TY - JOUR
T1 - Necessary and sufficient conditions for absolute stability
T2 - The case of second-order systems
AU - Margaliot, Michael
AU - Langholz, Gideon
PY - 2003/2
Y1 - 2003/2
N2 - We consider the problem of absolute stability of linear feedback systems in which the control is a sector-bounded time-varying nonlinearity. Absolute stability entails not only the characterization of the "most destabilizing" nonlinearity, but also determining the parametric value of the nonlinearity that yields instability of the feedback system. The problem was first formulated in the 1940s, however, finding easily verifiable necessary and sufficient conditions for absolute stability remained an open problem all along. Recently, the problem gained renewed interest in the context of stability of hybrid dynamical systems, since solving the absolute stability problem implies stability analysis of switched linear systems. In this paper, we introduce the concept of generalized first integrals and use it to characterize the "most destabilizing" nonlinearity and to explicitly construct a Lyapunov function that yields an easily verifiable, necessary and sufficient condition for absolute stability of second-order systems.
AB - We consider the problem of absolute stability of linear feedback systems in which the control is a sector-bounded time-varying nonlinearity. Absolute stability entails not only the characterization of the "most destabilizing" nonlinearity, but also determining the parametric value of the nonlinearity that yields instability of the feedback system. The problem was first formulated in the 1940s, however, finding easily verifiable necessary and sufficient conditions for absolute stability remained an open problem all along. Recently, the problem gained renewed interest in the context of stability of hybrid dynamical systems, since solving the absolute stability problem implies stability analysis of switched linear systems. In this paper, we introduce the concept of generalized first integrals and use it to characterize the "most destabilizing" nonlinearity and to explicitly construct a Lyapunov function that yields an easily verifiable, necessary and sufficient condition for absolute stability of second-order systems.
KW - Generalized first integral
KW - Hybrid systems
KW - Switched linear systems
UR - http://www.scopus.com/inward/record.url?scp=0037303811&partnerID=8YFLogxK
U2 - 10.1109/TCSI.2002.808219
DO - 10.1109/TCSI.2002.808219
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AN - SCOPUS:0037303811
SN - 1057-7122
VL - 50
SP - 227
EP - 234
JO - IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications
JF - IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications
IS - 2
ER -