Novel Delay-Dependent Stability Criteria for Discrete-Time Neural Networks with Time-Varying Delay

Thedelay-dependent stability problem is investigated for discrete-time neural networkswith time-varying delays. A newaugmented Lyapunov-Krasovskii functional (LKF) with single and double summation terms and several augmented vectors is proposed by decomposing the time-delay interval into two nonequidistant subintervals to derive less conservative stability conditions. Then, by using Wirtinger-based inequality, reciprocally, and extended reciprocally convex combination lemmas, tight estimations for sum terms in the forward difference of the LKF are given. Several zero equalities are introduced to further relax the existing results. Less conservative stability criteria are proposed in terms of linear matrix inequalities (LMIs). Finally, numerical examples are proposed to show the effectiveness and less conservativeness of the proposed method.


Introduction
During the past few decades, neural networks (NNs) have received great attention because of their wide applications in various fields such as image processing, signal processing, pattern recognition, associative memory, parallel computation, optimization, and error diagnosis [1,2].One of the most important questions in theoretical analysis of NNs is dynamical behaviors of the NNs, such as their stability, periodic oscillatory, and chaos.Among them, analysis of the stability has received much attention and various stability conditions have been obtained in [3,4].
It is well known that a time delay is inherent in various systems, including NNs, owing to the finite speed of signal transmission and conversion rate of the processors.Delays in a system may cause oscillation and divergence and further degrade the performance.It is of great importance to determine the admissible maximal delay bound such that the neural networks with a delay less than this upper bound remain stable.Hence, it is essential to investigate the stability of the neural networks with time delay [5][6][7][8][9][10].
Since most systems use a digital processor to acquire information from computers at discrete instants of time, it is essential to formulate discrete-time neural networks (DNNs) that are an analogue of continuous ones.Therefore, it is significant to study the dynamics of the DNNs and many results for the DNNs with time delay has received increasing attention; see [11][12][13][14][15][16] and references therein.
The stability criteria for the DNNs with time-varying delay can be classified into two categories: delay-independent stability criteria and delay-dependent ones [17,18].As is well known, delay-dependent stability criteria, which take advantage of the information on the size of time delays, are less conservative than delay-independent ones.The main aim of delay-dependent stability criteria is to get maximum delay bounds guaranteeing the addressed neural networks to be stable.In order to improve results regarding this problem, various methods have been applied to the delay-dependent category, such as augmented LKF method [13,[19][20][21][22], freeweighting matrix method [18,23], summation inequality method [16,[24][25][26][27], delay-partitioning method [5,28,29], and reciprocally convex approach [20,30,31].
It is known that the simple LKFs with the single and/or double sum terms lead to the criteria with high conservatism, while more augmented LKFs reduce conservatism but increase the number of decision variables.By constructing an augmented LKF with the sum terms of a state vector as well as the activation function, an improved stability condition for the asymptotic stability of DNNs was derived in [13] but with an enlarged number of decision variables.In recent years, the LKF including triple summation terms has also been applied to study the stability analysis of discrete-time DNNs for a further improvement of the results [19,32].For the purpose of reducing conservatism, the delay-partitioning LKFs, in which the delay interval was separated into several subintervals, were introduced [5,28,29].
Various free-weighting matrix-based stability criteria [18,23] were established to improve the results obtained by the Jensen-based summation inequality.However, introducing the free-weighting matrices may result in increased number of decision variables.
After the computational complexity became one of the crucial aspects of a research in the area of the system stability, the direct bounding method based on summation inequalities once again becomes the most popular method [17,19,32,33].Very recently, various types of the Wirtingerbased summation inequalities, tighter than the Jensen-based summation inequality, have been proposed for discrete-time linear time-delay systems [25,34,35] and have also been used for the study of the discrete-time DNNs [36].As one of useful methods to deal with the stability of delayed systems, the reciprocally convex approach was developed in [30] and has been extensively used to study the dynamical behaviours of time-delay systems since then (see [32,[37][38][39] and references therein).
Not long ago, an extended reciprocally convex combination lemma (ERCCL) has been developed [26,31] to replace the popular reciprocally convex combination lemma (RCCL).It has a potential to reduce the conservatism of the RCCLbased criteria without introducing any extra decision variable due to its advantage of reduced estimation gap using the same decision variables.
Motivated by the aforementioned discussions, in this paper, the problem of delay-dependent stability for DNNs with interval time-varying delay is considered.The objective of the paper is to derive simple but efficient stability conditions for DNNs with interval time-varying delay.The major contributions of this paper are as follows.Delay-decomposition method is used with new augmented Lyapunov-Krasovskii functional, which contains several augmented vectors in single and double summation terms with a free parameter  that divides interval . The value of parameter  is determined by solving linear matrix inequalities (LMIs) that are defined in the proposed stability criterion.In this way, a greater degree of freedom is enabled in estimating the stability of the DDN, which leads to a smaller conservative stability criterion.The convenient existing summations inequalities, as well as extended reciprocally convex combination lemma, have been used for calculation difference of KLF.In order to improve the stability criteria, the LKF is extended with an additional double summation term and zero equalities (ZEs) are introduced in calculation difference of LKF.As a result of applying the mentioned techniques, it is shown that the derived results are less conservative then the existing ones [24,31,40,41].
