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Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 793686, 9 pages

http://dx.doi.org/10.1155/2013/793686

## Wirtinger-Type Inequality and the Stability Analysis of Delayed Lur'e System

^{1}College of Computer Science and Information, GuiZhou University, Guiyang, Guizhou 550025, China^{2}School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, Guizhou 550004, China^{3}College 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.

#### Abstract

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.

#### 1. Introduction

Lur'e control system is an important nonlinear control system. Since the notion of absolute stability was first time introduced by Lur'e in [1], 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 [13]. Concerning the descriptor method for delayed system, an extensive work was developed by Fridman and Shaked in [14]. By employing linear matrix inequality and matrix decomposing technique, Cao and Zhong [6] 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 [12] and Yin et al. in [15], 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.

#### 2. Preliminaries

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 [16]). *For any positive definite symmetric constant matrix and scalar , such that the following integrations are well defined, then
*

Lemma 3 (see [17]). *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 [18], and double integral Jensen inequality [16], 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:
where

The time derivative of along the trajectory of system (1) is given as
where

Consider

Notice that
From Lemma 2, we have

Similarly,

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 [19], 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 [20], 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 [21], 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 [21], 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 [27], 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 [12], 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].

#### 5. Conclusions

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.

#### Acknowledgments

This work was supported by China Postdoctoral Science Foundation Grant (2012 M521718) and Soft Science Research Project in Guizhou Province (LKC2004).

#### References

- A. I. Lur'e,
*Some Nonliear Problem in the Theory of Automatic Control*, H.M. Stationary Office, London, UK, 1975. - Y. Wang, X. Zhang, and Y. He, “Improved delay-dependent robust stability criteria for a class of uncertain mixed neutral and Lur'e dynamical systems with interval time-varying delays and sector-bounded nonlinearity,”
*Nonlinear Analysis: Real World Applications*, vol. 13, no. 5, pp. 2188–2194, 2012. View at Publisher · View at Google Scholar · View at Scopus - S. Lun and Z. Tai, “Dissipativity for Lur'e distributed parameter control systems,”
*Applied Mathematics Letters*, vol. 25, no. 4, pp. 682–686, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. He and M. Wu, “Absolute stability for multiple delay general Lur'e control systems with multiple nonlinearities,”
*Journal of Computational and Applied Mathematics*, vol. 159, no. 2, pp. 241–248, 2003. View at Publisher · View at Google Scholar · View at MathSciNet - Q.-L. Han, “Absolute stability of time-delay systems with sector-bounded nonlinearity,”
*Automatica*, vol. 41, no. 12, pp. 2171–2176, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao and S. Zhong, “New delay-dependent condition for absolute stability of Lurie control systems with multiple time-delays and nonlinearities,”
*Applied Mathematics and Computation*, vol. 194, no. 1, pp. 250–258, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. I. Zečević, E. Cheng, and D. D. Šiljak, “Control design for large-scale Lur'e systems with arbitrary information structure constraints,”
*Applied Mathematics and Computation*, vol. 217, no. 3, pp. 1277–1286, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. A. Yakubovich, G. A. Leonov, and A. Kh. Gelig,
*Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities*, vol. 14 of*Series on Stability, Vibration and Control of Systems*, World Scientific, River Edge, NJ, USA, 2004. View at MathSciNet - F. Hao, “Absolute stability of uncertain discrete Lur'e systems and maximum admissible perturbed bounds,”
*Journal of the Franklin Institute*, vol. 347, no. 8, pp. 1511–1525, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Tai and S. Lun, “Absolutely exponential stability of Lur'e distributed parameter control systems,”
*Applied Mathematics Letters*, vol. 25, no. 3, pp. 232–236, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Huang, Z. Han, X. Cai, and L. Liu, “Adaptive full-order and reduced-order observers for the Lur'e differential inclusion system,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 7, pp. 2869–2879, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. M. Lee and J. H. Park, “Delay-dependent criteria for absolute stability of uncertain time-delayed Lur'e dynamical systems,”
*Journal of the Franklin Institute*, vol. 347, no. 1, pp. 146–153, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - J. G. Park, “A delay-dependent stability criterion for systems with un-certain linear state-delayed systems,”
*IEEE Transactions on Automatic Control*, vol. 35, pp. 878–877, 1999. - E. Fridman and U. Shaked, “${H}_{\infty}$-control of linear state-delay descriptor systems: an LMI approach,”
*Linear Algebra and its Applications*, vol. 351-352, pp. 271–302, 2002. View at Publisher · View at Google Scholar · View at MathSciNet - C. Yin, S.-m. Zhong, and W.-f. Chen, “On delay-dependent robust stability of a class of uncertain mixed neutral and Lur'e dynamical systems with interval time-varying delays,”
*Journal of the Franklin Institute*, vol. 347, no. 9, pp. 1623–1642, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Sun, G. P. Liu, and J. Chen, “Delay-dependent stability and stabilization of neutral time-delay systems,”
*International Journal of Robust and Nonlinear Control*, vol. 19, no. 12, pp. 1364–1375, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. El Ghaoui and H. Lebret, “Robust solutions to least-squares problems with uncertain data,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 18, no. 4, pp. 1035–1064, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Gu, “An integral inequality in the stability problem of time-delay systems,” in
*Proceedings of the 39th IEEE Confernce on Decision and Control (CDC '00)*, pp. 2805–2810, Sydney, Australia, December 2000. View at Scopus - B. J. Xu and X. X. Liao, “Absolute stability criteria of delay-dependent for Lurie
control systems,”
*Acta Automatica Sinica*, vol. 28, no. 2, pp. 317–320, 2002. View at MathSciNet - S. Zhou and J. Lam, “Robust stabilization of delayed singular systems with linear fractional parametric uncertainties,”
*Circuits, Systems, and Signal Processing*, vol. 22, no. 6, pp. 579–588, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Tian, S. Zhong, and L. Xiong, “Delay-dependent absolute stability of Lurie control systems with multiple time-delays,”
*Applied Mathematics and Computation*, vol. 188, no. 1, pp. 379–384, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. H. Nian, “Delay dependent conditions for absolute stability of Lur'e type control systems,”
*Acta Automatica Sinica*, vol. 25, no. 4, pp. 556–564, 1999. View at MathSciNet - D.-Y. Chen and W.-H. Liu, “Delay-dependent robust stability for Lurie control systems with multiple time-delays,”
*Control Theory and Applications*, vol. 22, no. 3, pp. 499–502, 2005. View at Scopus - B. Yang and M. Chen, “Delay-dependent criterion for absolute stability of Lurie type control systems with time delay,”
*Control Theory & Applications*, vol. 18, no. 6, pp. 929–931, 2001. View at Zentralblatt MATH · View at MathSciNet - Y. He, M. Wu, J.-H. She, and G.-P. Liu, “Robust stability for delay Lur'e control systems with multiple nonlinearities,”
*Journal of Computational and Applied Mathematics*, vol. 176, no. 2, pp. 371–380, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - S. Xu and G. Feng, “Improved robust absolute stability criteria for uncertain time-delay systems,”
*IET Control Theory and Applications*, vol. 1, no. 6, pp. 1630–1637, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. J. Choi, S. M. Lee, S. C. Won, and J. H. Park, “Improved delay-dependent stability criteria for uncertain Lur'e systems with sector and slope restricted nonlinearities and time-varying delays,”
*Applied Mathematics and Computation*, vol. 208, no. 2, pp. 520–530, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet