Delay-dependent stability and stabilization criteria for T–S fuzzy singular systems with interval time-varying delay by improved delay partitioning approach

This paper deals with the stability analysis and fuzzy stabilizing controller design for a class of Takagi–Sugeno fuzzy singular systems with interval time-varying delay and linear fractional uncertainties. By decomposing the delay interval into two unequal subintervals and seeking a appropriate ρ, a new Lyapunov–Krasovskii functional is constructed to develop the improved delay-dependent stability criteria, which ensures the considered system to be regular, impulse-free and stable. Furthermore, the desired fuzzy controller gains are also presented by solving a set of strict linear matrix inequalities. Compared with some existing results, the obtained ones give the result with less conservatism. Finally, some examples are given to show the improvement and the effectiveness of the proposed method.

So fuzzy singular model provides a new way to the analysis and synthesis of the nonlinear singular system and can be found in many applications, because it can combine the flexibility of fuzzy logic theory and fruitful linear singular system theory into a unified framework to approximate complex nonlinear singular systems, for details see Fridman (2002) and Lin et al. (2006). Meanwhile, time delays always exist in many dynamical systems and delays are the sources of poor stability and deteriorated performance of a system. Therefore, lots of stability analysis and control synthesis results (Wang et al. 2014;Mourad et al. 2013;Huang 2013;Han et al. 2012;Zhang et al. 2009) have been reported for T-S fuzzy singular systems with time-delay. It should be pointed out that almost all of the existing results on fuzzy systems with time delays, the maximum allowable delay bound has been used as an important performance index for measuring the conservatism of the obtained conditions.
On the other hand, in order to reduce the conservativeness of the delay-dependent criteria for fuzzy systems, input-ouput approach (Su et al. 2013;Zhao et al. 2013), delay partitioning method (Yang et al. 2015;Xia et al. 2014), convex combination technique (Su et al. 2014a;An and Wen 2011;Peng and Fei 2013;Park et al. 2011), and free weighting matrices approach (Souza et al. 2014;Liu et al. 2010;Tian et al. 2010) have been well used. The most noteworthy is the delay partitioning approach: the delay interval is divided into multiple uniform or non-uniform segments. It has been proved that less conservative results may be expected with the increasing of delay-partitioning segments. Recently, by non-uniformly dividing the time delay into multiple segments, An and Wen (2011) has established less conservative delay-dependent stability criteria than those in Li et al. (2009) using a convex way for uncertain T-S fuzzy systems with interval timevarying delay. Based on the input-output technique and delay partitioning approach, some new stability conditions of discrete-time T-S fuzzy systems with time delays have been proposed by applying scaled small-gain theorem in Su et al. (2013) , while an induced ℓ 2 performance is guaranteed. On the basis of delay-partitioning approach and new integral inequality established by reciprocally convex approach in Park et al. (2011) and Peng and Fei (2013) has developed less conservative stability criteria than those in Peng et al. (2011), Lien et al. (2007) and Tian and Chen (2006) for uncertain T-S fuzzy delay system. However, an important characteristic of fuzzy singular systems is the possible impulse behavior, which is harmful to the physical system and is undesired in system control. It's means that the aforementioned delay partitioning approach and the obtained results can not be directly applied to fuzzy singular system with additional algebraic constraints, because it requires considering not only stability, but also regularity and impulse-free conditions. Therefore, the motivation of this study is mainly focus on how to improve the delay partitioning approach and reduce the conservativeness of existing results for fuzzy singular systems because of its theoretical and practical significance.
More recently, some research works on stability analysis (Mourad et al. 2013;Zhang et al. 2009;Chadli et al. 2014;Wang et al. 2014) and controller design (Zhu et al. 2016;Ma et al. 2015;Zhao et al. 2015;Han et al. 2012) have been extended for T-S fuzzy singular systems with time-varying delay. In Zhang et al. (2009), the problems of delaydependent stability and H ∞ control were discussed utilizing model transformation techniques, but model transformation may lead to considerable conservativeness. In Han et al. (2012), the problems of sliding mode control for fuzzy descriptor systems were presented using delay partitioning approach, but the time-delay is constant. Using freeweight matrix method, Mourad et al. (2013) discussed the problems of delay-dependent stability and L 2 − L ∞ control, however, the free-weighting matrices may be redundant and increase the computational burden in case of stability analysis for deterministic delay systems. In Chadli et al. (2014), by using quadratic method, sufficient conditions on stability and stabilization were proposed in terms of LMI for uncertain T-S fuzzy singular systems. Based on delay partitioning approach, some less conservative stability and stabilization criteria for fuzzy singular systems with time-varying delay have been investigated in Wang et al. (2014). In Ma et al. (2015), a delay-central-point method was presented to develop less conservative delay-dependent conditions for memory dissipative control for fuzzy singular time-varying delay systems under actuator saturation.
