Structured system models Part 1. Controllability and observability indices

Abstract

The controllability indices for multivariable linear systems are ordinarily defined by selecting linearly independent columns of the controllability matrix. The usual two selection schemes result in Hermite indices and Kronecker indices. To express the information on uncontrollable modes, ‘augmented’ indices are introduced. The observability indices are treated in an analogous way. The relationships between the indices and a polynomial matrix representation of the system are studied. The augmented observability Hermite (Kronecker) indices are found to coincide with the degrees of the diagonal entries of the polynomial matrix having suitable diagonal blocks of upper triangular (canonical row proper) form. The relationships for controllability indices are analogous. As an application the augmented indices are utilized to construct observable or controllable state-space representations from non-observable or non-controllable representations.