Robust ℋ ∞ filtering design for Takagi–Sugeno fuzzy systems with time-varying delay via delta operator approach

Abstract

The problem of robust ℋ_∞ filtering design for Takagi–Sugeno fuzzy systems with time-varying delay via delta operator approach is investigated. The time-varying delay and parameter uncertainties are assumed to be of an internal-like type and a structured linear fractional form, respectively. Based on a Lyapunov–Krasovskii functional in delta domain, robust ℋ_∞ filter scheme is proposed. Then, a sufficient condition is established for the existence of the desired filter in terms of linear-matrix inequalities. A numerical example is provided to illustrate the design procedure of the present method.

Introduction

Takagi–Sugeno (T-S) fuzzy model has been well recognized as an effective approach to model and control nonlinear systems. Thus, significant research efforts have been devoted to investigate T-S fuzzy systems over the past two decades.^1–8 Recently, amount of attention have been focused on the stability analysis and controller design of networked control system (NCS) due to its great advantages, such as low cost, simple installation and maintenance, reduced weight, and power requirements. ⁹ Since time delay is frequently encountered in NCSs, it usually results in poor performance or instability in practical applications.^10–13 Therefore, time delay must be taken into account in networked T-S fuzzy systems. In addition, among various estimator schemes, the Kalman filter, and ℋ_∞ filter are the two main methods. Compared with the Kalman filter, the ℋ_∞ filtering method is to minimize signal estimation error for bounded disturbances or/and worse noises. Thus, the ℋ_∞ filtering method is more robust than the Kalman one. Moreover, the ℋ_∞ filter does not require the exact information for the statistics in the external noises, and it is of insensitivity to the exogenous statistics.^14,15 The advantages guarantee that the ℋ_∞ filter is very appropriate to many practical applications. As a result, robust ℋ_∞ filtering approach has been extensively investigated for T-S fuzzy systems with time delay.^16–19

On the other hand, with today’s exponentially growing computation power, in modern engineering applications, high-speed digital processing technology has became increasingly importance, such as flight control, nuclear reactor monitoring, and the integrated services digital network, which require a sampling rate at or above 1 MHz. Therefore, it is important for T-S fuzzy discrete-time systems with time delay to take high sampling rates into account. However, most of the research efforts for T-S fuzzy discrete-time models are focused on the shift operator method, which is of inherently ill-conditioned when datum is taken at high sampling rates. ²⁰ The delta operator model can be applied as a useful method for analyzing discrete-time systems and continuous-time systems into a consistent framework under high sampling rates.^21–24 As a result, the analysis and synthesis problems for T-S fuzzy discrete-time systems with time varying via delta operator approach are of important theoretical interest and practical value. More recently, the problem of robust ℋ_∞ control for T-S fuzzy systems with time delays via delta operator approach is reported in Yang et al. ²⁵ The work in Zhong et al. ²⁶ studied the problem of stability analysis and stabilization for T-S fuzzy delta operator systems with time-varying delays. Using delta operator approach, the problem of fault detection for T-S fuzzy systems with time-varying delays was considered in Li et al. ²⁷ The work in Xia et al. ²⁸ was devoted to robust sliding-mode control for time-delay systems with mismatched parametric uncertainties by delta operator method. To the best of our knowledge, there have been few results on the robust ℋ_∞ filtering design for T-S fuzzy discrete-time systems using delta operator approach, which motivates us to make an effort in this article.

In this article, we aims at studying robust ℋ_∞ filtering design for a class of uncertain T-S fuzzy systems with time-varying delay via delta operator approach. The state delay is considered as time varying and interval-like forms, so the available time-varying delay is of both the lower and upper bounds. The uncertainties are assumed to be of a structured linear fractional form. Based on a Lyapunov–Krasovskii functional (LKF), a sufficient condition for robust ℋ_∞ performance analysis is first established. Then, the ℋ_∞ filter gains can be obtained by solving a set of LMIs. Finally, a numerical example is given to illustrate the effectiveness and feasibility of our method.

This rest of this article is organized as follows. Section “Model description and problem formulation” gives the model description and problem formulation. Section “Robust fuzzy ℋ_∞ filtering analysis” is devoted to present the robust ℋ_∞ performance analysis. The robust ℋ_∞ filtering design scheme is proposed in section “Robust fuzzy ℋ_∞ filter design.” The simulation example is given in section “Simulation examples” to validate the advantage of the proposed method. Finally, conclusions are summarized in section “Conclusion.”

Notations: ${\mathbb{R}}^n$ and ${\mathbb{R}}^{n\times m}$ represent the n-dimensional Euclidean space and $n\times m$ matrices, respectively. $P>0$ and $P\ge 0$ denote that the matrix P is positive and semi-positive definite, respectively. $A^{-1}$ and $A^T$ denotes the inverse and transpose of the matrix A, respectively. $I_n$ represents an identity matrix with dimension n. $X>Y$ and $X\ge Y$ denote that the matrix $X-Y$ are positive and semi-positive definite, respectively. $\mathrm{diag}\{\cdots \}$ is a block diagonal matrix. The symbol “*” in a matrix stands for the transposed elements in the symmetric positions. $l_2[0,\infty )$ refers to the space of square-integrable vector functions over $[0,\infty )$ .

Model description and problem formulation

In the following, we consider a class of T-S fuzzy discrete-time systems in delta domain, which is represented by the following T-S fuzzy model.

Plant Rulei: IF ${\theta }_1(t)$ is $M_{i1}$ and ${\theta }_2(t)$ is $M_{i2}$ and … and ${\theta }_p(t)$ is $M_{\mathrm{ip}}$ , THEN

$$\{\begin{array}{c}\partial x(t)=(A_i+\Delta A_i)x(t)+(A_{\mathrm{di}}+\Delta A_{\mathrm{di}})x(t-n(t)T)+B_i\omega (t)\hfill \\ \multicolumn{1}{c}{y(t)=(C_i+\Delta C_i)x(t)+(C_{\mathrm{di}}+\Delta C_{\mathrm{di}})x(t-n(t)T)+D_i\omega (t)}\\ \multicolumn{1}{c}{z(t)=L_{i}x(t)}\\ \multicolumn{1}{c}{x(t)=\varphi (t),t\in -n_M,0],t=1,2,\dots ,r}\end{array}$$

(1)

where $x(t)\in {\mathbb{R}}^{n_1}$ denotes the system state vector, $\omega (t)\in {\mathbb{R}}^{n_2}$ is the disturbance input vector belonging to $l_2[0,\infty ]$ , $y(t)\in {\mathbb{R}}^{n_3}$ is the measurement output vector, $z(t)\in {\mathbb{R}}^{n_4}$ stands for the estimated signal vector, $M_{\mathrm{ij}}$ is the fuzzy set, r is the number of fuzzy rules, $\theta (t)=[{\theta }_1(t),{\theta }_2(t),\dots {\theta }_p(t)]$ are the premise variables which do not depend on the disturbance and the control variables, and T is a sampling period, $n(t)$ is a positive integer function, which represents the time-varying sample rates of the system (equation (1)) and satisfying the assumption as follows

$$n_m\le n(t)\le n_M$$

(2)

where $n_m$ and $n_M$ are two positive integers delegating the minimum and maximum sampling rates, respectively. $\varphi (t)$ is an initial value belonging to $[-n_{M}0]$ . $\partial x(t)$ is the delta operator of $x(t)$ , which is given by

$$\partial x(t)=\{\begin{array}{cc}\frac{d}{\mathrm{dt}}x(t),\hfill & T=0\hfill \\ \multicolumn{1}{c}{\frac{x(t+T)-x(t)}{T}}& T\ne 0\hfill \end{array}$$

(3)

Remark 1

For a linear system $\stackrel{·}{x}(t)=\overline{A}x(t)$ , it yields $A_q=\frac{1}{T}{\int }_0^T{e}^{\overline{A}t}\mathrm{dt}$ using the conventional q-operator approach. In that case, it is easy to see that $A_q$ becomes an identity matrix when the sampling period T tends to zero. When using the δ-operator one, it has $A_{\delta }=(A_q-I)/T$ . It can be directly obtain the property $A_{\delta }\to \overline{A}$ when the sampling period T tends to zero. ²⁹

Remark 2

It is easy to see that the equation $\partial x(t)=\stackrel{·}{x}(t)$ holds when $T=0$ , and the condition $\partial x(t)=x(t+T)-x(t)$ is equivalent the one using the q-operator approach. ³⁰