It is well-known that the challenges of deriving a less conservative result are to construct an appropriate LKF that includes more useful state information and to reduce the enlargement in bounding the derivative of LKF as much as possible. Inspired by the methods mentioned above, when revisiting the stability problem for T-S fuzzy singular systems with interval time-varying delay, we find that the existing works still leave plenty of room for improvement on the reduction of conservatism for the following reasons. (1) All the given stability conditions in Han et al. (2012), Mourad et al. (2013), and Wang et al. (2014) are not all in strict LMIs form due to equality constraints, which cannot be solved directly using standard LMI procedures; (2) In Wang et al. (2014) T (s)E T REẋ(s)ds, some useful time-varying delay-dependent integral items are ignored in the derivation of results; (3) More free-weighting matrices are employed to deduce the stabilization results in Yang et al. (2015) and Mourad et al. (2013), which have not considered the gain variations might be caused by the inaccuracies of controller implementation. The objective of this paper is to revisit the delay-dependent stability analysis and give new stabilization criteria by improved delay partitioning approach.
The main contributions of this paper lie in that, firstly, by seeking an appropriate ρ , a maximum admissible upper bound of the time delay can be obtained for T-S fuzzy singular systems with interval time-varying delay. The tunable parameter ρ which divide [τ 1 , τ 2 ] into two variable subintervals plays a crucial role in reducing the conservativeness of stability conditions. Secondly, new LKF is established by partitioning time delay [0, τ 1 ] into N segments, and the time-varying delay x(t − n N τ 1 ) is included in the LKF, which takes fully account of the relationship between the state vectors x(t − n N τ 1 ) and x(t − τ ρ ) . Thirdly, some new results on tighter bounding inequalities have been employed to reduce the enlargement in bounding the derivative of LKF when designing the controller with linear fractional uncertainties. Then, the newly developed conditions of stability and stabilization are expected to be less conservative than the previous ones.
The rest of this paper is organized as follows. The system description and some useful lemmas are presented in "Problem formulation" section. In "Main results" section, we show the results on stability conditions and fuzzy controller design scheme. In "Numerical examples" section, several numerical examples are given to demonstrate the effectiveness and merits of the proposed methods. Finally, a brief conclusion is drawn in "Conclusion".
Notations: Throughout the paper, R n denotes the n-dimensional real Euclidean space; I denotes the identity matrix; the superscripts T and −1 stand for the matrix transpose and inverse, respectively; notation X > 0(X ≥ 0) means that matrix X is real symmetric positive definite (positive semi-definite); � · � is the spectral norm. If not explicitly stated, all matrices are assumed to have compatible dimensions for algebraic operations. The symbol " * " stands for matrix block induced by symmetry.

Problem formulation
Consider a class of nonlinear singular system with interval time-varying delay, which can be represented by the following extended T-S fuzzy singular model: where x(t) ∈ R n is the state vector, u(t) ∈ R m is the control input vector. The fuzzy basis functions are given by , where M ij is fuzzy sets, M ij (ξ j (t)) represents the grade of membership of ξ j (t) in M ij . Here, it is easy to find that β i (ξ(t)) ≥ 0, (i = 1, 2, . . . , r), r j=1 β j (ξ(t)) > 0 and µ i (ξ(t)) ≥ 0, (i = 1, 2, . . . , r), r j=1 µ j (ξ(t)) = 1 for all t > 0, r is the number of IF-THEN rules. ξ 1 (t), . . . , ξ p (t) are the premise variables, which do not depend on the input variable u(t). φ i (t) is a vector-valued initial continuous function defined on the interval [−τ 2 , 0]. E ∈ R n×n is a constant matrix, which may be singular, that is, rank(E) = g ≤ n . A i , A τ i , B i are the constant real matrices of appropriate dimensions. �A i (t) and �A τ i (t) denote the norm-bounded parameter uncertainties in the system and are defined as: where M i , N 1i and N 2i are known matrices, F(t) is unknown time-varying matrix, which satisfies F T (t)F (t) ≤ I. The delay τ (t) in above systems is assumed to be interval time varying and satisfies where τ 1 , τ 2 and d are constants.
