Minimal Realizations of Linear Systems: The "Shortest Basis" Approach
Author(s)
Forney, G. David, Jr.
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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[superscript ⊥]. This approach seems conceptually simpler than that of classical minimal realization theory.
Date issued
2011-01Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer ScienceJournal
IEEE Transactions on Information Theory
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Citation
Forney, G. David. “Minimal Realizations of Linear Systems: The "Shortest Basis" Approach.” IEEE Transactions on Information Theory 57.2 (2011): 726–737.
Version: Author's final manuscript
ISSN
0018-9448