University of Alberta

Nonlinear State-Space Control Design for Displacement-Based Real-Time Testing of Structural Systems

by

Seyed Abdol Hadi Moosavi Nanehkaran

A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of

Master of Science

in

Structural Engineering

Department of Civil and Environmental Engineering

©Seyed Abdol Hadi Moosavi Nanehkaran

Spring 2011 Edmonton, Alberta

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otherwise reproduced in any material form whatsoever without the author's prior written permission.

Examining Committee Dr. Oya Mercan, Department of Civil Engineering and Environmental Dr. Alan Lynch, Department of Electrical and Computer Engineering Dr. Samer Adeeb, Department of Civil Engineering and Environmental

Abstract

This study presents the nonlinear design of a state space controller to

controlhydraulicactuatorsunderdisplacementcontrol,specificallyforreal‐

time pseudo‐dynamic testing applications. The proposed control design

processusesthenonlinearstatespacemodelofthedynamicsofthesystem

to be controlled; and utilizes state feedback linearization through a

transformationof the state variables.Comparisonsofnumerical simulation

resultsforlinearstate‐spaceandnonlinearstate‐spacecontrollersaregiven.

Alsorobustnessofthecontroldesignwithrespecttoidentifiedparametersis

investigated.Itisshownthatacontrollerwithimprovedperformancecanbe

designed using nonlinear state space control design techniques, provided

thatarepresentativemodelofthesystemisavailable.

Preface

Thereareseveraldynamictestingmethodsthathavebeenintroducedand

usedtoexamineand/orverifythedynamicperformanceofconventionaland

new structural systems. Some of thesemethods themselves are still under

research in an attempt to make themmore efficient and accurate. Among

these,pseudo‐dynamic(PSD)testingmethodandasitsextensionhybridPSD

testingmethod offer economical and practicalways to assess the dynamic

behaviourofstructuralsystems.Ifexecutedatfastrates(ideallyinreal‐time)

these testingmethods can handle load‐rate dependent structures (such as

theonesthathavedampersinstalledforseismichazardmitigationpurposes)

appropriately.

In aPSD test the equationofmotionof the test structure is solvedby a

direct step‐by step integration algorithm where the inertial and damping

forcecharacteristicsarekeptanalytical.Thesedisplacementsareimposedon

theteststructurebyhydraulicactuators;theresultingrestoringforcesfrom

the deformed structure are measured and fed back to the integration

algorithm for the computation of next step displacements. The method is

called hybrid PSDwhen thewhole structure is split into experimental and

analyticalsubstructurestoavoidfabricatingabigstructureinthelaboratory.

During a hybrid PSD test the command displacements generated by the

integration algorithm are imposed on the experimental and analytical

substructuresandtheresultsfrombotharecombinedandfedback.

Oneof themainchallenges inusingPSDmethod is thesensitivityof the

resultstotheexperimentalerrors.Thisisbecauseoftheclosedloopnature

of themethod; i.e. in each time step of the test procedure, the integration

algorithm uses themeasured information (whichmay be contaminated by

errors) from the previous step to generate the new displacements to be

applied.This fact implies that theerrors ineach stepareaffected from the

errorsofthepreviousstepswhichmaybecumulativelyaddedtogetheruntil

theendof thetest.AsaresultPSDmethodsuffers frompropagationof the

errorwhichatbestcausesaccuracyproblemsoratworstrendersthewhole

test unstable (i.e., the test needs to be aborted as the displacements grow

unboundedly).

From a stability (and also accuracy) point of view, the delay in the

measuredsignals (asopposedto lead,oramplitudeerrors)hasbeenfound

criticalinreal‐timetesting(MercanandRicles2008).Thedelaymainlyarises

due to the time it takes theactuators toreach thecommanddisplacements

issuedbytheintegrationalgorithm.

IngeneralaPIDcontroller(oramodifiedversion)isemployedtocontrol

the actuator displacements which may not provide acceptably accurate

tracking especially when large displacements need to be imposed at fast

ratesunderconsiderableload.Thismaybeduetothefactthatunderthese

typesofconditions,theservo‐hydraulicsystemnonlinearitiesareinvokedor

the test structure presents highly nonlinear behaviour. Hence, when

nonlinearities in the servo‐hydraulic system are invoked, theremight be a

needtousenonlinearstate‐spacecontrollertoaccountforthem.

Due to its matrix form, a state‐space control is preferable to a PID

controller in cases when physically coupled multiple degrees of freedom

needtobecontrolled.Themaineffortoftheresearchpresentedistodevelop

acontrolalgorithmbasedonadvancedcontroltheories(e.g.nonlinearstate‐

space) and assess its performance in comparison with a PID controller.

Simulations are done using a model of the servo‐hydraulic system that

accountsfornonlinearities.Thisstudyisintendedtoleadtheapplicationof

nonlinearstate‐spacecontrolstrategyinPSDtestingmethod.

Acknowledgement

Foremost, IwouldliketoexpressmysinceregratitudetomyadvisorDr.

OyaMercan for thecontinuoussupportofmyM.Sc.studyandresearch, for

herpersistence,interest,andhelp.Herguidancehelpedmeinallthetimeof

researchandwritingofthisthesis.Besidesmyadvisor,Iwouldliketothank

the rest of my thesis committee: Dr. Alan Lynch and Dr. Samer Adeeb for

theirinsightfulcomments,andhardquestions.

Last butnot the least Iwould like to thankmy family for their life‐time

support.

ContentsListofTables...........................................................................................................................1 

ListofFigures.........................................................................................................................2 

Acknowledgement................................................................................................................7 

Preface.......................................................................................................................................4 

1  Introduction.....................................................................................................................5 

1.1  General......................................................................................................................5 

1.2  SeismicTestingMethodsofStructures.......................................................5 

1.3  PSDandHybridPSDtestingmethod...........................................................7 

1.4  Researchgoalsandthesisorganisation....................................................13 

2  ModellingoftheServo‐SystemandIdentificationofsystemparameters

(systemID)..................................................................................................................................15 

2.1  ComponentsofaServo‐HydraulicSystem..............................................15 

2.1.1  FlowControlServo‐Valve......................................................................17 

2.1.2  LinearHydraulicActuator.....................................................................17 

2.1.3  DisplacementTransducer......................................................................17 

2.1.4  ServoController.........................................................................................17 

2.2  DynamicsofaServo‐HydraulicSystem....................................................18 

2.2.1  Servo‐valvedynamics..............................................................................18 

2.2.2  ActuatorChamberPressureDynamics............................................25 

2.2.3  PistonDynamics.........................................................................................26 

2.2.4  LinearApproximationoftheDynamics...........................................26 

2.3  SystemIdentification........................................................................................28 

3  ControlTheory.............................................................................................................30 

General................................................................................................................................30 

3.1  FeedbackControlofDynamicSystems.....................................................30 

3.2  LinearControlDesign(Basics).....................................................................33 

3.2.1  LaplaceTransformandTransferFunction.....................................33 

3.2.2  TheBlockDiagram....................................................................................37 

3.2.3  S‐plane,PolesandZeros.........................................................................38 

3.3  PIDControllerDesign.......................................................................................40 

3.4  State‐SpaceControllerDesign......................................................................41 

3.5  NonlinearStateSpaceControllerDesign.................................................44 

4  ImplementationofControlMethods...................................................................50 

General................................................................................................................................50 

4.1  DynamicModelofaServo‐HydraulicSystem........................................50 

LinearModel...............................................................................................................51 

NonlinearModel........................................................................................................53 

4.2  PIDControllerDesign.......................................................................................55 

4.2.1  ControllerTuning......................................................................................57 

4.3  LinearState‐SpaceControllerDesign........................................................58 

4.3.1  State‐variableformofequations........................................................58 

4.3.2  PolePlacement...........................................................................................60 

4.4  NonlinearState‐SpaceControllerDesign................................................61 

4.4.1  State‐variableformofequations........................................................61 

5  SimulationsResults....................................................................................................70 

General................................................................................................................................70 

5.1  NumericalValuesforServo‐HydraulicSystemandTestStructure

inLinearandNonlinearModels....................................................................................71 

5.2  ComparisonofControllersWithandWithoutSaturationLimits..72 

5.3  ImplementingtheNonlinearState‐SpacecontrollerinaPSDTest

Simulation...............................................................................................................................77 

5.4  RobustnessoftheControlDesign...............................................................85 

5.5  Conclusion.............................................................................................................86 

6  Testsetupdesign.........................................................................................................88 

AppendixA.............................................................................................................................94 

AppendixB  SomeonDifferentialGeometry(Lynch2009).........................96 

1  ChangesofCoordinatesorDiffeomorphisms........................................96 

2  VectorFields.........................................................................................................97 

3  DifferentialGeometryFunctionsUsedintheText...............................97 

Liebrackets..................................................................................................................97 

LieDerivative..............................................................................................................98 

4  Distributions.........................................................................................................99 

AppendixC  MatlabCodesandSimulinkModels...........................................100 

AppendixD  ConferencePaper...............................................................................108 

1

ListofTables

Table2‐1Nonlineardynamicsofaservo‐hydraulicsystemanditslinear

approximation

Table4‐1Linearizeddynamicsofaservo‐hydraulicsystem

Table4‐2Nonlineardynamicsofaservo‐hydraulic

Table5‐1Values for Parameters for Linear and Nonlinear Servo-Hydraulic

System Models

Table5‐2Approximatevaluesfortheelastomericdamperparameters

2

ListofFigures

Figure1‐1Structuresubjecttogroundaccelerationinarealearthquake

Figure1‐2PSDtestingmethod

Figure1‐3HybridPSDtestingmethod

Figure2‐1BlockdiagramofinnerloopinPSDtestmethod

Figure2‐2Crosssectionofatwo‐stageflowcontrolvalve

Figure2‐3Crosssectionofatwo‐stageflowcontrolvalveatoperation

Figure2‐4Turbulentflowthroughanorifice

Figure3‐1 anexampleforaunitstepresponseofasystem

Figure3‐2 Blockdiagramofaservohydraulicmodelincluding

disturbanceandnoise

Figure3‐3 amass‐damper‐springsystem

Figure3‐4 Threeexamplesofelementaryblockdiagrams

Figure3‐5 Timefunctionassociatedwithpointsinthe ‐plane(Franklin

etal.2010)

Figure3‐6 BlockdiagramofaPIDcontroller

Figure4‐1 Simulinkmodelforaservo‐hydraulicsystemwithlinearized

dynamics

3

Figure4‐2 Simulinkmodelforaservo‐hydraulicsystemwithnonlinear

dynamics

Figure4‐3 Simulinkmodelforaservo‐hydraulicsystemwithaPID

controller

Figure4‐4 Approximationoffunction

Figure5‐1 Stepresponsesforasystemwithoutsaturation

Figure5‐2 Stepresponsesforasystemwithsaturation

Figure5‐3 Responsetosinusoidinput

Figure5‐4 loadpressurevariationwithoutservo‐valvesaturation

Figure5‐5 loadpressurevariationwithservo‐valvesaturation

Figure5‐6 Servo‐valveopeningvariationwithoutservo‐valvesaturation

Figure5‐7 Servo‐valveopeningvariationwithservo‐valvesaturation

Figure5‐8 HybridPSDtestmethod(Mercan2007)

Figure5‐9 Simulinkmodelforreal‐timehybridPSDtestingwithMDOF

analyticalsubstructureandanelastomericdamperastheexperimental

substructure

Figure5‐10 Simulinkmodelforaservo‐hydraulicsubsystemwithaPID

controller

Figure5‐11 Simulinkmodelforaservo‐hydraulicsubsystemwithaNL

state‐spacecontroller

Figure5‐12 comparisonbetweencommandandmeasureddisplacement

(a)PIDcontroller(b)NLcontroller

Figure5‐13 comparisonbetweencommandandmeasureddisplacement

4

(a)PIDcontroller(b)NLcontroller

Figure5‐14 comparisonbetweenservo‐valvespoolopening(a)PID

controllerwithavelocityfeedforward(b)NLcontroller

Figure5‐15 effectof10%errorin ontheresponse

Figure6‐1ExperimentalsetupforPSDtest

Figure6‐2Expectedbaserotationatbaseusedindesign

5

1 Introduction

1.1 General

This chapter contains background information about seismic testing

methods.Themotivation andobjectiveof the research andorganizationof

thedissertationarealsodescribed.

