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Open Access

Peer-reviewed

Research Article

A novel criteria on exponentially passive analysis for Takagi-Sugeno fuzzy of neutral dynamic system with various time-varying delays

  • Janejira Tranthi,

    Roles Conceptualization, Data curation, Formal analysis, Methodology, Supervision, Writing – original draft

    Affiliation Department of Mathematics, Khon Kaen University, Khon Kaen, Thailand

    ⨯
  • Thongchai Botmart

    Roles Conceptualization, Data curation, Formal analysis, Funding acquisition, Methodology, Software, Supervision, Writing – original draft, Writing – review & editing

    thongbo@kku.ac.th

    Affiliation Department of Mathematics, Khon Kaen University, Khon Kaen, Thailand

    ⨯

Abstract

This paper is the first studying on designing exponentially passive analysis for T-S fuzzy of dynamic systems with various time-varying delays such as neutral, discrete, and distributed time-varying delays. Constructing the new Lyapunov-Krasovskii function and the Newton-Leibniz theory, the zero equations, and the matrix inequality techniques, the multiple delay-dependent criteria, with assuring exponentially passive on the discussed T-S fuzzy system, are defined in respect of linear matrix inequalities (LMIs) that can be checked easily using the LMI toolbox of MATLAB. Those approaches give less conservative, exponentially passive criteria for special cases of general stability of a neutral differential system. Furthermore, the results of this study are delay-dependent, which depend on the lower and upper bound with the time-varying delay. Lastly, some numerical examples illustrate the performance of our criteria based on the results obtained and summarize some of the previous achievements.

Citation: Tranthi J, Botmart T (2022) A novel criteria on exponentially passive analysis for Takagi-Sugeno fuzzy of neutral dynamic system with various time-varying delays. PLoS ONE 17(10): e0275057. https://doi.org/10.1371/journal.pone.0275057

Editor: Yiming Tang, Hefei University of Technology, CHINA

Received: January 1, 2022; Accepted: September 9, 2022; Published: October 7, 2022

Copyright: © 2022 Tranthi, Botmart. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper.

Funding: This research has received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation [grant number B05F640088].

Competing interests: The authors have declared that no competing interests exist.

1 Introduction

The research of Takagi and Sugeno created the Takagi-Sugeno (T-S) fuzzy system [1], which explained the time-delays frequently occurring in many dynamic systems, practically (e.g., biological systems neural, networks, metallurgical processes, and chemical processes). The researchers stated to handling with the synthesis and analysis problems of nonlinear systems can be proven by the fuzzy-logic theory. Especially, the T-S fuzzy model uses a set of IF-THEN rules built on linguistic variables and values by quantifying the semantics of linguistic values using a member function. In consequence, the analysis and class synthesis of non-linear systems, and many nonlinear analytical problems with traditional linear system theories were studied based on this fuzzy model of T-S. For instance, Zhang et al. [2] presented guarantee cost network control method of the T-S fuzzy systems with delay on the neural networks. Li et al. [3] were investigated the stability of an unstable randomized neural network for mixed-delayed neutral types. Moreover, Li et al. [4] demonstrated a stabilization and exponential stability analysis issue of T-S fuzzy systems under periodic sampling as well. Xu at al. [5] presents stability of uncertain systems, which the stability of the discrete singular fuzzy system at discrete time.

Time delays is of significance both in theory and application due to its detrimental effects on stability and performance of systems and its wide existence in practical dynamic processes. The cases of delays can be usually considered as time delay, multiple delays, interval delay, input and state delays and so on. All of them were discussed around two basic group, i.e. delay-independent and delay-dependent. The delay-dependent stability criterion are investigated with the extent upper bound of delay. Hence, the criterion of delay-dependent stabilization are proposed to guarantee that the delay system is stable for any value of time delay less than the provided upper bound. In different circumstances, the delay-independent stability criteria are proposed without consideration of the extent for time delay. In ordinary, the delay-dependent conditions are preferable than the delay-independent conditions while the effect of time delay is not acute. According to Zhu and Yang [6] illustrated Jensen’s inequality approach in synthesis the stability for continuous time systems with time-varying delay. The study all of delay, which defined by Lien [7], guarantee cost control for uncertain neutral system through the LMI system. Likewise, Chen et al. [8] applied guarantee cost control of T-S fuzzy system with input delays and state. The research of Lien et al. [9] supported the stability criterion of interval time-varying delay systems during the uncertain T-S fuzzy systems. According to Jiang and Han [10] researched the delay-dependent criterion of uncertain system with time-varying delays. However, the above mentioned, there is still room for further improvement: the fuzzy T-S method with delay-dependent based on latency to the possible extent of the thresholds for exponential stability and passivity performance.

