Delay-dependent robust $H_{\mathrm{\infty }}$ filter for T-S fuzzy time-delay systems with exponential stability

Abstract

This paper deals with the problem of delay-dependent robust $H_{\mathrm{\infty }}$ filter for T-S fuzzy time-delay systems with exponential stability. The purpose is to design filter parameters such that the filtering error system is exponentially stable and satisfies a prescribed $H_{\mathrm{\infty }}$ performance. In terms of linear matrix inequalities (LMIS), some sufficient conditions for the solvability of this problem are presented. Thanks to the new filter, the obtained stability criterion is less conservative than the existing ones. Finally, three examples are provided to demonstrate the effectiveness and the superiority of the proposed design methods.

1 Introduction

In the past several decades, robust filtering problem has received extensive attention of people. The current study of robust filtering mainly concentrated on two aspects: Kalman filter and $H_{\mathrm{\infty }}$ filter. Among them, the research of $H_{\mathrm{\infty }}$ filter is wider, and many important and interesting results have been proposed in terms of all kinds of approaches (see, for example, [1–3]). Actual industrial system such as the power grid, chemical processes, nuclear reactor and others often contain time-delay, and time-delay is the main factor that leads to system performance degradation and instability. Therefore, the research of the filtering problem for time-delay systems has important theoretical significance and application value.

In recent years, the research of the filtering for time-delay systems has made abundant achievements. Delay-dependent robust $H_{\mathrm{\infty }}$ and $L_2-L_{\mathrm{\infty }}$ filtering for a class of uncertain nonlinear time-delay systems was studied in [4]. $H_{\mathrm{\infty }}$ filtering of time-delay T-S fuzzy systems based on piecewise Lyapunov-Krasovskii functional was investigated in [5]. A new fuzzy $H_{\mathrm{\infty }}$ filter design for nonlinear continuous-time dynamic systems with time-varying delays was reported in [6]. Robust $H_{\mathrm{\infty }}$ filtering for a class of uncertain Lurie time-delay singular systems was studied in [7]. Delay-dependent $H_{\mathrm{\infty }}$ filtering for singular Markovian jump time-delay systems was studied in [8].

T-S fuzzy system has wide application in the network, economy, environment and other fields, it has attracted more and more concern of the scholars (see, for example, [9–11]). $H_{\mathrm{\infty }}$ filter has come to play an important role in fuzzy model during the past years, so the filtering of fuzzy system is especially important. Delay-dependent nonfragile robust $H_{\mathrm{\infty }}$ filtering of T-S fuzzy time-delay systems was investigated in [12]. An improved $H_{\mathrm{\infty }}$ filter design for nonlinear system with time-delay via T-S fuzzy models was studied in [13]. Exponential $H_{\mathrm{\infty }}$ filter design for uncertain Takagi-Sugeno fuzzy systems with time-delay was reported in [14]. New results on $H_{\mathrm{\infty }}$ filtering for fuzzy systems with interval time-varying delays was studied in [15]. Delay-dependent non-fragile $H_{\mathrm{\infty }}$ filtering for uncertain fuzzy systems based on switching fuzzy model and piecewise Lyapunov function was studied in [16]. But at present, the problem of delay-dependent robust $H_{\mathrm{\infty }}$ filter for T-S fuzzy time-delay systems with exponential stability has rarely been reported.

For T-S fuzzy time-delay systems with exponential stability, this paper discusses the design methods of delay-dependent robust $H_{\mathrm{\infty }}$ filter. First of all, it gave a criterion of exponential stability, and then discussed the conditions and design methods of delay-dependent robust $H_{\mathrm{\infty }}$ filter. The result of designed filter is exponential stability for the augmented system via LMI. Thanks to the new filter, the obtained criterion is less conservative than the existing ones. Finally, some numerical examples are given to show the effectiveness and the superiority of the proposed design methods.

2 System description and preliminary lemma

Consider the following T-S fuzzy time-delay system, which is described by plant Rule i: IF ${\epsilon }_1(t)$ is $M_{i1}$ and ${\epsilon }_2(t)$ is $M_{i2}\cdots {\epsilon }_p(t)$ is $M_{ip}$, THEN

$$\begin{array}{r}\dot{x}(t)=(A_{1i}+\mathrm{\Delta }{A}_{1i})x(t)+(A_{2i}+\mathrm{\Delta }{A}_{2i})x(t-d)+(B_i+\mathrm{\Delta }{B}_i)\omega (t),\\ y(t)=(C_{1i}+\mathrm{\Delta }{C}_{1i})x(t)+(C_{2i}+\mathrm{\Delta }{C}_{2i})x(t-d)+(D_i+\mathrm{\Delta }{D}_i)\omega (t),\\ z(t)=(L_i+\mathrm{\Delta }{L}_i)x(t),\\ x(t)=\varphi (t),t\in [-d,0],\end{array}$$

(1)

where $i\in R:=\{1,2,\dots ,r\}$, r is the number of IF-THEN rules. $x(t)\in R^n$ is the input vector, $\omega (t)\in R^m$ is the disturbance vector of the system which belongs to $L_2[0,+\mathrm{\infty })$, $y(t)\in R^q$ is the measurable output vector, $z(t)\in R^p$ is the signal vector to be estimated, $\varphi (t)$ is a compatible vector-valued initial function. $A_{1i}$, $A_{2i}$, $B_i$, $C_{1i}$, $C_{2i}$, $D_i$, $L_i$ are constant matrices with appropriate dimensions. ${\epsilon }_j(t)$ and $M_{ij}$ ($j=1,2,\dots ,p$) are the premise variables and the fuzzy sets. $d>0$ is the constant time delay. $\mathrm{\Delta }{A}_{1i}$, $\mathrm{\Delta }{A}_{2i}$, $\mathrm{\Delta }{B}_i$, $\mathrm{\Delta }{C}_{1i}$, $\mathrm{\Delta }{C}_{2i}$, $\mathrm{\Delta }{D}_i$, $\mathrm{\Delta }{L}_i$ are unknown matrices representing parametric uncertainties and are assumed to be of the form

$$\left[\begin{array}{ccc}\mathrm{\Delta }{A}_{1i}& \mathrm{\Delta }{A}_{2i}& \mathrm{\Delta }{B}_i\\ \mathrm{\Delta }{C}_{1i}& \mathrm{\Delta }{C}_{2i}& \mathrm{\Delta }{D}_i\end{array}\right]=\left[\begin{array}{c}{H}_{1i}\\ H_{2i}\end{array}\right]F_i[E_{1i}{E}_{2i}{E}_{3i}],\mathrm{\Delta }{L}_i=H_{3i}{F}_i{E}_{4i},$$

(2)

where $H_{1i}$, $H_{2i}$, $H_{3i}$, $E_{1i}$, $E_{2i}$, $E_{3i}$, $E_{4i}$ are known real constant matrices with appropriate dimensions, and $F_i$ is an unknown real time-varying matrix satisfying

$$F_i^T{F}_i\le I.$$

(3)

I is a unit matrix with appropriate dimensions. The parametric uncertainties $\mathrm{\Delta }{A}_{1i}$, $\mathrm{\Delta }{A}_{2i}$, $\mathrm{\Delta }{B}_i$, $\mathrm{\Delta }{C}_{1i}$, $\mathrm{\Delta }{C}_{2i}$, $\mathrm{\Delta }{D}_i$, $\mathrm{\Delta }{L}_i$ are said to be admissible if both (2) and (3) hold.

Remark 1 When $\mathrm{\Delta }{D}_i=\mathrm{\Delta }{E}_i=\mathrm{\Delta }{L}_i=0$, system (1) was studied in [14]. The system in this paper is a class of fuzzy time-delay systems broader than others.

