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## Archive of Issues

Russia Izhevsk
Year
2020
Volume
30
Issue
2
Pages
208-220
 Section Mathematics Title Spectrum assignment and stabilization of linear differential equations with delay by static output feedback with delay Author(-s) Zaitsev V.A.a, Kim I.G.a Affiliations Udmurt State Universitya Abstract A linear control system defined by a stationary differential equation with one lumped and one distributed delay is considered. In the system, the input is a linear combination of $m$ variables and their derivatives of order not more than $n-p$ and the output is a $k$-dimensional vector of linear combinations of the state and its derivatives of order not more than $p-1$. For this system, a spectrum assignment problem by linear static output feedback with delays is studied. Necessary and sufficient conditions are obtained for solvability of the arbitrary spectrum assignment problem by static output feedback controller of the same type as the system. Corollaries on stabilization of the system are obtained. Keywords linear differential equation, lumped delay, distributed delay, spectrum assignment, stabilization, static output feedback UDC 517.929, 517.977 MSC 93B55, 93B52, 93D20, 93C15, 93C05, 34H15 DOI 10.35634/vm200205 Received 1 May 2020 Language English Citation Zaitsev V.A., Kim I.G. Spectrum assignment and stabilization of linear differential equations with delay by static output feedback with delay, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 2, pp. 208-220. References Kharitonov V.L. Time-delay systems. Lyapunov functionals and matrices, Basel: Birkhäuser, 2013. https://doi.org/10.1007/978-0-8176-8367-2 Feng Q. Stability analysis and stabilization of linear systems with distributed delays, Doctoral Theses, University of Auckland, 2019. http://hdl.handle.net/2292/47377 Gouaisbaut F. Stability and stabilization of distributed time delay systems, Proceedings of the 44th IEEE Conference on Decision and Control, 2005. https://doi.org/10.1109/cdc.2005.1582351 Choon K.A. Stabilization of linear systems with distributed input delay using reduction transformation, Chinese Science Bulletin, 2011, vol. 56, no. 13, pp. 1413-1416. https://doi.org/10.1007/s11434-010-4152-x Goebel G., Munz U., Allgower F. Stabilization of linear systems with distributed input delay, Proceedings of the 2010 American Control Conference, 2010. https://doi.org/10.1109/acc.2010.5530430 Dolgii Y.F. Stabilization of linear autonomous systems of differential equations with distributed delay, Automation and Remote Control, 2007, vol. 68, no. 10, pp. 1813-1825. https://doi.org/10.1134/s0005117907100098 Moulay E., Dambrine M., Yeganefar N., Perruquetti W. Finite-time stability and stabilization of time-delay systems, Systems and Control Letters, 2008, vol. 57, no. 7, pp. 561-566. https://doi.org/10.1016/j.sysconle.2007.12.002 Zhou B. Stabilization of linear systems with multiple and distributed input delays, Truncated Predictor Feedback for Time-Delay Systems, Berlin-Heidelberg: Springer, 2014, pp. 45-80. https://doi.org/10.1007/978-3-642-54206-0_3 Michiels W., Niculescu S.-I. Stability and stabilization of time-delay systems. An eigenvalue-based approach, Philadelphia: SIAM, 2007. https://doi.org/10.1137/1.9780898718645 Manitius A.Z., Olbrot A.W. Finite spectrum assignment problem for systems with delays, IEEE Transactions on Automatic Control, 1979, vol. 24, issue 4, pp. 541-553. https://doi.org/10.1109/tac.1979.1102124 Watanabe K. Finite spectrum assignment and observer for multivarible systems with commensurate delay, IEEE Transactions on Automatic Control, 1986, vol. 31, issue 6, pp. 543-550. https://doi.org/10.1109/tac.1986.1104336 Wang Q.-G., Lee T.H., Tan K.K. Finite spectrum assignment for time-delay systems, London: Springer, 1998. https://doi.org/10.1007/978-1-84628-531-8 Metel'skii A.V. Finite spectrum assignment problem for a differential system of neutral type, Differential Equations, 2015, vol. 51, issue 1, pp. 69-82. https://doi.org/10.1134/s0012266115010073 Zaitsev V.A., Kim I.G. Finite spectrum assignment problem in linear systems with state delay by static output feedback, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 4, pp. 463-473. https://doi.org/10.20537/vm160402 Zaitsev V.A., Kim I.G. On finite spectrum assignment problem in bilinear systems with state delay, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 1, pp. 19-28. https://doi.org/10.20537/vm190102 Zaitsev V.A., Kim I.G., Khartovskii V.E. Finite spectrum assignment problem for bilinear systems with several delays, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 3, pp. 319-331. https://doi.org/10.20537/vm190303 Metel'skii A.V. Spectral reducibility of delay differential systems by a dynamic controller, Differential Equations, 2011, vol. 47, issue 11, pp. 1642-1659. https://doi.org/10.1134/s0012266111110115 Khartovskii V.E. Spectral reduction of linear systems of the neutral type, Differential Equations, 2017, vol. 53, issue 3, pp. 366-382. https://doi.org/10.1134/s0012266117030089 Borkovskaya I.M., Marchenko V.M. Modal control of systems with distributed delays, Automation and Remote Control, 1993, vol. 54, no. 8, pp. 1211-1222. https://zbmath.org/?q=an:0841.93024 Marchenko V.M., Borkovskaya I.M. Modal control of a system with distributed delay under the condition of incomplete information, Differential Equations, 1993, vol. 29, no. 11, pp. 1673-1680. https://zbmath.org/?q=an:0815.93065 Marchenko V.M. Control of systems with aftereffect in scales of linear controllers with respect to the type of feedback, Differential Equations, 2011, vol. 47, issue 7, pp. 1014-1028. https://doi.org/10.1134/s0012266111070111 Metel'skii A.V., Khartovskii V.E. Criteria for modal controllability of linear systems of neutral type, Differential Equations, 2016, vol. 52, issue 11, pp. 1453-1468. https://doi.org/10.1134/s0012266116110070 Khartovskii V.E. Criteria for modal controllability of completely regular differential-algebraic systems with aftereffect, Differential Equations, 2018, vol. 54, issue 4, pp. 509-524. https://doi.org/10.1134/s0012266118040080 Zaitsev V.A., Kim I.G. On arbitrary spectrum assignment in linear stationary systems with commensurate time delays in state variables by static output feedback, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 3, pp. 315-325. https://doi.org/10.20537/vm170303 Kim I.G., Zaitsev V.A. Spectrum assignment by static output feedback for linear systems with time delays in states, 2018 14th International Conference “Stability and Oscillations of Nonlinear Control Systems” (Pyatnitskiy's Conference) (STAB), 2018, pp. 1-4. https://doi.org/10.1109/stab.2018.8408365 Zaitsev V.A. Modal control of a linear differential equation with incomplete feedback, Differential Equations, 2003, vol. 39, no. 1, pp. 145-148. https://doi.org/10.1023/A:1025188512610 Zaitsev V., Kim I. Exponential stabilization of linear time-varying differential equations with uncertain coefficients by linear stationary feedback, Mathematics, 2020, vol. 8, issue 5, article 853. https://doi.org/10.3390/math8050853 Zaitsev V.A., Kim I.G. Arbitrary spectrum assignment by static output feedback for linear differential equations with state variable delays, IFAC-PapersOnLine, 2018, vol. 51, issue 32, pp. 810-814. https://doi.org/10.1016/j.ifacol.2018.11.446 Full text