Generalization of the separation principle beyond constant-gain state-feedback control

In the linear control community, the well-known separation principle states that, for a controller designed using an observer and a constant-gain state-feedback gain can be designed separately since the overall closed-loop system eigenvalues are the union of those due to the observer alone and those due to the state-feedback controller alone. In this note, we generalize the separation principle in two directions. First, we use eigenvalue calculations to show that the separation principle holds for dynamic controllers and not just constant-gain controllers; both dynamic state-feedback and dynamic output-feedback controllers are considered. Second, we use a simple linearity argument to argue that the separation principle holds, in fact, not just for the well-known observer structure, but for a whole class of asymptotic observers, including observers that are neither linear nor time-invariant. This paper includes radiotherapy control problem to illustrate the usefulness of these results.