Piecewise Convex Technique for the Stability Analysis of Delayed Neural Network

On the basis of the fact that the neuron activation function is sector bounded, this paper transforms the researched original delayed neural network into a linear uncertain system. Combined with delay partitioning technique, by using the convex combination between decomposed time delay and positive matrix, this paper constructs a novel Lyapunov function to derive new less conservative stability criteria. The benefit of the method used in this paper is that it can utilize more information on slope of the activations and time delays. To illustrate the effectiveness of the new established stable criteria, one numerical example and an application example are proposed to compare with some recent results.


Introduction
As a special class of nonlinear dynamical systems, neural networks (NNs) have attracted considerable attention due to their extensive applications in pattern recognition, signal processing, associative memories, combinatorial optimization, and many other fields.However, time delay is frequently encountered in NNs due to the finite switching speed of amplifier and the inherent communication time of neurons, especially in the artificial neural network.And it is often an important source of instability and oscillations.It has been shown that the existence of time delay can change the topology of neural networks, and then change the dynamic behavior of neural networks, such as oscillation and chaos.Thus, it is significant to introduce time delay into the neural network model.Additionally, stochastic disturbances and parameter uncertainties can also destroy the convergence of a neural network system.This makes the design or performance for the corresponding closed-loop systems become difficult.Therefore, the equilibrium and stability properties of NNs with time delay have been widely considered by many researchers.Up to now, various stability conditions have been obtained, and many excellent papers and monographs have been available (see [1][2][3][4][5][6][7][8]). So far, these obtained stability results are classified into two types: delay independent and delay dependent.Since sufficiently considered the information of time delays, delay-dependent criteria may be less conservative than delay-independent ones when the size of time delay is small, and much attention has been paid to the delay-dependent category [9][10][11][12].In order to utilize more information of time delay, delay interval is always divided into two or many subintervals with the same size [13,14].It has been shown that delay partitioning technique is effective, and the more delay subintervals are divided, the less conservatism of stable criterion may be.However, too many delay subintervals must increase the computational burden; how to balance these two contradictions is a very important issue.To solve this problem, weighting delay and convex analysis methods are widely employed [8,13].
Additionally, as pointed out by Li et al. [15], the choice of an appropriate Lyapunov-Krasovskii functional (LKF) and the utilization of neuron activation function's information are very important for deriving less conservative stability criteria.Thus, recently, many authors were devoted to propose a new

Stability Analysis
Consider a new class of Lyapunov functional candidate as follows: where Define (,   , ẋ Notice that We start with the case () ∈ [  , ], where (()) = 1.
Remark 7. Different from previous work, the LKF function in this paper is constructed by using the convex combination between decomposed time delay and positive matrix, which may reduce the conservatism of criterion.Remark 8. From the proof of Theorem 6, one can see that, by using the different combinations among Π () , Π () , and Π() , Π() , we can establish different stable criteria as follows.
Corollary 9.For given scalars   ≥ 0,   > 0, Remark 12. Since the existence of items Ψ    Ψ and Ψ    Ψ, the results established in Theorem 6 and Corollaries 9-11 are not LMI criteria.In order to use LMI toolbox in computing software, by using the lemma derived in [20], we further establish the following more practicable stable rules.
Theorem 13.For given scalars ) is globally asymptotically stable if there exist positive constants   ,   > 0 (,  = 1, 2, 3, 4) such that the following conditions hold: Remark 14.Similar to the analysis of Remark 12, the related practicable stable can also be established by using Corollaries 9-11, since the expressions are similar to Theorem 13, which are omitted here.

Illustrative Examples
In this section, two numerical examples are given to illustrate the effectiveness of the proposed method.Similar to [15], our purpose is to estimate the allowable upper bounds delay   under   = 0 such that the system (1) is globally asymptotically stable.For this example, when τ () = 0, the maximum allowable delay bound   is 1.4224 in [9], 1.9321 in [10], 3.5841 in [11], 3.6156 in [13], and 3.7327 in [12].Recently, by using delay-scope-dependent method, Li et al. improved the previous results further in [15] and gave out the maximum allowable delay bound   as 3.8363.Applying Theorem 13 in this paper, the maximum allowable delay bound is 3.9221 with   =   = 0.1, which means that, for this example, the result obtained in this paper is less conservative that those established in [9][10][11][12][13]15].Additionally, for this example, the computed variables in [12,15,21] are 130, 198, and 86, respectively.In Theorem 13, the computed variables are 150, which is less computationally demanding than in [21], but heavier than in [12,15].If  1 ,  2 ,  3 ,  1 ,  2 ,  3 ,  4 ,  5 , and  6 are all diagonal matrices, Theorem 13 established in this paper still holds.In this case, the computed variables in Theorem 13 are 96, which is less computationally demanding than in [12,21], but heavier than in [15].For the given initial value [6, 7, −5 − 8], when   = 3.9221, the simulation result can be seen in Figure 1.Simulation result shows that, for the given parameters in Example 1, system (1) is asymptotically stable.

An Application Example
Example 2. Consider the continuous pH neutralization of an acid stream by a highly concentrated basic stream, which can be expressed in the following form [13]: V ẏ () = − () −  () , pH =  2 tanh ( 1  ()) , (55 where V is the volume of the mixing tank, () is the strong acid,  is the acid flow rate, () is the manipulated variable representing the base flow rate, pH is the measured output signal, and  2 and  1 are some constants.
The purpose of this application is to find the maximum allowable upper bound of delay  for a feedback gain  with output feedback controller  = −×pH such that the closedloop system is asymptotically stable.In order to do this, we can rewrite system (55) in the following form: where ỹ() =  1 (), ( ỹ()) = tanh( ỹ()),  = /V,  =  1 w 2 /V,  =  1  2  3 /V.For this application problem, [16,17] gave out the maximum allowable upper bound of delay  as 17.4956 when the parameters are given as  = 5.8154, V = 1500.3732, 1 = 28.9860, 2 = −3.8500,and  3 = 2.56, and the feedback gain  is selected as  = 0.5022.Recently, by employing weighting-delay method, the maximum allowable upper bound of delay  is improved to 18.2871 in [13].Meanwhile, by using Theorem 13,  the maximum allowable upper bound of delay  is 18.7436.Namely, a little better result can be obtained by using our criteria.For a given initial value ỹ0 = 0.5, when   = 18.7436, the simulation result can be seen in Figure 2. Simulation result shows that, for the given parameters in Example 2, system (55) is asymptotically stable.

Conclusions
Combined with delay partitioning technique, by using the convex combination between decomposed time delay and positive matrix, this paper researches the stability problem of a class of delayed neural networks with interval time-varying delays.The benefit of the method used in this paper is that it can utilize more information on the slope of activations and time delays.Illustrative examples show that the new criteria derived in this paper are less conservative than some previous results obtained in the references cited therein.

Figure 1 :
Figure 1: The state variables of system (1) in Example 1.

Figure 2 :
Figure 2: The state variables of system (55) in Example 2.