In addition, $A_i,A_{\mathrm{di}},B_i,C_i,C_{\mathrm{di}},D_i$ , and $L_i$ are known system matrices; the uncertain matrices $\Delta A_i,\Delta A_{\mathrm{di}},\Delta C_i$ , and $\Delta C_{\mathrm{di}}$ are of the form

$$\left[\begin{array}{cc}\Delta A_i\hfill & \Delta A_{\mathrm{di}}\hfill \\ \multicolumn{1}{c}{\Delta C_i}& \Delta C_{\mathrm{di}}\hfill \end{array}\right]=\left[\begin{array}{c}{M}_{1i}\hfill \\ \multicolumn{1}{c}{M_{2i}}\end{array}\right]\Delta (t)[E_{1i}{E}_{2i}],i=1,2,\dots ,r$$

(4)

where $M_{1i},M_{2i},E_{1i}$ , and $E_{2i}$ are known matrices with appropriate dimensions, and $\Delta (t)$ is an unknown real-time-varying matrix, which satisfies

$$\Delta (t)=\Lambda (t)[I_{s_1}-J\Lambda (t){]}^{-1}$$

(5)

and

$$0<I_{s_1}-J{J}^T$$

(6)

where J is known matrix and $\Lambda (t)\in {\mathbb{R}}^{s_1\times s_2}$ is an unknown and time-varying matrix satisfying

$$\Lambda (t)\Lambda (t)\le I_{s_1}$$

(7)

Remark 3

It is noted that $n(t)$ of equation (2) presents the randomly sampling rates of $n_m$ out of $n_M$ . Multiplying equation (2) by a sampling period T, $n_{m}T$ and $n_{M}T$ represent the lower and upper boundaries for the considered time-varying delay, respectively. In such a way, $n(t)T$ is named an interval time-varying delay, which represents the state delay with time varying.^31,32

Remark 4

It is also noted that the class of parametric uncertainties is said to be admissible if equations (4)–(7) hold. It possesses a linear fractional representation. For the special case $J=0$ , the linear fractional form uncertainty reduces to the norm-bounded one. ³³

The following global T-S fuzzy delta operator system can be obtained

$$\{\begin{array}{c}\partial x(t)=\underset{i=1}{\sum ^r}{\lambda }_i(\theta (t))[(A_i+\Delta A_i)x(t)+(A_{\mathrm{di}}+\Delta A_{\mathrm{di}})x(t-n(t)T)+B_i\omega (t)]\hfill \\ \multicolumn{1}{c}{y(t)=\underset{i=1}{\sum ^r}{\lambda }_i(\theta (t))[(C_i+\Delta C_i)x(t)+(C_{\mathrm{di}}+\Delta C_{\mathrm{di}})x(t-n(t)T)+D_i\omega (t)]}\\ \multicolumn{1}{c}{z(t)=\underset{i=1}{\sum ^r}{\lambda }_i(\theta (t))L_{i}x(t)}\\ \multicolumn{1}{c}{x(t)=\varphi (t),t\in [-n_M,0],t=1,2,\dots ,r}\end{array}$$

(8)

where $\sum _{i=1}^r{\lambda }_i(\theta (t))=1$ , ${\lambda }_i(\theta (t))={\omega }_i(\theta (t))/\sum _{i=1}^r{\omega }_i(\theta (t))\ge 0$ , and ${\omega }_i(\theta (t))=\underset{j=1}{\overset{r}{\Pi }}{M}_{\mathrm{ij}}({\theta }_j(t))$ with $M_{\mathrm{ij}}({\theta }_j(t))$ represent the grade of membership of ${\theta }_j(t)$ in $M_{\mathrm{ij}}$ .

Given the T-S fuzzy delta operator system in equation (8), we are interesting in designing the robust fuzzy ℋ_∞ filter as follows.

Filter Rulei: IF ${\theta }_1(t)$ is $M_{i1}$ and ${\theta }_2(t)$ is $M_{i2}$ and … and ${\theta }_p(t)$ is $M_{\mathrm{ip}}$ , THEN

$$\{\begin{array}{c}\partial x_f(t)=A_{\mathrm{fi}}{x}_f(t)+B_{\mathrm{fi}}y(t)\hfill \\ \multicolumn{1}{c}{z_f(t)=C_{\mathrm{fi}}{x}_f(t)}\end{array}$$

(9)

where $x_f(t)\in {\mathbb{R}}^{n_5}$ is the filter state vector and $z_f(t)\in {\mathbb{R}}^{n_6}$ are the filter output vector. The matrices $A_{\mathrm{fi}},B_{\mathrm{fi}}$ , and $C_{\mathrm{fi}}$ are appropriately dimensioned filter gains to be designed. The overall fuzzy filter system can be expressed as

$$\{\begin{array}{c}\partial x_f(t)=\underset{i=1}{\sum ^r}{\lambda }_i(\theta (t))[A_{\mathrm{fi}}{x}_f(t)+B_{\mathrm{fi}}y(t)]\hfill \\ \multicolumn{1}{c}{z_f(t)=\underset{i=1}{\sum ^r}{\lambda }_i(\theta (t))C_{\mathrm{fi}}{x}_f(t)}\end{array}$$

(10)

Remark 5

It is noted that the fuzzy filter with synchronous premise variables is given in equation (10). However, in network-based perspective, some imperfect conditions, such as the sampling data control, time delays, and packet dropout, will induce the asynchronous premise variables between the system and the filter. It is also noted that when the asynchronous information is unavailable, the condition generally leads to a linear filter instead of a fuzzy one, which induces the design conservatism. ³⁴

Defining $\xi (t):=[x^T(t)x_f^T(t){]}^T,e(t):=z(t)-z_f(t)$ , we obtain the filtering error system as follows

$$\{\begin{array}{c}\partial \xi (t)=(\overline{A}(t)+\Delta \overline{A}(t))\xi (t)+({\overline{A}}_d(t)+\Delta {\overline{A}}_d(t))\xi (t-n(t)T)+\overline{D}(t)\omega (t)\hfill \\ \multicolumn{1}{c}{e(t)=\overline{L}(t)\xi (t)}\end{array}$$

(11)

where

$$\{\begin{array}{c}\overline{A}(t)=\underset{i=1}{\sum ^r}\underset{j=1}{\sum ^r}{\lambda }_i(\theta (t)){\lambda }_j(\theta (t))\left[\begin{array}{cc}{A}_j\hfill & 0\hfill \\ \multicolumn{1}{c}{B_{\mathrm{fi}}{C}_j}& A_{\mathrm{fi}}\hfill \end{array}\right]=\left[\begin{array}{cc}A(t)\hfill & 0\hfill \\ \multicolumn{1}{c}{B_f(t)C(t)}& A_f(t)\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{{\overline{A}}_d(t)=\underset{i=1}{\sum ^r}\underset{j=1}{\sum ^r}{\lambda }_i(\theta (t)){\lambda }_j(\theta (t))\left[\begin{array}{cc}{A}_{\mathrm{dj}}\hfill & 0\hfill \\ \multicolumn{1}{c}{B_{\mathrm{fi}}{C}_{\mathrm{dj}}}& 0\hfill \end{array}\right]=\left[\begin{array}{cc}{A}_d(t)\hfill & 0\hfill \\ \multicolumn{1}{c}{B_f(t)C_d(t)}& 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{\Delta \overline{A}(t)=\underset{i=1}{\sum ^r}\underset{j=1}{\sum ^r}{\lambda }_i(\theta (t)){\lambda }_j(\theta (t))\left[\begin{array}{cc}\Delta A_j\hfill & 0\hfill \\ \multicolumn{1}{c}{B_{\mathrm{fi}}\Delta C_j}& 0\hfill \end{array}\right]=\left[\begin{array}{cc}\Delta A(t)\hfill & 0\hfill \\ \multicolumn{1}{c}{B_f(t)\Delta C(t)}& 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{\Delta {\overline{A}}_d(t)=\underset{i=1}{\sum ^r}\underset{j=1}{\sum ^r}{\lambda }_i(\theta (t)){\lambda }_j(\theta (t))\left[\begin{array}{c}\Delta A_{\mathrm{dj}}\hfill \\ \multicolumn{1}{c}{B_{\mathrm{fi}}\Delta C_{\mathrm{dj}}}\end{array}\right]=\left[\begin{array}{cc}\Delta A_d(t)\hfill & 0\hfill \\ \multicolumn{1}{c}{B_f(t)\Delta C_d(t)}& 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{\overline{D}(t)=\underset{i=1}{\sum ^r}\underset{j=1}{\sum ^r}{\lambda }_i(\theta (t)){\lambda }_j(\theta (t))\left[\begin{array}{c}{B}_j\hfill \\ \multicolumn{1}{c}{B_{\mathrm{fi}}{D}_j}\end{array}\right]=\left[\begin{array}{c}B(t)\hfill \\ \multicolumn{1}{c}{B_f(t)D(t)}\end{array}\right]}\\ \multicolumn{1}{c}{\overline{L}(t)=\underset{i=1}{\sum ^r}\underset{j=1}{\sum ^r}{\lambda }_i(\theta (t)){\lambda }_j(\theta (t))[L_j-C_{\mathrm{fi}}]=[L(t)-C_f(t)]}\end{array}$$

