Improved Results on Delay-Dependent Robust <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.3443pt" id="M1" height="15.7437pt" version="1.1" viewBox="-0.0657574 -11.3994 26.6531 15.7437" width="26.6531pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-73" d="M828 650H570L564 622C646 614 650 609 634 524L604 366H253L281 524C296 608 308 615 382 622L388 650H132L126 622C211 615 214 609 198 524L121 122C106 42 102 35 23 28L17 0H276L282 28C196 35 191 40 206 122L243 324H597L559 123C544 42 537 35 452 28L446 0H710L716 28C632 35 628 43 643 122L719 524C735 610 741 615 822 622L828 650Z"/></g><g transform="matrix(.012,0,0,-0.012,13.299,4.134)"><path id="g50-30" d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z"/></g></svg> Control of Uncertain Neutral Systems with Mixed Time-Varying Delays

Zhang, Jingsha; Li, Yongke; Ma, Xiaolin; Lin, Zhilong; Wang, Changlong

doi:https://doi.org/10.1155/2021/6360923

Mathematical Problems in Engineering

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Advances in Delay Differential Equations and Applications

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 6360923 | https://doi.org/10.1155/2021/6360923

Improved Results on Delay-Dependent Robust Control of Uncertain Neutral Systems with Mixed Time-Varying Delays

Jingsha Zhang,¹Yongke Li,¹Xiaolin Ma,¹Zhilong Lin,¹and Changlong Wang¹

Academic Editor: Ramachandran Raja

Received31 Jul 2020

Revised30 Dec 2020

Accepted10 Mar 2021

Published13 Apr 2021

Abstract

In this paper, the problem of the delay-dependent robust control for a class of uncertain neutral systems with mixed time-varying delays is studied. Firstly, a robust delay-dependent asymptotic stability criterion is shown by linear matrix inequalities (LMIs) after introducing a new Lyapunov–Krasovskii functional (LKF). The LKF including vital terms is expected to obtain results of less conservatism by employing the technique of various efficient convex optimization algorithms and free matrices. Then, based on the obtained criterion, analyses for uncertain systems and controller design are presented. Moreover, on the analysis of the state-feedback controller, different from the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively, we can obtain the state-feedback gain directly by calculating the LMIs through the toolbox of MATLAB in this paper. Finally, the feasibility and validity of the method are illustrated by examples.

1. Introduction

Strictly speaking, some degrees of time delays are frequently encountered in almost all practical engineering systems such as [1–3], in which the existence of the delays may lower the performances of these engineering control systems and even result in the instability of such control systems. Then, it is an essential requirement to investigate the stability for systems with time delays. And quantities of results on stability for time-delay systems in actual applications have been obtained in the near past [4–6]. The stability analyses of the systems can be classified into delay-dependent ones and delay-independent ones. The stability analyses criteria in delay-dependent systems are always less conservative than the delay-independent ones. In addition, the stability criteria are developed by applying the LKF methodology. Otherwise, for some time-delay systems in practice such as air-feed system [7] and power converters system [8], the controllers are designed to make the systems more practical by employing advanced methods for the stability analyses. Though the time delay can be constant or varying, the former is a special case of the latter which is always more complex and typical. Thus, the stabilization problem of dynamical systems with time-varying delays has been taken into consideration.

On the other hand, there are also many typical practical models of time-delay neutral dynamical systems such as partial element equivalent circuit model [9] and population genetics and community ecology [10], which are described by neutral functional differential equations containing time delays not only in the state but also in its derivative. Obviously, studying neutral systems with mixed delays (NSMDs) is more challenging than the system with a delay only in the state. Results about stability analysis for NSMD have been obtained in recent years. In the following, some classical methods are shown for NSMD. In [11], delay-dependent conditions were improved via utilizing the LMI method. In [12], by applying a candidate LKF, less conservative criteria were proposed. Besides, to reduce conservatism, an approach of piece-wise delay was expressed in [13]. Advanced integral inequalities were used in [14] to get the delay-dependent stability conditions, which were less conservative. NSMD with uncertainties were observed, and then the robust stability analyses were discussed in [15]. As we can see in [16], compared with an inequality which might not cause a less conservative result, a tighter inequality based on a suitable LKF could get the advantage we want. From [11–16], we can see that the methods of LKF and integral terms are the mainstream ones. By constructing an LKF with inequality terms and applying the LMI method, sufficient conditions are obtained.

