The system under investigation is the commercially available HMS M-511. .... The first classification of the known control systems for HMS can be performed ...
CONTROL OF A HYBRID MICROPOSITIONING SYSTEM FOR USE IN INDUSTRY AND ROBOTICS Ph.D. Thesis
Candidate:
Mentor:
László Juhász, M.Sc.
Prof. Branislav Borovac, Ph.D.
Novi Sad, June 2011
УНИВЕРЗИТЕТ У НОВОМ САДУ ФАКУЛТЕТ ТЕХНИЧКИХ НАУКА 21000 НОВИ СА Д, Трг Д оситеја Обрадовића 6
КЉУЧНА ДОКУМЕНТАЦИЈСКА ИНФОРМАЦИЈА Редни број, РБР: Идентификациони број, ИБР: Тип документације, ТД:
Монографска публикација
Тип записа, ТЗ:
Текстуални штампани материјал
Врста рада, ВР:
Докторска теза
Аутор, АУ:
Мр. Ласло Јухас
Ментор, МН:
Др. Бранислав Боровац, ред. проф.
Наслов рада, НР:
Регулација хибридног микропозиционог система за примене у индустрији и роботици Енглески
Језик публикације, ЈП: Језик извода, ЈИ:
Енглески, Српски
Земља публиковања, ЗП:
Република Србија
Уже географско подручје, УГП:
Војводина
Година, ГО:
2011
Издавач, ИЗ:
Ауторски репринт
Место и адреса, МА:
Novi Sad, Srbija 7 / 199 / 195 / 24 / 199 / 0 / 17
Физички опис рада, ФО: (поглавља/страна/цитата/табела/слика/ графика/прилога) Научна област, НО:
Техничке науке
Научна дисциплина, НД:
Мехатроника
Предметна одредница/Кqучне речи, ПО:
микропозиционирање, хибридни системи, компензација нелинеарности, управљање y просторy стања, онлајн-оптимизација регулатора
УДК Чува се, ЧУ:
Библиотека Факултета Техничких Наука, Трг Доситеја Обрадовића 6, 21000 Нови Сад, Република Србија
Важна напомена, ВН: Извод, ИЗ:
Прецизно праћење задате трајекторије реда величине десетак сантиметара уз максимална дозвољена одступања од неколико стотина нанометара се често захтевају у роботици и индустријским применама. Нити класични позициони системи базирани на електричном мотору и навојном вретену, ни они засновани искључиво на пиезоелектричним актуаторима не могу да задовоље овакве захтеве. Један концепт за превазилажење неведеног проблема заснива се на хибридном систему, насталог редном комбинацијом ова два актуатора. Развој одговарајућег управљачког склопа за такав систем захтева нова решења. Ова теза обухвата развој одговарајућег хардверског и софтверског решења за практичну реализацију управљачког склопа, моделовање и идентификацију параметара система, истраживање и развој ефикасних метода за компензацију нелинеарности и поремећаја, проналажење одговарајуће структуре и саму реализацију регулатора повратне спреге, kao и развој алгоритама за поједностављено руковање и пуштање у погон. Коришћењем адаптивних алгоритама, дато је и једно решење за проблем подешавања параметара регулатора у случају промењених услова рада, као што је на пример промена објекта који се позиционира.
Датум прихватања теме, ДП: Датум одбране, ДО:
Факултет Техничких Наука: 29.09.2010, Универзитет у Н.С.: 21.10.2010
Чланови комисије, КО:
Председник:
Др. Душан Петровачки, ред. проф., ФТН, Нови Сад
Члан:
Dr. Jürgen Maas, prof., HS-OWL, Lemgo
Члан:
Др. Вељко Поткоњак, ред., проф., ЕТФ, Београд
Члан:
Др. Драган Шешлија, ред., проф., ФТН, Нови Сад
Члан:
Др. Ласло Нађ, ванр. проф., ФТН, Нови Сад
Члан, ментор:
Др. Бранислав Боровац, ред. проф., ФТН, Нови Сад
Потпис ментора
Oбразац Q2.НА.06-05- Издање 1
UNIVERSITY OF NOVI SAD FACULTY OF TECHNICAL SCIENCES 21000 NOVI SAD, Trg Dositeja Obradovića 6
KEY WORDS DOCUMENTATION
Accession number, ANO: Identification number, INO: Document type, DT:
Monographic publication
Type of record, TR:
Printed textual material
Contents code, CC:
Ph. D. Thesis
Author, AU:
László Juhász, M.Sc.
Mentor, MN:
prof. Branislav Borovac, Ph.D.
Title, TI:
Control of a hybrid micropositioning system for use in industry and robotics
Language of text, LT:
English
Language of abstract, LA:
English, Serbian
Country of publication, CP:
Republic of Serbia
Locality of publication, LP:
Vojvodina
Publication year, PY:
2011
Publisher, PB:
Author’s reprint
Publication place, PP:
Novi Sad, Serbia
Physical description, PD: (chapters/pages/ref./tables/pictures/graphs/ appendixes)
7 / 199 / 195 / 24 / 199 / 0 / 17
Scientific field, SF:
Engineering Sciences
Scientific discipline, SD:
Mechatronics
Subject/Key words, S/KW:
hybrid micropositioning system, nonlinearity compensation, state-feedback control, online control optimization
UC Holding data, HD:
Library of the Faculty of Technical Sciences, Trg Dositeja Obradovića 6, 21000 Novi Sad, Republic of Serbia
Note, N: Abstract, AB:
In robotics and many industrial applications, the availability of positioning systems capable to follow trajectory paths in the range of several centimetres, featuring at the same time a nanometre-range precision, is demanding. Pure piezoelectric stages and standard positioning systems with motor and spindle cannot meet such requirements. One concept for overcoming these problems consists of a hybrid positioning system built through the integration of a DCdrive in series with a piezoelectric actuator. The design of high-quality controller for such devices requires novel approaches and solutions. In this thesis, the control design for a hybrid micropositioning system is investigated. It comprises the design of the hardware and real-time software for the efficient plant control and accurate measurements, modelling and parameter identification, design of efficient disturbance and nonlinearity compensation, closed-loop control design and an appropriate commissioning scenario for the customer, featuring online control parameter adaptation. Experimental results achieved with the novel control system reveal significant improvement of the positioning quality compared to the factory controller.
Accepted by the Scientific Board on, ASB:
Faculty of Technical Sciences: 29.09.2010, University of Novi Sad: 21.10.2010
Defended on, DE: Defended Board, DB:
President:
prof. Dušan Petrovački, Ph.D., FTS, Novi Sad
Member:
prof. Jürgen Maas, Ph.D., HS-OWL, Lemgo
Member
prof. Veljko Potkonjak, Ph.D., FEE, Belgrade
Member:
prof. Dragan Šešlija, Ph.D., FTS, Novi Sad
Member:
prof. László Nagy, Ph.D., FTS, Novi Sad
Member, Mentor:
prof. Branislav Borovac, Ph.D., FTS, Novi Sad
Menthor's sign
Obrazac Q2.НА.06-05- Izdanje 1
IV
CONTENTS Preface and Acknowledgements ................................................................................................................................VII Abstract.................................................................................................................................................................... VIII Сажетак ...................................................................................................................................................................... IX Introduction ......................................................................................................................................................... 1
1. 1.1
Micropositioning systems with large positioning range based on one actuator.............................................. 1
1.2
Hybrid Micropositioning Systems .................................................................................................................. 3
1.2.1 System under investigation ................................................................................................................... 5 1.3 Control of Hybrid Micropositioning Systems: State of the Art ...................................................................... 7 1.4
Rapid Control Prototyping.............................................................................................................................. 9
1.5
Structure of the thesis ................................................................................................................................... 10
2.
Hardware and Software Framework for Controller Realization........................................................................ 11 2.1
Electrical parts and available ports of the M-511.HD................................................................................... 13
2.1.1 Electro-mechanical and electronic parts.............................................................................................. 13 2.1.2 Available signal and data ports ........................................................................................................... 13 2.2 Control hardware concept............................................................................................................................. 14 2.2.1 The selected RCP-system.................................................................................................................... 14 2.2.2 Hardware concept of the M-CTRL interface....................................................................................... 16 2.3 Hardware design of the M-CTRL Interface.................................................................................................. 17 2.3.1 Selection of the main hardware elements............................................................................................ 17 2.3.2 Practical realization ............................................................................................................................. 19 2.4 The software framework............................................................................................................................... 20 2.4.1 The M-CTRL blockset and library...................................................................................................... 21 2.4.2 The FPGA application framework ...................................................................................................... 23 2.5 Calibration and experimental validation ....................................................................................................... 30 3.
Plant Model and Parameter Identification ......................................................................................................... 34 3.1
Overall model of the HMS............................................................................................................................ 34
3.2
The mechanical subsystem ........................................................................................................................... 35
3.2.1 Modelling of the friction and backlash................................................................................................ 38 3.3 The electro-mechanical model of the DC drive ............................................................................................ 39 3.4
Modelling of the PEA drive.......................................................................................................................... 41
3.4.1 The piezoelectric effect ....................................................................................................................... 41 3.4.2 Piezoelectric actuator models.............................................................................................................. 42 3.4.3 Survey on hysteresis models ............................................................................................................... 45 3.4.4 Implementation of the MRC-based PEA model.................................................................................. 47 3.4.5 Identification of the PEA model parameters ....................................................................................... 50 3.4.6 Model of the piezo-amplifier............................................................................................................... 54 3.5 Linearization, state-space representation, and parameter identification of the mechanical sub-model ........ 58 3.5.1 State-space representation and TF of the linearized mechanical subsystem ....................................... 58 3.5.2 Parameter estimation of the mechanical sub-model ............................................................................ 59 3.6 Simulation and experimental results............................................................................................................. 63
4.
3.6.1 Validation of the parameterized PEA-model ......................................................................................64 3.6.2 Validation of the parameterized HMS-model ..................................................................................... 65 Disturbance and Nonlinearity Compensation .................................................................................................... 67
V 4.1
Compensation of the induced EMF by current control................................................................................. 67
4.2
Compensation of nonlinearities with local memory ..................................................................................... 70
4.3
PEA hysteresis compensation....................................................................................................................... 72
4.4
Friction compensation for the DC-drive ....................................................................................................... 75
4.5
Real-time implementation of the compensation measures............................................................................ 79
4.6
Experimental results ..................................................................................................................................... 80
5.
Closed-Loop Control for HMS.......................................................................................................................... 85 5.1
Overview of HMS control structures............................................................................................................ 85
5.2
Novel plant representation and overall control structure .............................................................................. 89
5.3
Estimation of the plant states ........................................................................................................................ 95
5.3.1 Model reduction and discretization ..................................................................................................... 96 5.4 Design of the controller for the DC-drive................................................................................................... 101 5.4.1 Quasi-continuous controller design................................................................................................... 102 5.4.2 Discrete controller design.................................................................................................................. 103 5.4.3 Ensuring steady-state error-free functionality and improved disturbance rejection .......................... 104 5.5 Trajectory generation and feed-forward control ......................................................................................... 105 5.6
Design of the PEA controller...................................................................................................................... 108
5.6.1 Simple state-feedback controller ....................................................................................................... 109 5.6.2 Cascaded state-feedback controller ................................................................................................... 112 5.7 Control-parameter adaptation by a positioned object of considerable mass ............................................... 114 5.7.1 The model behaviour for different positioned objects....................................................................... 115 5.7.2 Experimentally obtained plant responses versus model behaviour ................................................... 117 5.7.3 Model adaptation by involvement of the efficiently acting mass ...................................................... 119 5.8 Practical implementation of the control system .......................................................................................... 120 5.9
6.
Experimental validation.............................................................................................................................. 122
5.9.1 Trajectory-tracking results with a light object................................................................................... 122 5.9.2 Trajectory-tracking results with different positioned objects ............................................................ 132 Controller Commissioning and Adaptation ..................................................................................................... 135 6.1
Commissioning concept to accomplish through the manufacturer ............................................................. 136
6.2
Commissioning concept to accoplish through the customer....................................................................... 137
6.3
Model-based online control optimization by extremum control................................................................. 138
6.3.1 The online-calculated performance index ......................................................................................... 140 6.3.2 Online capable optimization algorithms............................................................................................ 141 6.3.3 Adaptation of the control parameter in real-time .............................................................................. 146 6.4 Practical realization and experimental results............................................................................................. 147 6.5
Example application: a two-degree-of-freedom nanopositioning system................................................... 154
7.
Conclusion....................................................................................................................................................... 157
8.
Appendices ...................................................................................................................................................... 158 8.1
SPI Communication Diagram..................................................................................................................... 158
8.2
Signal connection of LA-NP25, ADS8422 and ADS8406 ......................................................................... 160
8.3
Variable mapping tables of the DPMEM.................................................................................................... 161
8.4
FPGA and M-CTRL Interface signal mapping........................................................................................... 163
8.5
Estimated PEA parameters ......................................................................................................................... 164
8.6
Transfer-function coefficients of the linearized mechanical subsystem ..................................................... 164
8.7
Transfer-function coefficients of the simplified identification model ........................................................ 165
VI 8.8
Specified and estimated parameters of the hybrid micropositioning system under investigation............... 166
8.9
Acceleration of Simulink-simulation by use of the Real-time Workshop ................................................. 166
8.10 Approximation of the decoupling functions ............................................................................................... 169 8.11 Transfer function of the plant model for the PEA controller ...................................................................... 170 8.12 FPGA-implementation of the PEA controller............................................................................................. 171 8.13 Top-level view of the final FPGA-application ........................................................................................... 173 8.14 Further trajectory-tracking results achieved with a light object.................................................................. 175 8.15 Reference measurements with the factory controller PI C-702 .................................................................. 178 8.16 Approximations of the PEA TF for positioned object of 5 kg .................................................................... 179 8.17 Parameter of the extremum-seeking method .............................................................................................. 180 References...................................................................................................................................................................... i
VII
PREFACE AND ACKNOWLEDGEMENTS The theoretical and experimental research activities in scope of this thesis are performed at the Ostwestfalen-Lippe University of Applied Sciences in Lemgo, Germany within the research project “Adaptive Piezoelectric Hybrid Systems”. The project was founded by the Federal Ministry of Education and Research (BMBF) of Germany under grant number 17N2108 and conducted by the laboratory Control Engineering and Mechatronic Systems of the Ostwestfalen-Lippe University of Applied Sciences. Partners of the project were the company Physik Instrumente, Germany, and the Faculty of Technical Sciences in Novi Sad, Serbia. In the first line, I would like to thank to my supervisor Prof. Branislav Borovac from the University of Novi Sad for advising me to make a big step and start a new period of my life. Without his help, care and useful discussions during the research and the final documentation phase, you would perhaps never had this work in your hands. I am indebted to Prof. Jürgen Maas from the Ostwestfalen-Lippe University of Applied Sciences for the opportunity to participate in the research project and for his support during my stay at his laboratory. He deserves also my sincerely thanks for the very detailed reading and very useful suggestions for this manuscript. The excellent work atmosphere in his laboratory was enabled by the contribution of the colleagues as well. We worked on different projects and helped each other mutually. It was a pleasure to work with our students both from the Ostwestfalen-Lippe University of Applied Sciences and from the University of Novi Sad. They accomplished particular tasks from this research in scope of their master and bachelor thesis as well as praxis projects, helping me on this way to accelerate the overall progress. The interested reader may review the accompanying master-, bachelor- and praxis project documentations because these contain a lot of useful details which did not fit in the frame of this dissertation. My wife deserves my sincerest thanks for her endless understanding and patience, as well as for her great help by the proofreading and language corrections of the manuscript. I would like to express my deepest gratitude to my parents for their understanding and sacrificing support, what enabled me to finish my regular studies during the very hard times of 90’s. Finally, I would like to thank the interested reader for his/her time and attention.
VIII
ABSTRACT In robotics and many industrial applications like semiconductor production and optical inspection systems, the availability of positioning systems capable to follow trajectory paths in the range of several centimetres, featuring at the same time a nanometre-range precision, is demanding. Pure piezoelectric stages and standard positioning systems with motor and spindle are not able to meet such requirements, because of the small operation range and inadequacies like backlash and friction. One concept for overcoming these problems consists of a hybrid positioning system built through the integration of a DCdrive in series with a piezoelectric actuator. The wide range of potential applications enables a considerable market potential for such an actuator, but due to the high variety of possible positioned objects and dynamic requirements, the required modelling and control design complexity is significant. This complexity raises the need for novel approaches by the control design for such devices. For realization of a high-quality closed-loop control system, the control hardware and the real-time software, as well as the appropriate modelling and parameter identification approaches, and finally the design of efficient underlying control functionalities for disturbance compensation have to be considered as well. In this thesis, the control design for a hybrid micropositioning system consisting of a DC-motor, gearbox and piezoelectric actuators is investigated. It includes the design of the appropriate hardware and real-time software for efficient plant control and accurate measurement, modelling and parameter identification without the need for system disassembly, design of efficient disturbance compensation measures, the closed-loop control design based on a novel approach, as well as a proposed commissioning scenario at the customer site featuring online controller adaptation. Each of these tasks is preceded by a literature recherche and concluded by an experimental validation. The realized control system reveals a highly increased trajectory-tracking quality related to the factory controller. Results of the novel approach for hysteresis identification and compensation described in this thesis are comparable with recently published results, being at the same time significantly simpler in realization and thus capable for the hard real-time requirements. The implemented controller adaptation routine may be particularly important in case when a hybrid micropositioning system is sold off-the-shelf and the customer is not familiar with the control theory in detail. The application example with the twodegree-of-freedom nanopositioning system assembled using two hybrid micropositioning stages shows a number of further potential application possibilities for the industry and robotics.
IX
САЖЕТАК Наглим развојем мехатронике, у роботици као и у индустрији – нарочито оне везане за производњу и испитивање полупроводничких и оптичких компоненти – настала је потреба за микропозиционим системима који су у стању да веома прецизно прате задату путању. Захтевани опсег кретања при томе варира и до неколико десетина сантиметара, док, захтеви за максималну стационарну грешку позиционирања се крећу и до реда нанометара а за максимално одступање од задате путање често се захтевају вредности испод једног микрометра или чак неколико десетина нанометара. Задате брзине позиционирања при томе варирају у распону од неколико µm/s до неколико cm/s у зависности од конкретне примене. Међу примерима у роботици може се на пример напоменути синхронизовано померање огледала на великим астрономским телескопима у циљу фокусирања или компензације атмосферских сметњи . Јасно је, да ни системи искључиво засновани на пиезоелектричним актуаторима, као ни класични системи који се састоје од електричног мотора и навојног вретена не могу да задовоље наведене захтеве. Системи засновани на пиезоелектричним актуаторима са једне стране или имају веома ограничен радни опсег [7] или су ограничени самом брзином позиционирања [7]. Са друге стране, класични системи засновани на електричном мотору и механичком преносу преко навоја и вретена или они на бази флуида нису у стању да постигну захтевану тачност позиционирања због трења у зглобовима и зазора који се код оваквих система неминовно појављују. Због тога је неопходно извршити комбинацију ових система. У новије време су се појавили различити концепти микропозиционих система, који потенцијално могу да задовоље горе наведене захтеве. Chiang [18] је развио хибридни систем који се заснива на каскадној механичкој комбинацији хидрауличног и пиезоелектричног актуатора. Chen i Dwang [21] су представили приступ заснован на навојном вретену са куглицама које је комбиновано са пиезо-актуатором. Побољшан систем заснован на сличном концепту приказан је и у [22]. У радовима [27,28] дат је опис хибридног микропозиционог система који се састоји од каскадног механизма електричног погона са навојним вретеном и пиезоелектричног актуатора, намењен за микропозиционе системе са једним или више степени слобода кретања. Уз помоћ хибридног система за микропозиционирање могуће је постићи захтевану тачност позиционирања од неколико нанометара, као и праћење задате путање брзинама и до 100 mm/s уз максималну грешку одступања од задате трајекторије које може бити и испод једног микрометара. Радни опсег система при томе достиже и до 100 mm. Коришћењем позиционих сензора високе резолуције и одговарајућих закона управљања, омогућава се постизање веома добре поновљивости, брзог одзива и прецизног праћења задате путање. За такав микропозициони систем постоји велики број потенцијалних примена. Међутим, управо та разноликост примена која подразумева знатне разлике у захтеваној брзини кретања током позиционирања као и различите објекте који се позиционирају, намеће комплексне захтеве за пројектовање одговарајућег закона управљања. Основни задатак ове тезе се састојао у развоју одговарајућег закона управљања за један хибридни микропозициони систем. У склопу овог задатка извршен је развој одговарајућих хардверских и софтверских модула, моделовање и идентификација параметара система,
X
компензација нелинеарности, развој регулатора у повратној спрези, развој алгоритма за аутоматско подешавање параметара регулатора у случају промењених услова (на пример замена објекта који се позиционира) као и експериментална провера постигнутих резултата. Иако данас постоје напредни управљачки склопови за позиционе системе које се базирају на искључиво пиезоелектричним актуаторима, као и оне за системе који су искључиво базирани на класичним актуаторима (електрични погон са механичким преносом), управљање хибридним системима за микропозиционирање се и данас решава углавном кроз примену класичних PI/PID регулатора са notch филтрима [22,28]. Поред мана која се испољавају кроз смањен динамички одзив система који за последицу има погоршану тачност праћења задате трајекторије и смањену отпорност на сметње, значајан проблем код постојећих регулатора представља и проналажење оптималних параметара за дату конкретну примену. Показало се, да се применом концепта управљања у простору стања омогућава побољшан динамички одзив система, који је резултовао у знатном побољшању квалитета праћења трајекторије. Могућност једноставнијег одређивања оптималних параметара регулатора за конкретну примену говори такође у прилог развоја управљачког склопа у простору стања. Овај рад је подељен у неколико поглавља која одговарају фазама спроведеног истраживања. Свакој фази су претходиле претрага и преглед доступне литературе. Након сваке спроведене фазе истраживања (укључујући и развој и практичну реализацију), она је закључена експерименталном провером и оценом постигнутих резултата уз помоћ комерцијално доступног хибридног система за микропозиционирање фирме Physik Instrumente. Прва фаза истраживања одговара уводу тезе. Она се односи на испитивање ситуације, прелиминарну проверу могућих праваца истраживања и израду детаљних концепата рада за даље фазе. У другој фази истраживања која је описана у поглављу 2, развила се и оптимизовала хардверска и софтверска подршка неопходна за идентификацију параметара, имплементацију алгоритама управљања као и за експерименталне провере постигнутих резултата. Иако се некима ова фаза чини мање битним, она представља неопходну основу за даљи рад. Од квалитета хардверскософтверског решења умногоме зависи реализљивост и уопште могући квалитет реализованог управљачког закона. Иако су модерни рачунарски системи засновани на централно-процесорској јединици веома моћни у погледу времена извршавања комплексних операција и прорачуна, чак и код најмоћнијих микропроцесорских система постоје ограничења условљена самом процесорском архитектуром при извршавању задатака у реалном времену (када се захтевају кратка времена латенције или високе фреквенције одабирања). Погодна интеграција микропроцесорског система и Field Programmable Gate Array (FPGA) представља добро решење за поменуте проблеме. Хардверском и софтверском интеграцијом FPGA и класичног процесорског система омогућено је задовољавање услова за рад регулатора у реалном времену, као и знатно ефикасније искоришћење процесорске снаге. Реализовани систем [51] истовремено омогућава високе фреквенције одабирања за једноставније задатке са егзактним тајмингом уз могућнст имплементације сложених алгоритма визуелним програмирањем у програмском пакету Matlab/Simulink™. Трећа фаза истраживања посвећена је моделовању хибридног система за микропозиционирање и експерименталној идентификацији параметара модела. Њој одговара поглавље 3. Развијен је одговарајући математички модел хибридног система, који са једне стране верно описује процес, а са друге стране омогућава развој закона управљања. Модел садржи опис нелинеарних ефеката као што су механичко трење и хистерезис пиезоелектричних актуатора. Идентификација линеарних параметара модела је извршена коришћењем спектралних метода, док су нелинеарни елементи идентификовани у временском домену. Уз помоћ новоразвијених алгоритама омогућена је идентификација свих параметара механичког подсистема и пиезо-актуатора без потребе за растављањем самог урећаја [52]. Истраживање и реализација погодних алгоритама за компензацију нелинеарности и поремећаја су предмет четврте фазе истраживања, описана у четвртом поглављу тезе. Ефикасна компензација нелинеарности процеса спроведена у реалном времену (real-time compensation) игра значајну улогу при пројектовању квалитетног закона управљања. Развијене су нове методе за компензацију тих нелинеарности (пре свега, хистерезис пиезоактуатора) са ниским захтевима за меморију и процесорску снагу које су погодне за коришчење у реалном времену [52-54].
