New Relaxed Static Output Feedback Stabilization of T–S Fuzzy Systems with Time-Varying Delays

Abstract

This paper investigates the imperfect premise matching-based (IPMB) static output feedback (SOF) controller for T–S fuzzy systems with time-varying delays. Firstly, by employing integral inequality techniques, the membership-function-independent fuzzy SOF design methods in terms of LMIs are presented based on imperfect premise matching (IPM) strategy. The obtained stabilization conditions do not contain equality constraints, and the output matrices do not have rank constraints. Secondly, some suitable relaxation variables are employed, and more relaxed membership-function-dependent (MFD) stabilization conditions are obtained by considering the local boundary information of membership functions (MFs). Finally, two simulation examples are given to show the progressiveness of the proposed methods in this paper.

1. Introduction

Benefiting from the excellent characteristics, T–S fuzzy models can introduce the perfect linear systems analysis and synthesis methods into the control problem of nonlinear systems, which provide a significant method for the control of nonlinear systems [1,2,3,4,5,6,7,8,9]. Generally, time delays are often encountered in many dynamic systems, and may even lead to the performance degradation, oscillation, and instability of the systems [10,11,12]. Therefore, the issue of T–S fuzzy systems with time delays is widely considered, and many stability and stabilization results have been reported in [13,14,15,16,17,18,19,20,21,22,23,24]. In the above results, the state feedback laws were employed to stabilize the T–S fuzzy systems.

However, in many actual systems, the measurement of state variables is very difficult, sometimes even unmeasurable. Therefore, output feedback control approaches are usually employed. The SOF controller has the advantages of simple implementation and convenient application. In addition, many dynamic output feedback controllers can be redesigned as SOF controllers. Therefore, the research on SOF controllers has important theoretical and practical significance. However, when compared with the state feedback control, the output feedback control is more difficult to solve [25]. In the design of SOF controllers, due to the existence of cross terms between system matrices and control gain matrices, it is difficult to directly obtain less conservative stabilization conditions of time-delay systems in terms of LMIs. This is mainly because the necessary and sufficient conditions for SOF stabilization based on the Lyapunov method are essentially bilinear matrix inequality (BMI) problems. The SOF controllers were proposed for the T-S fuzzy system in [26,27,28]; however, the stabilization results were proposed in terms of BMIs. Equality constraints were introduced into the design conditions of SOF control in [29], but it inevitably brought great conservatism. A SOF controller was designed by using the cone complementarity linearization (CCL) algorithm in [30]. For a class of discrete generalized T–S fuzzy systems with time delays, a novel improved CCL algorithm was proposed in [31] to solve the fuzzy SOF controllers. In fact, the CCL method is based on iterative LMI. By equivalent transformation, the BMIs in the conditions of SOF controllers are transformed into LMIs containing a positive definite matrix and its inverse. It is worth noting that the iteration stop conditions of the CCL algorithm are relatively strict, which cannot guarantee the SOF controller satisfying the conditions, and the convergence speed is slow in the later stage. Based on some appropriate conservative treatment techniques, the design methods of SOF controllers for T–S fuzzy time-delay systems were presented based on LMIs technology in [32,33,34,35,36]. Further research is needed to obtain more relaxed stabilization results for T–S fuzzy time-delay systems through effective processing, especially the design methods of SOF controllers based on LMIs technology.

On the other hand, based on parallel distributed compensation (PDC) technology, the fuzzy SOF controllers were proposed in [26,27,28,29,30,31,32,33,34,35,36]. However, the PDC method may complicate the structure of fuzzy controllers [37]. To solve the above issue, IPM strategy was presented in [38], in which the MFs of fuzzy controllers can be selected arbitrarily. According to our best knowledge, the SOF control of T–S fuzzy time-delay systems based on IPM technology is almost in the blank field; only the fuzzy SOF controllers were designed based on IPM technology in [39,40,41]. In effect, the IPMB fuzzy SOF controllers proposed in [39,40] only considered the case of mismatched premise MFs, but did not consider the case of a different number of fuzzy rules. This does not fully reflect the design advantages of IPM strategy. In addition, the case of a constant time delay, rather than a time-varying delay, was considered in [40,41].

In this paper, we investigated the design of an IPMB fuzzy SOF controller for T–S fuzzy time-delay systems. The main contributions are summarized as follows:

(1) LMIs-based design methods of fuzzy SOF were presented based on the integral inequality technique. The obtained stabilization conditions did not contain equality constraints, and the output matrices did not have rank constraints.

(2) The local boundary information of MFs were considered, and the further relaxed MFD stabilization results were obtained.

(3) The premise MFs and number of fuzzy rules of the SOF controllers were allowed to be selected arbitrarily, which enhanced the flexibility of the controllers design.

2. Preliminaries

2.1. Fuzzy Time-Delay Models

Consider the following nonlinear system that can be represented by a T–S fuzzy model with time-varying delays, and the system has $\mathrm{p}$ plant rules:

Rule $i$: IF ${\theta }_1\left(x\left(t\right)\right)$ is ${\mathrm{M}}_1^i$ and ${\theta }_2\left(x\left(t\right)\right)$ is ${\mathrm{M}}_2^i$ and $\cdots$ and ${\theta }_{\chi }\left(x\left(t\right)\right)$ is ${\mathrm{M}}_{\chi }^i$, THEN

$$\{\begin{array}{l}\begin{array}{l}\dot{\mathit{x}}\left(t\right)={\mathbf{A}}_i\mathit{x}\left(t\right)+{\mathbf{A}}_{di}\mathit{x}\left(t-\tau \left(t\right)\right)+{\mathbf{B}}_i\mathit{u}\left(t\right)\\ \mathit{y}\left(t\right)={\mathbf{C}}_i\mathit{x}\left(t\right)\end{array}\hfill \\ \mathit{x}\left(t\right)=\mathbf{\varphi }\left(t\right),t\in \left[-{\mathsf{\tau }}_{\mathrm{M}},0\right]\hfill \end{array}$$

(1)

where ${\mathrm{M}}_{\alpha }^i$ denote the fuzzy sets, $\alpha =1,2,\cdots ,\chi$, $i=1,2,\cdots ,\mathrm{p}$; $x\left(t\right)\in {ℝ}^n$ denotes the state vector; $\mathit{u}\left(t\right)\in {ℝ}^m$ is the control input vector; $\mathit{y}\left(t\right)\in {ℝ}^q$ is the output vector; ${\mathbf{A}}_i$, ${\mathbf{A}}_{di}$, ${\mathbf{B}}_i$, ${\mathbf{C}}_i$ are constant real matrices; ${\theta }_1\left(x\left(t\right)\right),\dots ,{\theta }_{\chi }\left(x\left(t\right)\right)$ are the premise variables. The value $\mathbf{\varphi }\left(t\right)$ is the vector valued initial condition on $\left[-{\mathsf{\tau }}_{\mathrm{M}},0\right]$. The value $\tau \left(t\right)$ is time-varying delay and satisfies:

$$\{\begin{array}{l}0\le \tau \left(t\right)\le {\mathsf{\tau }}_{\mathrm{M}}\hfill \\ \dot{\tau }\left(t\right)\le \mu \hfill \end{array}$$

(2)

where ${\mathsf{\tau }}_{\mathrm{M}}$ and $\mathsf{\mu }$ are constants.

By fuzzy blending, the overall T–S fuzzy time-delay system can be expressed as in the following equations:

$$\{\begin{array}{l}\dot{\mathit{x}}\left(t\right)={\displaystyle \sum _{i=1}^{\mathrm{p}}}{m}_i\left(x\left(t\right)\right)\left({\mathbf{A}}_i\mathit{x}\left(t\right)+{\mathbf{A}}_{di}\mathit{x}\left(t-\tau \left(t\right)\right)+{\mathbf{B}}_i\mathit{u}\left(t\right)\right)\hfill \\ \mathit{y}\left(t\right)={\displaystyle \sum _{i=1}^{\mathrm{p}}}{m}_i\left(x\left(t\right)\right){\mathbf{C}}_i\mathit{x}\left(t\right)\hfill \end{array}$$

(3)

where $m_i\left(x\left(t\right)\right)=\frac{{\omega }_i\left(x\left(t\right)\right)}{{\displaystyle {\sum }_{i=1}^{\mathrm{p}}}{\omega }_i\left(x\left(t\right)\right)}$, and fuzzy weighting functions satisfy ${\displaystyle \sum _{i=1}^{\mathrm{p}}}{\omega }_i\left(x\left(t\right)\right)>0$, ${\displaystyle \sum _{i=1}^{\mathrm{p}}}{m}_i\left(x\left(t\right)\right)=1$.

2.2. Fuzzy SOF Controller under IPM

Based on the IPM technique, a fuzzy SOF controller with $\mathrm{c}$ rules can be organized as:

Rule $j$: IF ${\xi }_1\left(x\left(t\right)\right)$ is ${\mathrm{N}}_1^j$ and ${\xi }_2\left(x\left(t\right)\right)$ is ${\mathrm{N}}_2^j$ and $\cdots$ and ${\xi }_{\gamma }\left(x\left(t\right)\right)$ is ${\mathrm{N}}_{\gamma }^j$, THEN

$$\mathit{u}\left(t\right)={\mathbf{K}}_j\mathit{y}\left(t\right)$$

(4)

Therefore, the overall output of the fuzzy SOF controller can be represented as:

$$\mathit{u}\left(t\right)={\displaystyle \sum }_{j=1}^{\mathrm{c}}{h}_j\left(x\left(t\right)\right){\mathbf{K}}_j\mathit{y}\left(t\right)$$

(5)

where $h_j\left(x\left(t\right)\right)=\frac{{\varpi }_j\left(x\left(t\right)\right)}{{{\displaystyle \sum }}_{j=1}^{\mathrm{c}}{\varpi }_j\left(x\left(t\right)\right)}$, and fuzzy weighting functions satisfy ${\displaystyle \sum _{j=1}^{\mathrm{c}}{\varpi }_j\left(x\left(t\right)\right)>0}$, ${\displaystyle \sum _{j=1}^{\mathrm{c}}}{h}_j\left(x\left(t\right)\right)=1$. ${\mathbf{K}}_j\in {ℝ}^{m\times q}$, $j=1,2,\cdots ,\mathrm{c}$ are control gain matrices.

By substituting (5) into (3), the T–S fuzzy closed-loop system can be obtained:

$$\dot{\mathit{x}}\left(t\right)={\displaystyle \sum }_{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}\left(\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\mathit{x}\left(t\right)+{\mathbf{A}}_{di}\mathit{x}\left(t-\tau \left(t\right)\right)\right)$$

(6)

wherein, the closed-loop system can be represented as:

$$\dot{\mathit{x}}\left(t\right)=\left(\mathit{A}\left(t\right)+\mathit{B}\left(t\right)\mathit{K}\left(t\right)\mathit{C}\left(t\right)\right)\mathit{x}\left(t\right)+{\mathit{A}}_d\left(t\right)\mathit{x}\left(t-\tau \left(t\right)\right)$$

where

$$\mathit{A}\left(t\right)={\displaystyle \sum }_{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}{\mathbf{A}}_i$$

$$\mathit{B}\left(t\right)={\displaystyle \sum }_{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}{\mathbf{B}}_i$$

$$\mathit{C}\left(t\right)={\displaystyle \sum }_{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}{\mathbf{C}}_k$$

$$\mathit{K}\left(t\right)={\displaystyle \sum }_{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}{\mathbf{K}}_j$$

$${\mathit{A}}_d\left(t\right)={\displaystyle \sum }_{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}{\mathbf{A}}_{di}$$

In this paper, the main focus is to obtain the fuzzy SOF gain ${\mathbf{K}}_j$ based on the IPM technique so that the closed-loop system (6) is asymptotically stable.

