Reachability sets and the generalized h_2-norm of a linear discrete descriptor system
- Authors: Bubnova E.S.1
-
Affiliations:
- Lobachevsky State University of Nizhny Novgorod
- Issue: No 103 (2023)
- Pages: 78-93
- Section: Mathematical control theory
- URL: https://medbiosci.ru/1819-2440/article/view/363797
- DOI: https://doi.org/10.25728/ubs.2023.103.3
- ID: 363797
Cite item
Full Text
Abstract
The paper focuses on a linear discrete noncausal descriptor system on a finite horizon under consistent initial conditions and bounded external disturbances, i.e. a bounded l_2 norm. The notion of the generalized H_2-norm for a linear discrete descriptor system is introduced as the induced norm of the linear operator generated by the system under consideration. This norm is characterized in terms of difference projected generalized Lyapunov equation solutions. It is demonstrated that if the sum of the quadratic forms of the initial and final states and the sum of the quadratic forms of the disturbance over a finite time interval is bounded by a given value from above, the reachability set of this system is a time-varying ellipsoid whose matrix satisfies the difference projected generalized Lyapunov equation. It is established that the generalized H_2-norm of the system under non-zero initial conditions coincides with the value of the maximum half-axis of the reachability ellipsoidal set for a given output of the system. An example of a fourth-order descriptor system is provided as an illustration of the results. For this system a generalized H_2-norm is calculated and reachability sets are constructed. The paper demonstrates the results of numerical simulations and projections of reachability sets on the plane corresponding to the forward and backward subsystems.
About the authors
Elena Sergeevna Bubnova
Lobachevsky State University of Nizhny Novgorod
Author for correspondence.
Email: bubnova@itmm.unn.ru
Nizhny Novgorod
References
- Баландин Д. В., Бирюков Р. С., Коган М. М. Минимаксное управление уклонениями выходов линейной дискретной нестационарной системы // Автоматика и телемеханика. – 2019. – № 12. – С. 3–23.
- Баландин Д. В., Бирюков Р. С., Коган М. М. Оптимальное управление максимальными уклонениями выходов линейной нестационарной системы // Автоматика и телемеханика. – 2019. – № 10. – С. 37–61.
- Баландин Д. В., Бирюков Р. С., Коган М. М. Эллипсоидальные множества достижимости линейных нестационарных систем в задачах управления и оценивания // Дифференциальные уравнения. – 2019. – Т. 55, № 11. – С. 1485–1498.
- Баландин Д. В., Коган М. М. Управление и оценивание в линейных нестационарных системах на основе эллипсоидальных множеств достижимости // Автоматика и телемеханика. – 2020. – № 8. – С. 8–28.
- Белов А. А., Андрианова О. Г. Синтез субоптимальных анизотропийных регуляторов по состоянию для дескрипторных систем на основе линейных матричных неравенств // Автоматика и телемеханика. – 2016. – № 10. – С. 40–56.
- Белов А. А., Курдюков А. П. Дескрипторные системы и задачи управления. – М.: Физматлит, 2015. – 270 с.
- Бубнова Е. С., Бирюков Р. С. Обобщённая ℋ2-норма дескрипторных систем // Тезисы научной конференции «Летняя школа робототехники в Сириусе–2022». – С. 11–12.
- Andrianova O. G., Belov A. A. Robust Anisotropy-Based Control for Uncertain Descriptor Systems with Transient Response Constraints // IFAC-PapersOnLine. – 2018. – Vol. 51, No. 32. – P. 515–520.
- Belov A. A., Andrianova O. G. Robust state-feedback H∞ control for discrete-time descriptor systems with norm-bounded parametric uncertainties // Int. Journal of Systems Science. – 2019. – Vol. 50, No. 6. – P. 1303–1312.
- Cao Y., Feng Z., Liu Y. Real-time Reachable Set Estimation of Discrete-time Singular Systems // Int. Conf. on Information, Cybernetics, and Computational Social Systems. – 2021. – P. 127–130.
- Duan G. Analysis and Design of Descriptor Linear Systems. – Springer, 2010.
- Feng Z., Lam J. On reachable set estimation of singular systems // Automatica. – 2015. – Vol. 52. – P. 146–153.
- Feng Y., Yagoubi M. On State Feedback H∞ Control for Discrete-Time Singular Systems // IEEE Trans. on Automatic Control. – 2013. – Vol. 58, No. 10. – P. 2674–2679.
- Ishihara J. Y., Terra M. H., Sales R. M. The full information and state feedback H2 optimal controllers for descriptor systems // Automatica. – 2003. – Vol. 39. – P. 391–402.
- Lee L., Chen J. L. Strictly positive real lemma and absolute stability for discrete-time descriptor systems // IEEE Trans. on Circuits and Systems I: Fundamental Theory and Applications. – 2003. – Vol. 50, No. 6. – P. 788–794.
- Li J., Feng Z., Zhang C. Reachable Set Estimation for Discrete-Time Singular Systems // Asian Journal of Control. – 2017. – Vol. 19. – P. 1862–1870.
- Stykel T. Analysis and numerical solution of generalized Lyapunov equations: Ph.D. thesis, Institut für Mathematik, Technische Universität Berlin. – Berlin, 2002.
- Stykel T. On some norms for descriptor systems // IEEE Trans. on Automatic Control. – 2006. – Vol. 51, No. 5. – P. 842–847.
- Wilson D. A. Convolution and Hankel Operator Norms for Linear Systems // IEEE Trans. Autom. Control. – 1989. – Vol. 34. – P. 94–97.
- Zhang Z., Feng Z. Enclosing ellipsoid-based reachable set estimation for discrete-time singular systems // Int. Journal of Robust and Nonlinear Control. – 2022. – Vol. 32. – P. 9294–9306.
- Zhao Y., Liu Y., Ma Y. Robust finite-time sliding mode control for discrete-time singular system with time-varying delays // Journal of the Franklin Institute. – 2021. – Vol. 358. – P. 4848–4863.
Supplementary files
