Some remarks on non-assignable equilibria of linear time-invariant systems

In this note we investigate the following question: given a (finite dimensional) linear time-invariant system and a constant desired value for its output. Assume there is no assignable equilibrium point corresponding to the desired output. Does there exists a (possibly dynamic, nonlinear, time-varying) controller such that the output ultimately enters a prescribed neighborhood of its desired value preserving all internal signals bounded? Our main contribution is the establishment of necessary and sufficient conditions, given in terms of a (computable) bound on the “size” of the neighborhood, for the solution of the problem. Interestingly, there is no connection between this problem and the more familiar concepts of controllability and observability. However, we also prove that if the system has “non-assignable values” for the output then it is non minimum phase.