Enhanced Looped Lyapunov Functional for Sampled-Data Control for T-S Fuzzy Systems with Time Delay

Abstract

This paper addresses the problem of sampled-data control in T-S fuzzy systems with time delays. Initially, a negative-definite criterion is developed for the matrix containing membership functions with the help of the idea of a switching mechanism. Building upon this criterion, a membership-function-based looped Lyapunov functional is introduced. Furthermore, a sampling-time-dependent looped Lyapunov functional is constructed, incorporating crucial information about the sampling instance. Notably, this functional matrix is not constrained to satisfy positive-definiteness, and it varies with each sampling point. By utilizing the enhanced looped Lyapunov functional, less conservative stability conditions and a new controller criterion for the T-S fuzzy system with time delays are derived. This demonstrates that our proposed stability conditions allow for a larger sampling interval compared to existing work. Finally, the proposed method’s effectiveness is demonstrated through the validation of two simulation examples.

1. Introduction

T-S fuzzy systems integrate fuzzy logic and mathematical modeling to describe and control complex nonlinear systems. Their core idea is to use fuzzy rules to discretize a non-linear system into multiple linear subsystems [1,2,3]. Due to their ability to manage uncertainty and ambiguity, T-S fuzzy systems have been widely applied in automatic control, pattern recognition, and signal handling [4,5,6,7,8]. For example, the problem of stability and stabilization for a type of chaotic systems, using a T-S fuzzy model and non-uniform sampling, was investigated in [7]. An event-triggered filter for delayed T-S fuzzy systems was studied in [8].

With the fast advancement of digital computing technologies, sampled-data control has emerged as a dominant approach in the design of control systems [9,10]. In traditional continuous-time control, the controller must acquire system state information continuously, which is often challenging in practical applications, especially for complex modern control systems. Sampled-data control, by acquiring data at discrete time points and controlling based on these data, not only aligns with the operational mode of digital computers and microprocessors but also significantly reduces the system’s computational burden [11,12,13]. As a results, the study of sampled-data control for T-S fuzzy systems has received a lot of attention in the recent decade [14,15,16,17,18,19,20,21]. In [22], the dissipative control of T-S fuzzy systems was investigated, leveraging the novel looped functional. Dissipative stabilization of T-S fuzzy systems using an augmented looped functional was studied in [23]. Based on a novel sampling dependence function approach, stabilization problems for T-S fuzzy systems were considered in [24]. Furthermore, event-triggered control for T-S fuzzy systems based on non-periodic sampled data was investigated using a novel Lyapunov–Krasovskii functional and a dynamic partitioning method in [25]. It is worth noting that the above work focuses on efforts to increase the sampling data interval. However, developing advanced Lyapunov–Krasovskii functional approaches to further increase the sampling data interval still requires further investigation.

In practical applications of T-S fuzzy systems with sampled-data control, time delays are inevitable since network environments [26,27,28,29]. However, those delays often result in system oscillations or even instability. Therefore, the stability issue of T-S fuzzy systems with time delays is particularly important [30,31,32,33,34,35,36,37,38]. The tunable fuzzy dependence matrix method was developed to address the issue of exponential synchronization in T-S fuzzy neural networks, as discussed in [33]. In [34], robust control for sampled-data T-S fuzzy systems with time delays and uncertainties was explored by introducing a free weighting matrix to address the integral term. The approach involved transforming the coupled time-varying matrix inequalities into a set of decoupled matrix inequalities. The robustness of time-varying delay T-S fuzzy systems based on the control of memorized sampled data were investigated in [35] based on the piecewise Lyapunov–Krasovskii functional. The problem of sampled-data filtering for T-S fuzzy systems, subject to uncertainties and delays, was investigated based on a delayed product-type Lyapunov–Krasovskii functional in [37]. In order to minimize the effect of controller uncertainty, sampled-data control for the systems was investigated in [38]. It is important to note that previous Lyapunov–Krasovskii functionals have rarely adequately considered information about membership functions. Additionally, the previously constructed looped functional for delayed T-S fuzzy systems is independent of the sampling time. Therefore, there is still room for further research.

Building on the previous discussions, this work delves into the problem of sampled-data control for T-S fuzzy systems that incorporate time delays. By proposing a membership-function-based and sampling-moment-dependent enhanced looped Lyapunov functional, new and less conservative stability criteria and controller design criteria are developed. The key contributions of this work are sketched below: (1) A membership-function-based looped Lyapunov functional, aided by a negative-definite criterion for the matrix containing membership functions, is established, which considers not only the system state and delayed state but also the membership function. (2) A sampling-time-dependent looped Lyapunov–Krasovskii functional is developed, where the functional matrix is not constrained to satisfy positive-definiteness and varies with each sampling point. (3) Less conservative stability and controller conditions for the studied system are derived.

The rest of this paper is organized as follows. The system model and some necessary backgrounds are introduced in Section 2. The stability analysis and controller design for the T-S fuzzy system with time delays are presented in Section 3. In Section 4, the numerical simulations are provided. Finally, Section 5 concludes this work.

Notations: ${\mathbb{R}}^n$ represents the Euclidean space of dimension n, while ${\mathbb{R}}^{n\times n}$ denotes the collection of all $n\times n$ matrices. The set of $n\times n$ symmetric positive-definite matrices is indicated by ${\mathbb{S}}_n^{+}$. Additionally, $diag\{·\}$ stands for a block-diagonal matrix. ∗ signifies the symmetric term within a matrix. The operator $\left\{col\right\}·$ is used to represent either a column vector or a column matrix.

2. Preliminaries

Consider a delayed T-S fuzzy system [20]:

Model Rule i: If ${\delta }_1\left(t\right)$ is $M_{i1}$, ⋯, and ${\delta }_p\left(t\right)$ is $M_{ip}$, then

$$\begin{array}{c}\hfill \dot{x}\left(t\right)={\mathcal{A}}_{i}x\left(t\right)+{\mathcal{A}}_{di}x(t-\varrho )+{\mathcal{B}}_{i}u\left(t\right),\end{array}$$

(1)

where $x\left(t\right)\in {\mathbb{R}}^n$ and $u\left(t\right)\in {\mathbb{R}}^m$ are the state variable and the control input, respectively; ${\mathcal{A}}_i$, ${\mathcal{A}}_{di}$, and ${\mathcal{B}}_i$ denote known matrices with appropriate dimensions; $\varrho$ denotes delay; the terms ${\delta }_1\left(t\right),\dots ,{\delta }_p\left(t\right)$ and $M_{i1},\dots$, $M_{\mathrm{ip}}(i=1,\dots ,r)$ represent the premise variables and fuzzy sets. Utilizing the fuzzy inference approach, we have

$$\begin{array}{c}\hfill \dot{x}\left(t\right)=\sum _{i=1}^r{\kappa }_i\left(\delta \left(t\right)\right)\left({\mathcal{A}}_{i}x\left(t\right)+{\mathcal{A}}_{di}x(t-\varrho )+{\mathcal{B}}_{i}u\left(t\right)\right),\end{array}$$

(2)

where ${\kappa }_i\left(\delta \left(t\right)\right)$ with ${\sum }_{i=1}^r{\kappa }_i\left(\delta \left(t\right)\right)=1$ and ${\sum }_{i=1}^r{\dot{\kappa }}_i\left(\delta \left(t\right)\right)=0$ is the membership function, which meets

$$\begin{array}{c}\hfill {\kappa }_i\left(\delta \left(t\right)\right)={\displaystyle \frac{{\omega }_i\left(\delta \left(t\right)\right)}{{\sum }_{i=1}^r{\omega }_i\left(\delta \left(t\right)\right)}},{\omega }_i\left(\delta \left(t\right)\right)=\prod _{i=1}^p{M}_{ij}\left(\delta \left(t\right)\right),\end{array}$$

(3)

with $M_{ij}\left(\delta \left(t\right)\right)$ being the grade of membership for ${\delta }_j\left(t\right)$ in $M_{ij}$.

The system (2) is a control system that is regulated via a communication network and is characterized by the use of sampled data for closed-loop control. Since the system operates in a distributed environment, the control input signals are not acquired continuously but rely on sampling at certain time intervals. Specifically, the control inputs are generated through a zero-order hold mechanism that ensures that the control inputs remain constant during the time period between samples. In other words, the controller receives the state information of the system at each sampling moment $t_k$, computes the corresponding control input, and keeps it constant for the following time period (i.e., $t_k\le t<t_{k+1}$). This sequence of sampling moments is defined as $t_0<t_1<\cdots <t_k<t<t_{k+1}$, i.e., the control strategy of the system is updated based on discrete sampling moments, while the control inputs are maintained between two consecutive moments by a zero-order keeper after each sample. This regulation allows the system to update the state information at non-consecutive time points, while the keeper ensures a smooth application of the control signal during the time interval. Let $0<{\upsilon }_k\triangleq t_{k+1}-t_k\le h$, ensuring that time intervals between consecutive samples are bounded by h. Recognizing that additional perturbations in the controller could impact the system, a sampled-data controller is provided as

$$\begin{array}{c}\hfill u\left(t\right)=K_{j}x\left(t_k\right)+K_{\varrho j}x\left(t_k-\varrho \right).\end{array}$$

(4)

where $K_j,K_{\varrho j}$ denote controller gains and $t_k\le t<t_{k+1}$.

