Rectangular matrix pencils arise in many contexts in Linear Control Theory, for example the PBH Test for Controllability and the computation of transmission zeros. In this paper, we examine a method to find the distance from an arbitrary given pencil to a nearby rank-deficient one, in light of the fact that rectangular pencils are generically full-rank. We propose an experimental computational method that exhibits quadratic convergence to a local minimum of this distance function. This partially answers the question of the existence of a rank-deficient pencil in a neighborhood of a given pencil. We use the Algorithm to illustrate some limitations of previous algorithms to measure this distance.
Bibliographical noteFunding Information:
* This research was partially supported by NSF Grants DCR-8420935 and DCR-8519029.
- Algebraic structure
- Matrix pencils
- Numerical methods