Let $\epsilon (t)={[{\epsilon }_1(t){\epsilon }_2(t)\cdots {\epsilon }_p(t)]}^T$, through the use of ‘fuzzy blending’, the fuzzy system (1) can be inferred as follows:

$$\begin{array}{r}\dot{x}(t)=\sum _{i=1}^r{h}_i(\epsilon (t))[(A_{1i}+\mathrm{\Delta }{A}_{1i})x(t)+(A_{2i}+\mathrm{\Delta }{A}_{2i})x(t-d)+(B_i+\mathrm{\Delta }{B}_i)\omega (t)],\\ y(t)=\sum _{i=1}^r{h}_i(\epsilon (t))[(C_{1i}+\mathrm{\Delta }{C}_{1i})x(t)+(C_{2i}+\mathrm{\Delta }{C}_{2i})x(t-d)+(D_i+\mathrm{\Delta }{D}_i)\omega (t)],\\ z(t)=\sum _{i=1}^r{h}_i(\epsilon (t))[(L_i+\mathrm{\Delta }{L}_i)x(t)],\\ {\beta }_i(\epsilon (t))=\prod _{j=1}^p{M}_{ij}({\epsilon }_j(t)),h_i(\epsilon (t))=\frac{{\beta }_i(\epsilon (t))}{{\sum }_{i=1}^r{\beta }_i(\epsilon (t))},\end{array}$$

(4)

where $M_{ij}({\epsilon }_j(t))$ is the grade of membership of ${\epsilon }_j(t)$ in $M_{ij}$. It is easy to see that ${\beta }_i(\epsilon (t))\ge 0$ and ${\sum }_{i=1}^r{\beta }_i(\epsilon (t))\ge 0$. Hence, we have $h_i(\epsilon (t))\ge 0$ and ${\sum }_{i=1}^r{h}_i(\epsilon (t))=1$.

In this paper, we consider the following fuzzy filter:

$$\begin{array}{r}\dot{\hat{x}}(t)=\sum _{i=1}^r{h}_i(\epsilon (t))[A_{fi}\hat{x}(t)+B_{fi}y(t)+C_{fi}\hat{x}(t-d)],\\ \hat{z}(t)=\sum _{i=1}^r{h}_i(\epsilon (t))[L_{fi}\hat{x}(t)],i=1,2,\dots ,r,\end{array}$$

(5)

where $\hat{x}(t)\in R^k$ is the filter state vector, $\hat{z}(t)$ is the estimated vector, $A_{fi}$, $B_{fi}$, $C_{fi}$, $L_{fi}$ with compatible dimensions are matrices to be determined.

Remark 2 When $C_{fi}=0$, the fuzzy filter (5) was studied in [14] and [5]. This paper improves the function of the filter in [14] and [5], this makes the obtained result less conservative than the existing ones.

From (4) and (5), we obtain the filtering error system as follows:

$$\begin{array}{r}\dot{\tilde{x}}(t)=\sum _{i=1}^r\sum _{j=1}^r{h}_i(\epsilon (t))h_j(\epsilon (t))[A_{ij}\tilde{x}(t)+A_{d_{ij}}\tilde{x}(t-d)+B_{ij}\omega (t)],\\ e(t)=\sum _{i=1}^r\sum _{j=1}^r{h}_i(\epsilon (t))h_j(\epsilon (t))[L_{ij}\tilde{x}(t)],\end{array}$$

(6)

where

$$\begin{array}{c}e(t)=z(t)-\hat{z}(t),\tilde{x}(t)={[x^T(t){\hat{x}}^T(t)]}^T,\hfill \\ A_{ij}=\left[\begin{array}{cc}{A}_{1i}+\mathrm{\Delta }{A}_{1i}& 0\\ B_{fj}(C_{1i}+\mathrm{\Delta }{C}_{1i})& A_{fj}\end{array}\right],A_{d_{ij}}=\left[\begin{array}{cc}{A}_{2i}+\mathrm{\Delta }{A}_{2i}& 0\\ B_{fj}(C_{2i}+\mathrm{\Delta }{C}_{2i})& C_{fj}\end{array}\right],\hfill \\ B_{ij}=\left[\begin{array}{c}{B}_i+\mathrm{\Delta }{B}_i\\ B_{fj}(D_i+\mathrm{\Delta }{D}_i)\end{array}\right],L_{ij}=[L_i+\mathrm{\Delta }{L}_i-L_{fj}].\hfill \end{array}$$

Definition 1 The filtering error system (6) is said to be exponentially stable, if there exist scalars $\delta >0$ and $\epsilon >0$ such that $\parallel x(t)\parallel \le \delta {sup}_{-d\le \theta \le 0}\parallel \varphi (\theta )\parallel e^{-\epsilon t}$.

System (6) can be abbreviated as the following form:

$$\begin{array}{r}\dot{\tilde{x}}(t)=\tilde{A}\tilde{x}(t)+{\tilde{A}}_d\tilde{x}(t-d)+\tilde{B}\omega (t),\\ e(t)=\tilde{L}\tilde{x}(t),\end{array}$$

(7)

where

$$\begin{array}{r}\tilde{A}=\sum _{i=1}^r\sum _{j=1}^r{h}_i(\epsilon (t))h_j(\epsilon (t))A_{ij},{\tilde{A}}_d=\sum _{i=1}^r\sum _{j=1}^r{h}_i(\epsilon (t))h_j(\epsilon (t))A_{d_{ij}},\\ \tilde{B}=\sum _{i=1}^r\sum _{j=1}^r{h}_i(\epsilon (t))h_j(\epsilon (t))B_{ij},\tilde{L}=\sum _{i=1}^r\sum _{j=1}^r{h}_i(\epsilon (t))h_j(\epsilon (t))L_{ij}.\end{array}$$

The robust $H_{\mathrm{\infty }}$ filtering problem to be addressed in this paper is formulated as follows: given the T-S fuzzy time-delay system (4) and a prescribed level of noise attenuation $\gamma >0$, determine a filter with exponentially stable in the form of (5) such that the following requirements are satisfied:

• (a) The filtering error system (6) is exponentially stable,

• (b) Under zero initial conditions, (6) satisfies

  $${\parallel e(t)\parallel }_2<\gamma {\parallel \omega (t)\parallel }_2$$

  (8)

  for any nonzero $\omega (t)\in L_2[0,\mathrm{\infty })$ and all admissible uncertainties.

Lemma 1 [17]

Given a set of suited dimension real matrices E, F, H, Q is a symmetric matrix such that

$$Q+HFE+E^T{F}^T{H}^T<0$$

for all F satisfies $F^{\mathrm{T}}F\le I$ if and only if there exists a scalar $\epsilon >0$ such that

$$Q+\epsilon H{H}^T+{\epsilon }^{-1}{E}^{T}E<0.$$

Lemma 2 [18]

Suppose that $x(t)\in R^n$ is the vector function with a continuous derivative, if $\left[\begin{array}{ccc}U& M_1& M_2\\ \ast & Z_1& Z_2\\ \ast & \ast & Z_3\end{array}\right]\ge 0$, where $U,M_1,M_2,Z_1,Z_2,Z_3\in R^{n\times n}$, such that the following integration is well defined, then

$$-{\int }_{t-d}^t{\dot{x}}^T(s)U\dot{x}(s)ds\le {\left[\begin{array}{c}x(t)\\ x(t-d)\end{array}\right]}^T\left[\begin{array}{cc}{M}_1^T+M_1+d{Z}_1& -M_1^T+M_2+d{Z}_2\\ \ast & -M_2^T-M_2+d{Z}_3\end{array}\right]\left[\begin{array}{c}x(t)\\ x(t-d)\end{array}\right].$$

3 Main results

Theorem 1 For prescribed scalar $d>0$, the system (6) is exponentially stable, and (8) is satisfied if there exists symmetric positive definite matrix W, U and invertible matrix P such that

$$\begin{array}{r}\sum =\left[\begin{array}{ccccc}{\mathrm{\Sigma }}_{11}& {\mathrm{\Sigma }}_{12}& P^T\tilde{B}& {\tilde{A}}^T{H}^T& {\tilde{L}}^T\\ \ast & {\mathrm{\Sigma }}_{22}& 0& {\tilde{A}}_d^T{H}^T& 0\\ \ast & \ast & -{\gamma }^{2}I& {\tilde{B}}^T{H}^T& 0\\ \ast & \ast & \ast & -{(dU)}^{-1}& 0\\ \ast & \ast & \ast & \ast & -I\end{array}\right]<0,\\ {\mathrm{\Sigma }}_{11}=P^T\tilde{A}+{\tilde{A}}^{T}P+H^{T}WH+H^T(M_1^T+M_1+d{Z}_1)H,\\ {\mathrm{\Sigma }}_{12}=P^T{\tilde{A}}_d+H^T(-M_1^T+M_2+d{Z}_2)H,\\ {\mathrm{\Sigma }}_{22}=-H^{T}WH+H^T(-M_2^T-M_2+d{Z}_3)H.\end{array}$$

(9)

Proof First, we shall show the exponential stability of the system (6).

