Holdings: Algebraic and differential topology of robust stability

Algebraic and differential topology of robust stability:

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Bibliographic Details
Main Author:	Jonckheere, Edmond A. (Author)
Format:	Book
Language:	English
Published:	New York [u.a.] Oxford Univ. Press 1997
Subjects:	Algebraic topology Control theory Differential topology Algebraische Topologie Differentialtopologie Robuste Kontrolle
Online Access:	Inhaltsverzeichnis
Physical Description:	XLVI, 576 S. graph. Darst.
ISBN:	0195093011

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adam_text	ALGEBRAIC AND DIFFERENTIAL TOPOLOGY OF ROBUST STABILITY EDMOND A. JONCKHEERE UNIVERSITY OF SOUTHERN CALIFORNIA DEPARTMENT OF ELECTRICAL ENGINEERINGSYSTEMS AND CENTER FOR APPLIED MATHEMATICAL SCIENCES LOS ANGELES, CALIFORNIA WITH 79 PICTURES, COMPUTER GENERATED BY CHIH-YUNG CHENG, CHUNG-KUANG CHU, AND MURILO G. COUTINHO / NEW YORK OXFORD OXFORD UNIVERSITY PRESS 1997 CONTENTS LIST OF FIGURES XXXV LIST OF SYMBOLS XXXIX 1 PROLOGUE 1 1 SIMPLICIAL APPROXIMATION AND ALGORITHMS 2 ROBUST MULTIVARIABLE NYQUIST CRITERION 7 2.1 MULTIVARIABLE NYQUIST CRITERION 7 2.2 ROBUST MULTIVARIABLE NYQUIST CRITERION 9 -- 2.3 UNCERTAINTY SPACE 11 2.4 PUNCTURED UNCERTAINTY SPACES 14 2.5 COMPACTIFICATION OF IMAGINARY AXIS 15 2.6 HOROWITZ SUPERTEMPLATE APPROACH 16 2.6.1 NO IMAGINARY-AXIS OPEN-LOOP POLES 16 2.6.2 IMAGINARY-AXIS OPEN-LOOP POLES 17 2.7 CROSSOVER 19 2.8 MAPPING INTO OTHER SPACES 20 3 A BASIC TOPOLOGICAL PROBLEM 22 3.1 THE BOUNDARY PROBLEM 22 , 3.2 TOPOLOGY FOR BOUNDARY AND CONTINUITY * 23 3.3 MATHEMATICAL FORMULATION OF BOUNDARY PROBLEM 25 3.4 EXAMPLE (CONTINUOUS FRACTION CRITERION) 26 3.5 EXAMPLE (KHARITONOV) 27 3.6 EXAMPLE (REAL STRUCTURED SINGULAR VALUE) 29 3.7 EXAMPLE (BROUWER DOMAIN INVARIANCE) 32 3.8 EXAMPLE (COVERING MAP) 34 3.9 EXAMPLE (HOLOMORPHIC MAPPING) 36 3.10 EXAMPLE (PROPER MAPPING) 37 3.11 EXAMPLE (CONFORMAL MAPPING) 38 ,. 3.11.1 LOCALLY CONNECTED BOUNDARY 40 3.11.2 GENERAL BOUNDARY 41 3.12 EXAMPLES (HOROWITZ) 45 3.12.1 UNCERTAIN GAIN/REAL POLE 45 3.12.2 UNCERTAIN POLE/ZERO PAIR 46 3.13 EXAMPLE (FUNCTIONS ON POLYDISKS) 46 CONTENTS 3.14 SEVERAL COMPLEX VARIABLES 48 3.15 EXAMPLE (PLURISUBHARMONIC FUNCTIONS) 49 3.16 EXAMPLE (PROPER HOLOMORPHIC AND BIHOLOMORPHIC MAPS) 50 3.17 EXAMPLE (WHITNEY S ROOT SYSTEM) 