10. Compensators

We treat the two topics stabilization and compensator design in this chapter. Stabilization of an unstable IO behavior (the plant) by output feedback means to construct a suitable IO behavior (the stabilizing compensator or stabilizing controller) such that the feedback or closed-loop behavior is well-posed and T-stable for a suitable monoid T of stable polynomials in the sense of Chap. 9. In Sect. 10.1, we characterize the existence of a stabilizing compensator and then parameterize all of them. Our main emphasis is on the case where the compensator as well as the feedback behavior is proper IO behaviors, whereas the plant is not assumed proper. Compensator design means to construct stabilizing compensators that realize desired tasks like tracking, disturbance rejection, or model matching; and this is one of the key tasks in systems theory. We discuss the existence and parametrization of such restricted compensators in detail in Sect. 10.2. In the standard situations, we also show that the constructed compensators are robust. This signifies that the compensator is valid not only for the given plant, but also for all nearby ones. This obviously necessitates to topologize the set of all plants.