Notations.Throughout the paper, Z + denotes the set of positive integers and R  denotes the -dimensional Euclidean space and R × the set of all  ×  real matrices.For the positive integers  and  ( > ), Z[, ] denotes the set of all positive integers  satisfying  ≤  ≤ .  and 0 × denote  ×  identity matrix and  ×  zero matrix, respectively.  denotes the matrix transpose of  and * represent the elements below the main diagonal of a symmetric matrix.diag{, , . . ., } denotes the block-diagonal matrix with elements , , . . .,  in the diagonal entries and Sym{} =  +   .For any symmetric matrix  ∈ R × the notation  > 0 ( ≥ 0) means that  is positive definite (positive semidefinite) matrix and  < 0 ( ≤ 0) means that  is negative definite (negative semidefinite) matrix.For the matrices   ∈ R × ,  = 1, 2, . . ., , { 1 ,  2 , . . .,   } .Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
In this paper, the following assumption on the system (1) is made.
After applying Assumption 1, one can check that the function (⋅) with (0) = 0 satisfies for all  ̸ = .If = 0, we have for all  ̸ = 0.The following lemmas will be used in the sequel to establish the main results.
Lemma 2 (Jensen's inequality [42]).For any positive definite symmetric matrix  ∈ R × and two positive integers  and  > , the sum term ∑ −1 =   ()() is estimated as Lemma 3 (Wirtinger-based inequality [25]).For a given positive definite symmetric matrix , two positive integers  and  > , and any sequence of discrete-time variable  : Z[, ] → R  , the following inequality holds: where Remark 4. In [27], stability criteria for discrete-time delay systems are considered by using a new summation inequality with three slack matrix variables.It has been shown that inequality ( 8) is obtained as a special case of the new summation inequality with convenient selected values of the slack matrix variables.Therefore, by using the mentioned summation inequality, the conservatism can be reduced, but the introduction of three slack matrix variables can increase the computation complexity of the obtained stability criterion.
The reciprocally convex combination lemma (RCCL) plays an important role in the estimation of the forward difference of LKF.
Lemma 5 (reciprocally convex combination lemma [30,31,43]).For a real scalar  ∈ (0, 1), a symmetric matrix  > 0, and any matrix  satisfying the following inequality holds: An extended reciprocally convex combination lemma (ERCCL), which estimates the sum terms in the forward difference of the LKF tightly than RCCL, is presented as follows.
Lemma 6 (extended reciprocally convex combination lemma [31,40]).For a real scalar  ∈ (0, 1), a symmetric matrix  > 0, and any matrix , the following inequality holds: where Remark 7. Unlike RCCL, the condition ( 11) is not used in ERCCL.Thus, the matrix  in ERCCL can be chosen more freely then the one in RCCL such that the feasibility criterion based on ( 13) is better than one based on (11).Further, if (11) holds then   ≥ 0,  = 1, 2, and Thus, the terms (1−) and  2 , which appear in (13), reduce estimation gap between two sides of ( 13) and inequality (13) in ERCCL is tighter then inequality (12) in RCCL.Note that the RCCL and ERCCL require the same number of decision variables ( and ).
Remark 8.In [20], the specific reciprocally convex inequality is proposed.For comparison, we rewrite the result of Theorem 1 in [20] in a block form as where (15) immediately reduces to (13), which means that ERCCL (Lemma 6) is a special case of Theorem 1 in [20].During the calculation of the difference of LKF in (31), the inequality ( 13) is used instead of (15), because the same matrix  = Z2 appears twice on the block diagonal in (31), so that the condition  1 =  2 =  is automatically satisfied.Further, by applying inequality (13) to (32), only one matrix  was used instead of two ( 1 and  2 ), as in the case of application of inequality (15).Thereby, a computation complexity of the proposed stability criterion ( 13) is decreased, but there is a possibility of increasing conservatism (13) in regard to (15).Lemma 9 (see [40]).For a positive definite symmetric matrix  and matrices Υ and Ξ, the following statements are equivalent: (i) Ξ − Υ  Υ < 0 (ii) There exists a matrix Ψ with appropriate dimension such that
Remark 11.In Theorem 10, the Wirtinger-based summation inequality ( 8) is applied to summation with the constant lower and upper bound (ℎ 1 ∑ −1 =−ℎ 1   () 1 ()).However, in the case of the summation with the time-varying lower or upper bound combination of the Wirtinger-based inequality (9) and the reciprocally convex approach (13) is applied.
By introducing an augmented LKF and zero equations, a further improved stability condition of system (4) can be obtained as follows.
Remark 13.In order to improve the stability criterion proposed in Theorem 10, a new double summation term  6 () with vector  3 () = [  ()   ()]  and free matrix  is introduced in LKF.Further, in the proof of Theorem 12, two zero equalities with symmetric matrices  1 and  2 are introduced in Δ 6 ().In this way, the conservatism of the criterion is reduced.Remark 14.In order to reduce the number of decision variables, Lemma 9 is used twice.First, by using the lemma, nonaffine term with respect to ℎ() is transformed into the affine term Θ(ℎ(), ).On that occasion, additional matrix Ψ ∈ R 11×2 is introduced in (65) and the number of decision variables has increased significantly (for 22 2 ).Second, if we apply Lemma 9 on (66), then the matrix Ψ ∈ R 11×2 can be eliminated from Θ(ℎ  , ).In this way, the significant reduction of decision variables was performed.
Remark 15.The proposed stability criterion depends on the lower and upper bounds of time-delay, ℎ 1 and ℎ 2 .In order to compare our results with existing ones, a maximum allowable upper bound (MAUB) of the time-delay, ℎ 2max , is adopted, such that the concerned system is asymptotically stable for any delay size less than the MAUB.Note that a criterion that gives a lower value of MAUB is less conservative with respect to other criteria.