Before proceeding further, we will introduce some definitions and lemmas to be needed in the development of main results throughout this paper. Consider an unforced singular time-delay system described by Definition 1 (Xu et al. 2002) 1. The pair (E, A) is said to be regular if det(sE − A) is not identically zero. 2. The pair (E, A) is said to be impulse free if deg(det(sE − A)) = rank(E). (1) 3. The pair (E, A) is said to be asymptotically stable, if all roots of det(sE − A) = 0 lie inside the unit disk with center at the origin. 4. The delayed singular system (4) is said to be admissible if the pair (E, A) is regular, impulse free and asymptotically stable.

Delay-dependent admissibility
In this section, we suggest to develop a delay-dependent stability condition for the nominal unforced fuzzy singular system of (1), which can be written as In order to derive a maximum admissible upper bound of system (5), the delay interval [τ 1 , τ 2 ] is divided into two subintervals with unequal width as Case I: [τ 1 , τ ρ ] and Case II: Based on the Lyapunov-Krasovskii stability theorem, the following result is obtained.
Theorem 7 For the given scalars τ 1 , τ 2 , d and tuning parameter ρ, system (5) is admissible for any time-varying delay τ (t) satisfying (3) Proof The proof of this theorem is divided into two parts. The first one is concerned with the regularity and the impulse free characterizations, and the second one treats the stability property of system (5). Since rank(E) = g ≤ n, there must exist two invertible matrices G ∈ R n×n and H ∈ R n×n such that Similar to (14), we define Since � i < 0 and Q 1 > 0, S 1 > 0, we can formulate the following inequality easily: Then, pre-and post-multiplying ϒ i < 0 by H T and H, respectively, (16) yields Since Υ 11 and Υ 12 are irrelevant to the results of the following discussion, the real expression of these two variables are omitted here. From Eq. (17), it is easy to see that This implies that r i=1 µ i (ξ(t))Ã i22 is nonsingular. Therefore, the unforced fuzzy singular system (5) is regular and impulse free.
For the Case II, when τ ρ ≤ τ (t) ≤ τ 2 , the following equations are true: Then, the proof can be completed in a similar formulation to Case I and is omitted here for simplification. Therefore, if LMIs (6)-(7) hold, the fuzzy singular system (5) is admissible for the Cases I and II, respectively. This completes the proof.
For uncertain T-S fuzzy system of (5), the following result can be easily derived by applying Lemma 5 and Schur complement.
Corollary 8 For the given scalars τ 1 , τ 2 , d and ρ, the uncertain fuzzy system of (5) is robustly admissible for any time-varying delay τ (t) satisfying (3), if there exist matrices , and positive scalars ε 1i , some appropriate dimension matrices S, P 2 , P 3 and the constant matrix R satisfying E T R = 0 such that the following set of LMIs hold: we define different energy functional Q n (n = 1, 2, . . . , N ) in each different delay subinterval segment. Because the piecewise Lyapunov function candidates are much richer than the globally quadratic functions, so the obtained stability criteria based on this method can further reduce the conservativeness of analysis and synthesis.
Remark 10 Since the interval [τ 1 , τ 2 ] is divided into two unequal variable subintervals [τ 1 , τ ρ ] and [τ ρ , τ 2 ] in which ρ is a tunable parameter, it is clear that the LKF defined in Theorem 7 is more general and simple than Zhang et al. (2009) and Mourad et al. (2013) by seeking a appropriate ρ satisfying 0 < ρ < 1. For different ρ, the LKF matrices may be different and the LMIs also may be different in stability conditions, and thus compared with the methods using the same LKF matrices (Wang et al. 2014) or the uniformly dividing delay subintervals (Yang et al. 2015), the variable and different LKF matrices may lead to less conservativeness.

Remark 11
The decomposition method in Theorem 7 may increase the maximum allowable upper bounds on τ 2 for the fixed lower bound τ 1 , if one can set a suitable dividing point with relation to ρ. For seeking an appropriate ρ, a algorithm is given as follows: Step 1: For the given d, choose upper bound on δ satisfying (6)- (7), select this upper bound as initial value δ 0 of δ.
