Synthesis of static output feedback control laws subject to feasibility the kimura condition
- Authors: Mukhin A.V.1
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Affiliations:
- Lobachevsky State University
- Issue: No 117 (2025)
- Pages: 188-199
- Section: Control systems analysis and design
- URL: https://journals.rcsi.science/1819-2440/article/view/360563
- DOI: https://doi.org/10.25728/ubs.2025.117.9
- ID: 360563
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Abstract
A lot of control tasks, such as, for example, the synthesis of static output feedback, are expressed in the form of bilinear matrix inequalities. Solving such tasks based on iterative algorithms attended with considerable time, especially in the case of large-scale systems. If no solution is found for the initial values, then repeating calculations with different initial values does not guarantee success. The reason is the non-convexity of the feasible sets. The article investigates the possibility of bilinear matrix inequalities reducing to linear matrix inequalities by replacing of Lyapunov function matrix arbitrary to block-diagonal matrix. A sufficient condition for such a replacing is Kimura condition feasibility. It is proved that the necessary conditions for the linear matrix inequality feasibility are satisfied by two linear non-generate transformations of the system basis. To the synthesis problem investigating within the framework of linear matrix inequalities computational experiments were performed, in which 1000 linear systems were randomly generated. Based on the results of computational experiments, a hypothesis is proposed according to which the Kimura condition is a sufficient condition for bilinear matrix inequalities reducing to linear matrix inequalities with nonempty feasible sets.
About the authors
Aleksey Valer'evich Mukhin
Lobachevsky State University
Email: muhin-aleksei@yandex.ru
Nizhny Novgorod
References
1. БАЛАНДИН Д.В., КОГАН М. М. Синтез регуляторов на основе решения линейных матричных неравенств и алго-ритма поиска взаимно-обратных матриц // Автоматика и телемеханика. – 2005. – №1. – С. 82–99.2. ШУМАФОВ М.М. Стабилизация линейных систем управления. Проблема назначения полюсов. Обзор // Вестник СПбГУ. Математика. Механика. Астрономия. – 2019. – Т. 6(64). – Вып. 4. – С. 564–591.3. ЧАЙКОВСКИЙ М.М. Синтез анизотропийных субо-птимальных регуляторов заданного порядка на основе полуопределенного программирования и алгоритма по-иска взаимообратных матриц // Управление большими системами. – 2012. – Вып. 39. – С. 95–137.4. APKARIAN P., TUAN H. D. Robust control via concave minimization local and global algorithms // IEEE Trans. Au-tomat. Control. – 2000. – Vol. 45, No. 2. – P. 299–305.5. EL GHAOUI L., GAHINET P. Rank minimization under LMI constraints: a framework for output feedback problems // Proc. Eur. Control Conf. Groningen. – 1993. – P. 1176–1179.6. ELIAS A. A Novel relaxation of the static output feedback problem for a class of plants // Automatica. – 2023. – No. 158. – P. 111285–111290.7. ELIAS A., POCCINI J., PAPACHRISTODOULOU A. Static Output Feedback for a Certain Class of Systems of Order Four // European Control Conf. (ECC). – 2024. – P. 2586–2592.8. EREMENKO A., GABRIELOV A. Pole placement by static output feedback for generic linear system // SIAM J. Control Optim. – 2002. – Vol. 41. – P. 303–312.9. GOH K.-C., SAFONOV M.G., PAPAVASSILOPOLUS G.P. A global optimization approach for the BMN problem // Proc. of the IEEE Conf. on Decision and Control. – IEEE Press, Piscataway, NJ. – 1994. – P. 2009–2014.10. HASSIBI A., HOW J., BOYD S. A path following method for solving BMI problems in control // Proc. of American Con-trol Conf. – 1999. – Vol. 2. – P. 1385–1389.11. HENRION D., LOEFBERG J., KOCVARA M. et al. Solving polynomial static output feedback problems with PENBMI // Proc. Joint IEEE Conf. Decision Control and Europ. Control Conf. – 2005. – P. 7581–7586.12. IWASAKI T. The dual iteration for fixed order control // IEEE Trans. Automat. Control. – 1999. – Vol. 44, No. 4. – P. 783–788.13. IWASAKI T., SKELTON R. E. The XY-centering algorithm for the dual LMI problem: a new approach to fixed order control design // Int. Journal of Control. – 1995. – Vol .62, No. 6. – P. 1257–1272.14. IWASAKI T., SKELTON R.E. Parameterization of all stabi-lizing controllers via quadratic Lyapunov functions // J. Op-tim. Theory Appl. – 1995. – Vol. 77. – P. 291–307.15. KIMURA H. On pole placement by gain output feedback // IEEE Trans. Automat. Control. –1975. – Vol. 20. – P. 509–519.16. MUSHTAQ T., SEILER P., HEMATI M. S. On the convexity of static output feedback control synthesis for systems with lossless nonlinearities // Automatica. – 2024. – No. 159. – P. 111380–111385.17. RODRIGUES L. From LQR to Static Output Feedback: a New LMI Approach // Proc. of IEEE 61st Conf. on Decision and Control. – 2022. – P. 4878–4883.18. RӦBENACK K., VOSWINKEL R., FRANKE MIRCO et al. Stabilization by static output feedback: a quantifier elimina-tion approach // Proc. Int. Conf. Syst. Theory, Control, Computing (ICSTCC-2018). – 2018. – P. 715–721.19. SADABADI M.S., PEAUCELLE D. From static output feed-back to structured robust static output feedback: A survey // Annual Reviews in Control. – 2016. –Vol. 42. –P. 11–26.20. TOKER O., OZBAY H. On the NP hardness of solving bilin-ear matrix inequalities and simultaneous stabilization with static output feedback // Proc. of the American Control Conf. – 1995. – Vol.4. – P. 2525–2526.
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