Acoustic Emission Signal Processing Method and Modern Modeling Technology

Abstract

Using acoustic emission signal as the detection medium for particle characteristic parameters has the advantages of real-time, non-destructive, safe, and non-invasive flow field. In order to extract the rich information contained in the acoustic emission signal and establish the quantitative relationship between the acoustic emission signal and the particle characteristic parameters, it is necessary to carry out a series of mathematical processes on the acoustic emission signal in order to extract valuable features from it, and then take these features as the model variables and, through modern modeling methods, establish the quantitative relationship between the acoustic emission signal mode characteristics and the particle characteristic parameters. This chapter first introduces the application status and research progress of acoustic emission technology in chemical processes, then introduces the processing methods of acoustic emission signals, and finally focuses on the basic principles of wavelet (packet) analysis, the types of wavelet (packet) functions, the Mallat algorithm, signal wavelet (packet) noise reduction, and other basic theories, as well as the research progress of particle detection based on modern modeling technology of acoustic emission signals.

1. Introduction

Fourier analysis can only provide the frequency of the signal in the whole time domain, but it cannot provide the frequency information of the signal in a certain time period. Short-time Fourier transform divides the whole time domain into some small, equal time intervals, and then, Fourier analysis is used in each time period. It contains time and frequency information to a certain extent, but because the time interval cannot be adjusted, it is difficult to detect the short duration, the time when a pulse signal with high frequency occurs [1].

Wavelet are mathematical tools for analyzing time series or images [2].

The concept of wavelet transform was proposed first by J. Morlet, a French engineer engaged in petroleum signal processing, in 1974, but was not recognized by mathematicians at that time. In 1986, the famous mathematician Y. Meyer accidentally constructed a true wavelet basis and collaborated with S. Mallat to establish a unified method for constructing wavelet bases with multi-scale analysis [3, 4]. After that, wavelet analysis began to flourish. Among them, the Belgian female mathematician I. Daubechies’ “Ten Lessons on Wavelets” played an important role in promoting the popularization of wavelets [5]. Compared with the Fourier transform and window Fourier transform (Gabor transform), it is a local transform of time and frequency that can effectively extract information from signals. Through operational functions such as scaling and translation, it performs multi-scale analysis on functions or signals, solving many difficult problems that cannot be solved by the Fourier transform. Therefore, wavelet transform is known as the “mathematical microscope.” It is a milestone in the development history of harmonic analysis.

This section focuses on the basic theory of wavelet analysis and the method of signal time-frequency multiscale decomposition.

2. The basic principles of wavelet analysis $L^2\left(R\right)$ space

The function space mainly discussed in wavelet analysis is a real function space composed of square-integrable function $L^2\left(R\right)$. That is, $f\left(t\right)\in L^2\left(R\right)\iff {\int }_R{\left|f\left(t\right)\right|}^2\mathit{dt}<+\infty$, it is an infinite dimensional vector space. One of the main problems in wavelet analysis research is how to represent functions in space $L^2\left(R\right)$ using the dyadic contraction and translation of a basic wavelet function, that is: $f\left(t\right)={\sum }_{n\in Z}{b}_n{\varphi }_n\left(t\right)$.

2.1 Continue wavelet transform

Expanding the function $f\left(t\right)$ in any space $L^2\left(R\right)$ on a wavelet basis is called a continuous wavelet transform of the function $f\left(t.\right)$

Definition 1.1

Assume that the function $f\left(t\right)\in L^2\left(R\right),\psi \left(t\right)\in L^2\left(R\right),$ and $\psi \left(t\right)$ satisfies the permissibility condition:

$$C_{\psi }={\int }_{-\infty }^{+\infty }\frac{{\left|\widehat{\psi }\left(\omega \right)\right|}^2}{\left|\omega \right|}\mathit{d\omega }<+\infty$$E1

where $\widehat{\psi }\left(\omega \right)$ is the Fourier transform of the function $\psi \left(t\right)$, that is, $\widehat{\psi }\left(\omega \right)={\int }_{-\infty }^{+\infty }\psi \left(t\right)e^{-i2\mathit{\pi \omega t}}\mathit{dt}.$ The continuous wavelet transform of $f\left(t\right)$ is defined as:

$${\mathit{WT}}_x\left(a,b\right)=<f,{\psi }_{a,b}>=\frac{1}{\sqrt{a}}{\int }_{-\infty }^{+\infty }f\left(t\right){\psi }^{*}\left(\frac{t-b}{a}\right)\mathit{dt},a\ne 0$$E2

In the formula (2), “*” represents taking conjugation to a complex number, where

$${\psi }_{a,b}=\frac{1}{\sqrt{a}}\psi \left(\frac{t-b}{a}\right)a\ne 0$$E3

For the wavelet generating function $\psi \left(t\right)$, it can be seen from condition (1), $\widehat{\psi }\left(0\right)=0$ is necessarily (if $\widehat{\psi }\left(0\right)\ne 0$, then $C_{\psi }=\infty$, admissibility condition cannot be satisfied), so it follows that:

$$\widehat{\psi }\left(0\right)={\int }_{-\infty }^{+\infty }\psi \left(t\right)\mathit{dt}=0$$E4

That is, the algebraic sum of $\psi \left(t\right)$ and the area enclosed by the entire horizontal axis is zero, having bandpass properties. Its graph appears as an alternating positive and negative oscillating waveform on the horizontal axis, hence it is called “wavelet.”

Any linear transformation used for signal reconstruction should meet the requirement of complete reconstruction, and the same applies to wavelet transform. The reconstruction formula for wavelet transforms that meet the allowable conditions is:

$$f\left(t\right)=C_{\psi }^{-1}{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }\mathit{W}{\mathit{T}}_x\left(a,b\right){\mathit{\psi }}_{a,b}\left(t\right)\frac{\mathit{da}}{{\left|a\right|}^2}\mathit{db}$$E5

Continuous wavelets have the following properties [6]:

1. Linearity: the wavelet transform of a multi-component signal is equal to the sum of the wavelet transforms of each component, which can be expressed as follows by the formula:

   $$f\left(t\right)=\sum _{i=1}^n{f}_i\left(t\right)$$
   $$f\left(t\right)\leftrightarrow \mathit{WT},f_i\left(t\right)\leftrightarrow {\mathit{WT}}_i$$

   then: $\mathit{WT}={\sum }_{i=1}^n{\mathit{WT}}_i$

2. translation invariance (timeshift covariation):

   $$\mathit{iff}\left(t\right)\leftrightarrow W{T}_x\left(a,b\right),\mathit{\text{thenf}}\left(t-{\tau }_0\right)\leftrightarrow W{T}_x\left(a,b-{\tau }_0\right)$$

3. Time scale theorem (dilation covariance):

   if $f\left(t\right)⟷{\mathit{WT}}_x\left(a,b\right)$, then $f\left(\mathit{ct}\right)⟷\frac{1}{\sqrt{C}}{\mathit{WT}}_x\left(\mathit{ca},b\right)$, $c>0$

4. Self-similarity:

CWT (Continuous wavelet transform) transforms one-dimensional signals into two-dimensional space $f\left(t\right)\leftrightarrow W{T}_f.$ Therefore, there is redundant information in wavelet transform called redundancy. The continuous wavelet transforms corresponding to different scale parameters a and different translation parameters b are self-similarity. There is redundancy in information representation in continuous wavelet transform.

