A new stability criterion and its application to robust stability analysis for linear systems with distributed delays
Recently, the possibility to verify positive definiteness of Lyapunov–Krasovskii functionals only on a specific Razumikhin-type set of functions to conclude on the stability of linear time delay systems was shown. In this paper, for a class of systems with multiple concentrated and distributed delay...
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| Published in | Automatica (Oxford) Vol. 152; p. 110973 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.06.2023
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| Abstract | Recently, the possibility to verify positive definiteness of Lyapunov–Krasovskii functionals only on a specific Razumikhin-type set of functions to conclude on the stability of linear time delay systems was shown. In this paper, for a class of systems with multiple concentrated and distributed delays, we extend the result and prove that this specific set may be applied while verifying the negative definiteness of the functionals derivatives along the solutions as well. The result is applied in the robustness analysis with respect to uncertainties both in the system matrices and in the delays. It provides a simple way to prove the robust stability conditions due to obviating the need to work with indefinite derivatives of functionals along the solutions of a perturbed system. As a by-product, the estimates for an unstable eigenvalue of the system are derived, and the functionals defined on complex-valued initial functions are presented. |
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| AbstractList | Recently, the possibility to verify positive definiteness of Lyapunov–Krasovskii functionals only on a specific Razumikhin-type set of functions to conclude on the stability of linear time delay systems was shown. In this paper, for a class of systems with multiple concentrated and distributed delays, we extend the result and prove that this specific set may be applied while verifying the negative definiteness of the functionals derivatives along the solutions as well. The result is applied in the robustness analysis with respect to uncertainties both in the system matrices and in the delays. It provides a simple way to prove the robust stability conditions due to obviating the need to work with indefinite derivatives of functionals along the solutions of a perturbed system. As a by-product, the estimates for an unstable eigenvalue of the system are derived, and the functionals defined on complex-valued initial functions are presented. |
| ArticleNumber | 110973 |
| Author | Kudryakov, Dmitry A. Alexandrova, Irina V. |
| Author_xml | – sequence: 1 givenname: Dmitry A. surname: Kudryakov fullname: Kudryakov, Dmitry A. email: st068497@student.spbu.ru, d.a.kudryakov@gmail.com – sequence: 2 givenname: Irina V. surname: Alexandrova fullname: Alexandrova, Irina V. email: i.v.aleksandrova@spbu.ru |
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| Cites_doi | 10.1016/j.automatica.2017.02.034 10.1109/9.28025 10.1016/j.automatica.2018.01.012 10.1016/j.sysconle.2013.05.010 10.1016/j.ifacol.2015.09.383 10.1016/j.sysconle.2004.05.016 10.1080/002071797223406 10.1016/0022-247X(89)90166-2 10.1002/rnc.5244 10.1080/00207179.2016.1269950 10.1002/rnc.4879 10.1134/S0012266115130017 10.1016/S0005-1098(02)00195-4 10.1016/S0005-1098(96)80010-0 10.1109/TAC.2015.2487478 10.1016/j.sysconle.2006.01.006 10.1016/j.automatica.2014.10.074 10.1016/j.ifacol.2015.09.385 10.1016/0167-6911(92)90100-7 10.1016/0022-0396(78)90051-7 10.1016/j.automatica.2019.06.027 10.1016/j.ifacol.2018.07.214 10.1109/TAC.2020.2995404 10.1109/TAC.2010.2067510 10.1080/00207179.2017.1390261 10.1109/TAC.2002.806665 10.1016/j.automatica.2013.09.042 |
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| Keywords | Linear systems Stability Robust stability Lyapunov matrices Lyapunov–Krasovskii functionals Time delay Razumikhin condition |
