Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2019, Article ID 4969470, 10 pages
https://doi.org/10.1155/2019/4969470
Research Article

Novel Delay-Decomposing Approaches to Absolute Stability Criteria for Neutral-Type Lur’e Systems

School of Science, University of Science and Technology Liaoning, Anshan 114051, China

Correspondence should be addressed to Liang-Dong Guo; nc.ude.ltsu@ougdl

Received 27 August 2019; Accepted 9 November 2019; Published 2 December 2019

Academic Editor: Mingshu Peng

Copyright © 2019 Liang-Dong Guo 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 analysis for neutral-type Lur’e systems with time-varying delays is investigated. Novel delay-decomposing approaches are proposed to divide the variation interval of the delay into three unequal subintervals. Some new augment Lyapunov–Krasovskii functionals (LKFs) are defined on the obtained subintervals. The integral inequality method and the reciprocally convex technique are utilized to deal with the derivative of the LKFs. Several improved delay-dependent criteria are derived in terms of the linear matrix inequalities (LMIs). Compared with some previous criteria, the proposed ones give the results with less conservatism and lower numerical complexity. Two numerical examples are included to illustrate the effectiveness and the improvement of the proposed method.

1. Introduction

Over the last 30 years, time delay system has been one of the hottest research areas in control engineering for time delay often appears in many control systems either in the state, the control input, or the measurements [1, 2]. Since time delay frequently occurs in practical systems and is often the source of instability, there have been many results for stability of delayed systems [39]. Also stabilization [1012], filtering [13], and adaptive control [14] of time-delay systems have received considerable attention. The neutral systems often appear in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar [15]. A considerable number of studies related to this topic have been reported (see, for example, [1621], and the references therein).

Lur’e system is originally from a pilot robot [22], which is one of the important classes of nonlinear systems whose nonlinear element satisfies certain sector constraints. Many systems such as Chua’s circuits, Goodwin models, Swarm models, n-scroll attractors, and hyperchaotic attractors can be represented as Lur’e-type systems [23, 24]. In recent years, stability analysis of the neutral-type Lur’e system with time delays has attracted the attention of many researchers [2535]. The main purpose of stability analysis is to calculate the maximum allowable delay bounds (MADBs), which is a key index for judging the conservatism of stability criteria, such that the Lur’e system maintains absolute stability for any time delay less than the MADBs by using the LKF method. In [25], absolute stability of Lur’e systems with sector-bounded nonlinearities and constant delay was discussed by using a Lur’e-Postnikov function. To avoid involving a considerable number of free-weighting matrices and leading to a computationally expensive stability criterion in [26], the general free weighting matrix method was proposed and some improved delay-range-dependent stability criteria were obtained [27, 28]. By using integral inequality and the Wirtinger integral inequality method instead of free weighting matrix one, some improved delay-dependent robust stability criteria were derived in [29, 30], respectively. Also, by eliminating nonlinearity and reducing the number of free-weighting matrices, Lur’e systems with interval time-varying delays were discussed in [31]. By discretizing the delay interval into two segmentations with an unequal width, some delay-dependent sufficient conditions were given for the robust stability of neutral-type Lur’e systems in [32, 33]. By employing an augmented LKF method, reciprocally convex combination approach, and convex combination technique, several robust absolute stability criteria for uncertain neutral-type Lur’e systems with time-varying delays were presented in [34]. Very recently, by constructing a LKF including both double-integral terms and triple-integral terms, using the piecewise analysis method, Wirtinger-based integral inequality, and the reciprocally convex combination technique, some new stability criteria were obtained in [35]. In fact, one of the term in LKFs in [35] is the special case of the delay-partitioning approach, where . The delay-partitioning method is an effective one to reduce a criterion’s conservativeness, which is widely used in the stability analysis for various systems (see details in [3639]). However, when utilizing the delay-partitioning approach, the term is inevitably involved in LKFs, where and N is the delay-partitioning number. It is clear that the derived conditions become more complicated and the computational burden grows bigger when N increases. In addition, to relate to the Wirtinger-based integral inequality and deal with the derivative of triple-integral terms introduced in LKFs, it is ineluctable that the extra terms , , , and had to be introduced into the derivation process in [35], which leads to a sharp increase in the dimensions of the LMIs involved.

