2.3.1. PNN Principle

A PNN is a type of radial basis network that was first proposed by Dr. D.F. Speeht in 1989. The PNN is a supervised network classifier based on the Bayes minimum risk criterion [35]. As a feed-forward network, a PNN has the advantages of a fast training speed and simple parameter adjustment. Currently, PNNs are widely used in pattern classification [36]. Compared with other network classifiers, a PNN can not only guarantee real-time performance, but also produce classification and recognition results that are minimally influenced by complex parameter settings.

The signal sample vector can be represented as *X* = [*x*1, *x*2, ... , *xi*, ... , *xn*] with states *Y* = [*y*1, *y*2, ... , *yi*, ... , *yn*]. Then, the prior probability, posterior probability, and class-specific probability density functions for each state can be represented by *P*(*yi*), *P*(*yi*/*X*), and *P*(*X*/*yi*), respectively. For a given identification target, *P*(*yi*) is a known parameter, and *P*(*X*/*yi*) can be estimated using the Parzen function. The corresponding formula is as follows:

$$P(X/y\_i) = \frac{\sum\_{j=1}^{N\_i} \exp(-\frac{||X - x\_{ij}||^2}{2\sigma^2})}{N\_i (2\pi)^{\frac{d}{2}} \sigma^d} \tag{21}$$

where *Ni* is the total number of samples of the *i*th class, *d* is the dimensionality of the feature vector, *xij* is the *j*th sample of the *i*th class, and σ is the width of the Parzen function window, that is, the smoothing parameter.

The following formula is obtained from probabilistic and statistical theory.

$$P(y\_i/X) = P(X/y\_i)P(y\_i)/P(X) \tag{22}$$

If the possibility of misjudgment is not considered, the Bayes rule can be expressed as follows.

$$\exists \; j \neq i = 1, 2, 3, \dots, m,\\ \text{if } P(y\_i/X) > P(y\_j/X), X \in y\_j \tag{23}$$

However, because misjudgment can readily occur in real-world situations, it is necessary to introduce the risk coefficient λ*ij*, yielding the following risk function R for the decision conditions.

$$R(y\_i/X) = \sum\_{j=1}^{m} \lambda\_{ij} P(y\_j/X) \tag{24}$$

In summary, the Bayes minimum risk criterion can be expressed as follows.

$$\text{if } \mathcal{R}(y\_i/X) > \mathcal{R}(y\_j/X), X \in \mathcal{Y}\_j \tag{25}$$

In this paper, the minimum risk criterion is used as the basis of the feed-forward network that serves as the mill load state recognition model. By setting reasonable smoothing parameters, the network is trained on a set of sample feature vectors to estimate the probability densities of three distinct load states and enable the recognition of the mill load state.

#### 2.3.2. Principle of AEPSO

The optimization speed and position updating formulas of the traditional particle swarm optimization algorithm [37] are as follows:

$$\boldsymbol{V}\_{p}\boldsymbol{V}^{k+1} = \omega \boldsymbol{V}\_{p}^{k} + c\_{1}r\_{1}(\boldsymbol{W}\_{p} - \boldsymbol{X}\_{p}^{k}) + c\_{2}r\_{2}(\boldsymbol{W}\_{\mathcal{S}} - \boldsymbol{X}\_{p}^{k})\tag{26}$$

$$X\_p^{\;k+1} = X\_p^k + V\_p^{k+1} \tag{27}$$

where *k* is the number of iterations; ω is the inertial weight of the particle; *c*1, *c*<sup>2</sup> are the learning factors of the particle, of which the former is the individual factor and the latter is the global factor; and *r*1,*r*<sup>2</sup> are random numbers in the interval [0, 1], which make the particles independent and diverse.

