Nonlinear observer design techniques with observability functions

Two different observer design techniques for linear and nonlinear systems based on the so-called observability function, which is related to the Gramian matrix in the linear case, are proposed. Asymptotic and finite-time convergence results are provided. The effectiveness of the proposed approaches are demonstrated via numerical simulations. The asymptotically convergent observer design technique provide constructions similar to the Maximum-likelihood nonlinear filter despite the different perspectives.