In addition, passivity theory is another proficient tool for analyzing system stability. The passive theory is the main pointed to the system can keep the system’s internal stability which is the passive properties. So that, the problem of inactivity is therefore an important part of recent research. Then, the passive control uses the product of output and input as the power rating, which captures the attenuation properties of the system under the bounded input. In particular, passivity theory is more general than stability theory because it can be illustrated Lyapunov function under the theory of stabilization. This theory is used for issue of engineering i.e. electric circuits and heat energy systems. Nowadays many researchers have studied passive theory and passive control problems extensively, for instance, Zhang et al. [11]who studied the passive controller design issue with both state and input delays for a class of continuous-time T-S fuzzy systems. Another researcher such as Wu et al. [12] identified the problem of passive control for fuzzy network systems, considering the random uncertainty variable sampling interval and the delay caused by the fixed network. Similarly, Song and He [13] who researched the robust passive control is offered for a limited time for nonlinear systems with time-delay. The studied of Yu et al. [14] focused both passive analysis and passive control for erratic intermittent switching delay systems through a simple switching signal design. Likewise, Yotha et al. Improved delay-dependent approach to passivity analysis for uncertain neural networks with discrete interval and distributed time-varying delays [15]. So, it is challenging to solve exponentially passive for T-S fuzzy of neutral dynamic system with various time-varying delays.

Despite, in a specific physical system, mathematical models are described by functional differential equations of the neutral type. The neutral type of functional differential equation depends on the lag of the state and state derivatives. Approximately, neutral-type phenomena often appear in automatic control studies, chemical reactor, distributed network, the dynamic process such as steam pipes and water pipes. Also, population ecology, heat exchange, microwave oscillator, turbojet engine system, lossless transmission line, vibrating mass attached to an elastic band, etc. Likewise, the research of Zhou et al. [16] examined the problem of adaptive synchronization for neutral type random neural networks with Markovian switching parameters. Chen et al. [17] supported that stability of global exponential in mean squares and exponential stability are almost certain for randomly delayed neural networks, and in term of neutral differential system with stochastic effects stated by Arthi et al. [18]. Moreover, Zhu et al. [19] investigated the synthesis of stability neutral system with distributed and discrete time delays. According to [20–23] have illustrated the stability criteria for the neutral neural network with Marcovian jump parameters and mixed time delays. Therefore, passive analyzes for neutral neural networks have been discussed in the last few years. For instance, the studied of Balasubramaniam [24] demonstrated inertia analysis for neutral neural networks with Marcovian jump parameters and time leakage delay term. According to Samidurai [25] analyzed of passivity with mixed and leakage delays for neutral-type neural networks. Unfortunately, the exponential analysis of stability and the passivity performance of a neutral differential system with a time delay is of little concern, in nowadays.

According to the above discussion, this research the exponentially passive analysis was considered for the class of uncertainty neutral fuzzy differential systems generated by the Lyapunov-Krasovkii functional (LKF) method. Also, the systems created by stability theory and integral inequality techniques. The all of delays consist of discrete, neutral, and distributed delays that vary with time. In addition, this research offers a new approach to the resulting manipulation of exponential and inertial steady-states and more efficient compared to existing methods. The following topics to promote a clear understanding and objectives of this study are given

  • This study is the first ever exponentially passive analysis for Takagi-Sugeno fuzzy system of a dynamic system (1) consisting of time-varying, discrete, neutral, and distributed delays.
  • Especially, if Ci + ΔCi(t) = 0, Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0, The system (1) becomes the T-S fuzzy of differential system presented by Fang Liu, et al. [26], Li et al. [27], Lien et al. [7, 28] and Pin-Lin Liu [29].
  • Over and above, if Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0, Also, the system (1) becomes the T-S fuzzy of neutral differential system presented by Ding et al. [30]. Then, the system (1) is more advanced differential replica than the former times.
  • This study attained exponential stability of the T-S fuzzy system, where the upper boundary of delay was more effective than other studies. This present in the Examples 3, and Examples 5 with uncertain conditions.
  • Some methods and a new LKF have been presented to achieve the exponentially passive benchmarks of T-S fuzzy for uncertain dynamic systems with range discrete, neutral range and distributed time-delay.
  • Lastly, for the first time, an improved Wirtinger inequality, a new triple integral inequality, and zero equation together with convex combination approach are used in this work; as a result, we obtain more general results and maximum allowable delay bounds greater than in previous literature [7, 26–30].

