Filtering of interval type-2 fuzzy time-delay systems under a unified frame

Abstract

This study investigates the problems of filter design for nonlinear time-delay systems with parameter uncertainties under a unified frame. The nonlinear systems subjected to parameter uncertainties are represented by an interval type-2 (IT2) Takagi-Sugeno (T-S) fuzzy time-delay model. A novel IT2 fuzzy filter is designed by establishing a Lyapunov-Krasovskii function, so that the filtering error system can guarantee the passive, dissipative, H_∞ and L₂ - L_∞ performance indexes under a unified frame. The proposed criterion is presented via linear matrix inequalities (LMIs). Two numerical examples are given to demonstrate the capability of the suggested results.

Keywords

Interval type-2 T-S fuzzy systems time-delay filtering unified frame

1 Introduction

T-S fuzzy model is a powerful tool that describes complex global nonlinear systems using a set of simple local linear models. In recent years, the problems of stability analysis, controller design and filter design for T-S fuzzy systems have been developed, and a large number of theoretical results have been achieved. Common quadratic Lyapunov functions are frequently used to conduct stability analysis of T-S fuzzy systems [1 –4]. However, these functions lead to conservative results. Therefore, piecewise and fuzzy Lyapunov functions are proposed to avoid the conservativeness in stability analysis [5 –13]. Moreover, several results on controller design for T-S fuzzy systems have also been presented in [14 –16]. A novel output feedback controller for discrete-time systems based on observer were designed in [14]. A fuzzy state observer and a fuzzy controller for nonlinear systems were proposed, and the conditions of observability and controllability were achieved in [15]. Based on the piecewise Lyapunov functions, a new fuzzy controller for discrete-time fuzzy dynamic systems was designed in [16]. In recent years, the problems of filter design have attracted the interest of researchers. A novel approach to synthesize and generalize filters for discrete-time fuzzy dynamic systems on the basis of a piecewise Lyapunov function was proposed in [17]. The nonfragile H_∞ filter for continuous-time T-S fuzzy systems was designed in [18] under the assumption that it has two types of multiplicative gain variations. However, time delays are not considered in these articles.

Time delays cannot be avoided in various complex nonlinear systems, such as communication networks, power system and mechanical systems. The existence of time delays may lead to instability and even cause performance deterioration of control systems. Stability analysis and control design of dynamic time-delay systems have received considerable attention in recent years. Numerous results have been reported on the problem of stability analysis and controller design for T-S fuzzy time-delay systems in [19 –23]. The aforementioned results are obtained by type-1 fuzzy set theory. The type-1 T-S fuzzy model can perfectly deal with nonlinearities. However, the parameter uncertainties were ineffectively handled by the type-1 T-S fuzzy model [24].

In practical control systems, the parameter uncertainties are inescapable. An IT2 fuzzy system was proposed to represent nonlinear systems subjected to parameter uncertainties in [24] because of the limitations of the type-1 T-S fuzzy model in dealing with parameter uncertainties. Considerable results were reported which concerning IT2 T-S fuzzy systems in [25 –32]. From the IT2 fuzzy model, the problem of H_∞ model reduction for nonlinear systems with parameter uncertainties was proposed in [30]. The filtering of IT2 T-S fuzzy systems with intermittent measurements was proposed in [31]. The filter design for IT2 T-S fuzzy systems with D stability constraints was investigated in [32]. However, the problem of filter design for IT2 fuzzy systems with time delays was not considered in previous literature. Particularly, the filter design for IT2 fuzzy time-delay systems under a unified frame is still a challenging issue that inspired the present study.

This study focuses on investigating the passive, dissipative, H_∞, and L₂ - L_∞ filter design problems for IT2 fuzzy time-delay systems under a unified frame. By constructing a Lyapunov-Krasovskii functional, a novel criterion is developed via linear matrix inequalities (LMIs). In comparison with the existing results, the main advantages of this study are summarized as follows: (1) The problem of filter design for IT2 T-S fuzzy time-delay systems is first proposed under a unified frame. By establishing a Lyapunov-Krasovskii function, so that the filtering error system can guarantee the passive, dissipative, H_∞, and L₂ - L_∞ performance indexes. The proposed method generalizes the results in [32]. (2) The proposed IT2 fuzzy filter does not have the same premise membership functions as the IT2 fuzzy system, which enhances the design flexibility of the proposed filter. (3) The proposed conditions are presented in the form of LMIs using some advanced matrix inequalities.

The remainder organization of this paper is as follows: The problem formulation and preliminaries are discussed in Section 2. The main results and relevant lemmas are presented in Section 3. Two simulation examples that show the effectiveness of the results are described in Section 4. The last section provides the conclusion.

2 Problem formulation

Notation: The transpose of a matrix and its inverse are respectively denoted by superscript “T” and “-1”. Rⁿ denotes the n-dimensional Euclidean space. Sym{A} denotes A + A^T for simplicity. In symmetric block matrices, “*” denotes the transposed elements.

A nonlinear time-delay system with parameter uncertainties is described by the following IT2 T-S fuzzy model with p plant rules:

Plant rule i: IF f₁ (x (t)) is ${\tilde{M}}_{i 1}$ and...and f_r (x (t)) is ${\tilde{M}}_{ir}$ , THEN:

$\begin{matrix} \dot{x} (t) & = & A_{i} x (t) + A_{di} x (t - h) + D_{1 i} w (t), \\ z (t) & = & C_{i} x (t) + C_{di} x (t - h) + D_{2 i} w (t), \\ y (t) & = & E_{i} x (t) + E_{di} x (t - h) + D_{3 i} w (t), \end{matrix}$ (1) where ${\tilde{M}}_{i α}$ is an IT2 fuzzy set, f_α (x (t)) represents the premise variables, i = 1, 2, ⋯ p, α = 1, 2, ⋯ r, and r is a positive integer; x (t) ∈ Rⁿ, z (t) ∈ R^v, y (t) ∈ R^m, and w ∈ R^q are the system sate vector, control output, measure output, and disturbance input, respectively; and the known matrices A_i, A_di, D_1i, C_i, C_di, D_2i, E_i, E_di, and D_3i have the appropriate dimensions. The firing strength of the i-th rule is defined as: $W_{i} (x (t)) = [{\underline{w}}_{i} (x (t)), {\bar{w}}_{i} (x (t))], i = 1, 2, \dots p,$ where: $\begin{matrix} {\underline{w}}_{i} (x (t)) & = & \prod_{α}^{r} {\underline{μ}}_{{\tilde{M}}_{i α}} (f_{α} (x (t))) \geq 0, \\ {\bar{w}}_{i} (x (t)) & = & \prod_{α}^{r} {\bar{μ}}_{{\tilde{M}}_{i α}} (f_{α} (x (t))) \geq 0, \end{matrix}$ with ${\underline{μ}}_{{\tilde{M}}_{i α}} (f_{α} (x (t)))$ and ${\bar{μ}}_{{\tilde{M}}_{i α}} (f_{α} (x (t)))$ denoting the lower and upper membership grades, respectively. The IT2 T-S fuzzy time-delay system can be defined as:

$\begin{matrix} \dot{x} (t) & = & \sum_{i = 1}^{p} {\tilde{w}}_{i} (x (t)) \\ [A_{i} x (t) + A_{di} x (t - h) + D_{1 i} w (t)], \\ z (t) & = & \sum_{i = 1}^{p} {\tilde{w}}_{i} (x (t)) \\ [C_{i} x (t) + C_{di} x (t - h) + D_{2 i} w (t)], \\ y (t) & = & \sum_{i = 1}^{p} {\tilde{w}}_{i} (x (t)) \\ [E_{i} x (t) + E_{di} x (t - h) + D_{3 i} w (t)], \end{matrix}$ (2) where: $\begin{matrix} {\tilde{w}}_{i} (x (t)) & = & {\underline{a}}_{i} (x (t)) {\underline{w}}_{i} (x (t)) \\ + {\bar{a}}_{i} (x (t)) {\bar{w}}_{i} (x (t)) \geq 0, \forall i, \\ \sum_{i = 1}^{p} {\tilde{w}}_{i} (x (t)) = 1, \end{matrix}$ where ${\underline{a}}_{i} (x (t)) \in [0, 1]$ and ${\bar{a}}_{i} (x (t)) \in [0, 1]$ denote the nonlinear functions and satisfy ${\underline{a}}_{i} (x (t)) + {\bar{a}}_{i} (x (t)) = 1$ , ∀i.

The following definition gives the fuzzy H_∞ filter with p rules.