For any $t\ge d$, choose a Lyapunov functional candidate to be:

$$V({\tilde{x}}_t)={\tilde{x}}^T(t)P\tilde{x}(t)+{\int }_{t-d}^t{\tilde{x}}^T(s)H^{T}WH\tilde{x}(s)ds+{\int }_{-d}^0{\int }_{t+\theta }^t{\dot{\tilde{x}}}^T(s)H^{T}UH\dot{\tilde{x}}(s)dsd\theta ,$$

(10)

where W, U are symmetric positive definite matrices, and P is an invertible matrix to be determined, $H=[II]$, ${\tilde{x}}_t=\tilde{x}(t+\beta )$, $-d\le \beta \le 0$.

When $\omega (t)=0$, through Lemma 2, we get

$$\dot{V}({\tilde{x}}_t)\le {\left[\begin{array}{c}\tilde{x}(t)\\ \tilde{x}(t-d)\end{array}\right]}^T({\mathrm{\Gamma }}_1+{\mathrm{\Gamma }}_2)\left[\begin{array}{c}\tilde{x}(t)\\ \tilde{x}(t-d)\end{array}\right],$$

where

$${\mathrm{\Gamma }}_1=\left[\begin{array}{cc}{\mathrm{\Sigma }}_{11}& {\mathrm{\Sigma }}_{12}\\ \ast & {\mathrm{\Sigma }}_{22}\end{array}\right],{\mathrm{\Gamma }}_2=d\left[\begin{array}{c}{\tilde{A}}^T{H}^T\\ {\tilde{A}}_d^T{H}^T\end{array}\right]U{\left[\begin{array}{c}{\tilde{A}}^T{H}^T\\ {\tilde{A}}_d^T{H}^T\end{array}\right]}^T.$$

Now, applying the Schur complements, it is easy to see from (9) that there exists a scalar ${\delta }_0>0$ such that for any $t\ge d$,

$$\dot{V}({\tilde{x}}_t)\le -{\delta }_0{\parallel \tilde{x}(t)\parallel }^2.$$

(11)

Now, by (10) and (11), we have

$$\begin{array}{rl}\frac{d}{dt}[e^{\epsilon t}V({\tilde{x}}_t)]& =e^{\epsilon t}[\epsilon V(t)+\dot{V}({\tilde{x}}_t)]\\ \le e^{\epsilon t}[(\epsilon {\delta }_1-{\delta }_0){\parallel \tilde{x}(t)\parallel }^2+\epsilon {\delta }_2{\int }_{t-d}^t{\parallel \tilde{x}(s)\parallel }^{2}ds],\end{array}$$

where ${\delta }_1>0$, ${\delta }_2>0$.

Integrating both sides from 0 to $T>0$ gives

$$e^{\epsilon T}V({\tilde{x}}_T)-V({\tilde{x}}_0)\le (\epsilon {\delta }_1-{\delta }_0){\int }_0^T{e}^{\epsilon t}{\parallel \tilde{x}(t)\parallel }^{2}dt+\epsilon {\delta }_2{\int }_0^T{e}^{\epsilon t}dt{\int }_{t-d}^t{\parallel \tilde{x}(s)\parallel }^{2}ds.$$

Since

$$\begin{array}{rl}{\int }_0^T{e}^{\epsilon t}dt{\int }_{t-d}^t{\parallel \tilde{x}(s)\parallel }^{2}ds=& {\int }_{-d}^0{\parallel \tilde{x}(s)\parallel }^{2}ds{\int }_0^{s+d}{e}^{\epsilon t}dt+{\int }_0^{T-d}{\parallel \tilde{x}(s)\parallel }^{2}ds{\int }_s^{s+d}{e}^{\epsilon t}dt\\ +{\int }_{T-d}^T{\parallel \tilde{x}(s)\parallel }^{2}ds{\int }_s^T{e}^{\epsilon t}dt\\ \le & {\int }_{-d}^{0}d{e}^{\epsilon (s+d)}{\parallel \tilde{x}(s)\parallel }^{2}ds+{\int }_0^{T-d}d{e}^{t(s+d)}{\parallel \tilde{x}(s)\parallel }^{2}ds\\ +{\int }_{T-d}^{T}d{e}^{\epsilon (s+d)}{\parallel \tilde{x}(s)\parallel }^{2}ds\\ =& d{e}^{\epsilon d}{\int }_{-d}^T{e}^{\epsilon s}{\parallel \tilde{x}(s)\parallel }^{2}ds.\end{array}$$

Let the scalar $\epsilon >0$ small enough such that $\epsilon {\delta }_1-{\delta }_0+d\epsilon {\delta }_2{e}^{\epsilon d}\le 0$. Then, we get that there exists a scalar $\kappa >0$ such that

$$e^{\epsilon T}V({\tilde{x}}_T)\le V({\tilde{x}}_0)+[\epsilon {\delta }_1-{\delta }_0+d\epsilon {\delta }_2{e}^{\epsilon d}]{\int }_0^T{e}^{\epsilon t}{\parallel \tilde{x}(t)\parallel }^{2}dt\le \kappa \underset{-d\le \theta \le 0}{sup}{\parallel \varphi (\theta )\parallel }^2.$$

Taking into account that

$$V({\tilde{x}}_T)\ge {\lambda }_{min}(P){\parallel \tilde{x}(T)\parallel }^2.$$

It is not difficult to see that, for any $T>0$,

$$\parallel \tilde{x}(T)\parallel \le \sqrt{\frac{\kappa }{{\lambda }_{min}(P)}}\underset{-d\le \theta \le 0}{sup}\parallel \varphi (\theta )\parallel e^{-\epsilon T}.$$

Therefore, by Definition 1, the T-S fuzzy time-delay system (6) is exponentially stable.

Next, we show that for any nonzero $\omega (t)\in L_2[0,\mathrm{\infty })$, system (6) satisfies (8) under the zero initial condition. To this end, we introduce

$$J_T={\int }_0^T[e{(t)}^{T}e(t)-{\gamma }^2\omega {(t)}^T\omega (t)]dt,$$

where the scalar $T>0$. Consider the Lyapunov function of the augmented system (10) for $0\ne \omega (t)\in L_2[0,\mathrm{\infty })$, we have

$$\dot{V}({\tilde{x}}_t)\le {\left[\begin{array}{c}\tilde{x}(t)\\ \tilde{x}(t-d)\\ w(t)\end{array}\right]}^T({\mathrm{\Gamma }}_1^{\prime }+{\mathrm{\Gamma }}_2^{\prime })\left[\begin{array}{c}\tilde{x}(t)\\ \tilde{x}(t-d)\\ w(t)\end{array}\right],$$

where

$${\mathrm{\Gamma }}_1^{\prime }=\left[\begin{array}{ccc}{\mathrm{\Sigma }}_{11}& {\mathrm{\Sigma }}_{12}& P^T\tilde{B}\\ \ast & {\mathrm{\Sigma }}_{22}& 0\\ \ast & \ast & 0\end{array}\right],{\mathrm{\Gamma }}_2^{\prime }=d\left[\begin{array}{c}{\tilde{A}}^T{H}^T\\ {{\tilde{A}}_d}^T{H}^T\\ {\tilde{B}}^T{H}^T\end{array}\right]U{\left[\begin{array}{c}{\tilde{A}}^T{H}^T\\ {{\tilde{A}}_d}^T{H}^T\\ {\tilde{B}}^T{H}^T\end{array}\right]}^T.$$

It can be shown that for any nonzero $\omega (t)\in L_2[0,\mathrm{\infty })$ and $T>0$,

$$\begin{array}{rl}{J}_T& ={\int }_0^T[\dot{V}({\tilde{x}}_t)+e{(t)}^{\mathrm{T}}e(t)-{\gamma }^2\omega {(t)}^T\omega (t)]dt-V({\tilde{x}}_t)\\ \le {\int }_0^T[\dot{V}({\tilde{x}}_t)+e{(t)}^{\mathrm{T}}e(t)-{\gamma }^2\omega {(t)}^T\omega (t)]dt\\ \le {\int }_0^T{x}_{\alpha }^T({\mathrm{\Gamma }}_1^{″}+{\mathrm{\Gamma }}_2^{\prime })x_{\alpha }dt,\end{array}$$

where

$$x_{\alpha }^T=[{\tilde{x}}^T(t){\tilde{x}}^T(t-d){\omega }^T(t)],{\mathrm{\Gamma }}_1^{″}=\left[\begin{array}{ccc}{\mathrm{\Sigma }}_{11}+{\tilde{L}}^T\tilde{L}& {\mathrm{\Sigma }}_{12}& P^T\tilde{B}\\ \ast & {\mathrm{\Sigma }}_{22}& 0\\ \ast & \ast & -{\gamma }^{2}I\end{array}\right].$$

When $\sum <0$, we have $J_T<0$ for all $T>0$, which implies that ${\parallel e(t)\parallel }_2<\gamma {\parallel \omega (t)\parallel }_2$ for any nonzero $\omega (t)\in L_2[0,\mathrm{\infty })$. This completes the proof. □

Based on the sufficient conditions above, the design problem of robust $H_{\mathrm{\infty }}$ filter can be transformed into a problem of linear matrix inequality.

