1. Introduction

One of the main studies in the control field before controller design is analyzing the stability in the dynamics of the system. Two factors need to be considered when implementing dynamic systems; the first is the dynamic behavior of the equilibrium points of the system that considerably affect its applications. Therefore, ensuring the stability of the system is a basic premise. The second factor is the time delays that occur due to the limited transmission speed between elements of the systems [1,2]. Many practical systems, such as those in engineering, mathematics, statistics, economics, and biology research, have been modeled according to time-delay systems [3,4].

The stability of time-delay systems is always theoretically and practically a fundamental issue. Obtaining the maximum delay interval to check the conservatism of the stability criteria is the main goal to ensure asymptotic stability of the time-delay system as much as possible. The stability of time-delay systems has been widely studied, and various approaches have been proposed [5,6].

Characteristic equations [7] and eigenvalue analysis [8] were proposed to evaluate the maximum delay interval in a time-delay system. However, those methods are not suitable for a system with varying time delays. Another technique, the Lyapunov-Krasovskii functional (LKF) method, is a well-known and robust approach to handle the stability of a system with time delay [9,10]. An LKF method with more information usually achieves less conservative results. The conservativeness of this method depends on the choice of functional and the bounding methods on its derivative [11,12].

In general, the non-integral quadratic term, the activation function-based term, and the integral quadratic term are included in the LKF. In the derivation process, a tighter estimation for the time-derivative of the LKF has an important role in reducing the conservatism of the stability criterion. In order to reduce the conservatism, various techniques have been presented, such as the Jensen inequality [13], a Wirtinger-based inequality [14], the free-weighting matrix [15], and a free-matrix-based inequality [16]. The first- and second-order reciprocally convex methods were introduced in [17] for bounding the linear matrix inequality (LMI) derivation. Also, an optimal partitioning strategy and delay-dependent and delay-independent approaches were discovered [18,19]. Recently, the Bessel-Legendre inequality [20], the auxiliary function-based integral inequality [21], and generalized free-weight-matrix-based integral inequalities [22] have been proposed for tighter estimation in stability analysis. Moreover, current integral inequalities have been improved and extended [23-25] to ensure tighter estimation and improve stability criteria. The double integral inequality was proposed in [26]. However, there is still plenty of room to improve the stability criteria. Integral inequalities could be extended and further refined by considering more integral information about the state, or by introducing more free matrices.

In this paper, we propose an extension for the general free-weight-matrix-based integral inequality, which provides more flexibility and less conservatism in the estimation through the use of its independent free vector matrices. We introduce a new LKF term, $\frac{120}{(b-a)^{3}}\int _{a}^{b}\int _{s}^{b}\int _{u}^{b}\int _{v}^{b}x(r)\textit{drdvduds}$, into the LMI by using our proposed inequality. The double integral inequality introduced $x((b+a)/2)$ and $\int _{(b+a)/2}^{b}x(s)ds$ into the LMI. We demonstrate a less conservative criterion for stability of a time-delay system, which outperforms in comparison with the state-of-the-art model by employing those inequalities to evaluate the derivative of the Lyapunov-Krasovskii functional. Three widely used numerical examples are given to show the effectiveness of the proposed method. Finally, our results confirm the improvement via comparisons with recent results.

2. Problem Statement

In this section, we describe and prove our proposed integral inequality. The following lemmas will be used for further stability analysis. Lemma 1 proposes the extension for the general free-weight-matrix-based integral inequality.

$\textbf{Lemma 1.}$ For a positive definite matrix R, any appropriate dimension matrices $(L_{i}),(i=1,2,3,4)$, and integral function $(x(t)\colon [a,b]\rightarrow \mathrm{\mathbb{R}}^{n})$, the following inequality holds

where $\vartheta _{1},\vartheta _{2},\vartheta _{3}$, and $\vartheta _{4}$ are any vectors, and $\omega _{1},\omega _{2},\omega _{3},$ and $\omega _{4}$ are as described below.

Proof. Let us define the first three Legendre orthogonal polynomials

For any matrices $(L_{1}),(L_{2}),(L_{3})$, and $(L_{4})$, and positive definite matrix $(R)$,

inequality holds by the Schur complement.

Let $\varsigma ^{T}(s)=\left[\begin{array}{lllll} \vartheta _{1}^{T} & \lambda _{1}(s)\vartheta _{2}^{T} & \lambda _{2}(s)\vartheta _{3}^{T} & \lambda _{3}(s)\vartheta _{4}^{T} & x^{T}(s) \end{array}\right],$ so then, the $\int _{a}^{b}\varsigma ^{T}(s)\Lambda \varsigma (s)ds\geq 0$ inequality holds.

Hence, we get (1). This completes the proof.

$\textbf{Corollary 1.}$ For positive definite matrix R, any appropriate dimension matrices $(L_{i}),(i=1,2,3,4)$, and integral function $(x(t)\colon [a,b]\rightarrow \mathrm{\mathbb{R}}^{n})$, the following inequality holds

where $\theta _{1},\theta _{2},\theta _{3}$ and $\theta _{4}$ are any vectors, $\varpi _{1},\varpi _{2},\varpi _{3}$ and $\varpi _{4}$ are as described in below.

Proof. By replacing $x(t)$ with $\dot{x}(t)$ in Lemma 1, Corollary 1 can easily be obtained by simple integral calculation. So, the proof is omitted here.

