A canonical parameterization of the Kronecker form of a matrix pencil

Abstract

The Kronecker form is the classical canonical form for matrix pencils under strict equivalence transformation. Consider a matrix pencil whose entries are smooth functions of a parameter vector. The Kronecker form of the parameterized pencil will, in general, be a discontinuous function of the parameters. For a linear time-invariant control system these discontinuities correspond to a change in the finite and infinite invariant zero structure. Since many control strategies require knowledge of, or place restrictions on, the zero structure, these points of discontinuity are of considerable interest. In this paper a general approach to the study of such points is developed in the framework of singularity theory. We derive a ‘miniversal’ parameterization of a given pencil. That is, a parameterized family of pencils that: (i) includes the given pencil, (ii) is locally equivalent to any other family up to a change of parameters, and (iii) uses the fewest number of parameters to achieve this property. All ‘nearby’ zero structures can be obtained by varying parameter values in the miniversal parameterization. From all miniversal parameterizations, one having a particularly uncluttered representation is selected as canonical. However, some restrictions on the finite elementary divisors are then required.