Demonstrative Examples
In the section bellow, we provide three numerical examples to illustrate the effectiveness of the stability criteria proposed in this paper.
which was used to check the feasible region of stability criteria in [24,31,40,41].The activation function is in the form of () = tanh() and satisfies Assumption 1 with  − 1 =  − 2 = 0 and  + 1 =  + 2 = 1.By using Theorem 10, the MAUB ℎ 2max is computed with different lover bounds ℎ 1 and the obtained results are given in Table 1.The number of decision variables (NDVs) is also given to show computation complexity.From this table, we can see that our results are less conservative and/or have less decision variables than [24,31,40,41].
Example 2. Consider the DNN (4) with the following parameters [31]: Problems in Engineering 13 and activation functions that satisfy Assumption 1 with  − 1 =  − 2 = 0 and  + 1 =  + 2 = 1.In this example, the MAUB ℎ 2max is computed by using Theorems 10 and 12 for different lover bounds ℎ 1 and results are shown in Table 2. From this table, it can be seen that the proposed stability criteria, Theorems 10 and 12, provide significantly larger MAUB than the previous result [31], while the numbers of decision variables are slightly higher.
By applying Theorems 10 and 12 to the DNN (4) with parameters (73), the MAUB ℎ 2max is computed for different lover bounds ℎ 1 and results are shown in Table 3. From this table it can be seen that stability criterion proposed in Theorem 12 gives approximately the same results but with significantly smaller number of decision variables than [40].

Conclusion
In this study, a novel Lyapunov-Krasovskii functional with single and double summation terms and several augmented vectors is constructed by decomposing the time-delay interval into two nonequidistant subintervals.Using Wirtingerbased inequality, extended reciprocally convex approach, and

Example 1 .
Consider the DNN (4) with the following parameters:

Figure 2 :
Figure 2: State trajectories of the DNN.

Table MAUB ℎ
2max for given ℎ 1 in Example 3. zero equalities, new stability conditions are developed in the form of linear matrix inequalities.It is shown that the proposed results are less conservative than the existing ones.Several numerical examples are presented to show the effectiveness and less conservativeness of the proposed method. several