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## Archive of Issues

Russia Izhevsk
Year
2016
Volume
26
Issue
4
Pages
463-473
 Section Mathematics Title Finite spectrum assignment problem in linear systems with state delay by static output feedback Author(-s) Zaitsev V.A.a, Kim I.G.a Affiliations Udmurt State Universitya Abstract We consider a control system defined by a linear time-invariant system of differential equations with delay $$\dot x(t)=Ax(t)+A_1x(t-h)+Bu(t),\quad y(t)=C^*x(t),\quad t>0. \qquad\qquad (1)$$ We construct the controller for the system $(1)$ as linear output feedback $u(t)=Q_0 y(t)+Q_1 y(t-h)$. We study a finite spectrum assignment problem for the closed-loop system. One needs to construct gain matrices $Q_0$, $Q_1$ such that the characteristic quasipolynomial of the closed-loop system becomes a polynomial with arbitrary preassigned coefficients. We obtain conditions on coefficients of the system $(1)$ under which the criterion was found for solvability of the finite spectrum assignment problem. The obtained result extends to systems with several delays. Corollaries on stabilization by linear static output feedback with delay are obtained for system $(1)$ as well as for systems of type $(1)$ with several delays. Keywords linear delay systems, spectrum assignment, stabilization, output feedback UDC 517.929, 517.977 MSC 93B60, 93B55, 93B52, 93D20, 93C15, 93C05, 34H15 DOI 10.20537/vm160402 Received 1 October 2016 Language Russian Citation 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. References Krasovskii N.N., Osipov Yu.S. On the stabilization of motions of a plant with delay in a control system, Izv. Akad. Nauk SSSR, Tekh. Kibern., 1963, no. 6, pp. 3-15 (in Russian). Osipov Yu.S. Stabilization of control systems with delays, Differ. Uravn., 1965, vol. 1, no. 5, pp. 605-618 (in Russian). Bulatov V.I., Kalyuzhnaya T.S., Naumovich R.F. Control over spectrum of differential equations, Differ. Uravn., 1974, vol. 10, no. 11, pp. 1946-1952 (in Russian). Asmykovich I.K., Marchenko, V.M. Spectrum control in systems with delay, Automation and Remote Control, 1976, vol. 37, no. 7, pp. 975-984. Gabelaya A.G., Ivanenko V.I., Odarich O.N. Stabilizability of autonomous linear systems with a delay, Automation and Remote Control, 1976, vol. 37, no. 8, pp. 1145-1150. Asmykovich I.K., Marchenko V.M. Modal control of multiinput linear delayed systems, Automation and Remote Control, 1980, vol. 41, no. 1, pp. 1-5. Marchenko V.M. Modal control in systems with delay, Automation and Remote Control, 1988, vol. 49, no. 11, pp. 1449-1457. Khartovskii V.E., Pavlovskaya A.T. To the problem of modal control for linear systems of neutral type, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 4, pp. 146-155. DOI: 10.20537/vm130414 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. DOI: 10.1109/TAC.1979.1102124 Mondie S., Michiels W. Finite spectrum assignment of unstable time-delay systems with a safe implementation, IEEE Transactions on Automatic Control, 2003, vol. 48, issue 12, pp. 2207-2212. DOI: 10.1109/TAC.2003.820147 Metel’skii A.V. Complete damping of a linear autonomous differential-difference system by a controller of the same type, Differential Equations, 2012, vol. 48, issue 9, pp. 1219-1235. DOI: 10.1134/S0012266112090030 Metel’skii A.V. Finite spectrum assignment problem for a delay type system, Differential Equations, 2014, vol. 50, issue 5, pp. 689-699. DOI: 10.1134/S0012266114050115 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. DOI: 10.1134/S0012266115010073 Metel’skii A.V. Feedback control of the spectrum of differential-difference system, Automation and Remote Control, 2015, vol. 76, issue 4, pp. 560-572. DOI: 10.1134/S0005117915040025 Metel’skii A.V. Finite spectrum assignment and complete damping of a differential system of the neutral type by a single controller, Differential Equations, 2016, vol. 52, issue 1, pp. 92-110. DOI: 10.1134/S0012266116010080 Zaitsev V.A. Spectrum control in linear systems with incomplete feedback, Differential Equations, 2009, vol. 45, issue 9, pp. 1348-1357. DOI: 10.1134/S0012266109090109 Zaitsev V.A. Necessary and sufficient conditions in a spectrum control problem, Differential Equations, 2010, vol. 46, issue 12, pp. 1789-1793. DOI: 10.1134/S0012266110120128 Zaitsev V.A. Consistent systems and pole assignment: I, Differential Equations, 2012, vol. 48, issue 1, pp. 120-135. DOI: 10.1134/S001226611110120 Full text