Delay-range-dependent Stability Criteria for Takagi-sugeno Fuzzy Systems with Fast Time-varying Delays

The problem of delay-range-dependent stability for T-S fuzzy system with interval time-varying delay is investigated. The constraint on the derivative of the time-varying delay is not required, which allows the time delay to be a fast time-varying function. By developing delay decomposition approach, integral inequalities approach IIA, and Leibniz-Newton formula, the information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities LMIs which can be easily solved by various optimization algorithms. Simulation examples show resulting criteria outperform all existing ones in the literature. It is worth pointing out that our criteria are carried out more efficiently for computation and less conservatism of the proposed criteria.


Introduction
It is well known that time delay often appears in the dynamic systems, which is an important source of instability and degradation in the control performance.Fuzzy system in the form of Takagi-Sugeno T-S model has been paid considerable attention in the past two decades 1, 2 .It has been shown that the T-S model method gives an effective way to represent complex nonlinear systems by some simple local linear dynamic systems, and some analysis methods in the linear systems can be effectively extended to the T-S fuzzy systems.However, all the aforementioned criteria aim at time delay free T-S fuzzy systems.In practice, time delay, one of the instability sources in dynamical systems, is a common and complex phenomenon in many industrial and engineering systems such as chemical process, metallurgical processes, biological systems, rolling mill systems, and communication networks.As a result, stability analysis for T-S fuzzy systems with time delay is of great significance both in theory and in practice.Some approaches developed for general delay systems have been borrowed to deal with fuzzy systems with time delay.In recent years, the problems of stability and stabilization of the T-S fuzzy systems with time delay have attracted rapidly growing interests 3-21 .Among these references, great efforts have been focused on effective reduction of the conservation of the delayed T-S fuzzy model.Many effective methods, such as new bounding technique for cross-terms 7, 9 , augmented Lyapunov functional method 7 , and free-weighing matrix method 6, 10, 17-20 , have been proposed.We can see that the free-weighting matrix approach is used as a main tool to make the criteria less conservative in the literature, and only the lower and upper bounds of delay function h t are considered.When stability analysis of delayed systems is concerned, a very effective strategy is to apply the Gu's Lyapunov-Krasovskii functional discretization technique 22 .However, this discretization technique has been developed for linear systems subject to constant time delay.Besides, it is very hard to extend the stability analysis conditions obtained via this technique to control design since several products between decision variables will be generated, leading to nonconvex formulations.Therefore, lessconservative conditions for stability and control of T-S fuzzy systems subject to uncertain time delay are proposed based on a fuzzy weighting-dependent Lyapunov function, the Gu discretization technique 22 , and extra strategies to introduce slack matrix variables by 13, 23 .However, these results have conservatism to some extent, which exist room for further improvement.
The delay varying in an interval has strong application background, which commonly exists in many practical systems.The investigation for the systems with interval time-varying delay has caused considerable attention, see 13, 18, 24-26 and the references therein.In 15 , Lyapunov-Krasovskii function and augmented Lyapunov-Krasovskii function to construct uncorrelated augmented matrix UAM and to deal with cross terms in the UAM through improved Jessen's inequality.An improved delay-dependent criterion is derived in 18 by constructing a new Lyapunov functional and using free-weighting matrices.In 24 , a weighting delay method is used to deal with the stability of system with time varying delay.In 25, 26 , by developing a delay decomposition approach, the integral interval t − h, t is decomposed into t − h, t − αh and t − αh, t .Since a tuning parameter α is introduced, the information about x t − αh can be taken into full consideration; thus, the upper bound of t t−h ẋT s R ẋ s ds can be estimated more exactly no matter the delay derivative exists or not.However, it has been realized that too many free variables introduced in the free-weighting matrix method will complicate the system synthesis and consequently lead to a significant computational demand 7 .The problem of improving system performance while reducing the computational demand will be addressed in this paper.
The main contributions of this paper are highlighted as follows. 1 delay-dependent stability criteria are developed, which are an improvement over the latest results available from the open literature 3, 5-7, 9, 10, 12, 13, 15, 17-21, 23, 27 ; 2 theoretical proof is provided to show that the results in 6 are a special case of the results derived in this paper.The approach developed in this work uses the least number of unknown variables and consequently is the least mathematically complex and most computationally efficient.This implies that some redundant variables in the existing stability criteria can be removed while maintaining the efficiency of the stability conditions.With the present stability conditions, the computational burden is largely reduced; 3 since the delay decomposition approach is introduced in delay interval, it is clear that the stability results are based on the delay decomposition approach.When the positions of delay decomposition are varied, the stability results of the proposed criteria are also different.In order to obtain the optimal delay decomposition sequence, we proposed an implementation based on optimization methods.
Motivated by the above discussions, we propose new stability criteria for T-S fuzzy system with interval time-varying delay.By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration, and new delay-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities LMIs which can be easily solved by various optimization algorithms.Since the delay terms are concerned more exactly, less conservative results are presented.Moreover, the restriction on the change rate of time-varying delays is relaxed in the proposed criteria.The proposed stability conditions are much less conservative and are more general than some existing results.Numerical examples are given to illustrate the effectiveness of our theoretical results.

