Outlier-resistant observer-based control for interval type-2 T-S fuzzy systems under adaptive event-triggered protocols

Abstract

This paper is concerned with the outlier-resistant observer-based control problem for a class of interval type-2 (IT2) Takagi–Sugeno (T-S) fuzzy systems under adaptive event-triggered protocol (AETP). To improve the efficiency of resource utilization, AETP is mainly through an adjustable threshold parameter to dynamically schedule the data transmission, which occurs in the sensor-to-controller channel. In order to resist the abnormal measurement values, an output saturation composition is constructed in the observer. With the aid of Lyapunov theory, sufficient conditions are obtained to ensure the desired performance. Furthermore, the desired controller is designed for a class of IT2 T-S fuzzy systems. The combination of orthogonal decomposition and linear matrix inequalities is implemented to deal with the nonlinear coupling terms in the gain solving process. Finally, a numerical example is employed to justify the practicality and effectiveness of the proposed control scheme.

Introduction

During the past few decades, fuzzy control represented by Takagi–Sugeno (T-S) fuzzy control (Takagi and Sugeno, 1985, 1992) has become a very effective tool for handling the stability analysis and controller synthesis of nonlinear systems (Cuesta et al., 1999). Generally speaking, T-S fuzzy model (FM) is a global model constituted by a series of linear sub-models smoothly connected through nonlinear fuzzy weights, which can narrow the gap between the nonlinear and linear systems. For now, considerable results based on T-S FM for stability analysis and performance design have been reported in the literature (Xie et al., 2017; Zhao et al., 2009a). For instance, the fault-tolerant controller has been designed in Huang and Yang (2014), Makni et al. (2020) with actuator or sensor faults for T-S FM. In addition, the problem of fault detection is discussed about T-S FM in Zhao et al. (2009b) and Zhuang et al. (2015). It should be noted that the type-1 T-S FM is invalid in dealing with uncertainty of linear or nonlinear systems. Therefore, interval type-2 (IT2) fuzzy systems are presented to address this difficulty.

In traditional T-S fuzzy control, it is usually assumed that the fuzzy weights only contain certain information. However, in practical applications, uncertainties are often unavoidable in nonlinear systems, such as uncertain parameters, unpredictable variables, and unknown perturbations. Thus, Mendel (2014) describes the traditional T-S FM, which is used to stand for nonlinear system with parameter uncertainty, it may lead to uncertain information contained in its fuzzy weights. In this case, the parallel distributed compensation (PDC) strategy will not be available to design the fuzzy feedback controller (Arino and Sala, 2008; Lam and Leung, 2005). Although the non-PDC strategy can handle T-S fuzzy systems with uncertain membership functions (MFs), it cannot effectively use the uncertain information hidden in the MFs. Therefore, it usually generates greater conservativeness to the stability conditions. In addition, the upper and lower MFs of IT2 fuzzy sets mean the higher and lower bounds of uncertainty. Accordingly, it can effectively address the uncertain information in nonlinear systems. Moreover, IT2 T-S FM (Tang et al., 2019) is considered as the set of infinite type-1 T-S FM with determined MFs. Consequently, it has a stronger ability to handle uncertainty compared to the type-1 T-S FM. In Lam and Seneviratne (2008), the stability constraints of fuzzy control system have been analyzed by using the uncertainty information included in the uncertain domain of the IT2 fuzzy set. In Lam et al. (2014), the IT2 fuzzy controller was used to make the system stable in the case of the imperfect premise matching.

In recent years, increasing attentions have been attracted on the control problem of networked systems with limited communication capability (Sun et al., 2016; Zou et al., 2017). Since the constraint of communication network bandwidth and hardware, there are data collisions and network congestion (Zou et al., 2016) in network-based data transmission. Consequently, it inevitably leads to network-induced phenomena, for example, data transmission time-delays and data loss. In order to avoid such phenomena, it is necessary to establish network communication protocols for relieving network burden, saving communication energy, and alleviating data congestion. According to the communication protocol, the sensors or controllers use the network, transmit and receive data in accordance with the protocol rules. In addition, there are various effects on networked systems caused by different communication protocols.

Since the limited energy and heavy communication burden in network transmission, it is necessary to introduce an appropriate scheduling protocol to manage the necessary data transmission. In general, event-triggering scheme (Ding et al., 2015; Jia et al., 2014; Li et al., 2016; Zhang et al., 2017), the threshold is used to adopted as a constant. Because the threshold parameters cannot be adjusted flexibly, it will lead to the redundant sampling signals pass through the limited communication network. Therefore, the adaptive threshold parameter adjustment method (Peng et al., 2017, 2018; Wang et al., 2019) can effectively improve this situation, and has attracted extensive attention. As described in Zhang et al. (2019), the threshold parameters in adaptive event-triggered protocol (AETP) are related to the data transmission rate in the communication network. In some practical cases, that is not reasonable and conservative. In Li et al. (2021), a novel adaptive scheme is used to manage the data transmission. In addition, the impact of communication congestion is also mitigated. In Tang and Li (2020), an adaptive event-triggered model predictive control synthesis algorithm based on a state observer and an adaptive event-triggering scheme is proposed, aiming to solve the problems of bounded disturbances and the presence of data loss.

In the networked system, due to sensor network failures, malicious network attacks, and unknown environmental changes, the measured values may be anomalous, which will cause a sudden increase in the innovation of the measured values (large deviations), thus further degrade the desired control performance. Therefore, it is very important to go for an observer/estimator that is insensitive to the measured outliers in order to reduce the impact. Also, this has received special attention from a wide range of researchers, and some relevant results are in the literature (Alessandri and Zaccarian, 2018; Dai and so, 2018; Mu and Yuen, 2015; Park, 2017; Vecchia and Splett, 1994). For example, in Fu et al. (2020), Shen et al. (2020), a filter with an output saturation composition was constructed to resist the abnormal measurement values. Further, in Fu et al. (2021), a dynamic saturation constraint is utilized to suppress the effect of outliers on the system performance. Meanwhile, a comparison is made with a fixed saturation constraint in simulation, and the effect is clearly better than the latter. However, the corresponding results for observer-based control problems are very scarce. Therefore, it is one of the main motivations of this paper to propose an outlier-resistant observer to attenuate its effect on the system.

For the above discussion, we can conclude that it is necessary to further systematically investigate outlier-resistant observer-based control problem under AETP in particular. The key contributions of the article are emphasized as follows: (1) the outlier-resistant observer-based control problem is, for the first time, addressed for the IT2 T-S fuzzy systems under adaptive event-triggered protocol (AETP). (2) In the analysis and synthesis problems of the IT2 T-S fuzzy systems, a novel general framework is established in order to deal with the impact of innovation saturation and the adaptive event-triggered protocol (AETP). (3) The sufficient conditions are derived to guarantee the stability and the ${\ell }_2-{\ell }_{\infty }$ performance of the closed-loop system. Meanwhile, using the orthogonal decomposition method, a computationally appealing algorithm is proposed for designing the desired controller;

The specific contributions of this paper are divided into four non-divisions. The first part briefly introduces the background and current state of research, the advantages and disadvantages of interval type-1 T-S fuzzy control and IT2 T-S fuzzy control, and the advantages of AETPs. In the second part, this paper develops a control model for IT2 T-S fuzzy systems based on an outlier-resistant observer, introducing an innovative saturation mechanism to mitigate the effect of measurement outliers on the system, that is, the saturation function is insensitive to measurement outliers. In section “Main results,” the paper gives sufficient conditions to guarantee the stability and $l_2-l_{\infty }$ performance of the closed-loop system. Also, an algorithm for designing a computationally attractive expectation controller is proposed using the orthogonal decomposition method. In the last part of the article, simulations are performed to demonstrate the practicality and validity of the experimental results.

Notation: the notations applied in the article are quite standard unless otherwise indicated. ${\mathbb{R}}^{n_x}$ , ${\mathbb{R}}^{n_x\times n_y}$ represent the $n_x$ -dimensional Euclidean space and the set of all $n_x\times n_y$ real matrices, respectively. The $A^T$ and $A^{-1}$ mean transposition and inversion of A, which is a matrix, $A^{\perp }$ denotes its orthogonal basis. The identity matrix is represented by I, and 0 stands for the zero matrix with compatible dimensions. ${\ell }_2[0,+\infty )$ is the space of square-summable sequences. If not explicitly stated, matrices are assumed to have compatible dimensions.

