Hybrid Feedback Control for Exponential Stability and Robust H ∞ Control of a Class of Uncertain Neural Network with Mixed Interval and Distributed Time-Varying Delays

a Class Neural Network Abstract: This paper is concerned the problem of robust H ∞ control for uncertain neural networks with mixed time-varying delays comprising different interval and distributed time-varying delays via hybrid feedback control. The interval and distributed time-varying delays are not necessary to be differentiable. The main purpose of this research is to estimate robust exponential stability of uncertain neural network with H ∞ performance attenuation level γ . The key features of the approach include the introduction of a new Lyapunov–Krasovskii functional (LKF) with triple integral terms, the employment of a tighter bounding technique, some slack matrices and newly introduced convex combination condition in the calculation, improved delay-dependent sufﬁcient conditions for the robust H ∞ control with exponential stability of the system are obtained in terms of linear matrix inequalities (LMIs). The results of this paper complement the previously known ones. Finally, a numerical example is presented to show the effectiveness of the proposed methods.


Introduction
During the past decades, the problem of the reliable control has received much attention [1][2][3][4][5][6][7][8][9][10][11]. Neural networks have received considerable due to the effective use of many aspects such as signal processing, automatic control engineering, associative memories, parallel computation, fault diagnosis, combinatorial optimization and pattern recognition and so on [12][13][14]. It has been shown that the presence of time delay in a dynamical system is often a primary source of instability and performance degradation [15]. Many researchers have paid attentions to the problem of robust stability for uncertain systems with time delays [16][17][18][19]. The H ∞ controller can be used to guarantee closed loop system not only a robust stability but also an adequate level of performance. In practical control systems, actuator faults, sensor faults or some component faults may happen, which often lead to unsatisfactory performance, even loss of stability. Therefore, research on reliable control is necessary.
On the other hand, the H ∞ control of time-delay systems are practical and theoretical interest since time delay is often encountered in many engineering and industrial processes [20][21][22].
Most of works have been focused on the problem of designing a robust H ∞ controller that stabilizes linear uncertain systems with time-varying norm bounded parameter uncertainty in the state and input matrices. The problem of designing a robust reliable H ∞ controller for neural networks is considered in [23]. Ref [24] have studied the problem of delay dependent robust H ∞ control for a class of uncertain systems with distributed time-varying delays. The parameter uncertainties are supposed to be time-varying and norm bounded. The problem of H ∞ control design usually leads to solving an algebraic Lyaponov equation. It should be noted that some works have been dedicated to the problem of robust reliable control for nonlinear systems with time-varying delay [2,4,7]. However, to the best of the authors's knowledge, so far the research on robust reliable H ∞ control is still open problems, which are worth further investigations.
Motivated by above discussion, in this paper we have considered the problem of a robust H ∞ control for a class of uncertain systems with interval and distributed time-varying delays. The parameter uncertainties are supposed to be time varying and norm bounded. A sufficient condition for the H ∞ control is presented by LMIs approach using a new LKF with triple integral terms. The robust exponential stability of the system is established which satisfy a formulated H ∞ performance level for all admissible parameters uncertainties.
The main contributions of this paper are given as follows, • This research is the first time to study hybrid feedback control for exponential stability and robust H ∞ control of a class of uncertain neural network with mixed interval and distributed time-varying delays.
t t+s x(θ)dθds is first proposed to analyze the problem of robust H ∞ control for uncertain neural networks with mixed time-varying delays and augmented Lyapunov matrix P i (i = 1, 2, . . . , 6) do not need to be positive definiteness of the chosen LKF compared with [23]. • The problem of robust H ∞ control for uncertain neural networks with mixed timevarying delays comprising of interval and distributed time-varying delays, these delays are not necessarily differentiable. • For the neural networks system (1), the output z(t) contains the deterministic disturbance input w(t) and the feedback control u(t) which is more general and applicable than [23][24][25][26][27][28].
The rest of this paper is organized as follows. In Section 2, some notations, definitions and some well-known technical lemmas are given. Section 3 presents the H ∞ control for exponential stability and the robust H ∞ control for exponential stability. The numerical examples and their computer simulations are provided in Section 4 to indicate the effectiveness of the proposed criteria. Finally, this paper is concluded in Section 5.

