**New Results on Robust** H<sup>∞</sup> **Filter for Uncertain Fuzzy Descriptor Systems**

Wudhichai Assawinchaichote\*

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/47142

## **1. Introduction**

464 Fuzzy Controllers – Recent Advances in Theory and Applications

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The problem of filter design for descriptor systems system has been intensively studied by a number of researchers for the past three decades; see Ref.[1]-[6]. This is due not only to theoretical interest but also to the relevance of this topic in control engineering applications. Descriptor systems or so called singularly perturbed systems are dynamical systems with multiple time-scales. Descriptor systems often occur naturally due to the presence of small "parasitic" parameter, typically small time constants, masses, etc.

The main purpose of the singular perturbation approach to analysis and design is the alleviation of high dimensionality and ill-conditioning resulting from the interaction of slow and fast dynamics modes. The separation of states into slow and fast ones is a nontrivial modelling task demanding insight and ingenuity on the part of the analyst. In state space, such systems are commonly modelled using the mathematical framework of singular perturbations, with a small parameter, say *ε*, determining the degree of separation between the "slow" and "fast" modes of the system.

In the last few years, many researchers have studied the H<sup>∞</sup> filter design for a general class of linear descriptor systems. In Ref.[3], the authors have investigated the decomposition solution of H<sup>∞</sup> filter gain for singularly perturbed systems. The reduced-order H<sup>∞</sup> optimal filtering for system with slow and fast modes has been considered in Ref.[4]. Although many researchers have studied linear descriptor systems for many years, the H<sup>∞</sup> filtering design for nonlinear descriptor systems remains as an open research area. This is because, in general, nonlinear singularly perturbed systems can not be easily separated into slow and fast subsystems.

Fuzzy system theory enables us to utilize qualitative, linguistic information about a highly complex nonlinear system to construct a mathematical model for it. Recent studies show

<sup>\*</sup>W. Assawinchaichote is with the Department of Electronic and Telecommnunication Engineering, King Mongkut's University of Technology Thonburi, 126 Prachautits Rd., Bangkok 10140, Thailand.

©2012 Assawinchaichote, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 466 Fuzzy Controllers – Recent Advances in Theory and Applications New Results on Robust <sup>H</sup><sup>∞</sup> Filter for Uncertain Fuzzy Descriptor Systems <sup>3</sup>

that a fuzzy linear model can be used to approximate global behaviors of a highly complex nonlinear system; see for example, Ref.[7]-[19]. In this fuzzy linear model, local dynamics in different state space regions are represented by local linear systems. The overall model of the system is obtained by "blending" these linear models through nonlinear fuzzy membership functions. Unlike conventional modelling where a single model is used to describe the global behaviour of a system, the fuzzy modelling is essentially a multi-model approach in which simple sub-models (linear models) are combined to describe the global behaviour of the system.

**Assumption 1.**

*where Hji*

*or equal to γ if*

*C*ˆ

Δ*Ai* = *F*(*x*(*t*),*t*)*H*1*<sup>i</sup>*

Δ*C*1*<sup>i</sup>* = *F*(*x*(*t*),*t*)*H*4*<sup>i</sup>*

*for any known positive constant ρ.*

Next, let us recall the following definition.

 *Tf* 0

**3. Robust** H<sup>∞</sup> **fuzzy filter design**

*Eεx*˙(*t*) = ∑*<sup>r</sup>*

*z*(*t*) = ∑*<sup>r</sup>*

*y*(*t*) = ∑*<sup>r</sup>*

*E<sup>ε</sup>* ˙

*<sup>i</sup>*=<sup>1</sup> *μ<sup>i</sup>* 

*<sup>i</sup>*=<sup>1</sup> *μ<sup>i</sup>* 

*<sup>i</sup>*=<sup>1</sup> *μ<sup>i</sup>* 

*x*ˆ(*t*) = ∑*<sup>r</sup>*

*z*ˆ(*t*) = ∑*<sup>r</sup>*

(1) with *u*(*t*) = 0 as follows:

filter into two cases as follows.

*z*(*t*) − *z*ˆ(*t*)

*<sup>T</sup>*

*uncertainties. Furthermore, the following inequality holds:*

, Δ*B*1*<sup>i</sup>* = *F*(*x*(*t*),*t*)*H*2*<sup>i</sup>*

, Δ*C*2*<sup>i</sup>* = *F*(*x*(*t*), *t*)*H*5*<sup>i</sup>*

*and* Δ*D*21*<sup>i</sup>* = *F*(*x*(*t*),*t*)*H*7*<sup>i</sup>*

**Definition 1.** *Suppose γ is a given positive number. A system (1) is said to have an* L2*-gain less than*

*with x*(0) = 0*, where* (*z*(*t*) − *z*ˆ(*t*)) *is the estimated error output, for all Tf* ≥ 0 *and w*(*t*) ∈ L2[0, *Tf* ]*.*

Without loss of generality, in this section, we assume that *u*(*t*) = 0. Let us recall the system

[*C*1*<sup>i</sup>* + Δ*C*1*<sup>i</sup>*

[*C*2*<sup>i</sup>* + Δ*C*2*<sup>i</sup>*

*<sup>j</sup>*=<sup>1</sup> *μ*ˆ*iμ*ˆ*<sup>j</sup>* 

where *<sup>x</sup>*ˆ(*t*) ∈ �*<sup>n</sup>* is the filter's state vector, *<sup>z</sup>*<sup>ˆ</sup> ∈ �*<sup>s</sup>* is the estimate of *<sup>z</sup>*(*t*), *<sup>A</sup>*ˆ*ij*(*ε*), *<sup>B</sup>*<sup>ˆ</sup>

*<sup>i</sup>* are parameters of the filter which are to be determined, and *μ*ˆ*<sup>i</sup>* denotes the normalized

the inequality (3) holds. Clearly, in real control problems, all of the premise variables are not necessarily measurable. In this section, we then consider the designing of the robust H<sup>∞</sup> fuzzy

We are now aiming to design a full order dynamic H<sup>∞</sup> fuzzy filter of the form

*<sup>i</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>r</sup>*

*<sup>i</sup>*=<sup>1</sup> *<sup>μ</sup>*ˆ*iC*<sup>ˆ</sup>

time-varying fuzzy weighting functions for each rule (i.e., *<sup>μ</sup>*ˆ*<sup>i</sup>* <sup>≥</sup> 0 and <sup>∑</sup>*<sup>r</sup>*

[*Ai* + Δ*Ai*]*x*(*t*)+[*B*1*<sup>i</sup>* + Δ*B*1*<sup>i</sup>*

]*x*(*t*) 

]*x*(*t*)+[*D*21*<sup>i</sup>* + Δ*D*21*<sup>i</sup>*

*A*ˆ*ij*(*ε*)*x*ˆ(*t*) + *B*ˆ

 *dt* <sup>≤</sup> *<sup>γ</sup>*<sup>2</sup>

*z*(*t*) − *z*ˆ(*t*)

*, j* = 1, 2, ··· , 7 *are known matrix functions which characterize the structure of the*

, Δ*B*2*<sup>i</sup>* = *F*(*x*(*t*),*t*)*H*3*<sup>i</sup>*

New Results on Robust ∞ Filter for Uncertain Fuzzy Descriptor Systems 467

, Δ*D*12*<sup>i</sup>* = *F*(*x*(*t*),*t*)*H*6*<sup>i</sup>*

�*F*(*x*(*t*),*t*)� ≤ *ρ* (2)

 *Tf* 0

*wT*(*t*)*w*(*t*)*dt*

]*w*(*t*) 

*iy*(*t*) 

*ix*ˆ(*t*) (5)

]*w*(*t*) . (3)

(4)

*<sup>i</sup>* and

*<sup>i</sup>*=<sup>1</sup> *μ*ˆ*<sup>i</sup>* = 1), such that

,

What we intend to do in this paper is to design a robust H<sup>∞</sup> filter for a class of nonlinear descriptor systems with nonlinear on both fast and slow variables. First, we approximate this class of nonlinear descriptor systems by a Takagi-Sugeno fuzzy model. Then based on an LMI approach, we develop an H<sup>∞</sup> filter such that the L2-gain from an exogenous input to an estimate error is less or equal to a prescribed value. To alleviate the ill-conditioning resulting from the interaction of slow and fast dynamic modes, solutions to the problem are given in terms of linear matrix inequalities which are independent of the singular perturbation *ε*, when *ε* is sufficiently small. The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard, but also to nonstandard nonlinear descriptor systems.

This paper is organized as follows. In Section 2, system descriptions and definitions are presented. In Section 3, based on an LMI approach, we develop a technique for designing a robust H<sup>∞</sup> filter for the system described in section 2. The validity of this approach is demonstrated by an example from a literature in Section 4. Finally in Section 5, conclusions are given.

## **2. System descriptions**

In this section, we generalize the TS fuzzy system to represent a TS fuzzy descriptor system with parametric uncertainties. As in Ref.[19], we examine a TS fuzzy descriptor system with parametric uncertainties as follows:

$$\begin{aligned} \mathbf{E}\_{\varepsilon}\dot{\mathbf{x}}(t) &= \sum\_{i=1}^{r} \mu\_{i}(\boldsymbol{\nu}(t)) \left[ [A\_{i} + \Delta A\_{i}] \mathbf{x}(t) + [B\_{1\_{i}} + \Delta B\_{1\_{i}}] \mathbf{w}(t) + [B\_{2\_{i}} + \Delta B\_{2\_{i}}] \boldsymbol{u}(t) \right] \\ \mathbf{z}(t) &= \sum\_{i=1}^{r} \mu\_{i}(\boldsymbol{\nu}(t)) \left[ [\mathbf{C}\_{1\_{i}} + \Delta \mathbf{C}\_{1\_{i}}] \mathbf{x}(t) + [\mathbf{D}\_{12\_{i}} + \Delta \mathbf{D}\_{12\_{i}}] \boldsymbol{u}(t) \right] \\ \mathbf{y}(t) &= \sum\_{i=1}^{r} \mu\_{i}(\boldsymbol{\nu}(t)) \left[ [\mathbf{C}\_{2\_{i}} + \Delta \mathbf{C}\_{2\_{i}}] \mathbf{x}(t) + [\mathbf{D}\_{21\_{i}} + \Delta \mathbf{D}\_{21\_{i}}] \boldsymbol{w}(t) \right] \end{aligned} \tag{1}$$

where *E<sup>ε</sup>* = *I* 0 0 *εI* , *ε* > 0 is the singular perturbation parameter, *ν*(*t*)=[*ν*1(*t*) ··· *νϑ*(*t*)] is the premise variable vector that may depend on states in many cases, *μi*(*ν*(*t*)) denotes the normalized time-varying fuzzy weighting functions for each rule (i.e., *μi*(*ν*(*t*)) ≥ 0 and ∑*r <sup>i</sup>*=<sup>1</sup> *<sup>μ</sup>i*(*ν*(*t*)) = 1), *<sup>ϑ</sup>* is the number of fuzzy sets, *<sup>x</sup>*(*t*) ∈ �*<sup>n</sup>* is the state vector, *<sup>u</sup>*(*t*) ∈ �*<sup>m</sup>* is the input, *<sup>w</sup>*(*t*) ∈ �*<sup>p</sup>* is the disturbance which belongs to <sup>L</sup>2[0, <sup>∞</sup>), *<sup>y</sup>*(*t*) ∈ �� is the measurement and *<sup>z</sup>*(*t*) ∈ �*<sup>s</sup>* is the controlled output, the matrices *Ai*, *<sup>B</sup>*1*<sup>i</sup>* , *B*2*<sup>i</sup>* , *C*1*<sup>i</sup>* , *C*2*<sup>i</sup>* , *D*12*<sup>i</sup>* and *D*21*<sup>i</sup>* are of appropriate dimensions, and the matrices Δ*Ai*, Δ*B*1*<sup>i</sup>* , Δ*B*2*<sup>i</sup>* , Δ*C*1*<sup>i</sup>* , Δ*C*2*<sup>i</sup>* , Δ*D*12*<sup>i</sup>* and Δ*D*21*<sup>i</sup>* represent the uncertainties in the system and satisfy the following assumption.

**Assumption 1.**

2 Will-be-set-by-IN-TECH

that a fuzzy linear model can be used to approximate global behaviors of a highly complex nonlinear system; see for example, Ref.[7]-[19]. In this fuzzy linear model, local dynamics in different state space regions are represented by local linear systems. The overall model of the system is obtained by "blending" these linear models through nonlinear fuzzy membership functions. Unlike conventional modelling where a single model is used to describe the global behaviour of a system, the fuzzy modelling is essentially a multi-model approach in which simple sub-models (linear models) are combined to describe the global behaviour of the

What we intend to do in this paper is to design a robust H<sup>∞</sup> filter for a class of nonlinear descriptor systems with nonlinear on both fast and slow variables. First, we approximate this class of nonlinear descriptor systems by a Takagi-Sugeno fuzzy model. Then based on an LMI approach, we develop an H<sup>∞</sup> filter such that the L2-gain from an exogenous input to an estimate error is less or equal to a prescribed value. To alleviate the ill-conditioning resulting from the interaction of slow and fast dynamic modes, solutions to the problem are given in terms of linear matrix inequalities which are independent of the singular perturbation *ε*, when *ε* is sufficiently small. The proposed approach does not involve the separation of states into slow and fast ones and it can be applied not only to standard, but also to nonstandard

This paper is organized as follows. In Section 2, system descriptions and definitions are presented. In Section 3, based on an LMI approach, we develop a technique for designing a robust H<sup>∞</sup> filter for the system described in section 2. The validity of this approach is demonstrated by an example from a literature in Section 4. Finally in Section 5, conclusions

In this section, we generalize the TS fuzzy system to represent a TS fuzzy descriptor system with parametric uncertainties. As in Ref.[19], we examine a TS fuzzy descriptor system with

[*Ai* + Δ*Ai*]*x*(*t*)+[*B*1*<sup>i</sup>* + Δ*B*1*<sup>i</sup>*

is the premise variable vector that may depend on states in many cases, *μi*(*ν*(*t*)) denotes the normalized time-varying fuzzy weighting functions for each rule (i.e., *μi*(*ν*(*t*)) ≥ 0 and

*<sup>i</sup>*=<sup>1</sup> *<sup>μ</sup>i*(*ν*(*t*)) = 1), *<sup>ϑ</sup>* is the number of fuzzy sets, *<sup>x</sup>*(*t*) ∈ �*<sup>n</sup>* is the state vector, *<sup>u</sup>*(*t*) ∈ �*<sup>m</sup>* is the input, *<sup>w</sup>*(*t*) ∈ �*<sup>p</sup>* is the disturbance which belongs to <sup>L</sup>2[0, <sup>∞</sup>), *<sup>y</sup>*(*t*) ∈ �� is the measurement

]*x*(*t*)+[*D*12*<sup>i</sup>* + Δ*D*12*<sup>i</sup>*

]*x*(*t*)+[*D*21*<sup>i</sup>* + Δ*D*21*<sup>i</sup>*

, *ε* > 0 is the singular perturbation parameter, *ν*(*t*)=[*ν*1(*t*) ··· *νϑ*(*t*)]

]*w*(*t*)+[*B*2*<sup>i</sup>* + Δ*B*2*<sup>i</sup>*

]*u*(*t*) 

]*w*(*t*) 

, *B*2*<sup>i</sup>* , *C*1*<sup>i</sup>* , *C*2*<sup>i</sup>*

, Δ*C*1*<sup>i</sup>*

, Δ*C*2*<sup>i</sup>*

, Δ*B*2*<sup>i</sup>*

]*u*(*t*) 

, *D*12*<sup>i</sup>* and *D*21*<sup>i</sup>* are

, Δ*D*12*<sup>i</sup>* and Δ*D*21*<sup>i</sup>*

(1)

system.

are given.

where *E<sup>ε</sup>* =

∑*r*

nonlinear descriptor systems.

