Improved Delay-Dependent Stability and Stabilization Conditions of T-S Fuzzy Time-Delay Systems via an Augmented Lyapunov-Krasovskii Functional Approach

Abstract

This paper develops some improved stability and stabilization conditions of T-S fuzzy system with constant time-delay and interval time-varying delay with its derivative bounds available, respectively. These conditions are presented by linear matrix inequalities (LMIs) and derived by applying an augmented Lyapunov-Krasovskii functional (LKF) approach combined with a canonical Bessel-Legendre (B-L) inequality. Different from the existing LKFs, the proposed LKF involves more state variables in an augmented way resorting to the form of the B-L inequality. The B-L inequality is also applied in ensuring the positiveness of the constructed LKF and the negativeness of derivative of the LKF. By numerical examples, it is verified that the obtained stability conditions can ensure a larger upper bound of time-delay, the larger number of Legendre polynomials in the stability conditions can lead to less conservative results, and the stabilization condition is effective, respectively.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

T-S fuzzy time-delay systems are well-recognized by the integration of time-delay systems and fuzzy systems which are often employed to model several nonlinear systems in practice [1]. Such systems have two appealing advantages: T-S fuzzy system is capable of modeling the nonlinear systems [2] and the unavoidable time-delay phenomenon is explicitly incorporated in the system [1]. When both nonlinearity and time-delay phenomena are considered, T-S fuzzy system with time-delay indeed offers a feasible system representation. For such a system, much work has been done in the past few decades to achieve stability and performance conditions, delay-dependent ones in particular [3–8]. The presence of time-delay in the fuzzy system can be either constant or time-varying delay, and systems with time-varying delay are more difficult to be handled than constant time-delay case. In the light of flourishing research on linear time-delay systems, more and more interesting stability results have been published recently for T-S fuzzy systems with time-varying delays [3–8]. Since these delay-dependent stability and performance conditions are only sufficient conditions, the admissible maximum upper bound of time-delay computed by the conditions is commonly treated as an essential index to evaluate the conservatism of the conditions. Thus, a primary purpose of delay-dependent stability conditions is to search for the admissible maximum upper bound of time-delay as large as possible while ensuring the system stability and performance.

Recalling the existing delay-dependent stability results on T-S fuzzy time-delay systems, one can see that a LKF approach is prevalent and well-studied. The basic idea is to derive the condition by estimating the time-derivative of the constructed LKF which satisfies the conditions of the LKF theorem in [9]. Thus, the construction of the LKF and the estimation method of its derivative play a key role in developing less conservative stability conditions. In the derivative operation process of the LKF, various integral inequalities are established to produce an estimation as tight as possible, such as Jensen-type inequality [1, 3, 4], Wirtinger-type inequality [5, 10], B-L inequality [11–14], and reciprocally convex inequality with free weighting matrices [7, 15, 16]. Moreover, different types of LKFs such as fuzzy weighting-dependent LKFs [17, 18] and augmented LKFs [16, 19] are proposed for T-S fuzzy system. Recently, there are increasing works on exploring the connection or relationship between the process of integral inequality and the construction of the LKF [20], and they confirm that the augmented LKF approach can be considered as a competitive method [19] and the B-L inequality can cover Jensen-type inequality and Wirtinger-type inequality as special cases [11–13]. Accordingly, it is expected to develop some less conservative results for T-S fuzzy system with time-delay by updating the augmented LKF approach together with the B-L inequality, which mainly inspires the current research. More recently, for linear time-delay systems, the augmented LKF together with B-L inequalities are introduced to develop stability results [11–14]. However, this method in [11–14] is not presented for stabilization problem of T-S fuzzy systems with time-delay. More importantly, the augmented LKF in [13, 14, 16, 19] requires all Lyapunov matrix variables to be positive and the delay-induced convex combination in the augmented LKF is not sufficiently used, which leaves us much room for improvement.

In summary, the contributions of this paper are mainly in two aspects as follows.

(i)

Augmented LKFs based on the form of B-L inequality are proposed to achieve less conservative stability conditions of T-S fuzzy systems with constant or time-varying delay. In particular, for the time-varying delay case, free-weighting matrices are technically introduced in the process of the positive definiteness of the LKF and the negative definiteness of its derivative resorting to a reciprocally convex method.

(ii)

Both stability and stabilization conditions of T-S fuzzy systems with constant or time-varying delay are expressed by tractable LMIs. The advantages of the proposed stability conditions over than those in the existing literature and the design validity are shown in the examples.

Notation. ${\mathbb{R}}^n$ is the $n$-dimensional Euclidean space. ${\mathbb{S}}^n$ (respectively, ${\mathbb{S}}_{+}^n$) denotes a set of real symmetric (respectively, positive definite) matrices with $n\times n$ dimensional. For a square matrix $X$, $\mathrm{S}\mathrm{y}\mathrm{m}\{X\}$ means the sum of $X$ and $X^T$, i.e., $\mathrm{S}\mathrm{y}\mathrm{m}\{X\}=X+X^T$. For two positive integers $i,j\hspace{0.17em}\hspace{0.17em}(i\ge j)$, we define $\left(\begin{array}{c}i\\ j\end{array}\right)=i!/[(i-j)!j!]$. We use $col\{\dots \}$ to represent a column vector and $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{\dots \}$ to denote the block-diagonal matrix. The symmetric term in a symmetric matrix is denoted by “$\star$”. $Co\{q_{\mathrm{1}},q_{\mathrm{2}}\}$ is a polytope generated by two vertices $q_{\mathrm{1}}$ and $q_{\mathrm{2}}$.

2. System Description and Preliminaries

Consider the time-delay system with fuzzy rules as follows.

$Rule\hspace{0.17em}\hspace{0.17em}l$ . IF ${\delta }_{\mathrm{1}}(t)$ is $J_{l\mathrm{1}}$, ${\delta }_{\mathrm{2}}(t)$ is $J_{l\mathrm{2}},\cdots$, and ${\delta }_r(t)$ is $J_{lr}$, THEN$$\begin{array}{}\text{(1)}& \dot{x}\left(t\right)=A_{l\mathrm{0}}x\left(t\right)+A_{l\mathrm{1}}x\left(t-d\left(t\right)\right)+B_{l}u\left(t\right),\end{array}$$where $l\in \mathbb{T}\triangleq \{\mathrm{1,2},\dots ,s\}$, and ${\delta }_{\mathrm{1}}(t)$, ${\delta }_{\mathrm{2}}(t),\dots ,{\delta }_r(t)$ are the premise variables; $N_{l\mathrm{1}}$, $N_{l\mathrm{2}}$, $\dots$, $N_{lr}$ are the fuzzy sets; $x(t)\in {\mathbb{R}}^n$ and $u(t)\in {\mathbb{R}}^m$ represent the state vector and control input, respectively; $A_{l\mathrm{0}}$, $A_{l\mathrm{1}}$, and $B_l$ are known system matrices. $d(t)$ is the time-delay that can be either constant (let $d(t)\equiv d_{\mathrm{0}}$) or time-varying. For the time-varying case, it satisfies$$\begin{array}{}\text{(2)}& \left(d\left(t\right),\dot{d}\left(t\right)\right)\in {\mathcal{D}}_{\mathrm{1}}=\left[\mathrm{0},d_{\mathrm{0}}\right]\times \left[d_m,d_M\right].\end{array}$$where $d_{\mathrm{0}}$ is the upper bound of time-delay and $d_m$ and $d_M$ are the interval bounds of the derivative of $d(t)$. We define $x(t)=\varphi (t)$ where $t\in [-d_{\mathrm{0}},\mathrm{0}]$ as the initial condition of system (1).

According to the technique of fuzzy blending, system (1) can be inferred as the following overall system:$$\begin{array}{}\text{(3)}& \dot{x}\left(t\right)={\displaystyle \sum }_{l=\mathrm{1}}^s{o}_l\left(\delta \left(t\right)\right)\left[A_{l\mathrm{0}}x\left(t\right)+A_{l\mathrm{1}}x\left(t-d\left(t\right)\right)+B_{l}u\left(t\right)\right],\end{array}$$where $o_l(\delta (t))$ is the normalized membership function; $v_l(\delta (t))={\prod }_{m=\mathrm{1}}^r‍J_{lm}({\delta }_m(t))$ is the membership function of the fuzzy set $\{J_{lm}\}\hspace{0.17em}\hspace{0.17em}(m=\mathrm{1,2},\dots ,r)$, where $J_{lm}({\delta }_m(t))$ is the grade of membership of ${\delta }_m(t)$ in $J_{lm}$ and$$\begin{array}{c}\text{(4)}& o_l\left(\delta \left(t\right)\right)=\frac{v_l\left(\delta \left(t\right)\right)}{{\sum }_{l=\mathrm{1}}^s{v}_l\left(\delta \left(t\right)\right)},{\displaystyle \sum }_{l=\mathrm{1}}^s{o}_l\left(\delta \left(t\right)\right)=\mathrm{1}.\end{array}$$

Consider the following fuzzy controller for $l$th rule.

$Rule\hspace{0.17em}\hspace{0.17em}l$ . IF ${\delta }_{\mathrm{1}}(t)$ is $J_{l\mathrm{1}}$, ${\delta }_{\mathrm{2}}(t)$ is $J_{l\mathrm{2}},\cdots$, and ${\delta }_r(t)$ is $J_{lr}$, THEN$$\begin{array}{}\text{(5)}& u\left(t\right)=K_{l}x\left(t\right),\end{array}$$where $K_l$ ($l\in \mathbb{T}$) is the fuzzy controller gain. The fuzzy controller is given by$$\begin{array}{}\text{(6)}& u\left(t\right)={\displaystyle \sum }_{l=\mathrm{1}}^s{o}_l\left(\delta \left(t\right)\right)K_{l}x\left(t\right).\end{array}$$

Substituting (6) into (3), we obtain the following closed-loop system:$$\begin{array}{}\text{(7)}& \dot{x}\left(t\right)={\displaystyle \sum }_{l=\mathrm{1}}^s\hspace{0.17em}{\displaystyle \sum }_{m=\mathrm{1}}^s{o}_l{o}_m\left[\left(A_{l\mathrm{0}}+B_l{K}_m\right)x\left(t\right)+A_{l\mathrm{1}}x\left(t-d\left(t\right)\right)\right].\end{array}$$

To end this section, we give the following lemmas which will be used in deriving our main result.

Lemma 1 (see [14]).

