Algorithm design for structured systems: application to pole placement

Numerical algorithms for signal processing and control are quite often constructed by intuition. When the system to be designed contains algebraic or other invariants, then these constraints can be exploited to find appropriate transformations. The transformations in system theory are usually Lie groups. One has to find Lie groups which are consistent with the invariants. We show how this point of view can be applied to construct pole placement algorithms for symmetric and skew-symmetric realizations. However, Lie group theory only reveals the appropriate transformations but is not able to reduce the design process to a trivial task. The problem discussed here also shows this limitation.