Stabilizability Conditions for Switched Linear Systems
with Constant Input via Switched Observer
Takuya SOGA1, Naohisa OTSUKA2
1 Graduate School of Advanced Science and Technology, Tokyo Denki University,
Hatoyama-Machi, Hiki-Gun, Saitama, 350-0394, Japan
2 Division of Science, School of Science and Engineering, Tokyo Denki University,
Hatoyama-Machi, Hiki-Gun, Saitama, 350-0394, Japan
Abstract: In this paper, stabilizability conditions for switched linear systems with constant input via two types of switched rule which depends on the state of switched observer are presented. The obtained results provide stabilizability conditions via state feedback switched rule. Further, two illustrative numerical examples are also given.
Keywords: Switched Systems, Switched Observer, Stabilizability, Constant Input
CITE THIS PAPER AS:
Takuya SOGA, Naohisa OTSUKA, Stabilizability Conditions for Switched Linear Systems with Constant Input via Switched Observer, Studies in Informatics and Control, ISSN 1220-1766, vol. 22 (1), pp. 7-14, 2013. https://doi.org/10.24846/v22i1y201301
Switched system is one of the so-called hybrid systems which consist of a family of subsystems and a switching rule among them. The aspect of the switched system is found in various fields such as aircraft industry, mobile robot, animal world and Ethernet etc, . Further, the idea of switching has also been used to design intelligent control which is based on the switching between different controllers. An important problem in such switched systems is the stability problem with arbitrary switching and the stabilization problem via appropriate switching rule. Until now many results on stability and stabilization problems for various types of switched linear systems without input have been studied (e.g., ,,, , -).
In addition, it is also important to consider the case which contains the control input for practical applications. In particular, Deaecto et al.  gave some conditions for some equilibrium point to be globally asymptotically stable via state feedback switched rule. The conditions are related to continuous-time switched linear system with constant input. The results were applied to DC-DC converters control design. However, the same problems via switched observer which contains information of the outputs instead of the state for the switched systems have not been investigated.
The objective of this paper is to study conditions under which equilibrium points are globally asymptotically stable via the switched observer. The conditions are related to continuous-time switched linear systems with constant input. In Section 2 the main results of this paper are given. In Section 3 two illustrative numerical examples are shown. Finally, concluding remarks are given in Section 4.
- BRANICKY, M. S., Stability of Switched and Hybrid Systems, Proceedings of the 33rd IEEE Conference on Decision and Control, 1994, pp. 3498-3503.
- BRANICKY, M. S., Multiple Lyapunov Functions and Other Analysis Tools for Switched and Hybrid Systems, IEEE Transitions on Automatic Control, vol.43, no. 4, 1998, pp. 475-482.
- DAYAWANSA, W. P., C. F. MARTIN, A Converse Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching, IEEE Transitions on Automatic Control, vol. 44, no. 4, 1999, pp. 751-760.
- DEAECTO, G. S., J. C. GEROMEL, F. S. GARCIA, J. A. POMILIO, Switched Affine Systems Control Design with Application to DC-DC Converters, IET Control Theory and Applications, vol. 4, no. 7, 2010, pp. 1201-1210.
- FERON, E., Quadratic Stabilizibility of Switched Systems Via State and Output Feedback, Massachusetts Institute of Technology Technical report CICSP-468, 1996, pp. 1-13.
- GEROMEL, J. C., P. COLANERI, P. BOLZERN, Dynamic Output Feedback Control of Switched Linear Systems, IEEE Transcriptions on Automatic Control, vol. 53, no. 3, 2008, pp.720-733.
- KIM, M. K., L. SHAN, EDF-based Real-time Message Scheduling of Periodic Messages on a Master-Slave-based Synchronized Switched Ethernet, International Journal of Control and Automation, vol. 2, no. 4, 2009, pp. 25-34.
- LIBERZON, D., Basic Problems in Stability and Design of Switched Systems, IEEE Control Systems Magazine, vol. 19, 1999, pp. 59-70.
- LIBERZON, D., Switching in Systems and Control, Systems & Control: Foundation & Applications, Birkhäuser, 2003.
- LIN, H., P. J. ANTSAKLIS, A Necessary and Sufficient Condition for Robust Asymptotic Stabilizability of Continuous-time Uncertain Switched Linear Systems, Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, pp. 3690-3695.
- OTSUKA, N., T. SOGA, Quadratic Stabilizability for Polytopic Uncertain Continuous-time Switched Linear Systems Composed of Two Subsystems, International Journal of Control and Automation, vol. 3, no. 1, 2010, pp. 35-42.
- SAVKIN, A. V., R. J. EVANS, Hybrid Dynamical Systems: Controller and Sensor Switching Problems, Birkhäuser, 2002.
- SOGA, T., N. OTSUKA, Quadratic Stabilizability for Polytopic Uncertain Continuous-time Switched Linear Systems by Output Feedback, Proceedings of the 2010 American Control Conference, 2010, pp. 3920-3925.
- SOGA, T., N. OTSUKA, Quadratic Stabilizability for Polytopic Uncertain Continuous-time Switched Linear Systems via Switched Observer, Proceedings of the 19th Mediterranean Conference on Control and Automation, 2011, pp. 724-729.
- SUN, Z. D., S. S. GE, Stability Theory of Switched Dynamical Systems, Springer-Verlag, 2011.
- WANG, Y., Z. ZUO, Onquadratic Stabilizability of Linear Switched Systems with Polytopic Uncertainties, Proceedings of the 2005 IEEE International Conference on Systems, Man and Cybernetics, 2005, pp. 1640-1644.
- WICKS, M., P. PELETIES, R. DECARLO, Construction of Piecewise Lyapunov Functions for Stabilizing Switched Systems, Proceedings of the 33rd IEEE Conference on Decision and Control, 1994, pp. 3492-3497.
- WICKS, M., P. PELETIES, R. DECARLO, Switched Controller Synthesis for the Quadratic Stabilization of a Pair of Unstable Linear Systems, European Journal of Control, vol.4, no.2, 1998, pp. 140-147.
- ZHAI, G., Quadratic Stablizibility of Discrete-time Switched Systems Via State and Output Feedback, Proceedings of the 40th IEEE Conference on Decision and Control, 2001, pp. 2165-2166.
- ZHAI, G., H. LIN, P. J. ANTSAKLIS, Quadratic Stabilizability of Switched Systems with Polytopic Uncertainties, International Journal of Control, vol. 76, no. 7, 2003, pp. 747-753.