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 timedelay systems [3,4].
The stability of timedelay 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 timedelay
system as much as possible. The stability of timedelay 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 timedelay system. However,
those methods are not suitable for a system with varying time delays. Another technique,
the LyapunovKrasovskii functional (LKF) method, is a wellknown 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 nonintegral quadratic term, the activation functionbased term,
and the integral quadratic term are included in the LKF. In the derivation process,
a tighter estimation for the timederivative 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 Wirtingerbased inequality [14], the freeweighting matrix [15], and a freematrixbased inequality [16]. The first and secondorder reciprocally convex methods were introduced in [17] for bounding the linear matrix inequality (LMI) derivation. Also, an optimal partitioning
strategy and delaydependent and delayindependent approaches were discovered [18,19]. Recently, the BesselLegendre inequality [20], the auxiliary functionbased integral inequality [21], and generalized freeweightmatrixbased integral inequalities [22] have been proposed for tighter estimation in stability analysis. Moreover, current
integral inequalities have been improved and extended [2325] 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 freeweightmatrixbased
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, $\frac{120}{(ba)^{3}}\int _{a}^{b}\int _{s}^{b}\int _{u}^{b}\int _{v}^{b}x(r)\textit{drdvduds}$,
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 timedelay system, which outperforms in comparison with
the stateoftheart model by employing those inequalities to evaluate the derivative
of the LyapunovKrasovskii 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 freeweightmatrixbased 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 timedelay 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 functionbased 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, 2832, 34, 35]. We used the generalized timedelay 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 13 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 stateoftheart 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 stateoftheart results, which
demonstrates the effectiveness of our approach.
Table 1. MAUB $h_{M}$ in Example 1.
Methods

MAUBs

NoVs

[14]

6.059

16

[13]

6.165

45

[27]

6.12

137

[28]

6.1664

75

[29]

6.1719

106

[30]

6.1719

42

[34] case 1

6.1719

172

[35]

6.1719

45

[34] case 2

6.1724

921

Theorem 1

6.1723

176

Analytical bound

6.1725


Table 2. MAUB $h_{M}$ in Example 2.
Methods

MAUBs

NoVs

[27]

[0.200,1.877]

16

[14]

[0.2000,1.9504]

59

[31]

[0.2000,2.0402]

45

[29]

[0.2000,2.0412]

106

Theorem 1

[0.2000,2.0412]

176

Analytical bound

[0.2000,2.0412]


Table 3. MAUB $h_{M}$ in Example 3.
Methods

MAUBs

NoVs

[14]

0.126

16

[31]

0.126

59

[28]

0.577

75

[32]

0.675

45

Theorem 1

0.728

176

5. Conclusion
In this paper, we proposed an extension on the generalized freeweightmatrixbased
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.
ACKNOWLEDGMENTS
This work was supported in part by the National Research Foundation of Korea (NRF)
grant funded by the Korea government (MSIT) (No. 2020R1A2C2005612) and in part by
the Brain Research Program of the National Research Foundation (NRF) funded by the
Korean government (MSIT) (No. NRF2017M3C7A1044816).
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Author
Tuvshinbayar Chantsalnyam
Tuvshinbayar Chantsalnyam is a graduate student in the School of Electronics and
Information Engineering at Jeonbuk National University in Jeonju, Korea. He received
his BSc in electronics engineering and information technology from the Mongolian University
of Science and Technology, and received an MSc from Jeonbuk National University. He
is working on network system control, timedelay systems, neural networks, deep learning,
and bioinformatics.
JiHyoung Ryu received a BSc and PhD in electronic engineering from Jeonbuk National
University, South Korea, in 2005 and 2015, respectively. He is currently a Senior
Researcher with the Electronics and Telecommunications Research Institute, South
Korea. His current research interests include mobile robots, intelligent control and
control systems using artificial intelligence.
Kil To Chong received his Ph.D. in Mechanical Engineering from Texas A & M University
in 1995. Currently, he is a professor for the School of Electronics and Information
Engineering at Jeonbuk National University in Jeonju, Korea, and is head of the Advanced
Research Center of Electronics. His research interests are in the areas of bioinformatics,
computational biology, deep learning, medical image processing, and timedelay.