Rulej: If g₁ (x (t)) is ${\tilde{N}}_{j 1}$ and...and g_l (x (t)) is ${\tilde{N}}_{jl}$ , THEN:

$\begin{matrix} \dot{\hat{x}} (t) & = & A_{fj} \hat{x} (t) + B_{fj} y (t) \\ z_{f} (t) & = & C_{fj} \hat{x} (t), \end{matrix}$ (3) where ${\tilde{N}}_{j β}$ is an IT2 fuzzy set and g_β (x (t)) denote the premise variables, j = 1, 2, ⋯ p, β = 1, 2, ⋯ l, l is a positive integer; A_fj, B_fj and C_fj are the filter parameters to be designed. The firing strength of the j-th rule is defined as: $M_{j} (x (t)) = [{\underline{m}}_{j} (x (t)), {\bar{m}}_{j} (x (t))], j = 1, 2, \dots p,$ where: $\begin{matrix} {\underline{m}}_{j} (x (t)) = \prod_{β}^{l} {\underline{μ}}_{{\tilde{N}}_{j β}} (g_{β} (x (t))) \geq 0, \\ {\bar{m}}_{j} (x (t)) = \prod_{β}^{l} {\bar{μ}}_{{\tilde{N}}_{j β}} (g_{β} (x (t))) \geq 0, \end{matrix}$ with ${\underline{μ}}_{{\tilde{N}}_{j β}} (g_{β} (x (t))) \geq 0$ and ${\bar{μ}}_{{\tilde{N}}_{j β}} (g_{β} (x (t))) \geq 0$ denoting the lower and upper membership grades, respectively. The final form of the IT2 fuzzy filter is expressed as:

$\begin{matrix} \dot{\hat{x}} (t) & = & \sum_{j = 1}^{p} {\tilde{m}}_{j} (x (t)) [A_{fj} \hat{x} (t) + B_{fj} y (t)] \\ z_{f} (t) & = & \sum_{j = 1}^{p} {\tilde{m}}_{j} (x (t)) C_{fj} \hat{x} (t), \end{matrix}$ (4) where $\begin{matrix} {\tilde{m}}_{j} (x (t)) \\ = \frac{{\underline{β}}_{j} (x (t)) {\underline{m}}_{j} (x (t)) + {\bar{β}}_{j} (x (t)) {\bar{m}}_{j} (x (t))}{\sum_{k = 1}^{p} ({\underline{β}}_{k} (x (t)) {\underline{m}}_{k} (x (t)) + {\bar{β}}_{k} (x (t)) {\bar{m}}_{k} (x (t)))} \\ \geq 0, \forall j \\ \sum_{j = 1}^{p} {\tilde{m}}_{j} (x (t)) = 1, \end{matrix}$ ${\underline{β}}_{j} (x (t)) \in [0, 1]$ and ${\bar{β}}_{j} (x (t)) \in [0, 1]$ are predefined functions and satisfy ${\underline{β}}_{j} (x (t)) + {\bar{β}}_{j} (x (t)) = 1$ , ∀j.

From Equations (2) and (4), the error system with time delays is defined as:

$\begin{matrix} \dot{\bar{x}} (t) & = & \sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} \\ [{\bar{A}}_{ij} \bar{x} (t) + {\bar{A}}_{dij} \bar{x} (t - h) + {\bar{D}}_{1 ij} w (t)] \\ e (t) & = & \sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} \\ [{\bar{C}}_{ij} \bar{x} (t) + {\bar{C}}_{di} \bar{x} (t - h) + {\bar{D}}_{2 i} w (t)], \end{matrix}$ (5) where $\bar{x} (t) = [x^{T} (t) {\hat{x}}^{T} (t)]^{T}$ , e (t) = z (t) - z_f (t), ${\bar{A}}_{ij} = [\begin{matrix} A_{i} & 0 \\ B_{fj} E_{i} & A_{fj} \end{matrix}]$ , ${\bar{A}}_{dij} = [\begin{matrix} A_{di} & 0 \\ B_{fj} E_{di} & 0 \end{matrix}]$ , ${\bar{D}}_{1 ij} = [\begin{matrix} D_{1 i} \\ B_{fj} D_{3 i} \end{matrix}]$ , ${\bar{C}}_{ij} = [\begin{matrix} C_{i} & - C_{fj} \end{matrix}]$ , ${\bar{C}}_{di} = [\begin{matrix} C_{di} & 0 \end{matrix}]$ , ${\bar{D}}_{2 i} = D_{2 i}$ .

The following assumption and definition are introduced to develop the main results:

Assumption 1 [33]: Matrices Φ, Ψ₁, Ψ₂, and Ψ₃ satisfy the constraints below:

Φ=Φ^T ≥ 0, $Ψ_{1} = Ψ_{1}^{T} \leq 0$ , and $Ψ_{3} = Ψ_{3}^{T}$ ;

$(∥ {\bar{D}}_{2 i} ∥ + ∥ {\bar{C}}_{di} ∥) \cdot ∥ Φ ∥ = 0, \forall i$ ;

(∥ Ψ₁ ∥ + ∥ Ψ₂ ∥) ∥ Φ ∥ =0; and

${\bar{D}}_{2 i}^{T} Ψ_{1} {\bar{D}}_{2} i + {\bar{D}}_{2 i}^{T} Ψ_{2} + Ψ_{2}^{T} {\bar{D}}_{2 i} + Ψ_{3} > 0, \forall i$ .

Definition 1 [33]: For the given matrices Φ, Ψ₁, Ψ₂, and Ψ₃ that satisfied Assumption 1 , system (5) is believed to be extended dissipative if there has a scalar δ, such that the following inequality holds for any t > 0 and all w (t) ∈ L₂ [0, ∞): $\int_{0}^{t} J (t) dt - e^{T} (t) Φ e (t) \geq δ,$ (6)

where $J (t) = e^{T} (t) Ψ_{1} e (t) + 2 e^{T} (t) Ψ_{2} w (t) + w^{T} (t) Ψ_{3} w (t) .$

The inequality (6) represents the index performance.

Remark 1. Definition 1 contains several types of well-known performance indexes for the different values of Φ, Ψ₁, Ψ₂, and Ψ₃, including passive, dissipative, H_∞ and L₂ - L_∞ performances.

The purpose of this study is to design an IT2 fuzzy filter for IT2 fuzzy time delay systems under a unified frame, such that: (a) the filtering error system (5) is asymptotically stable when w (t) =0 and (b) the filtering error system (5) guarantees the performance index (6).

3 Main results

The following nomenclatures are used to simplify the vector and matrix representations: $\begin{matrix} ϑ_{1} (t) & = & \int_{t - h}^{t} \bar{x} (s) ds, \\ ϑ_{2} (t) & = & \int_{t - h}^{t} \int_{t - h}^{s} \bar{x} (u) duds, \\ η_{1} (t) & = & {[{\bar{x}}^{T} (t) ϑ_{1}^{T} (t) ϑ_{2}^{T} (t)]}^{T}, \\ {\dot{η}}_{1} (t) & = & {[{\dot{\bar{x}}}^{T} (t) {\dot{ϑ}}_{1}^{T} (t) {\dot{ϑ}}_{2}^{T} (t)]}^{T}, \\ ξ (t) & = & {[{\bar{x}}^{T} (t) {\bar{x}}^{T} (t - h) \frac{1}{h} ϑ_{1}^{T} (t) \frac{2}{h^{2}} ϑ_{2}^{T} (t)]}^{T}, \\ Γ_{ij} & = & [{\bar{A}}_{ij} {\bar{A}}_{dij} 0 0] . \end{matrix}$

Lemma 1. [34] Let x be a differentiable function: $[α, β] \to ℝ^{n}$ . For symmetric matrices $R (\in ℝ^{n \times n}) > 0$ , and $N_{1}, N_{2}, N_{3} \in ℝ^{4 n \times n}$ , the following inequality holds: $- \int_{α}^{β} {\dot{x}}^{T} (s) R \dot{x} (s) ds \leq ϑ^{T} Ω ϑ,$ (7) where $\begin{matrix} Ω & = & τ (N_{1} R^{- 1} N_{1}^{T} + \frac{1}{3} N_{2} R^{- 1} N_{2}^{T} + \frac{1}{5} N_{3} R^{- 1} N_{3}^{T}) \\ + Sym {N_{1} Π_{1} + N_{2} Π_{2} + N_{3} Π_{3}}, \\ Π_{1} & = & e_{1} - e_{2}, \\ Π_{2} & = & e_{1} + e_{2} - 2 e_{3}, \\ Π_{3} & = & e_{1} - e_{2} - 6 e_{3} + 6 e_{4}, \\ ϑ & = & [x^{T} (β) x^{T} (α) \frac{1}{τ} \int_{α}^{β} x^{T} (s) ds \\ {\frac{2}{τ^{2}} \int_{α}^{β} \int_{α}^{s} x^{T} (u) duds]}^{T}, \\ τ & = & β - α, \\ e_{i} & = & [0_{n \times (i - 1) n} I_{2 n} 0_{n \times (4 - i) n}], i = 1, 2, \dots 4 . \end{matrix}$