XI
Следећа фаза истраживања, описана у петом поглављу рада бави се као прво проналажењем погодне структуре закона управљања повратне спреге на основу линеаризованог модела система представљеног у облику са система више улаза и излаза (MIMO-system). Као следеће, одабрани су регулатори стања, чији развој се заснива на опису линеарног система у простору стања. Показано је да предложена управљачка структура омогућава далеко бољи квалитет праћења трајекторије од комерцијално доступног регулатора фирме Physik Instrumente. У закључној фази истраживања, описаној у шестом поглављу рада, развијен је алгоритам за поједностављање поступка подешавања регулатора. Такав алгоритам је користан за пуштање у рад и руковање хибридним микропозиционим системом код крајњег корисника. Наведени алгоритам користан је и у случају промењених услова рада (као што је промена масе позиционираног објекта). У закључку саме тезе дат је кратак резиме о постигнутим резултатима и напоменути су могући правци даљег истраживања.
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1. INTRODUCTION In robotics and various industry applications the availability of positioning systems capable to follow trajectory paths in the range of several centimetres featuring nanometre-range accuracy at the same time, is demanding [1]. The wide spectrum of such applications includes besides of atomic force and white-light microscopy [2], semiconductor manufacturing, precision optics alignment, wire feeding systems [3], microbiological cell manipulation, ultra-precision machine tools and micro robots, also further application fields like strategic defence, space technologies and astronomy [4], as well as laser nuclear ignition target production [1]. Thereby, the required positioning velocities vary – depending on the concrete application – between a few nanometres and several centimetres per second. Neither pure piezoelectric stages nor standard positioning systems with motor and spindle or fluidbased actuators are able to meet such requirements. On one hand, piezoelectric actuators (PEA) have highly limited operation range, usually only a few micrometres. The very small operating volume is limited by the change of the actuator dimensions, since one end of the actuator is fixed to the base. Using sequentially connected actuators (“stacked design”), the operating capacity can be enlarged, but it is still small. On the other hand, the dynamic positioning precisions of standard motor-spindle driven mechanism are seriously limited by inadequacies like friction and backlash. The correction of these structural problems using a feed-forward or feedback controller represents a complex task, which often cannot be solved in a satisfactory way. Therefore, different approaches were recently investigated in order to enable the construction of a micropositioning system with high accuracy and large operation range.
1.1 Micropositioning systems with large positioning range based on one actuator Various approaches for design of micropositioning systems with high accuracy and large positioning range are known. Actuators based on “recoil of an ejected mass” [5] use the effects of the stick-slip friction for the fine motion as shown in Fig. 1.1. The positioning process here consists of fine positioning steps, which can be repeated in order to achieve the desired positioning range. Such systems can have large (theoretically unlimited) positioning range and high precision. They are useful, among other applications, for positioning of printed boards [6]. The main drawback of these systems lies in the limited positioning speed. Furthermore, the small achievable hold and positioning forces, as well as the necessity for mounting of an additional mass (which is rapidly “rejected” and thereafter slowly “recoiled”) restrict the field of possible applications of these devices. A similar concept with a platform built from piezo-legs, where the inertia of the positioning system itself is used (instead of having an attached additional mass) is presented in [7] (Fig. 1.2). This system consists of a moving platform with three legs made of segmented PEA with a common inner and four separate outer electrodes (Fig. 1.2a). By the appropriate polarization of outer electrode(s) with respect to the inner one, due to the inverse piezo-effect, the tube wall changes its longitudinal dimension and
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the tube bends in the corresponding direction. The platform movement is performed by successive slow bending and quick stretching of the tubes (i.e. legs, see Fig. 1.2b). The range of motion is limited only by the length of the wires connecting it to the power source. Positioning accuracy below 0.2 µm for such a system is reported [7], whereas the achievable motion and hold forces are here limited by the weight of the platform and the friction coefficient between the legs and the basement. object to be moved piezo-actuator mass
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Phases of motion of the positioning system actuated by the method of “recoil of an ejected mass" [5]: motion to the left (a), motion to the right (b).
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Construction of the platform for micropositioning with piezo-legs [7] (a) and its motion phases: 1-initial position, 2-legs’ slow bending, 3-legs’ quick stretching (b).
The problem of the limited hold forces is also present by the impact drive reported in [8], where a combination of a PEA and a spring is utilized for the self-moving positioning system. By combination of two PEA controlled flexure hinges [9] with a moving slider, this problem can be solved, because the friction force may be regulated with the PEAs during the slider movement. Positioning systems based on inchworm actuators [10-14] follow a similar, but slightly different approach. Through an additional clamping mechanism (Fig. 1.3), the achievable positioning and hold forces by these actuators may be significantly increased. However, the positioning speed is here also highly limited. Nanopositioning systems based on a single actuator per degree of freedom (DOF) comprising planar electromagnetic drive together with air bearings is presented in [15]. The system features a wide positioning range of 250 mm with a standard deviation of the steady-state positioning noise less than 5 nm. The positioning control of this system is realized by a state-feedback controller with an added proportional-integral (PI) element [15].
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Operation principle of an inchworm-based positioning system [13].
A highly complex micropositioning system, designed by the Technical University Ilmenau [2,16] is commercially available. The system “Nanopositioning and Nanomeasuring Machine” bases on high quality stepper drives, piezoelectric actuators, air bearings, flexible couplings and drive screws. This system enables positioning with three degrees of freedom in the range of a 25 mm x 25 mm area having a positioning resolution of 0.1 nm.
1.2 Hybrid Micropositioning Systems A different approach is followed by systems which are built through a combination of two different actuators. The “coarse” actuator is responsible for the coarse motion. It has a large mass and long stroke, a large input range and longer response time. A second actuator (the “fine” actuator) is usually an unconventional one and realizes the fine motion. This actuator has a short stroke, a small mass, a small input range and fast response. Both actuators are connected mechanically and build an integrated positioning system. The final object position and velocity by such a hybrid micropositioning system (HMS) depend on the position, velocity and acceleration of both actuators. Exploiting the different features of the actuators can yield higher performance related to the performance of a positioning system having only one actuator. By a HMS, the “coarse” actuator enables the large positioning range, whereas the “fine” one is responsible for rapid response, as well as for compensation of dynamic disturbances and inadequacies involved by the coarse actuator. Thus, with a HMS a large operation range, by maintaining simultaneously excellent position accuracy can be principally achieved. Such devices can perform positioning with higher velocities (up to few centimetres per second) and manifest significant hold and positioning forces (hundreds or thousands of Newton). Besides the quality of the implemented controller, the type of mechanical connection between the coarse and fine actuator, as well as their properties (stiffness, response time, and nonlinearity) play a significant role regarding the achievable positioning dynamic and accuracy [17]. Recently, various hybrid micropositioning systems became reported. These may be classified according to the: • Coarse actuator type. Hybrid micropositioning systems based on o
conventional electrical drives with ball-screw,
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hydraulic actuators,
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pneumatic drives,
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electromagnetic motors, and
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voice-coil actuators are known.
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•
Fine actuator type. By the most systems, the fine actuators are realized with piezoelectric actuators. Systems with ultrasonic motors are known as well.
•
Mechanical arrangement between the coarse and the fine actuator. In most cases, the cascaded (or serial) connection is utilized, where the overall position is the sum of the individual displacements achieved by the coarse and the fine actuator. However, parallel arrangements are also known, where the overall actuator force is composed as a sum of the individual forces achieved by the coarse and the fine actuator.
•
Sensors and their arrangement. In most cases, only one sensor for the final position is available. However, systems with separate sensing of the course and fine actuator position by the cascaded actuator connection are also known.
A short summary on the related literature recherche is given as follows. •
In [18], a hybrid actuator system with cascaded mechanical coupling of a hydraulic and piezoelectric actuator was presented. The developed hydraulic-piezoelectric actuator is used for position control with medium/high loading of about 15 kg, large stroke of 200 mm and positioning precision of 0.1 µm. The system includes a hydraulic-servo cylinder and a PEA with operation range of 60 µm. The hydraulic-servo cylinder serves for coarse positioning with high speed and large stroke; the PEA positions in precision range for compensation of the influence of friction and further nonlinear effects involved by the hydraulic cylinder. The PEA is mounted on the hydraulic actuator; thus, the overall position is the sum of the displacement of the hydraulic actuator and the PEA. The steadystate positioning error with the implemented feedback controller is 0.1 µm, whereas the position settling time is between 0.5 and 1.02 seconds.
•
A hybrid micropositioning system based on a pneumatic coarse actuator and PEA for fine positioning, with position control for point-to-point positioning is presented in [19]. The target position is approached by the coarse actuator as first. When the the rough positioning accuracy within 10 µm is reached, the positioning control by the PZT actuator is subsequently carried out. The control algorithm will be iteratively repeated until the position error becomes less than 10 nm. The position response of the fine actuator by the initial error of 5 µm takes 1.76 s, whereas the time needed for the complete positioning task by the position reference change of 4 mm requires 2.68 s.
•
In [20], a hybrid micropositioning system for precise cutting based on the electrohydraulic coarse actuator and a PEA as fine actuator is described. With the master-slave controller structure, this device is capable for trajectory-tracking. The maximal trajectory-tracking errors are between 8 and 15 µm by a 10 Hz cam-signal of 6 mm magnitude.
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Chen and Dwang [21] presented an approach with a ball-screw drive mechanism featuring a piezoelectric nut for active ball-screw preload and fine control. The coarse positioning is done by a conventional servo motor and ball-screw, where the piezoelectric nut exerts a light ball-screw preload in order to reduce the friction torque and wear at high-speed motion. After the coarse positioning, a fine position adjustment is actuated by piezoelectric actuators to get very precise positioning. Published results show that the steady-state positioning error can be fine-adjusted to the nanometre level. The position control is realized with a PID controller and an additional pseudo-derivative feedback (PDF) loop. The main control issue is the high load-to-motor inertia ratio, what results in degraded dynamic performance when traditional PI/PID controllers are used.
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A HMS with improved mechanical design, based on a similar approach was presented in [22]. This device features a PEA integrated in the support-bearing unit of the ball-screw. Since the PEA provides not only fine motions to the table through the screw-supporting system but also preloads to support bearings, high stiffness is achieved with respect to pulling forces to the PEA. A dual feedback controller based on position loop realized with
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P-controller and a velocity loop realized with PI-controller is implemented. The controlled system achieves trajectory tracking errors in the range of 500 nm by tracking of a sinusoidal signal of 100 µm amplitude and 10 s period. However, by tracking of a sine wave with 500 µm amplitude and identical period, the control input of the PEA saturates, resulting in tracking errors of 2.5 µm range. •
In [23], a micropositioning system featuring 300 mm operating range and 0.69 nm control resolution is reported. The coarse motion is performed here by an electromagnetic motor whereas for the fine positioning a non-resonant type ultrasonic motor is used. This system can achieve high velocities in open-loop control configuration (up to 50 mm/s), but in this case the positioning accuracy is poor. A closed-loop control with conventional PID controller is applied for precise trajectory tracking, what results in increased accuracy and reduced positioning velocity.
•
A dual positioning stage featuring a lead-screw based DC servo drive and a flexure based PEA is presented in [24]. The system is equipped with separate position sensor both for the DC drive and the micro stage, and the steady-state positioning error is about 10 nm. The integrated control algorithm is realized using individual PID controller for the DC drive and the PEA, which are complementarily enabled or disabled in dependence on the actual position error. The presented controller enables hence accurate point-to-point positioning but no precise trajectory-following.
•
A precise positioning system utilizing a voice-coil motor actuator for the coarse motion and a PEA for fine positioning is presented in [25]. The reported static positioning accuracy is of 10 nm range whereas the typical positioning step takes up to 100 µm. The reported positioning control system supports only point-to-point (e.g. static) positioning.
•
The combination of the voice-coil motor and the PEA is also proposed for novel hard-disk drives with high tracks-per-inch density [26]. These systems also represent a kind of HMS, where the coarse and the fine motions are complex (combination of rotation and translation).
•
Liu [27] and Glöß [4,28], proposed the cascaded mechanical coupling of a DC-drive and a piezoelectric actuator for use in positioning systems with one or more degrees of freedom. A similar approach is proposed 2009 by Rong-Fong et. al [29].
Nowadays, commercially available HMS of this structure also exist, like the M-511.HD from Physik Instrumente ([30], Fig. 1.4). 1.2.1
System under investigation
The system under investigation is the commercially available HMS M-511.HD from Physik Instrumente. The internal structure of this system is displayed in Fig. 1.5. It consists of a DC-drive (M) connected to a spindle, a gearbox (G), and two moving masses (m1, m2) with a pair of identical piezoelectric actuators (PEA) in between. The DC drive moves the coupled masses m1 and m2 together, whereas the PEA achieves an additional motion of m2 related to m1. Through control of the voltage vPWM the angular position of the DC-drive, εΜ can be controlled. The achieved angular position εΜ is converted into a linear movement of the first moving mass m1 by the gearbox G of reduction ratio irt. Each PEA between the masses m1 and m2 is mounted in parallel with a spring denoted by c. The main function of the springs is to pre-stress the PEA enabling on this way symmetrical bipolar voltage operation range (Fig. 1.6). Furthermore, firm mechanical contact between the PEA and the positioning plate during PEA operation is ensured. By excitation of the PEA with identical voltage vPA, it is possible to achieve an additional motion of the second mass m2 relative to s1 (the position of the first mass m1). The object O (object to be positioned, not shown in the figure) is fixed to m2, and its linear position s2 is detected by a high-precision incremental optical encoder. The position information is available in digital form through a custom-specific synchronised serial data link interface (SPI).
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Fig. 1.4
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Photograph of the hybrid micropositioning system Physik Instrumente M-511.HD [30].
First moving mass (m1)
Positioning mass (m2)
Gearbox (G)
Position sensor for s2
Spindle
Position scale
Clutch DC-drive (M)
Springs (c)
Fig. 1.5
Piezoelectric stack actuators (PEA)
Drawing of the hybrid micropositioning system Physik Instrumente M-511.HD.
inner mass
Outer mass (positioning plate)
springs Piezoelectric actuators (PEA)
Fig. 1.6
Photograph of the micropositioning mechanism implemented in PI M-511.HD. The function of the springs is to pre-stress the PEA enabling a bipolar input voltage range.
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M
εM
irt
JM
m1 s1
Fig. 1.7
O
c
G
m2 PEA
vPA
s2
Schematic functional presentation of a hybrid micropositioning system.
An equivalent lumped model of the system is given in Fig. 1.7. The pairs of PEA and spring are represented here with a single equivalent actuator, connected in parallel to a single equivalent spring.
1.3 Control of Hybrid Micropositioning Systems: State of the Art Due to the wide spectrum of applications, there is a considerable market potential for HMS. However, the necessary efforts for the design of a suitable control law can reach significant complexity. This is on one hand because of the complex structure of HMS, consisting of different actuators and the mechanical connections between them. On the other hand, this complexity is also the consequence of the diversity of potential applications, resulting in a high variety of possible positioned objects and positioning velocities. Although nowadays advanced control concepts for pure piezoelectric and DC-motor driven systems exist, in recent publications the controller design for a HMS is performed mainly using classical frequency transfer function (TF) approaches. Mostly proportional-integral (PI) and proportionalintegral-derivative (PID) regulators are used. Furthermore, the mutual interaction between the coarse and the fine actuator are often neglected, and the nonlinearities (especially the hysteresis of the PEA) are by control design disregarded. This results in few cases in suboptimal control performance and increases the trajectory-tracking errors. The first classification of the known control systems for HMS can be performed according to their positioning capabilities: •
•
Systems described in [19,21,24,25,27] feature only static or quasi-static position control. Thus, a precise point-to-point positioning is enabled, but no dynamic trajectory-tracking is taken into account. The reference position is approached by the use of the coarse actuator and thereafter the control algorithm switches to the fine actuator in order to achieve the required accuracy. Dynamic position control – enabling the trajectory-tracking – is possible with systems described in [4,18,20,22,26,28].
As we are interested in trajectory-tracking, dynamic positioning capabilities are mandatory for the control system. Known systems featuring dynamic positioning may be classified according to the: •
1
Control system structure. All published solutions1 feature individual controllers for the coarse and fine actuator, which are either 1. accompanied with a proprietary decoupling network. For example, in [18] the control of the HMS is realized with two independent controllers and a complementary activation function, which distributes the position error onto the individual controllers. The output of the decoupling function depends only on the actual position error. When this error reaches the PEA range, the hydraulic servo cylinder’s controller is enabled, and the PEA controller is disabled. Alternatively, for small positioning errors, the PEA controller is enabled and the hydraulic servo controller is disabled. The process of the enabling and disabling of the appropriate
To the author’s knowledge.
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•
•
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controller is performed on a smooth way, by the decoupling function. Although such concepts are proven for functionality [31], it can be shown that this solution results in a suboptimal control performance, because one of the actuators is always disabled or works with a reduced performance [32,33]. 2. accompanied with a decoupling network realized by the optimization of the sensitivity functions [26]. By this method, the limited operation range of the PEA cannot be taken explicitly into account. 3. connected in a master-slave configuration [20,22] sometimes accompanied also with additional notch filters [4,34]. Individual controller type. All known solutions deal with traditional PI/PD/PID controller configurations. It can be shown, that the use of state-space methods may improve the dynamic response, what consequently enables better trajectory-tracking and positioning results [32,33] as well. Such an improvement could be particularly useful for the system presented in [22], where on this way the saturation of the PEA input and hence the large positioning error occurred by the tracking of the reference position signal may be avoided. Nonlinearity compensation. The hysteresis nonlinearity of the PEA is in all relevant publications for HMS disregarded. Indeed, in [22] its effect is noticed, but no further action was taken. Because the hysteresis behaviour is significant by PEA, it severely limits the system performance such as giving raise to undesirable inaccuracy or oscillations, even leading to instability. When the hysteresis nonlinearity is not taken into account by control design, it will act as an unmodeled phase lag whose presence will cause overshot, even instability, when in the closed-loop system no sufficient phase margin is provided [35]. Without the aid of an additional control technique to overcome this problem, it is only possible to obtain a limited positioning accuracy [36] and dynamical response [37]. A hysteresis compensation measure is also not implemented by the commercially available controller for the M-511.HD (C-702, [34]). Adaptation to changed conditions. A systematic or automated method for adaptation of the HMS controller parameter to changed circumstances – like the mass of the positioned object – is to the author’s knowledge not yet presented.
The commercially available controller C-702 for use with the M-511.HD device [34] deals also with separate feedback controller of PID and PI type for the DC-drive and the PEA. Here, additional notch filters are implemented in order to suppress undesired resonant peaks. One of the main drawbacks of such a solution is – besides of undesired phase shift involved by the notch filters – the problem of proper settings of the controller and notch filter’s parameter. Depending on the positioned object properties, appropriate controller and filter parameter values have to be determined by the customer. This requires the experience in automated control theory and involves also some trial-and-error iterations. This process results sometimes in poor positioning performance. A very similar PI/PID control concept with additional notch filters was followed by the high-end actuator [4] for the European Extremely Large Telescope (E-ELT) project. E-ELT aims to provide European astronomers with the largest optical-infrared telescope of the World. The reasons for lack of advanced controller structures designed for HMS include besides the significant plant complexity, also the specific properties of such systems. Particularly, often complex models are needed to describe the mutual interaction between the rough and the fine actuator. Finally, the limited amount of information about the actual inner states of the separate actuators available in real time makes the design of an advanced controller particularly difficult. (Usually a typical HMS has only one position sensor mounted near to the positioned object due to limited space and cost optimization reasons.) It is hence necessary to investigate and develop novel control based on a mathematical model for improvement of the trajectory-tracking quality by HMS. The main task of this thesis is the investigation, design and experimental validation of such advanced control approaches for hybrid micropositioning systems. In order to utilize the full potential of such a system, rapid control prototyping (RCP, [38]) techniques, briefly described in the next subchapter, will be used for the control design.
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Because the work presented in this thesis addresses beside the general problematic also various individual tasks (like the modelling of piezoelectric actuators, compensation of rate-independent nonlinearities, state feedback control, etc.), further discussions about known methods and reported results related to specific problems are given in the appropriate subchapter separately (see f.e. Subchapters 3.4.2, 3.4.3, 4.2, 5.1). For the practical controller implementation and validation, a dSPACE RCP system is used, which is accompanied with an FPGA (field-programmable gate array) -based interface, developed in scope of the project. This decision is made because the commercially available hybrid controller C-702 is not suitable for RCP techniques through its software (SW) and hardware (HW) architecture. Furthermore, by the use of an additional FPGA significantly better real-time features can be achieved (e.g. more exact timing, execution of high-frequency tasks on the FPGA etc.). Research results will be applied to and experimentally validated with the M-511.HD device from Physik Instrumente. Besides the achievement of requested positioning properties in different usecases, a further goal of the thesis is the design of an automated control parameter adjustment under changed conditions. Adaptive techniques will be utilized for this task.
1.4 Rapid Control Prototyping RCP is a technology which enables the fast controller development, rapid building of controller prototype and easy practical testing. It represents a particularly attractive field for research and development activities since early 90’s till today [39-41]. The use of RCP is particularly common in the automotive industry [42] and robotics [43], but this technique is recently involved in many other industry fields including mechatronic systems, as well [44,45]. dSPACE GmbH [46] is one of worldwide leading companies on the market of RCP systems. The competitors are among others, the xPC Target™ (MathWorks) [47], LabView RT™ [48] (National Instruments), and Opal RT™ [49] (Opal RT).
Create a preliminary Test the control design in offline simulation control design Create a theoretical model of the controlled object Specify I/O in the model Controlled Plant
Create real-time code for the model
Set up ControlDesk™ to acquire data, watch and change variables
Real-time code running on dSPACE hardware
Fig. 1.8
Controller development workflow with dSPACE RCP.
The workflow of the control development with the dSPACE RCP system is illustrated in Fig. 1.8. After the creation of the theoretical model and offline simulations with Simulink™, the user specifies the input-output (I/O) connections to the plant. The next step is the automated generation of the realtime code, which is executed on the RCP hardware. The online data acquisition and parameter tuning are performed from the host PC via Ethernet connection using the experimentation software. The controller design within the scope of this thesis will be implemented and experimentally tested using a dSPACE RCP system based on the DS1006 processor board, featuring an AMD OpteronTM CPU.
