Smooth observers are able to converge asymptotically to the actual value of the state, in the case where no measurement noise and no persistently acting perturbations are present. Under the same conditions continuous observers can converge in finite time. However, they are unable to converge if a perturbation/ uncertainty is present. In order to achieve finite time and exact convergence in the presence of perturbations, it is necessary to use discontinuous injection terms. In this chapter, some recent developments in this direction for second order systems will be presented and the results will be illustrated by means of simple examples. It will be also shown that by including non globally Lipschitz injection terms the convergence time of the observers can be made independent of the initial condition. The restriction to the two dimensional case is due to the fact that all proofs are done by means of Lyapunov functions, that are only available for planar systems. However, this has as advantage that the treatment is mainly tutorial, and provides on the one side an easy introduction to the topic, and on the other side it presents in the simplest case the main results that are (probably) valid for the general case. We hope to be able to provide a similar treatment of the general case in the near future.
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- On Discontinuous Observers for Second Order Systems: Properties, Analysis and Design
Jaime A. Moreno
- Springer Berlin Heidelberg