Now the following theorem is presented:

Theorem 1. Suppose A_fj, B_fi, and C_fj are known, ${\tilde{m}}_{j} - ρ_{j} {\tilde{w}}_{j} \geq 0$ hold for all j, and the given matrices Φ, Ψ₁, Ψ₂, and Ψ₃ satisfy Assumption 1, the system (5) meets the requirements of index and is guaranteed to be asymptotically stable, if there exist matrices Q > 0, R > 0, G, L = L^T, S = S^T, W = W^T, and $Λ_{i} = Λ_{i}^{T}, i = 1, 2 \dots, p$ such that the following LMIs hold: ${\bar{C}}_{ij}^{T} Φ {\bar{C}}_{ij} - L < 0$ (8) $Ξ_{ij} - Λ_{i} < 0$ (9) $ρ_{i} Ξ_{ii} - ρ_{i} Λ_{i} + Λ_{i} < 0$ (10) $\begin{matrix} ρ_{j} Ξ_{ij} - ρ_{j} Λ_{i} + Λ_{i} + ρ_{i} Ξ_{ji} \\ - ρ_{i} Λ_{j} + Λ_{j} < 0, i < j \end{matrix}$ (11) where $\begin{matrix} Ξ_{ij} & = & [\begin{matrix} Φ_{1 ij} + Φ_{2} - Φ_{4 ij} & Φ_{5 ij} & \sqrt{h} N_{1} & \sqrt{h} N_{2} & \sqrt{h} N_{3} \\ * & Φ_{6 ij} & 0 & 0 & 0 \\ * & * & - R & 0 & 0 \\ * & * & * & - 3 R & 0 \\ * & * & * & * & - 5 R \end{matrix}], \\ Φ_{1 ij} & = & Sym {Π_{4}^{T} \hat{P} Π_{5 ij}} + e_{1}^{T} {Qe}_{1} - e_{2}^{T} {Qe}_{2} \\ + h Γ_{ij}^{T} R Γ_{ij}, \\ Π_{4} & = & [e_{1}^{T} {he}_{3}^{T} \frac{h^{2}}{2} e_{4}^{T}]^{T}, \\ Π_{5 ij} & = & [Γ_{ij}^{T} e_{1}^{T} - e_{2}^{T} {he}_{3}^{T} - {he}_{2}^{T}]^{T}, \\ Φ_{2} & = & Sym {N_{1} Π_{1} + N_{2} Π_{2} + N_{3} Π_{3}}, \\ Φ_{4 ij} & = & e_{1}^{T} {\bar{C}}_{ij}^{T} Ψ_{1} {\bar{C}}_{ij} e_{1} + e_{1}^{T} {\bar{C}}_{ij}^{T} Ψ_{1} {\bar{C}}_{di} e_{2} \\ + e_{2}^{T} {\bar{C}}_{di}^{T} Ψ_{1} {\bar{C}}_{ij} e_{1} + e_{2}^{T} {\bar{C}}_{di}^{T} Ψ_{1} {\bar{C}}_{di} e_{2}, \\ Φ_{5 ij} & = & Π_{4}^{T} \hat{P} Π_{6 ij} + h Γ_{ij}^{T} R {\bar{D}}_{1 ij} - e_{1}^{T} {\bar{C}}_{ij}^{T} Ψ_{1} {\bar{D}}_{2 i} \\ - e_{2}^{T} {\bar{C}}_{di}^{T} Ψ_{1} {\bar{D}}_{2 i} - e_{1}^{T} {\bar{C}}_{ij}^{T} Ψ_{2} - e_{2}^{T} {\bar{C}}_{di}^{T} Ψ_{2}, \\ Π_{6 ij} & = & [{\bar{D}}_{1 ij}^{T} 0^{T} 0^{T}]^{T}, \\ Φ_{6 ij} & = & h {\bar{D}}_{1 ij}^{T} R {\bar{D}}_{1 ij} - {\bar{D}}_{2 i}^{T} Ψ_{1} {\bar{D}}_{2 i} - {\bar{D}}_{2 i}^{T} Ψ_{2} - Ψ_{2}^{T} {\bar{D}}_{2 i}^{T} - Ψ_{3}, \\ \hat{P} & = & [\begin{matrix} L & 0 & 0 \\ * & S & G \\ * & * & W \end{matrix}] . \end{matrix}$

In this sense, the value of scalar δ in Definition 1 is selected as -V (0).

Proof. Choose the following Lyapunov-Krasovskii functional:

$\begin{matrix} V (x_{t}) & = & η_{1}^{T} (t) \hat{P} η_{1} (t) + \int_{t - h}^{t} {\bar{x}}^{T} (s) Q \bar{x} (s) ds \\ + \int_{- h}^{0} \int_{t + θ}^{t} {\dot{\bar{x}}}^{T} (s) R \dot{\bar{x}} (s) dsd θ, \end{matrix}$ (12) where $\hat{P} = {\hat{P}}^{T} > 0$ . Then we have:

$\begin{matrix} \dot{V} (x_{t}) \\ = \sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} [ξ^{T} (t) Φ_{1 ij} ξ (t) \\ + w^{T} (t) Π_{6 ij}^{T} \hat{P} Π_{4} ξ (t) + ξ^{T} (t) Π_{4}^{T} \hat{P} Π_{6 ij} w (t) \\ + {hw}^{T} (t) {\bar{D}}_{1 ij}^{T} R Γ_{ij} ξ (t) + {hw}^{T} (t) {\bar{D}}_{1 ij}^{T} R {\bar{D}}_{1 ij} w (t) \\ + h ξ^{T} (t) Γ_{ij}^{T} R {\bar{D}}_{1 ij} w (t) - \int_{t - h}^{t} \dot{\bar{x}} (s)^{T} R \dot{\bar{x}} (s) ds] . \end{matrix}$ (13)

Thus, the following expression is obtained:

$\begin{matrix} \dot{V} (x_{t}) - J (t) & = & \sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} [ξ^{T} (t) (Φ_{1 ij} \\ - Φ_{4 ij}) ξ (t) + ξ^{T} (t) Φ_{5 ij} w (t) \\ + w^{T} (t) Φ_{5 ij}^{T} w (t) + w^{T} (t) Φ_{6 ij} w (t) \\ - \int_{t - h}^{t} \dot{\bar{x}} (s)^{T} R \dot{\bar{x}} (s) ds] . \end{matrix}$ (14)

Applying Lemma 1 to $- \int_{t - h}^{t} \dot{\bar{x}} (s)^{T} R \dot{\bar{x}} (s) ds$ , the following expression is obtained:

$\begin{matrix} \dot{V} (x_{t}) - J (t) \\ \leq {[\begin{matrix} ξ (t) \\ w (t) \end{matrix}]}^{T} (\sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} {\hat{Ξ}}_{ij}) [\begin{matrix} ξ (t) \\ w (t) \end{matrix}], \end{matrix}$ (15) where $\begin{matrix} {\hat{Ξ}}_{ij} & = & [\begin{matrix} Φ_{1 ij} + Φ_{2} + Φ_{3} - Φ_{4 ij} & Φ_{5 ij} \\ * & Φ_{6 ij} \end{matrix}], \\ Φ_{3} & = & N_{1} R^{- 1} N_{1}^{T} + \frac{1}{3} N_{2} R^{- 1} N_{2}^{T} + \frac{1}{5} N_{3} R^{- 1} N_{3}^{T} . \end{matrix}$

From Equation (15), if $\sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} {\hat{Ξ}}_{ij} < 0$ , then we have $\dot{V} (x_{t}) - J (t) < 0$ . Using the Schur complement of $\sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} {\hat{Ξ}}_{ij} < 0$ , the following expression is obtained: $\sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} Ξ_{ij} < 0,$ (16) where: $Ξ_{ij} = [\begin{matrix} Φ_{1 ij} + Φ_{2} - Φ_{4 ij} & Φ_{5 ij} & \sqrt{h} N_{1} & \sqrt{h} N_{2} & \sqrt{h} N_{3} \\ * & Φ_{6 ij} & 0 & 0 & 0 \\ * & * & - R & 0 & 0 \\ * & * & * & - 3 R & 0 \\ * & * & * & * & - 5 R \end{matrix}] .$

