The Scientific World Journal

Volume 2014, Article ID 278305, 6 pages

http://dx.doi.org/10.1155/2014/278305

## Stability Analysis of Nonlinear Systems with Slope Restricted Nonlinearities

Key Lab of Industrial Computer Control Engineering of Hebei Province, Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China

Received 22 August 2013; Accepted 14 November 2013; Published 28 January 2014

Academic Editors: H. Cakalli and S. Mohiuddine

Copyright © 2014 Xian 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

The problem of absolute stability of Lur’e systems with sector and slope restricted nonlinearities is revisited. Novel time-domain and frequency-domain criteria are established by using the Lyapunov method and the well-known Kalman-Yakubovich-Popov (KYP) lemma. The criteria strengthen some existing results. Simulations are given to illustrate the efficiency of the results.

#### 1. Introduction

Absolute stability of nonlinear systems has been investigated comprehensively for the past several decades [1–12]. It is well known that the Popov criterion and the circle criterion are two classical results with the forms of frequency-domain inequalities (FDIs), which are turned out to be equivalent to some linear matrix inequalities (LMIs). This not only gives the opportunity to use the powerful LMI toolbox [13] to study absolute stability, but also gives the opportunity to consider the controller design problems. In [14], absolute stability of single-input and single-output Lur’e systems with a sector and slope restricted nonlinearity is brought forward. It is pointed out that the slope restriction on the nonlinearity strengthens the Popov criterion by adding an additional term to the original FDI of the criterion. Much work [15–22] on the slope restricted and multivariable problem has been done by using a Lur’e-Postnikov function or an extended Lur’e-Postnikov function.

In this paper, both time-domain criterion and frequency-domain criterion for absolute stability of Lur’e systems with sector and slope restricted nonlinearities are presented based on the Lyapunov method and the KYP lemma. Some mathematical tools are used through the derivation of the absolute stability criterion. Compared with some existing results, the proposed results are less conservative. This should be owed to the effect of the slope restricted conditions on the nonlinearities. The rest of the paper is organized as follows. In Section 2, the system description and some preliminaries are presented. Time-domain and frequency-domain criteria for absolute stability of the system are given in Section 3. Numerical examples are given in Section 4 and some concluding remarks are given in Section 5.

Throughout this paper, the superscript means transpose of real matrices and conjugate transpose of complex matrices. For a Hermitian matrix , () denotes that is a positive definite (semidefinite) matrix and denotes that is a negative definite matrix. means for any real or complex square matrix .

#### 2. Problem Statement

Consider the following multi-input and multioutput Lur’e system where , , and are real matrices, , is the output, is piecewise continuously differentiable on , and are assumed to satisfy where , , , and . The inequalities (2) and (3) denote sector restriction and slope restriction on , respectively. Let , , , . Then , , , and . Setting , (3) is formulated as follows: The transfer function from to is denoted as .

System (1) is called to be absolutely stable if the equilibrium point is globally asymptotically stable for all nonlinear vector valued functions satisfying (2) and (3). In the following sections, less conservative absolute stability criteria including time-domain criterion and frequency-domain criterion for system (1) are given. Before studying these problems, first we introduce the KYP lemma and Schur complement. These lemmas will be used repeatedly in this paper to get our main results.

Lemma 1 (KYP lemma [23]). *Given that , , and symmetric matrix , with for , and the pair is controllable, the following two statements are equivalent.*(i)*, for all .*(ii)*There exists a matrix such that . The equivalence for strict inequalities holds even if is not controllable.*

*Lemma 2 (Schur complement [24]). The LMI is equivalent to one of the following statements:(i) and ;(ii) and .*

*3. Main Results*

*3. Main Results*

*We choose the following Lur’e-Postnikov function:
as the Lyapunov function, where and are necessary to be determined. It should be pointed out that is not necessary to be positive definite and are not necessary to be nonnegative.*

*Theorem 3. System (1) is absolutely stable for all satisfying (2) and (3) if is Hurwitzian and there exist diagonal matrices , , , and symmetric matrices such that the LMI is feasible:
where
*

*Proof. *We will demonstrate that the given conditions imply the negative definiteness of and the positive definiteness of .

Taking the derivative of along the trajectory of (1), we have
Conditions (2) and (4) for are equivalent to
For any and , , it follows
where and . Then
The given condition (6) guarantees the negative definiteness of the right hand of (11). Consequently, is negative definite.

Now we are only left to demonstrate that is positive definite. In (5), is only a symmetric matrix but not a positive definite matrix and may be a positive or negative number. Therefore, the proof of the positive definiteness of is a little difficult and complex. Without loss of generality, letting and , then has the following form:
where and . Since (2) implies , is satisfied. Then is positive definite if is positive definite, which is proved in what follows.

