Defect detection of metallic samples by electromagnetic tomography using closed loop fuzzy pid controlled iterative landweber method

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Defect detection of metallic samples by

electromagnetic tomography using closed-loop fuzzy PID-controlled iterative Landweber method

Pu Huang, Xiaofei Huang, Zhiying Li & Yuedong Xie

To cite this article: Pu Huang, Xiaofei Huang, Zhiying Li & Yuedong Xie (12 Jan 2024): Defect

detection of metallic samples by electromagnetic tomography using closed-loop fuzzy PID-controlled iterative Landweber method, Nondestructive Testing and Evaluation, DOI: 10.1080/10589759.2024.2304256

To link to this article: https://doi.org/10.1080/10589759.2024.2304256

Published online: 12 Jan 2024.

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Defect detection of metallic samples by electromagnetic tomography using closed-loop fuzzy PID-controlled iterative Landweber method

Pu Huang, Xiaofei Huang, Zhiying Li and Yuedong Xie

Key Laboratory of Precision Opto-mechatronics Technology of Education Ministry, School of

Instrumentation and Opto-Electronic Engineering, Beihang University, Beijing, China

ABSTRACT

Electromagnetic tomography (EMT) uses the mutual inductance of

the coil to visualise the conductivity distribution of interesting

regions Since the conductivity of defects and metal samples are

different, the metal samples with defects can be treated as binary-

valued material distributions This paper investigates the closed-

loop fuzzy proportional, integral and derivative (PID)-controlled

iterative Landweber method The whole method includes fuzzy

PID controller, the Landweber reconstruction method, and the

Dirichlet-to-Neumann map Specifically, the differential signal

between the mutual inductance of the coil and the feedback signal

is used as the input of the fuzzy PID controller The fuzzy controller

can automatically adjust three parameters (K p , K i and K d) of PID

controller Subsequently, the output of the PID controller can

serve as the input of the Landweber algorithm to reconstruct the

distribution of conductivity Furthermore, the Dirichlet-to-

Neumann map is used to calculate the mutual inductance, acting

as the feedback signal based on the reconstruction conductivity

distribution Finally, both the numerical simulation and

experi-ments are applied to verify the proposed method The results

indicate that the proposed method can reconstruct the image

with a clear edge, and the average correlation coefficient can

reach 0.792.

ARTICLE HISTORY

Received 27 September 2023 Accepted 2 January 2024

KEYWORDS

Electromagnetic tomography; defect detection; fuzzy PID controller; Landweber method; image reconstruction

1 Introduction

During the service process, metal materials are prone to defects due to corrosion, compression and wear, resulting in potential safety hazards Non-destructive testing can provide effective defect information without damaging the structure of metal mate-rials [1] Compared to other technologies, electromagnetic non-destructive testing has the advantages of non-contact and high sensitivity Traditional electromagnetic non- destructive testing equipment generally adopts a single sensor, and the sensor or metal plate needs to be moved during testing, which takes a long time and requires the corresponding mechanical devices [2–5] It not only easily causes detection errors but also cannot meet the requirements of real-time performance Compared with the single

CONTACT Yuedong Xie yuedongxie@buaa.edu.cn

sensor structure, the multi-sensor detection can improve the detection accuracy and efficiency by increasing the number of sensors [6,7]

Electromagnetic tomography (EMT) is a kind of non-destructive testing technology, which has the advantages of non-contact, visualisation and fast imaging [8,9] It is widely used in the field of defect detection, multiphase flow measurement, biomedical and other fields The EMT system mainly consists of sensor arrays, the data acquisition system and image reconstruction algorithm

