Finite-time stability and stabilization of interval type-2 fuzzy systems with time delay

Abstract

Finite-time stability and stabilization problems for a class of interval type-2 (IT2) fuzzy time-delay systems are studied. In this paper, we extend the concept of finite-time stability to IT2 fuzzy time-delay systems. Based on the Lyapunov stability method, integral inequality and some advanced matrix inequalities, a sufficient condition is proposed to guarantee finite-time stability of IT2 fuzzy time-delay systems. Then, by virtue of the results on finite-time stability and Finsler’s lemma, we propose an IT2 fuzzy state feedback controller which can guarantee the closed-loop system is finite time stable. The problem of finite time stabilization can be solved with con complementarity linearization iterative algorithm. Finally, two numerical examples are provided to verify the effectiveness of the proposed approach.

Keywords

IT2 fuzzy systems state feedback control finite-time time-delay

1 Introduction

Over the past few decades, Takagi-Sugeno (T-S) fuzzy model is gradually applied to the fields of system control, and many valuable results on stability analysis and control design have been presented [1 –16]. In recent years, Type-1 fuzzy model has been increasingly studied for nonlinear systems, and the existing studies have shown that fuzzy model can be used to handle complex nonlinear systems [17 –25]. Type-1 fuzzy model has many practical applications in the fields of nonlinear control, and the number of rules can be effectively reduced when fuzzy model is used to deal with multivariable systems. Some of the investigation on type-1 fuzzy control has been successfully applied to complex industrial control [26 –29]. As the considered systems become more complex and often have parameter uncertainty, it is difficult to describe the degree to which an object belongs to a fuzzy set with a certain membership value. In particular, in a highly uncertain environment, type-1 fuzzy set often cannot get good results. In order to improve the ability to deal with uncertainty, L.A. Zedeh proposed the concept of type-2 fuzzy set in 1975. The membership degree of type-2 fuzzy set is no longer the certain value in [0, 1], but it is type-1 fuzzy set. Therefore, type-2 fuzzy set greatly increases the freedom of design, and it can get better effects than type-1 fuzzy set in the case of high uncertainty ([30 –35]). Therefore, extending type-2 fuzzy logic to T-S fuzzy model enhances the ability of the system to deal with uncertainty and nonlinearity.

On the other hand, finite-time stability and finite-time stabilization have been gradually reported in recent years. The current research is mostly concerned about the Lyapunov asymptotic stability (LAS) of the system. But in practical applications, Lyapunov asymptotic stability may not be enough, and it is not acceptable that the system is asymptotically stable over an infinite time. Sometimes it is necessary to care about the performance of the system over a finite time interval, for example, missile and satellite system control and flight control. For these systems, we must concern with their stability over a finite time interval. Therefore, the concept of finite-time stability (FTS) has been developed. Note that, according to the existing results, there are two concepts of finite-time stability. One of the concepts is defined as, the state of the system does not always exceed a set area for a certain time interval once the bound of the initial condition of the system is given [36 –38]. The other one is said that the system trajectory would convergence and stay at the equilibrium point [39, 40]. In this paper, we consider the former concept.

Time delay is an important part of nonlinear system research, and widely exists in industrial systems. The presence of time-delay usually causes the poor performance and instability of control systems. It is necessary to carry on the controller design and the stability analysis to the time delay system. The finite-time stability of time-delay systems has been studied about twenty years ago, and we can find some valuable results [41 –43]. In recent years, with the development of linear matrix inequalities (LMIs) theory, some research results on the finite-time problem of time-delay systems have been massively published [44 –47].

The existing work on finite-time stabilization of fuzzy systems has been investigated in type-1 fuzzy model [48 –52]. However, in a highly uncertain environment, type-1 fuzzy model may get poor results. IT2 fuzzy models have been widely used to approximate nonlinear systems subject to parameter uncertainties [30 –35]. It has been shown that the parameter uncertainties can be captured by lower and upper membership functions of IT2 fuzzy model. Currently, the finite-time stability of IT2 fuzzy time-delay systems is still a challenging work. In this paper, we extend the concept of finite-time stability to IT2 fuzzy time-delay systems. In what follows, based on Lyapunov stability method, a new sufficient condition for finite-time stability of IT2 fuzzy time-delay systems is given in the form of LMIs. Then, IT2 fuzzy state feedback controller is designed based on the finite-time stability results. The controller gains can be obtained by con complementarity linearization iterative algorithm. The main contributions of this paper are summarized as follows.

The finite-time stability and stabilization problems for a class of IT2 fuzzy time-delay systems are studied for first time.

The IT2 fuzzy time-delay systems and IT2 fuzzy controller do not share the same premise membership functions, which enhances the degree of freedom for IT2 fuzzy finite-time controller.

Using con complementarity linearization iterative algorithm, the proposed IT2 fuzzy finite-time controller can be solved by linear convex optimization method.

The paper is organized as follows. In Section 2, the problem formulation and preliminaries are introduced. In Section 3, finite-time stability and stabilization conditions are derived by Lyapunov-Krasovskii function method. Two numerical examples are given to illustrate the effectiveness of the proposed approach in Section 4. The conclusion is drawn in the last section.

Natation: The following notations are used throughout this paper. The superscripts “T” and “-1” are denoted the transpose of a matrix and its inverse, respectively. $ℝ^{n}$ denotes the n-dimensional Euclidean space. $ℝ^{n \times m}$ is the set of all real matrices of dimension n × m. $𝕊^{n}$ and $𝕊_{+}^{n}$ are the set of symmetric and positive definite n × n matrices, respectively. sym {A} means A + A^T for simplicity. B > 0 denotes B is real symmetric and positive definite, and B > C means that the matrix B - C is positive definite. In symmetric block matrices, “*” denotes the transposed elements. λ_max (B) (λ_min (B)) means the maximum (minimum) of eigenvalues of a real symmetric matrix B. diag {⋯} denotes a block-diagonal matrix.

2 Problem statement and preliminaries

In this paper, we consider a nonlinear time-delay system described by the following IT2 fuzzy model:

Plant Rule i: IF f₁ (x (t)) is ${\tilde{M}}_{1}^{i}$ AND ⋯ AND f_p (x (t)) is ${\tilde{M}}_{p}^{i}$ ; THEN $\begin{matrix} \dot{x} (t) = A_{i} x (t) + A_{di} x (t - τ) + B_{i} u (t), \\ x (t) = φ (t), t \in [- τ, 0], \\ i \in 𝕊 : = {1, 2, \dots, r} \end{matrix}$ (1) where ${\tilde{M}}_{α}^{i}$ is an IT2 fuzzy set, α = 1, 2, …, p, f_α (x (t)) represents the premise variables, and p is a positive integer. $x (t) \in ℝ^{n}$ is the state, τ is time-delay, $u (t) \in ℝ^{m}$ is the control input, and $A_{i} \in ℝ^{n \times n}, A_{di} \in ℝ^{n \times n} and B_{i} \in ℝ^{n \times m}$ are the known system matrices. The firing strength of the ith rule is given by the following interval sets: $W_{i} (x (t)) = [ω –_{i} (x (t)) {\bar{ω}}_{i} (x (t))], i = 1, 2, \dots, r,$ where $ω –_{i} (x (t)) = \prod_{α = 1}^{p} u -_{{\tilde{M}}_{α}^{i}} (f_{α} (x (t))) \geq 0,$ ${\bar{ω}}_{i} (x (t)) = \prod_{α = 1}^{p} {\bar{u}}_{{\tilde{M}}_{α}^{i}} (f_{α} (x (t))) \geq 0,$ with $u -_{{\tilde{M}}_{α}^{i}} (f_{α} (x (t)))$ , ${\bar{u}}_{{\tilde{M}}_{α}^{i}} (f_{α} (x (t)))$ denoting the lower and upper membership grades, respectively.