For a matrix $T\in {\mathbb{S}}_{+}^n$, an integer $N\ge \mathrm{0}$, a matrix $W\in {\mathbb{R}}^{(N+\mathrm{1})n\times (N+\mathrm{1})n}$, two scalars $p$ and $q$ with $q>p$, and a vector function $x:[p,q]\to {\mathbb{R}}^{\mathrm{n}}$ such that the integrations below are well defined; then$$\begin{array}{}\text{(8)}& -{{\displaystyle \int }}_p^q{\dot{x}}^{\mathrm{T}}\left(s\right)T\dot{x}\left(s\right)ds\le {\xi }_N^{\mathrm{T}}\left[{\mathcal{M}}_N^{\mathrm{T}}{\mathcal{L}}_N^{\mathrm{T}}W+W^{\mathrm{T}}{\mathcal{L}}_N{\mathcal{M}}_N+\left(q-p\right)W^{\mathrm{T}}{\mathcal{T}}_N^{-\mathrm{1}}W\right]{\xi }_N,\end{array}$$where$$\begin{array}{c}\text{(9)}& {\mathcal{T}}_N=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{T,\mathrm{3}T,\dots ,\left(\mathrm{2}N+\mathrm{1}\right)T\right\},\text{(10)}& {\mathcal{L}}_N=\left[\begin{array}{cccc}I& \mathrm{0}& \cdots & \mathrm{0}\\ I& {\left(-\mathrm{1}\right)}^{\mathrm{1}}\left(\begin{array}{c}\mathrm{1}\\ \mathrm{1}\end{array}\right)\left(\begin{array}{c}\mathrm{1}+\mathrm{1}\\ \mathrm{1}\end{array}\right)& \cdots & \mathrm{0}\\ ⋮& ⋮& \ddots & ⋮\\ I& {\left(-\mathrm{1}\right)}^{\mathrm{1}}\left(\begin{array}{c}N\\ \mathrm{1}\end{array}\right)\left(\begin{array}{c}N+\mathrm{1}\\ \mathrm{1}\end{array}\right)& \cdots & {\left(-\mathrm{1}\right)}^N\left(\begin{array}{c}N\\ N\end{array}\right)\left(\begin{array}{c}N+N\\ N\end{array}\right)\end{array}\right],\text{(11)}& {\mathcal{M}}_N=\left[\begin{array}{cccccc}I& -I& \mathrm{0}& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& -I& I& \mathrm{0}& \cdots & \mathrm{0}\\ \mathrm{0}& -I& \mathrm{0}& \mathrm{2}I& \cdots & \mathrm{0}\\ ⋮& ⋮& ⋮& ⋮& \ddots & ⋮\\ \mathrm{0}& -I& \mathrm{0}& \mathrm{0}& \cdots & NI\end{array}\right],\text{(12)}& {\xi }_N=col\left\{x\left(q\right),x\left(p\right),{\iota }_{\mathrm{1}},\dots ,{\iota }_N\right\},{\iota }_k={{\displaystyle \int }}_p^q\frac{{\left(q-p\right)}^{k-\mathrm{1}}}{{\left(q-p\right)}^k}x\left(s\right)ds,\hspace{1em}k=\mathrm{1,2},\dots ,N.\end{array}$$

Lemma 2 (see [12]).

For given $Q_{\mathrm{1}}$ and $Q_{\mathrm{2}}$$\in$${\mathbb{S}}_{+}^n$, if one can find matrices $U_{\mathrm{1}},U_{\mathrm{2}}\in {\mathbb{R}}^{n\times n}$ and $H_{\mathrm{1}},H_{\mathrm{2}}\in {\mathbb{S}}^n$ such that$$\begin{array}{}\text{(13)}& \left[\begin{array}{cc}{Q}_{\mathrm{1}}& \mathrm{0}\\ \mathrm{0}& Q_{\mathrm{2}}\end{array}\right]-ϵ\left[\begin{array}{cc}{H}_{\mathrm{1}}& U_{\mathrm{1}}\\ U_{\mathrm{1}}^T& \mathrm{0}\end{array}\right]-\left(\mathrm{1}-ϵ\right)\left[\begin{array}{cc}\mathrm{0}& U_{\mathrm{2}}\\ U_{\mathrm{2}}^T& H_{\mathrm{2}}\end{array}\right]\ge \mathrm{0},\end{array}$$satisfied for $ϵ=\mathrm{0}$ and $\mathrm{1}$, then one has $$\begin{array}{}\text{(14)}& \left[\begin{array}{cc}\frac{\mathrm{1}}{ϵ}{Q}_{\mathrm{1}}& \mathrm{0}\\ \mathrm{0}& \frac{\mathrm{1}}{\mathrm{1}-ϵ}{Q}_{\mathrm{2}}\end{array}\right]\ge \left[\begin{array}{cc}{Q}_{\mathrm{1}}& \mathrm{0}\\ \mathrm{0}& Q_{\mathrm{2}}\end{array}\right]+\left(\mathrm{1}-ϵ\right)\left[\begin{array}{cc}{H}_{\mathrm{1}}& U_{\mathrm{2}}\\ U_{\mathrm{2}}^T& \mathrm{0}\end{array}\right]+ϵ\left[\begin{array}{cc}\mathrm{0}& U_{\mathrm{1}}\\ U_{\mathrm{1}}^T& H_{\mathrm{2}}\end{array}\right],\end{array}$$which holds for any $ϵ\in (\mathrm{0,1})$.

3. Stability Conditions of T-S Fuzzy Time-Delay Systems

In this section, we establish some asymptotic stability conditions of system (6) with two cases of time delay: constant delay and time-varying delay satisfying (2).

3.1. Constant Delay Case

Suppose that $d(t)\equiv d_{\mathrm{0}}>\mathrm{0}$. Choose the following augmented LKF candidate:$$\begin{array}{}\text{(15)}& {\mathcal{V}}_N^c\left(t,x_t\right)=x_{0N}^T\left(t\right)R_{0N}{x}_{0N}\left(t\right)+{{\displaystyle \int }}_{t-d_0}^t{\omega }^T\left(s\right)S_0\omega \left(s\right)ds+d_0{{\displaystyle \int }}_{-d_0}^0{{\displaystyle \int }}_{t+\theta }^t{\dot{x}}^T\left(s\right)T_0\dot{x}\left(s\right)ds\hspace{0.17em}d\theta ,\end{array}$$where$$\begin{array}{c}\text{(16)}& x_{\mathrm{0}N}\left(t\right)=col\left\{x\left(t\right),x\left(t-d_{\mathrm{0}}\right),d_{\mathrm{0}}{\varsigma }_{\mathrm{1}}\left(t\right),\dots ,d_{\mathrm{0}}{\varsigma }_N\left(t\right)\right\},{\varsigma }_k\left(t\right)=\frac{\mathrm{1}}{d_{\mathrm{0}}}{{\displaystyle \int }}_{t-d_{\mathrm{0}}}^t{\left(\frac{t-s}{d_{\mathrm{0}}}\right)}^{k-\mathrm{1}}x\left(s\right)ds,\hspace{1em}k=\mathrm{1},\dots ,N.\\ \omega \left(t\right)=col\left\{\dot{x}\left(t\right),x\left(t\right)\right\},\end{array}$$

and then we can derive the following stability condition.

Theorem 3.

For any given integer $N>\mathrm{0}$ and a positive constant $d_{\mathrm{0}}$, if there exist $R_{\mathrm{0}N}\in {\mathbb{S}}_{+}^{(N+\mathrm{2})n}$, $S_{\mathrm{0}}\in {\mathbb{S}}_{+}^{\mathrm{2}n}$, $T_{\mathrm{0}}\in {\mathbb{S}}_{+}^n$ and $W_{\mathrm{0}N}$ and $M_i\hspace{0.17em}\hspace{0.17em}(i=\mathrm{1,2})$ of appropriate dimensions such that for $l$, $m=\mathrm{1,2},\dots ,s$$$\begin{array}{}\text{(17)}& {\mathrm{\Xi }}_{Nlm}+{\mathrm{\Xi }}_{Nml}<\mathrm{0},\hspace{1em}l\le m\end{array}$$hold, then system (7) with constant delay is asymptotically stable, where $$\begin{array}{c}\text{(18)}& {\mathrm{\Xi }}_{Nlm}=\left[\begin{array}{cc}{\Psi }_{Nlm}& W_{\mathrm{0}N}^T\\ \star & -{\mathcal{T}}_{\mathrm{0}N}\end{array}\right]<\mathrm{0},{\Psi }_{Nlm}=\mathrm{S}\mathrm{y}\mathrm{m}\left\{{\psi }_{\mathrm{1}N}^{\mathrm{T}}{R}_{\mathrm{0}N}{\psi }_{\mathrm{2}N}\right\}+{\psi }_{\mathrm{3}}^{\mathrm{T}}{S}_{\mathrm{0}}{\psi }_{\mathrm{3}}-{\psi }_{\mathrm{4}}^{\mathrm{T}}{S}_{\mathrm{0}}{\psi }_{\mathrm{4}}+{\Lambda }_{\mathrm{0}N}^{\mathrm{T}}{W}_{\mathrm{0}N}+W_{\mathrm{0}N}^{\mathrm{T}}{\Lambda }_{\mathrm{0}N}+d_{\mathrm{0}}^{\mathrm{2}}{g}_{\mathrm{3}}^{\mathrm{T}}{T}_{\mathrm{0}}{g}_{\mathrm{3}}+\mathrm{S}\mathrm{y}\mathrm{m}\left\{\left(g_{\mathrm{1}}^{\mathrm{T}}{M}_{\mathrm{1}}+g_{\mathrm{3}}^{\mathrm{T}}{M}_{\mathrm{2}}\right){\mathcal{E}}_{lm}\right\},\\ {\mathcal{T}}_{\mathrm{0}N}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{T_{\mathrm{0}},\mathrm{3}{T}_{\mathrm{0}},\dots ,\left(\mathrm{2}N+\mathrm{1}\right)T_{\mathrm{0}}\right\},\\ {\mathcal{E}}_{lm}=\left(A_{l\mathrm{0}}+B_l{K}_m\right)g_{\mathrm{1}}+A_{l\mathrm{1}}{g}_{\mathrm{2}}-g_{\mathrm{3}},\\ {\psi }_{\mathrm{1}N}=col\left\{g_{\mathrm{1}},g_{\mathrm{2}},d_{\mathrm{0}}{g}_{\mathrm{5}},\dots ,d_{\mathrm{0}}{g}_{N+\mathrm{4}}\right\},\\ {\psi }_{\mathrm{2}N}=col\left\{g_{\mathrm{3}},g_{\mathrm{4}},g_{\mathrm{1}}-g_{\mathrm{2}},g_{\mathrm{5}}-g_{\mathrm{2}},\mathrm{2}{g}_{\mathrm{6}}-g_{\mathrm{2}},\dots ,\left(N-\mathrm{1}\right)g_{N+\mathrm{3}}-\left.g_{\mathrm{2}}\right)\right\},\\ {\psi }_{\mathrm{3}}=col\left\{g_{\mathrm{3}},g_{\mathrm{1}}\right\},\\ {\psi }_{\mathrm{4}}=col\left\{g_{\mathrm{4}},g_{\mathrm{2}}\right\},\\ {\Lambda }_{\mathrm{0}N}={\mathcal{L}}_N{\mathcal{M}}_{N}col\left\{g_{\mathrm{1}},g_{\mathrm{2}},g_{\mathrm{5}},\dots ,g_{N+\mathrm{4}}\right\},\end{array}$$with ${\mathcal{L}}_N$ and ${\mathcal{M}}_N$ being defined in Lemma 1, and $g_k=\left[\begin{array}{ccc}{\mathrm{0}}_{n\times (k-\mathrm{1})n}& I_{n\times n}& {\mathrm{0}}_{n\times (N+\mathrm{4}-k)n}\end{array}\right],\hspace{0.17em}\hspace{0.17em}k=\mathrm{1,2},\dots ,N+\mathrm{4}$.

Proof.