It is can be seen that $\sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} ({\tilde{w}}_{i} - {\tilde{m}}_{j}) Λ_{i} = 0$ , where $Λ_{i} = Λ_{i}^{T} (i = 1, 2, \dots, p)$ is an arbitrary matrix with the appropriate dimensions. Thus, the following expression is obtained:

$\begin{matrix} \sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} Ξ_{ij} \\ = \sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} ({\tilde{w}}_{i} - {\tilde{m}}_{j} + ρ_{j} {\tilde{w}}_{j} - ρ_{j} {\tilde{w}}_{j}) Λ_{i} \\ + \sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} Ξ_{ij} \\ = \sum_{i = 1}^{p - 1} \sum_{j = i + 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} (ρ_{j} Ξ_{ij} - ρ_{j} Λ_{i} + Λ_{i} + ρ_{i} Ξ_{ji} \\ - ρ_{i} Λ_{j} + Λ_{j}) + \sum_{i = 1}^{p} \sum_{j = i}^{p} {\tilde{w}}_{i}^{2} (ρ_{i} Ξ_{ii} - ρ_{i} Λ_{i} + Λ_{i}) \\ + \sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} ({\tilde{m}}_{j} - ρ_{j} {\tilde{w}}_{j}) (Ξ_{ij} - Λ_{i}) . \end{matrix}$ (17)

When ${\tilde{m}}_{j} - ρ_{j} {\tilde{w}}_{j} > 0$ for all j, Equations (9) to (11) show that:

$\begin{matrix} \dot{V} (x_{t}) - J (t) & < & {[\begin{matrix} ξ (t) \\ w (t) \end{matrix}]}^{T} (\sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} Ξ_{ij}) \\ [\begin{matrix} ξ (t) \\ w (t) \end{matrix}] \\ < & 0, \end{matrix}$ (18)

Thus, $J (t) > \dot{V} (x_{t}) (t), \forall t \geq 0$ holds. Then, we have:

$\begin{matrix} \int_{0}^{t} J (s) ds & > & V (x_{t}) - V (0) \\ = & η_{1}^{T} (t) \hat{P} η_{1} (t) + \int_{t - h}^{t} {\bar{x}}^{T} (s) Q \bar{x} (s) ds \\ + \int_{- h}^{0} \int_{t + θ}^{t} {\dot{\bar{x}}}^{T} (s) R \dot{\bar{x}} (s) dsd θ + δ \end{matrix}$ (19)

According to Definition 1 , the following inequality, which holds for any of the matrices Φ, Ψ₁, Ψ₂, and Ψ₃ that satisfy Assumption 1 , should be proven: $\int_{0}^{t} J (t) dt - e^{T} (t) Φ e (t) \geq δ .$ (20)

For this reason, two cases, i.e., ∥Φ ∥ =0 and ∥Φ ∥ ≠0, are considered, as follows:

∥Φ ∥ =0, inequality (20) always holds for any t ≥ 0 from (19).

∥Φ ∥ ≠0, under Assumption 1 , ∥Ψ₁ ∥ + ∥ Ψ₂ ∥ =0, $∥ {\bar{C}}_{di} ∥ = 0$ , and $∥ {\bar{D}}_{2 i} ∥ = 0$ show that Ψ₁ = Ψ₂ = 0, Ψ₃ > 0. Then, we have $e (t) = \sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} {\bar{C}}_{ij} \bar{x} (t)$ and J (t) = w^T (t) Ψ₃w (t) >0. Under the condition of $\hat{P} = [\begin{matrix} L & 0 & 0 \\ * & S & G \\ * & * & W \end{matrix}]$ , the equation $η_{1}^{T} (t) \hat{P} η_{1} (t) = ξ^{T} (t) Π_{4}^{T} \hat{P} Π_{4} ξ (t) > ξ^{T} (t) e_{1}^{T} {Le}_{1} ξ (t)$ is obtai-ned. From Equation (8), $L > {\bar{C}}_{ij}^{T} Φ {\bar{C}}_{ij}$ . Then, the following inequalities hold: $\begin{matrix} \int_{0}^{t} J (s) ds - e^{T} (t) Φ e (t) \\ = \int_{0}^{t} J (s) ds - \sum_{i = 1}^{n} \sum_{j = 1}^{n} {\tilde{w}}_{i} {\tilde{m}}_{j} {\bar{x}}^{T} (t) {\bar{C}}_{ij}^{T} Φ {\bar{C}}_{ij} \bar{x} (t) \\ = \int_{0}^{t} J (s) ds - \sum_{i = 1}^{n} \sum_{j = 1}^{n} {\tilde{w}}_{i} {\tilde{m}}_{j} ξ^{T} (t) e_{1}^{T} {\bar{C}}_{ij}^{T} Φ {\bar{C}}_{ij} e_{1} ξ (t) \\ > \int_{0}^{t} J (s) ds - ξ^{T} (t) e_{1}^{T} {Le}_{1} ξ (t) \\ > \int_{0}^{t} J (s) ds - ξ (t)^{T} Π_{4}^{T} \hat{P} Π_{4} ξ (t) \\ = \int_{0}^{t} J (s) ds - η_{1}^{T} (t) \hat{P} η_{1} (t) \\ > \int_{t - h}^{t} {\bar{x}}^{T} (s) Q \bar{x} (s) ds + \int_{- h}^{0} \int_{t + θ}^{t} {\dot{\bar{x}}}^{T} (s) R \dot{\bar{x}} (s) dsd θ \\ + δ \\ > δ, \forall t \geq 0 . \end{matrix}$

Given the two cases, i.e., ∥Φ ∥ =0 and ∥Φ ∥ ≠0, previously discussed, in the sense of Definition 1 , system (5) is believed to be extended dissipative.

Let w (t) ≡0, under Assumtion 1 that Ψ₁ ≤ 0, $\dot{V} (x_{t}) < 0$ , which indicates that system (5) is asymptotically stable when w (t) =0. The proof is completed.

Next, the IT2 T-S fuzzy filter is designed on the basis of Theorem 1 . From Assumption 1 , we know that Φ > 0 and Ψ₁ ≤ 0, so the matrices ${\tilde{Φ}}_{1}$ and ${\tilde{Ψ}}_{1}$ always exist and satisfy the following equation: $Ψ_{1} = - {\tilde{Ψ}}_{1}^{T} {\tilde{Ψ}}_{1}, Φ = {\tilde{Φ}}^{T} \tilde{Φ}$ (21)

Lemma 2. [35] For matrices T, M, L, and A with appropriate dimensions and scalar β, let there be the following condition is applied: $[\begin{matrix} T & * \\ LA & - β L - β L^{T} + β^{2} M \end{matrix}] < 0 .$ (22)

Then, T + A^TMA < 0 can be obtained.

The following theorem presents the design conditions of the IT2 T-S fuzzy filter.

