Research Article  Open Access
Improved Results on Robust Stability for Systems with Interval TimeVarying Delays and Nonlinear Perturbations
Abstract
This paper investigated delaydependent robust stability criteria for systems with interval timevarying delays and nonlinear perturbations. A delaypartitioning approach is used in this paper, the delayinterval is partitioned into multiple equidistant subintervals, a new LyapunovKrasovskii (LK) functional contains some tripleintegral terms, and augment terms are introduced on these intervals. Then, by using integral inequalities method together with freeweighting matrix approach, a new less conservative delaydependent stability criterion is formulated in terms of linear matrix inequalities (LMIs), which can be easily solved by optimization algorithms. Numerical examples are given to show the effectiveness and the benefits of the proposed method.
1. Introduction
Timedelay phenomena are ubiquitous in many practical systems such as communication systems, nuclear reactors, aircraft stabilization, and process control systems, which are often major sources of instability and poor performance. Hence, stability analysis and stabilization of systems with timedelays have received considerable attention in the past few decades [1–14]. Recently, many researchers pay attention to the interval timevarying delay, wherein the delay varies in a range for which the lower bound is not restricted to be zero. A typical example with interval time delay is the network control systems [2].
The basic framework for stability analysis and synthesis of stabilizing controllers is LK functional and linear matrix inequality (LMI). Under this framework, an important issue was to enlarge the feasible region of stability criteria. To derive the delaydependent stability conditions, many methods have been reported in the literature. For example, the freeweighting matrix method was used in [3–6]; Jensen’s integral inequality method was adopted in [7–13]. Recently, inspired by the discretized Lyapunov method, delaypartitioning approach was proposed in [14, 15], wherein the delayinterval was uniformly divided into multiple segments, choosing proper functional with different weighted matrices corresponding to different segments. A new technique called delaycentral point method was proposed in [16]. Based on the delaycentral point method and decomposition technique, [17] proposed a less conservative stability criterion for computing the maximum allowable bound of the delay range. As an extension of delaycentral point method, a new delaypartitioning approach was proposed in [18] for the uncertain stochastic systems with interval timevarying delay. Referring to the nonlinearities, as time delays, also can cause instability and poor performance of practical systems. Therefore, the stability problem of timedelay systems with nonlinear perturbations has received increasing attention [19–24]. A descriptor model transformation was employed in [19]. The freeweighting matrices approach was adopt in [20, 21]. Recently, a less conservative delaydependent stability criterion was provided in [22] by partitioning the delayinterval into two segments of equal length and evaluating the timederivative of a candidate LK functional in each segment of the delayinterval. The main advantage of the method [22] is that more information on the variation interval of the delay is employed, but we can employ information of timedelay much more if we partition the delayinterval into more segments.
Inspired by the idea of [18, 22], in this paper, we divide the variation interval of the delay into parts with equal length and construct a new LK functional with tripleintegral terms and augment terms for this delayinterval. Based on integral inequalities method together with freeweighting matrix approach, a new delaydependent stability criterion for the system is formulated in terms of linear matrix inequalities, which can be easily calculated by using MATLAB LMI control toolbox. Numerical examples are given to illustrate the effectiveness and less conservatism of the proposed method.
2. Problem Description and Preliminaries
Consider the following neutral system with mixed timevarying delays and nonlinear perturbations: where is the state vector, and are constant matrices with appropriate dimensions, and is a timevarying delay satisfying where and represent the lower and upper bounds of the timevarying delay , respectively, is the bound on the delayderivative, and initial condition is a continuous vectorvalued function. The functions and are unknown nonlinear perturbations with respect to the current state and in the delay state , respectively. They satisfy , , and where , are given constants; for simplicity we denote , .
Before moving on, we need the following instrumental lemma.
Lemma 1 (see [10]). For any scalar and constant matrix , , the following inequality holds: where is a freeweighting matrix with appropriate dimensions.
Lemma 2 (see [9]). For any constant matrix , , a scalar function and a vectorvalued function , such that the following integrations are well defined; then with and .
Lemma 3 (see [25]). Suppose , where . Then, for any constant matrices , , and with proper dimensions, the following matrix inequality holds, if and only if
3. Main Result
In this section, we study the delaydependent robust stability of system (1) based on the delaypartitioning approach.