1.2 SeismicTestingMethodsofStructures

Oneofthemaingoalsinseismicperformancetestingistoimposeloading

conditions on a test specimen that are representative of those that might

happen during a real earthquake. To achieve this goal, various forms of

earthquakeandstructuraldynamictestingmethodshavebeenthesubjectof

research.

Four experimental laboratory techniques are typically used in seismic

performancetestingofstructures:quasi‐statictesting,shakingtabletesting,

effectiveforcetesting(EFT)andpseudo‐dynamic(PSD)method

Inaquasi‐statictest,apredefinedcyclicdisplacementhistoryisappliedto

the structure or structural component under study and the behaviour is

observedandanalyzed.Thepredefineddisplacementhistory,ifnotselected

6

from some typically used displacement histories, is based on a response

computedfromadynamictimehistoryanalysispriortotesting.Thismethod

is commonly used and economical. However, it is limited in terms of

delivering the true earthquake response. This is because the model with

whichthepre‐testdynamicanalysisisperformedmaynotaccuratelypredict

thebehaviourand in turn, theresultingdisplacement timehistorymaynot

correspondtotherealearthquakeresponse.

Placing a structure on a shaking table and exerting a properly scaled

ground motion may be the most realistic method. In spite of this, due to

payloadrestrictionsofshaketables,shakingtabletestsareimplementedon

small‐scaleteststructures.Thisimpliesthatthegroundaccelerationneedsto

be scaled (compressed) accordingly. As a result the available time for

observing the behaviour will be very little during the test. Generally

speaking,despitethefactthattheshakingtablemayberepresentativeofthe

actual seismicbehaviour, thecombinedeffectsof theneed toconstruct the

complete structure, small scale test specimens, short observation time and

finallythecostlimittheuseofshakingtable.

Effective force testing (EFT) is a real‐time testingmethod. Thismethod

usesaforcecontrolapproachandcanbeemployedinreal‐timeearthquake

simulationoflargescalestructures.Knowingthestructuralmassandground

accelerationhistory,thecompleteforcehistorythatshouldbeappliedtothe

structure is calculated beforehand. Despite the conceptual simplicity, the

7

implementationof thismethodhasbeenobserved tobechallengingdue to

actuator‐structureinteraction(Dimigetal.1999).Toovercomethisproblem,

Zhao (2003) proposed a nonlinear velocity compensation scheme and

verified it through simulations and experimental studies under limited

conditions.

In the late 1970s and early 1980s PSD method was initiated as an

experimentaltechniqueinwhichthedisplacementresponseofstructuretoa

given ground acceleration is numerically calculated and quasi‐statically

imposedon the structure (Takanashi et al. 1975;Okada et al. 1980;Mahin

andWilliams1981;ShingandandMahin1983;MahinandShing1985).This

computed response is based on analytically predefined inertia and viscous

damping aswell as the experimentallymeasured structural resisting force.

Detailsfortheprocedurearegivenbelow.

1.3 PSDandHybridPSDtestingmethod

InaPSDtest, the teststructure is first idealizedasadiscreteparameter

system.Therebyforastructuresubjectedtogroundacceleration(Figure1‐1)

thegoverningequationsofmotioncanbeexpressedasa systemof second

orderordinarydifferentialequationswithrespecttotime.

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11

components and devices are metallic, friction, visco‐elastic, viscous fluid,

tuned mass, tuned liquid, elastomeric dampers and lead rubber bearings

whichare introduced intostructures forvibrationmitigationpurposes.For

assessing the capabilities of structures equipped with these devices, it is

necessarytoperformthePSDtestdynamicallyatratesapproachingtoreal‐

time.

In PSD testing method the results are sensitive to experimental errors.

Thisisbecausethemethodhasaclosedloopalgorithm;i.e.ineachtimestep

of the test procedure, the integration algorithm uses the measured

informationfromthepreviousstep(whichmaybecontaminatedbyerrors)

togeneratethenewdisplacementstobeapplied.Thismeansthattheerrors

in each step are affected from the errors of the previous steps which are

cumulativelyaddedtogetheruntil theendofthetest.HencePSDmethod is

pronetopropagationoftheerror,whichatbestcausesaccuracyproblemsor

atworstrendersthewholetestunstable(i.e.,thetestneedstobeabortedas

thedisplacementsgrowunboundedly).Mercan(2007)showedthatactuator

delay in following the command displacements in experimental setupmay

impairthedynamicstabilityofthetestsetupinareal‐timePSDtest.Thisis

alsotrueforreal‐timehybridPSDmethod.

Oneofthemainsourcesoftimedelayisthetimeittakestheactuatorto

reach the command displacement calculated by the integration algorithm

which cannot be done instantaneously. This is because it takes time to

12

convert energies (electrical to mechanical) in an experimental setup.

Therefore it is critical touse theavailableequipmentefficientlyso that the

corresponding time delay is minimum. This is done by using the control

theorytodesignservo‐hydrauliccontrollers.Thecommonlyusedcontroller

in PSD tests is the well‐known PID (proportional‐Integral‐Derivative)

controller.PIDcontrollersarepopularbecausetheycanbeadjusted(tuned)

tocontrol complexsystemswithout theneed for completeknowledgeover

thedominatingdynamics.

Conducting a fast (ideally real‐time)PSD test requires accurate actuator

control by means of sophisticated servo‐hydraulic strategies and reliable

computationschemethroughefficientintegrationalgorithms.

Thedynamicsofanelectrohydraulicservosystemishighlynonlinearand

involvessignandsquarerootfunctions.Howeverinmostindustrialcontexts

linearcontroltheoryisused.Althoughlinearizationaboutanoperatingpoint

decreases design effort, it degrades the performance at regions off the

operating point. There have been several works on controlling

electrohydraulicservosystemsusingadvancedcontrolmethods.

Lim(2002)applied linearizationandpoleplacement.YanadaandFuruta

(2007) combined linear theory with an adaptive approach. Kwon et al.

(2007) applied full‐state feedback linearization. Seo et al. (2007) used

feedback linearization to design controllers for displacement, velocity and

differentialpressurecontrolofarotationalhydraulicdriveandAyalewand

13

Jablokow(2007)usedpartial‐inputfeedbacklinearizationforthecontrolof

electrohydraulic servo systems. Mintsa et al. (2009) used feedback

linearization to design an adaptive control for electrohydraulic position

servo system with the objective of enhancing robustness with respect to

variationsofsupplypressure.

1.4 Researchgoalsandthesisorganisation

Inordertoimprovethetrackingcapability,andinturntheaccuracyofthe

overall real‐time PSD test results, this thesis investigates the design and

implementationofadvancedcontrolalgorithms.

As stated before, currently the majority of the controllers used in PSD

testing arePIDbased controllers. They are tuned according to a linearized

modelofthesystemdynamics(forsingleinputsingleoutputsystems),orby

ad‐hoc tuning; where the accurate window of operation is limited to the

linearrangeofthesystemdynamics.Inthecaseofamulti‐degreeoffreedom

test structure, state‐space design offers considerable reduction in control

designeffort(Mercanetal.2006).Theabovementionedmethods,astheyare

designedbasedonlinearmodelsofthesystem,maynotprovideacceptably

accurate tracking when the servo‐hydraulic system nonlinearities are

invokedortheteststructurepresentshighlynonlinearbehaviour.

Inthisstudythedesignofanonlinearstate‐spacecontrollerforrealtime

pseudo‐dynamic testing of structural systems is presented based on

advancedcontroltheoriesusinganonlinearmodelofthesystem.

14

After the introductioninthischapter,Chapter2considersthegoverning

dynamicsandequationsofaservo‐hydraulicsystem.Chapter3beginswith

basicsofcontrol theoryandafterwards linearandnonlinearcontroldesign

methodsareintroducedanddiscussed.Chapter4appliesthecontroldesign

methods introduced in Chapter 3 on the dynamics discussed in Chapter 2.

Finally,Chapter5 illustrates the implementationof thecontrolmethodsby

computersimulationandcomparesthebehaviourofdifferentcontrollers.

Chapter6discussesthedesignofasingledegreeoffreedomtestsetupto

investigate different aspects of a PSD test method (e.g. applying different

controllers).Aconferencepapertitled“ModificationofIntegrationAlgorithm

toAccountforLoadDiscontinuityinPseudo‐DynamicTesting”thatwasdone

alongwiththisresearchispresentedasanappendix.

15

2 Modelling of the Servo‐System and Identification of

systemparameters(systemID)

Toworkoutanewcontrolapproachwithanimprovedperformancefora

servo‐hydraulic system, the behaviour of the system has to be well

understoodthroughsimulationofrepresentativemodels.Thesemodelsare

based on the important dynamics that govern the behaviour and include

physicalparameterssomeofwhichmayalreadybeknown(e.g.pistonarea)

while others need to be identified through system identification. While

decidingforthelevelofaccuracy(complexity)ofthemodel,oneneedsalso

toconsideriftheparametersthatappearinthemodelareeasilyobtainable

throughsystemidentificationornot. Inthenextsectiondifferentpartsofa

servohydraulicsystemandthegoverningdynamicsareelaborated.

2.1 ComponentsofaServo‐HydraulicSystem

Apositioncontrolledservo‐hydraulicsystemused inaPSDtesttypically

consists of a hydraulic power supply, a flow control servo‐valve, a linear

actuator, a displacement transducer, and an electronic servo‐controller.

Figure2‐1showsablockdiagramthatrepresentshowtheabovementioned

parts are interconnected. The controller compares the command

displacement with the measured displacement coming from the

displacementtransducer(e.g.anLVDT)todeterminethepositionerrorand

16

thensendsoutacommandsignaltodrivetheflowcontrolservo‐valve.Infact

the command signal introduces a spool displacement in the servo‐valve to

adjust the flow of pressurized oil from the hydraulic power supply to the

linearactuatorchambersinordertomovetheactuatorpistontothedesired

position.

Figure2‐1BlockdiagramofinnerloopinPSDtestmethod

ThefollowinggivesaconciseexplanationofthepartsshowninFigure2‐1.

HydraulicPowersupply

A hydraulic power supply provides the pressurized fluid (oil) for the

hydraulic system. The level of oil pressure in the power supply is selected

considering several factors. Low pressure systems have less leakage but

physicallylargercomponentsareneededtoprovideaspecifiedforce.Onthe

otherhand,highpressuresystemsexperiencemoreleakagebuthavebetter

ServoController

LinearActuator

LoadFlowControlServo‐Valve

DisplacementTransducer

HydraulicPowerSupply

CommandDisplacement

MeasuredDisplacement

AppliedDisplacement

17

dynamicperformanceandhavesmaller(lighter)components. Inmanyhigh

performancesystems3000psi(210bar)isselectedforthesystempressure.

2.1.1 FlowControlServo‐Valve

An electro hydraulic flow control servo valve is a servo valve which is

designedtoproducehydraulicflowoutputproportionaltoelectricalcurrent

input.Themechanismanddynamicsarediscussedlater.

