Section
|
Mathematics
|
Title
|
On arbitrary spectrum assignment in linear stationary systems with commensurate time delays in state variables 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 commensurate delays in state
$$\dot x(t)=Ax(t)+\sum\limits_{j=1}^sA_jx(t-jh)+Bu(t),\quad y(t)=C^*x(t),\quad t>0. \qquad \qquad (1)$$
We construct a controller for the system $(1)$ as linear static output feedback $u(t)=\sum\limits_{\rho =0}^{\theta}Q_\rho y(t-\rho h)$. We study an arbitrary spectrum assignment problem for the closed-loop system. One needs to define a $\theta$ and to construct gain matrices $Q_{\rho}$, $\rho=0,\ldots,\theta$, such that the characteristic function of the closed-loop system with commensurate delays becomes a quasipolynomial with arbitrary preassigned coefficients. We obtain conditions on coefficients of the system $(1)$ under which the criterion is found for solvability of the problem of arbitrary spectrum assignment. Corollaries on stabilization by linear static output feedback with commensurate delays are obtained for the system $(1)$. An illustrative example is considered.
|
Keywords
|
linear delay systems, commensurate time delays, spectrum assignment problem, stabilization, static output feedback
|
UDC
|
517.929, 517.977
|
MSC
|
93B60, 93B55, 93B52, 93D20, 93C15, 93C05, 34H15
|
DOI
|
10.20537/vm170303
|
Received
|
20 April 2017
|
Language
|
Russian
|
Citation
|
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.
|
References
|
- Wang Q.-G., Lee T.H., Tan K.K. Finite spectrum assignment for time-delay systems, London: Springer, 1998, 124 p.
- Gu K., Niculescu S.-I. Survey on recent results in the stability and control of time-delay systems, Journal of Dynamic Systems, Measurement, and Control, 2003, vol. 125, issue 2, pp. 158-165. DOI: 10.1115/1.1569950
- Richard J.P. Time-delay systems: an overview of some recent advances and open problems, Automatica, 2003, vol. 39, issue 10, pp. 1667-1694. DOI: 10.1016/S0005-1098(03)00167-5
- Michiels W., Niculescu S.-I. Stability and stabilization of time-delay systems. An eigenvalue-based approach, Philadelphia: SIAM, 2007, 378 p.
- Sipahi R., Niculescu S.-I., Abdallah C.T., Michiels W., Gu K. Stability and stabilization of systems with time delay, IEEE Control Systems, 2011, vol. 31, issue 1, pp. 38-65. DOI: 10.1109/MCS.2010.939135
- Kamen E.W. Linear systems with commensurate time delays: stability and stabilization independent of delay, IEEE Transactions on Automatic Control, 1982, vol. 27, issue 2, pp. 367-375. DOI: 10.1109/TAC.1982.1102916
- Morse A.S. Ring models for delay-differential systems, Automatica, 1976, vol. 12, issue 5, pp. 529-531. DOI: 10.1016/0005-1098(76)90013-3
- Asmykovich I.K., Marchenko V.M. Spectrum control in systems with delay, Automation and Remote Control, 1976, vol. 37, no. 7, pp. 975-984.
- Olbrot A.W. Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays, IEEE Transactions on Automatic Control, 1978, vol. 23, issue 5, pp. 887-890. DOI: 10.1109/TAC.1978.1101879
- 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.
- Lee E.B., Zak S.H. On spectrum placement for linear time invariant delay systems, IEEE Transactions on Automatic Control, 1982, vol. 27, issue 2, pp. 446-449. DOI: 10.1109/TAC.1982.1102931
- Marchenko V.M. Modal control in systems with delay, Automation and Remote Control, 1988, vol. 49, no. 11, pp. 1449-1457.
- Manitius A., Olbrot A. Finite spectrum assignment problem for systems with delays, IEEE Transactions on Automatic Control, 1979, vol. 24, issue 4, pp. 541-552. DOI: 10.1109/TAC.1979.1102124
- Watanabe K., Ito M., Kaneko M., Ouchi T. Finite spectrum assignment problem for systems with delay in state variables, IEEE Transactions on Automatic Control, 1983, vol. 28, issue 4, pp. 506-508. DOI: 10.1109/TAC.1983.1103258
- Watanabe K., Ito M., Kaneko M. Finite spectrum assignment problem for systems with multiple commensurate delays in state variables, International Journal of Control, 1983, vol. 38, issue 5, pp. 913-926. DOI: 10.1080/00207178308933119
- Watanabe K., Ito M., Kaneko M. Finite spectrum assignment problem of systems with multiple commensurate delays in states and control, International Journal of Control, 1984, vol. 39, issue 5, pp. 1073-1082. DOI: 10.1080/00207178408933233
- 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. DOI: 10.1109/TAC.1986.1104336
- Watanabe K., Yamada K., Okuyama T., Takahashi K. A new algorithm for finite spectrum assignment of multivarible systems with time delays, Proceedings of the 34th IEEE Conference on Decision and Control, 1995, pp. 1489-1494. DOI: 10.1109/CDC.1995.480313
- 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. 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. 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
- Manitius A.Z., Manousiouthakis V. On spectral controllability of multi-input time-delay systems, Systems and Control Letters, 1985, vol. 6, issue 3, pp. 199-205. DOI: 10.1016/0167-6911(85)90041-6
- Spong M.W., Tarn T.J. On the spectral controllability of delay-differential equations, IEEE Transactions on Automatic Control, 1981, vol. 26, issue 2, pp. 527-528. DOI: 10.1109/TAC.1981.1102654
- Watanabe K., Ito M. A necessary condition for spectral controllability of delay systems on the basis of finite Laplace transforms, International Journal of Control, 1984, vol. 39, issue 2, pp. 363-374. DOI: 10.1080/00207178408933171
- Kharitonov V.L., Niculescu S.-I., Moreno J., Michiels W. Static output feedback stabilization: necessary conditions for multiple delay controllers, IEEE Transactions on Automatic Control, 2005, vol. 50, issue 1, pp. 82-86. DOI: 10.1109/TAC.2004.841137
- Bobtsov A.A. Output stabilization of nonlinear systems under delay conditions, Journal of Computer and Systems Sciences International, 2008, vol. 47, issue 2, pp. 179-186. DOI: 10.1134/S1064230708020020
- Bobtsov A.A., Kolyubin S.A., Pyrkin A.A. Stabilization of a nonlinear plant with input delay and sinusoidal perturbation, Automation and Remote Control, 2015, vol. 76, issue 1, pp. 16-23. DOI: 10.1134/S0005117915010026
- Zaitsev V.A., Kim I.G. Finite spectrum assignment problem in linear systems with state delay by static output feedback, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2016, vol. 26, issue 4, pp. 463-473. DOI: 10.20537/vm160402
- 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. Control of spectrum in bilinear systems, Differential Equations, 2010, vol. 46, issue 7, pp. 1071-1075. DOI: 10.1134/S0012266110070153
- 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
- Vaguina M.Yu., Kipnis M.M. Stability of the zero solution of delay differential equations, Mathematical Notes, 2003, vol. 74, issue 5-6, pp. 740-743. DOI: 10.1023/B:MATN.0000009007.19235.5a
|
Full text
|
|