The derivative of ${\mathcal{V}}_N^c(t,x_t)$ is given by$$\begin{array}{}\text{(19)}& {\dot{\mathcal{V}}}_N^c\left(t,x_t\right)={\dot{x}}_{0N}^T\left(t\right)R_{0N}{x}_{0N}\left(t\right)+x_{0N}^T\left(t\right)R_{0N}{\dot{x}}_{0N}\left(t\right)+{\omega }^T\left(t\right)S_0\omega \left(t\right)-{\omega }^T\left(t-d_0\right)S_0\omega \left(t-d_0\right)+d_0^2\dot{x}{\left(t\right)}^T{T}_0\dot{x}\left(t\right)-d_0{{\displaystyle \int }}_{t-d_0}^t{\dot{x}}^T\left(s\right)T_0\dot{x}\left(s\right)ds.\end{array}$$Denote $$\begin{array}{}\text{(20)}& \eta \left(t\right)=col\left\{x\left(t\right),x\left(t-d_{\mathrm{0}}\right),\dot{x}\left(t\right),\dot{x}\left(t-d_{\mathrm{0}}\right),{\varsigma }_{\mathrm{1}}\left(t\right),\dots ,{\varsigma }_N\left(t\right)\right\}.\end{array}$$Then$$\begin{array}{c}\text{(21)}& x_{\mathrm{0}N}={\psi }_{\mathrm{1}N}\eta \left(t\right),{\dot{x}}_{\mathrm{0}N}={\psi }_{\mathrm{2}N}\eta \left(t\right),\\ \omega \left(t\right)={\psi }_{\mathrm{3}}\eta \left(t\right),\\ \omega \left(t-d_{\mathrm{0}}\right)={\psi }_{\mathrm{4}}\eta \left(t\right).\end{array}$$Apply Lemma 1 to deal with the integral term in (19), and we have$$\begin{array}{}\text{(22)}& -d_0{{\displaystyle \int }}_{t-d_0}^t{\dot{x}}^T\left(s\right)T_0\dot{x}\left(s\right)ds\le d_0{\eta }^T\left(t\right)\left[{\Lambda }_{0N}^T{\tilde{W}}_{0N}+{\tilde{W}}_{0N}^T{\Lambda }_{0N}+d_0{\tilde{W}}_{0N}^T{\mathcal{T}}_{0N}^{-1}{\tilde{W}}_{0N}\right]\eta \left(t\right).\end{array}$$Let $W_{\mathrm{0}N}=d_{\mathrm{0}}{\tilde{W}}_{\mathrm{0}N}$. (22) can be rewritten as$$\begin{array}{}\text{(23)}& -d_0{{\displaystyle \int }}_{t-d_0}^t{\dot{x}}^T\left(s\right)T_0\dot{x}\left(s\right)ds\le {\eta }^T\left(t\right)\left[{\Lambda }_{0N}^T{W}_{0N}+W_{0N}^T{\Lambda }_{0N}+W_{0N}^T{\mathcal{T}}_{0N}^{-1}{W}_{0N}\right]\eta \left(t\right).\end{array}$$For any matrices $M_i$ ($i=\mathrm{1,2}$) with appropriate dimensions, the following equation holds for system (7)$$\begin{array}{}\text{(24)}& {\displaystyle \sum }_{l=1}^s\hspace{0.17em}{\displaystyle \sum }_{m=1}^s{o}_l{o}_m{\eta }^T\left(t\right)\left\{\left[g_1^T{M}_1+g_3^T{M}_2\right]\left[\left(A_{l0}+B_l{K}_m\right)g_1+A_{l1}{g}_2-g_3\right]\right\}\eta \left(t\right)=0.\end{array}$$

Then combining (19) with (23) and (24) yields$$\begin{array}{}\text{(25)}& {\dot{\mathcal{V}}}_N^c\left(t,x_t\right)\le {\displaystyle \sum }_{l=1}^s\hspace{0.17em}{\displaystyle \sum }_{m=1}^s{o}_l\left(\delta \left(t\right)\right)o_m\left(\delta \left(t\right)\right){\eta }^T\left(t\right)\left[{\Psi }_{Nlm}+W_{0N}^T{\mathcal{T}}_{0N}^{-1}{W}_{0N}\right]\eta \left(t\right)={\displaystyle \sum }_{l=1}^s\hspace{0.17em}{\displaystyle \sum }_{m=1}^s{o}_l\left(\delta \left(t\right)\right)o_m\left(\delta \left(t\right)\right){\eta }^T\left(t\right){\overline{\Psi }}_{Nlm}\eta \left(t\right).\end{array}$$If LMI (17) is satisfied, by Schur complement, one has ${\overline{\Psi }}_{Nlm}<\mathrm{0}$, which implies that ${\dot{\mathcal{V}}}_N^c(t,x_t)<\mathrm{0}$. The proof of Theorem 3 is completed.

3.2. Time-Varying Delay Case

For simplicity, we denote equations as follows: $$\begin{array}{c}\text{(26)}& {\nu }_k\left(t\right)=\frac{\mathrm{1}}{d\left(t\right)}{{\displaystyle \int }}_{t-d\left(t\right)}^t{\left(\frac{t-s}{d\left(t\right)}\right)}^{k-\mathrm{1}}x\left(s\right)ds,{\upsilon }_k\left(t\right)=\frac{\mathrm{1}}{d_{\mathrm{0}}-d\left(t\right)}{{\displaystyle \int }}_{t-d_{\mathrm{0}}}^{t-d\left(t\right)}{\left(\frac{t-d\left(t\right)-s}{d_{\mathrm{0}}-d\left(t\right)}\right)}^{k-\mathrm{1}}x\left(s\right)ds,\end{array}$$where $k=\mathrm{1,2},\dots ,N$.

In order to make full use of B-L inequality, we construct the following augmented LKF for system (3)$$\begin{array}{}\text{(27)}& {\mathcal{V}}_N^v\left(t,x_t\right)={\displaystyle \sum }_{i=\mathrm{1}}^{\mathrm{4}}{\mathcal{V}}_{Ni}^v\left(t,x_t\right),\end{array}$$with $$\begin{array}{c}\text{(28)}& {\mathcal{V}}_{N1}^v\left(t,x_t\right)=x_{1N}^T\left(t\right)R_{1N}{x}_{1N}\left(t\right),{\mathcal{V}}_{N2}^v\left(t,x_t\right)=d\left(t\right)x_{2N}^T\left(t\right)R_{2N}{x}_{2N}\left(t\right)+\left(d_0-d\left(t\right)\right)x_{3N}^T\left(t\right)R_{3N}{x}_{3N}\left(t\right),\\ {\mathcal{V}}_{N3}^v\left(t,x_t\right)={{\displaystyle \int }}_{t-d\left(t\right)}^t{\omega }^T\left(s\right)S_1\omega \left(s\right)ds+{{\displaystyle \int }}_{t-d_0}^{t-d\left(t\right)}{\omega }^T\left(s\right)S_2\omega \left(s\right)ds,\\ {\mathcal{V}}_{N4}^v\left(t,x_t\right)=d_0{{\displaystyle \int }}_{-d_0}^0{{\displaystyle \int }}_{t+\theta }^t{\dot{x}}^T\left(s\right)T_1\dot{x}\left(s\right)ds\hspace{0.17em}d\theta ,\end{array}$$where $R_{\mathrm{1}N}\in {\mathbb{S}}^{(\mathrm{2}N+\mathrm{3})n}$, $R_{\mathrm{2}N},R_{\mathrm{3}N}\in {\mathbb{S}}_{+}^{(N+\mathrm{3})n}$, $S_{\mathrm{1}},S_{\mathrm{2}}\in {\mathbb{S}}_{+}^{\mathrm{2}n},T_{\mathrm{1}}\in {\mathbb{S}}_{+}^n$, and $$\begin{array}{c}\text{(29)}& x_{\mathrm{1}N}\left(t\right)=col\left\{x\left(t\right),x\left(t\right)-x\left(t-d\left(t\right)\right),d\left(t\right){\nu }_{\mathrm{1}}\left(t\right),x\left(t-d\left(t\right)\right)-x\left(t-d_{\mathrm{0}}\right),\left(d_{\mathrm{0}}-d\left(t\right)\right){\upsilon }_{\mathrm{1}}\left(t\right),d\left(t\right){\nu }_{\mathrm{2}}\left(t\right),\dots ,d\left(t\right){\nu }_N\left(t\right),\left(d_{\mathrm{0}}-d\left(t\right)\right){\upsilon }_{\mathrm{2}}\left(t\right),\dots ,\left(d_{\mathrm{0}}-d\left(t\right)\right){\upsilon }_N\left(t\right)\right\},x_{\mathrm{2}N}\left(t\right)=col\left\{x\left(t\right),x\left(t-d\left(t\right)\right),x\left(t-d_{\mathrm{0}}\right),\nu \left(t\right)\right\},\\ x_{\mathrm{3}N}\left(t\right)=col\left\{x\left(t\right),x\left(t-d\left(t\right)\right),x\left(t-d_{\mathrm{0}}\right),\upsilon \left(t\right)\right\},\\ \nu \left(t\right)=col\left\{{\nu }_{\mathrm{1}}\left(t\right),{\nu }_{\mathrm{2}}\left(t\right),\dots ,{\nu }_N\left(t\right)\right\},\\ \upsilon \left(t\right)=col\left\{{\upsilon }_{\mathrm{1}}\left(t\right),{\upsilon }_{\mathrm{2}}\left(t\right),\dots ,{\upsilon }_N\left(t\right)\right\},\end{array}$$and $\omega (t)$ is defined in (16).

Firstly, we deal with the positive definiteness of ${\mathcal{V}}_N^v(t,x_t)$. Proposition 5 is proposed through which the positive definiteness of the LKF can be proved.

Remark 4.

The augmented term ${\mathcal{V}}_{N\mathrm{1}}^v(t,x_t)$ of the proposed LKF is constructed based on the form of B-L inequality, which contributes to reducing the conservatism of the conditions. As proved in [21], (i) the augmented LKF approach is useful for conservatism-reducing, (ii) for the same nonaugmented LKF, the use of Wirtinger-based inequality and Jensen-based inequality does not make a difference in conservatism-reducing, and (iii) for an augmented LKF, the use of Wirtinger-based inequality is better than Jensen-based inequality to obtain less conservative results. As these inequalities are the special cases of B-L inequality, the proposed augmented LKF is significant and effective in deriving the main results. For time-varying delay case, delay-product-type terms of LKF are introduced in [21]. Clearly, these terms in [21] are special cases of the terms in ${\mathcal{V}}_{N\mathrm{2}}^v(t,x_t)$, which are proposed based on the form of B-L inequality.

Proposition 5.

For a given scalar $d_{\mathrm{0}}>\mathrm{0}$ and any given integer $N>\mathrm{0}$, if there exist $R_{\mathrm{1}N}\in {\mathbb{S}}^{(\mathrm{2}N+\mathrm{3})n}$, $R_{\mathrm{2}N},R_{\mathrm{3}N}\in {\mathbb{S}}_{+}^{(N+\mathrm{3})n}$, $S_{\mathrm{1}},S_{\mathrm{2}}\in {\mathbb{S}}_{+}^{\mathrm{2}n},T_{\mathrm{1}}\in {\mathbb{S}}_{+}^n$ and $G\in {\mathbb{R}}^{\mathrm{2}n\times \mathrm{2}n}$ such that$$\begin{array}{}\text{(30)}& {\Theta }_N^{\mathrm{0}}=R_{\mathrm{1}N}+\frac{\mathrm{1}}{d_{\mathrm{0}}}\left[\begin{array}{ccc}{\mathrm{0}}_{n\times n}& {\mathrm{0}}_{n\times \mathrm{4}n}& {\mathrm{0}}_{n\times \left(\mathrm{2}N-\mathrm{2}\right)n}\\ \star & {\Theta }_{\mathrm{0}}& {\mathrm{0}}_{\mathrm{4}n\times \left(\mathrm{2}N-\mathrm{2}\right)n}\\ \star & \star & {\mathrm{0}}_{\left(\mathrm{2}N-\mathrm{2}\right)n\times \left(\mathrm{2}N-\mathrm{2}\right)n}\end{array}\right]>\mathrm{0},\text{(31)}& {\Theta }_{\mathrm{0}}=\left[\begin{array}{cc}{S}_{\mathrm{1}}& G\\ G^{\mathrm{T}}& S_{\mathrm{2}}\end{array}\right]\ge \mathrm{0}.\end{array}$$are satisfied, and then there exist ${\kappa }_{\mathrm{1}}>\mathrm{0}$ and ${\kappa }_{\mathrm{2}}>\mathrm{0}$ such that ${\mathcal{V}}_N^v(t,x_t)$ satisfies$$\begin{array}{}\text{(32)}& {\kappa }_{\mathrm{1}}{‖x\left(t\right)‖}^{\mathrm{2}}\le {\mathcal{V}}_N^v\left(t,x_t\right)\le {\kappa }_{\mathrm{2}}{‖x_t\left(\theta \right)‖}_{\mathbb{W}}^{\mathrm{2}},\end{array}$$where $x_t(\theta )=x(t+\theta )$ and ${\dot{x}}_t(\theta )=\dot{x}(t+\theta )$ with $\theta \in [-d_{\mathrm{0}},\mathrm{0}]$, and $\mathbb{W}$ represents the space of functions $x_t(\theta )$ and ${\dot{x}}_t(\theta )$ with ${‖x_t(\theta )‖}_{\mathbb{W}}^{\mathrm{2}}={\mathrm{m}\mathrm{a}\mathrm{x}}_{\theta \in [-d_{\mathrm{0}},\mathrm{0}]}{‖x_t(\theta )‖}^{\mathrm{2}}+{\int }_{-d_{\mathrm{0}}}^{\mathrm{0}}‍{‖x_t(\theta )‖}^{\mathrm{2}}d\theta$.

