Computation of the observer gain for extended Luenberger observers using automatic differentiation

For a given nonlinear system, the extended Luenberger observerprovides nearly exact error dynamics. In contrast to the normalform observer, the extended Luenberger observer exists evenif the associated integrability condition is violated. Up tonow, Lie derivatives and Lie brackets required by the designprocedure have been computed symbolically. Even for systemswith moderate size and complexity, one usually obtains extremelylarge expressions for the observer gain. The design of an extendedLuenberger observer based on symbolic differentiation is not feasible for complicated or large-scale systems. In this paperwe discuss a new approach to compute the observer gain. Ourapproach is based on a computation method for derivatives calledautomatic differentiation. In contrast to numeric differentiationby means of divided differences, automatic differentiationincurs no truncation errors.