Motivated by this mentioned above, the aim of this work is to revisit the stability analysis for the neutral-type Lur’e system. In this study, novel delay-decomposing approaches are proposed firstly. Different from the delay-partitioning approaches or the delay-decomposing ones in [32, 33, 3640], the interval of the state time delay is divided into three unequal subintervals , , and . In particular, to establish the relationship of the vectors such as , and , the novel terms and are introduced in LKFs in each of subintervals, which are more general than the ones in [28, 35], where and . It is worth mentioning that the merit of the proposed delay-decomposing method lies in that the dimensions of the LMIs involved are independent of the number of subinterval. In addition, to avoid introducing the extra vectors by Wirtinger-based integral inequality, the reciprocally convex combination method [41] and the integral inequality [42] are utilized to deal with the bounds of integral terms. Some novel LKFs related to the above inequalities are constructed on the obtained three subintervals. The presented stability criteria are given in terms of LMIs. Compared with the related literature, the conclusions of this paper have the advantages of less conservatism conservatism and the dimensions of the LMIs. Finally, two well-known numerical examples are given to demonstrate the effectiveness and less conservatism over the existing results.

Notation: in this paper, denotes n-dimensional Euclidean space and is the set of all real matrices. For symmetric matrices X and Y, the notation (respectively, ) means that the matrix is positive definite (respectively, nonnegative). denotes the block diagonal matrix. The subscript “T” denotes the transpose of the matrix. denotes the identity matrix.

2. Problem Statements and Preliminaries

Consider a class of Lur’e systems of neutral type with time-varying delays and sector-bound nonlinearities described as follows:where are constant matrices with appropriate dimensions. is the state vector, and is the output vector. The time delay is a time-varying continuous function satisfyingwhere are the known constant scalars.

is the nonlinear function and is assumed to satisfy the finite sector restriction:with known positive scalar or the infinite sector restriction:

The objective of this paper is to formulate the delay-dependent stability conditions of system (1). The following lemmas will play important roles in deriving the criteria.

Lemma 1 (see [41]). Let have positive values in an open subset D of . Then, the reciprocally convex combination of over D satisfiessubject to

Lemma 2 (see [42]). For any matrices with compatible dimensions, any continuous time vector function with compatible dimensions, and any scalar satisfying , the following integral inequality holds:if and .

Remark 1. Let and , the integral inequality reduces the one in [43].

3. Main Result

In this section, some new delay-dependent stability criteria are proposed for system (1).

Note and (). It is easy to see that holds. Then, we divide the time interval into three subintervals , and . In the following, we will propose some criteria for the three subintervals.

Now, we give the stability criteria for system (1) with conditions (2) and (4) when as follows.

Theorem 1. For given scalars , , , , and , , system (1) with conditions (2) and (4) is absolutely stable if there exist symmetric positive definite matrices , symmetric positive definite matrices , symmetric semi-positive definite matrices , positive diagonal matrices , , any matrix , and matrices , such that LMIs (5) and (6) hold when :where

Proof. For positive diagonal matrices and positive definite matrices , let us consider the following LKF candidates for the case :whereThe time derivative of along the trajectory of system (4) is given bywhereIf holds, one can compute out the following according to Lemma 1:where is given in (6).
Based on Lemma 2 (in which and ), when , the following inequality holds:and using Lemma 2 again (in which and ), when , one can concludeUnder the assumption on nonlinear function (4), the following inequality holdsIt is equivalent toRewrite the above as follows:whereThen, combining (3)–(15) leads toTherefore, if LMIs (5) and (6) hold, one has , which shows the absolute stability of system (1) subject to (2) and (4), when satisfies . This completes our proof.
For the case , we construct the LKFs as follows:whereand are defined in (7).
Then, the derivative of can be obtained asAccording to Lemma 1, if , the following inequality holds:where is given in (20).
In addition, by using Lemma 2 (in which and ), if , one can obtainThe other procedure is straightforward from the proof of Theorem 1; we can cope with the situation of and have the following.

Theorem 2. For given scalars , system (1) with conditions (2) and (4) is absolutely stable if there exist symmetric positive definite matrices , symmetric positive definite matrices , symmetric semi-positive definite matrices , positive diagonal matrices , , any matrix , and matrices , such that LMIs (19) and (20) hold for the case :whereand and K are defined in Theorem 1.
For the case , we construct the following LKF:whereand are defined in (7).
Taking the derivative of yieldsBy using Lemma 1, if , one can obtainwhere .
Handle the integral terms and in the same way as in (13) and (17), respectively. Following the same procedure, we can cope with the situation of and have the following.