$\textbf{Lemma 2.}$ [22] For a positive definite matrix $R$, and a differentiable vector function $x(t)\colon [a,b]\rightarrow \mathrm{\mathbb{R}}^{n}$, the following inequality holds:

where

with

Remark 1. The newly proposed integral inequality in Lemma 1 includes a fourth integral term that will contribute to stability analysis of time-delay systems.

Remark 2. The proposed inequality can be reduced to Lemma 5 in [33] by setting $\vartheta _{3},\vartheta _{4}=0$; on the other hand, if we set $\vartheta _{1},\vartheta _{2},\vartheta _{3}=\vartheta $ and $\vartheta _{4}=0$, the proposed inequality (3) is reduced to Lemma 2 in [25].

Remark 3. In Lemma 1 and Corollary 1, the free vectors $\vartheta _{1},\vartheta _{2},\vartheta _{3}$ and $\vartheta _{4}$ are independent of each other. Furthermore, this inequality requires less conservative criteria, since the vectors can be chosen independently and freely.

Remark 4. Lemma 2 includes new terms in the double integral inequality. Unlike an auxiliary function-based integral inequality, $x((b+a)/2)$ and $\int _{(b+a)/2}^{b}x(s)ds$ are included in the LMI.

3. Stability Analysis

Consider a system with discrete and distributed delay:

where $x(t)\in \mathrm{\mathbb{R}}^{n}$ is the state vector, $A,A_{d},A_{D}$ are constant matrices, and $h$ is the time delay satisfying $h\in [h_{min},h_{max}]; \phi (t)$ is the initial condition.

The following stability criterion is based on the proposed integral inequality.

$\textbf{Theorem 1.}$ For a given constant $h\in [h_{min},h_{max}]$ , the system (4) is asymptotically stable if there exist positive definite matrices $P,Q,R,S$ and any appropriate dimension matrices $M_{i}, i=1,2,3,4$, such that the following LMI holds

where

Proof. Define

Consider the LKF candidate as follows:

where

The time derivatives of $V_{i}(t),i=1,2,3,4$, along the trajectories of (6) can be derived as follows:

Applying Corollary 1 and Lemma 2 to the integral terms in (7)

where

Therefore, by combining (7), (8) and (9), we have

From Schur compliment, it is easy to see that the inequality $Υ +Υ _{4}<0$ is equivalent to (5), which guarantees the negative definition of $\dot{V}(t)$. This completes the proof.

4. Results

In this section, we list three numerical examples that are frequently utilized to guarantee the improvement and effectiveness of the stability criteria, compared with recently published results in [13, 14, 28-32, 34, 35]. We used the generalized time-delay system, shown and formulated in (4), for comparison with the recent results. The parameters of the examples are substituted into the LMI in Theorem 1. The matrices $P,Q,R,S$, and $M_{i}, i=1,2,3,4$ were declared as variables for the LMI. The dimensions of the final LMI in our work are 11${\times}$11.

Example 1. Consider system (4) with the following parameters

$A=\left[\begin{array}{ll} -2 & 0\\ 0.2 & -0.9 \end{array}\right],\,A_{d}=\left[\begin{array}{ll} -1 & 0\\ -1 & -1 \end{array}\right],\,A_{D}=\left[\begin{array}{ll} 0 & 0\\ 0 & 0 \end{array}\right]$.

Example 2. Consider system (4) with the following parameters

$A=\left[\begin{array}{ll} 0.2 & 0\\ 0.2 & 0.1 \end{array}\right],\,A_{d}=\left[\begin{array}{ll} 0 & 0\\ 0 & 0 \end{array}\right],\,A_{D}=\left[\begin{array}{ll} -1 & 0\\ -1 & -1 \end{array}\right]$.

Example 3. Consider system (4) with the following parameters

$A=\left[\begin{array}{ll} 0 & 1\\ -100 & -1 \end{array}\right],\,A_{d}=\left[\begin{array}{ll} 0 & 0.1\\ 0.1 & 0.2 \end{array}\right],\,A_{D}=\left[\begin{array}{ll} 0 & 0\\ 0 & 0 \end{array}\right]$.

Tables 1-3 illustrate the obtained maximum admissible upper bounds (MAUBs) and the number of variables (NoVs) in Example 1, Example 2, and Example 3, respectively, by Theorem 1. The results in Table 1 indicate that the MAUB obtained by Theorem 1 achieved a better result than most of the state-of-the-art results. It should be noted that the MAUB obtained by Theorem 1 is very close to the analytical bound that is obtained by eigenvalue analysis. The MAUB of the time delay in [34] is higher, but the number of variables is significantly greater, compared to our method. From Table~2, we can see that our result is similar to the analytical bound. In Table 3, the proposed method achieved a higher MAUB than recent results. It is obvious that the results of the proposed method outperformed the state-of-the-art results, which demonstrates the effectiveness of our approach.

5. Conclusion

In this paper, we proposed an extension on the generalized free-weight-matrix-based integral inequality with a fourth integral term. Furthermore, we used the double integral inequality proposed in [21] for stability analysis with an LKF to include the new terms. Finally, we have shown the advantage of our method by comparing it with recent results. Three numerical examples were used to guarantee the effectiveness of the proposed approach. Our future work will aim at discovering new approaches or integral inequalities to reduce the conservativeness of the stability condition for systems with a time delay. Furthermore, we want to extend our work to the stabilization of nonlinear network systems with time delay.