Observation and sliding mode observer for nonlinear fractional-order system with unknown input

•New results on the observability and left invertibility of nonlinear fractional-order systems.•Design of a step by step sliding mode observer with unknown input for fractional order systems.•Theoretical results on the finite-time convergence of the proposed observer.•Application to fault detection and estimation.

The main purpose of this paper is twofold. First, the observability and the left invertibility properties and the observable canonical form for nonlinear fractional-order systems are introduced. By using a transformation, we show that these properties can be deduced from an equivalent nonlinear integer-order system. Second, a step by step sliding mode observer for fault detection and estimation in nonlinear fractional-order systems is proposed. Starting with a chained fractional-order integrators form, a step by step first-order sliding mode observer is designed. The finite time convergence of the observer is established by using Lyapunov stability theory. A numerical example is given to illustrate the performance of the proposed approach.