Theorem 3. For given scalars , system (1) with conditions (2) and (4) is absolutely stable if there exist symmetric positive definite matrices , symmetric positive definite matrices , symmetric semi-positive definite matrices , positive diagonal matrices , , any matrices , and matrices , such that LMIs (23) and (24) hold for the case :whereand , K, and are defined in Theorems 1 and 2, respectively.

Remark 2. Theorems 13 present the absolute stability criteria for the Lur’e system. Unlike the delay-partitioning approach or delay-decomposing one used in [28, 32, 33, 35, 40], the interval of the state time delay has been divided into three subintervals , , and . Obviously, the range of the subintervals , , and is , , and . Thus, to know more about the time-varying , one can select the value of the parameter α as close as possible to 0 for the case or case and as close as possible to 0.5 for the case , respectively. The following examples in the work will illustrate the point.

Remark 3. The terms and have been introduced in LKFs in the work instead of the terms in [28, 35] and in [3639], where . And the relations between , and have been effectively expressed by the derivative of . One can easily see that or deduces into when . This is to say the established LKFs are more general than ones in [28, 35]. The idea is expected to reduce the conservatism of obtained criteria. On the other hand, it can effectively overcome the weakness of the computational burden growing bigger when delay-partitioning number increases in [3639].

Remark 4. In [35], the Wirtinger-based integral inequality and double-integral inequality were used to deal with bounds of the derivative of double-integral and triple-integral terms in LKFs. Thus, the extra terms , , , and were unavoidably introduced in the derivation process, which leads the dimensions of the LMIs involved to increase sharply. A detailed comparison is given in Example 1.

Remark 5. If function of system (1) satisfies sector condition (3), for any , one has the following:which is equivalent toThus, if function of system (1) satisfies sector condition (3), we have the following corollary.

Corollary 1. For given scalars , the system (1) with conditions (2) and (3) is absolutely stable if there exist symmetric positive definite matrices , , , with appropriate dimensions, positive diagonal matrices , , and matrices with appropriate dimensions and properties, such that LMIs (6) and (26) hold for the case , (3) and (28) hold for the case , and (20) and (28) hold for the case , respectively:where , , are defined in Theorems 13.

4. Illustrative Example

In this section, we will use two well-known numerical examples to show the effectiveness and benefits of our results.

Example 1. Consider the following nominal neutral-type Lur’e system (1) subject to (2) and (4) with the parameters:The purpose of this example is to compute MADBs of such that the neutral-type Lur’e system (1) remains stable for different and . For given , , and , the acceptable upper bounds of is 0.271, 2.277, and 2.652 when by using Corollary 1 () in our work, respectively. In order to make a comparison with some existing stability criteria, we calculate the MADBs and list them in Tables 13. Together with all derived MADBs listed in Tables 13, one can check that Corollary 1 can be superior over some present ones. The number of decision variables, maximal order of LMIs between our work, and the criteria in [29, 30, 33, 35] are listed in Table 4. It shows that our proposed method involves smaller decision variables or lower maximal order of LMIs than the relative ones. To confirm the obtained result, a simulation result is shown in Figure 1 when µ is unknown, .

Table 1: MADBs of for different , when for Example 1.
Table 2: MADBs of for different , when for Example 1.
Table 3: MADBs of for different , when for Example 1.
Table 4: Number of decision variables (NDV) and maximal order of LMIs (MOL) of different methods.
Figure 1: State responses of the system considered in Example 1.

Example 2. Consider Chua’s circuit example discussed in [35]:The nonlinear function . Let , and ; then, Chua’s circuit can be expressed as a Lur’e-type system withThe feedback nonlinear function belongs to .
Now, we calculate MADBs of . For different , the obtained results are given in Table 5. From this table, it is clear to see that Theorems 1 and 2 offer larger MADBs of than those methods in existing references.

Table 5: MADBs for different , when for Example 2.