Proof.

By using Jensen’s inequality to ${\mathcal{V}}_{N\mathrm{3}}^v(t,x_t)$, one has$$\begin{array}{}\text{(33)}& {{\displaystyle \int }}_{t-d\left(t\right)}^t{\omega }^T\left(s\right)S_1\omega \left(s\right)ds\ge \frac{1}{d\left(t\right)}{\eta }_1^T\left(t\right)S_1{\eta }_1\left(t\right),\text{(34)}& {{\displaystyle \int }}_{t-d_0}^{t-d\left(t\right)}{\omega }^T\left(s\right)S_2\omega \left(s\right)ds\ge \frac{1}{d_0-d\left(t\right)}{\eta }_2^T\left(t\right)S_2{\eta }_2\left(t\right),\end{array}$$where $$\begin{array}{c}\text{(35)}& {\eta }_{\mathrm{1}}\left(t\right)=\left[\begin{array}{c}x\left(t\right)-x\left(t-d\left(t\right)\right)\\ {{\displaystyle \int }}_{t-d\left(t\right)}^{t}x\left(s\right)ds\end{array}\right],{\eta }_{\mathrm{2}}\left(t\right)=\left[\begin{array}{c}x\left(t-d\left(t\right)\right)-x\left(t-d_{\mathrm{0}}\right)\\ {{\displaystyle \int }}_{t-d_{\mathrm{0}}}^{t-d\left(t\right)}x\left(s\right)ds\end{array}\right].\end{array}$$Then substituting (33) and (34) into (27), we obtain $$\begin{array}{}\text{(36)}& {\mathcal{V}}_N^v\left(t,x_t\right)\ge x_{1N}^T\left(t\right)\left(R_{1N}+{\overline{S}}_N\right)x_{1N}\left(t\right)+{\mathcal{V}}_{N2}^v\left(t,x_t\right)+{\mathcal{V}}_{N4}^v\left(t,x_t\right),\end{array}$$where $$\begin{array}{}\text{(37)}& {\stackrel{-}{S}}_N=\left[\begin{array}{cccc}{\mathrm{0}}_{n\times n}& {\mathrm{0}}_{n\times \mathrm{2}n}& {\mathrm{0}}_{n\times \mathrm{2}n}& {\mathrm{0}}_{n\times \left(\mathrm{2}N-\mathrm{2}\right)n}\\ \star & \frac{\mathrm{1}}{d\left(t\right)}{S}_{\mathrm{1}}& {\mathrm{0}}_{\mathrm{2}n\times \mathrm{2}n}& {\mathrm{0}}_{\mathrm{2}n\times \left(\mathrm{2}N-\mathrm{2}\right)n}\\ \star & \star & \frac{\mathrm{1}}{d_{\mathrm{0}}-d\left(t\right)}{S}_{\mathrm{2}}& {\mathrm{0}}_{\mathrm{2}n\times \left(\mathrm{2}N-\mathrm{2}\right)n}\\ \star & \star & \star & {\mathrm{0}}_{\left(\mathrm{2}N-\mathrm{2}\right)n\times \left(\mathrm{2}N-\mathrm{2}\right)n}\end{array}\right].\end{array}$$

If there exists a matrix $G\in {\mathbb{R}}^{\mathrm{2}n\times \mathrm{2}n}$ such that (31) is satisfied, similar to [12], using Lemma 2 with $H_{\mathrm{1}}=H_{\mathrm{2}}=\mathrm{0}$ and $U_{\mathrm{1}}=U_{\mathrm{2}}=G$ to $\left[\begin{array}{cc}\left(\mathrm{1}/d(t)\right)S_{\mathrm{1}}& {\mathrm{0}}_{\mathrm{2}n\times \mathrm{2}n}\\ \star & \left(\mathrm{1}/\left(d_{\mathrm{0}}-d(t)\right)\right)S_{\mathrm{2}}\end{array}\right]$, together with $R_{\mathrm{2}N}>\mathrm{0}$, ${\mathrm{R}}_{\mathrm{3}N}>\mathrm{0}$ and $T_{\mathrm{1}}>\mathrm{0}$, then one has ${\mathcal{V}}_N^v(t,x_t)\ge x_{\mathrm{1}N}^T(t){\Theta }_N^{\mathrm{0}}{x}_{\mathrm{1}N}(t)$. If (30) and $R_{\mathrm{2}N}>\mathrm{0}$, $R_{\mathrm{3}N}>\mathrm{0}$ and $T_{\mathrm{1}}>\mathrm{0}$ are satisfied, and there exists ${\kappa }_{\mathrm{1}}>\mathrm{0}$ such that ${\mathcal{V}}_N^v(t,x_t)\ge {\kappa }_{\mathrm{1}}{‖x(t)‖}^{\mathrm{2}}$, which verifies the first inequality of (32). Similar to [12], the proof of the second inequality in (32) can be obtained, and the specific steps are omitted for brevity. This ends the proof.

Remark 6.

Some extended reciprocally convex inequalities compared to Lemma 2 are proposed in [22, 23], and some improved results with less complexity [22] or less conservatism [23] are provided. It would be interesting to incorporate these inequalities with the proposed augmented LKF method to achieve some potential results in future work. The computing complexity of conditions and stochastic feature of systems [24] also could be studied.

Secondly, the negative definiteness of ${\dot{\mathcal{V}}}_N^v(t,x_t)$ is shown by the following proposition.

Proposition 7.

