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-catast...
| Main Author: | |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | en_US |
| Published: |
Institute of Electrical and Electronics Engineers (IEEE)
2012
|
| Online Access: | http://hdl.handle.net/1721.1/73076 |
| _version_ | 1811094050049097728 |
|---|---|
| author | Forney, G. David, Jr. |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Forney, G. David, Jr. |
| author_sort | Forney, G. David, Jr. |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T15:54:39Z |
| format | Article |
| id | mit-1721.1/73076 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T15:54:39Z |
| publishDate | 2012 |
| publisher | Institute of Electrical and Electronics Engineers (IEEE) |
| record_format | dspace |
| spelling | mit-1721.1/730762022-10-02T04:58:22Z Minimal Realizations of Linear Systems: The "Shortest Basis" Approach Forney, G. David, Jr. Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Forney, G. David, Jr. 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. 2012-09-20T17:42:03Z 2012-09-20T17:42:03Z 2011-01 2010-07 Article http://purl.org/eprint/type/JournalArticle 0018-9448 http://hdl.handle.net/1721.1/73076 Forney, G. David. “Minimal Realizations of Linear Systems: The "Shortest Basis" Approach.” IEEE Transactions on Information Theory 57.2 (2011): 726–737. en_US http://dx.doi.org/10.1109/tit.2010.2094811 IEEE Transactions on Information Theory Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf Institute of Electrical and Electronics Engineers (IEEE) arXiv |
| spellingShingle | Forney, G. David, Jr. Minimal Realizations of Linear Systems: The "Shortest Basis" Approach |
| title | Minimal Realizations of Linear Systems: The "Shortest Basis" Approach |
| title_full | Minimal Realizations of Linear Systems: The "Shortest Basis" Approach |
| title_fullStr | Minimal Realizations of Linear Systems: The "Shortest Basis" Approach |
| title_full_unstemmed | Minimal Realizations of Linear Systems: The "Shortest Basis" Approach |
| title_short | Minimal Realizations of Linear Systems: The "Shortest Basis" Approach |
| title_sort | minimal realizations of linear systems the shortest basis approach |
| url | http://hdl.handle.net/1721.1/73076 |
| work_keys_str_mv | AT forneygdavidjr minimalrealizationsoflinearsystemstheshortestbasisapproach |