Asymptotic and Probabilistic Perturbation Analysis of Controllable Subspaces

In this paper, we consider the sensitivity of the controllable subspaces of single-input linear control systems to small perturbations of the system matrices. The analysis is based on the strict component-wise asymptotic bounds for the matrix of the orthogonal transformation to canonical form derived by the method of the splitting operators. The asymptotic bounds are used to obtain probabilistic bounds on the angles between perturbed and unperturbed controllable subspaces implementing the Markoff inequality. It is demonstrated that the probability bounds allow us to obtain sensitivity estimates, which are much tighter than the usual deterministic bounds. The analysis is illustrated by a high-order example.