Realization theory of discrete-time dynamical systems:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2003
|
| Schriftenreihe: | Lecture notes in control and information sciences
296 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Literaturverz. S. [219] - 225 |
| Beschreibung: | X, 232 S. graph. Darst. |
| ISBN: | 3540406751 |
Internformat
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| 245 | 1 | 0 | |a Realization theory of discrete-time dynamical systems |c Tsuyoshi Matsuo ; Yasumichi Hasegawa |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2003 | |
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Datensatz im Suchindex
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| adam_text | TSUYOSHI MATSUO * YASUMICHI HASEGAWA REALIZATION THEORY OF DISCRETE-TIME
DYNAMICAL SYSTEMS WITH 14 FIGURES SPRINGER TABLE OF CONTENTS 1
INTRODUCTION 1 2 INPUT/OUTPUT MAPS 9 2.1 CONCATENATION MONOID: SET OF
EXPERIMENTS 9 2.2 INPUT RESPONSE MAP (INPUT/OUTPUT MAP WITH CAUSALITY)
10 2.3 HISTORICAL NOTES AND CONCLUDING REMARKS 14 3 GENERAL DYNAMICAL
SYSTEMS 15 3.1 REALIZATION THEORY OF GENERAL DYNAMICAL SYSTEMS 15 3.2
FINITE GENERAL DYNAMICAL SYSTEMS 18 3.3 CONTROL SYSTEMS AND
MULTI-EXPERIMENTS 23 3.4 HISTORICAL NOTES AND CONCLUDING REMARKS 26 3.5
APPENDIX 28 3.5.A [/-ACTIONS 28 3.5.B POINTED U-ACTIONS 30 3.5.C
[/-ACTIONS WITH A READOUT MAP 32 3.5.D GENERAL DYNAMICAL SYSTEMS 33 4
LINEAR REPRESENTATION SYSTEMS 38 4.1 REALIZATION THEORY OF LINEAR
REPRESENTATION SYSTEMS 38 4.2 FINITE DIMENSIONAL LINEAR REPRESENTATION
SYSTEMS 41 4.3 PARTIAL REALIZATION THEORY OF LINEAR REPRESENTATION
SYSTEMS . .48 4.4 HISTORICAL NOTES AND CONCLUDING REMARKS 50 4.5
APPENDIX 52 4.5.A REALIZATION THEOREM 52 4.5.A-1 LINEAR STATE STRUCTURE:
LINEAR U-ACTIONS 52 4.5.A-2 POINTED LINEAR U-ACTIONS 57 4.5.A-3 LINEAR
U-ACTIONS WITH A READOUT MAP 59 4.5.A-4 LINEAR REPRESENTATION SYSTEMS 60
4.5.A-5 SOPHISTICATED REALIZATION THEOREM 61 4.5.B FINITE DIMENSIONALITY
64 4.5.B-1 POINTED FINITE DIMENSIONAL LINEAR U-ACTIONS 64 4.5.B-2 FINITE
DIMENSIONAL LINEAR U-ACTIONS WITH A READOUT MAP .... 67 4.5.B-3 FINITE
DIMENSIONAL LINEAR REPRESENTATION SYSTEMS 69 4.5.B-4 EXISTENCE CRITERION
FOR LINEAR REPRESENTATION SYSTEMS 70 VIII TABLE OF CONTENTS 4.5.B-5
REALIZATION PROCEDURE FOR LINEAR REPRESENTATION SYSTEMS 71 4.5.C PARTIAL
REALIZATION 71 4.5.C-1 POINTED LINEAR U-ACTIONS 71 4.5.C-2 LINEAR
U-ACTIONS WITH A READOUT MAP 72 4.5.C-3 PARTIAL REALIZATION PROBLEM 73 5
AFFINE DYNAMICAL SYSTEMS 79 5.1 REALIZATION THEOREM OF AFFINE DYNAMICAL
SYSTEMS 79 5.2 FINITE DIMENSIONAL AFFINE DYNAMICAL SYSTEMS 84 5.3
HISTORICAL NOTES AND CONCLUDING REMARKS 91 5.4 APPENDIX 93 5.4.A
REALIZATION THEOREM 93 5.4.A-1 FUNDAMENTALS FOR AFFINE DYNAMICAL SYSTEMS
93 5.4.A-2 AFFINE STATE STRUCTURE: LINEAR [/-ACTIONS 94 5.4.A-3 AFFINE
STATE STRUCTURE WITH A READOUT MAP 98 5.4.A-4 AFFINE DYNAMICAL SYSTEMS
101 5.4.A-5 SOPHISTICATED REALIZATION THEOREM 103 5.4.B FINITE
DIMENSIONALITY 105 5.4.B-1 POINTED FINITE DIMENSIONAL LINEAR U-ACTIONS
105 5.4.B-2 FINITE DIMENSIONAL LINEAR U-ACTIONS WITH A READOUT MAP ...