Hence, the controller can be inferred as

$$\begin{array}{c}\hfill u\left(t\right)=\sum _{j=1}^r{\kappa }_j\left(\delta \left(t_k\right)\right)\left(K_{j}x\left(t_k\right)+K_{\varrho j}x\left(t_k-\varrho \right)\right).\end{array}$$

(5)

We can derive the subsequent overall system model as

$$\begin{array}{cc}\hfill \dot{x}\left(t\right)& =\sum _{i=1}^r\sum _{j=1}^r{\kappa }_i\left(\delta \left(t\right)\right){\kappa }_j\left(\delta \left(t_k\right)\right)\left[{\mathcal{A}}_{i}x\left(t\right)+{\mathcal{A}}_{di}x(t-\varrho )\right.\hfill \\ \hfill & \left.+{\mathcal{B}}_i{K}_{j}x\left(t_k\right)+{\mathcal{B}}_i{K}_{\varrho j}x\left(t_k-\varrho \right)\right].\hfill \end{array}$$

(6)

The objective of this work is to develop a less conservative stability condition and design an appropriate controller for the system (6). Prior to introducing the main findings, several lemmas are provided below.

Lemma 1

([39,40]). For differentiable vector $w\left(s\right)$ in $[d_1,d_2]\to {\mathbb{R}}^l$ and matrices $\Lambda \in {\mathbb{S}}_l^{+}$, $M_1$, $M_2\in {\mathbb{R}}^{m\times l}$, any vector $\varphi \left(t\right)\in {\mathbb{R}}^m$, such that

$$\begin{array}{cc}\hfill & (d_2-d_1){\int }_{d_1}^{d_2}{\dot{w}}^T\left(s\right)\Lambda \dot{w}\left(s\right)ds\ge {\Omega }_1^T\Lambda {\Omega }_1+3{\Omega }_2^T\Lambda {\Omega }_2,\hfill \\ \hfill & -{\int }_{d_1}^{d_2}{\dot{w}}^T\left(s\right)\Lambda \dot{w}\left(s\right)ds\le 2\varphi {\left(t\right)}^T\left(M_1{\Omega }_1+M_2{\Omega }_2\right)\varphi \left(t\right)+(d_2-d_1)\varphi {\left(t\right)}^T(M_1{\Lambda }^{-1}{M}_1^T\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & +M_2{\Lambda }^{-1}{M}_2^T)\varphi \left(t\right),\hfill \end{array}$$

(8)

where ${\Omega }_1=w\left(d_2\right)-w\left(d_1\right)$ and ${\Omega }_2=w\left(d_2\right)+w\left(d_1\right)-{\displaystyle \frac{2}{d_2-d_1}}{\int }_{d_1}^{d_2}$$w\left(s\right)ds$.

Next, we give a lemma for the derivation of double integral terms, which plays an important role in the LKF of the derivation in the main results.

Lemma 2.

For a function $\mathcal{F}\left(t\right)={\int }_{{\Im }_1}^{{\Im }_2}{\int }_{{\nu }_1(\Im ,t)}^{{\nu }_2(\Im ,t)}f(\Im ,\nu ,t)d\nu d\Im$ where ${\Im }_1,{\Im }_2$ are differentiable for t. ${\nu }_1(\Im ,t),{\nu }_2(\Im ,t)$ are continuous; thus, one has

$$\begin{array}{cc}& {\displaystyle \frac{d\mathcal{F}\left(t\right)}{dt}}={\dot{\Im }}_2{\int }_{{\nu }_1({\Im }_2,t)}^{{\nu }_2({\Im }_2,t)}f({\Im }_2,\nu ,t)d\nu -{\dot{\Im }}_1{\int }_{{\nu }_1({\Im }_1,t)}^{{\nu }_2({\Im }_1,t)}f({\Im }_1,\nu ,t)d\nu \hfill \\ & +{\int }_{{\Im }_1}^{{\Im }_2}{\displaystyle \frac{\partial {\nu }_2(\Im ,t)}{\Im t}}f(\Im ,{\nu }_2(\Im ,t),t)d\Im \hfill \\ & +{\int }_{{\Im }_1}^{{\Im }_2}{\int }_{{\nu }_1(\Im ,t)}^{{\nu }_2(\Im ,t)}{\displaystyle \frac{\partial f(\Im ,\nu ,t)}{\partial t}}d\nu d\Im -{\int }_{{\Im }_1}^{{\Im }_2}{\displaystyle \frac{\partial {\nu }_1(\Im ,t)}{\partial t}}f(\Im ,{\nu }_1(\Im ,t),t)d\Im .\hfill \end{array}$$

(9)

Proof of Lemma 2.

Let

$$\begin{array}{c}\hfill \mathcal{F}\left(t\right)={\int }_{{\Im }_1}^{{\Im }_2}{\mathcal{F}}_1(\Im ,t)d\Im ,{\mathcal{F}}_1(\Im ,t)={\int }_{{\nu }_1(\Im ,t)}^{{\nu }_2(\Im ,t)}f(\Im ,\nu ,t)d\nu ,\end{array}$$

(10)

taking the the derivative of $\mathcal{F}\left(t\right)$, one has

$$\begin{array}{c}\hfill {\displaystyle \frac{d\mathcal{F}\left(t\right)}{dt}}={\dot{\Im }}_2{\mathcal{F}}_1({\Im }_2,t)-{\dot{\Im }}_1{\mathcal{F}}_1({\Im }_1,t)+{\int }_{{\Im }_1}^{{\Im }_2}{\displaystyle \frac{\partial {\mathcal{F}}_1(\Im ,t)}{\partial t}}d\Im ,\end{array}$$

(11)

where

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial {\mathcal{F}}_1(\Im ,t)}{\partial t}}& ={\displaystyle \frac{\partial {\nu }_2(\Im ,t)}{\partial t}}f(\Im ,{\nu }_2(\Im ,t),t)-{\displaystyle \frac{\partial {\nu }_1(\Im ,t)}{\partial t}}f(\Im ,{\nu }_1(\Im ,t),t)+{\int }_{{\nu }_1(\Im ,t)}^{{\nu }_2(\Im ,t)}{\displaystyle \frac{\partial f(\Im ,\nu ,t)}{\partial t}}d\nu \hfill \end{array}.$$

Therefore, the proof is finished. □

Lemma 3.

For a membership function matrix ${\mathcal{P}}_{\kappa }={\sum }_{i=1}^r{\kappa }_i\left(\delta \left(t\right)\right)P_i$ with free matrices $P_i>0$, we obtain ${\dot{\mathcal{P}}}_{\kappa }\le 0$ if the following switched conditions are satisfied.

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{\kappa }}_j\left(\delta \left(t\right)\right)<0,P_j-P_r>0,\hfill \\ {\dot{\kappa }}_j\left(\delta \left(t\right)\right)\ge 0,P_j-P_r\le 0.\hfill \end{array}\right.\end{array}$$

(12)

where $j=1,\dots ,r-1$.

Proof of Lemma 3.

Taking the derivative of ${\mathcal{P}}_{\kappa }={\sum }_{i=1}^r{\kappa }_i\left(\delta \left(t\right)\right)P_i$, we obtain ${\dot{\mathcal{P}}}_{\kappa }={\sum }_{i=1}^r{\dot{\kappa }}_i$$\left(\delta \left(t\right)\right)P_i={\sum }_{j=1}^{r-1}{\dot{\kappa }}_j\left(\delta \left(t\right)\right)(P_j-P_r)$ based on Lemma 2. It is noted that ${\dot{\kappa }}_j\left(\delta \left(t\right)\right)$ can be negative or positive. When ${\dot{\kappa }}_j\left(\delta \left(t\right)\right)<0$, $P_j-P_r>0$ can ensure that ${\dot{\mathcal{P}}}_{\kappa }\le 0$ holds. When ${\dot{\kappa }}_j\left(\delta \left(t\right)\right)\ge 0$, $P_j-P_r\le 0$ can ensure that ${\dot{\mathcal{P}}}_{\kappa }\le 0$ holds. Therefore, if (12) holds, we can derive ${\dot{\mathcal{P}}}_{\kappa }\le 0$. □

Remark 1.

