Improved results on nonlinear perturbed T–S fuzzy systems with interval time-varying delays using a geometric sequence division method

This paper presents improved stability results by introducing a new delay partitioning method based on the theory of geometric progression to deal with T–S fuzzy systems in the appearance of interval time-varying delays and nonlinear perturbations. A common ratio \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha$$\end{document}α is applied to split the delay interval into multiple unequal subintervals. A modified Lyapunov–Krasovskii functional (LKF) is constructed with triple-integral terms and augmented factors including the length of every subintervals. In addition, the recently developed free-matrix-based integral inequality is employed to combine with the extended reciprocal convex combination and free weight matrices techniques for avoiding the overabundance of the enlargement when deducing the derivative of the LKF. Eventually, this developed research work can efficiently obtain the maximum upper bound of the time-varying delay with much less conservatism. Numerical results are conducted to illustrate the remarkable improvements of this proposed method.

criteria is more useful to produce less conservative results (Yang et al. 2015b;Senthilkumar and Mahanta 2010;Lam et al. 2007). Delay partitioning technique, alternatively known as a delay fractionizing method, was developed in Gouaisbaut and Peaucelle (2006). A number of research works have been developed to prove that delay partitioning approach can significantly enhance the stability conditions to obtain less conservatism as soon as the partitions get thinner (Yang et al. 2015a;Zhao et al. 2009;Wang et al. 2015). In Wang et al. (2015), a secondary partitioning method was proposed to further divide primarily separated intervals into a series of smaller segments, which illustrates good stability results. Nonetheless, the research development requires too many adjustable parameters. It thus cost extra computation burden.
In order to further achieve less conservative results, a number of inequalities methods have been proposed, such as Peng-Park's inequality, reciprocally convex combination, free-matrix-based inequality, etc, which are employed for the purpose of overabundance reduction of the enlargement of the Lyapunov functionals derivative (Sun et al. 2010;Gyurkovics 2015;Park et al. 2011Park et al. , 2015Peng and Han 2011;Zeng et al. 2015a). By introducing both augmented state and integral of the state over the period of the delay, these newly developed techniques can preserve extra items when dealing with the enlargement in bounding the derivative of the LKF comparing to the Jensen's inequality in Seuret and Gouaisbaut (2013). As a result, tighter bounding inequalities are obtained to reduce the conservatism.
In addition, the presence of nonlinearity can cause poor performance and even instability in engineering systems. Robust stability analysis with the effect of the nonlinear perturbation has been investigated with considerable attention (Zhang et al. 2010(Zhang et al. , 2015aRamakrishnan and Ray 2011). Because of process uncertainties and parameter variations, nonlinear perturbations commonly occur in both current and delayed states Ramakrishnan and Ray (2011). The previously developed techniques for such systems are rarely adaptive for the stability analysis with the appearance of nonlinear perturbations. In this paper, T-S fuzzy systems with interval time-varying delays and nonlinear perturbation are considered for stability analysis. Based on the geometric sequence division, some newly developed inequalities, free weight matrices techniques and the Finsler's Lemma are also employed for obtaining improved stability criteria. Main contributions of this work are: 1. Based on the recently developed geometric sequence division method on delay partitioning, improved stability criteria is presented. 2. Extended reciprocal convex combination(ERCC) is employed for the less enlargement of bounding the derivative of the augmented LKF which is able to reduce the overabundance when deal with the inequalities in the derivative of the LKF. 3. In terms of the system equation, free weight matrices techniques are applied to reduce the conservatism with respect to each fuzzy rule. Numerical examples are conducted to show that the improved stability conditions are obtained by comparing with some existing results.
Notations. R n and R n×m denote the n-dimensional Euclidean space and the set of all n × m real matrices, respectively. I(0) is the identity (zero) matrix with appropriate dimension; A T denotes the transpose, and He(A) = A + A T . The symbol * denotes the elements below the main diagonal of a symmetric block matrix. � • � is the Euclidean norm in R n . C([−τ b , 0], R n ) is the family of continuous functions ϕ from the interval [−τ b , 0] to R n with the norm �ϕ� τ = sup −τ ≤θ ≤0 �ϕ(θ)�. The notation A > (≥)B means that A − B is positive (semi-positive) definite.