51 3.17.1 NO DEGREE UNCERTAINTY 52 3.17.2 UNCERTAIN DEGREE 55 SIMPLICIAL APPROXIMATION 59 4.1 SIMPLEXES, COMPLEXES, AND POLYHEDRA 60 4.2 ABSTRACT COMPLEXES 66 4.3 ALEXANDROFF THEOREM 66 4.4 SIMPLICIAL APPROXIMATIONPOINT SET TOPOLOGY 68 4.5 SIMPLICIAL MAPALGEBRA 73 4.5.1 SIMPLICIAL THEORY 73 4.5.2 SEMISIMPLICIAL THEORY 79 4.6 COMPUTATIONAL ISSUES 81 4.7 RELATIVE SIMPLICIAL APPROXIMATION 83 S 4.8 CELL COMPLEXES AND CELLULAR MAPS 84 4.9 HISTORICAL NOTES 84 CARTESIAN PRODUCT OF MANY UNCERTAINTIES 8 6 5.1 PRISMATIC DECOMPOSITION 87 5.1.1 SIMPLICIAL UNCERTAINTY-FREQUENCY PRODUCT 87 5.1.2 SEMISIMPLICIAL CONCEPTUALIZATION 89 5.2 BOUNDARY OF CARTESIAN PRODUCT 90 5.2.1 SIMPLICIAL APPROACH 90 5.2.2 SEMISIMPLICIAL APPROACH 91 5.3 SIMPLICIAL COMBINATORICS OF CUBE 92 5.4 Q-TRIANGULATION 94 5.5 COMBINATORIAL EQUIVALENCE 95 5.6 FLATNESS 96 COMPUTATIONAL GEOMETRY 99 6.1 DELAUNAY TRIANGULATION OF TEMPLATE 100 6.2 SIMPLICIAL EDGE MAPPING 102 6.3 THE SIMPLICIALVIEW SOFTWARE 105 6.3.1 COARSE INITIAL REFINEMENT 105 6.3.2 VORONOI DIAGRAM AND DELAUNAY TRIANGULATION 105 6.3.3 REFINEMENT AND POINT LOCATION 106 6.3.4 CHECKING SIMPLICIAL PROPERTY 106 6.3.5 IDENTIFYING SIMPLEX CONTAINING THE ORIGIN 106 6.3.6 INVERSE IMAGE 107 6.4 NUMERICAL STABILITY, FLATNESS, AND CONDITIONING 107 CONTENTS 6.5 MAKING MAP (LOCALLY) SIMPLICIAL 107 6.6 PROCEDURE 108 7 PIECEWISE-LINEAR NYQUIST MAP 113 7.1 PIECEWISE-LINEAR NYQUIST MAP 113 7.1.1 A LINEAR PROGRAM 114 7.1.2 POLYHEDRAL CROSSOVER 116 7.2 FROM PIECEWISE-LINEAR TO SIMPLICIAL MAP 116 7.2.1 SIMPLICIAL APPROXIMATION TO PIECEWISE-LINEAR MAP 116 7.2.2 MAKING PIECEWISE-LINEAR MAP SIMPLICIAL 119 7.3 STRICT LINEAR COMPLEMENTARITY 121 8 GAME OF THE HEX ALGORITHM 124 * » 8.1 2-D HEX BOARD 124 ^ 8.2 N-D HEX BOARD 127 8.3 COMBINATORIAL EQUIVALENCE 127 8.4 TWO-DIMENSIONAL HEX GAME ALGORITHM 128 ^ - 8.4.1 PRIMAL 128 8.4.2 DUAL 129 8.4.3 COMPLEXITY 130 8.5 THREE-DIMENSIONAL HEX GAME ALGORITHM 131 8.5.1 DUAL 131 8.5.2 PRIMAL 132 8.6 HIGHER-DIMENSIONAL HEX GAMES 134 9 SIMPLICIAL ALGORITHMS 136 9.1 SIMPLICIAL ALGORITHMS OVER 2-D UNCERTAINTY SPACE 137 9.1.1 INTEGER LABELING 137 9.1.2 SEARCHINGFUNDAMENTAL GRAPH LEMMA 140 9.1.3 GRID REFINEMENT AND SPERNER S LEMMA 141 9.1.4 VECTOR LABELING 144 9.1.5 TEXTBOOK EXAMPLE 147 9.2 SIMPLICIAL ALGORITHMS OVER 3-D UNCERTAINTY SPACE 149 9.2.1 ALGORITHM 151 9.2.2 A 2-TORUS EXAMPLE 