A nine-fold canonical decomposition for linear systems

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Abstract
The zero structure for non-minimal proper systems in state-space form is investigated. The approach is ‘ geometric ’, and a complete characterization in geometric terms is given of the invariant, decoupling, system and transmission zeros, as defined by Rosenbrock. The first main result is a formula for the transmission zeros. Second, a ‘ canonical ’ lattice diagram is presented of a decomposition of the state space which can be viewed as the ‘ product ’ of the Kalman canonical decomposition and the Morse canonical decomposition. This decomposition gives a straightforward characterization of all zeros just mentioned in terms of spectral properties of subspaces under a certain class of feedback and injection mappings. Via this diagram a number of equivalent formulae for the transmission zeros are derived. The freedom in pole assignment leads to new characterizations for the invariant and system zeros in terms of greatest common divisors of characteristic polynomials. Finally, the relation is demonstrated between certain subspaces and some structural invariants, i.e. the zeros at infinity and the minimal indices of a polynomial basis for the kernel of the transfer function.