108 5.4.B-3 FINITE DIMENSIONAL AFFINE DYNAMICAL SYSTEMS 110 5.4.B-4
EXISTENCE CRITERION FOR AFFINE DYNAMICAL SYSTEMS 111 5.4.B-5 REALIZATION
PROCEDURE FOR AFFINE DYNAMICAL SYSTEMS 112 6 PSEUDO LINEAR SYSTEMS 113
6.1 INPUT RESPONSE MAPS WITH TIME-INVARIANCE 113 6.2 PSEUDO LINEAR
SYSTEMS 114 6.3 FINITE DIMENSIONAL PSEUDO LINEAR SYSTEMS 117 6.4 PARTIAL
REALIZATION OF PSEUDO LINEAR SYSTEMS 122 6.5 REAL-TIME PARTIAL
REALIZATION OF PSEUDO LINEAR SYSTEMS 125 6.6 HISTORICAL NOTES AND
CONCLUDING REMARKS 126 6.7 APPENDIX 128 6.7.A REALIZATION THEORY 128
6.7.A-1 DERIVING PSEUDO LINEAR SYSTEMS FROM AFFINE DYNAMICAL SYSTEMS 128
6.7.A-2 LINEAR STATE STRUCTURE: FREE MOTIONS 129 6.7.A-3 FREE MOTIONS
WITH AN AFFINE MAP 132 6.7.A-4 FREE MOTIONS WITH A READOUT MAP 135
6.7.A-5 PSEUDO LINEAR SYSTEMS 136 TABLE OF CONTENTS IX 6.7.A-6
SOPHISTICATED PSEUDO LINEAR SYSTEMS 137 6.7.B FINITE DIMENSIONALITY 140
6.7.B-1 FINITE DIMENSIONAL FREE MOTIONS WITH AN AFFINE MAP 140 6.7.B-2
FINITE DIMENSIONAL FREE MOTIONS WITH A READOUT MAP 143 6.7.B-3 FINITE
DIMENSIONAL PSEUDO LINEAR SYSTEMS 145 6.7.B-4 EXISTENCE CRITERION FOR
PSEUDO LINEAR SYSTEMS 146 6.7.B-5 REALIZATION PROCEDURE FOR PSEUDO
LINEAR SYSTEMS 146 6.7.C PARTIAL REALIZATION 147 6.7.C-1 FREE MOTIONS
WITH AN AFFINE MAP 147 6.7.C-2 FREE MOTION WITH A READOUT MAP 148
6.7.C-3 PARTIAL REALIZATION PROBLEM 149 6.7.C-4 REAL-TIME PARTIAL
REALIZATION PROBLEM 154 7 ALMOST LINEAR SYSTEMS 156 7.1 TIME-INVARIANT,
AFFINE INPUT RESPONSE MAP (INPUT RESPONSE MAPS WITH TIME-INVARIANCE AND
AFFINITY).... 156 7.2 ALMOST LINEAR SYSTEMS 157 7.3 FINITE DIMENSIONAL
ALMOST LINEAR SYSTEMS 160 7.4 PARTIAL REALIZATION THEORY OF ALMOST
LINEAR SYSTEMS 165 7.5 REAL-TIME PARTIAL REALIZATION THEORY OF ALMOST
LINEAR SYSTEMS 167 7.6 HISTORICAL NOTES AND CONCLUDING REMARKS 169 7.7
APPENDIX 170 7.7.A REALIZATION THEORY 171 7.7.A-1 DERIVING ALMOST LINEAR
