A typical separation process is formalized using the example of iron ore (magnetite quartzites) beneficiation technology. This study examines the suitability of various neural network structures as mathematical models for regression. The results of computer modeling for training using real indicators of magnetite quartzite beneficiation and reference models are presented. Comparison of the approximation results for different neural network bases is also included. The authors tested various samples of reference and noisy data. The best intellectual models are recommended for automating separation processes.
This study solving the problem of selecting optimal scientific approaches for formalizing
technological processes (TP) in separation technology, with the aim of automating control. A typical
example is the TP for beneficiation (separation or concentration) of iron ore (magnetite quartzites).
Numerous works of the authors [
A scheme is proposed [
The following notations are used in the scheme: Pi represents an apparatus or stage in the technological process (scheme PG); X represents the set of input actions that affect the control object. This set includes both external PG inputs to the process and inputs of individual devices Pi. The inputs can be controlled influences (i.e., control inputs) and uncontrolled influences (i.e., disturbances). Disturbances are sometimes measurable, but in other cases, they can only be assessed verbally (e.g., large, small, etc.). Y represents the set of outputs, which, similar to X, combines the outputs of the entire process and individual devices Pi. While all outputs are potentially controllable, in practice, only a subset of them are typically controlled. Not all connections between input signals and outputs can be determined using classical transfer functions (W(p)). This limitation is due to the dimension of the problem and the level of knowledge of the object under study. Although certain parameters (factors) may be known to exist, it may not always be possible to measure or evaluate them accurately enough. However, if a parameter is expected to have a significant effect on the dynamics of the process, it should be included in the consideration. If an accurate (quantitative) assessment is impossible, a fuzzy (linguistic) assessment can be assigned to a parameter. The implemented approach for developing technology for operational forecasting of concentration processes assumes the presence of implicit mutual influence among the technological parameters of the process and the characteristics of separation products.
To begin the analysis, we consider a scenario where the characteristics of the feedstock and all technological parameters are assumed to have a significant influence on the process outputs. The process model is represented as a directed graph, with the parameters as nodes and the arcs indicating their mutual influence. This representation of the technological process is similar to a neural network, where the nodes contain functions that convert signals.
It is known that to formalize the beneficiation TP under the conditions of a technological line, we need to consider a number of parameters that can be represented as a "black box" in classical cybernetics (see Fig. 2).
In Fig. 2 such additional notations are accepted: i = 1...Nr α = {α i },
is estimated raw ore grade; ξ = {ξ i } is specific gravity of every variety of ore; ρ = {ρ i } g = {gi } beneficiation stage; Ns – is quantity of stage; Q = {Q j }, j = 1...N s is number of
C = {C } j is parameters); corresponding amount of factors. Bm = {Bm j }, Bk = {Bk j }, Bs = {Bs j } are consumption of water to the mill, classifier and magnetic ρ s = {ρ } p j is a separation respectively; ρ k = {ρ k j } is a pulp density in the process of classification; β pp = {β pp j }= {β j } is an estimated grade in the industrial pulp density before magnetic separation; product; β õ = {β õ j } is loss of a commercial component in tails; β k is a quality of concentrate; γ = {γ j } is an output of useful component in an industrial product; k is an output of useful component in concentrate; ε = {ε }
j is an extraction of useful component in an industrial product; ∑k is an extraction of useful component in a concentrate. V = {ν 1,ν 2 ,ν 3,...,ν nv is a vector of input disturbing parameters } (input a priori information); U = {u1, u2 , u3,..., unu } is a control vector (control actions and/or regime Y = {y1, y2 , y3,..., yn }
y is a vector of output parameters of the system; nv, nu, ny are Q 0 Вm 1 В k 1 Вs 1
C 1
I stage Internal variables:
P m 1 ρ k 1 ρ s 1
V = {ν 1,ν 2,ν 3,...,ν nv } α ξ ρ g
It should be noted that the indices, such as α , β , γ , ε , can be monitored for several products (e.g., total iron and magnetic properties, etc.). Additionally, the factors α , ξ , ρ , g , d0 (Fig.2) can be considered a priori information, as they are determined in preceding technological processes, such as ore extraction from an open-pit mine or crushing in a crusher, and are not directly controlled (thus, they can be considered as disturbances). Other indices (Fig.2) are generated during the beneficiation process and can be regulated or adjusted based on the specific process conditions. Monitoring of these factors is performed, but not always with the necessary precision and accuracy (particularly for qualitative indicators). Therefore, the distribution of the state vector on input and output indices is conditional, as most of the parameters on output of the first stage will serve as input for the second stage, and so on.