2.3. Useful Lemmas

In order to obtain the main SOF stabilization results of this paper, we firstly give the following useful lemmas.

Lemma 1. 

Ref. [42]. For a positive definite matrix $\mathbf{R}>0$, and a differentiable function $x$: $\left[\mathrm{a},\mathrm{b}\right]\to {ℝ}^n$, the following inequality holds:

$${\displaystyle {\int }_{\mathrm{a}}^{\mathrm{b}}{\dot{\mathit{x}}}^{\mathrm{T}}\left(s\right)\mathbf{R}\dot{\mathit{x}}\left(s\right)}ds\ge \frac{1}{\mathrm{b}-\mathrm{a}}{\mathsf{\Omega }}^{\mathrm{T}}\mathbf{R}\mathsf{\Omega }$$

where

$$\mathsf{\Omega }=\mathit{x}\left(\mathrm{b}\right)-\mathit{x}\left(\mathrm{a}\right)$$

Lemma 2. 

Ref. [43]. Let $f_1,f_2,\cdots f_N$: ${ℝ}^m\to ℝ$ have positive values in an open subset $\mathrm{D}$ of ${ℝ}^m$. Then, the reciprocally convex combination of $f_i$ over $\mathrm{D}$ satisfies:

$$\left\{g_{i,j}:{\Re }^m\to \Re ,\hspace{0.17em}{g}_{j,i}\left(t\right)=g_{i,j}\left(t\right),\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{cc}{f}_i\left(t\right)& g_{i,j}\left(t\right)\\ g_{i,j}\left(t\right)& f_j\left(t\right)\end{array}\right]\ge 0\right\}$$

subject to

$$\underset{\left\{{\alpha }_i|{\alpha }_i>0,\hspace{0.17em}{\displaystyle {\sum }_i{\alpha }_i=1}\right\}}{\min }{\displaystyle \sum _i\frac{1}{{\alpha }_i}}{f}_i\left(t\right)={\displaystyle \sum _i{f}_i\left(t\right)}+\underset{g_{i,j}\left(t\right)}{\max }{\displaystyle \sum _{i\ne j}{g}_{i,j}\left(t\right)}$$

Lemma 3. 

Ref. [44]. For any vectors $\mathbf{X},\mathbf{Y}\in {ℝ}^m$, and a positive definite matrix $\mathbf{Q}\in {ℝ}^{n\times n}$, the following inequality holds:

$$2{\mathbf{X}}^{\mathrm{T}}\mathbf{Y}\le {\mathbf{X}}^{\mathrm{T}}\mathbf{Q}\mathbf{X}+{\mathbf{Y}}^{\mathrm{T}}{\mathbf{Q}}^{-1}\mathbf{Y}$$

Lemma 4. 

Ref. [45]. The following two inequalities are equivalent:

(1)

There exists a symmetric and positive-definite matrix $\mathbf{P}$ satisfying

$$\left[\begin{array}{cc}-\mathbf{P}& {\mathbf{A}}^{\mathrm{T}}\\ \mathbf{A}& -{\mathbf{P}}^{-1}\end{array}\right]<0$$

(2)

There exists a symmetric and positive-definite matrix $\mathbf{P}$ and matrix $\mathbf{Y}$ satisfying

$$\left[\begin{array}{cc}-\mathbf{P}& {\left(\mathbf{Y}\mathbf{A}\right)}^{\mathrm{T}}\\ \mathbf{Y}\mathbf{A}& Sym\left(-\mathbf{Y}\right)+\mathbf{P}\end{array}\right]<0$$

3. Main Results

3.1. Membership-Function-Independent Stabilization Conditions

In this subsection, based on IPM technology, the design methods of fuzzy the SOF controller in terms of LMIs are presented by combining with a reciprocally convex combination inequality.

Theorem 1. 

For given scalars ${\mathsf{\tau }}_{\mathrm{M}}>0$ and $\mathsf{\mu }$, the system (6) is asymptotically stable if there exist matrices $0<\mathbf{P}\in {ℝ}^{n\times n}$, $0<\mathbf{Q}\in {ℝ}^{n\times n}$, $0<\mathbf{S}\in {ℝ}^{n\times n}$, $0<\mathbf{R}\in {ℝ}^{n\times n}$, $0<\mathbf{J}\in {ℝ}^{2n\times 2n}$, and any matrices $\mathbf{M}\in {ℝ}^{n\times n}$, $\mathbf{X}\in {ℝ}^{m\times m}$, ${\mathbf{G}}_j\in {ℝ}^{m\times n}$, such that the following LMIs hold:

$$\left[\begin{array}{cc}\mathbf{R}& \mathbf{M}\\ {\mathbf{M}}^{\mathrm{T}}& \mathbf{R}\end{array}\right]>0$$

(7)

$${\mathit{\Sigma }}_{ijk}\left(\mathsf{\mu }\right)<0,\hspace{0.17em}\forall i=1,2,\cdots ,\mathrm{p};\hspace{0.17em}j=1,2,\cdots ,\mathrm{c};\hspace{0.17em}k=1,2,\cdots ,\mathrm{p}$$

(8)

The corresponding fuzzy SOF control gains are proposed by ${\mathbf{K}}_j={\mathbf{X}}^{-1}{\mathbf{G}}_j$, $j=1,2,\cdots ,\mathrm{c}$. where

$${\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)=\left[\begin{array}{cccc}{\mathit{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)& {\mathsf{\Gamma }}_{12i}& 0& 0\\ \ast & {\mathsf{\theta }}_{22i}+\mathbf{J}& {\mathsf{\theta }}_{23ijk}& 0\\ \ast & \ast & Sym\left\{-{\mathbf{B}}_i^{\mathrm{T}}{\mathbf{B}}_i\mathbf{X}\right\}& {\theta }_{34i}\\ \ast & \ast & \ast & -\mathbf{J}\end{array}\right]$$

$${\mathit{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)=\left[\begin{array}{cc}-\mathbf{S}-\mathbf{R}& \mathbf{R}-{\mathbf{M}}^{\mathrm{T}}\\ \ast & \left(1-\dot{\tau }\left(t\right)\right)\left(\mathbf{S}-\mathbf{Q}\right)+Sym\left\{\mathbf{M}\right\}-2\mathbf{R}\end{array}\right]$$

$${\mathsf{\Gamma }}_{12i}=\left[\begin{array}{cc}{\mathbf{M}}^{\mathrm{T}}& 0\\ Sym\left\{\mathbf{P}{\mathbf{A}}_i\right\}+\mathbf{R}-{\mathbf{M}}^{\mathrm{T}}& {\mathsf{\tau }}_{\mathrm{M}}{\left(\mathbf{R}{\mathbf{A}}_{di}\right)}^{\mathrm{T}}\end{array}\right]$$

$${\mathsf{\theta }}_{22i}=\left[\begin{array}{cc}Sym\left\{\mathbf{P}{\mathbf{A}}_i\right\}+\mathbf{Q}-\mathbf{R}& {\mathsf{\tau }}_{\mathrm{M}}{\left(\mathbf{R}{\mathbf{A}}_{di}\right)}^{\mathrm{T}}\\ \ast & -\mathbf{R}\end{array}\right]$$

$${\mathsf{\theta }}_{23ijk}=\left[\begin{array}{c}{\left({\mathbf{B}}_i^{\mathrm{T}}{\mathbf{B}}_i{\mathbf{G}}_j{\mathbf{C}}_k\right)}^{\mathrm{T}}\\ 0\end{array}\right]$$

$${\mathsf{\theta }}_{34i}=\left[\begin{array}{cc}{\mathbf{B}}_i^{\mathrm{T}}\mathbf{P}& {\mathbf{B}}_i^{\mathrm{T}}\mathbf{R}\end{array}\right]$$

Proof. 

Consider the following Lyapunov–Krasovskii functional candidate:

$$\begin{array}{l}V\left(t\right)={\mathit{x}}^{\mathrm{T}}\left(t\right)\mathbf{P}\mathit{x}\left(t\right)+{\displaystyle {\int }_{t-\tau (t)}^t{\mathit{x}}^{\mathbf{T}}\left(s\right)\mathbf{Q}\mathit{x}\left(s\right)}ds+{\displaystyle {\int }_{t-{\mathsf{\tau }}_{\mathrm{M}}}^{t-\tau (t)}{\mathit{x}}^{\mathrm{T}}\left(s\right)\mathbf{S}\mathit{x}\left(s\right)}ds\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\mathsf{\tau }}_{\mathrm{M}}{\displaystyle {\int }_{t-{\mathsf{\tau }}_{\mathrm{M}}}^t{\displaystyle {\int }_{\theta }^t{\dot{\mathit{x}}}^{\mathrm{T}}\left(s\right)}}\mathbf{R}\dot{\mathit{x}}\left(s\right)dsd\theta \end{array}$$

(9)

where $0<\mathbf{P}\in {ℝ}^{n\times n}$, $0<\mathbf{Q}\in {ℝ}^{n\times n}$, $0<\mathbf{S}\in {ℝ}^{n\times n}$, $0<\mathbf{R}\in {ℝ}^{n\times n}$ are the matrices to be solved.