The global fuzzy model can be expressed as $\dot{x} (t) = \sum_{i = 1}^{r} {\tilde{ω}}_{i} (x (t)) (A_{i} x (t) + A_{di} x (t - τ) + B_{i} u (t))$ (2) where $\begin{matrix} {\tilde{ω}}_{i} (x (t)) & = \frac{α -_{i} (x (t)) ω -_{i} (x (t)) + {\bar{α}}_{i} (x (t)) {\bar{ω}}_{i} (x (t))}{\sum_{j = 1}^{r} (α -_{j} (x (t)) ω -_{j} (x (t)) + {\bar{α}}_{j} (x (t)) {\bar{ω}}_{j} (x (t)))} \\ \geq 0, \forall i \end{matrix},$ ${\tilde{ω}}_{i} (x (t))$ is the normal membership functions with $\sum_{i = 1}^{r} {\tilde{ω}}_{i} (x (t)) = 1$ , and $α -_{i} (x (t))$ and ${\bar{α}}_{i} (x (t))$ are nonlinear functions with $0 ⩽ α -_{i} (x (t)) ⩽ 1, \forall i,$ $0 ⩽ {\bar{α}}_{i} (x (t)) ⩽ 1, \forall i,$ $α -_{i} (x (t)) + {\bar{α}}_{i} (x (t)) = 1, \forall i .$

To facilitate the next proof, we can simplify (2) into the overall fuzzy model: $\begin{matrix} \dot{x} (t) = A (t) x (t) + A_{d} (t) x (t - τ) + B (t) u (t), \\ x (t) = φ (t), t \in [- τ, 0] . \end{matrix}$ (3) where $\begin{matrix} A (t) & = \sum_{i = 1}^{r} {\tilde{ω}}_{i} (x (t)) A_{i}, A_{d} (t) = \sum_{i = 1}^{r} {\tilde{ω}}_{i} (x (t)) A_{di}, B (t) \\ = \sum_{i = 1}^{r} {\tilde{ω}}_{i} (x (t)) B_{i} \end{matrix}$

The mathematical description of FTS is given by the following definition.

Definition 1. (FTS). Systems (1) is said to be finite-time stable with respect to (c₁, γ, c₂, T) with c₂ > c₁ if $\begin{matrix} \forall t & \in [0, T] \\ {\begin{matrix} sup_{θ \in [- τ, 0]} φ^{T} (θ) φ (θ) ⩽ c_{1} \\ sup_{θ \in [- τ, 0]} {\dot{φ}}^{T} (θ) \dot{φ} (θ) ⩽ γ \end{matrix} \Rightarrow x^{T} (t) x (t) < c_{2} . \end{matrix}$ (4)

In this paper, we consider the following IT2 fuzzy controller form:

Rule j: IF θ₁ (x (t)) is ${\tilde{F}}_{1}^{j}$ AND … AND θ_p (x (t)) is ${\tilde{F}}_{p}^{j}$ ; THEN $u (t) = K_{j} x (t),$ where ${\tilde{F}}_{β}^{j}$ is an IT2 fuzzy set, β = 1, 2, …, p, j = 1, 2, …, r, and $K_{j} \in ℝ^{m \times n}$ , j = 1, 2, …, r, is the gain matrix of the state feedback controller in each rule. The firing strength of the jth rule is given by the following interval sets: $M_{j} (x (t)) = [\begin{matrix} m –_{j} (x (t)) & {\bar{m}}_{j} (x (t)) \end{matrix}], j = 1, 2, \dots, r,$ where ${\underline{m}}_{j} (x (t)) = \prod_{β}^{p} μ -_{{\tilde{F}}_{β}^{j}} (θ_{β} (x (t))) \geq 0,$ ${\bar{m}}_{j} (x (t)) = \prod_{β}^{p} {\bar{μ}}_{{\tilde{F}}_{β}^{j}} (θ_{β} (x (t))) \geq 0,$ with $μ -_{{\tilde{F}}_{β}^{j}} (θ_{β} (x (t)))$ , ${\bar{μ}}_{{\tilde{F}}_{β}^{j}} (θ_{β} (x (t)))$ denoting the lower and upper membership grades, respectively.

The overall state feedback controller is given by $u (t) = \sum_{j = 1}^{r} {\tilde{m}}_{j} (x (t)) K_{j} x (t) .$ (5) where $\begin{matrix} {\tilde{m}}_{j} (x (t)) & = \frac{β –_{j} (x (t)) m –_{j} (x (t)) + {\bar{β}}_{j} (x (t)) {\bar{m}}_{j} (x (t))}{\sum_{k = 1}^{r} [β –_{k} (x (t)) m -_{k} (x (t)) + {\bar{β}}_{k} (x (t)) {\bar{m}}_{k} (x (t))]} \\ \geq 0, \forall j, \end{matrix}$ ${\tilde{m}}_{j} (x (t))$ is the normal membership functions with $\sum_{i = 1}^{r} {\tilde{m}}_{j} (x (t)) = 1$ , and $β –_{j} (x (t))$ and ${\bar{β}}_{j} (x (t))$ are nonlinear functions with $0 ⩽ β –_{j} (x (t)) ⩽ 1, \forall j,$ $0 ⩽ {\bar{β}}_{j} (x (t)) ⩽ 1, \forall j,$ $β –_{j} (x (t)) + {\bar{β}}_{j} (x (t)) = 1, \forall j .$

Substituting (5) into system (2), we obtain the closed-loop of the nominal system as the following form $\begin{matrix} \dot{x} (t) = \sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{m}}_{j} [(A_{i} + B_{i} K_{j}) x (t) + A_{di} x (t - τ)] \\ x (t) = φ (t), t \in [- τ, 0], \\ i \in 𝕊 : = {1, 2, \dots, r} \end{matrix}$ (6)