In fact, the ultimate goal of EMT technology is to obtain the spatial distribution of materials with different conductivities There are two key problems, i.e the forward problem and the inverse problem, that need to be solved The inverse problem of EMT is to solve the distribution of conductivity according to the measured voltage and prior sensitivity matrix, that is image reconstruction [10–13] Due to the fact that the electromagnetic sensitive field of EMT exhibits nonlinear property, it is difficult to directly solve it Meanwhile, the EMT inverse problem belongs to the Fredholm integral equation of the first kind Its solution is ill-conditioned, which limits the development of EMT In practice, image reconstruction algorithms of EMT technol-ogy can be divided into iterative and non-iterative algorithms Non-iterative algo-rithms include linear back projection (LBP), Tikhonov regularisation and truncated singular value decomposition [14–16] The LBP algorithm is simple and has fast imaging speed, but the accuracy of image reconstruction is relatively low It is suitable for online rapid qualitative imaging, but cannot provide accurate quantitative infor-mation The regularisation method is an effective method to overcome the ill-posed problem of EMT However, the parameters of the regularisation method are selected based on experience The iterative algorithms mainly include Landweber iteration, algebraic reconstruction technique (ART), simultaneous iterative reconstruction tech-nique (SIRT) and so on [17–19] The ART and SIRT are commonly used algebraic iteration methods, which require a higher number of iterations to achieve better image reconstruction results The Landweber iterative algorithm is based on the principle of steepest descent and is the most commonly used iterative algorithm for solving the inverse problem of EMT However, it requires multiple iterations and has

a slow rate of convergence In recent years, Zhang et al investigated the compatible multi-template supervised descent method to monitor the structural information of CFRP (Carbon Fiber Reinforced Polymer) [20] Liu et al designed a novel L-type sensor and three-layer array eddy current sensor combined with LBP method to inspect the defect of the wheel [21,22] Moreover, Liu et al investigated image reconstruction algorithms combining deep learning and optimised fully connected net to learn image reconstruction of EMT [23] Wang et al investigated the sparse regularisation method to improve the EMT image reconstruction quality [24] Ma and Soleimani researched the dual-plane magnetic induction tomography method to locate the damage of composite parts [25,26] Besides that, Soleimani improved the reconstruction quality of EMT image using the Kalman filtering method [27] Teniou

et al proposed the constrained Landweber algorithm to improve image reconstruc-tion, which uses both boundary data and the foreground–background fractions [28] Wang et al developed a novel EMT system based on FPGA (Field Prog ram mable Gate Array), which uses TMR (Tunnel Magneto Resistance) sensors instead of tradi-tional coils [29,30] Meanwhile, the improved Landweber iterative algorithm is

investigated to improve the quality of image reconstruction [31] Tamburrinoa et al proposed non-iterative monotonic imaging algorithm for defect detection, and the method can be simplified using the geometric symmetry characteristics of the detected object [32]

In this paper, a closed-loop fuzzy proportional, integral and derivative (PID)- controlled iterative Landweber reconstruction method is proposed The whole recon-struction method includes a fuzzy PID controller, the Landweber method and the Dirichlet-to-Neumann map The differential signal between the measurement and the feedback signal is fed into the fuzzy controller, and the fuzzy controller can adjust the parameters of PID The output of PID acts as the input of the Landweber algorithm to reconstruct the distribution of conductivity Based on the distribution of conductivity, the boundary mutual inductance can be calculated by the Dirichlet-to-Neumann map, which is the feedback signal The closed-loop structure can improve the quality of reconstructed images The proposed method can achieve three-dimensional imaging for metallic defects, which is conducive to quantitative evaluation of defect size and thus avoids the occurrence of accidents in practice

2 Fundamental methods

In EMT systems, image reconstruction algorithms reconstruct the field distribution based on boundary measurement values and sensitivity matrices The factors that affect the quality of image reproduction mainly include two parts: software and hardware systems The hardware system mainly includes the rationality, accuracy and anti- interference ability of each part of the system design The software mainly includes image reconstruction algorithms, whose performance directly determines the final ima-ging quality and is the core of EMT

If EMT is approximated as a linear system, its forward problem can be expressed as Equation (1)

The greyscale value G of the reconstructed image can be obtained by Equation (2) if the inverse matrix of S is assumed to exist

However, the inverse matrix of S cannot be directly obtained in the inverse problem of