In the context of constructing stability analysis for T-S fuzzy systems, due to the unknown positive and negative characteristics of the membership function, the correlation matrix in the selected Lyapunov functional is typically independent of the membership function. This approach is consistent with studies such as those by [16,17,18,20,23]. In this paper, the Lyapunov functional is made to better encapsulate the membership function information since the interpretation of Lemma 3, which will lead to an advanced Lyapunov functional.

3. Main Results

This section presents the key findings, focusing on the stability analysis and controller design for the system (6). For simplicity, some variables and vectors are expressed, as follows:

$$\begin{array}{cc}& {\phi }_1=t-t_k,{\phi }_2=t_{k+1}-t,\hfill \\ & {\overline{\sigma }}_l=\left[\begin{array}{ccc}{0}_{n\times (l-1)n}\hfill & I_{n\times n}\hfill & 0_{n\times (6-l)n}\hfill \end{array}\right],l=1,\dots ,6,\hfill \\ & {\sigma }_l=\left[\begin{array}{ccc}{0}_{n\times (l-1)n}\hfill & I_{n\times n}\hfill & 0_{n\times (11-l)n}\hfill \end{array}\right],l=1,\dots ,11,\hfill \\ & {\varsigma }_1\left(t\right)=col\left\{x\left(t\right),x(t-\varrho ),{\int }_{t-\varrho }^{t}x\left(s\right)ds\right\},{\varsigma }_2\left(t\right)=col\{x\left(t\right),\dot{x}\left(t\right)\},\hfill \\ & {\psi }_1\left(t\right)=col\left\{{\phi }_2{\eta }_4\left(t\right),{\phi }_1{\eta }_5\left(t\right)\right\},{\psi }_2\left(t\right)=col\left\{{\eta }_4\left(t\right),{\eta }_5\left(t\right)\right\},\hfill \\ & {\psi }_3\left(t\right)=col\left\{x\left(t_k\right),x\left(t_{k+1}\right),x\left(t_k-\varrho \right),x\left(t_{k+1}-\varrho \right)\right\},\hfill \\ & {\psi }_4\left(t\right)=col\left\{{\psi }_3\left(t\right),{\eta }_1\left(t\right),{\eta }_2\left(t\right)\right\},\hfill \\ & {\eta }_1\left(t\right)={\displaystyle \frac{1}{{\phi }_1}}{\int }_{t_k}^{t}x\left(s\right)ds,{\eta }_2\left(t\right)={\displaystyle \frac{1}{{\phi }_2}}{\int }_t^{t_{k+1}}x\left(s\right)ds,{\eta }_3\left(t\right)={\displaystyle \frac{1}{\varrho }}{\int }_{t-\varrho }^{t}x\left(s\right)ds,\hfill \\ & {\eta }_4\left(t\right)=col\left\{x\left(t\right)-x\left(t_k\right),x(t-\varrho )-x\left(t_k-\varrho \right)\right\},\hfill \\ & {\eta }_5\left(t\right)=col\left\{x\left(t_{k+1}\right)-x\left(t\right),x\left(t_{k+1}-\varrho \right)-x(t-\varrho )\right\},\hfill \\ & \zeta \left(t\right)=col\left\{x\left(t\right),x(t-\varrho ),x\left(t_k\right),x\left(t_{k+1}\right),x\left(t_k-\varrho \right),x\left(t_{k+1}-\varrho \right),\dot{x}\left(t\right),\dot{x}(t-\varrho ),\right.\hfill \\ & \left.{\eta }_1\left(t\right),{\eta }_2\left(t\right),{\eta }_3\left(t\right)\right\}.\hfill \end{array}$$

3.1. Stability Condition

Theorem 1.

The system (6) is asymptotic stable, given constants $h,\varrho$, and gains matrices $K_{pj}$ and $K_{p\varrho j}(p=1,$$\dots ,2^{r-1})$, if there exist positive-definite matrices $P_i,S_i,H_i$, $D_1^{\ell }$, $D_2^{\ell }$ ($\ell =1,2$), symmetrical matrices $U_1^{\ell },U_2^{\ell }$, $Z^{\ell }$, and any matrices ${\Lambda }_{\iota }(\iota =1,2,3,4),X^{\ell }$, $M_{ij}^{\iota }$, such that

$$\begin{array}{cc}\hfill & {\dot{\mathcal{P}}}_{\kappa }\le 0,{\dot{\mathcal{S}}}_{\kappa }\le 0,{\dot{\mathcal{H}}}_{\kappa }\le 0,\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\Omega }_{ij}^0+h{\Omega }_{ij}^{21}& h{M}_{ij}^3& h{M}_{ij}^4\\ \ast & -h{\mathcal{U}}_2^1& 0\\ \ast & 0& -3h{\mathcal{U}}_2^1\end{array}\right]<0,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\Omega }_{ij}^0+h{\Omega }_{ij}^{31}& h{M}_{ij}^1& h{M}_{ij}^2\\ \ast & -h{\mathcal{U}}_1^2& 0\\ \ast & 0& -3h{\mathcal{U}}_1^2\end{array}\right]<0,\hfill \end{array}$$

(15)

where

$$\begin{array}{cc}& {\Omega }_{ij}^1=Sym\left\{{\Phi }_1^T{P}_i{\Phi }_2+{\Phi }_8^T{\mathcal{U}}_1{\Phi }_6+{\Phi }_8^{T}U{\mathcal{U}}_2{\Phi }_7+{\Phi }_{10}^T\mathcal{X}{\Phi }_4^b-{\Phi }_4^{a^T}\mathcal{X}{\Phi }_{10}+M_{ij}^1{\Lambda }_1\right.\hfill \\ & \left.+M_{ij}^2{\Lambda }_2+M_{ij}^3{\Lambda }_3+M_{ij}^4{\Lambda }_4\right\}+{\Phi }_3^T{S}_i{\Phi }_3-{\Phi }_4^T{S}_i{\Phi }_4+{\varrho }^2{\sigma }_7^T{H}_i{\sigma }_7-{\Phi }_0^T{H}_i{\Phi }_0\hfill \\ & -3{\Phi }_0^a{H}_i{\Phi }_0^a+{\kappa }_{ij},\hfill \\ & {\Omega }_{ij}^{21}=Sym\left\{{\Phi }_5^{a^T}{\mathcal{U}}_1{\Phi }_6+{\Phi }_5^{a^T}{\mathcal{U}}_2{\Phi }_7+{\Phi }_8^a{\mathcal{U}}_1{\Phi }_9+{\Phi }_{12}^{a^T}\mathcal{Z}{\Phi }_{11}\right\}\hfill \\ & +{\Phi }_{11}^T\mathcal{Z}{\Phi }_{11}+{\sigma }_7^T{\mathcal{D}}_1{\sigma }_7,\hfill \\ & {\Omega }_{ij}^{31}=Sym\left\{{\Phi }_5^{b^T}{\mathcal{U}}_1{\Phi }_6+{\Phi }_5^{b^T}{\mathcal{U}}_2{\Phi }_7+{\Phi }_8^{b^T}{\mathcal{U}}_1{\Phi }_9+{\Phi }_{12}^b\mathcal{Z}{\Phi }_{11}\right\}\hfill \\ & -{\Phi }_{11}^T\mathcal{Z}{\Phi }_{11}+{\sigma }_7^T{\mathcal{D}}_2{\sigma }_7,\hfill \\ & {\kappa }_{ij}=Sym\left\{{\kappa }_0\left(-{\sigma }_7+{\mathcal{A}}_i{\sigma }_1+{\mathcal{A}}_{di}{\sigma }_2+{\mathcal{B}}_i{K}_{pj}{\sigma }_3+{\mathcal{B}}_i{K}_{p\varrho j}{\sigma }_5\right)\right\}\hfill \end{array}$$