Stability Analysis
Consider a T-S fuzzy system with a time-varying delay, which is represented by a T-S fuzzy model, composed of a set of fuzzy implications, and each implication is expressed by a linear system model.The ith rule of this T-S fuzzy model is of the following form: Plant rule i : if z 1 t is M i1 , and . .., and where z 1 t , z 2 t , ..., z p t are the premise variables; M ij , i 1, 2, ..., r, j 1, 2, ..., p are the fuzzy sets; x t ∈ R n is the state; φ t is a vector-valued initial condition; A i and A di are constant real matrices with appropriate dimensions; the scalars ris the number of if-then rules; time delay, h t , is a time-varying delay.We will consider the following two cases for the timevarying delay.
Case 1. h t is a differentiable function satisfying where h 1 and h 2 are the lower and upper delay bounds, respectively; h 1 , h 2 , and h d are constants.Here h 1 the lower bound of delay may not be equal to 0, and when h d 0 we have h 1 h 2 .Both Cases 1 and 2 have considered the upper and nonzero lower delay bounds of the interval time-varying delay.Case 1 is a special case of Case 2. If the time-varying delay is differentiable and h d < 1, one can obtain a less conservative result using Case 1 than that using Case 2.
By fuzzy blending, the overall fuzzy model is inferred as follows: where z z 1 , z 2 , ..., z p ; w i : R p → 0, 1 , i 1, 2, ..., r, is the membership function of the system with respect to the plant rule i; θ i z t w i z t / r i 1 w i z t ; and It is assumed that w i z t ≥ 0, i 1, 2,...,r, r i 1 w i z t ≥ 0 for all t, so we have θ i z t ≥ 0, r i 1 θ i z t 1.In the following, we will develop some practically computable stability criteria for the system described 2.1a .The following lemmas are useful in deriving the criteria.First, we introduce the following technical Lemma 2.1 of integral inequality approach IIA .Lemma 2.1 see 11 .For any positive semidefinite matrices Then, one obtains Lemma 2.2 see 28 .The following matrix inequality x and S x depend on affine on x is equivalent to

2.8
In this paper, a new Lyapunov functional is constructed, which contains the information of the lower bound of delay h 1 and upper bound h 2 .The following Theorem 2.3 presents a delay-range-dependent result in terms of LMIs and expresses the relationships between the terms of the Leibniz-Newton formula.
Theorem 2.3.Under Case 1, for given scalars h 1 , h 2 , h d , and α 0 < α < 1 , System 2.4 subject to 2.2 is asymptotically stable if there exist symmetry positive-definite matrices P P T > 0, such that the following LMIs hold: where

2.11
Proof.If we can proof that Theorem 2.3 holds for two cases, that is, αδ ≤ h t ≤ h 2 and Construct a Lyapunov-Krasovskii functional candidate as ẋT s R 2 ẋ s ds dθ.

2.12
Calculating the derivative of 2.12 with respect to t > 0 along the trajectories of 2.1a and 2.1b leads to with the operator for the term ẋT t αδR 1 h 2 − αδ R 2 ẋ t as follows: Alternatively, the following equations are true: ẋT s Z 33 ẋ s ds.

2.15
By utilizing Lemma 2.1 and the Leibniz-Newton formula, we have 2.17

2.18
Substituting the above equations 2.14 -2.18 into 2.13 , we obtain where where

2.21
If 2 for a sufficiently small ε > 0. By the Schur complement of Lemma 2.2, Ω < 0 is equivalent to the following inequality and is true: where
For this case, the Lyapunov-Krasovskii functional candidate is chosen ẋT s R 2 ẋ s ds dθ,

2.24
where Choosing ξ T t x T t x T t − h t x T t − h 1 x T t − αδ and then using a proof process similar to that for Case 1, we derive the same condition 2.10a , 2.10b , 2.10c , and 2.10d as that for Case 1.This completes the proof.
When the information of the time derivative of delay is unknown, by eliminating Q 2 we have the following result from Theorem 2.3.

2.27
Proof.If the matrix Q 2 0 is selected in 2.10a , 2.10b , 2.10c , and 2.10d , this proof can be completed in a similar formulation to Theorem 2.3.
When h 1 0, Theorem 2.3 reduces to the following Corollary 2.5.

