×
Cheng, Daizhan

Stabilization of planar switched systems. (English) Zbl 1157.93482

Syst. Control Lett. 51, No. 2, 79-88 (2004).
Summary: This paper considers the problem of stabilization of single-input planar switched systems. We assume the switching law is observable, a formula is presented, which provides a necessary and sufficient condition for the system to be quadratically stabilizable. A set of linear inequalities are given to describe the set of all quadratic Lyapunov functions. The solvability and the control design technique are clearly described in a straightforward computation algorithm.

Cited in 49 Documents

MSC:

93D15 Stabilization of systems by feedback
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory

Keywords:

Switching system; Quadratic stabilization; Common quadratic Lyapunov function; Brunovskey canonical form
Cite Review PDF
Full Text: DOI

References:

[1] S. Boyd, L.E. Ghaoui, E. Feron, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Application Mathematics, SIAM, Philadelphia, 1994.; S. Boyd, L.E. Ghaoui, E. Feron, Linear Matrix Inequalities in System and Control Theory, SIAM Studies in Application Mathematics, SIAM, Philadelphia, 1994. · Zbl 0816.93004
[2] Branicky, M. S., Multiple Lyapunov functions and other analysis tools for switched hybrid systems, IEEE Trans. Automat. Control., 43, 4, 475-482 (1998) · Zbl 0904.93036
[3] Cheng, D.; Guo, L.; Huang, J., On quadratic Lyapunov functions, IEEE Trans. Automat. Control, 48, 5, 885-890 (2003) · Zbl 1364.93557
[4] Dayawansa, W. P.; Martin, C. F., A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE Trans. Automat. Control, 44, 4, 751-760 (1999) · Zbl 0960.93046
[5] Jeon, D.; Tomizuka, M., Learning hybrid force and position control of robot manipulators, IEEE Trans. Robotic. Automat., 9, 423-431 (1996)
[6] Ji, Y. D.; Chizeck, H. J., Jump linear quadratic Gaussian controlsteady-state solution and testable conditions, Control Theory Adv. Tech., 6, 3, 289-319 (1990)
[7] Liberzon, D.; Hespanha, J. P.; Morse, A. S., Stability of switched systemsa Lie-algebraic condition, Systems Control Lett., 37, 3, 117-122 (1999) · Zbl 0948.93048
[8] Liberzon, D.; Morse, A. S., Basic problems on stability and design of switched systems, IEEE Control Systems Mag., 34, 6, 1914-1946 (1999) · Zbl 1384.93064
[9] Ooba, T.; Funahashi, Y., Two conditions concerning common QLFs for linear systems, IEEE Trans. Automat. Control, 42, 5, 719-721 (1997) · Zbl 0892.93056
[10] Ooba, T.; Funahashi, Y., Stability robustness for linear state space models—a Lyapunov mapping approach, Systems Control Lett., 29, 191-196 (1997) · Zbl 0877.93087
[11] R.N. Shortenm, K.S. Narendra, A sufficient condition for the existence of a common Lyapunov function for two second-order linear systems, Proceedings of the 36th Conference on Decision and Control, San Diego, CA, 1997, pp. 3521-3522.; R.N. Shortenm, K.S. Narendra, A sufficient condition for the existence of a common Lyapunov function for two second-order linear systems, Proceedings of the 36th Conference on Decision and Control, San Diego, CA, 1997, pp. 3521-3522.
[12] R.N. Shortenm, K.S. Narendra, On the existence of a common Lyapunov function for linear stable switched systems, Proceedings of the 10th Yale Workshop on Adaptive and Learning Systems, Tampa, FL, 1998, pp. 130-140.; R.N. Shortenm, K.S. Narendra, On the existence of a common Lyapunov function for linear stable switched systems, Proceedings of the 10th Yale Workshop on Adaptive and Learning Systems, Tampa, FL, 1998, pp. 130-140.
[13] R.N. Shortenm, K.S. Narendra, Necessary and sufficient condition for the existence of a common quadratic Lyapunov function for M-stable linear second-order systems, Proceedings of 2000 ACC, Chicago, IL, 2000, pp.359-363.; R.N. Shortenm, K.S. Narendra, Necessary and sufficient condition for the existence of a common quadratic Lyapunov function for M-stable linear second-order systems, Proceedings of 2000 ACC, Chicago, IL, 2000, pp.359-363.
[14] Sira-Ranirez, H., Nonlinear P-I controller design for switch mode DC-to-DC power converters, IEEE Trans. Circuits Systems, 38, 4, 410-417 (1991)
[15] Sun, Z.; Zheng, D., On reachability and stabilization of switched linear systems, IEEE Trans. Automat. Control., 46, 2, 291-295 (2001) · Zbl 0992.93006
[16] Varaiya, P. P., Smart car on smart roadsproblems of control, IEEE Trans. Automat. Control., 38, 2, 195-207 (1993)
[17] Williams, S. M.; Hoft, R. G., Adaptive frequency domain control of PM switched power line conditioner, IEEE Trans. Power Electron., 6, 10, 665-670 (1991)
[18] Xue, F.; Guo, L., Necessary and Sufficient conditions for adaptive stabilizability of jump linear systems, Comm. Inform. Systems, 1, 2, 205-224 (2001) · Zbl 1059.93131
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.