Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems

•New Lemmas relating the Caputo Fractional Derivatives of general quadratic forms and the trace of the product of matrices.•Possibility to use general quadratics Lyapunov functions in the fractional-order extension of Lyapunov direct method.•Uniform stability for fractional systems using Lyapunov functions.•Stability analysis of Fractional Order Model Reference Adaptive Control (FOMRAC) schemes. This paper presents two new lemmas related to the Caputo fractional derivatives, when α∈0,1, for the case of general quadratic forms and for the case where the trace of the product of a rectangular matrix and its transpose appear. Those two lemmas allow using general quadratic Lyapunov functions and the trace of a matrix inside a Lyapunov function respectively, in order to apply the fractional-order extension of Lyapunov direct method, to analyze the stability of fractional order systems (FOS). Besides, the paper presents a theorem for proving uniform stability in the sense of Lyapunov for fractional order systems. The theorem can be seen as a complement of other methods already available in the literature. The two lemmas and the theorem are applied to the stability analysis of two Fractional Order Model Reference Adaptive Control (FOMRAC) schemes, in order to prove the usefulness of the results.