2.1.2 LinearHydraulicActuator

A hydraulic actuator converts hydraulic energy to mechanical force or

motion.Theyareimplementedwherelargeactuationforcesandfastmotion

arerequired.Governingdynamicsaregivenlater.

2.1.3 DisplacementTransducer

They generally come built‐in with actuators and are often attached

directlytothepistonrod.Therearevarioustypesoffeedbacktransducersin

useincludinginductivelinearvariabledifferentialtransformer(LVDT).Itisa

common practice to include external displacement transducers in the test

setuptocheckthemeasurementsoftheinternaltransducersandtoexclude

theactuatorsupportmotion.

2.1.4 ServoController

A controller continuously compares the command displacement against

the actuator position that is measured by a displacement transducer. The

resultofthiscomparisonisdisplacementerrorwhichisthenmanipulatedby

18

acontrol law inorder togenerateandsendacommandsignal to theservo

valve.

2.2 DynamicsofaServo‐HydraulicSystem

2.2.1 Servo‐valvedynamics

Servo‐valvesareusedinservo‐hydraulicsystemstoconverttheelectrical

command signal, coming from the controller, to a spool displacement. This

displacement alongwith the differential pressure between the servo‐valve

ports results in oil flow through valve control ports into and out of the

actuatorchamberenablingthemotionofthehydraulicpiston.

Amongseveral typesofservo‐valvesarethe flowcontrolservo‐valves in

which the control flow at constant load pressure is proportional to the

electricalinputcurrent(Thayer1958,revisedin1965).

Figure 2‐2 shows a two stage flow control servo valve. It is called two‐

stage as it has two portions containing a hydraulic amplifier. A hydraulic

amplifierisafluidvalvingdevicewhichactsasapoweramplifier,suchasa

slidingspoolandanozzleflapperandwillbeelaboratedinthischapter.

to

d

re

d

In a two‐

orque moto

eflection o

estrictsflui

isplacethe

arm(torqu

spool

Figure2‐2

stage servo

or coils crea

of armature

dflowthro

spool.

matureuemotor)

acch

topowersupply

Crosssection

o‐valve the

ates a mag

e‐flapper a

oughoneof

nozzle

to1stctuatorhamber

nofatwo‐stag

electrical c

netic force.

ssembly w

thetwono

toreturntank

geflowcontro

command s

. This magn

where the

ozzlesandr

a ppe

m

to2ndactuatorchamber

lvalve

ignal applie

netic force

resulting d

redirectsth

pairofermanent

magnets

flapper

19

ed to the

causes a

deflection

heflowto

co

o

sp

fl

b

a

st

le

th

Figu

Spool dis

ontrol port

ther contro

pringshow

lapper asse

ecomesequ

ssemblymo

tate of equ

evel(Moog2

In summa

he input c

ure2‐3Cross

placement

s and at th

ol port. Th

ninFigure

embly. Spo

ual to the t

ovesbackto

ilibrium un

2010).

ary, aspoin

current. Als

sectionofatw

opens the

he same tim

e displaced

2‐3thatcr

ol moveme

orque from

otheneutr

ntil the elec

ntedoutbef

so with co

wo‐stageflow

supply pre

me opens th

d spool app

reatesarest

ent continu

m themagne

alposition

ctrical comm

fore, the sp

onstant pre

a

controlvalve

essure port

he return ta

plies a forc

toringtorqu

ues until t

etic forcea

andthespo

mand signa

poolpositio

essure drop

deflectedarmature‐flap

assembly

deflectedcantileversp

e atoperation

t ( ) to on

ank port (

ce to the c

ueonthea

the restori

nd that isw

oolisheldo

al changes t

n ispropor

p across t

pper

dring

20

ne of the

) to the

cantilever

armature‐

ing force

when the

openina

to a new

rtional to

he valve

21

(constant loadpressure)theloadflowisproportionaltothespoolposition.

The governing dynamics of the servo valvewill be discussed in two parts;

valve spool dynamics which includes the relationship between the input

currentandthespooldisplacementandvalveflowdynamicswhichexplains

howthespooldisplacementrelatestotheflowfromthevalvetotheactuator

chambers.

2.2.1.1 ValveSpoolDynamics

Servo‐valves are complicated devices. Experience has shown that their

nonlinearandnon‐idealcharacteristicsmakeithardtotheoreticallyanalyze

servo‐valvedynamics in systemsdesign. Instead, it ismore convenientbut

alsoacceptablyaccuratetoapproximateservo‐valvedynamicswithsuitable

empirical transfer functions by using measured servo‐valve response

(Thayer 1958, revised in 1965). Depending on the frequency range of

interest the servo‐valve dynamics can be represented by a first order

transferfunction.

1

Eq.2‐1

Where , , and are servo‐valve spool opening (see Figure 2‐3),

differentialcurrentinputtoservo‐valve,servo‐valvestaticflowgainatzero

load pressure drop and apparent servo‐valve time constant. It should be

noted thaton the lefthand sideofEq. 2‐1 theLaplace transformsof spool

openingandinputcurrentareused.

2

se

T

fl

fl

n

2.2.1.2 Valv

Ascanbe

ervo‐valvea

Theseflows

lowthrough

2

, , a

lowingliqui

Rewriting

owthat

veFlowDyn

seeninthe

andtheactu

Figu

areclassifie

hanorifice(

2

and

idanddiffer

g the above

isthesame

2

2

namics

Figure2‐3,

uatorports.

ure2‐4Turbul

edasturbul

(seeFigure

aredischa

rentialpres

e turbulent

forallorifi

,fourflows

lentflowthrou

lentflows.E

2‐4)

argecoeffici

surebetwe

orifice flow

cesgives

canbereco

ughanorifice

Eq.2‐2repr

ient,orifice

enpoint a

w for Figur

ognizedbet

e

resentsthet

earea,dens

and respe

re 2‐3 assu

22

tweenthe

turbulent

Eq.2‐2

ityofthe

ectively.

uming for

Eq.2‐3

Eq.2‐4

23

2

Eq.2‐5

2

Eq.2‐6

Intheaboveequations, , istheorificeareaassociatedwith

flow , , and is thesupplypressure.Also and are the

pressuresateachoneoftheactuatorchambers.

The return pressure ( ) is assumed to be zero as it is usually much

smaller than the other pressures involved. If the return pressure is not

negligible, thesupplypressure in theaboveexpressionscanbe interpreted

assupplypressureminusreturnpressure.

Theareasoftheorificesarefunctionsofthespoolopening .Becausethe

valveorificesarematched,

and Eq.2‐7

Andbecausetheyaresymmetrical,

and Eq.2‐8

Itcanbeshownthatwhentheorificeareasarematchedandsymmetrical

and (Merritt1967) Eq.2‐9

24

CombiningEq.2‐3,Eq.2‐4,Eq.2‐7andEq.2‐9pressuresattwochambers

canberelatedtosupplypressureby,

Eq.2‐10

Bydefinitiontheloadpressureisthepressuredifferencebetweenthetwo

actuatorchambersandisexpressedas

Eq.2‐11

UsingEq.2‐10andEq.2‐11, and maybewrittenas

2

Eq.2‐12

2 Eq.2‐13

Load flow, which is the flow from the valve to one of the actuator

chambers,canbeexpressedas

Eq.2‐14

Finallyusingthederivedequationsforamatchedandsymmetricalvalve,

theloadflowmaybewrittenas

1 1

Eq.2‐15

25

For an ideal critical center valvewithmatched and symmetrical orifices

the leakage flows ( and when the spool displacement is positive) are

zerobecausethevalvegeometry isassumedideal.Ontheotherhandwhen

the spool displacement is negative and will be the leakage flows.

Considering the fact that the orifice areas are linear functions of the spool

openingastheproductof thespoolopeningandfullperipheryofthespool

( ),loadflowcanbeexpressedas(Merritt1967)

Eq.2‐16

where isdefinedas

1

Eq.2‐17

Eq 2‐16 is not differentiable due to sign function. Sign functionmay be

approximatedbysomesmoothfunctionandthenusedinthecontroldesign

(Chapter4)

2.2.2 ActuatorChamberPressureDynamics

The oil flow between valve and actuator chambers causes the actuators

piston tomove. Using conservation ofmass principle on both sides of the

actuatorchambers,theactuatorpressuredynamicscanbeexpressedas

4

Eq.2‐18

26

Inthisequation istheactuatorpistoncross‐sectionalarea; ispiston

velocityorthetimederivativeofpistondisplacement( ); istheleakage

coefficientofpiston; and areloadpressureasdefinedinEq.2‐11and

rateofloadpressurerespectively; isactuatorchambervolumeand isoil

modulus.

2.2.3 PistonDynamics

Writingtheforceequationofmotionandconsideringastaticfrictionand

theexternalforcefromatestspecimengives

Eq.2‐19

, and are piston mass, viscous damping coefficient of actuator

piston and static friction respectively and represents the effect of

stiffness,dampingandinertialforcesfromatestspecimen.

2.2.4 LinearApproximationoftheDynamics

Forcontrollerdesignpurposesalinearized,simplifiedmodelisnecessary

especiallywhenthecontrollerdesignwillbebasedonlinearcontroltheory.

Note that the valve spool dynamics (Eq. 2‐1) and actuator chamber

pressure dynamics (Eq. 2‐18) are already linear. Using Taylor series

expansion, Eq. 2‐15 (that represents flow‐pressure relationship) can be

linearizedaboutanoperatingpoint( 0)whileassumingzero

leakageflowandidealgeometry(Merritt1967)tobe

27

Eq.2‐20

and are called the flow gain coefficient and the flow pressure

coefficient respectively. And for the pistondynamics, in case the friction is

negligibleEq.2‐19canberewrittenas

Eq.2‐21

Table 2‐1 gives a summary of both nonlinear and linear equations

discussedsofar.

Table2‐1Nonlineardynamicsofaservo‐hydraulicsystemanditslinearapproximation

Dynamics Nonlinear LinearValvespool

1

(Eq.2‐1)

1

(Eq.2‐1)

Valveflow  

(Eq.2‐16) 

 

(Eq.2‐20) 

Actuator

chamber

pressure

4

 

(Eq.2‐18) 

4

 

(Eq.2‐18) 

Actuator

motion

 

(Eq.2‐19) 

(Eq.2‐21) 

 

28

2.3 SystemIdentification

It is a good idea to identify the system starting from the simplified

(linearized) model and finalize it by adding relevant nonlinearities and

additionaldynamicsasrequiredbythemeasuredresponse.

Grey‐boxmodeling option of the system ID toolbox ofMATLAB is ideal

when the model to be identified is known (e.g. derived from the first

principles) and the numerical values of the parameters that appear in this

modelneedtobeestimatedfrommeasureddata.

In identifying the servo‐hydraulic systemmentioned above the transfer

functionrelatingtheinputcurrentandthespoolopeningisknowntobe

1

Eq.2‐22

For a free standing actuator that have equal to zero, a transfer

functioncanbeestablishedusingEq.2‐18,Eq.2‐20,andEq.2‐21which in

turnwheniscombinedwithEq.2‐22willgiveadirecttransferfunctionfrom

inputcurrenttoactuatorpistondisplacement.

4

4 4 Eq.2‐23

where and 1 ⁄ .

MATLAB identification toolbox has a general built‐in systemmodel that

canbeadjustedtohavefourpolesandnozeroesas

29

1 2 1

Eq.2‐24

Theterm isconsideredtotakecareofanytimedelaythatmayexist

inthesystem.

