New power law inequalities for fractional derivative and stability analysis of fractional order systems

Three new power law inequalities for fractional derivative are proposed in this paper. We generalize the original useful power law inequality, which plays an important role in the stability analysis of pseudo state of fractional order systems. Moreover, three stability theorems of fractional order systems are given in this paper. The stability problem of fractional order linear systems can be converted into the stability problem of the corresponding integer order systems. For the fractional order nonlinear systems, a sufficient condition is obtained to guarantee the stability of the true state. The stability relation between pseudo state and true state is given in the last theorem by the final value theorem of Laplace transform. Finally, two examples and numerical simulations are presented to demonstrate the validity and feasibility of the proposed theorems.