On the Estimation of the Distance to Uncontrollability for Higher Order Systems
Emre Mengi · SIAM Journal on Matrix Analysis and Applications · 2008
A higher order dynamical system of order k is called controllable if the trajectory of the system as well as its first $k-1$ derivatives can be adjusted to pass through any given point at a finite time by choosing the input appropriately. The distance to uncontrollability is the norm of the smallest perturbation yielding an uncontrollable system. We derive a singular value minimization characterization for the distance to uncontrollability and present a trisection algorithm exploiting the singular value characterization. The algorithm is devised for low accuracy and depends on the extraction of the imaginary eigenvalues of even-odd matrix polynomials of degree $2k$ and size $2n$ with n denoting the size of the system. The well-studied first order distance to uncontrollability can be recovered as a special case.