Further results on stability analysis of time-varying delay systems via novel integral inequalities and improved Lyapunov-Krasovskii functionals

Xingyue Liu; Kaibo Shi; Xingyue Liu; Kaibo Shi

doi:10.3934/math.2022108

AIMS Mathematics

2022, Volume 7, Issue 2: 1873-1895. doi: 10.3934/math.2022108

Previous Article Next Article

Research article

Further results on stability analysis of time-varying delay systems via novel integral inequalities and improved Lyapunov-Krasovskii functionals

Xingyue Liu ¹,
Kaibo Shi ^{1,2,3
,
,}

1.
School of Electronic Information and Electrical Engineering, Chengdu University, Chengdu, 610106, China
2.
Engineering Research Center of Power Quality of Ministry of Education, Anhui University, Anhui University, Hefei 230601, China
3.
Institute of Electronic and Information Engineering of University of Electronic Science and Technology of China in Guangdong, 523808, China

Received: 26 July 2021 Accepted: 28 October 2021 Published: 03 November 2021
MSC : 34D20, 34E05

This work develops some novel approaches to investigate the stability analysis issue of linear systems with time-varying delays. Compared with the existing results, we give three innovation points which can lead to less conservative stability results. Firstly, two novel integral inequalities are developed to deal with the single integral terms with delay-dependent matrix. Secondly, a novel Lyapunov-Krasovskii functional with time-varying delay dependent matrix, rather than constant matrix is constructed. Thirdly, two improved stability criteria are established by applying the newly developed Lyapunov-Krasovskii functional and integral inequalities. Finally, three numerical examples are presented to validate the superiority of the proposed method.
- linear system,
- time-varying delay,
- delay dependent matrix,
- Lyapunov-Krasovskii functional,
- integral inequalities
Citation: Xingyue Liu, Kaibo Shi. Further results on stability analysis of time-varying delay systems via novel integral inequalities and improved Lyapunov-Krasovskii functionals[J]. AIMS Mathematics, 2022, 7(2): 1873-1895. doi: 10.3934/math.2022108

Related Papers:

Abstract

This work develops some novel approaches to investigate the stability analysis issue of linear systems with time-varying delays. Compared with the existing results, we give three innovation points which can lead to less conservative stability results. Firstly, two novel integral inequalities are developed to deal with the single integral terms with delay-dependent matrix. Secondly, a novel Lyapunov-Krasovskii functional with time-varying delay dependent matrix, rather than constant matrix is constructed. Thirdly, two improved stability criteria are established by applying the newly developed Lyapunov-Krasovskii functional and integral inequalities. Finally, three numerical examples are presented to validate the superiority of the proposed method.