(12)

The uncertain matrices $\Delta \overline{A}(t)$ and $\Delta {\overline{A}}_d(t)$ are of the form

$$[\Delta \overline{A}(t)\Delta {\overline{A}}_d(t)]={\overline{M}}_1(t)\Delta (t)[{\overline{E}}_1(t){\overline{E}}_2(t)]$$

(13)

where

$$\{\begin{array}{c}{\overline{M}}_1(t)=\underset{i=1}{\sum ^r}\underset{j=1}{\sum ^r}{\lambda }_i(\theta (t)){\lambda }_j(\theta (t))\left[\begin{array}{c}{M}_{1j}\hfill \\ \multicolumn{1}{c}{B_{\mathrm{fi}}{M}_{2j}}\end{array}\right]=\left[\begin{array}{c}{M}_1(t)\hfill \\ \multicolumn{1}{c}{B_f(t)M_2(t)}\end{array}\right]\hfill \\ \multicolumn{1}{c}{{\overline{E}}_1(t)=\underset{i=1}{\sum ^r}{\lambda }_i(\theta (t))\left[\begin{array}{cc}{E}_{1i}\hfill & 0\hfill \end{array}\right]=\left[\begin{array}{cc}{E}_1(t)\hfill & 0\hfill \end{array}\right],{\overline{E}}_2(t)=\underset{i=1}{\sum ^r}{\lambda }_i(\theta (t))\left[\begin{array}{cc}{E}_{2i}\hfill & 0\hfill \end{array}\right]=\left[\begin{array}{cc}{E}_2(t)\hfill & 0\hfill \end{array}\right]}\end{array}$$

(14)

Then, the robust ℋ_∞ filtering problem to be addressed in this article is stated as follows.

Robust ℋ_∞ filter design problem: Given the fuzzy delta operator system (equation (8)), the objective is to design a fuzzy filter of the form (equation (10)), which satisfies the following two requirements simultaneously:

1. The filtering error system (equation (11)) with $\omega (t)=0$ is asymptotically stable.

2. The ℋ_∞ performance is validated for all nonzero $\omega (t)\in l_2[0,\infty ]$ and for a given $\gamma >0$ under zero initial condition

$$J_{\infty }:=\underset{t=0}{\sum ^{\infty }}[e^T(t)e(t)-{\gamma }^2{\omega }^T(t)\omega (t)]0$$

(15)

Before moving on, we introduce the following lemmas, which are used to prove the main results proposed in this article.

Lemma 1

The property of delta operator: for any time function $x(t)$ and $y(t)$ , it holds that ³⁵

$$\partial (x(t)y(t))=\partial x(t)y(t)+x(t)\partial y(t)+T\partial x(t)\partial y(t)$$

where T denotes the sampled period.

Lemma 2

Given constant matrices $\tilde{X}$ , $\tilde{Y}$ , and $\tilde{Z}$ with $\tilde{X}={\tilde{X}}^T$ and $0<\tilde{Y}={\tilde{Y}}^T$ , then $\tilde{X}+{\tilde{Z}}^T{\tilde{Y}}^{-1}$ $\tilde{Z}<0$ if and only if ³³

$$\left[\begin{array}{cc}\tilde{X}\hfill & {\tilde{Z}}^T\hfill \\ \multicolumn{1}{c}{\tilde{Z}}& -\tilde{Y}\hfill \end{array}\right]<0\mathrm{or}\left[\begin{array}{cc}-\tilde{Y}\hfill & \tilde{Z}\hfill \\ \multicolumn{1}{c}{{\tilde{Z}}^T}& \tilde{X}\hfill \end{array}\right]<0$$

Lemma 3

Given matrices $Q,H,E$ , and $R=R^T>0$ of appropriate dimensions, where Q is symmetrical, then ³³

$$Q+\mathrm{HF}(t)E+E^T{F}^T(t)H^T<0$$

for all $F(t)$ satisfying the inequality $F^T(t)F(t)\le R$ if and only if there exists a positive scalar $\epsilon >0$ such that

$$Q+\epsilon H{H}^T+\frac{1}{\epsilon }{E}^T\mathrm{RE}<0$$

Lemma 4

For any positive semi-definite symmetric matrix W, two positive integers r and $r_0$ satisfying the inequality $r\ge r_0\ge 1$ , then it has ³⁶

$${[\underset{i=r_0}{\sum ^r}x(i)]}^{T}W[\underset{i=r_0}{\sum ^r}x(i)]\le (r-r_0+1)\underset{i=r_0}{\sum ^r}{x}^T(i)\mathrm{Wx}(i)$$

Robust fuzzy ℋ_∞ filtering analysis

In this section, based on an LKF in delta operator domain, a sufficient condition for the existence of an asymptotically stable filter with ℋ_∞ performance for the filtering error system in equation (11) is established as follows.

Lemma 5

Consider the system (equation (8)) and filter (equation (10)). Given a prescribed ℋ_∞ performance level $\gamma >0$ , integer scalars $n_M>n_m>0$ , and a positive scalar $\epsilon$ . If there exist matrix $0<\hat{P}={\hat{P}}^T\in {\mathbb{R}}^{(n_1+n_5)\times (n_1+n_5)}$ , $0<\hat{Q}={\hat{Q}}^T\in {\mathbb{R}}^{(n_1+n_5)\times (n_1+n_5)},0<{\hat{S}}_1={\hat{S}}_1^T\in {\mathbb{R}}^{(n_1+n_5)\times (n_1+n_5)},0<{\hat{S}}_2={\hat{S}}_2^T\in {\mathbb{R}}^{(n_1+n_5)\times (n_1+n_5)},0<{\hat{R}}_1={\hat{R}}_1^T\in {\mathbb{R}}^{(n_1+n_5)\times (n_1+n_5)},0<{\hat{R}}_2={\hat{R}}_2^T\in {\mathbb{R}}^{(n_1+n_5)\times (n_1+n_5)}$ , such that the following matrix inequality holds

$$\left[\begin{array}{ccc}{\Phi }_1(t)\hfill & {\Phi }_2(t)\hfill & {\Phi }_3(t)\hfill \\ \multicolumn{1}{c}{\star }& -\epsilon I\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & -\epsilon I\hfill \end{array}\right]<0$$

(16)

where

$${\Phi }_1(t)=\left[\begin{array}{ccccccc}{\Phi }_1(1,1)\hfill & \hat{P}\overline{A}(t)\hfill & \hat{P}{\overline{A}}_d(t)\hfill & 0\hfill & 0\hfill & \hat{P}\overline{D}(t)\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& {\Phi }_1(2,2)\hfill & \hat{P}{\overline{A}}_d(t)\hfill & \frac{{\hat{R}}_1}{n_{M}T}\hfill & \frac{{\hat{R}}_2}{n_{m}T}\hfill & \hat{P}\overline{D}(t)\hfill & \overline{L}{}^T(t)\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & -\hat{Q}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & -{\hat{S}}_1-\frac{{\hat{R}}_1}{n_{M}T}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & -{\hat{S}}_2-\frac{{\hat{R}}_2}{n_{m}T}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & \star \hfill & -{\gamma }^2\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & -I\hfill \end{array}\right]$$

$$\begin{array}{c}{\Phi }_1(1,1)=T\hat{P}+n_{M}T{\hat{R}}_1+n_{m}T{\hat{R}}_2-2\hat{P}\hfill \\ {\Phi }_1(2,2)=\hat{P}\overline{A}(t)+{\overline{A}}^T(t)\hat{P}+(n_{M}T-n_{m}T+1)\hat{Q}\hfill \\ -\frac{{\hat{R}}_1}{n_{M}T}+{\hat{S}}_1-\frac{{\hat{R}}_2}{n_{m}T}+{\hat{S}}_2+{\overline{L}}^T(t)\overline{L}(t)\hfill \end{array}$$

$${\Phi }_2(t)=\left[\begin{array}{c}\hat{P}{\overline{M}}_1(t)\hfill \\ \multicolumn{1}{c}{\hat{P}{\overline{M}}_1(t)}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right],{\Phi }_3(t)=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{\epsilon {\overline{E}}_1^T(t)}\\ \multicolumn{1}{c}{\epsilon {\overline{E}}_2^T(t)}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right]$$

(17)

then the filtering error system (equation (11)) is asymptotically stable with ℋ_∞ performance $\gamma$