151 9.2.3 EXAMPLE 152 9.2.4 ALGEBRAIC CURVE INTERPRETATION 154 9.3 RELATIVE UNCERTAINTY COMPLEX 156 9.4 SIMPLICIAL LABELING MAP 158 9.4.1 ABSTRACT LABEL COMPLEX 158 9.4.2 (STRONG) DEFORMATION RETRACT OF TEMPLATE 159 9.5 ALGORITHMINTEGER SEARCH 162 9.6 ALGORITHMVECTOR LABELING SEARCH 163 XXVI CONTENTS N HOMOLOGY OF ROBUST STABILITY 10 HOMOLOGY OF UNCERTAINTY AND OTHER SPACES 169 10.1 SIMPLICIAL HOMOLOGY 170 10.1.1 HOMOLOGY GROUPS 170 10.1.2 HOMOLOGY GROUP HOMOMORPHISM 173 10.1.3 COMPUTATION 175 10.2 SEMISIMPLICIAL HOMOLOGY 175 10.3 HOMOLOGY OF A CHAIN COMPLEX 176 10.4 HOMOTOPY INVARIANCE 176 10.4.1 CHAIN HOMOTOPY 177 10.4.2 ACYCLIC CARRIERS 177 10.4.3 INVARIANCE UNDER HOMOTOPY 178 10.4.4 HOMOTOPY EQUIVALENCE 179 ^ 10.5 HOMOLOGY OF PRODUCT OF UNCERTAINTY 179 10.5.1 EILENBERG-ZILBER THEOREM 180 10.5.2 KUNNETH THEOREM 182, / 10.5.3 REMARK 183 10.5.4 APPLICATIONUNCERTAINTY-FREQUENCY PRODUCT 183 10.5.5 APPLICATIONUNCERTAINTY TORUS 183 10.6 UNCERTAINTY MANIFOLDMAYER-VIETORIS SEQUENCE 184 10.6.1 APPLICATIONHOMOLOGY OF SPECIAL UNITARY UNCERTAINTY 185 10.7 RELATIVE HOMOLOGY SEQUENCE 186 10.8 MORE SOPHISTICATED HOMOLOGY COMPUTATION 187 11 HOMOLOGY OF CROSSOVER 189 11.1 COMBINATORIAL HOMOLOGY OF CROSSOVER 189 I 11.2 PROJECTING THE CROSSOVER 191! 12 COHOMOLOGY 192 12.1 SIMPLICIAL COHOMOLOGY 192 12.1.1 COHOMOLOGY GROUPS 193 12.1.2 COHOMOLOGY GROUP HOMOMORPHISM 194 12.1.3 CUP PRODUCT 194 12.1.4 COHOMOLOGY OF PRODUCT SPACE 195 12.2 DE RHAM COHOMOLOGY 195 I 12.2.1 COHOMOLOGY OF DIFFERENTIAL FORMS 196 12.2.2 PULL-BACK 197 12.2.3 WEDGE PRODUCT 197 13 TWISTED CARTESIAN PRODUCT OF UNCERTAINTIES 199 13.1 FIBER BUNDLE 200 13.1.1 BASIC DEFINITIONS AND CONCEPTS 201 CONTENTS XXVII 13.1.2 BUNDLE MORPHISMS 203 13.1.3 CLUTCHING 203 13.1.4 CROSS SECTION 204 13.1.5 BUNDLE INTERPRETATION OF KHARITONOV S THEOREM 204 13.1.6 PRINCIPAL BUNDLE 205 13.1.7 TANGENT BUNDLE 207 13.1.8 ELEMENTARY HOMOTOPY THEORY OF BUNDLES 208 13.1.9 FIBER BUNDLE INTERPRETATION OF DOLE2AL S THEOREM 210 13.2 SEMISIMPLICIAL BUNDLES 211 13.2.1 BACKGROUND 212 13.2.2 SEMISIMPLICIAL BUNDLE 214 13.2.3 TWISTED CARTESIAN PRODUCT 216 13.2.4 EXAMPLE 219 X 13.2.5 CROSS SECTION 221 13.2.6 COMMUTATIVITY 221 13.2.7 TWISTED TENSOR PRODUCT 222 13.3 NYQUIST FIBRATION 223 / 13.4 SEMISIMPLICIAL FIBRATION 227 13.5 SUMMARY 228 14 SPECTRAL SEQUENCE OF NYQUIST MAP 230 14.1 HOMOLOGY SPECTRAL