$$\begin{array}{cc}& {\kappa }_0={\sigma }_1^T{\Lambda }_1+{\sigma }_2^T{\Lambda }_2+{\sigma }_3^T{\Lambda }_3+{\sigma }_7^T{\Lambda }_4\hfill \\ & {\Phi }_0={\sigma }_1-{\sigma }_2,{\Phi }_0^a={\sigma }_1+{\sigma }_2-2{\sigma }_{11},{\Phi }_1=col\left\{{\sigma }_1,{\sigma }_2,\varrho {\sigma }_{11}\right\},\hfill \\ & {\Phi }_2=col\left\{{\sigma }_7,{\sigma }_8,{\sigma }_1-{\sigma }_2\right\},{\Phi }_3=col\left\{{\sigma }_1,{\sigma }_7\right\},{\Phi }_4=col\left\{{\sigma }_2,{\sigma }_8\right\},\hfill \\ & {\Phi }_4^a=col\left\{{\sigma }_1-{\sigma }_3,{\sigma }_2-{\sigma }_5\right\},{\Phi }_4^b=col\left\{{\sigma }_4-{\sigma }_1,{\sigma }_6-{\sigma }_2\right\},\hfill \\ & {\Phi }_5^a=col\left\{{\sigma }_7,{\sigma }_8,0,0\right\},{\Phi }_5^b=col\left\{0,0,-{\sigma }_7,-{\sigma }_8\right\},\hfill \\ & {\Phi }_6=col\left\{{\Phi }_4^a,{\Phi }_4^b\right\},{\Phi }_7=col\left\{{\sigma }_3,{\sigma }_4,{\sigma }_5,{\sigma }_6\right\},{\Phi }_8=col\left\{-{\Phi }_4^a,{\Phi }_4^b\right\},\hfill \\ & {\Phi }_8^a=col\left\{{\Phi }_4^a,0,0\right\},{\Phi }_8^b=col\left\{0,0,{\Phi }_4^b\right\},{\Phi }_9={\Phi }_5^a+{\Phi }_5^b,\hfill \\ & {\Phi }_{10}=col\left\{{\sigma }_7,{\sigma }_8\right\},{\Phi }_{11}=col\left\{{\Phi }_7,{\sigma }_9,{\sigma }_{10}\right\},{\Phi }_{12}^a=col\left\{0,0,0,0,{\sigma }_1-{\sigma }_9,0\right\},\hfill \\ & {\Phi }_{12}^b=col\left\{0,0,0,0,0,{\sigma }_{10}-{\sigma }_1\right\},{\Lambda }_1={\sigma }_1-{\sigma }_3,{\Lambda }_2={\sigma }_1+{\sigma }_3-2{\sigma }_9,\hfill \\ & {\Lambda }_3={\sigma }_4-{\sigma }_1,{\Lambda }_4={\sigma }_4+{\sigma }_1-2{\sigma }_{10}.\hfill \end{array}$$

Proof of Theorem 1.

A Lyapunov–Krasovskii functional is given as follows.

$$\begin{array}{c}\hfill V\left(t\right)=\sum _{r=1}^3{V}_r\left(t\right)\end{array}$$

(16)

and

$$\begin{array}{cc}\hfill V_1\left(t\right)=& {\varsigma }_1^T\left(t\right){\mathcal{P}}_{\kappa }{\varsigma }_1\left(t\right)+{\int }_{t-\varrho }^t{\varsigma }_2^T\left(\tau \right){\mathcal{S}}_{\kappa }{\varsigma }_2\left(\tau \right)d\tau +\varrho {\int }_{-\varrho }^0{\int }_{t+\delta }^t{\dot{x}}^T\left(\tau \right){\mathcal{H}}_{\kappa }\dot{x}\left(\tau \right)d\tau d\delta ,\hfill \\ \hfill V_2\left(t\right)=& 2{\psi }_1^T\left(t\right)\left({\mathcal{U}}_1{\psi }_2\left(t\right)+{\mathcal{U}}_2{\psi }_3\left(t\right)\right)+2{\eta }_4^T\left(t\right)\mathcal{X}{\eta }_5\left(t\right),\hfill \\ & +{\phi }_2{\phi }_1{\psi }_4^T\left(t\right)\mathcal{Z}{\psi }_4\left(t\right),\hfill \\ \hfill V_3\left(t\right)=& {\phi }_2{\int }_{t_k}^t{\dot{x}}^T\left(\tau \right){\mathcal{D}}_1\dot{x}\left(\tau \right)d\tau -{\phi }_1{\int }_t^{t_{k+1}}{\dot{x}}^T\left(\tau \right){\mathcal{D}}_2\dot{x}\left(\tau \right)d\tau ,\hfill \end{array}$$

where ${\mathcal{P}}_{\kappa }={\sum }_{i=1}^r{\kappa }_i\left(\delta \left(t\right)\right)P_i$, ${\mathcal{S}}_{\kappa }={\sum }_{i=1}^r{\kappa }_i\left(\delta \left(t\right)\right)S_i$, ${\mathcal{H}}_{\kappa }={\sum }_{i=1}^r{\kappa }_i\left(\delta \left(t\right)\right)H_i$, ${\mathcal{U}}_1=\chi U_1^1+(1-\chi ){\displaystyle \frac{{\upsilon }_k}{h}}{U}_1^2$, ${\mathcal{U}}_2=\chi U_2^1+(1-\chi )U_2^2$, $\mathcal{X}=\chi X^1+(1-\chi )X^2$, $\mathcal{Z}=\chi Z^1+(1-\chi )Z^2$, ${\mathcal{D}}_1=\chi D_1^1+(1-\chi )D_1^2$, and $\chi ={\displaystyle \frac{h-{\upsilon }_k}{h}}$.

Based on Lemma 3, the controller can be represented in the following form:

$$\begin{array}{cc}\hfill u_p\left(t\right)=& \sum _{j=1}^r{\kappa }_j\left(\delta \left(t_k\right)\right)\left[K_{pj}x\left(t_k\right)+K_{p\varrho j}x\left(t_k-\varrho \right)\right],\hfill \end{array}$$

(17)

where $p=1,\dots ,2^{r-1}$.

The derivative of $V\left(t\right)$ is given by

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(t\right)\le & 2{\varsigma }_1^T\left(t\right){\mathcal{P}}_{\kappa }{\dot{\eta }}_1\left(t\right)+{\varsigma }_2^T\left(t\right){\mathcal{S}}_{\kappa }{\varsigma }_2\left(t\right)-{\varsigma }_2^T(t-\varrho ){\mathbb{S}}_{\kappa }{\varsigma }_2(t-\varrho )+{\varrho }^2{\dot{x}}^T\left(t\right){\mathcal{H}}_{\kappa }\dot{x}\left(t\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & -\varrho {\int }_{t-\varrho }^t{\dot{x}}^T\left(\tau \right){\mathcal{H}}_{\kappa }\dot{x}\left(\tau \right)d\tau ,\hfill \\ \hfill {\dot{V}}_2\left(t\right)=& 2{\dot{\psi }}_1^T\left(t\right)\left({\mathcal{U}}_1{\psi }_2\left(t\right)+{\mathcal{U}}_2{\psi }_3\left(t\right)\right)+2{\psi }_1^T\left(t\right)\left({\mathcal{U}}_1{\dot{\psi }}_2\left(t\right)+{\mathcal{U}}_2{\dot{\psi }}_3\left(t\right)\right)\hfill \\ \hfill & +2{\dot{\eta }}_4^T\left(t\right)\mathcal{X}{\eta }_5\left(t\right)+2{\eta }_4^T\left(t\right)\mathcal{X}{\dot{\eta }}_5\left(t\right)-{\phi }_1{\psi }_4^T\left(t\right)\mathcal{Z}{\psi }_4\left(t\right)\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & +{\phi }_2{\psi }_4^T\left(t\right)\mathcal{Z}{\psi }_4\left(t\right)+2{\phi }_2{\phi }_1{\dot{\psi }}_4^T\left(t\right)\mathcal{Z}{\psi }_4\left(t\right),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill {\dot{V}}_3\left(t\right)\le & {\phi }_2{\dot{x}}^T\left(t\right){\mathcal{D}}_1\dot{x}\left(t\right)-{\int }_{t_k}^t{\dot{x}}^T\left(\tau \right){\mathcal{D}}_1\dot{x}\left(\tau \right)d\tau \hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & +{\phi }_1{\dot{x}}^T\left(t\right){\mathcal{D}}_2\dot{x}\left(t\right)-{\int }_t^{t_{k+1}}{\dot{x}}^T\left(\tau \right){\mathcal{D}}_2\dot{x}\left(\tau \right)d\tau .\hfill \end{array}$$

(21)

Then,

$$\begin{array}{cc}\hfill \dot{V}\left(t\right)\le & {\zeta }^T\left(t\right)\left\{(Sym\left\{{\Phi }_8^T({\mathcal{U}}_1{\Phi }_6+{\mathcal{U}}_2{\Phi }_7)+{\Phi }_1^T{\mathcal{P}}_{\kappa }{\Phi }_2-{\Phi }_4^{a^T}\mathcal{X}{\Phi }_{10}+{\Phi }_{10}^T\mathcal{X}{\Phi }_4^b\right\}\right.\hfill \\ \hfill & +{\Phi }_3^T{\mathcal{S}}_{\kappa }{\Phi }_3-{\Phi }_4^T{\mathbb{S}}_{\kappa }{\Phi }_4+{\varrho }^2{\sigma }_7^T{\mathcal{H}}_{\kappa }{\sigma }_7\hfill \\ \hfill & +{\phi }_2\left(Sym\left\{{\Phi }_5^{a^T}{\mathcal{U}}_1{\Phi }_6+{\Phi }_5^{a^T}{\mathcal{U}}_2{\Phi }_7+{\Phi }_8^{a^T}{\mathcal{U}}_1{\Phi }_9\right.+{\Phi }_{12}^{a^T}\mathcal{Z}{\Phi }_{11}\right\}\hfill \\ \hfill & \left.+{\Phi }_{11}^{T}Z{\Phi }_{11}+{\sigma }_7^T{\mathcal{D}}_1{\sigma }_7\right)\hfill \\ \hfill & +{\phi }_1\left(Sym\left\{{\Phi }_5^{b^T}{\mathcal{U}}_1{\Phi }_6+{\Phi }_5^{b^T}{\mathcal{U}}_2{\Phi }_7+{\Phi }_8^{b^T}{\mathcal{U}}_1{\Phi }_9+{\Phi }_{12}^{b^T}\mathcal{Z}{\Phi }_{11}\right\}\right.\hfill \\ \hfill & -{\Phi }_{11}^T\mathcal{Z}{\Phi }_{11}+{\sigma }_7^T{\mathcal{D}}_2{\sigma }_7\zeta \left(t\right)-\varrho {\int }_{t-\varrho }^t{\dot{x}}^T\left(\tau \right){\mathcal{H}}_{\kappa }\dot{x}\left(\tau \right)d\tau \hfill \\ \hfill & -{\int }_{t_k}^t{\dot{x}}^T\left(\tau \right){\mathcal{D}}_1\dot{x}\left(\tau \right)d\tau -{\int }_t^{t_{k+1}}{\dot{x}}^T\left(\tau \right){\mathcal{D}}_2\dot{x}\left(\tau \right)d\tau .\hfill \end{array}$$