As known, uncertainties can lead to poor performance in many time-delay systems. Therefore, much attention has been paid to the analysis of stability and stabilization for systems with parametric uncertainties. And robust control can be used for the analysis and design of the systems including uncertainties which are complex to handle in dynamical models. The problem of robust stabilization for a class of uncertain neutral time-delay dynamical systems with the uncertain matrix coefficients of both the delayed state and derivative of the delayed state is considered so that we can ensure the discussed systems stable under specific conditions. In addition, to overcome the external disturbances of actual operations in engineering models such as in [17, 18] and make them stable artificially, control is also an important issue. In [17], robust control was designed for PFC rectifiers. For the cooperative driving system with external disturbances and communication delays in the vicinity of traffic signals, robust control was studied in [18]. As presented in [19], the issue on robust control was discussed. Besides, control of neutral systems has been expressed in many papers up to now. In [20], the problem of robust control with state feedback was solved in the uncertain neutral system. For an uncertain neutral system in [21], performance for Markovian uncertain systems via the sliding mode control method was considered. Moreover, delay-dependent filtering conditions were proposed for uncertain neutral stochastic systems with Markovian switching and mode-dependent time delays [22]. By applying a state-feedback controller in [23], a neutral-type time-delay system was studied. And in another paper [24], the feedback controller was used to obtain the results.

Taking all the discussed elements into consideration, a representative system named the teleoperation system was introduced in [25]. The teleoperation system is composed of a master manipulated by a human operator and a slave operating in a remote environment through a communication channel. Time-varying delays existed in the long-range or flexible communication channel [26]; thus, the problem of stability of teleoperation systems for time-varying delays by neutral LMI techniques was discussed in [25]. Besides, due to that no real system is pure linear, time-varying model uncertainties considered as perturbations existed in real implementation of bilateral teleoperation [27], and then the study on robust control design for teleoperation systems was done in [28]. Moreover, the stability criteria above were all given in the form of LMIs, which can be used to compute the maximum values of delays. And in order to obtain the controller parameters directly by solving LMIs, state-feedback control law was found to stabilize the system with guaranteeing performance in [29]. Above all, in our paper, both time-varying delays and uncertainties are considered in the neutral model to analysis the robust control with state-feedback controller by the technique of LMIs. Different from the general delay system such as [16, 23], this paper also considers delays in the neutral terms. And unlike the delays in [11–15, 20, 24], the delays shown in both discrete and distributed forms in this article are all varying and different instead of constant or the same delay function. Then, the model derived from the actual application is more complex but more representative. Moreover, appropriate Lyapunov function is constructed to deal with the complicated model terms technically and get the control gain directly without applying the traditional method in [30].

Hence, motivated by the discussions above, the problem of delay-dependent robust control of uncertain neutral systems with mixed time-varying delays is explored in this article. After introducing an improved LKF and using the LMI method, a sufficient condition for the asymptotic stability of the system is established. Analyses on uncertainties and control are made then. Numerical examples are given to verify the results finally.

The main contributions of this paper are summered as follows:(1)In the LKF newly constructed, all terms are taken into account to obtain a larger delay bound.(2)The methods including not only free-weighting matrices but also efficient convex optimization algorithms are applied technically to get results of less conservatism.(3)Sufficient conditions are given all by LMIs, and the solutions can be solved by MATLAB Toolbox directly.(4)Though similar papers concerning the delay-dependent robust control of uncertain neutral systems with mixed time-varying delays have already been many, there are some differences between the current submission and latest works. To the extent of the authors’ knowledge, none of the existing works including all the terms of robust control, neutral systems, uncertainties, and mixed time-varying delays derived from actual engineering problems. Besides, the Lyapunov functional proposed in this article can get better conservatism with larger delay bounds when compared with these in [31–33], which is shown in Example 3. Finally, different from the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively, such as in [30], the controller gain illustrated in Theorem 4 of this paper can be obtained more easily and simply.