That is to say, the inverse transformation of wavelet transform is not unique. ${\psi }_{a,b}=\frac{1}{\sqrt{a}}\psi \left(\frac{t-b}{a}\right)$ is a family of super complete basis functions, they are linearly correlated. The measure of redundancy is called the regenerative kernel $K\left(a_0,b_0,a,b\right)$:

$$\begin{array}{l}K\left(a_0,b_0,a,b\right)=\frac{1}{C_{\psi }}{\int }_R{\psi }_{a,b}\left(t\right){\psi }_{a_0,b_0}^{*}\left(t\right)\mathit{dt}=\frac{1}{C_{\psi }}{\int }_R\frac{1}{\sqrt{a}}{\psi }_{a,b}^{*}\left(\frac{t-b}{a}\right){\psi }_{a_0,b_0}^{*}\left(\frac{t-b_0}{a_0}\right)\mathit{dt}\\ =\frac{1}{C_{\psi }}⟨{\psi }_{a,b}\left(t\right),{\psi }_{a_0,b_0}^{*}\left(t\right)⟩\end{array}$$E6

$C_{\psi }<\infty$ is needed in order to achieve the complete refactoring. This is also the permissibility condition mentioned in Eq. (1), also known as the complete reconstruction condition.

As the implementation of signal reconstruction is numerically stable, in addition to the complete reconstruction condition, it also requires the Fourier transform of wavelet ψ(t) satisfies the following “stability condition.”

$$A\le \sum _{j=-\infty }^{+\infty }{\left|\widehat{\psi }\left(2^{-j}\omega \right)\right|}^2\le B$$E7

$0<A\le B<\infty$.

Definition 2.2

If wavelet $\psi \left(t\right)$ satisfy the stability Condition (7), define a “Dual wavelet” $\stackrel{\sim }{\psi }\left(t\right).$ Its Fourier transform $\widehat{\stackrel{\sim }{\psi }}\left(t\right)$ is defined as:

$$\widehat{\stackrel{\sim }{\psi }}\left(\omega \right)=\frac{{\widehat{\psi }}^{*}\left(\omega \right)}{\sum _{j=-\infty }^{+\infty }{\left|\widehat{\psi }\left(2^{-j}\omega \right)\right|}^2}$$E8

${\widehat{\psi }}^{*}\left(\omega \right)$ is a conjugate function of $\widehat{\psi }\left(\omega \right)$.

2.2 Wavelet transform and adaptive time frequency window

Wavelet transform is similar to short-time Fourier transform. The difference is that the wavelet function is used as a window function. Define the time domain window radius of the mother wavelet function $\psi \left(t\right)$ of is $\mathit{\Delta t}$. The center of the window is $t^{*}$, and the frequency domain window radius $\mathit{\Delta \omega }$. The center of the frequency domain window is ${\omega }^{*}$, set the time domain window center of ${\psi }_{a,\tau }\left(t\right)$ to $t_{a,\tau }^{*}$. The center of the time domain window of ${\psi }_{a,\tau }\left(t\right)$ is $t_{a,\tau }^{*}$. The time domain window radius is $\Delta t_{a,\tau }$. The center of the frequency domain window is ${\omega }_{a,\tau }^{*}$. The frequency domain window radius is $\Delta {\omega }_{a,\tau }$. The center and radius of the time-frequency window of ${\psi }_{a,\tau }\left(t\right)\mathrm{and}\psi \left(t\right)$ have the following relationship expression:

$$\begin{array}{c}{t}_{a,\tau }^{*}={\mathit{at}}^{*}+\tau ,\Delta t_{a,\tau }=\mathit{a\Delta t},\\ {\omega }_{a,\tau }^{*}=\frac{1}{a}{\omega }^{*},\Delta {\omega }_{a,\tau }=\frac{1}{a}\mathit{\Delta \omega }\end{array}$$

It can be seen that the center and width of the time-frequency window of a continuous wavelet ${\psi }_{a,\tau }\left(t\right)$ will expand and contract with the change of scale a.

Wavelet transform uses a time-frequency window to show the time-frequency localization ability of wavelet transform, which is different from short-time Fourier transform. With the change of scale parameter a, the position of the wavelet transform time-frequency window on the phase plane is not only changing but also the shape of the window is changing, as shown in Figure 1. Figure 1(a) shows that as a time-frequency window function, when the wavelet function is widened (a increases) in the time domain, its frequency window width is narrowed, and the center of the frequency window is also smaller. Figure 1(b) shows that for the same time window center, as the frequency window center (frequency band center) moves up (at which point a decreases), the frequency window width (bandwidth) is widened and the time window width is compressed. Therefore, the shape of the wavelet transform time-frequency window varies with the scale parameter a, which can be analyzed based on the frequency of the signal ω. Adaptive adjustment: At low frequencies, the time resolution of wavelet transform is lower, while the frequency resolution is higher. At high frequencies, the time resolution of wavelet transform is higher, while the frequency resolution is lower. This adaptive characteristic of wavelet transform time-frequency window is also called “auto zoom function,” so it is convenient to realize multi-resolution time-frequency analysis for non-stationary signals by using wavelet transform.

Figure 1.

Schematic diagram of wavelet transform time-frequency window.

Although the center and width of the time window and frequency window of ${\psi }_{a,\tau }\left(t\right)$ vary with a and τ, the area (i.e., window) formed by the time-frequency window on the time-frequency phase plane does not vary with the parameters:

$$\Delta t_{a,\tau }\Delta {\omega }_{a,\tau }=\mathit{a\Delta t}·\frac{1}{a}\mathit{\Delta \omega }=\mathit{\text{ΔtΔω}}$$

The area of the time-frequency window of the short-time Fourier transform also has similar properties. This property is called the Heisenberg Uncertainty Theorem, which means the size of $\mathit{\Delta t}$ and $\mathit{\Delta \omega }$ is mutually constrained, and both cannot be arbitrarily small, so the resolution in both the time and frequency domains cannot be improved without limitation at the same time.

2.3 Typical wavelets

Compared with the standard Fourier transform, the wavelet functions used in wavelet analysis are non-unique, that is, there are various wavelet functions ψ(t) we can apply [6]. The diversity of wavelet analysis is a crucial issue in engineering applications, as selecting the optimal wavelet basis can yield different results when analyzing the same problem using different wavelet bases. At present, the quality of wavelet bases is mainly determined by the error between the results of signal processing using wavelet analysis methods and the theoretical results, thereby determining the quality of wavelet bases.