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1.5 Structure of the thesis Although modern central-processing-unit (CPU) based systems are powerful for complex calculations, the exclusive use of pure CPU-based systems may be inadequate when specific real-time requirements with highly precise timing or high-frequency tasks are required. A suitable integration of a CPU-based system with an FPGA-board promises the solution of those problems. Therefore, a novel solution, featuring an FPGA-based interface is designed [50,51]. Through integration of an FPGA with the CPU-based RCP-system, several advantages are achieved like increased dynamic bandwidth, decreased latencies and exact signal timing. The design of the FPGA-based interface is one of the main topics of Chapter 2. Besides of the hardware design, development of appropriate communication software for the RCP-system as well as the extendable software framework of FPGA-code is presented here. The experimental validation of the realized hardware and software solution is initially performed during the integration with the RCP-system. The realized HW/SW solution is later on permanently used during the parameter identification and control design. As a base for design of a high-quality control, an appropriate plant model has to be developed. Nonlinearities like friction of the DC-drive, as well as the hysteresis of PEA should be considered herein. Proper identification of the plant’s unknown parameter is also an essential task. Chapter 3 deals with the modelling and parameter identification of the HMS under investigation. Particular attention is paid to the presentation of a novel method, which enables the simple identification of the PEA parameter without the need for system disassembly [52]. For the design of the position control, the efficient real-time capable compensation of the plant nonlinearities is of particular importance. By the design of a feedback controller, one may also obtain satisfactory position-tracking results without compensation of nonlinear effects. However, by adequate combination of the feedback controller and a real-time capable nonlinearity-compensation measure, significant enhancement of the positioning quality can be achieved. When an appropriate compensation of the plant nonlinearities in real-time is possible, the feedback controller deals from its own point of view with a linear plant. In Chapter 4, efficient and real-time capable compensation methods for the plant nonlinearities (friction and hysteresis of the PEA) are investigated, practically realised, and experimentally validated [52-54]. The development of the closed-loop control structure and design for the HMS is presented in Chapter 5. Because state-feedback controller can achieve improved dynamics and robustness compared to traditional PI and PID controller, a state-space approach [33] is chosen. Algorithms for simplified commissioning together with a method for online controller parameter adaptation by changed conditions (like the change of the positioned mass) are presented in Chapter 6. Although the controller design and online parameter optimisation addresses a commercially available hybrid nanopositioning system, the achieved results are certainly suitable for other similar micropositioning systems as well. Finally, in Chapter 7 a short discussion of the achieved results and the further research potentials is given.
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2. HARDWARE AND SOFTWARE FRAMEWORK FOR CONTROLLER REALIZATION As mentioned in the Introduction, the use of the RCP-technology is designated for the control design. A dSPACE RCP-system is chosen with a DS1006 processor board because its outstanding CPU power [55]. However, at the project start the need for an additional interface is recognized. Only the main reasons for this decision will be listed here – today there are much more advantages known, enabled through the developed customizable interface. 1. At the system design phase, a particular bus-interface-master functionality was not covered with the portfolio of dSPACE I/O boards. However, this functionality was necessary for reading out of the measured position information from the HMS. 2. Besides, as mentioned in the Introduction, pure CPU-based systems are not always adequate for real-time applications in cases when accurate timing synchronization or high frequency task requirements have to be fulfilled. This is the particular case by the control of the HMS. 3. And finally, a new sensor should be added to the system: through measurement and control of the DC-drive’s current, improved dynamic response is expected. The current measurement and the associated control loop are designated to run with higher sampling rate related to the superimposed controller structures. A solution for these problems is found through the design of an intelligent FPGA-based interface, and its integration with the RCP-system using a dual-port-random-access-memory (DPMEM, [56]). Unlike to CPU-tasks, all FPGA processes run simultaneously without any mutual obstruction or resource conflicts. The number of such parallel running processes can be freely defined, according to the task requirements and available FPGA resources. Furthermore, the reaction of an FPGA application to an internal or external event may be set to as low as to few nanoseconds. This is significantly shorter from the interrupt latency of a “state-of-the-art” CPU-system. Using an FPGA, very precise and deterministic timing (up to the resolution of the FPGA clock signal) for all processes is achievable. Currently available FPGA-s have also significant drawbacks. The computation power is limited through the number of available logical blocks (gates, slices, multiplexers). The realization of complex calculations using an FPGA is not a trivial task either, because there is no native support for the floating-point arithmetic. Appropriate scalings, needed for the accurate fixed-point arithmetic have to be implemented manually. The same is true for the design of the proper timing and the inter-process communication. Because the features of the available tools for automatic FPGA code generation are also limited, FPGA devices are today less attractive for the control design compared to standalone RCP targets. The benefits of the suitable combination of an FPGA and an RCP-system are the solution both of the timing and latency issues, maintaining at the same time significantly more CPU-power, capability for complex floating-point calculations, and enabling the use of advanced automatic code generation tools.
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In order to utilize these possibilities, an FPGA-based interface is designed. The main designated tasks of the FPGA were at project start the management of the SPI bus-communication and preprocessing of the additional sensor data. The plant control signals are directly issued by the FPGA, enabling on this way precise I/O timing. The data exchange with the dSPACE-system is solved through the DPMEM (Fig. 2.1). The CPU resources of the RCP-system are more efficiently used, because very fast CPU interrupts may be avoided – these tasks are scheduled to the FPGA.
Reference input
CPU
Complex control Data for algorithms, acquisition superimposed controller. 1
Fig. 2.1
CPU data write
Dual-Port-RAM
CPU data read
FPGA data read
FPGA data write 2
FPGA Underlying controller, timecritical tasks, processing of measured data.
Plant control signals
PLANT
The controlled Plant plant including sensor actuators and signals sensors.
3
4
Integration of a CPU with an FPGA using a dual-port-random-access memory.
The concept of the integration of a generic CPU with a FPGA through the use of a DPMEM for data exchange is schematically shown in Fig. 2.1. The CPU (in our case the RCP-System, noted with “1”) is responsible for calculation of complex algorithms and superimposed controller functionalities. The FPGA (“3”) executes the underlying control functionalities which are executed with high sampling rates, issues control signals to the plant (“4”), reads in the sensor data, and performs all necessary preprocessing and statistical preparations (for example, calculation of the mean value of the oversampled input) on the measurement data. The data exchange between the CPU (which acts as a master) and the FPGA (the slave device) is solved through the DPMEM (“2”). Although this is not mandatory, it is useful to define separate segments which are in the DPMEM exclusively reserved for data write and read operations (all definitions are from CPU point of view). The implementation of an appropriate data-exchange and access-synchronization protocol is also advisable. The design of the new interface is performed partly in a top-down manner. After the analysis of the available solutions for the data exchange between the dSPACE-system and an FPGA-based interface, the main electrical and electro-mechanical parts and the available signal ports of the HMS system are reviewed. Based on the collected information, the required functionality is defined, followed by the selection of the appropriate dSPACE boards. At this point, the main elements of the M-CTRL interface were still generic. The definition of the detailed requirement specification is the next critical step in the design process. Various aspects, like the designated functionalities, the correct signal timing and the required signal quality, expected engineering efforts and the available budget, uncertainties and risks, should be considered herein. The system configurability and possibilities for future extensions should also be ensured. The selection of the main hardware modules and hardware parts is performed according to the detailed requirement specification. The main goal is the achievement of the expected performance, taking into account the minimization of the necessary efforts and risks. Finally, economic aspects as well as the free commercial availability of selected parts play also a significant role. The design of the hardware interface is thereafter performed regarding the requirement specification by the use of various optimization methods. Beside of the optimization for component placement, the minimization of electromagnetic disturbances [57,58] has to be considered during the creation of the printed circuit board (PCB) layouts and the signal routing. As next, an appropriate software framework is developed. It covers the appropriate data exchange routines for the RCP-system and the FPGA application which includes the data exchange and the basic I/O processes. The designed framework should represent an extendable system, ready for functional improvements both of hardware and software type. In this chapter, the HW design for the new interface (named “M-CTRL” as the abbreviation of “Motion-Control”) and the created basic SW framework is described as first. Thereafter, an experimental validation of the designed HW and SW is given.
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2.1 Electrical parts and available ports of the M-511.HD In this subchapter the relevant electro-mechanical and electronic parts of the HMS M-511.HD from Physik Instrumente are presented. As next, the available signal and data ports are described. This information is relevant for the design of the FPGA-based interface. 2.1.1
Electro-mechanical and electronic parts
The relevant electro-mechanical parts of the HMS PI M-511.HD are : - A couple of multilayer piezoelectric stack actuators of the type Physik Instrumente PICMA P885.50 [60], having maximal allowed input voltage range (without pre-stressing) of [-30 … 120] V. In the pre-stressed case (mounted into the HMS) the recommended voltage range is [-36 … +36] V, resulting in displacement range of [-4.2 … 4.2] µm. - Graphite-commutation DC micro-motor Faulhaber 3557-024CS [61] with permanent magnet and armature input voltage range of 24V. - Full H-bridge-circuit LMD18201 [62], enabling pulse-width modulated (PWM) signal for driving of the DC-motor. - Linear optical incremental encoder from Heidenhain [63] having 2 µm per-pulse resolution with analogue sine/cosine output. - GEMAC GC-IP1000 chip [64] containing an encoder sine/cosine interpolating circuit for enhancement of the position resolution and a Serial Peripheral Interface (SPI, [65]) slave port for the output of the position value. The resolution of the final position data (given as the value of the least significant bit (LSB) of the least significant word (LSW)) is 2 nm. - Two Hall-elements for sensing of the end-limit positions by the motion. One Hall-element is mounted on both the positive and negative end of the positioning range. - A reference lane and reference lane sensor. Enables the absolute referencing of the middleposition and hence the reproduction of a particular position or movement after system restart. The piezoelectric amplifier module Physic Instrumente M718E0022, dedicated for the used PEA type, is not part of the HMS. This module will be integrated into the M-CTRL interface. The input range of the piezoelectric amplifier is [-5 … +5] V and its amplification factor is 7.2. Thus, the recommended voltage range of the PEA of [-36 …36] V is completely covered. 2.1.2
Available signal and data ports
The commercially available hybrid micropositioning system features following input/output (I/O) signal and data ports [59]: - Serial Peripheral Interface (SPI, [65]) slave port, consisting of four differential signal line pairs ({SCLK, SCLK },{SEN, SEN } {SDI, SDI }, {SDO, SDO }). Through this port,
-
the information about the actual position can be read out using a SPI master device. The position information is acquired by the use of the incremental sensor from Heidenhain with sine/cosine interpolator featuring the GEMAC GC-IP1000 chip [64]. By system start-up, the measured position value is set to zero and all further measured position values are given relative to the start-up position. The required SPI clock frequency is 2.2 MHz. The measured position data is transmitted in two SPI cycles using 16-bit integer data format, whereas only 28 bits hold valid position information. The resulting maximal measurement range is 0.53687 m. The maximal effective position measurement rate is limited to 20 kHz. Reference lane information (REF). Digital output, holding information about the reference lane sensor, which changes its value in the middle position. The repeatability of this signal is better than 1 µm. Positive end position flag (pos. END). Digital output. Value is set to “1” when the positive end position is reached. Negative end position flag (neg. END). Digital output. Value is set to “1” when the negative end position is reached. Digital inputs for control of the H-bridge LMD18201 [62]. PWM signals for magnitude, sign, and brake inputs (PWM MAG, PWM SIGN, and PWM BRAKE) are required.
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Piezo-voltage (VCC Piezo). Analogue input of range: [-36V…+36V], connected directly to the PEA couple.
All digital signals are of CMOS (3.3V) signalising level. The set of the digital and analog I/O ports available by the HMS M-511.HD forms a mixed-signal SSI-bus2 [59]. The utilized SPI configuration contains one master and one slave device (the GEMAC GC-IP1000). The SPI signal connection is shown in Fig. 2.2. The communication through the SPI bus is organized in simultaneous command-and-response manner. The SPI master generates the signals SCLK, SCLK, SEN, SEN, and submits commands to slave using the master-out-slave-in (MOSI) signal connected to SDI port of the slave device. The slave device answers with SDO signal connected to the master-in-slave-out (MISO) port of the master device. The MOSI signal is set by the SPI master at the falling SCLK edge, whereas the value of the MOSI signal is evaluated at the rising SCLK edge. The timing diagram including SPI commands for communication with the GC-IP1000 is given in Appendix 8.1.
Fig. 2.2
SPI bus configuration with one master and one slave device
2.2 Control hardware concept The functional requirements for the control hardware and the base software framework are: - reading of the actual position information through SPI bus communication, - accurate measurement of the DC-current, - reading of the end position and reference lane sensor information, - control of the DC drive through PWM signals, - control of the piezo-amplifier through the analogue voltage reference, - measurement of the PEA voltage for testing and identification purposes. Regarding these requirements, the control hardware concept is created. It consists of the RCPsystem, the M-CTRL interface board, and the hybrid micropositioning system (Fig. 2.3). 2.2.1
The selected RCP-system
The chosen RCP-system features an Ethernet Host Interface (EHI), the DS1006 processor board, the DS4121 low-voltage-differential-signal (LVDS) interface card [66], and the DS551 plug-on device [67]. Model and controller calculations are performed on the DS1006 processor board. The hostconnection bus and the PHS-bus (Fig. 2.3, Fig. 2.4) are dSPACE-specific customized communication bus systems with low latency and high data throughput. The internal communication between dSPACE-boards through these busses is covered by the automated code generation process (Fig. 1.8). The data exchange between the processor board and the host-PC is solved through the EHI. This enables task synchronous measurement with data recording, as well as online parameter tuning. 2 The mixed-signal SSI-bus mentioned in this document should not be misconceived with the Synchronous Serial Interface (SSI) which is widely used in the industry.
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M-CTRL interface board
dSPACE RCP-System
Signal Signalconditioning conditioningand and additional additionalelements elements(A/D, (A/D,D/A, D/A, Piezo-Amplifier,PWM, Piezo-Amplifier,PWM,dig. dig.I/O) I/O)
DS DS4121 4121 PHS-bus
LVDS bus DS DS551 551
DS DS1006 1006
SSI bus
Hybrid micropositioning stage
FPGA FPGA -board -board
Host-conn. Host HostInterface Interface
Monitoring, Data recording, Parameter tuning
Ethernet cable
Fig. 2.3
PC with ControlDesk
The integrated hardware concept consists of the modular RCP-system, the M-CTRL interface board, and the hybrid micropositioning system.
The DS551 is a plug-on-device (POD) primary developed for time-critical function-bypassing tasks [39] with electronic control units (ECU, [39]) in automotive industry. This device features 16kB of fast dual-port reflective memory, which can be simultaneously accessed both from the DS1006 processor board via the DS4121 LVDS interface card and from the ECU via its standard address- and data-bus ports (Fig. 2.4). In the current design, these ports are connected to the FPGA on the M-CTRL interface. Besides the DPMEM-based data exchange, the DS551 features further advanced features, like hardware interrupt triggering in both directions, as well as sub-interrupt handling on the RCPsystem. LVDS-bus Cable (5m)
dSPACE-Box (PX10)
Address Bus [0…14]
LVDS Port 1
Host Interface bus
dSPACE dSPACE Host HostPC PC Connection Connection Board Board
PHS DS DS1006 1006 -bus Real-time Real-time Processor Processor Board Board Model (control law) calculation
DS DS4121 4121 Low Low Voltage Voltage Differential Differential Signal Signal (LVDS) (LVDS) Interface Interface Board Board LVDS Port 2 (free)
DS DS551 551 ECU ECU Interface Interface Plug-on Plug-on -Device -Device 16kB dual-port RAM memory
Data Bus [0…15] Chip Select Read Strobe Write Strobe
Monitoring, Data recording, Parameter tuning Ethernet cable
PC with ControlDesk
Fig. 2.4
The selected modular RCP System.
The timing requirements for the address and data bus of the DS551 board are given in Fig. 2.5. Data read and write directions are here defined from the ECU (e.g. FPGA) point of view. In one operation cycle a 16-bit integer can be written (Fig. 2.5a) or read (Fig. 2.5b) to/from the DPMEM. As one may note, the DS551 features data throughput of 40 MByte/s and – what is for current application from most importance – nanosecond-range latency times. The signal connection between the M-CTRL interface and the hybrid micropositioning system is realized through the SSI-bus described in Subchapter 2.1.
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>5ns
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>40ns
Chip select
Chip select
Write Strobe
Write Strobe
Read Strobe
Read Strobe
Adress Bus
address
Adress Bus
Data Bus
data
Data Bus
address data v PWM _ max , y C [k ] = ⎨v PWM _ max for y CP [k ] + y CI ⎪ y [k ] + y ′ [k ] else. CI ⎩ CP
(4.15)
Proportional part yCP [k ] = − d1 ⋅ K R ⋅ eC [k ]
Input filter
− d1 ⋅ K R
w′[k ] = w[k ] ⋅ (1 + d1 ) − d1 ⋅ w′[k − 1]
w[k ]
(1 + d1 )
eC [k ]
w′[k ] w′
K R ⋅ (1 + d1 )
yCP [k ]
vPWM _ min vPWM _ max
′ [k ] yCI
vPWM _ min vPWM _ max yk yC [k ]
d1 ~ iq [k ]
z −1
′ [k − 1] yCI
z −1
yCI [k ] = yCI [k − 1] + K R ⋅ (1 + d1 ) ⋅ eC [k ]
Integrator part
Fig. 4.2
Action diagram of the digital current controller.
The action diagram of the realized discrete PI controller featuring the AWC measure is shown in Fig. 4.2. The closed-loop system built by the DC-drive and the digital current controller (Fig. 4.1) can be approximately described by a first-order transfer function:
iq (s) =
i q* ( s ) 1 + TΣ ⋅ s
.
(4.16)
Thus, according to (3.10), the transfer function of the torque controlled DC-drive is: GTM ( s ) =
1 TM ( s ) = , * TM ( s ) 1 + TΣ ⋅ s
(4.17)
with time constant TΣ = 162 µs. More details about the practical realization and experimental testing of the current controller are given in Subchapters 4.5 and 4.6.
4.2 Compensation of nonlinearities with local memory In this subchapter, model-based compensation methods for plant nonlinearities with local memory for plants which can be expressed in form of a Hammerstein-model will be discussed. The presented algorithms are suitable for (but not limited to) PEA hysteresis compensation as well. Rate independent nonlinearities with local memory can be described in time domain by an operator N{ t,…}. The output p(t) depends on current and past values of input [r(t),…,r(t-τ)]:
p(t ) = N { r (t ),..., r (t − τ ), t }.
(4.18)
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An efficient real-time capable compensation of the plant non-linearity is by design of a control law from particular importance. Indeed, if the plant with rate independent nonlinearity can be presented as shown in Fig. 4.3, and the nonlinearity can be compensated in real time, control design methods available only for linear plants can be successfully applied. By appropriate compensation of nonlinearity N{t,…} by the compensator CN{t,…}, the feedback controller GC(s) deals from its own point of view with a linear plant GP(s) as shown in Fig. 4.3.
y (t )
u (t )
GC (s )
y * (t )
r (t )
C N {t ,...}
N {t ,...}
compensator
feedback controller
p (t )
y (t )
GP (s)
non-linear plant linear plant
Fig. 4.3
The controller of a non-linear plant with compensated non-linearity deals from its point of view with a linear one.
There are different possibilities for the compensation of the plant rate-independent nonlinearity. If an appropriate inverse operator N-1 can be found, the compensation may be realised directly in a feedforward manner as shown in Fig. 4.4a. Such compensation methods are reported in [108,111,113,116,126-129,138,143]. u(t ) u(t )
r (t )
ˆ −’1 N NN’
inverse-model based feed-forward compensator
p(t )
N N
ˆ N’ N N multiplicative local feedback compensator
b)
K
u(t ) +
r (t )
p(t ) u(t ) +
N N
-
d)
u(t ) +
1 / Kp
r (t )
-
N Kp
+
additive local feedback compensator for unitygain plant
p(t ) u(t ) + -
+
ˆ’ N’ N additive local feedback compensator with plant gain compensation
N
p(t )
+ +
ˆ’ N’ N
simplified multiplicative local feedback compensator
c)
r (t )
-
plant nonlinearity
N’ Nˆ
Fig. 4.4
plant nonlinearity
-
plant nonlinearity
p(t )
N
+
K
+
a)
e)
r (t )
+
r (t )
plant nonlinearity
N Kp
1 / Kp
+
p(t )
+
Nˆ plant nonlinearity
f)
additive local feedback compensator without plant gain compensation
plant nonlinearity
Non-linearity compensation methods: cascaded compensation using the inverse model of the plant (a), multiplicative local feedback compensator (b), simplified multiplicative local feedback compensator (c), additive local feedback compensator for unity-gain plant (d), additive local feedback compensator for non-unity gain plant with (e), and without (f) plant gain compensation.
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Founding an inverse operator is a complex task, or it is even impossible. In such cases a solution may be found with compensation approaches based on a local feedback structure. The multiplicative local feedback compensation (MLFC, Fig. 4.4b) contains a high gain value K (K>>1) connected in the direct branch. The mathematical model of the nonlinearity operator Nˆ is placed in the feedback. On this way a quasi-inversion of N can be achieved: p (t ) = N {r, t ,...t − τ },
r (t ) = u(t ) + K ⋅ (u (t ) − Nˆ {r, t ,...t − τ }) , (1 + K ) ⋅ u(t ) = r(t ) + K ⋅ Nˆ {r, t,...t − τ }),
(4.19)
with : K >> 1 ⇒ u(t ) ≈ Nˆ {r, t ,...t − τ } ⇒ r (t ) ≈ Nˆ {u, t ,...t − τ }, with : ( K >> 1 and Nˆ = N ) ⇒ p(t ) ≈ u (t ) ⇒ Y ( s ) ≈ G ( s ) ⋅ U ( s ). −1
P
The simplified multiplicative feedback compensation (SMLFC, Fig. 4.4c) uses a similar principle. A PEA hysteresis compensation structure based on SMLFC is reported in [144]. In practical realization, by both methods the value of K has to be chosen high enough to achieve satisfactory compensation quality. However, a too high value of K may cause convergence problems by the real-time application and increases the sensitivity for the modelling errors. If it is possible to express the plant nonlinearity in form of a sum of a linear gain and a non-linear operator N, the use of an additive local feedback compensation schema (ALFC Fig. 4.4d-f) is advisable. When the plant is separated into a unity-gain and the non-linear operator, the ALFC can be realized as shown in Fig. 4.4d. Here, the compensation quality depends only on the plant-model accuracy: r (t ) = u (t ) − Nˆ {r (t ),..., r (t − τ )} , p (t ) = r (t ) + Nˆ {r (t ),..., r (t − τ )} =
= u (t ) − Nˆ {r (t ),..., r (t − τ )}+ N {r (t ),..., r (t − τ )} ,
with Nˆ = N
⇒
(4.20)
p (t ) = u (t ) ⇒ Y ( s ) = G P ( s ) ⋅ U ( s ) .
In general, the software implementation of the compensator shown in Fig. 4.4d requires the solution of a fixed-point problem with memory. However, if the parameterized nonlinearity model fulfils the conditions ∂Nˆ {r (t ),..., r (t − τ )} ∈R ∧ ∂r (t )
∂Nˆ {r (t ),..., r (t − τ )} < 1, ∀{r (t ),..., r (t − τ )}, ∂r (t )
(4.21)
there exists an “attractive fixed point” [145] for any input signal r(t). In this case, the fixed-point problem can be discretized and replaced by a much simpler explicit difference equation:
r ( k ) = u(k ) − Nˆ {r ( k − 1),..., r ( k − n − 1)}.
(4.22)
Starting from the correct initial values, the compensator will work correctly if the sampling frequency is significantly higher than the maximal frequency of the reference input. By a non-unity gain, the ALFC with (Fig. 4.4e) or without the gain compensation can be realized (Fig. 4.4f), see also [53]. It can be shown, that the real-time implementation of an ALFC is also straightforward in cases when the influence of the plant nonlinearity is smaller than the influence of its linear part.
4.3 PEA hysteresis compensation The compensation of the rate-independent PEA hysteresis is a significant problem. Accordingly, this problem is the subject of a large number of publications and different approaches for its solution are known.