1.1. Notation

The notations throughout this paper are fairly standard. and denote, respectively, the matrix inverse and transpose of . and are the n-dimensional Euclidean space and the set of real matrices. means that is positive definite. is a unit matrix with compatible dimensions. denotes . Let , . expresses the Euclidean norm of function . refers to the space of square integral vectors.

2. Description of the Problem

Consider the NSMD with uncertainties:with , , , and as the state vector, the control input, the disturbance which belongs to , and the controlled output, respectively; and are given continuous initial condition functions. Continuous functions and are time-varying delays which satisfywhere are constants. and are matrices with uncertainties satisfyingwhere and are known constant matrices with suitable dimensions. The time-varying unknown matrices , and satisfy the following form:where and are known constant matrices with appropriate dimensions and as an unknown matrix function satisfies for any .

For system (1), the nominal form is given as follows:

For system (5), a state-feedback controller is designed bywhere is a constant matrix to be designed.

Combining (1) with (3) and the controller expression (6), we can get the closed-loop system of (1):

Remark 1. The mixed time-varying delays, respectively, distribute in the state and the derivative of the state, and the system could not always be stable with all values of delays. Then, we must find the bound of each time-varying delay, the larger the boundary and the better conservative system indicating that the broader time scope can be got to keep the system stable.

Remark 2. The two kinds of both discrete and distributed delays exist in the actual systems. And the large delay always causes the instability of the system. And either of the large delay can cause the instability of the system. Thus, considering the bounds of the discrete and distributed delays is a must.

Lemma 1. Schur complement [34]: given matrices with , then we have

Lemma 2. Jensen’s inequality [35]: for any scalar and any constant matrix , , the following inequality holds

Lemma 3. Cauchy matrix inequality [36]: let be a positive define matrix, then

Lemma 4 (see [37]). For any constant matrix , and free-weighting matrix , we havewhere

Lemma 5 (see [38]). Given matrices with suitable dimensions and satisfying , then for any constant , the inequality holds

3. Main Results

In this section, a new sufficient condition criterion of asymptotic stability for system (14) is presented by using the LKF method and LMI technique:

Theorem 1. For the given constants , and , system (14) is asymptotically stable if there exist positive-definite matrices and matrices with appropriate dimensions satisfying the following LMI:

Proof. Choose an LKF candidate aswherewithThe time derivative of is given byBy applying Lemma 4,Similarly,By using Lemma 2, we can getBased on the Newton–Leibniz formula, we haveAccording to Lemmas 2 and 3, we can getThen, we can obtainFrom system (14), it is easy to getCombing formulas (19) to (30), we haveBy Lemma 1, we can obtain . Obviously, system (14) is ensured to be asymptotically stable.

Remark 3. In the field of analysis of delay-dependent stability, the main purpose is to find the maximum delay bounds for guaranteeing stability. In the discussed LKF , the three terms , , and play an important role in reducing the conservatism of our results. More specifically, by taking the states , , as variables, more information can be sufficiently utilized on state variables, which can yield less conservatism. And as mentioned in [37], in order to derive an upper bound of in each intervals by Lemma 4, two different free-weighting matrices and are introduced at each interval. In addition, the integral terms in devote to a less result with Jensen’s inequality presented.

Remark 4. The LMI problem illustrated in Theorem 1 is a convex problem in relation to solution variables. And various convex optimization algorithms can solve the problem. In this paper, Matlab LMI Toolbox, as a useful software, for solving convex optimization problem, is used in which the interior-point algorithm was implemented.

