Necessary and sufficient stability condition for second-order switched systems: a phase function approach. (English) Zbl 1416.93158
Summary: To find a unified approach for the stability analysis of second-order switched system, the concept of phase function is proposed in this paper. First, the basic properties of phase function are explored. Following this concept and its properties, the phase-based stability criterion is investigated based on the Lyapunov theory, and a necessary and sufficient stability condition is obtained in the phase function approach. Moreover, the connection between phase-based stability conditions and algebraic condition of system matrices is also discussed. Finally, numerical examples are provided to exemplify the main result and make necessary comparisons with the existing methods.
MSC:
| 93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
References:
| [1] | Blanchini, F., Set invariance in control, Automatica,, 35, 11, 1747-1767 (1999) · Zbl 0935.93005 |
| [2] | Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A., Homogeneous Lyapunov functions for systems with structured uncertainties, Automatica,, 39, 6, 1027-1035 (2003) · Zbl 1079.93036 |
| [3] | Chesi, G.; Colaneri, P.; Geromel, J. C.; Middleton, R.; Shorten, R., A nonconservative LMI condition for stability of switched systems with guaranteed dwell time, IEEE Transactions on Automatic Control,, 5, 5, 1297-1302 (2012) · Zbl 1369.93524 |
| [4] | Godbehere, A. B.; Sastry, S. S., Stabilization of planar switched linear systems using polar coordinates, Proceedings of the 13th ACM international conference on hybrid systems: Computation and control, Stockholm, Sweden, 283-292 (2010), New York, NY: Association for Computing Machinery (ACM), New York, NY · Zbl 1360.93599 |
| [5] | Greco, L.; Tocchini, F.; Innocenti, M., A geometry-based algorithm for the stability of planar switching systems, International Journal of Systems Science,, 37, 11, 747-761 (2006) · Zbl 1136.34006 |
| [6] | Huang, Z. H.; Xiang, C.; Lin, H.; Lee, T. H., Necessary and sufficient conditions for regional stabilisability of generic switched linear systems with a pair of planar subsystems, International Journal of Control,, 83, 4, 694-715 (2010) · Zbl 1209.93133 |
| [7] | Khalil, H. K., Nonlinear Systems (2002), Upper Saddle River, NJ: Prentice-Hall, Upper Saddle River, NJ · Zbl 1003.34002 |
| [8] | Lin, H.; Antsaklis, P. J., Stability and stabilization of switched linear systems: A survey of recent results, IEEE Transactions on Automatic Control,, 54, 2, 308-322 (2009) · Zbl 1367.93440 |
| [9] | Lin, H.; Antsaklis, P. J., Hybrid dynamical systems: An introduction to control and verification, Foundations and Trends®in Systems and Control,, 1, 1, 1-172 (2014) |
| [10] | Margaliot, M.; Langholz, G., Necessary and sufficient condition for absolute stability: The case of second-order systems, IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications,, 50, 2, 227-234 (2003) · Zbl 1368.93601 |
| [11] | Naunheimer, H.; Bertsche, B.; Ryborz, J.; Novak, W., Automative transmissions: Fundamentals, selection, design and application (2011), Berlin: Springer-Verlag, Berlin |
| [12] | Polanski, A., Lyapunov function construction by linear programming, IEEE Transactions on Automatic Control,, 42, 7, 1013-1016 (1997) · Zbl 0885.93050 |
| [13] | Rhee, B.-J.; Won, S., A new fuzzy Lyapunov function approach for a Takagi-Sugeno fuzzy control system design, Fuzzy Sets and Systems,, 157, 1211-1228 (2006) · Zbl 1090.93025 |
| [14] | Serra, G. L. O., Frontiers in advanced control systems (2012), Rijeka: InTech, Rijeka |
| [15] | Shorten, R.; Narendra, K., Necessary and sufficient conditions for the existence of a common quadratic Lyapunov function for a finite number of stable second order linear time-invariant systems, International Journal of Adaptive Control and Signal Processing,, 16, 10, 709-728 (2002) · Zbl 1027.93037 |
| [16] | Tanaka, K.; Yoshida, H.; Ohtake, H.; Wang, H., A sum-of square approach to modeling and control of nonlinear dynamical systems with polynomial fuzzy systems, IEEE Transactions on Fuzzy Systems,, 17, 4, 911-922 (2009) |
| [17] | Wertz, J. R., Spacecraft attitude determination and control (1978), Dordrecht, Netherlands: Springer, Dordrecht, Netherlands |
| [18] | Wikipedia Contributors, Atan2.” Wikipedia, The Free Encyclopedia, Wikipedia, The Free Encyclopedia, (2016) |
| [19] | Xie, L.; Shishkin, S.; Fu, M., Piecewise Lyapunov functions for robust stability of linear time-varying systems, Systems & Control Letters,, 31, 3, 165-171 (1997) · Zbl 0901.93063 |
| [20] | Yang, Y.; Xiang, C.; Lee, T. H., Sufficient and necessary conditions for the stability of second-order switched linear systems under arbitrary switching, International Journal of Control,, 85, 12, 1977-1995 (2012) · Zbl 1253.93102 |
| [21] | Yang, Y.; Xiang, C.; Lee, T. H., Necessary and sufficient conditions for regional stabilisability of second-order switched linear systems with a finite number of subsystems, Automatica,, 50, 3, 931-939 (2014) · Zbl 1298.93297 |
| [22] | Yfoulis, C.; Shorten, R., A numerical technique for the stability analysis of linear switched systems, International Journal of Control,, 77, 11, 1019-1040 (2004) · Zbl 1069.93034 |
| [23] | Zelentsovsky, A. L., Nonquadratic Lyapunov functions for robust stability analysis of linear uncertain systems, IEEE Transactions on Automatic Control,, 39, 1, 135-138 (1994) · Zbl 0796.93101 |
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.
