• Title, Summary, Keyword: Lyapunov function

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Relaxed Stability Condition for Affine Fuzzy System Using Fuzzy Lyapunov Function (퍼지 리아푸노프 함수를 이용한 어파인 퍼지 시스템의 완화된 안정도 조건)

  • Kim, Dae-Young;Park, Jin-Bae;Joo, Young-Hoon
    • The Transactions of The Korean Institute of Electrical Engineers
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    • v.61 no.10
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    • pp.1508-1512
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    • 2012
  • This paper presents a relaxed stability condition for continuous-time affine fuzzy system using fuzzy Lyapunov function. In the previous studies, stability conditions for the affine fuzzy system based on quadratic Lyapunov function have a conservativeness. The stability condition is considered by using the fuzzy Lyapunov function, which has membership functions in the traditional Lyapunov function. Based on Lyapunov-stability theory, the stability condition for affine fuzzy system is derived and represented to linear matrix inequalities(LMIs). And slack matrix is added to stability condition for the relaxed stability condition. Finally, simulation example is given to illustrate the merits of the proposed method.

A NOTE ON THE EXISTENCE OF A LYAPUNOV FUNCTION

  • Goo, Yoon-Hoe
    • The Pure and Applied Mathematics
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    • v.5 no.2
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    • pp.143-147
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    • 1998
  • We show that a real valued function $\phi$ defined by $\phi (\chi)$ = (equation omitted) is a Lyapunov function of compact asymptotically stable set M.

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Study of the Robust Stability of the Systems with Structured Uncertainties using Piecewise Quadratic Lyapunov Function

  • Jo, Jang-Hyen
    • 제어로봇시스템학회:학술대회논문집
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    • pp.499-499
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    • 2000
  • The robust stability problems for nominally linear system with nonlinear, structured perturbations arc considered with Lyapunov direct method. The Lyapunov direct method has been utilized to determine the bounds for nonlinear, time-dependent functions which can be tolerated by a stable nominal system. In most cases quadratic forms are used either as components of vector Lyapunov function or as a function itself. The resulting estimates are usually conservative. As it is known, often the conservatism of the bounds we propose to use a piecewise quadratic Lyapunov function. An example demonstrates application of the proposed method.

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Robust Stable Conditions Based on the Quadratic Form Lyapunov Function (2차 형식 Lyapunov 함수에 기초한 강인한 안정조건)

  • Lee, Dong-Cheol;Bae, Jong-Il;Jo, Bong-Kwan;Bae, Chul-Min
    • Proceedings of the KIEE Conference
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    • pp.2212-2214
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    • 2004
  • Robust stable analysis with the system bounded parameteric variation is very important among the various control theory. This study is to investigate the robust stable conditions using the quadratic form Lyapunov function in which the coefficient matrix is affined linear system. The quadratic stability using the quadratic form Lyapunov function is not investigated yet. The Lyapunov unction is robust stable not to be dependent by the variable parameters, which means that the Lyapunov function is conservative. We suggest the robust stable conditions in the Lyapunov function in which the variable parameters are dependent in order to reduce the conservativeness of quadratic stability.

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H∞ Control of T-S Fuzzy Systems Using a Fuzzy Basis- Function-Dependent Lyapunov Function (퍼지 기저함수에 종속적인 Lyapunov 함수를 이용한 T-S 퍼지 시스템의 H∞ 제어)

  • Choi, Hyoun-Chul;Chwa, Dong-Kyoung;Hong, Suk-Kyo
    • Journal of Institute of Control, Robotics and Systems
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    • v.14 no.7
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    • pp.615-623
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    • 2008
  • This paper proposes an $H_{\infty}$ controller design method for Takagi-Sugeno (T-S) fuzzy systems using a fuzzy basis-function-dependent Lyapunov function. Sufficient conditions for the guaranteed $H_{\infty}$ performance of the T-S fuzzy control system are given in terms of linear matrix inequalities (LMIs). These LMI conditions are further used for a convex optimization problem in which the $H_{\infty}-norm$ of the closed-loop system is to be minimized. To facilitate the basis-function-dependent Lyapunov function approach and thus improve the closed-loop system performance, additional decision variables are introduced in the optimization problem, which provide an additional degree-of-freedom and thus can enlarge the solution space of the problem. Numerical examples show the effectiveness of the proposed method.

