Open Challenges on the Stability of Complex Systems: Insights of Nonlinear Phenomena with or without Delay
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New Results on Stability Analysis of Uncertain NeutralType Lur’e Systems Derived from a Modified LyapunovKrasovskii Functional
Abstract
This paper is concerned with the problem of the absolute and robustly absolute stability for the uncertain neutraltype Lur’e system with timevarying delays. By introducing a modified LyapunovKrasovskii functional (LKF) related to a delayproducttype function and two delaydependent matrices, some new delaydependent robustly absolute stability criteria are proposed, which can be expressed as convex linear matrix inequality (LMI) framework. The criteria proposed in this paper are less conservative than some recent previous ones. Finally, some numerical examples are presented to show the effectiveness of the proposed approach.
1. Introduction
In many real systems, time delay is often considered as the main cause of poor performance and even instability. The stability of timedelay systems is always a hot topic for researchers. As a result, to obtain stability criteria of timedelayed systems by using the Lyapunov theorem, the main efforts are concentrated on the following several directions; one is finding an appropriate positive definite functional with a negative definite time derivative along the trajectory of system, for example, LKF with delay partitioning approach [1, 2], LKF with augmented terms [3], LKF with tripleintegral and quadrupleintegral terms [4, 5], and so on. The other is reducing the upper bounds of the time derivative of LKF as much as possible by developing various inequality techniques, such as Jensen inequality [6], Wirtingerbased inequality [7], auxiliary function based inequality [8], and BesselLegendre inequality [9]. Besides, further to increase the freedom of solving LMIs, there are some other methods, for instance, the generalized zero equality [10, 11], the one or secondorder reciprocally convex combinations [12–15], the freeweightingmatrix approach [16], and so on.
In practical engineering applications, most systems are nonlinear. As is known to all, Lur’e system, which is composed of the feedback connection of the linear dynamical system and the nonlinearity satisfying the sectorbounded condition, can represent many deterministic nonlinear systems, for example, Chua’s Circuit and the Lorenz system [17]. Therefore, the study on the stability of Lur ’e systems becomes more and more popular [18–21]. Moreover, the paper [22] pointed out that many practical systems can be modeled as neutral timedelayed systems, in which not only the system states or outputs contain time delays, but also the derivative of the system states. Due to the theoretical and practical significance, the analysis of the robust stability of the timedelayed neutraltype Lur’e systems has attached great importance by many scholars [23–29], where many important robust stability criteria were given. However, the main improvement of stability criteria depends on the development of LKF and the update of inequality techniques based on linear systems. For example, recently, [29] improved the stability results of some previous ones by combining the extended double integral with Wirtingerbased inequalities technique; however, the range of delay with nonzero lower bound and the lower bound of the delay derivative are not involved; in [30], some less conservative stability criteria than some recent previous ones were derived for timedelayed Lur’e system via the secondorder BesselLegendre inequality approach, a novel inequality technique; in [21], some improved stability criteria for timedelayed neutraltype Lur’e system were given by constructing a novel LKF consisting of a quadratic term and integral terms for the timevarying delays and the nonlinearities, and so on. Recently, C. Zhang [31] considered the effect of the LKFs while discussing the relationship between the tightness of inequalities and the conservatism of criteria for linear systems. The results illustrate the integral inequality that makes the upper bound closer to the true value does not always deduce a less conservative stability condition if the LKF is not properly constructed. Particularly, another novel LKF was proposed by C. Zhang et al. [31, 32] with delayproducttype terms and . Compared with the general LKF, and were just symmetrical, not always positive definite, which can lead to a less conservative stability condition by extending the freedom for checking the feasibility of stable conditions based on LMI. Recently, to fully utilize the information of delay derivative, a new LKF was constructed by W. Kwon et al. [33] with delaydependent Lyapunov matrices and . W. Kwon et al. point that the stability conditions based on an LKF with delaydependent matrices are less conservative than those based on the LKF without delaydependent matrices. As mentioned above, the two types of LKFs only improve one class of Lyapunov matrices, respectively, that is, only for the Lyapunov matrix or the Lyapunov matrix . It is natural to wonder about whether can both classes of Lyapunov matrices be improved, simultaneously.
Inspired by the above analysis, the following ideas of reducing the conservation of the previous proposed stability criteria should be addressed:(i)A modified LKF with the above both classes of Lyapunov matrices, that is delayproducttype and delaydependent matrices, is constructed. Compared with the general LKFs in some previous published papers, such as [21, 28, 30], the Lyapunov matrices of the nonintegral item are just symmetrical, not always positive definite, which can extend the freedom for checking the feasibility of stable conditions based on LMI. And the delaydependent matrices of the singleintegral items are utilized, which can also further improve the utilization of time delay and its derivative information. In addition, the results proposed by [31–33] can be improved via the LKF modified in this paper due to the combination of the two types of LKFs.(ii)The double integral items of the modified LKF in this paper are decomposed into two subintervals, that is and , instead of being considered directly in [33], which further make full use of the information of timevarying delays , and their derivative . And the quadratic generalized freeweighting matrix inequality (QGFMI) technique can be used fully in each subinterval, which can further reduce the conservatism of the stability conditions.(iii)To deal with the delayderivativedependent singleintegral items feasibly, another double integral items of are also added to the LKF under the above two subintervals, instead of introducing a positive integral item, which is actually difficult to estimate, to the derivative of the LKF like [33].(iv)Indeed, the main result of [33] was not LMI due to the terms with even . The matrix inequalities of the stability criteria proposed in this paper are converted to LMIs via the properties of quadratic functions application, which can be solved easily by Matlab LMItoolbox. In conclusion, it is interesting and still challenging problem to address the above issues, which offers motivation to derive less conservative stability criteria for the timedelayed neutraltype Lur’e systems.
