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a Magnetic Suspension System with Uncertainties

Akiyoshi Hata

Scho ol of Information Science,

Japan Advanced Institute of Science and Technology

February 13, 1998

Keywords: structured uncertainty, , modeling, robustness analysis and synthesis,

magnetic susp ension systems.

This thesis deals with the robustness analysis and synthesisof a magnetic suspension

system againstuncertaintiesthat enters inthe feedbackform. Robustcontrol theory has

beengreatadvancesforthedesignofrobustuncertainty-tolerantfeedbackcontrolsystems,

and it has develop ed mathematical toolsfor the evaluation of stability and performance

intheworstcaseoversets ofsystems. Themagneticsuspensionsystemisoneofexamples

that applied to some physical systems, such as air plane, and so on. For the magnetic

suspension system, the characteristic of the magnetic force is extremely complex. Thus,

theanalysisofthisforceistoodiculttoexpresstheexactbehaviorofitbymathematical

models. The uncertainty is the dominant issue inmodels used for control system design,

and it is imp ortant fromthe above discussion for the controller designer totakeaccount

of the uncertainty. Previous researches for the magnetic suspension system have not

represented the uncertaintybetween the physicalsystemand the mo del, and havedealed

with the uncertaintiesas a disturbance.

Recently, although the researches using robust control have represented the uncer-

tainty with respect to this system, yet the uncertain model is a representation using the

unstructured uncertain mo del. Unstructured uncertainties are easy to deal with mathe-

maticallyandnumerically. Thereforetherepresentationoftheuncertaintyisconservative.

Eventhough robustness analysis and synthesisare simplied, the result isconservative.

Ifweallowmathematicalcomplexitytoincrease,especiallywhenitcomestocontroller

synthesis, structured uncertainties should reduce conservatism. For mathematical com-

plexity, the analysis and synthesis using the structured uncertainty is to be easy by the

CAD.

The rst contributionof this thesis isthe robustness analysis and synthesis of amag-

netic suspension system against the structured uncertainty, and the second contribution

Copyright c

1998byAkiyoshiHata

lar value () can be used to analyze the robustness of the system with respect to the

structured uncertainty. We haveseen that it may b e more natural tomodel some of the

uncertainty with real perturbation. While it is possible to simply treat these perturba-

tionsascomplexand proceedwith thecomplexanalysis, theresultscan beexpectedto

beconservative. Theseanalysisandsynthesisarecarriedoutbymixedusing-Analysis

and Synthesis To olbox and MATLAB.

Firstly, a nominal mathematical model with no uncertainty is derived, and the issue

of modeling the magnetic suspension system is discussed. We think of mathematical

models as our means of analyzing physical reality, and uncertainty is the gap that is

left between models and reality, the part of reality that is not accountfor in our model.

Thus it is imp ortant to discuss the mo deling for building the uncertain model. For the

magneticsuspensionsystem,weconsiderwhereuncertaintieswillsurelyariseinthisstep.

They arise at the linearization of the attraction force, the mass of the iron ball, and

the unmo deled dynamics of electromagnetic in the high frequency range. We apply a

mathematical model with the structureduncertainty tothe magnetic suspensionsystem.

Secondly,wedesignrobustcontrollersbythesynthesisagainstthederiveduncertain

model, and check the robustness. Furthermore, we design robust controllers against the

unstructured uncertain model, since we must make a comparative study of the former

andthe latter. On thedesignstep,there isatrade-ob etweenallowedperturbationsand

the desired p erformance. Through muchdesign giving consideration tothese, structured

uncertain mo dels can reduce conservatism more than unstructureduncertain models.

Thirdly,severalexperimentsare carriedoutinordertoverifyaphysicalsystem. First

of all, it is most important for these experiments to evaluate the robustness. In these

experiments,the uncertainty iscaused by moving steady gap between the electromagnet

andtheironball,sinceitisdiculttosupplyanironballwithadisturbanceforcedirectly.

Moving the gap means varying parameters of the linearization of electromagnetic force

and varying the nominalmodel. It corresponds tothe robuststability. Inaddition tothe

movement of steady gap, we give suspected-disturbances by adding a voltage signal to

a control input signal. It corresponds to the robust performance. In order to check the

decreaseofconservatism,weconsiderthebehavioroftheclosed-loopsystemwithdesigned

controllersat the criticalp oint. As a result, we investigatethe decreaseof conservatism.

However,itis necessarytoreduce conservatismmore, since the controllermayb e achieve

better robust stability.

Finally,toreduceconservatismmore,wediscussthatmixed synthesisshouldhavea

advantages morethan complex. Neverthelesssomeexperimentsresultedin our improved

synthesisbeing undesirable.

Wediscuss the uncertainty ofthe magneticsuspension system, andconclude that the

uncertainmodel withthestructureduncertaintyreducesconservatismbetweenaphysical

system andtheoreticalanalysisforthissystem. Addtothese,wewillobtainthefactthat

extremelycalculatedcomplexcanreduceconservatismmorethaninthecaseofmixed.

However,these decreaseof conservatismis a slightdierence. Therefore,representations

ofuncertaintiesasitexplicitlycapturethebehaviorofphysicalsystems,leavessomeroom

for consideration. Wemustalso studysynthesis that deal with real parameters.