Passivity-based control of Euler-Lagrange systems: mechanical, electrical, and electromechanical applications
Gespeichert in:
Format: | Buch |
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Sprache: | English |
Veröffentlicht: |
London
Springer
1998
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Schriftenreihe: | Communications and control engineering
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Schlagworte: | |
Online-Zugang: | Publisher description Table of contents only Contributor biographical information Inhaltsverzeichnis |
Beschreibung: | Includes bibliographical references (p. 515-537) and index |
Beschreibung: | XXXIV, 543 S. Ill., graph. Darst. 24 cm |
ISBN: | 1852330163 |
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245 | 1 | 0 | |a Passivity-based control of Euler-Lagrange systems |b mechanical, electrical, and electromechanical applications |c Romeo Ortega ... [et al.] |
264 | 1 | |a London |b Springer |c 1998 | |
300 | |a XXXIV, 543 S. |b Ill., graph. Darst. |c 24 cm | ||
336 | |b txt |2 rdacontent | ||
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490 | 0 | |a Communications and control engineering | |
500 | |a Includes bibliographical references (p. 515-537) and index | ||
650 | 4 | |a Passivity-based control | |
650 | 4 | |a Lagrange equations | |
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Datensatz im Suchindex
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adam_text | ROMEO ORTEGA, ANTONIO LORIA, PER JOHAN NICKLASSON AND HEBERTT
SIRA-RAMFREZ PASSIVITY-BASED CONTROL OF EULER-LAGRANGE SYSTEMS
MECHANICAL, ELECTRICAL AND ELECTROMECHANICAL APPLICATIONS SPRINGER
CONTENTS NOTATION XXXI 1 INTRODUCTION 1 1 FROM CONTROL ENGINEERING TO
MATHEMATICAL CONTROL THEORY AND BACK . . 1 2 A ROUTE TOWARDS
APPLICATIONS 3 3 WHY EULER-LAGRANGE SYSTEMS? 4 4 ON THE ROLE OF
INTERCONNECTION 7 5 WHY PASSIVITY? 8 6 WHAT IS PASSIVITY-BASED CONTROL?
10 7 SOME HISTORICAL REMARKS 12 7.1 EULER-LAGRANGE SYSTEMS AND NONLINEAR
DYNAMICS 12 7.2 PASSIVITY AND FEEDBACK STABILIZATION 12 2 EULER-LAGRANGE
SYSTEMS 15 1 THE EULER-LAGRANGE EQUATIONS 16 2 INPUT-OUTPUT PROPERTIES
19 2.1 PASSIVITY OF EL SYSTEMS 20 2.2 PASSIVITY OF THE ERROR DYNAMICS 22
2.3 OTHER PROPERTIES AND ASSUMPTIONS 24 2.4 PASSIVE SUBSYSTEMS
DECOMPOSITION 25 2.5 AN EL STRUCTURE-PRESERVING INTERCONNECTION 26 3