1. Haar wavelet

   $${\psi }_H\left(t\right)=\left\{\begin{array}{cc}1& 0\le t<1/2\\ -1& 1/2\le t<1\\ 0& \mathit{\text{else}}\end{array}\right.$$E9

   ${\psi }_H\left(t\right)$ is not only orthogonal to ${\left.{\psi }_H\left(2^{j}t\right)\right|}_{j\in Z}$ but also orthogonal to its own integer displacement, that is,:

   $$\int {\psi }_H\left(t\right){\psi }_H\left(2^{j}t\right)\mathrm{dt}=\int {\psi }_H\left(t\right){\psi }_H\left(t-n\right)\mathrm{dt}=0,j\in Z,n\in Z$$E10

2. Daubechies (DbN) wavelet

   Daubechies wavelet is a wavelet function constructed by the world-renowned wavelet analyst Inrid Daubechies. It can generally be abbreviated as DbN, where N is the order of the wavelet.

   The support region in wavelets $\psi \left(t\right)$ and scaling functions $\phi \left(t\right)$ is 2 N-1, and the vanishing moment of $\psi \left(t\right)$ is N. (If $\int t^p\psi \left(t\right)\mathrm{dt}=0$, $p=0,1,\cdots ,N-1,N\ge 1$ and $\int t^N\psi \left(t\right)\mathrm{dt}\ne t$, the vanishing moment is called N.) Except for N = 1, DbN does not exhibit symmetry (i.e., nonlinear phase). There is no explicit expression except for N = 1.

   The goal of Daubechies wavelets is to construct compactly supported orthogonal wavelets with high-order vanishing moments. The vanishing moment plays an important role in compression, denoising, and singularity detection. When there are singular points (mutation points) in the signal, under high-resolution conditions, the wavelet coefficient at the smooth point is very small, while at the singular point, the wavelet coefficient is large, making it possible to quickly determine the position of the singular point.

3. Symlet (symN) wavelets

4. Morlet wavelets

5. Meyer wavelets

6. Mexican hat(mexh) wavelets

7. Coiflet (coif N) wavelets

8. Biorthogonal (biorNr.Nd) wavelets

9. Reverse Bior wavelets

10. Dmeyer wavelets

11. Gaussian wavelets

12. Complex Gaussian wavelets

13. Complex Morlet wavelets

14. Complex Frequency B-Spline wavelets

15. Complex Shannon wavelets

3. The integrated technology of wavelet analysis and neural network for detecting the average particle size of fluidized bed

The acoustic measurement method for detecting the average particle size of materials in a fluidized bed has the advantages of non-invasive flow field and real-time online detection [7, 8]. However, the acoustic emission signal is a series of time series data. The data is difficult to associate with the chemical parameters in the fluidized bed [9, 10]. Considering that acoustic emission signals are emitted by a large number of particles colliding with each other, or particles colliding with the wall of a fluidized bed, they contain rich multi-scale information. Therefore, the acoustic emission signal can be decomposed at multiple scales, and then, features can be extracted from the information at each scale to form a pattern. Finally, it is associated with the detected object, that is, a certain chemical parameter, to form a soft sensing model for that parameter. There are several issues that need to be addressed. First, decomposition scales of acoustic emission signal, and then how to select and construct mode variables from the information of each scale. Second is how to extract features from these variables when there is a complex correlation between them, and third is how to locate the quantitative relationship between pattern features and the tested object when they are associated. To detect the average particle size of fluidized bed particles by acoustic emission signals, the following methods are proposed in this chapter.

First, multi-scale decomposition of acoustic emission signals is performed using wavelets or wavelet packets to obtain high-frequency and low-frequency detailed signals, energy patterns are constructed based on these signals. Next, principal component analysis (PCA) on the pattern variables to select suitable principal components as feature variables. Finally, establish a neural network model for detecting the average particle size of materials in a fluidized bed.

This chapter eliminates the complex collinearity between variables through multi-scale decomposition of acoustic emission signals and combining PCA screening mode features. When the neural network model is used to detect the average particle size of materials in a cold-model fluidized bed, the detection achieves high accuracy and good stability.

3.1 Artificial neural networks

Artificial neural networks are widely used in fields such as fitting, classification, clustering, feature mining, modeling of control and dynamic systems, and pattern recognition. The feedforward network is the most commonly used and common neural network to date.

People have proposed hundreds of artificial neural network models from different research perspectives, and there are three main types of existing neural networks: namely feedforward neural networks (FFNN), feedback networks (Feedback NN), and self-organizing neural networks (SOMs). In recent years, fuzzy neural networks have also developed rapidly. Fuzzy neural networks organically combine fuzzy technology with neural network technology, and combine the advantages of neural network and fuzzy theory, integrating learning, association, recognition, adaptation, and fuzzy information processing. Fuzzy neural networks have proposed various models and achieved fruitful application results to date. There are mainly feedforward fuzzy neural networks, T-S model fuzzy neural networks, fuzzy maximum minimum neural networks, fuzzy associative memory networks, etc.

3.2 RBF networks and algorithms

The radial basis function neural network (RBFN) was proposed by Powell in 1985 and is essentially a radial basis function method for multivariate interpolation [11, 12]. It is an artificial neural network with simple structure, simple training, wide application, and good generalization ability. It has been widely applied in many fields, especially in the practical applications of function approximation and pattern classification. In terms of structure and operation, RBF networks also belong to feedforward networks, but their neurons have specific settings and learning methods, which make them have specific performance. RBF networks outperform feedforward networks in terms of approximation ability, classification ability, and learning speed. Its advantage lies in using linear learning algorithms to complete the work done by previous nonlinear learning algorithms while maintaining the high accuracy and other characteristics of nonlinear algorithms. Therefore, it is an artificial neural network that has both the fast convergence characteristics of linear algorithms and the high accuracy characteristics of nonlinear algorithms.

Compared with ordinary feedforward networks, RBF networks perform RBF transformations on the input data at the hidden layer. It has been proven mathematically that through RBF transformation, nonlinear separable sample points in one space can be transformed into linearly separable sample points in another space. This is the theoretical basis for the superior performance of RBF networks over ordinary feedforward networks. There are two variants of radial basis neural networks: generalized regression networks (GRNN) and probabilistic neural networks (PNN). The former can be used for function approximation, while the latter can be used for classification.

3.2.1 The structure of RBF networks

The topology of RBF neural networks is a three-layer feedforward network, and the input layer does not perform any transformation on the input information. It only serves the purpose of transmitting data. The kernel function (action function) of hidden layer neurons is a Gaussian function that performs spatial mapping transformation on input information. The third layer is the output layer, which responds to the input mode. The action function of the output layer neurons is a linear function, and the output information of the hidden layer neurons is linearly weighted and output as the output result of the entire neural network. The topology structure of the RBF neural network based on the Gaussian kernel is shown in Figure 2. The transfer function of a radial basis function network is based on the distance between the input vector and the threshold vector as the independent variable. When the first input vector is passed to the hidden layer node, the output after processing is:

Figure 2.