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Electrically, the PEA behaves as a nonlinear capacitor, with the principal nonlinearity being a hysteresis in the charge-voltage characteristic. This electrical nonlinearity results from the same dielectric hysteresis as the electromechanical nonlinearity does. Thus, it happens that the relationship between the mechanical force (and displacement) of the actuator and the applied charge is roughly linear. For this reason, the control of the charge of the PEA (instead of its voltage) may appear to be the simplest solution. Such devices [111,146,147] are purely hardware-based. They can be realized with linear electronic parts. Although the compensation of hysteresis may be very efficient [147], the mentioned methods have some drawbacks as well: charge amplifiers and controllers are expensive and bulky, high internal voltages have to be generated and several problems regarding the drift currents have to be solved. The use of a second PEA in the feedback loop of an operational amplifier together with a charge amplifier utilizes a similar principle [148]. A novel approach combining current excitation by movement and voltage excitation during the steady-state is presented in [149]. Although the approach seems to be promising, results presented in [149] are limited to very low frequencies and slow positioning velocities. The software-based solutions are mostly used by voltage excitation of the PEA, and use a CPU (or FPGA). These approaches are mainly used by systems where the reference voltage is set by a D/A converter. The general operating principle is to predict (estimate) the voltage (or displacement) nonlinearity caused by the hysteresis loop, and correct the excitation voltage in order to achieve the linear dependence between the reference input and the PEA force (or displacement). Both modelbased [35,103,111-116,126,150,151] and model-free [152] approaches are known. Although there are publications with a feed-forward [35] or a local feedback [151] approach, most of the published model-based methods correspond to the more general nonlinearity-compensation principle shown in Fig. 4.4a. In this thesis, a different model-based approach is presented. For the hysteresis compensation, the parameterized MRC-model of the PEA is used. Because there no inverse hysteresis model in closed form is known, the compensation method from Fig. 4.4a cannot be applied directly. The other methods based on a local feedback loop (Fig. 4.4b-d) are, however, applicable. The general action diagram of the PEA hysteresis compensation measure using SMLFC is shown in Fig. 4.5, with x being the PEA displacement. Although such compensator reveals excellent simulation results [53], there are also significant drawbacks. The SMFLC measure contains an algebraic loop with gain of -K. Due to its high negative gain value, this algebraic loop cannot be resolved by including of a unit delay element. Instead, an appropriate iterative algebraic loop solver has to be implemented, or alternatively an appropriate low pass filter must be inserted in the SMLC structure. Both of these solutions have their drawbacks. An algebraic loop solver on one hand results in a computation overhead and a variable task turnaround-time. A low-pass filter on the other hand significantly limits the dynamical response of the system. Because the large gain value K, the SMLFC measure is also highly sensitive for modelling errors.
x
TEM
TEM ⋅ x +
+
C ⋅ vˆt
vP Fig. 4.5
MRC
qˆ c
vˆmrc
v Pc −
+
vˆt
+
C
z −1
K
−
v Pc
vP
vˆt
Action diagram of the hysteresis compensation measure using SMLFC.
The compensation measure designed by ALFC method does not have such an algebraic loop problem, and it is less sensitive to the model errors as well. When (4.22) is fulfilled (this is the usual case for the most PEA), the ALFC can be implemented as shown in Fig. 4.7. Here, by using the parameterized model of the MRC-element, the value of vmrc is calculated in the hysteresis compensator block (with its output quantity denoted as vˆmrc ) and added to the reference input voltage vPA (4.23)(4.24). The influence of the PA by the hysteresis compensation measure is not explicitly taken into
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account, because its influence in the intended frequency range is not significant. Furthermore, the PA influence is considered by the design of the state estimator in Subchapter 5.3.
vˆmrc = f ( x , v pc ) ≅ vmrc ,
(4.23)
vt = v pc − vmrc = v PA + vˆmrc − vmrc ≅ v PA .
(4.24)
The internal structure of the realized hysteresis compensator is given in Fig. 4.6, whereas in Fig. 4.7 the schematic view of its integration with the plant and the reference PEA voltage input is shown.
v pc x
+
TEM
TEM ⋅ x + qˆ = T ⋅ x + C p ⋅ vˆt +
−
vˆmrc
vˆmrc
Cp
C p ⋅ vˆt Fig. 4.6
MRC − Model
vˆt = v pc − vˆmrc
z −1
Action diagram of the realized ALFC hysteresis compensator.
x
Hysteresis compensator
vˆmrc v PA
v pc
+ +
Piezoelectric actuator +
vt
−
x
v mrc Fig. 4.7
Schematic view of the ALFC hysteresis compensation measure.
Due to the cancellation of the parasitic hysteresis voltage vmrc through the compensator unit, a linear dependence between the reference input voltage vPA and the generated PEA force Fp is achieved: Fp = TEM ⋅ vPA .
(4.25)
Thus, the model of compensated PEA can be presented by linear elements (Fig. 4.8).
q&
Fext
q&t
q&c v PA
Fig. 4.8
Cp
mp
qt = TEM ⋅ Δ s2
x
F p = TEM ⋅ v PA TEM
vt = vPA
Fp
c
d
The obtained linear electro-mechanical model of the PEA after compensation of the parasitic hysteresis voltage vmrc.
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In case of the HMS under investigation, the model part with the linear electro-mechanical model of the compensated PEA is shown in Fig. 4.9. Through the real-time plant linearization by the hysteresis compensation measure, the active PEA force is directly proportional to the PEA reference voltage. Fext q&
m2 + q&t
q& c v PA
Cp
qt = TEM ⋅ Δs2 TEM
v PA
F p = TEM ⋅ v PA
mP 2
s2
Fp c
d
Fp
m1 +
mP 2
s1
F1
Fig. 4.9
Part of the HMS electro-mechanical model with the compensated PEA.
Due to limited space, as well as for the reasons of cost optimization, the HMS does not have a separate sensor for the measurement of the PEA displacement. Thus, the dilatation of the PEA Δs2 = s2 − s1 cannot be measured during a trajectory-following task, where both the DC-drive and the PEA are simultaneously driven. This problem is solved by the design of a state estimator for the PEA [33], presented in the Subchapter 5.3. On this way, the hysteresis compensation measure does not requires any sensor information.
4.4 Friction compensation for the DC-drive The compensation of the nonlinear disturbances caused by friction is a significant mechatronic problem. Lot of publications deal with this issue. Different friction models are developed and various methods for compensation are known [86-91]. Because by the HMS specific restrictions (like the absence of a position sensor on the DC-drive’s rotor), requirements (functionality under changing conditions like temperature and load change) and conditions (spatial dependence of the friction force on the actual position) apply, a simpler friction compensation measure, featuring an online adaptation algorithm is designed. The investigation and the design of this measure for the case of the disabled PEA is described in [92]. These results can be generalized by estimation of the PEA force also for the case of enabled PEA controller. For this reason, this thesis gives only a brief overview of the work done and achieved results. As assessed in [92], the design of the friction compensator is possible using a simplified physical model. Through an experiment with constant DC-drive torque excitation by disabled PEA, it is noticed that the measured velocity of the mass m2 does not reveals any resonant peaks during the reversal of the torque’s sign (see Fig. 4.10 and Fig. 4.11, where v 2 ≡ s&2 ). Resonant peaks are reported by systems with elastic mass couplings [153]. Furthermore, Fig. 4.11 does not indicate any backlash [154]. According to these observations, it was found that a simple one-mass model shown in Fig. 4.12 is feasible for the friction compensation algorithm. The plant model used for the friction compensation is expressed through a linear and a non-linear part as shown in (4.26). The recent part (NF) summarizes the influence of the friction. The force Fz acting from the gearbox side is treated herein as a disturbance. By disabled PEA, FZ can be supposed to be zero and the angular velocity ε&M corresponds to the measureable velocity ( ε&M ≈ irt ⋅ s&2 ). When the piezoelectric actuators are enabled, the estimation of ε&M and FZ is performed using an appropriate state estimator which is described in Subchapter 5.3. The active torque of the DC-drive TM is estimated in both cases by taking into account the implemented current controller.
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DC-drive torque (Nm)
0.03 Ref. torque T *
0.02
M
Achiev. torque T
0.01
M
0 -0.01 -0.02 -0.03 0
0.2
0.4
0.6 Time (s)
0.8
1
1.2
0.2
0.4
0.6 Time (s)
0.8
1
1.2
2
Measured velocity v (m/s)
0.1 0.05 0 -0.05 -0.1 0
10
-1
10
-2
10
-3
10
-4
10
-5
2
|v (f)| (m/s)
Fig. 4.10 Results of the DC-drive torque experiment with sign reversal by disabled PEA.
0
50
100
150
200 250 300 Frequency (Hz)
350
400
450
500
Fig. 4.11 Fourier transformation absolute value of the velocity achieved by the DC torque experiment with sign reversal.
iq
ε M irt
TM M
TFM
JM
FZ
Fig. 4.12 The simplified plant model used for the friction compensation measure.
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The estimation of the friction torque TF is performed by the integration of a radial base function network (RBFN) into an extended Kalman-filter [95,155] as shown in Fig. 4.13. Kalman filters are Bayes optimal, minimum mean squared error estimators for linear systems with Gaussian noises. The design of the Kalman filter elements is done according to [156]. The procedure is described in [92]. The integration of the RBFN into the Kalman filter is solved using a prediction-correction algorithm based on [157]. Details to the integration are given in [92].
x& DC
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢0 1 ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ TF ⎥ 1 ⎥ 1 ⎢ ⎥ ⎢ ⎢ ⎢ ⎥ = 0 0 ⋅ x DC + 0 ⋅ u DC + − ⋅u − ⎢ (irt ⋅ J M ) ⎥ z ⎢ J M ⎥ ⎢ ⎢ ⎥ JM ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎢1⎥ 1⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢0 0 − ⎥ ⎢ ⎥ 0 TΣ ⎥⎦ ⎢⎣ ⎥ ⎢⎣ ⎥ ⎢⎣ ⎢⎣ TΣ ⎥⎦ { 1 44244 3 14 4244 3⎦ 123⎦ B DC
A DC
(4.26)
NF
BZ
y DC = [1 0 0]⋅ x DC 1424 3 C DC
with : x DC = [ε M
ε&M
TM ]T , u DC = TM* , u z = Fz , y DC = ε M ⋅ irt = s0 .
TM*
εˆM
Kalman-filter (state observer)
Neural network TˆFMNN (RBFN)
Fˆz
TˆFM
[εˆM , εˆ&M ]T
Fig. 4.13 Integration of the RBFN into a Kalman filter. The RBFN is a radial-basis neural network featuring a forward branch for information propagation (Fig. 4.14). The input vector u is distributed to all neurons in the hidden layer. Each neuron in the hidden layer features an activation function Φ(x,u). The output of the RBFN is built as sum of the hidden layer outputs, rated through the weighting function w. The array x contains the centre points where the activation functions reach their maximum. The activation functions are chosen in B-Spline form with slight modifications (marked with bold lines in Fig. 4.15), in order to enable the description of a simplified friction model shown in Fig. 4.16. The training of the RBFN is carried out by adaptation of the weighting-function w. For minimization of the estimation error, a normalized least-mean-square method [158] was used during the online training.
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Hidden layer Φ1 (x, u)
Input layer
wˆ 1
Output layer
u1
u2
wˆ N
w k = [wˆ 1
wk
Φ N (x, u)
wˆ 2 K wˆ N ]
yˆ k
∑
T
Φ k = [Φ1 Φ 2 K Φ N ] T u = [ε M ε&M ]
Training algorithm
T
ek
yk
Fig. 4.14 RBFN with online training algorithm. − ε&ME
ε&ME
v Φ N / 2−1 (x, ε&M ) Φ N / 2+1 ( x, ε&M )
x1
x3 x2
Φ N ( x, ε&M )
ε&M x N / 2 −3 xN / 2−1 x N / 2+1 xN / 2+3 xN xN + 2 xN / 2+ 2 x4 xN −1 xN / 2−2 xN +1
Fig. 4.15 Selected RBFN activation functions for the DC-motor’s angular velocity.
TF (ε&M ) TC
µ
1
− ε&ME
ε&M ε&ME
−µ
1
−TC
Fig. 4.16 Simplified friction model. The friction compensator is realized by a feed-forward measure: the estimation of the friction torque value is added to the reference torque TM*: * TMC = TM* + TˆFNN .
On this way, the torque lost by friction is approximately compensated.
(4.27)
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4.5 Real-time implementation of the compensation measures The disturbance and nonlinearity compensation measures are implemented using the hardware and software framework described in Chapter 2. The current controller for the DC-drive and the PEA hysteresis compensation are realized with the FPGA in form of simultaneously executed processes. The Xilinx System Generator™ and the Xilinx Blockset™ for Simulink™ are used. After all optimizations, the logical gate resources of the XS31500 were exhausted to more than 99%, whereas 30 from the 32 available multiplexers are used. The current controller is realized according to Fig. 4.2 and (4.14)-(4.15) with 40 kHz sampling frequency. The linearization of the PEA characteristic is performed according to the procedure described in Subchapter 4.3 using the parameter values from Table 8.4 and a sampling frequency of 100 kHz. As described in Chapter 1, due to limited space and for the reasons of cost optimization, the hybrid micropositioning system does not have a separate sensor for the measurement of the PEA displacement Δs2. Thus, the dilatation of the PEA cannot be measured during the trajectory-following task, where both the DC-drive and the PEA are simultaneously driven. This problem is solved by the design of a state estimator for the PEA ([33], described in more detail in Subchapter 5.3). In order to ensure the initial condition for the hysteresis model which reflects the PEA state, the signal from Fig. 3.20 is applied to the PEA, whenever the FPGA is powered on, before enabling the hysteresis compensation measure. A new identification of PEA model parameter is necessary only when the environmental conditions (temperature, load) change significantly. The overview of the FPGA top level layer after implementation of the current controller and the hysteresis compensator is given in Fig. 4.17. Further implementation details (FPGA process timing, data type selection and scaling, etc.) are given in [76,77]. The friction compensation measure is implemented in the RCP-system using Simulink and embedded Matlab-scripting. The sampling frequency is set to 10 kHz. The “fine tuning” of the RBFN can be enabled during the positioning process; on this way the adaptation to changed circumstances (changed friction force due temperature changes or wearing) is ensured.
Clock
External clock source (66 MHz)
divider
Clock Generator 66MHz
Synchronizer
4.4MHz 100kHz 40kHz 20kHz SPI_comm
SPI bus ref_pos, pos_end , neg_end
Synchro ensuring the data nizer integrity.
Read and SPI prepare position information ADS8406_ctrl ADS8422_ctrl
ADS8422 signal bus
DPMEM data bus
Read/Write DPMEM
DC-drive Current measurement with oversampling and averaging.
FPGA top-level software layer
PEA voltage measurement with oversampling and averaging.
Read/Write Data exchange DPMEM
DS551 (DPMEM)
address
with the RCP bus -system (according to Fig. control 2.10).
signals
Current controller and PWM PWM generation signal generation for the DC-drive
PWM signals
PEA Hysteresis compensator and state estimator.
Legend: Internal clock flow Internal data flow External signal flow
DAC7744_ctrl control of PEA voltage.
ADS8406 signal bus DAC7744 signal bus
Fig. 4.17 Top-level modules of the FPGA application after implementation of the current controller and the PEA hysteresis compensation measure.
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4.6 Experimental results The implemented nonlinearity and disturbance compensation measures were subject to thorough experimental tests. Some characteristic results are shown here in order to asses the quality of the developed algorithms. The response of the current-controller by the fixed rotor using a reference signal step by 1 A is shown in Fig. 4.18. In this case, the controller is all the time in the linear regime, because the normalized PWM value does not reach its limits of [-1…1]. The situation is different by a large reference input change. In Fig. 4.19, the response by fixed rotor in case of reference input step of 4 A is shown. Besides the fast time response, the functionality of the AWC-circuit is noticeable as well. 1.2
Current (A) / PWM-Value
1
0.8
0.6
0.4
0.2 Current controller output (PWM value) Measured current (i )
0
q
Reference current (i* ) q
-0.2 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (ms)
Fig. 4.18 The measured response of the current-controlled DC-drive by fixed rotor. 2.5 2
AWC
Current (A) / PWM-Value
1.5 1 0.5 0 -0.5 -1
Reference current (i* )
-1.5
q
Current controller output (PWM value)
-2
Measured current (i ) q
-2.5 -0.2
0
0.2
0.4
0.6
0.8
1
1.2
Time (ms)
Fig. 4.19 The measured response of the current-controlled DC-drive by fixed rotor for a large reference input.
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Current (A) / PWM-Value
The disturbance compensation quality can be observed in Fig. 4.20, where the experimental results for the moving motor are given. The maximal controller error for constant current input-reference of 500 mA is about 25 mA (5% of the reference value). Thus, the influence of the EMF in the DC-drive (ei) is successfully suppressed. 0.6 0.4 0.2 0
Reference current (i* ) q
-0.2
Measured current (i ) q
-0.4
Controller error (mA)
-0.1
Current controller output (PWM value) -0.05
0
0.05
0.1
0.15 Time (s)
0.2
0.25
0.3
0.35
0.4
-0.05
0
0.05
0.1
0.15 Time (s)
0.2
0.25
0.3
0.35
0.4
20 10 0 -10 -20 -0.1
Fig. 4.20 Behaviour of the current-controlled DC-drive by moving rotor. The experimental test of the hysteresis compensation measure is performed by excitation of PEA in open-loop control configuration with the voltage (vPA) commands issued by the RCP-System by disabled DC-drive. The measured position s2 is captured together with the voltage command (issued by the RCP-system) and the voltage applied to the PEA (calculated by the FPGA). Using various excitation signals, a number of measurements are performed for both enabled and disabled hysteresis compensation algorithm. Some representative results will be discussed here. A comparative figure of measured position response s2 by excitation of the non-compensated and compensated PEA with the identical triangular signal of decreasing amplitude is shown in Fig. 4.21. In the non-compensated case, a significant nonlinearity influence on the position response s2 can be observed. By defining the linearization error as: errcomp =
v PA ⋅ TEM − s2 c
(by low dynamic and disabled DC - drive we assumed : s2 ≈ Δs2 ),
(4.28)
its maximal value in the non-compensated case is 1.25 µm, whereas the NRMSE is 5.417%. In the compensated case, the calculated position error is significantly lower. The maximum error value here is less than 100 nm (1.19% of the maximal peak-to-peak displacement), whereas the NRMSE is 0.396%.
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Voltage in V
20
Excitation voltage v
PA
10 0 -10 -20
Displacement error in µm
Measured displacement in µm
8
10
12
14
16
18
20
22
24
5 No hysteresis compensation Hysteresis compensation enabled 0
-5
8
10
12
14
16
18
1
20
22
24
No hysteresis compensation Hysteresis compensation enabled
0.5 0 -0.5 -1 8
10
12
14
16 Time in s
18
20
22
24
Fig. 4.21 Response of the non-compensated and compensated piezoactuator to identical linear command signal with decreasing amplitude. Linearization results for the sine wave excitation with different frequencies are given in Table 4.2. They are comparable with those published in 2007 [138], obtained using the same type of the PEA. One may notice that the deviation between calculated and measured displacement rises with frequency. This is a consequence of the fact that the theoretical linearized value was calculated without taking into account the plant dynamics (see Eqs. (4.25) and (4.28)). Table 4.2 Hysteresis compensation results by sine-wave voltage reference Frequency (Hz) Reference voltage amplitude (V) PEA displacement amplitude (peak-to-peak) (µm) Maximal absolute displacement error (nm) Maximal relative displacement error (max error / peak-to-peak amplitude) (%) Displacement RMSE (nm) Displacement NRMSE (RMSE / peak-to-peak amplitude) (%)
1 21.60
3 21.60
6 21.60
10 21.60
30 21.60
7.788
7.656
7.564
7.512
7.396
158.005
98.515
128.928
177.828
341.608
2.029 62.603
1.287 47.411
1.704 65.324
2.367 92.203
4.619 208.011
0.804
0.619
0.864
1.227
2.812
An excellent agreement between the theoretical and experimental results can be observed in Fig. 4.22. In this diagram, the theoretically calculated displacements s2t are presented on the x-axis, whereas the y-axis shows the experimentally measured displacements obtained by applying the excitation signal from Fig. 4.21. In the compensated case, the linear dependence between the input reference voltage and the resulting PEA displacement can be observed, whereas in the noncompensated case, the influence of the hysteresis nonlinearity is noticeable.
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5 4
Measured Displacement s 2 (µm)
3 2 1 0 -1 -2 -3 -4 -5 -5
-4 -3 -2 -1 0 1 2 3 4 Theoretically calculated displacement s2t for input voltage vPA (s2t=vPA⋅TEM /c) (µm)
5
Fig. 4.22 Response of the non-compensated and compensated piezoactuator to identical linear command signal with decreasing amplitude.
30
Friction Torque (mNm)
20 10 0 -10 -20 -30 100 50 0 Angular Position (rad)
-50 -100 -150
-150
-100
-50
0
50
100
150
Angular Velocity (rad / s)
Fig. 4.23 Estimated friction torque versus angular velocity and angular position of the DC-drive. The dependence of the estimated friction torque on the angular velocity ε&M and the position of the DC-drive εM, is shown in Fig. 4.23. The non-linear dependence on the angular velocity is noticeable whereas the dependence on the angular position is not significant. For this reason, a simplified friction
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compensation algorithm is also developed, where only the angular velocity of the DC-drive is taken into account. On this way, a faster convergence of the online adaptation is achieved [92], resulting in the characteristic shown in Fig. 4.24. Further experimental results are given in [92]. 25 20
Estimated friction torque (mNm)
15 10 5 0 -5 -10 -15 -20 -25 -150
-100
-50
0 Angular velocity (rad/s)
50
100
150
Fig. 4.24 Estimated friction torque versus angular velocity. Through the real-time capable suppression of disturbances and nonlinearities, the HMS can be modelled with linear elements for the control design. The advantage of such a model is the possibility to successfully apply conventional control design methods that are available only for linear plants.
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5. CLOSED-LOOP CONTROL FOR HMS Achievable trajectory-tracking results depend highly on the quality of the employed hardware and low-level software, as well as on the underlying linearization and disturbance rejection algorithms. The designed closed-loop feedback controller plays by trajectory-tracking a key role in order to tap the full potential of the underlying system. After a short overview of the recently published control methods used by HMS systems, this chapter describes the design of a high-dynamic feedback controller for the HMS for the case of a light object5, by means of state-space controller design. The influence of the positioned object and the necessary control parameter adaptations are discussed as next. Finally, details about the practical implementation and the achieved experimental results are given.
5.1 Overview of HMS control structures Both the control of DC-drives and piezoelectric actuators are well discussed and a number of publications deal with these problems. The control of HMS containing coarse and fine actuators (in our case a DC-drive and PEA) is however still in research. Basically, each HMS represents a MISO (multiple inputs single output) system, whereas the PI M-511.HD has two mechanical control inputs (Fig. 3.30). Due to underlying current controller, the hysteresis compensation for the PEA, and the implemented friction compensation measure, the plant is also linear in terms of input references iq* and v P* . Thus, the dependence of the output s2 on the reference DC drive current iq* and the PEA reference voltage v P* can be also accurately described in frequency domain using the transfer functions G s2 / TM and Gs2 / Fp from Chapter 3: s2 ( s ) = Gs
2
/ i q*
( s ) ⋅ iq* ( s ) + Gs
2
/ v *P
( s ) ⋅ vP* ( s ),
where Gs Gs Gi
2
2
/ i q*
/ v *P
* q / iq
= GT
M
= GF =
p
/ i q*
/ v *P
⋅ Gs2 / TM = kmn ⋅ Gi
q
⋅ Gs2 / Fp = TEM ⋅ Gv
1 , and 1 + TΣ ⋅ s
Gv
* PA / v P
⋅ Gs2 / TM =
/ i q*
PA
/ v *P
=
⋅ Gs2 / Fp
1 ⋅ kmn ⋅ Gs2 / TM , 1 + TΣ ⋅ s 1 1 = ⋅ ⋅ TEM ⋅ Gs2 / F p , 1 + TPA1 ⋅ s 1 + TPA2 ⋅ s
(5.1)
1 1 . ⋅ 1 + TPA1 ⋅ s 1 + TPA2 ⋅ s
As one may notice, the influence of the frequency characteristics of the PEA amplifier and the current controller are in the frequency band [0…500 Hz] in which the HMS model are accurate not 5
E.g. for the case when the mass of the positioned object is not significant compared to the mass of the positioning plate m2.
