COMPEL 17,1/2/3. This work was supported by the Greek General Secretariat of Research and Technology through the PENED 94 research project.

Transcription

1 382 Non-destructive testing of layered structures using generalised radial basis function networks trained by the orthogonal least squares learning algorithm I.T. Rekanos, T.V. Yioultsis and T.D. Tsiboukis Department of Electrical and Computer Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece Keywords Conductivity profile, Electromagnetics, Least squares, Non-destructive testing Abstract The evaluation of the conductivity profile of layered metallic structures is performed via the inversion of the impedance of a circular air cored probe coil of rectangular cross section. The inversion approach is based on the implementation of generalised radial basis function neural networks. The choice of the size of the network and the evaluation of its weights are handled by the orthogonal least squares learning algorithm. The merits of the proposed method are illustrated in the light of two examples concerning non-destructive testing applications. COMPEL The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, Vol. 17 No. 1/2/3, 1998, pp MCB University Press, Introduction The application of neural networks to the inversion of electromagnetic field measurements for the estimation of the geometric or source parameters, that affect the forward electromagnetic field solution, has attracted a lot of interest over the last years. Actually, the most applied neural network architecture that appears in the literature is related to the multilayer perceptrons (MLPs)[1]. Recently, the implementation of radial basis function (RBF) networks for inverse electromagnetic problems has been investigated[2,3]. These networks offer an alternative to the multilayer perceptron neural networks. Their implementation is closely related to the Tikhonov sa regularisation theory and is much simpler and faster compared to the back propagation training algorithm of the MLPs. In this paper, the selection of the RBF network architecture and its training process is approached via the orthogonal least squares algorithm[4]. The applicability of the RBF networks to electromagnetic inversion is investigated in the case of impedance inversion in eddy current testing of layered metallic planar structures[3]. In a first example, the RBF network is trained to identify the unknown thickness of a metallic plate and the lift-off of This work was supported by the Greek General Secretariat of Research and Technology through the PENED 94 research project.

2 the probe coil. This lift-off can simulate a non-conducting coating like paint, that covers the metallic layer. A second example involves a two layer metallic structure separated by air, simulating a double layer skin of an airframe. The RBF network is trained to estimate the corrosion that takes place on the surfaces in the unreachable region between the metallic layers. The training is based on simulated data, involving the impedance of the probe coil, that is measured for various excitation frequencies. The validity of the inversion method is tested using a new set of analogous data. The data that compose the training and the testing sets are obtained numerically. Generalised radial basis function networks orthogonal least squares Let us assume two sets of N points x i R p and y i R q, i = 1,, N, that are related via the function f: R p R q. The objective of a generalised radial basis function network is to estimate the function f, using the knowledge of the pairs of vectors (x i, y i ). The basic assumption concerning the estimate, F(x i ) = Y i, of the function f, is that the values of y i are approximated by weighted sums of M radial basis functions. Thus: where w jm are the weights, u( ) the radial basis functions, and c m the centres of the functions. A very frequently used type of radial basis function is the Gaussian one: The crucial point during the construction of the network is the choice of its size, M, and the location of the centres, whereas the variances σ m of the Gaussian functions are given by: where d = max{ c i c j }. The original RBF network requires as many centres as data points, N, which is impractical. If the number of data is large, this could lead to ill-conditioning in the case where the centres are located close to each other. For this reason, the orthogonal least squares (OLS) algorithm has been applied for the construction of generalised RBF networks, where the number of centres is much lower than the number of available data points. The OLS method can be employed as a forward regression procedure to select an appropriate set of centres from a large set of candidates. At each step of the algorithm a new centre is selected, based on the requirement that the explained variance of the desired output is (1) (2) (3) Non-destructive testing of layered structures 383

3 384 maximised. The desired output of the RBF network can be written in a matrix form as follows: (4) where: y (j) = [y 1j y 2j y Nj ]T, Y (j) = [Y 1j Y 2j Y Nj ]T, U = [u 1 u 2 u M ], u m = [u m1 u m2 u mn ]T, u mi = u( x i c m ), w (j) = [w j1 w j2 w jm ]T, e (j) = ]e 1j e 2j e Nj ]T represents the error vector of the estimation, and I is the N N unitary matrix. By setting w j0 equal to the mean value of the jth component of the desired output, and d (j) = y (j) w j0 I we obtain: d = Uw + e 1 j q (5) The objective of the OLS algorithm is to transform the columns u m of the matrix U into a set of orthogonal basis vectors, in other words to decompose U into: (6) where the columns of V are orthogonal and A is an M M upper triangular matrix with units on its diagonal. This decomposition is obtained using the classical or the modified Gram-Schmidt orthogonalisation algorithm. Since the columns of V are orthogonal we have: and: where: According to (7), the summed variance of the output vector, with respect to all the q components, is given by: where: ( j) ( j) ( j ), The term: (7) (8) (9) (10) (11) (12)

