Observer-Based Exponential Stability Control of T-S Fuzzy Networked Systems with Varying Communication Delays

Abstract

This paper is concerned with the problem of dynamic output feedback exponential stability control of T-S fuzzy networked control systems (NCSs) with varying communication delays. First, with consideration of varying communication delays, a new model of the networked systems is established by using the T-S fuzzy method, and a state observer is designed to estimate the unknown control disturbance. Then, a delay-dependent exponential stability criterion of closed-loop systems is derived by means of iterative technique and multiple augmented Lyapnov functionals and the linear matrix inequality (LMI) method. Furthermore, an observer-based controller is explicitly constructed to realize exponential stability control for this class of NCSs. An iterative algorithm is developed to compute the controller’s matrix by means of the Cone Complementarity Linearization Method (CCLM). Lastly, the validity and feasibility of the proposed exponential stability criterion are confirmed via a numerical simulation example.

1. Introduction

Networked control systems (NCSs) are network-based feedback control systems that utilize network technology to connect dispersed devices, sensors, and actuators, enabling information exchange as well as greatly improving the automation and intelligence level of the systems. Since its proposal at the end of the 20th century, NCSs have attracted widespread attention from both academia and industry [1,2,3]. Compared with traditional control systems, NCSs have advantages such as flexible structure, easy scalability, and resource sharing, providing new ideas and methods for the design, implementation, and maintenance of the control systems. However, the research on NCSs also faces some challenges, such as the introduction of networks, which make it difficult to avoid latency and packet loss issues. These problems may lead to a decrease in controller performance and even instability.

In recent years, with the rapid development of control, network, and communication technologies, NCSs have gradually become a hot research topic [4,5,6]. The research content covers multiple aspects such as modeling, time delay, stability analysis, disturbance handling, controller design, and performance evaluation. In terms of modeling, researchers are committed to establishing more accurate NCS models to better describe the dynamic behavior of the systems. These models help us to understand the operating mechanism of the systems and provide a foundation for subsequent control.

Time delay and stability are key issues in NCSs. Researchers have conducted extensive research on time delay compensation, stability analysis, and other aspects to improve the robustness and stability of the systems. Disturbance handling is also one of the important directions in the research of NCSs. In practical applications, the systems are often affected by various external disturbances, such as noise, interference, etc. In order to improve the anti-interference ability of the systems, researchers have conducted in-depth research on disturbance suppression, filtering algorithms, and other aspects. In addition, network predictor design, traffic optimization, controller design, and scheduling are also hot topics in the research of NCSs. In summary, the research status of NCSs shows a trend of diversification and deepening, covering multiple key issues and research directions. With the continuous advancement of technology and the expansion of application fields, research on NCSs will continue to deepen, providing more efficient and intelligent control solutions for fields such as industrial automation and intelligent transportation [7,8]. Kundu et al. studied the allocation of communication resources in networked control systems. A method for finding periodic scheduling strategies has been proposed, under which the global asymptotic stability of each system in NCSs is maintained. A stable scheduling strategy is built by using loops that satisfy appropriate contractility conditions on the underlying weighted directed graph of NCSs [9]. An improved algorithm has been proposed to accelerate the convergence speed of the network systems. By using smaller initial values and faster decay rates, a more accurate upper limit of the system states can be obtained. The effectiveness of the improvement was demonstrated through simulation [10]. The finite-time stabilization problem of a class of discrete-time networked control systems with network-induced delay, as well as packet loss in feedback and forward channels, was studied. A new finite-time state feedback and output feedback stable controller is proposed using predictive control methods, and compensation is made for time delay and packet loss. A sufficient condition for finite-time stabilization is given for a given discrete-time networked control system [11]. Chen et al. studies the delay-dependent state feedback control problem of a class of networked control systems with nonlinear disturbances and two delay components. Based on the dynamic delay interval method and Wirdinger integral inequality, some improved delay-dependent stability analyses have been obtained. Subsequently, the results were extended to NCSs with time delay, and the corresponding stability analysis results and state feedback controllers were obtained [12].

Fuzzy systems are intelligent control systems based on fuzzy mathematics theory, which uses language rules and fuzzy logic reasoning to make decisions. Fuzzy systems have been widely applied in various industries and practical applications [13,14,15,16,17]. In the industrial field, fuzzy control is applied in various aspects such as industrial furnaces, petrochemical processes, robot systems, etc. The advantages of fuzzy control systems are that they do not require the establishment of an accurate mathematical model of the controlled object, have simple designs, are easy to implement, and have strong robustness. However, fuzzy systems also have some drawbacks, such as simple fuzzy processing of information, which may lead to a decrease in control accuracy and a deterioration in dynamic quality of the systems. The design of fuzzy control still lacks systematicity, making it difficult to effectively control complex systems. Li et al. studied the defense control problem of T-S fuzzy systems with multiple transmission channels for sampled data against asynchronous denial-of-service (DoS) attacks. A new exchange security control method has been proposed to tolerate asynchronous DoS attacks that act independently on each channel. In addition, by applying the segmented Lyapunov function method, the sufficient conditions dependent on the membership function were derived to ensure the exponential stability of the systems [18]. The robust fuzzy control problem of the nonlinear systems with actuator saturation limitation was studied. By using T-S fuzzy description, a fuzzy observer is established based on the sampled output affected by the network-induced delay. The saturation fuzzy control law is derived from the estimated states of the observer [19]. Zheng et al. studied the dynamic output feedback and H∞ stability analysis of a class of uncertain networked control systems with multiple time delays and external disturbances. A delay-dependent Lyapunov functional with double-product sub-items was designed, and the stability conditions in the form of LMI were obtained to ensure asymptotic stability of the closed-loop systems under the specified H∞ performance [20].

T-S fuzzy logic systems is the abbreviation of Takagi–Sugeno fuzzy logic systems, which quantifies the consequent of each fuzzy rule in pure fuzzy logic systems [21,22,23,24]. The T-S fuzzy model is an effective tool for describing nonlinear systems, which represents the dynamic characteristics of the entire system by dividing the system into a set of local models and applying a set of linear feedback control laws in each local model. This method is particularly suitable for NCSs and can effectively solve the stability problem of nonlinear systems [25,26,27,28,29]. Pan et al. investigated the design problem of a security controller based on resilient event-triggering for a nonlinear NCSs described by interval type-2 fuzzy model under non-periodic denial-of-service attacks. A resilient event-triggering strategy based on uncertain event-triggered variables is proposed for nonlinear NCSs. This strategy can transmit necessary data packets to the controller in unexpected situations [30]. Aslam1 et al. investigated the output tracking control problem of a class of T-S fuzzy nonlinear network control systems. A strategy based on event-triggered scheme is proposed to reduce bandwidth utilization in NCSs by utilizing inherent constraints, including communication latency and data transmission limitations. On this basis, a synchronous behavior-based operation model is proposed to make the design of linear controllers easier to implement [31].

However, the above research has mainly focused on the asymptotic stability and control design of fuzzy network systems, while research on exponential stability and dynamic output feedback control is relatively rare. This article is based on previous research and focuses on a class of T-S fuzzy NCSs with time-varying delays. By constructing a new Lyapunov functional and using linear matrix inequality tools to design an exponential stable state observer for the systems, the sufficient conditions for the simultaneous exponential stability of the closed-loop systems and error systems are obtained. The design strategy of the exponential stable controller is given by using the observation data.

2. Preliminaries

Consider the following fuzzy NCSs with varying communication delays given in Figure 1 with the following state delay plant:

Rule $i$:

If $z_1(t)$ is $M_1^i$ and $z_2(t)$ is $M_2^i$, and $z_n(t)$ is $M_n^i$,

Then $\dot{x}(t)=A_{i}x(t)+A_{di}x(t-d(t))+B_{i}u(t)$,

$$y(t)=C_{i}x(t), i=1,2,\dots ,q,$$

(1)

$$x(t)=\varphi (t), t\in [-\overline{d},0],$$

where $z(t)={[z_1(t) z_2(t)\dots z_n(t)]}^T$ is the premise variable, $x(t)\in R^n$ is the state, $q$ is the number of fuzzy rules, $M_k^i(i=1,2,\dots ,q ;k=1,2,\dots ,n)$ are the fuzzy sets, $u(t)\in R^m$ is the control input, $y(t)\in R^l$ is the controlled output, $A_i,A_{di}\in R^{n\times n}$, $B_i\in R^{n\times m}$, and $C_i\in R^{l\times n}$ are a set of known real matrices. $\varphi (t)={[{\varphi }_1(t) {\varphi }_2(t)\dots {\varphi }_n(t)]}^T$ $\in R^n$ is the initial condition of the state. $d(t)$ is the varying state delay.