LYAPUNOV STABILITY PROPERTIES 27 3.1 FULLY-DAMPED SYSTEMS 27 3.2
UNDERDAMPED SYSTEMS 28 4 EXAMPLES 30 XIU XIV CONTENTS 4.1 A
ROTATIONAL/TRANSLATIONAL PROOF MASS ACTUATOR 30 4.2 LEVITATED BALL 32
4.3 FLEXIBLE JOINTS ROBOTS 34 4.4 THE DUFFING SYSTEM 35 4.5 A MARINE
SURFACE VESSEL 36 5 CONCLUDING REMARKS 37 I MECHANICAL SYSTEMS 39 3
SET-POINT REGULATION 41 1 STATE FEEDBACK CONTROL OF FULLY-ACTUATED
SYSTEMS 42 1.1 A BASIC RESULT: THE PD CONTROLLER 42 1.2 AN INTRODUCTORY
EXAMPLE 44 1.3 PHYSICAL INTERPRETATION AND LITERATURE REVIEW 46 2 OUTPUT
FEEDBACK STABILIZATION OF UNDERACTUATED SYSTEMS . . . . . . . . 48 2.1
LITERATURE REVIEW 48 2.2 PROBLEM FORMULATION 48 2.3 EULER-LAGRANGE
CONTROLLERS 49 2.4 EXAMPLES 51 3 BOUNDED OUTPUT FEEDBACK REGULATION 61
3.1 LITERATURE REVIEW 61 3.2 PROBLEM FORMULATION 61 3.3 GLOBALLY
STABILIZING SATURATED EL CONTROLLERS 63 3.4 EXAMPLES 68 4 SET-POINT
REGULATION UNDER PARAMETER UNCERTAINTY 75 4.1 LITERATURE REVIEW 76 4.2
ADAPTIVE CONTROL 77 4.3 LINEAR PID CONTROL 79 4.4 NONLINEAR PID CONTROL
82 4.5 OUTPUT FEEDBACK REGULATION: THE PI 2 D CONTROLLER 85 5 CONCLUDING
REMARKS 91 CONTENTS XV 4 TRAJECTORY TRACKING CONTROL 93 1 STATE FEEDBACK
CONTROL OF FULLY-ACTUATED SYSTEMS 94 1.1 THE PD+ CONTROLLER 95 1.2 THE
SLOTINE AND LI CONTROLLER 96 2 ADAPTIVE TRAJECTORY TRACKING 97 2.1
ADAPTIVE CONTROLLER OF SLOTINE AND LI 97 2.2 A ROBUST ADAPTIVE
CONTROLLER 98 3 STATE FEEDBACK OF UNDERACTUATED SYSTEMS 100 3.1 MODEL
AND PROBLEM FORMULATION 100 3.2 LITERATURE REVIEW 101 3.3 A
PASSIVITY-BASED CONTROLLER 102 3.4 COMPARISON WITH BACKSTEPPING AND
CASCADED DESIGNS 104 3.5 A CONTROLLER WITHOUT JERK MEASUREMENTS 105 4
OUTPUT FEEDBACK OF FULLY-ACTUATED SYSTEMS 108 4.1 SEMIGLOBAL TRACKING
CONTROL OF ROBOT MANIPULATORS 109 4.2 DISCUSSION ON GLOBAL TRACKING 110
5 SIMULATION RESULTS 111 6 CONCLUDING REMARKS 113 5 ADAPTIVE DISTURBANCE
ATTENUATION: FRICTION COMPENSATION 115 1 ADAPTIVE FRICTION COMPENSATION
116 1.1 THE LUGRE FRICTION MODEL 117 1.2 DC MOTOR WITH FRICTION 119 1.3
ROBOT MANIPULATOR 122 1.4 SIMULATIONS 124 2 STATE-SPACE PASSIFIABLE
SYSTEMS WITH DISTURBANCES 127 2.1 BACKGROUND 127 2.2 A THEOREM FOR
PASSIFIABLE AFFINE NONLINEAR SYSTEMS 129 3 CONCLUDING REMARKS 131 XVI
CONTENTS II ELECTRICAL SYSTEMS 133 6 MODELING OF SWITCHED DC-TO-DC POWER
CONVERTERS 135 1 INTRODUCTION 135 2 LAGRANGIAN MODELING 137 2.1 MODELING
