  Symmetries of nonlinear control systems in state representation are
considered. To this end, a geometric approach to ordinary differential
equations is advocated. Invariant feedback laws for systems with Lie
symmetries, i.e. feedback laws that preserve the symmetry group of a considered
plant, can be constructed based on invariants of the considered group action.
Under minor technical assumptions suitable invariant tracking errors can be
determined by following a normalization procedure. The underlying local
geometric meaning of this procedure is discussed and it is shown how it can
also be applied in order to derive a local, reduced-order system
representation. Further, the idea of controlled symmetries, i.e. imposing
desired symmetry properties on a given control system by state feedback, is
discussed by outlining an exemplary control design for a predator-prey
bioreactor.