Figure 1. A typical networked control system.

Figure 1. A typical networked control system.

In Figure 1, ${\tau }_{sc}(t)$ and ${\tau }_{ca}(t)$ are the sensor–controller delay and controller–actuator delay, respectively. Then the communication delay is given by $\tau (t)={\tau }_{sc}(t)+{\tau }_{ca}(t)$.

Assumption 1.

Only the output of the systems is available.

Assumption 2.

$d(t)$ and $\tau (t)$ are time-varying under the following bounded condition.

$$d(t)\le \overline{d}, \overline{d}>0, \dot{d}(t)\le d, d\in [0.1), \tau (t)\le \overline{\tau },\overline{\tau }>0,\dot{\tau }(t)\le \tau ,\tau \in [0.1)$$

Using single-point fuzzification, a product inference engine, and the central fuzzy elimination method, the global fuzzy model of the systems (1) can be described as

$$\begin{array}{l}\dot{x}(t)={\displaystyle \sum _{i=1}^q{\mu }_i(z(t))[}{A}_{i}x(t)+A_{di}x(t-d(t))+B_{i}u(t-\tau (t))],\\ y(t)={\displaystyle \sum _{i=1}^q{\mu }_i(z(t))}{C}_{i}x(t),\\ x(t)=\varphi (t), t\in [-\widehat{d},0],\end{array}$$

(2)

where

$${\omega }_i(z(t))={\displaystyle \prod _{k=1}^n{M}_k^i(z_k(t))}, {\mu }_i(z(t))={\displaystyle \frac{{\omega }_i(z(t))}{{\displaystyle \sum _{i=1}^q{\omega }_i(z(t))}}}, \widehat{d}=\max \{\overline{d},\overline{\tau }\}.$$

where$M_k^i(z_k(t))$is the membership degree of$z_k(t)$corresponding to$M_k^i$, and${\omega }_i(z(t))$satisfies

$${\omega }_i(z(t))\ge 0, {\displaystyle \sum _{i=1}^q{\omega }_i(z(t))>0}, i=1,2,\dots ,q.$$

And we define

$$||\varphi (t)|{|}_{\widehat{d}}=\underset{\begin{array}{l}t\in [-\widehat{d},0]\\ i=1,2,\dots ,n\end{array}}{Sup}||{\varphi }_i(t)||.$$

In this paper, the following fuzzy observer of the systems (2) is designed

$$\begin{array}{l}\dot{\widehat{x}}(t)={\displaystyle \sum _{i=1}^q{\mu }_i(z(t))\{A_i\widehat{x}(t)+A_{di}\widehat{x}(t-d(t))+B_{i}u(t-\tau (t))+L_i[y(t)-\widehat{y}(t)]\}},\\ \widehat{y}(t)={\displaystyle \sum _{i=1}^q{\mu }_i(z(t))C_i\widehat{x}(t)},\\ \widehat{x}(t)=\psi (t), t\in [-\widehat{d} 0].\end{array}$$

(3)

where $\widehat{x}(t)$ is the state of the observer, and $L_i$ is a set of constant matrices to be determined.

The observer error is defined as

$$e(t)=x(t)-\widehat{x}(t).$$

(4)

With (2)–(4), the error systems can be obtained as

$$\begin{array}{l}\dot{e}(t)={\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))[(A_i-L_i{C}_j)e(t)+A_{di}e(t-d(t))]}},\\ e(t)=\eta (t), t\in [-\widehat{d},0],\end{array}$$

(5)

where $\eta (t)=\phi (t)-\psi (t)$.

And then, a fuzzy controller based on the above observer is designed as

$$u(t)={\displaystyle \sum _{i=1}^q{\mu }_i(z(t))K_i\widehat{x}(t).}$$

(6)

Using Equations (2), (4) and (6), the resultant closed-loop systems can be written as

$$\begin{array}{l}\dot{x}(t)={\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))[A_{i}x(t)+A_{di}x(t-d(t))+B_i{K}_{j}x(t-\tau (t))-B_i{K}_{j}e(t-\tau (t))]}},\\ x(t)=\varphi (t), t\in [-\widehat{d},0].\end{array}$$

(7)

Before establishing our main results, the following basic definition and Lemma are given.

Definition 1

([7]). For the systems (2), if there exist constants $\alpha >0$ and $\gamma \ge 1$ such that

$$||x(t)||\le \gamma ||\varphi (t)|{|}_{\widehat{d}}{\mathrm{e}}^{-\alpha t}, t\ge 0,$$

the systems (2) are exponentially stable.

Lemma 1

([8]). The linear matrix inequality

$$\left[\begin{array}{cc}Y(x)& W(x)\\ *& R(x)\end{array}\right]>0$$

is equivalent to

$$R(x)>0, Y(x)-W(x)R^{-1}(x)W^T(x)>0,$$

where the matrices$Y(x)=Y^T(x),R(x)=R^T(x)$depend on $x$.

Remark 1.

By taking the delay from sensor to controller and control to actuator as the control input delay, the fuzzy mathematical model of the network systems with varying delays is established using T-S fuzzy method. This model is more convenient to apply to practical networked control systems, such as networked industrial robot systems, industrial transmission systems, etc. The determination of the designed observer only requires the determination of coefficient matrices $L_i$.

3. Results

In this section, sufficient conditions for exponential stability of the error systems (5) and the closed-loop systems (7) are elaborated. To address this problem, we have adopted a dynamic output feedback controller such that the error systems (5) and the closed-loop systems (7) are exponentially stable.

3.1. Design of Fuzzy Observer

Theorem 1.

The error systems (5) is exponentially stable if there exist matrices$L_i\in R^{n\times l},S_{12}\in R^{n\times n}$, $i,j=1,2,\dots ,q$ and positive-definite matrices $P,Q,S,T\in R^{n\times n}$ and scalar $\alpha >0$ such that

$$\Omega =\left[\begin{array}{ccc}{\Omega }_{11}& {\Omega }_{12}& S_{12}\\ \ast & {\Omega }_{22}& T-S_{12}\\ \ast & \ast & -S{\mathrm{e}}^{-2\alpha \overline{d}}-T\end{array}\right]<0,$$

(8)

where

$$\begin{array}{l}{\Omega }_{11}=P(A_i-L_i{C}_j)+{(A_i-L_i{C}_j)}^T+Q+S+2\alpha P+{\overline{d}}^2{(A_i-L_i{C}_j)}^{T}T(A_i-L_i{C}_j)-T,\\ {\Omega }_{12}=P{A}_{di}+{\overline{d}}^2{(A_i-L_i{C}_j)}^{T}T{A}_{di}+T-S_{12},\\ {\Omega }_{22}=-(1-d)Q{\mathrm{e}}^{-2\alpha \overline{d}}+{\overline{d}}^2{A}_{di}^{T}T{A}_{di}-2T+S_{12}+S_{12}^T.\end{array}$$

Proof of Theorem 1.

Consider the Lyapunov–Krasovskii functional candidate as

$$\begin{array}{l}V(e(t))=e^T(t)Pe(t)+{\displaystyle {\int }_{t-d(t)}^t{e}^T(s)Q{\mathrm{e}}^{2\alpha (s-t)}e(s)}ds+{\displaystyle {\int }_{t-\overline{d}}^t{e}^T(s)S{\mathrm{e}}^{2\alpha (s-t)}e(s)}ds\\ +\overline{d}{\displaystyle {\int }_{-\overline{d}}^0{\displaystyle {\int }_{t+\theta }^t{\dot{e}}^T(s)T{\mathrm{e}}^{2\alpha (s-t)}\dot{e}(s)dsd\theta }}.\end{array}$$