Topological structure of RBF neural network based on Gaussian kernel.

$$o_j=exp\left(-{‖c_j-x_i‖}^2/{\sigma }_j^2\right)$$E11

where $c_j$ is the center vector of the Gaussian node of the neuron, and ${\sigma }_j$ is the width parameter (spread).

The output layer linearly weights the outputs of each node in the hidden layer as the output of RBFNN, that is, its activation function $y=x$ is a linear function, and the calculated output ${\overline{y}}_k$ corresponding to the kth input vector $x_k$ is:

$${\overline{y}}_k=w_{k0}+\sum _{j=1}^m{w}_{\mathrm{kj}}{o}_j$$E12

where $w_{\mathrm{kj}}$ is the connection weight, $w_{k0}$ is the bias, and $m$ is the number of nodes in the hidden layer.

3.2.2 Method for selecting RBF neural network centers

The key issue in the learning algorithm of RBF neural networks is the reasonable determination of the central parameters of hidden layer neurons. In existing learning algorithms, the central parameter (or initial value of the central parameter) is either directly selected from a given training sample set using a certain method or determined by clustering methods. Common methods include direct calculation (randomly selecting RBF centers), self-organized learning (selecting RBF centers), supervised learning (selecting RBF centers), and orthogonal least squares (selecting RBF centers).

K-means clustering is a clustering method that clusters according to the minimum distance. The idea of clustering is that for a p-dimensional input vector pattern, it can be seen as points in a p-dimensional Euclidean space. If the vectors representing each point are geometrically very close, they can be classified into the same class. Using Euclidean distance to measure their proximity:

$$‖x-c‖={\left[\sum _{i=1}^p{\left(x_i-c_i\right)}^2\right]}^{1/2}$$E13

where $x$ and $c$ are $p$-dimensional pattern vector

The main steps for determining the RBF center by the k-means clustering method are given by

① Initialize, set the number of categories $k_m$, assign initial values to the clustering centers of each category.

② Individual division: Calculate the distance between each individual and each cluster center according to Eq. (13), and divide each individual into different categories based on the principle of minimum distance.

③ Calculate new cluster centers: For the new classes established in step ②, their new center positions can be calculated as

$$c_j\left(l+1\right)=\frac{1}{N_j}{\sum }_{x^{\left(i\right)}\in S_j\left(l\right)}{x}^{\left(i\right)}1\le j\le k_m$$E14

where $l$ is the number of iterations, $c_j\left(l\right)$ is the clustering center value of the l-th class $j$, $x^{\left(i\right)}$ is $i$-th input individual, $S_j\left(l\right)$ is the entire class $j$ at l-th iteration, $N_j$ is the number of individuals belonging to class $S_j\left(l\right)$ in step ②.

Check convergence: If there is no further change in the clustering center in step ③,

convergence has been reached, which satisfies the formula $c_j\left(l+1\right)=c_j\left(l\right)$, else, back to step ②.

After determining the centers of each cluster, the centers of each radial basis function can be obtained. The extension constant can be set to

$$\text{spread}=\lambda \underset{i}{min}‖c_j-c_i‖$$E15

where $\lambda$ is the overlap coefficient.

3.2.3 Calculation of BRF network weight parameters

After obtaining the extension constants of each radial basis function center by the K-means clustering algorithm, the second step in the learning process is to use a supervised learning algorithm to obtain the weight of the output layer, the gradient descent method is often adopted.

Define an objective function as:

$$E=\frac{1}{2}\sum _{i=1}^P{e}_i^2$$E16

where $P$ is the number of individuals for training samples; $e_i$ is the error signal when inputting the ith individual, defined as:

$$e_i=y_i-{\overline{y}}_i=y_i-w_{i0}-\sum _{j=1}^m{w}_{\mathrm{ij}}exp\left(-{‖c_j-x_i‖}^2/{\sigma }_j^2\right)$$E17

where $y_i$ is the expected output corresponding to the $i$th input vector $x_i$, it is a mentor signal.

To minimize the objective function, the correction of each parameter is proportional to its negative gradient, that is,

$$\Delta c_j=-\eta \frac{\partial E}{\partial c_j}$$E18

$$\Delta {\sigma }_j=-\eta \frac{\partial E}{\partial {\sigma }_j}$$E19

$$\Delta w_j=-\eta \frac{\partial E}{\partial w_j}$$E20

where (18), (19), (20), $\eta$ is learning rate.

3.2.4 Learning algorithm of regularized RBF networks

For regularized RBF networks, the number of hidden layer nodes is equal to the number of input-training individuals. All training input individuals are at the center of the radial basis function. Each radial basis function has a uniform width distribution constant. The weight of the output layer is often calculated using the least mean square (LMS) algorithm. The input vector of the LMS algorithm is the output vector of the hidden node. The weight value can be initialized to any value.

3.3 Soft measurement of average particle size in fluidized beds by Sym8 WLA-PCA-MLFN

The multiphase reaction in gas-solid fluidized bed reactors is a typical spatiotemporal and multiscale problem. In order to obtain real, effective, and real-time information on the operation of the bed at the micro-scale, and to control the production process at the macro-scale, it is necessary to establish a correlation mechanism between the micro-system and the macro-system, so that changes in material performance parameters at the micro-scale can be reflected in a timely manner at the macro-scale. It is an important means of establishing such correlations by multi-scale methods. In the production of polyethylene, organic silicon monomer, and granular sodium percarbonate, the particle size and distribution of materials have a certain impact on the reaction rate and product properties. The particle size of materials is also related to reaction time, feeding speed, process parameters, etc., and is often a dynamic and complex process. For example, when conducting the synthesis reaction of dimethyl dichlorosilane in a fluidized bed, it is necessary to timely supplement the raw material with silicon powder and copper powder catalyst. Feeding at the appropriate time can avoid severe fluctuations in the bed temperature, making the process easy to control. The particle size and distribution of the material should be kept stable. If the particle size distribution is not reasonable, it may affect the reaction effect. It is difficult to conduct real-time online detection of material particle size in fluidized beds. As a new measurement method, acoustic measurement has the characteristic of real-time response to changes in reactor material particle size or concentration. It can be considered to use acoustic emission signals for online detection of material characteristics in the reactor. Acoustic emission signal detection has undergone rapid development since its application in the 1980s. But acoustic emission signals are a type of wave, often recorded as a series of temporal data points, and have sudden transients, often mixed with interference noise. How to effectively extract correlation mode information from acoustic emission signals is an urgent problem to be solved. The acoustic emission signal is related to various factors such as bed height, material composition, temperature, and empty bed gas velocity. For a stable fluidized bed, the acoustic emission signal is mainly influenced by the particle size and distribution of the material. Spectrum analysis, wavelet analysis, wavelet packet analysis, fractal feature analysis, or complexity analysis can all be used to analyze acoustic emission signals. Among them, wavelet analysis decomposes acoustic emission signals into low-frequency overview signals and high-frequency detail signals, which have significant advantages for analyzing nonlinear, non-stationary pulsating signals. Previous literature often focused on the spectrum of acoustic emission signals, which is relatively complex. The models established using this method are generally only suitable for qualitative analysis. Lack of quantitative indicators: In the construction of association patterns and feature selection, more concise methods can be considered.