Denote , , , and . Firstly, the given conditions imply that is Hurwitzian for any diagonal matrix satisfying . Actually, the matrix is Hurwitzian for in virtue of the given conditions. So we will demonstrate that is Hurwitzian for any diagonal matrix satisfying . We assume there exists a diagonal matrix satisfying such that the matrix is not Hurwitzian. On the one hand, a number satisfying can be found such that
holds for certain . Since is Hurwitzian, and are followed. The latter formula indicates that there exists a vector such that
where and . Then we derive
On the another hand, pre- and postmultiplying both sides of (6) by and , we have
where
By the Schur complement, (16) implies
From the KYP lemma, we derive that (18) holds if and only if
where and . Inequality (19) is equivalent to
in terms of the equalities and . Letting in (20) and pre- and postmultiplying both sides of the resulting inequality by and , it follows that
We can observe that (15) and (21) are contradictive, which means that the assumption is not true and is Hurwitzian for any diagonal matrix satisfying . Therefore, the matrix is Hurwitzian. Secondly, the given conditions imply that is positive definite. Actually, pre- and postmultiplying both sides of (16) by and yield
where
Inequality (22) implies . According to , , , , is followed. The matrix is Hurwitzian, which results in the positive definiteness of and . This completes the proof.

*It is found in the proof of Theorem 3, more exactly in inequality (16), that if (6) holds, then is Hurwitzian if and only if .*

*Theorem 4. System (1) is absolutely stable for all satisfying (2) and (3) if there exist diagonal matrices , , , symmetric matrices , such that and the LMI (6) holds.*

*Remark 5. *Theorem 3 is derived directly by using the time-domain method and can be used to study multi-input and multioutput Lur’e systems. Inequality (6) in Theorem 3 is in the form of LMI, which is easier to be solved by means of the LMI toolbox.

The LMI (6) can be transformed into an equivalent FDI. Thus, a frequency-domain criterion for (1) is given as follows.

*Theorem 6. System (1) is absolutely stable for all satisfying (2) and (3) if the matrix is Hurwitzian and there exist diagonal matrices , , such that the following frequency-domain inequality holds
*

*Proof. *Let , , and . Inequality (6) can be rewritten as
where
According to the KYP lemma, (25) is equivalent to
By simple computations, we have
where . Substituting (28) into (27), the equivalence between (6) and (24) is derived.

*Remark 7. *For the case , the FDI (24) reduces to
which corresponds to the FDI as given in Theorem in [4]. However, the results there only aim at single-input and single-output Lur’e systems.

If the slope restrictions on are removed, another absolute stability criterion is derived by choosing (5) as the Lyapunov function.

*Theorem 8. System (1) is absolutely stable for all satisfying (2) if the matrix is Hurwitzian and there exist diagonal matrices , , symmetric matrices , , such that the following LMI is feasible:
where , .*

*Proof. *The proof is similar to that of Theorem 3.

*Remark 9. *Theorem 8 gives absolute stability conditions for sector restricted Lur’e systems. In fact, the slope restricted condition (3) plays an important role in improving the condition of absolute stability. The forthcoming example shows that Theorem 3 is less conservative than Theorem 8.

Similar to Theorem 3, an equivalent frequency-domain criterion to Theorem 8 can be given as follows.

*Theorem 10. System (1) is absolutely stable for all satisfying (2) if the matrix is Hurwitzian and there exist diagonal matrices , such that the following FDI holds:
*

*Proof. *From the KYP lemma, (30) is equivalent to
The equivalence between (30) and (31) is derived from and .

*Remark 11. *Theorem 10 includes two particular cases. For the case , (31) is reduced to
Correspondingly, Theorem 10 is in the form of the circle criterion. For the case , (31) reduces to
Theorem 10 has the same form as the Popov criterion.

*4. Numerical Example*

*4. Numerical Example**In this section, a numerical example is presented to illustrate the effectiveness of the proposed results.*

*Consider Chua’s oscillator [25] with the following dimensionless equations
where , , , , , and are numbers. System (35) can be reformulated in the form of (1) with , , , , , and . The nonlinearity satisfies
Thus, and .*

*When , , , and are taken, system (35) is absolutely stable for by applying Theorem 3. However, we derive that system (35) is absolutely stable for and , respectively, by Theorem 8 and the Popov criterion. This shows that Theorem 3 is an improvement with respect to Theorem 8 and the Popov criterion, and the slope restrictions could improve the absolute stability condition. The states of system (35) with at the initial value are given in Figure 1, from which it is illustrated that system (35) is absolutely stable.*

*5. Conclusion*

*5. Conclusion**We have proposed new absolute stability criteria for Lur’e systems with sector and slope restricted nonlinearities from time-domain and frequency-domain points of view. The slope restrictions on nonlinearities improve the absolute stability conditions. We have shown that the criteria are less conservative than some existing results.*

*Conflict of Interests*

*Conflict of Interests**The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*Acknowledgments**This research is supported by the National Natural Science Foundation of China (61004050, 61172095) and the Natural Science Foundation of Scientific Research of Hebei Education Department (2009482).*