**2. System descriptions**

*Eεx*˙(*t*) = ∑*<sup>r</sup>*

*z*(*t*) = ∑*<sup>r</sup>*

*y*(*t*) = ∑*<sup>r</sup>*

 *I* 0 0 *εI* 

parametric uncertainties as follows:

*<sup>i</sup>*=<sup>1</sup> *μi*(*ν*(*t*))

*<sup>i</sup>*=<sup>1</sup> *μi*(*ν*(*t*))

*<sup>i</sup>*=<sup>1</sup> *μi*(*ν*(*t*))

and *<sup>z</sup>*(*t*) ∈ �*<sup>s</sup>* is the controlled output, the matrices *Ai*, *<sup>B</sup>*1*<sup>i</sup>*

of appropriate dimensions, and the matrices Δ*Ai*, Δ*B*1*<sup>i</sup>*

[*C*1*<sup>i</sup>* + Δ*C*1*<sup>i</sup>*

[*C*2*<sup>i</sup>* + Δ*C*2*<sup>i</sup>*

represent the uncertainties in the system and satisfy the following assumption.

$$\Delta A\_{i} = F(\mathbf{x}(t), t) H\_{1\_{i'}} \quad \Delta B\_{1\_{i}} = F(\mathbf{x}(t), t) H\_{2\_{i'}} \quad \Delta B\_{2\_{i}} = F(\mathbf{x}(t), t) H\_{3\_{i'}}$$

$$\Delta \mathbf{C}\_{1\_{i}} = F(\mathbf{x}(t), t) H\_{4\_{i'}} \quad \Delta \mathbf{C}\_{2\_{i}} = F(\mathbf{x}(t), t) H\_{5\_{i'}} \quad \Delta D\_{12\_{i}} = F(\mathbf{x}(t), t) H\_{6\_{i}}$$

$$\text{and } \Delta D\_{21\_{i}} = F(\mathbf{x}(t), t) H\_{7\_{i}}$$

*where Hji , j* = 1, 2, ··· , 7 *are known matrix functions which characterize the structure of the uncertainties. Furthermore, the following inequality holds:*

$$\|\|F(\mathbf{x}(t),t)\|\| \le \rho \tag{2}$$

*for any known positive constant ρ.*

Next, let us recall the following definition.

**Definition 1.** *Suppose γ is a given positive number. A system (1) is said to have an* L2*-gain less than or equal to γ if*

$$\int\_{0}^{T\_f} \left( z(t) - \hat{z}(t) \right)^T \left( z(t) - \hat{z}(t) \right) dt \le \gamma^2 \left[ \int\_{0}^{T\_f} w^T(t) w(t) dt \right] \tag{3}$$

*with x*(0) = 0*, where* (*z*(*t*) − *z*ˆ(*t*)) *is the estimated error output, for all Tf* ≥ 0 *and w*(*t*) ∈ L2[0, *Tf* ]*.*

## **3. Robust** H<sup>∞</sup> **fuzzy filter design**

Without loss of generality, in this section, we assume that *u*(*t*) = 0. Let us recall the system (1) with *u*(*t*) = 0 as follows:

$$\begin{aligned} \boldsymbol{E}\_{\varepsilon} \dot{\boldsymbol{x}}(t) &= \boldsymbol{\Sigma}\_{i=1}^{r} \mu\_{i} \left[ [\boldsymbol{A}\_{i} + \Delta \boldsymbol{A}\_{i}] \mathbf{x}(t) + [\boldsymbol{B}\_{1\_{i}} + \Delta \boldsymbol{B}\_{1\_{i}}] \boldsymbol{w}(t) \right] \\ \boldsymbol{z}(t) &= \boldsymbol{\Sigma}\_{i=1}^{r} \mu\_{i} \left[ [\boldsymbol{\mathsf{C}}\_{1\_{i}} + \Delta \boldsymbol{\mathsf{C}}\_{1\_{i}}] \mathbf{x}(t) \right] \\ \boldsymbol{y}(t) &= \boldsymbol{\Sigma}\_{i=1}^{r} \mu\_{i} \left[ [\boldsymbol{\mathsf{C}}\_{2\_{i}} + \Delta \boldsymbol{\mathsf{C}}\_{2\_{i}}] \mathbf{x}(t) + [\boldsymbol{D}\_{21\_{i}} + \Delta \boldsymbol{D}\_{21\_{i}}] \boldsymbol{w}(t) \right]. \end{aligned} \tag{4}$$

We are now aiming to design a full order dynamic H<sup>∞</sup> fuzzy filter of the form

$$\begin{array}{lcl}E\_{\varepsilon}\dot{\mathfrak{x}}(t) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} \mathfrak{h}\_{i} \mathfrak{h}\_{j} \Big[\hat{A}\_{ij}(\varepsilon)\mathfrak{x}(t) + \hat{B}\_{i}\mathfrak{y}(t)\Big] \\ \mathfrak{z}(t) &= \sum\_{i=1}^{r} \mathfrak{h}\_{i} \hat{\mathfrak{C}}\_{i} \mathfrak{x}(t) \end{array} \tag{5}$$

where *<sup>x</sup>*ˆ(*t*) ∈ �*<sup>n</sup>* is the filter's state vector, *<sup>z</sup>*<sup>ˆ</sup> ∈ �*<sup>s</sup>* is the estimate of *<sup>z</sup>*(*t*), *<sup>A</sup>*ˆ*ij*(*ε*), *<sup>B</sup>*<sup>ˆ</sup> *<sup>i</sup>* and *C*ˆ *<sup>i</sup>* are parameters of the filter which are to be determined, and *μ*ˆ*<sup>i</sup>* denotes the normalized time-varying fuzzy weighting functions for each rule (i.e., *<sup>μ</sup>*ˆ*<sup>i</sup>* <sup>≥</sup> 0 and <sup>∑</sup>*<sup>r</sup> <sup>i</sup>*=<sup>1</sup> *μ*ˆ*<sup>i</sup>* = 1), such that the inequality (3) holds. Clearly, in real control problems, all of the premise variables are not necessarily measurable. In this section, we then consider the designing of the robust H<sup>∞</sup> fuzzy filter into two cases as follows.

#### **3.1. Case I–***ν*(*t*) **is available for feedback**

The premise variable of the fuzzy model *ν*(*t*) is available for feedback which implies that *μ<sup>i</sup>* is available for feedback. Thus, we can select our filter that depends on *μ<sup>i</sup>* as follows:

$$\begin{array}{lcl}E\_{\varepsilon}\dot{\mathfrak{x}}(t) &= \sum\_{i=1}^{r} \sum\_{j=1}^{r} \mu\_{i}\mu\_{j} \left[\hat{A}\_{ij}(\varepsilon)\mathfrak{x}(t) + \hat{B}\_{i}\mathfrak{y}(t)\right] \\ \mathfrak{z}(t) &= \sum\_{i=1}^{r} \mu\_{i}\mathbb{C}\_{i}\mathfrak{x}(t). \end{array} \tag{6}$$

Before presenting our next results, the following lemma is recalled.

**Lemma 1.** *Consider the system (4). Given a prescribed* H<sup>∞</sup> *performance γ* > 0 *and a positive constant δ, if there exist matrices X<sup>ε</sup>* = *X<sup>T</sup> <sup>ε</sup> , Y<sup>ε</sup>* = *Y<sup>T</sup> <sup>ε</sup> ,* B*i*(*ε*) *and* C*i*(*ε*)*, i* = 1, 2, ··· ,*r, satisfying the following ε-dependent linear matrix inequalities:*

$$
\begin{bmatrix} X\_{\ell} & I \\ I & Y\_{\ell} \end{bmatrix} > 0 \tag{7}
$$

$$\begin{array}{ccc} X\_{\ell} > 0 \\ \dots & \dots \end{array} \tag{8}$$

*where*

*performance.*

*where E* =

*with*

� *I* 0 0 0 �

<sup>M</sup>*ij*(*ε*) = <sup>−</sup>*A<sup>T</sup>*

<sup>−</sup>*γ*−<sup>2</sup> �

*ε-independent linear matrix inequalities:*

*EX<sup>T</sup>*

*EY<sup>T</sup>*

*, D* =

Ψ11*ij* =

Ψ22*ij* =

*<sup>i</sup> <sup>E</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup>εE*−<sup>1</sup> *<sup>ε</sup> AiY<sup>ε</sup>* <sup>−</sup> �

<sup>1</sup>*<sup>i</sup>* <sup>+</sup> �

*<sup>X</sup>εE*−<sup>1</sup> *<sup>ε</sup> <sup>B</sup>*˜

*constant δ, if there exist matrices X*0*, Y*0*,* B0*<sup>i</sup> and* C0*<sup>i</sup>*

<sup>0</sup> <sup>=</sup> *<sup>X</sup>*0*E*, *<sup>X</sup><sup>T</sup>*

<sup>0</sup> <sup>=</sup> *<sup>Y</sup>*0*E*, *<sup>Y</sup><sup>T</sup>*

� 0 0 0 *I* � *,*

� *AiY<sup>T</sup>*

� *A<sup>T</sup> <sup>i</sup> <sup>X</sup><sup>T</sup>*

*C*˜ <sup>1</sup>*<sup>i</sup>* =

*and λ* =

� *Y*0*C*˜*<sup>T</sup>* 1*i* <sup>+</sup> <sup>C</sup>*<sup>T</sup>* 0*i D*˜ *<sup>T</sup>*

<sup>0</sup> <sup>+</sup> *<sup>Y</sup>*0*A<sup>T</sup>*

<sup>0</sup> + *X*0*Ai* + B0*<sup>i</sup>*

� *X*0*B*˜

*B*˜ <sup>1</sup>*<sup>i</sup>* <sup>=</sup> �

*<sup>D</sup>*˜ <sup>12</sup> = �

*<sup>D</sup>*˜ <sup>21</sup>*<sup>i</sup>* = �

∑ *i*=1

*r* ∑ *j*=1 � �*H<sup>T</sup>* 2*i H*2*<sup>j</sup>*

� *γρ <sup>δ</sup> <sup>H</sup><sup>T</sup>* 1*i γρ <sup>δ</sup> <sup>H</sup><sup>T</sup>* 5*i*

⎛

<sup>⎝</sup><sup>1</sup> <sup>+</sup> *<sup>ρ</sup>*<sup>2</sup> *<sup>r</sup>*

*<sup>i</sup>* <sup>+</sup> *<sup>γ</sup>*−2*B*˜

1*i B*˜ *T* <sup>1</sup>*<sup>j</sup>* (∗)*<sup>T</sup>*

<sup>1</sup>*<sup>i</sup>* + B0*<sup>i</sup>*

000 −

*δI I* 0 *B*1*<sup>i</sup>* 0

<sup>√</sup>2*λρH<sup>T</sup>* 4*i*

> √ 2*λI* �*T* ,

0 0 *δI D*21*<sup>i</sup> I*

<sup>12</sup>�*<sup>T</sup>* <sup>−</sup>*<sup>I</sup>*

*C*2*<sup>j</sup>* + *C<sup>T</sup>* 2*i* B*T* <sup>0</sup>*<sup>j</sup>* <sup>+</sup> *<sup>C</sup>*˜ *<sup>T</sup>* 1*i C*˜ <sup>1</sup>*<sup>j</sup>* (∗)*<sup>T</sup>*

*<sup>D</sup>*˜ <sup>21</sup>*<sup>j</sup>*

� ,

> <sup>√</sup>2*λC<sup>T</sup>* 1*i* �*T* ,

�

� <sup>+</sup> �*H<sup>T</sup>* 7*i H*7*<sup>j</sup>* � � ⎞ ⎠

�

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup><sup>ε</sup>*

� *<sup>E</sup>*−<sup>1</sup> *<sup>ε</sup> <sup>B</sup>*<sup>ˆ</sup>

**Remark 1.** *The LMIs given in Lemma 1 may become ill-conditioned when ε is sufficiently small, which is always the case for the descriptor systems. In general, these ill-conditioned LMIs are very difficult to solve. Thus, to alleviate these ill-conditioned LMIs, we have the following ε-independent well-posed LMI-based sufficient conditions for the uncertain fuzzy descriptor systems to obtain the prescribed* H<sup>∞</sup>

**Theorem 1.** *Consider the system (4). Given a prescribed* H<sup>∞</sup> *performance γ* > 0 *and a positive*

*I Y*0*E* + *DY*<sup>0</sup>

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup><sup>ε</sup>*

*Proof:* It can be shown by employing the same technique used in Ref.[18]-[19].

*X*0*E* + *DX*<sup>0</sup> *I*

� *<sup>E</sup>*−<sup>1</sup> *<sup>ε</sup> <sup>B</sup>*<sup>ˆ</sup> *iC*<sup>2</sup>*<sup>j</sup>*

*iD*˜ <sup>21</sup>*<sup>i</sup>* � *B*˜ *T* 1*j <sup>E</sup>*−<sup>1</sup> *<sup>ε</sup>* .

�

<sup>0</sup> *D* = *DX*0, *X*0*E* + *DX*<sup>0</sup> > 0 (18)

<sup>0</sup> *D* = *DY*0, *Y*0*E* + *DY*<sup>0</sup> > 0 (19)

�

�*<sup>T</sup>* <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>*

*<sup>Y</sup><sup>ε</sup>* <sup>−</sup> *<sup>C</sup>*˜*<sup>T</sup>* 1*i* � *C*˜ 1*j*

New Results on Robust ∞ Filter for Uncertain Fuzzy Descriptor Systems 469

*Y<sup>ε</sup>* + *D*˜ <sup>12</sup>*C*ˆ

*, i* = 1, 2, ··· ,*r, satisfying the following*

Ψ11*ii* < 0, *i* = 1, 2, ··· ,*r* (20) Ψ22*ii* < 0, *i* = 1, 2, ··· ,*r* (21)

�

1 2 ,

Ψ11*ij* + Ψ11*ji* < 0, *i* < *j* ≤ *r* (22) Ψ22*ij* + Ψ22*ji* < 0, *i* < *j* ≤ *r* (23)

> 0 (17)

*jYε* �

(24)

(25)

$$\begin{aligned} \mathbf{Y}\_{\varepsilon} &> \mathbf{0} \\ \Psi\_{11\_{\#}}(\varepsilon) &< \mathbf{0}, \quad \mathbf{i} = \mathbf{1}, \mathbf{2}, \dots, r \end{aligned} \tag{9}$$

$$\Psi\_{22\_{\#}}(\varepsilon) < 0, \quad \mathbf{i} = 1, 2, \dots, r \tag{11}$$

$$\Psi\_{11\_{\tilde{\eta}}}(\varepsilon) + \Psi\_{11\_{\tilde{\mu}}}(\varepsilon) < 0, \quad i < j \le r \tag{12}$$

$$\Psi\_{22\_{\parallel}}(\varepsilon) + \Psi\_{22\_{\parallel}}(\varepsilon) < 0, \quad i < j \le r \tag{13}$$

*where*

$$\Psi\_{11\_{[l]}}(\varepsilon) = \begin{pmatrix} \left(E\_{\varepsilon}^{-1}A\_{\mathrm{i}}Y\_{\varepsilon} + Y\_{\varepsilon}A\_{\mathrm{i}}^{T}E\_{\varepsilon}^{-1} + \gamma^{-2}E\_{\varepsilon}^{-1}\tilde{B}\_{\mathrm{i}}\tilde{B}\_{\mathrm{1}\_{l}}^{T}E\_{\varepsilon}^{-1} \; (\*)^{T}\right) \\\\ \left[Y\_{\varepsilon}\tilde{C}\_{\mathrm{1}\_{l}}^{T} + E\_{\varepsilon}^{-1}\mathcal{C}\_{\mathrm{i}}^{T}(\varepsilon)\tilde{D}\_{\mathrm{12}}^{T}\right]^{T} & -I \end{pmatrix} \tag{14}$$

$$\Psi\_{22\_{\tilde{\mathbb{Q}}}}(\varepsilon) = \begin{pmatrix} A\_i^T E\_\varepsilon^{-1} X\_\varepsilon + X\_\varepsilon E\_\varepsilon^{-1} A\_i + \mathcal{B}\_i(\varepsilon) \mathbf{C}\_{\mathbb{Q}} + \mathbf{C}\_{2\_l}^T \mathcal{B}\_j^T(\varepsilon) + \tilde{\mathbf{C}}\_{1\_l}^T \tilde{\mathbf{C}}\_{1\_l}(\ast)^T \\\left[ X\_\varepsilon E\_\varepsilon^{-1} \tilde{\mathcal{B}}\_{1\_l} + \mathcal{B}\_i(\varepsilon) \tilde{\mathcal{D}}\_{21\_l} \right]^T & -\gamma^2 I \end{pmatrix} \tag{15}$$

