Detectability Conditions and State Estimation for Linear Time-Varying and Nonlinear Systems

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.