2.30
Proof.Choose the following fuzzy Lyapunov-Krasovskii functional candidate to be where ẋT s R ẋ s ds dθ.

2.32
Then, taking the time derivative of V t with respect to t along the system 2.4 yield

2.33
Then the proof follows a linear similar to the proof of Theorem 2.3 and thus is omitted here.
Remark 2.6.In our Theorems 2.3, h d can be any value or unknown due to 13 .Therefore, Theorem 2.3 is applicable to both cases of fast and slow time varying delay.We will show the other characteristic of Theorem 2.3.When the distance between h 1 and h 2 is sufficiently small, the upper bound h 2 of delay for unknown h d will be very close to the upper bound for h d 0. This characteristic is not included in previous Lyapunov functional based work where the upper bound of delay for h d / 0 is always less than that for h d 0.
Remark 2.7.It is seen from the proof of Theorem 2.3 and Corollary 2.4 that the main characteristics of the method developed in this paper can be generalized as the following two steps.i Construct a Lyapunov function to integrated both lower and upper delay bounds, for example, αδ h 2 h 1 /2 in 2.12 and 2.24 .ii Employ Lemma 2.1 to deal with cross-product terms, for example, those in 2.15 -2.18 .It is also seen from the proof that neither model transformation nor free-weighting matrices have been employed to deal with the cross-product terms.Therefore, the stability criteria derived in this paper are expected to be less conservative.This will be demonstrated later through numerical examples.It is noted that although it has been observed that using αδ h 2 h 1 /2 in the constructed Lyapunov function can improve stability performance for many examples, theoretical evidence has not been found so far to explain the observations.
Based on that, a convex optimization problem is formulated to find the bound on the allowable delay time 0 ≤ h 1 ≤ h t ≤ h 2 which maintains the delay-dependent stability of the time delay system 2.4 .
Remark 2.8.It is interesting to note that h 1 , h 2 appears linearly in 2.10a and 2.26a .Thus, a generalized eigenvalue problem GEVP as defined in Boyd et al. 28 can be formulated to solve the minimum acceptable 1/h 1 or 1/h 2 and, therefore, the maximum h 1 or h 2 to maintain robust stability as judged by these conditions.
In this way, our optimization problem becomes a standard generalized eigenvalue problem, which can be then solved using GEVP technique.From this discussion, we have the following Remark 2.9.
Remark 2.9.Theorem 2.3 provides delay-dependent asymptotic stability criteria for the T-S fuzzy systems with an interval time-varying delay 2. 4  About how to seek an appropriate α satisfying 0 < α < 1, such that the upper bound h of delay h t subjecting to 2.29 is maximal, we give an algorithm as follows.
Algorithm 2.10 Maximizing h > 0 .Step 1: For given h d , choose an upper bound on h satisfying 2.29 , and then select this upper bound as the initial value h 0 of h.
Step 2: Set appropriate step lengths, h step and α step , for h and α, respectively.Set k as a counter, and choose k 1.Meanwhile, let h h 0 h step and the initial value α 0 of α equals to α step .
Step 3: Let α kα step , if inequality 2.29 is feasible, go to Step 4; otherwise, go to Step 5.
Remark 2.11.For Algorithm 2.10, the final h 0 is the desired maximum of the upper bound of delay h t satisfying 2.29 and α 0 is the corresponding value of α.

Illustrative Examples
To illustrate the usefulness of our results, this section will provide numerical examples.It will be shown that the proposed results can provide less conservative results that recent ones have given 3, 5-7, 9, 10, 12, 13, 15, 17-21, 23, 27 .It is worth pointing out that our criteria carried out more efficiently for computation.
Example 3.1.Consider a time-delayed fuzzy system without controlling input.The T-S fuzzy model of this fuzzy system is of the following form.Plant rules: and the membership functions for rule 1 and rule 2 are

3.5
Applying the criteria in 9, 10, 12, 17 and in this paper, the maximum values of h for the stability of system under considerations are listed in Table 1.It is easy to see that the stability criterion in this paper gives a much less conservative result than the ones in 9, 10, 12, 17 .
Furthermore, by taking the various h 1 h d 0.5 , and from Theorem 2.3, we obtain the maximum allowable delay bound MADB h 2 as shown in Table 2. From the above results of Table 2, if the h 1 increases, the delay time length increases. where 0.9 0 1 1.6 .3.9 Solution 3.5.By taking the parameter h d 0, using Corollary 2.5, the maximum value of delay time for the System 3.6 and 3.7 to be asymptotically stable is h ≤ 1.0245.By the criteria in 9, 16 , the system 3.6 and 3.7 is asymptotically stable for h ≤ 0.58 and h ≤ 0.6148, respectively.Hence, for this example, the criteria proposed here significantly improve the estimate of the stability limit compared with the result of 9, 16 .Employing the LMIs in 18-21 and those in Corollary 2.5 yields upper bounds on h 2 that guarantee the stability of system 3.6 and 3.7 for various h d , which are listed in Table 3, in which "-" means that the results where