SYSTEMS FROM PSEUDO LINEAR SYSTEMS 171 7.7.A-2 FREE MOTIONS WITH AN
AFFINE MAP 172 7.7.A-3 FREE MOTIONS WITH A READOUT MAP 176 7.7.A-4
ALMOST LINEAR SYSTEMS 178 7.7.A-5 SOPHISTICATED ALMOST LINEAR SYSTEMS
179 7.7.B FINITE DIMENSIONALITY 181 7.7.B-1 FINITE DIMENSIONAL FREE
MOTIONS WITH AFFINE MAP 181 7.7.B-2 FINITE DIMENSIONAL FREE MOTIONS WITH
A READOUT MAP 184 7.7.B-3 FINITE DIMENSIONAL ALMOST LINEAR SYSTEMS 186
7.7.B-4 EXISTENCE CRITERION FOR ALMOST LINEAR SYSTEMS 187 7.7.B-5
REALIZATION PROCEDURE FOR ALMOST LINEAR SYSTEMS 187 7.7.C PARTIAL
REALIZATION 188 7.7.C-1 FREE MOTIONS WITH AN AFFINE MAP 188 7.7.C-2 FREE
MOTIONS WITH A READOUT MAP 189 X TABLE OF CONTENTS 7.7.C-3 PARTIAL
REALIZATION PROBLEM 190 7.7.C-4 REAL-TIME PARTIAL REALIZATION OF ALMOST
LINEAR SYSTEMS 194 8 SO-CALLED LINEAR SYSTEMS (LINEAR SYSTEMS WITH A
NON-ZERO INITIAL STATE) 19 5 8.1 INPUT OUTPUT RELATIONS FOR SO-CALLED
LINEAR SYSTEMS 196 8.2 SO-CALLED LINEAR SYSTEMS AND ALMOST LINEAR
SYSTEMS 197 8.3 PARTIAL REALIZATION THEORY OF SO-CALLED LINEAR SYSTEMS
203 8.4 REAL-TIME PARTIAL REALIZATION THEORY OF SO-CALLED LINEAR SYSTEMS
207 8.5 HISTORICAL NOTES AND CONCLUDING REMARKS 208 8.6 APPENDIX 210
8.6.A SO-CALLED LINEAR SYSTEMS AND ALMOST LINEAR SYSTEMS 210 8.6.B
PARTIAL REALIZATION 211 8.6.C REAL-TIME PARTIAL REALIZATION 217
REFERENCES 219 INDEX 227
|
| adam_txt |
TSUYOSHI MATSUO * YASUMICHI HASEGAWA REALIZATION THEORY OF DISCRETE-TIME
DYNAMICAL SYSTEMS WITH 14 FIGURES SPRINGER TABLE OF CONTENTS 1
INTRODUCTION 1 2 INPUT/OUTPUT MAPS 9 2.1 CONCATENATION MONOID: SET OF
EXPERIMENTS 9 2.2 INPUT RESPONSE MAP (INPUT/OUTPUT MAP WITH CAUSALITY)
10 2.3 HISTORICAL NOTES AND CONCLUDING REMARKS 14 3 GENERAL DYNAMICAL
SYSTEMS 15 3.1 REALIZATION THEORY OF GENERAL DYNAMICAL SYSTEMS 15 3.2
FINITE GENERAL DYNAMICAL SYSTEMS 18 3.3 CONTROL SYSTEMS AND