The goal is to show that if condition (8) holds, then $\dot{V}(e(t))<-2\alpha V(e(t))$. Following the state trajectory of the systems (5), we have

$$\begin{array}{l}\dot{V}(e(t))={\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))[2e^T(t)P(A_i-L_i{C}_j)e(t)}}+2e^T(t)P{A}_{di}e(t-d(t))]\\ +e^T(t)Qe(t)-(1-\dot{d}(t))e^T(t-d(t))Q{\mathrm{e}}^{-2\alpha d(t)}e(t-d(t))-2\alpha {\displaystyle {\int }_{t-d(t)}^t{e}^T(s)Q{\mathrm{e}}^{2\alpha (s-t)}e(s)}ds\\ +e^T(t)Se(t)-e^T(t-\overline{d})S{\mathrm{e}}^{-2\alpha \overline{d}}e(t-\overline{d})-2\alpha {\displaystyle {\int }_{t-\overline{d}}^t{e}^T(s)S{\mathrm{e}}^{2\alpha (s-t)}e(s)}ds\\ +{\overline{d}}^2{\dot{e}}^T(t)T\dot{e}(s)-\overline{d}{\displaystyle {\int }_{t-\overline{d}}^t{\dot{e}}^T(s)T{\mathrm{e}}^{2\alpha (s-t)}\dot{e}(s)}ds-2\alpha \overline{d}{\displaystyle {\int }_{-\overline{d}}^0{\displaystyle {\int }_{t+\theta }^t{\dot{e}}^T(s)T{\mathrm{e}}^{2\alpha (s-t)}\dot{e}(s)dsd\theta }}\\ +2\alpha e^T(t)Pe(t)-2\alpha e^T(t)Pe(t).\end{array}$$

$$-\overline{d}{\displaystyle {\int }_{t-\overline{d}}^t{\dot{e}}^T(s)T{\mathrm{e}}^{2\alpha (s-t)}\dot{e}(s)}ds\le -{\left[\begin{array}{c}e(t)-e(t-d(t))\\ e(t-d(t))-e(t-\overline{d})\end{array}\right]}^T\left[\begin{array}{cc}T& S_{12}\\ \ast & T\end{array}\right]\left[\begin{array}{c}e(t)-e(t-d(t))\\ e(t-d(t))-e(t-\overline{d})\end{array}\right].$$

$$\begin{array}{l}{\overline{d}}^2{\dot{e}}^T(t)T\dot{e}(s)={\overline{d}}^2{\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))[e^T(t){(A_i-L_i{C}_j)}^{T}T(A_i-L_i{C}_j)e(t)}}\\ +2e^T(t){(A_i-L_i{C}_j)}^{T}T{A}_{di}e(t-d(t))+e^T(t-d(t))A_{di}^{T}T{A}_{di}e(t-d(t))].\end{array}$$

It can be obtained from reference [6]

$${\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))=1,}}$$

and therefore,

$$\begin{array}{l}\dot{V}(e(t))<{\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))\{2e^T(t)P(A_i-L_i{C}_j)e(t)}}+2e^T(t)P{A}_{di}e(t-d(t))\\ +e^T(t)Qe(t)-(1-d)e^T(t-d(t))Q{\mathrm{e}}^{-2\alpha \overline{d}}e(t-d(t))+e^T(t)Se(t)\\ -e^T(t-\overline{d})S{\mathrm{e}}^{-2\alpha \overline{d}}e(t-\overline{d})+{\overline{d}}^2[e^T(t){(A_i-L_i{C}_j)}^{T}T(A_i-L_i{C}_j)e(t)\\ +2e^T(t){(A_i-L_i{C}_j)}^{T}T{A}_{di}e(t-d(t))+e^T(t-d(t))A_{di}^{T}T{A}_{di}e(t-d(t))]\\ -{\left[\begin{array}{c}e(t)-e(t-d(t))\\ e(t-d(t))-e(t-\overline{d})\end{array}\right]}^T\left[\begin{array}{cc}T& S_{12}\\ \ast & T\end{array}\right]\left[\begin{array}{c}e(t)-e(t-d(t))\\ e(t-d(t))-e(t-\overline{d})\end{array}\right]\\ +2\alpha e^T(t)Pe(t)\}-2\alpha V(e(t))\\ ={\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))}{\zeta }^T\Omega }\zeta -2\alpha V(e(t)),\end{array}$$

where

$$\zeta (t)=\left[\begin{array}{c}e(t)\\ e(t-d(t))\\ e(t-\overline{d})\end{array}\right],$$

Substituting the matrix inequality (8) into the above equality, we have

$$\dot{V}(e(t))<-2\alpha V(e(t)),$$

then

$$V(e(t))<V(e(0)){\mathrm{e}}^{-2\alpha t}\le [{\lambda }_{\max }(P)+\overline{d}{\lambda }_{\max }(Q)+\overline{d}{\lambda }_{\max }(S)+{\overline{d}}^2{\lambda }_{\max }(T)]||\eta (t)|{|}_{\widehat{d}}^2{\mathrm{e}}^{-2\alpha t}.$$

(9)

It is easy to know from the expression of $V(e(t))$,

$$V(e(t))\ge {\lambda }_{\min }(P)||e(t)|{|}^2.$$

(10)

From (9), (10), it yields

$$||e(t)||<\sqrt{{\displaystyle \frac{{\lambda }_{\max }(P)+\overline{d}{\lambda }_{\max }(Q)+\overline{d}{\lambda }_{\max }(S)+{\overline{d}}^2{\lambda }_{\max }(T)}{{\lambda }_{\min }(P)}}}||\eta (t)|{|}_{\widehat{d}}{\mathrm{e}}^{-\alpha t}.$$

According to definition 1, the error systems (5) are exponentially stable. □

Remark 2.

A sufficient condition for exponential stability of the error systems (5) is given in Theorem 1. $\alpha$ is the exponential stability degree of the error systems (5). However, the sufficient condition (8) is not a linear matrix inequality and cannot be solved by the LMI toolbox in MATLAB2017.

Theorem 2.

The error systems (5) are exponentially stable if there exist matrices ${\overline{L}}_i\in R^{n\times l},S_{12}\in R^{n\times n}$, $i,j=1,2,\dots ,q$, positive-definite matrices $P,Q,S\in R^{n\times n}$, and scalar $\alpha >0$, such that

$$\Sigma =\left[\begin{array}{cccc}{\Sigma }_{11}& {\Sigma }_{12}& S_{12}& \overline{d}(A_i^{T}P-C_j^T{\overline{L}}_i^T)\\ \ast & {\Sigma }_{22}& P-S_{12}& \overline{d}{A}_{di}^{T}P\\ \ast & \ast & -S{\mathrm{e}}^{-2\alpha \overline{d}}-P& 0\\ \ast & \ast & \ast & -P\end{array}\right]<0,$$

(11)

where

$$\begin{array}{l}{\Sigma }_{11}=P{A}_i-{\overline{L}}_i{C}_j+A_i^{T}P-C_j^T{\overline{L}}_i^T+Q+S+2\alpha P-P,\\ {\Sigma }_{12}=P{A}_{di}+P-S_{12},\\ {\Sigma }_{22}=-(1-d)Q{\mathrm{e}}^{-2\alpha \overline{d}}-2P+S_{12}+S_{12}^T.\end{array}$$

The gain matrices of the fuzzy observer (7) can be obtained as $L_i=P^{-1}{\overline{L}}_i$.

Proof of Theorem 2 is omitted.

3.2. Design of Dynamic Output Feedback Fuzzy Controller

In this section, the design procedure of the controller in terms of LMIs is derived.

Theorem 3.

Under the proposed output feedback fuzzy controller (6), the error systems (5) and the closed-loop systems (7) are exponentially stable if there exist matrices${\overline{L}}_i\in R^{n\times l},S_{12},{\overline{S}}_{12},{\widehat{S}}_{12}\in R^{n\times n},K_j\in R^{m\times n}$,$i,j=1,2,\dots ,q$ and positive-definite matrices $P,Q,S,\overline{P},\overline{Q},\overline{S},\overline{R},\widehat{Q},\widehat{S},\widehat{R}\in R^{n\times n}$, as well as scalars$\alpha >0,\beta >0$, $\epsilon >0$, thus satisfying the condition (11) and the following inequality

$$\Theta =\left[\begin{array}{ccccc}{\Theta }_{11}& {\Theta }_{12}& {\overline{S}}_{12}& {\Theta }_{14}& {\widehat{S}}_{12}\\ \ast & {\Theta }_{22}& \overline{R}-{\overline{S}}_{12}& {\Theta }_{24}& 0\\ \ast & \ast & -\overline{S}{\mathrm{e}}^{-2\beta \overline{d}}-\overline{R}& 0& 0\\ \ast & \ast & \ast & {\Theta }_{44}& -{\widehat{S}}_{12}+\widehat{R}\\ \ast & \ast & \ast & \ast & -\widehat{S}{\mathrm{e}}^{-2\beta \overline{\tau }}-\widehat{R}\end{array}\right]<0,$$