Problem formulation and preliminaries

IT2 T-S fuzzy systems

An IT2 fuzzy model is considered with r rules whose the consequent is a linear dynamical system. Rule i is described as follows:

Plant Rule i: IF ${\nu }_1(k)$ is $G_1^i$ and ${\nu }_2(k)$ is $G_2^i$ and … and ${\nu }_{\theta }(k)$ is $G_{\theta }^i$ , THEN

$$\{\begin{array}{c}x(k+1)=A_{i}x(k)+E_i\sum _{n=1}^s{h}_{n}x(k-n)+B_{i}u(k)\hfill \\ \multicolumn{1}{c}{+D_{i}w(k)}\\ \multicolumn{1}{c}{y(k)=C_{i}x(k)}\\ \multicolumn{1}{c}{z(k)=L_{i}x(k)}\end{array}$$

(1)

where $G_{\varphi }^i$ represents IT2 fuzzy set of Rule i $(i=1,2,\cdots ,r)$ . The ${\nu }_{\varphi }(k)$ is the premise variable with $\varphi =1,2,\cdots ,\theta$ . r and $\theta$ denote two positive integers. $x(k)\in {\mathbb{R}}^{n_x}$ , $y(k)\in {\mathbb{R}}^{n_y}$ , and $z(k)\in {\mathbb{R}}^{n_z}$ stand for the system state, measurement output, and controlled output, respectively. $u(k)\in {\mathbb{R}}^{n_u}$ is the control input, and $w(k)\in ({\ell }_2[0,+\infty ),{\mathbb{R}}^{n_w})$ is the exogenous disturbance input. $A_i,E_i$ , $C_i$ , $B_i,D_i$ , and $L_i$ are known matrices with appropriate dimensions.

Next, the firing strength of Rule i is shown by the following intervals sets

$${\tilde{F}}_i(\nu (k))=[{\underset{¯}{F}}_i(\nu (k)),{\overline{F}}_i(\nu (k))]$$

where

$$\{\begin{array}{c}{\underset{¯}{F}}_i(\nu (k))=\underset{\varphi =1}{\overset{\theta }{\Pi }}{\underset{¯}{\beta }}_{G_{\varphi }^i}({\nu }_{\varphi }(k))\ge 0\hfill \\ \multicolumn{1}{c}{{\overline{F}}_i(\nu (k))=\underset{\varphi =1}{\overset{\theta }{\Pi }}{\overline{\beta }}_{G_{\varphi }^i}({\nu }_{\varphi }(k))\ge 0}\end{array}$$

$$\{\begin{array}{c}{\overline{\beta }}_{G_{\varphi }^i}({\nu }_{\varphi }(k))\ge {\underset{¯}{\beta }}_{G_{\varphi }^i}({\nu }_{\varphi }(k))\hfill \\ \multicolumn{1}{c}{{\overline{F}}_i(\nu (k))\ge {\underset{¯}{F}}_i(\nu (k))}\end{array}$$

${\underset{¯}{F}}_i(\nu (k))$ and ${\overline{F}}_i(\nu (k))$ represent the lower and upper membership grades, respectively. ${\underset{¯}{\beta }}_{G_{\varphi }^i}({\nu }_{\varphi }(k))$ and ${\overline{\beta }}_{G_{\varphi }^i}({\nu }_{\varphi }(k))\in [0,1]$ stand for the lower and upper MFs, respectively. Moreover, all the secondary grades of the IT2 fuzzy sets are indicated entirely by its own uncertainty footprint. Simplify $\sum _{i=1}^r{F}_i(\nu (k))=\sum _{F_i}$ . At the same time, the IT2 T-S fuzzy mode is denoted as follows

$$\{\begin{array}{c}x(k+1)=\sum _{F_i}[A_{i}x(k)+E_i\sum _{n=1}^s{h}_{n}x(k-n)\hfill \\ \multicolumn{1}{c}{+B_{i}u(k)+D_{i}w(k)]}\\ \multicolumn{1}{c}{y(k)=\sum _{F_i}{C}_{i}x(k)}\\ \multicolumn{1}{c}{z(k)=\sum _{F_i}{L}_{i}x(k)}\end{array}$$

(2)

where

$$F_i(\nu (k))={\underset{¯}{F}}_i(\nu (k)){\underset{¯}{\xi }}_i(\nu (k))+{\overline{F}}_i(\nu (k)){\overline{\xi }}_i(\nu (k))$$

$$\sum _{i=1}^r{F}_i(\nu (k))=1,0<F_i(\nu (k))<1$$

${\underset{¯}{\xi }}_i(\nu (k))$ and ${\overline{\xi }}_i(\nu (k))\in [0,1]$ are nonlinear weighting functions. They depend on the parameter uncertainties that satisfy ${\underset{¯}{\xi }}_i(\nu (k))+{\overline{\xi }}_i(\nu (k))=1$ .

• Remark 1. The parameter uncertainties which are present in the nonlinear system may lead to uncertainties of MFs. As a result, the weighting functions ${\underset{¯}{\xi }}_i(\nu (k))$ and ${\overline{\xi }}_i(\nu (k))$ possibly exist the parameter uncertainties and cause uncertain $F_i(\nu (k))$ . However, they are not necessary to be known in practical application, but they actually exist. Furthermore, it has been demonstrated that it is possible to reduce the conservativeness of desired controller by choosing reasonable values of the weighting coefficients in Lam et al. (2014). Specifically, we define ${\underset{¯}{\xi }}_i(\nu (k))$ and ${\overline{\xi }}_i(\nu (k))$ are associated with state variables as the nonlinear functions, but not fixed functions, in order to reduce the conservativeness.

AETP

The data transmission in this paper occurs from the sensor to the observer. To reduce energy consumption and alleviate data conflicts in this communication channel, the AETP is implemented to improve resource utilization. Meanwhile, AETP is discussed in this article as follows.

The event-triggered time instant sequence is $0\le l_0^g<l_1^g<\cdots <l_k^g<\cdots$ , which can be determined by

$$l_{k+1}^g=\underset{k}{\min }\{k>l_k^g|{\theta }_g^T(k){\mathrm{\Psi }}_g{\theta }_g(k)>{ϵ}_g(k)y_g^T(l_k^g){\mathrm{\Psi }}_g{y}_g(l_k^g)\}$$

(3)

where ${\theta }_g(k)=y_g(k)-y_g(l_k^g)$ , and $l_k^g$ is the latest transmission instant of the $g\mathrm{th}$ measurement component of $y(k)$ . ${\mathrm{\Psi }}_g>0$ is given weighted matrix. ${ϵ}_g(k)$ is the triggered threshold to be designed satisfying following adaptive law

$${ϵ}_g(k)=\min \{\max \{{ϵ}_d,\kappa {ϵ}_g(k-1)\},{ϵ}_D\}$$

(4)

where

$$\kappa =\{\begin{array}{c}0,\mathrm{if}\Vert y_g(k)\Vert \ge \Vert y_g(l_k^g)\Vert \hfill \\ \multicolumn{1}{c}{1-\frac{2\sigma }{\pi }\mathrm{atan}\left(\frac{\Vert y_g(k)\Vert -\Vert y_g(l_k^g)\Vert }{\Vert y_g(l_k^g)\Vert }\right),\mathrm{otherwise}.}\end{array}$$

(5)

where $\sigma >0$ reflect the change rate of ${ϵ}_g(k)$ . Meanwhile, ${ϵ}_d$ is the lower bound of ${ϵ}_g(k)$ and $ϵ(0)={ϵ}_d>0$ , and ${ϵ}_D$ is the higher bound of ${ϵ}_g(k)$ .