To facilitate the next proof, we introduce the following two lemmas, and some variables are used to simplify vector and matrix representations. $η_{1} (t) = {[\begin{matrix} x^{T} (t) & v_{1}^{T} (t) & v_{2}^{T} (t) \end{matrix}]}^{T},$ $v_{1} (t) = \int_{t - τ}^{t} x (s) ds, v_{2} (t) = \int_{t - τ}^{t} \int_{t - τ}^{s} x (u) duds,$ $ξ (t) = {[\begin{matrix} x^{T} (t) & x^{T} (t - τ) & \frac{1}{τ} v_{1}^{T} (t) & \frac{2}{τ^{2}} v_{2}^{T} (t) \end{matrix}]}^{T},$ $e_{i} = [\begin{matrix} 0_{n \times (i - 1)} & I_{n} & 0_{n \times (4 - i) n} \end{matrix}], i = 1, 2, \dots, 4 .$

Lemma 1 [53]. 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: $- \int_{α}^{β} {\dot{x}}^{T} (s) R \dot{x} (s) ds ⩽ υ^{T} \partial υ,$ where $\begin{matrix} \partial = & τ (N_{1} R^{- 1} N_{1}^{T} + \frac{1}{3} N_{2} R^{- 1} N_{2}^{T} + \frac{1}{5} N_{3} R^{- 1} N_{3}^{T}) \\ + Sym {N_{1} Π_{1} + N_{2} Π_{2} + N_{3} Π_{3}} \end{matrix}$ $\begin{matrix} Π_{1} = e_{1} - e_{2}, Π_{2} = e_{1} + e_{2} - 2 e_{3}, \\ Π_{3} = e_{1} - e_{2} - 6 e_{3} + 6 e_{4}, \end{matrix}$ $υ = {[\begin{matrix} x^{T} (β) & x^{T} (α) & \frac{1}{τ} \int_{α}^{β} x^{T} (s) ds & \frac{1}{τ^{2}} \int_{α}^{β} \int_{α}^{s} x^{T} (u) duds \end{matrix}]}^{T},$ $τ = β - α .$

Lemma 2. Let $x \in ℝ^{n}, A \in 𝕊^{n}, B \in ℝ^{m \times n}$ , rank {B} < n, $C \in 𝕊_{+}^{m}$ and $D \in ℝ^{n \times m} .$ The following statements are equivalent:

Equicalence 1 (Finsler’s lemma) [54]:

\begin{matrix} (i) x^{T} Ax < 0, \forall Bx = 0, x \neq 0, \\ (ii) B^{⊥^{T}} {AB}^{⊥} < 0, \\ (iii) A + DB + B^{T} D^{T} < 0 . \end{matrix}

3 Main results

In this section, we develop sufficient conditions for the finite-time stability of the nonlinear time-delay system described by the IT2 fuzzy model (1).

Theorem 1. Consider the open-loop IT2 fuzzy system (2) with u (t) =0, nonnegative scalar α, and time-delay τ. The open-loop system is FTS respect to (c₁, γ, c₂, T) , c₁ < c₂, if there exist positive definite matri ces Q, R, P₁₁, P₂₂, P₃₃, matrices N₁, N₂, N₃, and positive scalars β₁, β₂, β₃, β₄, β₅, β₆, such that the following LMI conditions hold: $Ω_{ii} < 0, i = 1, \dots, r,$ (7) $Ω_{ij} + Ω_{ji} < 0, 1 ⩽ i < j ⩽ r,$ (8) $\begin{matrix} β_{1} I < P_{11} < β_{2} I, 0 < P_{22} < β_{3} I, 0 < P_{33} < β_{4} I, \\ 0 < Q < β_{5} I, 0 < R < β_{6} I \end{matrix}$ (9) $[\begin{matrix} - c_{2} e^{- α T} β_{1} & \sqrt{c_{1}} β_{2} & τ \sqrt{c_{1}} β_{3} & \frac{1}{2} τ^{2} \sqrt{c_{1}} β_{4} & \sqrt{c_{1} τ} β_{5} & \frac{\sqrt{2}}{2} τ \sqrt{γ} β_{6} \\ * & - β_{2} & 0 & 0 & 0 & 0 \\ * & * & - β_{3} & 0 & 0 & 0 \\ * & * & * & - β_{4} & 0 & 0 \\ * & * & * & * & - β_{5} & 0 \\ * & * & * & * & * & - β_{6} \end{matrix}] < 0,$ (10) where $Ω_{ij} = [\begin{matrix} Φ_{ij} + Θ - α e_{1}^{T} P_{11} e_{1} & \sqrt{τ} N_{1} & \sqrt{τ} N_{2} & \sqrt{τ} N_{3} \\ * & - R & 0 & 0 \\ * & * & - 3 R & 0 \\ * & * & * & - 5 R \end{matrix}],$ (11) $Φ_{ij} = sym {Π_{4}^{T} P Π_{5}^{i}} + e_{1}^{T} {Qe}_{1} - e_{2}^{T} {Qe}_{2} + τ Γ_{i}^{T} R Γ_{j},$ $Π_{4} = {[\begin{matrix} e_{1}^{T} & τ e_{3}^{T} & \frac{τ^{2}}{2} e_{4}^{T} \end{matrix}]}^{T},$

$Π_{5}^{i} = {[\begin{matrix} Γ_{i}^{T} & e_{1}^{T} - e_{2}^{T} & τ e_{3}^{T} - τ e_{2}^{T} \end{matrix}]}^{T},$ $Θ = sym {N_{1} Π_{1} + N_{2} Π_{2} + N_{3} Π_{3}},$ $Γ_{i} = [\begin{matrix} A_{i} & A_{di} & 0 & 0 \end{matrix}],$ $P = diag (P_{11}, P_{22}, P_{33}) .$

Proof. Consider a Lyapunov-Krasovskii function candidate: $\begin{matrix} V (x (t)) & = η_{1}^{T} (t) P η_{1} (t) + \int_{t - τ}^{t} x^{T} (s) Qx (s) ds \\ + \int_{- τ}^{0} \int_{t + θ}^{t} {\dot{x}}^{T} (s) R \dot{x} (s) dsd θ \end{matrix}$ (12)

The time derivative of V (x (t)) is

$\begin{matrix} \dot{V} (x (t)) & = & \sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{ω}}_{j} \\ \times (ξ^{T} (t) Φ_{ij} ξ (t) - \int_{t - τ}^{t} {\dot{x}}^{T} (s) R \dot{x} (s) ds) \end{matrix}$ (13)

From Lemma 1, the following inequality holds: $\begin{matrix} - \int_{t - τ}^{t} {\dot{x}}^{T} (s) R \dot{x} (s) ds \\ ⩽ ξ^{T} (t) [τ N_{1} R^{- 1} N_{1}^{T} + \frac{τ}{3} N_{2} R^{- 1} N_{2}^{T} + \frac{τ}{5} N_{3} R^{- 1} N_{3}^{T} + \\ sym {N_{1} Π_{1} + N_{2} Π_{2} + N_{3} Π_{3}}] ξ (t) \\ = ξ^{T} (t) (Θ + Ψ) ξ (t) \end{matrix}$ (14) where $Ψ = τ N_{1} R^{- 1} N_{1}^{T} + \frac{τ}{3} N_{2} R^{- 1} N_{2}^{T} + \frac{τ}{5} N_{3} R^{- 1} N_{3}^{T} .$