The methods used in this section are as follows: First, wavelet analysis (WLA) is performed on the collected acoustic emission signals. Calculate the energy mode of the acoustic emission signal from the decomposed low-frequency profile signal and high-frequency detail signal. In order to eliminate the multicollinearity between variables, principal component analysis (PCA) was performed on the energy pattern, and principal components were extracted from the energy pattern. Then, using the principal component as the input and the average particle size in the fluidized bed as the output, a multi-layer feed-forward neural network (MLFN) is constructed to establish the quantitative relationship between the principal component of the acoustic emission signal energy mode and the average particle size. The experimental results show that when using this method to predict the average particle size of materials in the bed, the computational cost is not large and the accuracy is high.

3.3.1 The relationship between acoustic emission signals and fluidized bed particle size

There is a large amount of collision and friction between particles in the fluidized bed, as well as between particles and the container wall. The impact force generated by particles of different particle sizes when impacting the container wall or colliding with each other is significantly different, which will generate strong or weak acoustic emission signals with different frequencies and transmit them outward through the container wall and air in the form of elastic waves. Under the same other process conditions, the acoustic emission signals generated by particle groups of different particle sizes are different. Thus, acoustic emission signals can be used to detect the particle size and distribution of the fluidized bed.

For $n$ rigid spherical particles with particle size $d_p$ and mass $m$, the force generated by impact on the wall surface with area $\mathit{\Delta A}$ is given as [13]:

$$F\left(t\right)=\sum _{i=1}^{n}2{\mathit{mu}}_i\delta \left(t-t_i\right)$$E21

where t is time, $\delta \left(t\right)$ is Dirac delta function related to time t,$t_i$ is the time for the i-th particle to reach the wall, $u_i$ is the velocity at which the u-th particle vertically impacts the wall surface. There are $f_p·T$ impacts between particles and the wall surface within time $T$, $f_p$ is the average arrival rate of particles on the area $\mathit{\Delta A}$, that is, the frequency of particles hitting the wall. Therefore, the average force per unit time of these particles is:

$$\stackrel{\_\_\_\_\_\_\_}{F\left(t\right)}=\frac{{\int }_0^T\mathit{F}\left(t\right)\mathit{dt}}{T}=\frac{\mathit{2}\mathit{mu}{\int }_{\mathit{0}}^{\mathit{T}}\sum _{\mathit{i}=\mathit{1}}^{\mathit{n}}\delta \left(\mathit{t}-t_i\right)\mathit{dt}}{\mathit{T}}$$E22

where $u$ is the average velocity of particles impacting the wall vertically,

${\int }_0^T{\sum }_{i=1}^n\delta \left(t-t_i\right)\mathit{dt}=f_{p}T$, therefore:

$$\overline{F\left(t\right)}=2\mathit{mu}{f}_p$$E23

The resulting acoustic emission pressure is:

$$P_a=\frac{\eta ·\stackrel{\_\_\_\_\_\_\_}{F\left(t\right)}}{\mathit{\Delta A}}$$E24

In the equation, $\eta$ is the efficiency of converting impact pressure into sound pressure. Assuming that the concentration of particles colliding with the wall near the wall is $C$ (pieces/m³), which is inversely proportional to the square of the projected diameter of the particles $d_p^2$ on the wall, that is:

$$C=\xi /d_p^2$$E25

in the equation, $\xi$ is the proportional coefficient, and the frequency of particle impact on the wall is:

$$f_p=C·\frac{\mathit{\Delta Av}}{\mathit{\Delta A}}=C·u$$E26

Thus, the average AE flux of particles with particle size $d_p$ and mass $m$ impacting the wall surface per unit of time is

$$J=P_a\mathit{\Delta Au}=2\mathit{\xi \eta m}{u}^3/d_p^2$$E27

The AE energy of particles with particle size $d_p$ and mass m colliding with the wall at duration $T$ is

$$E={\int }_0^T\mathit{Jdt}={\int }_0^T\frac{\pi }{3}\mathit{\xi \eta }{\rho }_s{u}^3{d}_p\mathit{dt}$$E28

It can be seen that AE energy is a function of particle size $d_p$, material density ${\rho }_s$, and duration $T$. Since particles with different particle sizes have different AE energy, within a certain duration $T$, while keeping other parameters constant, changes in the average particle size and its distribution can be understood through acoustic emission signals,

In a fluidized bed reactor, the average particle size often changes continuously with the progress of the reaction process. The system maintains a dynamic balance. In the normal state, the characteristic mode of the acoustic emission signal will not undergo significant fluctuations. Once the balance of the system is disrupted, such as by fluctuations in particle size, agglomeration of materials in the reactor, the acoustic emission signal will also change accordingly.

Assuming that there are $J$ types of different particle sizes acting together on the wall, the percentage of particles with particle size $d_{p,j}$ is $x_j$. According to the principle of linear superposition of acoustic energy, the relationship between the acoustic energy $E_j$ generated by particles with particle size $d_{p,j}$ and the total sound energy is:

$$\sum _{j=1}^N\frac{E\left(d_{p,j}\right)}{E_{\mathrm{mix}}}{x}_j=1$$E29

where, $E_{\mathrm{mix}}={\sum }_{j=1}^{N}E\left(d_{p,j}\right)x_j$,

That is to say, the AE signals generated by mixed particle sizes can be seen as the sum of various acoustic emission signals with different particle sizes. Therefore, based on the AE signals, the average particle size and particle size distribution of particle groups under microscale conditions can be predicted.

3.3.2 Multi-scale decomposition of acoustic emission signals

The raw acoustic emission signals collected under specific process conditions can be regarded as a wide, stationary random time series. A complete description requires analysis from three aspects: amplitude domain, time domain, and frequency domain to extract features. This chapter first performs multi-scale decomposition of acoustic emission signals, then calculates the total energy of each detail signal, and extracts principal components as pattern features.