Theorem 4. For given positive scalars , , , , and system (1) with uncertainty (3) is asymptotically stable if there exist symmetric matrices and , , , , , , , any matrices , , , and scalars such that the following LMIs hold: where with
Proof. First, we decompose the delayinterval into equidistant subinterval, where is a given integer; that is,
Then, the LK functional corresponding to the timevariation is chosen as
with .
The timederivative of the LK functional along the trajectory of (1) is given by
Note that
From Lemmas 1 and 2, we have
From (3), we can obtain, for any scalars , ,
From the system (1), we have the following equation:
By substituting (17)~(20) in (15) and defining an augmented state vector,
Then the timederivative can be expressed as follows:
One can see that if ,
Then for some scalar , from which we conclude that system (1) is asymptotically stable according to LK stability theory [1].
Applying Lemma 3 to (23) yields the following inequality:
By Schur complement on (24), we can obtain (10) and (11) with .
Without loss of generality, when , construct the following LK functional:
with , and , , , , , , , are the same matrix variables used in . Now, define an augmented state vector:
Using the same method, we have the conditions (10) and (11) as that for . The proof is completed.
Remark 5. As an extension of the method used in [18, 22], we divide the delayinterval into subintervals, constructing a new LK functional that contains some tripleintegral terms and augment term for each delayinterval. This treatment makes us employ more information on the time delay and yields less conservative delayrange bounds.
Remark 6. If there is no perturbation, that is, , , then the stability problem of system (1) is reduced to analyze the stability of the system:
According to Theorem 4, we can obtain the following corollary for the delaydependent stability of system (27).
Corollary 7. For given positive scalars, , , , system (27) with uncertainty (10) is asymptotically stable if there exist real symmetric positive definitive matrices and , , , , , , , any matrices , , , and scalars such that the following LMIs hold: where with
4. Numerical Examples
Example 1. Consider the following neutral timedelay system with
For given values of , , and , we apply Theorem 4 to calculate the maximal allowable value that guarantees that the asymptotical stability of the system is listed in Table 1. From Table 1, it is easy to see that our proposed stability criterion gives much less conservative results than those in [21, 22].

Example 2. Consider the system (27) with the following matrices:
For given and , we calculate the allowable upper bound of that guarantees the asymptotical stability of system (27). Using different methods, computational results are obtained and these are listed in Table 2. From Table 2, it can be seen that our results are less conservative than the existing criteria.
5. Conclusion
This paper studies the problem of robust delaydependent stability for a class of linear systems with interval timevarying delay and nonlinear perturbations. Based on the delaypartitioning approach, the less conservative delaydependent stability conditions are derived. The reduction in the conservatism of the proposed stability criteria is mainly attributed to the new LK functional which contains some tripleintegral terms and augment terms for each divided segment. Numerical examples have illustrated the effectiveness of the proposed method.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The research is supported by the National Natural Science Foundation of China, Grant nos. 61304001 and 61304103.
References
 L. Wu and W. X. Zheng, “Passivitybased sliding mode control of uncertain singular timedelay systems,” Automatica, vol. 45, no. 9, pp. 2120–2127, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 W. Chen and L. Qiu, “Stabilization of networked control systems with multirate sampling,” Automatica, vol. 49, no. 6, pp. 1528–1537, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 Y. He, M. Wu, J.H. She, and G.P. Liu, “Parameterdependent Lyapunov functional for stability of timedelay systems with polytopictype uncertainties,” IEEE Transactions on Automatic Control, vol. 49, no. 5, pp. 828–832, 2004. View at: Publisher Site  Google Scholar  MathSciNet
