On functional observability and functional observer design

The objective of this paper is to introduce a new approach to functional observer design. It considers the design of functional observers with pole placement and their application to observer-based control problems. Initially, a new definition of functional observability is provided, followed by algebraic conditions for linear multivariable systems to be functionally observable. These conditions generalize those of observability and the Luenberger observer design in both full and reduced-order cases. The parameters of the functional observer are derived from the constrained Sylvester equation and the functional observability condition. Notably, the observability of the system under consideration is not required for the design of functional observers. The results are applied to the new design of observer-based control without the need to satisfy the observability condition.