Theorem 2 Given matrices Q, S, R, which Q, R are symmetric, and Q is negative definite for all uncertainties, the robust filtering issue is resolved for system (6) if there exist positive scalars ${\epsilon }_1>0$, ${\epsilon }_2>0$, symmetric positive definite matrices W, U, and invertible matrix $P=diag(P_1,P_2)$, such that the following LMIs holds:

$$\left[\begin{array}{ccc}{G^{\prime }}_{ii}& {\eta }_1& {\eta }_2^T\\ \ast & -{\epsilon }_{1}I& 0\\ \ast & \ast & -{\epsilon }_1^{-1}I\end{array}\right]<0,$$

(12)

$$\left[\begin{array}{ccc}{G^{\prime }}_{ij}& {\eta }_3& {\eta }_4^T\\ \ast & -{\epsilon }_{2}I& 0\\ \ast & \ast & -{\epsilon }_2^{-1}I\end{array}\right]<0,$$

(13)

where

$$\begin{array}{r}{G}_{ii}^{\prime }=\left[\begin{array}{ccccccc}{\theta }_{11}& {\theta }_{12}& {\theta }_{13}& {\theta }_{14}& {\theta }_{15}& {\theta }_{16}& L_i^T\\ \ast & {\theta }_{22}& {\theta }_{23}& {\theta }_{24}& {\theta }_{25}& A_{fi}^T& -L_{fj}^T\\ \ast & \ast & {\theta }_{33}& {\theta }_{34}& 0& {\theta }_{36}& 0\\ \ast & \ast & \ast & {\theta }_{44}& 0& C_{fi}^T& 0\\ \ast & \ast & \ast & \ast & {\theta }_{55}& {\theta }_{56}& 0\\ \ast & \ast & \ast & \ast & \ast & {\theta }_{66}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -I\end{array}\right],\\ {\eta }_1=\left[\begin{array}{ccccccc}{P}_1{H}_{1i}& 0& 0& 0& 0& 0& 0\\ M_{2i}{H}_{2i}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ H_{1i}+B_{fi}{H}_{2i}& 0& 0& 0& 0& 0& 0\\ 0& H_{3i}& 0& 0& 0& 0& 0\end{array}\right],\\ {\eta }_2=\left[\begin{array}{ccccccc}{E}_{1i}& 0& E_{2i}& 0& E_{3i}& 0& 0\\ E_{4i}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right],\\ G_{ij}^{\prime }=\left[\begin{array}{ccccccc}{\overline{\theta }}_{11}& {\overline{\theta }}_{12}& {\overline{\theta }}_{13}& {\overline{\theta }}_{14}& {\overline{\theta }}_{15}& {\overline{\theta }}_{16}& {\overline{\theta }}_{17}\\ \ast & {\overline{\theta }}_{22}& {\overline{\theta }}_{23}& {\overline{\theta }}_{24}& {\overline{\theta }}_{25}& {\overline{\theta }}_{26}& {\overline{\theta }}_{27}\\ \ast & \ast & {\overline{\theta }}_{33}& {\overline{\theta }}_{34}& 0& {\overline{\theta }}_{36}& 0\\ \ast & \ast & \ast & {\overline{\theta }}_{44}& 0& {\overline{\theta }}_{46}& 0\\ \ast & \ast & \ast & \ast & -2{\gamma }^{2}I& {\overline{\theta }}_{56}& 0\\ \ast & \ast & \ast & \ast & \ast & {\overline{\theta }}_{66}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -2I\end{array}\right],\\ {\eta }_3={\left[\begin{array}{cccccc}{P}_1{H}_{1i}& P_1{H}_{1j}& 0& 0& \cdots & 0\\ {\overline{\theta }}_{28}& {\overline{\theta }}_{29}& 0& 0& \cdots & 0\\ 0& 0& 0& 0& \cdots & 0\\ 0& 0& 0& 0& \cdots & 0\\ 0& 0& 0& 0& \cdots & 0\\ {\overline{\theta }}_{68}& {\overline{\theta }}_{69}& 0& 0& \cdots & 0\\ 0& 0& H_{3i}& H_{3j}& \cdots & 0\end{array}\right]}_{7\times 9},\\ {\eta }_4=\left[\begin{array}{ccccccc}{E}_{1i}& 0& E_{2i}& 0& E_{3i}& 0& 0\\ E_{1j}& 0& E_{2j}& 0& E_{3j}& 0& 0\\ E_{4i}& 0& 0& 0& 0& 0& 0\\ E_{4j}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\end{array}\right],\\ {\theta }_{11}=P_1{A}_{1i}+A_{1i}^T{P}_1+W+M_1^T+M_1+d{Z}_1,{\theta }_{12}=C_{1i}^T{M}_{2i}^T+W+M_1^T+M_1+d{Z}_1,\\ {\theta }_{13}=P_1{A}_{2i}-M_1^T+M_2+d{Z}_2,{\theta }_{14}=-M_1^T+M_2+d{Z}_2,{\theta }_{15}=P_1{B}_i,\\ {\theta }_{16}=A_{1i}^T+C_{1i}^T{B}_{fi}^T,{\theta }_{22}=M_{1i}+M_{1i}^T+W+M_1^T+M_1+d{Z}_1,\\ {\theta }_{23}=M_{2i}{C}_{2i}-M_1^T+M_2+d{Z}_2,{\theta }_{24}=M_{3i}-M_1^T+M_2+d{Z}_2,\\ {\theta }_{25}=M_{2i}{D}_i,{\theta }_{33}={\theta }_{34}=-W-M_2^T-M_2+d{Z}_3,\\ {\theta }_{36}=A_{2i}^T+C_{2i}^T{B}_{fi}^T,{\theta }_{44}=-W-M_2^T-M_2+d{Z}_3,\\ {\theta }_{55}=-{\gamma }^{2}I,{\theta }_{56}=B_i^T+D_i^T{B}_{fi}^T,{\theta }_{66}=-{(dU)}^{-1},\\ {\overline{\theta }}_{11}=P_1{A}_{1i}+A_{1i}^T{P}_1+P_1{A}_{1j}+A_{1j}^T{P}_1+2W+2(M_1^T+M_1+d{Z}_1),\\ {\overline{\theta }}_{12}=C_{1i}^T{M}_{2j}^T+C_{1j}^T{M}_{2i}^T+2W+2(M_1^T+M_1+d{Z}_1),\\ {\overline{\theta }}_{13}=P_1{A}_{2i}+P_1{A}_{2j}+2(-M_1^T+M_2+d{Z}_2),{\overline{\theta }}_{14}=2(-M_1^T+M_2+d{Z}_2),\\ {\overline{\theta }}_{15}=P_1{B}_i+P_1{B}_j,{\overline{\theta }}_{16}=A_{1i}^T+A_{1j}^T+C_{1i}^T{B}_{fj}^T+C_{1j}^T{B}_{fi}^T,\\ {\overline{\theta }}_{17}=L_i^T+L_j^T,{\overline{\theta }}_{22}=M_{1i}+M_{1j}+M_{1i}^T+M_{1j}^{\mathrm{T}}+2W+2(M_1^T+M_1+d{Z}_1),\\ {\overline{\theta }}_{23}=M_{2j}{C}_{2i}+M_{2i}{C}_{2j}+2(-M_1^{\mathrm{T}}+M_2+d{Z}_2),\\ {\overline{\theta }}_{24}=M_{3i}+M_{3j}+2(-M_1^T+M_2+d{Z}_2),\\ {\overline{\theta }}_{25}=M_{2j}{D}_i+M_{2i}{D}_j,{\overline{\theta }}_{26}=A_{fi}^T+A_{fj}^T,{\overline{\theta }}_{27}=-L_{fi}^T-L_{fj}^T,{\overline{\theta }}_{28}=M_{2j}{H}_{2i},\\ {\overline{\theta }}_{29}=M_{2i}{H}_{2j},{\overline{\theta }}_{33}={\overline{\theta }}_{34}=-2W+2(-M_2^T-M_2+d{Z}_3),\\ {\overline{\theta }}_{36}=A_{2i}^T+A_{2j}^T+C_{2i}^T{B}_{fj}^T+C_{2j}^T{B}_{fi}^T,\\ {\overline{\theta }}_{44}=-2W+2(-M_2^T-M_2+d{Z}_3),{\overline{\theta }}_{46}=C_{fi}^T+C_{fj}^T,\\ {\overline{\theta }}_{56}=B_i^{\mathrm{T}}+B_j^T+D_i^T{B}_{fj}^T+D_j^T{B}_{fi}^T,\\ {\overline{\theta }}_{66}=-2{(dU)}^{-1},{\overline{\theta }}_{68}=H_{1i}+B_{fj}{H}_{2i},{\overline{\theta }}_{69}=H_{1j}+B_{fi}{H}_{2j}.\end{array}$$