Assuming there is an acoustic emission signal sequence: $\left\{c_1^j,c_2^j,\cdots ,c_k^j,\cdots c_n^j\right\}$, where the superscript $j$ represents the observation scale, and the subscript represents the serial number, when $j=0$ it represents the original acoustic signal. At resolution $2^{-j}(j\in Z$, the discrete approximation of the signal can be represented as:

$$f_j\left(t\right)={\sum }_{j,k\in Z}{c}_k^j{\phi }_{j,k}\left(t\right)$$E30

${\phi }_{j,k}\left(t\right)$ is the scale function series under the resolution of a condition $2^{-j}$

According to the theory of wavelet decomposition, signals can be decomposed at multiple scales to calculate their approximate ${\widehat{A}}_{j+1}f$ and detailed information ${\widehat{D}}_{j+1}f$ under higher resolution conditions:

$$f_j\left(t\right)={\widehat{A}}_{j+1}f+{\widehat{D}}_{j+1}f={\sum }_k{c}_k^{j+1}{\phi }_{j+1,k}\left(t\right)+{\sum }_k{d}_k^{j+1}{\psi }_{j+1,k}\left(t\right)$$E31

where $d_k^{j+1}$ is wavelet coefficients on the $j+1$ scale, $\psi \left(t\right)$ is wavelet function, $\widehat{A},\widehat{D}$ is an operator for calculating low-frequency detail signals and high-frequency detail signals. Because of translational invariability and the orthogonality of expansion and contraction of $\phi$ and $\psi$, it can be calculated that

$$c_k^{j+1}=\sum _n{c}_n^j⟨{\varphi }_{j,n},{\varphi }_{j+1,k}⟩=\sum _n{c}_n^j{h}_{n-2k}^{*}$$E32

Similarly,

$$d_k^{j+1}=\sum _n{c}_n^j⟨{\varphi }_{j,n},{\psi }_{j+1,k}⟩=\sum _n{c}_n^j{g}_{n-2k}^{*}$$E33

Eqs. (32) and (33) are decomposition algorithms for wavelets. Where “*” represents the conjugation of complex numbers, ${\left\{h_k\right\}}_{k\in Z}$. It is the sequence of filter coefficients corresponding to the two scale equations of the orthogonal scaling function, which can be regarded as a low-pass filter, ${\left\{g_k\right\}}_{k\in Z}$ can be regarded as a high-pass filter. From the wavelet coefficients and scale coefficients at the $j+1$ scale, the scale coefficients at the $j$ scale can be obtained through reconstruction algorithms:

$$\begin{array}{c}{c}_k^j={\sum }_n{c}_n^{j+1}⟨{\phi }_{j+1,n},{\phi }_{j,k}⟩+{\sum }_n{d}_n^{j+1}⟨{\psi }_{j+1,n}{\phi }_{j,k}⟩\\ {\sum }_n{c}_n^{j+1}{h}_{n-2k}+{\sum }_n{d}_n^{j+1}{g}_{k-2n}\end{array}$$E34

The amplitude of the acoustic emission signal can also be regarded as energy, and when other process conditions are constant, it is mainly influenced by the particle size. Therefore, the energy value of the detail signal can reflect the size of the particle. For mixed-particle systems with different particle sizes, this difference can be reflected in detailed information through multi-scale decomposition of acoustic emission signals. The energy of the low-frequency profile and high-frequency detail signal after wavelet decomposition of the acoustic emission signal at the defined fs resolution can be represented by the following equation:

$${\epsilon }_a^{j+1}={\sum }_k\left|c_k^{j+1}\right|$$E35

$${\epsilon }_d^{j+1}={\sum }_k\left|d_k^{j+1}\right|$$E36

It can be seen from the multi-resolution that the sum of the scale coefficients of low-frequency signals and the absolute values of wavelet coefficients of high-frequency detail signals can be calculated as the energy mode of acoustic emission signals.

$$f_j=f_{j+1}+d_{j+1}=f_{j+2}+d_{j+2}+d_{j+1}=\cdots =f_p+d_p+d_{p-1}+\cdots +d_{j+1}$$

When the decomposition scale of the acoustic emission signal is $p$, a $p+1$ dimensional energy mode can be obtained: $\left({\epsilon }_a^p,{\epsilon }_d^1,{\epsilon }_d^2,\cdots ,{\epsilon }_d^p\right)$.

A certain quantitative relationship can be established between the average particle size and the energy mode of acoustic emission signals as follows:

$$y=y\left({\epsilon }^j\right)=y\left({\epsilon }_a^p,{\epsilon }_d^1,{\epsilon }_d^2,\cdots ,{\epsilon }_d^p\right)$$E37

In the equation above, the subscript $a$ represents low frequency and the subscript $d$ represents high frequency. P represents the maximum scale of discrete signal decomposition with a length of $2^n$, and $p\le n-1$. The sample data matrix composed of the energy mode of the $m$ observation of the acoustic emission signals generated by particles in a certain interval in a fluidized bed is:

$$X={\left(\begin{array}{cccc}{\epsilon }_{a,1}^p& {\epsilon }_{d,1}^1& \cdots & {\epsilon }_{d,1}^p\\ {\epsilon }_{a,2}^p& {\epsilon }_{d,2}^1& \cdots & {\epsilon }_{d,2}^p\\ ⋮& ⋮& ⋮& ⋮\\ {\epsilon }_{a,m}^p& {\epsilon }_{d,m}^1& \cdots & {\epsilon }_{d,m}^p\end{array}\right)}_{m\times \left(p+1\right)}$$E38

Considering the significant differences in signal energy among different details, it is usually necessary to standardize the standard deviation of the above energy mode matrix before use.

After p-scale decomposition of acoustic emission signals, the energy mode has a total of $p+1$ components, and there is often a complex collinearity between them. The energy mode is directly associated with the detection object, which not only has a large number of independent variables but may also cause interference with each other. To eliminate multicollinearity, methods such as principal component analysis can be used. At present, many data processing in process control and monitoring also follow this model, showing good application prospects.

3.4 Experiment and discussion

3.4.1 Collection of fluidized bed acoustic emission signals

The fluidized bed acoustic emission signal and data acquisition and analysis system (UNIAE2003) was developed by Professor Yang Yongrong’s research group at the Joint Chemical Reaction Engineering Research Institute of Zhejiang University. The data acquisition and analysis system includes acoustic sensors (AE sensors), amplifiers, A/D conversion cards, and computers, which can achieve multi-channel data acquisition. The device process is shown in Figure 3, and the sensors are tightly attached to the outer wall of the cold model fluidized bed, which can basically ignore other external noise. The sampling frequency of the data collection system is 500 kHz. To avoid interference and reduce the impact of external noise, hardware filters were used for sampling signals for differential filtering preprocessing.

Figure 3.

Schematic diagram of fluidized bed acoustic emission signal data processing system. 1-blower, 2-gas flow meter, 3-fluidized. Bed, 4-acoustic emission signal sensor, 5-signal amplifier, 6-signal processor, 7-computer host, 8-monitor.

The material used in the fluidized bed is granular polyethylene with different particle sizes. The operating gas speed is 0.6 m/s. The particles are divided into five particle size intervals: 1 ∼ 0.90, 0.90 ∼ 0.60, 0.60 ∼ 0.45, 0.45 ∼ 0.22, and 0.22 ∼ 0.180 mm. Twenty sampling observations are conducted on the acoustic emission signals of each particle size interval, and the length of the data recording points obtained from each observation is 16,384. This is the number of data points recorded by the recorder. Acoustic emission signals can also be collected under different bed heights, materials, and temperature conditions by this experimental equipment.

Multi-scale decomposition of the acoustic emission signal recorded by the instrument is used to construct the energy mode of the acoustic emission signal, and the wavelet types and decomposition scales should be screened. Perform principal component analysis on the energy mode components and select several principal components. Strive for good detection results.