MULTI-EXPERIMENTS 23 3.4 HISTORICAL NOTES AND CONCLUDING REMARKS 26 3.5
APPENDIX 28 3.5.A [/-ACTIONS 28 3.5.B POINTED U-ACTIONS 30 3.5.C
[/-ACTIONS WITH A READOUT MAP 32 3.5.D GENERAL DYNAMICAL SYSTEMS 33 4
LINEAR REPRESENTATION SYSTEMS 38 4.1 REALIZATION THEORY OF LINEAR
REPRESENTATION SYSTEMS 38 4.2 FINITE DIMENSIONAL LINEAR REPRESENTATION
SYSTEMS 41 4.3 PARTIAL REALIZATION THEORY OF LINEAR REPRESENTATION
SYSTEMS . .48 4.4 HISTORICAL NOTES AND CONCLUDING REMARKS 50 4.5
APPENDIX 52 4.5.A REALIZATION THEOREM 52 4.5.A-1 LINEAR STATE STRUCTURE:
LINEAR U-ACTIONS 52 4.5.A-2 POINTED LINEAR U-ACTIONS 57 4.5.A-3 LINEAR
U-ACTIONS WITH A READOUT MAP 59 4.5.A-4 LINEAR REPRESENTATION SYSTEMS 60
4.5.A-5 SOPHISTICATED REALIZATION THEOREM 61 4.5.B FINITE DIMENSIONALITY
64 4.5.B-1 POINTED FINITE DIMENSIONAL LINEAR U-ACTIONS 64 4.5.B-2 FINITE
DIMENSIONAL LINEAR U-ACTIONS WITH A READOUT MAP . 67 4.5.B-3 FINITE
DIMENSIONAL LINEAR REPRESENTATION SYSTEMS 69 4.5.B-4 EXISTENCE CRITERION
FOR LINEAR REPRESENTATION SYSTEMS 70 VIII TABLE OF CONTENTS 4.5.B-5
REALIZATION PROCEDURE FOR LINEAR REPRESENTATION SYSTEMS 71 4.5.C PARTIAL
REALIZATION 71 4.5.C-1 POINTED LINEAR U-ACTIONS 71 4.5.C-2 LINEAR
U-ACTIONS WITH A READOUT MAP 72 4.5.C-3 PARTIAL REALIZATION PROBLEM 73 5
AFFINE DYNAMICAL SYSTEMS 79 5.1 REALIZATION THEOREM OF AFFINE DYNAMICAL
SYSTEMS 79 5.2 FINITE DIMENSIONAL AFFINE DYNAMICAL SYSTEMS 84 5.3
HISTORICAL NOTES AND CONCLUDING REMARKS 91 5.4 APPENDIX 93 5.4.A
REALIZATION THEOREM 93 5.4.A-1 FUNDAMENTALS FOR AFFINE DYNAMICAL SYSTEMS
93 5.4.A-2 AFFINE STATE STRUCTURE: LINEAR [/-ACTIONS 94 5.4.A-3 AFFINE
STATE STRUCTURE WITH A READOUT MAP 98 5.4.A-4 AFFINE DYNAMICAL SYSTEMS
101 5.4.A-5 SOPHISTICATED REALIZATION THEOREM 103 5.4.B FINITE
DIMENSIONALITY 105 5.4.B-1 POINTED FINITE DIMENSIONAL LINEAR U-ACTIONS
105 5.4.B-2 FINITE DIMENSIONAL LINEAR U-ACTIONS WITH A READOUT MAP .