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significant, because the poles of their transfer functions are sTΣ = −1 / TΣ = −982.43 ⋅ 2 ⋅ π rad/s and sTPA1 = sTPA 2 = −1 / TPA1 = −1 / TPA2 = −3.729 ⋅ 103 ⋅ 2 ⋅ π rad/s . Thus, in the first approximation Eq. (5.1) can be rewritten in the form identical to the mechanical TF: Gs
2
Gs
* 2 / vP
/ iq*
≈ k mn ⋅ Gs2 / TM = Gs2 / iq ,
(5.2)
≈ TEM ⋅ Gs2 / Fp = Gs2 / v PA .
The diagram of these TF multiplied with the maximal input values, for the HMS under investigation, is given in Fig. 5.1. For the above mentioned reason, the investigations of suitable control concepts and the design of the control structure is performed using (5.2). In order to improve the accuracy of the control system, by the design of the controller for the DC-drive, the TF describing the influence of the electrical subsystem (5.1) is also considered. -20 G
s /i
*i
2 q
qmax
G
-40
s /v 2
*v PA
PAmax
-60
Magnitude (dB)
-80
-100
-120
-140
-160
-180 0 10
10
1
10
2
10
3
Frequency (Hz)
Fig. 5.1
Transfer functions of the linear plant model.
DISO (double-input/single-output) systems are a subset of MIMO systems, and thus design methods developed for MIMO compensator can be applied. These techniques include H ∞ , H2 and µ-synthesis [159]. A common disadvantage of these methods is the relatively high order of the compensators resulting from the design process. By the HMS under investigation, a further limitation is the small operation range of the PEA, which has to be considered during the control design. Therefore, certain approaches for appropriate control design using separate controller for the DCdrive and the PEA are investigated [31]. The main issue, besides the relatively high order of the transfer functions G s2 / iq and G s2 / v P , is the design of a suitable enabling structure for the control signal onto both actuators, which simultaneously regards their different dynamic properties and the very limited operating range of the PEA. The spatial-based activation-function approaches use the information about the positioning error. By several systems [19,21,24,25,27] the control loop is calculated either for the coarse or the fine actuator, depending on the difference between the reference and the measured position. Such an approach enables only precise static positioning. It can be applied only by systems where a coarse point-to-point reference tracking is sufficient. The control structure published in [18] uses a smooth function for the weighting of the individual controller inputs, in dependence on the actual positioning error. A similar approach is presented in [31], where the estimated actual PEA displacement is also taken into account. This control structure is shown in Fig. 5.2. The enabling structure consists of the
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activation function and an optional low-pass filter. Using the measured or estimated PEA displacement Δsˆ2 and the positioning error es2, the weighting variables kPEA and kDC are calculated by the appropriate weighting function fu. It can be, for example, a cosine function [18,31]. The individual controller inputs eDC and ePEA are calculated by multiplication of the position error es2 with the weighting variable. The controller for the DC-drive (CDC) and the PEA (CPEA) are designed using the individual plant transfer functions Gs2 / iq and G s2 / v P . Δsˆ2
1 0.8 0.6 0.4 0.2 0 -4µm
k PEA
k DC
Activation function Lowpass
s2*
es 2
X
0
k PEA = f u (es 2 + Δs2 ) ( eDC = es 2 ⋅ k DC ( ePEA = es 2 ⋅ k PEA
4µm
es 2 + Δs2
Controller
( k DC
eDC
( k PEA
s2 Enabling structure
Fig. 5.2
k DC = 1 − f u (es 2 + Δs2 )
iq*
C DC (s )
ePEA
X
Plant (HMS)
G HMS (s ) =
v *PA
C PEA (s )
Δsˆ2
s2
= [Gs2 / iq , Gs2 / vPA ]
Control structure with spatial control-signal enabling at the controller input.
An alternative structure with spatial activation function is shown in Fig. 5.3. Instead of the controller inputs, their outputs, iq* and v *PA , are multiplied with the weighting variables. Both controller inputs are here set to be equal to the positioning error es2. The optimal low-pass filtering of control signals may be implemented in both control schemes in order to solve potential stability problems. Although the presented control structure takes into account the limited operation range of the PEA and it was successfully tested [31], it suffers from several problems. The most significant one is the reduced efficiency, because either only one of actuators is fully active, or alternatively, both of them run with reduced power. Furthermore, nonlinearity is involved in the system through the activation function fu, making the control design more complex. Δsˆ2
1 0.8 0.6 0.4 0.2 0 -4µm
k PEA
es 2
k DC
Activation function
Controller
s2*
es 2
s2
Fig. 5.3
C DC (s )
CPEA (s )
~* iq
Lowpass X
k DC = 1 − f u (es 2 + Δs2 )
0
4µm
es 2 + Δs2
PA
PA
PEA
Plant (HMS)
( k DC
* v~PA
k PEA = f u (es 2 + Δs2 ) ~ ( iq* = iq* ⋅ k DC ( v * = v~* ⋅ k iq*
( k PEA X
* v PA
G HMS (s ) = = [Gs2 / iq , Gs2 / vPA ]
Δsˆ2
s2
Enabling structure
Control structure with spatial control-signal enabling at the controller output.
Yet another approach, presented in [160], deals with the distribution of the position error in the frequency domain. In general, a PQ design method can be used, as described in [160]. Such a method
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is used both by the dual-stage servo system for read/write head positioning in computer hard disk drives [26] as well as for the control of a piezo-electric dual-stage tape servo actuator [161]. A strongly simplified structure, designed according to such an approach is shown in Fig. 5.4. Such an approach is utilized in [31], whereas in [162] a slightly modified one is described. In [31], the error signal is split into a low-frequency and a high-frequency components es 2 _ LP and es 2 _ HP using a lowpass (FLP(s)) and a high-pass (FHP(s)) filter. The DC-controller – acting with the slower actuator – uses the low-frequency error-signal component, whereas the PEA-controller the high-frequency one. The advantage of this control concept is its linearity. But it also has some serious disadvantages. Through its architecture, the control system cannot explicitly take into account the very limited operation range of the PEA. Indeed, the cutoff-frequency of the high-pass filter influences PEA reference signal range implicitly. For optimal trajectory-tracking however, this frequency should be set in accordance with the PEA dynamics. Considering that the ratio of the DC-drive and the PEA operation ranges is more than 104, and reviewing of the TF functions from Fig. 5.1, it is likely that no satisfying frequency can be found. In [162], the frequency-based distribution is performed by the reference input. Although this method performs well in simulations [162], the practical implementation with a HMS may have the same drawbacks for similar reasons.
s2*
es 2
FLP (s)
FHP (s)
Fig. 5.4
es 2 _ LP
es 2 _ HP
CDC (s )
CPEA (s )
iq*
G HMS (s ) = v *PA
s2
= [Gs2 / iq , Gs2 / vPA ]
Control structure by frequency distribution of the position error.
The control structure actually used in the commercially available controller for hybrid micropositioning systems from Physik Instrumente [34] shown in Fig. 5.5, represents a version of the master-slave structure [160]. This method is an attempt to treat the MIMO controller design as a sequence of SISO designs under the assumption that there is very little interaction between the SISO loops. The structure uses the output of the fine actuator compensator as the reference input to the coarse actuator compensator. The idea is that the output of the fine actuator compensator is a good estimate of the relative position of coarse and fine actuator at low frequencies, and the coarse actuator control loop attempts to keep that relative position at zero. Unfortunately, there is usually more interaction between the DISO subsystems than the approximation warrants, and the performance of the closed-loop system degrade [160].
es 2 ePI
eDC
CDC (s )
v*DC Plant (HMS)
K PI
s2*
Fig. 5.5
es 2
CPEA (s )
* vPA
v*PA
G HMS (s )
s2
The control structure used by the commercially available HMS controller [34]. Because this device does not features the current measurement (and consequently no current controller is implemented), the DC-controller output is the PWM-voltage vDC.
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In this work, a novel approach will be followed. Although it has some similarities to the one published by Fujita et. al [22], the method presented here differs in several aspects and contains some enhancements. The first step according to this approach is the involvement of a coordinate transformation and the plant representation in MIMO-form. By insertion of a decoupling network, a structure of two virtually independent SISO-systems is obtained. The recent plant representation enables the independent control design both for the DC-drive and the PEA.
5.2 Novel plant representation and overall control structure A better insight in the mechanical sub-model described in Chapter 3 may be obtained, when a coordinate transformation is performed. By the conversion of the DC-drive’s rotational motion into a translational one by s0 = ε M / irt , the model shown in Fig. 5.6 can be derived.
cGl FDC d m0l Fig. 5.6
c
dGl
m0
d
m1
s0
m2
Fp
s1
s2
The mechanical sub-model after the coordinate transformation.
In the resulting model from Fig. 5.6, the force FDC corresponds to the DC-drive’s torque TM. The mass m0, the translational spring constant cGl, and the translational damping factors dGl and dm0l correspond furthermore to the inertia moment JM, the spring constant cG, and the damping factors dG and dMb of the original model:
FDC = TM ⋅ irt = kmn ⋅ iq ⋅ irt ,
(5.3)
m0 = J M ⋅ irt2 , cGl = cG ⋅ irt2 , d Gl = d G ⋅ irt2 , d m 0l = d Mb ⋅ irt2 .
All other elements, coordinates and the PEA force are the same as in Chapter 3. The state-space equations of the system can be calculated by the selection of states, inputs and outputs analogously to (3.43)-(3.45):
x t = [s0 u t = [ FDC
s1
s2
s&0
s&1
T s&2 ] ,
F p ]T ,
(5.4)
y t = [ s2 ], resulting in: ⎡ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ cGl − A t = ⎢ m0 ⎢ ⎢ cGl ⎢ m1 ⎢ ⎢ 0 ⎣⎢
0 0 0 cG m0 cGl + c − m1 c m2
0 0 0 0 c m1 c − m2
Ct = [0 0 1 0 0 0] .
1 0 0 d Gl + d M 0l − m0 d Gl m1 0
0 1 0 d Gl m0 d Gl + d − m1 d m2
⎤ ⎡ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢ 0 ⎥ ⎢ 1 0 ⎥ , B t = ⎢ m0 ⎥ ⎢ d ⎥ ⎢ 0 ⎢ m1 ⎥ ⎥ ⎢ d − ⎥ ⎢ 0 m2 ⎦⎥ ⎣⎢ 0 0 1
⎤ ⎥ ⎥ ⎥ ⎥ 0 ⎥ , ⎥ 1 − ⎥ m1 ⎥ 1 ⎥ ⎥ m2 ⎦⎥ 0 0 0
(5.5)
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By the enabled friction compensation measure, the damping element dm0l can be removed from the model and form Eq. (5.5), because its influence is compensated. Involving the new coordinate variables Δs, Δs1, and Δs2 defined by:
Δs = s2 − s0 , Δs1 = s1 − s0 , and Δs2 = s2 − s1 ,
(5.6)
and setting a virtual output vector as: y v = [ s0 , Δs ]T ,
(5.7)
⎡ 1 0 0 0 0 0⎤ Cv = ⎢ ⎥. ⎣ − 1 0 1 0 0 0⎦
(5.8)
the output matrix Cv becomes:
By the use of Eqs. (4.25) and (5.3), the state-space model can be re-written setting iq and vPA as input variables: u te = [iq
v PA ]T .
(5.9)
Because (4.25) and (5.3) are linear algebraic relations, only the input matrix is changed:
⎡ 0 ⎢ 0 ⎢ ⎢ 0 ⎢ irt ⋅ k mn B te = ⎢ m0 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎣⎢
⎤ ⎥ ⎥ ⎥ ⎥ 0 ⎥ . ⎥ TEM ⎥ − m1 ⎥ TEM ⎥ ⎥ m2 ⎦⎥ 0 0 0
(5.10)
The state-space equation system of the transformed model in terms of electrical input variables is hence given with (5.11). x& t = A t ⋅ x t + B te ⋅ u te ,
(5.11)
y v = Cv ⋅ x t .
The transfer-function matrix of this LTI system is obtained as:
⎡ Gs / i G v ( s ) = C v ⋅ ( s ⋅ I − A t ) −1 ⋅ B te = ⎢ 0 q G ⎣⎢ Δs / iq
Gs0 / v PA ⎤ , GΔs / v PA ⎥⎥ ⎦
(5.12)
and the input/output relations in the frequency domain are defined by: s0 ( s ) = Gs0 / iq ( s ) ⋅ iq ( s ) + Gs0 / vPA ( s ) ⋅ v PA ( s ), Δs ( s ) = GΔs / iq ( s ) ⋅ iq ( s ) + GΔs / vPA ( s ) ⋅ v PA ( s ).
(5.13)
As one may notice, due to physical constrains, s2 (t ) = s0 (t ) + Δs (t ) and hence s2 ( s ) = s0 ( s ) + Δs ( s ).
(5.14)
The model representations in DISO and MIMO-form are shown in Fig. 5.7. The advantage of the recent one for the control design becomes expressed, after the comparison of the individual transfer functions multiplied with the maximal input values is performed. It becomes apparent, that both the
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influence of the PEA onto s0, and the one of the DC drive onto Δs, for the given system are not significant in the frequency range of interest, and Gs0 / vPA ( jω ) tends toward zero in steady-state:
Gs0 / vPA ( jω ) ⋅ max( v PA ) Gs0 / iq ( jω ) ⋅ max( iq )
GΔs / iq ( jω ) ⋅ max( iq )
>m1,m2. In Fig. 5.38, a trajectory-tracking result by the use of the quasi-continuous controller designed in Sdomain is shown. The maximal positioning velocity was here 1 cm/s and the bandwidth of the
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controller is set to ωbwDC = 900 rad/s. The improvements introduced by the feed-forward measure are particularly noticeable during the motion start and stop. 0 Reference trajectory Position without FF Controller Position with FF Controller
Position (mm)
-20 -40 -60 -80 0
1
2
3
4
5 Time (s)
6
7
8
9
10
Positioning error (µm)
1 0.5 0 -0.5 -1
Positioning error without FF Controller |Error(no FF)|max = 1.664 µm. Positioning error with FF Controller |Error(with FF)|max = 1.180 µm.
-1.5 0
1
2
3
4
5 Time (s)
6
7
RMSE(no FF) = 0.259952 µm. RMSE(with FF) = 0.247890 µm. 8
9
10
Fig. 5.38 Trajectory-following results with the quasi-continuous DC-controller (PEA controller disabled). A trajectory-tracking result by the exclusive use of the digital position controller for the DC-drive designed in the Z-domain with enabled feed-forward measure and ωbwDC = 850 rad/s for the same maximal velocity of 1 cm/s is given in Fig. 5.39. The response of the digital controller compared to the quasi-continuously designed one is of better quality. 0
Position (mm)
-2 -4 -6 -8
Positioning error (µm)
-10
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1 Time (s)
1.2
1.4
1.6
1.8
2
0.5
0
-0.5 |Error(with FF)|max = 0.912 µm. RMSE(with FF) = 0.202389 µm. 1 Time (s)
1.2
1.4
1.6
1.8
2
Fig. 5.39 Trajectory-following results with the digital DC-controller (FF on, PEA controller off). The maximal and root-mean-squared errors by tracking of different trajectories by the solely use of the digital DC controller with ωbwDC = 850 rad/s and enabled feed-forward control are shown in Table 5.1.
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Table 5.1 Trajectory-tracking results with a light object using the digital DC-controller with feed-forward control from Fig. 5.22 by disabled PEA controller. Maximal velocity (mm/s)
Trajectory length (mm)
20 10 5 1 0.1 0.03 0.005 0.001
10 10 10 1 1 0.5 0.1 0.04
Positive direction Maximal Root-meandynamic squared tracking error (nm) error (nm) 1539.5 278.98 1050.9 231.71 1075.3 213.38 290.0 71.75 203.9 27.95 99.8 13.27 134.7 7.62 60.5 3.78
Negative direction Maximal Root-meandynamic tracking squared error (nm) error (nm) 1439.1 912.1 919.8 469.1 254.7 292.1 225.1 66.4
269.21 202.38 198.29 73.71 26.88 17.03 11.86 4.23
In Fig. 5.40 and Fig. 5.41 simulated and measured responses for position s2 of the controlled PEA, with respect to the position reference (Δs*), are compared in frequency domain for the case of a light object and disabled DC-controller. The sample time of PEA controller was set to 100 µs. The simulated and measured responses, achieved by the use of a traditional proportional-integral (PI) feedback controller designed with 40° phase margin, together with the response achieved with both the simple and the cascaded digital state-space controllers are displayed. The advantages of the state controllers regarding the bandwidth and resonance peaks are obvious. The agreement between the simulation and measurement is good. First trajectory-tracking results by both enabled quasi-continuous DC- and digital PEA-controller (both executed with 10 kHz sampling frequency and implemented in the RCP-system) without decoupling network and with no feed-forward control are shown in Fig. 5.42. Here, for position reference the trajectory-generator of the commercially available controller of the HMS is used. The bandwidth of the DC-controller is here set to 800 rad/s, whereas the one of the PEA controllers is 375·2·π rad/s. The drawback for the slight positioning-quality improvement observed by the cascaded PEA-controller in Fig. 5.42 is given through the significant complexity of the controller structure.
Magnitude (dB)
40 20
PI-Controller with 40° phase reserve Simple digital state-space controller (10 kHz sampling frequency) Cascaded digital state-space controller (10 kHz sampling frequency)
0 -20 -40 10
100 Frequency in Hz
1000
Phase (degrees)
0
-100
-200
-300 10
PI-Controller with 40° phase reserve Simple digital state-space controller (10 kHz sampling frequency) Cascaded digital state-space controller (10 kHz sampling frequency) 100 Frequency (Hz)
Fig. 5.40 Simulated frequency response of the closed loop control of the PEA.
1000
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Magnitude (dB)
40 PI-Controller with 40° phase reserve Simple digital state-space controller (10 kHz sampling frequency) Cascaded digital state-space controller (10 kHz sampling frequency)
20 0 -20 -40 10
100 Frequency (Hz)
1000
Phase (degrees)
0
-100
-200 PI-Controller with 40° phase reserve Simple digital state-space controller (10 kHz sampling frequency) Cascaded digitalstate-space controller (10 kHz sampling frequency)
-300 10
100 Frequency in Hz
1000
Fig. 5.41 Measured frequency response of the closed loop control of the PEA. 0 Reference trajectory Measured position with simple digital state controller Measured position with cascaded state controller
Position (mm)
-2 -4 -6 -8
0
0.5
1
1.5
Time (s)
Positioning error (µm)
0.4
Positioning error with simple digital state-space controller Positioning error with cascaded state-space controller
0.2 0 -0.2 -0.4 -0.6
|Error(SSC)|max = 0.730 µm. |Error(CSSC)|max = 0.641 µm.
-0.8 0
0.5
1
RMSE(SSC) = 0.278204 µm. RMSE(CSSC) = 0.237413 µm. 1.5
Time (s)
Fig. 5.42 Trajectory following results using the DC-controller with the simple and cascaded statefeedback PEA controller. No feed-forward control and no decoupling network were used. Trajectory-tracking results with a light object having the enabled decoupling network and feedforward control, by the use of the quasi-continuous and the digital DC-controller with the simple PEA state-controller (implemented in the FPGA with 100 kHz sampling frequency) are shown in Fig. 5.43Fig. 5.47 and in Table 5.2.
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Position (mm)
1 0.8 0.6 0.4
Reference position s
0.2
With quasicont. DC-contr. With discrete DC-contr.
0 0
0.5
1
1.5
2
2
100 50 0 -50 -100
2.5
0
0.5
1
2.5
1.5
2
2.5
0.3
q
0.8 0.6 0.4 0.2
0.2 0.1 0
0 0
0.5
1
1.5
2
2.5
0
0.5
1 Time (s)
PA
(V)
1 0.5
PEA voltage ref. v*
(mm/s2) ⋅⋅ 2
2
0.4
1
Time (s)
Acceleration reference s
1.5 Time (s)
DC current i (A)
⋅ 2
Velocity reference s (mm/s)
Time (s)
0 -0.5 -1 0
0.5
1
1.5
2
2.5
6 With quasicont. DC-contr. With discrete DC-contr.
4 2 0 -2
0
0.5
1
Time (s)
1.5
2
2.5
Time (s)
Fig. 5.43 Trajectory following results of the integrated control system from Fig. 5.37, using the quasi-continuous (red dotted line) and the discrete (blue dashed line) DC-controller with the simple digital state-feedback PEA controller. In Fig. 5.43, besides the reference trajectory values (position, velocity, acceleration) and the position error, the measured DC-current and the reference PEA voltage are shown. As one may notice, the PEA having higher dynamic potential is very useful by the correction of DC-drive positioning error. The PEA voltage of 5V corresponds to a displacement of about 600 nm; the trajectory following errors without use of the PEA would be hence significantly higher. The comparison of the transient error by the motion start depending on the selected decoupling type is shown in Fig. 5.44. The control system from Fig. 5.37 is used by the positioning velocity of 100 µm/s. By disabled decoupling there is a bias in the positioning error. Both the static and dynamic decoupling network suppressed this bias during the constant velocity phase, having the better suppression of the transient error by motion start through the dynamic decoupling.
Transient positioning error (nm)
80 60
40
No decoupling Static decoupling Dynamic decoupling
20 0
-20
Beginning of the movement at t=0 s. 0
0.1
0.2
0.3
0.4
0.5
Time (s)
Fig. 5.44 The transient tracking error by start of the motion by different decoupling network settings.
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Results of a trajectory-following task by slow motion (5 µm/s), using the simple discrete PEA statecontroller with the digital DC state-controller are shown in Fig. 5.45. Considering the outputs of the decoupling network, the importance of the decoupling element Gk21 at low velocities becomes obvious. An example of a high speed trajectory-tracking is given in Fig. 5.46. Further trajectoryfollowing results with the control systems from Fig. 5.37 and Fig. 5.36 are given in Appendix 8.14. 0.3 100
0.25 0.2
DC current (A)
Position (µm)
80 60 40
0.15 0.1
20 Reference trajectory Measured position
0 0
5
10 Time (s)
15
q
Output of G
k12
0 -0.05
20
0
5
10 Time (s)
15
20
2
6 PEA Voltage reference (V)
4 Positioning error (nm)
DC-controller output i~*
0.05
2 0 -2 -4 -6
RMSE = 2.1753 nm.
-8
PEA controller output v~ * PA
1.5
Output of G
k21
1 0.5 0
|Error|max = 9.1600 nm. 0
5
10 Time (s)
15
20
0
5
10 Time (s)
15
20
60
0.8
50
0.6 DC current (A)
Position (mm)
Fig. 5.45 Following of a slow trajectory (5 µm/s) with nanometre-accuracy, using the control system from Fig. 5.37 with the discrete DC-controller. The influence of the decoupling element Gk21 is here important.
40 30 20 Reference position Measured position
10 0
0
0.5
1 Time (s)
1.5
0.2 0 -0.2 -0.4
2
1.5
0
0.5
1 Time (s)
1.5
2
0
0.5
1 Time (s)
1.5
2
15
1 PEA Voltage (V)
Positioning error (µm)
0.4
0.5 0 -0.5 -1
10 5 0 -5 -10
|Error|max = 1562.500 nm
-1.5 0
0.5
1 Time (s)
RMSE = 167.781 nm 1.5
-15 2
Fig. 5.46 Following a long trajectory of 6 cm with maximal velocity of 5 cm/s using the control system from Fig. 5.36 with the discrete DC-controller.