(12)

where

$$\begin{array}{l}{\Theta }_{11}=\overline{P}{A}_i+A_i^T\overline{P}+\overline{Q}+\overline{S}+\widehat{Q}+\widehat{S}+2\beta \overline{P}+\epsilon \overline{P}+{\overline{d}}^2{A}_i^T\overline{R}{A}_i+{\overline{\tau }}^2{A}_i^T\widehat{R}{A}_i-\overline{R}-\widehat{R}\\ {\Theta }_{12}=\overline{P}{A}_{di}+{\overline{d}}^2{A}_i^T\overline{R}{A}_{di}+{\overline{\tau }}^2{A}_i^T\widehat{R}{A}_{di}+\overline{R}-{\overline{S}}_{12}\\ {\Theta }_{14}=\overline{P}{B}_i{K}_j+{\overline{d}}^2{A}_i^T\overline{R}{B}_i{K}_j+{\overline{\tau }}^2{A}_i^T\widehat{R}{B}_i{K}_j+\widehat{R}-{\widehat{S}}_{12}\\ {\Theta }_{22}=-(1-d)\overline{Q}{\mathrm{e}}^{-2\beta \overline{d}}+{\overline{d}}^2{A}_{di}^T\overline{R}{A}_{di}+{\overline{\tau }}^2{A}_{di}^T\widehat{R}{A}_{di}-2\overline{R}+{\overline{S}}_{12}+{\overline{S}}_{12}^T\\ {\Theta }_{24}={\overline{d}}^2{A}_{di}^T\overline{R}{B}_i{K}_j+{\overline{\tau }}^2{A}_{di}^T\widehat{R}{B}_i{K}_j\\ {\Theta }_{44}=-(1-\tau )\widehat{Q}{\mathrm{e}}^{-2\beta \overline{\tau }}+{\overline{d}}^2{K}_j^T{B}_i^T\overline{R}{B}_i{K}_j+{\overline{\tau }}^2{K}_j^T{B}_i^T\widehat{R}{B}_i{K}_j-2\widehat{R}+{\widehat{S}}_{12}+{\widehat{S}}_{12}^T.\end{array}$$

Proof of Theorem 3.

Consider the Lyapunov–Krasovskii functional candidate as

$$\begin{array}{l}V(x(t))=x^T(t)\overline{P}x(t)+{\displaystyle {\int }_{t-d(t)}^t{x}^T(s)\overline{Q}{\mathrm{e}}^{2\beta (s-t)}x(s)}ds+{\displaystyle {\int }_{t-\overline{d}}^t{x}^T(s)\overline{S}{\mathrm{e}}^{2\beta (s-t)}x(s)}ds\\ +\overline{d}{\displaystyle {\int }_{-\overline{d}}^0{\displaystyle {\int }_{t+\theta }^t{\dot{x}}^T(s)\overline{R}{\mathrm{e}}^{2\beta (s-t)}\dot{x}(s)dsd\theta }}+{\displaystyle {\int }_{t-\tau (t)}^t{x}^T(s)\widehat{Q}{\mathrm{e}}^{2\beta (s-t)}x(s)}ds\\ +{\displaystyle {\int }_{t-\overline{\tau }}^t{x}^T(s)\widehat{S}{\mathrm{e}}^{2\beta (s-t)}x(s)}ds+\overline{\tau }{\displaystyle {\int }_{-\overline{\tau }}^0{\displaystyle {\int }_{t+\theta }^t{\dot{x}}^T(s)\widehat{R}{\mathrm{e}}^{2\beta (s-t)}\dot{x}(s)dsd\theta }}.\end{array}$$

Following the state trajectory of the closed-loop systems (7), we obtain

$$\begin{array}{l}\dot{V}(x(t))={\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))[2x^T(t)\overline{P}{A}_{i}x(t)}}+2x^T(t)\overline{P}{A}_{di}x(t-d(t))\\ +2x^T(t)\overline{P}{B}_i{K}_{j}x(t-\tau (t))-2x^T(t)\overline{P}{B}_i{K}_{j}e(t-\tau (t))]\\ +x^T(t)\overline{Q}x(t)-(1-\dot{d}(t))x^T(t-d(t))\overline{Q}{\mathrm{e}}^{-2\beta d(t)}x(t-d(t))-2\beta {\displaystyle {\int }_{t-d(t)}^t{x}^T(s)\overline{Q}{\mathrm{e}}^{2\beta (s-t)}x(s)}ds\\ +x^T(t)\overline{S}x(t)-x^T(t-\overline{d})\overline{S}{\mathrm{e}}^{-2\beta \overline{d}}x(t-\overline{d})-2\beta {\displaystyle {\int }_{t-\overline{d}}^t{x}^T(s)\overline{S}{\mathrm{e}}^{2\beta (s-t)}x(s)}ds\\ +{\overline{d}}^2{\dot{x}}^T(t)\overline{R}\dot{x}(t)-\overline{d}{\displaystyle {\int }_{t-\overline{d}}^t{\dot{x}}^T(s)\overline{R}{\mathrm{e}}^{2\beta (s-t)}\dot{x}(s)}ds-2\beta \overline{d}{\displaystyle {\int }_{-\overline{d}}^0{\displaystyle {\int }_{t+\theta }^t{\dot{x}}^T(s)\overline{R}{\mathrm{e}}^{2\beta (s-t)}\dot{x}(s)dsd\theta }}\\ +x^T(t)\widehat{Q}x(t)-(1-\dot{\tau }(t))x^T(t-\tau (t))\widehat{Q}{\mathrm{e}}^{-2\beta \tau (t)}x(t-\tau (t))-2\beta {\displaystyle {\int }_{t-\tau (t)}^t{x}^T(s)\widehat{Q}{\mathrm{e}}^{2\beta (s-t)}x(s)}ds\\ +x^T(t)\widehat{S}x(t)-x^T(t-\overline{\tau })\widehat{S}{\mathrm{e}}^{-2\beta \overline{\tau }}x(t-\overline{\tau })-2\beta {\displaystyle {\int }_{t-\overline{\tau }}^t{x}^T(s)\widehat{S}{\mathrm{e}}^{2\beta (s-t)}x(s)}ds\\ +{\overline{\tau }}^2{\dot{x}}^T(t)\widehat{R}\dot{x}(t)-\overline{\tau }{\displaystyle {\int }_{t-\overline{\tau }}^t{\dot{x}}^T(s)\widehat{R}{\mathrm{e}}^{2\beta (s-t)}\dot{x}(s)}ds-2\beta \overline{\tau }{\displaystyle {\int }_{-\overline{\tau }}^0{\displaystyle {\int }_{t+\theta }^t{\dot{x}}^T(s)\widehat{R}{\mathrm{e}}^{2\beta (s-t)}\dot{x}(s)dsd\theta }}\\ +2\beta x^T(t)\overline{P}x(t)-2\beta x^T(t)\overline{P}x(t)\\ \le {\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))[2x^T(t)\overline{P}{A}_{i}x(t)}}+2x^T(t)\overline{P}{A}_{di}x(t-d(t))+2x^T(t)\overline{P}{B}_i{K}_{j}x(t-\tau (t))\\ -2x^T(t)\overline{P}{B}_i{K}_{j}e(t-\tau (t))]+x^T(t)\overline{Q}x(t)-(1-d)x^T(t-d(t))\overline{Q}{\mathrm{e}}^{-2\beta \overline{d}}x(t-d(t))\\ +x^T(t)\overline{S}x(t)-x^T(t-\overline{d})\overline{S}{\mathrm{e}}^{-2\beta \overline{d}}x(t-\overline{d})+{\overline{d}}^2{\dot{x}}^T(t)\overline{R}\dot{x}(t)-\overline{d}{\displaystyle {\int }_{t-\overline{d}}^t{\dot{x}}^T(s)\overline{R}{\mathrm{e}}^{2\beta (s-t)}\dot{x}(s)}ds\\ +x^T(t)\widehat{Q}x(t)-(1-\tau )x^T(t-\tau (t))\widehat{Q}{\mathrm{e}}^{-2\beta \overline{\tau }}x(t-\tau (t))+x^T(t)\widehat{S}x(t)\\ -x^T(t-\overline{\tau })\widehat{S}{\mathrm{e}}^{-2\beta \overline{\tau }}x(t-\overline{\tau })+{\overline{\tau }}^2{\dot{x}}^T(t)\widehat{R}\dot{x}(t)-\overline{\tau }{\displaystyle {\int }_{t-\overline{\tau }}^t{\dot{x}}^T(s)\widehat{R}{\mathrm{e}}^{2\beta (s-t)}\dot{x}(s)}ds\\ +2\beta x^T(t)\overline{P}x(t)-2\beta V(x(t)),\end{array}$$