108 5.4.B-3 FINITE DIMENSIONAL AFFINE DYNAMICAL SYSTEMS 110 5.4.B-4
EXISTENCE CRITERION FOR AFFINE DYNAMICAL SYSTEMS 111 5.4.B-5 REALIZATION
PROCEDURE FOR AFFINE DYNAMICAL SYSTEMS 112 6 PSEUDO LINEAR SYSTEMS 113
6.1 INPUT RESPONSE MAPS WITH TIME-INVARIANCE 113 6.2 PSEUDO LINEAR
SYSTEMS 114 6.3 FINITE DIMENSIONAL PSEUDO LINEAR SYSTEMS 117 6.4 PARTIAL
REALIZATION OF PSEUDO LINEAR SYSTEMS 122 6.5 REAL-TIME PARTIAL
REALIZATION OF PSEUDO LINEAR SYSTEMS 125 6.6 HISTORICAL NOTES AND
CONCLUDING REMARKS 126 6.7 APPENDIX 128 6.7.A REALIZATION THEORY 128
6.7.A-1 DERIVING PSEUDO LINEAR SYSTEMS FROM AFFINE DYNAMICAL SYSTEMS 128
6.7.A-2 LINEAR STATE STRUCTURE: FREE MOTIONS 129 6.7.A-3 FREE MOTIONS
WITH AN AFFINE MAP 132 6.7.A-4 FREE MOTIONS WITH A READOUT MAP 135
6.7.A-5 PSEUDO LINEAR SYSTEMS 136 TABLE OF CONTENTS IX 6.7.A-6
SOPHISTICATED PSEUDO LINEAR SYSTEMS 137 6.7.B FINITE DIMENSIONALITY 140
6.7.B-1 FINITE DIMENSIONAL FREE MOTIONS WITH AN AFFINE MAP 140 6.7.B-2
FINITE DIMENSIONAL FREE MOTIONS WITH A READOUT MAP 143 6.7.B-3 FINITE
DIMENSIONAL PSEUDO LINEAR SYSTEMS 145 6.7.B-4 EXISTENCE CRITERION FOR
PSEUDO LINEAR SYSTEMS 146 6.7.B-5 REALIZATION PROCEDURE FOR PSEUDO
LINEAR SYSTEMS 146 6.7.C PARTIAL REALIZATION 147 6.7.C-1 FREE MOTIONS
WITH AN AFFINE MAP 147 6.7.C-2 FREE MOTION WITH A READOUT MAP 148
6.7.C-3 PARTIAL REALIZATION PROBLEM 149 6.7.C-4 REAL-TIME PARTIAL
REALIZATION PROBLEM 154 7 ALMOST LINEAR SYSTEMS 156 7.1 TIME-INVARIANT,
AFFINE INPUT RESPONSE MAP (INPUT RESPONSE MAPS WITH TIME-INVARIANCE AND
AFFINITY). 156 7.2 ALMOST LINEAR SYSTEMS 157 7.3 FINITE DIMENSIONAL
ALMOST LINEAR SYSTEMS 160 7.4 PARTIAL REALIZATION THEORY OF ALMOST
LINEAR SYSTEMS 165 7.5 REAL-TIME PARTIAL REALIZATION THEORY OF ALMOST
LINEAR SYSTEMS 167 7.6 HISTORICAL NOTES AND CONCLUDING REMARKS 169 7.7
APPENDIX 170 7.7.A REALIZATION THEORY 171 7.7.A-1 DERIVING ALMOST LINEAR
SYSTEMS FROM PSEUDO LINEAR SYSTEMS 171 7.7.A-2 FREE MOTIONS WITH AN
AFFINE MAP 172 7.7.A-3 FREE MOTIONS WITH A READOUT MAP 176 7.7.A-4
ALMOST LINEAR SYSTEMS 178 7.7.A-5 SOPHISTICATED ALMOST LINEAR SYSTEMS
179 7.7.B FINITE DIMENSIONALITY 181 7.7.B-1 FINITE DIMENSIONAL FREE
MOTIONS WITH AFFINE MAP 181 7.7.B-2 FINITE DIMENSIONAL FREE MOTIONS WITH
A READOUT MAP 184 7.7.B-3 FINITE DIMENSIONAL ALMOST LINEAR SYSTEMS 186
7.7.B-4 EXISTENCE CRITERION FOR ALMOST LINEAR SYSTEMS 187 7.7.B-5
REALIZATION PROCEDURE FOR ALMOST LINEAR SYSTEMS 187 7.7.C PARTIAL
REALIZATION 188 7.7.C-1 FREE MOTIONS WITH AN AFFINE MAP 188 7.7.C-2 FREE
MOTIONS WITH A READOUT MAP 189 X TABLE OF CONTENTS 7.7.C-3 PARTIAL
REALIZATION PROBLEM 190 7.7.C-4 REAL-TIME PARTIAL REALIZATION OF ALMOST
LINEAR SYSTEMS 194 8 SO-CALLED LINEAR SYSTEMS (LINEAR SYSTEMS WITH A
NON-ZERO INITIAL STATE) 19 5 8.1 INPUT OUTPUT RELATIONS FOR SO-CALLED
LINEAR SYSTEMS 196 8.2 SO-CALLED LINEAR SYSTEMS AND ALMOST LINEAR
SYSTEMS 197 8.3 PARTIAL REALIZATION THEORY OF SO-CALLED LINEAR SYSTEMS
203 8.4 REAL-TIME PARTIAL REALIZATION THEORY OF SO-CALLED LINEAR SYSTEMS
207 8.5 HISTORICAL NOTES AND CONCLUDING REMARKS 208 8.6 APPENDIX 210
8.6.A SO-CALLED LINEAR SYSTEMS AND ALMOST LINEAR SYSTEMS 210 8.6.B
PARTIAL REALIZATION 211 8.6.C REAL-TIME PARTIAL REALIZATION 217
REFERENCES 219 INDEX 227 |
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| discipline | Informatik Mathematik |