For given scalars $d_{\mathrm{0}}>\mathrm{0}$, $d_m$ and $d_M$ and any given integer $N>\mathrm{0}$, there exists ${\kappa }_{\mathrm{3}}>\mathrm{0}$ such that the functional ${\mathcal{V}}_N^v(t,x_t)$ satisfies ${\dot{\mathcal{V}}}_N^v(t,x_t)\le -{\kappa }_{\mathrm{3}}{‖x(t)‖}^{\mathrm{2}}$ if there exist $R_{\mathrm{1}N}\in {\mathbb{S}}^{(\mathrm{2}N+\mathrm{3})n}$, $R_{\mathrm{2}N},R_{\mathrm{3}N}\in {\mathbb{S}}_{+}^{(N+\mathrm{3})n}$, $S_{\mathrm{1}},S_{\mathrm{2}}\in {\mathbb{S}}_{+}^{\mathrm{2}n},T_{\mathrm{1}}\in {\mathbb{S}}_{+}^n$, and $W_{\mathrm{1}N}$, $W_{\mathrm{2}N}$ and ${\tilde{M}}_i\hspace{0.17em}\hspace{0.17em}(i=\mathrm{1,2},\mathrm{3,4})$ of appropriate dimensions such that for $d_m\le \mu \le d_M$, the following LMIs hold for $l,m=\mathrm{1,2},\dots ,r$:$$\begin{array}{}\text{(38)}& {\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=\mu }\right)+{\tilde{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=\mu }\right)<\mathrm{0},\hspace{1em}l\le m\text{(39)}& {\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=\mu }\right)+{\tilde{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=\mu }\right)<\mathrm{0},\hspace{1em}l\le m\end{array}$$where $$\begin{array}{c}\text{(40)}& {\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0}}\right)=\left[\begin{array}{cc}{\tilde{\Psi }}_{Nlm}\left(\mathrm{0},\dot{d}\left(t\right)\right)& W_{\mathrm{2}N}^{\mathrm{T}}\\ \star & -{\mathcal{T}}_{\mathrm{1}N}\end{array}\right]<\mathrm{0},{\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}}}\right)=\left[\begin{array}{cc}{\tilde{\Psi }}_{Nlm}\left(d_{\mathrm{0}},\dot{d}\left(t\right)\right)& W_{\mathrm{1}N}^{\mathrm{T}}\\ \star & -{\mathcal{T}}_{\mathrm{1}N}\end{array}\right]<\mathrm{0},\\ {\tilde{\Psi }}_{Nlm}\left(d\left(t\right),\dot{d}\left(t\right)\right)={\Phi }_{\mathrm{1}N}+{\Phi }_{\mathrm{2}N}+{\Phi }_{\mathrm{3}}+{\Phi }_{\mathrm{4}N}+{\Phi }_{\mathrm{5}lm},\\ {\mathcal{T}}_{\mathrm{1}N}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{T_{\mathrm{1}},\mathrm{3}{T}_{\mathrm{1}},\dots ,\left(\mathrm{2}N+\mathrm{1}\right)T_{\mathrm{1}}\right\},\\ {\Phi }_{\mathrm{1}N}=\mathrm{S}\mathrm{y}\mathrm{m}\left\{{\phi }_{\mathrm{1}N}^{\mathrm{T}}{R}_{\mathrm{1}N}{\phi }_{\mathrm{2}N}\right\},\\ {\Phi }_{\mathrm{2}N}=\dot{d}\left(t\right)\left({\phi }_{\mathrm{3}N}^{\mathrm{T}}{R}_{\mathrm{2}N}{\phi }_{\mathrm{3}N}-{\phi }_{\mathrm{4}N}^{\mathrm{T}}{R}_{\mathrm{3}N}{\phi }_{\mathrm{4}N}\right)+\mathrm{S}\mathrm{y}\mathrm{m}\left\{{\phi }_{\mathrm{3}N}^{\mathrm{T}}{R}_{\mathrm{2}N}{\phi }_{\mathrm{5}N}+{\phi }_{\mathrm{4}N}^{\mathrm{T}}{R}_{\mathrm{3}N}{\phi }_{\mathrm{6}N}\right\}+d_{\mathrm{0}}^{\mathrm{2}}{\tilde{g}}_{\mathrm{6}}^{\mathrm{T}}{T}_{\mathrm{1}}{\tilde{g}}_{\mathrm{6}},\\ {\Phi }_{\mathrm{3}}={\phi }_{\mathrm{7}}^{\mathrm{T}}{S}_{\mathrm{1}}{\phi }_{\mathrm{7}}-\left(\mathrm{1}-\dot{d}\left(t\right)\right){\phi }_{\mathrm{8}}^{\mathrm{T}}\left(S_{\mathrm{1}}-S_{\mathrm{2}}\right){\phi }_{\mathrm{8}}-{\phi }_{\mathrm{9}}^{\mathrm{T}}{S}_{\mathrm{2}}{\phi }_{\mathrm{9}},\\ {\Phi }_{\mathrm{4}N}=\mathrm{S}\mathrm{y}\mathrm{m}\left\{{\Lambda }_{\mathrm{1}N}^{\mathrm{T}}{W}_{\mathrm{1}N}+{\Lambda }_{\mathrm{2}N}^{\mathrm{T}}{W}_{\mathrm{2}N}\right\},\\ {\Phi }_{\mathrm{5}lm}=\mathrm{S}\mathrm{y}\mathrm{m}\left\{\left[{\tilde{g}}_{\mathrm{1}}^{\mathrm{T}}{\tilde{M}}_{\mathrm{1}}+{\tilde{g}}_{\mathrm{2}}^{\mathrm{T}}{\tilde{M}}_{\mathrm{2}}+{\tilde{g}}_{\mathrm{3}}^{\mathrm{T}}{\tilde{M}}_{\mathrm{3}}+{\tilde{g}}_{\mathrm{4}}^{\mathrm{T}}{\tilde{M}}_{\mathrm{4}}\right]{\mathcal{F}}_{lm}\right\},\\ {\mathcal{F}}_{lm}=\left(A_{l\mathrm{0}}+B_l{K}_m\right){\tilde{g}}_{\mathrm{1}}+A_{l\mathrm{1}}{\tilde{g}}_{\mathrm{2}}-{\tilde{g}}_{\mathrm{6}},\\ {\phi }_{\mathrm{1}N}=col\left\{{\tilde{g}}_{\mathrm{1}},{\tilde{g}}_{\mathrm{1}}-{\tilde{g}}_{\mathrm{2}},d\left(t\right){\tilde{g}}_{\mathrm{7}},{\tilde{g}}_{\mathrm{2}}-{\tilde{g}}_{\mathrm{3}},\left(d_{\mathrm{0}}-d\left(t\right)\right){\tilde{g}}_{N+\mathrm{7}},d\left(t\right){\tilde{g}}_{\mathrm{8}},\dots ,d\left(t\right){\tilde{g}}_{N+\mathrm{6}},\left(d_{\mathrm{0}}-d\left(t\right)\right){\tilde{g}}_{N+\mathrm{8}},\dots ,\left(d_{\mathrm{0}}-d\left(t\right)\right){\tilde{g}}_{\mathrm{2}N+\mathrm{6}}\right\},\\ {\phi }_{\mathrm{2}N}=col\left\{{\tilde{g}}_{\mathrm{6}},{\tilde{g}}_{\mathrm{6}}-\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{\mathrm{4}},{\mathcal{l}}_{\mathrm{1}},\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{\mathrm{4}}-{\tilde{g}}_{\mathrm{5}},{\vartheta }_{\mathrm{1}},{\mathcal{l}}_{\mathrm{2}},\dots ,{\mathcal{l}}_N,{\vartheta }_{\mathrm{2}},\dots ,{\vartheta }_N\right\},\\ {\phi }_{\mathrm{3}N}=col\left\{{\tilde{g}}_{\mathrm{1}},{\tilde{g}}_{\mathrm{2}},{\tilde{g}}_{\mathrm{3}},{\tilde{g}}_{\mathrm{7}},\dots ,{\tilde{g}}_{N+\mathrm{6}}\right\},\\ {\phi }_{\mathrm{4}N}=col\left\{{\tilde{g}}_{\mathrm{1}},{\tilde{g}}_{\mathrm{2}},{\tilde{g}}_{\mathrm{3}},{\tilde{g}}_{N+\mathrm{7}},\dots ,{\tilde{g}}_{\mathrm{2}N+\mathrm{6}}\right\},\\ {\phi }_{\mathrm{5}N}=col\left\{d\left(t\right){\tilde{g}}_{\mathrm{6}},d\left(t\right)\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{\mathrm{4}},d\left(t\right){\tilde{g}}_{\mathrm{5}},{\lambda }_{\mathrm{1}},\dots ,{\lambda }_N\right\},\\ {\phi }_{\mathrm{6}N}=col\left\{\left(d_{\mathrm{0}}-d\left(t\right)\right){\tilde{g}}_{\mathrm{6}},\left(d_{\mathrm{0}}-d\left(t\right)\right)\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{\mathrm{4}},\left(d_{\mathrm{0}}-d\left(t\right)\right){\tilde{g}}_{\mathrm{5}},{\gamma }_{\mathrm{1}},\dots ,{\gamma }_N\right\},\\ {\phi }_{\mathrm{7}}=col\left\{{\tilde{g}}_{\mathrm{6}},{\tilde{g}}_{\mathrm{1}}\right\},\\ {\phi }_{\mathrm{8}}=col\left\{{\tilde{g}}_{\mathrm{4}},{\tilde{g}}_{\mathrm{2}}\right\},\\ {\phi }_{\mathrm{9}}=col\left\{{\tilde{g}}_{\mathrm{5}},{\tilde{g}}_{\mathrm{3}}\right\},\\ {\Lambda }_{\mathrm{1}N}={\mathcal{L}}_N{\mathcal{M}}_{N}col\left\{{\tilde{g}}_{\mathrm{1}},{\tilde{g}}_{\mathrm{2}},{\tilde{g}}_{\mathrm{7}},\dots ,{\tilde{g}}_{N+\mathrm{6}}\right\},\\ {\Lambda }_{\mathrm{2}N}={\mathcal{L}}_N{\mathcal{M}}_{N}col\left\{{\tilde{g}}_{\mathrm{2}},{\tilde{g}}_{\mathrm{3}},{\tilde{g}}_{N+\mathrm{7}},\dots ,{\tilde{g}}_{\mathrm{2}N+\mathrm{6}}\right\},\\ {\mathcal{l}}_k=\left\{\begin{array}{cc}-\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{\mathrm{2}}+{\tilde{g}}_{\mathrm{1}},\hfill & k=\mathrm{1}\hfill \\ -\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{\mathrm{2}}+\left(k-\mathrm{1}\right)\left({\tilde{g}}_{k+\mathrm{5}}-\dot{d}\left(t\right){\tilde{g}}_{k+\mathrm{6}}\right),\hfill & k=\mathrm{2,3},\dots ,N\hfill \end{array}\right.\\ {\vartheta }_k=\left\{\begin{array}{cc}-{\tilde{g}}_{\mathrm{3}}+\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{\mathrm{2}},\hfill & k=\mathrm{1}\hfill \\ -{\tilde{g}}_{\mathrm{3}}+\left(k-\mathrm{1}\right)\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{N+k+\mathrm{5}}+\left(k-\mathrm{1}\right)\dot{d}\left(t\right){\tilde{g}}_{N+k+\mathrm{6}},\hfill & k=\mathrm{2,3},\dots ,N\hfill \end{array}\right.\\ {\lambda }_k=\left\{\begin{array}{cc}-\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{\mathrm{2}}+{\tilde{g}}_{\mathrm{1}}-\dot{d}\left(t\right){\tilde{g}}_7,\hfill & k=\mathrm{1}\hfill \\ -\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{\mathrm{2}}+\left(k-\mathrm{1}\right){\tilde{g}}_{k+\mathrm{5}}-k\dot{d}\left(t\right){\tilde{g}}_{k+\mathrm{6}},\hfill & k=\mathrm{2,3},\dots ,N\hfill \end{array}\right.\\ {\gamma }_k=\left\{\begin{array}{cc}-{\tilde{g}}_{\mathrm{3}}+\left(\mathrm{1}-\dot{d}\left(t\right)\right){\tilde{g}}_{\mathrm{2}}+\dot{d}\left(t\right){\tilde{g}}_{N+\mathrm{7}},\hfill & k=\mathrm{1}\hfill \\ -{\tilde{g}}_{\mathrm{3}}+\left(\mathrm{1}-\dot{d}\left(t\right)\right)\left(k-\mathrm{1}\right){\tilde{g}}_{N+k+\mathrm{5}}+k\dot{d}\left(t\right){\tilde{g}}_{N+k+\mathrm{6}},\hfill & k=\mathrm{2,3},\dots ,N\hfill \end{array}\right.\\ {\tilde{g}}_k=\left[\begin{array}{ccc}{\mathrm{0}}_{n\times \left(k-\mathrm{1}\right)n}& I_{n\times n}& {\mathrm{0}}_{n\times \left(\mathrm{2}N+\mathrm{6}-k\right)n}\end{array}\right],\hspace{1em}k=\mathrm{1,2},\dots ,\mathrm{2}N+\mathrm{6}\end{array}$$and ${\mathcal{L}}_N,{\mathcal{M}}_N$ are defined in (10) and (11), respectively.

Proof.

For ${\mathcal{V}}_N^v(t,x_t)$ in (27), we compute its derivative and obtain$$\begin{array}{}\text{(41)}& {\dot{\mathcal{V}}}_N^v\left(t,x_t\right)={\displaystyle \sum }_{i=\mathrm{1}}^{\mathrm{4}}{\dot{\mathcal{V}}}_{Ni}^v\left(t,x_t\right),\end{array}$$where$$\begin{array}{c}\text{(42)}& {\dot{\mathcal{V}}}_{N1}^v\left(t,x_t\right)=\mathrm{S}\mathrm{y}\mathrm{m}\left\{x_{1N}^T\left(t\right)R_{1N}{\dot{x}}_{1N}\left(t\right)\right\},{\dot{\mathcal{V}}}_{N2}^v\left(t,x_t\right)=\dot{d}\left(t\right)x_{2N}^T\left(t\right)R_{2N}{x}_{2N}\left(t\right)-\dot{d}\left(t\right)x_{3N}^T\left(t\right)R_{3N}{x}_{3N}\left(t\right)\\ +\mathrm{S}\mathrm{y}\mathrm{m}\left\{d\left(t\right)x_{2N}^T\left(t\right)R_{2N}{\dot{x}}_{2N}\left(t\right)+\left(d_0-d\left(t\right)\right)x_{3N}^T\left(t\right)R_{3N}{\dot{x}}_{3N}\left(t\right)\right\},\\ {\dot{\mathcal{V}}}_{N3}^v\left(t,x_t\right)={\omega }^T\left(t\right)S_1\omega \left(t\right)-\left(1-\dot{d}\left(t\right)\right){\omega }^T\left(t-d\left(t\right)\right)\left(S_1-S_2\right)\omega \left(t-d\left(t\right)\right)-{\omega }^T\left(t-d_0\right)S_2\omega \left(t-d_0\right),\\ {\dot{\mathcal{V}}}_{N4}^v\left(t,x_t\right)=d_0^2{\dot{x}}^T\left(t\right)T_1\dot{x}\left(t\right)-d_0{{\displaystyle \int }}_{t-d_0}^t{\dot{x}}^T\left(s\right)T_1\dot{x}\left(s\right)ds.\end{array}$$Denote $$\begin{array}{c}\text{(43)}& \zeta \left(t\right)=col\left\{x\left(t\right),x\left(t-d\left(t\right)\right),x\left(t-d_{\mathrm{0}}\right),{\zeta }_{\mathrm{0}}\left(t\right)\right\},{\zeta }_{\mathrm{0}}\left(t\right)=col\left\{\dot{x}\left(t-d\left(t\right)\right),\dot{x}\left(t-d_{\mathrm{0}}\right),\dot{x}\left(t\right),\nu \left(t\right),\upsilon \left(t\right)\right\}.\end{array}$$We can rewrite ${\dot{\mathcal{V}}}_N^v(t,x_t)$ in (41) as follows:$$\begin{array}{}\text{(44)}& {\dot{\mathcal{V}}}_N^v\left(t,x_t\right)={\displaystyle \sum }_{l=1}^s{o}_l\left(\delta \left(t\right)\right){\zeta }^T\left(t\right)\left({\Phi }_{1N}+{\Phi }_{2N}+{\Phi }_3\right)\zeta \left(t\right)-d_0{{\displaystyle \int }}_{t-d\left(t\right)}^t{\dot{x}}^T\left(s\right)T_1\dot{x}\left(s\right)ds-d_0{{\displaystyle \int }}_{t-d_0}^{t-d\left(t\right)}{\dot{x}}^T\left(s\right)T_1\dot{x}\left(s\right)ds.\end{array}$$Using Lemma 1 to estimate the upper bounds of the above integral terms, respectively, we get$$\begin{array}{}\text{(45)}& -d_0{{\displaystyle \int }}_{t-d\left(t\right)}^t{\dot{x}}^T\left(s\right)T_1\dot{x}\left(s\right)ds\le d_0{\zeta }^T\left(t\right)\left[{\Lambda }_{1N}^T{\tilde{W}}_{1N}+{\tilde{W}}_{1N}^T{\Lambda }_{1N}+d\left(t\right){\tilde{W}}_{1N}^T{\mathcal{T}}_{1N}^{-1}{\tilde{W}}_{1N}\right]\zeta \left(t\right),\text{(46)}& -d_0{{\displaystyle \int }}_{t-d_0}^{t-d\left(t\right)}{\dot{x}}^T\left(s\right)T_1\dot{x}\left(s\right)ds\le d_0{\zeta }^T\left(t\right)\left[{\Lambda }_{2N}^T{\tilde{W}}_{2N}+{\tilde{W}}_{2N}^T{\Lambda }_{2N}+\left(d_0-d\left(t\right)\right){\tilde{W}}_{2N}^T{\mathcal{T}}_{1N}^{-1}{\tilde{W}}_{2N}\right]\zeta \left(t\right).\end{array}$$Setting $W_{\mathrm{1}N}=d_{\mathrm{0}}{\tilde{W}}_{\mathrm{1}N}$ and $W_{\mathrm{2}N}=d_{\mathrm{0}}{\tilde{W}}_{\mathrm{2}N}$. Then, according to (45) and (46), we yield$$\begin{array}{}\text{(47)}& \mathcal{Z}\left(t\right)\le {\zeta }^T\left(t\right)\left[\mathrm{S}\mathrm{y}\mathrm{m}\left\{{\Lambda }_{1N}^T{W}_{1N}+{\Lambda }_{2N}^T{W}_{2N}\right\}+\sigma W_{1N}^T{\mathcal{T}}_{1N}^{-1}{W}_{1N}+\left(1-\sigma \right)W_{2N}^T{\mathcal{T}}_{1N}^{-1}{W}_{2N}\right]\zeta \left(t\right),\end{array}$$where $\sigma =d(t)/d_{\mathrm{0}}$ and$$\begin{array}{}\text{(48)}& \mathcal{Z}\left(t\right)=-d_0{{\displaystyle \int }}_{t-d\left(t\right)}^t{\dot{x}}^T\left(s\right)T_1\dot{x}\left(s\right)ds-d_0{{\displaystyle \int }}_{t-d_0}^{t-d\left(t\right)}{\dot{x}}^T\left(s\right)T_1\dot{x}\left(s\right)ds.\end{array}$$Similar to the proof of Theorem 3, we introduce the following equation:$$\begin{array}{}\text{(49)}& {\displaystyle \sum }_{l=1}^s\hspace{0.17em}{\displaystyle \sum }_{m=1}^s{o}_l{o}_m{\zeta }^T\left(t\right)\left\{\left[{\tilde{g}}_1^T{\tilde{M}}_1+{\tilde{g}}_2^T{\tilde{M}}_2+{\tilde{g}}_3^T{\tilde{M}}_3+{\tilde{g}}_6^T{\tilde{M}}_4\right]\left[\left(A_{l0}+B_l{K}_m\right){\tilde{g}}_1+A_{l1}{\tilde{g}}_2-{\tilde{g}}_6\right]\right\}\zeta \left(t\right)=0.\end{array}$$Thus, according to (44), (45), (46), (47), (48), and (49) we have$$\begin{array}{}\text{(50)}& {\dot{\mathcal{V}}}_N^v\left(t,x_t\right)<{\displaystyle \sum }_{l=1}^s\hspace{0.17em}{\displaystyle \sum }_{m=1}^s{o}_l{o}_m{\zeta }^T\left(t\right)\left\{{\tilde{\Psi }}_{Nlm}\left(d\left(t\right),\dot{d}\left(t\right)\right)+\sigma W_{1N}^T{\mathcal{T}}_{1N}^{-1}{W}_{1N}+\left(1-\sigma \right)W_{2N}^T{\mathcal{T}}_{1N}^{-1}{W}_{2N}\right\}\zeta \left(t\right)={\displaystyle \sum }_{l=1}^s\hspace{0.17em}{\displaystyle \sum }_{m=1}^s{o}_l{o}_m{\zeta }^T\left(t\right){\widehat{\Psi }}_{Nlm}\left(d\left(t\right),\dot{d}\left(t\right)\right)\zeta \left(t\right).\end{array}$$It can be clearly seen that the matrix ${\widehat{\Psi }}_{Nlm}(d(t),\dot{d}(t))$ is a convex combination in $d(t)$ and $\dot{d}(t)$. Similar to the two-dimensional convex combination method in [16], if LMIs (38) and (39) are satisfied, one can easily derive ${\widehat{\Psi }}_{Nlm}(d(t),\dot{d}(t))<\mathrm{0}$ for $(d(t),\dot{d}(t))\in {\mathcal{D}}_{\mathrm{1}}$. The proof of Proposition 7 is completed.