4 represents the reduction of the approximation error after the choice of the centre c m for the construction of the network. The vector v m and its corresponding centre c m should be selected in such a way that the term z m is maximised during the mth iteration of the orthogonalisation algorithm. Thus, if ρ is the threshold of the approximation error, the centres are selected from the large set of candidates, by successive decompositions of U, until: Finally, the weights of the RBF network are given by: (13) (14) Non-destructive testing of layered structures 385 Applications The general configuration of the non-destructive problems that have been investigated consists of an air-cored probe coil located over a layered conducting structure (Figure 1). The complex impedance of the coil, Z, is a function of the parameters of the structure and the coil itself: Z = F(r 1, r 2, l 1, l 2, σ 1, µ 1, d 1, σ 2, µ 2, d 2,, σ L, µ L, f ), (15) where r 1 is the inner radius of the coil, r 2 its outer radius, l 1 the lift-off, l 2 the length of the coil, σ n, µ n, d n the conductivity, the relative permeability and the thickness respectively of the nth layer and f the excitation frequency. The impedance difference of the probe coil Z produced due to presence of the metallic layers can be evaluated numerically [3]. In the first application, the layered structure was composed of only one layer. The parameters were set (r 1 = 5.35mm, r 2 = 7.7mm, l 2 = 2.3mm, σ =1.39MS/m, µ = µ 0 ), whereas l 1 was allowed to vary from 0 to 1.5mm and d from 0.5 to 3.0mm. For a variety of frequencies (10, 20, 40KHz), training sets of 1,000 input-output Figure 1. Geometric configuration

5 386 vectors were created. The input and output vectors were [Re{ Z} Im{ Z}] T and [l 1 d] T respectively. The threshold of the approximation error, ρ, was set equal to Testing sets of new 500 input-output vectors were used to validate the generalisation ability of the RBF networks. Results of the RBF network construction procedure and its generalisation ability are presented in Table I. For an excitation frequency equal to 10KHz, the estimated versus the real values of the outputs over the testing set are illustrated in Figures 2 and 3. In the second application, the layered structure consisted of two metallic layers 1 and 3, separated by air (layer 2). The thicknesses of the conducting Table I. Results of the training and the generalisation procedures for the two applications First application Second application Excitation frequency f (KHz) Number of centres M Mean square error Figure 2. thickness d Figure 3. lift-off l 1

6 layers d 1, d 3 are supposed to decrease due to corrosion. The corrosion takes place at the lower surface of layer 1 and at the upper surface of layer 3. The known parameters were r 1 = 5.35mm, r 2 = 7.7mm, l 1 = 0.5mm, l 2 = 2.3mm, σ = 35.4MS/m, µ = µ 0, whereas the initial thicknesses of the layers (without corrosion) were d 1 = 2mm, d 2 = 1mm, d 3 = 2mm. The corrosions δd 1, δd 3 of the conducting layers could vary from 0 to 0.5d 1 and 0 to 0.5d 3 respectively. Similarly, training and testing sets were created for a variety of excitation frequencies (0.5, 1.0, 5.0KHz). The output vectors had the form of [δd 1 δd 3 ] T, whereas the error threshold, ρ, was set equal to Results of the training and validation procedures are shown in Table I. For an excitation frequency of 0.5KHz, estimated versus real values of the corrosions, over the testing set, are illustrated in Figures 4 and 5. Conclusions The generalised radial basis function networks have been applied to the evaluation of the conductivity profile of layered metallic structures. The Non-destructive testing of layered structures 387 Figure 4. corrosion δ d 1 Figure 5. corrosion δ d 3

7 388 networks were constructed using the orthogonal least squares algorithm that enhances their efficiency. The proposed approach could provide a robust tool for other classes of inverse problems as well. References 1. Hoole, S.R.H., Artificial neural networks in the solution of inverse electromagnetic field problems, IEEE Trans. Magnetics, Vol. 29 No. 2, March 1993, pp Ishikawa, T. and Matsunami, M., An optimization method based on radial basis function, IEEE Trans. Magnetics, Vol. 33 No. 2, March 1997, pp Rekanos, I.T. and Tsiboukis, T.D., Electromagnetic field inversion based on generalized radial basis functions, The 4th Int. Workshop on Optimization and Inverse Problems in Electromagnetism, Brno, Czech Republic, June 1996, p Chen, S., Cowan, C.F.N. and Grant, P.M., Orthogonal least squares learning algorithm for radial basis function networks, IEEE Trans. Neural Networks, Vol. 2 No. 2, March 1991, pp