(22)

By employing Lemma 1, one has

$$\begin{array}{cc}\hfill & -\varrho {\int }_{t-\varrho }^t{\dot{x}}^T\left(\tau \right){\mathcal{H}}_{\kappa }\dot{x}\left(\tau \right)d\tau \le {\zeta }^T\left(t\right)\left(-{\Phi }_0^T{\mathcal{H}}_{\kappa }{\Phi }_0-3{\Phi }_0^{a^T}{\mathcal{H}}_{\kappa }{\Phi }_0^a\right)\zeta \left(t\right),\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & -{\int }_{t_k}^t{\dot{x}}^T\left(\tau \right){\mathcal{D}}_1\dot{x}\left(\tau \right)d\tau \le {\zeta }^T\left(t\right){\delta }_1\left(t\right)\zeta \left(t\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & -{\int }_t^{t_{k+1}}{\dot{x}}^T\left(\tau \right){\mathcal{D}}_2\dot{x}\left(\tau \right)d\tau \le {\zeta }^T\left(t\right){\delta }_2\left(t\right)\zeta \left(t\right),\hfill \end{array}$$

(25)

where

$$\begin{array}{cc}\hfill {\delta }_1\left(t\right)=& {\phi }_1\left(M_{ij}^1{\mathcal{D}}_1^{-1}{M}_{ij}^{1}T+{\displaystyle \frac{1}{3}}{M}_{ij}^2{\mathcal{D}}_1^{-1}{M}_{ij}^{2T}\right)+Sym\left\{M_{ij}^1{\Lambda }_1+M_{ij}^2{\Lambda }_2\right\},\hfill \\ \hfill {\delta }_2\left(t\right)=& {\phi }_2\left(M_{ij}^3{\mathcal{D}}_2^{-1}{M}_{ij}^{3T}+{\displaystyle \frac{1}{3}}{M}_{ij}^4{\mathcal{D}}_2^{-1}{M}_{ij}^{4T}\right)+Sym\left\{M_{ij}^3{\Lambda }_3+M_{ij}^4{\Lambda }_4\right\}.\hfill \end{array}$$

For any matrices ${\Lambda }_1,{\Lambda }_2,{\Lambda }_3$, and ${\Lambda }_4$, the following equation is given:

$$\begin{array}{cc}\hfill & 2\sum _{i=1}^r\sum _{j=1}^r{\kappa }_i\left(\delta \left(t\right)\right){\kappa }_j\left(\delta \left(t_k\right)\right)\left[x^T\left(t\right){\Lambda }_1+x^T(t-\varrho ){\Lambda }_2+x^T\left(t_k\right){\Lambda }_3+{\dot{x}}^T\left(t\right){\Lambda }_4\right]\hfill \\ \hfill & \times \left[-\dot{x}\left(t\right)+{\mathcal{A}}_{i}x\left(t\right)+{\mathcal{A}}_{di}x(t-\varrho )+{\mathcal{B}}_i{K}_{pj}x\left(t_k\right)+{\mathcal{B}}_i{K}_{p\varrho j}x\left(t_k-\varrho \right)\right]=0.\hfill \end{array}$$

(26)

To sum up, we obtain

$$\begin{array}{c}\hfill \dot{V}\left(t\right)\le \sum _{i=1}^r\sum _{j=1}^r{\kappa }_i\left(\delta \left(t\right)\right){\kappa }_j\delta \left(t_k\right){\zeta }^T\left(t\right){\tilde{\Omega }}_{ij}\zeta \left(t\right),\end{array}$$

(27)

where

$$\begin{array}{cc}& {\tilde{\Omega }}_{ij}={\Omega }_{ij}^1+{\phi }_2{\Omega }_{ij}^2+{\phi }_1{\Omega }_{ij}^3,\hfill \\ & {\Omega }_{ij}^2={\Omega }_{ij}^{21}+M_{ij}^3{\mathcal{D}}_2^{-1}{M}_{ij}^{3T}+{\displaystyle \frac{1}{3}}{M}_{ij}^4{\mathcal{D}}_2^{-1}{M}_{ij}^{4T},\hfill \\ & {\Omega }_{ij}^3={\Omega }_{ij}^{31}+M_{ij}^1{\mathcal{D}}_1^{-1}{M}_{ij}^{1T}+{\displaystyle \frac{1}{3}}{M}_{ij}^2{\mathcal{D}}_1^{-1}{M}_{ij}^{2T}.\hfill \end{array}$$

By a simple calculation, we have

$$\begin{array}{c}\hfill {\tilde{\Omega }}_{ij}={\displaystyle \frac{{\phi }_2}{{\upsilon }_k}}\left({\Omega }_{ij}^1+{\upsilon }_k{\Omega }_{ij}^2\right)+{\displaystyle \frac{{\phi }_1}{{\upsilon }_k}}\left({\Omega }_{ij}^1+{\upsilon }_k{\Omega }_{ij}^3\right).\end{array}$$

(28)

Therefore, $\dot{V}\left(t\right)<0$ if conditions (14) and (15) hold. □

Remark 2.

Compared to the Lyapunov–Krasovskii functionals constructed in [16,17,18,19,20,23,24,25,26,31], our approach leverages the membership function information as proposed in Lemma 3 to construct $V_1\left(t\right)$. Although $V_1\left(t\right)$ introduces additional constraints, it allows for a more comprehensive utilization of the membership function information. This, in turn, aids in deriving a stability criterion with reduced conservatism. The enhanced utilization of membership function data in our methodology demonstrates a significant improvement in capturing the dynamic characteristics of the system, thereby facilitating a less conservative stability analysis.

Remark 3.

Inspired by the looped functional in [12,13,31,32,33], new kinds of looped functionals $V_2\left(t\right)$ and $V_3\left(t\right)$ are constructed. The main advantages lie in the following two aspects: Firstly, the enhanced looped Lyapunov–Krasovskii functional provides additional details, not only concerning the states $x\left(t_k-\varrho \right)$ and $x\left(t_{k+1}-\varrho \right)$ but also the delay-dependent states $x\left(t_k-\varrho \right)$ and $x\left(t_{k+1}-\varrho \right)$. Secondly, unlike the looped functional in [33,34,35,36,37,38], we introduce the sampling-dependent matrix function ${\mathcal{U}}_1=\chi U_1^1+(1-\chi )U_1^2$, ${\mathcal{U}}_2=\chi U_2^1+(1-\chi )U_2^2$, $\mathcal{X}=\chi X^1+(1-\chi )X^2$, $\mathcal{Z}=\chi Z^1+(1-\chi )Z^2$, ${\mathcal{D}}_1=\chi D_1^1+(1-\chi )D_1^2$ to connect the augmentation vectors, and if $U_1^1=U_1^2$, $U_2^1=U_2^2$, $X^1=X^2$, $Z^1=Z^2$, $D_1^1=D_1^2$, $D_2^1=D_2^2$, then the functional simplifies to a conventional looped form. This simplification highlights the enhanced benefits of our proposed enhanced looped Lyapunov functional, which offers a stability criterion that is less conservative.

Remark 4.

In Theorem 1, alongside the incorporation of an advanced looped Lyapunov functional, it is crucial to emphasize the inclusion of additional free matrices. This enhancement stems from the integration of membership-function-based terms, which play a pivotal role in accurately estimating the integral component. As a result, this methodology provides a greater degree of freedom, ultimately yielding a more refined and less conservative stability criterion for T-S fuzzy systems. This advancement not only strengthens the theoretical foundation but also broadens the applicability of the stability results, facilitating more effective control design in complex systems.