Remark 1 This study constructs the suitable Lyapunov-Krasovskii functional, which consists of single, double, triple, and quadruple integral terms containing information about the lower and upper bounds of the delays σ2, τ2 and a state x(t). Furthermore, the LKF contains new triple integral terms as follows: and new quadruple integral terms that do not appear in [7, 26–30]. These improvement techniques enhance to get better results.

Henceforward, this study is divided into 5 Section: Section 2, the generalization for neutral differential of fuzzy replica is defined, and definitions and lemmas. Section 3, the exponentially passive criteria for the generalized fuzzy of dynamical system and will present a special case of the generalized fuzzy of neutral differential system. Section 4 will illustration the numerical examples to indicate the exponentially passive for the common fuzzy of dynamical systems. This includes the special case of the general phase value system for the neutral dynamic system. Lastly, Section 5.

2 Problem statement and preliminaries

Consider Takagi-Sugeno fuzzy of the neutral dynamic system with time-varying delays of the ensuing form:

Rule i: if κ1(t) imply μi1 and … and κP(t) imply μip hence where μij, i = 1, 2, …, r, j = 1, 2, …, p implies the set for fuzzy, implies the state vector, stands for the external inputs, is the output of the system, Ai, Bi, Ci, implies constant matrices, constant r implies the amount of IF-Then rule, κ1(t), κ(t), …, κP(t) implies premise variables. τ(t), σ(t) and h(t) implies neutral discrete and distributed interval time-varying delays, successively, agreeable Furthermore, ΔAi(t), ΔBi(t), ΔCi(t), ΔDi(t) and ΔEi(t) implies the terms of uncertain on system and specify where F, H1i, H2i, H3i, H4i and H5i are known constant matrices and G(t) is a real-unknown matrix function, agreeable, when I is a suitable dimension identity matrices. By fuzzy blending, the entire fuzzy replica is compiled as following: (1) where implies membership function for system which agreeable the rule i, and It is observed as to the fuzzy weighting function ρi(θ(t)) agreeable

Remark 2 In the uncertain fuzzy differential system, the interval time delay (σ1 ≤ σ(t) ≤ σ2) is considered to be longer than the constant time delay (σ(t) = σ2) and bounded time-varying delay (0 ≤ σ(t) ≤ σ2). Then the system (1) is more general.

Definition 1 [31] The system (1) is exponentially passive from input u(t) to out put z(t), if there is a Lyapunov function V(t) and positive real number k satisfy: for all u(t), all initial condition X(t0).

Lemma 1 [32] Let any is positive definite constant matrices, 0 ≤ g1 ≤ g(t) ≤ g2, vector function hence the integration connected are defined, so

Lemma 2 [33, 34] Let is positive definite matrix, for any continuously differentiable function , the following inequality holds: where

Lemma 3 [35] Let is positive definite matrix, for any continuously differentiable function , the following inequality holds: where

Lemma 4 [36] Let is positive definite matrix, for any continuously differentiable function , the following inequality holds: where

Lemma 5 [37] Give L = LT, J, S and Q(t) agreeable QT(t)Q(t) ≤ I are matrices that suitable dimensions, hence the inequality as ensuing: is real, if it’s tantamount the following inequality holds for any ε > 0,

Lemma 6 [38] (Jensen’s Inequality) Let A is positive definite matrix, and is vector function hence the inequality as ensuing:

Lemma 7 [39] (Schur complement) For constant matrices M1, M2 and M3 with suitable dimensions, when and , hence if and only if

3 Main results

Theorem 1 For given constants σ1, σ2, τ1, τ2, h1, h2 ≥ 0 system (1) with certain terms is exponentially passive. If there are real positive definite matrices L1, Q1, Q2, Q3, R1, R2, R3, R4, Z1, Z2, Z3, W1, W2, W3, W4, U1, U2, U3, U4, S1, S2, S3, S4 and a positive λ agreeable the ensuing LMI holds for k = 1, 2, …, m: (2) where

proof This study focal point in the following Lyapunov-Krasovskii function of the system (1) where