Then, we have $\dot{V} (x (t)) ⩽ ξ^{T} (t) ((\sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{ω}}_{j} Φ_{ij}) + Θ + Ψ) ξ (t)$ (15)

From (7) and (8), and adding membership function information, we have $\sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{ω}}_{j} Ω_{ij} < 0$ (16)

Using Schur complement to (16), we have $(\sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{ω}}_{j} Φ_{ij}) + Θ + Ψ - α e_{1}^{T} P_{11} e_{1} < 0$ (17)

Pre and post multiplying (17) by ξ^T (t) and ξ (t), we have

$\begin{matrix} ξ^{T} (t) ((\sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{ω}}_{j} Φ_{ij}) + Θ + Ψ) \\ \times ξ (t) < α x^{T} (t) P_{11} x (t) \end{matrix}$ (18)

Combining (15) and (18), we have $\dot{V} (x (t)) < α x^{T} (t) P_{11} x (t) < α V (x (t))$ (19)

Integrating (19) from 0 to T, with t ∈ [0, T] , we obtain $V (x (t)) < e^{α T} V (x (0))$ (20)

Using (9), for the right side of the inequality (20), we have $\begin{matrix} V (x (0)) & < λ_{max} (P_{11}) c_{1} + λ_{max} (P_{22}) c_{1} τ^{2} \\ + \frac{1}{4} λ_{max} (P_{33}) c_{1} τ^{4} \\ + λ_{max} (Q) c_{1} τ + \frac{1}{2} λ_{max} (R) γ τ^{2} \end{matrix}$ (21)

Combining (19), (20) and (21), we have $\begin{matrix} x^{T} (t) x (t) < \\ \frac{λ_{max} (P_{11}) c_{1} + λ_{max} (P_{22}) c_{1} τ^{2} + \frac{1}{4} λ_{max} (P_{33}) c_{1} τ^{4} + λ_{max} (Q) c_{1} τ + \frac{1}{2} λ_{max} (R) γ τ^{2}}{e^{- α T} λ_{min} (P_{11})} \end{matrix}$ (22)

Using Schur complement to (10), we have

$\begin{matrix} - c_{2} e^{- α T} β_{1} + β_{2} c_{1} + β_{3} c_{1} τ^{2} + \frac{1}{4} β_{4} c_{1} τ^{4} \\ + β_{5} c_{1} τ + \frac{1}{2} β_{6} γ τ^{2} < 0 \end{matrix}$ (23)

Combining (9), (22) and (23), we obtain $\begin{matrix} x^{T} (t) x (t) < \\ \frac{λ_{max} (P_{11}) c_{1} + λ_{max} (P_{22}) c_{1} τ^{2} + \frac{1}{4} λ_{max} (P_{33}) c_{1} τ^{4} + λ_{max} (Q) c_{1} τ + \frac{1}{2} λ_{max} (R) γ τ^{2}}{e^{- α T} λ_{min} (P_{11})} \\ < \frac{β_{2} c_{1} + β_{3} c_{1} τ^{2} + \frac{1}{4} β_{4} c_{1} τ^{4} + β_{5} c_{1} τ + \frac{1}{2} β_{6} γ τ^{2}}{e^{- α T} β_{1}} \\ < \frac{c_{2} e^{- α T} β_{1}}{e^{- α T} β_{1}} = c_{2} \end{matrix}$ (24)

According to the Definition 1, system (2) is finite-time stable with respect to (c₁, γ, c₂, T). The proof is completed.

Remark 1. IT2 T-S fuzzy system can effectively describe nonlinear systems subject to parameter uncertainties. It can be seen from Theorem 1 that the proposed stability conditions can be solved by MATLAB LMI toolbox. Therefore, Theorem 1 provides a relatively simple way to handle the finite-time stability of nonlinear systems subject to parameter uncertainties.

Now, based on the above results, we present the design of state feedback finite-time controller. Sufficient conditions for the finite-time stability of closed-loop system (6) are given in theorem 2 by considering the information of membership functions.

Theorem 2. Given nonnegative scalar α and σ, there exists an IT2 fuzzy state feedback controller in the form of $u (t) = \sum_{j = 1}^{r} {\tilde{m}}_{j} (x (t)) K_{j} x (t)$ such that the closed-loop system (6) is FTS with respect to (c₁, γ, c₂, T) , c₁ < c₂, if there exist positive define symmetric matrices $\hat{Q}, \hat{R}, G$ , ${\hat{P}}_{11}, {\hat{P}}_{22}, {\hat{P}}_{33}$ , matrices ${\hat{N}}_{1}, {\hat{N}}_{2}, {\hat{N}}_{3}, Z_{j}$ , $Λ_{j} = Λ_{j}^{T}$ , and positive scalars β₁, β₂, β₃, β₄, β₅, β₆, under the conditions ${\tilde{m}}_{j} - ρ_{j} {\tilde{ω}}_{j} \geq 0 (0 < ρ_{j} < 1)$ , for all j = 1, 2, …, r, such that condition (10) and the following conditions hold: ${\hat{Ω}}_{ij} - Λ_{i} < 0, i, j = 1, 2, \dots, r$ (25) $ρ_{i} {\hat{Ω}}_{ii} - ρ_{i} Λ_{i} + Λ_{i} < 0, i = 1, 2, \dots, r$ (26) $ρ_{j} {\hat{Ω}}_{ij} + ρ_{i} {\hat{Ω}}_{ji} - ρ_{j} Λ_{i} - ρ_{i} Λ_{j} + Λ_{i} + Λ_{j} ⩽ 0, i < j$ (27) $\begin{matrix} β_{1} I < G^{- 1} {\hat{P}}_{11} G^{- 1} < β_{2} I, 0 < G^{- 1} {\hat{P}}_{22} G^{- 1} < β_{3} I, \\ 0 < G^{- 1} {\hat{P}}_{33} G^{- 1} < β_{4} I, 0 < G^{- 1} {\hat{QG}}^{- 1} < β_{4} I, \\ 0 < G^{- 1} {\hat{RG}}^{- 1} < β_{6} I \end{matrix}$ (28) where