EMT due to the ill-condition The image reconstruction algorithm can also be seen as

a process of approximating the inverse matrix of S, so the imaging accuracy is limited to

some extent [33]

The Landweber iterative algorithm has good imaging accuracy, which is widely applied to image reconstruction for EMT The Landweber iterative algorithm transforms the original problem into an optimisation problem, which can be expressed as Equation (3)

min f ðGÞ ¼1

2kSG Uk

2

(3) Equation (3) can be converted to finding the minimum value of equation (4)

f ðGÞ ¼1

2ðSG UÞ

T

ðSG UÞ ¼1

2ðG

T

S T SG 2G T S T U þ U T UÞ (4) Therefore, the Landweber algorithm takes the negative direction of the gradient as the optimisation search direction, and the iterative formula can be expressed as,

G0¼S T U

G kþ1¼G k α k S TðSG k UÞ

�

(5)

3 Fuzzy PID-controlled iterative Landweber method

Figure 1 illustrates the sketch map of the proposed fuzzy PID-controlled iterative Landweber method for image reconstruction in EMT The mutual inductances of coils are measured, and the matrix form of the discretized boundary map (Dirichlet-to- Neumann map) is established The measured signal is compared with the feedback signal and is further fed into the fuzzy controller, which can be used to adjust the parameters of PID to yield an input for the Landweber method The Landweber method reconstructs the image based on the measured mutual inductances Subsequently, the reconstructed image is normalised to obtain a feedback Dirichlet-to Neumann map for comparison with the measured signal The iteration termination is determined by the difference between the reconstructed conductivity distribution of the current iteration and the previous one When the difference of the reconstructed conductivity distribution is less than the threshold, the reconstructed conductivity distribution can be the output The closed-loop structure can ensure the convergence of the iterations, and the proposed robust method can be achieved for conductivity distribution

3.1 Fuzzy PID controller

The proposed reconstruct algorithm adopts fuzzy PID controller to reduce diver-gence between the reconstructed image and the measured Dirichlet-to-Neumann image The fuzzy PID control utilises fuzzy logic to optimise the parameters of the PID controller in real time based on certain fuzzy rules The fuzzy PID controller

Figure 1 The sketch map of the proposed PID-controlled iterative Landweber method

can overcome the disadvantage of traditional PID parameters that cannot be adjusted in real time Specifically, the deviation is input into the controller and is fuzzificated into the fuzzy set using the membership function Fuzzy reasoning is applied by following fuzzy rules to yield a fuzzy set of output Finally, it is defuzzified to update the PID coefficients The three parameters of the PID con-troller are updated during each iteration The input and output of the fuzzy PID controller can be expressed as,

Δq N�NðiÞ ¼ K pðiÞ½p N�NðiÞ p N�Nði 1Þ� þ K IðiÞp N�NðiÞ

þK DðiÞ½p N�NðiÞ 2p N�Nði 1Þ þ p N�Nði 2Þ� (6)

where i is the number of iterations K pðiÞ, K IðiÞ and K DðiÞ denote the proportional,

integral and derivative parameters of PID controller in ith iteration p N�NðiÞ is the

difference between the measured mutual inductance matrix M measure and the feedback

mutual inductance matrix M fb, which is the input of the fuzzy PID controller

Δq N�NðiÞ ¼ q N�NðiÞ q N�Nði 1Þ is the output of the fuzzy PID controller

The input of fuzzy PID controller are e1ðiÞ and e2ðiÞ, which are related with p N�NðiÞ and

can be expressed as,

e1ðiÞ ¼ pk N�NðiÞk2 (7)

e2ðiÞ ¼ e1ðiÞ 2e1ði 1Þ þ e1ði 2Þ (8)

In fact, the Gaussian membership function has the characteristics of continuity and smoothness, which can improve the accuracy of the membership function and make the output of the fuzzy controller more accurate Meanwhile, the Gaussian membership function can effectively solve ambiguity and uncertainty, such as noise interference Hence, the Gaussian membership function is adopted, and the Gaussian membership values for these two inputs are,