*with*

$$
\tilde{B}\_{1\_i} = \begin{bmatrix} \delta I \ I \ \mathbf{0} \ \bar{B}\_{1\_i} \ \mathbf{0} \end{bmatrix}\_i,
$$

$$
\tilde{C}\_{1\_i} = \begin{bmatrix} \frac{\gamma \rho}{\delta} H\_{1\_i}^T \ \frac{\gamma \rho}{\delta} H\_{5\_i}^T \ \sqrt{2} \lambda \rho H\_{4\_i}^T \ \sqrt{2} \lambda C\_{1\_i}^T \end{bmatrix}^T,
$$

$$
\tilde{D}\_{12} = \begin{bmatrix} 0 \ 0 \ 0 \ -\sqrt{2} \lambda I \end{bmatrix}^T,
$$

$$
\tilde{D}\_{21\_i} = \begin{bmatrix} 0 \ 0 \ \delta I \ D\_{21\_i} \ I \end{bmatrix}
$$

$$
and \quad \lambda = \left( 1 + \rho^2 \sum\_{i=1}^r \sum\_{j=1}^r \left[ ||H\_{2\_i}^T H\_{2\_j}|| + ||H\_{7\_i}^T H\_{7\_j}|| \right] \right)^{\frac{1}{2}},
$$

*then the prescribed* H<sup>∞</sup> *performance γ* > 0 *is guaranteed. Furthermore, a suitable filter is of the form (6) with*

$$\begin{array}{lcl}\hat{A}\_{ij}(\varepsilon) &= E\_{\varepsilon} \left[Y\_{\varepsilon}^{-1} - X\_{\varepsilon}\right]^{-1} \mathcal{M}\_{ij}(\varepsilon) Y\_{\varepsilon}^{-1} \\ \hat{B}\_{i} &= E\_{\varepsilon} \left[Y\_{\varepsilon}^{-1} - X\_{\varepsilon}\right]^{-1} \mathcal{B}\_{i}(\varepsilon) \\ \hat{\mathsf{C}}\_{i} &= \mathcal{C}\_{i}(\varepsilon) E\_{\varepsilon}^{-1} Y\_{\varepsilon}^{-1} \end{array} \tag{16}$$

*where*

4 Will-be-set-by-IN-TECH

The premise variable of the fuzzy model *ν*(*t*) is available for feedback which implies that *μ<sup>i</sup>* is

**Lemma 1.** *Consider the system (4). Given a prescribed* H<sup>∞</sup> *performance γ* > 0 *and a positive constant*

*<sup>i</sup> <sup>E</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>+</sup> *<sup>γ</sup>*−2*E*−<sup>1</sup> *<sup>ε</sup> <sup>B</sup>*˜

*<sup>i</sup>* (*ε*)*D*˜ *<sup>T</sup>* 12

*δI I* 0 *B*1*<sup>i</sup>* 0

000 −

*then the prescribed* H<sup>∞</sup> *performance γ* > 0 *is guaranteed. Furthermore, a suitable filter is of the form*

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup><sup>ε</sup>*

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup><sup>ε</sup>*

<sup>√</sup>2*λρH<sup>T</sup>* 4*i*

0 0 *δI D*21*<sup>i</sup> I*

<sup>√</sup>2*λ<sup>I</sup>*

�−<sup>1</sup>

�−<sup>1</sup> B*i*(*ε*)

<sup>1</sup>*<sup>i</sup>* <sup>+</sup> <sup>B</sup>*i*(*ε*)*D*˜ <sup>21</sup>*<sup>j</sup>*

� ,

<sup>+</sup> *<sup>E</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>C</sup>*<sup>T</sup>*

*<sup>i</sup> <sup>E</sup>*−<sup>1</sup> *<sup>ε</sup> <sup>X</sup><sup>ε</sup>* <sup>+</sup> *<sup>X</sup>εE*−<sup>1</sup> *<sup>ε</sup> Ai* <sup>+</sup> <sup>B</sup>*i*(*ε*)*C*2*<sup>j</sup>* <sup>+</sup> *<sup>C</sup><sup>T</sup>*

� *<sup>X</sup>εE*−<sup>1</sup> *<sup>ε</sup> <sup>B</sup>*˜

*B*˜ <sup>1</sup>*<sup>i</sup>* <sup>=</sup> �

*D*˜ <sup>12</sup> = �

*<sup>D</sup>*˜ <sup>21</sup>*<sup>i</sup>* <sup>=</sup> �

∑ *i*=1

�

�

*<sup>i</sup>* <sup>=</sup> <sup>C</sup>*i*(*ε*)*E*−<sup>1</sup> *<sup>ε</sup> <sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>*

*r* ∑ *j*=1 � �*H<sup>T</sup>* 2*i H*2*<sup>j</sup>*

� *γρ <sup>δ</sup> <sup>H</sup><sup>T</sup>* 1*i γρ <sup>δ</sup> <sup>H</sup><sup>T</sup>* 5*i*

⎛

*A*ˆ

*B*ˆ

*C*ˆ

<sup>⎝</sup><sup>1</sup> <sup>+</sup> *<sup>ρ</sup>*<sup>2</sup> *<sup>r</sup>*

*ij*(*ε*) = *E<sup>ε</sup>*

*<sup>i</sup>* = *E<sup>ε</sup>*

�

*ij*(*ε*)*x*ˆ(*t*) + *B*ˆ

*iy*(*t*) �

*<sup>ε</sup> ,* B*i*(*ε*) *and* C*i*(*ε*)*, i* = 1, 2, ··· ,*r, satisfying the following*

> 0 (7)

*X<sup>ε</sup>* > 0 (8) *Y<sup>ε</sup>* > 0 (9) Ψ11*ii*(*ε*) < 0, *i* = 1, 2, ··· ,*r* (10) Ψ22*ii*(*ε*) < 0, *i* = 1, 2, ··· ,*r* (11)

*<sup>E</sup>*−<sup>1</sup> *<sup>ε</sup>* (∗)*<sup>T</sup>*

*<sup>j</sup>* (*ε*) + *<sup>C</sup>*˜*<sup>T</sup>* 1*i C*˜ <sup>1</sup>*<sup>j</sup>* (∗)*<sup>T</sup>*

�*<sup>T</sup>* <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>*

1 2 ,

⎞

⎠ (14)

�

(15)

(16)

�*<sup>T</sup>* <sup>−</sup>*<sup>I</sup>*

2*i* B*T*

<sup>√</sup>2*λC<sup>T</sup>* 1*i* �*T* ,

�*T* ,

�

� <sup>+</sup> �*H<sup>T</sup>* 7*i H*7*<sup>j</sup>* � � ⎞ ⎠

<sup>M</sup>*ij*(*ε*)*Y*−<sup>1</sup> *<sup>ε</sup>*

Ψ11*ij*(*ε*) + Ψ11*ji*(*ε*) < 0, *i* < *j* ≤ *r* (12) Ψ22*ij*(*ε*) + Ψ22*ji*(*ε*) < 0, *i* < *j* ≤ *r* (13)

> 1*i B*˜ *T* 1*j*

*ix*ˆ(*t*). (6)

*<sup>j</sup>*=<sup>1</sup> *μiμ<sup>j</sup>* � *A*ˆ

available for feedback. Thus, we can select our filter that depends on *μ<sup>i</sup>* as follows:

*<sup>i</sup>*=<sup>1</sup> <sup>∑</sup>*<sup>r</sup>*

*<sup>i</sup>*=<sup>1</sup> *<sup>μ</sup>iC*<sup>ˆ</sup>

**3.1. Case I–***ν*(*t*) **is available for feedback**

*δ, if there exist matrices X<sup>ε</sup>* = *X<sup>T</sup>*

Ψ11*ij*(*ε*) =

Ψ22*ij*(*ε*) =

⎛ ⎝ �

� *A<sup>T</sup>*

*C*˜ <sup>1</sup>*<sup>i</sup>* =

*and λ* =

*where*

*with*

*(6) with*

*ε-dependent linear matrix inequalities:*

*E<sup>ε</sup>* ˙

*x*ˆ(*t*) = ∑*<sup>r</sup>*

Before presenting our next results, the following lemma is recalled.

*<sup>ε</sup> , Y<sup>ε</sup>* = *Y<sup>T</sup>*

*<sup>E</sup>*−<sup>1</sup> *<sup>ε</sup> AiY<sup>ε</sup>* + *<sup>Y</sup>εA<sup>T</sup>*

� *YεC*˜*<sup>T</sup>* 1*i*

� *X<sup>ε</sup> I I Y<sup>ε</sup>*

*z*ˆ(*t*) = ∑*<sup>r</sup>*

$$\begin{split} \mathcal{M}\_{\vec{l}\vec{j}}(\varepsilon) &= -A\_{\vec{l}}^{T} \mathbb{E}\_{\varepsilon}^{-1} - X\_{\vec{\varepsilon}} \mathbb{E}\_{\varepsilon}^{-1} A\_{\vec{l}} Y\_{\varepsilon} - \left[ Y\_{\varepsilon}^{-1} - X\_{\vec{\varepsilon}} \right] \mathbb{E}\_{\vec{\varepsilon}}^{-1} \mathbb{\hat{B}}\_{\vec{l}} \mathbb{C}\_{\mathcal{Q}\_{\vec{l}}} Y\_{\varepsilon} - \tilde{\mathsf{C}}\_{1\_{\vec{l}}}^{T} \left[ \tilde{\mathsf{C}}\_{1\_{\vec{l}}} Y\_{\varepsilon} + \tilde{\mathsf{D}}\_{12} \hat{\mathsf{C}}\_{\vec{j}} Y\_{\varepsilon} \right] \\ & \quad - \gamma^{-2} \left\{ X\_{\vec{\varepsilon}} \mathbb{E}\_{\varepsilon}^{-1} \tilde{\mathsf{B}}\_{1\_{\vec{l}}} + \left[ Y\_{\vec{\varepsilon}}^{-1} - X\_{\vec{\varepsilon}} \right] \mathbb{E}\_{\varepsilon}^{-1} \hat{\mathsf{B}}\_{\vec{l}} \tilde{\mathsf{D}}\_{21\_{\vec{l}}} \right\} \tilde{\mathsf{B}}\_{1\_{\vec{l}}}^{T} \mathbb{E}\_{\varepsilon}^{-1} . \end{split}$$

*Proof:* It can be shown by employing the same technique used in Ref.[18]-[19].

**Remark 1.** *The LMIs given in Lemma 1 may become ill-conditioned when ε is sufficiently small, which is always the case for the descriptor systems. In general, these ill-conditioned LMIs are very difficult to solve. Thus, to alleviate these ill-conditioned LMIs, we have the following ε-independent well-posed LMI-based sufficient conditions for the uncertain fuzzy descriptor systems to obtain the prescribed* H<sup>∞</sup> *performance.*

**Theorem 1.** *Consider the system (4). Given a prescribed* H<sup>∞</sup> *performance γ* > 0 *and a positive constant δ, if there exist matrices X*0*, Y*0*,* B0*<sup>i</sup> and* C0*<sup>i</sup> , i* = 1, 2, ··· ,*r, satisfying the following ε-independent linear matrix inequalities:*

$$
\begin{bmatrix} X\_0 E + D X\_0 & I \\ I & Y\_0 E + D Y\_0 \end{bmatrix} > 0 \tag{17}
$$

$$EX\_0^T = X\_0 E, \quad X\_0^T D = DX\_0, \quad X\_0 E + DX\_0 > 0 \tag{18}$$

$$EY\_0^T = \mathbf{Y}\_0 \mathbf{E}\_\prime \quad \mathbf{Y}\_0^T \mathbf{D} = D\mathbf{Y}\_0 \quad \mathbf{Y}\_0 \mathbf{E} + D\mathbf{Y}\_0 > \mathbf{0} \tag{19}$$

$$\Psi\_{11\_{\tilde{\mu}}} < 0, \quad i = 1, 2, \dots, r \\ \tag{20}$$

$$\Psi\_{22\_{\shortparallel}} < 0, \quad i = 1, 2, \dots, r \tag{21}$$

$$\Psi\_{11\_{\parallel}} + \Psi\_{11\_{\parallel}} < 0, \quad i < j \le r \tag{22}$$

$$\Psi\_{22\_{\parallel}} + \Psi\_{22\_{\parallel}} < 0, \quad i < j \le r \tag{23}$$

$$\begin{aligned} \text{where } E &= \begin{pmatrix} I & 0 \\ 0 & 0 \end{pmatrix}, D = \begin{pmatrix} 0 \ 0 \\ 0 \ I \end{pmatrix}, \\ \Psi\_{11\_{\tilde{\eta}}} &= \begin{pmatrix} A\_i Y\_0^T + Y\_0 A\_i^T + \gamma^{-2} \tilde{B}\_{1i} \tilde{B}\_{1\_j}^T \ (\*)^T \\ \left[ Y\_0 \tilde{\mathcal{C}}\_{1\_i}^T + \mathcal{C}\_{0\_i}^T \tilde{D}\_{12}^T \right]^T & -I \end{pmatrix} \\ \mathbf{w}\_{\infty\_{\tilde{\eta}}} &= \begin{pmatrix} A\_i^T X\_0^T + X\_0 A\_i + \mathcal{B}\_{0\_i} \mathcal{C}\_{2\_j} + \mathcal{C}\_{2\_i}^T \mathcal{B}\_{0\_j}^T + \tilde{\mathcal{C}}\_{1\_i}^T \tilde{\mathcal{C}}\_{1\_j} \ (\*)^T \end{pmatrix} \end{aligned} \tag{24}$$

$$\Psi\_{22\_{\langle\rangle}} = \begin{pmatrix} A\_i^1 \, \mathbf{X}\_0^1 + \mathbf{X}\_0 \mathbf{A}\_i + \mathcal{B}\_{0\_i} \mathbf{C}\_{2\_{\langle\rangle}} + \mathbf{C}\_{2\_{\langle\rangle}}^1 \mathcal{B}\_{0\_{\langle\rangle}}^1 + \mathbf{C}\_{1\_i}^1 \mathbf{C}\_{1\_{\langle\rangle}} \ (\*)^\, \\\left[ \mathbf{X}\_0 \mathcal{B}\_{1\_i} + \mathcal{B}\_{0\_i} \mathcal{D}\_{21\_{\langle\rangle}} \right]^T & -\gamma^2 I \end{pmatrix} \tag{25}$$

*with*

$$\tilde{B}\_{1\_i} = \left[\delta I \stackrel{\scriptstyle}{I} \begin{matrix} I \ \delta I \ \bar{B}\_{1\_i} \ \mathbf{0} \end{matrix} \right],$$

$$\tilde{C}\_{1\_i} = \left[\begin{array}{cc} \frac{\gamma \rho}{\delta} H\_{1\_i}^T & \frac{\gamma \rho}{\delta} H\_{5\_i}^T \ \sqrt{\Delta} \lambda \rho H\_{4\_i}^T & \sqrt{\Delta} \lambda C\_{1\_i}^T \end{array} \right]^T,$$

$$\tilde{D}\_{12} = \left[\begin{array}{cc} 0 \ 0 \ 0 \ -\sqrt{2} \lambda I \end{array} \right]^T,$$

$$\mathcal{D}\_{21\_i} = \left[\begin{array}{cc} 0 \ 0 \ \delta I \ D\_{21\_i} \ I \end{array} \right]$$

$$and \quad \lambda = \left(1 + \rho^2 \sum\_{i=1}^r \sum\_{j=1}^r \left[\left|| H\_{2\_i}^T H\_{2\_j} \| + \left|| H\_{7\_i}^T H\_{7\_j} \| \right| \right] \right)^{\frac{1}{2}}\right).$$

#### 6 Will-be-set-by-IN-TECH 470 Fuzzy Controllers – Recent Advances in Theory and Applications New Results on Robust <sup>H</sup><sup>∞</sup> Filter for Uncertain Fuzzy Descriptor Systems <sup>7</sup>

*then there exists a sufficiently small ε*ˆ > 0 *such that for ε* ∈ (0,*ε*ˆ]*, the prescribed* H<sup>∞</sup> *performance γ* > 0 *is guaranteed. Furthermore, a suitable filter is of the form (6) with*

$$\begin{aligned} \hat{A}\_{\hat{\imath}\jmath}(\varepsilon) &= \left[Y\_{\varepsilon}^{-1} - X\_{\varepsilon}\right]^{-1} \mathcal{M}\_{0\_{\hat{\imath}\jmath}}(\varepsilon) Y\_{\varepsilon}^{-1} \\ \hat{B}\_{\hat{\imath}} &= \left[Y\_{0}^{-1} - X\_{0}\right]^{-1} \mathcal{B}\_{0\_{\hat{\imath}}} \\ \hat{C}\_{\hat{\imath}} &= \mathcal{C}\_{0\_{\hat{\imath}}} Y\_{0}^{-1} \end{aligned} \tag{26}$$

Substituting (28) and (29) into the left hand side of (32), we get

*ε*(*X*<sup>2</sup> + *Y*−<sup>1</sup>

⎡ ⎣

*<sup>X</sup>*<sup>1</sup> <sup>−</sup> *<sup>Y</sup>*−<sup>1</sup>

*Y*−<sup>1</sup> <sup>0</sup> − *X*<sup>0</sup>

*iY<sup>T</sup>* <sup>0</sup> <sup>+</sup> *<sup>ε</sup>C*<sup>ˆ</sup>

<sup>B</sup>*i*(*ε*) = �

<sup>C</sup>*i*(*ε*) = *<sup>C</sup>*<sup>ˆ</sup>

respectively and the *ε*-dependent linear matrices are

⎛

⎜⎝

⎛

⎜⎝

� *Y*˜*C*˜*<sup>T</sup>* 1*i* <sup>+</sup> <sup>C</sup>*<sup>T</sup> εi D*˜ *<sup>T</sup>* 12*<sup>j</sup>* �*T* 0

*A<sup>T</sup>*

*ψ*11*ij* = *ε*

*ψ*22*ij* = *ε*

*<sup>X</sup>*<sup>1</sup> <sup>−</sup> *<sup>Y</sup>*−<sup>1</sup>

<sup>1</sup> *<sup>Y</sup>*2*Y*−<sup>1</sup>

*<sup>X</sup>*<sup>1</sup> <sup>−</sup> *<sup>Y</sup>*−<sup>1</sup>

<sup>1</sup> *<sup>ε</sup>*(*X*<sup>2</sup> <sup>+</sup> *<sup>Y</sup>*−<sup>1</sup>

<sup>1</sup> 0

<sup>0</sup> *<sup>X</sup>*<sup>3</sup> <sup>−</sup> *<sup>Y</sup>*−<sup>1</sup>

<sup>1</sup> <sup>&</sup>gt; 0 and *<sup>X</sup>*<sup>3</sup> <sup>−</sup> *<sup>Y</sup>*−<sup>1</sup>

Using (35) and the Schur complement, it can be shown that there exists a sufficiently small

Next, employing (28), (29) and (30), the controller's matrices given in (16) can be re-expressed

= C0*<sup>i</sup>* + *ε*C*ε<sup>i</sup>*

.

