## Minimal realizations of linear systems: The "shortest basis" approach

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Forney Jr, G. David

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Given a controllable discrete-time linear system C, a shortest basis for C is
a set of linearly independent generators for C with the least possible lengths.
A basis B is a shortest basis if and only if it has the predictable span
property (i.e., has the predictable delay and degree properties, and is
non-catastrophic), or alternatively if and only if it has the subsystem basis
property (for any interval J, the generators in B whose span is in J is a basis
for the subsystem C_J). The dimensions of the minimal state spaces and minimal
transition spaces of C are simply the numbers of generators in a shortest basis
B that are active at any given state or symbol time, respectively. A minimal
linear realization for C in controller canonical form follows directly from a
shortest basis for C, and a minimal linear realization for C in observer
canonical form follows directly from a shortest basis for the orthogonal system
C^\perp. This approach seems conceptually simpler than that of classical
minimal realization theory.

Comment: 20 pages. Final version, to appear in special issue of IEEE Transactions on Information Theory on "Facets of coding theory: From algorithms to networks," dedicated to Ralf Koetter

Comment: 20 pages. Final version, to appear in special issue of IEEE Transactions on Information Theory on "Facets of coding theory: From algorithms to networks," dedicated to Ralf Koetter

##### Keywords

Computer Science - Information Theory, Electrical Engineering and Systems Science - Systems and Control, Mathematics - Optimization and Control