3.4.2 Energy mode and principal component analysis of acoustic emission signals

The original acoustic emission signal sequence is processed according to the following steps:

3.4.2.1 Select decomposition wavelet

Many wavelets or wavelet packets that can be selected to decompose AE signals so as to obtain energy patterns. There is no clear regulation on which wavelet is more suitable for decomposing them. When selecting wavelet types, the minimum error generated by signal decomposition reconstruction can be taken as the indicator. The best decomposed wavelet (or wavelet packet) should have the minimum reconstruction error.

The original acoustic emission signal with a length of 16,384 is decomposed at 6 scales using different wavelets, and then reconstructed using the same wavelet. The absolute error sum is used as the judgment indicator, as shown in Eq. (39):

$${\mathrm{Err}}_w=\sum _{i=1}^n\left|C_i^0-{\stackrel{\sim }{C}}_i\right|$$E39

where $n=16384$, $C_i^0$ is the original sequence of acoustic emission signals, ${\stackrel{\sim }{C}}_i$ is a reconstructed sequence of acoustic emission signals. The subscript i represents the serial number of the acoustic emission signal. Examine the Haar wavelet, Daubechies wavelet series, and Sym wavelet series. It was found that the reconstruction error of Sym8 wavelet is the smallest. Therefore, Sym8 wavelet is chosen as the wavelet for decomposing acoustic emission signals.

3.4.2.2 Energy mode construction and preprocessing

A 9-dimensional energy pattern was obtained by Sym8 wavelet 8-scale decomposition on the acoustic emission signal. Table 1 shows the energy patterns obtained at five different particle sizes (partial). From the data in the table, it can be seen that the energy of high-frequency detail signals is much greater than that of low-frequency detail signals, with a difference of one order of magnitude. At this point, logarithms can be taken for the energy mode data so that the data is on the same order of magnitude.

	Σ|d1|	Σ|d2|	Σ|d3|	Σ|d4|	Σ|d5|	Σ|d6|	Σ|d7|	Σ|d8|	Σ|a8|
1 ∼ 0.90 mm	1.2271	1.1938	1.1605	1.1500	1.1003	0.6694	0.1671	0.0471	0.0471
	1.2926	1.2603	1.2318	1.2209	1.1588	0.7235	0.2219	0.0885	0.0885
	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
0.90 ∼ 0.60 mm	1.0690	1.0516	1.0250	1.0090	0.9508	0.6064	0.2298	0.0602	0.0602
	1.1206	1.0988	1.0695	1.0542	1.0058	0.6553	0.2370	0.0634	0.0634
	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
0.60 ∼ 0.45 mm	1.4479	1.2481	0.8552	0.6495	0.4915	0.3104	0.1160	0.0505	0.0505
	1.5587	1.3527	0.9110	0.6915	0.4862	0.2869	0.1199	0.0544	0.0544
	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
0.45 ∼ 0.22 m	0.2171	0.2007	0.1788	0.1535	0.1083	0.0555	0.0483	0.0488	0.0488
	0.2218	0.2040	0.1771	0.1444	0.1061	0.0549	0.0477	0.0481	0.0481
	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮
0.22 ∼ 0.180 mm	0.1487	0.1267	0.1179	0.1126	0.1026	0.0453	0.0404	0.0402	0.0402
	0.1398	0.1182	0.1078	0.1014	0.0902	0.0514	0.0440	0.0435	0.0435
	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮	⋮

Table 1.

Energy patterns obtained from 8-scale decomposition of acoustic emission signals(×1.0 e⁻³).

Due to the fact that both high-frequency and low-frequency detail signals are decomposed from the original acoustic emission signal, there exists a strong multicollinearity. Principal component analysis of the data is required.

The purpose of principal component analysis is to convert multiple existing indicators into fewer linearly unrelated comprehensive indicators. For a p-dimensional random vector $x=\left(x_1,x_2,\cdots ,x_p\right)$, if there is a complex correlation between its variables, they can be summarized by $\mathrm{mk}\left(\mathit{mk}\le p\right)$ “comprehensive variables” or “principal components,” which are linear combinations of the original p variables.

Under the principle of less loss of useful information, replace more original multidimensional variables with fewer comprehensive variables to achieve dimensionality reduction of high-dimensional data. The principle of selecting the number of principal components usually ensures that the sum of the variance contribution ratio (SVCR) reaches or exceeds 90%, as shown in Eq. (40). SVCR expresses the amount of information that $\mathit{mk}$ principal components account for all features of $x_1,x_2,\cdots ,x_p$. Table 2 lists the results of principal component analysis of the energy mode of acoustic emission signals after 8-scale decomposition. The variance occupied by the following three principal components is 0.0017%, so it is not listed.

Particle size/principal component	$f_1$	$f_2$	$f_3$	$f_4$	$f_5$	$f_6$
1 ∼ 0.90 mm	−1.7612	−1.3737	−1.3862	−0.5974	−0.0494	0.0086
	−3.8690	1.0179	0.3630	−0.1481	0.0058	−0.0074
	⋮	⋮	⋮	⋮	⋮	⋮
0.90 ∼ 0.60 mm	−1.9344	−0.2136	−0.9412	0.2943	0.0443	−0.0063
	−2.3201	−0.0994	−0.8812	0.2670	−0.0128	0.0113
	⋮	⋮	⋮	⋮	⋮	⋮
0.60 ∼ 0.45 mm	−0.2694	−1.4987	0.3921	0.1166	−0.0365	0.0284
	−0.5988	−1.4909	0.7059	0.1738	0.0594	−0.0148
	⋮	⋮	⋮	⋮	⋮	⋮
0.45 ∼ 0.22 m	2.9133	0.5259	0.1801	0.0004	0.0319	−0.0333
	2.9517	0.4738	0.1563	0.0036	0.0229	−0.0244
	⋮	⋮	⋮	⋮	⋮	⋮
0.22 ∼ 0.18 mm	3.4536	0.1576	−0.2595	−0.0546	0.0078	0.0263
	3.3436	0.3753	−0.1152	−0.0024	−0.0217	0.0232
	⋮	⋮	⋮	⋮	⋮	⋮

Table 2.

Principal components of energy patterns obtained from scale decomposition (partial).

$$\text{SVCR}=\sum _{i=1}^{\mathrm{mk}}{\lambda }_i/\sum _{i=1}^p{\lambda }_i=\sum _{i=1}^{\mathrm{mk}}{\lambda }_i/p$$E40

In the equation, ${\lambda }_i$ is the eigenvalues of the sample covariance matrix, ${\lambda }_1\ge {\lambda }_2\ge \cdots \ge {\lambda }_p\ge 0$

3.4.2.3 Determine decomposition scale

Acoustic emission signals of different size particles were multi-scale decomposed by Sym8 wavelet, and then, the energy of the corresponding scale acoustic emission signals was obtained. After range normalization processing and principal component analysis, the variance of each principal component is shown in Table 3. It can be seen that the ratio of the maximum variance to the minimum variance of the principal components is quite large, indicating that multicollinearity is very serious. Considering that the excessively high decomposition scale leads to an increase in computational time. The decomposition scale is too small, and it is difficult to grasp the rich information contained in each detail. Therefore, the decomposition scale is determined to be 6, resulting in a 7-dimensional energy pattern. Among the seven principal components, the first three principal components account for 99.99% of the total variance, which is sufficient to extract all information. Therefore, the first three principal components are selected as inputs for the neural network.