� *B*ˆ *<sup>i</sup>* + *ε* � *<sup>N</sup><sup>ε</sup>* <sup>−</sup> *<sup>X</sup>*˜ � *B*ˆ *i* Δ

*iY*˜ *<sup>T</sup>* <sup>Δ</sup>

Substituting (28), (29), (30) and (36) into (14) and (15), and pre-post multiplying by � *<sup>E</sup><sup>ε</sup>* <sup>0</sup>

where the *ε*-independent linear matrices Ψ11*ij* and Ψ22*ij* are defined in (24) and (25),

*<sup>i</sup>* (∗)*<sup>T</sup>*

⎞

*C*2*<sup>j</sup>* + *C<sup>T</sup>* 2*i* B*T <sup>ε</sup><sup>j</sup>* (∗)*<sup>T</sup>*

*<sup>D</sup>*˜ <sup>21</sup>*<sup>j</sup>* �*T*

*AiY*˜ *<sup>T</sup>* + *YA*˜ *<sup>T</sup>*

*<sup>i</sup> <sup>X</sup>*˜ <sup>+</sup> *<sup>X</sup>*˜ *TAi* <sup>+</sup> <sup>B</sup>*ε<sup>i</sup>*

<sup>1</sup>*<sup>i</sup>* + B*ε<sup>i</sup>*

Employing (20)-(22) and knowing the fact that for any given negative definite matrix W, there exists an *ε* > 0 such that W + *εI* < 0, one can show that there exists a sufficiently small *ε*ˆ > 0 such that for *ε* ∈ (0,*ε*ˆ], (10)-(13) hold. Since (7)-(13) hold, using Lemma 1, the inequality (3)

� *X*˜ *B*˜

Note that the *ε*-dependent linear matrices tend to zero when *ε* approaches zero.

<sup>3</sup> )*<sup>T</sup> <sup>ε</sup>*(*X*<sup>3</sup> <sup>−</sup> *<sup>Y</sup>*−<sup>1</sup>

3

⎤

<sup>1</sup> *<sup>Y</sup>*2*Y*−<sup>1</sup> <sup>3</sup> ) ⎤

New Results on Robust ∞ Filter for Uncertain Fuzzy Descriptor Systems 471

= B0*<sup>i</sup>* + *ε*B*ε<sup>i</sup>*

Ψ11*ij* + *ψ*11*ij* and Ψ22*ij* + *ψ*22*ij* (37)

0

⎞

⎦ . (33)

⎦ > 0. (34)

<sup>3</sup> > 0. (35)

⎟⎠ (38)

⎟⎠ . (39)

(36)

0 *I* � ,

<sup>3</sup> )

⎡ ⎣

The Schur complement of (17) is

According to (34), we learn that

as follows:

holds.

we, respectively, obtain

*ε*ˆ > 0 such that for *ε* ∈ (0,*ε*ˆ], (7) holds.

*where*

$$\mathcal{M}\_{0\_{\parallel}}(\varepsilon) = -A\_{i}^{T} - \mathbf{X}\_{\varepsilon}A\_{i}\mathbf{Y}\_{\varepsilon} - \left[\mathbf{Y}\_{\varepsilon}^{-1} - \mathbf{X}\_{\varepsilon}\right]\boldsymbol{\tilde{\mathcal{B}}}\_{i}\mathbf{C}\_{2\_{\parallel}}\mathbf{Y}\_{\varepsilon} - \boldsymbol{\tilde{\mathcal{C}}}\_{1\_{\parallel}}^{T}\left[\boldsymbol{\tilde{\mathcal{C}}}\_{1\_{\parallel}}\mathbf{Y}\_{\varepsilon} + \boldsymbol{\tilde{\mathcal{D}}}\_{12}\boldsymbol{\tilde{\mathcal{C}}}\_{j}\mathbf{Y}\_{\varepsilon}\right]$$

$$-\gamma^{-2}\left\{\mathbf{X}\_{\varepsilon}\boldsymbol{\tilde{\mathcal{B}}}\_{1\_{\parallel}} + \left[\mathbf{Y}\_{\varepsilon}^{-1} - \mathbf{X}\_{\varepsilon}\right]\boldsymbol{\tilde{\mathcal{B}}}\_{i}\boldsymbol{\tilde{\mathcal{D}}}\_{21\_{\parallel}}\right\}\boldsymbol{\tilde{\mathcal{B}}}\_{1\_{\parallel}}^{T}$$

$$\mathbf{X}\_{\varepsilon} = \left\{\mathbf{X}\_{0} + \boldsymbol{\varepsilon}\boldsymbol{\tilde{\mathcal{X}}}\right\}\boldsymbol{E}\_{\varepsilon} \text{ and } \ Y\_{\varepsilon}^{-1} = \left\{Y\_{0}^{-1} + \boldsymbol{\varepsilon}\boldsymbol{N}\_{\varepsilon}\right\}\boldsymbol{E}\_{\varepsilon} \tag{27}$$

$$D\left(\mathbf{X}\_{0}^{T} - \mathbf{X}\_{0}\right)\text{ and } \mathbf{N}\_{\varepsilon} = D\left((Y\_{0}^{-1})^{T} - Y\_{0}^{-1}\right).$$

*with X*˜ = *D X<sup>T</sup>* <sup>0</sup> − *X*<sup>0</sup> *and N<sup>ε</sup>* = *D* (*Y*−<sup>1</sup> <sup>0</sup> )*<sup>T</sup>* <sup>−</sup> *<sup>Y</sup>*−<sup>1</sup> 0

*Proof:* Suppose the inequalities (17)-(19) hold, then the matrices *X*<sup>0</sup> and *Y*<sup>0</sup> are of the following forms:

$$X\_0 = \begin{pmatrix} X\_1 \ X\_2 \\ 0 \ X\_3 \end{pmatrix} \quad \text{and} \quad Y\_0 = \begin{pmatrix} Y\_1 \ Y\_2 \\ 0 \ Y\_3 \end{pmatrix}.$$

with *X*<sup>1</sup> = *X<sup>T</sup>* <sup>&</sup>gt; 0, *<sup>X</sup>*<sup>3</sup> <sup>=</sup> *<sup>X</sup><sup>T</sup>* <sup>&</sup>gt; 0, *<sup>Y</sup>*<sup>1</sup> <sup>=</sup> *<sup>Y</sup><sup>T</sup>* <sup>&</sup>gt; 0 and *<sup>Y</sup>*<sup>3</sup> <sup>=</sup> *<sup>Y</sup><sup>T</sup>* > 0. Substituting *X*<sup>0</sup> and *Y*<sup>0</sup> into (27), respectively, we have

$$X\_{\varepsilon} = \left\{ X\_0 + \varepsilon \vec{X} \right\} \mathbf{E}\_{\varepsilon} = \begin{pmatrix} X\_1 & \varepsilon X\_2 \\ \varepsilon X\_2^T & \varepsilon X\_3 \end{pmatrix} \tag{28}$$

$$Y\_{\varepsilon}^{-1} = \left\{ Y\_0^{-1} + \varepsilon N\_{\varepsilon} \right\} E\_{\varepsilon} = \begin{pmatrix} Y\_1^{-1} & -\varepsilon Y^{-1} Y\_2 Y\_3^{-1} \\ -\varepsilon (Y^{-1} Y\_2 Y\_3^{-1})^T & \varepsilon Y\_3^{-1} \end{pmatrix}. \tag{29}$$

Clearly, *X<sup>ε</sup>* = *X<sup>T</sup> <sup>ε</sup>* , and *<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* = (*Y*−<sup>1</sup> *<sup>ε</sup>* )*T*. Knowing the fact that the inverse of a symmetric matrix is a symmetric matrix, we learn that *Y<sup>ε</sup>* is a symmetric matrix. Using the matrix inversion lemma, we can see that

$$Y\_{\varepsilon} = E\_{\varepsilon}^{-1} \left\{ Y\_0 + \varepsilon \tilde{Y} \right\} \tag{30}$$

where *Y*˜ = *Y*0*Nε*(*I* + *εY*0*Nε*)−1*Y*0. Employing the Schur complement, one can show that there exists a sufficiently small *ε*ˆ such that for *ε* ∈ (0, *ε*ˆ], (8)-(9) holds.

Now, we need to show that

$$
\begin{pmatrix} X\_{\varepsilon} & I \\ & I & Y\_{\varepsilon} \end{pmatrix} > 0. \tag{31}
$$

By the Schur complement, it is equivalent to showing that

$$X\_{\varepsilon} - Y\_{\varepsilon}^{-1} > 0. \tag{32}$$

Substituting (28) and (29) into the left hand side of (32), we get

$$\begin{bmatrix} \mathbf{X}\_1 - \mathbf{Y}\_1^{-1} & \varepsilon (\mathbf{X}\_2 + \mathbf{Y}\_1^{-1} \mathbf{Y}\_2 \mathbf{Y}\_3^{-1})\\\\ \varepsilon (\mathbf{X}\_2 + \mathbf{Y}\_1^{-1} \mathbf{Y}\_2 \mathbf{Y}\_3^{-1})^T & \varepsilon (\mathbf{X}\_3 - \mathbf{Y}\_3^{-1}) \end{bmatrix}. \tag{33}$$

The Schur complement of (17) is

6 Will-be-set-by-IN-TECH

*then there exists a sufficiently small ε*ˆ > 0 *such that for ε* ∈ (0,*ε*ˆ]*, the prescribed* H<sup>∞</sup> *performance*

−<sup>1</sup>

−<sup>1</sup> B0*<sup>i</sup>*

> *B*ˆ *iC*<sup>2</sup>*<sup>j</sup>*

 *B*ˆ *iD*˜ <sup>21</sup>*<sup>i</sup> B*˜ *T* 1*j*

 *Y*−<sup>1</sup> <sup>0</sup> + *εN<sup>ε</sup>*

and *Y*<sup>0</sup> =

<sup>1</sup> <sup>&</sup>gt; 0 and *<sup>Y</sup>*<sup>3</sup> <sup>=</sup> *<sup>Y</sup><sup>T</sup>*

<sup>−</sup>*ε*(*Y*−1*Y*2*Y*−<sup>1</sup>

*<sup>ε</sup>* , and *<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* = (*Y*−<sup>1</sup> *<sup>ε</sup>* )*T*. Knowing the fact that the inverse of a symmetric matrix

<sup>M</sup>0*ij*(*ε*)*Y*−<sup>1</sup> *<sup>ε</sup>*

*<sup>Y</sup><sup>ε</sup>* <sup>−</sup> *<sup>C</sup>*˜ *<sup>T</sup>* 1*i C*˜ 1*j*

 *Y*<sup>1</sup> *Y*<sup>2</sup> 0 *Y*<sup>3</sup> <sup>1</sup> <sup>−</sup>*εY*−1*Y*2*Y*−<sup>1</sup>

<sup>3</sup> )*<sup>T</sup> <sup>ε</sup>Y*−<sup>1</sup>

*Y<sup>ε</sup>* + *D*˜ <sup>12</sup>*C*ˆ

*jY<sup>ε</sup>* 

*E<sup>ε</sup>* (27)

<sup>3</sup> > 0. Substituting *X*<sup>0</sup> and *Y*<sup>0</sup> into

3

> 0. (31)

*<sup>X</sup><sup>ε</sup>* <sup>−</sup> *<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>&</sup>gt; 0. (32)

3

(26)

(28)

(30)

. (29)

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup><sup>ε</sup>*

*Y*−<sup>1</sup> 0

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup><sup>ε</sup>*

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup><sup>ε</sup>*

<sup>0</sup> )*<sup>T</sup>* <sup>−</sup> *<sup>Y</sup>*−<sup>1</sup> 0 *.*

 *X*<sup>1</sup> *εX*<sup>2</sup> *εX<sup>T</sup>* <sup>2</sup> *εX*<sup>3</sup>

*Y*−<sup>1</sup>

is a symmetric matrix, we learn that *Y<sup>ε</sup>* is a symmetric matrix. Using the matrix inversion

 *Y*<sup>0</sup> + *εY*˜ 

where *Y*˜ = *Y*0*Nε*(*I* + *εY*0*Nε*)−1*Y*0. Employing the Schur complement, one can show that there

*<sup>Y</sup><sup>ε</sup>* = *<sup>E</sup>*−<sup>1</sup> *<sup>ε</sup>*

*X<sup>ε</sup> I*

*I Y<sup>ε</sup>*

*Proof:* Suppose the inequalities (17)-(19) hold, then the matrices *X*<sup>0</sup> and *Y*<sup>0</sup> are of the following

*<sup>E</sup><sup>ε</sup> and Y*−<sup>1</sup> *<sup>ε</sup>* =

*Y*−<sup>1</sup> <sup>0</sup> − *X*<sup>0</sup>

*γ* > 0 *is guaranteed. Furthermore, a suitable filter is of the form (6) with*

*B*ˆ

*C*ˆ

<sup>M</sup>0*ij*(*ε*) = <sup>−</sup>*A<sup>T</sup>*

*X<sup>ε</sup>* = 

<sup>1</sup> <sup>&</sup>gt; 0, *<sup>X</sup>*<sup>3</sup> <sup>=</sup> *<sup>X</sup><sup>T</sup>*

*X<sup>ε</sup>* = 

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* =

 *Y*−<sup>1</sup> <sup>0</sup> + *εN<sup>ε</sup>*

<sup>−</sup>*γ*−<sup>2</sup> *XεB*˜ <sup>1</sup>*<sup>i</sup>* <sup>+</sup>

*X*<sup>0</sup> + *εX*˜

*and N<sup>ε</sup>* = *D*

*X*<sup>0</sup> =

*X*<sup>0</sup> + *εX*˜

exists a sufficiently small *ε*ˆ such that for *ε* ∈ (0, *ε*ˆ], (8)-(9) holds.

