This paper investigates the problem of designing an adaptive high gain observer for a class of MIMO non uniformly observable systems involving some unknown constant parameters. The considered systems are not necessarily linear with respect to the unknown parameters up to a well defined parameter structure involving the parameter characteristic indices. The observer design model is elaborated from the state system dynamics augmented with the dynamics of the unknown parameters. An adaptive observer whose gain is derived from the resolution of a Lyapunov differential equation is proposed and its exponential convergence is established under an appropriate persistent excitation property up to the classical high gain state observer design assumptions. Moreover, it is shown that the equations of the observer can be put under an adaptive form emphasizing thereby its versatility to include several available adaptive observers. The main steps of the observer design and its convergence properties are illustrated through a typical problem involving an induction motor where one aims at estimating the mechanical speed, the load torque and rotor fluxes as well as the rotor inductance and resistance from the measurements of the stator currents. (C) 2018 Elsevier Ltd. All rights reserved.