Abstract
This paper is concerned with the delay-dependent stability of systems with distributed time-varying delays. The novelty relies on the use of some new inequalities which are less conservative than some existing inequalities. A less conservative stability criterion is obtained by constructing some new augmented Lyapunov–Krasovskii functionals, which are given in terms of linear matrix inequalities. The effectiveness of the presented criterion is demonstrated by two numerical examples.
1. Introduction
Consider the systems with distributed time-varying delays:where is the system state, are constant matrices, and is the time-varying delay satisfying
Since time delays occur in many dynamic systems, stability analysis of the time delay system [1–5] has become a hot topic in the past few decades. Due to the representation of linear systems with time-varying delays, the delay-dependent stability analysis via the LKF method has attracted much attention. The conservatism of the LKF method comes from two aspects: the construction of the LKF and the bound on its derivative. Selecting the LKF is crucial to derive less conservative criteria. An augmented LKF [6] is proposed to reduce the conservatism in the early literature. Recently, a new augmented LKF [7] is introduced by employing the information of a second-order Bessel–Legendre inequality. It is necessary to take the derivative of the LKF to derive a stability criterion. The difficulty lies in the bounds of the integrals that arise in the derivative of the LKF. There are two main methods for dealing with such integrals: the free-weighting matrix method [8] and the integral inequality method. The integral inequality method includes various integral inequalities, such as Jensen inequality [9–11], Wirtinger-based inequality [12–15], free matrix-based inequality [16, 17], auxiliary function-based inequality [18], relaxed integral inequality [19], and Bessel–Legendre inequality [20]. Very recently, the improved inequality-based functions approach [21] is proposed to derive less conservative results for systems with time-varying delays. However, when estimating , is only estimated as . are the same as in Lemma 2. Then, a new integral inequality was proposed in [22] to further reduce the conservatism. But the integral inequality can only deal with the constant time delay. On the other hand, stability analysis for systems with distributed delays is of both practical and theoretical importance. Then, it is desirable to extend the system model to include distributed delays. In recent years, the stability analysis of systems with distributed delays has been received considerable attention [23–27]. But only the authors in [25, 26] consider the systems with distributed time-varying delays.
This paper is concerned with the delay-dependent stability of systems with distributed time-varying delays. Based on some new inequalities and some new augmented LKFs, a less conservative stability criterion is obtained in terms of LMIs. Our paper has two characteristics: (1) the integral is estimated as , which includes those in [9, 13, 20] as special cases. , is the same as in Lemma 2. (2) An augmented LKF which contains more information about is proposed to reduce the conservatism. The effectiveness of the presented criterion is demonstrated by two numerical examples.
Throughout this paper, the set denotes the set of symmetric matrices and the set denotes the set of symmetric positive definite matrices. For any square matrix , we define .
2. Main Results
In this section, the following lemmas are introduced to derive the main results.
Lemma 1 (see [20]). For any matrices , , , and , the inequalityholds, where
Lemma 2 (see [22]). For a matrix and any continuously differentiable function , the inequalityholds, where
Lemma 3 (see [28]). Suppose that are the constant matrices of appropriate dimensions, , , thenholds if and only if the following inequalities hold:
Based on Lemmas 1–3, a novel stability criterion is derived for system (1) with distributed time-varying delays.
Theorem 1. For given scalars , if there exist matrices , , , such that the LMIholds for , i.e.,then, system (1) is asymptotically stable, whereand is defined as for .
Proof. Introduce an LKF candidate aswhereCalculate the derivative of along the solution of system (1) as follows:According to (18)–(22), we can obtainwhereLet , then , applying Lemma 2, and we haveFor any matrices and applying Lemma 1, we can obtainFrom (23)–(26), we getBy Lemma 2 [20], if the LMI (10) is true for , , then holds for all and . By Lemma 3, LMI (10) holds if and only if LMIs (11)–(14) hold. This completes the proof.
Remark 1. The integral in [9, 13, 20] is estimated as , , respectively. In this paper, the integral is estimated as , which includes those in [9, 13, 20] as special cases. So our method can yield less conservative results.
Remark 2. An augmented LKF which contains more information about time-varying delay which was proposed to reduce the conservatism. is added as a state vector, which may yield less conservative criteria.
3. Numerical Examples
Two numerical examples are given to demonstrate advantages of the proposed criterion.
Example 1. Consider system (1) withFor different , Table 1 presents the allowable upper bound of , which guarantees the stability of system (1). Table 1 shows that our method produces the larger upper bound than those in [7, 12, 13, 16, 17, 21]. In this sense, our stability criterion is less conservative than those in [7, 12, 13, 16, 17, 21].
Example 2. Consider system (1) withFor different , Table 2 presents the allowable upper bound of , which guarantees the stability of system (1). Table 2 shows that our method produces the larger upper bound than those in [7, 11, 13, 15, 16, 21]. In this sense, our stability criterion is less conservative than those in [7, 11, 13, 15, 16, 21].
4. Conclusions
This paper focus on delay-dependent stability analysis for systems with distributed time-varying delays. The novelty relies on the use of some new inequalities which are less conservative than some existing inequalities. A less conservative stability criterion is obtained by constructing some new augmented LKFs. The effectiveness of the presented criterion is demonstrated by two numerical examples. In addition, the proposed method can be applied to stability analysis of other dynamic systems such as fuzzy systems with time-varying delay and neutral systems with time-varying delay.
Data Availability
No additional data are available for this paper.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
This work was supported by the Innovative Groups of Education Department of Guizhou Province (Qian jiao he KY [2016]046); High-level Innovative Talents in Guizhou Province (Zun shi ke he rencai [2016]13).