Proof From Theorem 1, the sufficient condition of solving robust $H_{\mathrm{\infty }}$ filtering problem is matrix inequality (9) holds. Then we have

$$\sum _{i=1}^r\sum _{j=1}^r{h}_i(\epsilon (t))h_j(\epsilon (t))G_{ij}=\sum _{i=1}^r{h_i}^2(\epsilon (t))G_{ii}+\sum _{i<j}^r{h}_i(\epsilon (t))h_j(\epsilon (t))(G_{ji}+G_{ij})<0,$$

where

$$\begin{array}{r}{G}_{ij}=\left[\begin{array}{ccccc}{\mathrm{\Xi }}_{11}& {\mathrm{\Xi }}_{12}& P^T{B}_{ij}& A_{ij}^T{H}^T& L_{ij}^T\\ \ast & {\mathrm{\Xi }}_{22}& 0& A_{d_{ij}}^T{H}^T& 0\\ \ast & \ast & -{\gamma }^{2}I& B_{ij}^T{H}^T& 0\\ \ast & \ast & \ast & -{(dU)}^{-1}& 0\\ \ast & \ast & \ast & \ast & -I\end{array}\right],\\ {\mathrm{\Xi }}_{11}=P^T{A}_{ij}+A_{ij}^{T}P+H^{T}WH+H^T(M_1^T+M_1+d{Z}_1)H,\\ {\mathrm{\Xi }}_{12}=P^T{A}_{d_{ij}}+H^T(-M_1^T+M_2+d{Z}_2)H,{\mathrm{\Xi }}_{22}=H^T(-W-M_2^T-M_2+d{Z}_3)H.\end{array}$$

However, $h_i(\epsilon (t))\ge 0$ and ${\sum }_{i=1}^r{\sum }_{j=1}^r{h}_i(\epsilon (t))h_j(\epsilon (t))=1$. So matrix inequality (9) holds as long as

$$\begin{array}{r}{G}_{ii}<0,i=1,2,\dots ,r,\\ G_{ji}+G_{ij}<0,i<j\le r.\end{array}$$

(14)

When $i=j$, from (14), adapt $G_{ii}$, we have $G_{ii}=G_{ii}^{\prime }+G_{ii}^{″}$, where

$$\begin{array}{r}{G}_{ii}^{\prime }=\left[\begin{array}{ccccccc}{\theta }_{11}& {\theta }_{12}& {\theta }_{13}& {\theta }_{14}& {\theta }_{15}& {\theta }_{16}& L_i^T\\ \ast & {\theta }_{22}& {\theta }_{23}& {\theta }_{24}& {\theta }_{25}& A_{fi}^T& -L_{fj}^T\\ \ast & \ast & {\theta }_{33}& {\theta }_{34}& 0& {\theta }_{36}& 0\\ \ast & \ast & \ast & {\theta }_{44}& 0& C_{fi}^T& 0\\ \ast & \ast & \ast & \ast & {\theta }_{55}& {\theta }_{56}& 0\\ \ast & \ast & \ast & \ast & \ast & {\theta }_{66}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -I\end{array}\right],\\ G_{ii}^{″}=\left[\begin{array}{ccccccc}{\theta }_{11}^{\prime }& {\theta }_{12}^{\prime }& {\theta }_{13}^{\prime }& 0& {\theta }_{15}^{\prime }& {\theta }_{16}^{\prime }& \mathrm{\Delta }{L}_i^{\mathrm{T}}\\ \ast & 0& {\theta }_{23}^{\prime }& 0& {\theta }_{25}^{\prime }& 0& 0\\ \ast & \ast & 0& 0& 0& {\theta }_{36}^{\prime }& 0\\ \ast & \ast & \ast & 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & 0& {\theta }_{56}^{\prime }& 0\\ \ast & \ast & \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & 0\end{array}\right],\\ {\theta }_{11}=P_1{A}_{1i}+A_{1i}^T{P}_1+W+M_1^T+M_1+d{Z}_1,{\theta }_{12}=C_{1i}^T{B}_{fi}^{T}P+W+M_1^T+M_1+d{Z}_1,\\ {\theta }_{13}=P_1{A}_{2i}-M_1^T+M_2+d{Z}_2,{\theta }_{14}=-M_1^T+M_2+d{Z}_2,{\theta }_{15}=P_1{B}_i,\\ {\theta }_{16}=A_{1i}^T+C_{1i}^T{B}_{fi}^T,{\theta }_{22}=P_2^T{A}_{fi}+A_{fi}^T{P}_2+W+M_1^T+M_1+d{Z}_1,\\ {\theta }_{23}=P_2^T{B}_{fi}{C}_{2i}-M_1^T+M_2+d{Z}_2,{\theta }_{24}=P_2^T{C}_{fi}-M_1^T+M_2+d{Z}_2,\\ {\theta }_{25}=P_2^T{B}_{fi}{D}_i,{\theta }_{33}={\theta }_{34}=-W-M_2^T-M_2+d{Z}_3,{\theta }_{36}=A_{2i}^T+C_{2i}^T{B}_{fi}^T,\\ {\theta }_{44}=-W-M_2^T-M_2+d{Z}_3,{\theta }_{55}=-{\gamma }^{2}I,\\ {\theta }_{56}=B_i^T+D_i^T{B}_{fi}^T,{\theta }_{66}=-{(dU)}^{-1},\\ {\theta }_{11}^{\prime }=P_1\mathrm{\Delta }{A}_{1i}+\mathrm{\Delta }{A}_{1i}^T{P}_1,{\theta }_{12}^{\prime }=\mathrm{\Delta }{C}_{1i}^T{B}_{fi}^T{P}_2,{\theta }_{13}^{\prime }=P_1\mathrm{\Delta }{A}_{2i},\\ {\theta }_{15}^{\prime }=P_1\mathrm{\Delta }{B}_i,{\theta }_{16}^{\prime }=\mathrm{\Delta }{A}_{1i}^T+\mathrm{\Delta }{C}_{1i}^T{B}_{fi}^T,\\ {\theta }_{23}^{\prime }=P_2^T{B}_{fi}\mathrm{\Delta }{C}_{2i},{\theta }_{25}^{\prime }=P_2^T{B}_{fi}\mathrm{\Delta }{D}_i,\\ {\theta }_{36}^{\prime }=\mathrm{\Delta }{A}_{2i}^T+\mathrm{\Delta }{C}_{2i}^T{B}_{fi}^T,{\theta }_{56}^{\prime }=\mathrm{\Delta }{B}_i^T+\mathrm{\Delta }{D}_i^T{B}_{fi}^T.\end{array}$$

Based on (2), we obtain that

$$G_{ii}^{″}={\eta }_1{F}_i{\eta }_2+{\eta }_2^T{F}_i^T{\eta }_1^T.$$

Through Lemma 1, we can get that $G_{ii}<0$ ($i=1,2,\dots ,r$) is equivalent to

$$G_{ii}^{\prime }+{\epsilon }_1^{-1}{\eta }_1{\eta }_1^T+{\epsilon }_1{\eta }_2^T{\eta }_2<0.$$

Via the Schur complements, we obtain that

$$\left[\begin{array}{ccc}{G^{\prime }}_{ii}& {\eta }_1& {\eta }_2^T\\ \ast & -{\epsilon }_{1}I& 0\\ \ast & \ast & -{\epsilon }_1^{-1}I\end{array}\right]<0.$$

Let $M_{1i}=P_2^T{A}_{fi}$, $M_{2i}=P_2^T{B}_{fi}$, $M_{3i}=P_2^T{C}_{fi}$, (12) is completed.