$$\begin{array}{c}\begin{array}{l}-\overline{d}{\displaystyle {\int }_{t-\overline{d}}^t{\dot{x}}^T(s)\overline{R}{\mathrm{e}}^{2\beta (s-t)}\dot{x}(s)}ds\le -{\left[\begin{array}{c}x(t)-x(t-d(t))\\ x(t-d(t))-x(t-\overline{d})\end{array}\right]}^T\left[\begin{array}{cc}\overline{R}& {\overline{S}}_{12}\\ \ast & \overline{R}\end{array}\right]\left[\begin{array}{c}x(t)-x(t-d(t))\\ x(t-d(t))-x(t-\overline{d})\end{array}\right].\\ -\overline{\tau }{\displaystyle {\int }_{t-\overline{\tau }}^t{\dot{x}}^T(s)\widehat{R}{\mathrm{e}}^{2\beta (s-t)}\dot{x}(s)}ds\le -{\left[\begin{array}{c}x(t)-x(t-\tau (t))\\ x(t-\tau (t))-x(t-\overline{\tau })\end{array}\right]}^T\left[\begin{array}{cc}\widehat{R}& {\widehat{S}}_{12}\\ \ast & \widehat{R}\end{array}\right]\left[\begin{array}{c}x(t)-x(t-\tau (t))\\ x(t-\tau (t))-x(t-\overline{\tau })\end{array}\right].\end{array}\\ \\ \begin{array}{l}{\overline{d}}^2{\dot{x}}^T(t)\overline{R}\dot{x}(s)={\overline{d}}^2{\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))[x^T(t)A_i^T\overline{R}{A}_{i}x(t)+2x^T(t)A_i^T\overline{R}{A}_{di}x(t-d(t))}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+2x^T(t)A_i^T\overline{R}{B}_i{K}_{j}x(t-\tau (t))-2x^T(t)A_i^T\overline{R}{B}_i{K}_{j}e(t-\tau (t))\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+x^T(t-d(t))A_{di}^T\overline{R}{A}_{di}x(t-d(t))+2x^T(t-d(t))A_{di}^T\overline{R}{B}_i{K}_{j}x(t-\tau (t))\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-2x^T(t-d(t))A_{di}^T\overline{R}{B}_i{K}_{j}e(t-\tau (t))+x^T(t-\tau (t))K_j^T{B}_i^T\overline{R}{B}_i{K}_{j}x(t-\tau (t))\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-2x^T(t-\tau (t))K_j^T{B}_i^T\overline{R}{B}_i{K}_{j}e(t-\tau (t))+e^T(t-\tau (t))K_j^T{B}_i^T\overline{R}{B}_i{K}_{j}e(t-\tau (t))].\end{array}\end{array}$$

$$\begin{array}{c}\begin{array}{l}{\overline{\tau }}^2{\dot{x}}^T(t)\widehat{R}\dot{x}(s)={\overline{\tau }}^2{\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))[x^T(t)A_i^T\widehat{R}{A}_{i}x(t)+2x^T(t)A_i^T\widehat{R}{A}_{di}x(t-d(t))}}\\ +2x^T(t)A_i^T\widehat{R}{B}_i{K}_{j}x(t-\tau (t))-2x^T(t)A_i^T\widehat{R}{B}_i{K}_{j}e(t-\tau (t))\\ +x^T(t-d(t))A_{di}^T\widehat{R}{A}_{di}x(t-d(t))+2x^T(t-d(t))A_{di}^T\widehat{R}{B}_i{K}_{j}x(t-\tau (t))\end{array}\\ \\ \begin{array}{l}-2x^T(t-d(t))A_{di}^T\widehat{R}{B}_i{K}_{j}e(t-\tau (t))+x^T(t-\tau (t))K_j^T{B}_i^T\widehat{R}{B}_i{K}_{j}x(t-\tau (t))\\ -2x^T(t-\tau (t))K_j^T{B}_i^T\widehat{R}{B}_i{K}_{j}e(t-\tau (t))+e^T(t-\tau (t))K_j^T{B}_i^T\widehat{R}{B}_i{K}_{j}e(t-\tau (t))].\end{array}\end{array}$$

Therefore,

$$\begin{array}{l}\dot{V}(x(t))\le {\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))}}[{\xi }^T(t)\Theta \xi (t)-2x^T(t)\overline{P}{B}_i{K}_{j}e(t-\tau (t))\\ -2x^T(t)A_i^T\overline{R}{B}_i{K}_{j}e(t-\tau (t))-2x^T(t-d(t))A_{di}^T\overline{R}{B}_i{K}_{j}e(t-\tau (t))\\ -2x^T(t-\tau (t))K_j^T{B}_i^T\overline{R}{B}_i{K}_{j}e(t-\tau (t))\\ +e^T(t-\tau (t))K_j^T{B}_i^T\overline{R}{B}_i{K}_{j}e(t-\tau (t))-2x^T(t)A_i^T\widehat{R}{B}_i{K}_{j}e(t-\tau (t))\\ -2x^T(t-d(t))A_{di}^T\widehat{R}{B}_i{K}_{j}e(t-\tau (t))-2x^T(t-\tau (t))K_j^T{B}_i^T\widehat{R}{B}_i{K}_{j}e(t-\tau (t))\\ +e^T(t-\tau (t))K_j^T{B}_i^T\widehat{R}{B}_i{K}_{j}e(t-\tau (t))-\epsilon x^T(t)\overline{P}x(t)-2\beta V(x(t))]\\ <{\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))}}[{\xi }^T(t)\Theta \xi (t)-\epsilon x^T(t)\overline{P}x(t)\\ +e^T(t-\tau (t))K_j^T{B}_i^T\overline{R}{B}_i{K}_{j}e(t-\tau (t))\\ +e^T(t-\tau (t))K_j^T{B}_i^T\widehat{R}{B}_i{K}_{j}e(t-\tau (t))-2\beta V(x(t))],\end{array}$$

where $\epsilon >0$ is a scalar, and

$$\xi (t)=\left[\begin{array}{c}x(t)\\ x(t-d(t))\\ x(t-\overline{d})\\ x(t-\tau (t))\\ x(t-\overline{\tau })\end{array}\right]$$

If the matrix inequality (12) holds, the above inequality can be changed as

$$\begin{array}{l}\dot{V}(x(t))<-\epsilon x^T(t)\overline{P}x(t)+{\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))}}[e^T(t-\tau (t))K_j^T{B}_i^T\overline{R}{B}_i{K}_{j}e(t-\tau (t))\\ +e^T(t-\tau (t))K_j^T{B}_i^T\widehat{R}{B}_i{K}_{j}e(t-\tau (t))]-2\beta V(x(t))\\ \le -\epsilon {\lambda }_{\min }(\overline{P})||x(t)|{|}^2+{\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))}}||K_j^T{B}_i^T(\overline{R}+\widehat{R})B_i{K}_j||||e(t-\tau (t))|{|}^2\\ -2\beta V(x(t))\\ =-(\epsilon {\lambda }_{\min }(\overline{P})||x(t)|{|}^2-{\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))}}||K_j^T{B}_i^T(\overline{R}+\widehat{R})B_i{K}_j||||e(t-\tau (t))|{|}^2)\\ -2\beta V(x(t)),\end{array}$$

(13)

When $\epsilon {\lambda }_{\min }(\overline{P})||x(t)|{|}^2- {\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))}}||K_j^T{B}_i^T(\overline{R}+\widehat{R})B_i{K}_j||||e(t-\tau (t))|{|}^2\le 0$, we have

$$||x(t)||\le \sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))}}||K_j^T{B}_i^T(\overline{R}+\widehat{R})B_i{K}_j||}{\epsilon {\lambda }_{\min }(\overline{P})}}}||e(t-\tau (t))||.$$

(14)

According to Theorem 2, when matrix inequality (11) holds, the error systems (5) are exponentially stable, that is, the error $e(t)$ is exponentially stable, and the state $x(t)$ of the closed-loop systems (7) is also exponentially stable according to inequality (14).