Problem statements and preliminaries
Considering nonlinear perturbed T-S fuzzy systems with interval time-varying delays, for each l = 1, 2, . . . r (r is the number of the plant rules), the lth rule of this fuzzy model with r plant rules are described as follows.
, M ls (s = 1, 2, . . . , p) are premise variables and the related fuzzy sets, respectively. A l , B l , C l , D l are the constant matrices with appropriate dimensions. τ (t) is the time-varying delay. f (x(t), t) and g(x(t − τ (t)), t) are unknown nonlinear perturbations with respect to the current state x(t) and the delayed state Then the fuzzy model can be inferred as: where r is the number of fuzzy implications, h l (t) = W l (t) For W l (t) ≥ 0, h l (t) ≥ 0 and r l=1 h l (t) = 1 thus holds. The time-varying delay τ (t) is considered as the following two cases:

Case 1 τ (t) is a differentiable function satisfying
Case 2 τ (t) is a continuous function satisfying where τ a , τ b , µ are constants. (2) where γ ≥ 0, β ≥ 0 are known scalars, F and G are known constant matrices, ∀x ∈ R n , and f and g are the short expressions of f (x(t), t) and g(x(t − τ (t)), t), respectively.
A few lemmas are introduced for stability analysis as follows.
Lemma 1 (Han 2003(Han , 2005 For n × n matrix Q > 0, scalar τ > 0, vector-valued function ẋ : [−τ , 0] −→ R n such that the following integrations are well defined, it holds that Lemma 2 (Zeng et al. 2015, Free-matrix-based integral inequality) Let x be a differentiable function : [a, b] → R n , Z ∈ R n×n and W 1 , W 3 ∈ R 3n×3n be symmetric matrices, and W 2 ∈ R 3n×3n , N 1 , N 2 ∈ R 3n×n satisfying this condition it holds: 1 =ē 1 −ē 2 , 2 = 2ē 3 −ē 1 −ē 2 ,ē 1 = I 0 0 ,ē 2 = 0 I 0 ,ē 3 = 0 0 I . Remark 1 By introducing both augmented state and integral of the state over the period of the delay, the well known Wirtinger-based inequality was developed with less conservatism comparing to the Jensen's inequality in Seuret and Gouaisbaut (2013) to reduce enlargement in bounding the derivative of the LKF inequalities. However, due to the unadjustable parameters, the tightest upper bound is rarely to be determined in this development. In fact, this Wirtinger-based inequality is the special case of free-matrixbased integral inequality (8) by setting Particularly, a set of slack variables inequality in this inequality can be flexibly adjusted, which provide remarkable extra freedom for the purpose of conservatism reduction.

Main results
The stability criteria of T-S fuzzy systems in the presences of interval time-varying delays and nonlinear perturbations are analyzed in this section. In terms of the geometric sequence division method, a new delay partitioning technique is proposed in Fig. 1.
α is a real positive number, and δ i is the length of the ith subinterval which equals to α q−i . The following expressions are used for notational simplification. The augmented vector is defined as, where Next, the new delay dependent stability criteria is presented for the T-S fuzzy system described in (2).

Theorem 1 Given a positive integer m, and δ
The Lyapunov-Krasovskii functional is as follows: The derivative of the Lyapunov functional V (x t , k) | τ (t)∈I k along the trajectory of the perturbed T-S fuzzy system described in (2) is given as: where The derivative of the second term of the V 2 (x t ) is derived as Thus, The derivative of V 3 (x t , k) is deduced as For the case of τ (t) ∈ I k (1 ≤ k ≤ m), the second term in (19) is deduced as follows Applying Lemma 2 to deal with (20), it is obtained In the case of i = k, applying Jensen's inequality and the extended ERCC in Lemma 3, it is given as, Then , it follows from (19-22) that The derivative of V 4 (x t ) is presented as By using Lemma 1, the last two terms of (24) are deduced as Thus (24) implies that Referring to (5), for any scalars 1 ≥ 0, 2 ≥ 0 , the nonlinear perturbations can be derived as According to the system in (1), with N 1 and N 2 are defined as N 1 = r l=1 h l (t) N 1l and N 2 = r l=1 h l (t) N 2l , and N 1l , N 2l are constant matrices. Then it is given as = ξ T (t)� l,6 ξ(t) Therefore, the following inequality holds Using the augmented vector (11) with the simplification expression (10), the T-S fuzzy system (2) is represented as where Ŵ l is defined in Theorem 1. Hence, the asymptotic stability condition for the T-S fuzzy system (2) with interval time-varying delays and nonlinear perturbations is expressed as Consequently, by means of the Lemma 4, there exists a matrix Y with appropriate dimensions such that the (31) is equivalent to As a result, the derivatives of the newly proposed Lyapunov functionals is deduced as V (x t , k) | τ (t)∈I k < 0. It means V (x t , k) | τ (t)∈I k < ρ�x(t)� 2 for sufficiently small ρ > 0.
Hence the T-S fuzzy system in (2) is globally asymptotically stable. This completes the proof.
Proof The same Lyapunov-Krasovskii functional candidate (14) for system (33) is selected for stability analysis. The augment vector (11) is modified as where η(t), η 1 (t) and η 2 (t) are defined in Theorem 1. Then following the similar process of the proof of Theorem 1, the asymptotic stability condition for the T-S system (33) is equivalent to This completes the proof.
Corollary 1 Given a positive integer m, and δ i = α q−i . Considering τ (t) is a continuous function in (4). Then the system (2) is asymptotically stable if there exist symmetric positive definite matrices Z i , Q i , R 2i , R 3i ∈ R n×n (i = 1, 2 . . . , m), P = P ij (m+1)×(m+1) ∈ R (m+1)n×(m+1)n , symmetric matrices W 1 , W 3 ∈ R 3n×3n , and J ∈ R n×n , matrices W 2 ∈ R 3n×3n , N 1 , N 2 ∈ R 3n×n , and Y ∈ R (3m+6)n×n , such that the following LMIs hold where k,l is deduced from k,l by replacing 2 as Proof For the T-S fuzzy system (2) with interval time-varying delays, modify the Lyapunov functionals (14) by setting Q = 0, i.e., Then following the similar process of the proof of Theorem 1, the asymptotic stability condition for the T-S system (2) is equivalent to This completes the proof.
Remark 3 Both lower and upper bounds of the time-varying delay τ (t) are concerned in Cases 1 and 2. Actually, it is pointed out that Case 1 is a special case of Case 2, which means less conservative results can be obtained by using Case 1 instead of Case 2 in the case of a differentiable function of τ (t). Nonetheless, if τ (t) is not differentiable, Case 2 is able to overcome this issue Peng and Han (2011).
Remark 4 Considering a unit common ratio, i.e. α = 1 , which means the length of each subinterval is equivalent. Then previous developed research works using the equivalent partition method Hui et al. (2015), Wang and Shen (2012), Zhao et al. (2009) can be considered as a special case of this proposed approach. Therefore, the developed partitioning method is more generalized.