Add derivative with V(x(t)) in accordance direction of result for system (1) is specific as: where (3) (4)

Lemma 2 is used to obtain

(5)

Lemma 1 and Lemma 6, are used to obtain (6)

Lemma 3 is used to obtain (7)

Lemma 4 is used to obtain (8)

From the Newton-Leibniz formula, it can be expressed as (9)

Combining Eqs (3)–(9), it can be expressed as In addition, we possess hence

It is can be concluded the following inequality by (3)–(9) and z(t) Therefore, the system (1) is guaranteed to be exponentially passive from Definition 1. The proof is completed.

Based on Theorem 1, we can perform the robust stability analysis for system (1) with uncertainty.

Theorem 2 For scalars σ1, σ2, τ1, τ2, h1, h2 ≥ 0 system (1) with uncertain terms is exponentially passive. If there are matrices L1, Q1, Q2, Q3, R1, R2, R3, R4, Z1, Z2, Z3, W1, W2, U1, U2 > 0 and a positive λ satisfying the ensuing LMI holds: where

proof Replacing Ai, Bi, Ci, Di and Ei with Ai + FG(t)H1i, Bi + FG(t)H2i, Ci + FG(t)H3i, Di + FG(t)H4i and Ei + FG(t)H5i in (2), respectively, (10)

Since the lemma 5, there are some real numbers λ > 0 to result in system (10) true that lead to following inequality: (11) From Lemma 7, Eq (11) is equivalent to Eq (2). The proof is completed. Now the system (1) when Ei+ ΔEi(t) = 0 is demonstrated.

Corollary 1 For given constants σ1, σ2, τ1, τ2, h1, h2 ≥ 0 system (1) with uncertain terms is exponential stable. If there are real positive definite matrices L1, Q1, Q2, Q3, R1, R2, R3, R4, Z1, Z2, Z3, W1, W2, W3, W4, U1, U2, U3, U4, S1, S2, S3, S4 and a positive λ agreeable the ensuing LMI holds for k = 1, 2, …, m: (12) where Then the system (1) when Ei + ΔEi(t) = 0 is exponential stability.

After that, this study shall present the delay-dependent condition of the passivity and exponential stability for system (1) when Ci + ΔCi(t) = Di + ΔDi(t) = 0.

Theorem 3 For given a constant σ2 ≥ 0, system (1) where Ci + ΔCi(t) = Di + ΔDi(t) = 0 with uncertain terms is exponentially passive. If there are real positive definite matrices L1, R1, Z1, W1, U1, S1, S2 and a positive λ agreeable the following LMI holds for k = 1, 2, …, m: (13) where

proof This study focal point in the following Lyapunov-Krasovskii function of the system (1) where Ci + ΔCi(t) = Di + ΔDi(t) = 0 where

Abovementioned by Theorem 1 and Theorem 2, this study obtain the exponentially passive for delay-dependent criteria of systems (1) when Ci + ΔCi(t) = Di + ΔDi(t) = 0.

Now the system (1) when Ci + ΔCi(t) = Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0 is demonstrated.

Remark 3 If Ci + ΔCi(t) = Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0 the fuzzy replica (1) become the T-S fuzzy of neutral differential system presented by [7, 26–29].

Corollary 2 For given a constant σ2 ≥ 0, system (1) when Ci + ΔCi(t) = Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0 with uncertain terms is exponentially. If there are matrices L1, R1, Z1, W1, U1, S1, S2 > 0 and a positive λ agreeable the LMI for k = 1, 2, …, m: where

Then the system (1) when Ci + ΔCi(t) = Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0 is exponentially stability.

Remark 4 According to Corollary 2 that using Lemmas 2, 3 and Lemma 4 yielded fewer conservative outcomes than other results, [7, 26–29] which illustrate in Table 3. Even, these lemmas contain a large number of free weighting matrices, that could bring about their more calculation intricately.

After that, this study shall present the delay-dependent condition of the passivity and exponential stability for system (1) when Di + ΔDi(t) = 0.

