×

Conjugate Lyapunov functions for saturated linear systems. (English) Zbl 1125.93439

Summary: Based on a recent duality theory for linear differential inclusions (LDIs), the condition for stability of an LDI in terms of one Lyapunov function can be easily derived from that in terms of its conjugate function. This paper uses a particular pair of conjugate functions, the convex hull of quadratics and the maximum of quadratics, for the purpose of estimating the domain of attraction for systems with saturation nonlinearities. To this end, the nonlinear system is locally transformed into a parametertized LDI system with an effective approach which enables optimization on the parameter of the LDI along with the optimization of the Lyapunov functions. The optimization problems are derived for both the convex hull and the max functions, and the domain of attraction is estimated with both the convex hull of ellipsoids and the intersection of ellipsoids. A numerical example demonstrates the effectiveness of this paper’s methods.

MSC:

93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
34A60 Ordinary differential inclusions
34D20 Stability of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aizerman, M. A.; Gantmacher, F. R., Absolute stability of regulator systems (1964), Holden-Day: Holden-Day San Francisco, CA · Zbl 0123.28401
[2] Barabanov, N.E., 1995. Stability of inclusions of linear type. Proceedings of American control conference (pp. 3366-3370).; Barabanov, N.E., 1995. Stability of inclusions of linear type. Proceedings of American control conference (pp. 3366-3370).
[3] Blanchini, F., Nonquadratic Lyapunov functions for robust control, Automatica, 31, 451-461 (1995) · Zbl 0825.93653
[4] Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V., Linear matrix inequalities in systems and control theory (1994), SIAM Studies in Applied Mathematics: SIAM Studies in Applied Mathematics Philadelphia, PA: SIAM · Zbl 0816.93004
[5] Brayton, R. K.; Tong, C. H., Stability of dynamical systems: a constructive approach, IEEE Transactions on Circuits and Systems, 26, 224-234 (1979) · Zbl 0413.93048
[6] Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A., Homogeneous Lyapunov functions for systems with structured uncertainties, Automatica, 39, 1027-1035 (2003) · Zbl 1079.93036
[7] Dayawansa, W. P.; Martin, C. F., A converse Lyapunov theorem for a class of dynamical systems which undergo switching, IEEE Transactions on Automatic Control, 44, 751-760 (1999) · Zbl 0960.93046
[8] Goebel, R., Hu, T., Teel, A. R., 2005. Dual matrix inequalities in stability and performance analysis of linear differential/difference inclusions. In Current trends in nonlinear systems and control. Basel: Birkhauser, to appear.; Goebel, R., Hu, T., Teel, A. R., 2005. Dual matrix inequalities in stability and performance analysis of linear differential/difference inclusions. In Current trends in nonlinear systems and control. Basel: Birkhauser, to appear.
[9] Goebel, R., Teel, A. R., Hu, T., Lin, Z., 2005. Conjugate convex Lyapunov functions for dual linear differential inclusions. IEEE Transactions on Automatic Control, to appear.; Goebel, R., Teel, A. R., Hu, T., Lin, Z., 2005. Conjugate convex Lyapunov functions for dual linear differential inclusions. IEEE Transactions on Automatic Control, to appear. · Zbl 1366.34030
[10] Hassibi, A., How, J., Boyd, S., 1999. A path-following method for solving BMI problems in control. Proceedings of American control conference (pp. 1385-1389).; Hassibi, A., How, J., Boyd, S., 1999. A path-following method for solving BMI problems in control. Proceedings of American control conference (pp. 1385-1389).
[11] Hu, T.; Lin, Z., Composite quadratic Lyapunov functions for constrained control systems, IEEE Transactions on Automatic Control, 48, 440-450 (2003) · Zbl 1364.93108
[12] Hu, T.; Lin, Z.; Chen, B. M., An analysis and design method for linear systems subject to actuator saturation and disturbance, Automatica, 38, 351-359 (2002) · Zbl 0991.93044
[13] Hu, T.; Lin, Z.; Chen, B. M., Analysis and design for linear discrete-time systems subject to actuator saturation, Systems & Control Letters, 45, 97-112 (2002) · Zbl 0987.93027
[14] Jarvis-Wloszek, Z., Packard, A. K., 2002. An LMI method to demonstrate simultaneous stability using non-quadratic polynomial Lyapunov functions. Proceedings of the IEEE conference on decision and control (pp. 287-292), Las Vegas, NV.; Jarvis-Wloszek, Z., Packard, A. K., 2002. An LMI method to demonstrate simultaneous stability using non-quadratic polynomial Lyapunov functions. Proceedings of the IEEE conference on decision and control (pp. 287-292), Las Vegas, NV.
[15] Johansson, M.; Rantzer, A., Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Transactions on Automatic Control, 43, 555-559 (1998) · Zbl 0905.93039
[16] Molchanov, A. P., Criteria of asymptotic stability of differential and difference inclusions encountered in control theory, Systems & Control Letters, 13, 59-64 (1989) · Zbl 0684.93065
[17] Narendra, K. S.; Taylor, J., Frequency domain methods for absolute stability (1973), Academic Press: Academic Press New York
[18] Power, H. M.; Tsoi, A. C., Improving the predictions of the circle criterion by combining quadratic forms, IEEE Transactions on Automatic Control, 28, 65-67 (1973) · Zbl 0264.93013
[19] Rockafellar, R. T., Convex analysis (1970), Princeton University Press: Princeton University Press Princeton, NJ · Zbl 0229.90020
[20] Xie, L.; Shishkin, S.; Fu, M., Piecewise Lyapunov functions for robust stability of linear time-varying systems, Systems & Control Letters, 31, 165-171 (1997) · Zbl 0901.93063
[21] Zelentsovsky, A. L., Non-quadratic Lyapunov functions for robust stability analysis of linear uncertain systems, IEEE Transactions on Automatic Control, 39, 135-138 (1995) · 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.