Differentiating the derivative of $V\left(t\right)$ along the trajectory of the system (6) leads to

$$\begin{array}{l}\dot{V}\left(t\right)=2{\mathit{x}}^{\mathrm{T}}\left(t\right)\mathbf{P}\dot{\mathit{x}}\left(t\right)+{\mathit{x}}^{\mathrm{T}}\left(t\right)\mathbf{Q}\mathit{x}\left(t\right)-\left(1-\dot{\tau }\left(t\right)\right){\mathit{x}}^{\mathrm{T}}\left(t-\tau \left(t\right)\right)\mathbf{Q}\mathit{x}\left(t-\tau \left(t\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(1-\dot{\tau }\left(t\right)\right){\mathit{x}}^{\mathrm{T}}\left(t-\tau \left(t\right)\right)\mathbf{S}\mathit{x}\left(t-\tau \left(t\right)\right)-{\mathit{x}}^{\mathrm{T}}\left(t-{\mathsf{\tau }}_{\mathrm{M}}\right)\mathbf{S}\mathit{x}\left(t-{\mathsf{\tau }}_{\mathrm{M}}\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mathsf{\tau }}_{\mathrm{M}}^2{\dot{\mathit{x}}}^{\mathrm{T}}\left(t\right)\mathbf{R}\dot{\mathit{x}}\left(t\right)-{\mathsf{\tau }}_{\mathrm{M}}{\displaystyle {\int }_{t-{\mathsf{\tau }}_{\mathrm{M}}}^t{\dot{\mathit{x}}}^{\mathrm{T}}\left(s\right)\mathbf{R}\dot{\mathit{x}}\left(s\right)ds}\\ \hspace{1em}\hspace{1em}={\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}(2{\mathit{x}}^{\mathrm{T}}\left(t\right)\mathbf{P}(\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\mathit{x}\left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mathbf{A}}_{di}\mathit{x}\left(t-\tau \left(t\right)\right))+{\mathit{x}}^{\mathrm{T}}\left(t\right)\mathbf{Q}\mathit{x}\left(t\right)-\left(1-\dot{\tau }\left(t\right)\right){\mathit{x}}^{\mathrm{T}}\left(t-\tau \left(t\right)\right)\mathbf{Q}\mathit{x}\left(t-\tau \left(t\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(1-\dot{\tau }\left(t\right)\right){\mathit{x}}^{\mathrm{T}}\left(t-\tau \left(t\right)\right)\mathbf{S}\mathit{x}\left(t-\tau \left(t\right)\right)-{\mathit{x}}^{\mathrm{T}}\left(t-{\mathsf{\tau }}_{\mathrm{M}}\right)\mathbf{S}\mathit{x}\left(t-{\mathsf{\tau }}_{\mathrm{M}}\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\mathsf{\tau }}_{\mathrm{M}}^2{\left(\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\mathit{x}\left(t\right)+{\mathbf{A}}_{di}\mathit{x}\left(t-\tau \left(t\right)\right)\right)}^{\mathrm{T}}\mathbf{R}\left(\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\mathit{x}\left(t\right)+{\mathbf{A}}_{di}\mathit{x}\left(t-\tau \left(t\right)\right)\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\mathsf{\tau }}_{\mathrm{M}}{\displaystyle {\int }_{t-\tau \left(t\right)}^t{\dot{\mathit{x}}}^{\mathrm{T}}\left(s\right)\mathbf{R}\dot{\mathit{x}}\left(s\right)}ds-{\mathsf{\tau }}_{\mathrm{M}}{\displaystyle {\int }_{t-{\mathsf{\tau }}_{\mathrm{M}}}^{t-\tau \left(t\right)}{\dot{\mathit{x}}}^{\mathrm{T}}\left(s\right)\mathbf{R}\dot{\mathit{x}}\left(s\right)}ds\end{array}$$

(10)

The integral terms in (10) are estimated by employing the Jensen integral inequality in Lemma 1, and the following inequalities are obtained:

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\mathsf{\tau }}_{\mathrm{M}}{\displaystyle {\int }_{t-\tau \left(t\right)}^t{\dot{\mathit{x}}}^{\mathrm{T}}\left(s\right)\mathbf{R}\dot{\mathit{x}}\left(s\right)ds}\\ \le -\frac{{\mathsf{\tau }}_{\mathrm{M}}}{\tau \left(t\right)}{\left(\mathit{x}\left(t\right)-\mathit{x}\left(t-\tau \left(t\right)\right)\right)}^{\mathrm{T}}\mathbf{R}\left(\mathit{x}\left(t\right)-\mathit{x}\left(t-\tau \left(t\right)\right)\right)\\ =-\frac{{\mathsf{\tau }}_{\mathrm{M}}}{\tau \left(t\right)}{\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right){\left({\mathbf{e}}_1-{\mathbf{e}}_2\right)}^{\mathrm{T}}\mathbf{R}\left({\mathbf{e}}_1-{\mathbf{e}}_2\right)\xi \left(t\right)\end{array}$$

(11)

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\mathsf{\tau }}_{\mathrm{M}}{\displaystyle {\int }_{t-{\mathsf{\tau }}_{\mathrm{M}}}^{t-\tau \left(t\right)}{\dot{\mathit{x}}}^{\mathrm{T}}\left(s\right)\mathbf{R}\dot{\mathit{x}}\left(s\right)ds}\\ \le -\frac{{\mathsf{\tau }}_{\mathrm{M}}}{{\mathsf{\tau }}_{\mathrm{M}}-\tau \left(t\right)}{\left(\mathit{x}\left(t-\tau \left(t\right)\right)-\mathit{x}\left(t-{\mathsf{\tau }}_{\mathrm{M}}\right)\right)}^{\mathrm{T}}\mathbf{R}\left(\mathit{x}\left(t-\tau \left(t\right)\right)-\mathit{x}\left(t-{\tau }_M\right)\right)\\ =-\frac{{\mathsf{\tau }}_{\mathrm{M}}}{{\mathsf{\tau }}_{\mathrm{M}}-\tau \left(t\right)}{\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right){\left({\mathbf{e}}_2-{\mathbf{e}}_3\right)}^{\mathrm{T}}\mathbf{R}\left({\mathbf{e}}_2-{\mathbf{e}}_3\right)\xi \left(t\right)\end{array}$$

(12)

where

$$\xi \left(t\right)={\left[\begin{array}{ccc}{\mathit{x}}^{\mathrm{T}}\left(t\right)& {\mathit{x}}^{\mathrm{T}}\left(t-\tau \left(t\right)\right)& {\mathit{x}}^{\mathrm{T}}\left(t-{\mathsf{\tau }}_{\mathrm{M}}\right)\end{array}\right]}^{\mathrm{T}}$$

$${\mathbf{e}}_{\sigma }=\left[\begin{array}{ccc}{0}_{n\times \left(\sigma -1\right)n}& {\mathbf{I}}_n& 0_{n\times \left(3-\sigma \right)n}\end{array}\right],\sigma =1,2,3$$

According to Lemma 2, if there exists a matrix $\mathbf{M}\in {ℝ}^{n\times n}$ satisfying matrix inequality (7), then we can obtain

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-{\mathsf{\tau }}_{\mathrm{M}}{\displaystyle {\int }_{h-{\mathsf{\tau }}_{\mathrm{M}}}^t{\dot{\mathit{x}}}^{\mathrm{T}}\left(s\right)\mathbf{R}\dot{\mathit{x}}\left(s\right)ds}\\ =-{\mathsf{\tau }}_{\mathrm{M}}{\displaystyle {\int }_{t-\tau \left(t\right)}^t{\dot{\mathit{x}}}^{\mathrm{T}}\left(s\right)\mathbf{R}\dot{\mathit{x}}\left(s\right)}ds-{\mathsf{\tau }}_{\mathrm{M}}{\displaystyle {\int }_{t-{\mathsf{\tau }}_{\mathrm{M}}}^{t-\tau \left(t\right)}{\dot{\mathit{x}}}^{\mathrm{T}}\left(s\right)\mathbf{R}\dot{\mathit{x}}\left(s\right)}ds\\ \le {\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right){\left(\begin{array}{c}{\mathbf{e}}_1-{\mathbf{e}}_2\\ {\mathbf{e}}_2-{\mathbf{e}}_3\end{array}\right)}^{\mathrm{T}}\left[\begin{array}{cc}\mathbf{R}& \mathbf{M}\\ {\mathbf{M}}^{\mathrm{T}}& \mathbf{R}\end{array}\right]\left(\begin{array}{c}{\mathbf{e}}_1-{\mathbf{e}}_2\\ {\mathbf{e}}_2-{\mathbf{e}}_3\end{array}\right)\xi \left(t\right)\end{array}$$

(13)

Combining Formulas (10) and (13), and we can obtain

$$\dot{V}\left(t\right)\le {\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right)\left[{\Xi }_{ijk}\left(\dot{\tau }\left(t\right)\right)+{\mathbf{\gamma }}_{ijk}^{\mathrm{T}}{\mathbf{R}}^{-1}{\mathbf{\gamma }}_{ijk}\right]\xi \left(t\right)$$

(14)

where

$${\Xi }_{ijk}\left(\dot{\tau }\left(t\right)\right)=\left[\begin{array}{ccc}Sym\left\{\mathbf{P}\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\right\}+\mathbf{Q}-\mathbf{R}& Sym\left\{\mathbf{P}{\mathbf{A}}_i\right\}+\mathbf{R}-\mathbf{M}& \mathbf{M}\\ \ast & \left(1-\dot{\tau }\left(t\right)\right)\left(\mathbf{S}-\mathbf{Q}\right)+Sym\left\{\mathbf{M}\right\}-2\mathbf{R}& \mathbf{R}-\mathbf{M}\\ \ast & \ast & -\mathbf{S}-\mathbf{R}\end{array}\right]$$

$${\mathbf{\gamma }}_{ijk}=\left[\begin{array}{ccc}{\mathsf{\tau }}_{\mathrm{M}}\mathbf{R}\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)& {\mathsf{\tau }}_{\mathrm{M}}\mathbf{R}{\mathbf{A}}_{di}& 0\end{array}\right]$$

At this time, if inequality ${\Xi }_{ijk}\left(\dot{\tau }\left(t\right)\right)+{\mathbf{\gamma }}_{ijk}^{\mathrm{T}}{\mathbf{R}}^{-1}{\mathbf{\gamma }}_{ijk}<0$ holds, the closed-loop system (6) is asymptotically stable. According to the Schur complement lemma, inequality ${\Xi }_{ijk}\left(\dot{\tau }\left(t\right)\right)+{\mathbf{\gamma }}_{ijk}^{\mathrm{T}}{\mathbf{R}}^{-1}{\mathbf{\gamma }}_{ijk}<0$ is equivalent to the following inequality:

$$\begin{array}{l}[\begin{array}{ccc}Sym\left\{\mathbf{P}\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\right\}+\mathbf{Q}-\mathbf{R}& Sym\left\{\mathbf{P}{\mathbf{A}}_i\right\}+\mathbf{R}-\mathbf{M}& \mathbf{M}\\ \ast & \left(1-\dot{\tau }\left(t\right)\right)\left(\mathbf{S}-\mathbf{Q}\right)+S\mathit{y}m\left\{\mathbf{M}\right\}-2\mathbf{R}& \mathbf{R}-\mathbf{M}\\ \ast & \ast & -\mathbf{S}-\mathbf{R}\\ \ast & \ast & \ast \end{array}\\ \begin{array}{c}{\mathsf{\tau }}_{\mathrm{M}}{\left(\mathbf{R}\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\right)}^{\mathrm{T}}\\ {\mathsf{\tau }}_{\mathrm{M}}{\left(\mathbf{R}{\mathbf{A}}_{di}\right)}^{\mathrm{T}}\\ 0\\ -\mathbf{R}\end{array}]<0\end{array}$$

(15)

