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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 793686, 9 pages
Wirtinger-Type Inequality and the Stability Analysis of Delayed Lur'e System
1College of Computer Science and Information, GuiZhou University, Guiyang, Guizhou 550025, China
2School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, Guizhou 550004, China
3College of Science, Guizhou University, Guiyang, Guizhou 550025, China
Received 11 May 2013; Revised 26 July 2013; Accepted 27 July 2013
Academic Editor: M. De la Sen
Copyright © 2013 Zixin Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This paper proposes a new delay-depended stability criterion for a class of delayed Lur'e systems with sector and slope restricted nonlinear perturbation. The proposed method employs an improved Wirtinger-type inequality for constructing a new Lyapunov functional with triple integral items. By using the convex expression of the nonlinear perturbation function, the original nonlinear Lur'e system is transformed into a linear uncertain system. Based on the Lyapunov stable theory, some novel delay-depended stability criteria for the researched system are established in terms of linear matrix inequality technique. Three numerical examples are presented to illustrate the validity of the main results.
Lur'e control system is an important nonlinear control system. Since the notion of absolute stability was first time introduced by Lur'e in , the problem of the absolute stability of Lur'e control system has been widely studied for several decades (see [2–6]). However, because of the existence of time delays, stochastic disturbances, parameter uncertainties, and so on, the convergence of Lur'e system may often be destroyed. This makes the design or performance for the corresponding closed-loop systems become difficult. Therefore, the stability analysis of delayed Lur'e system becomes very important. Up to now, various stability conditions have been obtained, and many excellent papers and monographs have been available (see [7–12]).
Recently, a great deal of effort has been done to the stability analysis of delayed Lur'e system with sector and slope restricted nonlinearities. To enlarge the feasibility region of the stability criteria, by introducing variables in cross-term, Park researched a new bounding technique in . Concerning the descriptor method for delayed system, an extensive work was developed by Fridman and Shaked in . By employing linear matrix inequality and matrix decomposing technique, Cao and Zhong  researched the absolute stability problem of Lur’e control systems with multiple time delays and nonlinearities and established some improved delay-dependent criteria. In order to further reduce stability criterion's conservatism, sector bounds and slope bounds are employed to a Lyapunov-Krasovskii functional through convex representation of the nonlinearities so that some new improved criteria were established by Lee and Park in  and Yin et al. in , respectively.
On the other hand, these previous works only focused on the relationship between and or between and . One natural question is whether there exists a relationship among , , , , and . This idea motivates this study. By using the Jensen integral inequality, we first establish some improved vector Wirtinger-type inequalities. On the basis of these new established inequalities, a new Lyapunov functional including triple integral items is proposed, and some less conservative delay-dependent stability criteria are derived. Finally, three numerical examples are presented to illustrate the validity of the main results.
Notation. The notations are used in our paper except where otherwise specified. are real and -dimension real number sets, respectively; denotes the block diagonal matrix. is identity matrix; * represents the elements below the main diagonal of a symmetric block matrix; real matrix <0 denotes is a positive-definite (negative-definite) matrix.
Consider the following delayed Lur'e system: where denotes the state vector; is the output vector; ; are constant known matrices of appropriate dimensions. The delay is assumed to satisfy denotes the nonlinear function in feedback path, which has the following form: which satisfies a sector condition with belonging to sector , where are known constant scalars; that is,
Notice that the nonlinear function can be written as a convex combination of the sector bounds as follows: where satisfying . Namely, , where is an element of a convex hull . Set , where . Obviously, . Define , then nonlinear function can be expressed as , where satisfies . And the system (1) can be rewritten as the following delayed uncertain system:
Remark 1. Different from previous work [6, 9, 10, 12, 15], in this paper, by using the convex expression , we transform the original nonlinear system (1) into a linear uncertain system (6). As a result, the stability problem of nonlinear Lur'e system (1) can be transformed into the robust stability problem of linear uncertain system (6).
Let and . For further discussion, the following lemmas are needed.
Lemma 2 (see ). For any positive definite symmetric constant matrix and scalar , such that the following integrations are well defined, then
Lemma 3 (see ). Given symmetric matrix and any real matrices of appropriate dimensions, for all satisfying if and only if there exists such that where .
On the basis of Jensen integral inequality, we first give out an improved Wirtinger-type vector inequality as follows.
Lemma 4. Let have continuous derived function on interval . Assume that ; then for any -matrix , the following inequality holds:
Proof. Since , one can get that , This completes the proof.
On the basis of Lemma 4, we further give out some improved Wirtinger-type inequalities as follows.
Lemma 5. Let have continuous derived function on interval . Then for any -matrix , scalar , the following inequality holds:
Proof. Set . Notice that ; from Lemma 4, we have Additionally, from Jensen inequality, one can get Thus, we have Similarly, let ; from inequality (16), Lemmas 2, and 4, one can obtain This completes the proof.