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fullname: Bellman – year: 1975 ident: 10.1016/j.automatica.2023.110973_b7 contributor: fullname: Desoer – volume: 53 start-page: 395 issue: 5 year: 2004 ident: 10.1016/j.automatica.2023.110973_b24 article-title: Exponential estimates for time delay systems publication-title: Systems & Control Letters doi: 10.1016/j.sysconle.2004.05.016 contributor: fullname: Kharitonov – volume: 68 start-page: 923 issue: 4 year: 1997 ident: 10.1016/j.automatica.2023.110973_b14 article-title: Discretized LMI set in the stability problem of linear uncertain time-delay systems publication-title: International Journal of Control doi: 10.1080/002071797223406 contributor: fullname: Gu – volume: 142 start-page: 83 year: 1989 ident: 10.1016/j.automatica.2023.110973_b15 article-title: Generalization of Liapunov’s theorem in a linear delay systems publication-title: Journal of Mathematical Analysis and Applications doi: 10.1016/0022-247X(89)90166-2 contributor: fullname: Huang – volume: 20 start-page: 315 issue: 3 year: 1956 ident: 10.1016/j.automatica.2023.110973_b27 article-title: On the application of the second Lyapunov method for equations with time delays [in Russian] publication-title: Prikladnaya Matematika i Mekhanika contributor: fullname: Krasovskii – volume: 30 start-page: 8299 issue: 18 year: 2020 ident: 10.1016/j.automatica.2023.110973_b18 article-title: Robust stability analysis for linear systems with distributed delays: A time-domain approach publication-title: International Journal of Robust and Nonlinear Control doi: 10.1002/rnc.5244 contributor: fullname: Juárez – volume: 91 start-page: 2588 issue: 11 year: 2018 ident: 10.1016/j.automatica.2023.110973_b9 article-title: Approximation of delay Lyapunov matrices publication-title: International Journal of Control doi: 10.1080/00207179.2016.1269950 contributor: fullname: Egorov – volume: 30 start-page: 2204 issue: 6 year: 2020 ident: 10.1016/j.automatica.2023.110973_b19 article-title: Dynamic predictor for systems with state and input delay: A time-domain robust stability analysis publication-title: International Journal of Robust and Nonlinear Control doi: 10.1002/rnc.4879 contributor: fullname: Juárez – volume: 8 start-page: 50 issue: 1 year: 2021 ident: 10.1016/j.automatica.2023.110973_b11 article-title: On the robust stability of linear systems with delay [in Russian] publication-title: Control Processes and Stability contributor: fullname: Evtina – volume: 29 start-page: 564 issue: 3 year: 1965 ident: 10.1016/j.automatica.2023.110973_b33 article-title: Quadratic Lyapunov functionals for systems with delay [in Russian] publication-title: Prikladnaya Matematika i Mekhanika contributor: fullname: Repin – volume: 51 start-page: 1675 issue: 13 year: 2015 ident: 10.1016/j.automatica.2023.110973_b23 article-title: Predictor-based controls: The implementation problem publication-title: Differential Equations doi: 10.1134/S0012266115130017 contributor: fullname: Kharitonov – volume: 39 start-page: 15 issue: 1 year: 2003 ident: 10.1016/j.automatica.2023.110973_b26 article-title: Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems publication-title: Automatica doi: 10.1016/S0005-1098(02)00195-4 contributor: fullname: Kharitonov – volume: 32 start-page: 1723 issue: 12 year: 1996 ident: 10.1016/j.automatica.2023.110973_b34 article-title: Further results on stability of ẋ(t)=Ax(t)+Bx(t−τ) publication-title: Automatica doi: 10.1016/S0005-1098(96)80010-0 contributor: fullname: Tissir – volume: 20 start-page: 500 issue: 4 year: 1956 ident: 10.1016/j.automatica.2023.110973_b32 article-title: On the stability of systems with delay [in Russian] publication-title: Prikladnaya Matematika i Mekhanika contributor: fullname: Razumikhin – volume: 61 start-page: 1987 year: 2016 ident: 10.1016/j.automatica.2023.110973_b6 article-title: Necessary stability conditions for delay systems with multiple pointwise and distributed delays publication-title: IEEE Transactions on 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| SubjectTerms | Linear systems Lyapunov matrices Lyapunov–Krasovskii functionals Razumikhin condition Robust stability Stability Time delay |
| Title | A new stability criterion and its application to robust stability analysis for linear systems with distributed delays |
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