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Table 5.2 Trajectory-tracking results with the integrated control system from Fig. 5.37. Maximal velocity (mm/s)
Trajectory length (mm)
50 20 10 5 1 0.1 0.03 0.005 0.001
20 10 10 10 1 1 0.5 0.1 0.04
Digital DC-controller (TsDC=100 µs, ωbwDC = 650 rad/s) and simple statefeedback PEA-controller (FPGA, TsPEA =10 µs, ωbwPEA = 375·2·π rad/s)
Quasi-continuous DCcontroller (TsDC =100 µs, ωbwDC = 900 rad/s) and simple statefeedback PEA-controller (FPGA, TsPEA =10 µs, ωbwPEA = 375·2·π rad/s) Maximal RMSE tracking (nm) error (nm) 1135 209.82 693 111.29 628 119.69 402 97.65 126 41.93 62 29.23 60 42.44 48 26.95 48 33.75
Maximal tracking error (nm) 810 717 684 400 96 24 12 10 10
RMSE (nm) 89.21 90.70 96.14 69.92 39.72 7.63 2.31 2.17 5.55
Table 5.2 clearly indicates the advantage of the digital DC-controller over the quasi-sontinuous one. Accurate tracking of a periodic position reference is required in many industrial applications. In Fig. 5.47, trajectory-tracking results for a periodic piecewise linear position reference of 200 µm amplitude, 800 µm/s positioning speed, and 1 s period are given by different control configurations. 200 150 Ref. position (µm)
100 50 0 -50 -100 -150 -200
0
0.2
0.4
0.6
0.8 1 Time (s)
1.2
1.4
1.6
1.8
5000 DC only (1) DC with FF (2) DC and PEA (3) DC with FF and PEA (4)
Transient pos. error (nm)
4000 3000 2000 1000 0 -1000 -2000
(1):|Error|
= 5319 nm
(1): RMSE = 625.832 nm
-3000
(2):|Error|
= 1818 nm
(2): RMSE = 214.572 nm
(3):|Error|
= 4794 nm
(3): RMSE = 297.244 nm
(4):|Error|
= 417 nm
max max
-4000
max max
-5000 0
0.2
0.4
0.6
0.8 1 Time (s)
(4): RMSE = 54.273 nm 1.2
1.4
1.6
1.8
Fig. 5.47 Tracking of a piecewise linear position-reference by different control configurations.
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Transient pos. error (nm)
Ref. position (µm)
The maximal transient error occurs by the reversal of the movement direction, where the maximal acceleration is set to 50 cm/s2. This error is by the solely use of the DC-controller without the feedforward measure 5.319 µm in the movement reversal points, caused through the high positioning dynamics. By enabled PEA-controller, the positioning RMSE is reduced significantly, but the peak error is still high. This is reasonable, as the PEA operating range is only +/- 4.2 µm. The main advantage of the feed-forward measure is in Fig. 5.47 obvious. By the enabled feed-forward control, the reduction of the peak transient error to 1.818 µm by the disabled PEA-controller and to 0.417 µm in case of the full control system configuration from Fig. 5.37 is observed. Another measurement result by a sine-wave position-reference and different control configurations is shown in Fig. 5.48. The position signal amplitude was 100 µm and its period set to 10 s. 100
0
-100
0
1
2
3
4
5 Time (s)
6
7
8
9
0
1
2
3
4
5 Time (s)
6
7
8
9
400 200 0 -200 -400
DC only DC with FF DC and PEA DC with FF and PEA
0.1 0 -0.1
PEA voltage (V)
DC current (A)
0.2
0
1
2
3
4
5 Time (s)
6
7
8
9
0
1
2
3
4
5 Time (s)
6
7
8
9
2 0 -2 -4 -6 -8
a)
Transient pos. error (nm)
10
DC only DC with FF DC and PEA DC with FF and PEA
5
0
-5
-10 7.2
7.4
7.6
7.8 Time (s)
8
8.2
8.4
b) Fig. 5.48 Tracking of a sine-wave position-signal by the use of different control configurations (a) and the error at the movement direction change with the enabled PEA-controller (b).
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From Fig. 5.48a it is obviously, that the positive effect of the feed-forward measure is significant when the PEA-controller is disabled. The PEA-controller is able to suppress the transient error widely caused by the reversal of the movement direction. Thus, by the enabled PEA controller, the effect of the feed-forward-measure is smaller, but it is still detectable during the reversal of the movement direction (see Fig. 5.48b). The advantage of the PEA is significant during the whole movement period, and offers the largest benefit for the positioning quality. A comparison of the transient position RMSE as well as the maximal transient error by the sinewave position reference of 100 µm amplitude for different periods and control configurations is shown in Fig. 5.49. Here, the discrete controller for the DC-drive and the FPGA-based controller for the PEA are used. Each measurement is performed at least three times to verify the repeatability. In Fig. 5.49a the averaged RMSE, whereas in Fig. 5.49b the averaged maximal transient error values are visualized using logarithmic axes. 1500
DC only DC with FF DC and PEA DC with FF and PEA
200
Averaged maximal error (nm)
100 Averaged RMSE (nm)
DC only DC with FF DC and PEA DC with FF and PEA
1000
50 40 30 20
500 300 200
100
50 40
10
30 1
2
5 Sine-wawe period (s)
10
20
1
2
5 Sine-wawe period (s)
10
20
a) b) Fig. 5.49 Average RMSE (a) and maximal transient error (b) by a sine-wave position reference of 100 µm amplitude and different periods by various control configurations. For comparison, trajectory-tracking experiments using the commercially available controller C-702 from Physic Instrumente with the optimized settings for a light object are performed. The maximal positioning velocity supported by this controller is 1 cm/s. In Fig. 5.50, a typical result is shown, for a trajectory with length of 1 mm and positioning speed of 1 mm/s. The maximal transient tracking error is here 2216 nm, the RMSE 396 nm. Further results are given in Appendix 8.15. The overview of the averaged RMSE and maximal transient error values versus the positioning velocity for the different control configurations is given in Fig. 5.51 and Fig. 5.52. Here, the average control quality of all performed trajectory-tracking experiments with: - the state-feedback control structure from Fig. 5.37 (data from Appendix 8.14), - the state-feedback controller for the DC-drive with feed-forward measure from Fig. 5.22, - and the factory controller C-702 with the optimized parameter settings (see Appendix 8.15) are compared using logarithmic axes both for the trajectory velocity and the averaged RMSE. As one may notice, the state-space control structure from Fig. 5.37 shows the best performance; but even the state-feedback DC-controller with feed-forward measure (by disabled PEA controller) performs better than the factory controller C-702 (where both actuators are simultaneously utilized). The most performance enhancement of the space-feedback control structure is achieved in the low-velocity range where up to 100 times smaller RMSE are measured related to the C-702, whereas in the highvelocity range the reduction of the RMSE is about four times. From Fig. 5.52 one may notice the main advantage of the control system from Fig. 5.37 over the simpler configuration using the DC statecontroller with feed-forward measure from Fig. 5.22: the maximal transient error is in the first case significantly reduced, particularly by the low-velocity trajectories. This error occurs by such trajectories usually when the movement starts and ends, while the PEA is capable to suppress it more efficiently as the DC-drive.
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0.8
DC voltage (V)
Position (mm)
1
0.6 0.4 0.2
0
0.4
0.6 0.8 Time (s)
1
-1
1.2
0.2
0.4
0.6 0.8 Time (s)
1
1.2
0
0.2
0.4
0.6 0.8 Time (s)
1
1.2
30 PEA voltage (V)
RMSE = 0.369 µm. 1 0 -1 -2
0
40
|Error|max = 2.216 µm.
2 Position error (µm)
0.2
1 0
Reference position Measured position
0
2
20 10 0 -10
0
0.2
0.4
0.6 0.8 Time (s)
1
1.2
-20
Fig. 5.50 Trajectory-tracking result using the factory controller C-702. 2000
Average maximal transient error (nm)
1000 500 400 300 200 100 70 60 50 40 30 DC with FF and PEA DC with FF Factory controller C-702
20
10 0.001
0.005
0.03
0.1 0.5 1 Positioning velocity (mm/s)
5
10
20
50
Fig. 5.51 Comparison diagram of the average maximal transient trajectory-tracking error achieved with the control system from Fig. 5.37 and the commercially available controller C-702. The summary of the achieved results in this subchapter may be formulated as: • The functionality of the novel control structure described in this chapter is successfully validated. A comparison with the commercial HMS-controller C-702 shows the advantage of the proposed control structure through decreasing both the maximal as well as the rootmean-squared positioning errors.
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400 300
Averaged RMSE (nm)
200
100 70 60 50 40 30 20
10 DC with FF and PEA DC with FF Factory controller C-702 0.001
0.005
0.03
0.1 0.5 1 Positioning velocity (mm/s)
5
10
20
50
Fig. 5.52 Comparison diagram of the average trajectory-tracking RMSE achieved with the control system from Fig. 5.37 and the commercially available controller C-702. • • •
•
•
• • •
5.9.2
The enhanced control of the DC-drive plays a significant role in the improvement of the overall control quality. Even without the use of the PEA is possible to achieve position accuracies in sub-micrometer range, which are better as the ones achieved with the C-702. The advantage of the feed-forward measure for the DC-controller is particularly observable in case of a reversal of the movement direction, as well as during the start and stop of the movement. The optimal bandwidth of the DC-controller is slightly reduced when the PEA-controller is enabled. This is due to the approximations made in modelling and control design. On the other hand, the advantages of the PEA-controller and the decoupling network are significant when a light object is positioned, as seen in Fig. 5.43-Fig. 5.52 and Table 5.2. The positive effect of the developed friction compensation measure was not detectable by enabled PEA-controller. This is most probably the consequence of the higher dynamics of the PEA-controller. Disturbances, occurred through the DC-drive and gearbox friction are eliminated using the controlled PEA in a more efficient way. The quality improvements achieved through both the dynamic and the static decoupling network are observable by all trajectories. The improvements of the static network are particularly expressed during a low-velocity movement, whereas the dynamic network offers additional enhancements during the acceleration phases. In cases when the transient errors by motion start and stop are not from importance, the use of the static network may be also a good solution, as it requires less computation. Controller designed in Z-domain showed better positioning quality by the same bandwidth both by the DC-drive and the PEA, especially by low-velocity positioning tasks. The drawback for the slight quality improvement of the cascaded state-feedback controller is the significantly higher complexity of the controller realization. The FPGA-implementation of the state-estimator and the PEA-controller with 100 kHz sampling frequency resulted in an increase of the positioning accuracy related to the one achieved with the exclusive use of the RCP-system using 10 kHz sampling frequency. Trajectory-tracking results with different positioned objects
In order to validate the considerations from Subchapter 5.7, several measurements are performed with different positioned objects; only some characteristic ones will be shown here. The control
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parameters are calculated according to description in Subchapters 5.3-5.6 by substitution of m2 by m2eff. The appropriate values for m2eff and ωbwPEA are obtained by observation of the recorded frequency responses and set manually. According to results from the previous chapter, the digital state-feedback DC-drive controller with the feed-forward measure and the simple digital state-feedback PEAcontroller are selected for the validation. By the positioned object of 2.3 kg mass, the optimal values of m2eff and ωbwPEA are found as 0.85 kg and 262 Hz respectively. Trajectory-tracking results by maximal positioning velocity of 5 mm/s are shown in Fig. 5.53. The maximal positioning error was here 927.5 nm, whereas the RMSE had the value of 239.268 nm. For the positioned object of 5 kg mass, the optimal values of m2eff and ωbwPEA are found as 1.8 kg and 200 Hz respectively. Trajectory-tracking results by maximal positioning velocity of 100 µm/s are shown in Fig. 5.54. The maximal positioning error was here 173.5 nm, whereas the RMSE had the value of 39.66 nm. In case of the positioned object of 8 kg, the optimal values of m2eff and ωbwPEA are found as 2.1 kg and 120 Hz respectively. Trajectory-tracking results by maximal positioning velocity of 1 cm/s are shown in Fig. 5.55. The maximal positioning error was here 1.8 µm, whereas the RMSE had the value of 545.98 nm. 10 0.4 DC current (A)
6 4 2
0
0.5
1
1.5 Time (s)
2
0
-0.5 |Error|max = 927 nm 0.5
1
1.5 Time (s)
RMSE = 239.26 nm 2
0.2 0.1
2.5
0.5
0
0.3
0
p
Positioning error (µm)
0
Reference position Measured position
PEA voltage reference v* (V)
Position (mm)
8
2.5
0
0.5
1
1.5 Time (s)
2
2.5
0
0.5
1
1.5 Time (s)
2
2.5
10 5 0 -5 -10
Fig. 5.53 Trajectory-following task by maximal velocity of 5 mm/s with an object of 2.3 kg mass. The parameter m2eff was here set to 0.85 kg, whereas for ωbwPEA the optimal value is found as 262 Hz. The achieved experimental results confirmed the considerations from Subchapter 5.7. The summary of the foundings in this subchapter is formulated as: • The advantage of the PEA is detectable as long as the bandwidth of the PEA-controller is significantly higher than the one of the DC-controller. When the bandwidths of the both controller are similar, the use of the PEA-controller does not offer any advantage anymore. Moreover, its use in such a case becomes disadvantageous. This is reasonable, as the both controller working with a similar bandwidth may interfere and disturb each other. • A more efficient way for the adaptation of m2eff and ωbwPEA has to be found. For an easy commissioning by different basements and HMS mountings, this should be enabled online and in an automated way.
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Position (mm)
3
2
1 Reference position Measured position 0
5
10
15
20 25 Time (s)
30
0.05
35
0
5
10
15
20 25 Time (s)
30
35
0
5
10
15
20 25 Time (s)
30
35
5
p
0.15 Positioning error (µm)
0.1
PEA voltage reference v* (V)
0
0.15
0.1 0.05 0 -0.05 -0.1 |Error|max = 173.525 nm
-0.15 0
5
10
15
RMSE = 39.659 nm
20 25 Time (s)
30
4 3 2 1 0 -1
35
Fig. 5.54 Trajectory-following task by maximal velocity of 100 µm/s with an object of 5 kg mass. The parameter m2eff was here set to 1.8 kg, the optimal value for ωbwPEA is found as 200 Hz. 0
Reference position Measured position
-5
DC current (A)
Position (mm)
0
-10 -15
-0.1 -0.2 -0.3 -0.4
0
0.5
1
1.5 2 Time (s)
2.5
3
3.5
1
10
0.5
5
PEA Voltage (V)
Positioning error (µm)
-20
0 -0.5 -1
0
0.5
1
1.5 2 Time (s)
2.5
3
3.5
0
0.5
1
1.5 2 Time (s)
2.5
3
3.5
0 -5 -10 -15
-1.5
|Error|max = 1840.400 nm RMSE = 545.980 nm 0
0.5
1
1.5 2 Time (s)
2.5
3
3.5
-20
Fig. 5.55 Trajectory-following task by maximal velocity of 1 cm/s with an object of 8 kg mass. The parameter m2eff was here set to 2.1 kg, the optimal value for ωbwPEA is found as 150 Hz. According to the achieved results, the further research considering the commissioning and the online controller parameter optimization is performed using the discrete DC-state-feedback controller including the feed-forward measure, and the simple discrete PEA controller. Because the FPGA resources are fully exhausted with the implementation of the state estimator and the PEA statefeedback controller, the extension of these modules with the online variable parameter in the FPGA was not possible. For this reason, in the following works the system configuration from Fig. 5.36, having both the state estimator and the PEA state-feedback controller implemented in the RCP-system with 10 kHz sampling frequency, is used.
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6. CONTROLLER COMMISSIONING AND ADAPTATION In the previous chapter, the design of the state-feedback controller structure is presented. The control design was initially performed for a fixed/constant object and it was as first validated for the case when the mass of the positioned object mobj is not significant compared to the mass of the positioning plate m2. Although such a configuration is frequently used by various applications (like scanning microscopy and further), in some other applications the mass of the positioned object is considerable and under change. In Chapter 5, the appropriate control parameter set for the case of a considerable object mass is obtained by offline and online tests, using the frequency response for the parameter settings. However, end-users prefer simple commissioning- and control-parameter adaptation algorithms, which does not propose a deep knowledge of the automated control theory. Both after the manufacturing of a new type of HMS, as well as by the initial start-up at the customer site the (sub-) optimal control parameter should be set to ensure the best possible positioning quality. One may distinguish between the following three commissioning cases: − commissioning of a new HMS type at the manufacturer’s site, − commissioning of a HMS by the customer with the support of the manufacturer by a given trajectory-tracking task, and − commissioning of the HMS at the customer site without the support of the manufacturer. In the first two cases, offline parameter identification and control optimization are possible; the manufacturer is familiar with the HMS, versant in the control theory and possesses the necessary hardware and software tools. In case of the commissioning at the customer site without the manufacturer’s support, simplified methods and automated control optimizations are preferred, because the customer may not be familiar with the HMS and perhaps unacquainted with the control theory. Hence, the commissioning should be performed in a simplified and automated way, by the use of online adaptations to achieve optimal trajectory-tracking quality. In this chapter, a commissioning scenario to be performed through the manufacturer and a second one for the commissioning by the customer are proposed as first. The detailed descriptions of the individual commissioning steps and its practical realization are given in [176]. The commissioning by the customer requires online control adaptation. The changes involved by the positioned object of a significant mass were analysed in Subchapter 5.7; these considerations are used as preparation for the design of such a method. A model-based approach for the optimization of control parameter is given as next, followed by the overview and the implementation of an appropriate real-time capable optimization algorithm. The practical implementation for experimental validations is performed using the control configuration from Fig. 5.36. Here, the digital statefeedback controllers are chosen both for the control of the DC-drive and the PEA. Experimental results obtained by the application of the proposed scenario “commissioning concept to accoplish through the customer” by different positioned objects and a useful application for optical commissioning, microassemby, and robotic involving a 2DOF HMS-system conclude this chapter.
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6.1 Commissioning concept to accomplish through the manufacturer The proposed commissioning scenarios performed through the manufacturer are very similar to each other. The scenario for the commissioning of a new type of HMS is schematically shown in Fig. 6.1 and consists of steps 1-4 and 6. The commissioning starts with step 1 by offline parameter identification for the nominal use-case (for example a light object for positioning and a standard mounting/base) as described in Chapter 3. The next step is the parameterization and setup of the disturbance and nonlinearity compensation measures (the current controller and the PEA hysteresis compensator) according to Chapter 4. In step 3, follows the selection of the individual controller types for the DC-drive and the PEA and the determination of their optimal control parameters (step 4) based on the SISO models derived in Chapter 5. In case of state-feedback controllers, these parameters can be calculated through pole-assignment method as it is done in Chapter 5. The fine tuning of the control parameter (e.g. controller bandwidth) and the decoupling network magnitude is performed by trajectory-tracking experiments (step 6). This is necessary because the decoupling networks are sensitive to the modelling errors. After the fine tuning is concluded, the control system is optimized for the nominal use-case (f.e. a light object, standard mounting and commonly used positioning velocities). Start
1
Offline parameter identification of the HMS electro-mechanical model
Offline identification as described in Chapter 3
2
Parameterization and setup of the current controller and hysteresis compensator for PEA
As described in Chapter 4
3
Selection of the controller types for the DC-drive and the PEA.
4
Setup of control system parameters (controller, decoupling network, state estimator).
5
6
Compare the achievable bandwidth of the individual controller. Decision if the PEAcontroller should be used.
Fine tuning of the control parameter and validation of the closed-loop system.
Through the control system structure, both controller can be selected separately and parameterized based on the respective SISO-plant Pole assignment (for state-feedback controller as designed in Chapter 5) or loop-shaping (for traditional controller) may be used.
This step is performed only during the commissioning at the customer site by a certain positioned object.
The final setting for the new device or the particular (commissioning at the customer site through the manufacturer) case.
Control system optimized End
Fig. 6.1
Proposed scenario for the HMS commissioning through the manufacturer.
The proposed commissioning scenario for a certain use-case at the customer site has one additional task: after the setup of individual controller, the achievable bandwidths of both controllers are compared (step 5). When the bandwidth of the PEA-controller is not significantly higher than the bandwidth of the DC-drive controller, there will be probably no significant benefits of the PEA use. In such cases, it is reasonable to disable the PEA-controller (see summary from Subchapter 5.9.2) or a simpler micropositioning system with DC-drive, spindle and gearbox (without a PEA) may be used.
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Commissioning concept to accoplish through the customer
By the commissioning through the customer it is supposed that the HMS is already optimized for a light object, a standard mounting and common positioning velocities. The disturbance and nonlinearity compensation measures are already set up. These settings will be valid in most cases. However, the proper parameters of the control system must be adapted to the particular positioned object, the HMS mounting and chosen positioning velocities. Because the customer may be not versant in automatic control theory (and furthermore not equipped with the necessary hardware and software tools), the identification methods used in Chapter 3 cannot be used here. Instead, the adaptation of the control parameter should be performed online and in an automated way, supported by an appropriate commissioning process. A proposed scenario for the commissioning through the customer is shown in Fig. 6.2. After the commissioning start, the first step is the setup of the nonlinear compensation measure. The automated re-identification of the PEA hysteresis and re-parameterization of the hysteresis compensator as described in [52] may be performed optionally. The next step is the online parameter adaptation of the DC-drive controller, followed by the fine tuning (optional) and the assessment of the achievable trajectory-tracking quality. During the steps 2 and 3, the PEA controller, the decoupling network, and the state-estimator are disabled, and the controlled variable is the measured position s2. In the next step, the PEA controller and the state estimator are enabled. The decoupling network remains switched off because of its sensitivity to model parameter errors (the control configuration from Fig. 5.10 is used). Alternatively, the static decoupling network may be used9. During the commissioning step 4, the online adaptation of the PEA-controller is done, followed by the enabling and the automated fine tuning of the dynamic decoupling network in step 5. Start by change of the positioned object or in case of bad positioning quality.
1
2
Online adaptation of the parameter of the DC-drive controller by disabled PEA controller, (see Subchapter 6.3)
3
Fine tuning and validation through trajectory-tracking experiments
4
Online adaption of the PEA controller parameter by both enabled DC- and PEA-controller as described in Subchapter 6.3
Before this task, the bandwidth of the DC-controller is slightly reduced (f.e. 5%). The control configuration from Fig. 5.10 together with the state estimator is used (the decoupling network is disabled). Alternatively, the configuration with the static decoupling may be used here.
5
Enabling and online adaptation of the dynamic decoupling. Validation of trajectory-tracking quality.
The dynamic decoupling network is switched on and adapted. The trajectory-tracking quality is assessed by the particular customer application. By higher mass and velocities, the assesment of the positioning quality with the static decoupling is also recommended.
No Switch off PEA controller and decoupling network
Finish
Fig. 6.2 9
The online re-identification of the PEA parameter as described in chapter 3 is necessary only when the environmental conditions or load are significantly changed.
Initialization and setup of the current controller and the hysteresis compensator.
Use PEA for positioning? 6 7
Yes
Finish
In this step only the DC-drive controller with the feed-forward measure and the trajectory generator are enabled The controller parameter may be fine tuned (optional). The trajectory-tracking quality during the particular customer application is assessed by disabled PEA controller. RMSE and maximal transient error are recorded.
The decision is made based on the comparison of the trajectory-tracking quality (e.g. RMSE or maximal transient error depending on the application) achieved in step 3 and 5.(see summary from Subchapter 5.9.2). By heavy positioned objects and large velocities, the advantages of the PEA may diminish. In such cases the use of the control configuration from step 3 may be advantageous.
Proposed scenario for the HMS commissioning through the customer.
However, according to the achieved experience the enabled decoupling network decelerates the controller adaptation.