Step 2: Set step lengths, δ step and ρ step for δ and ρ, respectively. Set k as a counter and choose k = 1. Meanwhile, let δ = δ 0 + δ step and the initial value ρ 0 of ρ equals ρ step .

Remark 12
In order to further reduce the enlargement of the derivative of LKF, inspired by Liu (2013), a new integral inequality is employed to estimate the integral term, which will be helpful to increase the maximum admissible upper bound of time delay. Moreover, when the information of the time-derivative of delay is unknown or the time delay is not differentiable, just let S 1 = 0 and proceed in a similar way as the previous proof, some new stability criteria can be obtained from Theorem 7. Due to limited space, no more tautology here.

Fuzzy controller design
In this section, based on Theorem 7, we will proposed a design method of fuzzy controller. Consider the controller gain variations might be caused by the inaccuracies of controller implementation, we employ the following controller form with PDC scheme: where K i are the local gain matrices to be determined, and �K i (t) is the controller gain perturbations and satisfies where M ai and N ai are known matrices, and F a (t) is an unknown time-varying matrix satisfying F T a (t)F a (t) ≤ I. Then, the resulting closed-loop system from (1) and (32) can be written as The aim of this section is to design a state feedback controller in the form of (32) with the gain perturbations satisfying (33), such that the closed-loop system (34) is regular, impulse-free, and asymptotically stable.
Proof For the uncertain closed-loop T-S fuzzy singular system (34), replacing A i and (5), respectively. Then, according to (2) and (33), the condition (6) and other matrix elements such as ij are defined in Theorem 7. By Lemma 5, we get from (46) that where scalars ε 1ij > 0 and ε 2ij > 0. Then, by Schur complement, inequality (50) equals to where In order to obtain the control gain matrix, take P 3 = P 2 , where is the designing parameter and define the following matrices variables: Then, pre-and post-multiplying both sides of inequality (51) with diag{X T , . . . , X T , I, I, I, I} and its transpose, respectively, we can obtain the conditions (35) and (36), which means that the closed-loop fuzzy singular system (34) is regular, impulse-free and stable under fuzzy control (32). This completes the proof.
Remark 14 Different from the work in Su et al. (2013) concerned with dynamic output controller design for discrete-time T-S fuzzy delay systems, this study is mainly focused on the state feedback controller design for T-S fuzzy singular systems with time-varying delay while the gain variations may be caused by the inaccuracies of controller implementation. In addition, the input-output technique (Su et al. 2013;Zhao et al. 2013) is employed to reduce the conservativeness in stability analysis, however, the model transformation of the original system will result in approximation error. In this study, only need to select a appropriate ρ in the new constructed LKF, less conservative stability and stabilization conditions can be directly obtained. In Examples 1-3, the comparison results with input-output approach in Su et al. (2013) and other methods to deal with time delays are presented to illustrate the advantages of the proposed approach.
Remark 15 It should be mentioned that the main character of delay partitioning approach lies in that when the number of subintervals N is increased, the conservatism of the result decreases. Meanwhile, the computational complexity increases, see Yang et al. (2015), Wang et al. (2014) and Peng and Fei (2013). Therefore, the choice of the number of subintervals N generally depends on the tradeoff between the conservatism reduction and the computational burden. However, according to the examples presented in the next section, we can see that our results (N = 1) used less partitioning segments is much better than the one in Wang et al. (2014) (N = 2), Peng and Fei (2013) (N = 3) and Yang et al. (2015) (N = 3), It is means that the presented approach has higher computational efficiency, especially when the number of delay partitioning segments is large.

Numerical examples
In this section, four examples are given to demonstrate the effectiveness of the proposed approaches. The first three examples are presented to show the improvement of our results over the existing ones. The last example is used to demonstrate the applicability of the controller design method.
Example 16 Consider the following time-delayed nonlinear system: which can be exactly expressed as a nominal T-S delayed system with the following rules: where the membership functions for above rule 1 and rule 2 are h 1 (θ(t)) = sin 2 (θ(t)), h 2 (θ(t)) = cos 2 (θ(t)) with θ(t) = x 1 (t), and the system matrices are: For this example, because the time-derivative of delay τ (t) is unknown and the considered systems is nonsingular, we set S 1 = 0, E = I 2×2 in Theorem 7 and choose the delay interval segmentation parameter ρ = 0.7 in Case I, ρ = 0.3 in Case II, respectively. The upper delay bounds τ 2 derived by the input-output method (Zhao et al. 2013), convex combination technique Peng and Fei 2013), free weighting matrices approach (Tian et al. 2009;Souza et al. 2014) and the improved delay partitioning method proposed in this paper are tabulated in Table 1 under different values of τ 1 . It is seen from Table 1 that the results obtained from Theorem 7 of this paper are significantly better than those obtained from the other methods. When the system matrices of rule 2 are given as Lien et al. (2007) with the improvement of this paper is shown in Table 2. It can be concluded that the obtained results in our method are less conservative than those of Souza et al. (2014), Peng et al.