Neural networks are successfully used for the synthesis of such control systems for dynamic objects
[
Within the framework of ongoing research, the use of backpropagation networks seems to be the
most promising. Networks of this type generally have significantly shorter training times than
backpropagation networks, which allows them to respond more quickly to changes in the conditions of
the enrichment process, such as fluctuations in feedstock characteristics, process parameters, or
equipment wear. The counter-propagation neural network combines two well-known algorithms:
Kohonen's self-organizing map [
The analysis of the potential of using neural networks for creating models for express analysis of production processes has led to the determination of the neural network model structure. This was achieved by analyzing the technological scheme of the flotation department, taking into account the study's previous results and the conceptual principles of the technology for modeling production processes, such as mineral ore enrichment, which were already adopted within the project.
The parameters used in the model are classified into three groups: benchmarks, control parameters, and indicators. Benchmarks include the characteristics of the input and output products of the flow sheet. The model considers 15 control parameters, which are the parameters that can be influenced to change the conditions for the implementation of the technological process and the values of control indicators. The control parameters in the model include temperature, humidity, pressure, liquid level, voltage, current, etc. Overall, approximately 27 indicative parameters can be considered.
Layer 0 neurons do not perform calculations but serve as branching points. Each layer 0 neuron is
connected to every neuron in layer 1 (Kohonen's layer), and each neuron in layer 1 is connected to every
neuron in layer 2 (Grossberg's fracture). Each connection link has its own weight associated with it.
The weights wi of connections of layers 0 and 1 form a matrix of weights W, and the weights VJ of
connections of neurons in layers 1 and 2 form a matrix of weights V. The weight values are adjusted in
the network training mode, when a priori known vectors of inputs X and outputs Y are fed into the
model (Fig. 1). In the predictive mode, the input vector X, which is generated based on the current
monitoring data, is fed into the model, and the output vector Y is generated by the network. The output
of each layer neuron is simply the sum of the weighted inputs. The Kohonen layer uses a competitive
learning process in which the neuron with the highest weighted sum of inputs is selected as the winner.
The output of this neuron is assigned the value "1", and the outputs of the remaining neurons of the
Kohonen layer are assigned the value "0". The Grossberg layer functions in a similar way - its outputs
are determined by the weighted sum of the corresponding inputs from the Kohonen layer. But, since
only one neuron of the Kohonen layer has the value “1” set at the output, then in fact each neuron of
the Grossberg layer only outputs the value of the weight that connects this neuron with the only
nonzero Kohonen neuron. In essence, the Kohonen layer classifies input vectors into similar groups, thereby
providing the definition of regions of the multidimensional input space that map to a small
neighbourhood of the same “point” in the output space. This is achieved by adjusting the weights of the
Kohonen layer, which ensures that the same neuron of this layer is activated by the corresponding input
vectors. Before training starts, all weights of the network are assigned some random values. During the
learning process, the weight vectors change by "tracking" a small group of input vectors. Training ends
when the required pattern of outputs is formed at the output of the neural network. The training of the
Grossberg layer is carried out by adjusting only those weights that are associated with a Kohonen neuron
that has a non-zero output. The amount of weight correction is proportional to the difference between
the weight and the desired output of the Grossberg neuron to which it is connected. The use of a neural
network model assumes an a priori classification of the states of the system (beneficiation process) into
a finite number of options. Each state in which there is a violation of the procedural characteristics of
the process is associated with a set of corrective actions that involve specific changes in control
parameters. Both expert assessments and formal classification methods, such as factor and cluster
analysis, can be used for classification. The values of output vectors Y are used as the main classification
criterion. To determine the current state of the process, a comparison is made between the output of the
neural network model and the stored vectors in the system's database that determine the selected states
of the enrichment process. If the database indicates that the identified state corresponds to a violation
of the regulatory characteristics, then the system retrieves recommendations from the database for
correcting the state. If there is an appropriate actuator, the launch of corrective actions can be automated.
The developed neural network model of the flotation process was implemented and studied in the
Matlab environment [
To evaluate the efficiency of radial basis networks and multilayer perceptrons, consider the problem
of approximating the function shown in [
x 1 d (x, y) = 3(1− x)2 exp(−x2 − ( y +1)2 ) −10( − x3 − y5 exp(−x2 − y2 ) − exp(−(x +1)2 − y2 ) (2) 5 2 , where changing variables within -3 ≤ x≤ 3 and -3 ≤ y≤ 3. The graph of this function is shown in Fig.3.
Based on a training set of 625 data groups ([x, y], d) generated with a uniform distribution of
variables x and y in their domains of definition, a 2-36-1 network structure (2 input neurons, 36
Gaussian-type radial neurons and one output linear neuron). We also used here a hybrid learning
algorithm similar to the technique [
10 5 -4 -3 -2 -1 x 0 1 2 3
The process of building a neural network model can be divided into 5 main stages (Fig. 5).