Remark 5. The proposed criteria in Theorem 1 may give larger stability regions in compassion with some of the existing criteria. This claim is due to the following characteristics of the proposed stability analysis scheme: in [16], the relationship between the augmented LKF and the tightness of inequalities was explored. It has been shown that a tighter inequality does not always lead to a less conservative criterion. Nevertheless, a suitable augmented LKF and a tighter inequality, both of which are indispensable, can effectively reduce conservatism. In this paper, an augmented LKF combined with more cross terms can be well dealt with Lemma 4 technically. In addition, double- and triple-integral terms help to further increase the stability region.

Remark 6. (i) The system modeled in (1) is more representative than the ones in [11, 16, 36, 37]. For example, if , the neutral items are linear, then system (1) here can reduce to the neutral model without uncertainties in [11]. Besides, when , the system becomes the general uncertain delay time model in [16, 37]. If , then the general delay time systems without uncertainties in [16, 36] are got. And if , the varying delay terms are turned into constant delay ones. (ii) In the system of this article, all the coefficient matrices of the states and the derivative are functional matrices with uncertainties, different from the known constant matrices or few uncertain matrices in the existing literature. In the majority of the papers including all the cited papers in this paper, uncertainties are neglected more or less. However, in this article, uncertainties are all considered in the states and the derivative vector. (iii) Most articles about the content of neutral time-delay systems are concerned with the stability problem such as [9, 11–15], and our article not only solves the stability problem but also contains the control strategies.

Remark 7. If and , we can get a special case of system (14):By applying Theorem 1, solutions of system (32) and the upper delay bounds comparing with other results are shown in Example 1.

Theorem 2. For a prescribed scalar and given constants , system (5) with is robustly asymptotically stable with performance if there exist positive-definite matrices , , and matrices with appropriate dimensions , satisfying the LMI:where are the same terms as in Theorem 1.

Proof. Introduce the following performance index under the zero initial condition:For any nonzero disturbance , we can seeBased on the method which is similar to the one utilized in Theorem 1, we can conclude , which means . That is to say, system (5) with is robustly asymptotically stable with performance .

Remark 8. To overcome the external disturbances in actual operations of engineering models and make them stable artificially, considering the problem of robust control is a must. The study [18] has shown the artificial applications of robust control under external disturbances.

Remark 9. As a special case of system (5), the following time-varying system is considered:Let , then the stability analysis can be got for system (36) by Theorem 2, and the conservatism of this method is shown in Example 2.

Theorem 3. For a prescribed scalar and given constants , system (1) with is robustly asymptotically stable with performance if there exist positive-definite matrices , , and matrices with appropriate dimensions , and satisfying the following LMI:wherewith the same terms in Theorem 2.

Proof. in Theorem 2 are substituted by , respectively, and then we can obtainAccording to Lemma 5, there exists satisfyingApplying Lemma 1, obviously, (40) is equal to (37) and the theorem is proved.

Remark 10. The uncertain terms are solved in Theorem 3, which makes the system less complex to deal with.

Remark 11. The method applied with parameters in Example 3 is proved to be less conservative.
In the following theorem, we will put our focus on the synthesis of the controller for system (1) and the goal is to construct a state-feedback control law (6). On the basis of the results of the previous section, closed-loop system (7) is analyzed. The following Theorem 4 gives an LMI-based computational procedure to determine a state-feedback controller.

Theorem 4. For given constants and a small enough scalar , system (1) is robustly asymptotically stable with performance if there exist positive-definite matrices , , and matrices with appropriate dimensions , , and satisfying the following LMI:with the same terms in Theorem 3.

Proof. Substituting in Theorem 3 by , it is easy to getwhereThere must be a small enough scalar subject towhich equals toThen, we can obtain (41). Thus, the theorem is proved.

Remark 12. To get the controller gain in Theorem 4, the method utilized in this paper is faster and easier than the traditional method which multiplies the matrix inequality left and right by some matrix and its transpose, respectively.

Remark 13. The feasibility of the method mentioned in Theorem 4 is shown in Example 4.