Now, we establish the stability conditions of the T-S fuzzy system (7) based on Propositions 5 and 7 as follows.

Theorem 8.

For given scalars $d_{\mathrm{0}}>\mathrm{0}$, $d_m$ and $d_M$ and any given integer $N>\mathrm{0}$, if there exist $R_{\mathrm{1}N}\in {\mathbb{S}}^{(\mathrm{2}N+\mathrm{3})n}$, $R_{\mathrm{2}N},R_{\mathrm{3}N}\in {\mathbb{S}}_{+}^{(N+\mathrm{3})n}$, $S_{\mathrm{1}},S_{\mathrm{2}}\in {\mathbb{S}}_{+}^{\mathrm{2}n},T_{\mathrm{1}}\in {\mathbb{S}}_{+}^n$, $G\in {\mathbb{R}}^{\mathrm{2}n\times \mathrm{2}n}$ and ${\tilde{M}}_i$ ($i=\mathrm{1,2},\mathrm{3,4}$), $W_{\mathrm{1}N}$ and $W_{\mathrm{2}N}$ of appropriate dimensions such that the LMIs (30), (31) and the LMIs as follows $$\begin{array}{c}\text{(51)}& {\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=d_m}\right)+{\tilde{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=d_m}\right)<\mathrm{0},\hspace{1em}l\le m{\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=d_M}\right)+{\tilde{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=d_M}\right)<\mathrm{0},\hspace{1em}l\le m\\ {\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=d_m}\right)+{\tilde{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=d_m}\right)<\mathrm{0},\hspace{1em}l\le m\\ {\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=d_M}\right)+{\tilde{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=d_M}\right)<\mathrm{0}.\hspace{1em}l\le m\end{array}$$are satisfied for $m,l=\mathrm{1,2},\dots ,s$, then system (7) with time-varying delay $d(t)$ subject to (2) is asymptotically stable.

For the case of time-varying delay, [12, 14] propose a refined delay set ${\mathcal{D}}_{\mathrm{2}}$ which has been proven to derive less conservative results than delay set ${\mathcal{D}}_{\mathrm{1}}$. Taking the refined delay set ${\mathcal{D}}_{\mathrm{2}}=Co\{(\mathrm{0,0}),(\mathrm{0},d_M),(d_{\mathrm{0}},d_m),(d_{\mathrm{0}},\mathrm{0})\}$ into account and using Theorem 8, we obtain a stability condition as follows.

Theorem 9.

For given scalars $d_{\mathrm{0}}>\mathrm{0}$, $d_m$ and $d_M$ and any given integer $N>\mathrm{0}$, if there exist $R_{\mathrm{1}N}\in {\mathbb{S}}^{(\mathrm{2}N+\mathrm{3})n}$, $R_{\mathrm{2}N},R_{\mathrm{3}N}\in {\mathbb{S}}_{+}^{(N+\mathrm{3})n}$, $S_{\mathrm{1}},S_{\mathrm{2}}\in {\mathbb{S}}_{+}^{\mathrm{2}n},T_{\mathrm{1}}\in {\mathbb{S}}_{+}^n$, $G\in {\mathbb{R}}^{\mathrm{2}n\times \mathrm{2}n}$ and ${\tilde{M}}_i$ ($i=\mathrm{1,2},\mathrm{3,4}$), $W_{\mathrm{1}N}$ and $W_{\mathrm{2}N}$ of appropriate dimensions such that (30), (31) are satisfied and the following LMIs hold for $l,m=\mathrm{1,2},\dots ,s$, then system (7) is asymptotically stable for $(d(t),\dot{d}(t))\in {\mathcal{D}}_{\mathrm{2}}$$$\begin{array}{c}\text{(52)}& {\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=\mathrm{0}}\right)+{\tilde{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=\mathrm{0}}\right)<\mathrm{0},\hspace{1em}l\le m{\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=d_M}\right)+{\tilde{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=d_M}\right)<\mathrm{0},\hspace{1em}l\le m\\ {\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=d_m}\right)+{\tilde{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=d_m}\right)<\mathrm{0},\hspace{1em}l\le m\\ {\tilde{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=\mathrm{0}}\right)+{\tilde{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=\mathrm{0}}\right)<\mathrm{0}.\hspace{1em}l\le m\end{array}$$

4. Fuzzy Controller Design

In this section, controller design conditions will be given by two cases of the time delay.

4.1. Constant Delay Case

Based on Theorem 3, the fuzzy control gains of system (7) with constant delay can be derived from the following theorem.

Theorem 10.

For given scalars $d_{\mathrm{0}}>\mathrm{0}$, $\varrho$, and a given integer $N>\mathrm{0}$, if there exist ${\overline{R}}_{\mathrm{0}N}\in {\mathbb{S}}_{+}^{(N+\mathrm{2})n}$, ${\overline{S}}_{\mathrm{0}}\in {\mathbb{S}}_{+}^{\mathrm{2}n}$, ${\overline{T}}_{\mathrm{0}}\in {\mathbb{S}}_{+}^n$ and ${\overline{H}}_m$ ($m=\mathrm{1,2},\dots ,s$), ${\overline{W}}_{\mathrm{0}N}$ and $\overline{X}$ of appropriate dimensions such that the following LMIs hold for $l,m=\mathrm{1,2},\dots ,s$, then the closed-loop system (7) with constant delay $d(t)\equiv d_{\mathrm{0}}$ is asymptotically stable $$\begin{array}{}\text{(53)}& {\overline{\mathrm{\Xi }}}_{Nlm}+{\overline{\mathrm{\Xi }}}_{Nml}<\mathrm{0},\hspace{1em}l\le m\end{array}$$where $$\begin{array}{c}\text{(54)}& {\overline{\mathrm{\Xi }}}_{Nlm}=\left[\begin{array}{cc}{\overline{\Psi }}_{Nlm}& {\stackrel{-}{W}}_{\mathrm{0}N}^{\mathrm{T}}\\ \star & -{\overline{\mathcal{T}}}_{\mathrm{0}N}\end{array}\right]<\mathrm{0},{\overline{\mathcal{T}}}_{\mathrm{0}N}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{{\overline{T}}_{\mathrm{0}},\mathrm{3}{\overline{T}}_{\mathrm{0}},\dots ,\left(\mathrm{2}N+\mathrm{1}\right){\overline{T}}_{\mathrm{0}}\right\},\\ {\overline{\Psi }}_{Nlm}=\mathrm{S}\mathrm{y}\mathrm{m}\left\{{\psi }_{\mathrm{1}N}^{\mathrm{T}}{\overline{R}}_{\mathrm{0}N}{\psi }_{\mathrm{2}N}\right\}+{\psi }_{\mathrm{3}}^{\mathrm{T}}{\overline{S}}_{\mathrm{0}}{\psi }_{\mathrm{3}}-{\psi }_{\mathrm{4}}^{\mathrm{T}}{\overline{S}}_{\mathrm{0}}{\psi }_{\mathrm{4}}+{\Lambda }_{\mathrm{0}N}^{\mathrm{T}}{\overline{W}}_{\mathrm{0}N}+{\overline{W}}_{\mathrm{0}N}^{\mathrm{T}}{\Lambda }_{\mathrm{0}N}+d_{\mathrm{0}}^{\mathrm{2}}{g}_{\mathrm{3}}^{\mathrm{T}}{\overline{T}}_{\mathrm{0}}{g}_{\mathrm{3}}+\mathrm{S}\mathrm{y}\mathrm{m}\left\{\left(\varrho g_{\mathrm{1}}^{\mathrm{T}}+g_{\mathrm{3}}^{\mathrm{T}}\right){\overline{\mathcal{E}}}_{lm}\right\},\\ {\overline{\mathcal{E}}}_{lm}=\left(A_{l\mathrm{0}}{\overline{X}}^{\mathrm{T}}+B_l{\overline{H}}_m\right)g_{\mathrm{1}}+A_{l\mathrm{1}}{\overline{X}}^{\mathrm{T}}{g}_{\mathrm{2}}-{\overline{X}}^{\mathrm{T}}{g}_{\mathrm{3}},\end{array}$$with ${\psi }_{\mathrm{1}N}$, ${\psi }_{\mathrm{2}N}$, ${\psi }_{\mathrm{3}}$, ${\psi }_{\mathrm{4}}$, and ${\Lambda }_{\mathrm{0}N}$ being defined in Theorem 3. Then, the fuzzy control gains can be calculated by $K_m={\overline{H}}_m{\overline{X}}^{\hspace{0.17em}-\mathrm{T}}\hspace{0.17em}\hspace{0.17em}(m=\mathrm{1,2},\dots ,s)$.