Let $\Delta ={\left[\begin{array}{cccc}{\tilde{\mathbf{e}}}_3& {\tilde{\mathbf{e}}}_2& {\tilde{\mathbf{e}}}_1& {\tilde{\mathbf{e}}}_4\end{array}\right]}^{\mathrm{T}}$, where ${\tilde{\mathbf{e}}}_{\sigma }=\left[\begin{array}{ccc}{0}_{n\times \left(\sigma -1\right)n}& {\mathbf{I}}_n& 0_{n\times \left(4-\sigma \right)n}\end{array}\right],\sigma =1,2,3,4$. By pre-multiplying the side of (15) with $\Delta$ and post-multiplying the side of (15) with ${\Delta }^{\mathrm{T}}$, we can obtain

$$\begin{array}{l}[\begin{array}{ccc}-\mathbf{S}-\mathbf{R}& \mathbf{R}-{\mathbf{M}}^{\mathrm{T}}& {\mathbf{M}}^{\mathrm{T}}\\ \ast & \left(1-\dot{\tau }\left(t\right)\right)\left(\mathbf{S}-\mathbf{Q}\right)+Sym\left\{\mathbf{M}\right\}-2\mathbf{R}& Sym\left\{\mathbf{P}{\mathbf{A}}_i\right\}+\mathbf{R}-{\mathbf{M}}^{\mathrm{T}}\\ \ast & \ast & Sym\left\{\mathbf{P}\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\right\}+\mathbf{Q}-\mathbf{R}\\ \ast & \ast & \ast \end{array}\\ \begin{array}{c}0\\ {\mathsf{\tau }}_{\mathrm{M}}{\left(\mathbf{R}{\mathbf{A}}_{di}\right)}^{\mathrm{T}}\\ {\mathsf{\tau }}_{\mathrm{M}}{\left(\mathbf{R}\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\right)}^{\mathrm{T}}\\ -\mathbf{R}\end{array}]<0\end{array}$$

(16)

For the convenience of proof, inequality (16) is rewritten as follows:

$$\left[\begin{array}{cc}{\mathit{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)& {\mathsf{\Gamma }}_{12i}\\ \ast & {\mathsf{\Gamma }}_{22ijk}\end{array}\right]<0$$

(17)

where

$${\mathit{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)=\left[\begin{array}{cc}-\mathbf{S}-\mathbf{R}& \mathbf{R}-{\mathbf{M}}^{\mathrm{T}}\\ \ast & \left(1-\dot{\tau }\left(t\right)\right)\left(\mathbf{S}-\mathbf{Q}\right)+Sym\left\{\mathbf{M}\right\}-2\mathbf{R}\end{array}\right]$$

$${\mathsf{\Gamma }}_{12i}=\left[\begin{array}{cc}{\mathbf{M}}^{\mathrm{T}}& 0\\ Sym\left\{\mathbf{P}{\mathbf{A}}_i\right\}+\mathbf{R}-{\mathbf{M}}^{\mathrm{T}}& {\mathsf{\tau }}_{\mathrm{M}}{\left(\mathbf{R}{\mathbf{A}}_{di}\right)}^{\mathrm{T}}\end{array}\right]$$

$${\mathsf{\Gamma }}_{22ijk}=\left[\begin{array}{cc}Sym\left\{\mathbf{P}\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\right\}+\mathbf{Q}-\mathbf{R}& {\mathsf{\tau }}_{\mathrm{M}}{\left(\mathbf{R}\left({\mathbf{A}}_i+{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)\right)}^{\mathrm{T}}\\ \ast & -\mathbf{R}\end{array}\right]$$

Rewrite ${\mathsf{\Gamma }}_{22ijk}$ as follows:

$${\mathsf{\Gamma }}_{22ijk}=\left[\begin{array}{cc}Sym\left\{\mathbf{P}{\mathbf{A}}_i\right\}+\mathbf{Q}-\mathbf{R}& {\mathsf{\tau }}_{\mathrm{M}}{\mathbf{A}}_i{}^{\mathrm{T}}\mathbf{R}\\ \ast & -\mathbf{R}\end{array}\right]\hspace{0.17em}+\left[\begin{array}{cc}\mathbf{P}{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k& 0\\ \mathbf{R}{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k& 0\end{array}\right]\hspace{0.17em}+\left[\begin{array}{cc}{\left(\mathbf{P}{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)}^{\mathrm{T}}& {\left(\mathbf{P}{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k\right)}^{\mathrm{T}}\\ 0& 0\end{array}\right]$$

(18)

Let ${\mathbf{K}}_j={\mathbf{X}}^{-1}{\mathbf{G}}_j$, where $\mathbf{X}\in {ℝ}^{m\times m}$ is an arbitrary invertible matrix. ${\mathbf{G}}_j\in {ℝ}^{m\times q}$ is an arbitrary matrix, and the following equation can be obtained:

$$\left[\begin{array}{cc}\mathbf{P}{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k& 0\\ \mathbf{R}{\mathbf{B}}_i{\mathbf{K}}_j{\mathbf{C}}_k& 0\end{array}\right]=\left[\begin{array}{cc}\mathbf{P}{\mathbf{B}}_i& 0\\ \mathbf{R}{\mathbf{B}}_i& 0\end{array}\right]\left[\begin{array}{cc}{\mathbf{X}}^{-1}{\mathbf{G}}_j{\mathbf{C}}_k& 0\\ 0& 0\end{array}\right]$$

Defining ${\mathbf{W}}_i=\left[\begin{array}{c}\mathbf{P}{\mathbf{B}}_i\\ \mathbf{R}{\mathbf{B}}_i\end{array}\right]$, ${\mathbf{T}}_{jk}=\left[\begin{array}{cc}{\mathbf{X}}^{-1}{\mathbf{G}}_j{\mathbf{C}}_k& 0\end{array}\right]$, and we rewrite matrix ${\mathsf{\Gamma }}_{22ijk}$ as:

$${\mathsf{\Gamma }}_{22ijk}={\mathsf{\theta }}_{22i}+Sym\left\{{\mathbf{W}}_i{\mathbf{T}}_{jk}\right\}$$

(19)

From (17) and (19), one has

$$\left[\begin{array}{cc}{\mathit{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)& {\mathsf{\Gamma }}_{12i}\\ \ast & {\mathsf{\Gamma }}_{22ijk}\end{array}\right]=\left[\begin{array}{cc}{\mathsf{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)& {\mathsf{\Gamma }}_{12i}\\ \ast & {\mathsf{\theta }}_{22i}\end{array}\right]+Sym\left\{\left[\begin{array}{c}{0}_{2n\times 2n}\\ {\mathbf{I}}_{2n}\end{array}\right]{\mathbf{W}}_i{\mathbf{T}}_{jk}\left[\begin{array}{cc}{0}_{2n\times 2n}& {\mathbf{I}}_{2n}\end{array}\right]\right\}<0$$

(20)

According to Lemma 3, there exists a positive definite matrix $\mathbf{J}\in {ℝ}^{2n\times 2n}$, such that the following inequality holds:

$$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Sym\left\{\left[\begin{array}{c}{0}_{2n\times 2n}\\ {\mathbf{I}}_{2n}\end{array}\right]{\mathbf{W}}_i{\mathbf{T}}_{jk}\left[\begin{array}{cc}{0}_{2n\times 2n}& {\mathbf{I}}_{2n}\end{array}\right]\right\}\\ \le \left[\begin{array}{c}{0}_{2n\times 2n}\\ {\mathbf{I}}_{2n}\end{array}\right]{\left[\begin{array}{cc}{\mathbf{X}}^{-1}{\mathbf{G}}_j{\mathbf{C}}_k& 0\end{array}\right]}^{\mathrm{T}}{\left[\begin{array}{c}\mathbf{P}{\mathbf{B}}_i\\ \mathbf{R}{\mathbf{B}}_i\end{array}\right]}^{\mathrm{T}}{\mathbf{J}}^{-1}\left[\begin{array}{c}\mathbf{P}{\mathbf{B}}_i\\ \mathbf{R}{\mathbf{B}}_i\end{array}\right]\left[\begin{array}{cc}{\mathbf{X}}^{-1}{\mathbf{G}}_j{\mathbf{C}}_k& 0\end{array}\right]\left[\begin{array}{cc}{0}_{2n\times 2n}& {\mathbf{I}}_{2n}\end{array}\right]\\ \hspace{0.17em}\hspace{0.17em}+\left[\begin{array}{c}{0}_{2n\times 2n}\\ {\mathbf{I}}_{2n}\end{array}\right]\mathbf{J}\left[\begin{array}{cc}{0}_{2n\times 2n}& {\mathbf{I}}_{2n}\end{array}\right]\end{array}$$

(21)

Combining (20) and (21), we can obtain:

$$\begin{array}{l}\left[\begin{array}{cc}{\mathit{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)& {\mathsf{\Gamma }}_{12i}\\ \ast & {\mathsf{\Gamma }}_{22ijk}\end{array}\right]\le \left[\begin{array}{cc}{\mathit{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)& {\mathsf{\Gamma }}_{12i}\\ \ast & {\mathsf{\theta }}_{22i}\end{array}\right]+\left[\begin{array}{c}{0}_{2n\times 2n}\\ {\mathbf{I}}_{2n}\end{array}\right]{\left[\begin{array}{cc}{\mathbf{X}}^{-1}{\mathbf{G}}_j{\mathbf{C}}_k& 0\end{array}\right]}^{\mathrm{T}}{\left[\begin{array}{c}\mathbf{P}{\mathbf{B}}_i\\ \mathbf{R}{\mathbf{B}}_i\end{array}\right]}^{\mathrm{T}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\times {\mathbf{J}}^{-1}\left[\begin{array}{c}\mathbf{P}{\mathbf{B}}_i\\ \mathbf{R}{\mathbf{B}}_i\end{array}\right]\left[\begin{array}{cc}{\mathbf{X}}^{-1}{\mathbf{G}}_j{\mathbf{C}}_k& 0\end{array}\right]\left[\begin{array}{cc}{0}_{2n\times 2n}& {\mathbf{I}}_{2n}\end{array}\right]+\left[\begin{array}{c}{0}_{2n\times 2n}\\ {\mathbf{I}}_{2n}\end{array}\right]\mathbf{J}\left[\begin{array}{cc}{0}_{2n\times 2n}& {\mathbf{I}}_{2n}\end{array}\right]<0\end{array}$$

(22)

According to the Schur complement lemma, inequality (22) is equivalent to the following inequality:

$$\left[\begin{array}{ccc}{\mathit{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)& {\mathsf{\Gamma }}_{12i}& 0\\ \ast & {\mathsf{\theta }}_{22i}+\mathbf{J}& {\Delta }_{23jk}\\ \ast & \ast & {\mathsf{\theta }}_{33i}\end{array}\right]<0$$

(23)

where

$${\mathsf{\theta }}_{22i}=\left[\begin{array}{cc}Sym\left\{\mathbf{P}{\mathbf{A}}_i\right\}+\mathbf{Q}-\mathbf{R}& {\mathsf{\tau }}_{\mathrm{M}}{\left(\mathbf{R}{\mathbf{A}}_{di}\right)}^{\mathrm{T}}\\ \ast & -\mathbf{R}\end{array}\right]$$

$${\Delta }_{23jk}=\left[\begin{array}{c}{\left({\mathbf{X}}^{-1}{\mathbf{G}}_j{\mathbf{C}}_k\right)}^{\mathrm{T}}\\ 0\end{array}\right]$$

$${\mathsf{\theta }}_{33i}=-{\left({\left[\begin{array}{c}\mathbf{P}{\mathbf{B}}_i\\ \mathbf{R}{\mathbf{B}}_i\end{array}\right]}^{\mathrm{T}}{\mathbf{J}}^{-1}\left[\begin{array}{c}\mathbf{P}{\mathbf{B}}_i\\ \mathbf{R}{\mathbf{B}}_i\end{array}\right]\right)}^{-1}$$