SEQUENCE 234 14.1.1 GRADUATION AND FILTRATION 234 14.1.2 FILTRATION OF CYCLE AND BOUNDARY GROUPS 236 14.1.3 FILTRATION OF HOMOLOGY GROUP 237 14.1.4 SUCCESSIVE APPROXIMATION 239 14.1.5 INITIALIZATION 240 14.1.6 CONVERGENCE 241 14.2 EXAMPLE (SPECTRAL SEQUENCE OF A MATRIX) 241 14.3 SPECTRAL SEQUENCE OF GEOMETRIC SYSTEM THEORY 244 14.4 COHOMOLOGY SPECTRAL SEQUENCE 245 14.5 DIHOMOLOGY SPECTRAL SEQUENCE 247 14.6 LERAY-SERRE SPECTRAL SEQUENCE 250 14.7 SEMISIMPLICIAL SERRE SPECTRAL SEQUENCE OF NYQUIST MAP 252 14.7.1 TWISTED CARTESIAN PRODUCT 253 14.7.2 TWISTED TENSOR PRODUCT 253 14.8 EILENBERG-MOORE SPECTRAL SEQUENCE 253 14.8.1 CONTRACTIBLE TEMPLATEOPEN-LOOP STABLE CASE 255 14.8.2 UNCONTRACTIBLE TEMPLATE 256 XXVIII CONTENTS III HOMOTOPY OF ROBUST STABILITY 15 HOMOTOPY GROUPS AND SEQUENCES 259 15.1 HOMOTOPY GROUPS 259 15.2 HOMOTOPY GROUP HOMOMORPHISM 261 15.3 HOMOTOPY GROUPS OF SPHERES 261 15.4 BASIC OBSTRUCTION RESULT 262 15.5 HOMOTOPY SEQUENCE OF NYQUIST FIBRATION 262 15.5.1 EXACT HOMOTOPY SEQUENCE 263 15.6 COROLLARIES OF EXACT HOMOTOPY SEQUENCE 264 15.7 HISTORICAL NOTES 265 16 OBSTRUCTION TO EXTENDING THE NYQUIST MAP 266 16.1 STATEMENT OF NYQUIST EXTENSION PROBLEM 267 16.2 OBSTRUCTION TO EXTENDING A GENERAL MAP 269 16.2.1 PATH CONNECTEDNESS OF RANGE 270 16.2.2 ABSOLUTE OBSTRUCTION TO EXTENSION 270/ 16.2.3 RELATIVE OBSTRUCTION TO EXTENSION 272 16.3 OBSTRUCTION TO EXTENDING NYQUIST MAP 272 16.3.1 ABSOLUTE RESULTS 273 16.3.2 RELATIVE RESULTS 274 16.4 WEAK CONVERSE 275 16.5 COMPUTATION OF HOMOTOPY CLASS 276 16.5.1 PIECEWISE-LINEAR NYQUIST MAP 276 16.5.2 COMPARISON WITH PART I 278 16.5.3 RELATIVE RESULTS 279 16.6 HOMOTOPY EXTENSION 279 16.7 HOMOTOPY EXTENSION AND EDGE TESTS 281 16.8 APPENDIXOBSTRUCTION TO CROSS SECTIONING 282 17 HOMOTOPY CLASSIFICATION OF NYQUIST MAPS 284 17.1 FUNDAMENTAL CLASSIFICATION RESULT 284 17.2 CLASSIFICATION OF MAPS TO SPHERES - 285 17.3 ELEMENTARY PROOF OF MAIN RESULT 285 17.4 COHOMOLOGY OF PRODUCT OF UNCERTAINTY 289 17.4.1 MULTIVARIABLE PHASE MARGIN 290 17.4.2 SPECIAL ORTHOGONAL PERTURBATION 290 17.5 FORMAL CLASSIFICATION 290 18 BROUWER DEGREE OF NYQUIST MAP 292 18.1 ORIENTATION 293 18.2 COMBINATORIAL DEGREE 294 18.3 ANALYTICAL DEGREE 296 CONTENTS 18.4 HOMOLOGICAL DEGREE OF MAPS BETWEEN SPHERES 298 18.5 SIMPLE EXAMPLES 298 18.5.1 DEGREE OF A LINEAR MAP 298 18.5.2 DEGREE OF A HOLOMORPHIC