Hybrid implementation of observers in initial coordinates with a finite number of approximate inversions and global convergence

In this paper, we assume we are given an asymptotic observer whose dynamics are not written in the plant's coordinates and whose implementation requires the inversion of an injective immersion at each time. To avoid these costly computations, we propose a method to write the observer dynamics directly in the plant's coordinates by extending the injective immersion into a diffeomorphism and inverting its Jacobian. This is done by combining, in a hybrid way, those dynamics with an independent practical observer (maybe of smaller dimension), which is used to reset the estimate whenever it leaves the diffeomorphism domain where the Jacobian is invertible. This latter operation may necessitate to inverse an injective map, but we show that it happens only a finite number of times during the transient, and this inversion does not need to be exact: it can be done thanks to a minimization on a rough grid. The obtained observer is proved to be globally asymptotically convergent and robust to noise. Its performances are illustrated on a bioreactor model.