${\hat{Ω}}_{ij} = [\begin{matrix} \bar{Φ} + \bar{Θ} + Ξ_{ij} - α {\hat{e}}_{1}^{T} {\hat{P}}_{11} {\hat{e}}_{1} & \sqrt{τ} {\hat{N}}_{1} & \sqrt{τ} {\hat{N}}_{2} & \sqrt{τ} {\hat{N}}_{3} \\ * & - \hat{R} & 0 & 0 \\ * & * & - 3 \hat{R} & 0 \\ * & * & * & - 5 \hat{R} \end{matrix}],$ ${\hat{e}}_{i} = [0_{n \times (i - 1) n} I_{n} 0_{n \times (5 - i) n}], i = 1, 2, \dots, 5$ , $\bar{Φ} = sym {{\hat{Π}}_{5}^{T} \hat{P} {\hat{Π}}_{4}} + {\hat{e}}_{2}^{T} \hat{Q} {\hat{e}}_{2} - {\hat{e}}_{3}^{T} \hat{Q} {\hat{e}}_{3} + τ {\hat{e}}_{1}^{T} \hat{R} {\hat{e}}_{1},$ $\bar{Θ} = sym {{\hat{N}}_{1} {\hat{Π}}_{1} + {\hat{N}}_{2} {\hat{Π}}_{3} + {\hat{N}}_{3} {\hat{Π}}_{3}},$ $\begin{matrix} Ξ_{ij} & = & sym {(σ {\hat{e}}_{1}^{T} + {\hat{e}}_{2}^{T}) [(A_{i} G + B_{i} Z_{j}) {\hat{e}}_{2} \\ + A_{di} G {\hat{e}}_{3} - G {\hat{e}}_{1}]}, \end{matrix}$ ${\hat{Π}}_{1} = {\hat{e}}_{2} - {\hat{e}}_{3}, {\hat{Π}}_{2} = {\hat{e}}_{2} + {\hat{e}}_{3} - 2 {\hat{e}}_{4},$ ${\hat{Π}}_{3} = {\hat{e}}_{2} - {\hat{e}}_{3} - 6 {\hat{e}}_{4} + 6 {\hat{e}}_{5},$ ${\hat{Π}}_{4} = {[\begin{matrix} {\hat{e}}_{2}^{T} & τ {\hat{e}}_{4}^{T} & \frac{τ^{2}}{2} {\hat{e}}_{5}^{T} \end{matrix}]}^{T},$ ${\hat{Π}}_{5} = {[\begin{matrix} {\hat{e}}_{1}^{T} & {\hat{e}}_{2}^{T} - {\hat{e}}_{3}^{T} & τ {\hat{e}}_{4}^{T} - τ {\hat{e}}_{3}^{T} \end{matrix}]}^{T} .$

Then, the state feedback finite-time fuzzy controller gain can be obtained by K_j = Z_jG^-1.

Proof. Define ${\bar{N}}_{1}, {\bar{N}}_{2}, {\bar{N}}_{3} \in ℝ^{5 n \times n},$ $\hat{ξ} (t) = [\begin{matrix} {\dot{x}}^{T} (t) & x^{T} (t) & x^{T} (t - τ) & \frac{1}{τ} v_{1}^{T} (t) & \frac{2}{τ^{2}} v_{2}^{T} (t) \end{matrix}] .$

Using the similar proof procedure with Theorem 1, we have $\begin{matrix} \dot{V} (x (t)) & = ({\hat{ξ}}^{T} (t) \hat{Φ} \hat{ξ} (t) - \int_{t - τ}^{t} {\dot{x}}^{T} (s) R \dot{x} (s) ds) \\ ⩽ {\hat{ξ}}^{T} (t) (\hat{Φ} + \hat{Θ} + \hat{Ξ}) \hat{ξ} (t) \end{matrix}$ (29) where $\hat{Φ} = sym {{\hat{Π}}_{5}^{T} P {\hat{Π}}_{4}} + {\hat{e}}_{2}^{T} Q {\hat{e}}_{2} - {\hat{e}}_{3}^{T} Q {\hat{e}}_{3} + τ {\hat{e}}_{1}^{T} R {\hat{e}}_{1},$ $\hat{Θ} = sym {{\bar{N}}_{1} {\hat{Π}}_{1} + {\bar{N}}_{2} {\hat{Π}}_{3} + {\bar{N}}_{3} {\hat{Π}}_{3}},$ $\hat{Ξ} = τ {\bar{N}}_{1} R^{- 1} {\bar{N}}_{1}^{T} + \frac{τ}{3} {\bar{N}}_{2} R^{- 1} {\bar{N}}_{2}^{T} + \frac{τ}{5} {\bar{N}}_{3} R^{- 1} {\bar{N}}_{3}^{T} .$

From the idea of [55], we have the following equation: $\begin{matrix} \sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{m}}_{j} {\hat{Ω}}_{ij} \\ = \sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i}^{2} (ρ_{i} {\hat{Ω}}_{ii} - ρ_{i} Λ_{i} + Λ_{i}) + \sum_{i = 1}^{r - 1} \sum_{j = r + 1}^{r} {\tilde{ω}}_{i} {\tilde{ω}}_{j} \\ \times (ρ_{j} {\hat{Ω}}_{ij} - ρ_{j} Λ_{i} + Λ_{i} + ρ_{i} {\hat{Ω}}_{ji} - ρ_{i} Λ_{j} + Λ_{j}) \\ + \sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} ({\tilde{m}}_{j} - ρ_{j} {\tilde{ω}}_{j}) ({\hat{Ω}}_{ij} - Λ_{i}) \end{matrix}$ (30)

From (25), (26) and (27), we have $\sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{m}}_{j} {\hat{Ω}}_{ij} < 0$ (31)

Using Schur complement to (31),we have $\begin{matrix} \bar{Φ} + \bar{Θ} + \sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{m}}_{j} Ξ_{ij} - α e_{1}^{T} {\hat{P}}_{11} e_{1} + τ {\hat{N}}_{1} {\hat{R}}^{- 1} {\hat{N}}_{1}^{T} \\ + \frac{τ}{3} {\hat{N}}_{2} {\hat{R}}^{- 1} {\hat{N}}_{2}^{T} + \frac{τ}{5} {\hat{N}}_{3} {\hat{R}}^{- 1} {\hat{N}}_{3}^{T} < 0 \end{matrix}$ (32)

Define $D = σ {\hat{e}}_{1}^{T} G^{- 1} + {\hat{e}}_{2}^{T} G^{- 1},$

$C = [\begin{matrix} G {\hat{e}}_{1}^{T} & G {\hat{e}}_{2}^{T} & G {\hat{e}}_{3}^{T} & G {\hat{e}}_{4}^{T} & G {\hat{e}}_{5}^{T} \end{matrix}]^{T},$

$\hat{P} = diag (G G G) Pdiag (G G G)$ , $\hat{Q} = GQG$ , $\hat{R} = GRG$ , ${\hat{N}}_{1} = C {\bar{N}}_{1} G$ , ${\hat{N}}_{2} = C {\bar{N}}_{2} G$ , and ${\hat{N}}_{3} = C {\bar{N}}_{3} G$ .