FUNCTION 299 18.5.3 DEGREE OF REAL POLYNOMIAL MAP 300 18.5.4 APPLICATION TO ROBUST STABILITY 300 18.6 APPLICATION (INDEX OF VECTOR FIELD) 300 18.7 COHOMOLOGICAL DEGREE OF MAPS TO SPHERES 301 18.7.1 DEGREE OF NYQUIST-RELATED MAP 302 18.7.2 DEGREE OF MAPS FROM MANIFOLDS TO SPHERES 303 18.7.3 A COUNTEREXAMPLE 304 18.8 DEGREE PROOF OF SUPERSTRONG SPERNER LEMMA 305 18.9 DEGREE OF MAPS BETWEEN PSEUDOMANIFOLDS 306 18.10 HOMOTOPY COLLAPSE OF TEMPLATE 30 ( 18.11 CONTINUATION OR EMBEDDING METHODS 308 18.12 HISTORICAL NOTES 311 19 HOMOTOPY OF MATRIX RETURN DIFFERENCE MAP 313 19.1 MATRIX RETURN DIFFERENCE 314 19.2 GENERAL LINEAR VERSUS UNITARY GROUPS 314 19.3 HOMOTOPY GROUPS OF GL 316 19.3.1 STABLE HOMOTOPY (2N L N) 31 6 19.3.2 UNSTABLE HOMOTOPY (2N L N) 317 19.4 DEGREE (STABLE HOMOTOPY CASE) 319 19.4.1 2N, = N 320 19.4.2 2N, N 320 19.5 DIFFERENTIAL DEGREE 321 19.5.1 COHOMOLOGY OF GENERAL LINEAR GROUP 322 19.5.2 DE RHAM COHOMOLOGY OF DIFFERENTIAL FORMS 326 19.5.3 INVARIANT DIFFERENTIAL FORMS ON GL 327 19.5.4 INVARIANT DIFFERENTIAL FORMS ON U 332 19.5.5 EXAMPLE (SO GROUP) 337 19.5.6 PULL-BACK 338 19.5.7 DEGREE 338 19.5.8 CONNECTION WITH ANALYTICAL DEGREE 339 19.6 EXAMPLE (THE PRINCIPLE OF ARGUMENT) 341 19.7 EXAMPLE (MAPPING INTO SO) 342 19.7.1 DEGREE 1 MAP 342 19.7.2 DEGREE 2 MAP 343 19.8 EXAMPLE (BROUWER DEGREE) 344 19.9 EXAMPLE (MCMILLAN DEGREE) 344 19.10 OBSTRUCTION TO EXTENDING GL-VALUED NYQUIST MAP 345 XXX CONTENTS 20 K-THEORY OF ROBUST STABILIZATION 347 20.1 RETURN DIFFERENCE OPERATOR 349 20.1.1 OPEN-LOOP STABLE, DISCRETE-TIME SYSTEMS 349 - . 20.1.2 TOEPLITZ OPERATORS 350 20.1.3 OPEN-LOOP UNSTABLE SYSTEMS 350 20.1.4 CLOSED-LOOP STABILITY 351 20.2 INDEX OF FREDHOLM OPERATORS 352 20.3 INDEX OF FREDHOLM TOEPLITZ OPERATORS 356 20.4 INDEX OF FREDHOLM FAMILY 357 20.4.1 CONSTANT COKERNEL DIMENSION 358 20.4.2 VECTOR BUNDLE FORMULATION 360 20.5 K-GROUP 362 20.5.1 COMPLEX BUNDLE OVER UNCERTAINTY SPACE 362 20.5.2 EQUIVALENT BUNDLES 362 20.5.3 TRIVIAL BUNDLE 363 V 20.5.4 WHITNEY SUM 363 20.5.5 GROTHENDIECK CONSTRUCTION 364 20.5.6 K-GROUP 365 20.5.7 K-GROUP HOMOMORPHISM 365 20.6 INDEX OF UNCERTAIN RETURN DIFFERENCE OPERATOR 365 20.7 OPEN-LOOP UNSTABLE RETURN DIFFERENCE OPERATOR 369 20.8 REDUCED K-GROUPS 370 20.9 UNITARY APPROACH TO K-THEORY 371 20.9.1 CHERN CLASSES AND CHARACTER 373 20.10 HIGHER K-GROUPS AND BOTT PERIODICITY 374 