Proof

Choose the following LKF

$$V(t)=V_1(t)+V_2(t)+V_3(t)+V_4(t)+V_5(t)$$

(18)

where

$$V_1(t)={\xi }^T(t)\hat{P}\xi (t)$$

$$V_2(t)=T\underset{i=1}{\sum ^{n(t)}}{\xi }^T(t-\mathrm{iT})\hat{Q}\xi (t-\mathrm{iT})$$

$$\begin{array}{cc}\multicolumn{1}{c}{V_3(t)=}& T\underset{i=1}{\sum ^{n_M}}{\xi }^T(t-\mathrm{iT}){\hat{S}}_1\xi (t-\mathrm{iT})+T\hfill \\ \underset{i=1}{\sum ^{n_m}}{\xi }^T(t-\mathrm{iT}){\hat{S}}_2\xi (t-\mathrm{iT})\hfill \end{array}$$

$$V_4(t)=T^2\underset{i=n_m}{\sum ^{n_M}}\underset{j=1}{\sum ^i}{\xi }^T(t-\mathrm{jT})\hat{Q}\xi (t-\mathrm{jT})$$

$$\begin{array}{cc}\multicolumn{1}{c}{V_5(t)=}& \sum _{i=1}^{n_M}\underset{j=1}{\sum ^i}{e}^T(t-\mathrm{jT}){\hat{R}}_{1}e(t-\mathrm{jT})\hfill \\ +\underset{i=1}{\sum ^{n_m}}\underset{j=1}{\sum ^i}{e}^T(t-\mathrm{jT}){\hat{R}}_{2}e(t-\mathrm{jT})\hfill \end{array}$$

and $e(j)=\xi (j)-\xi (j+T)$ , so that $\partial \xi (j)=-e(j)/T$ and $e(t-\mathrm{jT})=\xi (t-\mathrm{jT})-\xi (t-(j-1)T)$ hold.

Using the delta operator manipulations of $V_1(t)$ along the trajectory of system (equation (11)) and using Lemma 1, it has

$$\begin{array}{cc}\multicolumn{1}{c}{\partial V_1(t)}& ={\partial }^T(\xi (t))\hat{P}\xi (t)+{\xi }^T(t)\hat{P}\partial (\xi (t))+T{\partial }^T(\xi (t))\hat{P}\partial (\xi (t))\hfill \\ =2{\xi }^T(t)\hat{P}[(\overline{A}(t)+\Delta \overline{A}(t))\xi (t)+({\overline{A}}_d(t)\hfill \\ +\Delta {\overline{A}}_d(t))\xi (t-n(t)T)+\overline{D}(t)\omega (t)]\hfill \\ +T{\partial }^T(\xi (t))\hat{P}\partial (\xi (t))\hfill \end{array}$$

(19)

Similarly, we have

$$\begin{array}{c}\multicolumn{1}{c}{\partial V_2(t)=\frac{1}{T}\times T[\underset{i=1}{\sum ^{n(t)}}{\xi }^T(t-(i-1)T)\hat{Q}\xi (t-(i-1)T)-\underset{i=1}{\sum ^{n(t)}}{\xi }^T(t-\mathrm{iT})\hat{Q}\xi (t-\mathrm{iT})]}\\ \multicolumn{1}{c}{\le {\xi }^T(t)\hat{Q}\xi (t)-{\xi }^T(t-n(t)T)\hat{Q}\xi (t-n(t)T)+T\underset{i=n_m}{\sum ^{n_M}}{\xi }^T(t-\mathrm{iT})\hat{Q}\xi (t-\mathrm{iT})}\end{array}$$

(20)

$$\begin{array}{cc}\multicolumn{1}{c}{\partial V_3(t)=}& \frac{1}{T}\times T[\underset{i=1}{\sum ^{n_M}}{\xi }^T(t-(i-1)T){\hat{S}}_1\xi (t-(i-1)T)-\underset{i=1}{\sum ^{n_M}}{\xi }^T(t-\mathrm{iT}){\hat{S}}_1\xi (t-\mathrm{iT})]\hfill \\ +\frac{1}{T}\times T[\underset{i=1}{\sum ^{n_m}}{\xi }^T(t-(i-1)T){\hat{S}}_2\xi (t-(i-1)T)-\underset{i=1}{\sum ^{n_m}}{\xi }^T(t-\mathrm{iT}){\hat{S}}_2\xi (t-\mathrm{iT})]\hfill \\ \multicolumn{1}{c}{=}& {\xi }^T(t){\hat{S}}_1\xi (t)-{\xi }^T(t-n_{M}T){\hat{S}}_1\xi (t-n_{M}T)\hfill \\ +{\xi }^T(t){\hat{S}}_2\xi (t)-{\xi }^T(t-n_{m}T){\hat{S}}_2\xi (t-n_{m}T)\hfill \end{array}$$

(21)

$$\begin{array}{c}\partial V_4(t)=T\underset{i=n_m}{{\displaystyle \sum }^{n_M}}\left[\underset{j=1}{{\displaystyle \sum }^i}{\xi }^T(t-(j-1)T)\widehat{Q}\xi (t-(j-1)T)-\underset{j=1}{{\displaystyle \sum }^i}{\xi }^T(t-jT)\widehat{Q}\xi (t-jT)\right]\hfill \\ =(n_{M}T-n_{m}T+T){\xi }^T(t)\widehat{Q}\xi (t)-T\underset{i=n_m}{{\displaystyle \sum }^{n_M}}{\xi }^T(t-iT)\widehat{Q}\xi (t-iT)\hfill \end{array}$$

(22)

Based on the delta operator manipulation of $V_5(t)$ and using Lemma 4, we have

$$\begin{array}{cc}\multicolumn{1}{c}{\partial V_5(t)=}& \frac{1}{T}[\underset{i=1}{\sum ^{n_M}}\underset{j=1}{\sum ^i}{e}^T(t-(j-1)T){\hat{R}}_{1}e(t-(j-1)T)-\underset{i=1}{\sum ^{n_M}}\underset{j=1}{\sum ^i}{e}^T(t-\mathrm{jT}){\hat{R}}_{1}e(t-\mathrm{jT})]\hfill \\ +\frac{1}{T}[\underset{i=1}{\sum ^{n_m}}\underset{j=1}{\sum ^i}{e}^T(t-(j-1)T){\hat{R}}_{2}e(t-(j-1)T)-\underset{i=1}{\sum ^{n_m}}\underset{j=1}{\sum ^i}{e}^T(t-\mathrm{jT}){\hat{R}}_{2}e(t-\mathrm{jT})]\hfill \\ \le n_{M}T{\partial }^T\xi (t){\hat{R}}_1\partial \xi (t)-\frac{1}{n_{M}T}{[\xi (t-n_{M}T)-\xi (t)]}^T{\hat{R}}_1[\xi (t-n_{M}T)-\xi (t)]\hfill \\ +n_{m}T{\partial }^T\xi (t){\hat{R}}_2\partial \xi (t)-\frac{1}{n_{m}T}{[\xi (t-n_{m}T)-\xi (t)]}^T{\hat{R}}_2[\xi (t-n_{m}T)-\xi (t)]\hfill \end{array}(23)$$

(23)

Define the matrix $P=P^T>0$ , we have the following equation from (equation (11))

$$\begin{array}{c}\multicolumn{1}{c}{0=-2{\partial }^T(\xi (t))\hat{P}[\partial \xi (t)-(\overline{A}(t)+\Delta \overline{A}(t))\xi (t)-({\overline{A}}_d(t)+\Delta {\overline{A}}_d(t))\xi (t-n(t)T)-\overline{D}(t)\omega (t)]}\end{array}$$

(24)

Substituting equations (19)–(24) into $\partial V(t)$ , we have

$$\partial V(t)\le {\Upsilon }_1^T(t){\Sigma }_1{\Upsilon }_1(t)$$

(25)

where

$$\{\begin{array}{c}{\Upsilon }_1(t)={[{(\partial \xi (t))}^T{\xi }^T(t){\xi }^T(t-n(t)T){\xi }^T(t-n_{M}T){\xi }^T(t-n_{m}T){\omega }^T(t)]}^T\hfill \\ \multicolumn{1}{c}{{\Sigma }_1=\left[\begin{array}{cccccc}{\Sigma }_1(1,1)\hfill & {\Sigma }_1(1,2)\hfill & {\Sigma }_1(1,3)\hfill & 0\hfill & 0\hfill & \hat{P}\overline{D}(t)\hfill \\ \multicolumn{1}{c}{\star }& {\Sigma }_1(2,2)\hfill & {\Sigma }_1(2,3)\hfill & \frac{{\hat{R}}_1}{n_{M}T}\hfill & \frac{{\hat{R}}_2}{n_{m}T}\hfill & \hat{P}\overline{D}(t)\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & -\hat{Q}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & -{\hat{S}}_1-\frac{{\hat{R}}_1}{n_{M}T}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & -{\hat{S}}_2-\frac{{\hat{R}}_2}{n_{m}T}\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & \star \hfill & 0\hfill \end{array}\right]}\\ \multicolumn{1}{c}{{\Sigma }_1(1,1)=T\hat{P}+n_{M}T{\hat{R}}_1+n_{m}T{\hat{R}}_2-2\hat{P}}\\ \multicolumn{1}{c}{{\Sigma }_1(1,2)=\hat{P}\overline{A}(t)+\hat{P}\Delta \overline{A}(t)}\\ \multicolumn{1}{c}{{\Sigma }_1(1,3)={\Sigma }_1(2,3)=\hat{P}{\overline{A}}_d(t)+\hat{P}\Delta {\overline{A}}_d(t)}\\ \multicolumn{1}{c}{{\Sigma }_1(2,2)=\hat{P}\overline{A}(t)+\hat{P}\Delta \overline{A}(t)+{\overline{A}}^T(t)\hat{P}+\Delta {\overline{A}}^T(t)\hat{P}}\\ \multicolumn{1}{c}{+(n_{M}T-n_{m}T+T+1)\hat{Q}-\frac{{\hat{R}}_1}{n_{M}T}+{\hat{S}}_1-\frac{{\hat{R}}_2}{n_{m}T}+{\hat{S}}_2}\end{array}$$