Remark 6. Compared with traditional Wirtinger-type integral inequality, Jensen integral inequality , and double integral Jensen inequality , Lemma 5 gives out a transitional form among , , , , and . These inequality relationships can be used as a handy tool to deal with the stability problem.
3. Main Results
In this section, we attempt to establish some new practically computable stability criteria for system (1). By constructing a new Lyapunov functional including triple integral items, we obtain the following stability result.
Theorem 7. For given scalars , , , , , system (1) is globally asymptotically stable if there exist positive definite diagonal matrices , , , symmetric positive definite matrices , , , , and arbitrary matrices of appropriate dimensions such that the following condition holds: where , ,
Proof. Choose a new class of Lyapunov functional candidate as follows:
The time derivative of along the trajectory of system (1) is given as where
Notice that From Lemma 2, we have
Namely, Additionally, for arbitrary matrices , of appropriate dimensions, we have Hence, , where , and . Since , thus . From Lemma 3, if the conditions in Theorem 7 are satisfied, then, ; by Lyapunov stable theory, the delayed Lur'e system (1) is asymptotically stable, which completes the proof.
Remark 8. Different from previous work, in the proof of Lemma 5, we establish the relationship among , , and . On the basis of this new relationship, in Theorem 7, items , and are introduced as the state of ; this may reduce criterion's conservatism.
For further use of the information of nonlinear function , let us define , where , . Since , there must exist such that . Set , . From the result obtained in , can be rewritten as , where denotes the th column vectors of the identity matrix, and , where satisfies . This means that system (1) can be rewritten as where satisfies . From Theorem 7, we can get the following result.
Theorem 9. For given scalars , , , system (1) is globally asymptotically stable if there exist positive definite diagonal matrices , , , symmetric positive definite matrices , , , , and arbitrary matrices of appropriate dimensions such that the following condition holds: where , ,
Remark 10. Due to the existence of items and , the results established in Theorems 7 and 9 are not LMI criteria. In order to overcome this flaw, by using the lemma derived in , we further establish the following more practicable stable rules.
Corollary 11. For given scalars , , , system (1) is globally asymptotically stable if there exist positive definite diagonal matrices , , , symmetric positive definite matrices , , , , positive scalar , and arbitrary matrices of appropriate dimensions such that the following condition holds:
Corollary 12. For given scalars , , , system (1) is globally asymptotically stable if there exist positive definite diagonal matrices , , , symmetric positive definite matrices , , , , positive scaler , and arbitrary matrices of appropriate dimensions such that the following condition holds:
4. Numerical Example
In order to show the effectiveness of the technique proposed in this paper, we revisit the example in , and compare our criteria with existing delay-dependent criteria.
Example 1. In order to compare with preview results easily, consider the delayed system (1) with parameters given by Time-varying delay is a constant.
This system has been investigated in [19, 22–24]. And the maximum values of for the stability of system (1) are , , , and , respectively. In order to deduce the conservatism of those criteria established in [19, 22–24], Tian et al. derived a new improved result by decomposing matrix as in , where and gave out the maximum delay bound as . However, one can see that, when time-varying delay is not a constant, then the results established in [19, 21–24] are invalid. If , . Obviously, nonlinear function satisfies , , and . If , . Let ; one can get that the results obtained in Corollaries 11 and 12 are feasibility. This mean that the results obtained in this paper are more general and less conservative than those in [19, 21–24].
Example 2. Consider the system described in (1) with parameters given by , , .
This system has been investigated in [25, 26], and the maximum value of for the stability of system (1) are , and , respectively. In , by using the sector bounds and slope bounds to the Lyapunov-Krasovskii functional through convex representation of the nonlinearities, Choi et al. improved the upper bound of to 1.1131. Let ; by using the results obtained in Corollaries 11 and 12, one can get the maximum values of are 1.2103 and 1.2314, respectively. This means that the results obtained in this paper are less conservative than those in [25–27].
Example 3. Consider the system described in (1) with parameters given by . Time-varying delay is a constant.
Obviously, the bounds of the sector nonlinearity are . For this system, the maximum values of for the stability of system (1) are and in [5, 26], respectively. By using various convex optimization techniques, the results obtained in [5, 26] were further improved by Lee and Park in , and . From Corollary 11, we find the maximum allowable time delay bound can be 2.5812, which means that the result established in Corollary 11 is less conservative than the ones obtained in [5, 26].
Combined with Lyapunov stable theory and double integral inequality, this paper researches a class of delayed Lur'e systems with interval time-varying delays. Different from previous work on this topic, this paper first establishes some new vector Wirtinger-type inequalities. Then, by using the property of convex function, the original nonlinear Lur'e system is transformed into a linear uncertain system. At last, by constructing a new Lyapunov functional including triple integral items, some new less conservative delay-dependent stability criteria are established. Numerical examples show that the new criteria derived in this paper are less conservative than some previous results obtained in the references cited therein.
This work was supported by China Postdoctoral Science Foundation Grant (2012 M521718) and Soft Science Research Project in Guizhou Province (LKC2004).
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