Remark 14. In this paper, we focus on the improvement of the robust stability analysis of control for delay-dependent uncertain neutral systems with mixed time-varying delays which shows in the following aspects. First, based on a new LKF, robust stabilization criteria for time-varying delays are established. Second, by introducing free-weighting matrices and using the technique of convex combination, sufficient conditions are derived in terms of LIMs. And it is convenient for us to get all the solutions by MATLAB toolbox directly.

Remark 15. In order to guarantee the solutions of the discussed system asymptotically stable, we develop a continuous state-feedback controller scheme with a rather simple structure to make a closed loop of the system. And when the parameters are given, we can know the value of the controller gain whether to be feasible or not by MATLAB toolbox directly. It is easy to know that the larger control gain is, the less conservative of Theorem 4 is. However, a larger control gain means higher control cost. Therefore, we need to choose the control gain as small as possible on the basis of satisfying the inequalities in Theorem 4.

4. Numerical Examples

In this section, some examples are presented to demonstrate that the proposed methods are feasible and effective.

Example 1. Consider neutral-type system (32) with parameters Case I: for , by LMI Toolbox, the main solutions are Because of the large number and big dimensions, the solution to and is omitted here. The results of the state response of the neutral system with parameters in case I are, respectively, depicted in Figure 1. Case II: by taking parameter , the admissible upper bounds of for different are listed in Table 1 by Theorem 1. From the comparison, we can see that the methods proposed in this paper are less conservative.As we can see in Table 1, the results of delay bound to guarantee stability with two different were presented in [12, 31, 39] and the best results were in [31]. However, by applying Theorem 1, we can obtain larger values than [31].

Remark 16. The example above has shown that our obtained methods not only can solve the normal stability problem as a special case proposed in this paper but also can get results which are better than some existing results.

Example 2. Consider time-varying delay system (36) with matrix parametersBy applying the conditions of Theorem 2, Table 2 reports the maximum upper bound for different with , and simulation results are conducted to show the differences between some typical different parameters in Figure 2.
In Table 2, comparison of maximum allowable delay bound for different values of is listed. With the increase of , the maximum allowable delay bound becomes larger. And it also shows that Theorem 2 provides improved delay bounds.

Example 3. Consider the uncertain neutral systemwhereFor with different values of and with different values of , Tables 3 and 4 give out the related comparative results, respectively. From messages of the tables, it is easy to get the conclusion that the maximum allowable delay bounds increase along with the decrease of or . Moreover, we can see that the results obtained in Theorem 3 are less conservative than those in [11, 31–33, 39, 42, 43, 45, 46], which illustrates the effectiveness of the methods we proposed.

Example 4. Let us consider system (1) with the parametersLet , according to the strict LIMs of Theorem 4 by LMI toolbox in MATLAB, we can obtain the state-feedback gain:and the value .

5. Conclusion

In this work, the robust stability and stabilization for the delay-dependent uncertain neutral system with mixed time-varying delays have been studied in detail. Based on an LKF newly constructed and methods of free-weighting matrices and various efficient convex optimization algorithms, several new robust stability and stabilization criteria are expressed in terms of LMIs. Especially, a desired state-feedback controller gain can be obtained directly by MATLAB, and it can lead to a less conservative result. Finally, numerical examples are shown to illustrate the superiority and feasibility of the proposed method.

The weakness of the obtained results is that terms in the constructed Lyapunov function should be handled technically to make the system stable, and then appropriate analysis techniques and methods must be provided. Thus, dealing with a bit too many terms is a challenge. And another weakness is that the nonlinear disturbances have not been contained in our modeled systems. Thus, dealing with nonlinear disturbances in the neutral-type system is in our future work. Besides, the stability problem of neutral-type neural networks with time-varying delay has been more popular recently. Hence, we will also focus on the stability content of different types of delayed neutral-type neural networks and the neural networks with time delays: (i) fixed-time stabilization of neutral-type inertial neural networks with discrete and distributed time-varying delays; (ii) stability criteria Hopfield neural networks of neutral-type with multiple delays.

Data Availability

No dara were used to support the findings of this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This paper was supported by the National Natural Science Foundation of China (no. 61305076).