Proof.

Pre- and postmultiplying both sides of (17) with $\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{\overline{X},\overline{X},\dots ,\overline{X}{\}}_{(N+\mathrm{4})n}$ and its transpose, introducing $\overline{X}=M_{\mathrm{2}}^{-\mathrm{1}}$, $M_{\mathrm{1}}=\varrho M_{\mathrm{2}}$, and defining ${\overline{X}}_{\mathrm{1}}{R}_{\mathrm{0}N}{\overline{X}}_{\mathrm{1}}^T={\overline{R}}_{\mathrm{0}N}$, ${\overline{X}}_{\mathrm{2}}{S}_{\mathrm{0}}{\overline{X}}_{\mathrm{2}}^T={\overline{S}}_{\mathrm{0}}$, ${\overline{X}}_{\mathrm{3}}{\mathcal{T}}_{\mathrm{0}N}{\overline{X}}_{\mathrm{3}}^T={\overline{\mathcal{T}}}_{\mathrm{0}N}$, $\overline{X}{T}_{\mathrm{0}}{\overline{X}}^T={\overline{T}}_{\mathrm{0}}$, ${\overline{X}}_{\mathrm{1}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{\overline{X},\dots ,\overline{X}{\}}_{(N+\mathrm{2})n}$, ${\overline{X}}_{\mathrm{2}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{\overline{X},\overline{X}\}$ and ${\overline{X}}_{\mathrm{3}}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\{\overline{X},\dots ,\overline{X}{\}}_{(N+\mathrm{1})n}$, we obtain the LMI (53). This completes the proof.

4.2. Time-Varying Delay Case

For the case of $(d(t),\dot{d}(t))\in {\mathcal{D}}_{\mathrm{2}}$, the next theorem is given to obtain the fuzzy control gains.

Theorem 11.

For a given integer $N>\mathrm{0}$, given parameters $d_{\mathrm{0}}>\mathrm{0}$, $d_m$ and $d_M$, and given scalars ${\rho }_i\hspace{0.17em}\hspace{0.17em}(i=\mathrm{1,2},\mathrm{3})$, if there exist ${\widehat{R}}_{\mathrm{1}N}\in {\mathbb{S}}^{(\mathrm{2}N+\mathrm{3})n}$, ${\widehat{R}}_{\mathrm{2}N},{\widehat{R}}_{\mathrm{3}N}\in {\mathbb{S}}_{+}^{(N+\mathrm{3})n}$, ${\widehat{S}}_{\mathrm{1}},{\widehat{S}}_{\mathrm{2}}\in {\mathbb{S}}_{+}^{\mathrm{2}n},{\widehat{T}}_{\mathrm{1}}\in {\mathbb{S}}_{+}^n$, $\widehat{G}\in {\mathbb{R}}^{\mathrm{2}n\times \mathrm{2}n}$ and ${\widehat{H}}_m$ ($m=\mathrm{1,2},\dots ,s$), $\widehat{X}$, ${\widehat{W}}_{\mathrm{1}N}$ and ${\widehat{W}}_{\mathrm{2}N}$ of appropriate dimensions such that the following LMIs hold for $l$, $m=\mathrm{1,2},\dots ,s$, then the closed-loop system (7) with time-varying delay $(d(t),\dot{d}(t))\in {\mathcal{D}}_{\mathrm{2}}$ is asymptotically stable $$\begin{array}{c}\text{(55)}& {\widehat{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=\mathrm{0}}\right)+{\widehat{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=\mathrm{0}}\right)<\mathrm{0},\hspace{1em}l\le m{\widehat{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=d_M}\right)+{\widehat{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0},\dot{d}\left(t\right)=d_M}\right)<\mathrm{0},\hspace{1em}l\le m\\ {\widehat{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=d_m}\right)+{\widehat{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=d_m}\right)<\mathrm{0},\hspace{1em}l\le m\\ {\widehat{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=\mathrm{0}}\right)+{\widehat{\mathrm{\Xi }}}_{Nml}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}},\dot{d}\left(t\right)=\mathrm{0}}\right)<\mathrm{0},\hspace{1em}l\le m\end{array}$$where $$\begin{array}{c}\text{(56)}& {\widehat{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=\mathrm{0}}\right)=\left[\begin{array}{cc}{\widehat{\Psi }}_{Nlm}\left(\mathrm{0},\dot{d}\left(t\right)\right)& {\widehat{W}}_{\mathrm{2}N}^{\mathrm{T}}\\ \star & -{\widehat{\mathcal{T}}}_{\mathrm{1}N}\end{array}\right]<\mathrm{0},{\widehat{\mathrm{\Xi }}}_{Nlm}\left({\left.d\left(t\right),\dot{d}\left(t\right)\right|}_{d\left(t\right)=d_{\mathrm{0}}}\right)=\left[\begin{array}{cc}{\widehat{\Psi }}_{Nlm}\left(d_{\mathrm{0}},\dot{d}\left(t\right)\right)& {\widehat{W}}_{\mathrm{1}N}^{\mathrm{T}}\\ \star & -{\widehat{\mathcal{T}}}_{\mathrm{1}N}\end{array}\right]<\mathrm{0},\\ {\widehat{\Psi }}_{Nlm}\left(d\left(t\right),\dot{d}\left(t\right)\right)={\widehat{\Phi }}_{\mathrm{1}N}+{\widehat{\Phi }}_{\mathrm{2}N}+{\widehat{\Phi }}_{\mathrm{3}}+{\widehat{\Phi }}_{\mathrm{4}N}+{\widehat{\Phi }}_{\mathrm{5}lm},\\ {\widehat{\mathcal{T}}}_{\mathrm{1}N}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}\left\{{\widehat{T}}_{\mathrm{1}},\mathrm{3}{\widehat{T}}_{\mathrm{1}},\dots ,\left(\mathrm{2}N+\mathrm{1}\right){\widehat{T}}_{\mathrm{1}}\right\},\\ {\widehat{\Phi }}_{\mathrm{1}N}=\mathrm{S}\mathrm{y}\mathrm{m}\left\{{\phi }_{\mathrm{1}N}^{\mathrm{T}}{\widehat{R}}_{\mathrm{1}N}{\phi }_{\mathrm{2}N}\right\},\\ {\widehat{\Phi }}_{\mathrm{2}N}=\dot{d}\left(t\right)\left({\phi }_{\mathrm{3}N}^{\mathrm{T}}{\widehat{R}}_{\mathrm{2}N}{\phi }_{\mathrm{3}N}-{\phi }_{\mathrm{4}N}^{\mathrm{T}}{\widehat{R}}_{\mathrm{3}N}{\phi }_{\mathrm{4}N}\right)+\mathrm{S}\mathrm{y}\mathrm{m}\left\{{\phi }_{\mathrm{3}N}^{\mathrm{T}}{\widehat{R}}_{\mathrm{2}N}{\phi }_{\mathrm{5}N}+{\phi }_{\mathrm{4}N}^{\mathrm{T}}{\widehat{R}}_{\mathrm{3}N}{\phi }_{\mathrm{6}N}\right\}+d_{\mathrm{0}}^{\mathrm{2}}{\tilde{g}}_{\mathrm{6}}^{\mathrm{T}}{\widehat{T}}_{\mathrm{1}}{\tilde{g}}_{\mathrm{6}},\\ {\widehat{\Phi }}_{\mathrm{3}}={\phi }_{\mathrm{7}}^{\mathrm{T}}{\widehat{S}}_{\mathrm{1}}{\phi }_{\mathrm{7}}-\left(\mathrm{1}-\dot{d}\left(t\right)\right){\phi }_{\mathrm{8}}^{\mathrm{T}}\left({\widehat{S}}_{\mathrm{1}}-{\widehat{S}}_{\mathrm{2}}\right){\phi }_{\mathrm{8}}-{\phi }_{\mathrm{9}}^{\mathrm{T}}{\widehat{S}}_{\mathrm{2}}{\phi }_{\mathrm{9}},\\ {\widehat{\Phi }}_{\mathrm{4}N}=\mathrm{S}\mathrm{y}\mathrm{m}\left\{{\Lambda }_{\mathrm{1}N}^{\mathrm{T}}{\widehat{W}}_{\mathrm{1}N}+{\Lambda }_{\mathrm{2}N}^{\mathrm{T}}{\widehat{W}}_{\mathrm{2}N}\right\},\\ {\widehat{\Phi }}_{\mathrm{5}lm}=\mathrm{S}\mathrm{y}\mathrm{m}\left\{\left[{\tilde{g}}_{\mathrm{1}}^{\mathrm{T}}{\rho }_{\mathrm{1}}+{\tilde{g}}_{\mathrm{2}}^{\mathrm{T}}{\rho }_{\mathrm{2}}+{\tilde{g}}_{\mathrm{3}}^{\mathrm{T}}{\rho }_{\mathrm{3}}+{\tilde{g}}_{\mathrm{6}}^{\mathrm{T}}\right]{\widehat{\mathcal{F}}}_{lm}\right\},\\ {\widehat{\mathcal{F}}}_{lm}=\left(A_{l\mathrm{0}}{\widehat{X}}^{\mathrm{T}}+B_l{\widehat{H}}_m\right){\tilde{g}}_{\mathrm{1}}+A_{l\mathrm{1}}{\widehat{X}}^{\mathrm{T}}{\tilde{g}}_{\mathrm{2}}-{\widehat{X}}^{\mathrm{T}}{\tilde{g}}_{\mathrm{6}}.\end{array}$$

The proof of Theorem 11 is similar to the one of Theorem 10 which is omitted here for brevity. Moreover, the fuzzy control gains are given by ${\widehat{K}}_m={\widehat{H}}_m{\widehat{X}}^{-T}\hspace{0.17em}\hspace{0.17em}(m=\mathrm{1,2},\dots ,s)$.

5. Numerical Examples

In this section, we give two numerical examples to verify the effectiveness of the proposed methods, where Example 1 is widely used for the comparison of the admissible delay upper bound computed by the delay-dependent stability conditions of this paper and some existing results, and Example 2 is employed to confirm the validity of the controller design conditions.

5.1. Example 1

Consider system (3) with $s=\mathrm{2}$, $u(t)=\mathrm{0}$ and system matrices as follows: $$\begin{array}{c}\text{(57)}& A_{\mathrm{10}}=\left[\begin{array}{cc}-\mathrm{0.2}& \mathrm{0}\\ \mathrm{0}& -\mathrm{0.9}\end{array}\right],A_{\mathrm{11}}=\left[\begin{array}{cc}-\mathrm{1}& \mathrm{0}\\ -\mathrm{1}& -\mathrm{1}\end{array}\right],\\ A_{\mathrm{20}}=\left[\begin{array}{cc}-\mathrm{1}& \mathrm{0.5}\\ \mathrm{0}& -\mathrm{1}\end{array}\right],\\ A_{\mathrm{21}}=\left[\begin{array}{cc}-\mathrm{1}& \mathrm{0}\\ \mathrm{0.1}& -\mathrm{1}\end{array}\right].\end{array}$$

5.1.1. Constant Delay Case

When $d(t)\equiv d_{\mathrm{0}}$, using Theorem 3 for different $N\in \{\mathrm{1,2},\mathrm{3,4}\}$ and the results of some recent literature, the obtained admissible maximum time-delay upper bounds $d_{\mathrm{0}}$ are listed in Table 1 for comparison. Meanwhile, the computing complexity of the methods is compared by listing the number of decision variables. We can clearly see that for an integer $N\ge \mathrm{1}$, Theorem 3 proposed in this paper can provide a larger upper bound $d_{\mathrm{0}}$ than some existing literature, e.g., $\mathrm{1.6341}(s)$ in [25] and $\mathrm{1.9538}(s)$ in [26], which verifies that Theorem 3 is less conservative. But the computing complexity of Theorem 3 is larger than others.