Numerical example
In this section, numerical examples are conducted to investigate the stability of the T-S fuzzy systems in (2) and (33).
Considering the lower bound of the time-varying delay τ a = 0, different values of delay derivative rate µ are selected to obtain the upper bound of τ b for comparisons in Table 1.
In Table 1, considering different values of µ, the comparisons of the maximum upper bounds τ b are given for τ a = 0. According to results in Lian et al. (2016), it is clearly to show that for µ = 0, 0.1, 0.5 this proposed method can dramatically increase the (40) r l=1 h l ξ T (t) � k,l + He(YŴ l ) ξ(t) < 0 (41) upper bound of the time varying delay when selecting the partitioning number m = 3 . Figure 2 illustrates that with respect to the newly conducted maximum value of τ b the state response still converges to zero, which means the T-S fuzzy system (41) is globally asymptotically stable. Considering τ (t) to be a continuous function, as it is given in (4), i.e., µ is unknown. Then upper bound of the τ b in this proposed work is compared with some other research results shown in the right column of Table 1.
In Figures 2, 3, simulation performance illustrates that under the maximum tolerant delay τ b shown in Table 1 the T-S fuzzy system (41) is asymptotical stable.
For a given lower bound of τ a = 0 in Theorem 1, considering different values of µ as well as the unknown µ in Corollary 1, the upper bounds of τ b in this proposed work are obtained in Table 2.
In the presence of nonlinear perturbations, under a fixed value of delay derivative and the unknown µ, the upper bound of delays are conducted in Table 2. It is shown that the proposed method works well in the perturbed T-S fuzzy system (2). By means of the simulation results in Table 2, selecting µ = 0.5, τ b = 1.94 and unknown µ, τ b = 1.72 the state responses of the T-S system (41) are conducted in Figs. 4, 5.
Remark 5 By comparing with the results in Lian et al. (2016), Zeng et al. (2015b), Liu et al. (2010), less conservative results are obtained for the nominal T-S fuzzy system. Simulation results are conducted to demonstrate the remarkable improvements of the proposed method. The proposed geometric progression technique for delay partition can deal with the time-varying delayed T-S fuzzy systems with nonlinear perturbations with excellent stability criteria.
Remark 6 Tables 1 and 2 demonstrate that the maximum value of τ b drops down when µ increases. In addition, the upper bound of time-varying delay τ (t) becomes bigger as soon as the partitioning segment gets finer. Figures 2, 3, 4 and 5 display that the convergence time of the state response rises up in the case of an unknown delay derivative µ. x 1 (t), τ b = 1.98 x 2 (t), τ b = 1.98 Fig. 3 The state response of system (41) with unknown µ

Conclusions
In this paper, a novel delay partitioning method using the geometric sequence division is proposed for stability analysis of the perturbed T-S fuzzy system with interval timevarying delays. Recently developed inequalities and new modified Lyapunov functionals are introduced in this work. Numerical examples are given to demonstrate that less conservative results can be obtained in this design by comparing with some previously developed approaches.