According to Lemma 4, inequality (23) is equivalent to the following inequality:

$$\left[\begin{array}{ccc}{\mathit{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)& {\mathsf{\Gamma }}_{12i}& 0\\ \ast & {\mathsf{\theta }}_{22i}+\mathbf{J}& {\Delta }_{23jk}{\mathbf{X}}^{\mathrm{T}}{\mathbf{B}}_i^{\mathrm{T}}{\mathbf{B}}_i\\ \ast & \ast & Sym\left\{-{\mathbf{B}}_i^{\mathrm{T}}{\mathbf{B}}_i\mathbf{X}\right\}-{\mathsf{\theta }}_{33i}^{-1}\end{array}\right]<0$$

(24)

Employing Schur complement lemma for inequality (24) again, and inequality (24) is equivalent to the following inequality:

$${\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)=\left[\begin{array}{cccc}{\mathit{\Gamma }}_{11}\left(\dot{\tau }\left(t\right)\right)& {\mathsf{\Gamma }}_{12i}& 0& 0\\ \ast & {\mathsf{\theta }}_{22i}+\mathbf{J}& {\Delta }_{23jk}{\mathbf{X}}^{\mathrm{T}}{\mathbf{B}}_i^{\mathrm{T}}{\mathbf{B}}_i& 0\\ \ast & \ast & Sym\left\{-{\mathbf{B}}_i^{\mathrm{T}}{\mathbf{B}}_i\mathbf{X}\right\}& {\mathsf{\theta }}_{34i}\\ \ast & \ast & \ast & -\mathbf{J}\end{array}\right]<0$$

(25)

where

$${\mathsf{\theta }}_{34i}=\left[\begin{array}{cc}{\mathbf{B}}_i^{\mathrm{T}}\mathbf{P}& {\mathbf{B}}_i^{\mathrm{T}}\mathbf{R}\end{array}\right]$$

Moreover, we can obtain:

$$\dot{V}\left(t\right)\le {\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{m}_i\left(\mathit{x}\left(t\right)\right)h_j\left(\mathit{x}\left(t\right)\right)m_k\left(\mathit{x}\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right){\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)\xi \left(t\right)<0$$

(26)

Therefore, from (7) and (8), we can obtain $\dot{V}\left(t\right)<0$. This proof is completed. □

3.2. MFD Stabilization Conditions

To obtain the less conservative stabilization conditions, the piecewise membership function (PMF) concept is introduced firstly based on the results in [46].

Suppose ${\mathsf{\Gamma }}_l$, $l=1,2,\cdots ,\mathsf{\vartheta }$ represent the connected state subspaces of the operating domain $\mathsf{\Gamma }$, such that $\mathsf{\Gamma }={\cup }_{l=1}^{\mathsf{\vartheta }}{\mathsf{\Gamma }}_l$. Defining a vector $\mathsf{{\rm X}}=\left[x_1\hspace{0.17em}\cdots \hspace{0.17em}{x}_{\mathsf{\eta }}\right]$ and considering $\mathsf{{\rm X}}\in {\mathsf{\Gamma }}_l$, we have the infimum and supremum of $x_r$ as ${\underset{\_}{x}}_{r{i}_{r}l}$ and ${\overline{x}}_{r{i}_{r}l}$, $r=1,2,\cdots ,\mathsf{\eta }$, $i_r=1,\hspace{0.17em}2$, respectively, such that ${\underset{\_}{x}}_{r{i}_{r}l}\le x_r\left(t\right)\le {\overline{x}}_{r{i}_{r}l}$. The vertices of the subspaces ${\mathsf{\Gamma }}_l$ are denoted as $x_{i_1\cdots i_{\mathsf{\eta }}l}=\left[x_{1i_{1}l}\hspace{0.17em}\cdots \hspace{0.17em}{x}_{\mathsf{\eta }{i}_{\mathsf{\eta }}l}\right]$, and the PMF is defined as:

$$\widehat{\omega }\left(x\left(t\right)\right)={\displaystyle \sum _{l=1}^{\mathsf{\vartheta }}{\displaystyle \sum _{i_1=1}^2\cdots {\displaystyle \sum _{i_{\mathsf{\eta }}=1}^2{\displaystyle \prod _{r=1}^{\mathsf{\eta }}{\nu }_{r{i}_{r}l}\left(x_r\left(t\right)\right)\widehat{\omega }\left(x_{i_1\cdots i_{\mathsf{\eta }}l}\right)}}}}$$

(27)

and the function ${\nu }_{r{i}_{r}l}\left(x_r\left(t\right)\right)$ has the following characteristics:

(1) $0\le {\nu }_{r{i}_{r}l}\left(x_r\left(t\right)\right)\le 1$, ${\displaystyle \sum _{i_r=1}^2{\nu }_{r{i}_{r}l}\left(x_r\left(t\right)\right)}=1$ for all $r$, $l$, and $x\in {\mathsf{\Gamma }}_l$, otherwise, ${\nu }_{r{i}_{r}l}\left(x_r\left(t\right)\right)=0$;

(2) ${\displaystyle \sum _{l=1}^{\mathsf{\vartheta }}{\displaystyle \sum _{i_1=1}^2\cdots {\displaystyle \sum _{i_{\mathsf{\eta }}=1}^2{\displaystyle \prod _{r=1}^{\mathsf{\eta }}{\nu }_{r{i}_{r}l}\left(x_r\left(t\right)\right)=1}}}}$.

Theorem 2. 

For given scalars ${\mathsf{\tau }}_{\mathrm{M}}>0$ and $\mathsf{\mu }$, the system (6) is asymptotically stable if there exist matrices $0<\mathbf{P}\in {ℝ}^{n\times n}$, $0<\mathbf{Q}\in {ℝ}^{n\times n}$, $0<\mathbf{S}\in {ℝ}^{n\times n}$, $0<\mathbf{R}\in {ℝ}^{n\times n}$, $0<{\mathbf{F}}_{ijk}\in {ℝ}^{\left(6n+m\right)\times \left(6n+m\right)}$, $\mathbf{E}={\mathbf{E}}^{\mathrm{T}}\in {ℝ}^{\left(6n+m\right)\times \left(6n+m\right)}$, $0<\mathbf{J}\in {ℝ}^{2n\times 2n}$ and any matrices $\mathbf{M}\in {ℝ}^{n\times n}$, $\mathbf{X}\in {ℝ}^{m\times m}$, ${\mathbf{G}}_j\in {ℝ}^{m\times n}$, such that the following LMIs hold:

$$\left[\begin{array}{cc}\mathbf{R}& \mathbf{M}\\ {\mathbf{M}}^{\mathrm{T}}& \mathbf{R}\end{array}\right]>0$$

(28)

$${\mathit{\Sigma }}_{ijk}\left(\mathsf{\mu }\right)+\mathbf{E}-{\mathbf{F}}_{ijk}<0;\hspace{0.17em}\forall i=1,2,\cdots ,\mathrm{p};\hspace{0.17em}j=1,2,\cdots ,\mathrm{c};\hspace{0.17em}k=1,2,\cdots ,\mathrm{p}$$

(29)

$$\begin{array}{l}{\displaystyle \sum _{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\left({\widehat{\omega }}_{ijk}\left(x_{i_1{i}_2\cdots i_{n}l}\right){\mathit{\Sigma }}_{ijk}\left(\mathsf{\mu }\right)+\Delta {\underset{\_}{\omega }}_{ijk}\left({\mathit{\Sigma }}_{ijk}\left(\mathsf{\mu }\right)+\mathbf{E}\right)+\left(\Delta {\overline{\omega }}_{ijk}-\Delta {\underset{\_}{\omega }}_{ijk}\right){\mathbf{F}}_{ijk}\right)<0}}}\\ \hspace{1em}\hspace{1em}\forall i_1,i_2,\cdots ,i_n\in \left\{1,2\right\};\hspace{0.17em}l=1,2,\cdots ,\mathsf{\vartheta }\hspace{1em}\end{array}$$

(30)

The fuzzy SOF control gains are proposed by ${\mathbf{K}}_j={\mathbf{X}}^{-1}{\mathbf{G}}_j$, $j=1,2,\cdots ,\mathrm{c}$.

Wherein the PMF ${\widehat{\omega }}_{ijk}\left(x\left(t\right)\right)$ is defined in (31), $x_{i_1{i}_2\cdots i_{n}l}$, $l=1,2,\cdots ,\mathsf{\vartheta }$ denote the apexes of ${\mathsf{\Gamma }}_l$, $\mathsf{\vartheta }$ and represents the number of partitioned state subspaces of PMF. $\Delta {\overline{\omega }}_{ijk}$, $\Delta {\underset{\_}{\omega }}_{ijk}$ are the supremum and infimum of $\Delta {\omega }_{ijk}\left(x\left(t\right)\right)$, respectively. Moreover, some other matrices are defined as Theorem 1.

Proof. 

Construct the following PMF according to the definition (28):

$${\widehat{\omega }}_{ijk}\left(x\left(t\right)\right)={\displaystyle \sum _{l=1}^{\mathsf{\vartheta }}{\displaystyle \sum _{i_1=1}^2\cdots {\displaystyle \sum _{i_{\mathsf{\eta }}=1}^2{\displaystyle \prod _{r=1}^{\mathsf{\eta }}{\nu }_{r{i}_{r}l}\left(x_r\left(t\right)\right){\widehat{\omega }}_{ijk}\left(x_{i_1\cdots i_{\mathsf{\eta }}l}\right)}}}}$$

(31)

In the following proof, PMF ${\widehat{\omega }}_{ijk}\left(x\left(t\right)\right)$ is introduced to approximate the MFs ${\omega }_{ijk}\left(x\left(t\right)\right)=m_i\left(x\left(t\right)\right)h_j\left(x\left(t\right)\right)m_k\left(x\left(t\right)\right)$, and we can obtain some further relaxed MFD stabilization conditions.