(26)

Next, a straightforward computation gives

$$\begin{array}{c}{\hat{J}}_{\infty }:=\underset{t=0}{\sum ^{\infty }}[e^T(t)e(t)-{\gamma }^2{\omega }^T(t)\omega (t)+\partial V(t)]\hfill \\ \multicolumn{1}{c}{\le \underset{t=0}{\sum ^{\infty }}{\Upsilon }_1^T(t){\Sigma }_2{\Upsilon }_1(t)}\end{array}$$

(27)

where

$$\{\begin{array}{c}{\Sigma }_2=\left[\begin{array}{cccccc}{\Sigma }_1(1,1)\hfill & {\Sigma }_1(1,2)\hfill & {\Sigma }_1(1,3)\hfill & 0\hfill & 0\hfill & \hat{P}\overline{D}(t)\hfill \\ \multicolumn{1}{c}{\star }& {\Sigma }_2(2,2)\hfill & {\Sigma }_1(2,3)\hfill & \frac{{\hat{R}}_1}{n_{M}T}\hfill & \frac{{\hat{R}}_2}{n_{m}T}\hfill & \hat{P}\overline{D}(t)\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & -\hat{Q}\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & -{\hat{S}}_1-\frac{{\hat{R}}_1}{n_{M}T}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & -{\hat{S}}_2-\frac{{\hat{R}}_2}{n_{m}T}\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & *\hfill & -{\gamma }^2\hfill \end{array}\right]\hfill \\ \multicolumn{1}{c}{{\Sigma }_2(2,2)=\hat{P}\overline{A}(t)+\hat{P}\Delta \overline{A}(t)+{\overline{A}}^T(t)\hat{P}+\Delta {\overline{A}}^T(t)\hat{P}+(n_{M}T-n_{m}T+T+1)\hat{Q}}\\ \multicolumn{1}{c}{-\frac{{\hat{R}}_1}{n_{M}T}+{\hat{S}}_1-\frac{{\hat{R}}_2}{n_{m}T}+{\hat{S}}_2+{\overline{L}}^T(t)\overline{L}(t)}\end{array}$$

(28)

Applying the Schur complement and S-procedure (Lemmas 2 and 3), it shows that equation (16) is equivalent to ${\Sigma }_2<0$ ; thus, we obtain ${\hat{J}}_{\infty }:=\sum _{t=0}^{\infty }[e^T(t)e(t)-{\gamma }^2{\omega }^T(t)\omega (t)]+\sum _{t=0}^{\infty }\partial V(t)<0$ . Obviously, $\sum _{t=0}^{\infty }\partial V(t)>0$ , which implies that (equation (15)) holds, such that ℋ_∞ performance is verified.

In addition, it is easy to see from equation (16) that the delta operator manipulations of $V(t)$ along the solution of equation (11) with $\omega (t)=0$ guarantees $\partial V(t)<0$ , which shows the asymptotic stability of system (equation (11)) with $\omega (t)=0.$ The proof is completed.

Robust fuzzy ℋ_∞ filter design

Lemma 5 provides a sufficient condition for existing an asymptotically stable filter with ℋ_∞ performance $\gamma$ . However, there exist some coupling matrix variables in Lemma 5. In this section, motivated by Yang et al., ³⁷ we will use decoupling technique. In this way, we give the following results.

Theorem 1

Consider the system (equation (8)) and filter (equation (10)). Given a prescribed ℋ_∞ performance index $\gamma >0$ and integer scalars $n_M>n_m>0.$ The filtering error system (11) is asymptotically stable with ℋ_∞ performance $\gamma$ if there exist matrices $0<X=X^T$ , $0<Y=Y^T,S_1^{11},S_1^{12},S_1^{22},S_2^{11},S_2^{12},S_2^{22},Q_1$ , $Q_2,Q_3,R_1^{11},R_1^{12},R_1^{22},R_2^{11},R_2^{12},R_2^{22}$ of appropriate dimensions, and a positive scalar $\epsilon$ , such that the following LMIs hold

$$\begin{array}{c}\left[\begin{array}{cc}{S}_1^{11}\hfill & S_1^{12}\hfill \\ \multicolumn{1}{c}{*}& S_1^{22}\hfill \end{array}\right]>0,\left[\begin{array}{cc}{R}_1^{11}\hfill & R_1^{12}\hfill \\ \multicolumn{1}{c}{*}& R_1^{22}\hfill \end{array}\right]>0,\left[\begin{array}{cc}{Q}_1\hfill & Q_2\hfill \\ \multicolumn{1}{c}{*}& Q_3\hfill \end{array}\right]>0\hfill \\ \left[\begin{array}{cc}{S}_2^{11}\hfill & S_2^{12}\hfill \\ \multicolumn{1}{c}{*}& S_2^{22}\hfill \end{array}\right]>0,\left[\begin{array}{cc}{R}_2^{11}\hfill & R_2^{12}\hfill \\ \multicolumn{1}{c}{*}& R_2^{22}\hfill \end{array}\right]>0,X-Y>0\hfill \end{array}$$

(29)

and

$${\overline{\Phi }}_{\mathrm{ii}}<0,(1\le i\le r)$$

(30)

and

$${\overline{\Phi }}_{\mathrm{ij}}+{\overline{\Phi }}_{\mathrm{ji}}<0,(1\le i<j\le r)$$

(31)

where

$${\overline{\Phi }}_{\mathrm{ij}}=\left[\begin{array}{ccccc}{\Phi }_{\mathrm{ij}\left(1\right)}\hfill & {\Phi }_{\mathrm{ij}\left(2\right)}\hfill & {\Phi }_{\mathrm{ij}\left(3\right)}\hfill & {\Phi }_{\mathrm{ij}\left(4\right)}\hfill & {\Phi }_{\mathrm{ij}\left(5\right)}\hfill \\ \multicolumn{1}{c}{\star }& -{\gamma }^2\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & -I\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & -\epsilon I\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & -\epsilon I\hfill \end{array}\right]$$

$${\Phi }_{\mathrm{ij}\left(1\right)}=\left[\begin{array}{cccccccccc}{\Phi }_{\mathrm{ij}\left(1\right)}^{(1,1)}\hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(1,2)}\hfill & Y{A}_j\hfill & Y{A}_j\hfill & Y{A}_{\mathrm{dj}}\hfill & Y{A}_{\mathrm{dj}}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& {\Phi }_{\mathrm{ij}\left(1\right)}^{(2,2)}\hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(2,3)}\hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(2,4)}\hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(2,5)}\hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(2,6)}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(3,3)}\hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(3,4)}\hfill & Y{A}_{\mathrm{dj}}\hfill & Y{A}_{\mathrm{dj}}\hfill & \frac{R_1^{11}}{n_{M}T}\hfill & \frac{R_1^{12}}{n_{M}T}\hfill & \frac{R_2^{11}}{n_{m}T}\hfill & \frac{R_2^{12}}{n_{m}T}\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(4,4)}\hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(4,5)}\hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(4,6)}\hfill & \frac{{(R_1^{12})}^T}{n_{M}T}\hfill & \frac{R_1^{22}}{n_{M}T}\hfill & \frac{{(R_2^{12})}^T}{n_{m}T}\hfill & \frac{R_2^{22}}{n_{m}T}\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & -Q_1\hfill & -Q_2\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & \star \hfill & -Q_3\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(7,7)}\hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(7,8)}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(8,8)}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(9,9)}\hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(9,10)}\hfill \\ \multicolumn{1}{c}{\star }& \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & \star \hfill & {\Phi }_{\mathrm{ij}\left(1\right)}^{(10,10)}\hfill \end{array}\right]$$