Theorem 2. Given the scalars α and β, under the conditions of ${\tilde{m}}_{j} - ρ_{j} {\tilde{w}}_{j} \geq 0$ , the system (5) meets the requirements of index and is guaranteed to be asymptotically stable, if there exist appropriate matrices Q > 0, R > 0, G, $L_{1} = L_{1}^{T}$ , $L_{2} = L_{2}^{T}$ , S = S^T, W = W^T, ${\bar{Λ}}_{i} = {\bar{Λ}}_{i}^{T}$ , ${\bar{A}}_{fj}$ , ${\bar{B}}_{fj}$ , and ${\bar{C}}_{fj}$ , such that the following LMIs hold: $\hat{P} > 0,$ (23) $[\begin{matrix} - L & {\bar{C}}_{ij}^{T} {\tilde{Φ}}^{T} \\ * & - I \end{matrix}] < 0,$ (24) ${\bar{Ξ}}_{ij} - {\bar{Λ}}_{i} < 0,$ (25) $ρ_{i} {\bar{Ξ}}_{ii} - ρ_{i} {\bar{Λ}}_{i} + {\bar{Λ}}_{i} < 0,$ (26) $\begin{matrix} ρ_{j} {\bar{Ξ}}_{ij} - ρ_{j} {\bar{Λ}}_{i} + {\bar{Λ}}_{i} \\ + ρ_{i} {\bar{Ξ}}_{ji} - ρ_{i} {\bar{Λ}}_{j} + {\bar{Λ}}_{j} < 0, i < j \end{matrix}$ (27) where $\begin{matrix} \hat{P} = [\begin{matrix} \begin{matrix} L & 0 & 0 \\ * & S & G \\ * & * & W \end{matrix} \end{matrix}], L = [\begin{matrix} \begin{matrix} L_{1} & L_{2} \\ * & α L_{2} \end{matrix} \end{matrix}], \\ {\bar{Ξ}}_{ij} & = & [\begin{matrix} Θ_{1 ij} & Θ_{2 ij} & Θ_{3 ij} & Θ_{4 ij} & \sqrt{h} N_{1} & \sqrt{h} N_{2} & \sqrt{h} N_{3} \\ * & Θ_{5 ij} & Θ_{6 ij} & Θ_{7 ij} & 0 & 0 & 0 \\ * & * & - I & 0 & 0 & 0 & 0 \\ * & * & * & - β L - β L^{T} + β^{2} M & 0 & 0 & 0 \\ * & * & * & * & - R & 0 & 0 \\ * & * & * & * & * & - 3 R & 0 \\ * & * & * & * & * & * & - 5 R \end{matrix}], \\ Θ_{1 ij} & = & Sym {e_{1}^{T} [\begin{matrix} L_{1} A_{i} + {\bar{B}}_{fj} E_{i} & {\bar{A}}_{fj} & L_{1} A_{di} + {\bar{B}}_{fj} E_{i} & 0 & 0 & 0 & 0 & 0 \\ L_{2} A_{i} + α {\bar{B}}_{fj} E_{i} & α {\bar{A}}_{fj} & L_{2} A_{di} + α {\bar{B}}_{fj} E_{di} & 0 & 0 & 0 & 0 & 0 \end{matrix}] \\ + {he}_{3}^{T} [S (e_{1} - e_{2}) + G ({he}_{3} - {he}_{2})] \\ + \frac{h^{2}}{2} e_{4}^{T} [G^{T} (e_{1} - e_{2}) + W ({he}_{3} - {he}_{2})]} \\ + e_{1}^{T} {Qe}_{1} \\ - e_{2}^{T} {Qe}_{2} + Sym {N_{1} Π_{1} + N_{2} Π_{2} + N_{3} Π_{3}}, \\ Θ_{2 ij} & = & e_{1}^{T} [\begin{matrix} \begin{matrix} L_{1} D_{1 i} + {\bar{B}}_{fj} D_{3 i} \\ L_{2} D_{1 i} + α {\bar{B}}_{fj} D_{3 i} \end{matrix} \end{matrix}] - e_{1}^{T} {\bar{C}}_{ij}^{T} Ψ_{2} - e_{2}^{T} {\bar{C}}_{di}^{T} Ψ_{2}, \\ Θ_{3 ij} & = & e_{1}^{T} {\bar{C}}_{ij}^{T} {\tilde{Ψ}}_{1} + e_{2}^{T} {\bar{C}}_{di}^{T} {\tilde{Ψ}}_{1}, \\ Θ_{4 ij} & = & {[\begin{matrix} \begin{matrix} L_{1} A_{i} + {\bar{B}}_{fj} E_{i} & {\bar{A}}_{fj} & L_{1} A_{di} + {\bar{B}}_{fj} E_{i} & 0 & 0 & 0 & 0 & 0 \\ L_{2} A_{i} + α {\bar{B}}_{fj} E_{i} & α {\bar{A}}_{fj} & L_{2} A_{di} + α {\bar{B}}_{fj} E_{di} & 0 & 0 & 0 & 0 & 0 \end{matrix} \end{matrix}]}^{T}, \\ Θ_{5 ij} & = & - {\bar{D}}_{2 i}^{T} Ψ_{2} - Ψ_{2}^{T} {\bar{D}}_{2 i} - Ψ_{3}, \\ Θ_{6 ij} & = & {\bar{D}}_{2 i}^{T} {\tilde{Ψ}}_{1}^{T}, \\ Θ_{7 ij} & = & {[\begin{matrix} \begin{matrix} L_{1} D_{1 i} + {\bar{B}}_{fj} D_{3 i} \\ L_{2} D_{1 i} + α {\bar{B}}_{fj} D_{3 i} \end{matrix} \end{matrix}]}^{T} . \end{matrix}$

The IT2 filter parameters are expressed as: $A_{fj} = L_{2}^{- 1} {\bar{A}}_{fj}, B_{fj} = L_{2}^{- 1} {\bar{B}}_{fj}, C_{fj} = {\bar{C}}_{fj} .$ (28)

Proof. Inequality (24) holds by using the Schur complement of inequality (8). Matrix ${\bar{Λ}}_{ij}$ is defined with the appropriate dimensions. From the inequalities of (15) and (18), $\sum_{i = 1}^{p} \sum_{j = 1}^{p} {\tilde{w}}_{i} {\tilde{m}}_{j} {\hat{Ξ}}_{ij} < 0$ can ensure that $\dot{V} (x_{t}) - J (t) < 0$ . Under the condition of $Ψ_{1} = - {\tilde{Ψ}}_{1}^{T} {\tilde{Ψ}}_{1}$ , ${\hat{Ξ}}_{ij}$ can be expressed as:

$\begin{matrix} {\hat{Ξ}}_{ij} & = & [\begin{matrix} Φ_{1 ij} + Φ_{2} + Φ_{3} & Φ_{4 ij} \\ * & Φ_{5 ij} \end{matrix}] \\ + [\begin{matrix} e_{1}^{T} {\bar{C}}_{ij}^{T} {\tilde{Ψ}}_{1}^{T} + e_{2}^{T} {\bar{C}}_{di}^{T} {\tilde{Ψ}}_{1}^{T} \\ {\bar{D}}_{2 i}^{T} {\tilde{Ψ}}_{1}^{T} \end{matrix}] [\begin{matrix} {\tilde{Ψ}}_{1} {\bar{C}}_{ij} e_{1} + {\tilde{Ψ}}_{1} {\bar{C}}_{di} e_{2} & {\tilde{Ψ}}_{1} {\bar{D}}_{2 i} \end{matrix}] \\ < 0, \end{matrix}$ (29) where $\begin{matrix} Φ_{4 ij} & = & Π_{4}^{T} \hat{P} Π_{6} + h Γ_{ij}^{T} R {\bar{D}}_{1 ij} - e_{1}^{T} {\bar{C}}_{1 ij} Ψ_{2}, \\ - e_{2}^{T} {\bar{C}}_{di} Ψ_{2} \\ Φ_{5 ij} & = & h {\bar{D}}_{1 ij}^{T} R {\bar{D}}_{1 ij} - {\bar{D}}_{2 i}^{T} Ψ_{2} - Ψ_{2}^{T} {\bar{D}}_{2 i} - Ψ_{3} . \end{matrix}$

Using the Schur complement of Equation (29), the following expression is obtained: $[\begin{matrix} Φ_{1 ij} + Φ_{2} + Φ_{3} & Φ_{4 ij} & e_{1}^{T} {\bar{C}}_{ij}^{T} {\tilde{Ψ}}_{1}^{T} + e_{2}^{T} {\bar{C}}_{di}^{T} {\tilde{Ψ}}_{1}^{T} \\ * & Φ_{5 ij} & {\bar{D}}_{2 i}^{T} {\tilde{Ψ}}_{1}^{T} \\ * & * & - I \end{matrix}] < 0 .$ (30)

Then, the inequality (30) can be written as:

$\begin{matrix} [\begin{matrix} Φ_{1 ij}^{'} + Φ_{2} + Φ_{3} & Φ_{4 ij}^{'} & e_{1}^{T} {\bar{C}}_{ij}^{T} {\tilde{Ψ}}_{1}^{T} + e_{2}^{T} {\bar{C}}_{di}^{T} {\tilde{Ψ}}_{1}^{T} \\ * & Φ_{5 ij}^{'} & {\bar{D}}_{2 i}^{T} {\tilde{Ψ}}_{1}^{T} \\ * & * & - I \end{matrix}] \\ + [\begin{matrix} Γ_{ij}^{T} \\ {\bar{D}}_{1 ij}^{T} \\ 0 \end{matrix}] hR [\begin{matrix} Γ_{ij} & {\bar{D}}_{1 ij} & 0 \end{matrix}] < 0, \end{matrix}$ (31) where $\begin{matrix} Φ_{1 ij}^{'} & = & Sym {Π_{4}^{T} \hat{P} Π_{5 ij}} + e_{1}^{T} {Qe}_{1} - e_{2}^{T} {Qe}_{2}, \\ Φ_{4 ij}^{'} & = & Π_{4}^{T} \hat{P} Π_{6} - e_{1}^{T} {\bar{C}}_{1 ij} Ψ_{2} - e_{2}^{T} {\bar{C}}_{di} Ψ_{2}, \\ Φ_{5 ij}^{'} & = & - {\bar{D}}_{2 i}^{T} Ψ_{2} - Ψ_{2}^{T} {\bar{D}}_{2 i} - Ψ_{3} . \end{matrix}$