Theidentificationprocedurebasicallystartswithexcitingthesystemwith

somepredefinedinputsandloggingtheresponse.Thereisnorestrictionfor

selecting the inputs but usually step and sinusoid inputs with different

frequencies are used. Then the logged data is analyzed by MATLAB

identificationtoolboxandvaluesfortransferfunctionparametersarefound.

ComparingEq.2‐23andEq.2‐24andconsideringthefactthat

hasa

verysmallvalueandcanbeneglected(Zhangetal.2005),itcanbeshown

4

Eq.2‐25

4

Eq.2‐26

1 Eq.2‐27

Eq.2‐28

30

3 ControlTheory

General

In a servo‐hydraulic system the controller calculates the displacement

errorandusesitasaninputtoacontrollaw.Theoutputofthecontrollawis

the command signal to the servo valve. In this chapter some principles of

feedback control theory is given. Then classic control design, linear state‐

space control design and nonlinear state‐space control design are

introduced.

3.1 FeedbackControlofDynamicSystems

In thecontextof thisstudyacontroller isdesignedtogivethe following

characteristics to the system; (1) the ability to follow command

displacements(tracking);(2)theabilitytomaintainthesystemstability;(3)

the ability to reduce the sensitivity of the system to external disturbances

(disturbancerejection).Somecriterianeedtobedefined inordertohavea

basisof comparisonbetween theperformancesofdifferent controllers.For

this purpose specific input signals like step functions and sinusoidal

functionsareused.Figure3‐1showsatypicalresponseofasystemtoaunit

st

a

re

e

ch

p

d

th

a

ca

w

tepinput.A

n input it

eachedafte

For the ca

ssential fa

haracteristi

erformance

elaytaking

he resisting

mplitudeer

auses inacc

willoccur.

Aphysicalsy

cannot inst

rexhibiting

Figure3‐1

ase of aPS

ctor for th

ics and ste

ecriteria.M

thesystem

g force) has

rror.Phase

curacyand

transienrespons

ystemstore

tantly follo

gatransient

anexample

D test, othe

he system

eady‐state

Mercan(200

mtoapplyth

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errorintrod

if it goesbe

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w it. Instea

tresponse.

eforaunitste

er than stab

to be op

error are

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hecommand

re of a detr

ducesaneq

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hereforewh

ad a steady

epresponseo

bility requir

perational,

also consid

dthatphase

ddisplacem

rimental eff

quivalentne

itical value,

adystatesponse

steady stateerror

henitissub

y state con

fasystem

rement,wh

transient

dered as c

eerror(i.e.

mentandto

fect compar

egativedam

dynamic in

e

31

bjectedto

ndition is

hich is an

response

controller

the time

measure

red to an

mpingthat

nstability

32

Figure 3‐2 gives a block diagram of the system including the possible

disturbancesandnoises.Itshouldbenoticedthatexternaldisturbancesmay

alsobepresentinindividualcomponents(e.g.,theservovalve)butinorder

tobeconciseinFigure3‐2,onlytheresultantofthedisturbancesisshown.

Figure3‐2 Blockdiagramofaservohydraulicmodelincludingdisturbanceandnoise

Thedynamicsofasystemcanbedefinedbyaset(system)ofdifferential

equations which are obtained from principles of physics. For a quick

approximateanalysisandforcontrollerdesignpurposesusinglinearcontrol

theory, linear approximation of the differential equations (whenever they

entailnonlinearterms) isused.Ontheotherhand, torepresent thesystem

dynamics more realistically computer simulations including system

nonlinearities can be performed. Moreover, in the event that the linear

controllersthatarebasedonlinearmodelsarenotefficient,nonlineardesign

offeedbackcontrolmaybeused.

ServoController

LinearActuator

LoadFlowControlServo‐Valve

DisplacementTransducer

HydraulicPowerSupply

CommandDisplacement

MeasuredDisplacement

AppliedDisplacement

Disturbance

Noise

33

3.2 LinearControlDesign(Basics)

Oneoftheattributesofalineartime‐invariantsystemisthatitobeysthe

principle of superposition. This principle states that if the system has an

input that can be expressed as a sum of signals then the response of the

system can be expressed as the sum of the individual responses to each

signal. In control engineering the dynamics of the system are typically

studied using root locus (in s‐plane), frequency response or state‐space

methods. The first two methods are based on Laplace transform and are

mainly used in classical control analysis or design. State‐space based

methodsareusedinmoderncontroldesign.

3.2.1 LaplaceTransformandTransferFunction

The Laplace transform is the mathematical tool that transforms

differentialequationsintoanalgebraicformwhichareeasiertomanipulate.

Compared to the Fourier transformwhich is informative about the steady

state response, the Laplace transform yields complete response

characteristics (both transient and steady state response) (Franklin et al.

2010).

Theunilateral (onesided)Laplace transform fora timedomain function

like is

Eq.3‐1

a

fu

in

1

p

ex

2

A

tr

Where i

nd is th

unctionsin

nAppendix

.

The inte

erforming

xemplarwh

).

Theequat

Assumingze

ransformof

is a comple

he imagina

timedomai

A.therefor

rpretation

Laplace t

hereamass

Fig

tionofmoti

eroinitialco

fbothsides

exvariable

ry part of

in,theirLap

ethereisn

of the d

transformat

s‐damper‐s

gure3‐3 am

onforthea

onditions(

ofEq.3‐2

of the form

the compl

placetransfo

oneedtop

dynamic be

tion) is pr

pringsyste

mass‐damper

aboveoscilla

0 0,

m

lex variabl

formsareta

erformthe

ehaviour in

rovided us

em isconsid

r‐springsystem

atorintime

0 0)an

. is the

e. For ele

abulatedas

integration

n s‐domai

sing the f

dered(seeF

m

edomainis

dtakingthe

34

realpart

ementary

provided

ninEq.3‐

n (upon

following

Figure5‐

Eq.3‐2

eLaplace

35

⇒

Eq.3‐3

FromEq.3‐3 canbesolvedas

Eq.3‐4

If,forexample, isagivenforcetimefunctionlikeaunit‐stepfunction

(1 ),whichisdefinedas

10, 01, 0.

Eq.3‐5

AgainlookingatthetableinAppendixA,itisknownthat

11

Eq.3‐6

Therefore the response of the system to a unit step input in Laplace

domain

1

Eq.3‐7

Togettheresponseintimedomain,inverseLaplacetransformcanbeused

anditisdefinedas

36

12

Eq.3‐8

Where isachosenvaluetotherightofallthesingularitiesof inthe

‐plane. ‐planeistheplaneonwhichthe ‐typevariables( )canbe

shown.In fact,Eq.3‐8 isseldomused. Instead,complexLaplacetransforms

arebrokendownintosimplerexpressionsthatarelistedinthetablesalong

with their corresponding time responses (AppendixA). For example, if the

numerical values of the physical properties are such that Eq. 3‐7 can be

writtenas

1 3 2

Eq.3‐9

Using the partial‐fraction expansion technique Eq. 3‐10 can be broken

downintosimplerexpressions

12 1

1

122

Eq.3‐10

Using thematching time functions from Appendix A, the corresponding

time functionof eachcomponent canbe foundand the total time response

for willbethesumofthesetimefunctions.Hence

12

12

0Eq.3‐11

37

3.2.2 TheBlockDiagram

AtransferfunctionisdefinedastheratiooftheLaplacetransformofthe

output to the Laplace transform of the input. Inmany control systems the

dynamicequationscanbewrittenso that theircomponentsdonot interact

exceptbyhavingtheinputofonetransferfunctionastheoutputofanother

one. The dynamics of a system having multiple components are easier to

representinablockdiagramformwhereeachblockrepresentsthetransfer

function of one component (e.g. G1(s), G2(s) in Figure 3‐4) and the input‐

outputrelationshipsbetweentheblocksareshownbylinesandarrows.The

resulting transfer function for thewhole canbeobtainedbyblockdiagram

algebra. This method is often easier and more informative than algebraic

manipulation. Some examples for block diagrams and their equivalent

algebraicinput‐outputrelationshipsareshowninFigure3‐4.

Figure3‐4 Threeexamplesofelementaryblockdiagrams

R(s) Y(s)

R(s) Y(s)+

+

R(s)+ Y(s)

1

(a) (b)

(c)

38

Figure3‐4(c) illustratesanegative feedbackarrangement that isusedto

compare the output of a system with the command input to perform a

tracking task. This is also referred to as closed‐loop control as opposed to

open‐loop control. Open‐loop control is generally simpler and does not

introducestabilityproblems.Althoughfeedbackcontrolismorecomplicated

andmayhavestability issues, ithas thepotential toachieveamuchbetter

performance.Moreover, if the process is naturally (in open‐loop) unstable,

feedbackcontrol is theonlypossibility toattainastablesystemthatmeets

anyperformancecriteria(Franklinetal.2010).

3.2.3 S‐plane,PolesandZeros

Inthedesignandanalysisofacontrolsystem,thetransferfunctionofthe

system gives useful information about the system characteristics including

itsfrequencyresponse.

Therootsofthenumeratorofthesystemtransferfunctionarecalledzeros

of the systemwhich correspond to the locations in the ‐planewhere the

transferfunctioniszero.Therootsofthedenominatorofthesystemtransfer

function are called the poles of the system. Apparently, the poles are the

locations in the ‐plane where the magnitude of the transfer function

becomesinfinite.Forexample,inthepreviousexamplethetransferfunction

hadnozeroandthreepoles.Thezerosandpolesmaybecomplexquantities

andtheirlocationcanbedisplayedinacomplexplane,whichisreferredtoas

the ‐plane.

n

re

d

g

a

p

w

ex

m

The poles

atural or u

esponseisd

omain asso

eneral rule

ssociatedw

olescloser

withpositive

xponential

Figure3‐5

A feedba

modifyingth

s of the sys

unforced be

determined

ociated wit

poles farth

withnatural

to the imag

erealvalue

functionsw

Timefunct

ack control

hesystem’s

STABLE

stem determ

ehaviour of

dbythepole

th poles at

her to the le

lsignals tha

ginaryaxis.

s(inrighth

whichareun

tionassociate

ller improv

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mine its sta

f the syste

es.Figure3

different l

eft in the

atdecay fas

.Alsoas in

half‐plane,i.

nstable.

edwithpoints

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ons.

ability prop

m. Basicall

‐5showsth

locations in

‐plane (LHP

sterthanth

dicated in t

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dynamic re

UNSTABL

perties and

ly the shap

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esponse by

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39

also the

pe of the

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ane. As a

3‐5)are

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al.2010).

y mainly

40

3.3 PIDControllerDesign

ThefactthatPIDcontrollersareabletocontrolcomplexsystemswithoutthe

need forprecise identificationof theirdynamicshasmade thempopular in

control applications. A PID controller controls the dynamic error based on

themagnitude, history and rate of the calculated error. The corresponding

transferfunctionofaPIDcontrollerhasthreeterms.

Eq.3‐12

Intheaboveequation istheerrorsignaland iscontrolleroutput.

, and arethegains(coefficients)correspondingtoeachoftheterms.

InfactdesigningaPIDcontrolleristodecideonacombinationofthesethree

gains to get the desired system behaviour. Figure 3‐6 shows the block

diagramofaPIDcontroller.

Figure3‐6 BlockdiagramofaPIDcontroller

Therefore the transfer function for a PID controller is as below, which

introducesanextrapoletothesystem.

E(s) + U(s)

+

+

41

Eq.3‐13

APIDcontrollermaybedesignedusingtheroot locusmethod, frequency

responsemethodorcanbetunedexperimentallyviaanin‐situapproachsuch

asusingZieglerNicholstuningrules(Franklinetal.2010).

Rootlocusmethodstudiestheeffectofanyoneparameterthatentersthe

equation linearly to modify the location of system’s poles in the ‐plane.