When $\epsilon {\lambda }_{\min }(\overline{P})||x(t)|{|}^2- {\displaystyle \sum _{i=1}^q{\displaystyle \sum _{j=1}^q{\mu }_i(z(t)){\mu }_j(z(t))}}||K_j^T{B}_i^T(\overline{R}+\widehat{R})B_i{K}_j||||e(t-\tau (t))|{|}^2>0$, the inequality (13) can be changed as

$$\dot{V}(x(t))<-2\beta V(x(t)),$$

and therefore,

$$\begin{array}{l}V(x(t))<V(x(0)){\mathrm{e}}^{-2\beta t}\le [{\lambda }_{\max }(\overline{P})+\overline{d}{\lambda }_{\max }(\overline{Q})+\overline{d}{\lambda }_{\max }(\overline{S})+{\overline{d}}^2{\lambda }_{\max }(\overline{R})\\ +\overline{\tau }{\lambda }_{\max }(\widehat{Q})+\overline{\tau }{\lambda }_{\max }(\widehat{S})+{\overline{\tau }}^2{\lambda }_{\max }(\widehat{R})]||\varphi |{|}_{\widehat{d}}^2{\mathrm{e}}^{-2\beta t}.\end{array}$$

(15)

It is easy to know from the expression of $V(x(t))$ that

$$V(x(t))\ge {\lambda }_{\min }(\overline{P})||x(t)|{|}^2.$$

(16)

Using (15) and (16), it is concluded that

$$\begin{array}{l}||x(t)||\\ <\sqrt{{\displaystyle \frac{{\lambda }_{\max }(\overline{P})+\overline{d}{\lambda }_{\max }(\overline{Q})+\overline{d}{\lambda }_{\max }(\overline{S})+{\overline{d}}^2{\lambda }_{\max }(\overline{R})+\overline{\tau }{\lambda }_{\max }(\widehat{Q})+\overline{\tau }{\lambda }_{\max }(\widehat{S})+{\overline{\tau }}^2{\lambda }_{\max }(\widehat{R})}{{\lambda }_{\min }(\overline{P})}}}||\varphi |{|}_{\widehat{d}}{\mathrm{e}}^{-\beta t},\end{array}$$

meaning the closed-loop systems (7) are exponentially stable. It is also known from Theorem 2 that when the matrix inequality (11) holds, the error systems (5) and the closed-loop systems (7) are exponentially stable.

In summary, when the matrix inequalities (11) and (12) hold, the error systems (5) and the closed-loop systems (7) are exponentially stable. □

Remark 3.

In Theorem 3, the sufficient conditions (11) and (12) are not linear matrix inequalities, which cannot be solved by the LMI toolbox in MATLAB. Next, we will use Lemma 1 to transform the inequalities (11) and (12) into linear matrix inequalities.

Theorem 4.

Under the proposed output feedback fuzzy controller (6), the error systems (5) and the closed-loop systems (7) are exponentially stable if there exist matrices${\overline{L}}_i\in R^{n\times l},S_{12},{\tilde{S}}_{12},{\stackrel{⌢}{S}}_{12}\in R^{n\times n},{\overline{K}}_j\in R^{m\times n}$, $i,j=1,2,\dots ,q$ and positive-definite matrices $P,Q,S,X,\tilde{Q},\tilde{S},\stackrel{⌢}{Q},\stackrel{⌢}{S}\in R^{n\times n}$, as well as scalars$\alpha >0,\beta >0$, $\epsilon >0$, thus satisfying the condition (11) and the following linear matrix inequality

$$\Xi =\left[\begin{array}{ccccccc}{\Xi }_{11}& {\Xi }_{12}& {\tilde{S}}_{12}& {\Xi }_{14}& {\stackrel{⌢}{S}}_{12}& \overline{d}X{A}_i^T& \overline{\tau }X{A}_i^T\\ \ast & {\Xi }_{22}& {\Xi }_{23}& 0& 0& \overline{d}X{A}_{di}^T& \overline{\tau }X{A}_{di}^T\\ \ast & \ast & {\Xi }_{33}& 0& 0& 0& 0\\ \ast & \ast & \ast & {\Xi }_{44}& {\Xi }_{45}& \overline{d}{\overline{K}}_j^T{B}_i^T& \overline{\tau }{\overline{K}}_j^T{B}_i^T\\ \ast & \ast & \ast & \ast & {\Xi }_{55}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -X& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -X\end{array}\right]<0,$$

(17)

where

$$\begin{array}{l}{\Xi }_{11}=A_{i}X+X{A}_i^T+\tilde{Q}+\tilde{S}+\stackrel{⌢}{Q}+\stackrel{⌢}{S}+2\beta X+\epsilon X-2X\\ {\Xi }_{12}=A_{di}X+X-{\tilde{S}}_{12}\\ {\Xi }_{14}=B_i{\overline{K}}_j+X-{\stackrel{⌢}{S}}_{12}\\ {\Xi }_{22}=-(1-d){\mathrm{e}}^{-2\beta \overline{d}}\tilde{Q}-2X+{\tilde{S}}_{12}+{\tilde{S}}_{12}^T\\ {\Xi }_{23}=X-{\tilde{S}}_{12}\\ {\Xi }_{33}=-{\mathrm{e}}^{-2\beta \overline{d}}\tilde{S}-X\\ {\Xi }_{44}=-(1-\tau ){\mathrm{e}}^{-2\beta \overline{\tau }}\stackrel{⌢}{Q}-2X+{\stackrel{⌢}{S}}_{12}+{\stackrel{⌢}{S}}_{12}^T\\ {\Xi }_{45}=-{\stackrel{⌢}{S}}_{12}+X\\ {\Xi }_{55}=-{\mathrm{e}}^{-2\beta \overline{\tau }}\stackrel{⌢}{S}-X.\end{array}$$

The gain matrices of the fuzzy controller (6) can be obtained as $K_j={\overline{K}}_j\overline{P}$.

Proof of Theorem 4.

Let

$$\begin{array}{l}\Upsilon ={\left[\begin{array}{ccccc}\overline{d}{\overline{R}}^T{A}_i& \overline{d}{\overline{R}}^T{A}_{di}& 0& \overline{d}{\overline{R}}^T{B}_i{K}_j& 0\end{array}\right]}^T,\\ {\rm Z}={\left[\begin{array}{ccccc}\overline{\tau }{\widehat{R}}^T{A}_i& \overline{\tau }{\widehat{R}}^T{A}_{di}& 0& \overline{\tau }{\widehat{R}}^T{B}_i{K}_j& 0\end{array}\right]}^T.\end{array}$$

As a result,

$$\Theta ={\Theta }_0+\Upsilon {\overline{R}}^{-1}{\Upsilon }^T+{\rm Z}{\widehat{R}}^{-1}{{\rm Z}}^T$$

where

$${\Theta }_0=\left[\begin{array}{ccccc}{\Theta }_{011}& \overline{P}{A}_{di}+\overline{R}-{\overline{S}}_{12}& {\overline{S}}_{12}& \overline{P}{B}_i{K}_j+\widehat{R}-{\widehat{S}}_{12}& {\widehat{S}}_{12}\\ \ast & {\Theta }_{022}& \overline{R}-{\overline{S}}_{12}& 0& 0\\ \ast & \ast & -\overline{S}{\mathrm{e}}^{-2\beta \overline{d}}-\overline{R}& 0& 0\\ \ast & \ast & \ast & {\Theta }_{044}& -{\widehat{S}}_{12}+\widehat{R}\\ \ast & \ast & \ast & \ast & -\widehat{S}{\mathrm{e}}^{-2\beta \overline{\tau }}-\widehat{R}\end{array}\right]$$

in which

$$\begin{array}{l}{\Theta }_{011}=\overline{P}{A}_i+A_i^T\overline{P}+\overline{Q}+\overline{S}+\widehat{Q}+\widehat{S}+2\beta \overline{P}+\epsilon \overline{P}-\overline{R}-\widehat{R}\\ {\Theta }_{022}=-(1-d)\overline{Q}{\mathrm{e}}^{-2\beta \overline{d}}-2\overline{R}+{\overline{S}}_{12}+{\overline{S}}_{12}^T\\ {\Theta }_{044}=-(1-\tau )\widehat{Q}{\mathrm{e}}^{-2\beta \overline{\tau }}-2\widehat{R}+{\widehat{S}}_{12}+{\widehat{S}}_{12}^T\end{array}$$

