×

zbMATH — the first resource for mathematics

Stability analysis of nonlinear quadratic systems via polyhedral Lyapunov functions. (English) Zbl 1219.93088
Summary: Quadratic systems play an important role in the modeling of a wide class of nonlinear processes (electrical, robotic, biological, etc.). For such systems it is mandatory not only to determine whether the origin of the state space is locally asymptotically stable, but also to ensure that the operative range is included into the convergence region of the equilibrium. Based on this observation, this paper considers the following problem: given the zero equilibrium point of a nonlinear quadratic system, assumed to be locally asymptotically stable, and a certain polytope in the state space containing the origin, determine whether this polytope belongs to the domain of attraction of the equilibrium. The proposed approach is based on polyhedral Lyapunov functions, rather than on the classical quadratic Lyapunov functions. An example shows that our methodology may return less conservative results than those obtainable with previous approaches.

MSC:
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
93C10 Nonlinear systems in control theory
93C20 Control/observation systems governed by partial differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Branch, M.A.; Coleman, T.; Grace, A., Optimization toolbox 3, user’s guide, (2007), The Mathworks, Inc. Natick, MA
[2] Amato, F., Cosentino, C., & Merola, A. (2006). On the region of asymptotic stability of nonlinear quadratic systems. In Proc. of the IEEE Mediterranean symposium in control and automation, Ancona, Italy. · Zbl 1138.93028
[3] Amato, F.; Cosentino, C.; Merola, A., On the region of attraction of nonlinear quadratic systems, Automatica, 43, 2119-2123, (2007) · Zbl 1138.93028
[4] Barber, C.B.; Dobkin, D.P.; Huhdanpaa, H.T., The quickhull algorithm for convex hulls, ACM transactions on mathematical software, 22, 469-483, (1996) · Zbl 0884.65145
[5] Bitsoris, G., & Athanasopoulos, A. (2008). Constrained stabilization of bilinear discrete-time systems using polyhedral Lyapunov functions. In Proc. of the 17th IFAC world congress, Seoul. · Zbl 1205.93093
[6] Blanchini, F., Constrained control for uncertain linear systems, Journal of optimization theory and applications, 71, 3, 465-483, (1991) · Zbl 0794.49015
[7] Blanchini, F., Nonquadratic Lyapunov functions for robust control, Automatica, 31, 3, 451-461, (1995) · Zbl 0825.93653
[8] Brayton, R.K.; Tong, C.H., Stability of dynamical systems: a constructive approach, IEEE transactions on automatic control, 26, 224-234, (1979) · Zbl 0413.93048
[9] Brayton, R.K.; Tong, C.H., Constructive stability and asymptotic stability of dynamical systems, IEEE transactions on circuits and systems, 27, 11, (1980), 1121-1130 · Zbl 0458.93047
[10] Chesi, G., Estimating the domain of attraction via union of continous families of Lyapunov estimates, Systems and control letters, 56, 4, (2007), 326-333 · Zbl 1109.37012
[11] Chesi, G., Estimating the domain of attraction for non-polynomial systems via LMI optimizations, Automatica, 45, 6, 1536-1541, (2009) · Zbl 1166.93355
[12] Chesi, G.; Garulli, A.; Tesi, A.; Vicino, A., LMI-based computation of optimal quadratic Lyapunov functions for odd polynomial systems, International journal of nonlinear and robust control, 15, 35-49, (2005) · Zbl 1056.93059
[13] Chesi, G., Tesi, A., Vicino, A., & Genesio, R. (1999). On convexification of some minimum distance problems. In Proc. of the ECC’99 European control conference, Karlsruhe, Germany. · Zbl 0934.93028
[14] Khalil, H.K., Nonlinear systems, (1992), MacMillan · Zbl 0626.34052
[15] Lorenz, F., Deterministic non-periodic flow, Journal of atmospheric science, 20, 130-141, (1963) · Zbl 1417.37129
[16] Merola, A.; Amato, F.; Cosentino, C., An insight into tumor dormancy equilibrium via the analysis of its domain of attraction, Biomedical signal processing and control, 3, 212-219, (2008)
[17] Molchanov, A.P.; Pyatnitskii, E.S., Lyapunov functions specifying necessary and sufficient conditions of absolute stability of nonlinear nonstationary control system, Automation and remote control, 47, (1986) · Zbl 0626.93051
[18] Murray, J.D., Mathematical biology, (2002), Springer New York
[19] Rockafellar, R.T., Convex analysis, (1972), Princeton University Press Princeton, NJ · Zbl 0224.49003
[20] Rössler, O.E., Chaotic behavior in simple reaction system, Zeitschrift für naturforsch A, 31, 259-264, (1976)
[21] Tesi, A.; Villoresi, F.; Genesio, R., On the stability domain estimation via a quadratic Lyapunov function: convexity and optimality properties for polynomial systems, IEEE transactions on automatic control, 41, 1650-1657, (1996) · Zbl 0870.34057
[22] Tibken, B. (2000). Estimation of the domain of attraction for polynomials systems via LMIs. In Proc. of the 39th IEEE international conference on decision and control, Sidney, Australia.
[23] Ziegler, G.M., Lectures on polytopes, (1998), Springer NY · Zbl 0898.52006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.