By the above analysis, the latest triggering measurement output vector $y(l_k)$ can be indicated as

$$y(l_k)=[y_1^T(l_k^1)y_2^T(l_k^2)\cdots y_{n_y}^T(l_k^{n_y}){]}^T$$

and similarly, $y(k)$ can also be expressed as

$$y(k)=[y_1^T(k)y_2^T(k)\cdots y_{n_y}^T(k){]}^T$$

Therefore, $\theta (k)$ is written as

$$\theta (k)=y(k)-y(l_k)$$

• Remark 2. Notice that the value of $\kappa$ in equation (5) has two situations: one is $\kappa =0$ and the other is $\kappa \in [1,1+\sigma )$ . Moreover, when $\Vert y_g(k)\Vert <\Vert y_g(l_k^g)\parallel$ with $k>0$ , it can be derived that ${ϵ}_g(k)>{ϵ}_g(k-1)$ . In this case, the triggered threshold keeps increasing, which will lead to a lower communication frequency. In addition, the smaller ${ϵ}_g(k)$ can make a higher transmission frequency, which will create data congestion and cause network burden. Furthermore, if the parameter $\kappa =0$ , we can know that ${ϵ}_g(k)={ϵ}_d$ , the adaptive transmission scheme (3) will be regarded as the traditional event-triggered protocol. Meanwhile, the adaptive scheme discussed in equation (3) will be periodic by selecting parameters ${ϵ}_d={ϵ}_D=0$ . The threshold parameters of adaptive event-triggering are more sensitive in the regulation process compared to traditional static event-triggering, which is used in this paper. An innovative adaptive scheme is introduced to manage the data transfer, which not only ignores the inconvenient effects of inflexible point threshold adjustment, but also improves to some extent the unfavorable factors such as data flow congestion, thus making the system have a better performance.

Outlier-resistant observer-based controller

In this paper, innovation saturation is considered in the observer to get desired control performance, which has the same MFs with plant. Therefore, the overall IT2 T-S fuzzy observer is shown

$$\{\begin{array}{c}\hat{x}(k+1)=\sum _{F_i}[A_i\hat{x}(k)+E_i\sum _{n=1}^s{h}_n\hat{x}(k-n)\hfill \\ \multicolumn{1}{c}{+H_i\psi (y(l_k)-C_i\hat{x}(k))+B_{i}u(k)]}\\ \multicolumn{1}{c}{\hat{z}(k)=\sum _{F_i}{L}_i\hat{x}(k)}\end{array}$$

(6)

where $\hat{x}(k)$ and $\hat{z}(k)$ are the estimate of $x(k)$ and $z(k)$ , respectively. $H_i(i=1,2,\dots ,r)$ denotes the observer gain matrix to be designed. The saturation function $\psi (·)$ is indicated as

$$\psi (v)=[\psi (v_1)\psi (v_2)\cdots \psi (v_{n_y}){]}^T$$

which satisfies

$$(\psi (v)-\chi v{)}^T(\psi (v)-v)\le 0$$

(7)

where

$$\begin{array}{c}v=[v_1{v}_2\cdots v_{n_y}{]}^T\hfill \\ \chi =\mathrm{diag}\{{\chi }_1,{\chi }_2,\dots ,{\chi }_{n_y}\},0\le \chi \le I\hfill \\ \psi (v_t)=\mathrm{sign}(v_t)\min \{v_t^M,|v_t|\},t=1,2,\dots ,n_y\hfill \end{array}$$

in which $v_t^M$ denotes the $t\mathrm{th}$ element of the saturation level vector $v^M$ . Note that, without loss of generality, the saturation level is taken as unity.

• Remark 3. It should be noted that the innovation $y(l_k)-C_i\hat{x}(k)$ is constrained to a predetermined range subject to the saturation function $\psi (·)$ . This avoids the deviation of the innovation anomaly from its usual range due to the abnormal measurement value. In addition, the saturation level $v^M$ can be determined based on engineering practice, and the saturation function is limited by the sector bounded condition (7), which facilitates the subsequent performance analysis. Moreover, the introduction of the outlier-resistant structure improves the reliability of the proposed observer. In particular, the designed observer (6) becomes a conventional Luenberger observer when $v^M$ tends to infinity. Meanwhile, to address the outlier problem, this paper introduces an observer with an innovative saturation mechanism that not only has many of the advantages of the Luenberger observer but also mitigates the impact of measurement outliers on the system, that is, the saturation function is insensitive to measurement outliers and is more helpful for observation.

In this paper, we employ a different MF from the plant to fuzzy controller parameters, which can improve the flexibility of the design.

Controller Rule c: IF $q_1(k)$ is $Q_1^c$ and $q_2(k)$ is $Q_2^c$ and … and $q_n(k)$ is $Q_n^c$ , THEN

$$u(k)=K_c\hat{x}(k)$$

(8)

where $K_c$ $(c=1,2,\dots ,r)$ is the controller gain matrix to be determined. Next, the firing strength of Rule c is shown by the intervals sets as follows

$${\tilde{\mathrm{\Theta }}}_c(q(k))=[{\underset{¯}{\mathrm{\Theta }}}_c(q(k)),{\overline{\mathrm{\Theta }}}_c(q(k))]$$

where

$$\{\begin{array}{c}{\underset{¯}{\mathrm{\Theta }}}_c(q(k))=\underset{p=1}{\overset{n}{\Pi }}{\underset{¯}{\mu }}_{Q_p^c}(q(k))\ge 0\hfill \\ \multicolumn{1}{c}{{\overline{\mathrm{\Theta }}}_c(q(k))=\underset{p=1}{\overset{n}{\Pi }}{\overline{\mu }}_{Q_p^c}(q(k))\ge 0}\end{array}$$

$$\{\begin{array}{c}{\overline{\mu }}_{Q_p^c}(q(k))\ge {\underset{¯}{\mu }}_{Q_p^c}(q(k))\hfill \\ \multicolumn{1}{c}{{\overline{\mathrm{\Theta }}}_c(q(k))\ge {\underset{¯}{\mathrm{\Theta }}}_c(q(k))}\end{array}$$

where $Q_p^c$ $(p=1,2,\dots ,n)$ is fuzzy set. n is the number of controller premise variables $q(k)$ . ${\underset{¯}{\mathrm{\Theta }}}_c(q(k))$ and ${\overline{\mathrm{\Theta }}}_c(q(k))$ represent the lower and upper membership grades, respectively. ${\underset{¯}{\mu }}_{Q_p^c}(q(k)))$ and ${\overline{\mu }}_{Q_p^c}(q(k))\in [0,1]$ stand for the lower and upper MFs, respectively. The global IT2 controller is shown by

$$u(k)=\sum _{c=1}^r{\mathrm{\Theta }}_c(q(k))K_c\hat{x}(k)$$

(9)

where

$${\mathrm{\Theta }}_c(q(k))={\underset{¯}{\mathrm{\Theta }}}_c(q(k)){\underset{¯}{d}}_i(q(k))+{\overline{\mathrm{\Theta }}}_c(q(k)){\overline{d}}_i(q(k))$$

$$\sum _{i=1}^r{\mathrm{\Theta }}_c(q(k))=1,0<{\mathrm{\Theta }}_c(q(k))<1$$

${\underset{¯}{d}}_i(q(k))$ and ${\overline{d}}_i(q(k))\in [0,1]$ are nonlinear weighting functions. They depend on the parameter uncertainties that satisfy ${\underset{¯}{d}}_i(q(k))+{\overline{d}}_i(q(k))=1$ .