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As shown in Subchapter 5.7, the bandwidth of the model validity for the transfer function Gs2 / v PA ( s ) (describing the PEA influence to the object position s2) is reduced by the increase of the positioned mass (Fig. 5.29). Thus, the PEA controller bandwidth must be also decreased to preserve the stability. At the same time, the sensitivity of the control system to disturbances (like the surface and spindle roughness, sensor interpolation errors etc.) increases. For these reasons, the advantage of the PEA may diminish, particularly in case of heavy objects and high positioning velocities. In such cases, it may be advantageous to use the static decoupling network and in cases when the PEA controller bandwith becomes comparable to the DC-controller bandwidth, to switch off the PEA controller and use the configuration from step 3 for trajectory tracking. This decision is done during the commissioning step 6, based on the trajectory-tracking qualities achieved in steps 3 and 5. The adaptation of the both controllers and the decoupling network can be raised also on automated way, as soon as the deterioration of the positioning quality is detected. From Fig. 6.2 it is obviously, that the online adaptation of the DC-drive, the PEA controller and the decoupling network is necessary. One possible solution for this problem is given in the next chapter.
6.3 Model-based online control optimization by extremum control Different adaptive control design approaches are known for the control problem by the partially unknown or changing plant dynamic. An adaptive controller is formed by combining an on-line parameter estimator, which provides estimates of unknown parameters at each instant, with a control law that is motivated by the known parameter case. The way in which the parameter estimator is combined with the control law defines the three mostly used fundamental approaches [188]. In the first approach, referred to Model Reference Adaptive Systems (MRAS), the desired behaviour of the system is specified by a model, and the parameters of the controller are adjusted based on the model error, which is the difference between the outputs of the closed-loop system and the model [189]. The second approach, known as model model Identification Adaptive Systems (MIAS), consists basically of a recursive identification algorithm that finds the parameters of the system, accompanied with a controller design algorithm that on the basis of the estimated parameters calculates the controller parameters to satisfy the demands [190]. The third approach is known as Parameter Scheduling. The controller structure consists by parameter scheduling of a feedback loop and a controller with adjustable gains. This approach supposes the knowledge about the plant’s properties in dependence on its measured (or estimated) state. By the design of the online adaptation algorithm, the number of the parameters which have to be optimized has a significant impact on the adaptation convergence [190]. For this reason, instead of a simultaneous optimization all of the controller parameters individually (the state feedback controller, the state estimator, and the decoupling network), in this work the approach based on the plant’s physical model from Subchapter 5.2 is chosen. This ensures the reflection with the physical model as well, resulting in better insight during the online optimization process. According to the investigations in Subchapter 5.7, the value of m2eff is the dominant parameter under changing, and hence to be identified. Because the physical models from Fig. 5.14 and Fig. 5.17 describe the real plant accurately only up to a certain frequency range (depending also on the positioned object mass), the bandwidths of the controller for the DC-drive and PEA (defined through ωbwDC and ωbwPEA) should be limited. To achieve the optimal control quality, the appropriate value sets {mΣ, ωbwDC} for the DC-drive controller and {m2eff, ωbwPEA} for the PEA-controller have to be found, where mΣ = m0 + m1 + m2 eff , with m0 >> m1 ∧ m0 >> m2 eff
and therefore :
mΣ ≈ m0 .
(6.1)
The adaptation of the control parameters for the DC-drive is performed during the commissioning step 2 in Fig. 6.2. In this task, the PEA controller, the decoupling network and the state estimator are disabled and the measured position s2 is set as controlled variable. The DC-controller is adjusted in the real time according to the optimization of {mΣ, ωbwDC}.
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During the adaptation of the PEA-controller parameter (performed in step 4 in Fig. 6.2), the individual parameter of the controller, and the state estimator are adjusted in real time according to the optimization of {m2eff, ωbwPEA}, while the dynamic decoupling network remains disabled until the optimal value set is found. This is necessary, because of its sensitivity to the model errors [159]. The adaptation of the decoupling gains k12 and k21 performed in step 5 in Fig. 6.2 is required through its high sensitivity. Preferably, the parameter set {k12, k21} is adapted online as well. As the optimization problems for the DC-drive, the PEA controller and the decoupling network gains are very similar to each other, only the optimization of the PEA controller parameter is discussed here in detail. The adaptation of the DC-controller and the decoupling gains is performed using the same algorithm, only some optimization parameters are modified [176]. As by the trajectory-tracking the positioning RMSE is frequently used as a quality-criterion, and a rapid change of the positioned mass is not expected, here an adaptation algorithm is chosen which does not belongs to the former mentioned fundamental ones. In this work, a slightly modified form of the general extremum control method [189] is applied. A distinction between classical adaptive control and extremum seeking (ES) is that the later is not model-based. Extremum seeking is applicable in situations where there is nonlinearity in the control problem, and the nonlinearity has a local minimum or a maximum. The nonlinearity may be in the plant, as a physical nonlinearity, possibly manifesting itself through an equilibrium map, or it may be in the control objective, added to the system through a cost functional of an optimization problem. Therefore, one can use extremum seeking both for tuning a set point to achieve an optimal value of the output, and for tuning parameters of a feedback law. The parameter space can be multivariable. Extremum control is related to optimization techniques; many of the ideas have been transferred from numerical optimization. The first documented use of ES is Leblanc’s 1922 application to electric railway systems [191]. There was a great interest in extremum control in the 1950s and 1960s and commercial products were put on market as well. For instance, the first computer controller system installed in the process industry was motivated by the possibility of optimizing the set-point of the controllers [189]. The simplified diagram of an extremum control system is shown in Fig. 6.3. Generally, the process can work in open loop or in closed loop as depicted in the figure. The most important feature is, that the process is assumed to be nonlinear in the sense that at least the performance is a nonlinear function of the reference signal. The output used in the search algorithm is some measure of the performance of the system – for instance, efficiency. The conventional regulator can use this signal, but it is more common for the regulator to use some other output of the process [189].
Search algorithm Setpoint
Reference -
Fig. 6.3
Controller
Performance
Process
Output
A simplified block diagram of an extremum control system.
The HMS control structure extended with the extremum control optimization algorithm is shown in Fig. 6.4. The goal of the optimization is to keep the trajectory-tracking RMSE as small as possible. The search algorithm in Fig. 6.3 consists hence of three main elements. The performance index eSErs2 is built in real-time using the present and past values of the dynamic tracking error es2. The online parameter optimization routine minimizes this index by optimizing of the value set {m2eff, ωbwPEA}. The controller re-design and update algorithm re-calculates the new parameter of the discrete PEA controller, the discrete state estimator, and the decoupling network after every optimization step in real time. All these elements are continuously updated, but the decoupling network is switched on first after the optimization process is finished.
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For the realization of the online control parameter optimization algorithm using the extremum control method, three problems have be solved: 1. define a suitable performance index10 which can be calculated in real-time, 2. select and implement an appropriate online-capable parameter optimization algorithm, 3. solve the re-calculation and updating of the controller, state estimator and decoupling-network parameter in real-time. In following subchapters these individual points are discussed for the adaptation of the PEAcontroller and for each task a solution is given. The adaptation of the DC-drive controller and the decoupling gains is performed in the same way.
es 2 Performance index
eSErs 2 Parameter optimization
{m2 eff , ωbwPEA } Controller and estimator parameter re-calculation and update
s2*
Main controller
Decoupling network
sˆ0 ,..., Δs&ˆ1
iq* , v *PA
Plant (HMS)
~ s2
iq* , v *PA State estimator
~ s2
Integrated control system Fig. 6.4
6.3.1
Schematic view of the online control optimization of the PEA controller by extremum control method. The online-calculated performance index
The solution of the offline optimization-problem usually supposes the minimization of a quadratic performance index [136]. For the online optimization, it is more advantageous to chose the index to be a “smooth function” e.g. to rule out the measurement noise and transient disturbances. Furthermore, because the calculation of this index is performed within the task with the sample time of the position measurement, it should not require lot of memory or CPU resources. For this reason, the performance index eSErs2 is defined in form of a discrete recursive root-meansquared-error (RRMSE): eSErs2 (k ) = eSEr s2 (k − 1) ⋅
λ −1 + ( s2* (k ) − ~ s2 (k ))2 . λ
(6.2)
The calculation of eSErs2 according to (6.2) requires only few mathematical operations and a very little memory. The forgetting factor λ for calculating the error index eSErs2 has to be set high enough to suppress the measurement noise and the influence of transient disturbances on the one hand. On the other hand a value chosen to high results in a slower optimization process. As the measurement sample time is 100 µs, the value λ=250 turned out as an appropriate choice. 10
By some authors referred also as “cost function“ or “performance measure”.
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Online capable optimization algorithms
By the online optimization of the control parameter the utilized optimization algorithm plays an important role. The optimal parameter search routine should start by safe default values and converge toward optimal ones. As analysed by simulations, the control system’s sensitivity to estimated value pair {m2eff, ωbwPEA} rises rapidly over a specific value. This is illustrated in Fig. 6.5, where simulated values of the RRMSE are displayed for different estimated values m2eff having for ωbwPEA a value of about 2·π·150 1/s (a), and in the second case (b) the ωbwPEA is varied by a simulated estimation error of 50 g for m2eff. From Fig. 6.5 it is obviously, that “too big” steps in a wrong direction may hence render the control system to instability. Some classical optimization algorithms like the binary search [192], and the random optimization [192] are for not appropriate this reason. 1.4 1.38
without additional object with object m =950g 2eff
1
with object m2eff =1.45kg
with object m
2eff
RRMSE (µm)
1.36 RRMSE (µm)
1.2
without additional object with object m2eff =950g
1.34 1.32
=1.45kg
0.8
0.6
1.3 0.4
1.28 1.26 0.2
0.4
0.6
0.8 1 ^ m2eff (kg)
a) Fig. 6.5
1.2
1.4
1.6
0.2 150
200
250
ωˆ bwPEA ⋅ 2 ⋅ π
300
350
400
( Hz )
b)
Simulated results for RRMSE value versus estimation quality of meff for ωbwPEA =2·π·150 1/s (a), and versus ωbwPEA by 50 g estimation error in m2eff (b).
In [170] and [193], the suitability of different minimum-search algorithms for the online control parameter optimization is investigated. Following methods were investigated: - The simple hill-climbing algorithm [192], - gradient descent methods [192], - the Nelder-Mead (downhill-simplex) method [192], - real-time optimization by extremum-seeking control11 [194]. The listed optimization algorithms are updated for real-time capability, and in [170] thoroughly tested by simulations and experiments with the HMS. It has been found, that in principal all these algorithms can be used for the application, but there are significant differences in the optimization speed, convergence, and robustness. The qualitative results from [170] and [193] are presented in Table 6.1. In this work, the extremum-seeking method is chosen. The decision for this method is made in the first line because the good robustness, convergence and stability noticed in [170]. A further advantage of this method is that it offers a “native support” for the multivariable optimization. As in [170] the optimizations of m2eff and ωbwPEA were performed individually in separate sequences, in this work both parameters should be optimized simultaneously. A short overview of the extremum-seeking control with remarks about its application for the HMS is given in the following subchapter.
11 It is important to distinguish between the terms “extremum control” related to the adaptive system, and the “extremum-seeking control” which describes the real-time capable optimization routine.
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Table 6.1: Comparison of online-capable optimization algorithms for the application Algorithm Hill-climbing (unary search) Gradient descent (steepest descent) Nelder-Mead (simplex) Extremum-seeking control
Optimization Speed
Convergence
Robustness
System Stability
Poor
Poor
Poor
Average
Good
Good
Average
Average
Good
Good
Good
Average
Average
Very good
Very good
Very good
Extremum-seeking control The action diagram of the extremum-seeking control for a continuous SISO control scheme is shown in Fig. 6.6. The function which is to be minimized is denoted with f (θ ) and in general case, it can be approximated locally with: f (θ ) = f * + f ′′ > 0.
f ′′ (θ − θ * ) 2 , 2
where
(6.3)
Any C2 function f(θ) can be approximated locally by Eq. (6.3). The assumption f " > 0 is made without loss of generality. If f " > 0, we just replace k (k > 0) in Figure Fig. 6.6 with -k. The purpose of the algorithm is to make θ -θ * as small as possible, so that the output f(θ) is driven to its minimum f *. The perturbation signal a ⋅ sin(ωt ) fed into the plant helps to get a measure of gradient information of the map f(θ). An elementary intuitive explanation as to how the scheme “works” is described here according to [194], whereas the rigorous analysis can be found in [194]. f*
θ*
θ
θˆ
a sin(ω t )
Fig. 6.6
−
k s
y
f (θ )
ζ
s s+h sin(ω t )
Action-diagram of the extremum-seeking control [194].
~ When the optimization error of the unknown optimal parameter θ * is denoted with θ : ~ θ = θ * − θˆ,
(6.4)
it follows, ~
θ − θ * = a sin(ω ⋅ t ) − θ , which is substituted into (6.3) and results in system output:
(6.5)
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f ′′ ~ [θ − a sin(ω ⋅ t )]2 . 2
(6.6)
Expanding the expression and using the basic trigonometrical identity 2 ⋅ sin 2 (ω ⋅ t ) = 1 − cos(2ω ⋅ t )
(6.7)
one gets
y= f*+
f ′′ ~ 2 a 2 ⋅ f ′′ a 2 ⋅ f ′′ ~ θ + cos(2ω ⋅ t ). − af ′θ′ sin(ω ⋅ t ) − 4 2 4
(6.8)
s applied to the system output serves to remove f *. Namely: ( s + h) ~ s f ′′ ⋅ θ 2 a 2 ⋅ f ′′ ~ ⋅ [ y ( s )]} ≈ − a ⋅ f ′′ ⋅ θ sin(ω ⋅ t ) − L−1{ cos(2ω ⋅ t ) . 2 s+h 4
The high-pass filter
(6.9)
This signal is then “demodulated” by multiplication with sin(ω ⋅ t ) and we get for ζ : ~ f ′′ ⋅ θ 2 ~ a 2 ⋅ f ′′ ζ ≈ sin(ω ⋅ t ) − a ⋅ f ′′ ⋅ θ sin 2 (ω ⋅ t ) − cos(2ω ⋅ t ) ⋅ sin(ω ⋅ t ). 2 4
(6.10)
Applying (6.7) again together with a second trigonometrical identity 2 ⋅ cos(2ω ⋅ t ) ⋅ sin(ω ⋅ t ) = sin(3ω ⋅ t ) − sin(ω ⋅ t ) ,
(6.11)
yields:
ζ ≈−
f ′′ ~ a 2 ⋅ f ′′ ~ a 2 ⋅ f ′′ f ′′ ~ [sin(ω ⋅ t ) − sin(3ω ⋅ t )] + θ 2 sin(ω ⋅ t ). θ+ θ ⋅ cos(2ω ⋅ t ) + 2 2 8 2
(6.12)
Noting that θ * is constant, ~&
&
θ = −θˆ,
(6.13)
follows: ⎤ a 2 ⋅ f ′′ f ′′ ~ ⎧ k ⎫ ⎡ a ⋅ f ′′ ~ a ⋅ f ′′ ~ θ + θ cos(2ω ⋅ t ) + [sin(ω ⋅ t ) − sin(3ω ⋅ t )] + θ 2 sin(ω ⋅ t )⎥. 2 2 8 2 ⎦
~
θ ≈ L−1 ⎨ ⎬ * ⎢− ⎩s⎭ ⎣
(6.14)
The last term can be neglected because it is quadratic and we are interested only in local analysis: ~
⎤ a 2 ⋅ f ′′ ⎧ k ⎫ ⎡ a ⋅ f ′′ ~ a ⋅ f ′′ ~ θ + θ cos(2ω ⋅ t ) + [sin(ω ⋅ t ) − sin(3ω ⋅ t )]⎥. 2 2 8 ⎦
θ ≈ L−1 ⎨ ⎬ * ⎢− ⎩s⎭ ⎣
(6.15)
The last two terms in (6.15) are high-frequency signals. They will be greatly attenuated when passing through the integrator. Hence, they can be neglected, getting: ~
⎧ k ⎫ ⎡ a ⋅ f ′′ ~ ⎤ θ⎥ 2 ⎩s⎭ ⎣ ⎦
θ ≈ L−1 ⎨ ⎬ * ⎢−
k ⋅ a ⋅ f ′′ ~ ~& → θ ≈− θ. 2
(6.16)
~ Since k ⋅ f ′′ > 0 , this is a stable system. Thus, we can conclude that θ → 0 or in terms of the original problem, θˆ(t ) converges within a small distance to θ *. It is important to note, that the approximations made here hold only when ω is large in qualitative sense related to k, a, h and f ′′ . The multivariable extremum-seeking case is discussed in detail in [194] and [195]. Analogous to the single parameter case, f(θ) is let here be a function of the form:
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f (θ) = f * (t ) + (θ − θ* (t ))T ⋅ P(θ − θ* (t )) ,
(6.17)
where Plxl=PT is a positive definite matrix, θ = [θ1 ,..., θl ]T and θ* (t ) = [θ1* (t ) ,..., θl* (t )]T . Any vector function f(θ) with a quadratic minimum can be approximated by (6.17), and hence the parameter optimization is enabled. As explained in [195], for tracking of l parameters, it is principally enough to have l/2 frequencies. But it is also possible to use a different frequency for each of the l parameters [194]. As shown in [195], the design difficulty in general increases with the dimension l. The implementation of the extremum-seeking algorithm for the real-time optimization of {m2eff, ωbwPEA} is shown in Fig. 6.7. The same structure can be used for the optimization of {m0, ωbwDC} and {k12, k21} as well. Here, two separate excitation frequencies are chosen; on this way the same structure enables both the simultaneous optimization of m2eff and ωbwPEA, as well as the sequential individual optimization described in [170]. Control param. re-calc., update
mˆ 2 eff
k − 1 s
a1 ⋅ sin(ω1 ⋅ t )
ωˆ bwPEA a2 ⋅ sin(ω2 ⋅ t ) Fig. 6.7
Integrated control structure
HMS (plant)
ζ m2 eff
Perform. index
eSErs2
s s + h1
sin(ω1 ⋅ t )
−
k2 s
ζ ωbwPEA
s s + h2
sin(ω2 ⋅ t )
Implementation of the extremum-seeking algorithm for the optimization of {m2eff, ωbwPEA}.
The parameters ω1, h1, k1, a1, ω2, h2, k2, and a2 are set empirically, taking into account the dynamic of the performance index and the hints from [194]. The determined optimal values are given in Appendix 8.17. Simulation results of the real-time parameter optimization of the PEA-controller, using the model of the controlled HMS for m2eff =0.84 kg are given in Fig. 6.8. The phase-plane representation of the adaptation is shown furthermore in Fig. 6.9.
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ωbwPEA optimization (rad/s)
100
Position (mm)
80 60 40 20
Position reference Simulated position
1400 1300 1200 1100 1000 900
0 0
2
4
6 Time (s)
8
800
10
2
2
4
6 Time (s)
8
10
0
2
4
6 Time (s)
8
10
1 Transient position error Position RMSE
0.9 optimization (kg)
1.5 Position error (µm)
0
1
0.7 0.6
m
2eff
0.5
0.8
0
0.5 0.4 0.3
-0.5
Fig. 6.8
0
2
4
6 Time (s)
8
0.2
10
Simulation results for optimization of m2eff and ωbwPEA by extremum-seeking algorithm.
ω
bwPEA
optimization (rad/s)
1500 1400 1300 1200 1100 1000 900 800 0.2
0.3
0.4
0.5 m
Position error (µm)
2eff
0.7
0.8
0.9
1
Parameter optimized
0.9 0.8 0.7 0.6 0.5 0.8
0.7
0.6
m
2eff
Fig. 6.9
0.6 optimization (kg)
optimization (kg)
0.5
0.4
950
1000
1050
1100
1150
1200
1250
1300
1350
ωbwPEA optimization
Simulation results for optimization of m2eff and ωbwPEA by the extremum-seeking algorithm.
The applied positioning velocity for the simulations was set to 10 mm/s. A good convergence of the applied extremum-seeking method was observed; the estimated value of m2eff after ten seconds reaches the value m2eff = 0.84 kg, which was set in the HMS model.
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Adaptation of the control parameter in real-time
The final element in the control optimization structure consists of the real-time capable recalculation of the PEA controller as well as the state estimator parameters. Both two tasks consist of several sub-elements. For the re-calculation of the PEA controller’s parameter in real-time following actions are needed: 1. online reduction of the plant model for the PEA controller design on m2eff change, 2. real-time capable discretization of the online reduced model, 3. re-calculation of the control parameter based on the updated discrete Butterworth polynomial coefficients for the given bandwidth ωbwPEA, whereas for the re-calculation of the state-estimator parameter in real-time the required actions are: 1. the online reduction of the plant model for state estimator on m2eff change, 2. and the real-time capable discretization of the online reduced model. These tasks are represented in block-diagram form in Fig. 6.10. Indeed, the first two elements of both tasks are similar, having a slight difference in the plant model used for the PEA controller design (5.53) and the state estimator (5.21). For this reason, they will be discussed together. One of challenging issues by the realization of the real-time capable model reduction is the solution of (5.25) for calculation of the eigenvalue vector. As in Subchapter 5.3.1 stated, the closed form is possible because the system is of 4th order. But such a solution consists of thousands of operations, which is not suitable for real-time execution in a fast task. The numerical solution of (5.25) either requires a variable number of iterations or returns with variable accuracy. Both are inacceptable: variable number of iterations represents a hazard for the real-time capability; variable accuracy may impact the control quality. For this reason, the model reduction and the discretization of the reduced model are merged into one block as shown in Fig. 6.10 and the values of ΦPEAr, HPEAr, CPEAr, Φr, Hr, and Cr are calculated offline and stored in a LUT for the whole input range of meff. A similar method is utilized for the real-time calculation of the discrete Butterworth polynomial as well. Because there is only one input port in each block, simple one-dimensional look-up tables can be used. The memory requirements are hence neglectable, by great acceleration in the task execution speed. The input range of [0 … 5] kg with equidistant resolution of 10 g is selected and a linear interpolation is used between the stored values. More details on the implementation of the look-up-tables are given in [170]. The automated re-calculation of the PEA controller and the state-estimator parameter is performed by embedded M-scripting. The “recalculate flags” are involved to save CPU-resources: if there is no change in m2eff and/or ωbwPEA, then it is no need for the parameter re-calculation, rather the formerly calculated values can be used. x PEAr = [Δs1 , Δs&1 , Δs2 ]T Δs*
m2 eff
Discrete statefeedback controller with automatically recalculation of control parameter
ΦPEAr Model reduction and H PEAr discretization using LUTs CPEAr
1+2
recalculate flag
Calculation of the p0
ωbwPEA discrete Butterworth p1
polynomial by LUTs p 2
3
recalculate flag
a)
3
vPA*
ud = [v*p , iq* ]T
m2eff
Model reduction Φr and discretization Hr using LUTs
1+2
Discrete statexˆ r = [Δs1 , Δs&1 , Δs2 ]T estimator with variable parameter
Cr
b)
Fig. 6.10 Adaptation of the PEA controller (a) and the state-estimator (b) parameter in real-time. By the online adaptation of the PEA controller, the decoupling network poles of Gk12 and Gk21 have to be determined and the neighbouring ones cancelled in order to get Gk12App and Gk21App in real-time. Like in the previous case, the use of look-up tables turned out as the most suitable way. The static gains k12 and k21 can be directly re-calculated as they are available in closed form in dependence of m2 (respectively m2eff, see (5.18)).
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The adaptation of the DC-drive controller using the extremum-seeking control method is simpler, because the denominator polynomial for the critical damping behaviour is available in closed form (see (5.49)) and there is no need for online model reduction, [176]. The adaptation of the decoupling network gains is also simpler. Here, only the previously calculated parameter set {k12, k21} is optimized, [176].
6.4 Practical realization and experimental results The practical realization of the commissioning scenario shown in Fig. 6.2 with automated controlparameter adaptation described in Subchapter 6.3 bases on the experimental system shown in Fig. 5.36. The online control-parameter optimization algorithms are realized using the RCP-system, having the adaptation frequency set to 10 kHz. The friction compensation measure for the DC-drive is disabled, as its benefit by the enabled PEA-controller was not noticeable. The proposed commissioning scenario from Subchapter 6.2 is realized using the ControlDesk automation feature including Python scripting. The proposed commissioning scenario outlined in Subchapter 6.2 together with the implemented online adaptation routines in Subchapter 6.3 is validated with different positioned objects and trajectories. A large number of experiments are performed. However, here only representative results are discussed in the following. More detailed results are given in [176]. To compare the achieved control quality after the parameter adaptation with the one of the factory controller, the measurement set with the positioned object of 8 kg mass is performed using the C-702 as well. All measurements are performed with the HMS mounted to a vibration-isolated test rig depicted in Fig. 6.11.