Let $\Delta {\omega }_{ijk}\left(x\left(t\right)\right)={\omega }_{ijk}\left(x\left(t\right)\right)-{\widehat{\omega }}_{ijk}\left(x\left(t\right)\right)$, wherein, $\Delta {\overline{\omega }}_{ijk}$ and $\Delta {\underset{\_}{\omega }}_{ijk}$ represent the supremum and infimum of $\Delta {\omega }_{ijk}\left(x\left(t\right)\right)$, respectively. Meanwhile, the equation ${\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\Delta {\omega }_{ijk}\left(x\left(t\right)\right)}}=0$ is obtained by employing the characteristics of the PMF. Furthermore, a slack matrix $\mathbf{E}={\mathbf{E}}^{\mathrm{T}}\in {ℝ}^{\left(6n+m\right)\times \left(6n+m\right)}$ is introduced to reduce conservatism, and we can obtain:

$${\displaystyle \sum }_{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\Delta {\omega }_{ijk}\left(x\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right)\mathbf{E}\xi \left(t\right)=0$$

(32)

Combining (26) and (32), we can obtain:

$$\begin{array}{l}\dot{V}\left(t\right)\le {\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{\omega }_{ijk}\left(x\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right){\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)\xi \left(t\right)\\ \hspace{1em}\hspace{1em}={\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{\widehat{\omega }}_{ijk}\left(x\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right){\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)\xi \left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\Delta {\omega }_{ijk}\left(x\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right){\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)\xi \left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\Delta {\omega }_{ijk}\left(x\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right)\mathbf{E}\xi \left(t\right)\\ \hspace{1em}\hspace{1em}={\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{\widehat{\omega }}_{ijk}\left(x\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right){\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)\xi \left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\left(\Delta {\omega }_{ijk}\left(x\left(t\right)\right)-\Delta {\underset{\_}{\omega }}_{ijk}+\Delta {\underset{\_}{\omega }}_{ijk}\right)}}{\xi }^{\mathrm{T}}\left(t\right)\left({\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)+\mathbf{E}\right)\xi \left(t\right)\\ \hspace{1em}\hspace{1em}={\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{\widehat{\omega }}_{ijk}\left(x\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right){\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)\xi \left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\left(\Delta {\omega }_{ijk}\left(x\left(t\right)\right)-\Delta {\underset{\_}{\omega }}_{ijk}\right)}}{\xi }^{\mathrm{T}}\left(t\right)\left({\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)+\mathbf{E}\right)\xi \left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\Delta {\underset{\_}{\omega }}_{ijk}}}{\xi }^{\mathrm{T}}\left(t\right)\left({\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)+\mathbf{E}\right)\xi \left(t\right)}\end{array}$$

(33)

Introducing another slack matrix $0<{\mathbf{F}}_{ijk}\in {ℝ}^{\left(6n+m\right)\times \left(6n+m\right)}$, $i=1,2,\cdots ,\mathrm{p}$, $j=1,2,\cdots ,\mathrm{c}$, $k=1,2,\cdots ,\mathrm{p}$, and we can obtain the following inequality based on (33):

$$\begin{array}{l}\dot{V}\left(t\right)\le {\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}{\widehat{\omega }}_{ijk}\left(x\left(t\right)\right)}}{\xi }^{\mathrm{T}}\left(t\right){\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)\xi \left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\Delta {\underset{\_}{\omega }}_{ijk}}}{\xi }^{\mathrm{T}}\left(t\right)\left({\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)+\mathbf{E}\right)\xi \left(t\right)}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\left(\Delta {\omega }_{ijk}\left(x\left(t\right)\right)-\Delta {\underset{\_}{\omega }}_{ijk}\right)}}{\xi }^{\mathrm{T}}\left(t\right)\left({\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)+\mathbf{E}-{\mathbf{F}}_{ijk}\right)\xi \left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=1}^{\mathrm{p}}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\left(\Delta {\omega }_{ijk}\left(x\right)-\Delta {\underset{\_}{\omega }}_{ijk}\right)}}{\xi }^{\mathrm{T}}\left(t\right){\mathbf{F}}_{ijk}\xi \left(t\right)\end{array}$$

(34)

Expanding the PMF ${\widehat{\omega }}_{ijk}\left(x\left(t\right)\right)$ in (34) according to the definition (31), and we have:

$$\begin{array}{l}\dot{V}\left(t\right)\le {\displaystyle \sum _{l=1}^{\mathsf{\vartheta }}{\displaystyle \sum _{i_1=1}^2\cdots {\displaystyle \sum _{i_{\mathsf{\eta }}=1}^2{\displaystyle \prod _{r=1}^{\mathsf{\eta }}{\nu }_{r{i}_{r}l}\left(x_r\left(t\right)\right)\hspace{0.17em}{\xi }^{\mathrm{T}}\left(t\right)}}}}[{\displaystyle \sum _{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}({\widehat{\omega }}_{ijk}\left(x_{i_1{i}_2\cdots i_{n}l}\right){\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)}}}\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\Delta {\underset{\_}{\omega }}_{ijk}\left({\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)+\mathbf{E}\right)+\left(\Delta {\overline{\omega }}_{ijk}-\Delta {\underset{\_}{\omega }}_{ijk}\right){\mathbf{F}}_{ijk}]\xi \left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+{\displaystyle \sum _{i=1}^{\mathrm{p}}{\displaystyle \sum _{j=1}^{\mathrm{c}}{\displaystyle \sum _{k=1}^{\mathrm{p}}\left(\Delta {\omega }_{ijk}\left(x\left(t\right)\right)-\Delta {\underset{\_}{\omega }}_{ijk}\right){\xi }^{\mathrm{T}}\left(t\right)\left({\mathit{\Sigma }}_{ijk}\left(\dot{\tau }\left(t\right)\right)+\mathbf{E}-{\mathbf{F}}_{ijk}\right)\xi \left(t\right)}}}\end{array}$$

(35)

If LMIs (28)–(30) hold, we have $\dot{V}\left(t\right)<0$; then, the closed-loop system (6) is asymptotically stable. This proof is completed. □

Remark 1. 

Different from the results in [29], the MFD stabilization conditions based on LMIs are presented in Theorem 2, in which the obtained stabilization conditions do not contain equality constraints, and the input and output matrices do not have rank constraints. Therefore, the proposed design methods of fuzzy SOF controllers reduce the conservatism of stabilization results and have better application prospects. In addition, the stabilization conditions of the fuzzy SOF controller given in [30] are nonconvex and cannot be solved directly by MATLAB LMIs toolbox.

4. Simulation Examples

In this subsection, two simulation examples are given to illustrate the progressiveness and effectiveness of the proposed IPMB fuzzy SOF controllers.

Example 1. 

Consider the following complex nonlinear system given in [47]:

$$\{\begin{array}{l}\begin{array}{l}{\dot{\mathit{x}}}_1\left(t\right)={\mathit{x}}_1\left(t\right)+{\mathit{x}}_2\left(t\right)+\sin {\mathit{x}}_3\left(t\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}-0.1{\mathit{x}}_4\left(t\right)+\left({\mathit{x}}_1^2\left(t\right)+1\right)\mathit{u}\left(t\right)\\ {\dot{\mathit{x}}}_2\left(t\right)={\mathit{x}}_1\left(t\right)-2{\mathit{x}}_2\left(t\right)\\ {\dot{\mathit{x}}}_3\left(t\right)={\mathit{x}}_1\left(t\right)+{\mathit{x}}_1^2\left(t\right){\mathit{x}}_2\left(t\right)-0.3{\mathit{x}}_3\left(t\right)\\ {\dot{\mathit{x}}}_4\left(t\right)=\sin {\mathit{x}}_3\left(t\right)-{\mathit{x}}_4\left(t\right)\\ {\mathit{y}}_1\left(t\right)=x_{\mathit{2}}\left(t\right)+\left({\mathit{x}}_1^2\left(t\right)+1\right){\mathit{x}}_4\left(t\right)\end{array}\hfill \\ {\mathit{y}}_2\left(t\right)={\mathit{x}}_1\left(t\right)\hfill \end{array}$$

(36)

wherein $\mathrm{a}$, $\mathrm{b}$ are positive real numbers, and ${\mathit{x}}_1\left(t\right)\in \left[-\mathrm{a},\hspace{0.17em}\mathrm{a}\right]$, ${\mathit{x}}_3\left(t\right)\in \left[-\mathrm{b},\hspace{0.17em}\mathrm{b}\right]$.

The above nonlinear time-delay system (36) can be approximated by the following T–S fuzzy models with four plant rules:

• Rule 1: IF ${\mathit{x}}_1\left(t\right)$ is ${\mathrm{M}}_1^1$ and ${\mathit{x}}_3\left(t\right)$ is ${\mathrm{M}}_3^1$, THEN $\{\begin{array}{l}\dot{\mathit{x}}\left(t\right)={\mathbf{A}}_1\mathit{x}\left(t\right)+{\mathbf{B}}_1\mathit{u}\left(t\right)\hfill \\ \mathit{y}\left(t\right)={\mathbf{C}}_1\mathit{x}\left(t\right)\hfill \end{array}$
• Rule 2: IF ${\mathit{x}}_1\left(t\right)$ is ${\mathrm{M}}_1^1$ and ${\mathit{x}}_3\left(t\right)$ is ${\mathrm{M}}_3^2$, THEN $\{\begin{array}{l}\dot{\mathit{x}}\left(t\right)={\mathbf{A}}_2\mathit{x}\left(t\right)+{\mathbf{B}}_2\mathit{u}\left(t\right)\hfill \\ \mathit{y}\left(t\right)={\mathbf{C}}_2\mathit{x}\left(t\right)\hfill \end{array}$
• Rule 3: IF ${\mathit{x}}_1\left(t\right)$ is ${\mathbf{M}}_1^2$ and ${\mathit{x}}_3\left(t\right)$ is ${\mathbf{M}}_3^1$, THEN $\{\begin{array}{l}\dot{\mathit{x}}\left(t\right)={\mathbf{A}}_3\mathit{x}\left(t\right)+{\mathit{B}}_3\mathit{u}\left(t\right)\hfill \\ \mathit{y}\left(t\right)={\mathbf{C}}_3\mathit{x}\left(t\right)\hfill \end{array}$
• Rule 4: IF ${\mathit{x}}_1\left(t\right)$ is ${\mathbf{M}}_1^2$ and ${\mathit{x}}_3\left(t\right)$ is ${\mathbf{M}}_3^2$, THEN $\{\begin{array}{l}\dot{\mathit{x}}\left(t\right)={\mathbf{A}}_{4}x\left(t\right)+{\mathit{B}}_4\mathit{u}\left(t\right)\hfill \\ \mathit{y}\left(t\right)={\mathbf{C}}_4\mathit{x}\left(t\right)\hfill \end{array}$

where

$${\mathbf{A}}_1=\left[\begin{array}{cccc}1& 1& 1& -0.1\\ 1& -2& 0& 0\\ 1& {\mathrm{a}}^2& -0.3& 0\\ 0& 0& 1& -1\end{array}\right], {\mathbf{A}}_2=\left[\begin{array}{cccc}1& 1& \frac{\sin \mathrm{b}}{\mathrm{b}}& -0.1\\ 1& -2& 0& 0\\ 1& {\mathrm{a}}^2& -0.3& 0\\ 0& 0& \frac{\sin \mathrm{b}}{\mathrm{b}}& -1\end{array}\right], {\mathbf{A}}_3=\left[\begin{array}{cccc}1& 1& 1& -0.1\\ 1& -2& 0& 0\\ 1& 0& -0.3& 0\\ 0& 0& 1& -1\end{array}\right]$$

$${\mathbf{A}}_4=\left[\begin{array}{cccc}1& 1& \frac{\sin \mathrm{b}}{\mathrm{b}}& -0.1\\ 1& -2& 0& 0\\ 1& 0& -0.3& 0\\ 0& 0& \frac{\sin \mathrm{b}}{\mathrm{b}}& -1\end{array}\right], {\mathbf{B}}_1={\mathbf{B}}_2=\left[\begin{array}{c}1+{\mathrm{a}}^2\\ 0\\ 0\\ 0\end{array}\right], {\mathbf{B}}_3={\mathbf{B}}_4=\left[\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right]$$

$${\mathbf{C}}_1={\mathbf{C}}_2=\left[\begin{array}{cccc}0& 1& 0& 1+a^2\\ 1& 0& 0& 0\end{array}\right], {\mathbf{C}}_3={\mathbf{C}}_4=\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 0& 0\end{array}\right]$$

