Yang, Haifeng; Xie, Guangming; Chu, Tianguang; Wang, Long Commuting and stable feedback design for switched linear systems. (English) Zbl 1090.93029 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 64, No. 2, 197-216 (2006). Summary: Commuting and stable feedback design for switched linear systems is investigated. This problem is formulated as to build up suitable state feedback controller for each subsystem such that the closed-loop systems are not only asymptotically stable but also commuting each other. A new concept, common admissible eigenvector set (CAES), is introduced to establish necessary/sufficient conditions for commuting and stable feedback controllers. For second-order systems, a necessary and sufficient condition is established. Moreover, a parametrization of the CAES is also obtained. The motivation comes from stabilization of switched linear systems which consist of a family of LTI systems and a switching law specifying the switching between them, where if all the subsystems are stable and commuting each other, then the total system is stable under arbitrary switching. Cited in 5 Documents MSC: 93C57 Sampled-data control/observation systems 93B52 Feedback control 93D15 Stabilization of systems by feedback Keywords:Switched linear systems; Commuting and stable feedback design; Common admissible eigen-vector set PDFBibTeX XMLCite \textit{H. Yang} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 64, No. 2, 197--216 (2006; Zbl 1090.93029) Full Text: DOI References: [1] Beldiman, O.; Bushnell, L., Stability, linearization and control of switched systems, (Proceedings of 1999 American Control Conference (1999)), 2950-2954 [2] Branicky, M. S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Autom. Control, 43, 475-482 (1998) · Zbl 0904.93036 [3] C.-T. Chen, Linear system theory and design, Holt Rhinehart And Winston, New York, 1984, pp. 214-217.; C.-T. Chen, Linear system theory and design, Holt Rhinehart And Winston, New York, 1984, pp. 214-217. [4] Daafouz, J.; Millerioux, G.; Iung, C., About poly-quadratic stability and switched systems, (Proceedings of 2002 IFAC World Congress (2002)), 1-6 [5] Daafouz, J.; Riedinger, P.; Iung, C., Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach, IEEE Trans. Autom. Control, 47, 11, 1883-1887 (2002) · Zbl 1364.93559 [6] Ezzine, J.; Haddad, A. H., Controllability and observability of hybrid systems, Int. J. Control, 49, 2045-2055 (1989) · Zbl 0683.93011 [7] Fang, L.; Lin, H.; Antsaklis, P. J., Stabilization and performance analysis for a class of switched systems, (Proceedings of 43rd IEEE Conference Decision Control (2004)), 3265-3270 [8] Ge, S. S.; Sun, Z.; Lee, T. H., Reachability and controllability of switched linear discrete-time systems, IEEE Trans. Autom. Control, 46, 9, 1437-1441 (2001) · Zbl 1031.93028 [9] Hespanha, J. P.; Morse, A. S., Stability of switched systems with average dwell-time, (Proceedings of 38th Conference Decision Control (1999)), 2655-2660 · Zbl 0108.23602 [10] Horn, R. A.; Johnson, C. R., Matrix Analysis (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0576.15001 [11] Hu, B.; Xu, X.; Antsaklis, P. J.; Michel, A. N., Robust stabilizing control laws for a class of second-order switched systems, Syst. Control Lett., 38, 197-207 (1999) · Zbl 0948.93013 [12] Kaileth, T., Linear Systems (1980), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ [13] Li, Z.; Wen, C. Y.; Soh, Y. C., Stabilization of a class of switched systems via designing switching laws, IEEE Trans. Autom. Control, 46, 665-670 (2001) · Zbl 1001.93065 [14] Liberzon, D.; Hespanha, HJ. P.; Morse, A. S., Stability of switched systems: a Lie-algebraic condition, Syst. Control Lett., 37, 117-122 (1999) · Zbl 0948.93048 [15] Liberzon, D.; Morse, A. S., Basic problems in stability and design of switched systems, IEEE Control Syst. Mag., 19, 59-70 (1999) · Zbl 1384.93064 [16] Narendra, K. S.; Balakrishnan, J., A common Lyapunov function for stable Lti systems with commuting A-matrices, IEEE Trans. Autom. Control, 39, 2469-2471 (1994) · Zbl 0825.93668 [17] Peleties, P.; Decarlo, R. A., Asymptotic stability of M-switched systems using Lyapunov-like functions, (Proceedings of 1991 American Control Conference (1991)), 1679-1684 [18] Petterson, S.; Lennartson, B., Stability and robustness for hybrid systems, (Proceedings of 35th IEEE Conference Decision Control (1996)), 1202-1207 [19] Schinkel, M.; Wang, Y.; Hunt, K., Stable and robust state feedback design for hybrid systems, (Proceedings of 2000 American Control Conference (2000)), 215-219 [20] Shorten, R. N.; Narendra, K. S., On the stability and existence of common Lyapunov functions for stable linear switching systems, (Proceedings of 37th Conference Decision Control (1998)), 3723-3724 [21] Sun, Z.; Ge, S. S.; Lee, T. H., Controllability and reachability criteria for switched linear systems, Automatica, 38, 5, 775-786 (2002) · Zbl 1031.93041 [22] Sun, Z.; Zheng, D., On stabilization of switched linear control systems, IEEE AC, 46, 2, 291-295 (2001) · Zbl 0992.93006 [23] Wicks, M. A.; Peleties, P.; Decarlo, R. A., Switched controller synthesis for the quadratic stabilization of a pair of unstable linear systems, Eur. J. Control, 4, 140-147 (1998) · Zbl 0910.93062 [24] Wonham, W. M., Linear Multivariable Control: A Geometric Approach (1985), Springer: Springer New York · Zbl 0393.93024 [25] Xie, G.; Wang, L., Necessary and sufficient conditions for controllability of switched linear systems, (Proceedings of 2002 American Control Conference (2002)), 1897-1902 · Zbl 0531.93011 [26] Xie, G.; Wang, L., Controllability and stabilizability of switched linear-systems, Syst. Control Lett., 48, 2, 135-155 (2002) · Zbl 1134.93403 [27] Xie, G.; Wang, L.; Wang, Y., Controllability of periodically switched linear systems with delay in control, (Proceedings of Fifteenth International Symposium on the Mathematical Theory of Networks and Systems, University of Notre Dame (2002)), 1-15 [28] Xie, G.; Wang, L., Reachability realization and stabilizability of switched linear discrete-time systems, J. Math. Anal. Appl., 280, 2, 209-220 (2003) · Zbl 1080.93015 [29] Xie, G.; Zheng, D.; Wang, L., Controllability of switched linear systems, IEEE AC, 47, 8, 1401-1405 (2002) · Zbl 1364.93075 [30] Xu, X.; Antsaklis, P. J., Design of stabiling control laws for second-order switched systems, (Proceedings of 14th IFAC World Congress C (1999)), 181-186 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.