ABSTRACT
N. Kazantzis–C. Kravaris and G. Kreisselmeier–R. Engel have suggested two apparently different approaches for constructing observers for nonlinear systems. This chapter shows that these approaches are closely related, leading to observers with linear error dynamics in transformed variables. It provides 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. The Kreisselmeier–Engel construction applies to Lipschitz continuous systems and defines an observer whose dimension is at least as large as that of the system. The former requires the solution of a PDE and the latter requires multiple solutions to an ordinary differential equation followed by quadratures. From an implementation point of view, the former is easier as the PDE can be solved approximately by a finite power series; but this solution is only local as is the resulting observer.