Model Description and Mathematic Preliminaries
The following notation will be used in this paper: R and R + denote the set of real numbers and the set of nonnegative real numbers, respectively. R n denotes the n−dimensional space. R n×r denotes the set of n × r real matrices. C[[− , 0], R n ] denotes the space of all continuous vector functions mapping [− , 0] into R n where ∈ R + . A T and A −1 denote the transpose and the inverse of matrix A, respectively. A is symmetric if A = A T , λ(A) denotes all the eigenvalue of A, λ max (A) = max{Re λ : λ ∈ λ(A)}, λ min (A) = min{Re λ : λ ∈ λ(A)}, A > 0 or A < 0 denotes that the matrix A is a symmetric and positive definite or negative definite matrix. If A, B are symmetric matrices, A > B means that A − B is positive definite matrix, I denotes the identity matrix with appropriate dimensions. The symmetric term in the matrix is denoted by * . The following norms will be used: || · || refer to the Euclidean vector norm; ||φ|| c = sup t∈[− ,0] ||φ(t)|| stands for the norm of a function φ(·) ∈ C[[− , 0], R n ].
The following definition and lemma are necessary in the proof of the main results: 29]). Given α > 0. The zero solution of system (1), where u(t) = 0, w(t) = 0, is α−stable if there is a positive number N > 0 such that every solution of the system satisfies ||x(t, φ)|| ≤ N||φ|| c e −αt , ∀t ≤ 0.

Lemma 1 ([30], Cauchy inequality).
For any symmetric positive definite matrix N ∈ M n×n and x, y ∈ R n we have ±2x T y ≤ x T Nx + y T N −1 y.

Lemma 2 ([30]
). For a positive definite matrix Z ∈ R n×n , and two scalars 0 ≤ r 1 < r 2 and vector function x : [r 1 , r 2 ] → R n such that the following integrals are well defined, one has x T (s)Zx(s)ds.

Lemma 3 ([31]
). For any positive definite symmetric constant matrix P and scalar τ > 0, such that the following integrals are well defined, one has

Lemma 4 ([32]
). For given matrices H, E, and F with F T F ≤ I and a scalar > 0, the following inequality holds:

Stability Analysis
In this section, we will present stability criterion for system (3).
Consider a Lyapunov-Krasovskii functional candidate as where
Let us set

Remark 3.
It is noted that the previous works [23][24][25][26][27][28] consider the Lyapunov martices P 1 , P 3 and P 6 which are positive definite. In our paper, we remove this restriction by applying the method of x t ) and V 13 (t, x t ) as shown in the proof of Proposition 1. Hence, P 1 , P 3 and P 6 are only real matrices. It can be seen that our paper are more applicable and less conservative than aforementioned works.
and matrices P 1 = P T 1 , P 3 = P T 3 , P 6 = P T 6 , P 2 , P 4 , P 5 such that the following LMIs hold: where Moreover, stabilizing feedback control is given by and the solution of the system satisfies Proof. Choosing the Lyapunov-Krasovskii functional candidate as (8), It is easy to check that We take the time-derivative of V i along the solutions of system (3) By Lemmas 1 and 2, we have h(x(s))ds 2u and the Leibniz-Newton formula gives Denote Next, when Using Lemma 2, we get By reciprocally convex with a = h 2 −h(t) , the following inequality holds: which implies Then, we can get from (22) By using Lemmas 2 and 3, we obtain x(s)ds .
By using the following identity relation: h(x(s))ds we have By Lemmas 1 and 2, we get h(x(s))ds

