Asymptotic achievability for linear time invariant state space systems

We consider here the problem of finding a controller such that when interconnected to the plant, we obtain a system which is asymptotically equivalent to a desired system. Here ‘asymptotic equivalence’ is formalized as ‘asymptotic bisimilarity’. Intuitively speaking, two systems are asymptotically bisimilar if the difference between their outputs decays to zero with time. We give necessary and sufficient conditions for the existence of such a controller. These conditions can be verified computationally using standard algorithms in linear geometric control. The systems we consider are linear time invariant input-state-output systems.