Asymptotic orders of reachability in perturbed linear systems

A framework for studying asymptotic orders of reachability in perturbed linear, time-invariant systems is developed. The systems of interest are defined by matrices that have asymptotic expansions in powers of a perturbation parameter epsilon about the point 0. The reachability structure is exposed by means of the Smith form of the reachability matrix. This approach is used to provide insight into the kinds of inputs needed to reach weakly reachable target states, into the structure of high-gain feedback for pole placement, and into the types of inputs that steer trajectories arbitrarily close to almost (A,B)-invariant subspaces and almost (A,B)-controllability subspaces.<>