20.11 INDEX FOR FREDHOLM TOEPLITZ FAMILY 376 20.12 ATIYAH-HIRZEBRUCH SPECTRAL SEQUENCE 378 20.13 KO-THEORY OF REAL PERTURBATION 379 20.14 KR-THEORY OF REAL PERTURBATION 380 20.15 CONNECTION WITH ALGEBRAIC K-THEORY 384 IV DIFFERENTIAL TOPOLOGY OF ROBUST STABILITY 21 COMPACT DIFFERENTIABLE UNCERTAINTY MANIFOLDS 391 21.1 COMPACT DIFFERENTIABLE UNCERTAINTY MANIFOLD 393 21.2 SINGULARITY ANALYSIS OF NYQUIST MAP 395 21.2.1 VARIATIONAL INTERPRETATION OF TEMPLATE BOUNDARY 395 21.2.2 BASIC FACTS OF MORSE THEORY 399 * 21.2.3 DEGENERACY PHENOMENA 401 21.2.4 THREE APPROACHES TO SINGULARITY ANALYSIS 403 21.3 NYQUIST CURVE AS CRITICAL VALUE PLOT 403 21.4 NASH FUNCTIONS 403 CONTENTS XX 21.5 SARD S THEOREM 406 21.6 CRITICAL VALUES PLOTS 407 .. 21.6.1 THE PROBLEM 407 21.6.2 CLASSIFICATION OF CRITICAL POINTS BY THEIR CODIMENSIONS 408 21.6.3 ISOTOPY 412 21.6.4 STRATIFICATION OF SPACE OF DIFFERENTIABLE FUNCTIONS 415 21.6.5 STRATIFICATION OF THE SPACE OF MORSE FUNCTIONS 417 21.6.6 LOCAL PROPERTIES OF FAMILY 418 21.6.7 GLOBAL PROPERTIES OF FAMILY 419 21.6.8 EFFECT OF VARIATION OF CERTAIN PARAMETERS 422 21.7 LOOPS OF CRITICAL POINTS 422 21.8 DEGREE APPROACH TO CRITICAL POINTS 426 21.9 VECTOR FIELD APPROACH TO CRITICAL POINTS 427^ 21.10 QUADRATIC DIFFERENTIAL OF NYQUIST MAP 428 21.11 THOM-BOARDMAN SINGULARITY SETS 429 21.12 THE CASE OF TWO UNCERTAIN PARAMETERS 431 21.13 TEMPLATE BOUNDARY REVISITED 433 . 21.14 EXAMPLE I 433 21.15 EXAMPLE II 439 21.16 CELL DECOMPOSITION 440 SINGULARITY OVER STRATIFIED UNCERTAINTY SPACE 450 22.1 (WHITNEY) STRATIFIED UNCERTAINTY SPACE 451 * . 22.2 STRATIFIED MORSE THEORY 454 22.3 BOUNDARY SINGULARITY 456 22.4 APPLICATION TO MAPPING THEOREMS 459 STRUCTURAL STABILITY OF CROSSOVER 460 23.1 JET SPACE 461 23.2 WHITNEY TOPOLOGY 463 23.2.1 C» CASE 463 23.2.2 C* AND C CASES 464 23.3 (ELEMENTARY) TRANSVERSALITY 466 23.4 SINGULARITY SETS REVISITED 467 23.4.1 ITERATED JACOBI EXTENSIONS 468 * . 23.4.2 JACOBI EXTENSION DEFINITION OF SINGULARITY SETS 468 23.4.3 JACOBIAN OF A SET OF FUNCTIONS 469 23.5 UNIVERSAL SINGULARITY SETS 469 * . * 23.5.1 TOTAL JACOBI EXTENSION 470 ; 23.5.2 UNIVERSAL SINGULARITY SETS 470 23.5.3 (STRONG) TRANSVERSALITY 471 23.6 STABILITY OF NYQUIST MAP 472 23.7 INFINITESIMAL STABILITY 474 XXXII CONTENTS 23.8 LOCAL INFINITESIMAL STABILITY 475 23.9 STABILITY OF WHITNEY FOLD AND CUSP 479 23.10 EXAMPLE (SIMPLE) 479 -23.11 EXAMPLE (WHITNEY FOLD) 480 23.12 EXAMPLE (PHASE MARGIN) 481 23.13 EXAMPLE (UNCERTAIN DEGREE) 482 23.14 EXAMPLE (POLE/ZERO CANCELLATION) 483 23.15 STRUCTURAL STABILITY OF CROSSOVER 484 23.16 THE COUNTER-EXAMPLE 487 V ALGEBRAIC GEOMETRY OF CROSSOVER 24 GEOMETRY OF CROSSOVER 495 24.1 CROSSOVER AS A REAL ALGEBRAIC SET 495 24.2 TRIANGULATION OF REAL ALGEBRAIC SETS 497 24.3 LOCAL EULER CHARACTERISTIC OF REAL ALGEBRAIC CROSSOVER SET 498 24.4 BETTI NUMBERS OF REAL ALGEBRAIC CROSSOVER SET 498 24.5 ALGEBRAIC CROSSOVER CURVE 501 24.6 EXAMPLE 501 25 GEOMETRY OF STABILITY BOUNDARY 504 25.1 TARSKI*SEIDENBERG ELIMINATION 504 25.2 COMPLEXITY 505 25.3 EXAMPLE 505 VI EPILOGUE 26 EPILOGUE 511 VII APPENDICES A HOMOLOGICAL ALGEBRA OF GROUPS 515 A.I ABELIAN GROUPS AND HOMOMORPHISMS 515 A.2 CHAIN COMPLEXES 517 A. 3 TENSOR PRODUCT 517 A.4 CATEGORIES AND FUNCTORS 517 A.5 EXACT SEQUENCES 518 A.6 FREE RESOLUTION 519 A.7 CONNECTING MORPHISM 520 A.8 TORSION PRODUCT 522 A.9 UNIVERSAL COEFFICIENT THEOREM 523 A.LOKIINNETH FORMULA 523 CONTENTS XXXIII MATRIX ANALYSIS OF INTEGRAL HOMOLOGY GROUPS 525 B.I MATRIX COMPUTATION OF HOMOLOGY GROUPS 525 B.2 HOPF TRACE THEOREM 527 HOMOLOGICAL ALGEBRA OF MODULES 530 C.I MODULES 530 C.I.I MODULES AND PROJECTIVE MODULES 530 C.I.2 PROJECTIVE RESOLUTION 531 C.I.3 TENSOR PRODUCT 532 C.1.4 HIGHER TORSION PRODUCTS 532 C.2 ALGEBRA 533 C.3 DIFFERENTIAL GRADED MODULES 534 C.3.1 DG MODULES 534 C.3.2 DG ALGEBRA 534 C.3.3 DG MODULE OVER DG ALGEBRA 535 C.3.4 TENSOR PRODUCT 535 C.3.5 TORSION PRODUCT 535 ALGEBRAIC SINGULARITY THEORY 537 D.I WEIERSTRASS PREPARATION THEOREM 537 D.I.I EXAMPLE (ROOT-LOCUS BREAKAWAY POINT) 538 D.1.2 PROOFS 539 D.2 MALGRANGE PREPARATION THEOREM 540 D.2.1 PROOFS 541 D.3 JETS AND GERMS 544 D.4 RINGS AND IDEALS OF FUNCTIONS 545 D.5 FORMAL INVERSE FUNCTION THEOREM 549 D.6 LOCAL RING OF A MAP 550 D.7 MODULES OVER RINGS OF FUNCTIONS 552 D.8 GENERALIZED MALGRANGE PREPARATION THEOREM 552 D.9 JACOBI IDEAL, CODIMENSION, AND DETERMINACY 554 D.10 UNIVERSAL UNFOLDING 555 BIBLIOGRAPHY 557 INDEX 571
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id	DE-604.BV011671408
illustrated	Illustrated
indexdate	2024-07-09T18:13:46Z
institution	BVB
isbn	0195093011
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-007867961
oclc_num	33667308
open_access_boolean
owner	DE-703 DE-11