Applying Lemma 1, the inequality (12) is equivalent to

$$\Pi =\left[\begin{array}{ccccccc}{\Pi }_{11}& {\Pi }_{12}& {\overline{S}}_{12}& {\Pi }_{14}& {\widehat{S}}_{12}& \overline{d}{A}_i^T\overline{R}& \overline{\tau }{A}_i^T\widehat{R}\\ \ast & {\Pi }_{22}& \overline{R}-{\overline{S}}_{12}& 0& 0& \overline{d}{A}_{di}^T\overline{R}& \overline{\tau }{A}_{di}^T\widehat{R}\\ \ast & \ast & {\Pi }_{33}& 0& 0& 0& 0\\ \ast & \ast & \ast & {\Pi }_{44}& -{\widehat{S}}_{12}+\widehat{R}& \overline{d}{K}_j^T{B}_i^T\overline{R}& \overline{\tau }{K}_j^T{B}_i^T\widehat{R}\\ \ast & \ast & \ast & \ast & {\Pi }_{55}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -\overline{R}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -\widehat{R}\end{array}\right]<0$$

(18)

where

$$\begin{array}{l}{\Pi }_{11}=\overline{P}{A}_i+A_i^T\overline{P}+\overline{Q}+\overline{S}+\widehat{Q}+\widehat{S}+2\beta \overline{P}+\epsilon \overline{P}-\overline{R}-\widehat{R}\\ {\Pi }_{12}=\overline{P}{A}_{di}+\overline{R}-{\overline{S}}_{12}\\ {\Pi }_{14}=\overline{P}{B}_i{K}_j+\widehat{R}-{\widehat{S}}_{12}\\ {\Pi }_{22}=-(1-d)\overline{Q}{\mathrm{e}}^{-2\beta \overline{d}}-2\overline{R}+{\overline{S}}_{12}+{\overline{S}}_{12}^T\\ {\Pi }_{33}=-\overline{S}{\mathrm{e}}^{-2\beta \overline{d}}-\overline{R}\\ {\Pi }_{44}=-(1-\tau )\widehat{Q}{\mathrm{e}}^{-2\beta \overline{\tau }}-2\widehat{R}+{\widehat{S}}_{12}+{\widehat{S}}_{12}^T\\ {\Pi }_{55}=-\widehat{S}{\mathrm{e}}^{-2\beta \overline{\tau }}-\widehat{R}\end{array}$$

Pre- and post-multiplying (18) by $diag\{{\overline{P}}^{-1},{\overline{P}}^{-1},{\overline{P}}^{-1},{\overline{P}}^{-1},{\overline{P}}^{-1},{\overline{R}}^{-1},{\widehat{R}}^{-1},\}$, we obtain evidence that the inequality (18) is equivalent to

$$\Gamma =\left[\begin{array}{ccccccc}{\Gamma }_{11}& {\Gamma }_{12}& {\overline{P}}^{-1}{\overline{S}}_{12}{\overline{P}}^{-1}& {\Gamma }_{14}& {\overline{P}}^{-1}{\widehat{S}}_{12}{\overline{P}}^{-1}& \overline{d}{\overline{P}}^{-1}{A}_i^T& \overline{\tau }{\overline{P}}^{-1}{A}_i^T\\ \ast & {\Gamma }_{22}& {\Gamma }_{23}& 0& 0& \overline{d}{\overline{P}}^{-1}{A}_{di}^T& \overline{\tau }{\overline{P}}^{-1}{A}_{di}^T\\ \ast & \ast & {\Gamma }_{33}& 0& 0& 0& 0\\ \ast & \ast & \ast & {\Gamma }_{44}& {\Gamma }_{45}& \overline{d}{\overline{P}}^{-1}{K}_j^T{B}_i^T& \overline{\tau }{\overline{P}}^{-1}{K}_j^T{B}_i^T\\ \ast & \ast & \ast & \ast & {\Gamma }_{55}& 0& 0\\ \ast & \ast & \ast & \ast & \ast & -{\overline{R}}^{-1}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -{\widehat{R}}^{-1}\end{array}\right]<0$$

(19)

where

$$\begin{array}{l}{\Gamma }_{11}=A_i{\overline{P}}^{-1}+{\overline{P}}^{-1}{A}_i^T+{\overline{P}}^{-1}\overline{Q}{\overline{P}}^{-1}+{\overline{P}}^{-1}\overline{S}{\overline{P}}^{-1}+{\overline{P}}^{-1}\widehat{Q}{\overline{P}}^{-1}+{\overline{P}}^{-1}\widehat{S}{\overline{P}}^{-1}+2\beta {\overline{P}}^{-1}\\ +\epsilon {\overline{P}}^{-1}-{\overline{P}}^{-1}\overline{R}{\overline{P}}^{-1}-{\overline{P}}^{-1}\widehat{R}{\overline{P}}^{-1}\\ {\Gamma }_{12}=A_{di}{\overline{P}}^{-1}+{\overline{P}}^{-1}\overline{R}{\overline{P}}^{-1}-{\overline{P}}^{-1}{\overline{S}}_{12}{\overline{P}}^{-1}\\ {\Gamma }_{14}=B_i{K}_j{\overline{P}}^{-1}+{\overline{P}}^{-1}\widehat{R}{\overline{P}}^{-1}-{\overline{P}}^{-1}{\widehat{S}}_{12}{\overline{P}}^{-1}\\ {\Gamma }_{22}=-(1-d){\mathrm{e}}^{-2\beta \overline{d}}{\overline{P}}^{-1}\overline{Q}{\overline{P}}^{-1}-2{\overline{P}}^{-1}\overline{R}{\overline{P}}^{-1}+{\overline{P}}^{-1}{\overline{S}}_{12}{\overline{P}}^{-1}+{\overline{P}}^{-1}{\overline{S}}_{12}^T{\overline{P}}^{-1}\\ {\Gamma }_{23}={\overline{P}}^{-1}\overline{R}{\overline{P}}^{-1}-{\overline{P}}^{-1}{\overline{S}}_{12}{\overline{P}}^{-1}\\ {\Gamma }_{33}=-{\mathrm{e}}^{-2\beta \overline{d}}{\overline{P}}^{-1}\overline{S}{\overline{P}}^{-1}-{\overline{P}}^{-1}\overline{R}{\overline{P}}^{-1}\\ {\Gamma }_{44}=-(1-\tau ){\mathrm{e}}^{-2\beta \overline{\tau }}{\overline{P}}^{-1}\widehat{Q}{\overline{P}}^{-1}-2{\overline{P}}^{-1}\widehat{R}{\overline{P}}^{-1}+{\overline{P}}^{-1}{\widehat{S}}_{12}{\overline{P}}^{-1}+{\overline{P}}^{-1}{\widehat{S}}_{12}^T{\overline{P}}^{-1}\\ {\Gamma }_{45}=-{\overline{P}}^{-1}{\widehat{S}}_{12}{\overline{P}}^{-1}+{\overline{P}}^{-1}\widehat{R}{\overline{P}}^{-1}\\ {\Gamma }_{55}=-{\mathrm{e}}^{-2\beta \overline{\tau }}{\overline{P}}^{-1}\widehat{S}{\overline{P}}^{-1}-{\overline{P}}^{-1}\widehat{R}{\overline{P}}^{-1}\end{array}$$

Defining

$$\begin{array}{l}X={\overline{P}}^{-1},\overline{R}=\widehat{R}=\overline{P},\tilde{Q}={\overline{P}}^{-1}\overline{Q}{\overline{P}}^{-1},\tilde{S}={\overline{P}}^{-1}\overline{S}{\overline{P}}^{-1},\stackrel{⌢}{Q}={\overline{P}}^{-1}\widehat{Q}{\overline{P}}^{-1},\stackrel{⌢}{S}={\overline{P}}^{-1}\widehat{S}{\overline{P}}^{-1},\\ {\tilde{S}}_{12}={\overline{P}}^{-1}{\overline{S}}_{12}{\overline{P}}^{-1},{\overline{K}}_j=K_j{\overline{P}}^{-1},{\stackrel{⌢}{S}}_{12}={\overline{P}}^{-1}{\widehat{S}}_{12}{\overline{P}}^{-1}\end{array}$$

the inequality (19) is equivalent to the inequality (17). □

Remark 4.

The sufficient conditions (11) and (17) are in terms of linear matrix inequality, which can be obtained by the LMI toolbox in MATLAB. The exponential stability $\alpha$ and $\beta$ of the error systems and the closed-loop systems can be obtained, and the maximum values of delay $\overline{d},\overline{\tau }$ can be optimized.

Remark 5.

In order to optimize the exponential stability degree $\alpha ,\beta$, we can select different values of $\alpha ,\beta$, and solve the conditions (11) and (17) repeatedly, so as to obtain the optimization results.