5. Conclusions

This paper has investigated the absolute stability analysis for neutral-type Lur’e systems with time-varying delays. Based on the new delay-decomposition approaches in combination with the integral inequality and reciprocally convex technique, several improved stability criteria have been derived by constructing some appropriate LKFs on the subintervals. The merit of the obtained stability criteria lies in the significant less conservativeness and lower computational complexity than some existing ones. Finally, two examples have been given to demonstrate the effectiveness and less conservatism of the proposed method. In the future works, we will be dedicated to study the stability analysis for systems with infinite delays and devote to the study of output feedback, tracking control and filtering of the neutral-type Lur’e systems with time-varying delays based on the method proposed in this paper.

Data Availability

All data generated or analyzed during this study are included in this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grants 61503058 and 61773013.

References

  1. J.-K. Hale and S.-M. Lunel, Introduction to Functional Differential Equations, Springer-Verlag, Berlin, Germany, 1993.
  2. J.-H. Park, H. Lee, Y. Liu, and J. Chen, Dynamic Systems with Time Delays: Stability and Control, Springer-Nature, Berlin, Germany, 2019.
  3. O.-M. Kwon, J.-H. Park, and S.-M. Lee, “Augmented Lyapunov functional approach to stability of uncertain neutral systems with time-varying delays,” Applied Mathematics and Computation, vol. 207, no. 1, pp. 202–212, 2009. View at Publisher · View at Google Scholar · View at Scopus
  4. X.-M. Zhang, Q.-L. Han, A. Seuret, and F. Gouaisbaut, “An improved reciprocally convex inequality and an augmented Lyapunov-Krasovskii functional for stability of linear systems with time-varying delay,” Automatica, vol. 84, pp. 221–226, 2017. View at Publisher · View at Google Scholar · View at Scopus
  5. C.-K. Zhang, Y. He, L. Jiang, M. Wu, and Q.-G. Wang, “An extended reciprocally convex matrix inequality for stability analysis of systems with time-varying delay,” Automatica, vol. 85, pp. 481–485, 2017. View at Publisher · View at Google Scholar · View at Scopus
  6. J.-H. Park and S. Won, “A note on stability of neutral delay-differential systems,” Journal of the Franklin Institute, vol. 336, no. 3, pp. 543–548, 1999. View at Publisher · View at Google Scholar · View at Scopus
  7. E. Fridman, “New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems,” Systems & Control Letters, vol. 43, no. 4, pp. 309–319, 2001. View at Publisher · View at Google Scholar · View at Scopus
  8. K. Liu, E. Fridman, K. H. Johansson, and Y. Xia, “Generalized Jensen inequalities with application to stability analysis of systems with distributed delays over infinite time-horizons,” Automatica, vol. 69, pp. 222–231, 2016. View at Publisher · View at Google Scholar · View at Scopus
  9. K. Liu, A. Seuret, Y. Xia, F. Gouaisbaut, and Y. Ariba, “Bessel-Laguerre inequality and its application to systems with infinite distributed delays,” Automatica, vol. 109, pp. 1–7, 2019. View at Publisher · View at Google Scholar
  10. C. Hua, S. Wu, and X. Guan, “Stabilization of T-S fuzzy system with time delay under sampled-data control using a new looped-functional,” IEEE Transactions on Fuzzy Systems, p. 1, 2019. View at Publisher · View at Google Scholar
  11. G.-Zhang, J.-Xia, W.-H. Zhang, J.-S. Zhao, Q. Sun, and H.-S. Zhang, “State feedback control for stochastic Markovian jump delay systems based on LaSalle-type theorem,” Journal of the Franklin Institute, vol. 335, no. 5, pp. 2179–2196, 2018. View at Google Scholar
  12. G.-M. Zhang, J.-W. Xia, and W.-H. Zhang, “Normalization design for delayed singular Markovian jump systems based on system transformation technique,” International Journal of Systems Science, vol. 49, no. 8, pp. 1603–1614, 2018. View at Google Scholar
