Designing stable neural identifier based on Lyapunov method

The stability of learning rate in neural network identifiers and controllers is one of the challenging issues, which attract many researchers’ interest in neural networks. This paper suggests adaptive gradient descent algorithm with stable learning laws for modified dynamic neural network (MDNN) and studies the stability of this algorithm. Also, stable learning algorithm for parameters of MDNN is proposed. By the proposed method, some constraints are obtained for learning rate. Lyapunov stability theory is applied to study the stability of the proposed algorithm. The Lyapunov stability theory guaranteed the stability of the learning algorithm. In the proposed method, the learning rate can be calculated online and will provide an adaptive learning rate for the MDNN structure. Simulation results are given to validate the results.


Introduction
In recent decades, soft computing is frequently used in business and industry.Artificial neural networks are essential parts of any computing software [4].The most widely used neural network architecture is the multilayer feed forward neural network.The most popular approach for training the multilayer feed-forward neural network (FNN) is the backpropagation (BP) algorithm based on gradient descent (GD) method.Determining an appropriate learning rate for this algorithm is important.The training algorithms (learning rules) could be defined as "a procedure for modifying the weights and biases of a network in order to train the network to perform some tasks" [6,11].The network model having a good function approximation capability through the training samples can well reflect the complex nonlinear relationship between objects [17].However, one problem inherent within them is their convergence to local minima and the user set acceleration rates and inertia factor parameters that are sensitive to the learning process [1][2][3].The FNNs with the BP learning algorithm have been used successfully in pattern recognition, optimization, classification, modeling, identification and controlling [13,31].However, the problems of the slow convergence rate, local minimum and instability are the most challenging issues in this algorithm.
In recent decades, many efforts have been made to improve the convergence of the BP algorithm.There are some works to improve BP algorithm in order to have online training [9,[19][20][21].For this algorithm determining, an appropriate learning rate is necessary, so that the learning process become stable.If the learning rate is large, learning may happen rapidly, but it may also become unstable.To ensure stable learning, the learning rate is small enough.Small learning rate may also lead to a long training time.These problems are inherent to the basic learning rule of FNN that are based on GD optimization methods [15,30].The convergence properties of such algorithms are discussed in [5,7,12,15,16,18], and [22].Learning algorithms based on GD includes real-time recurrent learning (RTRL), ordered derivative learning and so on [1].
Derivative-based methods have the advantage of fast convergence, but they tend to converge to local minima [2].In addition, due to their dependence on the analytical derivatives, they are limited to specific objective functions, inferences, and MFs [2].Some papers [1][2][3]25,26] have investigated the stability of fuzzy neural networks.The popular method for stability analysis is Lyapunov stability.Also, in [8,10,27,28] Lyapunov stability theorem is considered.The learning algorithm in neural and fuzzy neural networks not only has the role of updating parameters but also has influence on stability and convergence.The stability and convergence of learning algorithms are rarely investigated in the papers.In this study, the stability of learning algorithm is addressed in dynamic neural networks.
In this paper, the main concern is using Lyapunov stability approach for determining stable learning rate in system identification via modified dynamic neural network.The GD training the parameters of update rule for MDNN is considered.
The rest of article is organized as follows: in section 2, the structure of the dynamic neural network is discussed.In section 3, MDNN learning algorithm applied to process.Simulations and results for three nonlinear systems are presented in section 4. Section 5 presents conclusions.

Dynamic neural network
In the feed forward artificial neural, a neuron receives its inputs from other neurons.The weight sum of these signals is the input to the activation function.The resulting value of the activation function is the output of a neuron.This output is branched out to other processing units.This simple model of the artificial neuron ignores many of the characteristics of its biological counterpart.For example, it does not take into account time delays that affect the dynamics of the system [23].The dynamic neural network (DNN) for the first time proposed by Gupta [24].The basic structure of dynamic neuron (DN) is shown in figure 1.Each neuron composed of two units: the inhibitory (negative) unit and excitatory (positive) unit.The inhibitory units received the summation of positive inputs, a delay of own outputs and abstraction a delay of excitatory outputs by multiple to determined weights.The excitatory units received the summation of negative inputs, a delay of own outputs and abstraction a delay of initiatory outputs by multiple to the determined weights [29].
The final output of the neuron can be written as follow: where,

and inhibitory units (
I n e t ), respectively and can be written as:

a X t a n e t t b n e t t
(2) 1 1 And: where, E X , I X are the positive and negative inputs, respectively and the parameters of 0 a , 11 a , 12 a , 0 b , 11 b , 12 b are the weights of DN.