By the Schur complement, it is equivalent to showing that

*where*

*with X*˜ = *D*

with *X*<sup>1</sup> = *X<sup>T</sup>*

Clearly, *X<sup>ε</sup>* = *X<sup>T</sup>*

lemma, we can see that

Now, we need to show that

forms:

 *X<sup>T</sup>* <sup>0</sup> − *X*<sup>0</sup> 

(27), respectively, we have

*A*ˆ*ij*(*ε*) =

*<sup>i</sup>* =

*<sup>i</sup>* = C0*<sup>i</sup>*

*<sup>i</sup>* <sup>−</sup> *<sup>X</sup>εAiY<sup>ε</sup>* <sup>−</sup>

 (*Y*−<sup>1</sup>

*X*<sup>1</sup> *X*<sup>2</sup>

<sup>3</sup> <sup>&</sup>gt; 0, *<sup>Y</sup>*<sup>1</sup> <sup>=</sup> *<sup>Y</sup><sup>T</sup>*

 *E<sup>ε</sup>* =

> *E<sup>ε</sup>* =

0 *X*<sup>3</sup>

$$
\begin{bmatrix} X\_1 - Y\_1^{-1} & 0 \\ & 0 & X\_3 - Y\_3^{-1} \end{bmatrix} > 0. \tag{34}
$$

According to (34), we learn that

$$X\_1 - Y\_1^{-1} > 0 \qquad \text{and} \qquad X\_3 - Y\_3^{-1} > 0. \tag{35}$$

Using (35) and the Schur complement, it can be shown that there exists a sufficiently small *ε*ˆ > 0 such that for *ε* ∈ (0,*ε*ˆ], (7) holds.

Next, employing (28), (29) and (30), the controller's matrices given in (16) can be re-expressed as follows:

$$\begin{split} \mathcal{B}\_{i}(\varepsilon) &= \left[Y\_{0}^{-1} - X\_{0}\right] \mathring{\mathcal{B}}\_{i} + \varepsilon \left[N\_{\varepsilon} - \breve{X}\right] \mathring{\mathcal{B}}\_{i} \stackrel{\Delta}{=} \mathcal{B}\_{0\_{i}} + \varepsilon \mathcal{B}\_{\varepsilon\_{i}} \\ \mathcal{C}\_{i}(\varepsilon) &= \mathring{\mathcal{C}}\_{i} Y\_{0}^{T} + \varepsilon \mathring{\mathcal{C}}\_{i} \breve{Y}^{T} \stackrel{\Delta}{=} \mathcal{C}\_{0\_{i}} + \varepsilon \mathcal{C}\_{\varepsilon\_{i}}. \end{split} \tag{36}$$

Substituting (28), (29), (30) and (36) into (14) and (15), and pre-post multiplying by � *<sup>E</sup><sup>ε</sup>* <sup>0</sup> 0 *I* � , we, respectively, obtain

$$\Psi\_{11\_{\vert\rangle}} + \psi\_{11\_{\vert\rangle}} \text{ and } \Psi\_{22\_{\vert\rangle}} + \psi\_{22\_{\vert\vert}} \tag{37}$$

where the *ε*-independent linear matrices Ψ11*ij* and Ψ22*ij* are defined in (24) and (25), respectively and the *ε*-dependent linear matrices are

$$\Psi\_{11\_{\tilde{l}}} = \varepsilon \begin{pmatrix} A\_{\dot{l}} \tilde{Y}^T + \tilde{Y} A\_{\dot{l}}^T & (\*)^T \\\\ \left[ \tilde{Y} \tilde{\mathfrak{C}}\_{1\_{\tilde{l}}}^T + \mathcal{C}\_{\varepsilon \dot{l}}^T \mathcal{D}\_{12\_{\tilde{l}}}^T \right]^T & 0 \end{pmatrix} \tag{38}$$

$$\Psi\_{\mathfrak{V}\mathfrak{Z}\_{ij}} = \varepsilon \begin{pmatrix} A\_i^T \mathcal{X} + \mathcal{X}^T A\_i + \mathcal{B}\_{\varepsilon\_l} \mathbb{C}\_{\mathbf{2}\_{j}} + \mathbb{C}\_{\mathbf{2}\_{l'}}^T \mathcal{B}\_{\varepsilon\_{l'}}^T (\*)^T \\\\ \left[ \tilde{\mathbf{X}} \tilde{\mathcal{B}}\_{1\_l} + \mathcal{B}\_{\varepsilon\_l} \tilde{\mathcal{D}}\_{\mathbf{2}\_{l'}} \right]^T & 0 \end{pmatrix} . \tag{39}$$

Note that the *ε*-dependent linear matrices tend to zero when *ε* approaches zero.

Employing (20)-(22) and knowing the fact that for any given negative definite matrix W, there exists an *ε* > 0 such that W + *εI* < 0, one can show that there exists a sufficiently small *ε*ˆ > 0 such that for *ε* ∈ (0,*ε*ˆ], (10)-(13) hold. Since (7)-(13) hold, using Lemma 1, the inequality (3) holds.

#### **3.2. Case II–***ν*(*t*) **is unavailable for feedback**

The fuzzy filter is assumed to be the same as the premise variables of the fuzzy system model. This actually means that the premise variables of fuzzy system model are assumed to be measurable. However, in general, it is extremely difficult to derive an accurate fuzzy system model by imposing that all premise variables are measurable. In this subsection, we do not impose that condition, we choose the premise variables of the filter to be different from the premise variables of fuzzy system model of the plant. In here, the premise variables of the filter are selected to be the estimated premise variables of the plant. In the other words, the premise variable of the fuzzy model *ν*(*t*) is unavailable for feedback which implies *μ<sup>i</sup>* is unavailable for feedback. Hence, we cannot select our filter which depends on *μi*. Thus, we select our filter as (5) where *μ*ˆ*<sup>i</sup>* depends on the premise variable of the filter which is different from *μi*. Let us re-express the system (1) in terms of *μ*ˆ*i*, thus the plant's premise variable becomes the same as the filter's premise variable. By doing so, the result given in the previous case can then be applied here. Note that it can be done by using the same technique as in subsection. After some manipulation, we get

$$\begin{aligned} E\_{\varepsilon} \dot{\mathbf{x}}(t) &= \boldsymbol{\Sigma}\_{i=1}^{r} \boldsymbol{\hat{\mu}}\_{i} \Big[ [\boldsymbol{A}\_{i} + \Delta \bar{\boldsymbol{A}}\_{i}] \mathbf{x}(t) + [\boldsymbol{B}\_{1\_{i}} + \Delta \bar{\boldsymbol{B}}\_{1\_{i}}] \boldsymbol{w}(t) \\ \boldsymbol{z}(t) &= \boldsymbol{\Sigma}\_{i=1}^{r} \boldsymbol{\hat{\mu}}\_{i} \Big[ [\mathbf{C}\_{1\_{i}} + \Delta \bar{\mathbf{C}}\_{1\_{i}}] \mathbf{x}(t) \Big] \\ \boldsymbol{y}(t) &= \boldsymbol{\Sigma}\_{i=1}^{r} \boldsymbol{\hat{\mu}}\_{i} \Big[ [\mathbf{C}\_{2\_{i}} + \Delta \bar{\mathbf{C}}\_{2\_{i}}] \mathbf{x}(t) + [\boldsymbol{D}\_{21\_{i}} + \Delta \boldsymbol{D}\_{21\_{i}}] \boldsymbol{w}(t) \Big] \end{aligned} \tag{40}$$

**Theorem 2.** *Consider the system (4). Given a prescribed* H<sup>∞</sup> *performance γ* > 0 *and a positive*

*I Y*0*E* + *DY*<sup>0</sup>

�

<sup>0</sup> *D* = *DX*0, *X*0*E* + *DX*>00 (42)

New Results on Robust ∞ Filter for Uncertain Fuzzy Descriptor Systems 473

<sup>0</sup> *D* = *DY*0, *Y*0*E* + *DY*>00 (43)

�

<sup>√</sup>2*λ*¯ *<sup>C</sup><sup>T</sup>* 1*i* �*T* ,

�*T* ,

�

� <sup>+</sup> �*H*¯ *<sup>T</sup>* 7*i H*¯ 7*j* � � ⎞ ⎠

<sup>M</sup>0*ij*(*ε*)*Y*−<sup>1</sup> *<sup>ε</sup>*

*Y<sup>ε</sup>* − *C* ˜¯ *T* 1*i* � *C* ˜¯ 1*j*

� *Y*−<sup>1</sup> <sup>0</sup> + *εN<sup>ε</sup>*

�*<sup>T</sup>* <sup>−</sup>*γ*<sup>2</sup> *<sup>I</sup>*

*X*0*E* + *DX*<sup>0</sup> *I*

*, i* = 1, 2, ··· ,*r, satisfying the following*

>0 (41)

Ψ11*ii* < 0, *i* = 1, 2, ··· ,*r* (44) Ψ22*ii* < 0, *i* = 1, 2, ··· ,*r* (45)

�

1 2 ,

*<sup>Y</sup><sup>ε</sup>* <sup>+</sup> *<sup>D</sup>*˜¯ <sup>12</sup>*C*<sup>ˆ</sup>

� *Eε* *jY<sup>ε</sup>* � (48)

Ψ11*ij* + Ψ11*ji* < 0, *i* < *j* ≤ *r* (46) Ψ22*ij* + Ψ22*ji* < 0, *i* < *j* ≤ *r* (47)

*constant δ, if there exist matrices X*0*, Y*0*,* B0*<sup>i</sup> and* C0*<sup>i</sup>*

<sup>0</sup> <sup>=</sup> *<sup>X</sup>*0*E*, *<sup>X</sup><sup>T</sup>*

� 0 0 0 *I* � *,*

� *AiY<sup>T</sup>*

� *A<sup>T</sup> <sup>i</sup> <sup>X</sup><sup>T</sup>*

*C* ˜¯ <sup>1</sup>*<sup>i</sup>* =

*and λ*¯ =

<sup>M</sup>0*ij*(*ε*) = <sup>−</sup>*A<sup>T</sup>*

<sup>−</sup>*γ*−<sup>2</sup> � *XεB* ˜¯ <sup>1</sup>*<sup>i</sup>* <sup>+</sup> �

*X<sup>ε</sup>* = � � *Y*0*C* ˜¯*T* 1*i* <sup>+</sup> <sup>C</sup>*<sup>T</sup>* 0*i D*˜¯ *T*

<sup>0</sup> <sup>+</sup> *<sup>Y</sup>*0*A<sup>T</sup>*

<sup>0</sup> + *X*0*Ai* + B0*<sup>i</sup>*

� *X*0*B* ˜¯ <sup>1</sup>*<sup>i</sup>* + B0*<sup>i</sup>*

*B* ˜¯ <sup>1</sup>*<sup>i</sup>* <sup>=</sup> �

*<sup>D</sup>*˜¯ <sup>12</sup> <sup>=</sup> �

*<sup>D</sup>*˜¯ <sup>21</sup>*<sup>i</sup>* <sup>=</sup> �

*r* ∑ *j*=1 � �*H*¯ *<sup>T</sup>* 2*i H*¯ 2*j*

2 *r* ∑ *i*=1

� *γρ*¯ *<sup>δ</sup> <sup>H</sup>*¯ *<sup>T</sup>* 1*i γρ*¯ *<sup>δ</sup> <sup>H</sup>*¯ *<sup>T</sup>* 5*i*

⎛ ⎝1 + *ρ*¯

*B*ˆ

*C*ˆ

*and N<sup>ε</sup>* = *D*

*γ* > 0 *is guaranteed. Furthermore, a suitable filter is of the form (***??***) with*

*A*ˆ*ij*(*ε*) = �

*<sup>i</sup>* = �

*<sup>i</sup>* <sup>−</sup> *<sup>X</sup>εAiY<sup>ε</sup>* <sup>−</sup> �

*X*<sup>0</sup> + *εX*˜

� (*Y*−<sup>1</sup>

*<sup>i</sup>* = C0*<sup>i</sup>*

*<sup>i</sup>* <sup>+</sup> *<sup>γ</sup>*−2*<sup>B</sup>* ˜¯ 1*i B* ˜¯ *T* <sup>1</sup>*<sup>j</sup>* (∗)*<sup>T</sup>*

<sup>12</sup>�*<sup>T</sup>* <sup>−</sup>*<sup>I</sup>*

*C*2*<sup>j</sup>* + *C<sup>T</sup>* 2*i* B*T* <sup>0</sup>*<sup>j</sup>* + *C* ˜¯ *T* 1*i C* ˜¯ <sup>1</sup>*<sup>j</sup>* (∗)*<sup>T</sup>*

*δI I* 0 *B*1*<sup>i</sup>* 0

000 −

*then there exists a sufficiently small ε*ˆ > 0 *such that for ε* ∈ (0,*ε*ˆ]*, the prescribed* H<sup>∞</sup> *performance*

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup><sup>ε</sup>*

*Y*−<sup>1</sup> 0

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup><sup>ε</sup>*

*<sup>Y</sup>*−<sup>1</sup> *<sup>ε</sup>* <sup>−</sup> *<sup>X</sup><sup>ε</sup>*

<sup>0</sup> )*<sup>T</sup>* <sup>−</sup> *<sup>Y</sup>*−<sup>1</sup> 0 � *.*

�

*Y*−<sup>1</sup> <sup>0</sup> − *X*<sup>0</sup>

<sup>√</sup>2*λ*¯ *<sup>ρ</sup>*¯*H*¯ *<sup>T</sup>* 4*i*

0 0 *δI D*21*<sup>i</sup> I*

�−<sup>1</sup>

�−<sup>1</sup> B0*<sup>i</sup>*

> � *B*ˆ *iC*<sup>2</sup>*<sup>j</sup>*

� *B*ˆ *iD*˜¯ <sup>21</sup>*<sup>i</sup>* � *B* ˜¯ *T* 1*j*

*<sup>E</sup><sup>ε</sup> and Y*−<sup>1</sup> *<sup>ε</sup>* =

<sup>√</sup>2*λ*¯ *<sup>I</sup>*

*<sup>D</sup>*˜¯ <sup>21</sup>*<sup>j</sup>*

� ,

<sup>0</sup> <sup>=</sup> *<sup>Y</sup>*0*E*, *<sup>Y</sup><sup>T</sup>*

�

*ε-independent linear matrix inequalities:*

*EX<sup>T</sup>*

*where E* =

*with*

*where*

*with X*˜ = *D*

� *X<sup>T</sup>* <sup>0</sup> − *X*<sup>0</sup> �

� *I* 0 0 0 � *EY<sup>T</sup>*

*, D* =

Ψ11*ij* =

Ψ22*ij* =

where

$$
\Delta\bar{A}\_{i} = \bar{\mathsf{F}}(\mathbf{x}(t), \hat{\mathbf{x}}(t), t)\bar{H}\_{1\_{i}}, \quad \Delta\bar{B}\_{1\_{i}} = \bar{\mathsf{F}}(\mathbf{x}(t), \hat{\mathbf{x}}(t), t)\bar{H}\_{2\_{i}}, \quad \Delta\bar{B}\_{2\_{i}} = \bar{\mathsf{F}}(\mathbf{x}(t), \hat{\mathbf{x}}(t), t)\bar{H}\_{3\_{i}}, \quad \Delta\bar{B}\_{2\_{i}} = \bar{\mathsf{F}}(\mathbf{x}(t), \hat{\mathbf{x}}(t), t)\bar{H}\_{3\_{i}}, \quad \Delta\bar{B}\_{3\_{i}} = \bar{\mathsf{F}}(\mathbf{x}(t), \hat{\mathbf{x}}(t), t)\bar{H}\_{5\_{i}}, \quad \Delta\bar{B}\_{4\_{i}} = \bar{\mathsf{F}}(\mathbf{x}(t), \hat{\mathbf{x}}(t), t)\bar{H}\_{5\_{i}}, \quad \Delta\bar{B}\_{5\_{i}} = \bar{\mathsf{F}}(\mathbf{x}(t), \hat{\mathbf{x}}(t), t)\bar{H}\_{7\_{i}}
$$
 
$$
\text{and} \quad \Delta\bar{D}\_{21\_{i}} = \bar{\mathsf{F}}(\mathbf{x}(t), \hat{\mathbf{x}}(t), t)\bar{H}\_{7\_{i}}
$$

with

$$
\bar{H}\_{1\_{l}} = \left[H\_{1\_{l}}^{T}A\_{1}^{T} \cdots A\_{r}^{T}H\_{1\_{1}}^{T} \cdots H\_{1\_{r}}^{T}\right]^{T}, \ \ \bar{H}\_{2\_{l}} = \left[H\_{2\_{l}}^{T}B\_{1\_{1}}^{T} \cdots B\_{1\_{r}}^{T}H\_{2\_{1}}^{T} \cdots H\_{2\_{r}}^{T}\right]^{T},
$$

$$
\bar{H}\_{3\_{l}} = \left[H\_{3\_{l}}^{T}B\_{2\_{l}}^{T} \cdots B\_{2\_{r}}^{T}H\_{3\_{1}}^{T} \cdots H\_{3\_{r}}^{T}\right]^{T}, \ \ \bar{H}\_{4\_{l}} = \left[H\_{4\_{l}}^{T}C\_{1\_{1}}^{T} \cdots C\_{1\_{r}}^{T}H\_{4\_{1}}^{T} \cdots H\_{4\_{r}}^{T}\right]^{T},
$$

$$
\bar{H}\_{5\_{l}} = \left[H\_{5\_{l}}^{T}C\_{2\_{l}}^{T} \cdots C\_{2\_{r}}^{T}H\_{5\_{1}}^{T} \cdots H\_{5\_{r}}^{T}\right]^{T}, \ \ \ \bar{H}\_{6\_{l}} = \left[H\_{6\_{l}}^{T}D\_{12\_{l}}^{T} \cdots D\_{12\_{r}}^{T}H\_{6\_{1}}^{T} \cdots H\_{6\_{r}}^{T}\right]^{T},
$$

$$
\bar{H}\_{7\_{l}} = \left[H\_{7\_{l}}^{T}D\_{21\_{l}}^{T} \cdots D\_{21\_{l}}^{T}H\_{7\_{l}}^{T} \cdots H\_{7\_{r}}^{T}\right]^{T} \text{ and}
$$