Table 1

Maximum Delay Bound For Constant Delay.

Method	$d_{\mathrm{0}}(s)$	Number of variables
Corollary 1 in [25]	1.6341	$\mathrm{4.5}{n}^{\mathrm{2}}+\mathrm{2.5}n$
Theorem 1 in [26]	1.9538	$\mathrm{6}{n}^{\mathrm{2}}+\mathrm{3}n$
Theorem 1 in [18]	2.0477	$\mathrm{14.5}{n}^{\mathrm{2}}+\mathrm{4.5}n$
Theorem 3 (N=1)	2.0291	$\mathrm{19}{n}^{\mathrm{2}}+\mathrm{3}n$
Theorem 3 (N=2)	2.0477	$\mathrm{30.5}{n}^{\mathrm{2}}+\mathrm{3.5}n$
Theorem 3 (N=3)	2.0481	$\mathrm{45}{n}^{\mathrm{2}}+\mathrm{4}n$

5.1.2. Time-Varying Delay Case

Set the lower and upper bounds of the derivative of time delay to be $d_m=-\mathrm{0.1}$ and $d_M=\mathrm{0.1}$. Considering $(d(t),\dot{d}(t))\in {\mathcal{D}}_{\mathrm{1}}$, one can compute the admissible upper bound $d_{\mathrm{0}}=\mathrm{2.0291}(s)$ by Theorem 8 with $N=\mathrm{1}$. Using the existing results in [27–30], the obtained maximum delay bounds are listed in Table 2 for comparison. The results for $N=\mathrm{2}$, $\mathrm{3}$ and $\mathrm{4}$ are also shown in Table 2. From Table 2, it is clear that Theorem 8 produces a larger delay upper bound than Theorem 2 in [27] and others listed in Table 2, which implies that Theorem 8 it is less conservative.

Table 2

Maximum Delay Bound with $d_M=-d_m=\mathrm{0.1}$.

Method	$d_{\mathrm{0}}(s)$	Number of variables
Theorem 2 in [27]	1.4847	$\mathrm{59}{n}^{\mathrm{2}}+\mathrm{12}n$
Theorem 2 in [28]	1.5332	$\mathrm{23}{n}^{\mathrm{2}}+\mathrm{5}n$
Theorem 1 in [29] (m=3)	1.8090	$$\mathrm{16.5}{n}^{\mathrm{2}}+\mathrm{7.5}n$$
Theorem 1 in [30]	1.8447	$\mathrm{49}{n}^{\mathrm{2}}+\mathrm{5}n$
Theorem 8 (N=1)	1.9491	$\mathrm{73}{n}^{\mathrm{2}}+\mathrm{9}n$
Theorem 8 (N=2)	1.9690	$\mathrm{122}{n}^{\mathrm{2}}+\mathrm{11}n$
Theorem 8 (N=3)	1.9700	$\mathrm{185}{n}^{\mathrm{2}}+\mathrm{13}n$
Theorem 8 (N=4)	1.9711	$\mathrm{262}{n}^{\mathrm{2}}+\mathrm{15}n$

Assume that $(d(t),\dot{d}(t))\in {\mathcal{D}}_{\mathrm{2}}$. Then we use Theorem 9 for different $N\in \{\mathrm{1,2},\mathrm{3,4}\}$. The obtained maximum delay bounds are given in Table 3, from which one can see that Theorem 9 delivers a larger delay upper bound than Theorem 8. Tables 2 and 3 also show that both Theorems 8 and 9 ensure a larger $d_{\mathrm{0}}$ with increasing $N$.

Table 3

Maximum Delay Bound with $d_M=-d_m=\mathrm{0.1}$.

Method	$d_{\mathrm{0}}(s)$	Number of variables
Theorem 9 ($N=\mathrm{1}$)	2.0291	$\mathrm{73}{n}^{\mathrm{2}}+\mathrm{9}n$
Theorem 9 ($N=\mathrm{2}$)	2.0477	$\mathrm{122}{n}^{\mathrm{2}}+\mathrm{11}n$
Theorem 9 ($N=\mathrm{3}$)	2.0481	$\mathrm{185}{n}^{\mathrm{2}}+\mathrm{13}n$

Furthermore, to check that the system can tolerate the time-delay limited by the proposed results, we employ the simulation by Matlab. In simulation, let $o_{\mathrm{1}}(x_{\mathrm{1}}(t))=\mathrm{1}/\left(\mathrm{1}+e^{-\mathrm{2}{x}_{\mathrm{1}}(t)}\right)$ and $o_{\mathrm{2}}(x_{\mathrm{1}}(t))=e^{-\mathrm{2}{x}_{\mathrm{1}}(t)}/\left(\mathrm{1}+e^{-\mathrm{2}{x}_{\mathrm{1}}(t)}\right)$. Then we choose $d_{\mathrm{0}}=\mathrm{2.0481}(s)$ and set the initial condition $\varphi (t)={\left[\begin{array}{cc}\mathrm{1.5}& \mathrm{1}\end{array}\right]}^T$ for $t\in [-d_{\mathrm{0}},\mathrm{0}]$. We depict the states $x_{\mathrm{1}}(t)$ and $x_{\mathrm{2}}(t)$ by Figure 1, which confirms asymptotically stable responses.

[figure omitted; refer to PDF]

5.2. Example 2

Consider system (7) with s=2 and system matrices as follows: $$\begin{array}{c}\text{(58)}& A_{\mathrm{10}}=\left[\begin{array}{cc}\mathrm{0}& \mathrm{0.6}\\ \mathrm{0}& \mathrm{1}\end{array}\right],A_{\mathrm{11}}=\left[\begin{array}{cc}\mathrm{0.5}& \mathrm{0.9}\\ \mathrm{0}& \mathrm{2}\end{array}\right],\\ B_{\mathrm{1}}=\left[\begin{array}{c}\mathrm{1}\\ \mathrm{1}\end{array}\right],\\ A_{\mathrm{20}}=\left[\begin{array}{cc}\mathrm{1}& \mathrm{0}\\ \mathrm{1}& \mathrm{0}\end{array}\right],\\ A_{\mathrm{21}}=\left[\begin{array}{cc}\mathrm{0.9}& \mathrm{0}\\ \mathrm{1}& \mathrm{1.6}\end{array}\right],\\ B_{\mathrm{2}}=\left[\begin{array}{c}\mathrm{1}\\ \mathrm{1}\end{array}\right].\end{array}$$

Obviously, when $d_{\mathrm{0}}=\mathrm{0}$ and without any control input, both the subsystems of the above system are unstable, because $A_{\mathrm{10}}+A_{\mathrm{11}}$ and $A_{\mathrm{20}}+A_{\mathrm{21}}$ are not Hurwitz matrices; that is, it is an unstable system. Now we demonstrate that the designed controllers by Theorems 10 and 11 are effective, respectively.

5.2.1. Constant Delay Case

Using Theorem 10 with $N=\mathrm{2}$, $\varrho =\mathrm{0.5}$, the admissible maximum upper bound of constant delay can be computed to be $d_{\mathrm{0}}=\mathrm{1.1}(s)$, and the corresponding control gains are as follows:$$\begin{array}{c}\text{(59)}& K_{\mathrm{1}}=\left[\begin{array}{cc}\mathrm{67.6465}& -\mathrm{194.6495}\end{array}\right],K_{\mathrm{2}}=\left[\begin{array}{cc}\mathrm{65.3394}& -\mathrm{189.3471}\end{array}\right].\end{array}$$According to the feasibility of Theorem 10, one can say that the controller (6) with the proposed gains (59) ensures the stability of the closed-loop system.

Similar to Example 1, we let $o_{\mathrm{1}}(x_{\mathrm{1}}(t))=\mathrm{1}/\left(\mathrm{1}+e^{-\mathrm{2}{x}_{\mathrm{1}}(t)}\right)$ and $o_{\mathrm{2}}(x_{\mathrm{1}}(t))=e^{-\mathrm{2}{x}_{\mathrm{1}}(t)}/\left(\mathrm{1}+e^{-\mathrm{2}{x}_{\mathrm{1}}(t)}\right)$ and set the initial condition $\varphi (t)={\left[\begin{array}{cc}\mathrm{1.5}& \mathrm{1}\end{array}\right]}^T$ for $t\in [-d_{\mathrm{0}},\mathrm{0}]$. The state responses of the closed-loop system under the controller (6) with gains (59) are demonstrated by Figure 2 which shows a positive result. From Figure 2, one can say that closed-loop system under the controller (6) with gains (59) is asymptotically stable. Then it can be concluded that Theorem 10 is effective.

[figure omitted; refer to PDF]

5.2.2. Time-Varying Delay Case

Let $h_M=-h_m=\mathrm{0.1}$. Then, according to Theorem 11 with $N=\mathrm{1}$, ${\rho }_{\mathrm{1}}=\mathrm{0.5}$, ${\rho }_{\mathrm{2}}=\mathrm{0.2}$ and ${\rho }_{\mathrm{3}}=\mathrm{0.2}$, one can deliver the admissible maximum bound $d_{\mathrm{0}}=\mathrm{1.3176}(s)$ while the control gains are$$\begin{array}{c}\text{(60)}& K_{\mathrm{1}}=\left[\begin{array}{cc}\mathrm{148.7390}& -\mathrm{420.4591}\end{array}\right],K_{\mathrm{2}}=\left[\begin{array}{cc}\mathrm{148.6048}& -\mathrm{422.3894}\end{array}\right].\end{array}$$From the feasibility of Theorem 11, it can be considered that the closed-loop system can be stabilized by the controller (6) with the proposed gains (60).

For further confirmation, the normalized membership functions are set as $o_{\mathrm{1}}(x_{\mathrm{1}}(t))=\mathrm{1}/\left(\mathrm{1}+e^{-\mathrm{2}{x}_{\mathrm{1}}(t)}\right)$ and $o_{\mathrm{2}}(x_{\mathrm{1}}(t))=e^{-\mathrm{2}{x}_{\mathrm{1}}(t)}/\left(\mathrm{1}+e^{-\mathrm{2}{x}_{\mathrm{1}}(t)}\right)$. Besides, we introduce the initial condition $\varphi (t)={\left[\begin{array}{cc}\mathrm{1.5}& \mathrm{1}\end{array}\right]}^T$ for $t\in [-d_{\mathrm{0}},\mathrm{0}]$ and choose the time-varying delay $d\left(t\right)=\mathrm{1.2176}+\mathrm{0.1}\sin (t)$. The state responses in Figure 3 shows the positive effect of the controller (6) with gains (60) on the stabilization of the closed-loop system, which demonstrates the effectiveness of Theorem 11.

[figure omitted; refer to PDF]

6. Conclusion

In this paper, we have proposed an augmented LKF approach to derive some improved stability and stabilization conditions of fuzzy system with constant delay and interval time-varying delay with its derivative bounds available, respectively. In particular, the proposed LKF have been constructed on the basis of the form of the B-L inequality. Two numerical examples have illustrated the improvements of the obtained conditions comparing with some existing recently results and the design validity of fuzzy controllers.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported in part by the Natural Science Foundation of China under Grant 61773238, the Fundamental Research Funds of Shandong University, and the Program for the Outstanding Innovative Teams of Higher Learning Institutions of Shanxi.

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