Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

Dong Hwan Lee; Jin Bae Park; Young Hoon Joo

doi:10.1016/j.sysconle.2011.06.003

Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

Dong Hwan Lee, Jin Bae Park, Young Hoon Joo

Department of Electrical and Electronic Engineering

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.

Original language	English
Pages (from-to)	778-785
Number of pages	8
Journal	Systems and Control Letters
Volume	60
Issue number	9
DOIs	https://doi.org/10.1016/j.sysconle.2011.06.003
Publication status	Published - 2011 Sep

Bibliographical note

Funding Information:
The authors would like to thank the Associate Editor and the anonymous Reviewers for their careful reading and constructive suggestions. This work was supported by the National Research Foundation (NRF) of Korea grant funded by the Korea goverment(MEST) (No.: R01-2008-000-20844-0 ).

All Science Journal Classification (ASJC) codes

Control and Systems Engineering
Computer Science(all)
Mechanical Engineering
Electrical and Electronic Engineering

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title = "Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems",

abstract = "It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.",

author = "Lee, {Dong Hwan} and Park, {Jin Bae} and Joo, {Young Hoon}",

note = "Funding Information: The authors would like to thank the Associate Editor and the anonymous Reviewers for their careful reading and constructive suggestions. This work was supported by the National Research Foundation (NRF) of Korea grant funded by the Korea goverment(MEST) (No.: R01-2008-000-20844-0 ). ",

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AU - Joo, Young Hoon

N1 - Funding Information: The authors would like to thank the Associate Editor and the anonymous Reviewers for their careful reading and constructive suggestions. This work was supported by the National Research Foundation (NRF) of Korea grant funded by the Korea goverment(MEST) (No.: R01-2008-000-20844-0 ).

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N2 - It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.

AB - It has already been recognized that looking for a positive definite Lyapunov function such that a high-order linear differential inequality with respect to the Lyapunov function holds along the trajectories of a nonlinear system can be utilized to assess asymptotic stability when the standard Lyapunov approach examining only the first derivative fails. In this context, the main purpose of this paper is, on one hand, to theoretically unveil deeper connections among existing stability conditions especially for linear time-invariant (LTI) systems, and from the other hand to examine the effect of the higher-order time-derivatives approach on the stability results for uncertain polytopic LTI systems in terms of conservativeness. To this end, new linear matrix inequality (LMI) stability conditions are derived by generalizing the concept mentioned above, and through the development, relations among some existing stability conditions are revealed. Examples illustrate the improvement over the quadratic approach.

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Fundamental connections among the stability conditions using higher-order time-derivatives of Lyapunov functions for the case of linear time-invariant systems

Abstract

Bibliographical note

All Science Journal Classification (ASJC) codes

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