Remark 6.

Theorem 4 gives the sufficient conditions (11) and (17) based on the existence of the dynamic output feedback fuzzy controller.

4. Simulation

To verify the effectiveness and superiority of the control design method proposed in this paper, we consider the following networked control systems with varying communication delays

$$\begin{array}{l}\dot{x}(t)={\displaystyle \sum _{i=1}^3{\mu }_i(z(t))[}{A}_{i}x(t)+A_{di}x(t-d(t))+B_{i}u(t-\tau (t))],\\ y(t)={\displaystyle \sum _{i=1}^3{\mu }_i(z(t))}{C}_{i}x(t),\\ x(t)=\varphi (t), t\in [-\widehat{d},0],\end{array}$$

where

$$\begin{array}{l}{A}_1=\left[\begin{array}{ccc}-3& 0& 1\\ 1& -1& 1\\ -2& 3& -2\end{array}\right],A_2=\left[\begin{array}{ccc}-0.1& 1& 0.1\\ 0& -2& 0.5\\ -0.1& 0.5& -3\end{array}\right],A_3=\left[\begin{array}{ccc}-2& 0.2& 0.1\\ 0.1& -0.2& 0.2\\ -0.5& 0.1& -0.6\end{array}\right],\\ A_{d1}=\left[\begin{array}{ccc}-1& 0& 1\\ 1& -2& 0\\ -2& 3& -3\end{array}\right],A_{d2}=\left[\begin{array}{ccc}-0.1& 0& 0.5\\ 2& -1& 1\\ -0.2& 0.3& -3\end{array}\right],A_{d3}=\left[\begin{array}{ccc}-2& 0.2& 0.1\\ 0.1& -0.2& 0.1\\ -1.5& 0.3& -0.6\end{array}\right],\\ B_1=\left[\begin{array}{c}1\\ 1\\ 0.3\end{array}\right],B_2=\left[\begin{array}{c}0.2\\ 0.1\\ 2\end{array}\right],B_3=\left[\begin{array}{c}1\\ 2\\ 1\end{array}\right],C_1=\left[\begin{array}{ccc}0.1& 1& 0.3\end{array}\right],C_2=\left[\begin{array}{ccc}0.2& -0.4& 3\end{array}\right],\end{array}$$

$$\begin{array}{l}{C}_3=\left[\begin{array}{ccc}2& 0.5& 0.3\end{array}\right],{\mu }_1(z(t))=\sin (t),{\mu }_2(z(t))=\cos (t),{\mu }_3(z(t))=1-\sin (t)-\cos (t),\\ \overline{d}=0.5,\overline{\tau }=0.4,\widehat{d}=0.5.\end{array}$$

The state feedback controller can be obtained by using the algorithm in [20] as follows:

$$u(t)=\left[\begin{array}{ccc}2.3285& -2.0478& 3.7652\end{array}\right]x(t).$$

On the other hand, using the proposed approach in this paper, by solving the linear matrix inequalities (11) and (17), we obtain the coefficient matrices of the controller and state observer as follows:

$$\begin{array}{l}{\overline{K}}_1=\left[\begin{array}{ccc}0.3276& -0.2735& 1.3287\end{array}\right],{\overline{K}}_2=\left[\begin{array}{ccc}-1.6527& -0.3276& -1.6287\end{array}\right],\\ {\overline{K}}_3=\left[\begin{array}{ccc}1.2746& -1.4205& 3.4189\end{array}\right],{\overline{L}}_1={\left[\begin{array}{ccc}0.2135& 1.4586& 1.6254\end{array}\right]}^T,\\ {\overline{L}}_2={\left[\begin{array}{ccc}2.5314& -1.2067& 2.4655\end{array}\right]}^T,{\overline{L}}_3={\left[\begin{array}{ccc}0.4256& 0.1259& 1.1563\end{array}\right]}^T,\\ X=\left[\begin{array}{ccc}2.3372& 0.1527& 0.1472\\ 0.1527& 2.5526& 0.2338\\ 0.1472& 0.2338& 1.7624\end{array}\right].\end{array}$$

After calculation, the dynamic output feedback controller is as follows:

$$\begin{array}{l}u(t)=\left[\begin{array}{cc}-2.1267\sin (t)+1.7952\cos (t)& 2.8775\sin (t)-0.3312\cos (t)\end{array}\right.\\ \left.-1.5661\sin (t)-0.1421\cos (t)\right]\widehat{x}(t).\end{array}$$

When we select the initial value condition such that

$$\varphi (0)=\psi (0)=\left[\begin{array}{c}-3\\ 5\\ 4\end{array}\right],$$

and the sampling time $k=1s$, the response curves of the systems states $x_1(t),x_2(t),x_3(t)$ are obtained as shown in Figure 2, Figure 3 and Figure 4.

Figure 2 shows the variation in systems state $x_1(t)$ under two algorithms. The solid line is the response diagram of the systems state $x_1(t)$ under the control action designed by Theorem 4. The dotted line is the response diagram of the systems state $x_1(t)$ under the control action designed by the method given in [20]. It is easy to see from the graph that the solid line converges to 0 faster than the dotted line. The smoothness and overshoot of solid lines are significantly better than those of dotted lines.

Figure 3 shows the variation in systems state $x_2(t)$ under two algorithms. The solid line is the response diagram of the systems state $x_2(t)$ under the control action designed by Theorem 4. The dotted line is the response diagram of the systems state $x_2(t)$ under the control action designed by the method given in [20]. The solid line converges to a small neighborhood of origin within 7 s. The dotted line converges to a small neighborhood of origin only after 10 s, and the convergence speed is slow. The overshoots of the two lines are small, and the amplitude does not exceed 0.1.

Figure 4 shows the variation in systems state $x_3(t)$ under two algorithms. The solid line is the response diagram of the systems state $x_3(t)$ under the control action designed by Theorem 4. The dotted line is the response diagram of the systems state $x_3(t)$ under the control action designed by the method given in [20]. The solid line converges to a small neighborhood of origin within 5 s. The curve is smooth enough, the overshoot is small, and the amplitude does not exceed 0.1. The dotted line converges to a small neighborhood of origin only after 8 s and the convergence speed is slow. The smoothness of the curve is poor, the overshoot is large, and the amplitude reaches 2.

In order to further compare the advantages and disadvantages of the two algorithms, we introduce the following Integral Absolute Error (IAE) performance function to conduct a performance comparison analysis of the systems

$$IAE={\displaystyle {\int }_0^{\infty }|e(t)|dt}.$$

After calculation, the IAE performance index function changes, as shown in the following figure.

Figure 5 shows the changes in the IAE performance index of the systems under the two different algorithms. The solid line is the change curve of the IAE performance index of the systems under the algorithm in this paper. The dotted line is the change curve of the IAE performance index of the systems under the algorithm in [20]. The dotted line reaches the peak value 0.081 in 1.9 s. The solid line reaches the peak value 0.061 in 1.7 s. Overall, the slope of the dotted line is relatively large, and the upward speed is fast. The solid line is relatively flat and has good stability.

In summary, the fuzzy controller design method proposed in this article is superior to the method presented in [20].

5. Discussion

The contribution of this paper is to design an exponentially stable state observer and dynamic output feedback controller for T-S fuzzy network systems with both state delay and communication delay by using linear matrix inequality method. A less conservative and easy-to-solve method is proposed to reduce the impact of state delay and network communication delay on system performance. The innovation of this paper lies in the design of an exponentially stable state observer and the provision of design conditions for dynamic output feedback control in the form of linear matrix inequalities. It is worth mentioning that our research results can be extended to other types of network systems, such as those with distributed delay, rapidly changing delay, or random delay, which will be our next research work.

6. Conclusions

This paper designs an exponential stable observer and dynamic output feedback control for T-S fuzzy network systems with both state delay and communication delay. The main tasks are as follows: (I) Simultaneously considering the existence of time-varying state delay and network communication delay, T-S fuzzy method is used to establish a mathematical model of the network systems that is more in line with engineering practice. (II) Based on Lyapunov stability theory, an exponential stable state observer is designed by constructing a new Lyapunov functional. (III) Using matrix inequality transformation techniques, the nonlinear stability condition is transformed into a condition in the form of a linear matrix inequality, and a design method for exponential stability control of the systems is given. Unfortunately, the research results are focused on time delay network systems based on T-S models, and studying more complex nonlinear network systems, especially stochastic network systems, will be another topic for us in the future.