Strong versions of impulsive controllability and sampled observability

Abstract

We give a simple proof of the (perhaps not so) well known fact that exponential polynomials of order $k$ with real exponents have at most $k-1$ real zeros. We deduce several results that relate to impulsive controllability and sampled observability of finite-dimensional linear time-invariant dynamical systems. We prove that the initial state of a continuous-time linear time-invariant dynamical system of dimension $n$ can be uniquely reconstructed from the sampled output regardless of its sampling time sequence of length $n$ if and only if the system is observable and all the eigenvalues of the system matrix are real. This result thus characterizes sampled observability with arbitrary sampling times. Likewise, we prove that the system is controllable by means of $n$ impulses regardless of when they occur if and only if the system is controllable and all the eigenvalues of the system matrix are real. We also show that, if the system is observable and all the eigenvalues of the system matrix are real, then there exists an initial state such that the times where the output crosses a prescribed threshold are prescribed $m$ times with $m<n$.