Pre and post multiplying (32) by C^-1 and C^-T, we have $H + DT + T^{T} D^{T} < 0$ (33) where $H = {\hat{Φ}}_{1} + \hat{Θ} + \hat{Ξ} - α {\hat{e}}_{2}^{T} P_{11} {\hat{e}}_{2}$ , $T = \sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{m}}_{j} [(A_{i} + B_{i} K_{j}) {\hat{e}}_{2} + A_{di} {\hat{e}}_{3} - {\hat{e}}_{1}] .$

Moreover, we have $\begin{matrix} \sum_{i = 1}^{r} \sum_{j = 1}^{r} {\tilde{ω}}_{i} {\tilde{m}}_{j} [(A_{i} + B_{i} K_{j}) {\hat{e}}_{2} + A_{di} {\hat{e}}_{3} - {\hat{e}}_{1}] \hat{ξ} (t) \\ = T \hat{ξ} (t) = 0 \end{matrix}$ (34)

Using Lemma 2 to (33) and (34), we have ${\hat{ξ}}^{T} (t) ({\hat{Φ}}_{1} + \hat{Θ} + \hat{Ξ} - α {\hat{e}}_{2}^{T} P_{11} {\hat{e}}_{2}) \hat{ξ} (t) < 0$ (35)

Combining (29) and (35), we have $\dot{V} (x (t)) < α x^{T} (t) P_{11} x (t)$ (36)

Replacing P₁₁, P₂₂, P₃₃, Q and R in (9) with $G^{- 1} {\hat{P}}_{11} G^{- 1}$ , $G^{- 1} {\hat{P}}_{22} G^{- 1}$ , $G^{- 1} {\hat{P}}_{33} G^{- 1}$ , $G^{- 1} {\hat{QG}}^{- 1}$ and $G^{- 1} {\hat{RG}}^{- 1}$ , we get (28).

In the following, using the similar proof procedure with (20)–(24), we can obtain that the closed-loop system (6) is finite-time stable with respect to (c₁, γ, c₂, T). The proof is completed.

Remark 2. The condition (28) in Theorem 2 cannot be converted to LMI. Thus, it is difficult to solve the Theorem 2 by MATLAB LMI toolbox. Thus, it is necessary to develop effective and simple algorithm to solve the problem of finite-time state feedback controller.

Remark 3. Using Schur complement to (28), we have the following inequality: $\begin{matrix} [\begin{matrix} β_{1}^{- 1} I & G^{T} \\ G & {\hat{P}}_{11} \end{matrix}] > 0, [\begin{matrix} {\hat{P}}_{11}^{- 1} & G^{- T} \\ G^{- 1} & β_{2} I \end{matrix}] > 0, [\begin{matrix} {\hat{P}}_{22}^{- 1} & G^{- T} \\ G^{- 1} & β_{3} I \end{matrix}] > 0, \\ [\begin{matrix} {\hat{P}}_{33}^{- 1} & G^{- T} \\ G^{- 1} & β_{4} I \end{matrix}] > 0, [\begin{matrix} {\hat{Q}}^{- 1} & G^{- T} \\ G^{- 1} & β_{5} I \end{matrix}] > 0, [\begin{matrix} {\hat{R}}^{- 1} & G^{- T} \\ G^{- 1} & β_{6} I \end{matrix}] > 0 . \end{matrix}$ (37)

Inspired by the idea of [56, 57], we introduce a con complementarity linearization iterative algorithm(CCLA) to solve the Theorem 2. In fact, We can use the following nonlinear minimization problem to replace the conditions (10), (25)–(27) and (37) in Theorem 2.

Problem 1. $\begin{matrix} min {trace (GU + {\hat{QV}}_{1} + {\hat{RV}}_{2} + {\hat{P}}_{11} V_{3} + {\hat{P}}_{22} V_{4} \\ + {\hat{P}}_{33} V_{5}) + β_{1} μ} \end{matrix}$

subject to (10), (25)–(27) and $\begin{matrix} [\begin{matrix} μ I & G \\ G & {\hat{P}}_{11} \end{matrix}] > 0, [\begin{matrix} V_{3} & U \\ U & β_{2} I \end{matrix}] > 0, [\begin{matrix} V_{4} & U \\ U & β_{3} I \end{matrix}] > 0, \\ [\begin{matrix} V_{5} & U \\ U & β_{4} I \end{matrix}] > 0, [\begin{matrix} V_{1} & U \\ U & β_{5} I \end{matrix}] > 0, [\begin{matrix} V_{2} & U \\ U & β_{6} I \end{matrix}] > 0, \\ [\begin{matrix} β_{1} & 1 \\ 1 & μ \end{matrix}] > 0, [\begin{matrix} {\hat{P}}_{11} & I \\ I & V_{3} \end{matrix}] > 0, [\begin{matrix} {\hat{P}}_{22} & I \\ I & V_{4} \end{matrix}] > 0, \\ [\begin{matrix} {\hat{P}}_{33} & I \\ I & V_{5} \end{matrix}] > 0, [\begin{matrix} \hat{Q} & I \\ I & V_{1} \end{matrix}] > 0, [\begin{matrix} \hat{R} & I \\ I & V_{2} \end{matrix}] > 0, [\begin{matrix} G & I \\ I & U \end{matrix}] > 0 . \end{matrix}$ (38)

If the solution of Problem 1 is 6n + 1, then the conditions (10), (25)–(27) and (37) are solvable. In the following analysis, we give an iterative algorithm to solve Problem 1.

Algorithm 1.

Step 1. Select two nonnegative scalars α, σ so that there is a feasible solution for (10), (25)–(27) and (38).

Step 2. Choose a set of feasible solutions $(G, Z_{j}, {\hat{P}}_{11}, {\hat{P}}_{22}, {\hat{P}}_{33}, \hat{R}, \hat{Q}, U, V_{1}, V_{2}, V_{3},$ V₄, V₅, μ, β₁, β₂, β₃, β₄, β₅, β₆, α, σ) ⁰ that satisfy (10), (25)–(27) and (38). Select the appropriate convergence accuracy Δ > 0. If (39) holds, then the controller gain can be obtained by $K_{j} = Z_{j}^{0} (G^{0})^{- 1}$ . Otherwise, execute step3. $\begin{matrix} | trace (G^{0} U^{0} + {\hat{Q}}^{0} V_{1}^{0} + {\hat{R}}^{0} V_{2}^{0} + {\hat{P}}_{11}^{0} V_{3}^{0} + {\hat{P}}_{22}^{0} V_{4}^{0} \\ + {\hat{P}}_{33}^{0} V_{5}^{0}) + β_{1}^{0} μ^{0} - 6 n - 1 | < Δ \end{matrix}$ (39)

Step 3. Set k = 0 and the maximum number of iterations N. Solve the LMI problem: $\begin{matrix} min {trace (G^{k} U + U^{k} G + {\hat{Q}}^{k} V_{1} + V_{1}^{k} \hat{Q} + {\hat{R}}^{k} V_{2} \\ + V_{2}^{k} \hat{R} + {\hat{P}}_{11}^{k} V_{3} V_{3}^{k} {\hat{P}}_{11} + {\hat{P}}_{22}^{k} V_{4} + V_{4}^{k} {\hat{P}}_{22} \\ + {\hat{P}}_{33}^{k} V_{5} + V_{5}^{k} {\hat{P}}_{33}) + β_{1}^{k} μ + μ^{k} β_{1})} \end{matrix}$ (40)subject to (10), (25) (27) and (38), and set $\begin{matrix} (G, {\hat{P}}_{11}, {\hat{P}}_{22}, {\hat{P}}_{33}, \hat{R}, \hat{Q}, U, V_{1}, V_{2}, V_{3}, \\ {V_{4}, V_{5}, μ, β_{1})}^{k + 1} \\ = (G, {\hat{P}}_{11}, {\hat{P}}_{22}, {\hat{P}}_{33}, \hat{R}, \hat{Q}, U, V_{1}, V_{2}, \\ {V_{3}, V_{4}, V_{5}, μ, β_{1})}^{k} \end{matrix}$

