Finite time converging input observers for nonlinear second-order systems

We propose a discontinuous observer able to estimate in finite time both unmeasured states and the unknown input of a SISO nonlinear second order system. The only requirements are the observability of both the states and the unknown input, and that the unknown input is a uniformly Lipschitz time function. We illustrate the method in a biological reactor that models the functioning of a Waste Water Treatment Plant (WWTP), for which the problem of estimating the unknown input is of practical relevance. For concreteness of the presentation we restrict the treatment to second order systems, but the idea can be extended to a more general context.