  This work proposes a detectability condition for linear time-varying systems
based on the exponential dichotomy spectrum. The condition guarantees the
existence of an observer, whose gain is determined only by the unstable modes
of the system. This allows for an observer design with low computational
complexity compared to classical estimation approaches. An extension of this
observer design to a class of nonlinear systems is proposed and local
convergence of the corresponding estimation error dynamics is proven. Numerical
results show the efficacy of the proposed observer design technique.