Define $e(k)=x(k)-\hat{x}(k)$ and $\varkappa (k)=z(k)-\hat{z}(k)$ . For convenience, simplify $\sum _{i=1}^r\sum _{c=1}^r{F}_i(\nu (k)){\mathrm{\Theta }}_c(q(k))=\sum _{F_i{\mathrm{\Theta }}_c}$ . We can obtain the following argument system

$$\{\begin{array}{c}\eta (k+1)=\sum _{F_i{\mathrm{\Theta }}_c}[{\mathcal{A}}_{\mathrm{ic}}\eta (k)+{\mathcal{E}}_i\sum _{n=1}^s{h}_n\eta (k-n)\hfill \\ \multicolumn{1}{c}{+{\mathcal{D}}_{i}w(k)+{\mathcal{H}}_i\psi ({\vartheta }_i(k))]}\\ \multicolumn{1}{c}{y(k)=\sum _{F_i}{\overline{C}}_i\eta (k)}\\ \multicolumn{1}{c}{\varkappa (k)=\sum _{F_i}{\mathcal{L}}_i\eta (k)}\end{array}$$

(10)

where

$$\begin{array}{c}\eta (k)={\left[\begin{array}{cc}{x}^T(k)\hfill & e^T(k)\hfill \end{array}\right]}^T,{\mathcal{C}}_i=\left[\begin{array}{cc}0\hfill & C_i\hfill \end{array}\right]\hfill \\ {\overline{C}}_i=\left[\begin{array}{cc}{C}_i\hfill & 0\hfill \end{array}\right]\hfill \\ {\mathcal{H}}_i=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-H_i}\end{array}\right],{\vartheta }_i(k)={\mathcal{C}}_i\eta (k)-\theta (k)\hfill \\ {\mathcal{A}}_{\mathrm{ic}}=\left[\begin{array}{cc}{A}_i+B_i{K}_c\hfill & -B_i{K}_c\hfill \\ \multicolumn{1}{c}{0}& A_i\hfill \end{array}\right],{\mathcal{E}}_i=\left[\begin{array}{cc}{E}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& E_i\hfill \end{array}\right]\hfill \\ {\mathcal{D}}_i=\left[\begin{array}{c}{D}_i\hfill \\ \multicolumn{1}{c}{D_i}\end{array}\right],{\mathcal{L}}_i=\left[\begin{array}{cc}0\hfill & L_i\hfill \end{array}\right]\hfill \end{array}$$

Remark 4. It can be seen that the plant and the observer share the same premise variables and MFs, which are different from the controller. Accordingly, they each have their own separate design. To some extent, this increases the flexibility of the controller design.

Definition 1. The augmented system (10) with $w(k)=0$ is exponentially stable if there exist constants $\delta >0$ and $\epsilon \in (0,1)$ satisfy

$$\Vert \eta (k){\Vert }^2\le \delta {\epsilon }^k\underset{i\in [-s,0]}{max}\Vert \eta (i){\Vert }^2$$

(11)

This paper aims to look for an observer-based controller with the form (9) which can satisfy both of the following requirements.

1. The argument system (10) is exponentially stable.

2. For a given scalar $\gamma >0$ , the following ${\ell }_2-{\ell }_{\infty }$ performance is satisfied for $\eta (0)=0$ and all nonzero $w(k)$

$$\Vert \varkappa (k){\Vert }_{\infty }<\gamma \sqrt{\sum _{k=0}^{\infty }\Vert w(k){\Vert }^2}$$

(12)

where $\Vert \varkappa (k){\Vert }_{\infty }=\mathrm{su}{\mathrm{p}}_k\sqrt{{\varkappa }^T(k)\varkappa (k)}$ .

Main results

Exponentially stability and ${\ell }_2-{\ell }_{\infty }$ performance analysis

In this section, we focus on analyzing the exponentially stable and ${\ell }_2-{\ell }_{\infty }$ performance of the augmented system (10) under innovation saturation and adaptive event trigger protocol.

Theorem 1. With the given gain matrices $K_c$ , $H_i$ and performance index $\gamma >0$ . The system (10) is exponentially stable with $w(k)=0$ while achieving the ${\ell }_2-{\ell }_{\infty }$ performance constraint (12) if there exist positive definite matrices P, L and positive scalars ${\varrho }_1$ , ${\varrho }_2$ such that

$$\{\begin{array}{c}{\mathrm{\Sigma }}_{\mathrm{ic}}<0,i=c=1,\dots ,r\hfill \\ \multicolumn{1}{c}{{\mathrm{\Sigma }}_{\mathrm{ic}}+{\mathrm{\Sigma }}_{\mathrm{ci}}\le 0,1\le i<c\le r}\end{array}$$

(13)

$$U=\left[\begin{array}{cc}-P\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathcal{L}}_i}& -I\hfill \end{array}\right]<0$$

(14)

where

$$\begin{array}{c}{\mathrm{\Sigma }}_{\mathrm{ic}}=\left[\begin{array}{cc}{\mathrm{\Sigma }}_1\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathrm{\Sigma }}_2}& -P^{-1}\hfill \end{array}\right]\\ {\mathrm{\Sigma }}_1=\left[\begin{array}{ccccc}{\mathrm{\Sigma }}_1^{11}\hfill & *\hfill & *\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{0}& {\mathrm{\Sigma }}_1^{22}\hfill & *\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathrm{\Sigma }}_1^{31}}& 0\hfill & {\mathrm{\Sigma }}_1^{33}\hfill & *\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathrm{\Sigma }}_1^{41}}& 0\hfill & {\mathrm{\Sigma }}_1^{43}\hfill & {\mathrm{\Sigma }}_1^{44}\hfill & *\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & -{\gamma }^{2}I\hfill \end{array}\right]\\ {\mathrm{\Sigma }}_2=\left[\begin{array}{ccccc}{\mathcal{A}}_{\mathrm{ic}}\hfill & {\mathcal{E}}_i\hfill & {\mathcal{H}}_i\hfill & 0\hfill & {\mathcal{D}}_i\hfill \end{array}\right]\\ \mathrm{\Psi }=\mathrm{diag}\{{\mathrm{\Psi }}_1,{\mathrm{\Psi }}_2,\dots ,{\mathrm{\Psi }}_{n_y}\}\end{array}$$

$$\begin{array}{c}{\mathrm{\Sigma }}_1^{11}=-P+\overline{h}L+{\varrho }_2{\epsilon }_D{\overline{C}}_i^T\mathrm{\Psi }{\overline{C}}_i-{\varrho }_1{\mathcal{C}}_i^T{\chi }^T{\mathcal{C}}_i\\ {\mathrm{\Sigma }}_1^{31}=\frac{{\varrho }_1}{2}(\chi +I){\mathcal{C}}_i,{\mathrm{\Sigma }}_1^{41}={\varrho }_1\chi {\mathcal{C}}_i-{\varrho }_2{\epsilon }_D{\mathrm{\Psi }}^T{\overline{C}}_i\\ {\mathrm{\Sigma }}_1^{22}=-\frac{1}{\overline{h}}L,{\mathrm{\Sigma }}_1^{33}=-{\varrho }_{1}I,{\mathrm{\Sigma }}_1^{43}=-\frac{{\varrho }_1}{2}({\chi }^T+I),\overline{h}=\sum _{n=1}^s{h}_n\\ {\mathrm{\Sigma }}_1^{44}={\varrho }_2({\epsilon }_D-1)\mathrm{\Psi }-{\varrho }_1{\chi }^T\end{array}$$

Proof. To get the purpose simply, the following Lyapunov–Krasovskii function is selected

$$\begin{array}{c}V(k)=V_1(k)+V_2(k)\hfill \\ ={\eta }^T(k)P\eta (k)+\sum _{n=1}^s{h}_n\sum _{v=k-n}^{k-1}{\eta }^T(v)L\eta (v)\hfill \end{array}$$

(15)

By following the trajectory of the augmented system (10), then, the difference of the $V(k)$ yields

$$\begin{array}{c}\mathrm{\Delta }{V}_1(k)\hfill \\ ={\eta }^T(k+1)P\eta (k+1)-{\eta }^T(k)P\eta (k)\hfill \\ =\sum _{F_i{\mathrm{\Theta }}_c}[{\eta }^T(k){\mathcal{A}}_{\mathrm{ic}}^{T}P{\mathcal{A}}_{\mathrm{ic}}\eta (k)\hfill \\ +\sum _{n=1}^s{h}_n{\eta }^T(k-n){\mathcal{E}}_i^{T}P{\mathcal{E}}_i\sum _{n=1}^s{h}_n\eta (k-n)\hfill \\ +{\psi }^T({\vartheta }_i(k)){\mathcal{H}}_i^{T}P{\mathcal{H}}_i\psi ({\vartheta }_i(k))+2{\eta }^T(k){\mathcal{A}}_{\mathrm{ic}}^{T}P{\mathcal{E}}_i\sum _{n=1}^s{h}_n\eta (k-n)\hfill \\ +2\sum _{n=1}^s{h}_n{\eta }^T(k-n){\mathcal{E}}_i^{T}P{\mathcal{H}}_i\psi ({\vartheta }_i(k))\hfill \\ +2{\eta }^T(k){\mathcal{A}}_{\mathrm{ic}}^{T}P{\mathcal{H}}_i\psi ({\vartheta }_i(k))]-{\eta }^T(k)P\eta (k)\hfill \end{array}$$

