Email this article CONSTRUCTING DIAGONAL LYAPUNOV–KRASOVSKII FUNCTIONALS FOR A CLASS OF POSITIVE DIFFERENTIAL-ALGEBRAIC SYSTEMS
- Authors: Aleksandrov A.Y.1
-
Affiliations:
- Saint Petersburg State University
- Issue: Vol 60, No 5 (2024)
- Pages: 579–589
- Section: ORDINARY DIFFERENTIAL EQUATIONS
- URL: https://journals.rcsi.science/0374-0641/article/view/259930
- DOI: https://doi.org/10.31857/S0374064124050013
- EDN: https://elibrary.ru/LBUEJV
- ID: 259930
Cite item
Open Access
Access granted
Subscription Access
Abstract
A coupled system describing the interaction of a differential subsystem with nonlinearities of a sector type and a linear difference subsystem is considered. It is assumed that the system is positive. A diagonal Lyapunov–Krasovskii functional is constructed and conditions are determined under which the absolute stability of the investigated system can be proved with the aid of such a functional. In the case of nolinearities of the power form, estimates for the convergence rate of solution to the origin are derived. The stability analysis of the corresponding system with parameter switching is fulfiled. Sufficient conditions guaranteeing the asymptotic stability of the zero solution for any admissible switching law are obtained.
About the authors
A. Yu. Aleksandrov
Saint Petersburg State University
Email: a.u.aleksandrov@spbu.ru
Russia
References
- Niculescu, S.-I. Delay Effects on Stability. A Robust Control Approach / S.-I. Niculescu. — Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milano ; Paris ; Singapur ; Tokyo : Springer, 2001. — 388 p.
- Fridman, E. Introduction to Time-Delay Systems: Analysis and Control / E. Fridman. — Basel : Birkh¨auser, 2014. — 362 p.
- Pepe, P. A new Lyapunov–Krasovskii methodology for coupled delay differential and difference equations / P. Pepe, Z.-P. Jiang, E. Fridman // Int. J. Control. — 2007. — V. 81, № 1. — P. 107–115.
- Rasvan, V. Oscillations in lossless propagation models: a Liapunov–Krasovskii approach / V. Rasvan, S.-I. Niculescu // IMA J. Math. Control Inform. — 2002. — V. 19. — P. 157–172.
- Gu, K. Lyapunov–Krasovskii functional for uniform stability of coupled differential-functional equations / K. Gu, Y. Liu // Automatica. — 2009. — V. 45. — P. 798–804.
- Метельский, А.В. Синтез регуляторов успокоения решения вполне регулярных дифференциально-алгебраических систем с запаздыванием / А.В. Метельский, В.Е. Хартовский // Дифференц. уравнения. — 2017. — Т. 53, № 4. — С. 547–558.
- Щеглова, А.А. Робастная устойчивость дифференциально-алгебраических уравнений произвольного индекса неразрешенности / А.А. Щеглова, А.Д. Кононов // Автоматика и телемеханика. — 2017. — № 5. — С. 36–55.
- Щеглова, А.А. К вопросу о сверхустойчивости интервального семейства дифференциальноалгебраических уравнений / А.А. Щеглова // Автоматика и телемеханика. — 2021. — № 2. — С. 55–70.
- Pepe, P. On the stability of coupled delay differential and continuous time difference equations / P. Pepe, E.I. Verriest // IEEE Trans. on Automatic Control. — 2003. — V. 48, № 8. — P. 1422–1427.
- Ngoc, P.H.A. Stability of coupled functional differential-difference equations / P.H.A. Ngoc // Int. J. Control. — 2020. — V. 93, № 8. — P. 1920–1930.
- Shen, J. Positivity and stability of coupled differential-difference equations with time-varying delays / J. Shen, W.X. Zheng // Automatica. — 2015. — V. 57. — P. 123–127.
- Briat, C. Stability and performance analysis of linear positive systems with delays using input–output methods / C. Briat // Int. J. Control. — 2017. — V. 91, № 7. — P. 1669–1692.
- Aleksandrov, A.Y. Absolute stability and Lyapunov–Krasovskii functionals for switched nonlinear systems with time-delay / A.Y. Aleksandrov, O. Mason // J. Franklin Institute. — 2014. — V. 351, № 8. — P. 4381–4394.
- Kazkurewicz, E. Matrix Diagonal Stability in Systems and Computation / E. Kazkurewicz, A. Bhaya. — Boston : Birkh¨auser, 1999. — 267 p.
- Mason, O. Diagonal Riccati stability and positive time-delay systems / O. Mason // Systems and Control Letters. — 2012. — V. 61. — P. 6–10.
- Aleksandrov A. Diagonal Riccati stability and applications / A. Aleksandrov, O. Mason // Linear Algebra and its Appl. — 2016. — V. 492. — P. 38–51.
- Александров, А.Ю. О диагональной устойчивости позитивных систем с переключениями и запаздыванием / А.Ю. Александров, О. Мейсон // Автоматика и телемеханика. — 2018. — № 12. — С. 16–33.
- Liberzon, D. Basic problems in stability and design of switched systems / D. Liberzon, A.S. Morse // IEEE Control Systems Magazine. — 1999. — V. 19, № 5. — P. 59–70.
- Pastravanu, O.C. Max-type copositive Lyapunov functions for switching positive linear systems / O.C. Pastravanu, M.-H. Matcovschi // Automatica. — 2014. — V. 50. — P. 3323–3327.
- Aleksandrov, A.Y. On the existence of a common Lyapunov function for a family of nonlinear positive systems / A.Y. Aleksandrov // Systems and Control Letters. — 2021. — V. 147. — Art. 1048324.
- Aleksandrov, A.Y. On the existence of diagonal Lyapunov–Krasovskii functionals for a class of nonlinear positive time-delay systems / A.Y. Aleksandrov // Automatica. — 2024. — V. 160. — Art. 111449.