Theorem 4 For given constants σ2, τ2 ≥ 0 systems (1) where Di + ΔDi(t) = 0 with uncertain is exponentially passive. If there are positive real symmetric matrices L1, R1, R2, Q1, Q2, Z1, Z2, W1, W2, U1, U2 and a positive λ agreeable the LMI for k = 1, 2, …, m: (14) where

proof This study focal point in the following Lyapunov-Krasovskii function of the system (1) when Di + ΔDi(t) = 0. where

Abovementioned by Theorem 1 and Theorem 2, this study attain the exponentially passive synthesis of delay-dependent condition for systems (1) when Di + ΔDi(t) = 0.

Acquired from Corollary 3, the purpose of this study is for the consequences of uncertainty for T-S fuzzy system (1) when Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0.

Remark 5 If Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0, the uncertainty fuzzy replica (1) become the T-S fuzzy of neutral differential system presented by [30].

Corollary 3 For given constants σ2, τ2 ≥ 0 systems (1) where Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0 with uncertain is exponentially stability. If there are symmetric matrices L1, R1, R2, Q1, Q2, Z1, Z2, W1, W2, U1, U2 > 0 and a positive λ agreeable the LMI for k = 1, 2, …, m as ensuing: where

Then the system (1) when Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0 is exponentially stability.

Remark 6 According to Corollary 3 that using Lemmas 2, 3 and Lemma 4 yielded fewer conservative outcomes than other results, [30] which illustrate in Table 6. Even, these lemmas contain a large number of free weighting matrices, that could bring about their more calculation intricately.

4 Numerical simulation

In this part, the number of sample figures illustrate the performance of our key solution, by comparison of the largest allowable bound σ and the convergent rate α. The LMI control toolbox in MATLAB is used to find all the threshold possibilities.

Example 1 Analyze the uncertainty neutral of T-S fuzzy dynamic system by the parameters as following: where LMI (2), is solved where and to obtain set of parameters for guarantee the exponentially passive as following:

In this example, we used to discuss the exponentially passive of the T-S fuzzy for neutral differential system (1). For dissimilar values α, τd, σd in example 1 that are shown in Table 1, the maximum allowable bounds of σ2 are got by solving the LMIs in Theorem 1 and Theorem 2 with the MATLAB control toolbox.

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Table 1. The maximum allowable bounds of σ2 with Example 1.

https://doi.org/10.1371/journal.pone.0275057.t001

Example 2 Analyze the uncertainty neutral of T-S fuzzy dynamic system by the parameters as following where

LMI (12), is solved where and to obtain set of parameters for guarantee the exponentially as following:

In this example, we used to discuss the exponential stability criteria of the T-S fuzzy for neutral differential system (1). For dissimilar values α, τd, σd in example 2 that are shown in Table 2, the maximum allowable bounds of σ2 are got by solving the LMIs in Theorem 1 with the MATLAB control toolbox.

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Table 2. The maximum allowable bounds of σ2 with Example 2.

https://doi.org/10.1371/journal.pone.0275057.t002

Example 3 Analyze the uncertainty of T-S fuzzy dynamic system presented in [7, 26–29] by the parameters as following where The purpose of example 3 is compare the maximum allowable bounds for tolerable delays of σ(t) which ensure the exponential stability with the fuzzy convergent rate α of the T-S fuzzy dynamic system above. Based on Table 3, the results present-day available for comparison purposes are recorded. This present the proposed method is less conservative than the immemorial method.

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Table 3. The maximum allowable bounds of σ2 for σd and α of Example 3.

https://doi.org/10.1371/journal.pone.0275057.t003

Fig 1 gives the state trajectory of the T-S fuzzy dynamical system (1) where Ci + ΔCi(t) = 0 Di + ΔDi(t) = 0 and Ei + ΔEi(t) = 0 with parameters in Example 3 where u(t) = 0 and the initial condition [x1(t), x2(t)]T = [−0.1 cos(t), 0.1 cos(t)]T, which shows that the T-S fuzzy for dynamical system is stable

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Fig 1. State trajectory of T-S fuzzy dynamical system for Example 3.

https://doi.org/10.1371/journal.pone.0275057.g001

Example 4 Analyze the uncertainty neutral of T-S fuzzy dynamic system by the parameters as following: Where LMI (13) is solved where .

In example 4, we used to discuss the stability criterion and passivity performance of the T-S fuzzy for dynamical system (1) where Ci + ΔCi(t) = 0 and Di + ΔDi(t) = 0. LMIs in Theorem 3 is solved by the MATLAB control toolbox to obtain the largest allowable bounds of σ2 for dissimilar values of σd, α in example 4 are shown in Table 4.