(16)

$$\begin{array}{c}\mathrm{\Delta }{V}_2(k)\hfill \\ =\sum _{n=1}^s{h}_n\sum _{v=k+1-n}^k{\eta }^T(v)L\eta (v)-\sum _{n=1}^s{h}_n\sum _{v=k-n}^{k-1}{\eta }^T(v)L\eta (v)\hfill \\ =\overline{h}{\eta }^T(k)L\eta (k)-\sum _{n=1}^s{h}_n{\eta }^T(k-n)L\eta (k-n)\hfill \\ \le \overline{h}{\eta }^T(k)L\eta (k)-\frac{1}{\overline{h}}\sum _{n=1}^s{h}_n{\eta }^T(k-n)L\sum _{n=1}^s{h}_n\eta (k-n)\hfill \end{array}$$

(17)

Summing equations (16) and (17), therefore, we can obtain

$$\mathrm{\Delta }V(k)\le \sum _{F_i{\mathrm{\Theta }}_c}{\varsigma }^T(k){\Xi }_{1\mathrm{ic}}\varsigma (k)$$

(18)

where

$$\begin{array}{c}\varsigma (k)={\left[\begin{array}{ccc}{\eta }^T(k)\hfill & \sum _{n=1}^s{h}_n{\eta }^T(k-n)\hfill & {\psi }^T({\vartheta }_i(k))\hfill \end{array}\right]}^T\\ {\Xi }_{1\mathrm{ic}}=\left[\begin{array}{ccc}{\Xi }_1^{11}\hfill & {\mathcal{A}}_{\mathrm{ic}}^{T}P{\mathcal{E}}_i\hfill & {\mathcal{A}}_{\mathrm{ic}}^{T}P{\mathcal{H}}_i\hfill \\ \multicolumn{1}{c}{*}& {\Xi }_1^{22}\hfill & {\mathcal{E}}_i^{T}P{\mathcal{H}}_i\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\mathcal{H}}_i^{T}P{\mathcal{H}}_i\hfill \end{array}\right]\\ {\Xi }_1^{11}={\mathcal{A}}_{\mathrm{ic}}^{T}P{\mathcal{A}}_{\mathrm{ic}}-P+\overline{h}L\end{array}$$

$${\Xi }_1^{22}={\mathcal{E}}_i^{T}P{\mathcal{E}}_i-\frac{1}{\overline{h}}L$$

Take equation (3) into account and we can get as follows

$${\theta }_g^T(k){\mathrm{\Psi }}_g{\theta }_g(k)-{ϵ}_g(k)y_g^T(l_k^g){\mathrm{\Psi }}_g{y}_g(l_k^g)<0$$

which can be written in compact form and satisfy the following condition

$${\theta }^T(k)\mathrm{\Psi }\theta (k)-{ϵ}_D{y}^T(l_k)\mathrm{\Psi }y(l_k)<0$$

(19)

Moreover, on the basis of equations (7) and (19), we know that

$$\begin{array}{c}\mathrm{\Delta }V(k)\hfill \\ \le \sum _{F_i{\mathrm{\Theta }}_c}\{{\varsigma }^T(k){\Xi }_{1\mathrm{ic}}\varsigma (k)\hfill \\ -{\varrho }_1[{(\psi ({\vartheta }_i(k))-\chi {\vartheta }_i(k))}^T(\psi ({\vartheta }_i(k))-{\vartheta }_i(k))]\hfill \\ -{\varrho }_2[{\theta }^T(k)\mathrm{\Psi }\theta (k)-ϵ(k)y^T(l_k)\mathrm{\Psi }y(l_k)]\}\hfill \\ \le \sum _{F_i{\mathrm{\Theta }}_c}\{{\varsigma }^T(k){\Xi }_{1\mathrm{ic}}\varsigma (k)-{\varrho }_1[{\psi }^T({\vartheta }_i(k))\psi ({\vartheta }_i(k))\hfill \\ -{\vartheta }_i^T(k)({\chi }^T+I)\psi ({\vartheta }_i(k))+{\vartheta }_i^T(k){\chi }^T{\vartheta }_i(k)]\hfill \\ -{\varrho }_2[{\theta }^T(k)\mathrm{\Psi }\theta (k)-{ϵ}_D{y}^T(l_k)\mathrm{\Psi }y(l_k)]\}\hfill \\ =\sum _{F_i{\mathrm{\Theta }}_c}\{{\varsigma }^T(k){\Xi }_{1\mathrm{ic}}\varsigma (k)-{\varrho }_1[{\psi }^T({\vartheta }_i(k))\psi ({\vartheta }_i(k))\hfill \\ -{\eta }^T(k){\mathcal{C}}_i^T({\chi }^T+I)\psi ({\vartheta }_i(k))+{\theta }^T(k)({\chi }^T+I)\psi ({\vartheta }_i(k))\hfill \\ +{\eta }^T(k){\mathcal{C}}_i^T{\chi }^T{\mathcal{C}}_i\eta (k)+{\theta }^T(k){\chi }^T\theta (k)\hfill \\ -2{\eta }^T(k){\mathcal{C}}_i^T{\chi }^T\theta (k)]-{\rho }_2[{\theta }^T(k)(1-{ϵ}_D)\mathrm{\Psi }\theta (k)\hfill \\ -{ϵ}_D{\eta }^T(k){\overline{C}}_i^T\mathrm{\Psi }{\overline{C}}_i\eta (k)+2{ϵ}_D{\eta }^T(k){\overline{C}}_i^T\mathrm{\Psi }\theta (k)]\}\hfill \\ =\sum _{F_i{\mathrm{\Theta }}_c}\{{\varsigma }^T(k){\Xi }_{1\mathrm{ic}}\varsigma (k)+{\varrho }_1{\eta }^T(k){\mathcal{C}}_i^T({\chi }^T+I)\psi ({\vartheta }_i(k))\hfill \\ -{\varrho }_1{\psi }^T({\vartheta }_i(k))\psi ({\vartheta }_i(k))-{\varrho }_1{\theta }^T(k)({\chi }^T+I)\psi ({\vartheta }_i(k))\hfill \\ +{\eta }^T(k)({\varrho }_2{\epsilon }_D{\overline{C}}_i^T\mathrm{\Psi }{\overline{C}}_i-{\varrho }_1{\mathcal{C}}_i^T{\chi }^T{\mathcal{C}}_i)\eta (k)\hfill \\ +{\theta }^T(k)[{\varrho }_2({\epsilon }_D-1)\mathrm{\Psi }-{\varrho }_1{\chi }^T]\theta (k)\hfill \\ +2{\eta }^T(k)({\varrho }_1{\mathcal{C}}_i^T{\chi }^T-{\varrho }_2{\epsilon }_D{\overline{C}}_i^T\mathrm{\Psi })\theta (k)\}\hfill \\ =\sum _{i=1}^r\sum _{i=c}^r{F}_i{\mathrm{\Theta }}_c{\zeta }^T(k){\Xi }_{2\mathrm{ic}}\zeta (k)\hfill \\ +\sum _{i=1}^r\sum _{i<c}^r{F}_i{\mathrm{\Theta }}_c{\zeta }^T(k)({\Xi }_{2\mathrm{ic}}+{\Xi }_{2\mathrm{ci}})\zeta (k)\hfill \end{array}$$