Stability analysis of learning algorithm
Suppose an MDNN as an identifier shown in figure 2.  Assume that the parameters of the MDNN are changed with time.
In figure 3, The cost function for the training algorithm is defined as: where,  

 
Ok is the final output of MDNN.Weights of output layer are updated by GD method as follows: where,    is the learning rate parameters of the output layer, and we have: From ( 8) and ( 9), it will be inferred that: And the updating rule for hidden layer parameters by GD method is as follows: where, the weights are updated as follows: Ok ek ak (14) From ( 13) and ( 14), it will be inferred that: It can also be written as:   (21) In this paper, candidate Lyapunov function is a function of error which is associated with GD based learning algorithms.This means when Lyapunov function converges to zero GD based learning algorithm is converged to zero too.Now a discrete Lyapunov function is considered as follows: Then, the variation of Lyapunov function on each iteration will be: e k e k e k e k e k e k e k (23) The variation of error can be approximated by: where, the   .tr is the trace of matrices.From (10) it will be concluded that: From ( 15) and ( 21), 1  obtained as follows: Let .F be Frobenius norm.Thus, using ( 16) to ( 20) and ( 21), the following equations are obtained: So, it can be written as: From ( 23) and (32), it will be inferred that: Then: From (35) we choose the learning rates as follows: (37)

Simulation and results
In this section, the proposed algorithm in sections 3 is simulated on three nonlinear systems as examples 1, 2 and 3.
In each example, there are 1000 random numbers which divide to training and test data sets.Dynamic neural network is used as identifier as illustrated in figure 2.
Example 1: Identification of a nonlinear dynamical system.In this example, the nonlinear plant with multiple time-delays is described as [3]: where, () uk and () yk are the system input and output, respectively.Where: For simulation of this nonlinear system, a neural network with the structure depicted in figure 2 is employed, where n is assumed to be 5, and p is taken as 15.In this neural network, it has been assumed that 2 f is a linear function, and 1 f is a symmetric sigmoid function defined as below: where, net is the weighted sum of the inputs.For comparison, the mean square error (MSE) criterion has been used.In the simulation results, figure 4 indicates convergence with fulfillment of the stability conditions.
Example 2: This system of equation is as follows [14]: where, () uk and () yk are the system input and output, respectively.For simulation of this nonlinear system, a neural network with the structure depicted in figure 2 is employed, where n is assumed to be 2, and p is taken as 10.In the simulation results, figure 5 indicates convergence with fulfillment of the stability conditions.simulation of this nonlinear system, a neural network with the structure depicted in figure 2 is employed, where n is assumed to be 3, and p is taken as 15.In the simulation results, figure 6 indicates convergence with fulfillment of the stability conditions.

Conclusions
In this paper, for permanent learning of modified dynamic neural network parameters online, Lyapunov stability theory is employed.In this learning algorithm, the associated parameters are trained according to descending gradient.Taking advantage of Lyapunov stability theory, some regions are defined, and through selection of the permanent training rate out of these regions, it can be guaranteed that the learning algorithm is stable throughout the identification process.The results of the obtained theory have been simulated for three examples.The simulation results suggest that in the training process, the system is stable and the convergence rate is desirable.This procedure can be employed in neural controllers.

Figure 2 .
Figure 2. Modified dynamic neural network as the identifier.Details of the MDNN network are illustrated in the figure 3.
output of DN neuron.DN structure of neurons is shown in figure1.This network has n inputs in the input layer, p DN in the hidden layer and one conventional neuron in the output layer.are nonlinear activation functions.According to figure3it is obvious that:

0 11 12 0 11 12 a
a a b b b   is the weight vector and,

Figure 4 .
Figure 4. Learning rate is smaller than the upper limit bound.

Figure 5 .Example 3 :
Figure 5. Learning rate is smaller than the upper limit bound.

Figure 6 .
Figure 6.Learning rate is smaller than the upper limit bound.