References

H. Liu, D. Li, Z. Zuo, and Y. Zhong, “Robust attitude control for quadrotors with input time delays,” Control Engineering Practice, vol. 58, pp. 142–149, 2017.
View at: Publisher Site | Google Scholar
F. N. Koumboulis and N. D. Kouvakas, “Mobile robots in singular time-delay form—modeling and control,” Journal of the Franklin Institute, vol. 353, no. 1, pp. 160–179, 2016.
View at: Publisher Site | Google Scholar
J. Wang and M. Krstic, “Delay-compensated control of sandwiched ODE-PDE-ODE hyperbolic systems for oil drilling and disaster relief,” Automatica, vol. 120, Article ID 109131, 2020.
View at: Publisher Site | Google Scholar
Y. Zhang, S. Wang, D.-b. Pan, and G. Zhang, “Stability analysis for a new lattice hydrodynamic model with time-varying delay in sensing traffic flux,” Physica A: Statistical Mechanics and Its Applications, vol. 561, Article ID 125269, 2021.
View at: Publisher Site | Google Scholar
Q. Ye, R. Wang, Y. Cai, and M. Chadli, “The stability and accuracy analysis of automatic steering system with time delay,” ISA Transactions, vol. 104, pp. 278–286, 2020.
View at: Publisher Site | Google Scholar
G. Zhang, “Study on varying time delay on traffic stability in a novel lattice hydrodynamic model,” Physica A: Statistical Mechanics and Its Applications, vol. 505, pp. 1103–1112, 2018.
View at: Publisher Site | Google Scholar
J. Liu, W. Luo, X. Yang, and L. Wu, “Robust model-based fault diagnosis for pem fuel cell air-feed system,” IEEE Transactions on Industrial Electronics, vol. 63, no. 5, pp. 3261–3270, 2016.
View at: Publisher Site | Google Scholar
J. Liu, S. Laghrouche, and M. Wack, “Observer-based higher order sliding mode control of power factor in three-phase ac/dc converter for hybrid electric vehicle applications,” International Journal of Control, vol. 87, no. 6, pp. 1117–1130, 2014.
View at: Publisher Site | Google Scholar
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 II: Express Briefs, vol. 51, no. 12, pp. 685–689, 2004.
View at: Publisher Site | Google Scholar
W. Xiong and H. Jiang, “Research of population with impulsive perturbations based on dynamics of a neutral delay equation and ecological quality system,” Saudi Journal of Biological Sciences, vol. 23, no. 1, pp. S78–S82, 2016.
View at: Publisher Site | Google Scholar
F. Qiu, B. Cui, and Y. Ji, “Further results on robust stability of neutral system with mixed time-varying delays and nonlinear perturbations,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 895–906, 2010.
View at: Publisher Site | Google Scholar
K. Ramakrishnan and G. Ray, “Robust stability criteria for uncertain neutral systems with interval time-varying delay,” Journal of Optimization Theory and Applications, vol. 149, no. 2, pp. 366–384, 2011.
View at: Publisher Site | Google Scholar
J. Cheng, H. Zhu, S. Zhong, and G. Li, “Novel delay-dependent robust stability criteria for neutral systems with mixed time-varying delays and nonlinear perturbations,” Applied Mathematics and Computation, vol. 219, no. 14, pp. 7741–7753, 2013.
View at: Publisher Site | Google Scholar
R. Mohajerpoor, L. Shanmugam, H. Abdi, R. Rakkiyappan, S. Nahavandi, and J. H. Park, “Improved delay-dependent stability criteria for neutral systems with mixed interval time-varying delays and nonlinear disturbances,” Journal of the Franklin Institute, vol. 354, no. 2, pp. 1169–1194, 2017.
View at: Publisher Site | Google Scholar
P. Singkibud and K. Mukdasai, “On robust stability for uncertain neutral systems with non-differentiable interval time-varying discrete delay and nonlinear perturbations,” Asian-European Journal of Mathematics, vol. 11, no. 1, Article ID 1850007, 2018.