3.2. Controller Scheme

In this subsection, we give the controller design scheme.

Theorem 2.

The system (6) is asymptotic stable, given constants $h,\varrho$, and ${\Im }_v(v=1,2,3)$, if there exist positive-definite matrices ${\tilde{P}}_i,{\tilde{S}}_i,{\tilde{H}}_i$, ${\tilde{D}}_1^{\ell }$, ${\tilde{D}}_2^{\ell }$ ($\ell =1,2$), symmetrical matrices ${\tilde{U}}_1^{\ell },{\tilde{U}}_2^{\ell }$, ${\tilde{Z}}^{\ell }$, and any matrices Λ, ${\tilde{K}}_{pj}$ and ${\tilde{K}}_{p\varrho j}\left(p=1,\dots ,2^{r-1}\right)$, ${\tilde{X}}^{\ell }$, ${\tilde{M}}_{ij}^{\iota }$, such that

$$\begin{array}{cc}\hfill & {\dot{\tilde{\mathcal{P}}}}_{\kappa }\le 0,{\dot{\tilde{\mathcal{S}}}}_{\kappa }\le 0,{\dot{\tilde{\mathcal{H}}}}_{\kappa }\le 0,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\tilde{\Omega }}_{ij}^0+h{\tilde{\Omega }}_{ij}^{21}& h{\tilde{M}}_{ij}^3& h{\tilde{M}}_{ij}^4\\ \ast & -h{\tilde{\mathcal{U}}}_2^1& 0\\ \ast & 0& -3h{\tilde{\mathcal{U}}}_2^1\end{array}\right]<0,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\tilde{\Omega }}_{ij}^0+h{\tilde{\Omega }}_{ij}^{31}& h{\tilde{M}}_{ij}^1& h{\tilde{M}}_{ij}^2\\ \ast & -h{\tilde{\mathcal{U}}}_1^2& 0\\ \ast & 0& -3h{\tilde{\mathcal{U}}}_1^2\end{array}\right]<0,\hfill \end{array}$$

(31)

where

$$\begin{array}{cc}& {\tilde{\Omega }}_{ij}^1=Sym\left\{{\Phi }_1^T{\tilde{P}}_i{\Phi }_2+{\Phi }_8^T{\tilde{\mathcal{U}}}_1{\Phi }_6+{\Phi }_8^T{\tilde{\mathcal{U}}}_2{\Phi }_7+{\Phi }_{10}^T\tilde{\mathcal{X}}{\Phi }_4^b-{\Phi }_4^{a^T}\tilde{\mathcal{X}}{\Phi }_{10}+{\tilde{M}}_{ij}^1{\Lambda }_1\right.\hfill \\ & \left.+{\tilde{M}}_{ij}^2{\Lambda }_2+{\tilde{M}}_{ij}^3{\Lambda }_3+{\tilde{M}}_{ij}^4{\Lambda }_4\right\}+{\Phi }_3^T{\tilde{S}}_i{\Phi }_3-{\Phi }_4^T{\tilde{S}}_i{\Phi }_4+{\varrho }^2{\sigma }_7^T{\tilde{H}}_i{\sigma }_7-{\Phi }_0^T{\tilde{H}}_i{\Phi }_0\hfill \\ & -3{\Phi }_0^a{\tilde{H}}_i{\Phi }_0^a+{\tilde{\kappa }}_{ij},\hfill \\ & {\tilde{\Omega }}_{ij}^{21}=Sym\left\{{\Phi }_5^{a^T}{\tilde{\mathcal{U}}}_1{\Phi }_6+{\Phi }_5^{a^T}{\tilde{\mathcal{U}}}_2{\Phi }_7+{\Phi }_8^a{\tilde{\mathcal{U}}}_1{\Phi }_9+{\Phi }_{12}^{a^T}\tilde{\mathcal{Z}}{\Phi }_{11}\right\}\hfill \end{array}$$

$$\begin{array}{cc}& +{\Phi }_{11}^T\tilde{\mathcal{Z}}{\Phi }_{11}+{\sigma }_7^T{\tilde{\mathcal{D}}}_1{\sigma }_7,\hfill \\ & {\tilde{\Omega }}_{ij}^{31}=Sym\left\{{\Phi }_5^{b^T}{\tilde{\mathcal{U}}}_1{\Phi }_6+{\Phi }_5^{b^T}{\tilde{\mathcal{U}}}_2{\Phi }_7+{\Phi }_8^{b^T}{\tilde{\mathcal{U}}}_1{\Phi }_9+{\Phi }_{12}^b\tilde{\mathcal{Z}}{\Phi }_{11}\right\}\hfill \\ & -{\Phi }_{11}^T\tilde{\mathcal{Z}}{\Phi }_{11}+{\sigma }_7^T{\tilde{\mathcal{D}}}_2{\sigma }_7,\hfill \\ & {\tilde{\kappa }}_{ij}=Sym\left\{{\tilde{\kappa }}_0\times \left(-{\Lambda }^T{\sigma }_7+{\mathcal{A}}_i{\Lambda }^T{\sigma }_1+{\mathcal{A}}_{di}{\Lambda }^T{\sigma }_2+{\mathcal{B}}_i{\overline{K}}_j{\sigma }_3+{\mathcal{B}}_i{\overline{K}}_{\varrho j}{\sigma }_5\right)\right\},\hfill \\ & {\tilde{\kappa }}_0={\sigma }_1^T+{\Im }_1{\sigma }_2^T+{\Im }_2{\sigma }_3^T+{\Im }_3{\sigma }_7^T.\hfill \end{array}$$

Moreover, the sampled-data control gains are $K_{pj}={\tilde{K}}_{pj}{\left({\Lambda }^T\right)}^{-1}$ and $K_{p\varrho j}={\tilde{K}}_{p\varrho j}{\left({\Lambda }^T\right)}^{-1}$.

Proof of Theorem 2.

Define

$$\begin{array}{cc}& {\Lambda }_1={\Lambda }^{-1},{\Lambda }_2={\Im }_1{\Lambda }^{-1},{\Lambda }_3={\Im }_2{\Lambda }^{-1},{\Lambda }_4={\Im }_3{\Lambda }^{-1},K_j={\tilde{K}}_j{\left({\Lambda }^T\right)}^{-1},K_{\varrho j}={\tilde{K}}_{\varrho j}{\left({\Lambda }^T\right)}^{-1},\hfill \\ & {\tilde{\mathcal{P}}}_{\kappa }=diag\{\Lambda ,\Lambda ,\Lambda \}{\mathcal{P}}_{\kappa }diag{\{\Lambda ,\Lambda ,\Lambda \}}^T,\hfill \\ & {\tilde{\mathcal{S}}}_h=diag\{\Lambda ,\Lambda \}{\mathcal{S}}_{\kappa }diag{\{\Lambda ,\Lambda \}}^T,{\tilde{\mathcal{H}}}_{\kappa }=\Lambda {\mathcal{H}}_{\kappa }{\Lambda }^T,{\tilde{\mathcal{D}}}_1=\Lambda {\mathcal{D}}_1{\Lambda }^T,{\tilde{\mathcal{D}}}_2=\Lambda {\mathcal{D}}_2{\Lambda }^T,\hfill \\ & {\tilde{\mathcal{U}}}_1=diag\{\Lambda ,\Lambda ,\Lambda ,\Lambda \}{\mathcal{U}}_{1}diag{\{\Lambda ,\Lambda ,\Lambda ,\Lambda \}}^T,\hfill \\ & {\tilde{\mathcal{U}}}_2=diag\{\Lambda ,\Lambda ,\Lambda ,\Lambda \}{\mathcal{U}}_{2}diag{\{\Lambda ,\Lambda ,\Lambda ,\Lambda \}}^T,\hfill \\ & \tilde{\mathcal{X}}=diag\{\Lambda ,\Lambda \}\mathcal{X}diag\left\{{\Lambda }^T,{\Lambda }^T\right\},\hfill \\ & \tilde{\mathcal{Z}}=diag\{\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda \}\mathcal{Z}diag{\{\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda \}}^T,\hfill \\ & {\tilde{M}}_{ij}^{\iota }=Q_1^T{M}_{ij}^{\iota }{\Lambda }^T,\hfill \\ & Q_1=diag\{\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda ,\Lambda \},Q_2=diag\left\{Q_1,I\right\},Q_3=diag\left\{Q_1,\Lambda ,\Lambda ,I\right\}.\hfill \end{array}$$

By pre- and postmultiplying (14) and (15) by $Q_3$ and $Q_3^T$, we can easily derive the inequalities (30) and (31). The proof is completed. □

Remark 5.

