After constructing a nonlinear observer and applying state feedback depending on the estimated states to asymptotically stabilize the system, we introduce new inputs of bounded norm to the system. We then show that the parameters for the observer, with sufficiently good initial error estimates, can be chosen so that the error between states and estimates goes to zero. Particular remarks are appropriate for the class of globally feedback linearizable systems, where the bounds on the norm of the initial states and the amplitude of the new inputs can be as large as desired.