Let $G_{ji}+G_{ij}=G_{ij}^{\prime }+G_{ij}^{″}$, where

$$\begin{array}{r}{G}_{ij}^{\prime }=\left[\begin{array}{ccccccc}{\overline{\theta }}_{11}& {\overline{\theta }}_{12}& {\overline{\theta }}_{13}& {\overline{\theta }}_{14}& {\overline{\theta }}_{15}& {\overline{\theta }}_{16}& {\overline{\theta }}_{17}\\ \ast & {\overline{\theta }}_{22}& {\overline{\theta }}_{23}& {\overline{\theta }}_{24}& {\overline{\theta }}_{25}& {\overline{\theta }}_{26}& {\overline{\theta }}_{27}\\ \ast & \ast & {\overline{\theta }}_{33}& {\overline{\theta }}_{34}& 0& {\overline{\theta }}_{36}& 0\\ \ast & \ast & \ast & {\overline{\theta }}_{44}& 0& {\overline{\theta }}_{46}& 0\\ \ast & \ast & \ast & \ast & {\overline{\theta }}_{55}& {\overline{\theta }}_{56}& 0\\ \ast & \ast & \ast & \ast & \ast & {\overline{\theta }}_{66}& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & -2I\end{array}\right],\\ G_{ij}^{″}=\left[\begin{array}{ccccccc}{{\overline{\theta }}^{\prime }}_{11}& {{\overline{\theta }}^{\prime }}_{12}& {{\overline{\theta }}^{\prime }}_{13}& 0& {{\overline{\theta }}^{\prime }}_{15}& {{\overline{\theta }}^{\prime }}_{16}& {{\overline{\theta }}^{\prime }}_{17}\\ \ast & 0& {{\overline{\theta }}^{\prime }}_{23}& 0& {{\overline{\theta }}^{\prime }}_{25}& 0& 0\\ \ast & \ast & 0& 0& 0& {{\overline{\theta }}^{\prime }}_{36}& 0\\ \ast & \ast & \ast & 0& 0& 0& 0\\ \ast & \ast & \ast & \ast & 0& {{\overline{\theta }}^{\prime }}_{56}& 0\\ \ast & \ast & \ast & \ast & \ast & 0& 0\\ \ast & \ast & \ast & \ast & \ast & \ast & 0\end{array}\right],\\ {\overline{\theta }}_{11}=P_1{A}_{1i}+A_{1i}^T{P}_1+P_1{A}_{1j}+A_{1j}^T{P}_1+2W+2(M_1^T+M_1+d{Z}_1),\\ {\overline{\theta }}_{12}=C_{1i}^T{B}_{fj}^T{P}_2+C_{1j}^T{B}_{fi}^T{P}_2+2W+2(M_1^T+M_1+d{Z}_1),\\ {\overline{\theta }}_{13}=P_1{A}_{2i}+P_1{A}_{2j}+2(-M_1^T+M_2+d{Z}_2),{\overline{\theta }}_{14}=2(-M_1^T+M_2+d{Z}_2),\\ {\overline{\theta }}_{15}=P_1{B}_i+P_1{B}_j,{\overline{\theta }}_{16}=A_{1i}^T+A_{1j}^T+C_{1i}^T{B}_{fj}^T+C_{1j}^T{B}_{fi}^T,\\ {\overline{\theta }}_{17}=L_i^T+L_j^T,{\overline{\theta }}_{22}=P_2^T{A}_{fi}+P_2^T{A}_{fj}+A_{fi}^T{P}_2+A_{fj}^T{P}_2+2W+2(M_1^T+M_1+d{Z}_1),\\ {\overline{\theta }}_{23}=P_2^T{B}_{fj}{C}_{2i}+P_2^T{B}_{fi}{C}_{2j}+2(-M_1^T+M_2+d{Z}_2),\\ {\overline{\theta }}_{24}=P_2^T{C}_{fi}+P_2^T{C}_{fj}+2(-M_1^T+M_2+d{Z}_2),{\overline{\theta }}_{25}=P_2^T{B}_{fj}{D}_i+P_2^T{B}_{fi}{D}_j,\\ {\overline{\theta }}_{26}=A_{fi}^T+A_{fj}^T,{\overline{\theta }}_{27}=-L_{fi}^T-L_{fj}^T,{\overline{\theta }}_{33}={\overline{\theta }}_{34}=-2W+2(-M_2^T-M_2+d{Z}_3),\\ {\overline{\theta }}_{36}=A_{2i}^T+A_{2j}^T+C_{2i}^T{B}_{fj}^T+C_{2j}^T{B}_{fi}^T,{\overline{\theta }}_{44}=-2W+2(-M_2^T-M_2+d{Z}_3),\\ {\overline{\theta }}_{46}=C_{fi}^T+C_{fj}^T,{\overline{\theta }}_{55}=-2{\gamma }^{2}I,{\overline{\theta }}_{56}=B_i^T+B_j^T+D_i^T{B}_{fj}^T+D_j^T{B}_{fi}^T,\\ {\overline{\theta }}_{66}=-2{(dU)}^{-1},{{\overline{\theta }}^{\prime }}_{11}=P_1\mathrm{\Delta }{A}_{1i}+\mathrm{\Delta }{A}_{1i}^T{P}_1+P_1\mathrm{\Delta }{A}_{1j}+\mathrm{\Delta }{A}_{1j}^T{P}_1,\\ {{\overline{\theta }}^{\prime }}_{12}=\mathrm{\Delta }{C}_{1i}^T{B}_{fj}^T{P}_2+\mathrm{\Delta }{C}_{1j}^T{B}_{fi}^T{P}_2,{{\overline{\theta }}^{\prime }}_{13}=P_1\mathrm{\Delta }{A}_{2i}+P_1\mathrm{\Delta }{A}_{2j},{{\overline{\theta }}^{\prime }}_{15}=P_1\mathrm{\Delta }{B}_i+P_1\mathrm{\Delta }{B}_j,\\ {{\overline{\theta }}^{\prime }}_{16}=\mathrm{\Delta }{A}_{1i}^T+\mathrm{\Delta }{A}_{1j}^T+\mathrm{\Delta }{C}_{1i}^T{B}_{fj}^T+\mathrm{\Delta }{C}_{1j}^T{B}_{fi}^T,{{\overline{\theta }}^{\prime }}_{17}=\mathrm{\Delta }{L}_i^T+\mathrm{\Delta }{L}_j^T,\\ {{\overline{\theta }}^{\prime }}_{23}=P_2^T{B}_{fj}\mathrm{\Delta }{C}_{2i}+P_2^T{B}_{fi}\mathrm{\Delta }{C}_{2j},{{\overline{\theta }}^{\prime }}_{25}=P_2^T{B}_{fj}\mathrm{\Delta }{D}_i+P_2^T{B}_{fi}\mathrm{\Delta }{D}_j,\\ {{\overline{\theta }}^{\prime }}_{36}=\mathrm{\Delta }{A}_{2i}^T+\mathrm{\Delta }{A}_{2j}^T+\mathrm{\Delta }{C}_{2i}^T{B}_{fj}^T+\mathrm{\Delta }{C}_{2j}^T{B}_{fi}^T,{{\overline{\theta }}^{\prime }}_{56}=\mathrm{\Delta }{B}_i^T+\mathrm{\Delta }{B}_j^T+\mathrm{\Delta }{D}_i^T{B}_{fj}^T+\mathrm{\Delta }{D}_j^T{B}_{fi}^T.\end{array}$$

Based on (2), we obtain that

$$G_{ij}^{″}={\eta }_3\overline{F}{\eta }_4+{\eta }_4^T{\overline{F}}^T{\eta }_3^T,$$

where

$$\overline{F}=\left[\begin{array}{c}{F}_i\\ F_j\end{array}\right].$$

Through Lemma 1, we can get that $G_{ji}+G_{ij}<0$ ($i<j\le r$) is equivalent to

$$G_{ij}^{\prime }+{\epsilon }_2^{-1}{\eta }_3{\eta }_3^T+{\epsilon }_2{\eta }_4^T{\eta }_4<0.$$

Via the Schur complement, we obtain that

$$\left[\begin{array}{ccc}{G^{\prime }}_{ij}& {\eta }_3& {\eta }_4^T\\ \ast & -{\epsilon }_{2}I& 0\\ \ast & \ast & -{\epsilon }_2^{-1}I\end{array}\right]<0.$$

Let $M_{1i}=P_2^T{A}_{fi}$, $M_{2i}=P_2^T{B}_{fi}$, $M_{3i}=P_2^T{C}_{fi}$, (13) is completed.