  13. G.-M. Zhuang, S.-Y. Xu, B.-Y. Zhang, H.-L. Xu, and Y. Chu, “RobustH∞deconvolution filtering for uncertain singular Markovian jump systems with time-varying delays,” International Journal of Robust and Nonlinear Control, vol. 26, no. 12, pp. 2564–2585, 2016. View at Publisher · View at Google Scholar · View at Scopus
  14. C.-C. Hua, G.-P. Liu, L. Li, and X.-P. Guan, “Adaptive fuzzy prescribed performance control for nonlinear switched time-delay systems with unmodeled dynamics,” IEEE Transactions on Fuzzy Systems, vol. 26, no. 4, pp. 1934–1945, 2018. View at Publisher · View at Google Scholar · View at Scopus
  15. D. Yue and Q.-L. Han, “A delay-dependent stability criterion of neutral systems and its application to a partial element equivalent circuit model,” IEEE Transactions on Circuits and Systems, vol. 6, no. 12, pp. 5438–5442, 2004. View at Google Scholar
  16. Q.-L. Han, “A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays,” Automatica, vol. 40, no. 10, pp. 1791–1796, 2004. View at Publisher · View at Google Scholar · View at Scopus
  17. Y. He, M. Wu, J.-H. She, and G.-P. Liu, “Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays,” Systems & Control Letters, vol. 51, no. 1, pp. 57–65, 2004. View at Publisher · View at Google Scholar · View at Scopus
  18. O.-M. Kwon, J.-H. Park, and S.-M. Lee, “On delay-dependent robust stability of uncertain neutral systems with interval time-varying delays,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 843–853, 2008. View at Publisher · View at Google Scholar · View at Scopus
  19. L.-D. Guo, H. Gu, and D.-Q. Zhang, “Robust stability criteria for uncertain neutral system with interval time varying discrete delay,” Asian Journal of Control, vol. 12, no. 6, pp. 739–745, 2010. View at Publisher · View at Google Scholar · View at Scopus
  20. G.-M. Zhuang, S.-Y. Xu, J.-W. Xia, Q. Ma, and Z.-Q. Zhang, “Non-fragile delay feedback control for neutral stochastic Markovian jump systems with time-varying delays,” Applied Mathematics and Computation, vol. 355, pp. 21–32, 2019. View at Publisher · View at Google Scholar · View at Scopus
  21. G.-M. Zhuang, J.-W. Xia, J.-E. Feng, B.-Y. Zhang, J.-W. Lu, and Z. Wang, “Admissibility analysis and stabilization for neutral descriptor hybrid systems with time-varying delays,” Nonlinear Analysis: Hybrid Systems, vol. 33, pp. 311–321, 2019. View at Publisher · View at Google Scholar
  22. A. I. Lur’e, Some Nonlinear Problems in the Theory of Automatic Control, H. M. Stationery Office, London, UK, 1957.
  23. M. E. Yal’in, J. A. K. Suykens, and J. Vandewalle, “Master-slave synchronization of Lure systems with time-delay,” International Journal of Bifurcation and Chaos, vol. 11, no. 6, pp. 1707–1722, 2001. View at Google Scholar
  24. C.-Y. Yang, Q.-L. Zhang, J. Sun, and T.-Y. Chai, “Lur’e Lyapunov function and absolute stability criterion for Lur’e singularly perturbed systems,” IEEE Transactions on Automatic Control, vol. 56, no. 11, pp. 2666–2671, 2011. View at Publisher · View at Google Scholar · View at Scopus
  25. X. Liu, J.-Z. Wang, Z.-S. Duan, and L. Huang, “New absolute stability criteria for time-delay Lur’e systems with sector-bounded nonlinearity,” International Journal of Robust Nonlinear Control, vol. 20, no. 6, pp. 659–672, 2010. View at Google Scholar
  26. 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. 27, no. 5, pp. 515-516, 2010. View at Google Scholar
  27. K. Ramakrishnan and G. Ray, “Improved delay-range-dependent robust stability criteria for a class of Lur’e systems with sector-bounded nonlinearity,” Journal of the Franklin Institute, vol. 348, no. 8, pp. 1769–1786, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. K. Ramakrishnan and G. Ray, “An improved delay-dependent stability criterion for a class of Lur’e systems of neutral type,” Siam Journal on Applied Dynamical Systems, vol. 134, no. 1, pp. 1–6, 2012. View at Publisher · View at Google Scholar · View at Scopus