Example 17 Consider the following uncertain fuzzy system with two rules: wherė   Yang et al. (2015), Zeng et al. (2014), Liu et al. (2010) and Lien et al. (2007), the computed upper bounds that guarantee the robust stability of the considered system are summarized in Table 3. It can be concluded that the result proposed in this paper is better than the aforementioned results.
In addition, compared with the results in Yang et al. (2015), assume that i = 2, there are (13n(n + 1)/2) + 7n 2 (N = 3) scalar decision variables and six LMIs in their Theorem 1. However, different from the delay interval [τ 1 , τ 2 ] is divided into multiple segments, we divide the delay interval into two unequal subintervals by seeking a appropriate ρ. Thus, only (10n(n + 1)/2) + 4n 2 (N = 1) scalar decision variables and four LMIs are required to improve the results. Especially, when N is increased, less number of decision variables and LMIs may reduce the mathematical complexity and computational load.
Example 18 Consider a continuous fuzzy singular system composed of two rules and the following system matrices: In order to compare with the existing results, supposing that τ (t) satisfies (3) and with τ 1 = 2. Then, setting ρ = 0.45, ρ = 0.95 in Cases I and II, respectively. Table 4 presents a comparison results with various d, which show that the stability condition in Theorem 7 give less conservative results than those in Wang et al. (2014), Mourad et al. (2013) and Zhang et al. (2009). It is worth mention that the stability conditions in the aforementioned works are not in strict LMIs form due to equality constraints. However, by introducing the variable R, much better results are obtained by solving strict LMIs via the existing numerical convex optimization method. The range of θ (t) is assumed to satisfy |θ(t)| < ϕ, ϕ = 2. u(t) is the control input. τ (t) = 0.85 + 0.05sin(10t) is the time-varying delay (thus, τ 1 = 0.8, τ 2 = 0.9, d = 0.5). For the simulation purpose, the system parameter is given as a = 0.3, b = 0.5, e = 0.2, c = 1 , c τ = 0.8. As in Lin et al. (2006), we introduce new variables x 2 (t) =θ(t) and x 3 (t) =θ(t). The system is described by Then this system can be expressed exactly by the following fuzzy singular form with respect to uncertainties described by (2): where The membership functions can be chosen as Here, we set ρ = 0.5, = 1 and assume that the parameters uncertainty matrices in �A i (t) and �A τ i (t) in (2) are given as follows: In this example, considering the case of controller gain variation in the form of (33), the parameters are given as Then, according to Theorem 13 and by solving LMIs (35)-(43) with (44), we can obtain the feasible solution for Case I (N = 1) as follows: (due to space consideration, we do not list all the matrices here) Then, the feedback controller gains are designed as µ 1 (t) = x 2 2 (t) ϕ 2 + 2 , µ 2 (t) = 1 + cos(x 1 (t)) ϕ 2 + 2 , µ 3 (t) = φ 2 − x 2 2 (t) + 1 − cos(x 1 (t)) ϕ 2 + 2 Similarly, according to Theorem 13 and by solving LMIs (35)-(43) with (45), we can obtain that the feedback controller gains in Case II are designed as: Then, let the initial condition be x 1 (t) = 1, x 2 (t) = −1, and the unknown matrix function F (t) = F a (t) = sin(t). The simulation results are given in Figs. 1, 2, 3, 4 and 5. Figures 1 and 2 plots the state trajectories of the closed-loop system with the obtained feedback gain matrices in Case I and Case II, respectively. The phase portraits of closed system are given in Figs. 3, 4 and 5. From the simulation result, it can be seen that the designed fuzzy controller not only makes the closed-loop system states converge to zero, but also effectively attenuate the uncertainty as expected. The phase portrait of closed-system states x 2 (t) and x 3 (t) in Cases I and II