*<sup>F</sup>*¯(*x*(*t*), *<sup>x</sup>*ˆ(*t*), *<sup>t</sup>*) = *F*(*x*(*t*),*t*) (*μ*<sup>1</sup> − *μ*ˆ1) ··· (*μ<sup>r</sup>* − *μ*ˆ*r*) *F*(*x*(*t*),*t*)(*μ*<sup>1</sup> − *μ*ˆ1) ··· *F*(*x*(*t*),*t*)(*μ<sup>r</sup>* − *μ*ˆ*r*) . Note that �*F*¯(*x*(*t*), *<sup>x</sup>*ˆ(*t*), *<sup>t</sup>*)� ≤ *<sup>ρ</sup>*¯ where *<sup>ρ</sup>*¯ <sup>=</sup> {3*ρ*<sup>2</sup> <sup>+</sup> <sup>2</sup>} 1 <sup>2</sup> . *ρ*¯ is derived by utilizing the concept of vector norm in the basic system control theory and the fact that *μ<sup>i</sup>* ≥ 0, *μ*ˆ*<sup>i</sup>* ≥ 0, ∑*r <sup>i</sup>*=<sup>1</sup> *<sup>μ</sup><sup>i</sup>* <sup>=</sup> 1 and <sup>∑</sup>*<sup>r</sup> <sup>i</sup>*=<sup>1</sup> *μ*ˆ*<sup>i</sup>* = 1.

Note that the above technique is basically employed in order to obtain the plant's premise variable to be the same as the filter's premise variable; e.g. [17]. Now, the premise variable of the system is the same as the premise variable of the filter, thus we can apply the result given in Case I. By applying the same technique used in Case I, we have the following theorem.

**Theorem 2.** *Consider the system (4). Given a prescribed* H<sup>∞</sup> *performance γ* > 0 *and a positive constant δ, if there exist matrices X*0*, Y*0*,* B0*<sup>i</sup> and* C0*<sup>i</sup> , i* = 1, 2, ··· ,*r, satisfying the following ε-independent linear matrix inequalities:*

$$
\begin{bmatrix} X\_0 E + DX\_0 & I \\ I & Y\_0 E + DY\_0 \end{bmatrix} \tag{41}
$$

$$EX\_0^T = X\_0 E\_\prime \quad X\_0^T D = DX\_0 \cdot X\_0 E + DX\sharp \tag{42}$$

$$EY\_0^T = Y\_0 E, \quad Y\_0^T D = D Y\_0, \quad Y\_0 E + D Y \mathfrak{H} \tag{43}$$

$$\Psi\_{11\_{\tilde{\mu}}} < 0, \quad i = 1, 2, \dots, r \tag{44}$$

$$\Psi\_{22\_{\text{il}}} < 0, \quad \mathbf{i} = 1, 2, \dots, r \tag{45}$$

$$\Psi\_{11\_{\tilde{\mu}}} + \Psi\_{11\_{\tilde{\mu}}} < 0, \quad i < j \le r \tag{46}$$

$$\Psi\_{22\_{\parallel}} + \Psi\_{22\_{\parallel}} < 0, \quad i < j \le r \tag{47}$$

$$\begin{aligned} \text{where } \mathbf{E} = \begin{pmatrix} I & 0 \\ 0 & 0 \end{pmatrix}, D &= \begin{pmatrix} 0 \ 0 \\ 0 \ I \end{pmatrix}, \\ \mathbf{Y}\_{11\_{\mathcal{I}}} &= \begin{pmatrix} A\_{i}Y\_{0}^{T} + Y\_{0}A\_{i}^{T} + \gamma^{-2}\tilde{\mathcal{B}}\_{1\_{i}}\tilde{\mathcal{B}}\_{1\_{\mathcal{I}}}^{T} \ (\*)^{T} \\ \left[Y\_{0}\tilde{\mathcal{C}}\_{1\_{\mathcal{I}}}^{T} + \mathcal{C}\_{0\_{i}}^{T}\tilde{D}\_{12}^{T}\right]^{T} & -I \end{pmatrix} \\ \mathbf{Y}\_{22\_{\mathcal{I}}} &= \begin{pmatrix} A\_{i}^{T}X\_{0}^{T} + \mathcal{X}\_{0}A\_{i} + \mathcal{B}\_{0\_{i}}\mathcal{C}\_{2\_{\mathcal{I}}} + \mathcal{C}\_{2\_{i}}^{T}\mathcal{B}\_{0\_{\mathcal{I}}}^{T} + \tilde{\mathcal{C}}\_{1\_{i}}^{T}\tilde{\mathcal{C}}\_{1\_{\mathcal{I}}} \ (\*)^{T} \\ \left[\mathcal{X}\_{0}\tilde{\mathcal{B}}\_{1\_{\mathcal{I}}} + \mathcal{B}\_{0\_{i}}\tilde{D}\_{21\_{\mathcal{I}}}\right]^{T} & -\gamma^{2}I \end{pmatrix}. \end{aligned}$$

*with*

8 Will-be-set-by-IN-TECH

The fuzzy filter is assumed to be the same as the premise variables of the fuzzy system model. This actually means that the premise variables of fuzzy system model are assumed to be measurable. However, in general, it is extremely difficult to derive an accurate fuzzy system model by imposing that all premise variables are measurable. In this subsection, we do not impose that condition, we choose the premise variables of the filter to be different from the premise variables of fuzzy system model of the plant. In here, the premise variables of the filter are selected to be the estimated premise variables of the plant. In the other words, the premise variable of the fuzzy model *ν*(*t*) is unavailable for feedback which implies *μ<sup>i</sup>* is unavailable for feedback. Hence, we cannot select our filter which depends on *μi*. Thus, we select our filter as (5) where *μ*ˆ*<sup>i</sup>* depends on the premise variable of the filter which is different from *μi*. Let us re-express the system (1) in terms of *μ*ˆ*i*, thus the plant's premise variable becomes the same as the filter's premise variable. By doing so, the result given in the previous case can then be applied here. Note that it can be done by using the same technique

[*Ai* + <sup>Δ</sup>*A*¯*i*]*x*(*t*)+[*B*1*<sup>i</sup>* + <sup>Δ</sup>*B*¯

1*i* ]*x*(*t*) 

2*i*

<sup>1</sup>*<sup>i</sup>* = *<sup>F</sup>*¯(*x*(*t*), *<sup>x</sup>*ˆ(*t*), *<sup>t</sup>*)*H*¯ <sup>2</sup>*<sup>i</sup>*

<sup>2</sup>*<sup>i</sup>* = *<sup>F</sup>*¯(*x*(*t*), *<sup>x</sup>*ˆ(*t*), *<sup>t</sup>*)*H*¯ <sup>5</sup>*<sup>i</sup>*

, *<sup>H</sup>*¯ <sup>2</sup>*<sup>i</sup>* =

, *<sup>H</sup>*¯ <sup>4</sup>*<sup>i</sup>* =

, *<sup>H</sup>*¯ <sup>6</sup>*<sup>i</sup>* =

concept of vector norm in the basic system control theory and the fact that *μ<sup>i</sup>* ≥ 0, *μ*ˆ*<sup>i</sup>* ≥ 0,

Note that the above technique is basically employed in order to obtain the plant's premise variable to be the same as the filter's premise variable; e.g. [17]. Now, the premise variable of the system is the same as the premise variable of the filter, thus we can apply the result given in Case I. By applying the same technique used in Case I, we have the following theorem.

<sup>21</sup>*<sup>r</sup> <sup>H</sup><sup>T</sup>*

<sup>211</sup> ··· *<sup>D</sup><sup>T</sup>*

 *H<sup>T</sup>* <sup>2</sup>*<sup>i</sup> <sup>B</sup><sup>T</sup>*

 *H<sup>T</sup>* <sup>4</sup>*<sup>i</sup> <sup>C</sup><sup>T</sup>*

 *H<sup>T</sup>* <sup>6</sup>*<sup>i</sup> <sup>D</sup><sup>T</sup>*

<sup>71</sup> ··· *<sup>H</sup><sup>T</sup>* 7*r T*

*F*(*x*(*t*),*t*) (*μ*<sup>1</sup> − *μ*ˆ1) ··· (*μ<sup>r</sup>* − *μ*ˆ*r*) *F*(*x*(*t*),*t*)(*μ*<sup>1</sup> − *μ*ˆ1) ··· *F*(*x*(*t*),*t*)(*μ<sup>r</sup>* −

and <sup>Δ</sup>*D*¯ <sup>21</sup>*<sup>i</sup>* = *<sup>F</sup>*¯(*x*(*t*), *<sup>x</sup>*ˆ(*t*), *<sup>t</sup>*)*H*¯ <sup>7</sup>*<sup>i</sup>*

1*i* ]*w*(*t*)

> ]*w*(*t*)

<sup>2</sup>*<sup>i</sup>* = *<sup>F</sup>*¯(*x*(*t*), *<sup>x</sup>*ˆ(*t*), *<sup>t</sup>*)*H*¯ <sup>3</sup>*<sup>i</sup>*

<sup>21</sup> ··· *<sup>H</sup><sup>T</sup>* 2*r T* ,

<sup>41</sup> ··· *<sup>H</sup><sup>T</sup>* 4*r T* ,

> <sup>61</sup> ··· *<sup>H</sup><sup>T</sup>* 6*r T*

<sup>2</sup> . *ρ*¯ is derived by utilizing the

, <sup>Δ</sup>*D*¯ <sup>12</sup>*<sup>i</sup>* = *<sup>F</sup>*¯(*x*(*t*), *<sup>x</sup>*ˆ(*t*), *<sup>t</sup>*)*H*¯ <sup>6</sup>*<sup>i</sup>*

<sup>1</sup>*<sup>r</sup> <sup>H</sup><sup>T</sup>*

<sup>1</sup>*<sup>r</sup> <sup>H</sup><sup>T</sup>*

<sup>12</sup>*<sup>r</sup> <sup>H</sup><sup>T</sup>*

(40)

,

]*x*(*t*)+[*D*21*<sup>i</sup>* + <sup>Δ</sup>*D*¯ <sup>21</sup>*<sup>i</sup>*

, Δ*B*¯

<sup>11</sup> ··· *<sup>B</sup><sup>T</sup>*

<sup>11</sup> ··· *<sup>C</sup><sup>T</sup>*

<sup>121</sup> ··· *<sup>D</sup><sup>T</sup>*

1

and

**3.2. Case II–***ν*(*t*) **is unavailable for feedback**

as in subsection. After some manipulation, we get

*Eεx*˙(*t*) = ∑*<sup>r</sup>*

*z*(*t*) = ∑*<sup>r</sup>*

*y*(*t*) = ∑*<sup>r</sup>*

<sup>1</sup> ··· *<sup>A</sup><sup>T</sup>*

<sup>21</sup> ··· *<sup>B</sup><sup>T</sup>*

<sup>21</sup> ··· *<sup>C</sup><sup>T</sup>*

*<sup>i</sup>*=<sup>1</sup> *μ*ˆ*<sup>i</sup>* = 1.

*<sup>H</sup>*¯ <sup>7</sup>*<sup>i</sup>* = *H<sup>T</sup>* <sup>7</sup>*<sup>i</sup> <sup>D</sup><sup>T</sup>*

<sup>Δ</sup>*A*¯*<sup>i</sup>* = *<sup>F</sup>*¯(*x*(*t*), *<sup>x</sup>*ˆ(*t*), *<sup>t</sup>*)*H*¯ <sup>1</sup>*<sup>i</sup>*

<sup>1</sup>*<sup>i</sup>* = *<sup>F</sup>*¯(*x*(*t*), *<sup>x</sup>*ˆ(*t*), *<sup>t</sup>*)*H*¯ <sup>4</sup>*<sup>i</sup>*

where

with

*μ*ˆ*r*) 

∑*r*

Δ*C*¯

*<sup>H</sup>*¯ <sup>1</sup>*<sup>i</sup>* = *H<sup>T</sup>* <sup>1</sup>*<sup>i</sup> <sup>A</sup><sup>T</sup>*

*<sup>H</sup>*¯ <sup>3</sup>*<sup>i</sup>* = *H<sup>T</sup>* <sup>3</sup>*<sup>i</sup> <sup>B</sup><sup>T</sup>*

*<sup>H</sup>*¯ <sup>5</sup>*<sup>i</sup>* = *H<sup>T</sup>* <sup>5</sup>*<sup>i</sup> <sup>C</sup><sup>T</sup>*

*F*¯(*x*(*t*), *x*ˆ(*t*), *t*) =

*<sup>i</sup>*=<sup>1</sup> *<sup>μ</sup><sup>i</sup>* <sup>=</sup> 1 and <sup>∑</sup>*<sup>r</sup>*

*<sup>i</sup>*=<sup>1</sup> *μ*ˆ*<sup>i</sup>* 

*<sup>i</sup>*=<sup>1</sup> *μ*ˆ*<sup>i</sup>* 

*<sup>i</sup>*=<sup>1</sup> *μ*ˆ*<sup>i</sup>* 

, Δ*B*¯

, Δ*C*¯

*<sup>r</sup> <sup>H</sup><sup>T</sup>*

<sup>2</sup>*<sup>r</sup> <sup>H</sup><sup>T</sup>*

<sup>2</sup>*<sup>r</sup> <sup>H</sup><sup>T</sup>*

. Note that �*F*¯(*x*(*t*), *<sup>x</sup>*ˆ(*t*), *<sup>t</sup>*)� ≤ *<sup>ρ</sup>*¯ where *<sup>ρ</sup>*¯ <sup>=</sup> {3*ρ*<sup>2</sup> <sup>+</sup> <sup>2</sup>}

<sup>11</sup> ··· *<sup>H</sup><sup>T</sup>* 1*r T*

<sup>31</sup> ··· *<sup>H</sup><sup>T</sup>* 3*r T*

<sup>51</sup> ··· *<sup>H</sup><sup>T</sup>* 5*r T*

[*C*1*<sup>i</sup>* + <sup>Δ</sup>*C*¯

[*C*2*<sup>i</sup>* + <sup>Δ</sup>*C*¯

$$\begin{aligned} \tilde{\boldsymbol{B}}\_{1\_i} &= \left[ \boldsymbol{\delta I} \boldsymbol{I} \; \boldsymbol{I} \; \boldsymbol{0} \; \boldsymbol{B}\_{1\_i} \; \boldsymbol{0} \right] \; \\ \tilde{\boldsymbol{C}}\_{1\_i} &= \left[ \begin{array}{c} \underline{\gamma} \bar{\boldsymbol{\rho}} \boldsymbol{H}\_{1\_i}^T \; \frac{\boldsymbol{\gamma} \bar{\boldsymbol{\rho}}}{\boldsymbol{\delta}} \boldsymbol{H}\_{5\_i}^T \sqrt{2} \bar{\boldsymbol{\lambda}} \boldsymbol{\rho} \boldsymbol{H}\_{4\_i}^T \; \sqrt{2} \boldsymbol{\lambda} \boldsymbol{C}\_{1\_i}^T \right]^T \; \\ \tilde{\boldsymbol{D}}\_{12} &= \left[ \boldsymbol{0} \; \boldsymbol{0} \; \boldsymbol{0} \; - \sqrt{2} \bar{\boldsymbol{\lambda}} \boldsymbol{I} \right]^T \; \\ \tilde{\boldsymbol{D}}\_{21\_i} &= \left[ \boldsymbol{0} \; \boldsymbol{0} \; \boldsymbol{\delta} \; \boldsymbol{I} \; \boldsymbol{D}\_{21\_i} \; \boldsymbol{I} \right] \end{aligned} $$
 
$$\text{and } \bar{\boldsymbol{\lambda}} = \left( \boldsymbol{1} + \bar{\boldsymbol{\rho}}^2 \sum\_{i=1}^r \sum\_{j=1}^r \left[ ||\boldsymbol{H}\_{2\_i}^T \bar{\boldsymbol{H}}\_{2\_j} \boldsymbol{||} + ||\boldsymbol{H}\_{7\_i}^T \bar{\boldsymbol{H}}\_{7\_j} \boldsymbol{\|} \right] \right)^{\frac{1}{2}} \; \boldsymbol{I} $$