In (31), if $\hat{T} = [\begin{matrix} Φ_{1 ij}^{'} + Φ_{2} + Φ_{3} & Φ_{4 ij}^{'} & e_{1}^{T} {\bar{C}}_{ij}^{T} {\tilde{Ψ}}_{1}^{T} + e_{2}^{T} {\bar{C}}_{di}^{T} {\tilde{Ψ}}_{1}^{T} \\ * & Φ_{5 ij}^{'} & {\bar{D}}_{2 i}^{T} {\tilde{Ψ}}_{1}^{T} \\ * & * & - I \end{matrix}]$ , $\hat{A} = [\begin{matrix} Γ_{ij} & {\bar{D}}_{1 ij} & 0 \end{matrix}]$ , and M = hR, then inequality (31) can be rewritten as $\hat{T} + {\hat{A}}^{T} M \hat{A} < 0$ . From Lemma 2, the following expression can be derived:

$\begin{matrix} [\begin{matrix} \begin{matrix} Φ_{1 ij}^{'} + Φ_{2} + Φ_{3} & Φ_{4 ij}^{'} & e_{1}^{T} {\bar{C}}_{ij}^{T} {\tilde{Ψ}}_{1} + e_{2}^{T} {\bar{C}}_{di}^{T} {\tilde{Ψ}}_{1} & Γ_{ij}^{T} L^{T} \\ * & Φ_{5 ij}^{'} & {\bar{D}}_{2 i}^{T} {\tilde{Ψ}}_{1}^{T} & {\bar{D}}_{ij}^{T} L^{T} \\ * & * & - I & 0 \\ * & * & * & - β L - β L^{T} + β^{2} M \end{matrix} \end{matrix}] \\ < 0 . \end{matrix}$ (32)

Using the Schur complement of Equation (32), the following inequality holds:

$\begin{matrix} [\begin{matrix} \begin{matrix} Φ_{1 ij}^{'} + Φ_{2} & Φ_{4 ij}^{'} & Θ_{3 ij} & Θ_{4 ij} & \sqrt{h} N_{1} & \sqrt{h} N_{2} & \sqrt{h} N_{3} \\ * & Φ_{5 ij}^{'} & Θ_{6 ij} & Θ_{7 ij} & 0 & 0 & 0 \\ * & * & - I & 0 & 0 & 0 & 0 \\ * & * & * & - β L - β L^{T} + β^{2} M & 0 & 0 & 0 \\ * & * & * & * & - R & 0 & 0 \\ * & * & * & * & * & - 3 R & 0 \\ * & * & * & * & * & * & - 5 R \end{matrix} \end{matrix}] \\ < 0, \end{matrix}$ (33) where

$\begin{matrix} Θ_{3 ij} & = & e_{1}^{T} {\bar{C}}_{ij}^{T} {\tilde{Ψ}}_{1} + e_{2}^{T} {\bar{C}}_{di}^{T} {\tilde{Ψ}}_{1}, \\ Θ_{4 ij} & = & Γ_{ij}^{T} L^{T}, \\ Θ_{6 ij} & = & {\bar{D}}_{2 i}^{T} {\tilde{Ψ}}_{1}^{T}, \\ Θ_{7 ij} & = & {\bar{D}}_{ij}^{T} L^{T} . \end{matrix}$

Thus, Equation (33) can be rewritten as:

$\begin{matrix} {\bar{Ξ}}_{ij} & = & [\begin{matrix} \begin{matrix} Θ_{1 ij} & Θ_{2 ij} & Θ_{3 ij} & Θ_{4 ij} & \sqrt{h} N_{1} & \sqrt{h} N_{2} & \sqrt{h} N_{3} \\ * & Θ_{5 ij} & Θ_{6 ij} & Θ_{7 ij} & 0 & 0 & 0 \\ * & * & - I & 0 & 0 & 0 & 0 \\ * & * & * & - β L - β L^{T} + β^{2} M & 0 & 0 & 0 \\ * & * & * & * & - R & 0 & 0 \\ * & * & * & * & * & - 3 R & 0 \\ * & * & * & * & * & * & - 5 R \end{matrix} \end{matrix}] \\ < 0, \end{matrix}$ (34)

where $\begin{matrix} Θ_{1 ij} & = & Φ_{1 ij}^{'} + Φ_{2} = Sym {Π_{4}^{T} \hat{P} Π_{5 ij}} + e_{1}^{T} {Qe}_{1} \\ - e_{2}^{T} {Qe}_{2} + Φ_{2}, \\ Θ_{2 ij} & = & Φ_{4 ij}^{'} = Π_{4}^{T} \hat{P} Π_{6} - e_{1}^{T} {\bar{C}}_{1 ij} Ψ_{2} - e_{2}^{T} {\bar{C}}_{di} Ψ_{2} . \\ Θ_{5 ij} & = & Φ_{5 ij}^{'} = - {\bar{D}}_{2 i}^{T} Ψ_{2} - Ψ_{2}^{T} {\bar{D}}_{2 i} - Ψ_{3}, \end{matrix}$

Expanded $Π_{4}^{T} \hat{P} Π_{5 ij}$ and $Π_{4}^{T} \hat{P} Π_{6}$ , the following expression is obtained: $\begin{matrix} Π_{4}^{T} \hat{P} Π_{5 ij} & = & e_{1}^{T} L Γ_{ij} + {he}_{3}^{T} [S (e_{1} - e_{2}) \\ + G ({he}_{3} - {he}_{2})] + \frac{h^{2}}{2} e_{4}^{T} [G^{T} (e_{1} - e_{2}) \\ + W ({he}_{3} - {he}_{2})], \\ Π_{4}^{T} \hat{P} Π_{6} & = & e_{1}^{T} L {\bar{D}}_{1 ij}, \\ L {\bar{A}}_{ij} & = & [\begin{matrix} L_{1} & L_{2} \\ * & α L_{2} \end{matrix}] [\begin{matrix} A_{i} & 0 \\ B_{fj} E_{i} & A_{fj} \end{matrix}], \\ L {\bar{A}}_{dij} & = & [\begin{matrix} L_{1} & L_{2} \\ * & α L_{2} \end{matrix}] [\begin{matrix} A_{di} & 0 \\ B_{fj} E_{di} & 0 \end{matrix}], \\ L Γ_{ij} & = & [\begin{matrix} L {\bar{A}}_{ij} & L {\bar{A}}_{dij} & 0 & 0 \end{matrix}], \\ L {\bar{D}}_{1 ij} & = & [\begin{matrix} L_{1} & L_{2} \\ L_{2} & α L_{2} \end{matrix}] [\begin{matrix} D_{1 i} \\ B_{fj} D_{3 i} \end{matrix}] . \end{matrix}$

Let $L_{2} A_{fi} = {\bar{A}}_{fi}$ , $L_{2} B_{fi} = {\bar{B}}_{fi}$ , Θ_1ij, Θ_2ij, Θ_3ij, Θ_4ij, Θ_5ij can be derived as followings: $\begin{matrix} Θ_{1 ij} \\ = Sym {e_{1}^{T} [\begin{matrix} L_{1} A_{i} + {\bar{B}}_{fj} E_{i} & {\bar{A}}_{fj} & L_{1} A_{di} + {\bar{B}}_{fj} E_{i} & 0 & 0 & 0 & 0 & 0 \\ L_{2} A_{i} + α {\bar{B}}_{fj} E_{i} & α {\bar{A}}_{fj} & L_{2} A_{di} + α {\bar{B}}_{fj} E_{di} & 0 & 0 & 0 & 0 & 0 \end{matrix}] \\ + {he}_{3}^{T} [S (e_{1} - e_{2}) + G ({he}_{3} - {he}_{2})] \\ + \frac{h^{2}}{2} e_{4}^{T} [G^{T} (e_{1} - e_{2}) + W ({he}_{3} - {he}_{2})]} \\ + e_{1}^{T} {Qe}_{1} \\ - e_{2}^{T} {Qe}_{2} + Sym {N_{1} Π_{1} + N_{2} Π_{2} + N_{3} Π_{3}}, \end{matrix}$