Step 4. If (41) holds, then the controller gain can be obtained by $K_{j} = Z_{j}^{k + 1} (G^{k + 1})^{- 1}$ . Otherwise set k = k + 1, if k < N, return to step 3, then break the loop. $\begin{matrix} | trace (G^{k + 1} U^{k + 1} + {\hat{Q}}^{k + 1} V_{1}^{k + 1} + {\hat{R}}^{k + 1} V_{2}^{k + 1} + {\hat{P}}_{11}^{k + 1} V_{3}^{k + 1} \\ + {\hat{P}}_{22}^{k + 1} V_{4}^{k + 1} + {\hat{P}}_{33}^{k + 1} V_{5}^{k + 1}) + β_{1}^{k + 1} μ^{k + 1} - 6 n - 1 | < Δ \end{matrix}$ (41)

Remark 4. Using Algorithm 1, the design of finite-time state feedback controller can be easily facilitated by MATLAB LMI toolbox. Therefore, Theorem 2 provides a convenient solution to design finite-time state feedback controller for nonlinear systems subject to parameter uncertainties.

Remark 5. In the framework of type-1 fuzzy model, the problem of finite-time stabilization has been investigated in [48 –52]. Once the considered systems are subject to parameter uncertainties, the membership function of fuzzy model may become uncertain. In such case, the existing finite-time stabilization methods based on type-1 fuzzy logic would be invalid. This study gives a novel solution in the framework of IT2 fuzzy systems.

4 Illustrative examples

In this section, two numerical examples are provided to verify the effectiveness of the proposed approach.

Example 1. In this example, consider the IT2 fuzzy time delay system (2) with system matrices (42). $\begin{matrix} A_{1} = [\begin{matrix} 0 & 1 \\ - 10.88 & - 2 \end{matrix}], A_{2} = [\begin{matrix} 0 & 1 \\ - 5 & - 2 \end{matrix}], \\ A_{d 1} = [\begin{matrix} 0.1 & 0 \\ 0 & 0 \end{matrix}] \\ A_{d 2} = [\begin{matrix} 0.1 & 0 \\ 0 & 0 \end{matrix}], B_{1} = B_{2} = {[\begin{matrix} 1 & 1 \end{matrix}]}^{T} . \end{matrix}$ (42)

The membership functions of the IT2 fuzzy model are chosen as $\begin{matrix} ω -_{1} (x_{1} (t)) = 0.0765 x_{1}^{2}, {\bar{ω}}_{1} (x_{1} (t)) = 0.05102 + 0.1224 x_{1}^{2}, \\ ω -_{2} (x_{1} (t)) = 1 - {\bar{ω}}_{1} (x_{1} (t)), {\bar{ω}}_{2} (x_{1} (t)) = 1 - ω -_{1} (x_{1} (t)), \end{matrix}$ ${\tilde{ω}}_{i} (x (t)) = α -_{i} (x (t)) ω -_{i} (x (t)) + {\bar{α}}_{i} (x (t)) {\bar{ω}}_{i} (x (t)) \geq 0,$ and x₁ (t) ∈ [-2, 2] where $α -_{i} (x (t))$ and ${\bar{α}}_{i} (x (t))$ are nonlinear parameters satisfying $α -_{i} (x (t)) \in [0, 1]$ and ${\bar{α}}_{i} (x (t)) = 1 - α -_{i} (x (t))$ .

Now consider the IT2 fuzzy state feedback controller, and the membership functions of IT2 fuzzy state feedback controller in (5) are selected as $\begin{matrix} m -_{1} (x_{1}) & = & 1 - \frac{1}{1 + e^{\frac{- x_{1} + 0.25}{2}}}, \\ {\bar{m}}_{1} (x_{1}) & = & 1 - \frac{1}{1 + e^{\frac{- x_{1} - 0.25}{2}}}, \end{matrix}$ $\begin{matrix} m -_{2} (x_{1}) & = & 1 - {\bar{m}}_{1} (x_{1}), \\ {\bar{m}}_{2} (x_{1}) & = & 1 - m -_{1} (x_{1}), β –_{j} = {\bar{β}}_{j} = 0.5 . \end{matrix}$

Our purpose is to find feedback gains such that the closed-loop system is finite-time stable. Let τ = 0.4, c₁ = 0.4, γ = 9, c₂ = 0.8, T = 50, ρ₁ = 0.1626, ρ₂ = 0.4661, and using algorithm 1 for α = 10^-5 and σ = 0.4, we obtain the following controller gains: $\begin{matrix} K_{1} = [\begin{matrix} - 8.7864 & - 1.7715 \end{matrix}]; \\ K_{2} = [\begin{matrix} - 8.7918 & - 1.7743 \end{matrix}]; \end{matrix}$ (43)

Under the initial condition of $φ^{T} (t) = [\begin{matrix} 0.4 & 0 \end{matrix}]$ , t ∈ [- τ, 0] and c₁, γ, c₂, T, the state responses of the closed-loop system are shown in Fig. 1. From Fig. 1, it can been seen that the values of state variables convergence and stay at the equilibrium point when t → T. Figure 2 shows the norm of the state vector of closed-loop system. From Fig. 2, we can see that x^T (t) x (t) < c₂, ∀ t ∈ [0, T], so we can say the closed-loop system is finite-time stable with respect to (0.4, 9, 0.8, 50).

Fig.1

The state response of the closed-loop system.

Fig.2

The norm of the state vector of the closed-loop system.

Example 2. Consider an inverted pendulum model [58] ${\begin{matrix} {\dot{x}}_{1} (t) = x_{2} (t) \\ {\dot{x}}_{2} (t) = \frac{g sin (x_{1} (t)) - {am}_{p} {Lx}_{2} (t)^{2} sin (2 x_{1} (t)) / 2 - a cos (x_{1} (t)) u (t)}{4 L / 3 - {am}_{p} L {cos}^{2} (x_{1} (t))} \end{matrix}$ where $x_{1} (t) \in [- \frac{5 π}{12}, \frac{5 π}{12}]$ , x₂ (t) ∈ [- 5, 5], g = 9.8, L = 0.5. m_p ∈ [2, 3] and m_c ∈ [8, 16] are uncertain parameters, and a = 1/(m_p + m_c).