In Theorems 1 and 2, asymptotic stability depends on the existence of matrices that satisfy specific linear matrix inequality (LMI) conditions. These matrices can be computed using MATLAB’s (2023a) LMI toolbox. Moreover, to maintain the positivity of the membership function matrix and ensure its derivatives are negative, we introduce a switching mechanism based on Lemma 3 in Theorem 2. The largest sampling interval is defined according to Theorem 2 as $h={min}_{1\le p\le 2^{r-1}}\left\{h_p\right\}$. For instance, when ${\dot{\kappa }}_j\left(\delta \left(t\right)\right)<0$, the controller changes to $K_{1j}$ and $K_{1\varrho j}$. Conversely, when ${\dot{\kappa }}_j\left(\delta \left(t\right)\right)\ge 0$, it switches to $K_{2j}$ and $K_{2\varrho j}$. Therefore, the ultimate maximum sampling interval can be expressed as $h=min\left\{h_1,h_2\right\}$. Furthermore, adjusting the gain of the switching controller enables the controller to stabilize the system outlined in Equation (6).

To further show the superiority of the developed approach, we let ${\mathcal{A}}_{di}=0,\varrho =0$, and $K_{\varrho j}=0$, and the system (6) reduces

$$\begin{array}{c}\hfill \dot{x}\left(t\right)=\sum _{i=1}^r\sum _{j=1}^r{\kappa }_i\left(\delta \left(t\right)\right){\kappa }_j\left(\delta \left(t_k\right)\right)\left({\mathcal{A}}_{i}x\left(t\right)+{\mathcal{B}}_i{K}_{j}x\left(t_k\right)\right).\end{array}$$

(32)

Hence, we derive the following corollary.

Corollary 1.

The system (32) is asymptotic stable, given constants h, ϱ, and ${\Im }_v(v=1,2)$, if there exist positive-definite matrices ${\overline{P}}_i$, $bar{D}_2^{\ell }$ ($\ell =1,2$), symmetrical matrices ${\overline{U}}_1^{\ell },{\overline{U}}_2^{\ell }$, ${\overline{Z}}^{\ell }$, and any matrices Λ, ${\overline{K}}_{pj}$$\left(p=1,\dots ,2^{r-1}\right)$, ${\overline{X}}^{\ell }$, ${\overline{M}}_{ij}^{\iota }$, such that

$$\begin{array}{cc}\hfill & {\dot{\overline{\mathcal{P}}}}_{\kappa }\le 0,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\overline{\Omega }}_{ij}^0+h{\overline{\Omega }}_{ij}^{21}& h{\overline{M}}_{ij}^3& h{\overline{M}}_{ij}^4\\ \ast & -h{\overline{\mathcal{U}}}_2^1& 0\\ \ast & 0& -3h{\overline{\mathcal{U}}}_2^1\end{array}\right]<0,\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & \left[\begin{array}{ccc}{\overline{\Omega }}_{ij}^0+h{\overline{\Omega }}_{ij}^{31}& h{\overline{M}}_{ij}^1& h{\overline{M}}_{ij}^2\\ \ast & -h{\overline{\mathcal{U}}}_1^2& 0\\ \ast & 0& -3h{\overline{\mathcal{U}}}_1^2\end{array}\right]<0,\hfill \end{array}$$

(35)

where

$$\begin{array}{cc}& {\overline{\Omega }}_{ij}^1=Sym\left\{{\overline{\sigma }}_1^T{\overline{P}}_i{\overline{\sigma }}_4+{\overline{\Phi }}_8^T{\overline{\mathcal{U}}}_1{\overline{\Phi }}_6+{\overline{\Phi }}_8^T{\overline{\mathcal{U}}}_2{\overline{\Phi }}_7+{\overline{\Phi }}_{10}^T\overline{\mathcal{X}}{\overline{\Phi }}_4^b-{\overline{\Phi }}_4^{a^T}\overline{\mathcal{X}}{\overline{\Phi }}_{10}+{\overline{M}}_{ij}^1{\Lambda }_1\right.\hfill \\ & \left.+{\overline{M}}_{ij}^2{\Lambda }_2+{\overline{M}}_{ij}^3{\Lambda }_3+{\overline{M}}_{ij}^4{\Lambda }_4\right\}+{\overline{\kappa }}_{ij},\hfill \\ & {\overline{\Omega }}_{ij}^{21}=Sym\left\{{\overline{\Phi }}_5^{a^T}{\overline{\mathcal{U}}}_1{\overline{\Phi }}_6+{\overline{\Phi }}_5^{a^T}{\overline{\mathcal{U}}}_2{\overline{\Phi }}_7+{\overline{\Phi }}_8^a{\overline{\mathcal{U}}}_1{\overline{\Phi }}_9+{\overline{\Phi }}_{12}^{a^T}\overline{\mathcal{Z}}{\overline{\Phi }}_{11}\right\}\hfill \\ & +{\overline{\Phi }}_{11}^T\overline{\mathcal{Z}}{\overline{\Phi }}_{11}+{\sigma }_7^T{\overline{\mathcal{D}}}_1{\sigma }_7,\hfill \\ & {\overline{\Omega }}_{ij}^{31}=Sym\left\{{\overline{\Phi }}_5^{b^T}{\overline{\mathcal{U}}}_1{\overline{\Phi }}_6+{\overline{\Phi }}_5^{b^T}{\overline{\mathcal{U}}}_2{\overline{\Phi }}_7+{\overline{\Phi }}_8^{b^T}{\overline{\mathcal{U}}}_1{\overline{\Phi }}_9+{\overline{\Phi }}_{12}^b\overline{\mathcal{Z}}{\overline{\Phi }}_{11}\right\}\hfill \\ & -{\overline{\Phi }}_{11}^T\overline{\mathcal{Z}}{\overline{\Phi }}_{11}+{\sigma }_7^T{\overline{\mathcal{D}}}_2{\sigma }_7,\hfill \\ & {\overline{\kappa }}_{ij}=Sym\left\{{\overline{\kappa }}_0\left(-{\overline{\Lambda }}^T{\overline{\sigma }}_4+{\mathcal{A}}_i{\overline{\Lambda }}^T{\overline{\sigma }}_1+{\mathcal{B}}_i{\overline{K}}_{pj}{\overline{\sigma }}_2\right)\right\}\hfill \\ & {\overline{\kappa }}_0={\overline{\sigma }}_1^T+{\Im }_1{\overline{\sigma }}_2^T+{\Im }_2{\overline{\sigma }}_4^T,{\overline{\Phi }}_4^a={\overline{\sigma }}_1-{\overline{\sigma }}_2,{\overline{\Phi }}_4^b={\overline{\sigma }}_3-{\overline{\sigma }}_1,\hfill \\ & {\overline{\Phi }}_5^a=col\left\{{\overline{\sigma }}_4,0\right\},{\Phi }_5^b=col\left\{0,-{\overline{\sigma }}_4\right\},{\overline{\Phi }}_6=col\left\{{\overline{\sigma }}_2-{\overline{\sigma }}_3,{\overline{\sigma }}_4-{\overline{\sigma }}_2\right\},\hfill \\ & {\overline{\Phi }}_7=col\left\{{\overline{\sigma }}_3,{\overline{\sigma }}_4\right\},{\overline{\Phi }}_8=col\left\{{\overline{\sigma }}_3-{\overline{\sigma }}_2,{\overline{\sigma }}_4-{\overline{\sigma }}_2\right\},{\overline{\Phi }}_8^a=col\left\{{\overline{\sigma }}_3-{\overline{\sigma }}_3,0\right\},\hfill \\ & {\overline{\Phi }}_8^b=col\left\{0,{\overline{\sigma }}_4-{\overline{\sigma }}_2\right\},{\overline{\Phi }}_{11}=col\left\{{\overline{\sigma }}_3,{\overline{\sigma }}_4,{\overline{\sigma }}_6,{\overline{\sigma }}_7\right\},\hfill \\ & {\overline{\Phi }}_9=col\left\{{\overline{\sigma }}_5,-{\overline{\sigma }}_5\right\},{\overline{\Phi }}_{12}^a=col\left\{0,0,{\overline{\sigma }}_1-{\overline{\sigma }}_5,0\right\},\hfill \\ & {\overline{\Phi }}_{12}^b=col\left\{0,0,0,{\overline{\sigma }}_6-{\overline{\sigma }}_1\right\},{\overline{\Lambda }}_1={\overline{\sigma }}_1-{\overline{\sigma }}_2,{\overline{\Lambda }}_2={\overline{\sigma }}_1+{\overline{\sigma }}_2-2{\overline{\sigma }}_5,\hfill \\ & {\overline{\Lambda }}_3={\overline{\sigma }}_3-{\overline{\sigma }}_1,{\overline{\Lambda }}_4={\overline{\sigma }}_3+{\overline{\sigma }}_1-2{\overline{\sigma }}_6.\hfill \end{array}$$

Moreover, the control gains for the system are presented as follows $K_{pj}={\overline{K}}_{pj}{\left({\Lambda }^T\right)}^{-1}$.