The parameters of the robust $H_{\mathrm{\infty }}$ filter are

$$A_{fi}=P_2^{-T}{M}_{1i},B_{fi}=P_2^{-T}{M}_{2i},C_{fi}=P_2^{-T}{M}_{3i},L_{fi}.$$

 □

4 Numerical example

Example 1 Consider system (6) described by

$$\begin{array}{r}{A}_{11}=\left[\begin{array}{cc}-1.2& -0.8\\ 1& -2\end{array}\right],A_{21}=\left[\begin{array}{cc}-0.5& 0.2\\ 1& 0\end{array}\right],B_1=\left[\begin{array}{cc}0.3& 0.2\\ 0.4& -0.1\end{array}\right],\\ C_{11}=\left[\begin{array}{cc}0.1& 0.3\\ 0.1& 0.2\end{array}\right],C_{21}=\left[\begin{array}{cc}-0.2& 0.4\\ 0.1& 0.2\end{array}\right],D_1=\left[\begin{array}{cc}0.1& 0.2\\ 0.1& 0.1\end{array}\right],\\ L_1=\left[\begin{array}{cc}1.5& 0.1\\ 0.1& -0.2\end{array}\right],A_{12}=\left[\begin{array}{cc}-1.6& 0.2\\ 0.7& -1\end{array}\right],A_{22}=\left[\begin{array}{cc}0.2& 0.1\\ 0.1& -0.3\end{array}\right],\\ B_2=\left[\begin{array}{cc}0.7& 0.1\\ 0.2& -0.1\end{array}\right],C_{12}=\left[\begin{array}{cc}0.1& 0.4\\ 0.3& 0.2\end{array}\right],C_{22}=\left[\begin{array}{cc}-0.3& 0.4\\ 0.1& 0.2\end{array}\right],\\ D_2=\left[\begin{array}{cc}0.2& 0.3\\ 0.1& 0.4\end{array}\right],L_2=\left[\begin{array}{cc}-0.1& 0.2\\ 0.3& 0.1\end{array}\right],H_{11}=\left[\begin{array}{c}0.1\\ -0.2\end{array}\right],H_{12}=\left[\begin{array}{c}-0.2\\ 0.1\end{array}\right],\\ H_{21}=\left[\begin{array}{c}-0.2\\ 0.2\end{array}\right],H_{22}=\left[\begin{array}{c}0.2\\ -0.1\end{array}\right],H_{31}=\left[\begin{array}{c}-0.2\\ 0.1\end{array}\right],H_{32}=\left[\begin{array}{c}-0.2\\ 0.3\end{array}\right],\\ E_{11}=[\begin{array}{cc}-0.2& 0.1\end{array}],E_{21}=[\begin{array}{cc}0.2& -0.2\end{array}],E_{31}=[\begin{array}{cc}-0.2& 0.1\end{array}],\\ E_{41}=[\begin{array}{cc}-0.2& 0.2\end{array}],E_{12}=[\begin{array}{cc}-0.2& 0.1\end{array}],E_{22}=[\begin{array}{cc}-0.1& -0.2\end{array}],\\ E_{32}=[\begin{array}{cc}-0.3& 0.2\end{array}],E_{42}=[\begin{array}{cc}-0.2& 0.3\end{array}].\end{array}$$

The normalized membership functions of the first subsystem are

$$h_1(x(t))=\frac{1+cos(x(t))}{2},h_2(x(t))=\frac{1-cos(x(t))}{2}.$$

Given $d=8$, $\gamma =0.25$. We can obtain the filter parameters as follows:

$$\begin{array}{r}{A}_{f1}=\left[\begin{array}{cc}0.9585& 0.3257\\ 0.3600& 0.3510\end{array}\right],B_{f1}=\left[\begin{array}{cc}90.6474& 170.4250\\ -199.3139& -298.7841\end{array}\right],\\ C_{f1}=\left[\begin{array}{cc}-0.6731& -0.5042\\ -0.8152& -0.5380\end{array}\right],L_{f1}=\left[\begin{array}{cc}0.7716& -0.0043\\ 0.4243& -0.0258\end{array}\right],\\ A_{f2}=\left[\begin{array}{cc}-23.4914& -18.7283\\ -30.3710& -25.2203\end{array}\right],B_{f2}=\left[\begin{array}{cc}-74.1374& -86.4398\\ 616.2515& 655.3905\end{array}\right],\\ C_{f2}=\left[\begin{array}{cc}4.4241& 4.4659\\ 4.8130& 6.1622\end{array}\right],L_{f2}=\left[\begin{array}{cc}-0.7979& -0.0079\\ -0.4663& 0.0086\end{array}\right].\end{array}$$

Example 2 In order to show the advantage of the proposed method, we consider the T-S fuzzy time-delay system as the system show in simulation example in [14] with two rules.

Plant Rule 1: IF $x_1(t)$ is $u_1$ (e.g., small) THEN

$$\begin{array}{r}\dot{x}(t)=[A_1+\mathrm{\Delta }{A}_1(t)]x(t)+[A_{d1}+\mathrm{\Delta }{A}_{d1}(t)]x(t-1.5)+D_1\omega (t),\\ y(t)=[C_1+\mathrm{\Delta }{C}_1(t)]x(t)+[C_{d1}+\mathrm{\Delta }{C}_{d1}(t)]x(t-1.5)+E_1\omega (t),\\ z(t)=L_{1}x(t),\end{array}$$

and

Plant Rule 2: IF $x_1(t)$ is $u_2$ (e.g., big) THEN

$$\begin{array}{r}\dot{x}(t)=[A_2+\mathrm{\Delta }{A}_2(t)]x(t)+[A_{d2}+\mathrm{\Delta }{A}_{d2}(t)]x(t-1.5)+D_2\omega (t),\\ y(t)=[C_2+\mathrm{\Delta }{C}_2(t)]x(t)+[C_{d2}+\mathrm{\Delta }{C}_{d2}(t)]x(t-1.5)+E_2\omega (t),\\ z(t)=L_{2}x(t),\end{array}$$

where

$$\begin{array}{r}{A}_1=\left[\begin{array}{cc}-1.2& -0.8\\ 1& -2\end{array}\right],A_2=\left[\begin{array}{cc}-1.6& 0.2\\ 0.7& -1\end{array}\right],A_{d1}=\left[\begin{array}{cc}-0.5& 0.2\\ 1& 0\end{array}\right],\\ A_{d2}=\left[\begin{array}{cc}0.6& 0.2\\ 0.5& -0.8\end{array}\right],D_1=\left[\begin{array}{c}1\\ -0.2\end{array}\right],D_2=\left[\begin{array}{c}0.3\\ 0.1\end{array}\right],\\ C_1=[10],C_2=[0.5-0.6],C_{d1}=[-0.80.6],\\ C_{d2}=[-0.21],E_1=0.3,E_2=-0.6,\\ L_1=[1-0.5],L_2=[-0.20.3],\\ M_{11}=\left[\begin{array}{c}0\\ -0.5\end{array}\right],M_{21}=0.8,N_{11}=[00.3],N_{21}=[0.20],\\ M_{12}=\left[\begin{array}{c}0\\ 0.3\end{array}\right],M_{22}=0.6,N_{12}=[0.50],N_{22}=[0-0.2].\end{array}$$

The normalized membership functions of the first subsystem are

$$h_1(x_1(t))=\{\begin{array}{ll}1,& x_1<-1,\\ \frac{1}{2}-\frac{1}{2}{x}_1,& |x_1|\le 1,\\ 0,& x_1>1,\end{array}\text{and}{h}_2(x_1(t))=\{\begin{array}{ll}0,& x_1<-1,\\ \frac{1}{2}+\frac{1}{2}{x}_1,& |x_1|\le 1,\\ 1,& x_1>1.\end{array}$$

Filter Rule 1: IF $x_1(t)$ is $u_1$ THEN

$$\begin{array}{r}\dot{\hat{x}}(t)=A_{f1}\hat{x}(t)+B_{f1}y(t),\\ \hat{z}(t)=L_{f1}\hat{x}(t),\end{array}$$

and

Filter Rule 2: IF $x_1(t)$ is $u_2$ THEN

$$\begin{array}{r}\dot{\hat{x}}(t)=A_{f2}\hat{x}(t)+B_{f2}y(t),\\ \hat{z}(t)=L_{f2}\hat{x}(t).\end{array}$$

