Delay-dependent and order-dependent LMI-based sliding mode [formula omitted] control for variable fractional order uncertain differential systems with time-varying delay and external disturbance

Abstract

This paper concentrates on the sliding mode H∞ control for variable fractional order uncertain differential systems with time-varying delay and external disturbance by using the delay-dependent and order-dependent LMI approach. Firstly, by designing a suitable variable fractional order sliding mode surface, an equivalent control law has been given and the new state-space model has been introduced. Secondly, by using a Lyapunov–Krasovskii function and constructing an appropriate variable fractional order inequality, a novel delay-dependent and order-dependent stability theorem of the nominal systems without external disturbance is proposed. Then, based on the above stability conditions and the H∞ performance, the delay-dependent and order-dependent stability conditions for the systems with external disturbance are discussed. Moreover, the reachability of the sliding mode surface has been presented. In order to stabilize the systems with uncertainties, the controller gain matrices are also derived with the help of the proposed stability criteria. All the results are in the form of linear matrix inequalities. Finally, two numerical examples are provided to verify the effectiveness of the introduced theoretical formulation.