This paper mainly analyzes and studies the stability of uncertain neutraltype Lur’e systems with mixed timevarying delays. Some less conservative delaydependent absolute stability criteria and robust absolute stability criteria than some previous ones are derived via a modified LKF application. In the end, four popular numerical examples are given to illustrate that this method improves some existing methods and achieves good results in stability. The structure of this paper is as follows: Section 1 describes the research background and research topic status and defines the scope of the study of this article; Section 2 describes the main research questions, including some necessary definitions, assumptions, and lemmas; Section 3 presents the main results, including theorems and corollaries; in Section 4 the discussions and simulations based on numerical examples are given; Section 5 summarizes the whole thesis.
Notation. () represents a positive (negative) definite matrix. and represent an identity matrix and a zero matrix with the corresponding dimensions, respectively. denotes the symmetric terms in a block matrix and denotes a blockdiagonal matrix. are block entry matrices with , where is the dimension of the vector . denotes is the function of and . .
2. Problem Formulation
Consider the following neutraltype Lur’e system with mixed timevarying delays:where and are the state and output vectors of the system, respectively. , , , , and are real constant matrices with appropriate dimensions; is an valued continuous initial functional specified on or with known positive scalars , , and . is the nonlinear functional in the feedback path. The timevarying delays and are continuoustime functional and satisfy the following two types of conditions: C. 1. C. 2.
where , , , , , and are constants.
The nonlinear functional in the feedback path is given bysatisfying the finite sector condition:or the infinite sector condition:where .
, , and denote realvalued matrix functions representing parameter uncertainties, which are assumed to satisfywhere , , , and are known constant matrices with appropriate dimensions, and is an unknown matrix with Lebesguemeasurable elements and satisfies
This paper mainly analyzes and studies the stability of uncertain neutraltype Lur’e system (1) under conditions (2), (3), (5), (6), (7), and (8) based on Lyapunov stability theory. For neutraltype systems, the assumption that [41] is required, where denotes the spectral radius of . To obtain the main results of this paper, the following definition and lemmas are important.
Definition 1 (robustly absolute stability). The uncertain neutraltype Lur’e system described by (1) is said to be robustly absolutely stable in the sector (or ), if the system is asymptotically stable for any nonlinear function satisfying (5) or (6) and all admissible uncertainties.
Lemma 2 (see [15]). For given vectors , and positive real scalars satisfying , symmetric positive definite matrix , and any matrix , the following inequality holdswhere , .
Lemma 3 (QGFMI [33]). For any given matrices , , a positive definite matrix and a continuous differentiable function , the following inequality holdswhere is an any vector, and .
Lemma 4 (see [42]). For a given quadratic function , where (), , if the following inequalities holdone has , for all .
Proof. The proof is similar to lemma 2 of [42]. First, in the case of , is a convex function. So, (i) and (ii) guarantee , . Next, for , is a concave function. So, . Then from (iii) and from (ii) guarantee that , for all . This completes the proof.
Remark 5. It is interesting to note that, in Lemma 4, when , inequalities (11) can be rewritten in those of lemma 2 in [42]. Hence the established Lemma 4 covers the lemma in [42].
Lemma 6 (see [43]). Given matrices , , and , the following inequalityholds for any satisfying , if and only if there exists a scalar such that
Remark 7. Recently, [29] improved the stability results of the uncertain neutraltype Lur’e system (1) by combining the extended double integral with Wirtingerbased inequalities technique. In practice, it is known that the range of delay with nonzero lower bound is often encountered, and such systems are referred to as interval timedelay systems. So, both the range of delay with zero lower bound and that with nonzero lower bound are considered in this paper. In addition, the lower bound of the delay derivative is also involved in this paper, which is not mentioned in [29].
3. Main Results
3.1. Absolute Stability Criteria for Nominal Form
In this section, we will investigate the robustly absolute stability problem of the system (1). First, we give an absolute stability criterion for nominal form of system (1) without uncertainties described as
For the sake of simplicity on matrix representation, the notations of several symbols and matrices are defined as Box 1 of Appendix A. The following theorem will give an absolute stability criterion for Lur’e system (14) satisfying the conditions C. 1 and (5).

Theorem 8. The system (14) satisfying the conditions (2) and (5) is absolutely stable for given values of , , , and , if there exist symmetric matrices , , , positive definite matrices , , , and any matrices , , , such that the following LMIs hold for :where the related notations are defined in Box 3 of Appendix B.
Proof. Construct an LKF candidate aswithwhere notations of several symbols and matrices can be found in Boxes 1 and 3 of Appendixes A and B.
First step, because the positive definiteness of the Lyapunov matrices , , and is not required, the positive definiteness of the LKF (22) should be proved. The and dependent terms can be rewritten aswhere .
Based on , and Jensen’s inequality, the term can be estimated asAccording to and Lemma 2, we can obtain the following inequality from (24) and (25)It follows from (15)(16), (22), (24), (25), and (26) and , , thatThus, the LKF (22) is positive definite.
Second step, the time derivative of with respect to time along the trajectory of the system (14) is as follows:For additional symmetric matrices , , , , and the following zero equations are satisfiedTaking the zero inequalities in and , we have the following integral terms.It follows from Lemma 3 with an augmented vector that