Novel Stability Analysis for Uncertain Neutral-Type Lur ’ e Systems with Time-Varying Delays Using New Inequality

This paper considers the delay-dependent stability analysis of neutral-type Lur’e systems with time-varying delays and sector bounded nonlinearities. First of all, using constructed function methods, a new Jensen-like inequality is introduced to obtain less conservative results. Second, a new class of Lyapunov-Krasovskii functional (LKF) is constructed according to the characteristic of the considered systems. Third, combining with the new inequality and reciprocal convex approach and some other inequality techniques, the new less conservative robust stability criteria are shown in the form of linear matrix inequalities (LMIs). Finally, three examples demonstrate the feasibility and the superiority of our methods.

Delay-dependent stability criteria were presented for nominal and uncertain neutral-type Lur' e systems with constant time delays and sector bounded nonlinearities in [27].The robust stability problems for neutral-type Lur' e systems with time-varying delays were considered because time delays vary always depending on time in [12][13][14][15][16][17][18][19][20][21][27][28][29][30].The freeweighting matrix method was applied to get less conservative stability criteria and to deal with the robust stability problems for neutral-type Lur' e systems with time-varying delays in [15,16].However, the free-weighting matrix method brings more variables which make the computation quite complex.So, it is the right time to improve the disadvantage of the freeweighting matrix method and to get less conservative stability criteria for neutral-type Lur' e systems with time-varying delays.Some new robust stability criteria were proposed without using the general free-weighting matrix method which are less conservative and easier to calculate than some previous ones in [15,19].Reference [31] can reduce conservatism by reducing the conservatism of the Jensenlike inequality.Motivated by [31], developing the Jensenlike inequality with double-integral term may reduce the conservativeness.As a result, less conservative criteria may be also got by constructing new integral inequalities used in LKF.
This paper studies the stability for a class of neutral-type Lur' e systems with time-varying delays and sector bounded nonlinearities.To investigate the neutral-type Lur' e system, this paper introduces a new triple-integral inequality used in the following LKFs and gets less conservative criteria.The LKF contains not only double-integral terms but also triple-integral terms.Using some effective techniques, such as a novel integral inequality, a piecewise analysis method, and the reciprocally convex combination inequality, instead of the general free-weighting matrix method, the delaydependent stability criteria derived in the form of LMIs are less conservative than some existing results in other papers.The effectiveness and the less conservatism of stability criteria proposed in this paper are demonstrated by using numerical examples in Section 4.

Problem Statement and Preliminaries
Consider a class of Lur' e systems of uncertain neutral type with time-varying delays and sector-bound nonlinearities described by the following equation: where () ∈   and () ∈   stand for the output and state vectors, respectively.() ∈   is a real-valued continuous initial function on [− max (ℎ, ), 0].,  1 , , , and  are known real constant matrices with appropriate dimensions.Δ(), Δ 1 (), and Δ() represent the time-varying uncertainty parameters.(()) ∈   is the nonlinear function such as where   () is the th component of the output vector (), and each term   (  ()) ( = 1, 2, . . ., ) satisfies the finite sector condition shown in Figure 1(a) [32] with known positive scalars   or the infinite sector condition shown in Figure 1(b) [32]   ( Δ(), Δ 1 (), and Δ() are assumed to satisfy the following condition: where ,   ,  1 , and   are known constant matrices with appropriate dimensions, and the uncertainty time-varying matrix () satisfies The time-varying delays ℎ() and () are continuous functions satisfying the following conditions: This paper investigates the delay-dependent stability of Lur' e system (1) satisfying conditions (3) (or (4)) and ( 5)-( 7) to gain less conservative robust stability criteria by using a new inequality and constructing a new LKF.Throughout this paper, the results will be acquired by assuming that all the eigenvalues of  are inside the unit circle [33].The following lemmas are useful in deriving criteria.
Lemma 1 (see [34]   Lemma 2 (see [31]).For a given matrix  > 0, any differentiable function  in [, ] →   , the following inequality holds: where Remark 3. It is clear that this new inequality encompasses the Jensen inequality.It is also worth noting that it plays an important role in getting the derivative of the LKF.Hence, by using optimization theory and improving the Jensen-like inequality, the double-integral inequality would be got in the following lemma.
Lemma 4. For any matrix  > 0, scalars ℎ 2 ≥ ℎ 1 ≥ 0, and any continuous function  in  →   the following inequality holds: where Proof.Recall that the objectives of the present paper lemma are to acquire new lower bounds of the integral +   ()()  and to compare with the Jensenlike inequality.For this reason, the appropriate function () is given as the following form: where (, ) =  − (1/2) −  is constructed to satisfy and to get rid of the effect of cross terms.It is easy to obtain Mathematical Problems in Engineering Then, based on the formulas above, the following equality holds: For a given symmetric positive definite matrix  > 0, then, the left-hand side of formula ( 16) above is positive definite.It is obvious that Lemma 4 is correct.
Remark 5. Compared with Lemma 1 in [35], the inequality in Lemma 4 has two advantages.First, since  > 0, the second term in the right side of the inequality in Lemma 4 is definite negative.It thus implies that the Jensen-like inequality in [35] has been included in the inequality proposed above.Second, it is also worth noting that this improvement is allowed by using extra signals Remark 6.It is worth paying attention to the fact that the construction of function  is the key to Lemma 4. ∫ 0 −ℎ ∫  +    = 0 must be sure when we construct the function  to get rid of crossing items and to make the second term in the right side of our inequality be definite negative.
Remark 7. In fact, the result of Lemma 4 is less conservative than formula (3) in literature [36].When ℎ 1 ̸ = 0, the result of Lemma 4 is different from inequality (16) in literature [37].What is more, under the special scaling condition of the left side of the inequality in Lemma 4, the more conservativeness of inequality (16) in literature [37] would be easily concluded.To some extent, the conditions of Lemma 4 are wider than inequality (16) in literature [37] in terms of integral domain.Actually, with the help of delay decomposition technology, + ()  can be scaled properly with our inequality in Lemma 4. In our future research, we will try to use formulas in the literature [37] which may be more flexible than the Lemma 4 to reduce the conservatism of stability criteria in another case.