In order to prove the effectiveness of the methods proposed in this paper, the other system matrices are selected in [48]:

$${\mathbf{A}}_{d1}={\mathbf{A}}_{d2}={\mathbf{A}}_{d3}={\mathbf{A}}_{d4}=\left[\begin{array}{cccc}0.1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$$

The premise MFs of fuzzy models are as the same as those defined in [48]:

$$m_1\left(x\left(t\right)\right)={\mathrm{M}}_1^1{\mathrm{M}}_3^1, m_2\left(x\left(t\right)\right)={\mathrm{M}}_1^1{\mathrm{M}}_3^2, m_3\left(x\left(t\right)\right)={\mathrm{M}}_1^2{\mathrm{M}}_3^1, m_4\left(x\left(t\right)\right)={\mathrm{M}}_1^2{\mathrm{M}}_3^2$$

where

$${\mathrm{M}}_1^1=\frac{x_1^2}{{\mathrm{a}}^2}, {\mathrm{M}}_1^2=1-{\mathrm{M}}_1^1, {\mathrm{M}}_3^1=\{\begin{array}{l}\frac{\mathrm{b}\sin x_3-x_3\sin \mathrm{b}}{x_3\left(\mathrm{b}-\sin \mathrm{b}\right)},\hspace{1em}{x}_3\ne 0\hfill \\ 1,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{x}_3\ne 0\hfill \end{array}, {\mathrm{M}}_3^2=1-{\mathrm{M}}_3^1$$

Based on IPM technology, fuzzy SOF controllers with two plant rules are designed as follows:

• Rule 1: IF ${\xi }_1\left(x\left(t\right)\right)$ is ${\mathrm{N}}_1{}^1$, THEN $\mathit{u}\left(t\right)={\mathbf{K}}_1\mathit{y}\left(t\right)$
• Rule 2: IF ${\xi }_2\left(x\left(t\right)\right)$ is ${\mathrm{N}}_1{}^2$, THEN $\mathit{u}\left(t\right)={\mathbf{K}}_2\mathit{y}\left(t\right)$

Based on IPM technology, the following simpler MFs are selected for the fuzzy SOF controllers:

$$h_1\left(x_1\right)=\frac{1}{1+\exp \left(-2x_1\right)}, h_2\left(x_1\right)=1-h_1\left(x_1\right)$$

Since the product of MFs ${\omega }_{ijk}\left(x\right)$, $i=1,2,3,4$, $j=1,\hspace{0.17em}2$, $k=1,2,3,4$ depends on $x_1$ and $x_3$, and we construct the following PMF ${\widehat{\omega }}_{ijk}\left(x\right)={\displaystyle \sum _{l=1}^{\mathsf{\vartheta }}{\displaystyle \sum _{i_1=1}^2{\displaystyle \sum _{i_3=1}^2{\nu }_{1i_{1}l}\left(x_1\right){\nu }_{3i_{1}l}\left(x_3\right){\widehat{\omega }}_{ijk}\left(x_{i_1{i}_{3}l}\right)}}}$. Firstly, we divide the state spaces $x_1\left(t\right)\in \left[-\mathrm{g},\hspace{0.17em}\mathrm{h}\right]$, $x_3\left(t\right)\in \left[-\mathrm{g},\hspace{0.17em}\mathrm{h}\right]$ into $\mathsf{\vartheta }$ connected state subspaces, and denote the l-th subspace as $\left(\frac{\mathrm{g}+\mathrm{h}}{\mathsf{\vartheta }}\right)\left(l-\frac{\mathsf{\vartheta }}{2}-1\right)\le x_s\le \left(\frac{\mathrm{g}+\mathrm{h}}{\mathsf{\vartheta }}\right)\left(l-\frac{\mathsf{\vartheta }}{2}\right)$, $s=1,3$, $l=1,2,\cdots ,\mathsf{\vartheta }$, $\mathrm{g},\mathrm{h}\in R^{+}$. In addition, ${\nu }_{11l}\left(x_1\right)=1-\frac{x_1-x_{2l}}{x_{1l}-x_{2l}}$, ${\nu }_{12l}\left(x_1\right)=1-{\nu }_{11l}\left(x_1\right)$, ${\nu }_{31l}\left(x_3\right)=1-\frac{x_3-x_{2l}}{x_{1l}-x_{2l}}$ and ${\nu }_{32l}\left(x_3\right)=1-{\nu }_{31l}\left(x_3\right)$ are selected, where $x_{1l}=\left(\frac{\mathrm{g}+\mathrm{h}}{\mathsf{\vartheta }}\right)\left(l-\frac{\mathsf{\vartheta }}{2}-1\right)$, $x_{2l}=\left(\frac{\mathrm{g}+\mathrm{h}}{\mathsf{\vartheta }}\right)\left(l-\frac{\mathsf{\vartheta }}{2}\right)$. Especially, $\mathrm{g}=\mathrm{h}=10$, $\mathsf{\vartheta }=20$ are selected, respectively. In

Table 1. The infimum and supremum of $\Delta {\omega }_{ijk}\left(x\right)$ with $\mathsf{\vartheta }=20$.

$$\mathbf{The} \mathbf{Infimum} \mathbf{of} \Delta {\mathit{\omega }}_{\mathit{i}\mathit{j}\mathit{k}}\left(\mathit{x}\right)$$		$$\mathbf{The} \mathbf{Supremum} \mathbf{of} \Delta {\mathit{\omega }}_{\mathit{i}\mathit{j}\mathit{k}}\left(\mathit{x}\right)$$
$\Delta {\underset{\_}{\omega }}_{111}=-7.7231\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{112}=-6.7881\times {10}^{-2}$	$\Delta {\overline{\omega }}_{111}=7.9312\times {10}^{-2}$	$\Delta {\overline{\omega }}_{112}=6.5367\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{113}=-3.7058\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{114}=-4.5041\times {10}^{-2}$	$\Delta {\overline{\omega }}_{113}=2.0982\times {10}^{-2}$	$\Delta {\overline{\omega }}_{114}=9.3016\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{121}=-4.5582\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{122}=-8.4183\times {10}^{-2}$	$\Delta {\overline{\omega }}_{121}=2.6978\times {10}^{-2}$	$\Delta {\overline{\omega }}_{122}=9.7881\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{123}=-6.3980\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{124}=-1.6824\times {10}^{-2}$	$\Delta {\overline{\omega }}_{123}=5.3882\times {10}^{-3}$	$\Delta {\overline{\omega }}_{124}=6.5743\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{211}=-3.6015\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{212}=-5.5514\times {10}^{-2}$	$\Delta {\overline{\omega }}_{211}=4.5137\times {10}^{-2}$	$\Delta {\overline{\omega }}_{212}=2.5983\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{213}=-3.6342\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{214}=-9.7648\times {10}^{-2}$	$\Delta {\overline{\omega }}_{213}=6.5739\times {10}^{-2}$	$\Delta {\overline{\omega }}_{214}=5.1385\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{221}=-8.4417\times {10}^{-4}$	$\Delta {\underset{\_}{\omega }}_{222}=-7.8991\times {10}^{-2}$	$\Delta {\overline{\omega }}_{221}=1.5766\times {10}^{-2}$	$\Delta {\overline{\omega }}_{222}=5.1464\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{223}=-8.6359\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{224}=-5.9734\times {10}^{-2}$	$\Delta {\overline{\omega }}_{223}=8.1625\times {10}^{-4}$	$\Delta {\overline{\omega }}_{224}=8.6592\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{311}=-3.6658\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{312}=-2.6971\times {10}^{-2}$	$\Delta {\overline{\omega }}_{311}=4.5122\times {10}^{-2}$	$\Delta {\overline{\omega }}_{312}=9.2563\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{313}=-7.4128\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{314}=-9.7881\times {10}^{-2}$	$\Delta {\overline{\omega }}_{313}=3.7558\times {10}^{-2}$	$\Delta {\overline{\omega }}_{314}=2.5619\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{321}=-4.5613\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{322}=-8.7516\times {10}^{-2}$	$\Delta {\overline{\omega }}_{321}=1.0562\times {10}^{-2}$	$\Delta {\overline{\omega }}_{322}=7.2648\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{323}=-5.8815\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{324}=-2.5097\times {10}^{-2}$	$\Delta {\overline{\omega }}_{323}=8.5473\times {10}^{-2}$	$\Delta {\overline{\omega }}_{324}=4.2669\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{411}=-6.3984\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{412}=-1.8547\times {10}^{-2}$	$\Delta {\overline{\omega }}_{411}=9.5618\times {10}^{-2}$	$\Delta {\overline{\omega }}_{412}=7.2651\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{413}=-7.6635\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{414}=-7.1564\times {10}^{-2}$	$\Delta {\overline{\omega }}_{413}=6.5871\times {10}^{-2}$	$\Delta {\overline{\omega }}_{414}=9.5134\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{421}=-4.6238\times {10}^{-2}$	$\Delta {\underset{\_}{\omega }}_{422}=-2.1657\times {10}^{-4}$	$\Delta {\overline{\omega }}_{421}=5.6736\times {10}^{-2}$	$\Delta {\overline{\omega }}_{422}=8.6951\times {10}^{-2}$
$\Delta {\underset{\_}{\omega }}_{423}=-3.8749\times {10}^{-4}$	$\Delta {\underset{\_}{\omega }}_{424}=-8.3346\times {10}^{-2}$	$\Delta {\overline{\omega }}_{423}=8.2561\times {10}^{-2}$	$\Delta {\overline{\omega }}_{424}=3.6182\times {10}^{-2}$

, we can obtain the infimum and supremum of $\Delta {\omega }_{ijk}\left(x\right)$ based on the formula $\Delta {\omega }_{ijk}\left(x\right)={\omega }_{ijk}\left(x\right)-{\widehat{\omega }}_{ijk}\left(x\right)$.

For the purpose of comparison, we cannot obtain a feasibility region for this example by employing Theorem 1, which demonstrates that the MFD analysis approach can effectively reduce conservatism. Therefore, further relaxed MFD stabilization results can be obtained by considering the information of MFs. Compared with the membership-function-independent method, the MFD approach can obtain less conservative results.

As in [47,48], we selected $\mathrm{a}=1.4$, $\mathrm{b}=0.7$. Let ${\mathsf{\tau }}_{\mathrm{M}}=2$, $\mathsf{\mu }=0.2$, and by employing MATLAB LMIs toolbox, the IPMB fuzzy SOF controllers are given according to Theorem 2 as follows:

$${\mathbf{K}}_1=\left[-1.5285-2.5338\right], {\mathbf{K}}_2=\left[-1.9137-3.3069\right]$$

We selected time-varying delay $\tau \left(t\right)=1+\sin \left(\frac{t}{5}\right)$ and initial condition $\mathbf{\varphi }\left(0\right)={\left[\begin{array}{cccc}-1.2& 0.5& 0.7& -0.6\end{array}\right]}^{\mathrm{T}}$. Figure 1 depicts the state response of the closed-loop system. Obviously, the fuzzy SOF controllers designed in Theorem 2 can guarantee that the system (36) is asymptotically stable.

Example 2. 