OF SWITCHED NETWORKS 137 2.2 A VARIATIONAL ARGUMENT 138 2.3 GENERAL
LAGRANGIAN MODEL: PASSIVITY PROPERTY 140 2.4 EXAMPLES 145 3 HAMILTONIAN
MODELING 157 3.1 CONSTITUTIVE ELEMENTS 158 3.2 LC CIRCUITS 160 3.3
EXAMPLES 161 4 AVERAGE MODEIS OF PWM REGULATED CONVERTERS 168 4.1
GENERAL ISSUES ABOUT PULSE-WIDTH-MODULATION 169 4.2 EXAMPLES 171 4.3
SOME STRUCTURAL PROPERTIES 176 5 CONCLUSIONS 180 7 PASSIVITY-BASED
CONTROL OF DC-TO*DC POWER CONVERTERS 181 1 INTRODUCTION 181 2 PBC OF
STABILIZING DUTY RATIO 182 2.1 THE BOOST CONVERTER 183 2.2 THE
BUCK-BOOST CONVERTER 187 2.3 SIMULATION RESULTS 188 3 PASSIVITY BASED
SLIDING MODE STABILIZATION 191 3.1 INTRODUCTION 191 3.2 SLIDING MODE
CONTROL OF THE BOOST CONVERTER 192 3.3 PASSIVITY-BASED SLIDING
CONTROLLER 198 4 ADAPTIVE STABILIZATION 206 4.1 CONTROLLER DESIGN 206
4.2 SIMULATION RESULTS 211 5 EXPERIMENTAL COMPARISON OF SEVERAL
NONLINEAR CONTROLLERS 213 CONTENTS XVII 5.1 FEEDBACK CONTROL LAWS 213
5.2 EXPERIMENTAL CONFIGURATION 219 5.3 EXPERIMENTAL RESULTS 221 5.4
CONCLUSIONS 236 III ELECTROMECHANICAL SYSTEMS 241 8 NESTED-LOOP
PASSIVITY-BASED CONTROL: AN ILLUSTRATIVE EXAMPLE 243 1 INTRODUCTION 244
1.1 MODEL AND CONTROL PROBLEM 245 2 PASSIVITY-BASED CONTROL WITH TOTAL
ENERGY-SHAPING 246 3 NESTED-LOOP PASSIVITY-BASED CONTROL 247 3.1 CONTROL
STRUCTURE 248 3.2 PASSIVITY-BASED CONTROLLER DESIGN 249 4
OUTPUT-FEEDBACK PASSIVITY-BASED CONTROL 253 5 COMPARISON WITH FEEDBACK
LINEARIZATION AND BACKSTEPPING 254 5.1 FEEDBACK-LINEARIZATION CONTROL
255 5.2 INTEGRATOR BACKSTEPPING CONTROL 256 5.3 COMPARISON OF THE
SCHEMES 257 5.4 SIMULATION RESULTS 259 5.5 CONCLUSIONS AND FURTHER
RESEARCH 262 9 GENERALIZED AC MOTOR 265 1 INTRODUCTION 265 1.1 AC MOTORS
265 1.2 REVIEW OF PREVIOUS WORK 268 1.3 OUTLINE OF THE REST OF THIS
CHAPTER 279 2 LAGRANGIAN MODEL AND CONTROL PROBLEM 280 2.1 THE
EULER-LAGRANGE EQUATIONS FOR AC MACHINES 281 2.2 CONTROL PROBLEM
FORMULATION 283 2.3 REMARKS TO THE MODEL 284 2.4 EXAMPLES 287 CONTENTS 3
A PASSIVITY-BASED APPROACH FOR CONTROLLER DESIGN 288 3.1 PASSIVE
SUBSYSTEMS FEEDBACK DECOMPOSITION 288 3.2 DESIGN PROCEDURE 289 4 A
GLOBALLY STABLE TORQUE TRACKING CONTROLLER 289 4.1 STRICT PASSIFIABILITY
VIA DAMPING INJECTION 290 4.2 CURRENT TRACKING VIA ENERGY-SHAPING 292
4.3 FROM CURRENT TRACKING TO TORQUE TRACKING 294 4.4 PBC FOR ELECTRIC
MACHINES 297 5 PBC OF UNDERACTUATED ELECTRICAL MACHINES REVISITED 302