$$\begin{array}{c}{\Phi }_{\mathrm{ij}\left(2\right)}=\left[\begin{array}{c}Y{B}_j\hfill \\ \multicolumn{1}{c}{{\Phi }_{\mathrm{ij}\left(2\right)}^{(2,11)}}\\ \multicolumn{1}{c}{Y{B}_j}\\ \multicolumn{1}{c}{{\Phi }_{\mathrm{ij}\left(2\right)}^{(4,11)}}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right],{\Phi }_{\mathrm{ij}\left(3\right)}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{{\Phi }_{\mathrm{ij}\left(3\right)}^{(3,12)}}\\ \multicolumn{1}{c}{L_j^T}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right],{\Phi }_{\mathrm{ij}\left(4\right)}=\left[\begin{array}{c}Y{M}_{1j}\hfill \\ \multicolumn{1}{c}{{\Phi }_{\mathrm{ij}\left(4\right)}^{(2,13)}}\\ \multicolumn{1}{c}{Y{M}_{1j}}\\ \multicolumn{1}{c}{{\Phi }_{\mathrm{ij}\left(4\right)}^{(4,13)}}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right],{\Phi }_{\mathrm{ij}\left(5\right)}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{\epsilon E_{1j}^T}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{\epsilon E_{2j}^T}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \end{array}$$

$$\begin{array}{c}{\Phi }_{\mathrm{ij}}^{(1,1)}=(T-2)Y+n_M{\mathrm{TR}}_1^{11}+n_m{\mathrm{TR}}_2^{11},{\Phi }^{(1,2)}=(T-2)Y+n_M{\mathrm{TR}}_1^{12}+n_m{\mathrm{TR}}_2^{12}\hfill \\ {\Phi }^{(2,2)}=(T-2)X+n_M{\mathrm{TR}}_1^{22}+n_m{\mathrm{TR}}_2^{22},{\Phi }_{\mathrm{ij}}^{(2,3)}=X{A}_j+{\hat{B}}_{\mathrm{fi}}{C}_j+{\hat{A}}_{\mathrm{fi}}\hfill \\ {\Phi }_{\mathrm{ij}}^{(2,4)}=X{A}_j+{\hat{B}}_{\mathrm{fi}}{C}_j,{\Phi }_{\mathrm{ij}}^{(2,5)}=X{A}_{\mathrm{dj}}+{\hat{B}}_{\mathrm{fi}}{C}_{\mathrm{dj}},{\Phi }_{\mathrm{ij}}^{(2,6)}=X{A}_{\mathrm{dj}}+{\hat{B}}_{\mathrm{fi}}{C}_{\mathrm{dj}}\hfill \\ {\Phi }_{\mathrm{ij}}^{(2,11)}=X{B}_j+{\hat{B}}_{\mathrm{fi}}{D}_j,{\Phi }_{\mathrm{ij}}^{(2,13)}=X{M}_{1j}+{\hat{B}}_{\mathrm{fi}}{M}_{2j}\hfill \\ {\Phi }_{\mathrm{ij}}^{(3,3)}=Y{A}_j+A_j^{T}Y+(n_{M}T-n_{m}T+T+1)Q_1-\frac{R_1^{11}}{n_{M}T}+S_1^{11}-\frac{R_2^{11}}{n_{m}T}+S_2^{11}\hfill \\ {\Phi }_{\mathrm{ij}}^{(3,4)}=Y{A}_j+A_j^{T}X+C_j^T{\hat{B}}_{\mathrm{fi}}^T+{\hat{A}}_{\mathrm{fi}}^T+(n_{M}T-n_{m}T+T+1)Q_2\hfill \\ -\frac{R_1^{12}}{n_{M}T}+S_1^{12}-\frac{R_2^{12}}{n_{m}T}+S_2^{12},{\Phi }_{\mathrm{ij}}^{(3,12)}=L_j^T-{\hat{C}}_{\mathrm{fi}}^T,\hfill \\ {\Phi }_{\mathrm{ij}}^{(4,4)}=X{A}_j+{\hat{B}}_{\mathrm{fi}}{C}_j+A_j^{T}X+C_j^T{\hat{B}}_{\mathrm{fi}}^T++(n_{M}T-n_{m}T+T+1)Q_3\hfill \\ -\frac{R_1^{22}}{n_{M}T}+S_1^{22}-\frac{R_2^{22}}{n_{m}T}+S_2^{22},{\Phi }_{\mathrm{ij}}^{(4,5)}=X{A}_{\mathrm{dj}}+{\hat{B}}_{\mathrm{fi}}{C}_{\mathrm{dj}}\hfill \\ {\Phi }_{\mathrm{ij}}^{(4,6)}=X{A}_{\mathrm{dj}}+{\hat{B}}_{\mathrm{fi}}{C}_{\mathrm{dj}},{\Phi }_{\mathrm{ij}}^{(4,11)}=X{B}_j+{\hat{B}}_{\mathrm{fi}}{D}_j,{\Phi }^{(4,13)}=X{M}_{1j}+{\hat{B}}_{\mathrm{fi}}{M}_{2j}\hfill \\ {\Phi }^{(7,7)}=-S_1^{11}-\frac{R_1^{11}}{n_{M}T},{\Phi }^{(7,8)}=-S_1^{12}-\frac{R_1^{12}}{n_{M}T}\hfill \\ {\Phi }^{(8,8)}=-S_1^{22}-\frac{R_1^{22}}{n_{M}T},{\Phi }^{(9,9)}=-S_2^{11}-\frac{R_2^{11}}{n_{m}T}\hfill \\ {\Phi }^{(9,10)}=-S_2^{12}-\frac{R_2^{12}}{n_{m}T},{\Phi }^{(10,10)}=-S_2^{22}-\frac{R_2^{22}}{n_{m}T}\hfill \end{array}$$

(32)

In this case, the admissible robust ℋ_∞ filtering parameters in (equation (10)) are given by

$$A_{\mathrm{fi}}=S^{-1}{\hat{A}}_{\mathrm{fi}}{Y}^{-1}{W}^{-T},B_{\mathrm{fi}}=S^{-1}{\hat{B}}_{\mathrm{fi}},C_{\mathrm{fi}}={\hat{C}}_{\mathrm{fi}}{Y}^{-1}{W}^{-T}$$

(33)

where S and W are two nonsingular matrices satisfying

$$\mathrm{SW}=I-X{Y}^{-1}$$

(34)

Proof

First, equation (14) can be rewritten as

$$\Phi (t)=\underset{i=1}{\sum ^r}{\lambda }_i^2(\theta (t)){\Phi }_{\mathrm{ii}}+\underset{i<j}{\sum ^r}{\lambda }_i(\theta (t)){\lambda }_j(\theta (t))({\Phi }_{\mathrm{ij}}+{\Phi }_{\mathrm{ji}})<0$$

(35)

Then, from $X-Y>0$ , it implies that $I-X{Y}^{-1}$ is nonsingular. Hence, we partition $\hat{P}$ and $H_1$ as ³⁸

$$\hat{P}=\left[\begin{array}{cc}X\hfill & S\hfill \\ \multicolumn{1}{c}{S^T}& W^{-1}{Y}^{-1}(X-Y)Y^{-1}{W}^{-T}\hfill \end{array}\right],H_1=\left[\begin{array}{cc}I\hfill & I\hfill \\ \multicolumn{1}{c}{W^{T}Y}& 0\hfill \end{array}\right]$$

(36)

From $S{W}^T=I-X{Y}^{-1}$ , we have

$$H_1^T\hat{P}=\left[\begin{array}{cc}Y\hfill & 0\hfill \\ \multicolumn{1}{c}{X}& S\hfill \end{array}\right],\hat{P}{H}_1=\left[\begin{array}{cc}Y\hfill & X\hfill \\ \multicolumn{1}{c}{0}& S^T\hfill \end{array}\right]$$

(37)

