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Minimal Realizations of Linear Systems: The “Shortest Basis” Approach

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1 Author(s)
G. David Forney ; Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA, USA

Given a 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 CJ). 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. This approach seems conceptually simpler than that of classical minimal realization theory.

Published in:

IEEE Transactions on Information Theory  (Volume:57 ,  Issue: 2 )