TypicallythatoneparameterischosenfromoneofthePIDgains(wherethe

othersareexpressedinrelationtothatone).

Theuseoffrequencyresponsemethodsismorecommoninthedesignof

feedback control systems for industrial applications. One of the reasons is

that with frequency response method it is easy to use experimental

information fordesignpurposes. Frequency responsemethodsutilizeBode

plotsthatportraythesteadystateresponseofasystemtosinusoidalinput.

Whenarepresentativeanalyticalmodelforthesystemisnotavailable,a

PIDcontrollercanstillbeusedbyusingexperimentaltuningapproacheslike

ZieglerNichols(Franklinetal.2010).

3.4 State‐SpaceControllerDesign

Studying thesystemdynamic instatespace formhas the followingmain

advantages(Franklinetal.2010):

42

Having the differential equations in state‐variable form gives a

compact standard form where multi‐input multi‐output systems

canbestudiedeasilyeveninthepresenceofnonlinearities.

Incontrasttotransferfunctionwhichrelatesonlytheinputtothe

outputanddoesnotshowtheinternalbehaviourofthesystem,the

state form connects the internal variables to external inputs and

outputs. This keeps the internal information at hand, which at

timesisimportant.

The state‐variable representation of a continuous linear time‐invariant

open‐loopdynamicsystemcanbeexpressedas

Eq.3‐14

where the 1 column vector is called the state (vector) of the

system, is the 1 inputvector, is the 1outputvector, is

the system matrix, is the input matrix, is the the

output matrix, is the direct transmittance matrix; and, as can be

understood fromabove, , , and are thedimensionsof state, input and

outputvectors,respectively.

As a result of the freedom in choosing the state vector, state space

representationofasystemisnotunique.However, foragivensystemthey

areequivalentintermsoftheinput‐outputrelationship.

43

The eigenvalues ofmatrix are the roots of the characteristic equation

(i.e., the roots of the denominator polynomial (poles) of the open‐loop

transferfunctionforasingle‐inputsingle‐outputsystem).

In state‐spacemethodmoving the closed‐loop pole locations to desired

locationsisaccomplishedthroughafullstatefeedback.Forthestate‐variable

systemdescribedabove,withfullstatefeedback,theinputvectorbecomes

Eq.3‐15

Where is the 1 reference input vector and is an gain

matrix. For example if the reference input is zero (sucha controller is

calledaregulator)fortheclosedloopsystemdynamics becomes

Eq.3‐16

Inthiscase,theeigenvaluesofmatrix (rootsof 0)are

the closed‐loop poles. It can be shown that the closed‐loop poles of the

systemcanbeplacedanywhereinthecomplexplaneaslongasallthestates

arecontrollable(How2007).Thisiscalledpole‐placementmethod.Thereis

a function called place in the commercially available software package

MATLAB which calculates matrix to have the closed loop poles of the

systemmovetothedesiredlocations.

Inthecaseoftrackingareferenceinput( 0),thenonzeroreference

input needs to be introducedproperly for a goodperformance in tracking.

44

This is doneby scaling the reference input and then combining itwith full

statefeedbacktogettheproperinputvector(How2007).

Eq.3‐17

where

Eq.3‐18

Eq.3‐17ensuresthat forastepinputtherewillbenosteadystateerror

aftertransientbehaviour.

Onewaytoselectthelocationofclosedlooppolesistoconsidertreating

thesystemasasecondordersystembyselectingapairofdominantpoles,

with the remaining poles having a real part corresponding to sufficiently

dampedmodes.Thiswillresultinasystemwhichissimilartoasecondorder

system(How2007).

3.5 NonlinearStateSpaceControllerDesign

Inalloftheaforementionedmethodsthedynamicsystemtobecontrolled

was assumed to be linear. Most dynamic systems have some sort of

nonlinearity. In some cases the nonlinearities may safely be ignored or

linearized about anoperatingpoint.However there are systemswhere the

nonlinearities cannot be ignored or the range of operation is beyond the

limitswherelinearapproximationsarevalid.Inordertodesignacontroller

forsuchsystemsnonlineartechniquesneedtobeused.

45

Instead of using linear approximations of the dynamics as done in

Jacobian linearization, feedback linearization is a nonlinear control design

approach which algebraically transforms nonlinear system dynamics into

linear ones and permits the subsequent application of linear control

techniques.

Thesenonlineartechniquesmakeuseofdifferentialgeometryconcepts.In

thefollowingsectionswhereveranewdifferentialgeometricconceptisused

it is defined briefly, and also a more detailed summary of differential

geometryisprovidedinAppendixB.

A single inputnonlinear system in theneighbourhoodof an equilibrium

point, , corresponding to 0 i.e. . can be expressed in state‐

variableformas

Eq.3‐19

orsimply

Eq.3‐20

InEq.3‐20 and areassumedtobesmoothvectorfieldsand .

Avectorfieldisamapthatassignseach avector ofthesamesize.So

forexample,if isastatevectorofsize 1,

46

⋮ Eq.3‐21

Aswill be elaborated later there arenecessary and sufficient conditions

underwhichthesystemdefinedbyequationEq.3‐20istransformableintoa

linear controllable system by nonlinear feedback and coordinate

transformation. This problem is called feedback linearization. Feedback

linearization is viewed as a generalization of pole placement for linear

systems.

The nonlinear single input system in Eq. 3‐20 is said to be locally state

feedbacklinearizableifitislocallyfeedbackequivalenttoalinearsystemin

Brunovskycontrollerform(MarinoandTomei1995)whichis

Eq.3‐22

where the state vector and input in the new coordinate are and

respectivelyand

0 1 0 ⋯ 00 0 1 ⋯ 0⋮ ⋮ ⋮ ⋱ ⋮0 0 0 ⋯ 10 0 0 ⋯ 0

Eq.3‐23

47

00⋮01

Eq.3‐24

where the state vector and input in the new coordinate are and

respectively.

InordertobeabletocastasetofequationsinBrunovskycontrollerform,

the theorem of feedback linearization needs to be satisfied. This theorem

indicatesthatthesingleinputsysteminEq.3‐20with statesislocallystate

feedbacklinearizableifandonlyifinaneighbourhoodoforigin:

(i) thedistributionspan ,… , isofrank ,and

(ii) the distribution span ,… , is involutive and of

constantrank 1.

Expressions like are iterative forms of Lie bracket which is a

functionindifferentialgeometryactingontwovectorfieldslike and .Lie

bracket function is illustrated in Appendix B. Also an exact mathematical

explanationofdistributionanditsrankisgiveninAppendixB.Howeverasa

simpleexplanation,condition(i)requiresthatthespacegeneratedfromthe

indicated vector fields has a dimension of whichmeans the vector fields

have to be linearly independent. Condition (ii) requires that the space

generated from the indicated vector fields has a dimension of 1.

Moreoveradistributionis involutiveif,givenanytwovectorfields and

48

belonging to that distribution, their Lie bracket, , , also belongs to the

distribution.

Theabovetwoconditionsguaranteetheexistenceofafunction : →

such that in the neighbourhood of origin, the following conditions are

satisfied.

⟨ , ⟩ 0

⟨ , ⟩ 0, 0 2

Eq.3‐25

Intheaboveexpressions iscalledgradientof andisdefinedas

, , … , Eq.3‐26

andinnerproductisdefinedas

⟨ , ⟩ . Eq.3‐27

HavingsolvedtheconditionsinEq.3‐25for ,thetransformationfrom

to willbe

, , … , , , … , Eq.3‐28

where the expression is called the Lie derivative of function

alongthevectorfield .ThisoperatorisalsodefinedinAppendixB

49

HencethedynamicsysteminEq.3‐20i.e.

Eq.3‐29

transformsinto

, 1 1

Eq.3‐30

Eq.3‐31

The system expressed above is dynamically equivalent to the original

system which means they have identical poles. is the input of the

transformedsystemand (input for theoriginalsystem)canbecalculated

fromitusingequationEq.3‐31.Sothestatefeedbackis

Eq.3‐32

50

4 ImplementationofControlMethods

General

InthischapterthecontroltechniquesintroducedinChapter3areutilized.

As discussed before an actuator delay resulting in a time delay in

experimentalsubstructurecanimpairthedynamicstabilityandaccuracyof

the system. Control theory is used to design servo‐hydraulic controller to

minimizeactuatordelay.

In thisstudya linearizedmodelof thesystemdynamicswasused in the

designof the controllerusing linear controldesign techniques.Anonlinear

model of the system whose parameters were obtained through system

identification (Mercan2007)wasused in thenonlinear state space control

designandalso in thenumerical simulations.This isbecause thenonlinear

model is accepted toprovideamore realistic representationof the system

dynamics.

4.1 DynamicModelofaServo‐HydraulicSystem

The dynamic equations that govern a servo‐hydraulic system were

explainedinChapter2andtwoversionsofthemodel(linearandnonlinear)

51

were introduced. These dynamics are modelled in commercial software

packageMATLAB/SimulinktosimulatetheinnerloopofaPSDtestsetup.

LinearModel

Equations for the linearmodel presented in Chapter 2 are summarized

here.

Table4‐1Linearized dynamicsofaservo‐hydraulicsystem

Valvespool(in

domain) 1

(Eq.2‐1)

Valveflow(Eq.2‐

20) 

Actuatorchamber

pressure 4 

(Eq.2‐

18) 

Actuatormotion (Eq.2‐21) 

Figure 4‐1 shows the Simulinkmodel for a servo‐hydraulic systemwith

the above dynamics that is connected to a linear single‐degree of freedom

teststructure.Thisstructurehasamassof ,dampingof andstiffnessof .

Asitisshowninthefigurethisaddsaloaddynamicstotheaboveequations

as

Eq.4‐1

Figure4‐1

Simulinkmodelforaservo‐hydraulicsystemwithlinearizeddynamics

52

53

NonlinearModel

Equationsforthenonlinearmodelare

Table4‐2Nonlineardynamicsofaservo‐hydraulic

Valvespool(in

domain)1

(Eq.2‐1)

Valveflow  (Eq.2‐

16) 

Actuatorchamberpressure

4 

(Eq.2‐

18) 

Actuatormotion 

(Eq.2‐

19) 

Figure4‐2showstheSimulinkmodelforaservo‐hydraulicsystemofthe

abovedynamics that is connected to a linear single‐degreeof freedom test

structure.

ThelinearmodelwillbeusedfordesigningaPIDandalinearstatespace

controllerandthenonlinearmodelwillbeusedtodesignanonlinearstate

spacecontroller.

Figure4‐2

Simulinkmodelforaservo‐hydraulicsystemwithnonlineardynamics

54

55

4.2 PIDControllerDesign

Figure4‐3showstheSimulinkmodel foraservo‐hydraulicsystemalong

with a PID controller. As it can be seen, the servo‐hydraulic system is

represented by a blockwhich has an input as a command signal,which is

issuedbythecontroller,andfouroutputsthatmaybemeasuredduringatest

andarenamelypistondisplacement,pistonvelocity,loadpressureandvalve

opening. In this study the nonlinear models introduced in Figure 4‐2 was

usedandembeddedtothisblock.

The following gives a summary of the roles of each PID gains

(Ahmadizadeh2007).

Proportional( )–Thisistohandlethepresentrequirements.Theerror

ismultiplied by .Hence, the greater the proportional gain, themore the

servo‐valve opens for a given error. There is a trade‐off for selecting an

appropriate .Although a largeproportional gainmaydecrease the error

resultinginaclosertrackingofreferencesignalandreducedresponsetime,

itdecreasesthestabilitymarginofthesystemandincreasesthefrequencyof

theoscillationinthetransientresponse.

