The article shows that neural networks can be effectively used to identify the parameters of dynamic systems. The main attention in the paper is paid to modelling and practical results obtained in the MatLab Neural Network Toolbox environment. The use of the feedforward network and the Elman recurrent network is discussed. The simulation results show that the identification of dynamic systems using neural networks is most effective when the experimental data on the system contains internal redundancy or are incomplete.
Neural network models have internal regularizing properties that allow one to obtain small errors of generalization [4-6]. The regularizing properties of neural networks are especially useful to take into account in situations where the experimental data about the system contain internal redundancy. Redundancy allows to represent a collection of data with a model that contains fewer parameters than there is data. Thus, the neural network model compresses experimental information, eliminating noise components and emphasizing continuous dependences.1
2021 Copyright for this paper by its authors. Thus, the regularizing methods are reduced to optimizing the error functional:
N EG = (G( x ) − y ( ) ) 2 + G ,
where is a regularizing functional, - is a non-negative regularization constant.
To solve the identification problem, it is proposed to use a multilayer perceptron (Fig.1). Control based on a multilayer neural network allows solving poorly formalized control problems for complex dynamic objects in cases where a priori models and algorithms are unknown [7].
wi zj
wj xm b1 bi bm An m-dimensional feature vector {xi, i=1,2,…, m} is fed to the network input.
A fully connected structure is used between the input and hidden layers, as well as between the hidden and output layers, where m is the dimension of the original data, m = n+1, n is the number of unknown coefficients; р is the number of neurons on the hidden layer; xi is the vector of inputs, i = 1..m; wi is the weight coefficients between the input and hidden layers, i = 0,1,…,m , wj is the weight coefficients between hidden and output layers, k = 0,1,…,l. zj is the value of the output of the j-th neuron of the hidden layer, j = 1..l, b is the value of the output neuron of the network (network output); f1(x) is the activation function of neurons of the hidden layer; f2(x) is the activation function of the neuron of the output layer.
The sigmoidal activation function f(x): with the derivative: = 1(∑ =0 ); ( ) = 1+ 1 (− )
, ′( ) = ( )(1 − ( )), is used as the activation function of the neurons of the hidden layer.
Let’s consider the case when the dynamical system is described by the second-order equations with unknown coefficients:
2 2 2 + 1 + 0 = ( ). form:
where 1, 2, . . . , are unknown coefficients of the differential equations, is a vector of inputs of the neural network, is the expected (measured) value at the output of the network.
Then the system of linear equations (SLE) with unknowns describes the state of the system to the {
, 1 1 + 2 2+. . . + −
To determine the architecture of the neural network, it is necessary to first identify the control object. For this, the Plant Identification panel from the Control Systems section of the Simulink neural block library is used (Fig.2). Using the control elements of the Plant Identification panel, the architecture of the neural network, the parameters of the training sequence and the parameters of training, as well as the assessment of the quality of this process, are set. multilayer perceptrons using the backpropagation method [8-10].
In the course of scientific research, to build a model of the process of identifying system parameters using neural networks in the MatLab environment, the Feed-forward backprop network with the trainlm training function (Fig. 3) was used, which returns weights and biases using the Levenberg–Marquardt algorithm (LMA), which based on the method of Mean Squared Error (MSE).
The function learning quality - mse from MatLab is determined by the root mean square error. In the network adaptation mode, the learningdm function of correction of weights and biases was used as a tuning function. Sufficient computational accuracy was obtained using the gradient optimization algorithm with an iterative component and a purelin linear activation function.
To estimate the required number of synaptic weights, the formula is applicable:
1+ 2 ≤ ≤ ( + 1) ( + + 1) + , where is the dimension of the input signal, number of training sample elements.
- is the dimension of the output signal, - is the lue10-10 B g n i n i raT10-15 10-20 10-250
The obtained result - the coefficients of the differential equation (roots of the SLE) - is displayed in the Data: network1_outputs dialog box (Fig. 5).
The value of the computational error - in the Data: network1_errors window (Fig. 6)
To solve this problem, a comparative study of the results was carried out using the models of the Feedforward network and the Elman's recurrent network, characterized by partial recurrence in the form of feedback between the hidden and the input layer [11]. Elman's network is implemented using tansig function. It is advisable to use Elman's networks in control systems of moving objects, for example in the construction of technical vision systems [12].
In conclusion, this study confirms a technique for solving SLEs using neural networks in the MatLab Neural Network Toolbox. A series of model experiments and calculations allow us to make a conclusion that the minimum calculation error and high computational accuracy are achieved for neural networks even with a small number of layers when using a network model with error backpropagation. Using Elman's networks in control systems of moving objects, for example in the construction of technical vision systems is advisable.
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