Minimal Realizations of Linear Systems: The "Shortest Basis" Approach

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-catas...

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Bibliographic Details
Published in:IEEE transactions on information theory 2011-02, Vol.57 (2), p.726-737
Main Author: Forney, G David
Format: Article
Language:English
Subjects:
Block codes
Canonical forms
Control systems
Delay
Generators
Intervals
Linear systems
minimal realizations
Observers
Symbols
Trajectory
Vectors
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Summary: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.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2010.2094811