Figure4‐3

Simulinkmodelforaservo‐hydraulicsystemwithaPIDcontroller

56

57

Integral ( )–Theerror is integrated(addedup)overaperiodof time,

multipliedbyaconstant andaddedtothecontrolsignal.Awell‐tunedPI

controller will converge to the reference signal (zero steady‐state error),

leadingtoareducederrorbetweencommandandfeedback.

Integral ( ) – This is to handle the future requirements. The first

derivativeoftheerrorovertimeiscalculated,multipliedby andaddedto

the control signal.Basically, this termcontrols the response to a change in

thesystem.Inpracticeduetonoisesthatenterthemeasuredsignalinareal

PSDtest,acontrollerwithoutaderivativetermmaybeused.

4.2.1 ControllerTuning

Tuning a controller is adjusting its parameters to get a desired system

response.Thegoalistohaveasystemthathasastableandfastresponse(i.e.,

ashortresponsetime)withasmallsteady‐stateerror.

One of the Ziegler‐Nichols tuning methods is called ultimate sensitivity

methodandisusedinthisstudy.Inthismethodthecriteriaforadjustingthe

parameters are based on evaluating the amplitude and frequency of the

oscillationsofthesystematthelimitofstabilityratherthantakingthestep

response. To use the method, the proportional gain is increased until the

system becomes marginally stable and continuous oscillations just begin,

withamplitudelimitedbythesaturationoftheactuator.Thecorresponding

gainisdefinedas (knownasultimategain)andtheperiodofoscillationis

(knownasultimateperiod). shouldbemeasuredwhentheamplitudeof

58

oscillation is as small as possible. Then the tuning parameters for a PI

controller are selected as 0.45 and 1.2. Experience has

shownthatthecontrollersettingsaccordingtoZiegler‐Nicholsrulesprovide

acceptable closed‐loop response for many systems. Ziegler‐Nichols gives a

goodstartingpoint for thecontrollerparametersandthe fine tuningof the

controller is still needed to achieve a desired behaviour. Details for this

methodcanbefoundin(Franklinetal.2010).

Tuning thenonlinearmodel inSimulinkresults in 80, 0.5and

0.Andclosed‐looppolesofthesystemcanbecalculatedasfollowing.

242, 141, , 21.9 31.2 , 0.00625

AsshowninEquation3‐13,thetransferfunctionofaPIDcontrolleraddsa

poleatorigintotheopenlooptransferfunctioninEquation2‐23.Here,inthe

closed loop transfer function this extra pole moves a bit off the origin

( 0.00625).

4.3 LinearState‐SpaceControllerDesign

4.3.1 State‐variableformofequations

As pointed out before, for a given system, depending on the states

selected,theremaybemultiplestatespacerepresentations.Usingthelinear

equationsintroducedinTable4‐1,fourstatesandoneinputcanbechosen.

59

, Eq.4‐2

Alternatively,iftheservo‐valveisassumedtohaveafastresponsetothe

inputsignalcomparedtotherestofthesystem;itsdynamicscanbeomitted

withoutcompromisingthetrackingcapability.

, Eq.4‐3

Considering these three states, the state‐variable representation

(Equation3‐14)ofthesystemwillbe

0 1 0

04 4

004

1 0 0

Eq.4‐4

Andifallfourstatesareconsidered

60

0 1 0 0

0

04 4 4

0 0 01

000

Eq.4‐5

Itshouldbenoticedthatthevalvespooldynamics(Equation2‐1)intime

domainis

1 Eq.4‐6

4.3.2 PolePlacement

Boththree‐stateandfour‐statepresentationofthesystemcanbeusedto

designthecontroller.Forthefour‐statecase,polesareselectedequaltothe

poles corresponding to thepreviouslydesigned systemwithPIDcontroller

ignoringthepoleassociatedtoPIDitself,i.e.

242, 141, , 21.9 31.2

61

Function place.m of commercial software packageMATLAB is used to find

matrix and thenusingequation3‐18matrix is calculated to introduce

thesignalinputas

Eq.4‐7

Appendix C contains the MATLAB code which is used to calculate these

matrices.

If the servo‐valve spool dynamics is ignored (assumed to be faster than

otherpartsof thesystem)consideringonly threestates, thecorresponding

poleneeds tobe ignored too.As illustratedbefore, thepoles far left in the

complexplanearerelatedtofastresponses.Thereforepole 242isthe

one associated to servo‐valve spool dynamics. Consequently, the selected

polestodesignathree‐statecontrollerare

141, , 21.9 31.2

4.4 NonlinearState‐SpaceControllerDesign

4.4.1 State‐variableformofequations

State‐variableformofanonlinearsystemneedstobewrittenintheform

of

Eq.4‐8

62

Itcanbeshownthattheonlywaytowritedownthenonlinearequations

intheaboveformistoconsiderallfourstates.Sohaving

, Eq.4‐9

state‐variableformofthedynamicsaccordingtoTable4‐2willbe

4 sign

000 Eq.4‐10

Tofollowthecalculationsthefollowingnotationwillbeusedinstead

4 sign

000 Eq.4‐11

ItwasstatedinChapter3that and mustbesmoothvectorfieldsbut

the sign function in the third term of does not allow differentiation

at 0.Figure4‐4showsthatforalargecoefficient signfunctioncanbe

approximated by using an Arc Tan. Mintsa et al. (2009) used an equation

involvingexponentialtermstoapproximatethesignfunction.

63

Figure4‐4 Approximationoffunction

HencethedynamicsinEq.4‐11canbeapproximatedas

000

Eq.4‐12

and inEq.4‐12aresmoothvectorfieldswhichcanbeusedtodesigna

nonlinearcontroller.

As it was stated in Chapter 3, for a system to be locally state feedback

linearizable two conditions must be satisfied. These conditions are now

checkedfortheabovenonlinearsystem.

-0.002 -0.001 0.001 0.002

-1.0

-0.5

0.5

1.0

2100

21000 2

106

64

Condition(i) thedistributionspan , … , isofrank .

000

000⋆

, Eq.4‐13

00

4

2

2

00⋆⋆

Eq.4‐14

Andinthesamemanner,

0⋆⋆⋆

Eq.4‐15

⋆⋆⋆⋆

Eq.4‐16

Intheaboveequations“⋆”representsanonzeroexpression.Since

span

000⋆

,

00⋆⋆

,

0⋆⋆⋆

,

⋆⋆⋆⋆

Eq.4‐17

65

hasarankequalto4,thefirstconditionissatisfied.

Condition(ii) thedistributionspan , … , is involutive

andofconstantrank 1.

Itisalreadyknownthat

span

000⋆

,

00⋆⋆

,

0⋆⋆⋆

Eq.4‐18

hasarankof3.Howeverthedistributionneedstobeinvolutive.

Itcanbeshownthat

,

00⋆0

Eq.4‐19

,

0000

Eq.4‐20

,

0⋆⋆0

Eq.4‐21

Allabovevector fieldsbelongtothedistribution inEquation4‐15which

meansthedistributionisinvolutiveandthesecondconditionisalsosatisfied.

66

Therefore the nonlinear system is locally state feedback linearizable and

thereisafunctionlike : → suchthat

⟨ , ⟩ 0

⟨ , ⟩ 0, 0 2

Eq.4‐22

Theaboveconditionsareexaminedinmoredetailinthefollowing.

1. ⟨ , ⟩ 0,whichgives

, , ,

000 0 ⇒ 0 ⇒ 0 Eq.4‐23

2. ⟨ , ⟩ 0,whichinthesamemannergives

0 Eq.4‐24

3. ⟨ , ⟩ 0,whichalsointhesamemannergives

0 Eq.4‐25

4. ⟨ , ⟩ 0,whichgives

0 Eq.4‐26

67

It should benoted that the function is notunique (Marino andTomei

1995).

Function maybeselectedtobe

Eq.4‐27

AswasintroducedinEquation3‐25,coordinatetransformationisdefined

as

, , , , , , Eq.4‐28

So

Eq.4‐29

1 0 0 0 Eq.4‐30

0 1 0 0 Eq.4‐31

Eq.4‐32

68

Thistransformationofcoordinatesletsthesystemdynamicstobewritten

inBrunovskycontrollerformas

Eq.4‐33

where

Eq.4‐34

andintheaboveequation

Eq.4‐35

and

69

4

12 ArcTan

2 ArcTan 4

Eq.4‐36

Thereforethestatefeedbackwillbecalculatedas

Eq.4‐37

Chapter 5 illustrates implementation of the above coordinate

transformationinaMatlab‐Simulinkmodelalongwithlinearcontrollers.

70

5 SimulationsResults

General

This chapter presents simulation results for the inner loop of PSD test

models i.e. the control of servo‐hydraulic‐test structure system. Simulink,

which isdevelopedbyMathWorks, is an interactive graphical environment

formodeling, simulatingandanalyzingdynamic systemsand itworkswith

MATLAB.TheSimulinkmodelspresented in this chapter include linearized

andnonlinearmodelofaservo‐hydraulicsystemdiscussedinChapter2and

the controllers discussed and implemented in Chapter 3 and Chapter 4

includingPID,linearstate‐spaceandnonlinearstate‐spacecontrollers.These

simulations only consider the servo‐hydraulic system connected to a test

structure.Tobetterunderstandtheproposedcontrolmethod,foracomplete

simulation of a PSD test, the outer‐loop needs to be introducedwhere the

integration algorithm and also the analytical substructuremodel (if it is a

hybridPSDtest)reside.

71

5.1 Numerical Values for Servo‐Hydraulic System and Test

StructureinLinearandNonlinearModels

The values of the parameters for linear and nonlinear formulation of

dynamics of a servo‐hydraulic system are given in Table 5‐1. These

numerical values are obtained through system identification for a servo‐

hydraulic system used in another PSD test research program (Zhang et al.

2005)andareassumedtoberepresentative.Forthestructureselection,the

important thing foranSDOFare thenaturalperiod(shouldbearound1or

1.3sec),damping(around2to5%). InaPSDtest, it isdesiredtokeepthe

inertialanddampingforcesanalytical.

Table5‐1Values for Parameters for Linear and Nonlinear Servo-Hydraulic System Models

parameter value mass of test structure 1025kg 

viscous damping coefficient of test structure 500 N sec/m 

stiffness test structure 5.00 N/m 

mass of actuator piston 1025kg

viscous damping coefficient of actuator piston 356.18 N sec/m

actuator piston cross section area 0.0808m

4

actuator chamber volume 5.65 m /Pa

oil modulus supply pressure of the hydraulic system 207 Pa 

Γ 1 ⟶orifice coeffiecient

Γ ⟶valve opening gradient ⟶hydraulic oil density

2.62 m /s/Pa .  

servo-valve flow-pressure coefficient 1.5 m / Pa sec

servo-valve flow gain 0.035 m /sec

actuator leakage coefficient 3 E m / Pa sec

servo-valve gain 0.9613

servo-valve time constant 0.004sec

, natural frequency of the test structure 69.8 1 sec⁄

damping ratio of the test structure 0.00349

72

5.2 Comparison of Controllers With and Without Saturation

Limits

Numerical simulations were performed to compare the tracking

capabilities of the PID, linear state‐space and nonlinear state‐space

controllers. As explained in Chapter 4, all the controllersweredesigned to

have the same dynamic characteristics. This was done by figuring out the

polelocationsfromthetunedPIDcontrollerandusingthemfordesigningthe

linear and nonlinear state‐space controllers. Linear state space controller

uses the linearized state space representation (Table 4‐1) in the pole

placement procedure. On the other hand, the nonlinear controller design

accounts for the nonlinearity in servo‐valve flow‐pressure relationship

(Table4‐2). Itutilizesacoordinatetransformationtoexpressthesystemin

Brunovskyformwhichprovidesanequivalent linearsystemrepresentation

forthesystemdynamics.Itshouldbenotedthatthecalculatedinputtothis

equivalentsystem( )hastobetransformedbacktotheinputoftheoriginal

system ( ) using the corresponding expression in Equation 4‐34. While

performing thenumerical simulations, thedynamics of the servo‐hydraulic

test structure system was represented by the nonlinear model as it is

presumedtobemorerepresentativeoftherealbehaviour.Figure5‐1shows

step responses of the system with linear and nonlinear state‐space

controllers. Itwas observed that the PID responsewas almost identical to

linear state‐space controller which was expected as both were based on

linearizedrepresentationofthesystem.Thereforeonlythelinearstate‐space

re

o

co

o

d

st

w

w

p

st

esponse is

penasmuc

ontrolleris

f the respo

esign has s

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Figure 5‐2

withthediffe

when the s

erforming b

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shown. In

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observedto

onse. Moreo

some steady

error.