At the first stage, 2 sets of training samples ([x, y], d) containing random noise (errors) were generated in order to simulate the data obtained in the study of a stochastic process or physical phenomenon [12-14]. The first group, dʹ, contained highly noisy data (noise at the level of 20%), the second, dʹʹ, contained weakly noisy data (noise at the level of 2%). These datasets were generated in the following way. The function d(x,y) is tabulated within -3 ≤ x≤ 3 and - 3 ≤ y≤ 3 with a step of 0.25, and a table of values of the function d is compiled. A certain value ε is added to each value of the function, obtained using a random number generator with a uniform distribution. The expression is used to evaluate − p max( d )
p max( d ) ≤ ε ≤ 2 2 , where p is the noise level in fractions of a unit. Thus, the sets of points [x, y, dʹ] and [x, y, dʹʹ] imitate the results of observations of some physical process containing measurement errors, on the basis of which the process d(x, y) itself will be modelled in a real problem for the researcher unknown.
At the second stage, the data were normalized in the range [-1…1]. In this case, there is no need to divide the total sample into training and testing sets, since the required function d(x,y) is used for testing. (3)
In Fig. 6a, the function is modeled on highly noisy data with standard deviation S2=1,806 (22%) multilayer perceptron. S2 neural network – 1,63 (20%). The deviation of the neural network from the noisy signal is S2 =0,844 (9%).
In Fig. 6b function contains weak interference S2 =0,185 (2%) and approximated by a multilayer perceptron. S2 neural network – 0,179 (2%). The deviation of the neural network from the noisy signal is S2 =0,11 (1%).
The third stage involves choosing the type of neural network from two suitable for solving the approximation problem: a multilayer perceptron and a radial basic network. In this experiment, both types are used to compare their effectiveness. The architecture of the multilayer perceptron was defined as follows: the network consists of two hidden layers, each containing 8 neurons. The RBF network contains 1 hidden layer, and the number of neurons in this layer grows during the learning process.
At the fourth stage, two specified types of neural networks were trained on each of the data sets. The Neural Toolbox of the Matlab package was used for building and training the neural networks.
At the final stage, response surfaces of the neural network models were built (Fig. 6, 7). The standard deviation, S2, was used as the error metric for evaluating the resulting models. 10 5 -5 -2 -4 -3 -2 -1 x 0 1 2 3 2 0 y -2 -4 -4 -2 x 0 2 4
In Fig. 7a function contains noise, S2 =1,88 (23%) and it is approximated by a radial basis network. S2 of neural network is 1,68 (20%). Deviation of a neural network from a noisy signal is S2 =0,85 (10%). On Fig. 7b function contains noise, S2 =1,185 (2%) and it is approximated by a radial basis network. S2 of neural network is 0,184 (2%). Deviation of a neural network from a noisy signal is S2 =0,185 (2%).
Both types of neural networks considered were able to build a regression model of a noisy signal. In all cases, the networks demonstrated the ability to filter noise similarly to [15-16]. Although the RBF network showed a slightly larger deviation (error) in all cases, the difference from the multilayer perceptron (Fig. 8) is small and insignificant. The radial basis network has an advantage over the multilayer perceptron because it does not require an expert to determine the number of layers and neurons [17, 18]. In the case of a radial basis network, the number of neurons increases during the learning process to achieve a given model accuracy. However, the number of neurons in the RBF network is significantly greater (in this study, by an order of magnitude) than in the perceptron, which slows down working with the RBF network. When predicting the behavior of a function outside the learning range, it is more beneficial to use a multilayer perceptron because it has the ability to extrapolate a function. Numerous studies by the authors [19-22] in various areas of applied neural network technologies confirm similar results.
Regarding the neural network architectures for approximation and regression analysis, multilayer perceptrons and radial basis networks are both applicable. While each type has its advantages and disadvantages in dependency recovery tasks, both effectively approximate complex functions by learning from noisy data. Multilayer perceptrons have shown good results in processing experimental data, including multidimensional data, making it possible to model patterns hidden within them. In terms of training, a three-layer perceptron based on a linear activation function for the output neuron and hidden layers with a hyperbolic activation function showed the best results in terms of convergence of the learning process and prediction accuracy. Therefore, the aim of this study has been fully achieved.
Future research will be focused on identifying optimal methods for training these neurostructures in real-time. This will include classical gradient algorithms based on error backpropagation (first and second orders [23]) and non-iterative approaches [24]. We also plan to consider alternative neural network architectures, such as recurrent and dynamic structures [25]. These findings will be essential for solving problems related to structural and parametric identification, as well as intelligent control of the separation process (beneficiation or concentration). Based on a priori estimates, this approach has demonstrated sufficient effectiveness in mining plant conditions.