The topological structure of singular observer-based compensators

Existing results in standard control theory of regular observer-based compensators (OBC's) are surveyed. Conditions for an arbitrary compensator to be observer-based are presented. The set of scalar OBC's is completely characterized. The recently developed singular OBC structure is introduced. As with the regular case, the separation principle holds when a singular OBC is incorporated in the closed-loop system. Various categories of OBC's are classified based on the structure of both regular and singular OBC's.;New results are obtained concerning various properties of OBC 's. It is shown that the set of OBC's is neither bounded, convex, nor open for any plant. The interior of the set of regular OBC's is explored based on stable invariant subspaces of a closed-loop matrix. The result shows that certain sets of OBC's have dense interior. Based on applying a certain open map, it is shown that there is a direct relationship between density of regular OBC's and density of singular ones. An important topological property, path-connectedness, is studied for all categories. Various conditions on the closed-loop pencil may also be imposed when singular OBC's are considered. In this regard, path-connectedness is still preserved in some cases. The results show that singular OBC's have enormous mathematical advantages over their regular counterparts when path-connectedness comes into consideration.