$\begin{matrix} Θ_{2 ij} & = & e_{1}^{T} [\begin{matrix} \begin{matrix} L_{1} D_{1 i} + {\bar{B}}_{fj} D_{3 i} \\ L_{2} D_{1 i} + α {\bar{B}}_{fj} D_{3 i} \end{matrix} \end{matrix}] - e_{1}^{T} {\bar{C}}_{ij}^{T} Ψ_{2} - e_{2}^{T} {\bar{C}}_{di}^{T} Ψ_{2}, \\ Θ_{3 ij} & = & e_{1}^{T} {\bar{C}}_{ij}^{T} {\tilde{Ψ}}_{1} + e_{2}^{T} {\bar{C}}_{di}^{T} {\tilde{Ψ}}_{1}, \\ Θ_{4 ij} & = & {[\begin{matrix} \begin{matrix} L_{1} A_{i} + {\bar{B}}_{fj} E_{i} & {\bar{A}}_{fj} & L_{1} A_{di} + {\bar{B}}_{fj} E_{i} & 0 & 0 & 0 & 0 & 0 \\ L_{2} A_{i} + α {\bar{B}}_{fj} E_{i} & α {\bar{A}}_{fj} & L_{2} A_{di} + α {\bar{B}}_{fj} E_{di} & 0 & 0 & 0 & 0 & 0 \end{matrix} \end{matrix}]}^{T} \\ Θ_{5 ij} & = & - {\bar{D}}_{2 i}^{T} Ψ_{2} - Ψ_{2}^{T} {\bar{D}}_{2 i} - Ψ_{3}, \\ Θ_{6 ij} & = & {\bar{D}}_{2 i}^{T} {\tilde{Ψ}}_{1}^{T}, \\ Θ_{7 ij} & = & {[\begin{matrix} \begin{matrix} L_{1} D_{1 i} + {\bar{B}}_{fj} D_{3 i} \\ L_{2} D_{1 i} + α {\bar{B}}_{fj} D_{3 i} \end{matrix} \end{matrix}]}^{T} . \end{matrix}$

In the inequalities (9) to (11), if Ξ_ij is replaced with ${\bar{Ξ}}_{ij}$ and Λ_ij is replaced with ${\bar{Λ}}_{ij}$ , then inequalities (26) to (28) hold. If inequalities (24) to (28) hold, then problem of filter design is completed and the filter matrices can be designed as: $A_{fj} = L_{2}^{- 1} {\bar{A}}_{fj}, B_{fj} = L_{2}^{- 1} {\bar{B}}_{fj}, C_{fj} = {\bar{C}}_{fj} .$

This statement completes the proof.

Remark 2. From Theorem 2 , IT2 fuzzy filter is designed for IT2 fuzzy time-delay systems under a unified frame. The proposed criterion is presented in the form of LMIs. Note that the proposed IT2 fuzzy filter is used to the constant time-delay systems. Further work will focus on the related academic research for IT2 fuzzy systems with time-varying delay.

4 Numerical examples

In this section, two numerical examples are presented to illustrate the effectiveness of the proposed results.

Example 1. Consider the two-rule IT2 fuzzy system in [36], and the system matrices are listed as follows: $\begin{matrix} A_{1} & = & [\begin{matrix} - 2.1 & 0.1 \\ 1 & - 2 \end{matrix}], A_{2} = [\begin{matrix} - 1.9 & 0 \\ - 0.2 & - 1.1 \end{matrix}], \\ A_{d 1} & = & [\begin{matrix} - 1.1 & 0.1 \\ - 0.8 & - 0.9 \end{matrix}], A_{d 2} = [\begin{matrix} - 0.9 & 0 \\ - 1.1 & - 1.2 \end{matrix}], \\ D_{11} & = & [\begin{matrix} 1 \\ - 0.2 \end{matrix}], D_{12} = [\begin{matrix} 0.3 \\ 0.1 \end{matrix}], C_{1} = [\begin{matrix} 1 & 0 \end{matrix}], \\ C_{2} & = & [\begin{matrix} 0.5 & - 0.6 \end{matrix}], C_{d 1} = [\begin{matrix} - 0.8 & 0.6 \end{matrix}], \\ C_{d 2} & = & [\begin{matrix} - 0.2 & 1 \end{matrix}], D_{21} = 0.3, D_{22} = - 0.6, \\ E_{1} & = & [\begin{matrix} 1 & - 0.5 \end{matrix}], E_{2} = [\begin{matrix} - 0.2 & 0.3 \end{matrix}], \\ E_{d 1} & = & [\begin{matrix} 0.1 & 0 \end{matrix}], E_{d 2} = [\begin{matrix} 0 & 0.2 \end{matrix}], \\ D_{31} & = & 0, D_{32} = 0 . \end{matrix}$

The membership functions of the IT2 fuzzy model are selected as:

${\tilde{w}}_{1} (x_{1}) = 1 - \frac{1}{1 + e^{- x_{1} + k}}, (k \in [1, 5]$ is the uncertain parameter),

${\tilde{w}}_{2} (x_{1}) = 1 - {\tilde{w}}_{1} (x_{1})$ .

The membership functions of the IT2 fuzzy filter in the form (4) are defined as: $\begin{matrix} {\underline{β}}_{j} = {\bar{β}}_{j} = 0.5, {\underline{m}}_{1} (x_{1}) = 1 - \frac{1}{1 + e^{\frac{- x_{1} + 0.25}{2}}}, \\ {\bar{m}}_{1} (x_{1}) = 1 - \frac{1}{1 + e^{\frac{- x_{1} - 0.25}{2}}}, \\ {\underline{m}}_{2} (x_{1}) = 1 - {\bar{m}}_{1} (x_{1}), {\bar{m}}_{2} (x_{1}) = 1 - {\underline{m}}_{1} (x_{1}) . \end{matrix}$

Only the L₂ - L_∞ performance is considered in this example because of the limited space. In the LMI conditions of Theorem 2 , let Φ = I, Ψ₁ = Ψ₂ = 0, Ψ₃ = γ²I. Given the scalars α = 10, β = 1, and h = 3 denotes the retardation time. By calculation, we can obtain the value of ρ_i as ρ₁ = 0.0719, ρ₂ = 0.9817. The minimized L₂ - L_∞ performance index is γ = 3.2706. The L₂ - L_∞ filter parameters in Equation (4) are as follows: $\begin{matrix} A_{f 1} & = & [\begin{matrix} - 0.7150 & - 0.0484 \\ - 0.0536 & - 0.7165 \end{matrix}], \\ B_{f 1} & = & [\begin{matrix} 0.1985 \\ - 0.3374 \end{matrix}], \\ C_{f 1} & = & [\begin{matrix} - 0.0865 & 0.1202 \end{matrix}], \\ A_{f 2} & = & [\begin{matrix} - 0.7011 & - 0.0001 \\ - 0.0092 & - 0.6731 \end{matrix}], \\ B_{f 2} & = & [\begin{matrix} 0.1937 \\ - 0.3237 \end{matrix}], \\ C_{f 2} & = & [\begin{matrix} - 0.0865 & 0.1202 \end{matrix}] . \end{matrix}$

The disturbance is given as follows: $w (t) = {\begin{matrix} 0.1 sin (2 t), t \leq 55, \\ 0, else . \end{matrix}$ (35)

Under the conditions of x₁ (t) =1, x₂ (t) = -2, ${\hat{x}}_{1} (t) = 1$ and ${\hat{x}}_{2} (t) = - 2$ , the state responses x (t) and $\hat{x} (t)$ of filtering error system (5) are shown in Figs. 1 and 2. Figure 3 shows the error response e (t) under the conditions of x₁ (t) =0, x₂ (t) =0, ${\hat{x}}_{1} (t) = 0$ and ${\hat{x}}_{2} (t) = 0$ . From Figs. 1, 2 and 3, it is shown that the designed filter can handle the IT2 fuzzy time-delay system well. On the other hand, the proposed method in this paper can capture the uncertainties in membership functions well. The effectiveness of the presented results has been verified from Figs. 1, 2 and 3.

Fig.1

State response of x₁ (t), ${\hat{x}}_{1} (t)$ .

Fig.2

State response of x₂ (t), ${\hat{x}}_{2} (t)$ .

Fig.3

Error response.

