Invariance and System Theory: Algebraic and Geometric Aspects
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845 Allen Tannenbaum
Invariance and System Theory: Algebraic and Geometric Aspects
Springer-Verlag Berlin Heidelberg New York 1981
Author Allen Tannenbaum Oepartment of Theoretical Mathematics The We izmann Institute of Sc ience Rehovotllsrael
AMS Subject Classifications (1980): 14020,14025, 93B15, 93B25, 93015 ISBN 3-540-10565-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10565-4 Springer-Verlag New York Heidelberg Berl in Th is work is subject to copyright. All rights are reserved, wh ether the whole or part of the material is conce rned, speci fica lly those of translation, repr int ing, re-use of illu strations, broad casting , reproduction by photo copy ing mac hine or similar means, and storag e in data banks. Under § 54 of the German Copy right Law where co pies are made for other than private use, a fee is payable to "Verwertunqsqes ellsc haft Wort", Mun ich.
© by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and bind ing : Beltz Offsetdruck, Hemsb ach/Bergstr. 2141 /3140-543210
To 'Rina
TABLE OF CONTENTS Introducti on Not a t ion a n d Te r mi no l ogy Part I.
Part I I .
Pa r t
I II.
Part IV .
Par t V.
Part VI.
Pa rt VII.
Some Basic Al geb r a i c Geomet ry
1
1. 2. 3. 4. 5. 6. 7.
1 3
Affine Geometry Pro j ect i v e Geometry Reg u l a r Mapping s Th e Constructibility Theo r em o f Cheval l ey Sheave s and Ve cto r Bundles The Grassman Va r i e ty Some Remarks on Algebraic Geometry Ov e r a No n Al g ebrai c ally Closed F ie l d
6 9 10 14 16
Some Bas i c Sys tem Theo ry
18
1. Dynami c al Sys tems 2 . Co n t ro ll ability and Reachabil i t y 3 . Constructibility and Ob s erv abi l i t y 4. State Variables 5. Transfe r Functions
18 22 24 26 32
Invariant Theory a nd Or bit Space P r oblems
36
1. 2. 3. 4. 5.
36 38 42 44 46
Algeb r a i c Groups On t h e Moduli o f En d omo r p h i sms Quo t i e n t s Reduc tive Groups a nd Hi lbert 's 1 4th Pr o b lem Richardso n 's Cri t e rion
Gl oba l Mod u li o f Linear Time-Inva r i a nt Dynamical Syst e ms
48
1. Comp le te Re achab i l i ty and Pre- S t ab i l ity 2 . Construction of the Quo tien t Space o f Comp l e t e l y Reachable Pairs 3. Moduli of Linear Time-Inv ariant Dy nami c al Systems 4. The Geomet ric St r uct ure of t he Modul i Sp a c e 5 . Th e Glob a l Modu l i o f Comp lete ly Reach able Matr ix Triple s 6 . Some Open P r oblems
48
Local Mod uli of Linear Time -In v a r i a nt Dyn amica l Sy stems
65
1. 2. 3. 4.
68 71 74
Vers a l De f o rm a t ions o f Matrix Pa i rs Th e Co n t r o l Ca non ica l Fo rm On the Cons t ructio n o f Holo morphic Canonical Forms Ve rsal De fo rmat ions of Matrix Trip l es
53 54 57 59 63
66
Algebrai c Re a l i z a t i o n Theo ry
76
1. Input/Output Maps and Ab s tract Realiz ation Theory 2 . Hankel Matr ices 3 . Realizations o f Rational Matrice s 4. Partial Reali zations 5 . Sy s t e ms o ve r Ri ngs 6 . polynomi a l Systems
76 80 84 87 89 96
On t he Geomet r y of Ra tional Tran s f er Functions
1 04
1. 2. 3. 4.
1 04 108 11 3 116
Cauchy I n d i
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