Creating a nilpotent pencil via deadbeat

Abstract

We consider the problem of finding a square low-rank correction (λC − B)F to a given square pencil (λE − A) such that the new pencil λ(E − CF) − (A − BF) has all its generalised eigenvalues at the origin. We give necessary and sufficient conditions for this problem to have a solution and we also provide a constructive algorithm to compute F when such a solution exists. We show that this problem is related to the deadbeat control problem of a discrete-time linear system and that an (almost) equivalent formulation is to find a square embedding that has all its finite generalised eigenvalues at the origin.

Keywords:

• deadbeat control
• generalised state–space systems
• numerical methods

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work of the first author is partly supported by the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme under contract IAP07-11, initiated by the Belgian State, Science Policy Office, and by the Short Term Mobility program 2015 of the National Research Council of Italy. The work of the second author is partly supported by the GNCS INdAM project Strategie risolutive per sistemi lineari di tipo KKT con uso di informazioni strutturali. The scientific responsibility rests with its authors.