Through Theorem 2 of this article, we can obtain ${\gamma }_{min}=0.0271$, it is less than the minimal level 0.4721 in simulation example in [14], this clearly shows the superiority of the results derived in this paper to those obtained from [14], and the filter parameters as follows:

$$\begin{array}{r}{A}_{f1}=\left[\begin{array}{cc}-2.9870& -0.7508\\ 1.8256& -1.5045\end{array}\right],A_{f2}=\left[\begin{array}{cc}-1.1822& 0.3462\\ 1.1459& -1.8463\end{array}\right],\\ B_{f1}=\left[\begin{array}{c}-0.4195\\ 1.3158\end{array}\right],B_{f2}=\left[\begin{array}{c}-0.2797\\ 1.1891\end{array}\right],\\ L_{f1}=[\begin{array}{cc}-0.6002& 0.3459\end{array}],L_{f2}=[\begin{array}{cc}-0.2737& -0.3206\end{array}].\end{array}$$

Using the filter in (5), we can get ${\gamma }_{min}=0.0190$, it is less than the minimal level 0.0271 with $C_{fi}=0$ in this paper, and the filter parameters as follows:

$$\begin{array}{r}{A}_{f1}=\left[\begin{array}{cc}-1.0003& -0.3517\\ 1.8344& -2.5099\end{array}\right],A_{f2}=\left[\begin{array}{cc}-1.1829& 0.3544\\ 1.6542& -3.8510\end{array}\right],\\ B_{f1}=\left[\begin{array}{c}-0.4196\\ 1.3157\end{array}\right],B_{f2}=\left[\begin{array}{c}-0.2799\\ 1.1890\end{array}\right],\\ C_{f1}=\left[\begin{array}{cc}-1.6825& -2.6491\\ -0.0680& -0.0098\end{array}\right],C_{f2}=\left[\begin{array}{cc}5.0149& 2.5331\\ -0.0682& -0.0121\end{array}\right],\\ L_{f1}=[-0.60300.3481],L_{f2}=[-0.2755-0.3215].\end{array}$$

Therefore, we can see the advantage of the proposed method in this paper.

Example 3 In order to show the advantage of the proposed method, we consider the time-delay T-S fuzzy system as the system show in example 2 in [5] with two rules.

$R_1$: if $x_1(t)$ is about 0, then

$$\begin{array}{r}\dot{x}(t)=(A_1+\mathrm{\Delta }{A}_1)x(t)+(A_{d1}+\mathrm{\Delta }{A}_{d1})x(t-d)+(B_1+\mathrm{\Delta }{B}_1)w(t),\\ y(t)=(C_1+\mathrm{\Delta }{C}_1)x(t)+(C_{d1}+\mathrm{\Delta }{C}_{d1})x(t-d)+(D_1+\mathrm{\Delta }{D}_1)w(t),\\ z(t)=(L_1+\mathrm{\Delta }{L}_1)x(t),\end{array}$$

$R_2$: if $x_1(t)$ is about $\pm \frac{\pi }{2}$, then

$$\begin{array}{r}\dot{x}(t)=(A_2+\mathrm{\Delta }{A}_2)x(t)+(A_{d2}+\mathrm{\Delta }{A}_{d2})x(t-d)+(B_2+\mathrm{\Delta }{B}_2)w(t),\\ y(t)=(C_2+\mathrm{\Delta }{C}_2)x(t)+(C_{d2}+\mathrm{\Delta }{C}_{d2})x(t-d)+(D_2+\mathrm{\Delta }{D}_2)w(t),\\ z(t)=(L_2+\mathrm{\Delta }{L}_2)x(t),\end{array}$$

where

$$\begin{array}{r}{A}_1=\left[\begin{array}{cc}1& 0.1\\ -0.5& 1\end{array}\right],A_2=\left[\begin{array}{cc}1& 0.5\\ -0.1& 1\end{array}\right],A_{d1}=\left[\begin{array}{cc}0& 0.2\\ 0& 0.1\end{array}\right],\\ A_{d2}=\left[\begin{array}{cc}0& 0.3\\ 0& 0.6\end{array}\right],B_1=\left[\begin{array}{c}0\\ 1\end{array}\right],B_2=\left[\begin{array}{c}1\\ 0\end{array}\right],C_1=[10],\\ C_2=[0.5-0.6],C_{d1}=[-0.20.6],C_{d2}=[-0.20.6],\\ L_1=[1-0.5],L_2=[-0.20.3],D_1=0.3,D_2=-0.6,\\ H_{11}=\left[\begin{array}{c}0\\ -0.5\end{array}\right],H_{12}=\left[\begin{array}{c}0\\ 0.3\end{array}\right],H_{21}=0.8,H_{22}=0.6,\\ H_{L1}=0.2,H_{L2}=0.1,E_{11}=[00.3],\\ E_{12}=[0.50],E_{21}=0.1,E_{22}=0.2,E_{31}=[0.10],\\ E_{32}=[0.20],E_{d1}=[0.20],E_{d2}=[0-0.2].\end{array}$$

The normalized membership functions of the first subsystem are

$$\begin{array}{r}{h}_1(x_1(t))=(1-\frac{1}{1+exp(-7(x_1(t)-\pi /4))})\times \left(\frac{1}{1+exp(-7(x_1(t)-\pi /4))}\right),\\ h_2(x_1(t))=h_1(x_1(t)).\end{array}$$

Filter $R_1$: if $x_1(t)$ is ${\mu }_1$ then

$$\begin{array}{r}\dot{\hat{x}}(t)=A_{e1}\hat{x}(t)+B_{e1}y(t),\\ \hat{Z}(t)=L_{e1}\hat{x}(t).\end{array}$$

Filter $R_2$: if $x_1(t)$ is ${\mu }_2$ then

$$\begin{array}{r}\dot{\hat{x}}(t)=A_{e2}\hat{x}(t)+B_{e2}y(t),\\ \hat{Z}(t)=L_{e2}\hat{x}(t).\end{array}$$

Given $d=0.5$, through Theorem 2 of this article, we can obtain minimum $\gamma =0.0178$, and it is less than the minimal level 0.2453 in example 2 in [5], this clearly shows the superiority of the results derived in this paper to those obtained from [5], and the filter parameters as follows:

$$\begin{array}{r}{A}_{e1}=\left[\begin{array}{cc}-0.9836& 0.3306\\ -1.7660& -3.0998\end{array}\right],A_{e2}=\left[\begin{array}{cc}-0.8186& 0.4682\\ -1.7634& -4.4274\end{array}\right],B_{e1}=\left[\begin{array}{c}0.2093\\ 0.1007\end{array}\right],\\ B_{e2}=\left[\begin{array}{c}-0.0132\\ 0.4979\end{array}\right],L_{e1}=[3.46264.3515],L_{e2}=[-0.1314-1.0629].\end{array}$$

Using the filter in (5), we can get ${\gamma }_{min}=0.0144$, it is less than the minimal level 0.0178 with $C_{fi}=0$ in this paper, therefore, using of the new filter, the obtained criterion is less conservative than those without the new filter, and the filter parameters are as follows:

$$\begin{array}{r}{A}_{f1}=\left[\begin{array}{cc}-0.9852& 0.3396\\ -1.7748& -3.1052\end{array}\right],A_{f2}=\left[\begin{array}{cc}-0.8193& 0.4764\\ -1.7667& -4.4321\end{array}\right],\\ B_{f1}=\left[\begin{array}{c}0.2099\\ 0.1027\end{array}\right],B_{f2}=\left[\begin{array}{c}-0.0138\\ 0.4980\end{array}\right],\\ C_{f1}=\left[\begin{array}{cc}-1.5825& -2.6501\\ -0.0599& -0.0089\end{array}\right],C_{f2}=\left[\begin{array}{cc}4.9954& 2.6001\\ -0.0710& -0.0098\end{array}\right],\\ L_{f1}=[3.46544.3537],L_{f2}=[-0.1332-1.0648].\end{array}$$

Therefore, we can see the advantage of the proposed method in this paper.

5 Conclusion

In the paper, the problem of robust $H_{\mathrm{\infty }}$ filter with the exponential stability is investigated. Some criterion is proposed to ensure the considered system to be exponentially stable, and it satisfies a prescribed $H_{\mathrm{\infty }}$ performance in terms of LMIs. Thanks to the new filter, the obtained criterion is less conservative than the existing ones. Numerical examples show the effectiveness of the proposed methods.