A Relaxed Stabilization Condition for Discrete T-S Fuzzy Model under Imperfect Premise Matching (불완전한 전반부 정합 하에서의 이산 T-S 퍼지 모델에 대한 완화된 안정화 조건)

  • Lim, Hyeon Jun;Joo, Young Hoon;Park, Jin Bae
    • Journal of Korean Institute of Intelligent Systems
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    • v.27 no.1
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    • pp.59-64
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    • 2017
  • In this paper, a controller for discrete Takagi-Sugeno(T-S) fuzzy model under imperfect premise matching is proposed. Most of previous papers have obtained the stabilization condition using common quadratic Lyapunov function. However, the stabilization condition may be conservative due to the typical disadvantage of the common quadratic Lyapunov function. Hence, in order to solve this problem, we propose the stabilization condition of discrete T-S fuzzy model using fuzzy Lyapunov function. Finally, the proposed approach is verified by the simulation experiments.

Switching Control for End Order Nonlinear Systems by Avoiding Singular Manifolds (특이공간 회피에 의한 2차 비선형 시스템의 스위칭 제어기 설계)

  • Yeom, D.H.;Im, K.H.;Choi, J.Y.
    • Proceedings of the KIEE Conference
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    • pp.315-318
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    • 2003
  • This paper proposes a switching control method applicable to any affine, 2nd order nonlinear system with single input. The key contribution is to develop a control design method which uses a piecewise continuous Lyapunov function non-increasing at every discontinuous point. The proposed design method requires no restrictions except full state availability. To obtain a non-increasing, piecewise continuous Lyapunov function, we change the sign of off-diagonal term s of the positive definite matrix composing the former Lyapunov function according to the sign of the Inter-connection term. And we use the solution of inequalities which guarantee each Lyapunov function is non-increasing at any discontinuous point.

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ON STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS BY LYAPUNOV'S DIRECT METHOD

  • Park, Jong-Yeoul;Phat, Vu-Ngoc;Jung, Il-Hyo
    • Journal of the Korean Mathematical Society
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    • v.37 no.5
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    • pp.805-821
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    • 2000
  • The paper deals with asymtotic stabillity of nonlinear nonautinomous systems by Lyapunov's direct method. The proposed Lyapunov-like function V(t, x) needs not be continuous in t and Lipschitz in x in a Banach space. The class of systems considered is allowed to be nonautonomous and infinite-dimensional and we relax the boundedness, the Lipschitz assumption on the system and the definite decrescent condition on the Lyapunov function.

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Derivation of a Group of Lyapunov Functions reflecting Damping Effects and its Application (댐핑 영향을 반영하는 Lyapunov 함수 그룹의 유도 및 응용)

  • Moon, Y.H.;Choi, B.K.;Roh, T.H.;Lee, T.S.;Lee, Y.S.
    • Proceedings of the KIEE Conference
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    • pp.195-198
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    • 1995
  • Most of the theorems of nonlinear stability is based on the Lyapunov stability theory. The Lyapunov function method is the most well-known and provides precise and rigorous theoretical backgrounds. However, tile conventional approach to direct stability analysis has been performed without taking account of damping effects. For accurate stability analysis of nonlinear systems, it is required to consider the damping effects. This paper presents a new method to derive a group of Lyapunov functions to reflect the damping effects by considering the integral relationships of the system governing equations. This method tan be utilized as a powerful tool to determine the region of attraction.

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A study on Voltage Stability Analysis by the Lyapunov Direct Method in 2-bus Systems (Lyapunov 직접법에 의한 2모선 개통의 전압안정도 해석)

  • Moon, Young-Hyun;Lee, Tae-Shik;Kim, Baik;Lee, Eung-Hyuk;Park, Neung-Soo
    • Proceedings of the KIEE Conference
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    • pp.106-108
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    • 1993
  • This paper gives derivations of voltage collapse conditions by using the Lyapunov function, which yields the exactly same results with the sensitivity analysis approach. On the other hand, the voltage collapse phenomenon is interpreted in the physical sense, and causes of the static voltage unstability and the dynamic voltage unstability are analyzed with the use of proposed Lyapunov function. In addition a new method is developed to calculate the power capacity limit of transmission lines with respect to voltage stability. This paper shows that it is able to analyze the voltage collapse with the Lyapunov direct method in the simple 2-bus system and proposes that the method works well by observing the variation of Lyapunov function as the load is increased in the system.

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