.

The sample selection and update strategy based on the global, approximate error measure of section 7.3.1 is iteratively performed until the mismatch error floor is reached. The error floor, typically of the order of 10−3 indicates that the maximum mismatch error between all

overlapping models is less than the floor, in this case 10−3_.

The output of the adaptive sampling TFE module is thus 1 or several FMs defined over the frequency band of interest, which accurately describes the system response for the parameter concerned. The accuracy is of course determined by the chosen error floor and the total num- ber of iterations required to reach this floor; it is thus dependent on the degree of resonance of the response.

7.3.3 Validation on a Simulated Structure

1 2 3 4 5 6 7 8 9 10 0 0.2 0.4 0.6 0.8 1 1.2 frequency units Real Compo nent

Original Generating Model Fitting Model M1 Fitting Model M2

Figure 7.4: The original GM with continuous fitting models M1 and M2 superimposed

The error measures of section 7.3.1 are developed using comparisons of overlapping models over some frequency range. The simulated transfer function of section 7.2.1 was used as the platform for this evaluation. The known, real frequency information is at fi = 3.5 + i0.5 for

i = 0, 1, . . . , 14. For discussion purposes, consider 2 models, M1 and M2, characterized by the

frequency samples fi at which the parameter information is used to construct the FMs:

M1 = {3.5 4 4.5 5 5.5 6}

M₂ = {3.5 4 4.5 5 5.5 6 6.5 7 7.5 8}

The overlapping region is the closed interval [3.5 6]. Figure 7.4 depicts the approximation of fitting models M and M to the actual GM, where both fitting models are evaluated

3.5 4 4.5 5 5.5 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 frequency units Mis match Erro r Mismatch Error ∆MM12 GM Exact Error ∆GM12 GM Exact Error ∆GM21

Figure 7.5: The mismatch error function ∆M M12 is compared with the exact, point-wise approximations of ∆GM12 and ∆GM21

over the range [1 10]. Both the exact point-wise and global approximate error measures of section 7.3.1 are evaluated. In figure 7.5, the green (–) and red (-.) curves are ∆GM12 and

∆GM₂₁ respectively, while the black (–) curve is ∆M M₁₂. For the mismatch error function, ∆M M12, the next sample is required at frequency 5.27. Hence, the new models have samples

-

M1 = {3.5 4 4.5 5 5.27 5.5 6}

M₂ = {3.5 4 4.5 5 5.27 5.5 6 6.5 7 7.5 8}

Similarly, for two models overlapping the range [8.5 11], the error measures are given in figure 7.6. The fitting models generated after the first iteration have significantly improved accuracy (comparing figures 7.4 and 7.7). Performing a further iteration, accuracy is still further improved. As at the second iteration, the mismatch error function is depicted in figure 7.8. The impact of the effectiveness of this algorithm is obvious by comparing the mismatch error function of the initial models (figure 7.5) with those of the FMs after 2 iterations (figure 7.8).

8.5 9 9.5 10 10.5 11 0 0.2 0.4 0.6 0.8 1 1.2 frequency units Mis match E rr or Mismatch Error ∆MM12 GM Exact Error ∆GM12 GM Exact Error ∆GM21

Figure 7.6: The error measures of the ∆GM12, ∆GM21 and ∆M M12 after 1 iteration

3 3.5 4 4.5 5 5.5 6 6.5 0 0.2 0.4 0.6 0.8 1 1.2 Frequency units R eal c omponent

Original Generating Model Fitting Model M1 Fitting Model M2

Figure 7.7: The original GM with continuous fitting models M1 and M2 superimposed, after 1 iteration

3.5 4 4.5 5 5.5 6 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 frequency units Mis match Error Mismatch Error ∆MM12 GM Exact Error ∆GM12 GM Exact Error ∆GM21

Figure 7.8: The error measures of the ∆GM12, ∆GM21 and ∆M M12after 2 iterations