From (19)-(30), we obtaiṅ
where Using the Schur complement lemma, pre-multiplying and post-multiplying M 1 , M 2 , M 3 and M 4 by P 1 and P 1 respectively, the inequality M 1 , M 2 , M 3 and M 4 are equivalent to Ξ 1 < 0, Ξ 2 < 0, Ξ 3 < 0 and Ξ 4 < 0 respectively, and from the inequality (31) it follows that Letting w(t) = 0, and since we finally obtain from the inequality (32) thaṫ Integrating both sides of (33) from 0 to t , we obtain Taking the condition (18) into account, we have Then, the solution ||x(t, φ)|| of the system (3) satisfies which implies that the zero solution of the closed-loop system is α−stable. To complete the proof of the theorem, it remains to show the γ−optimal level condition (ii). For this, we consider the following relation: From (32) we obtain thaṫ Observe that the value of ||z(t)|| 2 is defined as Submitting the estimation ofV(t, x t ) and ||z(t)|| 2 , we obtain Hence, from (38) it follows that Letting t → ∞, and setting c 0 = λ 2 γ , we obtain that This completes the proof of the theorem.
For neural networks with parameter uncertainties, we consider the following systeṁ x(s)ds, where ∆A, ∆B, ∆C and ∆D are the unknown matrices, denoting the uncertainties of the concerned system and satisfying the following equation: where E A , E B , E C and E D are known matrices,F(t) is an unknown, real and possibly time-varying matrix with Lebesgue measurable elements and satisfies Then, we have the following theorem.

Theorem 2.
Given α > 0, The H ∞ control of system (39) has a solution if there exist symmetric positive definite matrices Q 1 , Q 2 , Q 3 , R 1 , R 2 , S 1 , S 2 , S 3 , W 1 , W 2 , W 3 , Z 1 , Z 2 , Z 3 , diagonal matrices U > 0, U 2 > 0, U 3 > 0, and matrices P 1 = P T 1 , P 3 = P T 3 , P 6 = P T 6 , P 2 , P 4 , P 5 such that the following LMI hold: Moreover, stabilizing feedback control is given by and the solution of the system satisfies Proof. We choose the similar Lyapunov-Krasovskii functional in Theorem 1, where matrices A, B, C and D in (19) and (29) are replaced by A + NF(t)E A , B + NF(t)E B , C + NF(t)E C and D + NF(t)E D , respectively. By Lemmas 1 and 2, we have h(x(s))ds From (46), we get where ζ T (t) = ξ T (t) t t−d h T (x(s))ds . By using the Schur complement lemma, the inequality (47) and (48) are equivalent to Ω 1 < 0 and Ω 2 < 0 respectively. By the similar proof of Theorem 1, so the proof is completed.

Remark 4.
The time delay in this paper is identified as a continuous function which serve on a given interval that the lower and upper bounds for the time-varying delay exist. Moreover, the time delay function is not necessary to be differentiable. In some previous works, the time delay function needs to be differentiable which are shown in [24][25][26][33][34][35][36][37].

Numerical Examples
In this section, we provide two numerical examples with their simulations to demonstrate the effectiveness of our results. Example 1. Consider neural networks (3) with parameters as follows: From the conditions (14)- (17)  The feedback control is given by Moreover, the solution x(t, φ) of the system satisfies ||x(t, φ)|| ≤ 1.2300e −0.01t ||φ|| c .

Example 2.
Consider neural networks (39) with parameters as follows: From the conditions (42) The feedback control is given by u(t) = B 1 P

Conclusions
In this paper, the problem of a robust H ∞ control for a class of uncertain systems with interval and distributed time-varying delays was investigated. It is assumed that the interval and distributed time-varying delays are not necessary to be differentiable. Firstly, we considered an H ∞ control for exponential stability of neural network with interval and distributed time-varying delays via hybrid feedback control and a robust H ∞ control for exponential stability of uncertain neural network with interval and distributed time-varying delays via hybrid feedback control. Secondly, by using a novel Lyapunov-Karsovskii functional that the Lyapunov matrix P i (i = 1, 2, . . . , 6) do not need to be positive definiteness, the employment of a tighter bounding technique, some slack matrices and newly introduced convex combination condition in the calculation, improved delaydependent sufficient conditions for the robust H ∞ control with exponential stability of the system are obtained. Finally, numerical examples have been given to illustrate the effectiveness of the proposed method. The results in this paper improve the corresponding results of the recent works. In the future work, the derived results and methods in this work are expected to be applied to other systems, for example, H ∞ state estimation of neural networks, exponential passivity of neural networks, neutral-type neural networks, stochastic neural networks, T-S fuzzy neural networks, and so on [24,[38][39][40][41]. Funding: The first author was financially supported by the Science Achievement Scholarship of Thailand (SAST). The second author was financially supported by Khon Kaen University. The fourth author was supported by the Unit of Excellence in Mathematical Biosciences (FF64-UoE042) supported by the University of Phayao.

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.