The above inverted pendulum model is subject to parameter uncertainties, and thus the existing type-1 fuzzy finite-time stabilization methods [48 –52] cannot be applied in this example. Using the modelling method in [59], a four-rule IT2 T-S fuzzy model in the form of (2) is employed to describe the inverted pendulum. The system matrices in (2) are defined as

$\begin{matrix} A_{1} = A_{2} = [\begin{matrix} 0 & 1 \\ 10.0078 & 0 \end{matrix}], \\ A_{3} = A_{4} = [\begin{matrix} 0 & 1 \\ 18.4800 & 0 \end{matrix}], \\ A_{d 1} = A_{d 2} = A_{d 3} = A_{d 4} = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}], \\ B_{1} = B_{3} = [\begin{matrix} 0 \\ - 0.1765 \end{matrix}], \\ B_{2} = B_{4} = [\begin{matrix} 0 \\ - 0.0261 \end{matrix}] . \end{matrix}$ (44)

The lower and upper membership functions for each rule in this example are defined in Table 1. And ${\tilde{ω}}_{i} (x (t)) = α -_{i} (x (t)) ω -_{i} (x (t)) + {\bar{α}}_{i} (x (t)) {\bar{ω}}_{i} (x (t)) \geq 0, \forall i,$ where $α -_{i} (x (t))$ and ${\bar{α}}_{i} (x (t))$ are nonlinear parameters, $α -_{i} (x (t)) \in [0, 1]$ , ${\bar{α}}_{i} (x (t)) = 1 -$ $α -_{i} (x (t))$ .

Table 1

Lower and upper membership functions of the IT2 T-S fuzzy model of inverted pendulum in example 2

Lower membership functions	Upper membership functions
$u -_{{\tilde{M}}_{1}^{1}} (x_{1} (t)) = 1 - e^{\frac{- x_{1} (t)^{2}}{1.2}}$	${\bar{u}}_{{\tilde{M}}_{1}^{1}} (x_{1} (t)) = 1 - 0.23 e^{\frac{- x_{1} (t)^{2}}{0.25}}$
$u -_{{\tilde{M}}_{1}^{2}} (x_{1} (t)) = 1 - e^{\frac{- x_{1} (t)^{2}}{1.2}}$	${\bar{u}}_{{\tilde{M}}_{1}^{2}} (x_{1} (t)) = 1 - 0.23 e^{\frac{- x_{1} (t)^{2}}{0.25}}$
$u -_{{\tilde{M}}_{1}^{3}} (x_{1} (t)) = 0.23 e^{\frac{- x_{1} (t)^{2}}{0.25}}$	$u -_{{\tilde{M}}_{2}^{3}} (x_{1} (t)) = 0.5 e^{\frac{- x_{1} (t)^{2}}{0.25}}$
$u -_{{\tilde{M}}_{1}^{4}} (x_{1} (t)) = 0.23 e^{\frac{- x_{1} (t)^{2}}{0.25}}$	${\bar{u}}_{{\tilde{M}}_{1}^{4}} (x_{1} (t)) = e^{\frac{- x_{1} (t)^{2}}{1.2}}$
$u -_{{\tilde{M}}_{2}^{1}} (x_{1} (t)) = 0.5 e^{\frac{- x_{1} (t)^{2}}{0.25}}$	${\bar{u}}_{{\tilde{M}}_{2}^{1}} (x_{1} (t)) = e^{\frac{- x_{1} (t)^{2}}{1.5}}$
$u -_{{\tilde{M}}_{2}^{2}} (x_{1} (t)) = 1 - e^{\frac{- x_{1} (t)^{2}}{1.5}}$	${\bar{u}}_{{\tilde{M}}_{2}^{2}} (x_{1} (t)) = 1 - 0.5 e^{\frac{- x_{1} (t)^{2}}{0.25}}$
$u -_{{\tilde{M}}_{2}^{3}} (x_{1} (t)) = 0.5 e^{\frac{- x_{1} (t)^{2}}{0.25}}$	${\bar{u}}_{{\tilde{M}}_{2}^{3}} (x_{1} (t)) = e^{\frac{- x_{1} (t)^{2}}{1.5}}$
$u -_{{\tilde{M}}_{2}^{4}} (x_{1} (t)) = 1 - e^{\frac{- x_{1} (t)^{2}}{1.5}}$	${\bar{u}}_{{\tilde{M}}_{2}^{4}} (x_{1} (t)) = 1 - 0.5 e^{\frac{- x_{1} (t)^{2}}{0.25}}$

Let ρ₁ = ρ₂ = ρ₃ = ρ₄ = 0.1, τ = 0.2, c₁ = 0.4, γ = 9, c₂ = 0.53, T = 100, and using algorithm 1 for α = 10^-5 and σ = 10^-2, we obtain the controller gains given in (45). $\begin{matrix} K_{1} = [\begin{matrix} 2499 . 3339 & 3295.3284 \end{matrix}]; \\ K_{2} = [\begin{matrix} 2499.3418 & 3295.3406 \end{matrix}]; \\ K_{3} = [\begin{matrix} 2499.3310 & 3295.3239 \end{matrix}]; \\ K_{4} = [\begin{matrix} 2499.3343 & 3295 . 3289 \end{matrix}]; \end{matrix}$ (45)

With the controller gains (45), the closed-loop system is finite-time stable with respect to (0.4, 9, 0.53, 100). Figure 3 shows the state response of closed-loop system. Obviously, the values of state variables convergence and stay at the equilibrium point when t → Tunder the initial condition $φ^{T} (t) = [\begin{matrix} \frac{π}{6} & 0 \end{matrix}], t \in [- τ, 0]$ . The norm of the state vector of the closed-loop system is shown in Fig. 4. From Fig. 4, we have x^T (t) x (t) < c₂, ∀ t ∈ [0, T], so the designed IT2 fuzzy controller can ensure the closed-loop system is finite-time stable with respect to (0.4, 9, 0.53, 100).

Fig.3

The state response of the closed-loop system.

Fig.4

The norm of the state vector of the closed-loop system.

5 Conclusions

In this paper, we extend the concept of finite-time stability to IT2 fuzzy time-delay systems. The finite-time stability and stabilization issues for a class of IT2 fuzzy time-delay systems are investigated. Based on the Lyapunov stability method and some advanced properties of matrix inequalities, a sufficient condition is proposed to guarantee finite-time stability of IT2 fuzzy time-delay systems. This study gives a novel solution to handle the finite-time control problem of nonlinear systems subject to parameter uncertainties in the framework of IT2 fuzzy systems. The state feedback fuzzy controller that guarantee the closed-loop system is finite-time stable, is also designed by solving con complementarity linearization iterative algorithm. The numerical examples illustrate the effectiveness of the proposed approach. In the future work, we will focus on output feedback finite-time stabilization issues and finite-time stability of IT2 fuzzy systems with time-varying delays.

Footnotes

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61703291) and the Applied Basic Research Program of Science and Technology Department of Sichuan Province, China (2016JY0085).

References

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