(20)

where

$$\begin{array}{c}\zeta (k)=[{\eta }^T(k)\sum _{n=1}^s{h}_n{\eta }^T(k-n){\psi }^T({\vartheta }_i(k)){\theta }^T(k)]\hfill \\ {\Xi }_{2\mathrm{ic}}=\left[\begin{array}{cccc}{\Xi }_2^{11}\hfill & {\mathcal{A}}_{\mathrm{ic}}^{T}P{\mathcal{E}}_i\hfill & {\Xi }_2^{13}\hfill & {\Xi }_2^{14}\hfill \\ \multicolumn{1}{c}{*}& {\Xi }_1^{22}\hfill & {\mathcal{E}}_i^{T}P{\mathcal{H}}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\Xi }_2^{33}\hfill & {\Xi }_2^{34}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & {\mathrm{\Sigma }}_1^{44}\hfill \end{array}\right]\hfill \\ {\Xi }_2^{11}={\Xi }_1^{11}+{\varrho }_2{\epsilon }_D{\overline{C}}_i^T\mathrm{\Psi }{\overline{C}}_i-{\varrho }_1{\mathcal{C}}_i^T{\chi }^T{\mathcal{C}}_i\hfill \\ {\Xi }_2^{13}={\mathcal{A}}_{\mathrm{ic}}^{T}P{\mathcal{H}}_i+\frac{{\varrho }_1}{2}{\mathcal{C}}_i^T({\chi }^T+I)\hfill \\ {\Xi }_2^{14}={\varrho }_1{\mathcal{C}}_i^T{\chi }^T-{\varrho }_2{\epsilon }_D{\overline{C}}_i^T\mathrm{\Psi }\hfill \\ {\Xi }_2^{33}={\mathcal{H}}_i^{T}P{\mathcal{H}}_i-{\varrho }_{1}I,{\Xi }_2^{34}=-\frac{{\varrho }_1}{2}(\chi +I)\hfill \end{array}$$

By the Schur complement lemma, it follows immediately from equation (13) that $\mathrm{\Delta }V(k)<0$ . Taking the above analysis, we can clearly demonstrate exponentially stable via making a similar analysis to Wang et al. (2009). In what follows, let’s push the process forward and make ${\ell }_2-{\ell }_{\infty }$ performance analysis of the argument system (10) for $w(k)\ne 0$ . For this purpose, $J(k)$ is constructed as follows

$$\begin{array}{l}J(k)=V(k)-{\gamma }^2\sum _{f=0}^{k-1}{w}^T(f)w(f)\\ =\sum _{f=0}^{k-1}(\mathrm{\Delta }V(f)-{\gamma }^2{w}^T(f)w(f))-V(0)\\ \le \sum _{f=0}^{k-1}\{\sum _{F_i{\mathrm{\Theta }}_c}[{\zeta }^T(f){\Xi }_{2ic}\zeta (f)+w^T(f){\mathcal{D}}_i^{T}P{\mathcal{D}}_{i}w(f)\\ +2{\eta }^T(f){\mathcal{A}}_{ic}^{T}P{\mathcal{D}}_{i}w(f)+2{\psi }^T(\vartheta (f)){\mathscr{H}}_i^{T}P{\mathcal{D}}_{i}w(f)\\ +2\sum _{n=1}^s{h}_n{\eta }^T(f-n){ℰ}_i^{T}P{\mathcal{D}}_{i}w(f)]-{\gamma }^2{w}^T(f)w(f)\}\\ =\sum _{f=0}^{k-1}\{\sum _{i=1}^r\sum _{i=c}^r{F}_i{\mathrm{\Theta }}_c{\xi }^T(f){\mathrm{\Gamma }}_{ic}\xi (f)+\sum _{i=1}^r\sum _{i<c}^r{F}_i{\mathrm{\Theta }}_c{\xi }^T(f)({\mathrm{\Gamma }}_{ic}+{\mathrm{\Gamma }}_{ci})\xi (f)\}\end{array}$$

(21)

where

$$\begin{array}{c}\xi (k)={\left[\begin{array}{cc}{\zeta }^T(k)\hfill & w^T(k)\hfill \end{array}\right]}^T\hfill \\ {\mathrm{\Gamma }}_{\mathrm{ic}}=\left[\begin{array}{ccccc}{\Xi }_2^{11}\hfill & {\mathcal{A}}_{\mathrm{ic}}^{T}P{\mathcal{E}}_i\hfill & {\Xi }_2^{13}\hfill & {\Xi }_2^{14}\hfill & {\mathrm{\Gamma }}_{15}\hfill \\ \multicolumn{1}{c}{*}& {\Xi }_1^{22}\hfill & {\mathcal{E}}_i^{T}P{\mathcal{H}}_i\hfill & 0\hfill & {\mathrm{\Gamma }}_{25}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\Xi }_2^{33}\hfill & {\Xi }_2^{34}\hfill & {\mathrm{\Gamma }}_{35}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & {\mathrm{\Sigma }}_1^{44}\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & {\mathrm{\Gamma }}_{55}\hfill \end{array}\right]\hfill \\ {\mathrm{\Gamma }}_{15}={\mathcal{A}}_{\mathrm{ic}}^{T}P{\mathcal{D}}_i,{\mathrm{\Gamma }}_{25}={\mathcal{E}}_i^{T}P{\mathcal{D}}_i\hfill \\ {\mathrm{\Gamma }}_{35}={\mathcal{H}}_i^{T}P{\mathcal{D}}_i,{\mathrm{\Gamma }}_{55}={\mathcal{D}}_i^{T}P{\mathcal{D}}_i-{\gamma }^{2}I\hfill \end{array}$$

Equation $(21)<0$ can be determined by equation (13) because of the following

$${\mathrm{\Gamma }}_{\mathrm{ic}}={\mathrm{\Sigma }}_1+{\mathrm{\Sigma }}_2^{T}P{\mathrm{\Sigma }}_2<0$$

Thus, we can know that

$${\eta }^T(k)P\eta (k)=V_1(k)<V(k)<{\gamma }^2\sum _{f=0}^{k-1}{w}^T(f)w(f)$$

Furthermore, it can be derive from equation (14)

$${\varkappa }^T(k)\varkappa (k)={\eta }^T(k){\mathcal{L}}_i^T{\mathcal{L}}_i\eta (k)<{\eta }^T(k)P\eta (k)$$

Therefore

$${\varkappa }^T(k)\varkappa (k)<{\gamma }^2\sum _{f=0}^{k-1}{w}^T(f)w(f)$$

(22)

According to the nature of the inequality, the following changes are made on both sides of equation (22)

$$\mathrm{su}{\mathrm{p}}_k\sqrt{{\varkappa }^T(k)\varkappa (k)}<\gamma \sqrt{\sum _{f=0}^{\infty }\Vert w(f){\Vert }^2}$$

The proof of Theorem 1 is completed.

Observer and controller design under adaptive event trigger protocol

• Theorem 2. With the given performance index $\gamma >0$ . The system (10) is exponentially stable with $w(k)=0$ while achieving the ${\ell }_2-{\ell }_{\infty }$ performance constraint (12) if there exist positive definite matrices $P_1$ , $P_2$ and L, and gain matrices ${\stackrel{`}{K}}_c$ , ${\stackrel{`}{H}}_i$ , matrix $\mathrm{\Lambda }$ and positive scalars ${\varrho }_1$ , ${\varrho }_2$ such that

$$\{\begin{array}{c}{\mathrm{\Upsilon }}_{\mathrm{ic}}<0,i=c=1,\dots ,r\hfill \\ \multicolumn{1}{c}{{\mathrm{\Upsilon }}_{\mathrm{ic}}+{\mathrm{\Upsilon }}_{\mathrm{ci}}\le 0,1\le i<c\le r}\end{array}$$

(23)

$$U=\left[\begin{array}{cc}-P\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathcal{L}}_i}& -I\hfill \end{array}\right]<0$$

(24)

Furthermore, the gain matrix of the controller and observer is given by

$$K_c={\mathrm{\Lambda }}_{11}^{-1}{\stackrel{`}{K}}_c,H_i=P_2^{-1}{\stackrel{`}{H}}_i$$