5.1 REALIZATION OF THE PBC VIA BP TRANSFORMABILITY 302 5.2 A GEOMETRIC
PERSPECTIVE 304 6 EXAMPLES 305 7 CONCLUSIONS 307 7.1 SUMMARY 307 7.2
OPEN ISSUES 308 VOLTAGE-FED INDUCTION MOTORS 311 1 INDUCTION MOTOR MODEL
312 1.1 DYNAMIC EQUATIONS 312 1.2 SOME CONTROL PROPERTIES OF THE MODEL
313 1.3 COORDINATE TRANSFORMATIONS 315 1.4 REMARKS TO THE MODEL 318 1.5
CONCLUDING REMARKS 320 2 PROBLEM FORMULATION 320 3 A NESTED-LOOP PBC 321
3.1 A SYSTEMS INVERSION PERSPECTIVE OF THE TORQUE TRACKING PBC 323 3.2
OBSERVER-LESS PBC FOR INDUCTION MOTORS 327 3.3 REMARKS TO THE CONTROLLER
331 3.4 INTEGRAL ACTION IN STATOR CURRENTS 333 3.5 ADAPTATION OF STATOR
PARAMETERS 334 3.6 A FUNDAMENTAL OBSTACLE FOR ROTOR RESISTANCE
ADAPTATION . . . . 335 3.7 A D^-IMPLEMENTATION 337 CONTENTS XIX 3.8
DEFINITIONS OF DESIRED ROTOR FLUX NORM 338 3.9 SIMULATION RESULTS 340 4
A PBC WITH TOTAL ENERGY-SHAPING 342 4.1 FACTORIZATION OF WORKLESS FORCES
343 4.2 PROBLEM FORMULATION 344 4.3 IDEAL CASE WITH FUELL STATE FEEDBACK
344 4.4 OBSERVER-BASED PBC FOR INDUCTION MOTORS 346 4.5 REMARKS TO THE
CONTROLLER 348 4.6 A DG-IMPLEMENTATION 349 4.7 SIMULATION RESULTS 351
4.8 CONCLUDING REMARKS 353 5 FIELD-ORIENTED CONTROL AND FEEDBACK
LINEARIZATION 353 5.1 RATIONALE OF FIELD-ORIENTED CONTROL 354 5.2 STATE
ESTIMATION OR REFERENCE VALUES 357 5.3 SHORTCOMINGS OF FOC 358 5.4
FEEDBACK LINEARIZATION 361 6 EXPERIMENTAL RESULTS 363 6.1 EXPERIMENTAL
SETUP 363 6.2 OUTLINE OF EXPERIMENTS 369 6.3 OBSERVER-LESS CONTROL 370
6.4 OBSERVER-BASED CONTROL 375 6.5 COMPARISON WITH FOC 376 6.6
CONCLUDING REMARKS 379 11 CURRENT-FED INDUCTION MOTORS 381 1 MODEL OF
THE CURRENT-FED INDUCTION MOTOR 383 2 FIELD ORIENTATION AND FEEDBACK
LINEARIZATION 385 2.1 DIRECT FIELD-ORIENTED CONTROL 385 2.2 INDIRECT
FIELD-ORIENTED CONTROL 386 2.3 OBSERVER-BASED FEEDBACK-LINEARIZING
CONTROL 387 2.4 REMARKS TO OBFL AND FOC 390 3 PASSIVITY-BASED CONTROL OF
CURRENT-FED MACHINES 392 3.1 PBC IS DOWNWARD COMPATIBLE WITH FOC 392 3.2
STABILITY OF INDIRECT FOC FOR KNOWN PARAMETERS 393 4 EXPERIMENTAL
COMPARISON OF PBC AND FEEDBACK LINEARIZATION 394 4.1 EXPERIMENTAL SETUP
395 4.2 SELECTION OF FLUX REFERENCE IN EXPERIMENTS 398 4.3 SPEED
TRACKING PERFORMANCE 399 4.4 ROBUSTNESS AND DISTURBANCE ATTENUATION 401
4.5 CONCLUSION 402 5 ROBUST STABILITY OF PBC 403 5.1 GLOBAL BOUNDEDNESS
404 5.2 COORDINATE CHANGES AND UNIQUENESS OF EQUILIBRIUM 405 5.3 LOCAL