*then there exists a sufficiently small ε*ˆ > 0 *such that for ε* ∈ (0,*ε*ˆ]*, the prescribed* H<sup>∞</sup> *performance γ* > 0 *is guaranteed. Furthermore, a suitable filter is of the form (***??***) with*

$$\begin{array}{lcl}\hat{A}\_{ij}(\varepsilon) = \left[Y\_{\varepsilon}^{-1} - X\_{\varepsilon}\right]^{-1} \mathcal{M}\_{0\_{lj}}(\varepsilon)Y\_{\varepsilon}^{-1} \\ \hat{B}\_{i} &= \left[Y\_{0}^{-1} - X\_{0}\right]^{-1} \mathcal{B}\_{0\_{l}} \\ \hat{\mathcal{C}}\_{l} &= \mathcal{C}\_{0\_{l}}Y\_{0}^{-1} \end{array} \tag{48}$$

*where*

$$\begin{split} \mathcal{M}\_{\mathbf{0}\_{\tilde{\mathbf{y}}}}(\varepsilon) &= -A\_{\tilde{\mathbf{y}}}^{T} - \mathcal{X}\_{\varepsilon}A\_{\bar{\mathbf{}}}Y\_{\varepsilon} - \left[Y\_{\varepsilon}^{-1} - \mathcal{X}\_{\varepsilon}\right]\mathcal{B}\_{i}\mathcal{C}\_{\mathbf{2}\_{\tilde{\mathbf{y}}}}Y\_{\varepsilon} - \tilde{\mathcal{C}}\_{\mathbf{1}\_{i}}^{T}\left[\tilde{\mathcal{C}}\_{1\_{j}}Y\_{\varepsilon} + \tilde{\mathcal{D}}\_{12}\hat{\mathcal{C}}\_{j}Y\_{\varepsilon}\right] \\ & \qquad - \gamma^{-2}\left\{\mathcal{X}\_{\varepsilon}\tilde{\mathcal{B}}\_{1\_{i}} + \left[Y\_{\varepsilon}^{-1} - \mathcal{X}\_{\varepsilon}\right]\mathcal{B}\_{i}\tilde{\mathcal{D}}\_{21\_{i}}\right\}\tilde{\mathcal{B}}\_{1\_{j}}^{T} \\ \mathcal{X}\_{\varepsilon} &= \left\{\mathcal{X}\_{0} + \varepsilon\tilde{\mathcal{X}}\right\}\mathcal{E}\_{\varepsilon} \text{ and } \ Y\_{\varepsilon}^{-1} = \left\{Y\_{0}^{-1} + \varepsilon N\_{\varepsilon}\right\}\mathcal{E}\_{\varepsilon} \end{split}$$
 with  $\tilde{\mathcal{X}} = D\left(X\_{0}^{T} - X\_{0}\right)$  and  $N\_{\varepsilon} = D\left((Y\_{0}^{-1})^{T} - Y\_{0}^{-1}\right)$ .

*Proof:* It can be shown by employing the same technique used in the proof for Theorem 1.

**Plant Rule 1:** IF *x*1(*t*) is *M*1(*x*1(*t*)) THEN

**Plant Rule 2:** IF *x*1(*t*) is *M*2(*x*1(*t*)) THEN

<sup>1</sup> (*t*) *<sup>x</sup><sup>T</sup>*

*A*<sup>1</sup> = 

where *x*(*t*)=[*x<sup>T</sup>*

*A*ˆ <sup>11</sup>(*ε*) =

*A*ˆ <sup>21</sup>(*ε*) =

*B*ˆ <sup>1</sup> = 1.5835 3.2008

*C*ˆ <sup>1</sup> = *z*(*t*) = *C*<sup>11</sup> *x*(*t*),

*z*(*t*) = *C*<sup>12</sup> *x*(*t*),

<sup>2</sup> (*t*)]*T*, *<sup>w</sup>*(*t*)=[*w<sup>T</sup>*

bounded within 10% of their nominal values given in (49), we have

1 with *ε* = 100 *μ*H, *γ* = 0.6 and *δ* = 1, we obtain the following results:

, *B*ˆ

, *C*ˆ

−0.1 10 −1 −1

*C*<sup>1</sup> = 1 0 0 1

Deficient". *Case I-ν*(*t*) *are available for feedback*

 <sup>−</sup>0.0674 <sup>−</sup>0.3532 <sup>−</sup>30.7181 <sup>−</sup>4.3834

 <sup>−</sup>0.0928 <sup>−</sup>0.3138 <sup>−</sup>34.7355 <sup>−</sup>3.8964

<sup>−</sup>1.7640 <sup>−</sup>0.8190

*y*(*t*) = *C*<sup>21</sup> *x*(*t*) + *D*211*w*(*t*).

*y*(*t*) = *C*<sup>22</sup> *x*(*t*) + *D*212*w*(*t*)

, *A*<sup>2</sup> =

*Eεx*˙(*t*)=[*A*<sup>1</sup> + Δ*A*1]*x*(*t*) + *B*11*w*(*t*), *x*(0) = 0,

*Eεx*˙(*t*)=[*A*<sup>2</sup> + Δ*A*2]*x*(*t*) + *B*12*w*(*t*), *x*(0) = 0,

<sup>2</sup> (*t*)]*T*,

, *B*<sup>11</sup> = *B*<sup>12</sup> =

 0 0 0 0.1 ,

> 1 0 0 *ε* .

0.1 0 ,

New Results on Robust ∞ Filter for Uncertain Fuzzy Descriptor Systems 475

 <sup>−</sup>0.0674 <sup>−</sup>0.3532 <sup>−</sup>30.7181 <sup>−</sup>4.3834

 <sup>−</sup>0.0928 <sup>−</sup>0.3138 <sup>−</sup>34.7355 <sup>−</sup>3.8964

.

4.5977 <sup>−</sup>0.8190

,

,

−4.6 10 −1 −1

, *<sup>C</sup>*<sup>21</sup> = *<sup>C</sup>*<sup>22</sup> = *<sup>J</sup>*, *<sup>D</sup>*<sup>21</sup> =

 0 0 0 1 .

<sup>1</sup> (*t*) *<sup>w</sup><sup>T</sup>*

Δ*A*<sup>1</sup> = *F*(*x*(*t*), *t*)*H*<sup>11</sup> , Δ*A*<sup>2</sup> = *F*(*x*(*t*), *t*)*H*<sup>12</sup> and *E<sup>ε</sup>* =

*H*<sup>11</sup> = *H*<sup>12</sup> =

Now, by assuming that �*F*(*x*(*t*),*t*)� ≤ *ρ* = 1 and since the values of *R* are uncertain but

Note that the plot of the membership function Rules 1 and 2 is the same as in Figure 2. By employing the results given in Lemma 1 and the Matlab LMI solver, it is easy to realize that *ε* < 0.006 for the fuzzy filter design in Case I and *ε* < 0.008 for the fuzzy filter design in Case II, the LMIs become ill-conditioned and the Matlab LMI solver yields the error message, "Rank

In this case, *x*1(*t*) = *ν*(*t*) is assumed to be available for feedback; for instance, *J* = [1 0]. This implies that *μ<sup>i</sup>* is available for feedback. Using the LMI optimization algorithm and Theorem

<sup>12</sup>(*ε*) =

<sup>2</sup> = 1.2567 3.8766 ,

<sup>2</sup> =

, *A*ˆ

, *A*ˆ22(*ε*) =

## **4. Example**

Consider the tunnel diode circuit shown in Figure 1 where the tunnel diode is characterized by

$$i\_D(t) = 0.01v\_D(t) + 0.05v\_D^3(t).$$

Assuming that the inductance, *L*, is the parasitic parameter and letting *x*1(*t*) = *vC*(*t*) and

**Figure 1.** Tunnel diode circuit.

*x*2(*t*) = *iL*(*t*) as the state variables, we have

$$\begin{array}{l} \text{C} \dot{\mathbf{x}}\_1(t) = -0.01 \mathbf{x}\_1(t) - 0.05 \mathbf{x}\_1^3(t) + \mathbf{x}\_2(t) \\ L \dot{\mathbf{x}}\_2(t) = -\mathbf{x}\_1(t) - R \mathbf{x}\_2(t) + 0.1 w\_2(t) \\ y(t) = f \mathbf{x}(t) + 0.1 w\_1(t) \\ z(t) = \begin{bmatrix} \mathbf{x}\_1(t) \\ \mathbf{x}\_2(t) \end{bmatrix} \end{array} \tag{49}$$

where *w*(*t*) is the disturbance noise input, *y*(*t*) is the measurement output, *z*(*t*) is the state to be estimated and *J* is the sensor matrix. Note that the variables *x*1(*t*) and *x*2(*t*) are treated as the deviation variables (variables deviate from the desired trajectories). The parameters of the circuit are *C* = 100 *mF*, *R* = 10 ± 10% Ω and *L* = *ε H*. With these parameters (49) can be rewritten as

$$\begin{array}{l} \dot{\mathbf{x}}\_1(t) = -0.1\mathbf{x}\_1(t) + 0.5\mathbf{x}\_1^3(t) + 10\mathbf{x}\_2(t) \\ \dot{\mathbf{x}}\_2(t) = -\mathbf{x}\_1(t) - (10 + \Delta R)\mathbf{x}\_2(t) + 0.1w\_2(t) \\ y(t) = J\mathbf{x}(t) + 0.1w\_1(t) \\ z(t) = \begin{bmatrix} \mathbf{x}\_1(t) \\ \mathbf{x}\_2(t) \end{bmatrix}. \end{array} \tag{50}$$

For the sake of simplicity, we will use as few rules as possible. Assuming that |*x*1(*t*)| ≤ 3, the nonlinear network system (50) can be approximated by the following TS fuzzy model:

**Plant Rule 1:** IF *x*1(*t*) is *M*1(*x*1(*t*)) THEN

10 Will-be-set-by-IN-TECH

Consider the tunnel diode circuit shown in Figure 1 where the tunnel diode is characterized

Assuming that the inductance, *L*, is the parasitic parameter and letting *x*1(*t*) = *vC*(*t*) and

L <sup>D</sup>

+

−

*Cx*˙1(*t*) = <sup>−</sup>0.01*x*1(*t*) <sup>−</sup> 0.05*x*<sup>3</sup>

*y*(*t*) = *Jx*(*t*) + 0.1*w*1(*t*)

 *x*1(*t*) *x*2(*t*)

*<sup>x</sup>*˙1(*t*) = <sup>−</sup>0.1*x*1(*t*) + 0.5*x*<sup>3</sup>

*y*(*t*) = *Jx*(*t*) + 0.1*w*1(*t*)

 .

 *x*1(*t*) *x*2(*t*)

*z*(*t*) =

*z*(*t*) =

*Lx*˙2(*t*) = −*x*1(*t*) − *Rx*2(*t*) + 0.1*w*2(*t*)

where *w*(*t*) is the disturbance noise input, *y*(*t*) is the measurement output, *z*(*t*) is the state to be estimated and *J* is the sensor matrix. Note that the variables *x*1(*t*) and *x*2(*t*) are treated as the deviation variables (variables deviate from the desired trajectories). The parameters of the circuit are *C* = 100 *mF*, *R* = 10 ± 10% Ω and *L* = *ε H*. With these parameters (49) can be

*εx*˙2(*t*) = −*x*1(*t*) − (10 + Δ*R*)*x*2(*t*) + 0.1*w*2(*t*)

For the sake of simplicity, we will use as few rules as possible. Assuming that |*x*1(*t*)| ≤ 3, the nonlinear network system (50) can be approximated by the following TS fuzzy model:

*<sup>D</sup>*(*t*).

<sup>v</sup> <sup>v</sup> <sup>c</sup>

<sup>1</sup>(*t*) + *x*2(*t*)

<sup>1</sup>(*t*) + 10*x*2(*t*)

D

(49)

(50)

C

i

*iD*(*t*) = 0.01*vD*(*t*) + 0.05*v*<sup>3</sup>

*Proof:* It can be shown by employing the same technique used in the proof for Theorem 1.

**4. Example**

R

*x*2(*t*) = *iL*(*t*) as the state variables, we have

**Figure 1.** Tunnel diode circuit.

rewritten as

i i <sup>L</sup> c

by

$$\begin{aligned} E\_{\varepsilon} \dot{\mathbf{x}}(t) &= [A\_1 + \Delta A\_1] \mathbf{x}(t) + B\_{1\_1} w(t), \quad \mathbf{x}(0) = \mathbf{0}, \\ z(t) &= \mathbf{C}\_{1\_1} \mathbf{x}(t), \\ y(t) &= \mathbf{C}\_{2\_1} \mathbf{x}(t) + D\_{21\_1} w(t). \end{aligned}$$

**Plant Rule 2:** IF *x*1(*t*) is *M*2(*x*1(*t*)) THEN

$$\begin{aligned} E\_{\varepsilon} \dot{\mathbf{x}}(t) &= [A\_2 + \Delta A\_2] \mathbf{x}(t) + B\_{1\_2} w(t), \quad \mathbf{x}(0) = \mathbf{0}, \\ z(t) &= \mathbf{C}\_{1\_2} \mathbf{x}(t), \\ y(t) &= \mathbf{C}\_{2\_2} \mathbf{x}(t) + D\_{21\_2} w(t) \end{aligned}$$

where *x*(*t*)=[*x<sup>T</sup>* <sup>1</sup> (*t*) *<sup>x</sup><sup>T</sup>* <sup>2</sup> (*t*)]*T*, *<sup>w</sup>*(*t*)=[*w<sup>T</sup>* <sup>1</sup> (*t*) *<sup>w</sup><sup>T</sup>* <sup>2</sup> (*t*)]*T*,

$$A\_{1} = \begin{bmatrix} -0.1 & 10 \\ -1 & -1 \end{bmatrix}, \ A\_{2} = \begin{bmatrix} -4.6 & 10 \\ -1 & -1 \end{bmatrix}, \ B\_{1\_{1}} = B\_{1\_{2}} = \begin{bmatrix} 0 & 0 \\ 0 & 0.1 \end{bmatrix},$$

$$\mathbf{C}\_{1} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}, \ \mathbf{C}\_{2\_{1}} = \mathbf{C}\_{2\_{2}} = I\_{\prime} \ \ D\_{21} = \begin{bmatrix} 0.1 \ 0 \end{bmatrix},$$

$$\Delta A\_{1} = F(\mathbf{x}(t), t)H\_{1\_{1}} \ \Delta A\_{2} = F(\mathbf{x}(t), t)H\_{1\_{2}} \ \text{and} \ \ E\_{\varepsilon} = \begin{bmatrix} 1 \ 0 \\ 0 \ \varepsilon \end{bmatrix}.$$

Now, by assuming that �*F*(*x*(*t*),*t*)� ≤ *ρ* = 1 and since the values of *R* are uncertain but bounded within 10% of their nominal values given in (49), we have

$$H\_{1\_1} = H\_{1\_2} = \begin{bmatrix} 0 \ 0 \\ 0 \ 1 \end{bmatrix}.$$

Note that the plot of the membership function Rules 1 and 2 is the same as in Figure 2. By employing the results given in Lemma 1 and the Matlab LMI solver, it is easy to realize that *ε* < 0.006 for the fuzzy filter design in Case I and *ε* < 0.008 for the fuzzy filter design in Case II, the LMIs become ill-conditioned and the Matlab LMI solver yields the error message, "Rank Deficient". *Case I-ν*(*t*) *are available for feedback*

In this case, *x*1(*t*) = *ν*(*t*) is assumed to be available for feedback; for instance, *J* = [1 0]. This implies that *μ<sup>i</sup>* is available for feedback. Using the LMI optimization algorithm and Theorem 1 with *ε* = 100 *μ*H, *γ* = 0.6 and *δ* = 1, we obtain the following results:

$$\begin{aligned} \hat{A}\_{11}(\varepsilon) &= \begin{bmatrix} -0.0674 & -0.3532 \\ -30.7181 & -4.3834 \end{bmatrix}, & \hat{A}\_{12}(\varepsilon) &= \begin{bmatrix} -0.0674 & -0.3532 \\ -30.7181 & -4.3834 \end{bmatrix}, \\\ \hat{A}\_{21}(\varepsilon) &= \begin{bmatrix} -0.0928 & -0.3138 \\ -34.7355 & -3.8964 \end{bmatrix}, & \hat{A}\_{22}(\varepsilon) &= \begin{bmatrix} -0.0928 & -0.3138 \\ -34.7355 & -3.8964 \end{bmatrix}, \\\ \hat{B}\_{1} &= \begin{bmatrix} 1.5835 \\ 3.2008 \end{bmatrix}, & \hat{B}\_{2} &= \begin{bmatrix} 1.2567 \\ 3.8766 \end{bmatrix}, \\\ \hat{C}\_{1} &= \begin{bmatrix} -1.7640 & -0.8190 \end{bmatrix}, & \hat{C}\_{2} &= \begin{bmatrix} 4.5977 & -0.8190 \end{bmatrix}. \end{aligned}$$

The resulting fuzzy filter is

0

<sup>0</sup> (*z*(*t*)−*z*ˆ(*t*))*T*(*z*(*t*)−*z*ˆ(*t*))*dt*

<sup>0</sup> *<sup>w</sup>T*(*t*)*w*(*t*)*dt*

.