Consider the following truck-trailer system, an actual control system with time-varying delays given in [49,50]:

$$\{\begin{array}{l}\begin{array}{l}{\dot{x}}_1\left(t\right)=-\mathrm{a}\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}{x}_1\left(t\right)-\left(1-\mathrm{a}\right)\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}{x}_1\left(t-\tau \left(t\right)\right)+\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{l}{\mathrm{t}}_0}u\left(t\right)\\ {\dot{x}}_2\left(t\right)=\mathrm{a}\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}{x}_1\left(t\right)+\left(1-\mathrm{a}\right)\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}{x}_1\left(t-\tau \left(t\right)\right)\end{array}\hfill \\ {\dot{x}}_3\left(t\right)=\frac{\mathrm{v}\overline{\mathrm{t}}}{{\mathrm{t}}_0}\sin \left(x_2\left(t\right)+\mathrm{a}\frac{\mathrm{v}\overline{\mathrm{t}}}{2\mathrm{L}}{x}_1\left(t\right)+\left(1-\mathrm{a}\right)\frac{\mathrm{v}\overline{\mathrm{t}}}{2\mathrm{L}}{x}_1\left(t-\tau \left(t\right)\right)\right)\hfill \end{array}$$

(37)

where $x_1\left(t\right)$ is the angular difference between the truck and trailer, $x_2\left(t\right)$ is the angle of the trailer relative to the horizontal position, and $x_3\left(t\right)$ is the vertical position of the rear end of the trailer; $\mathrm{l}=2.8$ is the length of truck, $\mathrm{L}=5.5$ is the length of trailer, and $v=-1.0$ is constant speed of the tractor in the reverse direction. The model parameters are given as $\mathrm{a}=0.7$, $\overline{\mathrm{t}}=2.0$, ${\mathrm{t}}_0=0.5$.

Let $\theta \left(x\left(t\right)\right)=x_2\left(t\right)+\mathrm{a}\frac{\mathrm{v}\overline{\mathrm{t}}}{2\mathrm{L}}{x}_1\left(t\right)+\left(1-\mathrm{a}\right)\frac{\mathrm{v}\overline{\mathrm{t}}}{2\mathrm{L}}{x}_1\left(t-\tau \left(t\right)\right)$, and $x\left(t\right)={\left[\begin{array}{ccc}{x}_1\left(t\right)& x_2\left(t\right)& x_3\left(t\right)\end{array}\right]}^{\mathrm{T}}$. For the above nonlinear system with time-varying delays, we can employ the following fuzzy model with two fuzzy rules to approximate, and the specific expressions are as follows:

• Rule 1: IF $\theta \left(x\left(t\right)\right)$ is 0 rad, THEN

  $$\{\begin{array}{l}\dot{\mathit{x}}\left(t\right)={\mathbf{A}}_1\mathit{x}\left(t\right)+{\mathbf{A}}_{d1}\mathit{x}\left(t-\tau \left(t\right)\right)+{\mathbf{B}}_1\mathit{u}\left(t\right)\hfill \\ \mathit{y}\left(t\right)={\mathbf{C}}_1\mathit{x}\left(t\right)\hfill \end{array}$$
• Rule 2: IF $\theta \left(x\left(t\right)\right)$ is $-\mathsf{\pi }$ rad or $\mathsf{\pi }$ rad, THEN

  $$\{\begin{array}{l}\dot{\mathit{x}}\left(t\right)={\mathbf{A}}_2\mathit{x}\left(t\right)+{\mathbf{A}}_{d2}\mathit{x}\left(t-\tau \left(t\right)\right)+{\mathbf{B}}_2\mathit{u}\left(t\right)\hfill \\ \mathit{y}\left(t\right)={\mathbf{C}}_2\mathit{x}\left(t\right)\hfill \end{array}$$

where

$${\mathbf{A}}_1=\left[\begin{array}{ccc}-\mathrm{a}\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}& 0& 0\\ \mathrm{a}\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}& 0& 0\\ \mathrm{a}\frac{{\mathrm{v}}^2{\overline{\mathrm{t}}}^2}{2\mathrm{L}{\mathrm{t}}_0}& \frac{\mathrm{v}\overline{\mathrm{t}}}{{\mathrm{t}}_0}& 0\end{array}\right], {\mathbf{A}}_{d1}=\left[\begin{array}{ccc}-\left(1-\mathrm{a}\right)\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}& 0& 0\\ \left(1-\mathrm{a}\right)\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}& 0& 0\\ \left(1-\mathrm{a}\right)\frac{{\mathrm{v}}^2{\overline{\mathrm{t}}}^2}{2\mathrm{L}{\mathrm{t}}_0}& 0& 0\end{array}\right], {\mathbf{B}}_1={\mathbf{B}}_2={\left[\begin{array}{ccc}\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{l}{\mathrm{t}}_0}& 0& 0\end{array}\right]}^{\mathrm{T}}$$

$${\mathbf{A}}_2=\left[\begin{array}{ccc}-\mathrm{a}\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}& 0& 0\\ \mathrm{a}\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}& 0& 0\\ \mathrm{a}\frac{\mathrm{d}{\mathrm{v}}^2{\overline{\mathrm{t}}}^2}{2\mathrm{L}{\mathrm{t}}_0}& \frac{\mathrm{d}\mathrm{v}\overline{\mathrm{t}}}{{\mathrm{t}}_0}& 0\end{array}\right], {\mathbf{A}}_{d2}=\left[\begin{array}{ccc}-\left(1-\mathrm{a}\right)\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}& 0& 0\\ \left(1-\mathrm{a}\right)\frac{\mathrm{v}\overline{\mathrm{t}}}{\mathrm{L}{\mathrm{t}}_0}& 0& 0\\ \left(1-\mathrm{a}\right)\frac{\mathrm{d}{\mathrm{v}}^2{\overline{\mathrm{t}}}^2}{2\mathrm{L}{\mathrm{t}}_0}& 0& 0\end{array}\right], {\mathbf{C}}_1={\mathbf{C}}_2=\left[\begin{array}{ccc}7& -2& 0.03\end{array}\right]$$

As in [50], take $\mathrm{d}=\frac{10{\mathrm{t}}_0}{\mathsf{\pi }}$, and the following MFs are employed:

$$m_1\left(x\right)=1-\left(1-\frac{1}{1+\exp \left(-3\left(x_1-\frac{1}{2}\mathsf{\pi }\right)\right)}\right)\left(\frac{1}{1+\exp \left(-3\left(x_1+\frac{1}{2}\mathsf{\pi }\right)\right)}\right)$$

Based on IPM technology, the fuzzy SOF controllers with two plant rules are designed as follows:

• Rule 1: IF ${\xi }_1\left(x\left(t\right)\right)$ is ${\mathrm{N}}_1{}^1$, THEN $\mathit{u}\left(t\right)={\mathbf{K}}_1\mathit{y}\left(t\right)$
• Rule 2: IF ${\xi }_2\left(x\left(t\right)\right)$ is ${\mathrm{N}}_1{}^2$, THEN $\mathit{u}\left(t\right)={\mathbf{K}}_2\mathit{y}\left(t\right)$

Based on IPM technology, the following simpler MFs are selected for the fuzzy SOF controllers:

$$h_1\left(x_1\right)=\frac{1}{1+\exp \left(-2x_1\right)}, h_2\left(x_1\right)=1-h_1\left(x_1\right)$$

Similar to example 1, the infimum and supremum of $\Delta {\omega }_{ijk}\left(x\right)$, $i=1,2$, $j=1,\hspace{0.17em}2$, $k=1,2$ with $\mathsf{\vartheta }=20$ are obtained as shown in

Table 2. The infimum and supremum of $\Delta {\omega }_{ijk}\left(x\right)$ with $\mathsf{\vartheta }=20$.

$$\mathbf{The} \mathbf{Infimum} \mathbf{of} \Delta {\mathit{\omega }}_{\mathit{i}\mathit{j}\mathit{k}}\left(\mathit{x}\right)$$		$$\mathbf{The} \mathbf{Supremum} \mathbf{of} \Delta {\mathit{\omega }}_{\mathit{i}\mathit{j}\mathit{k}}\left(\mathit{x}\right)$$
$\Delta {\underset{\_}{\omega }}_{111}=-5.3678\times {10}^{-3}$	$\Delta {\underset{\_}{\omega }}_{112}=-3.5846\times {10}^{-3}$	$\Delta {\overline{\omega }}_{111}=3.5628\times {10}^{-3}$	$\Delta {\overline{\omega }}_{112}=2.9583\times {10}^{-3}$
$\Delta {\underset{\_}{\omega }}_{121}=-8.1629\times {10}^{-3}$	$\Delta {\underset{\_}{\omega }}_{122}=-4.8617\times {10}^{-3}$	$\Delta {\overline{\omega }}_{121}=5.6592\times {10}^{-3}$	$\Delta {\overline{\omega }}_{122}=6.2369\times {10}^{-3}$
$\Delta {\underset{\_}{\omega }}_{211}=-5.6659\times {10}^{-3}$	$\Delta {\underset{\_}{\omega }}_{212}=-2.8631\times {10}^{-3}$	$\Delta {\overline{\omega }}_{211}=8.2683\times {10}^{-3}$	$\Delta {\overline{\omega }}_{212}=7.2856\times {10}^{-3}$
$\Delta {\underset{\_}{\omega }}_{221}=-1.6852\times {10}^{-3}$	$\Delta {\underset{\_}{\omega }}_{222}=-3.1537\times {10}^{-3}$	$\Delta {\overline{\omega }}_{221}=8.1227\times {10}^{-3}$	$\Delta {\overline{\omega }}_{222}=7.3329\times {10}^{-3}$

.

As in [49], we selected ${\mathsf{\tau }}_{\mathrm{M}}=2.2$, $\mathsf{\mu }=0.2$, employing MATLAB LMIs toolbox, the IPMB fuzzy SOF controllers are given according to Theorem 2 as follows:

$${\mathbf{K}}_1=0.7156, {\mathbf{K}}_2=1.2584$$

In addition, as in [50], we selected time-varying delay $\tau \left(t\right)=2+0.2\sin t$ and initial condition $\mathbf{\varphi }\left(0\right)={\left[\begin{array}{ccc}0.5\pi & 0.75\pi & -5\end{array}\right]}^{\mathrm{T}}$. Figure 2 depicts the state response of the closed-loop system. Obviously, the fuzzy SOF controllers designed in Theorem 2 can guarantee that the system (37) is asymptotically stable.

Furthermore, as shown in Figure 2, when the same time-delay terms are selected, the fuzzy SOF controllers we designed can ensure that the system (37) is asymptotically stable at $t=40s$, while the system (37) is asymptotically stable at $t=140s$ by applying the methods in [49]. Therefore, under the same conditions, the fuzzy SOF controllers designed in this paper can make the system reach a stable state in a shorter time. It further shows that the methods in this paper are less conservative.

5. Conclusions

For a class of T–S fuzzy systems with time-varying delays, the design methods of SOF controllers are presented based on IPM technology. Firstly, based on the integral inequality technique, new LMI-ed stabilization conditions are proposed by selecting an appropriate Lyapunov–Krasovskii functional. The premise MFs and number of fuzzy rules of the IPMB SOF controllers can be selected arbitrarily. Secondly, further relaxed MFD stabilization results are obtained by considering the information of MFs. Finally, two simulation examples were given to demonstrate the progressiveness of the proposed IPMB fuzzy SOF controllers.