Decomposition scale	3	4	5	6	7	8
${\lambda }_1$	3.8709	4.7243	5.55	6.4723	7.2595	7.7735
${\lambda }_2$	0.12901	0.27542	0.44917	0.51595	0.59732	0.83072
${\lambda }_3$	8.051e-5	0.0001945	0.000689	0.011085	0.13799	0.29968
${\lambda }_4$	1.87e-35	5.054e-5	0.000117	0.0005371	0.0045857	0.090915
${\lambda }_5$		3.85e-35	4.817e-5	0.0001136	0.00049631	0.0045248
${\lambda }_6$			3.59e-36	4.4803e-5	0.000111	0.0004932
${\lambda }_7$				1.046e-34	4.3571e-5	0.0001101
${\lambda }_8$					5.4742e-32	4.269e-5
${\lambda }_9$						1.475e-31

Table 3.

Variance of principal components under different decomposition scales.

3.4.2.4 Construction of neural network model

During each experiment, among 100 individuals, 18 individuals were selected from samples of each granularity (calculated principal components) as training sample data, 1 as validation data, and 1 as data for the test network. A total of 90 individuals were used for training, 5 for validation, and 5 for testing. Adopting a cross-over approach. The purpose of validation is to prevent the network from being overtrained (overfitting). During the experiment, the three principal components extracted were used as inputs to the network, and the average particle size was used as the output of the network. The number of output neurons is 1, and the number of input neurons is 3. The network has three layers, and the hidden layer transfer function is “tansig,” which is a hyperbolic tangent type transfer function. The output layer transfer function is “purelin,” which is a linear transformation function. The L-M (Levenberg Marquardt) algorithm is used for network training, with the aim that the L-M algorithm does not need to calculate the Hessian matrix and has a fast convergence speed. The target error during training is set to 10⁻³. According to the principle that the number of hidden layer nodes is roughly twice the number of input layer nodes [14], this chapter determines that the number of hidden layer nodes is 6, so the network structure is 3-6-1. The stopping condition for training is that the training error is less than the set target value.

The error function used for validation is specified as follows:

As for acoustic emission signals generated at specific particle sizes, compare the predicted particle size ${\widehat{y}}_i$ calculated by the network with the actual average particle size $y_i$ for the acoustic emission signals generated at specific particle sizes. Calculate the mean square error

$$\text{RMSE}=\sqrt{\frac{1}{N}{\sum }_{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}$$E41

N is the number of individuals used for validation.

4. Results and discussion

Fitting error refers to randomly selecting five individuals (one for each granularity) from the training sample, using the trained network for prediction, and calculating the absolute value of relative error. Prediction error is the absolute value of the relative error obtained when using a trained network to predict test samples. Each particle size has 20 individuals, and 20 fitting and prediction experiments are conducted. The results are listed in Table 4. From Table 4, it can be seen that for larger particles, the fitting and prediction accuracy are relatively high when predicting particle size based on principal component analysis. However, for small particles, the prediction accuracy is relatively low. It is possible that the signal-to-noise ratio of small particles is slightly lower than that of large particles. The average prediction accuracy of all acoustic emission signal samples is 94.25%, and the average fitting accuracy is 97.7%. This situation indicates that in actual production processes, the average particle size of materials in the bed can be analyzed and predicted online from the acoustic emission signals emitted by the fluidized bed.

Particle size range /mm		1 ∼ 0.90 mm		0.90 ∼ 0.60 mm		0.60 ∼ 0.45 mm		0.45 ∼ 0.22 mm		0.22 ∼ 0.180 mm
Sample number		fitting	prediction	fitting	prediction	fitting	prediction	fitting	prediction	fitting	prediction
Fitting and prediction accuracy	1	0.98635	0.97046	0.98634	0.96996	0.9798	0.97795	0.98102	0.9129	0.9833	0.89281
	2	0.94695	0.90697	0.98911	0.97954	0.97596	0.98421	0.98099	0.89347	0.98865	0.89233
	3	0.98677	0.97687	0.98711	0.98291	0.97701	0.98174	0.98496	0.90512	0.98647	0.89963
	4	0.96576	0.97951	0.95559	0.9701	0.97583	0.97254	0.98137	0.90389	0.98075	0.90041
	5	0.98868	0.93139	0.96554	0.98004	0.97788	0.96411	0.97755	0.89583	0.97233	0.89578
	6	0.97757	0.97436	0.98926	0.97579	0.97662	0.97618	0.98935	0.89707	0.97311	0.89718
	7	0.98867	0.97806	0.92952	0.97207	0.97772	0.97132	0.98367	0.90033	0.98327	0.89986
	8	0.93635	0.98478	0.98913	0.98287	0.978	0.9661	0.97717	0.89994	0.98195	0.89378
	9	0.98555	0.98633	0.95537	0.97242	0.97688	0.97724	0.98303	0.89962	0.98296	0.90367
	10	0.97805	0.97313	0.98672	0.98114	0.97833	0.97791	0.98428	0.89221	0.97792	0.9003
	11	0.98705	0.98619	0.98674	0.98292	0.98046	0.97928	0.97333	0.90851	0.98614	0.90075
	12	0.98602	0.9242	0.9874	0.96456	0.97779	0.97367	0.89982	0.89766	0.98604	0.90798
	13	0.96688	0.8971	0.97626	0.98975	0.97698	0.97291	0.97592	0.90047	0.9814	0.89611
	14	0.98439	0.97782	0.98573	0.98178	0.97837	0.97397	0.99054	0.90144	0.98879	0.90775
	15	0.98841	0.97475	0.98642	0.98025	0.97708	0.97413	0.98429	0.9046	0.97678	0.90528
	16	0.98572	0.98082	0.98841	0.97116	0.97534	0.97609	0.94056	0.90255	0.97972	0.89917
	17	0.9906	0.97829	0.99008	0.97371	0.97773	0.98342	0.9878	0.90123	0.98374	0.90157
	18	0.98874	0.96725	0.9873	0.96814	0.97699	0.9756	0.98739	0.893	0.97851	0.9071
	19	0.96582	0.95985	0.98665	0.96866	0.97659	0.97825	0.9816	0.90485	0.92117	0.90164
	20	0.97635	0.9077	0.97634	0.97053	0.9798	0.9554	0.96642	0.90797	0.90247	0.90238
mean value		0.97803	0.960792	0.979251	0.97592	0.97755	0.97460	0.97555	0.901133	0.974774	0.900274

Table 4.

The fitting and prediction accuracy of Sym8-WLA-PCA-MLFN for particle size.