*Case I, and* 0.25 *H, i.e., ε* ∈ (0, 0.25] *H in Case II.*

*Tf*

*where <sup>γ</sup>* <sup>=</sup> <sup>√</sup>

0.05

0.1

0.15

Ratio of the filter error energy to the disturbance energy

0.2

0.25

0.3

where

*Tf*

*E<sup>ε</sup>* ˙ *x*ˆ(*t*) =

*z*ˆ(*t*) =

2 ∑ *i*=1

2 ∑ *i*=1 *μ*ˆ*iC*ˆ *ix*ˆ(*t*)

2 ∑ *j*=1

*μ*ˆ*iμ*ˆ*jA*ˆ*ij*(*ε*)*x*ˆ(*t*) +

*μ*ˆ1 = *M*1(*x*ˆ1(*t*)) and *μ*ˆ2 = *M*2(*x*ˆ1(*t*)).

0 50 100 150 200

Time (sec)

**Remark 2.** *The ratios of the filter error energy to the disturbance input noise energy are depicted in Figure 3 when ε* = 100 *μH. The disturbance input signal, w*(*t*)*, which was used during the simulation is the rectangular signal (magnitude 0.9 and frequency 0.5 Hz). Figures 4(a) - 4(b), respectively, show the responses of x*1(*t*) *and x*2(*t*) *in Cases I and II. Table I shows the performance index γ with different values of ε in Cases I and II. After* 50 *seconds, the ratio of the filter error energy to the disturbance input noise energy tends to a constant value which is about* 0.02 *in Case I and* 0.08 *in Case II. Thus, in Case I*

*value* 0.6*. From Table 9.1, the maximum value of ε that guarantees the* L2*-gain of the mapping from the exogenous input noise to the filter error energy being less than* 0.6 *is* 0.30 *H, i.e., ε* ∈ (0, 0.30] *H in*

0.02 <sup>=</sup> 0.141 *and in Case II where <sup>γ</sup>* <sup>=</sup> <sup>√</sup>0.08 <sup>=</sup> 0.283*, both are less than the prescribed*

**Figure 3.** The ratio of the filter error energy to the disturbance noise energy:

2 ∑ *i*=1 *μ*ˆ*iB*ˆ *iy*(*t*)

New Results on Robust ∞ Filter for Uncertain Fuzzy Descriptor Systems 477

Case I Case II

**Figure 2.** Membership functions for the two fuzzy set.

Hence, the resulting fuzzy filter is

$$\begin{aligned} E\_{\varepsilon} \dot{\mathfrak{X}}(t) &= \sum\_{i=1}^{2} \sum\_{j=1}^{2} \mu\_{i} \mu\_{j} \hat{A}\_{ij}(\varepsilon) \mathfrak{X}(t) + \sum\_{i=1}^{2} \mu\_{i} \hat{B}\_{i} \mathfrak{y}(t) \\\\ \hat{z}(t) &= \sum\_{i=1}^{2} \mu\_{i} \hat{\mathbf{C}}\_{i} \mathfrak{X}(t) \end{aligned}$$

where

$$
\mu\_1 = M\_1(\alpha\_1(t)) \text{ and } \mu\_2 = M\_2(\alpha\_1(t)).
$$

*Case II: ν*(*t*) *are unavailable for feedback*

In this case, *x*1(*t*) = *ν*(*t*) is assumed to be unavailable for feedback; for instance, *J* = [0 1]. This implies that *μ<sup>i</sup>* is unavailable for feedback. Using the LMI optimization algorithm and Theorem 2 with *ε* = 100 *μ*H, *γ* = 0.6 and *δ* = 1, we obtain the following results:

$$
\begin{aligned}
\hat{A}\_{11}(\varepsilon) &= \begin{bmatrix} -2.3050 & -0.4186 \\ -32.3990 - 4.4443 \end{bmatrix}, & \hat{A}\_{12}(\varepsilon) &= \begin{bmatrix} -2.3050 & -0.4186 \\ -32.3990 - 4.4443 \end{bmatrix}, \\
\hat{A}\_{21}(\varepsilon) &= \begin{bmatrix} -2.3549 & -0.3748 \\ -32.4539 & -3.9044 \end{bmatrix}, & \hat{A}\_{22}(\varepsilon) &= \begin{bmatrix} -2.3549 & -0.3748 \\ -32.4539 & -3.9044 \end{bmatrix}, \\
\hat{B}\_{1} &= \begin{bmatrix} -0.3053 \\ 3.9938 \end{bmatrix}' & \hat{B}\_{2} = \begin{bmatrix} -0.3734 \\ 5.1443 \end{bmatrix}' \\
\hat{C}\_{1} &= \begin{bmatrix} 4.3913 & -0.1406 \end{bmatrix}, & \hat{C}\_{2} = \begin{bmatrix} 1.9832 & -0.1406 \end{bmatrix}.
\end{aligned}
$$

The resulting fuzzy filter is

$$\begin{aligned} E\_{\varepsilon} \dot{\mathfrak{X}}(t) &= \sum\_{i=1}^{2} \sum\_{j=1}^{2} \hat{\mu}\_{i} \hat{\mu}\_{j} \hat{A}\_{ij}(\varepsilon) \mathfrak{X}(t) + \sum\_{i=1}^{2} \hat{\mu}\_{i} \hat{B}\_{i} \mathfrak{y}(t), \\ \hat{\varepsilon}(t) &= \sum\_{i=1}^{2} \hat{\mu}\_{i} \hat{\mathfrak{C}}\_{i} \mathfrak{X}(t) \end{aligned}$$

where

12 Will-be-set-by-IN-TECH

1

2

M (x )

1

x

 <sup>−</sup>2.3050 <sup>−</sup>0.4186 −32.3990 −4.4443

 <sup>−</sup>2.3549 <sup>−</sup>0.3748 −32.4539 −3.9044

.

 ,

1.9832 <sup>−</sup>0.1406

 ,

 ,

0

1 −3 3

*μiμjA*ˆ*ij*(*ε*)*x*ˆ(*t*) +

*μ*<sup>1</sup> = *M*1(*x*1(*t*)) and *μ*<sup>2</sup> = *M*2(*x*1(*t*)).

In this case, *x*1(*t*) = *ν*(*t*) is assumed to be unavailable for feedback; for instance, *J* = [0 1]. This implies that *μ<sup>i</sup>* is unavailable for feedback. Using the LMI optimization algorithm and

, *A*ˆ <sup>12</sup>(*ε*) =

, *A*ˆ <sup>22</sup>(*ε*) =

<sup>2</sup> = 

<sup>2</sup> =

−0.3734 5.1443

Theorem 2 with *ε* = 100 *μ*H, *γ* = 0.6 and *δ* = 1, we obtain the following results:

, *B*ˆ

, *C*ˆ

2 ∑ *i*=1 *μiB*ˆ *iy*(*t*)

1

M (x )

**Figure 2.** Membership functions for the two fuzzy set.

*E<sup>ε</sup>* ˙ *x*ˆ(*t*) =

*z*ˆ(*t*) =

2 ∑ *i*=1

2 ∑ *i*=1 *μiC*ˆ *ix*ˆ(*t*)

2 ∑ *j*=1

Hence, the resulting fuzzy filter is

*Case II: ν*(*t*) *are unavailable for feedback*

−0.3053 3.9938

 <sup>−</sup>2.3050 <sup>−</sup>0.4186 −32.3990 −4.4443

 <sup>−</sup>2.3549 <sup>−</sup>0.3748 −32.4539 −3.9044

4.3913 <sup>−</sup>0.1406

where

*A*ˆ <sup>11</sup>(*ε*) =

*A*ˆ <sup>21</sup>(*ε*) =

*B*ˆ <sup>1</sup> = 

*C*ˆ <sup>1</sup> = 1

$$
\hat{\mu}\_1 = M\_1(\pounds\_1(t)) \text{ and } \pounds\_2 = M\_2(\pounds\_1(t)).
$$

**Figure 3.** The ratio of the filter error energy to the disturbance noise energy: *Tf* <sup>0</sup> (*z*(*t*)−*z*ˆ(*t*))*T*(*z*(*t*)−*z*ˆ(*t*))*dt Tf* <sup>0</sup> *<sup>w</sup>T*(*t*)*w*(*t*)*dt* .

**Remark 2.** *The ratios of the filter error energy to the disturbance input noise energy are depicted in Figure 3 when ε* = 100 *μH. The disturbance input signal, w*(*t*)*, which was used during the simulation is the rectangular signal (magnitude 0.9 and frequency 0.5 Hz). Figures 4(a) - 4(b), respectively, show the responses of x*1(*t*) *and x*2(*t*) *in Cases I and II. Table I shows the performance index γ with different values of ε in Cases I and II. After* 50 *seconds, the ratio of the filter error energy to the disturbance input noise energy tends to a constant value which is about* 0.02 *in Case I and* 0.08 *in Case II. Thus, in Case I where <sup>γ</sup>* <sup>=</sup> <sup>√</sup> 0.02 <sup>=</sup> 0.141 *and in Case II where <sup>γ</sup>* <sup>=</sup> <sup>√</sup>0.08 <sup>=</sup> 0.283*, both are less than the prescribed value* 0.6*. From Table 9.1, the maximum value of ε that guarantees the* L2*-gain of the mapping from the exogenous input noise to the filter error energy being less than* 0.6 *is* 0.30 *H, i.e., ε* ∈ (0, 0.30] *H in Case I, and* 0.25 *H, i.e., ε* ∈ (0, 0.25] *H in Case II.*

The performance index *γ ε* Fuzzy Filter in Case I Fuzzy Filter in Case II

New Results on Robust ∞ Filter for Uncertain Fuzzy Descriptor Systems 479

0.0001 0.141 0.283 0.1 0.316 0.509 0.25 0.479 0.596 0.26 0.500 > 0.6 0.30 0.591 > 0.6 0.31 > 0.6 > 0.6

The problem of designing a robust H<sup>∞</sup> fuzzy *ε*-independent filter for a TS fuzzy descriptor system with parametric uncertainties has been considered. Sufficient conditions for the existence of the robust H<sup>∞</sup> fuzzy filter have been derived in terms of a family of *ε*-independent LMIs. A numerical simulation example has been also presented to illustrate the theory

*Department of Electronic and Telecommunication Engineering at King Mongkut's University of*

[1] H. K. Khalil, "Feedback control of nonstandard singularly perturbed systems," *IEEE*

[2] Z. Gajic and M. Lim, "A new filtering method for linear singularly perturbed systems,"

[3] X. Shen and L. Deng, "Decomposition solution of H<sup>∞</sup> filter gain in singularly perturbed

[4] M.T. Lim and Z. Gajic, "Reduced-Order H<sup>∞</sup> optimal filtering for systems with slow and

[5] P. Shi and V. Dragan, "Asymptotic H<sup>∞</sup> control of singularly perturbed system with parametric uncertainties," *IEEE Trans. Automat. Contr.*, vol. 44, pp. 1738–1742, 1999. [6] P. V. Kokotovic, H. K. Khalil, and J. O'Reilly, *Singular Perturbation Methods in Control:*

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**Table 1.** The performance index *γ* of the system with different values of *ε*.

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systems," *Signal Processing*, vol. 55, pp. 313–320, 1996.

*Analysis and Design*, London: Academic Press, 1986.

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**5. Conclusion**

development.

**Author details**

**6. References**

Wudhichai Assawinchaichote

*Technology Thonburi, Bangkok, Thailand*

**Figure 4.** The histories of the state variables, *x*1(*t*) and *x*2(*t*).


478 Fuzzy Controllers – Recent Advances in Theory and Applications New Results on Robust <sup>H</sup><sup>∞</sup> Filter for Uncertain Fuzzy Descriptor Systems <sup>15</sup> New Results on Robust ∞ Filter for Uncertain Fuzzy Descriptor Systems 479

**Table 1.** The performance index *γ* of the system with different values of *ε*.

## **5. Conclusion**

14 Will-be-set-by-IN-TECH

x 1 (t)

Case I: fuzzy estimated x1

Case II: fuzzy estimated x1

(t)

(t)

0 5 10 15

Time (sec)

0 5 10 15

Case I: fuzzy estimated x2

Case II: fuzzy estimated x2

(t)

(t)

x 2 (t)

Time (sec)

(b) The histories of *x*2(*t*)

(a) The histories of *x*1(*t*)

−0.2

0 0.02 0.04 0.06

The state variable, x2(t)

−0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02

**Figure 4.** The histories of the state variables, *x*1(*t*) and *x*2(*t*).

−0.1

0

0.1

The state variable, x1(t)

0.2

0.3

The problem of designing a robust H<sup>∞</sup> fuzzy *ε*-independent filter for a TS fuzzy descriptor system with parametric uncertainties has been considered. Sufficient conditions for the existence of the robust H<sup>∞</sup> fuzzy filter have been derived in terms of a family of *ε*-independent LMIs. A numerical simulation example has been also presented to illustrate the theory development.

## **Author details**

Wudhichai Assawinchaichote

*Department of Electronic and Telecommunication Engineering at King Mongkut's University of Technology Thonburi, Bangkok, Thailand*

## **6. References**

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© 2012 Manai and Benrejeb, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**Robust Stabilization for Uncertain** 

Yassine Manai and Mohamed Benrejeb

http://dx.doi.org/10.5772/48075

**1. Introduction** 

performances.

Additional information is available at the end of the chapter

investigated with non-quadratic Lyapunov functions [4]-[6], [7].

class gives less conservative stability criteria than independent ones.

conservatism entailed in the previous results using quadratic function.

**Takagi-Sugeno Fuzzy Continuous Model with** 

Fuzzy control systems have experienced a big growth of industrial applications in the recent decades, because of their reliability and effectiveness. Many researches are investigated on the Takagi-Sugeno models [1], [2] and [3] last decades. Two classes of Lyapunov functions are used to analysis these systems: quadratic Lyapunov functions and non-quadratic Lyapunov ones which are less conservative than first class. Many researches are

Recently, Takagi–Sugeno fuzzy model approach has been used to examine nonlinear systems with time-delay, and different methodologies have been proposed for analysis and synthesis of this type of systems [1]-[11], [12]-[13]. Time delay often occurs in many dynamical systems such as biological systems, chemical system, metallurgical processing system and network system. Their existences are frequently a cause of infeasibility and poor

The stability approaches are divided into two classes in term of delay. The fist one tries to develop delay independent stability criteria. The second class depends on the delay size of the time delay, and it called delay dependent stability criteria. Generally, delay dependent

Two classes of Lyapunov-Razumikhin function are used to analysis these systems: quadratic Lyapunov-Razumikhin function and non-quadratic Lyapunov- Razumikhin ones. The use of first class brings much conservativeness in the stability test. In order to reduce the

As the information about the time derivatives of membership function is considered by the PDC fuzzy controller, it allows the introduction of slack matrices to facilitate the stability

and reproduction in any medium, provided the original work is properly cited.

**Time-Delay Based on Razumikhin Theorem** 