References

[1]	R. Abolpour, M. Dehghani, H. A. Talebi, Stability analysis of systems with time-varying delays using overlapped switching lyapunov krasovskii functional, J. Franklin Inst., 357 (2020), 10844–10860. doi: 10.1016/j.jfranklin.2020.08.018. doi: 10.1016/j.jfranklin.2020.08.018
[2]	T. H. Lee, J. H. Park, S. Y. Xu, Relaxed conditions for stability of time-varying delay systems, Automatica, 75 (2017), 11–15. doi: 10.1016/j.automatica.2016.08.011. doi: 10.1016/j.automatica.2016.08.011
[3]	L. Wu, X. Su, P. Shi, J. Qiu, A new approach to stability analysis and stabilization of discrete-time TS fuzzy time-varying delay systems, IEEE T. Syst. Man. Cy. B, 41 (2011), 273–286. doi: 10.1109/TSMCB.2010.2051541. doi: 10.1109/TSMCB.2010.2051541
[4]	X. Liao, G. Chen, E. N. Sanchez, Delay-dependent exponential stability analysis of delayed neural networks: An LMI approach, Neural Netw., 15 (2002), 855–866. doi: 10.1016/S0893-6080(02)00041-2. doi: 10.1016/S0893-6080(02)00041-2
[5]	E. Tian, C. Peng, Delay-dependent stability analysis and synthesis of uncertain T-S fuzzy systems with time-varying delay, Fuzzy Sets Syst., 157 (2006), 544–559. doi: 10.1016/j.fss.2005.06.022. doi: 10.1016/j.fss.2005.06.022
[6]	J. Chen, J. H. Park, S. Xu, Stablity analysis for delayed neural networks with an improved general free-matrix-based integral-inequality, IEEE T. Neur. Net. Lear., 31 (2019), 675–684. doi: 10.1109/TNNLS.2019.2909350. doi: 10.1109/TNNLS.2019.2909350
[7]	M. Wu, Y. He, J. H. She, G. P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems, Automatic, 40 (2004), 1435–1439. doi: 10.1016/j.automatica.2004.03.004. doi: 10.1016/j.automatica.2004.03.004
[8]	M. Liu, Y. He, M. Wu, J. Shen, Stability analysis of systems with two additive time-varying delay components via an improved delay interconnected Lyapunov-Krasovskii functional, J. Franklin. Inst., 356 (2019), 3457–3473. doi: 10.1016/j.jfranklin.2019.02.006. doi: 10.1016/j.jfranklin.2019.02.006
[9]	W. Qian, Y. Gao, Y. Chen, J. Yang, The stability analysis of time-varying delayed systems based on new augmented vector method, J. Franklin. Inst., 356 (2019), 1268–1286. doi: 10.1016/j.jfranklin.2018.10.027. doi: 10.1016/j.jfranklin.2018.10.027
[10]	B. Wu, C. Wang, A generalized multiple-integral inequality and its application on stability analysis for time-varying delay system, J. Franklin. Inst., 356 (2019), 4026–4042. doi: 10.1016/j.jfranklin.2019.02.003. doi: 10.1016/j.jfranklin.2019.02.003
[11]	K. Shi, Y. Tang, X. Liu, S. Zhong, Secondary delay-partition approach on robust performance analysis for uncertain time-varying Lurie nonlinear control system, Optim. Contr. Appl. Met., 38 (2017), 1208–1226. doi: 10.1002/oca.2326. doi: 10.1002/oca.2326
[12]	K. Shi, Y. Tang, X. Liu, S. Zhong, Non-fragile sampled-data robust synchronization of uncertain delayed chaotic lur'e systems with randomly occurring controller gain fluctuation, ISA Trans., 66 (2017), 185–199. doi: 10.1016/j.isatra.2016.11.002. doi: 10.1016/j.isatra.2016.11.002
[13]	K. Shi, X. Liu, H. Zhu, S. Zhong, Y. Liu, C. Yin, Novel integral inequality approach on master-slave synchronization of chaotic delayed lur'e systems with sampled-data feedback control, Nonlinear Dynam., 83 (2016), 1259–1274. doi: 10.1007/s11071-015-2401-x. doi: 10.1007/s11071-015-2401-x
[14]	K. Shi, Y. Tang, S. Zhong, C. Yin, X. Huang, W. Wang, Nonfragile asynchronous control for uncertain chaotic Lurie network systems with Bernoulli stochastic process, Int. J. Robust Nonlin., 28 (2018), 1693–1714. doi: 10.1002/rnc.3980. doi: 10.1002/rnc.3980
[15]	K. Shi, J. Wang, Y. Tang, S. Zhong, Reliable asynchronous sampled-data filtering of T-S fuzzy uncertain delayed neural networks with stochastic switched topologies, Fuzzy Set. Syst., 381 (2020), 1–25. doi: 10.1016/j.fss.2018.11.017. doi: 10.1016/j.fss.2018.11.017
[16]	D. X. Peng, X. D. Li, R. Rakkiyappan, Y. H. Ding, Stabilization of stochastic delayed systems: Event-triggered impulsive control, Appl. Math. Comput., 401 (2021), 126054. doi: 10.1016/j.amc.2021.126054. doi: 10.1016/j.amc.2021.126054
[17]	Y. S. Zhao, X. D. Li, J. D. Cao, Global exponential stability for impulsive systems with infinite distributed delay based on flexible impulse frequency, Appl. Math. Comput., 386 (2020), 125467. doi: 10.1016/j.amc.2020.125467. doi: 10.1016/j.amc.2020.125467
[18]	T. D. Wei, X. Xie, X. D. Li, Input-to-state stability of delayed reaction-diffusion neural networks with multiple impulses, AIMS Math., 6 (2021), 5786–5800. doi: 10.3934/math.2021342. doi: 10.3934/math.2021342
[19]	N. Olgac, R. Sipahi, An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems, IEEE T. Automat. Contr., 47 (2002), 793–797. doi: 10.1109/TAC.2002.1000275. doi: 10.1109/TAC.2002.1000275
[20]	O. M. Kwon, M. J. park, J. H. Park, S. M. Lee, E. J. Cha, Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality, J. Frankl. Inst., 351 (2014), 5386–5398. doi: 10.1016/j.jfranklin.2014.09.021. doi: 10.1016/j.jfranklin.2014.09.021