ASYMPTOTIC STABILITY 409 5.4 GLOBAL EXPONENTIAL STABILITY 410 6 OFF-LINE
TUNING OF PBC 415 6.1 PROBLEM FORMULATION 416 6.2 CHANGE OF COORDINATES
417 6.3 LOCAL STABILITY 418 6.4 A PERFORMANCE EVALUATION METHOD BASED ON
PASSIVITY 420 6.5 ILLUSTRATIVE EXAMPLES 425 7 DISCRETE-TIME
IMPLEMENTATION OF PBC 429 7.1 THE EXACT DISCRETE-TIME MODEL OF THE
INDUCTION MOTOR 431 7.2 ANALYSIS OF DISCRETE-TIME PBC 432 7.3 A NEW
DISCRETE-TIME CONTROL ALGORITHM 433 7.4 DISCUSSION OF DISCRETE-TIME
CONTROLLER 435 7.5 EXPERIMENTAL RESULTS 435 8 CONCLUSIONS AND FURTHER
RESEARCH 438 12 FEEDBACK INTERCONNECTED SYSTEMS: ROBOTS WITH AC DRIVES
441 1 INTRODUCTION 442 1.1 CASCADED SYSTEMS 442 1.2 ROBOTS WITH AC
DRIVES 445 2 GENERAL PROBLEM FORMULATION 446 CONTENTS XXI 3 ASSUMPTIONS
448 3.1 REALIZABILITY OF THE CONTROLLER 448 3.2 OTHER ASSUMPTIONS 450 4
PROBLEM SOLUTION 451 4.1 PROOF OF THEOREM 12.7 451 5 APPLICATION TO
ROBOTS WITH AC DRIVES 455 5.1 MODEL 455 5.2 GLOBAL TRACKING CONTROLLER
457 6 SIMULATION RESULTS 461 7 CONCLUDING REMARKS 464 13 OTHER
APPLICATIONS AND CURRENT RESEARCH 467 1 OTHER APPLICATIONS 468 2 CURRENT
RESEARCH 469 2.1 POWER ELECTRONICS 469 2.2 POWER SYSTEMS 470 2.3
GENERATION OF STORAGE FUNCTIONS FOR FORCED EL SYSTEMS 470 2.4
PERFORMANCE 471 A DISSIPATIVITY AND PASSIVITY 475 1 CIRCUIT EXAMPLE 476
2 2 AND 2 E SPACES 477 3 PASSIVITY AND FINITE-GAIN STABILITY 477 4
FEEDBACK SYSTEMS 479 5 INTERNAL STABILITY AND PASSIVITY 480 6 THE
KAIMAN-YAKUBOVICH-POPOV LEMMA 481 B DERIVATION OF THE EULER-LAGRANGE
EQUATIONS 483 1 GENERALIZED COORDINATES AND VELOCITIES 483 2 HAMILTON S
PRINCIPLE 487 3 PROM HAMILTON S PRINCIPLE TO THE EL EQUATIONS 488 4 EL
EQUATIONS FOR NON-CONSERVATIVE SYSTEMS 489 5 LIST OF GENERALIZED
VARIABLES 489 XXII CONTENTS 6 HAMILTONIAN FORMULATION 489 C BACKGROUND
MATERIAL 493 D PROOFS 495 1 PROOFS FOR THE PI 2 D CONTROLLER 495 1.1
PROPERTIES OF THE STORAGE TISIQ, Q,D) 495 1.2 LYAPUNOV STABILITY OF THE
PI 2 D 497 2 PROOF OF POSITIVE DEFINITENESS OF F(Q P ) DEFINED IN (3.43)
498 3 THE BP TRANSFORMATION 500 3.1 PROOF OF PROPOSITION 9.20 500 3.2 A
LEMMA ON THE BP TRANSFORMATION 502 4 PROOF OF EQS. (10.41) AND (10.77)
503 4.1 A THEOREM ON POSITIVITY OF A BLOCK MATRIX 503 4.2 PROOF OFEQ.
(10.77) 503 4.3 PROOF OFEQ. (10.41) 506 5 DERIVATION OF EQS. (10.55) AND
(10.56) 507 5.1 DERIVATION OF EQ. (10.55) 507 5.2 DERIVATION OF EQ.
