Abstract
In this paper, we address the stabilization problem for linear periodically time-varying switched systems. Using discretization technique, we derive new conditions for the global stabilizability in terms of the solution of matrix inequalities. An algorithm for finding stabilizing controller and switching strategy is presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A.S. Matveev and A. V. Savkin. Qualitative Theory of Hybrid Dynamical Systems[M]. Boston: Birkhauser, 2000.
A.S. Morse, ed., Control Using Logic-Based Switching[M]. London: Springer-Verlag, 1997.
Savkin A. V. and R J. Evans. Hybrid Dynamical Systems: Controller and Sensor Switching Problems[M]. Boston: Birkhauser, 2002.
D. Liberzon and A.S Morse. Basic problems in stability and design of switched systems[J]. IEEE Control Systems Magazine, 1999,19(5): 59–69.
A. Baddou and A. Benzaouia. Dynamic improvement for linear constrained control continuous-time systems with a state observer[J]. J. of Dynamical and Control Systems, 2001,7(1): 1–14.
Phat V.N.. Robust stability and stabilizability of uncertain linear hybrid systems with state delays[J]. IEEE Trans. on Circuits and Systems II: Analog and Digital Signal Processing, 2005,52(2): 94–98.
Wicks M.A., P. Peleties and R.A. DeCarlo. Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems[J]. European J. of Control, 1998,4(2): 140–147.
S. Boyd, L.El. Ghaoui, E. Feron and V. Balakrishnan. Linear Matrix Inequalities in Systems and Control Theory[M]. Philadelphia: SIAM, 1994.
F. Callier F. and C.A. Desoer. Linear System Theory[M]. Hongkong: Springer-Verlag, 1992.
Shorten R.N and K.S. Narendra. On the stability and existence of common Lyapunov functions for stable switching systems[C] // Proc. of the 37th IEEE Conf. on Decision and Control. Piscataway, New Jercy: IEEE Press, 1998,12: 3723–3724.
Sun H. and J. Zhao. Control Lyapunov functions for switched control systems[C] // Proc. of American Control Conf., Piscataway, NJ: IEEE Press, 2001: 1890–1891.
S. Bittanti, P. Colaneri and G. Guardabassi. Periodic solutions of periodic Riccati equations[J]. IEEE Trans. on Automatic Control, 1984,29(7): 665–667.
M.S. Chen and Y.Z. Chen. Static output feedback control for periodically time-varying systems[J]. IEEE Trans. on Automatic Control, 1999,44(2): 218–222.
Phat V.N., Stabilization of linear continuous time-varying systems with state delays in Hilbert spaces[J]. Electronic J. of Differential Equations, 2001, 2001(67): 1–13.
M.A. Hammami. On the stability of nonlinear control systems with uncertainty[J]. J. of of Dynamical and Control Systems, 2001,7(2): 171–179.
H. Kano and T. Nishimura. Controllability, stabilizability and matrix Riccati equations for periodic systems[J]. IEEE Trans. on Automatic Control, 1985,30(4): 1129–1131.
Phat V.N.. New stabilization criteria for linear time-varying systems with state delays and norm-bounded uncertainties[J]. IEEE Trans. on Automatic Control, 2002,47(12): 2095–2098.
Son N.K. and Su N.V.. Linear periodic control systems: controllability with restrained controls[J]. Applied Mathematics and Optimization, 1986,14(2): 173–185.
Phat V.N.. Constrained Control Problems of Discrete Processes[M]. Singapore: World Scientific Publishing Co. Pte. Ltd., 1996.
Uhlig F.. A recurring theorem about pairs of quadratic forms and extensions[J]. Linear Algebra and Its Applications, 1979,25(2): 219–237.
A.J. Laub. Schur techniques for solving Riccati equations[J]. IEEE Trans. Automatic Control, 1979,24(6): 913–921.
Wonham W.M.. Linear Multivariable Control: A Geometric Approach[M]. Berlin: Springer-Verlag, 1979.
Author information
Authors and Affiliations
Additional information
This work was supported by the Basic Program in Natural Sciences, Vietnam and Thai Research Fund Grant, Thailand
Phat VU N. received the B.S. and Ph.D degrees in Mathematics at the former USSR Azerbaidzan State University in 1975 and 1984, respectively. In 1995 he received D.Sc. in Mathematics at the Institute of Mathematics, Polish Academy of Science, Poland. He has held research positions at the Pusan National University, South Korea in 1998–1999 and University of New South Wales, Department of EE&T in 2001–2003. He is currently Professor of Institute of Mathematics, Hanoi, Vietnam. He was an Associate Editor of Optimization, Acta Mathematica Vietnamica and is currently Associate Editor of Nonlinear Functional Analysis and Applications, Advances in Nonlinear Variational Inequalities and a member of the technical committee of Nonlinear Control, IFAC. His research interests include control and stability problems, qualitative theory of differential and functional equations, variational inequalities, and optimization techniques.
Satiracoo PAIROTE received the B.S. degree in Mathematics at the Mahidol University, Bangkok, Thailand in 1998; M.S. and Ph.D. degrees in Mathematics at the University of Warwick, England in 1999 and 2003, respectively. He is currently a lecturer at the Department of Mathematics, Mahidol University, Bangkok, Thailand. His research interests include linear control theory, qualitative theory of differential/difference equations, bifurcation problems of nonlinear systems.
Rights and permissions
About this article
Cite this article
Phat, V.N., Pairote, S. Global stabilization of linear periodically time-varying switched systems via matrix inequalities. J. Control Theory Appl. 4, 26–31 (2006). https://doi.org/10.1007/s11768-006-5261-6
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s11768-006-5261-6