Main Results
To accept easily the robust stability problem of system (1), firstly, we investigate the stability criterion of the nominal form: The stability of system (17) can be analyzed via timeindependent functions of the form where and define Remark 8.It is worth noticing that the construction of LKF can be improved by adding to get less conservative stability criteria for any positive definite matrices  1 and  2 .However, our methods have reduced the conservatism of the existing results in the instance.Hence, this paper dose not add  1 and  2 to control the number of variables.
Remark 9. A new Lyapunov-Krasovskii functional is constructed in Section 3 to obtain a new delay-dependent stability criterion, which includes the relationship among the states, the nonlinear function, and the derivative of the states.In order to fully reduce the conservative of the condition, the constant delay ℎ is decomposed into ℎ/2 and thus ( 18) is obtained.Particularly, the function  1 is constructed to eliminate the effects among different forms of the states; the function  2 is constructed to eliminate the relationship among the states with different delays; to consider the time-varying delay the function  3 is constructed; and our Lyapunov-Krasovskii functional contains double-integral (function  4 ) and tripleintegral (function  5 ) terms which yield less conservative delay-dependent stability criteria.Therefore, more information for the state is employed in constructing the more general Lyapunov-Krasovskii functionals, which may lead to reduced conservatism.
If nominal system (17) satisfies conditions (3) and ( 7) the following theorem can be got.
Theorem 14. System (1) satisfying conditions ( 3) and ( 5)-( 7) is robustly absolutely stable for given values ℎ ≥ 0, ℎ  ,   < 1, and   > 0 ( = 1, 2, . . ., ), if there exist appropriate dimensional matrices  1 , as well as ( 22)- (24), where and scalars   > 0 such that the following LMIs hold: as well as ( 22)-( 24),  = 1, 2.  Jensen inequality [35] is always presented as Obviously, Lemma 4 not only can obtain a more accurate bound than the Jensen inequality but also can reduce the number of decision variables.What is more, Lemma 4 can dispose the integral terms with ℎ 1 ̸ = 0. Therefore, Lemma 4 is used to derive our criterion, and the effectiveness of our method will be demonstrated by some existing numerical examples.

Examples
In this section, the effectiveness of our approaches will be shown by the following three numerical examples.
Example 1 (see [16,[19][20][21]).Consider Chua's circuit example discussed in [16,[19][20][21].Chua's circuit is a simple nonlinear electronic circuit design.It can show the standard chaos theory.It was published by professor Leo Chua in 1983.The ease of making this circuit makes it a ubiquitous example of chaotic systems in the real world, leading somebody to declare it as a model of a chaotic system.The system equation is given by and parameters  0 = −1/7,  1 = 2/7,  = 9,  = 14.28, and  = 1.The system can be represented in normal Lur' e system framework (17) with The feedback nonlinear function belongs to  [0,1] .The MADBs of ℎ of system (49) for different ℎ  have been shown in Table 1 with the existing results in [16,[19][20][21].Table 1 shows us that the MADBs obtained by using Theorem 10 in this paper are better than the existing results in [16,[19][20][21].

Conclusions
In this paper, we have studied the stability for a class of neutral-type Lur' e systems with time-varying delays and sector bounded nonlinearities by using a new LKF.The LKF contains not only double-integral terms but also tripleintegral terms.Using some effective techniques, such as a novel integral inequality, a piecewise analysis method, and the reciprocally convex combination inequality, instead of the general free-weighting matrix method, the delaydependent stability criteria derived in the form of LMIs are less conservative than some existing results.The effectiveness and the less conservatism of stability criteria proposed in this paper are demonstrated by using standard numerical examples.

) Remark 13 .
Absolute stability criteria Theorem 10 and Corollary 12 proposed in this paper are less conservative than existing results which are shown by the following examples mentioned in the Section 4. The main reasons are the combination of introducing new inequality in Lemma 4 and the method of the exchange of integral order, based on the new constructed Lyapunov functionals.

Remark 16 .
Clearly, robust stability criteria Theorem 14 and Corollary 15 proposed in this paper would be less conservative, because of using the new inequality and some other techniques which are similar to Theorem 10 and Corollary 12.