Now, let us define

$$\begin{array}{c}{\hat{S}}_1=\left[\begin{array}{cc}{\hat{S}}_1^{11}\hfill & {\hat{S}}_1^{12}\hfill \\ \multicolumn{1}{c}{*}& {\hat{S}}_1^{22}\hfill \end{array}\right],{\hat{R}}_1=\left[\begin{array}{cc}{\hat{R}}_1^{11}\hfill & {\hat{R}}_1^{12}\hfill \\ \multicolumn{1}{c}{*}& {\hat{R}}_1^{22}\hfill \end{array}\right],\hat{Q}=\left[\begin{array}{cc}{\hat{Q}}_1\hfill & {\hat{Q}}_2\hfill \\ \multicolumn{1}{c}{*}& {\hat{Q}}_3\hfill \end{array}\right]\hfill \\ {\hat{S}}_2=\left[\begin{array}{cc}{\hat{S}}_2^{11}\hfill & {\hat{S}}_2^{12}\hfill \\ \multicolumn{1}{c}{*}& {\hat{S}}_2^{22}\hfill \end{array}\right],{\hat{R}}_2=\left[\begin{array}{cc}{\hat{R}}_2^{11}\hfill & {\hat{R}}_2^{12}\hfill \\ \multicolumn{1}{c}{*}& {\hat{R}}_2^{22}\hfill \end{array}\right],\hfill \\ {\hat{H}}_1=\mathrm{diag}\{H_1,H_1,H_1,H_1,I,I,I,I\}\hfill \end{array}$$

(38)

Pre- and post-multiplying ${\Phi }_{\mathrm{ii}}$ of equation (35) by ${\hat{H}}_1^T$ and ${\hat{H}}_1,$ respectively, yields equation (30) with changes of variables as

$$\begin{array}{c}{\hat{H}}_1^T{\hat{S}}_1{\hat{H}}_1=\left[\begin{array}{cc}{S}_1^{11}\hfill & S_1^{12}\hfill \\ \multicolumn{1}{c}{*}& S_1^{22}\hfill \end{array}\right],{\hat{H}}_1^T{\hat{R}}_1{\hat{H}}_1=\left[\begin{array}{cc}{R}_1^{11}\hfill & R_1^{12}\hfill \\ \multicolumn{1}{c}{*}& R_1^{22}\hfill \end{array}\right],{\hat{A}}_{\mathrm{fi}}=S{A}_{\mathrm{fi}}{W}^{T}Y\hfill \\ {\hat{H}}_1^T{\hat{S}}_2{\hat{H}}_1=\left[\begin{array}{cc}{S}_2^{11}\hfill & S_2^{12}\hfill \\ \multicolumn{1}{c}{*}& S_2^{22}\hfill \end{array}\right],{\hat{H}}_1^T{\hat{R}}_2{\hat{H}}_1=\left[\begin{array}{cc}{R}_2^{11}\hfill & R_2^{12}\hfill \\ \multicolumn{1}{c}{*}& R_2^{22}\hfill \end{array}\right],{\hat{B}}_{\mathrm{fi}}=S{B}_{\mathrm{fi}}\hfill \\ {\hat{H}}_1^T\hat{Q}{\hat{H}}_1=\left[\begin{array}{cc}{Q}_1\hfill & Q_2\hfill \\ \multicolumn{1}{c}{*}& Q_3\hfill \end{array}\right],{\hat{C}}_{\mathrm{fi}}=C_{\mathrm{fi}}{W}^{T}Y\hfill \end{array}$$

Then, following the similar line of above process ${\Phi }_{\mathrm{ij}}$ and ${\Phi }_{\mathrm{ji}}$ , which are similar to ${\Phi }_{\mathrm{ii}}$ , yields equation (31). The proof is completed.

Remark 6

Theorem 1 obtains the robust ℋ_∞ filtering design by solving a set of LMIs. For the system (equation (8)) with $\Delta (t)=0$ , the corresponding robust ℋ_∞ filtering design result is readily obtained by removing the terms with parameter uncertainties from Theorem 1.

Corollary 1

Consider the system (equation (8)) with $\Delta (t)=0$ and filter (equation (10)). Given a prescribed ℋ_∞ performance level $\gamma >0$ and integer scalars $n_M>n_m>0$ . The filtering error system (equation (11)) is asymptotically stable with ℋ_∞ performance $\gamma$ if there exist matrices $0<X=X^T$ , $0<Y=Y^T,S_1^{11},S_1^{12},S_1^{22},S_2^{11},S_2^{12},S_2^{22},Q_1,Q_2,Q_3,R_1^{11},R_1^{12},R_1^{22},R_2^{11},R_2^{12},R_2^{22}$ of appropriate dimensions, such that the following LMIs hold

$$\begin{array}{c}\left[\begin{array}{cc}{S}_1^{11}\hfill & S_1^{12}\hfill \\ \multicolumn{1}{c}{*}& S_1^{22}\hfill \end{array}\right]>0,\left[\begin{array}{cc}{R}_1^{11}\hfill & R_1^{12}\hfill \\ \multicolumn{1}{c}{*}& R_1^{22}\hfill \end{array}\right]>0,\left[\begin{array}{cc}{Q}_1\hfill & Q_2\hfill \\ \multicolumn{1}{c}{*}& Q_3\hfill \end{array}\right]>0\hfill \\ \left[\begin{array}{cc}{S}_2^{11}\hfill & S_2^{12}\hfill \\ \multicolumn{1}{c}{*}& S_2^{22}\hfill \end{array}\right]>0,\left[\begin{array}{cc}{R}_2^{11}\hfill & R_2^{12}\hfill \\ \multicolumn{1}{c}{*}& R_2^{22}\hfill \end{array}\right]>0,X-Y>0\hfill \end{array}$$

(40)

and

$${\hat{\Phi }}_{\mathrm{ii}}<0,(1\le i\le r)$$

(41)

and

$${\hat{\Phi }}_{\mathrm{ij}}+{\hat{\Phi }}_{\mathrm{ji}}<0,(1\le i<j\le r)$$

(42)

where

$$\begin{array}{c}{\hat{\Phi }}_{\mathrm{ij}}=\left[\begin{array}{ccc}{\hat{\Phi }}_{\mathrm{ij}(1)}\hfill & {\hat{\Phi }}_{\mathrm{ij}(2)}\hfill & {\hat{\Phi }}_{\mathrm{ij}(3)}\hfill \\ \multicolumn{1}{c}{*}& -{\gamma }^2\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -I\hfill \end{array}\right]\hfill \\ {\hat{\Phi }}_{\mathrm{ij}(1)}=\left[\begin{array}{cccccccccc}{\Phi }_{\mathrm{ij}}^{(1,1)}\hfill & {\Phi }_{\mathrm{ij}}^{(1,2)}\hfill & Y{A}_j\hfill & Y{A}_j\hfill & Y{A}_{\mathrm{dj}}\hfill & Y{A}_{\mathrm{dj}}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& {\Phi }_{\mathrm{ij}}^{(2,2)}\hfill & {\Phi }_{\mathrm{ij}}^{(2,3)}\hfill & {\Phi }_{\mathrm{ij}}^{(2,4)}\hfill & {\Phi }_{\mathrm{ij}}^{(2,5)}\hfill & {\Phi }_{\mathrm{ij}}^{(2,6)}\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & {\Phi }_{\mathrm{ij}}^{(3,3)}\hfill & {\Phi }_{\mathrm{ij}}^{(3,4)}\hfill & Y{A}_{\mathrm{dj}}\hfill & Y{A}_{\mathrm{dj}}\hfill & \frac{R_1^{11}}{n_{M}T}\hfill & \frac{R_1^{12}}{n_{M}T}\hfill & \frac{R_2^{11}}{n_{m}T}\hfill & \frac{R_2^{12}}{n_{m}T}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & {\Phi }_{\mathrm{ij}}^{(4,4)}\hfill & {\Phi }_{\mathrm{ij}}^{(4,5)}\hfill & {\Phi }_{\mathrm{ij}}^{(4,6)}\hfill & \frac{{(R_1^{12})}^T}{n_{M}T}\hfill & \frac{R_1^{22}}{n_{M}T}\hfill & \frac{{(R_2^{12})}^T}{n_{m}T}\hfill & \frac{R_2^{22}}{n_{m}T}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & -Q_1\hfill & -Q_2\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & -Q_3\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & {\Phi }_{\mathrm{ij}}^{(7,7)}\hfill & {\Phi }_{\mathrm{ij}}^{(7,8)}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & {\Phi }_{\mathrm{ij}}^{(8,8)}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & {\Phi }_{\mathrm{ij}}^{(9,9)}\hfill & {\Phi }_{\mathrm{ij}}^{(9,10)}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & *\hfill & {\Phi }_{\mathrm{ij}}^{(10,10)}\hfill \end{array}\right]\hfill \\ {\hat{\Phi }}_{\mathrm{ij}(2)}=\left[\begin{array}{c}Y{B}_j\hfill \\ \multicolumn{1}{c}{{\Phi }_{\mathrm{ij}}^{(2,11)}}\\ \multicolumn{1}{c}{Y{B}_j}\\ \multicolumn{1}{c}{{\Phi }_{\mathrm{ij}}^{(4,11)}}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right],{\hat{\Phi }}_{\mathrm{ij}(3)}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{{\Phi }_{\mathrm{ij}}^{(3,12)}}\\ \multicolumn{1}{c}{L_j^T}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\\ \multicolumn{1}{c}{0}\end{array}\right]\hfill \end{array}$$

(43)

In this case, the admissible robust ℋ_∞ filtering parameters in equation (10) are given by

$$A_{\mathrm{fi}}=S^{-1}{\hat{A}}_{\mathrm{fi}}{Y}^{-1}{W}^{-T},B_{\mathrm{fi}}=S^{-1}{\hat{B}}_{\mathrm{fi}},,C_{\mathrm{fi}}={\hat{C}}_{\mathrm{fi}}{Y}^{-1}{W}^{-T}$$

(44)

where S and W are two nonsingular matrices satisfying

$$\mathrm{SW}=I-X{Y}^{-1}$$

(45)

Simulation examples

This section provides a simulation example to validate the robust ℋ_∞ filter design approach developed in the previous sections.