| discipline_str_mv | Informatik Mathematik |
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| id | DE-604.BV022124495 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:16:24Z |
| indexdate | 2024-07-09T20:50:59Z |
| institution | BVB |
| isbn | 3540406751 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015339170 |
| oclc_num | 52729046 |
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| physical | X, 232 S. graph. Darst. |
| publishDate | 2003 |
| publishDateSearch | 2003 |
| publishDateSort | 2003 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in control and information sciences |
| series2 | Lecture notes in control and information sciences Engineering online library |
| spelling | Matsuo, Tsuyoshi Verfasser aut Realization theory of discrete-time dynamical systems Tsuyoshi Matsuo ; Yasumichi Hasegawa Berlin [u.a.] Springer 2003 X, 232 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in control and information sciences 296 Engineering online library Literaturverz. S. [219] - 225 Systèmes échantillonnés Discrete-time systems Realisierung (DE-588)4294084-9 gnd rswk-swf Zeitdiskretes System (DE-588)4127297-3 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s DE-604 Zeitdiskretes System (DE-588)4127297-3 s Realisierung (DE-588)4294084-9 s Hasegawa, Yasumichi Verfasser aut Lecture notes in control and information sciences 296 (DE-604)BV005848579 296 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015339170&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Matsuo, Tsuyoshi Hasegawa, Yasumichi Realization theory of discrete-time dynamical systems Lecture notes in control and information sciences Systèmes échantillonnés Discrete-time systems Realisierung (DE-588)4294084-9 gnd Zeitdiskretes System (DE-588)4127297-3 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4294084-9 (DE-588)4127297-3 (DE-588)4013396-5 |
| title | Realization theory of discrete-time dynamical systems |
| title_auth | Realization theory of discrete-time dynamical systems |
| title_exact_search | Realization theory of discrete-time dynamical systems |
| title_exact_search_txtP | Realization theory of discrete-time dynamical systems |
| title_full | Realization theory of discrete-time dynamical systems Tsuyoshi Matsuo ; Yasumichi Hasegawa |
| title_fullStr | Realization theory of discrete-time dynamical systems Tsuyoshi Matsuo ; Yasumichi Hasegawa |
| title_full_unstemmed | Realization theory of discrete-time dynamical systems Tsuyoshi Matsuo ; Yasumichi Hasegawa |
| title_short | Realization theory of discrete-time dynamical systems |
| title_sort | realization theory of discrete time dynamical systems |
| topic | Systèmes échantillonnés Discrete-time systems Realisierung (DE-588)4294084-9 gnd Zeitdiskretes System (DE-588)4127297-3 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Systèmes échantillonnés Discrete-time systems Realisierung Zeitdiskretes System Dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015339170&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV005848579 |
| work_keys_str_mv | AT matsuotsuyoshi realizationtheoryofdiscretetimedynamicalsystems AT hasegawayasumichi realizationtheoryofdiscretetimedynamicalsystems |