owner_facet	DE-703 DE-11
physical	XLVI, 576 S. graph. Darst.
publishDate	1997
publishDateSearch	1997
publishDateSort	1997
publisher	Oxford Univ. Press
record_format	marc
spelling	Jonckheere, Edmond A. Verfasser aut Algebraic and differential topology of robust stability Edmond A. Jonckheere New York [u.a.] Oxford Univ. Press 1997 XLVI, 576 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algebraic topology Control theory Differential topology Algebraische Topologie (DE-588)4120861-4 gnd rswk-swf Differentialtopologie (DE-588)4012255-4 gnd rswk-swf Robuste Kontrolle (DE-588)4232797-0 gnd rswk-swf Algebraische Topologie (DE-588)4120861-4 s Robuste Kontrolle (DE-588)4232797-0 s DE-604 Differentialtopologie (DE-588)4012255-4 s GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007867961&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Jonckheere, Edmond A. Algebraic and differential topology of robust stability Algebraic topology Control theory Differential topology Algebraische Topologie (DE-588)4120861-4 gnd Differentialtopologie (DE-588)4012255-4 gnd Robuste Kontrolle (DE-588)4232797-0 gnd
subject_GND	(DE-588)4120861-4 (DE-588)4012255-4 (DE-588)4232797-0
title	Algebraic and differential topology of robust stability
title_auth	Algebraic and differential topology of robust stability
title_exact_search	Algebraic and differential topology of robust stability
title_full	Algebraic and differential topology of robust stability Edmond A. Jonckheere
title_fullStr	Algebraic and differential topology of robust stability Edmond A. Jonckheere
title_full_unstemmed	Algebraic and differential topology of robust stability Edmond A. Jonckheere
title_short	Algebraic and differential topology of robust stability
title_sort	algebraic and differential topology of robust stability
topic	Algebraic topology Control theory Differential topology Algebraische Topologie (DE-588)4120861-4 gnd Differentialtopologie (DE-588)4012255-4 gnd Robuste Kontrolle (DE-588)4232797-0 gnd
topic_facet	Algebraic topology Control theory Differential topology Algebraische Topologie Differentialtopologie Robuste Kontrolle
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007867961&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT jonckheereedmonda algebraicanddifferentialtopologyofrobuststability

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