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Table 4. The maximum allowable bounds of σ2 for σd and α of Example 4.

https://doi.org/10.1371/journal.pone.0275057.t004

Fig 2 gives the state trajectory of the T-S fuzzy for dynamical system (1) where Ci + ΔCi(t) = 0 and Di + ΔDi(t) = 0 with parameter in Example 4 where the initial condition [x1(t), x2(t)]T = [−0.1 cos(t), 0.1 cos(t)]T, which shows that the T-S fuzzy for dynamical system is stable.

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Fig 2. State trajectory of T-S fuzzy dynamical system for Example 4.

https://doi.org/10.1371/journal.pone.0275057.g002

Example 5 Analyze the uncertainty neutral of T-S fuzzy dynamic system by the parameters as ollowing: where LMI (14) is solved where and .

In this example, we used to discuss the stability criterion and passivity performance of the T-S fuzzy for neutral differential system (1) where Di + ΔDi(t) = 0. LMIs in Theorem 4 is solved by MATLAB control toolbox to obtain the largest allowable bounds of σ2 for dissimilar values of τd, α, σd in example 5 are shown in Table 5.

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Table 5. The maximum allowable bounds of σ2 for τd, σd and α of Example 5.

https://doi.org/10.1371/journal.pone.0275057.t005

Example 6 Analyze the uncertainty of T-S fuzzy dynamic system presented in [30] by the parameters as following: where

The purpose of example 6 is compare the largest allowable bounds for tolerable delays of σ(t) which ensure the exponential stability with the fuzzy convergent rate α = 0.23 of the T-S fuzzy dynamic system above. Based on Table 6, the results present-day available for comparison purposes are recorded. This present the proposed method is less conservative than the immemorial method.

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Table 6. The maximum allowable bounds of σ2 for τ of Example 6.

https://doi.org/10.1371/journal.pone.0275057.t006

Remark 7 New results on robust exponential stability of Takagi–Sugeno fuzzy for neutral differential systems with mixed time-varying delays [40] only focus on exponential stability for neutral differential equations of the uncertain Takagi—Sugeno fuzzy system. This paper studies exponential stable and exponentially passive neutral differential equations of the uncertain Takagi—Sugeno fuzzy system for further improvement. Furthermore, we consider mixed interval time-varying delays: mixed interval discrete time-varying delay, interval distributed time-varying delay, and interval neutral time-varying delay, i.e., σ(t) is interval discrete time-varying delay which satisfies 0 ≤ σ1 ≤ σ(t)≤σ2, h(t) is interval distributed time-varying delays which satisfies 0 ≤ h1 ≤ h(t)≤h2, and τ(t) is interval neutral time-varying delay which satisfy 0 ≤ τ1 ≤ τ(t)≤τ2.

5 Conclusion

This study rectifies the exponentially passive analysis of the neutral difference of the uncertain Takagi-Sukeno fuzzy system with neutral, discrete, and distributed interval time-varying delay. By spending the Newton-Leibniz formulas, Lyapunov-Krasovskii Functions (LKF), zero equations, and matrix inequality techniques. The form of linear matrix inequalities (LMIs) is constructed from the exponentially passive, in which the numerical efficiency can be verified. Hence, this study shows the example of numbers to demonstrate the effectiveness of our theoretical results and to illustrate that our results are less conservative than the results available in other works: according to Corollary 2, we get the upper bounds of the time-varying delay σ2 for various σd and α. Summarize them in Table 3 for comparison with the results obtained in [7, 26–29]. It is concluded that our results have the upper bounds of the time-varying delay σ2 at the amount of 1.1121 for σd = 0.9 and α = 0.9. Moreover, in Corollary 3, we get the upper bounds of the time-varying delay σ2. Summarize them in Table 6 for comparison with the results obtained in [30]. It is concluded that our results have the upper bounds of the time-varying delay σ2 at the amount of 0.5021 for τ = 0.2450 and α = 0.23. We can see that our obtained results are less conservative than some existing results. In future work, the derived results and methods in this paper are expected to be applied to other systems such as fuzzy generalized complex-valued, guaranteed cost, pinning control of neural networks for impulsive effects on stability and passivity analysis, and so on [41–45].

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