  29. W.-Y. Duan, B.-Z. Du, Z.-F. Liu, and Y. Zou, “Improved stability criteria for uncertain neutral-type Lur’e systems with time-varying delays,” Journal of the Franklin Institute, vol. 351, no. 9, pp. 4538–4554, 2014. View at Publisher · View at Google Scholar · View at Scopus
  30. Y.-J. Liu, S.-M. Lee, O.-M. Kwon, and J.-H. Park, “Robust delay-dependent stability criteria for time-varying delayed Lur’e systems of neutral type,” Circuits, Systems, and Signal Processing, vol. 34, no. 5, pp. 1481–1497, 2015. View at Publisher · View at Google Scholar · View at Scopus
  31. Y.-T. 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
  32. W.-Y. Duan, B.-Z. Du, J. You, and Y. Zou, “Improved robust stability criteria for a class of Lur’e systems with interval time-varying delays and sector-bounded nonlinearity,” International Journal of Systems Science, vol. 46, no. 5, pp. 944–954, 2015. View at Publisher · View at Google Scholar · View at Scopus
  33. W.-Y. Duan, X.-R. Fu, Z.-F. Liu, and X.-D. Yang, “Improved robust stability criteria for time-delay Lur’e system,” Asian Journal of Control, vol. 19, no. 1, pp. 139–150, 2016. View at Publisher · View at Google Scholar · View at Scopus
  34. Y.-T. Wang, Y. Xue, and X. Zhang, “Less conservative robust absolute stability criteria for uncertain neutral-type Lur’e systems with time-varying delays,” Journal of the Franklin Institute, vol. 353, no. 4, pp. 816–833, 2016. View at Publisher · View at Google Scholar · View at Scopus
  35. Y.-M. Wang, L.-L. Xiong, Y.-K. Li, H.-Y. Zhang, and C. Peng, “Novel stability analysis for uncertain neutral-type Lur’e systems with time-varying delays using new inequality,” Mathematical Problems in Engineering, vol. 2017, Article ID 5731325, p. 13, 2017. View at Google Scholar
  36. X.-Y. Meng, J. Lam, B.-Z. Du, and H.-J. Gao, “A delay-partitioning approach to the stability analysis of discrete-time systems,” Automatica, vol. 46, no. 3, pp. 610–614, 2010. View at Publisher · View at Google Scholar · View at Scopus
  37. Z. Yan, H.-J. Gao, J. Lam et al., “Stability and stabilization of delayed T-S fuzzy systems: a delay partitioning approach,” IEEE Transactions on Fuzzy Systems, vol. 17, no. 4, pp. 750–762, 2009. View at Google Scholar
  38. O.-M. Kwon, M.-J. Park, J.-H. Park, S.-M. Lee, and E.-J. Cha, “New delay-partitioning approaches to stability criteria for uncertain neutral systems with time-varying delays,” Journal of the Franklin Institute, vol. 349, no. 9, pp. 2799–2823, 2012. View at Publisher · View at Google Scholar · View at Scopus
  39. P. Mahmoudabadi, M. Shasadeghi, and J. Zarei, “New stability and stabilization conditions for nonlinear systems with time-varying delay based on delay-partitioning approach,” ISA Transactions, vol. 70, pp. 46–52, 2017. View at Publisher · View at Google Scholar · View at Scopus
  40. L.-D. Guo, X.-H. He, and J.-J. He, “New delay-decomposing approaches to stability criteria for delayed neural networks,” Neurocomputing, vol. 189, pp. 123–129, 2016. View at Publisher · View at Google Scholar · View at Scopus
  41. P. Park, J.-W. Ko, and C. Jeong, “Reciprocally convex approach to stability of systems with time-varying delays,” Automatica, vol. 47, no. 1, pp. 235–238, 2011. View at Publisher · View at Google Scholar · View at Scopus
  42. L.-D. Guo, H. Gu, J. Xing, and X.-Q. He, “Asymptotic and exponential stability of uncertain system with interval delay,” Applied Mathematics and Computation, vol. 218, no. 19, pp. 9997–10006, 2012. View at Publisher · View at Google Scholar · View at Scopus
  43. P.-L. Liu, “Improved delay-range-dependent robust stability for uncertain systems with interval time-varying delay,” ISA Transactions, vol. 53, no. 6, pp. 1731–1738, 2014. View at Publisher · View at Google Scholar · View at Scopus
  44. K.-B. Shi, H. Zhu, S.-M. Zhong, Y. Zeng, and Y.-P. Zhang, “Improved delay-dependent robust stability criteria for a class of uncertain neutral type Lur’e systems with discrete and distributed delays,” Mathematical Problems in Engineering, vol. 2014, Article ID 980351, p. 14, 2014. View at Google Scholar