Luenberger Observers for Nonautonomous Nonlinear Systems

We show how the (initial) Luenberger methodology presented by Luenberger in “Observing the state of a linear system,” IEEE Transactions on Military Electronics, vol. 8, pp. 74-80, 1964, for linear systems can be used to design causal observers for controlled nonlinear systems. Their implementation relies on the resolution of a time-varying PDE, the solutions of which transform the dynamics into linear asymptotically stable ones. We prove the existence and injectivity (after a certain time) of such transformations, under standard observability assumptions such as differential observability or backward-distinguishability. We show on examples how this PDE can be solved and how the observability assumptions can be checked. Also, we show that similarly to the high gain framework, it is possible to use a time-independent transformation when the system is observable for any input and strongly differentially observable of order the dimension of the system.