μ1ði; hÞ ¼ e

e1 a1ðhÞ σ1ðhÞ

� �2

(9)

μ2ði; hÞ ¼ e

e2 a2ðhÞ σ2ðhÞ

� �2

(10)

where μ1ði; hÞ and μ2ði; hÞ are the membership function of the two fuzzy PID controller

inputs e1ðiÞ and e2ðiÞ in the hth fuzzy value a1ðhÞ and a2ðhÞ are the central value of the hth Gaussian curve of e1ðiÞ and e2ðiÞ Besides that, σ1ðhÞ and σ2ðhÞ are the standard

deviations of the hth Gaussian curve.

The fuzzy rules are adopted to adjust the parameters of PID controller, i.e K p , K I and K D

Firstly, the range of input of fuzzy controller (e1ðiÞ and e2ðiÞ) and PID parameters can be

estimated by FEM (Finite Element Method) numerical solution The fuzzy sets should be able to cover the entire range of inputs and outputs Furthermore, the number of rules can be determined to cover all possible input conditions A large number of fuzzy rules can lead to overfitting, while a small number of rules may reduce the control accuracy Finally, according to the performance of the system, the parameters of fuzzy control

rules, such as membership function parameters, need to be adjusted using FEM numer-ical solution

Assuming that the hth fuzzy value contains a total of m combinations, the membership degree of the hth fuzzy value is

μ outði; hÞ ¼

Xm j¼1

μ1ði; hðjÞÞμ2ði; hðjÞÞ

Pm j¼1

e

e1 a1ðhÞ σ1ðhÞ

� �2

e

e2 a2ðhÞ σ2ðhÞ

where μ outði; hÞ is the membership value of output parameters, and μ1ði; hðjÞÞ and

μ2ði; hðjÞÞ denote the membership value of e1ðiÞ and e2ðiÞ in the hðjÞ fuzzy value.

The centroid method is used for defuzzification, which can be expressed as,

μ outðiÞ ¼X

n h¼1

μ outði; hÞλðhÞ

,

Xn h¼1

μ outði; hÞ (12)

where n is the number of fuzzy values and λðhÞ is the hth fuzzy value of output The

μ outðiÞ is the output of the fuzzy controller in the ith iteration Hence, K pðiÞ, K IðiÞ and

K DðiÞ of PID controller in the ith iteration can be calculated as,

K PðiÞ ¼ Pn

h¼1

μ Poutði; hÞλ PðhÞ

�

Pn h¼1

μ outði; hÞ

K IðiÞ ¼ P

n h¼1

μ Ioutði; hÞλ IðhÞ

�

Pn h¼1

μ outði; hÞ

K DðiÞ ¼ P

n h¼1

μ Doutði; hÞλ DðhÞ

�

Pn h¼1

μ outði; hÞ

8

>

>

>

>

>

>

(13)

where μ Poutði; hÞ, μ Ioutði; hÞ and μ Doutði; hÞ are the hth membership values of PID

con-troller parameters K pðiÞ, K IðiÞ and K DðiÞ in the ith iteration λ PðhÞ, λ IðhÞ and λ DðhÞ are

the corresponding fuzzy values

3.2 Iterative Landweber method based on a fuzzy PID controller in the closed-loop control

The Dirichlet-to-Neumann map can be expressed as a mutual inductance matrix

mea-sured on the coil M measure when the conductivity distribution of sensing region is σðzÞ

@ n

�

�

@Ω

(14)

where z ¼ x þ yi is the coordinate of point ðx; yÞ @Ω is the boundary of sensing region, and n is the unit normal vector of region boundary.

