TY - JOUR
T1 - A generalization of Kasovskii-LaSalle theorem for nonlinear time-varying systems
T2 - Converse results and applications
AU - Lee, Ti Chung
AU - Jiang, Zhong Ping
N1 - Funding Information:
Manuscript received November 24, 2003; revised July 28, 2004 and March 28, 2005. Recommended by Associate Editor M. Reyhanoglu. This work was supported in part by the NSC, Taiwan, R.O.C., under Contract NSC-91-2213-E-159-004, and by the National Science Foundation Grants ECS-0093176 and INT-9987317. Part of this work was done when the first author was visiting the Polytechnic University, Brooklyn, New York.
PY - 2005/8
Y1 - 2005/8
N2 - This paper presents a practically applicable characterization of uniform (global) asymptotic stability (UAS and UGAS) for general nonlinear time-varying systems, under certain output-dependent conditions in the spirit of the Krasovskii-LaSalle theorem. The celebrated Krasovskii-LaSalle theorem is extended from two directions. One is using the weak zero-state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time-derivative of the Lyapunov function is the zero set. Another one is using an almost bounded output-energy condition to relax the assumption that the time derivative of the Lyapunov function is negative semi-definite. Then, the UAS and UGAS properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly Lyapunov stable systems. The proposed conditions turn out to be also necessary under some mild assumptions and thus, give a new characterization of UGAS (and UAS). Through an equivalence relation, the proposed detectability condition can also be verified in terms of usual PE condition. To validate the proposed results, the obtained stability criteria are applied to a class of time-varying passive systems and to revisit a tracking control problem of nonholonomic chained systems. For the latter, under certain persistency of excitation conditions, the K-exponential stability is achieved based on our approach.
AB - This paper presents a practically applicable characterization of uniform (global) asymptotic stability (UAS and UGAS) for general nonlinear time-varying systems, under certain output-dependent conditions in the spirit of the Krasovskii-LaSalle theorem. The celebrated Krasovskii-LaSalle theorem is extended from two directions. One is using the weak zero-state detectability property associated with reduced limiting systems of the system in question to generalize the condition that the maximal invariance set contained in the zero locus of the time-derivative of the Lyapunov function is the zero set. Another one is using an almost bounded output-energy condition to relax the assumption that the time derivative of the Lyapunov function is negative semi-definite. Then, the UAS and UGAS properties of the origin can be guaranteed by employing these two improved conditions related to certain output function for uniformly Lyapunov stable systems. The proposed conditions turn out to be also necessary under some mild assumptions and thus, give a new characterization of UGAS (and UAS). Through an equivalence relation, the proposed detectability condition can also be verified in terms of usual PE condition. To validate the proposed results, the obtained stability criteria are applied to a class of time-varying passive systems and to revisit a tracking control problem of nonholonomic chained systems. For the latter, under certain persistency of excitation conditions, the K-exponential stability is achieved based on our approach.
KW - Detectability
KW - Nonholonomic systems
KW - Nonlinear time-varying systems
KW - Reduced limiting systems
KW - Uniform asymptotic stability
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U2 - 10.1109/TAC.2005.852567
DO - 10.1109/TAC.2005.852567
M3 - Article
AN - SCOPUS:26244435293
SN - 0018-9286
VL - 50
SP - 1147
EP - 1163
JO - IEEE Transactions on Automatic Control
JF - IEEE Transactions on Automatic Control
IS - 8
ER -