Abstract
This paper investigates fractional order Barbalat’s lemma and its applications for the stability of fractional order nonlinear systems with Caputo fractional derivative at first. Then, based on the relationship between Caputo fractional derivative and Riemann-Liouville fractional derivative, fractional order Barbalat’s lemma with Riemann-Liouville derivative is derived. Furthermore, according to these results, a set of new formulations of Lyapunov-like lemmas for fractional order nonlinear systems are established. Finally, an example is presented to verify the theoretical results in this paper.
Highlights
Asymptotic stability analysis of non-autonomous systems is generally much harder than that of autonomous systems, since it is usually very difficult to build a Lyapunov function with a negative definite derivative
A fractional order Barbalat’s Lemma will be introduced at first, some Fractional order Lyapunov-like Lemmas will be derived for fractional order nonlinear systems
Caputo fractional operator plays an important role in the fractional systems, since the initial conditions for fractional differential equations with Caputo derivatives take on the same form as for integer-order differential equations, which have well understood physical meanings
Summary
Asymptotic stability analysis of non-autonomous systems is generally much harder than that of autonomous systems, since it is usually very difficult to build a Lyapunov function with a negative definite derivative. In [27], adaptive sliding mode approach was applied to synchronize two different fractional-order chaotic systems, which the integer order Barbalat’s Lemma was used. The relationship between uniform continuity of a function and the boundedness of its fractional derivative will be built, a fractional order Barbalat’s Lemma described by fractional derivative will be proposed. Both Caputo operator and RiemannLiouville operator will be discussed. A set of new formulations of Lyapunov-like stability lemmas for fractional order nonlinear systems are established
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