(25)

where

$$\begin{array}{c}{\mathrm{\Upsilon }}_{\mathrm{ic}}=\left[\begin{array}{cc}{\mathrm{\Sigma }}_1\hfill & *\hfill \\ \multicolumn{1}{c}{{\mathrm{\Upsilon }}_2}& {\mathrm{\Upsilon }}_3\hfill \end{array}\right],{\mathrm{\Upsilon }}_2=\left[\begin{array}{ccccc}{\stackrel{`}{\mathcal{A}}}_{\mathrm{ic}}\hfill & {\stackrel{`}{\mathcal{E}}}_i\hfill & {\stackrel{`}{\mathcal{H}}}_i\hfill & 0\hfill & {\stackrel{`}{\mathcal{D}}}_i\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{{\stackrel{`}{\mathcal{A}}}_{\mathrm{ic}}=\left[\begin{array}{cc}\mathrm{\Lambda }{\mathrm{\Theta }}_i{A}_i+{\hat{K}}_c\hfill & -{\hat{K}}_c\hfill \\ \multicolumn{1}{c}{0}& P_2{A}_i\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\stackrel{`}{\mathcal{E}}}_i=\left[\begin{array}{cc}\mathrm{\Lambda }{\mathrm{\Theta }}_i{E}_i\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& P_2{E}_i\hfill \end{array}\right],{\hat{K}}_c={\left[{\stackrel{`}{K}}_c^{T}0\right]}^T}\\ \multicolumn{1}{c}{{\mathrm{\Theta }}_i={\left[B_i{(B_i^T{B}_i)}^{-1}{(B_i^T)}^{\perp }\right]}^T}\\ \multicolumn{1}{c}{\mathrm{\Lambda }=\left[\begin{array}{cc}{\mathrm{\Lambda }}_{11}\hfill & {\mathrm{\Lambda }}_{12}\hfill \\ \multicolumn{1}{c}{0}& {\mathrm{\Lambda }}_{22}\hfill \end{array}\right],P=\mathrm{diag}\{P_1,P_2\}}\\ \multicolumn{1}{c}{{\mathrm{\Upsilon }}_3=\mathrm{diag}\{P_1-\mathrm{\Lambda }{\mathrm{\Theta }}_i-{\mathrm{\Theta }}_i^T{\mathrm{\Lambda }}^T,-P_2\}}\\ \multicolumn{1}{c}{{\stackrel{`}{\mathcal{H}}}_i=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-{\stackrel{`}{\mathcal{H}}}_i}\end{array}\right],{\stackrel{`}{\mathcal{D}}}_i=\left[\begin{array}{c}\mathrm{\Lambda }{\mathrm{\Theta }}_i{D}_i\hfill \\ \multicolumn{1}{c}{P_2{D}_i}\end{array}\right]}\end{array}$$

Proof. Let $B_i=B_c$ and pre and post multiplying the inequality (13) by diag $\{I,I,I,I,I,\mathrm{\Lambda }{\mathrm{\Theta }}_i,P_2\}$ and its transposition, we have

$$\{\begin{array}{c}{\stackrel{\vee }{\mathrm{\Sigma }}}_{\mathrm{ic}}<0,i=c=1,\dots ,r\hfill \\ \multicolumn{1}{c}{{\stackrel{\vee }{\mathrm{\Sigma }}}_{\mathrm{ic}}+{\stackrel{\vee }{\mathrm{\Sigma }}}_{\mathrm{ci}}\le 0,1\le i<c\le r}\end{array}$$

(26)

where

$$\begin{array}{c}{\stackrel{\vee }{\mathrm{\Sigma }}}_{\mathrm{ic}}=\left[\begin{array}{cc}{\mathrm{\Sigma }}_1\hfill & *\hfill \\ \multicolumn{1}{c}{{\stackrel{\vee }{\mathrm{\Sigma }}}_2}& {\stackrel{\vee }{\mathrm{\Sigma }}}_3\hfill \end{array}\right],{\stackrel{\vee }{\mathrm{\Sigma }}}_2=\left[\begin{array}{ccccc}\stackrel{\vee }{{\stackrel{`}{\mathcal{A}}}_{\mathrm{ic}}}\hfill & {\stackrel{`}{\mathcal{E}}}_i\hfill & \stackrel{\vee }{{\mathcal{H}}_i}\hfill & 0\hfill & {\stackrel{`}{\mathcal{D}}}_i\hfill \end{array}\right]\hfill \\ \stackrel{\vee }{{\mathcal{A}}_{\mathrm{ic}}}=\left[\begin{array}{cc}\mathrm{\Lambda }{\mathrm{\Theta }}_i{A}_i+\mathrm{\Lambda }{\mathrm{\Theta }}_i{B}_i{K}_c\hfill & -\mathrm{\Lambda }{\mathrm{\Theta }}_i{B}_i{K}_c\hfill \\ \multicolumn{1}{c}{0}& P_2{A}_i\hfill \end{array}\right]\hfill \\ {\stackrel{\vee }{\mathrm{\Sigma }}}_3=\mathrm{diag}\{-\mathrm{\Lambda }{\mathrm{\Theta }}_i{P}_1^{-1}{\mathrm{\Theta }}_i^T{\mathrm{\Lambda }}^T,-P_2\}\hfill \\ \stackrel{\vee }{{\mathcal{H}}_i}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-P_2{H}_i}\end{array}\right]\hfill \end{array}$$

It is clear that

$$\begin{array}{c}\mathrm{\Lambda }{\mathrm{\Theta }}_i+{\mathrm{\Theta }}_i^T{\mathrm{\Lambda }}^T-\mathrm{\Lambda }{\mathrm{\Theta }}_i{P}_1^{-1}{\mathrm{\Theta }}_i^T{\mathrm{\Lambda }}^T-P_1\hfill \\ =-(\mathrm{\Lambda }{\mathrm{\Theta }}_i-P_1)P_1^{-1}(\mathrm{\Lambda }{\mathrm{\Theta }}_i-P_1{)}^T\le 0\hfill \end{array}$$

which implies that

$$-\mathrm{\Lambda }{\mathrm{\Theta }}_i{P}_1^{-1}{\mathrm{\Theta }}_i^T{\mathrm{\Lambda }}^T\le P_1-\mathrm{\Lambda }{\mathrm{\Theta }}_i-{\mathrm{\Theta }}_i^T{\mathrm{\Lambda }}^T$$

(27)

Accounting for equation (27) and making use of the variable substitution

$${\stackrel{`}{K}}_c={\mathrm{\Lambda }}_{11}{K}_c,{\stackrel{`}{H}}_i=P_2{H}_i$$

(28)

Furthermore, we conclude that equation (26) is ensured by equation (23). Therefore, the remaining proof proceeds immediately from Theorem 1.

Remark 5. The observer-based controller design problem is solved by adopting the orthogonal decomposition proposed in Dong and Yang (2008). Specifically, the matrix ${\mathrm{\Theta }}_i=[B_i{(B_i^{\mathrm{T}}{B}_i)}^{-1}{(B_i^{\mathrm{T}})}^{\perp }{]}^{\mathrm{T}}$ and a special constructed free matrix $\mathrm{\Lambda }$ are denoted to get over the difficulty from the products $P_1{B}_i{K}_c$ in Theorem 1. In addition, we know that the main results have established in Theorem 2, which provides sufficient conditions for designing the desired observer-based controller under the AETP.

Illustrative example

In this section, we give the following simulation example to verify the effectiveness of the proposed fuzzy controller and adaptive event trigger protocol in the IT2 T-S fuzzy system.

Now, take into account the following fuzzy model with two rules in the form of equation (1) as follows

$$\{\begin{array}{c}x(k+1)=\sum _{i=1}^2{F}_i(x_1(k))[A_{i}x(k)+E_i\sum _{n=1}^s{h}_{n}x(k-n)\hfill \\ \multicolumn{1}{c}{+B_{i}u(k)+D_{i}w(k)]}\\ \multicolumn{1}{c}{y(k)=\sum _{i=1}^2{F}_i(x_1(k))C_{i}x(k)}\\ \multicolumn{1}{c}{z(k)=\sum _{i=1}^2{F}_i(x_1(k))L_{i}x(k)}\end{array}$$

(29)