 Y. He, Q.G. Wang, L. Xie, and C. Lin, “Further improvement of freeweighting matrices technique for systems with timevarying delay,” IEEE Transactions on Automatic Control, vol. 52, no. 2, pp. 293–299, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 M. Wu, Y. He, J.H. She, and G.P. Liu, “Delaydependent criteria for robust stability of timevarying delay systems,” Automatica, vol. 40, no. 8, pp. 1435–1439, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 P. Park and J. W. Ko, “Stability and robust stability for systems with a timevarying delay,” Automatica, vol. 43, no. 10, pp. 1855–1858, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. Wu and W. X. Zheng, “Weighted ${H}_{\infty}$ model reduction for linear switched systems with timevarying delay,” Automatica, vol. 45, no. 1, pp. 186–193, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. Shao, “New delaydependent stability criteria for systems with interval delay,” Automatica, vol. 45, no. 3, pp. 744–749, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 X.M. Zhang and Q.L. Han, “New LyapunovKrasovskii functionals for global asymptotic stability of delayed neural networks,” IEEE Transactions on Neural Networks, vol. 20, no. 3, pp. 533–539, 2009. View at: Publisher Site  Google Scholar
 O. M. Kwon, J. H. Park, and S. M. Lee, “An improved delaydependent criterion for asymptotic stability of uncertain dynamic systems with timevarying delays,” Journal of Optimization Theory and Applications, vol. 145, no. 2, pp. 343–353, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 K. Ramakrishnan and G. Ray, “Robust stability criteria for uncertain neutral systems with interval timevarying delay,” Journal of Optimization Theory and Applications, vol. 149, no. 2, pp. 366–384, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. Wu, X. Yang, and H.K. Lam, “Dissipativity analysis and sysnthesis systems for discretetime TS fuzzy stochastic systems with timevaryng delay,” IEEE Transactions on Fuzzy Systems, vol. 22, no. 2, pp. 380–394, 2014. View at: Publisher Site  Google Scholar
 R. Rakkiyappan, P. Balasubramaniam, and R. Krishnasamy, “Delay dependent stability analysis of neutral systems with mixed timevarying delays and nonlinear perturbations,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2147–2156, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Q.L. Han, “Improved stability criteria and controller design for linear neutral systems,” Automatica, vol. 45, no. 8, pp. 1948–1952, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Q.L. Han, “A discrete delay decomposition approach to stability of linear retarded and neutral systems,” Automatica, vol. 45, no. 2, pp. 517–524, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 C. Peng and Y.C. Tian, “Improved delaydependent robust stability criteria for uncertain systems with interval timevarying delay,” IET Control Theory & Applications, vol. 2, no. 9, pp. 752–761, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 K. Ramakrishnan and G. Ray, “Robust stability criteria for uncertain linear systems with interval timevarying delay,” Journal of Control Theory and Applications, vol. 9, no. 4, pp. 559–566, 2011. View at: Publisher Site  Google Scholar  MathSciNet
 C. Wang and Y. Shen, “Delay partitioning approach to robust stability analysis for uncertain stochastic systems with interval timevarying delay,” IET Control Theory & Applications, vol. 6, no. 7, pp. 875–883, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 Q.L. Han, “Robust stability for a class of linear systems with timevarying delay and nonlinear perturbations,” Computers & Mathematics with Applications, vol. 47, no. 89, pp. 1201–1209, 2004. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Z. Zuo and Y. Wang, “New stability criterion for a class of linear systems with timevarying delay and nonlinear perturbations,” IEE Proceedings. Control Theory & Applications, vol. 153, no. 5, pp. 623–626, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 W. Zhang, X.S. Cai, and Z.Z. Han, “Robust stability criteria for systems with interval timevarying delay and nonlinear perturbations,” Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 174–180, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 K. Ramakrishnan and G. Ray, “Delayrangedependent stability criterion for interval timedelay systems with nonlinear perturbations,” International Journal of Automation and Computing, vol. 8, no. 1, pp. 141–146, 2011. View at: Publisher Site  Google Scholar
 P. Balasubramaniam and G. Nagamani, “A delay decomposition approach to delaydependent passivity analysis for interval neural networks with timevarying delay,” Neurocomputing, vol. 74, no. 10, pp. 1646–1653, 2011. View at: Publisher Site  Google Scholar
 L. Wu, X. Su, and P. Shi, “Sliding mode control with bounded ${L}_{2}$ gain performance of Markovian jump singular timedelay systems,” Automatica, vol. 48, no. 8, pp. 1929–1933, 2012. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 D. Yue, E. Tian, and Y. Zhang, “A piecewise analysis method to stability analysis of linear continuous/discrete systems with timevarying delay,” International Journal of Robust and Nonlinear Control, vol. 19, no. 13, pp. 1493–1518, 2009. View at: Publisher Site  Google Scholar  MathSciNet
Copyright
Copyright © 2014 Xin Zhou 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.