[21]	C. K. Zhang, Y. He, L. Jiang, M. Wu, Notes on stability of time-delay systems: bounding inequalities and augmented Lyapunov-Krasovskii functionals, IEEE T. Automat. Contr., 62 (2016), 5331–5336. doi: 10.1109/TAC.2016.2635381. doi: 10.1109/TAC.2016.2635381
[22]	T. H. Lee, J. H. Park, A novel Lyapunov functional for stability of time-varying delay systems via matrix-refined-function, Automatic, 80 (2017), 239–242. doi: 10.1016/j.automatica.2017.02.004. doi: 10.1016/j.automatica.2017.02.004
[23]	T. H. Lee, J. H. Park, Stability analysis of sampled-data systems via free-matrix-based time-dependent discontinuous Lyapunov approach, IEEE T. Automat. Contr., 62 (2017), 3653–3657. doi: 10.1109/TAC.2017.2670786. doi: 10.1109/TAC.2017.2670786
[24]	K. Gu, V. L. Kharitonov, J. Chen, Stability of time-delay systems, Birkhauser, 2003.
[25]	A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatic, 49 (2013), 2860–2866. doi: 10.1016/j.automatica.2013.05.030. doi: 10.1016/j.automatica.2013.05.030
[26]	H. B. Zeng, Y. He, M. Wu, J. She, Free-matrix-based integral inequality for stability analysis of systems with time-varying delay, IEEE T. Automat. Contr., 60 (2015), 2768–2772. doi: 10.1109/TAC.2015.2404271. doi: 10.1109/TAC.2015.2404271
[27]	P. G. Park, W. I. Lee, S. Y. Lee, Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Frankl. Inst., 352 (2015), 1378–1396. doi: 10.1016/j.jfranklin.2015.01.0041. doi: 10.1016/j.jfranklin.2015.01.0041
[28]	A. Seuret, F. Gouaisbaut, Hierarthy of LMI conditions for the stability analysis of time-delay systems, Syst. Control Lett., 81 (2015), 1–7. doi: 10.1016/j.sysconle.2015.03.007. doi: 10.1016/j.sysconle.2015.03.007
[29]	C. K. Zhang, Y. He, L. Jiang, M. Wu, H. B. Zeng, Stability analysis of systems with time-varying delay via relaxed integral inequalities, Syst. Control Lett., 92 (2016), 52–61. doi: 10.1016/j.sysconle.2016.03.002. doi: 10.1016/j.sysconle.2016.03.002
[30]	H. B. Zeng, Y. He, M. Wu, J. H. She, New results on stability analysis for systems with discrete distributed delay, Automatica, 60 (2015), 189–192. doi: 10.1016/j.automatica.2015.07.017. doi: 10.1016/j.automatica.2015.07.017
[31]	E. Gyurkovics, G. Szabo-Varga, K. Kiss, Stability analysis of linear systems with interval time-varying delays utilizing multiple integral inequalities, Appl. Math. Comput., 311 (2017), 164–177. doi: 10.1016/j.amc.2017.05.004. doi: 10.1016/j.amc.2017.05.004
[32]	E. Gyurkovics, T. Takacs, Multiple integral inequalities and stability analysis of time delay systems, Syst. Control Lett., 96 (2016), 72–80. doi: 10.1016/j.sysconle.2016.07.002. doi: 10.1016/j.sysconle.2016.07.002
[33]	M. J. Park, O. M. Kwon, J. H. Park, S. M. Lee, E. J. Cha, Stability of time-delay systems via wirtinger-based double integral inequality, Automatica, 55 (2015), 204–208. doi: 10.1016/j.automatica.2015.03.010. doi: 10.1016/j.automatica.2015.03.010
[34]	K. Liu, A. Seuret, Y. Xia, Stability analysis of systems with time-varying delays via the second-order Bessel-Legendre inequality, Automatica, 76 (2017), 138–142. doi: 10.1016/j.automatica.2016.11.001. doi: 10.1016/j.automatica.2016.11.001
[35]	J. H. Kim, Further improvement of Jensen inequality and application to stability of time-delayed systems, Automatica, 64 (2016), 121–125. doi: 10.1016/j.automatica.2015.08.025. doi: 10.1016/j.automatica.2015.08.025
[36]	A. Seuret, F. Gouaisbaut, Stability of linear systems with time-varying delays using Bessel-Legendre inequalities, IEEE Trans. Automat. Contr., 63 (2018), 225–232. doi: 10.1109/TAC.2017.2730485. doi: 10.1109/TAC.2017.2730485
[37]	H. B. Zeng, X. G. Liu, W. Wang, A generated free-matrix-based integral inequality for stability analysis of time-varying delay systems, Appl. Math. Comput., 354 (2019), 1–8. doi: 10.1016/j.amc.2019.02.009. doi: 10.1016/j.amc.2019.02.009
[38]	Y. He, Q. G. Wang, C. Lin, Delay-range-dependent stability for systems with time-varying delay, Automatica, 43 (2007), 371–376. doi: 10.1016/j.automatica.2006.08.015. doi: 10.1016/j.automatica.2006.08.015
[39]	N. Li, Y. H. Sun, Z. N. Wei, G. Q. Sun, Delay-dependent stability criteria for power system based on wirtinger integral inequality, Autom. Electr. Power Syst., 41 (2017), 108–113. doi: 10.7500/AEPS20160418002. doi: 10.7500/AEPS20160418002
[40]	P. Park, J. Ko, Stability and robust stability for systems with a time-varying delay, Automatica, 43 (2007), 1855–1858. doi: 10.1016/j.automatica.2007.02.022. doi: 10.1016/j.automatica.2007.02.022
[41]	J. H. Kim, Note on stability of linear systems with time-varying delay, Automatica, 47 (2011), 2118–2121. doi: 10.1016/j.automatica.2011.05.023. doi: 10.1016/j.automatica.2011.05.023
[42]	X. M. Zhang, Q. L. Han, A. Seuret, F. Gouaisbaut, An improved reciprocally convex inequality and an augmented Lyazpunov-Krasovskii functional for stability of linear systems with time-varying delay, Automatica, 84 (2017), 1–6. doi: 10.1016/j.automatica.2017.04.048. doi: 10.1016/j.automatica.2017.04.048

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)