Figure5‐1

2 shows th

erencethat

servo‐valve

better than

sephases.

displacement(m)

Figure 5‐1

ed (i.e., the

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over, in the

y state erro

Steprespo

he step resp

ittakessat

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n the linear

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e steady sta

or while th

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73

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74

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75

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76

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77

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78

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79

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Figure5‐10

Simulinkmodelforaservo‐hydraulicsubsystem

withaPIDcontroller

80

Figure5‐11

Simulinkmodelforaservo‐hydraulicsubsystem

withaNLstate‐spacecontroller

81

v

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Table 5-2 Appr

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82

proximate

ontroller.

ticalpole

following

Table5‐1

m 

ontroller.

ainst the

PIDcontroller

time(sec)

time(sec

(b)

)

c)

o

g

ex

v

v

sh

sp

For both

verlapping

iveninFigu

Figure5‐1

It can be

xtent. This

elocity feed

elocity feed

hows that s

pacecontro

displacement(m)

displacement(m)

controllers

in this tim

ure5‐13.

13 compariso

seen that t

much of

d forward t

d forward

spool openi

ollerisused

, command

me scale. He

onbetweenco(b

the NL cont

improveme

to PID cont

may distor

ing has am

.

d andmeasu

ence a narr

(a)

(b)

ommandandmb)NLcontroll

troller is im

ent could a

troller but a

rt the spoo

much smoot

ured displa

ower view

measureddispler

mproving th

also be ach

as shown in

ol opening

ther behavi

measured

command

measured

comman

acements ar

of the resp

placement(a)

he tracking

hieved by

n 5‐14 (a)

response.

iourwhenN

e

d

83

re almost

ponses is

)PIDcontrolle

g to some

adding a

adding a

5‐14 (b)

NL state‐

time(sec)

time(sec

er

c)

Figure5‐1

Valveopening(m)

Valveopening(m)

14 comparisoonbetweensefeedfo

(a)

(b)

ervo‐valvesporward(b)NL

oolopening(acontroller

a)PIDcontrolle

84

erwithaveloc

time(sec)

time(sec)

city

5

p

w

ex

p

v

e

ro

p

p

m

S

a

5.4 Robus

There is

arameters

whatisreally

xamined w

arameters.

ariationsin

nhancedfee

obustness o

hysical par

ercent of o

model rema

upply pres

ccumulator

displacement(m)

stnessofth

always the

used in the

yinthesys

with respec

Mintsa et

nsupplypre

edback‐line

of the prop

rameters in

over/under

ained intact

ssure is co

rs.

Figu

heContro

e possibilit

e controller

tem.Thero

ct to pote

al. (2009)

essurebyde

earization‐b

posed contr

n the contr

restimation

t. , ,

nsidered a

ure5‐15 effe

No

lDesign

ty that the

r block are

obustnessof

ential error

addressed

esigningaL

asedcontro

rol design,

roller block

error whi

and ar

almost cons

ectof10%erro

10%erro

error

identified

inaccurate

fthecontro

rs in the

the robust

Lyapunovap

ollaw.Here

the values

k were con

le their va

re the para

stant due

orin onthe

t

or

values of

e and differ

oldesignne

identified

tness regar

pproachtod

eintoinvest

s of some i

taminated

alues in the

ameters con

to the pre

eresponse

time(sec)

85

physical

rent from

eedstobe

physical

rding the

derivean

tigatethe

identified

by some

e system

nsidered.

esence of

86

Figure5‐15comparesthecasewhenthevalue for in thecontroller is

overestimatedby10%withthecasewhenthereisnoerror.Itisreadilyseen

that 10% of error has made the system unstable. A similar graph was

conducted for 5% error (not shown in figure for sake of clarity) and the

control design was able to handle it without stability problems. Similar

comparisons were done for negative percent errors (underestimation)

wherein the accuracy did not change much and there were no instability

problems.

Conducting the same procedure for , and showed that the

proposed control design can easily handle errors within 20% of the real

value.

5.5 Conclusion

Thecontroller’sabilitytotrackthecommanddisplacementsaccuratelyis

crucialintheoverallstabilityandaccuracyofthereal‐timePSDtesting.For

thedisplacementcontrolloop,itwasshownthatacontrollerwithimproved

performance can be designed using nonlinear state space control design

techniques,providedthatarepresentativemodelofthesystemisavailable.

Simulationsweredone fora servo‐hydraulic systemattached toa linear

structure for cases with and without saturation in servo‐valve and it was

concludedthataNLcontrollercanimprovethetrackingcapabilities.

87

Moreover,theproposedNLcontrollerwascomparedtoPIDcontrollerby

implementing it in a real‐time PSD test simulation to show that it can

decreasethetimelagofthemeasureddisplacement.

Robustnessofthenonlinearcontroldesignwithrespecttosomeidentified

physicalparameterswasinvestigatedanditwasobservedthaterrorswithin

reasonablerangecanbeeasilyhandledintermsofinstabilitywithoutlossof

accuracy.

Future research is needed on application of the nonlinear controller to

multi‐input, multi‐output system control together with a comprehensive

numericalstudyandexperimentalvalidation.

88

6 Testsetupdesign

The design of the test specimen considers the displacement and force

limitationsof theexistingactuators.Theseactuatorshaveastrokerangeof

5"(12.7cm)andhaveaforcelimitof5.5Kip(24.5kN).

Theaimistodesignatestsetupthatexhibitsnonlinearbehaviourwithin

aboveactuatorslimitations.Figure3‐1showsasketchofthetestsetup.This

single‐degree‐of‐freedomsystemconsistsofashortcolumnwithwideflange

sectionthat ishingedtoasupportbymeansofaplateandaclevis.As it is

seen in the figure, replaceable steel coupons are used to providemoment

resistance for the support. Since the moment resistance capacity of the

coupons is smaller than the columns, the nonlinear behaviour (yielding)

initiates in the coupons and the column is expected to remain elastic.

Therefore, after each nonlinear PSD test, the sacrificial coupons can be

replacedeasilyatalowcost.

h

h

co

h

co

cr

Looking a

ave been u

ingecanbe

22

Twoof th

ompression

ingeandth

oupons. i

rosssection

Fi

at the defle

used, the int

workedou

hecoupons

n.Intheabo

hecoupons

is themodu

nalareaofo

igure6‐1Expe

cted shape

ternal force

utasthepro

willexperi

oveequatio

inthedirec

ulusof elast

onecoupon.

erimentalsetu

and assum

e in the cou

oductofthei

ence tensio

n istheh

ctionofapp

ticityof the

upforPSDtes

ming that tw

upons ( ) a

irdeflection

onandtheo

horizontald

pliedforce.

e couponm

st

wo pairs of

at either sid

nandaxials

other twow

istancebetw

istheleng

material and

coupo

o

89

f coupons

de of the

stiffness.

Eq.6‐1

willbe in

weenthe

gthofthe

d is the

n

y

th

n

ca

2

Also it is

ieldingforc

2

is th

hebeamand

As menti

onlinearity

alculatedby

2

s obvious t

ce,so

hemaximum

d isyield

oned befor

.Therotati

yequating

⇒

Figure6

that the fo

m force tha

ingstresso

re the cou

onoftheba

inEq.6‐1

‐2Expectedb

rce in the

at twocoup

ofthecoupo

upons are

aseatwhich

to inE

baserotationa

coupons c

ons canpro

onsteel.

intended t

hthecoupo

Eq.6‐2.

atbaseusedin

cannot exce

ovideaton

to experien

onsyield(

ndesign

90

eed their

Eq.6‐2

ne sideof

nce some

)canbe

Eq.6‐3

91

Inordertobeabletoobservenonlinearresponseintheexperimentaltest

set‐up,thedesignbaserotation willbesethigherthantheyieldrotation

.

1 Eq.6‐4

Ontheotherhand,becauseoftheactuatorstrokelimit,testsetupcannot

movemorethan5”(12.7cm)toeitherside.HenceaccordingtotheFigure3‐

1

sin 5" Eq.6‐5

orapproximately

5" ⇒

5" ⇒ 5" Eq.6‐6

Alsoforasingle‐degreeoffreedomsystemsuchastheoneinFigure3‐1,

ignoringtheself‐weightof thesystem,staticequilibriumofmomentsabout

point gives

⇒ Eq.6‐7

Andtheforceappliedbytheactuatorcannotexceed5.5Kip 24.5kN .So,

92

5.5Kip Eq.6‐8

whichaccordingtoEq.6‐2gives

2 5.5Kip ⇒

1

5.5 Kip2

Eq.6‐9

CombiningEq.6‐6withEq.6‐9showsthatthecolumnneedstobeinthe

followingrange.

2 5.5Kip

5"

Eq.6‐10

Itwasdecidedtouseawide‐flangesectionof 6 20forthecolumnand

for the coupons a diameter of 0.25" 0.635cm)was selected that gives an

area of 98.2 10 in 0.634cm . The clevises available in the

structurallaboratoryattheUniversityofAlbertadictatesacouponlengthof

8.13" 20.65cm .Thedistance between the coupons, as indicated in the

Figure3‐1,.waschosentobe 6" 15.24cm .

Assuming 34ksi 234 , 29000ksi 2 10 and 3,

Eq3‐10becomes

93

234ksi 98.2 10 in 6"

5.6kips5"

6" 29000 ksi3 8.13" 34 ksi

⇒ 7.16" 1000"

Eq.6‐11

Ascanbeobservedforheightofthecolumnthereisawiderangetoselect

from.Howeveroneshouldbeawarethattheupperlimitiscorrespondingto

3. That implies that any height less than 1000" corresponds to bigger

valuesof .Accordingtothespaceavailableinthestructurallabaheightof

50" 127cm waschosentobeused.

Buckling of the coupons also needs to be considered. Knowing that the

endsofthecouponsarefixed,bucklingloadofacouponis

29000ksi 40.25′′2

0.58.13′′3.32kips 14.8kN

whiletheyieldingforceis

340.254

1.67kips 7.43kN

Thismeansthatthecouponswillyieldlongbeforebuckling.

94

AppendixA

TableofLaplaceTransforms

Number , 0

1 1

2 1

1

3 1

4 2!

5 !

