Sliding-mode observers for systems with unknown inputs: A high-gain approach

The problem of designing sliding-mode observers for systems with unknown inputs is considered when the so-called observer matching condition is not satisfied. This condition severely restricts the applicability of sliding-mode observers. To circumvent the observer matching condition, thereby broadening the class of systems for which sliding-mode observers can be constructed, a possible method is to generate auxiliary outputs that are then used to construct the sliding-mode observer. High-order sliding-mode exact differentiators can be used to obtain auxiliary outputs. A proof of the asymptotic stability of the state estimation error is provided. However, the resulting overall observer architecture is complex. In this paper, high-gain approximate differentiators are proposed to generate the estimates of auxiliary outputs instead. The resulting architecture is much simpler than the architecture involving high-order sliding-mode exact differentiators. It is shown that the state estimation error is uniformly ultimately bounded with respect to a ball whose radius can be controlled by design parameters. The performance of the high-gain approximate differentiator based sliding-mode observer is demonstrated to be comparable to that of the observer that uses high-order sliding-mode exact differentiators. The use of the presented observers to reconstruct the unknown inputs is also analyzed and then illustrated by numerical examples.