View at: Publisher Site | Google Scholar
C.-K. Zhang, Y. He, L. Jiang, and M. Wu, “Notes on stability of time-delay systems: bounding inequalities and augmented Lyapunov-Krasovskii functionals,” IEEE Transactions on Automatic Control, vol. 62, no. 10, pp. 5331–5336, 2017.
View at: Publisher Site | Google Scholar
F. Tahami, H. M. Ahmadian, and A. Moallem, “Robust control design for PFC rectifiers,” in Proceedings of the 7th International Conference on Power Electronics and Drive Systems, IEEE, Bangkok, Thailand, November 2007.
View at: Google Scholar
C. Dong, M. Zhao, D. H. Sun et al., “Robust control of cooperative driving system with external disturbances and communication delays in the vicinity of traffic signals,” Physica A: Statal Mechanics and Its Applications, vol. 542, Article ID 123385, 2020.
View at: Google Scholar
P. Tang and Y. Ma, “Exponential stabilization and sampled-date H_∞ control for uncertain T-S fuzzy systems with time-varying delay,” Journal of the Franklin Institute, vol. 356, no. 9, pp. 4859–4887, 2019.
View at: Publisher Site | Google Scholar
J. D. Chen, “LMI-based robust H_∞ control of uncertain neutral systems with state and input delays,” Journal of Optimization Theory and Applications, vol. 126, no. 3, pp. 553–570, 2005.
View at: Publisher Site | Google Scholar
J. Xie, Y. Kao, and J. H. Park, “ performance for neutral-type Markovian switching systems with general uncertain transition rates via sliding mode control method,” Nonlinear Analysis: Hybrid Systems, vol. 27, pp. 416–436, 2018.
View at: Publisher Site | Google Scholar
M. Hua, H. Tan, J. Chen, and J. Fei, “Robust delay-range-dependent non-fragile H_∞ filtering for uncertain neutral stochastic systems with Markovian switching and mode-dependent time delays,” Journal of the Franklin Institute, vol. 352, no. 3, pp. 1318–1341, 2015.
View at: Publisher Site | Google Scholar
A. Shariati, H. D. Taghirad, and B. Labibi, “Delay-dependent control linear systems with time-varying delay using proportional-derivative state feedback,” in Proceedings of the 18th Mediterranean Conference on Control Automation (MED), Marrakesh, Morocco, June 2010.
View at: Google Scholar
M. A. Gomez, W. Michiels, and S. Mondié, “Design of delay-based output-feedback controllers optimizing a quadratic cost function via the delay Lyapunov matrix,” Automatica, vol. 107, pp. 146–153, 2019.
View at: Publisher Site | Google Scholar
E. Delgado, M. Daz-Cacho, and A. Barreiro, “Stability of teleoperation systems for time-varying delays by neutral LMI techniques,” Mathematical Problems in Engineering, vol. 2012, Article ID 467629, 17 pages, 2012.
View at: Publisher Site | Google Scholar
J.-P. Richard, “Time-delay systems: an overview of some recent advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667–1694, 2003.
View at: Publisher Site | Google Scholar
P. F. Hokayem and M. W. Spong, “Bilateral teleoperation: an historical survey,” Automatica, vol. 42, no. 12, pp. 2035–2057, 2006.
View at: Publisher Site | Google Scholar
B. Zhang, A. Kruszewski, J.-P. Richard et al., “ robust control design for teleoperation systems,” IFAC Proceedings Volumes, vol. 45, no. 13, pp. 666–671, 2012.
View at: Publisher Site | Google Scholar
H. Du, “H_∞ state‐feedback control of bilateral teleoperation systems with asymmetric time‐varying delays,” IET Control Theory & Applications, vol. 7, no. 4, pp. 594–605, 2013.
View at: Publisher Site | Google Scholar
Y. Sun, N. Li, M. Shen, Z. Wei, and G. Sun, “Robust H_∞ control of uncertain linear system with interval time-varying delays by using Wirtinger inequality,” Applied Mathematics and Computation, vol. 335, pp. 1–11, 2018.
View at: Publisher Site | Google Scholar
