Hybrid implementation of observers in plant’s coordinates with a finite number of approximate inversions and global convergence

We assume we are given a continuous-time 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 methods to write the observer dynamics directly in the plant’s coordinates by extending the injective immersion into a diffeomorphism, inverting its Jacobian, and using a hybrid mechanism to guarantee completeness of solutions. The obtained observers areproved to recover the same performances in terms of convergence and robustness to noise as theinitial observer, and require only a finite number of approximate inversions. This methodology appliesto a broad class of nonlinear observers including (low-power) high gain and Luenberger designs, andis illustrated on a Van der Pol oscillator with unknown parameters