Consider a mass–spring–damper mechanical system, the system equations are given by

$$m\stackrel{··}{\theta }\left(t\right)+d(\stackrel{·}{\theta }\left(t\right))\stackrel{·}{\theta }\left(t\right)+k\theta \left(t\right)=\varphi f\left(t\right)$$

where $\theta \left(t\right)$ denotes the relative position of the mass and $f(t)$ is the external force.

In this simulation, the mass is chosen as $m=1\mathrm{kg}$ ; the stiffness of the springs is $k=0.2\mathrm{N}/\mathrm{m}$ ; the input coefficients are $\varphi =1$ . We choose two local models, that is, by linearizing the mass–spring–damper mechanical system around the origin, $\stackrel{·}{\theta }(t)=(\pm 0.8,0)$ and $\stackrel{·}{\theta }(t)=(\pm 1.0,0)$ . Then, by discretization of the T-S fuzzy system with sampling period T = 0.01 s, we obtain the T-S fuzzy delta operator model as follows.

Plant Rulei: IF ${\theta }_1(t)$ is $M_{i1}$ and ${\theta }_2(t)$ is $M_{i2}$ and … and ${\theta }_p(t)$ is $M_{\mathrm{ip}}$ , THEN

$$\{\begin{array}{c}\partial x(t)=(A_i+\Delta A_i)x(t)+(A_{\mathrm{di}}+\Delta A_{\mathrm{di}})x(t-\mathrm{nT})+B_i\omega (t)\hfill \\ \multicolumn{1}{c}{y(t)=(C_i+\Delta C_i)x(t)+(C_{\mathrm{di}}+\Delta C_{\mathrm{di}})x(t-\mathrm{nT})+D_i\omega (t)}\\ \multicolumn{1}{c}{z(t)=L_{i}x(t),i=\{1,2\}}\end{array}$$

where the system parameters are as follows:

$$\begin{array}{c}\left[\begin{array}{ccc}{A}_1\hfill & {A_d}_1\hfill & B_1\hfill \\ \multicolumn{1}{c}{C_1}& L_1\hfill & D_1\hfill \end{array}\right]=\left[\frac{\begin{array}{cc}-1.2\hfill & -0.8\hfill \\ \multicolumn{1}{c}{1}& -2\hfill \end{array}}{\begin{array}{cc}1\hfill & 0\hfill \end{array}}|\frac{\begin{array}{cc}-0.5\hfill & 0.2\hfill \\ \multicolumn{1}{c}{1}& 0\hfill \end{array}}{\begin{array}{cc}1\hfill & -0\hfill \end{array}.5}|\frac{\begin{array}{c}1\hfill \\ \multicolumn{1}{c}{-0.2}\end{array}}{0.3}\right]\hfill \\ \left[\begin{array}{ccc}{A}_2\hfill & {A_d}_2\hfill & B_2\hfill \\ \multicolumn{1}{c}{C_2}& L_2\hfill & D_2\hfill \end{array}\right]=\left[\frac{\begin{array}{cc}-1.6\hfill & 0.2\hfill \\ \multicolumn{1}{c}{0.7}& -1\hfill \end{array}}{\begin{array}{cc}0.5\hfill & -0.6\hfill \end{array}}|\frac{\begin{array}{cc}0.6\hfill & 0.2\hfill \\ \multicolumn{1}{c}{0.5}& -0.8\hfill \end{array}}{\begin{array}{cc}-0.2\hfill & 0\hfill \end{array}.3}|\frac{\begin{array}{c}0.3\hfill \\ \multicolumn{1}{c}{0.1}\end{array}}{-0.6}\right]\hfill \end{array}$$

and a sampling period $T=0.01$ .

The parameter uncertainties are assumed to satisfy equations (4)–(6) with the parameters as follows

$$M_{11}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{-0.5}\end{array}\right],M_{21}=0.08,E_{11}=[00.03],E_{21}=[0.020]$$

$$M_{12}=\left[\begin{array}{c}0\hfill \\ \multicolumn{1}{c}{0.3}\end{array}\right],M_{22}=0.06,E_{12}=[0.050],E_{22}=[0-0.02]$$

$$\Lambda (t)=\sin (t),J=0.2.$$

The membership functions are given as follows

$${\lambda }_1(x_1(t))=\{\begin{array}{c}\frac{2-x_1(t)}{2}\mathrm{for}0<x_1(t)\le 2\hfill \\ \frac{2+x_1(t)}{2}\mathrm{for}-2\le x_1(t)\le 0\hfill \\ 0\mathrm{for}{x}_1(t)<-2\hfill \end{array}$$

and

$${\lambda }_2(x_1(t))=1-{\lambda }_1(x_1(t))$$

The disturbance input vector $\omega (t)$ is assumed to satisfy

$$\omega (t)=0.1e^{-0.03\mathrm{tT}}\sin (\mathrm{tT}),t=1,2,\dots$$

Now, our aim is to design a robust fuzzy filter of the form (equation (8)), which guarantees the stability of the filtering error system (equation (10)) and ℋ_∞ performance index ${\gamma }_{min}.$ Given matrix

$$S=\left[\begin{array}{cc}1\hfill & -0.5\hfill \\ \multicolumn{1}{c}{0.8}& 1\hfill \end{array}\right]$$

and assume that the sampling rates with time-varying $n(t)$ satisfy (equation (2)) with $n_m=60$ and $n_M=80$ . The minimum performance level ${\gamma }_{min}=2.4249$ can be obtained by applying Theorem 1. The desired robust ℋ_∞ filter parameters are computed as follows

$$\begin{array}{l}{A}_{f1}=\left[\begin{array}{cc}-1.4932\hfill & -0.3856\hfill \\ 1.1385\hfill & -0.2665\hfill \end{array}\right]\mathrm{,}{B}_{f1}=\left[\begin{array}{c}-0.3165\hfill \\ 0.1515\hfill \end{array}\right]\mathrm{,}\hfill \\ C_{f1}=\left[\begin{array}{cc}-1.2732\hfill & 0.2561\hfill \end{array}\right]\hfill \\ A_{f2}=\left[\begin{array}{cc}-0.8192\hfill & 0.1050\hfill \\ \multicolumn{1}{c}{1.0479}& -0.7562\hfill \end{array}\right],B_{f2}=\left[\begin{array}{c}-0.1410\hfill \\ \multicolumn{1}{c}{0.5188}\end{array}\right],\hfill \\ C_{f2}=\left[\begin{array}{cc}-0.3514\hfill & -0.3656\hfill \end{array}\right]\hfill \end{array}$$

With the above solution, given the initial conditions $x(0)=[\pi /4,0{]}^T$ and $x_f(0)=[0,0{]}^T$ , the randomly sampling rates $n(t)$ with time varying are shown in Figure 1, and simulation results of the state responses of $x(t)$ and $x_f(t),$ the responses of $z(t)$ , $z_f(t)$ , and $w(t)$ , and the filtering error signal $e(t)=z(t)-z_f(t)$ are shown in Figures 2, 3, and 4, respectively. Figure 5 shows the response of the ℋ_∞ performance $\gamma$ under zero initial conditions; it can be seen that $\gamma$ is less than 0.4, which is less than the prescribed performance index ${\gamma }_{min}=2.4249$ obtained from Theorem 1. Therefore, the proposed robust fuzzy ℋ_∞ filter is satisfactory.

Figure 1.

The randomly sampling rates $n(t)$ .

Figure 2.

State responses of $x(t)$ and $x_f(t)$ .

Figure 3.

Responses of $z(t)$ , $z_f(t)$ , and $w(t)$ .

Figure 4.

Error response of $e(t)$ .

Figure 5.

Response of the ℋ_∞ performance $\gamma$ .

Conclusion

This article investigated the problem of robust ℋ_∞ filtering design for T-S fuzzy systems with interval time-varying delay using delta operator approach. Based on a LKF in delta domain, one ℋ_∞ filtering design scheme was proposed. A numerical example was used to validate the effectiveness of the theoretical results obtained.