Apart from that, M0 denotes the mutual inductance matrix of coil when the conductivity

of sensing region Ω is conductivity of metal samples The difference signal

ΔM ¼ M measure M0, i.e variation of mutual inductance matrices, is the input of the

reconstruction algorithm In addition, the Dirichlet-to-Neumann map can be calculated

by the finite element model

Since the reconstructed image of metallic samples with defects can be regarded as binary image, if only step function is used to achieve binarization, it will lead to closed-loop instability Considering that the output of sigmoid function ranges from 0 to 1 and the sigmoid function has the characteristic of smooth output, it can be applied to binariza-tion of the reconstructed conductivity distribubinariza-tion The sigmoid funcbinariza-tion can be writ-ten as,

G BðiÞ ¼ 1

where G BðiÞ and G NðiÞ are the reconstructed conductivity distribution before and after

normalisation a is the coefficient of the sigmoid function.

Figure 2 depicts the flow chart of the proposed closed-loop PID reconstruction algorithm, and specific details are as follows:

(1) Mutual inductance matrices of metal samples M0 and measured field M measure are

obtained by EMT sensor Furthermore, the difference signal ΔM ¼ M measure M0

is calculated, which is the input of the proposed method Meanwhile, the para-meters of fuzzy PID controller are initialised

(2) The ith feedback mutual inductance matrices M fb can be calculated according to

ith reconstructed conductivity distribution using the Dirichlet-to-Neumann map

The corresponding mutual inductance variation ΔM0 is obtained by subtracting

M0 from M fb Above all, the ith mutual inductance difference matrix p N�NðiÞ

between the ΔM and ΔM0 is acquired, which is input to the fuzzy PID controller

and outputs Δq N�NðiÞ based on Equation (6).

(3) The Landweber algorithm is applied to reconstruct conductivity distribution, and

the conductivity distribution G BðiÞ can be normalised by the sigmoid function

The iterations can be terminated until the k G BðiÞ G Bði 1Þ k2 is smaller than

threshold T, and the final reconstructed conductivity distribution G Bð Þi is output

4 Simulation and discussion

In this section, the simulations are carried out to determine the parameters and evaluate the proposed method As shown in Figure 3, the EMT sensor adopts nine planar coils, which are arranged in a structure of 3 × 3 The sensor is located above the copper samples with defects The air region surrounds the sample and the sensor to ensure the boundary conditions The parameters of sensor array are listed in Table 1 The materials of samples are adopted as copper in simulations

As shown in Figure 4, four typical different defect distributions are involved to verify the performance of the proposed reconstruction method Specifically, four typical conductivity distributions are single central defect, single non-central defect, two edge defects and four edge defects The thickness of the samples is

5 mm, and the depth of defect is 2 mm In addition, the red region in Figure 4 is copper with 58 MS/m The blue region is defect, and the conductivity of defect is

0.1 S/m The conductivity of copper is 58 MS/m, and the conductivity of defect (air) is close to 0 MS/m Hence, the conductivity of defects is about 10 orders of magnitude smaller than that of copper In other words, as long as the conductiv-ity of the defect is set small enough, the impact on the reconstruction results is negligible The position and number of defects are different, which includes single central defect, single non-central defects, two defects, and four defects In order

to evaluate the quality of the reconstructed image, the correlation coefficient

(Cor) is applied and defined as,

Figure 2 The flow chart of the proposed closed-loop PID reconstruction algorithm

Cor ¼

Pn

i¼1ðG RðiÞ G�RÞðG TðiÞ G�TÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pn

i¼1ðG RðiÞ G�RÞ2Pn i¼1ðG TðiÞ G�TÞ2

where G R represents the grey value of the reconstructed image and G T is the actual

grey value of the image n denotes the number of pixels in the image In addition, � G R and

�

G T are the average grey values of G R and G T, respectively

The correlation coefficient ranges from 0 to 1 The closer the correlation coeffi-cient of the reconstructed image is to 1, the better the quality of the reconstruc-tion image If the correlareconstruc-tion coefficient is much less than 1, the quality of reconstruction will be worse

Figure 3 The planer EMT sensor in FEM

Table 1 The dimension of the coil in the EMT sensor

Inner and outer radii of the coil 3 mm/5 mm

Spacing between coils 10 mm

Figure 4 Four typical conductivity distributions