The model parameters are given below

$$A_1=\left[\begin{array}{cc}0.68\hfill & 0.47\hfill \\ \multicolumn{1}{c}{0.5}& 0.3\hfill \end{array}\right],E_1=\left[\begin{array}{cc}0.01\hfill & 0.05\hfill \\ \multicolumn{1}{c}{0.03}& 0.02\hfill \end{array}\right]$$

$$B_1=B_2=\left[\begin{array}{c}0.15\hfill \\ \multicolumn{1}{c}{0.12}\end{array}\right],D_1=\left[\begin{array}{c}0.02\hfill \\ \multicolumn{1}{c}{0.01}\end{array}\right]$$

$$\begin{array}{c}{C}_1=\left[\begin{array}{cc}0.1\hfill & 0.13\hfill \\ \multicolumn{1}{c}{0.5}& 0.1\hfill \end{array}\right],L_1=\left[\begin{array}{cc}0.1\hfill & 0.2\hfill \\ \multicolumn{1}{c}{0.03}& 0.05\hfill \end{array}\right]\\ A_2=\left[\begin{array}{cc}0.57\hfill & 0.5\hfill \\ \multicolumn{1}{c}{0.58}& 0.33\hfill \end{array}\right],D_2=\left[\begin{array}{c}0.02\hfill \\ \multicolumn{1}{c}{0.01}\end{array}\right],E_2=\left[\begin{array}{cc}0.02\hfill & 0.03\hfill \\ \multicolumn{1}{c}{0.04}& 0.02\hfill \end{array}\right]\\ C_2=\left[\begin{array}{cc}0.06\hfill & 0.23\hfill \\ \multicolumn{1}{c}{0.4}& 0.1\hfill \end{array}\right],L_2=\left[\begin{array}{cc}0\hfill & 0.2\hfill \\ \multicolumn{1}{c}{0.52}& 0.01\hfill \end{array}\right]\end{array}$$

where the MFs of the plant are given

$$\{\begin{array}{c}{\underset{¯}{F}}_1(x_1)=1-e^{-x_1^2/1.4}\hfill \\ \multicolumn{1}{c}{{\overline{F}}_1(x_1)=1-1-0.23e^{-x_1^2/0.35}}\\ \multicolumn{1}{c}{{\underset{¯}{F}}_2(x_1)=1-{\overline{F}}_1(x_1)}\\ \multicolumn{1}{c}{{\overline{F}}_2(x_1)=1-{\underset{¯}{F}}_1(x_1)}\end{array}$$

The MFs of the controller are given

$$\{\begin{array}{c}{\underset{¯}{\mathrm{\Theta }}}_1(x_1)=1-\frac{1}{1+e^{x_1+4-1}}\hfill \\ \multicolumn{1}{c}{{\overline{\mathrm{\Theta }}}_1(x_1)=1-\frac{1}{1+e^{x_1+4+1}}}\\ \multicolumn{1}{c}{{\underset{¯}{\mathrm{\Theta }}}_2(x_1)=1-{\overline{\mathrm{\Theta }}}_1(x_1)}\\ \multicolumn{1}{c}{{\overline{\mathrm{\Theta }}}_2(x_1)=1-{\underset{¯}{\mathrm{\Theta }}}_1(x_1)}\end{array}$$

where $x_1=x_1(k)$ represents the first element of the system state.

In this example, we define the weighting functions under the fuzzy rules. ${\underset{¯}{\xi }}_1(x_1)={\underset{¯}{d}}_1(x_1)=\mathrm{si}{\mathrm{n}}^2(x_1)$ and ${\overline{\xi }}_1(x_1)={\overline{d}}_1(x_1)=1-\mathrm{si}{\mathrm{n}}^2(x_1)$ , ${\underset{¯}{\xi }}_2(x_1)={\underset{¯}{d}}_2(x_1)=\mathrm{co}{\mathrm{s}}^2(x_1)$ , and ${\overline{\xi }}_2(x_1)={\overline{d}}_2(x_1)=1-\mathrm{co}{\mathrm{s}}^2(x_1)$ .

We choose the distributed time-delays $h_n=2^{-(n+1)}$ and the constant $s=3$ . The exogenous disturbance is taken as $w(k)=0.03\sin (k)/k$ , and $\gamma$ is selected to be 0.95. The sector-bounded parameter in equation (7) is chosen as $\chi =\mathrm{diag}\{0.55,0.65\}$ . The triggered thresholds parameters in equation (4) are given as ${ϵ}_d=ϵ(0)=0.15$ , ${ϵ}_D=0.42$ , and $\sigma =0.52$ . Moreover, the event-triggered weight matrix is selected as $\mathrm{\Psi }=\mathrm{diag}\{0.2,0.1\}$ .

Using the MATLAB software with the YALMIP 3.0, we can solve equations (23) and (24) and get the following solutions

$$P=\left[\begin{array}{cccc}15.8184\hfill & -5.9427\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-5.9427}& 16.0468\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 11.4277\hfill & -2.7136\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -2.7136\hfill & 13.9790\hfill \end{array}\right]$$

$$L=\left[\begin{array}{cccc}4.9478\hfill & -6.2009\hfill & -0.7518\hfill & 1.1925\hfill \\ \multicolumn{1}{c}{-6.2009}& 9.2963\hfill & 1.0276\hfill & -1.6612\hfill \\ \multicolumn{1}{c}{-0.7518}& 1.0276\hfill & 4.6371\hfill & -5.6087\hfill \\ \multicolumn{1}{c}{1.1925}& -1.6612\hfill & -5.6087\hfill & 8.2275\hfill \end{array}\right]$$

$$\mathrm{\Lambda }=\left[\begin{array}{cc}2.4440\hfill & -18.9524\hfill \\ \multicolumn{1}{c}{0}& 22.1318\hfill \end{array}\right],{\varrho }_1=1.19,{\varrho }_2=1.5$$

$${\stackrel{`}{K}}_1={\left[\begin{array}{c}-3.0525\hfill \\ \multicolumn{1}{c}{-2.7747}\end{array}\right]}^T,{\stackrel{`}{K}}_2={\left[\begin{array}{c}-1.7417\hfill \\ \multicolumn{1}{c}{-2.7024}\end{array}\right]}^T$$

$${\stackrel{`}{H}}_1=\left[\begin{array}{cc}-0.6381\hfill & 5.3531\hfill \\ \multicolumn{1}{c}{5.3531}& 2.5342\hfill \end{array}\right],{\stackrel{`}{H}}_2=\left[\begin{array}{cc}2.4929\hfill & 3.7995\hfill \\ \multicolumn{1}{c}{3.7995}& 3.2097\hfill \end{array}\right]$$

Therefore, the gain matrices are summarized by equation (25) as follows

$$K_1={\left[\begin{array}{c}-1.2490\hfill \\ \multicolumn{1}{c}{-1.1353}\end{array}\right]}^T,K_2={\left[\begin{array}{c}-0.7127\hfill \\ \multicolumn{1}{c}{-1.1057}\end{array}\right]}^T$$

$$H_1=\left[\begin{array}{cc}0.1539\hfill & 0.5362\hfill \\ \multicolumn{1}{c}{0.4128}& 0.2854\hfill \end{array}\right],H_2=\left[\begin{array}{cc}0.2964\hfill & 0.4057\hfill \\ \multicolumn{1}{c}{0.3293}& 0.3084\hfill \end{array}\right]$$

Figure 1 shows the system structure. The simulation results are presented in Figures 2–6. Specifically, Figure 2 shows the state trajectory of the system (10) without control, and it is clearly unstable. Moreover, Figure 3 plots the dynamics of the closed-loop system (10) with outlier-resistant control. Figure 4 depicts the system dynamics without outlier-resistant observer-based control (traditional Luenberger observer). Obviously, Figure 2 is more advantageous in terms of resistance to outliers. Figures 5 and 6 represent the triggering instant under AETP and traditional event-triggering mechanisms (with a fixed threshold), respectively. It can be seen that the triggering number of the former is significantly less than that of the latter, indicating that the AETP proposed in this paper is effective.

Figure 1.

Structure of the control system.

Figure 2.

Evolution of the system state $x(k)$ without control.

Figure 3.

Evolution of the system state $x(k)$ with outlier-resistant control.

Figure 4.

Evolution of the system state $x(k)$ with control.

Figure 5.

Adaptive event trigger protocol time instants for different components of $y(k)$ .

Figure 6.

traditional event-triggered scheme time instants for different components of $y(k)$ .

Conclusion

In this paper, we have studied outlier-resistant observer-based control problem for the IT2 T-S fuzzy discrete-time systems with distributed delays under AETP. Supported by Lyapunov stability theory, the sufficient conditions are derived to guarantee the stability of the controlled fuzzy system as well as ensure the prescribed ${\ell }_2-{\ell }_{\infty }$ performance. Moreover, the relevant control problem is reduced to a linear convex optimization problem using orthogonal decomposition and LMI technique. The numerical simulation results indicate the effectiveness and reliability of the proposed control method. It needs to be noted that the primary results of this paper can be promoted to other fields, such as the occurrence of more complex network-induced phenomena or dynamic saturation constraint. It is useful to research our future research topics.