Example 2. The mass-spring-damping system shown in Fig. 4 is considered [37]. The following expression can be obtained according to Newton’s law: $m \ddot{x} + c \ddot{x} + \hat{k} x + \hat{k} a^{2} x^{3} = u (t),$ where m is the mass, $F_{2} = c \dot{x}$ is the friction force, and $F_{1} = \hat{k} x + \hat{k} a^{2} x^{3}$ denotes the restoring force of the spring and u (t) stands for the external control input. x denotes the displacement from a reference point. Define $x (t) = [\begin{matrix} x_{1} (t) \\ x_{2} (t) \end{matrix}] = [\begin{matrix} x \\ \dot{x} \end{matrix}]$ and $\bar{f} (t) = \frac{- \hat{k} - \hat{k} a^{2} x_{1}^{2} (t)}{m}$ , and let a = 0.3m^-1, m = 1kg, c = 2N · m/s, and x₁ (t) ∈ [-2, 2]. Suppose $\hat{k} \in [5, 8] N / m$ , we have x₁ (t) =0, ${\bar{f}}_{\max} = - 5$ with $\hat{k} = 5$ and x₁ (t) = ±2, ${\bar{f}}_{\min} = - 10.88$ with $\hat{k} = 8$ , respectively. In this example, the assumption is u (t) =0. Meanwhile, the disturbance input w (t) is considered. From the modeling method in [37], an IT2 fuzzy model in the form of Equation (1) is used to represent the mass-spring-damping system. The matrices for the IT2 fuzzy system are obtained as:

Fig.4

Mass-Spring-Damping System.

$A_{1} = [\begin{matrix} 0 & 1 \\ {\bar{f}}_{\min} & - \frac{c}{m} \end{matrix}]$ , $A_{2} = [\begin{matrix} 0 & 1 \\ {\bar{f}}_{\max} & - \frac{c}{m} \end{matrix}]$ , $A_{d 1} = [\begin{matrix} 0.1 & 0 \\ 0 & 0 \end{matrix}]$ , $A_{d 2} = [\begin{matrix} 0.1 & 0 \\ 0 & 0 \end{matrix}]$ , $D_{11} = [\begin{matrix} - 0.03 \\ - 0.02 \end{matrix}]$ , $D_{12} = [\begin{matrix} 0.01 \\ 0.01 \end{matrix}]$ , $C_{1} = [\begin{matrix} 0.1 & 0.1 \end{matrix}]$ , $C_{2} = [\begin{matrix} 0.1 & 0.1 \end{matrix}]$ , $C_{d 1} = [\begin{matrix} - 0.8 & 0.6 \end{matrix}]$ , $C_{d 2} = [\begin{matrix} - 0.2 & 1 \end{matrix}]$ , D₂₁ = -0.7, D₂₂ = 0.2, $E_{1} = [\begin{matrix} 1 & 0 \end{matrix}]$ , $E_{2} = [\begin{matrix} 1 & 0 \end{matrix}]$ , $E_{d 1} = [\begin{matrix} 0.1 & 0 \end{matrix}]$ , $E_{d 2} = [\begin{matrix} 0 & 0.2 \end{matrix}]$ , D₃₁ = 0, D₃₂ = 0.

The membership functions of the IT2 fuzzy model are expressed as:

${\underline{w}}_{1} (x_{1}) = \frac{- \bar{f} + {\bar{f}}_{\max}}{{\bar{f}}_{\max} - {\bar{f}}_{\min}}$ , with $\hat{k} = 5$ ,

${\bar{w}}_{1} (x_{1}) = \frac{- \bar{f} + {\bar{f}}_{\max}}{{\bar{f}}_{\max} - {\bar{f}}_{\min}}$ , with $\hat{k} = 8$ ,

${\underline{w}}_{2} (x_{2}) = \frac{\bar{f} - {\bar{f}}_{\min}}{{\bar{f}}_{\max} - {\bar{f}}_{\min}}$ , with $\hat{k} = 8$ ,

${\bar{w}}_{2} (x_{2}) = \frac{\bar{f} - {\bar{f}}_{\min}}{{\bar{f}}_{\max} - {\bar{f}}_{\min}}$ , with $\hat{k} = 5$ ,

${\hat{w}}_{i} (x_{1}) = \underline{ζ} (x_{i}) \underline{w} (x_{i}) + \bar{ζ} (x_{i}) \bar{w} (x_{i})$ , $\underline{ζ} (x_{i}) \in [0, 1]$ is uncertain parameters,

$\bar{ζ} (x_{i}) = 1 - \underline{ζ} (x_{i})$ , ${\tilde{w}}_{i} (x_{1}) = \frac{{\hat{w}}_{i} (x_{1})}{\sum_{s = 1}^{p} {\hat{w}}_{s} (x_{1})}$ .

The membership functions of the IT2 fuzzy filter are same as Example 1.

Only the L₂ - L_∞ performance is considered in this example because of the limited space. In the LMI conditions of Theorem 2 , let Φ = I, Ψ₁ = Ψ₂ = 0, Ψ₃ = γ²I. Given the scalars α = 1, β = 10, and h = 8 denotes the retardation time. By calculation, we can obtain the value of ρ_i as ρ₁ = 0.1626, ρ₂ = 0.4661. Then the minimized L₂ - L_∞ performance index is γ = 1.1526. The L₂ - L_∞ filter parameters in Equation (4) are as follows: $\begin{matrix} A_{f 1} & = & [\begin{matrix} - 1.1152 & 0.8917 \\ - 6.9021 & - 1.5817 \end{matrix}], \\ B_{f 1} & = & [\begin{matrix} - 0.4088 \\ - 0.0964 \end{matrix}], \\ C_{f 1} & = & [\begin{matrix} - 0.0059 & - 0.0264 \end{matrix}], \\ A_{f 2} & = & [\begin{matrix} - 1.0974 & 0.9068 \\ - 6.7723 & - 1.5810 \end{matrix}], \\ B_{f 2} & = & [\begin{matrix} - 0.4100 \\ - 0.11041 \end{matrix}], \\ C_{f 2} & = & [\begin{matrix} - 0.0059 & - 0.0264 \end{matrix}] . \end{matrix}$

The disturbance is given as follows: $w (t) = {\begin{matrix} e^{- t} cos (t), t \leq 55, \\ 0, else . \end{matrix}$ (36)

Under the conditions of x₁ (t) =1, x₂ (t) = -2, ${\hat{x}}_{1} (t) = 1$ and ${\hat{x}}_{2} (t) = - 2$ , the state responses x (t) and $\hat{x} (t)$ of filtering error system (5) are shown in Figs. 5 and 6. Figure 7 shows the error response e (t) under the conditions of x₁ (t) =0, x₂ (t) =0, ${\hat{x}}_{1} (t) = 0$ and ${\hat{x}}_{2} (t) = 0$ . From Figs. 5, 6 and 7, it is shown that the designed IT2 fuzzy filter is effective to the mass-spring-damping system. On the other hand, the proposed method in this paper can commendably deal with the uncertainties in membership functions. The advantage of the presented results has been demonstrated by Figs. 5, 6 and 7.

Fig.5

State response of x₁ (t), ${\hat{x}}_{1} (t)$ .

Fig.6

State response of x₂ (t), ${\hat{x}}_{2} (t)$ .

Fig.7

Error response.

Remark 3. The proposed approach is more suitable for handling nonlinear systems subjected to parameter uncertainties compared with the existing type-1 fuzzy filtering methods in [18, 36]. It should be noted that the type-1 fuzzy approaches in [18, 36] cannot be applied in Examples 1 and 2 because the membership functions of T-S fuzzy model in Examples 1 and 2 include uncertain parameters. Moreover, the proposed approach contains several types of the well-known performance indices. Compared with [32], time delays are first considered.

5 Conclusion

The problem of filter design for IT2 fuzzy time-delay systems with parameter uncertainties is investigated under a unified frame. A novel IT2 fuzzy filter is designed by establishing a Lyapunov-Krasovskii function, so that the filtering error system can guarantee the passive, dissipative, H_∞ and L₂ - L_∞ performance indexes in a unified frame. A new criterion is presented in the form of LMIs. Two numerical examples are provided to demonstrate the merits of the proposed approach. In the future directions, we will work to reduce the conservatism of the results derived, and generalize the proposed method into IT2 fuzzy systems with time-varying delays.

Footnotes

Acknowledgments

This work is supported by the Applied Basic Research Program of Science and Technology Department of Sichuan Province, China (2016JY0085) and the National Natural Science Foundation of China (51477146).

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