(10.56) 508 6 BOUNDEDNESS OF ALL SIGNALS FOR INDIRECT FOC 510 6.1 PROOF
OF PROPOSITION 11.10 510 BIBLIOGRAPHY 515 INDEX 539
|
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id | DE-604.BV035376072 |
illustrated | Illustrated |
indexdate | 2024-07-09T21:32:26Z |
institution | BVB |
isbn | 1852330163 |
language | English |
lccn | 98023128 |
oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017180384 |
open_access_boolean | |
physical | XXXIV, 543 S. Ill., graph. Darst. 24 cm |
publishDate | 1998 |
publishDateSearch | 1998 |
publishDateSort | 1998 |
publisher | Springer |
record_format | marc |
series2 | Communications and control engineering |
spelling | Passivity-based control of Euler-Lagrange systems mechanical, electrical, and electromechanical applications Romeo Ortega ... [et al.] London Springer 1998 XXXIV, 543 S. Ill., graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier Communications and control engineering Includes bibliographical references (p. 515-537) and index Passivity-based control Lagrange equations Passives System (DE-588)4202764-0 gnd rswk-swf Rückkopplung (DE-588)4050851-1 gnd rswk-swf Nichtlineares dynamisches System (DE-588)4126142-2 gnd rswk-swf Ljapunov-Stabilitätstheorie (DE-588)4167992-1 gnd rswk-swf Lagrange-Bewegungsgleichungen (DE-588)4166460-7 gnd rswk-swf Reglerentwurf (DE-588)4177447-4 gnd rswk-swf Nichtlineares dynamisches System (DE-588)4126142-2 s Lagrange-Bewegungsgleichungen (DE-588)4166460-7 s Ljapunov-Stabilitätstheorie (DE-588)4167992-1 s Passives System (DE-588)4202764-0 s Rückkopplung (DE-588)4050851-1 s Reglerentwurf (DE-588)4177447-4 s DE-604 Ortega, Romeo 1954- Sonstige (DE-588)120106574 oth http://www.loc.gov/catdir/enhancements/fy0812/98023128-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0812/98023128-t.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0815/98023128-b.html Contributor biographical information GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017180384&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
spellingShingle | Passivity-based control of Euler-Lagrange systems mechanical, electrical, and electromechanical applications Passivity-based control Lagrange equations Passives System (DE-588)4202764-0 gnd Rückkopplung (DE-588)4050851-1 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd Ljapunov-Stabilitätstheorie (DE-588)4167992-1 gnd Lagrange-Bewegungsgleichungen (DE-588)4166460-7 gnd Reglerentwurf (DE-588)4177447-4 gnd |
subject_GND | (DE-588)4202764-0 (DE-588)4050851-1 (DE-588)4126142-2 (DE-588)4167992-1 (DE-588)4166460-7 (DE-588)4177447-4 |
title | Passivity-based control of Euler-Lagrange systems mechanical, electrical, and electromechanical applications |
title_auth | Passivity-based control of Euler-Lagrange systems mechanical, electrical, and electromechanical applications |
title_exact_search | Passivity-based control of Euler-Lagrange systems mechanical, electrical, and electromechanical applications |
title_full | Passivity-based control of Euler-Lagrange systems mechanical, electrical, and electromechanical applications Romeo Ortega ... [et al.] |
title_fullStr | Passivity-based control of Euler-Lagrange systems mechanical, electrical, and electromechanical applications Romeo Ortega ... [et al.] |
title_full_unstemmed | Passivity-based control of Euler-Lagrange systems mechanical, electrical, and electromechanical applications Romeo Ortega ... [et al.] |
title_short | Passivity-based control of Euler-Lagrange systems |
title_sort | passivity based control of euler lagrange systems mechanical electrical and electromechanical applications |
title_sub | mechanical, electrical, and electromechanical applications |
topic | Passivity-based control Lagrange equations Passives System (DE-588)4202764-0 gnd Rückkopplung (DE-588)4050851-1 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd Ljapunov-Stabilitätstheorie (DE-588)4167992-1 gnd Lagrange-Bewegungsgleichungen (DE-588)4166460-7 gnd Reglerentwurf (DE-588)4177447-4 gnd |
topic_facet | Passivity-based control Lagrange equations Passives System Rückkopplung Nichtlineares dynamisches System Ljapunov-Stabilitätstheorie Lagrange-Bewegungsgleichungen Reglerentwurf |
url | http://www.loc.gov/catdir/enhancements/fy0812/98023128-d.html http://www.loc.gov/catdir/enhancements/fy0812/98023128-t.html http://www.loc.gov/catdir/enhancements/fy0815/98023128-b.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017180384&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
work_keys_str_mv | AT ortegaromeo passivitybasedcontrolofeulerlagrangesystemsmechanicalelectricalandelectromechanicalapplications |