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.