State observers with first-/second-order sliding-mode for nonlinear systems with bounded noises

A full-order sliding-mode state observer for a class of nonlinear continuous-time dynamic systems is proposed and conditions for the stability of the estimation error in the absence of noises are provided. If the system is affected by bounded disturbances, under such conditions the existence of an attractive invariant set for the estimation error is ensured. The design of the observer can be made via LMI techniques. A second-order sliding-mode differentiator is introduced, which makes available the first time derivative of the measurable output in finite time. This artificially constructed measurement is exploited to reduce the invariant set of the estimation error.