To address the nonlinear problem of ball mill load identification, the AEPSO algorithm introduces a nonlinear adaptive time-varying inertial weight.

$$
\omega\_t = \omega\_{\text{start}} - (\omega\_{\text{start}} - \omega\_{\text{end}}) \times \exp(-\frac{1}{1 + 2t/t\_{\text{max}}}) \tag{28}
$$

For the learning factors *c*1, *c*<sup>2</sup> of particles, the traditional particle colony algorithm usually sets *c*<sup>1</sup> = *c*<sup>2</sup> = 2, but this approach ignores the phase difference of the algorithm during training. The AEPSO algorithm adopts the strategy of managing the learning factor in segment, and the formula is as follows.

$$\begin{cases} \ c\_1 = 2.5, c\_2 = 1.5 & t < t\_{\text{max}/2} \\ \ c\_1 = 1.5, c\_2 = 2.5 & t \ge t\_{\text{max}/2} \end{cases} \tag{29}$$

To enhance the adaptability of particle swarm optimization after iteration, the AEPSO algorithm introduces the local search operator β in Equation (13). The revised formula is as follows:

$$X\_p^{\ k+1} = X\_p^k + \beta \times V\_p^{k+1} \tag{30}$$

where β = *rand*( )[*rand*()+ 0.5] and *rand*( ) is a random number in [0, 1].

#### 2.3.3. Optimization of the PNN by AEPSO

The smoothing parameter σ in the PNN has a considerable influence on the training effect. The improper selection of the σ value makes it easy to misjudge the recognition of the mill load state. Therefore, this paper uses an AEPSO algorithm to optimize the smoothing parameters of the PNN so that the optimized network can effectively identify the state of the mill load. The specific steps in the algorithm are as follows.


The network structure of the AEPSO\_PNN includes four parts: the input layer, the mode layer, the summation layer, and the output layer, as shown in Figure 4.

As Figure 5 shows, the training step of the load state identification model of a ball mill based on the AEPSO\_PNN is as follows.


**Figure 5.** Adaptive evolution particle swarm optimization probabilistic neural network (AEPSO\_PNN) network structure diagram.

#### **3. Design of the Load State Identification Method for a Ball Mill**

Based on the research on the EWT algorithm, MFE theory, and the PNN clustering algorithm combined with the characteristics of ball mill vibration signals, a feature extraction algorithm for vibration signals is proposed based on modified EWT, MFE, and AEPSO\_PNN classification. The specific steps in the algorithm are as follows.


$$\rho\_{xy} = \frac{\sum\_{i=1}^{N} (x\_i - \overline{x})(y\_i - \overline{y})}{\sqrt{\sum\_{i=1}^{N} (x\_i - \overline{x})^2} \sqrt{\sum\_{i=1}^{N} (y\_i - \overline{y})^2}} \tag{31}$$

The correlation coefficient threshold is calculated as

$$\mu\_h = \frac{\max(\mu\_i)}{10 \times \max(\mu\_i) - 3} \tag{32}$$

where μ*<sup>h</sup>* is the threshold, μ*<sup>i</sup>* is the correlation coefficient between the *i*th AM-FM component and the original signal, and max is the maximum correlation coefficient value. Each AM-FM component for which the value of the correlation coefficient with the original signal is greater than the threshold μ*<sup>h</sup>* is retained as a sensitive AM-FM component. Each AM-FM component for which the correlation coefficient is smaller than the threshold μ*<sup>h</sup>* is removed as a spurious component.


Thus, the overall flow of the ball mill load identification model that is proposed in this paper based on the modified EWT, MFE, and AEPSO\_PNN classification methods can be summarized as shown in Figure 6.

**Figure 6.** Algorithm flow based on the modified EWT, MFE, and AEPSO\_PNN classification methods.

#### **4. Experimental Analysis of Mill Load State Recognition**

#### *4.1. Data Collection*

To verify the method proposed in this paper, a grinding experiment was performed using a Φ305 × 305 mm Bond index experimental ball mill. The experimental device is shown in Figure 7. The material used in the experiment was tungsten ore from a mine in Jiangxi, China, with a Protodyakonov scale of hardness of 14–18, a density of 1.8 t/m3, and five grades of particle sizes: 1–3 mm, 3–6 mm, 6–9 mm, 9–11 mm, and >11 mm. The experimental parameters considered were the fill rate, powder-to-ball ratio, and grinding concentration. The vibration signal acquisition system of the mill cylinder consisted of a DH5922N dynamic data acquisition instrument and a DH131 acceleration sensor, which were used to collect the signals of various load parameters under three different load conditions. According to the literature, the mill load was divided into the following states: the underloaded state, corresponding to a fill rate of 10–20%; the normal load state, corresponding to a fill rate of 20–40%; and the overloaded state, corresponding to a fill rate of 40–60% [38].

**Figure 7.** Experimental device.