T. Wang, T. Li, G. Zhang, and S. Fei, “Further triple integral approach to mixed-delay-dependent stability of time-delay neutral systems,” ISA Transactions, vol. 70, pp. 116–124, 2017.
View at: Publisher Site | Google Scholar
W. Chen, S. Xu, Y. Li, and Z. Zhang, “Stability analysis of neutral systems with mixed interval time-varying delays and nonlinear disturbances,” Journal of the Franklin Institute, vol. 357, no. 6, pp. 3721–3740, 2020.
View at: Publisher Site | Google Scholar
K. Ramakrishnan and G. Ray, “An improved delay-dependent robust stability criterion for uncertain neutral systems with time-varying delays,” Communications in Computer and Information Science, vol. 140, pp. 11–18, 2011.
View at: Publisher Site | Google Scholar
B. Boyd, L. E. Ghoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadephia, PA, USA, 1994.
M. S. Ali, J. Yogambigai, S. Saravanan, and S. Elakkia, “Stochastic stability of neutral-type Markovian-jumping BAM neural networks with time varying delays,” Journal of Computational and Applied Mathematics, vol. 349, pp. 142–156, 2019.
View at: Publisher Site | Google Scholar
H.-B. Zeng, X.-G. Liu, and W. Wang, “A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems,” Applied Mathematics and Computation, vol. 354, pp. 1–8, 2019.
View at: Publisher Site | Google Scholar
O. M. Kwon, J. H. Park, and S. M. Lee, “An improved delay-dependent criterion for asymptotic stability of uncertain dynamic systems with time-varying delays,” Journal of Optimization Theory and Applications, vol. 145, no. 2, pp. 343–353, 2010.
View at: Publisher Site | Google Scholar
S. Rajavel, R. Samidurai, R. Samidurai, S. A. J. Kilbert, J. Cao, and A. Alsaedi, “Non-fragile mixed H and passivity control for neural networks with successive time-varying delay components,” Nonlinear Analysis: Modelling and Control, vol. 23, no. 2, pp. 159–181, 2018.
View at: Publisher Site | Google Scholar
K.-W. Yu and C.-H. Lien, “Stability criteria for uncertain neutral systems with interval time-varying delays,” Chaos, Solitons & Fractals, vol. 38, no. 3, pp. 650–657, 2008.
View at: Publisher Site | Google Scholar
S. Y. Xu, J. Lam, and Y. Zou, “New results on delay-dependent robust control for systems with time-varying delay,” Automatica, vol. 42, pp. 343–348, 2006.
View at: Google Scholar
Z. Zhao, W. Wang, and B. Yang, “Delay and its time-derivative dependent robust stability of neutral control system,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1326–1332, 2007.
View at: Publisher Site | Google Scholar
Q.-L. Han and L. Yu, “Robust stability of linear neutral systems with nonlinear parameter perturbations,” IEE Proceedings—Control Theory and Applications, vol. 151, no. 5, pp. 539–546, 2004.
View at: Publisher Site | Google Scholar
L. Guo, H. Gu, and D. 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 Site | Google Scholar
K. Ramakrishnan and G. Ray, “An improved delay-dependent stability criterion for neutral systems with mixed time-delays and nonlinear perturbations,” in Proceedings of 2010 Annual IEEE India Conference, Kolkata, India, December 2010.
View at: Google Scholar
R. Rakkiappan, P. Balasubramanium, and R. Krishnaswamy, “Delay-dependent stability analysis of neutral systems with mixed time-varying delays and nonlinear perturbations,” Journal of Computational and Applied Mathematics, vol. 235, pp. 2147–2156, 2011.
View at: Google Scholar
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 Site | Google Scholar

Copyright

Copyright © 2021 Jingsha Zhang 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.

PDF Download Citation

Download other formats

Order printed copies

Views

211

Downloads

509

Citations