Positioned object
Vibration-isolated test rig
Hybrid micropositioning system
Optical plate
Fig. 6.11 Photograph of the HMS with a positioned object mounted onto a vibration-isolated test rig. The adaptation of the parameter set {mΣ, ωbwDC} for the DC-drive controller with a positioned object of 5 kg mass and maximal trajectory velocity of 100 µm/s is shown in Fig. 6.12. The algorithm is started by the time t=1 s (e.g. after the constant velocity motion is reached) with safe default initial values and reaches the optimal bandwidth in 5 seconds, whereas the changes in mΣ are very small, as it was expected.
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ωbwDC optimization (rad/s)
2
Position (mm)
1.5 1 0.5 Position reference Measured position
0 -0.5
0
5
10
450 400 350 300
15
Adaptation start 0
5
Time (s)
15
10
15
85.8 Transient position error Position RMSE
1
m optimization (kg)
0 -1
Σ
Position error (µm)
2
Adaptation start -2
10 Time (s)
0
5
10
85.6 85.4 85.2 85
Adaptation start 15
0
5
Time (s)
Time (s)
Fig. 6.12 Adaptation of the parameter set {mΣ, ωbwDC} for the DC-drive controller with a positioned object of 5 kg mass and positioning velocity of 100 µm/s. After the adaptation of {mΣ, ωbwDC} the control quality by the solely use of the DC-drive is illustrated in Fig. 6.13.
Position (mm)
1 0.8 0.6 0.4 Reference position Measured position
0.2
Transient positioning error (nm)
0
0
5
10
15
20 Time (s)
25
30
35
600 400 200 0 -200
|Error|max = 642.650 nm
-400 0
5
10
15
RMSE = 70.724 nm 20 Time (s)
25
30
35
Fig. 6.13 Trajectory-tracking result by the solely use of the DC-drive controller after the parameter adaptation. The positioning velocity is 30 µm/s and the positioned object is of 5 kg mass. The adaptation of {m2eff, ωbwPEA} by disabled decoupling network for the same positioned mass is illustrated in Fig. 6.14, whereas in Fig. 6.15 the adaptation by the enabled static decoupling is shown.
6. CONTROLLER COMMISSIONING AND ADAPTATION ωbwPEA optimization (rad/s)
Position reference Measured position
0.15 0.1 0.05 0
0
5
10
20
900 800 700 600 500
25
Transient position error Position RMSE
1.5 1
0
5
10
15 Time (s)
20
25
0
5
10
15 Time (s)
20
25
0.7 0.65 0.6 0.55
2
0.5 0 -0.5
1000
0.75
2 Position error (µm)
15 Time (s)
1100
m optimization (kg)
Position (mm)
0.2
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Adaptation start 0
5
10
15 Time (s)
20
0.5 0.45
25
Fig. 6.14 Adaptation of the parameter set {m2eff, ωbwPEA} for the PEA-controller using a positioned object of 5 kg mass by disabled decoupling network.
0.6
900 800
bwPEA
0.4
1000
optimization (rad/s)
Position reference Measured position
0.2
700
ω
Position (mm)
0.8
5
600
15
5
10 Time (s)
15
5
10 Time (s)
15
0.8
1.5
Transient position error Position RMSE
1 0.5 0 -0.5
2
Position error (µm)
10 Time (s)
Adaptation start
-1
m optimization (kg)
0
0.7 0.6 0.5 0.4
-1.5 5
10 Time (s)
15
Fig. 6.15 Adaptation of the parameter set {m2eff, ωbwPEA} for the PEA-controller using a positioned object of 5 kg mass by enabled static decoupling. The relatively large transient positioning error bias in Fig. 6.14 is suppressed after the enabling and adaptation of the dynamic decoupling network (see Fig. 5.44 for reference). Although the convergence of the PEA-controller adaptation by the enabled static decoupling network offers better results, this method shows an increased sensitivity for disturbances and often ends in a suboptimal value for ωbwPEA. During the optimization of {m2eff, ωbwPEA}, the value of k21 changes in time (see Eqs. (5.18) and (5.19)), resulting in a variable dynamic feedback path inside of the control system.
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The adaptation of the relative value of k21 by the positioning velocity of 30 µm/s, after the optimal PEA-controller parameters are found, is shown in Fig. 6.16. Actually, here the relative magnitude of the parameter set {k12, k21} is adapted, but the changes in k12 were not significant.
Position (µm)
300 200 100
Position reference Measured position
0 2
3
4
5 Time (s)
6
7
8
9
10
5 Time (s)
6
7
8
9
10
Adaptation start 0.25
0.2 0
k
Position error (nm)
1
0.3
21
optimization (relative gain)
0
1
2
3
4
300
Transient position error Position RMSE
200 100 0 -100
0
1
2
3
4
5 Time (s)
6
7
8
9
10
Fig. 6.16 Adaptation of the relative gain of k21 by optimized DC- and PEA-controller. The positioned object is of 5 kg mass and the positioning velocity is 30 µm/s. The validation of the optimized control through a trajectory-following task by unchanged positioning velocity is illustrated in Fig. 6.17.
Position (µm)
500 400 300 200 Position reference Measured position
100
Position error (nm)
0
0
2
4
6
8 Time (s)
10
12
14
16
100
0
-100
|Error| 0
2
max
= 158 nm 4
RMSE = 20.45 nm 6
8 Time (s)
10
12
14
16
Fig. 6.17 Validation of the optimized control system by a trajectory-tracking task by the positioning velocity of 30 µm/s.
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In this example, through comparison of the results depicted in Fig. 6.13 and Fig. 6.17 it is obviously that the use of the PEA-controller for the given mass and trajectory speed is advantageous. The commissioning ends hence immediately after step 6 in Fig. 6.2. The proposed commissioning scenario shown in Fig. 6.2 is validated with various test objects and trajectory velocities. A short excerpt of the recorded data and a summary of the achieved results are given. In Fig. 6.18 the trajectory-tracking result in case of a positioned object of 2.3 kg mass from Fig. 6.11 by the positioning velocity of 30 µm/s is shown. In this case, during the commissioning it was shown that the use of the PEA-controller is advantageous. The maximal transient trajectory-tracking error is 19 nm, whereas the RMSE is as low as 4.46 nm. 250 Position (µm)
200 150 100 50
Position reference Measured position
0 0
Position error (nm)
20
1
|Error|
2
max
3
4 Time (s)
5
6
7
8
5
6
7
8
= 19 nm, RMSE = 4.46 nm
10 0 -10 -20
0
1
2
3
4 Time (s)
Fig. 6.18 Trajectory-tracking results with the optimized control system by the positioned mass of 2.3 kg and positioning velocity of 30 µm/s. By the positioned object of 8 kg mass having the positioning velocity of 1 mm/s, the PEA could not offer any enhancement in the positioning quality. In Fig. 6.19, the trajectory-tracking results by the disabled PEA- and optimized DC-controller with the feed-forward measure are shown.
Position (mm)
0 Position reference Measured position
-2 -4 -6 -8 -10
0
2
4
6 Time (s)
8
10
Position error (nm)
1000 500 0 -500 |Error| -1000
0
2
4
6 Time (s)
max
= 935 nm, RMSE = 212.50 nm 8
10
Fig. 6.19 Trajectory-tracking results after the commissioning using the positioned mass of 8 kg by the positioning velocity of 1 mm/s. The PEA-controller was here switched off.
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The best achieved trajectory-tracking result with the C-702 for the same object, positioning velocity and movement direction is given in Fig. 6.20. The maximal transient error is here 5.178 µm and the trajectory-tracking RMSE is 830 nm.
Position (mm)
0
Reference position Measured position
-2 -4 -6 -8 -10
Position error (nm)
0
5000 4000 3000 2000 1000 0 -1000 -2000 -3000 -4000 -5000
2
4
6 Time (s)
|Error|max = 5178 nm. 0
2
4
8
10
12
RMSE = 830 nm. 6 Time (s)
8
10
12
Fig. 6.20 The best achieved trajectory-tracking result by the use of the C-702 controller with the positioned object of 8 kg by the positioning velocity of 1 mm/s. A summary and comparison of the achieved trajectory-tracking quality after the commissioning and controller adaptation for various positioned objects and trajectory velocities is given in Fig. 6.21 and Fig. 6.22. 5000
Average maximal transient trajectory-tracking error (nm)
3000 2000
1000
500 400 300 200 SSC structure, m =2.3kg obj
SSC structure, m =5kg
100
obj
DC+FF (PEA disabled), m =5kg obj
70 60 50 40
SSC structure, m =8kg obj
DC+FF (PEA disabled), m =8kg obj
C-702, param. set #1,m =8kg obj
30
C-702, param. set #2,m =8kg obj
0.005
0.03
0.1 0.5 Positioning velocity (mm/s)
1
5
10
Fig. 6.21 Maximal transient trajectory-tracking errors by different objects and positioning velocities.
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In Fig. 6.21, the average maximal transient trajectory-tracking error with different positioned objects (mobj=2.3 kg, 5 kg and 8 kg) and movement velocities (from 5 µm/s up to 1 cm/s) is visualised. Each of the measurements is repeated at least six times to check its reliability. By the commissioning with the test object of 2.3 kg mass, the use of the PEA controller was advantageous for all positioning velocities and it was enabled all the time. By the test object of 5 kg mass, for the positioning velocities above 1 mm/s, the PEA was switched off as its benefit was not more noticeable. By the further velocity increase, the use of the PEA even reduces the trajectory-tracking quality. According to the commissioning scenario, the measurements for higher velocities are therefore done using the DCcontroller with the feed-forward measure only, e.g. by disabled PEA-controller (in Fig. 6.21 this area is marked with dotted line). A similar situation occurred by the test object of 8 kg mass, where the PEA-controller was disabled above the positioning velocities of 100 µm/s. For comparison of the achievable control quality, the trajectory-tracking experiments are performed in the same way using the factory controller C-702 as well. With the C-702 controller only the measurement using the test object of 8 kg mass is performed. As one may notice, the performance of the optimized state-feedback control system is significantly better than the performance of the factory controller, even in cases when the PEA-controller is switched off. Furthermore, as expected, the maximal transient error in general rises with the positioning velocity and the positioned object’s mass. 2000
Average root mean square trajectory-tracking error (nm)
1000
500 400 300 200
100 70 60 50 40 30 SSC structure, m =2.3kg obj
20
SSC structure, m =5kg obj
DC+FF (PEA disabled), m =5kg obj
10
SSC structure, m =8kg obj
DC+FF (PEA disabled), m =8kg obj
C-702, param. set #1,m =8kg obj
C-702, param. set #2,m =8kg obj
0.005
0.03
0.1 0.5 Positioning velocity (mm/s)
1
5
10
Fig. 6.22 Trajectory-tracking RMSE by different object masses and positioning velocities. In Fig. 6.22, the trajectory-tracking RMSE for the same positioned objects and trajectory velocities is given. The dependence of the RMSE on the positioning velocity and the positioned object displays the expected behaviour, with a slight deviation revealed for the positioned object of 5 kg. According to the achieved positioning results, one may notice that the positioning quality with the online-optimized state-feedback control system illustrated in Fig. 5.36 is significantly better than the achievable positioning quality with the factory controller C-702.
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6.5 Example application: a two-degree-of-freedom nanopositioning system Potential applications of a HMS include optical commissioning and microassembly. For both application fields, planar translational positioning is often required. As by HMS a one-degree-offreedom motion can be realized, two HMS are necessary to achieve a two-degree-of-freedom (2DOF) trajectory-tracking (e.g. the planar translational motion). The mechanical realization of such a nanopositioning system – by fixing of the base of the second HMS (y-direction) onto the positioning plate of the first HMS (x-direction) under 90° angle – is shown in Fig. 6.23.
HMS #2 HMS #1
y x
Fig. 6.23 Mechanical realization of the 2-DOF nanopositioning system with two HMS and the chosen coordinate system. As the projection of the mutual forces of the two HMS are approximately zero and the accelerations are small, a separate control of the individual HMS is possible. The overview of the realized control system for the 2DOF nanopositioning system is shown in Fig. 6.24. The 2DOF-capable trajectory generator and the individual HMS-controllers are realized with the dSPACE RCP-system. As the DS4121 board features two LVDS channels, it was possible to connect simultaneously two M-CTRL interfaces to it. The HMS-controllers are realized according to Fig. 5.12 and Fig. 5.36 with the RCPsystem and feature the online adaptation capabilities described in the previous subchapters. The friction estimators and compensators were removed from the RCP application in order to save some CPU-resources. On this way, the parallel execution and adaptation of the both SSC-control structures with the sampling frequency of 10 kHz was enabled. M-CTRL interface #1 s2* x , s&2* x , &s&2* x
2DOF 2DOF trajectory trajectory * * * generator generator s2 y , s&2 y , &s&2 y
SSC-structure SSC-structure for forHMS HMS#1 #1 (x-direction) (x-direction)with with online-adaption online-adaption SSC-structure SSC-structure for forHMS HMS#2 #2 (y-direction) (y-direction)with with online-adaption online-adaption
dSPACE-system (DS1006 + DS4121)
DPMEMx
DPMEMy
- -SPI SPIcommunication communication vPAx , vPWMx - -current currentcontroller controller s2 x , iDCx - -hyst. hyst.compensator compensator
HMS HMS#1 #1 x-direction x-direction
v ,v - -SPI SPIcommunication communication PAy PWMy - -current currentcontroller controller s2 y , iDCy - -hyst. hyst.compensator compensator
HMS HMS#2 #2 y-direction y-direction
M-CTRL interface #2
Fig. 6.24 Overview of the control system for the 2DOF nanopositioning stage.
6. CONTROLLER COMMISSIONING AND ADAPTATION
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250
y-position (µm)
200 150 100 50
0
50
100 150 x-position (µm)
200
250
20 10 0 -10 -20 -30 -100
50 0 -50 -100
Transient trajectory-tracking error y (nm)
Trajectory-tracking error y (nm)
0
Reference position Measured position
Transient trajectory-tracking error x (nm)
The commissioning and parameter adaptation for the both HMS-controllers is performed simultaneously, during the tracking of a 2DOF-trajectory. The capability of the 2DOF HMS is validated using linear and circular trajectories with a light object for positioning. A short excerpt of the achieved experimental results is shown here. For the trajectory velocities up to 100 µm/s, the PEA-controller was enabled in both systems as its benefit was considerable by both the linear and the circular trajectories. In Fig. 6.25 the results for the tracking of a linear 2DOF trajectory with the positioning speed of 30 µm/s are given. The maximal transient error in the x-direction was 146 nm, with the positioning RMSE being 28.93 nm. The errors in the y-direction are smaller12: the maximal error is 36 nm, the trajectory-tracking RMSE 7.76 nm. Results for a circular positioning reference with 40 µm diameter and 2 s period are shown in Fig. 6.26.
-50 0 50 Trajectory-tracking error x (nm)
|Error|max = 146 nm 0
2
4
RMSE = 28.93 nm 6
8
Time (s)
20 10 0 -10 -20 -30
|Error|max = 36 nm 0
2
4
RMSE = 7.76 nm 6
8
Time (s)
Reference position x Measured position x
10 0 -10 -20
Trajectory-tracking error y (nm)
-20
-10
0 x-position (µm)
10
20
50 0 -50 -100 -150 -300
-200
-100 0 100 200 Trajectory-tracking error x (nm)
300
Transient trajectory-tracking error y (nm)
y-position (µm)
20
Transient trajectory-tracking error x (nm)
Fig. 6.25 Tracking results by a linear 2DOF trajectory using the positioning velocity of 30 µm/s. 300 200 100 0 -100 -200 |Error|max = 396 nm
-300 0
0.5
1 Time (s)
RMSE = 105.58 nm 1.5
50 0 -50 -100 -150
|Error|max = 185 nm 0
0.5
1 Time (s)
RMSE = 21.08 nm 1.5
Fig. 6.26 Tracking results for a circular 2DOF trajectory with 40 µm diameter and 2 s period. By the circular trajectory, both the maximal transient error and the RMSE are higher related to the 12
Such result is to be expected, because the first HMS scheduled to the x-direction carries the second HMS scheduled to the y-direction.
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y-position (µm)
1500
1000
500 Reference position Measured position 0
500
1000 x-position (µm)
1500
600 400 200 0 -200 -400 -1000
-500 0 500 Trajectory-tracking error x (nm)
1000 500 0 -500
1000
|Error|max = 1350 nm
-1000
Transient trajectory-tracking error y (nm)
Trajectory-tracking error y (nm)
0
Transient trajectory-tracking error x (nm)
linear one. This is the consequence of the larger maximal velocities (up to 62.83 µm/s in both x and y directions) as well as of the continuously changing accelerations. Having the trajectory speed over 100 µm/s, the PEA-controller of the first HMS did not reveal any measurable benefit. This is reasonable, because the first HMS which is scheduled for the x-direction carries the second HMS (scheduled to the y-direction), whereas the mass of a single HMS is about 5.4 kg. Trajectory-tracking experiments with higher positioning velocities were hence carried out with disabled PEA-controller by the first HMS. The use of the PEA-controller of the second HMS was advantageous for all trajectories. In Fig. 6.27 the experimental results for trajectory velocity of 1 mm/s are shown. The positioning RMSE were here 335.35 nm and 169.36 nm for the x- and y-direction.
0
0.5
1 Time (s)
RMSE = 335.35 nm 1.5
600 400 200 0 -200 -400
|Error|max = 699 nm 0
0.5
1 Time (s)
RMSE = 169.64 nm 1.5
10
Transient trajectory-tracking error x (µm)
5
Transient trajectory-tracking error y (µm)
Fig. 6.27 Tracking results by a linear 2DOF trajectory using the positioning velocity of 1 mm/s. 10 Reference position Measured position
y-position (mm)
5 0 -5 -10
Trajectory-tracking error y (µm)
-10
-5
0 x-position (mm)
5
3 2 1 0 -1 -2 -3 -5
0 Trajectory-tracking error x (µm)
5
0
|Error|max = 5608 nm
-5 0
0.5
1 Time (s)
RMSE = 1449.05 nm 1.5
3 2 1 0 -1 -2 -3
|Error|max = 3912 nm 0
0.5
1 Time (s)
RMSE = 665.55 nm 1.5
Fig. 6.28 Tracking results for a circular 2DOF trajectory with 2 cm diameter and 2 s period. In Fig. 6.28 an extreme trajectory-tracking task is shown. Here, the circular trajectory with 2 cm diameter and 2 s period is followed. Having in mind that maximal velocities up to 3.14 cm/s and continuously changing accelerations up to 10 cm/s2 were required during the trajectory-tracking, the achieved result can be classified as very good.
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7. CONCLUSION Hybrid micropositioning systems play a significant role in many applications in the industry and robotic. Despite this fact, the control design for such systems is still in early development phase. A unified systematic approach for the control design for such systems does not exist yet. Only specific solutions, applicable to a particular HMS are published. This work presents several novel approaches for the modelling, parameter identification, control design and its adaptation for a HMS. Besides some theoretical considerations and simulations, experimental results are given using the HMS from Physik Instrumente. The validation of the proposed control structure is performed by trajectory-tracking using different positioning velocities and positioned objects. Although the controller design and online parameter optimization address a commercially available hybrid nanopositioning system, the achieved results are certainly suitable for other similar micropositioning systems as well. The main contributions of this research are the novel parameter identification and hysteresis compensation algorithm for the PEA, the novel structure of the overall control consisting of a state estimator, decoupling network and the individual DC- and PEA-controllers, as well as the implementation of the model-based control parameter adaptation by the extremum-seeking method. The usefulness of these approaches is confirmed through the significantly enhanced positioning quality by trajectory-tracking, related to the achievable quality by the factory controller. Acquired results are significantly better as the achievable ones by the HMS factory controller, even when the PEA-controller is disabled. This points out the importance of the current control by the DC-drive, as well as the advantages of state-feedback controller against traditional PI/PID controller. The proposed commissioning scenario enables the user (which may be not familiar with the control theory) the systematic and automated selection of the optimal control parameters. On this way, the usability of the HMS in industrial application and robotic is enhanced. The example with a 2DOF-nanopositioning system assembled from two single HMS shows further application potentials of such a nanopositioning system with high-quality positioning control. The control hardware concept consisting of a CPU-system scheduled for the complex calculations and an FPGA dedicated for the direct plant control, using a DPMEM for the data exchange, seems to be confirmed through the industrial development as well. In the meantime, intelligent interfaces utilizing the same principle became also commercially available. One of the structural problems of the HMS under investigation is related to its mechanical construction. Because of the low gearbox stiffness (related to the stiffness of the PEA pair), the efficient utilization of the PEA by large loads and high positioning dynamics is reduced. Thus, in cases when a heavy object with high velocity has to be positioned, the contribution of the PEA to the positioning quality diminishes. In further research, on the one hand the investigation of possibilities to increase the control-system robustness to external disturbances may be done. It would be also interesting to check the achievable control performance by other adaptive control of the PEA – for example using a MRAC approach. The design of a dedicated MIMO-controller for a HMS with more degrees of freedom would certainly enhance the trajectory-tracking capability. On the other hand, there exists also potential to optimize the mechanical construction of the HMS by increasing the stiffness of the transmission line.
8. APPENDICES
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8. APPENDICES 8.1
SPI Communication Diagram
The timing diagram for the GC-IP1000 is given in Fig. 8.1, whereas TOSZ is the oscillator base period.
Fig. 8.1
SPI timing diagram of the GEMAC GC-IP1000 [64]
For the basic communication (reading of the actual position information) following command sentence is eligible. Further commands for configuration, etc. are available, but they use is not necessary. The MOSI signal (connected to SDI) should have following binary values during the read sequence: - 0b10000000 00000000 -> command: “send me the first word of the position value”, - 0b10010000 00000000 -> command: “send me the next word”, - 0b10100000 00000000 -> command: “no operation”.
8. APPENDICES
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The MISO signal (connected to SDO) contains the answer (given here in binary form): - 0b00000000 00000000 -> response: zeros, don’t care values (no meaning), - 0bxxxxxxxx xxxxxxxx -> response: the first 16 bits of the position value, - 0bxxxxxxxx xxxxxxxx -> response: the second 16-bit world of the position value. The SDO return values consist in this communication sequence of 3 data words á 16 bit. The first return word is always 0b0000000000000000. This word should not be interpreted. The second data word contains the Least Significant 16 bit-Word (LSW) of the position data. The first bit received in this data word is the Most Significant Bit (MSB), the last one is the Least Significant Bit (LSB) of the data word (and of the complete 32-bit value as well). The third data word contains the Most Significant 16-bit Word (MSW) of the position data. The first bit is the MSB of this word (and the MSB of the 32-bit value as well), whereas the last received bit represents the LSB of this word. A communication sequence performed using the hybrid controller C-702, recorded with a digital scope Agilent Technologies Series 6000 is shown in Fig. 8.2. Although the sequence period here is 100µs (sampling rate of 10 kHz), it can be performed using 50µs period (20 kHz sampling rate) as well. SCLK (V)
4 2 0
0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
5
10
15
20
25
30
0
5
10
15 Time (µs)
20
25
30
SEN (V)
4 2 0
MOSI (V)
4 2 0
MISO (V)
4 2 0
Fig. 8.2
Recorded SPI command sequence generated by the hybrid controller C-702.
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8.2 Signal connection of LA-NP25, ADS8422 and ADS8406
C10 tantal 10µF/20V=
C125
R45
2,2k
R3
182Ohm
