Non-existence of finite-time stable equilibria in fractional-order nonlinear systems
We note that in the literature it is often taken for granted that for fractional-order system without delays, whenever the system trajectory reaches the equilibrium, it will stay there. In fact, this is the well-known phenomenon of finite-time stability. However, in this paper, we will prove that fo...
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| Published in | Automatica (Oxford) Vol. 50; no. 2; pp. 547 - 551 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Kidlington
Elsevier Ltd
01.02.2014
Elsevier |
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| Online Access | Get full text |
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| Abstract | We note that in the literature it is often taken for granted that for fractional-order system without delays, whenever the system trajectory reaches the equilibrium, it will stay there. In fact, this is the well-known phenomenon of finite-time stability. However, in this paper, we will prove that for fractional-order nonlinear system described by Caputo’s or Riemann–Liouville’s definition, any equilibrium cannot be finite-time stable as long as the continuous solution corresponding to the initial value problem globally exists. In addition, some examples of stability analysis are revisited and linear Lyapunov function is used to prove the asymptotic stability of positive fractional-order nonlinear systems. |
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| AbstractList | We note that in the literature it is often taken for granted that for fractional-order system without delays, whenever the system trajectory reaches the equilibrium, it will stay there. In fact, this is the well-known phenomenon of finite-time stability. However, in this paper, we will prove that for fractional-order nonlinear system described by Caputo's or Riemann-Liouville's definition, any equilibrium cannot be finite-time stable as long as the continuous solution corresponding to the initial value problem globally exists. In addition, some examples of stability analysis are revisited and linear Lyapunov function is used to prove the asymptotic stability of positive fractional-order nonlinear systems. |
| Author | Shen, Jun Lam, James |
| Author_xml | – sequence: 1 givenname: Jun surname: Shen fullname: Shen, Jun email: junshen2009@gmail.com – sequence: 2 givenname: James surname: Lam fullname: Lam, James email: james.lam@hku.hk |
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| Keywords | Finite-time stability Fractional-order nonlinear system Equilibria Linear Lyapunov function Initial value problem Non linear control Delay system Non linear system Positive system Modeling Fractional derivative Asymptotic stability Time interval Lyapunov function Non existence |
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| Snippet | We note that in the literature it is often taken for granted that for fractional-order system without delays, whenever the system trajectory reaches the... |
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| SubjectTerms | Applied sciences Asymptotic properties Computer science; control theory; systems Control system analysis Control theory. Systems Delay Dynamical systems Equilibria Exact sciences and technology Finite-time stability Fractional-order nonlinear system Initial value problems Linear Lyapunov function Lyapunov functions Nonlinear dynamics Stability System theory Trajectories |
| Title | Non-existence of finite-time stable equilibria in fractional-order nonlinear systems |
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