Proof of Corollary 1.

By letting ${\mathcal{S}}_{\kappa }=0$, ${\mathcal{H}}_{\kappa }=0$ and the augmentation vector $\zeta \left(t\right)$ become $\overline{\zeta }\left(t\right)=col\left\{x\left(t\right),x\left(t_k\right),x\left(t_{k+1}\right),\dot{x}\left(t\right),{\eta }_1\left(t\right),{\eta }_2\left(t\right)\right\}$, the proof of Corollary 1 can be easily derived from Theorems 1 and 2. □

4. Simulation Case Study

This section utilizes two examples to show the effectiveness of the suggested approach.

Example 1.

Consider the following nonlinear system [19,38]:

$$\left\{\begin{array}{c}{\dot{x}}_1\left(t\right)=-\wp {\displaystyle \frac{v\hslash }{W{t}_0}}{x}_1\left(t\right)-(1-\wp ){\displaystyle \frac{v\hslash }{W{t}_0}}{x}_1(t-\varrho )+{\displaystyle \frac{v\hslash }{ϵt_0}}u\left(t\right),\hfill \\ {\dot{x}}_2\left(t\right)=\wp {\displaystyle \frac{v\hslash }{W{t}_0}}{x}_1\left(t\right)+(1-\wp ){\displaystyle \frac{v\hslash }{W{t}_0}}{x}_1(t-\varrho ),\hfill \\ {\dot{x}}_3\left(t\right)={\displaystyle \frac{v\hslash }{W{t}_0}}sin\left(x_2\left(t\right)+{\displaystyle \frac{\wp v\hslash }{2W}}{x}_1\left(t\right)+(1-\wp ){\displaystyle \frac{v\hslash }{2W}}{x}_1(t-\varrho )\right).\hfill \end{array}\right.$$

Let $ϵ=2.8,W=5.5,\wp =0.7,v=-1,\hslash =2.0$, and $t_0=0.5$. Then, the system parameters are given by

$$\begin{array}{cc}& {\mathcal{A}}_1=\left[\begin{array}{ccc}-\wp {\displaystyle \frac{v\hslash }{W{t}_0}}& 0& 0\\ \wp {\displaystyle \frac{v\hslash }{W{t}_0}}& 0& 0\\ \wp {\displaystyle \frac{v^2{t}^{*2}}{2W{t}_0}}& {\displaystyle \frac{v\hslash }{t_0}}& 0\end{array}\right],{\mathcal{A}}_{d1}=\left[\begin{array}{ccc}-(1-\wp ){\displaystyle \frac{v\hslash }{W{t}_0}}& 0& 0\\ (1-\wp ){\displaystyle \frac{v\hslash }{W{t}_0}}& 0& 0\\ (1-\wp ){\displaystyle \frac{v^2{t}^{*2}}{2W{t}_0}}& 0& 0\end{array}\right],\hfill \\ & {\mathcal{A}}_2=\left[\begin{array}{ccc}-\wp {\displaystyle \frac{v\hslash }{W{t}_0}}& 0& 0\\ \wp {\displaystyle \frac{v\hslash }{W{t}_0}}& 0& 0\\ \wp {\displaystyle \frac{v^2{t}^{*2}}{2W{t}_0}}& {\displaystyle \frac{dv\hslash }{t_0}}& 0\end{array}\right],{\mathcal{A}}_{d2}=\left[\begin{array}{ccc}-(1-\wp ){\displaystyle \frac{v\hslash }{W{t}_0}}& 0& 0\\ (1-\wp ){\displaystyle \frac{v\hslash }{W{t}_0}}& 0& 0\\ (1-\wp ){\displaystyle \frac{d{v}^2{t}^{*2}}{2W{t}_0}}& 0& 0\end{array}\right],\hfill \\ & {\mathcal{B}}_1={\mathcal{B}}_2={\left[{\displaystyle \frac{v\hslash }{ϵt_0}},0,0\right]}^T.\hfill \end{array}$$

where $d=10t_0/\pi$. Moreover, the membership functions are given as follows:

$${\kappa }_1\left(t\right)={\displaystyle \frac{1}{(1+\left.exp\left(x_1\left(t\right)\right)+0.5\right)}},{\kappa }_2\left(t\right)=1-{\kappa }_1\left(t\right).$$

We selected $\varrho =0.17,K_{\varrho j}=0,{\Im }_1=0.75,{\Im }_2=2$, and ${\Im }_3=0.5$. The maximum sampling intervals (MSIs) obtained using [19,38] and Theorem 2 are listed in

Table 1. The MSI h for various approaches.

Theorem 2 [19]	Theorem 2 [38]	Theorem 2
0.1820	0.3339	0.4511

. From Table 1, it is evident that the stability criterion derived using the proposed method is significantly less conservative, highlighting the merit of the proposed method. Furthermore, the controller gain derived from Theorem 2 and verified through double-command simulations in MATLAB is as follows:

$$\begin{array}{cc}& K_{11}=K_{12}=\left[\begin{array}{ccc}3.2761\hfill & -2.1157\hfill & 0.5755\hfill \end{array}\right],\hfill \\ & K_{21}=K_{22}=\left[\begin{array}{ccc}4.1754\hfill & -1.9943\hfill & 0.3219\hfill \end{array}\right].\hfill \end{array}$$

With the initial condition set as $x\left(0\right)={[4,1,-6]}^T$, the state response curves and control inputs are shown in Figure 1 and Figure 2. As indicated by Figure 1, the system attains converge, affirming the efficacy of both the controller and the methodology introduced in this paper.

Example 2.

Consider the following Rosser’s system [41]:

$$\left\{\begin{array}{c}{\dot{x}}_1\left(t\right)=-x_2\left(t\right)-x_3\left(t\right),\hfill \\ {\dot{x}}_2\left(t\right)=x_1\left(t\right)+J_1{x}_2\left(t\right),\hfill \\ {\dot{x}}_3\left(t\right)=J_2{x}_1\left(t\right)-\left(\gamma -x_1\left(t\right)\right)x_3\left(t\right)+u\left(t\right).\hfill \end{array}\right.$$

Then, with $x_1\left(t\right)\in [\gamma -d,\gamma +d]$, the system matrices are given as

$$\begin{array}{cc}& {\mathcal{A}}_1=\left[\begin{array}{ccc}0& -1& -1\\ 1& J_1& 0\\ J_2& 0& -d\end{array}\right],{\mathcal{A}}_2=\left[\begin{array}{ccc}0& -1& -1\\ 1& J_1& 0\\ J_2& 0& d\end{array}\right],\hfill \\ & {\mathcal{B}}_1={\mathcal{B}}_2={\left[\begin{array}{ccc}0\hfill & 0\hfill & 1\hfill \end{array}\right]}^T,\hfill \end{array}$$

where $J_1=0.3,J_2=0.5,\gamma =5$, and $d=10$. Moreover, the membership functions are given as follows:

$${\kappa }_1\left(t\right)={\displaystyle \frac{\gamma +d-x_1\left(t\right)}{2d}},{\kappa }_2\left(t\right)=1-{\kappa }_1\left(t\right)$$

The MSIs determined using [18,19,21,24] and Corollary 1 are presented in

Table 2. The MSI h for various approaches.

Corollary 1 [18]	Corollary 1 [19]	Theorem 3 [24]	Corollary 1 [21]	Corollary 1
0.0959	0.1147	0.1173	0.1292	0.1733

. Based on the data in Table 2, it can be concluded that the MSIs derived from the method proposed in this paper are significantly larger, underscoring the superiority of our approach. Additionally, the controller gain derived from Corollary 1 and verified through double-command simulations in MATLAB is as follows:

$$\begin{array}{cc}& K_{11}=K_{12}=\left[\begin{array}{ccc}11.1122\hfill & 1.5575\hfill & -8.2936\hfill \end{array}\right],\hfill \\ & K_{21}=K_{22}=\left[\begin{array}{ccc}10.7566\hfill & 1.6673\hfill & -8.5529\hfill \end{array}\right].\hfill \end{array}$$

With the initial condition set as $x\left(0\right)={[-7,-1,6]}^T$, Figure 3 and Figure 4 display the state response curves and control inputs. As illustrated in Figure 3, the system achieves converge.

5. Conclusions

A novel approach to sampled-data control for T-S fuzzy systems with time delay has been presented in this paper. A negative-definite criterion was proposed for the matrix-containing membership functions. Based on this, a membership-function-based looped Lyapunov functional was introduced. Moreover, a sampling-time-dependent looped Lyapunov functional was developed. Then, less conservative stability and controller conditions for the studied system were derived. Finally, through two simulation examples, the effectiveness and superiority of the proposed approach were validated. It is crucial to note that studying communication channel failures and network attacks is very significant for stabilizing T-S fuzzy systems. Therefore, further research on the effects of more complex communication networks on T-S fuzzy systems is necessary in the future.