Computing rank-deficiency of rectangular matrix pencils

Computing whether a system is close to uncontrollable is numerically difficult task. In this paper some early results on a new simple experimental approach are presented. This method is based on reducing the problem to a rank problem for a certain matrix pencil. This method exhibits locally quadratic convergence to a local minimum of a function that yields the distance (in State-Space sense) to the nearest uncontrollable system from a given system. The entire computation take place in state space for numerical stability. We use this algorithm to compute the distance for certain examples, and use the examples to show some severe limitations on the popular Staircase Algorithm.