*References*

*References*

- V. M. Popov, “Absolute stability of nonlinear systems of automatic control,”
*Automation and Remote Control*, vol. 22, no. 8, pp. 857–875, 1961. View at Google Scholar - L. T. Grujić, “On absolute stability and the aizerman conjecture,”
*Automatica*, vol. 17, no. 2, pp. 335–349, 1981. View at Google Scholar · View at Scopus - W. M. Haddad and D. S. Bernstein, “Parameter-dependent Lyapunov functions and the discrete-time Popov criterion for robust analysis,”
*Automatica*, vol. 30, no. 6, pp. 1015–1021, 1994. View at Publisher · View at Google Scholar · View at Scopus - 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 Scopus - R. Medina, “Absolute stability of discrete-time systems with delay,”
*Advances in Difference Equations*, vol. 2008, Article ID 396504, 10 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - Q. L. Han, “A new delay-dependent absolute stability criterion for a class of nonlinear neutral systems,”
*Automatica*, vol. 44, no. 1, pp. 272–277, 2008. View at Publisher · View at Google Scholar · View at Scopus - J. Cao, S. Zhong, and Y. Hu, “Delay-dependent condition for absolute stability of Lurie control systems with multiple time delays and nonlinearities,”
*Journal of Mathematical Analysis and Applications*, vol. 338, no. 1, pp. 497–504, 2008. View at Publisher · View at Google Scholar · View at Scopus - H. Wang, A. Xue, and R. Lu, “Absolute stability criteria for a class of nonlinear singular systems with time delay,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 70, no. 2, pp. 621–630, 2009. View at Publisher · View at Google Scholar · View at Scopus - B. Zhang, J. Lam, S. Xu, and Z. Shu, “Absolute exponential stability criteria for a class of nonlinear time-delay systems,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 3, pp. 1963–1976, 2010. View at Publisher · View at Google Scholar · View at Scopus - 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 Scopus - C. A. C. Gonzaga, M. Jungers, and J. Daafouz, “Stability analysis of discrete-time Lur’e systems,”
*Automatica*, vol. 48, no. 9, pp. 2277–2283, 2012. View at Google Scholar - D. Wang and F. Liao, “Absolute stability of Lurie direct control systems with time-varying coefficients andmultiple nonlinearities,”
*Applied Mathematics and Computation*, vol. 219, no. 9, pp. 4465–4473, 2013. View at Google Scholar - P. Gahinet, A. Nemirovski, A. J. Laub, and M. Chilali,
*LMI Control Toolbox Users Guide*, The Math Works, Natick, Mass, USA, 1995. - V. A. Yakubocivh, “The method of matrix inequalities in the stability theory of nonlinear control systems: II,”
*Automatica and Telemechanic*, vol. 26, pp. 577–592, 1965. View at Google Scholar - J. A. K. Suykens, J. Vandewalle, and B. de Moor, “An absolute stability criterion for the Lur'e problem with sector and slope restricted nonlinearities,”
*IEEE Transactions on Circuits and Systems I*, vol. 45, no. 9, pp. 1007–1009, 1998. View at Publisher · View at Google Scholar · View at Scopus - S. M. Lee, J. H. Park, and O. M. Kwon, “Improved asymptotic stability analysis for Lur'e systems with sector and slope restricted nonlinearities,”
*Physics Letters A*, vol. 362, no. 5-6, pp. 348–351, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. M. Lee, O. M. Kwon, and J. H. Park, “Delay-independent absolute stability for time-delay Lur'e systems with sector and slope restricted nonlinearities,”
*Physics Letters A*, vol. 372, no. 22, pp. 4010–4015, 2008. 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 Scopus - S. M. Lee and J. H. Park, “Robust stabilization of discrete-time nonlinear Lur'e systems with sector and slope restricted nonlinearities,”
*Applied Mathematics and Computation*, vol. 200, no. 1, pp. 429–436, 2008. View at Publisher · View at Google Scholar · View at Scopus - D. H. Ji, J. H. Park, and S. C. Won, “Master-slave synchronization of Lur'e systems with sector and slope restricted nonlinearities,”
*Physics Letters A*, vol. 373, no. 11, pp. 1044–1050, 2009. View at Publisher · View at Google Scholar · View at Scopus - J. Carrasco, W. P. Heath, and A. Lanzon, “Equivalence between classes of multipliers for slope-restrited nonlinearities,”
*Automatica*, vol. 49, no. 6, pp. 1732–1740, 2013. View at Google Scholar - G. A. Leonov, D. V. Ponomarenko, and V. B. Smirnova,
*Frequency-Domain Methods for Nonlinear Analysis: Theory and Applications*, World Scientific, Singapore, 1996. - A. Rantzer, “On the Kalman-Yakubovich-Popov lemma,”
*Systems and Control Letters*, vol. 28, no. 1, pp. 7–10, 1996. View at Publisher · View at Google Scholar · View at Scopus - S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan,
*Linear Matrix Inequality in System and Control Theory*, Society for Industrial and Applied Mathematics, Philadelphia, 1994. - R. Martinez-Guerra, D. M. Corona-Fortunio, and J. L. Mata-Machuca, “Synchronization of chaotic Liouvillian systems: an application to Chua's oscillator,”
*Applied Mathematics and Computation*, vol. 219, no. 23, pp. 10934–10944, 2013. View at Google Scholar

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