Recently, Kazantzis–Kravaris and Kreisselmeier–Engel have suggested two apparently different approaches for constructing observers for nonlinear systems. We show that these approaches are closely related, leading to observers with linear error dynamics in transformed variables. In particular, we give the sufficient conditions for the existence of smooth solutions to the Kazantzis–Kravaris partial differential equation (KK PDE). These methods can be used for systems that exhibit chaotic behavior.