3.13
Solution 3.7.By using Corollary 2.4, the maximum allowable delay bound MADB can be calculated as h 2 2.3725.The results for stability conditions in different methods are compared in Table 4.It can be seen that the delay-dependent stability condition in this paper is less conservative than earlier reported ones in the literature 5, 7, 10, 15, 18, 27 .Compared with Guan and chen 7 who used 5 LMI variables, Yoneyama 27 employed 20 LMI variables to get better stability results.To obtain improved stability results than those in 15, 18, 27 , we need 8 variables in Corollary 2.4 the same as 5, 10 .It is also seen from Table 4 that the larger r is the more unknown LMI variables are required in 18, 27 .However, the unknown number of LMI variables is independent of r in the results of this paper.It can be shown that the delay-dependent stability condition in this paper is the best performance. where

3.16
Solution 3.10.The objective is to determine the maximum value of constant time-delay h h 2 h 1 0 for which the system is stable.

3.20
Solution 3.12.Considering a constant time delay, by using Corollary 2.4, the maximum allowable delay bound MADB can be calculated as h 2 0.4130 h 1 0, α 0.5 .By the criteria in 13, 23 , the systems 3.17 and 3.18 is asymptotically stable for any h that satisfies h ≤ 0.322 and h ≤ 0.4060, respectively.Hence, for this example, the criteria proposed here significantly improve the estimate of the stability limit compared with the result of 13, 23 .

Conclusion
In this paper, we have dealt with the stability problem for T-S fuzzy systems with interval time-varying delay.By constructing a Lyapunov-Krasovskii functional, the supplementary requirement that the time derivative of time-varying delays must be smaller than one is released in the proposed delay-range-dependent stability criterion.By developing a delay decomposition approach, the information of the delayed plant states can be taken into full consideration, and new delay-range-dependent sufficient stability criteria are obtained in terms of linear matrix inequalities LMIs which can be easily solved by various optimization algorithms.Since the delay term is concerned more exactly, it is less conservative and more computationally efficient than those obtained from existing methods.Thus, the present method could largely reduce the computational burden in solving LMIs.Both theoretical and numerical comparisons have been provided to show the effectiveness and efficiency of the present method.Numerical examples are given to illustrate the effectiveness of our theoretical results.
in terms of solvability of LMIs 28 .Based on them, we can obtain the maximum allowable delay bound MADB 0 ≤ h 1 ≤ h t ≤ h 2 such that 2.4 is stable by solving the following convex optimization problem: Solution 3.2.For system 3.1 and 3.2 , by taking the parameter h d 0 and α 0.6, we get the Corollary 2.5 which remains feasible for any delay time h ≤ 2.2459.In case of maximum allowable delay bound MADB h 2.2459, solving Corollary 2.5 yields the following set of feasible solutions:

Table 1 :
Computation MADB h for varying h d in Example 3.1.

Table 2 :
MADB h 2 h d 0.5 for various h 1 in Example 3.1.Similar to Algorithm 2.10, an algorithm for seeking an appropriate α such that the upper bound of delay 0 ≤ h 1 ≤ h t ≤ h 2 , subjecting to 2.10a , 2.10b , 2.10c , and 2.10d is maximal can be easily obtained.

Table 3 :
MADB h for various h d in Example 3.4.to the corresponding cases.It can be seen from Table3that Corollary 2.5 in this paper yields the least conservative stability test than other approaches, showing the advantage of the stability result in this paper.

Table 4 :
MADB h 2 for unknown h d h 1 0 in Example 3.6 for different methods Number of variables: N v ; Number of fuzzy rules: r .Remark 3.8.Similar to Algorithm 2.10, we can also find an appropriate scalar α, such that the upper bound of delay 0 ≤ h 1 ≤ h t ≤ h 2 , subjecting to 2.26a , 2.26b , 2.26c , and 2.26d reaches the maximum.
Table 5 compares works based on common quadratic functionals 7, 18-20, 23 with the fuzzy functional of Corollary 2.4.It is clear by inspecting Table 5 that Corollary 2.4 provides the largest time delays.For comparison, Table 6 also lists the maximum allowable delay bound MADB h obtained from the criterion 3 .It is clear that Corollary 2.5 gives much better results than those obtained by 3 .It is illustrated that the proposed stability criteria are effective in comparison to earlier and newly published results existing in the literature.

Table 5 :
MADB h h 2 h 1 0 for different methods in Example 3.9.

Table 6 :
MADB h for different h d for Example 3.9.Example 3.11.Consider a nominal time-delay fuzzy system.The T-S fuzzy model of this fuzzy system is of the following form.Rule 1: If x 1 t is M 1 , then