6 1

7 1

8 1

11

9 1

10 11

11

95

12 1

13

1 1

14

15 sin

16 cos

17 cos

18

sin

19 1 cos sin

96

AppendixB SomeonDifferentialGeometry(Lynch2009)

1 ChangesofCoordinatesorDiffeomorphisms

A nonlinear change of coordinates is a function defined on

⊆ andmappingto . It must have the properties

1. is an invertible function, i.e., there must exist an inverse

function suchthat

, ∀ ∈

2. Both and are mappings.

When ⊆ thechangeofcoordinatesiscalledglobal.Whenthisisnot

thecase, thechangeofcoordinates issaid local.Asufficientcondition fora

mapping tobe a local changeof coordinates is given by the Inverse Function

Theorem (Marino and Tomei 1995). Note that Condition 2 is required, i.e., that

the change of coordinate is , since these coordinates changes will be applied to

the state of a system, and the systems are required to be expressed in the new

coordinatesalsotobe orsmooth.Aninfinitedegreeofsmoothnessisnot

usually required, but it is assumed to avoid keeping track of the degree of

97

smoothness. The invertibility Condition 1 allows to uniquely recover the original

state coordinate from the new one. In addition to applying a change of coordinates

to a system’s state, transformation of a system’s output variable or time is also

possible.

2 VectorFields

Avector field on ⊆ isa mappingfrom to . It is customary to

write vector fields using one of two notations. The first notation represents vector

fields as column vectors. The basis used to represent the vector field is implied

and this might lead to confusion.

⋮, , … ,

Alternately

where is the th unit tangent vector in the ‐coordinates. The latter

notation provides more information in that the basis used to express the

vectorfieldisstatedexplicitly.

3 DifferentialGeometryFunctionsUsedintheTextLiebracketsLie brackets between two vector fields and is another vector field

whichisdefinedinlocalcoordinatesas

98

,

IteratedLiebracketsaredefinedas

, , 1

LieDerivative

Liederivativeoffunction alongvectorfield istheinnerproductof

thegradientof definedas

, , … ,

andvectorfield ,therefore

⟨ , ⟩

andtheiterativeformofitisexplainedas

⟨ , ⟩, 1

SequentialformofLieDerivativeisdefinedas

⟨ , ⟩

99

4 Distributions

A ‐dimensionaldistribution∆definedon ⊆ isamapwhichassignsto

each ∈ ,a ‐dimensionalsubspaceof (ormoreprecisely the tangent

space ) such that for all ∈ there exists a neighbourhood ⊆

containing and smoothvectorfieldssuchthat

1. , … , arelinearlyindependent∀ ∈

2.∆ , … , forall ∈ .

NotethatinCondition1,“linearindependence”istheusualdefinitionon

fromlinearalgebra.InCondition2thespanontheRHSinvolvesreal linear

combinations of the constant vectors , … , in , that is, it is a

subspace of . In many cases we use so‐called generating vector fields

, 1 to define a distribution, however, the are only one

choiceofbasisforthesubspace∆ andotherscanbechosen.Forexample

if ∆ , then we also have∆

, .

Thedimensionof adistribution∆,denotedasdim ∆ is a functionof

and is equal to the dimension of the subspace ∆ . A distribution is said

nonsingularon ifithasconstantdimensionon .Innonlinearcontrol,we

oftenassumetherelevantdistributionsarenon‐singular.

Givena ‐dimensionaldistribution∆definedon anda vector field on

⊆ ,wesay belongsto∆if

∈ ∆ , ∀ ∈

Formoreinformationpleaserefertothereference.

100

AppendixC MatlabCodesandSimulinkModels

TypicalMatlab Code to define the parameters and design the controller

(poleplacement)usingthe“place”functionofMatlab.

clear all; clc; % Structure m=1000; c=500; k=5000000; % Both systems mp=1025; bp=356.18e3; %ap=0.0808; ap=0.1095; %v4b=5.65e-10; v4b=4e-11; cp=3e-11; tm=0.004; xvmax=2.73e-3; %xvmax=2.73e+3; % Linear kc=1.5e-11; kq=0.035; % Nonlinear ps=207e5; n=2.617e-3; %xvmax=2.73e-3; kv=1; wv=1/0.004; % PID kp=80; kd=0;

101

ki=0.5; %State-space design for linear model with 3 states a1=[0 1 0; -k/(m+mp) -(bp+c)/(m+mp) ap/(m+mp); 0 -ap/v4b -(kc+cp)/v4b]; b1=[0; 0; kq/v4b]; c1=[1 0 0]; p=[-123,-26.8+29.9*1i,-26.8-29.9*1i]; kcl1=place(a1,b1,p); nbar1=-inv(c1*((a1-b1*kcl1)\b1)); %State-space design for linear model with 4 a2=[0 1 0 0; -k/(m+mp) -(bp+c)/(m+mp) ap/(m+mp) 0; 0 -ap/v4b -(kc+cp)/v4b kq/v4b; 0 0 0 -wv]; b2=[0; 0; 0; kv*wv]; c2=[1 0 0 0]; p=[-242,-21.9+31.2*1i,-21.9-31.2*1i,-141]; kcl2=place(a2,b2,p); nbar2=-inv(c2*((a2-b2*kcl2)\b2)); % Nonlinear state-space design for non-linear model with 4 states a3=[0 1 0 0; 0 0 1 0; 0 0 0 1; 0 0 0 0]; b3=[0; 0; 0; 1]; c3=[1 0 0 0]; p=[-242,-21.9+31.2*1i,-21.9-31.2*1i,-141]; kcl3=place(a3,b3,p); nbar3=-inv(c3*((a3-b3*kcl3)\b3));

102

Functiondefinitionfornonlinearbehaviouroftheservo‐valve.

function flowB = fcn(xv,PL) xv;PL; %%% this fcn calculates the flow considering the leakage Cd=0.58; rho=700; %kg/m^3, mass density PS=207e5; %Pa , supply pressure d=0.038; %m, spool diameter % xvmax=2.73e-3; %m max spool opening xvmax=2.73e-3; %%%% leakage properties of the individual valve %xvlap=6/100; xvlap=0; %Aeffnull=2.056e-6; %m^2 Aeffnull=0; xvlapm=-1*xvlap; %% default A1 and A4 A1=0; A4=0; %%% define the orifice areas if xv <= xvlapm A1=0; A4=((pi*d*xvmax-2*Aeffnull)/(1-xvlap))*abs(xv)-(((pi*d*xvmax-2*Aeffnull)/(1-xvlap))*xvlap-2*Aeffnull); end if xv >= xvlap A4=0; A1=((pi*d*xvmax-2*Aeffnull)/(1-xvlap))*abs(xv)-(((pi*d*xvmax-2*Aeffnull)/(1-xvlap))*xvlap-2*Aeffnull); end if xv > xvlapm && xv <=0 A1= -(Aeffnull/xvlap)*abs(xv)+Aeffnull; A4= (Aeffnull/xvlap)*abs(xv)+Aeffnull; end if xv > 0 && xv < xvlap A1= (Aeffnull/xvlap)*abs(xv)+Aeffnull; A4= -(Aeffnull/xvlap)*abs(xv)+Aeffnull; end %%% calculate the flow flowB=(Cd*A1*((PS-PL)/rho)^0.5)-(Cd*A4*((PS+PL)/rho)^0.5);

FigureC‐1

SimulinkmodelfornonlineardynamicsofaServo‐hydraulic

103

FigureC1

SimulinkmodelfornonlineardynamicsofaServo

hydraulic

FigureC‐2

Servo‐hydraulicsystem

withaPIDcontroller

104

FigureC2

ServohydraulicsystemwithaPIDcontroller

FigureC‐3

Servo‐hydraulicsystem

withalinearstate‐spacecontroller

105

FigureC3

Servohydraulicsystemwithalinearstatespacecontroller

FigureC‐4

Servo‐hydraulicsystem

withanonlinearstate‐spacecontroller

106

FigureC4

Servohydraulicsystemwithanonlinearstatespacecontroller

FigureC‐5

nonlinearstate‐spacecontroller

107

FigureC5

nonlinearstatespacecontroller

108

AppendixD ConferencePaper

109

MODIFICATIONS OF INTEGRATION ALGORITHMS TO ACCOUNT FOR LOAD DISCONTINUITY IN PSEUDODYNAMIC TESTING

S. Hadi Moosavi1, Oya Mercan2

ABSTRACT When there is a sudden change in the loading (e.g., rectangular pulse), the discretized version of the load history will involve an artificial impulse, which manifests itself as an amplitude distortion in the structural response obtained by the numerical solution of the equation of motion. An approach to account for load discontinuity by modifying existing integration algorithms used in the solution of force equation of motion is introduced in this paper. Modified versions of four different algorithms, namely Central Difference, Newmark Explicit, α-method with a fixed number of iterations, and Rosenbrock-W integration algorithms are presented. The general approach in modifying an integration algorithm to account for load discontinuity is discussed and the improved accuracy of these modified algorithms is presented through numerical simulations.

Introduction

Pseudodynamic (PSD) test method is a displacement based experimental technique that can be used to determine the behavior of structural systems subjected to dynamic loading. In a PSD test, a direct step by step integration algorithm generates the command displacements by solving the force equation of motion. These displacements are imposed on the test structure by a servo-hydraulic system, and using the measured restoring force feedback from the deformed test structure, the integration algorithm computes the subsequent command displacements. For load-rate insensitive structures, PSD testing method can be applied in slow time (using an expanded time axis), or for structures that exhibit load-rate dependent vibration characteristics, it can be applied at fast rates (ideally in real-time). Both the slow-time and real-time PSD testing have been successfully applied for seismic loading (Mahin 1985, Nakashima 1999), but if the loading history has a sharp discontinuity as in the case of pulse loading (see Fig. 1-a), the numerical solution of the force equation of motion will have an amplitude distortion which may render the PSD test results inaccurate. This is due to the extra impulse (shaded area in Fig. 1-b) introduced during the discretization of the loading history.

To circumvent this problem, the use of step by step solution of the momentum equation of motion was suggested and the resulting improved accuracy was verified through numerical simulations (Chang, 2001, 2002, 2007a) and experiments (Chang, 1998). In this approach, the force equation of motion is replaced by its integral form, which is the momentum equation of motion. As a result of the time integration of the load that appears on the right hand side of the momentum equation, provided that the area under the load history is computed correctly, the discontinuity in the load history is eliminated. Although the success of the momentum approach

110

has been presented for Newmark explicit integration algorithm, replacing the force equilibrium equation with the momentum equation may not be a trivial task if one wishes to use an integration algorithm customized (e.g., unconditionally stable, implicit, real-time compatible) for particular testing needs. Other than the momentum approach, in an attempt to obtain an accurate solution in the presence of load discontinuity, Chang, (2007b) also proposed the use of a single time step immediately after the discontinuity that is much smaller than the discretization step size. This small step was recommended to be one-hundredth of the discretization step size or smaller. Especially within the context of real-time PSD testing, a variable time step is detrimental for it would result in inaccurate velocities as the displacement commands are typically imposed using a digital controller with a constant clock speed.

The study presented here introduces an approach, which is referred to as limit approach,

to account for the load discontinuity by modifying a given integration algorithm that solves the force equation of motion in its final form. Depending on the way a particular integration algorithm is formulated, these modifications generally involve updated force and/or acceleration values at the time of discontinuity. In the paper, the general approach (i.e., the limit approach) that introduces modifications to a given integration algorithm is discussed and then implemented to derive modified versions of the Central difference, Newmark explicit, α-method with a fixed number of iterations, and Rosenbrock-W integration algorithms. Both α-method and Rosenbrock-W algorithms are suitable for real-time testing, where the former is an implicit scheme. For each of the four integration algorithms considered in this paper, a summary of the original formulation is provided together with its modified version. The improved accuracy of the modified algorithms is presented through numerical simulations.

The Limit Approach

The limit approach starts by defining an intermediate step just after the load

discontinuity between times (where the discontinuity takes place) and 1 (see Fig. 1-c). In order to be able to set it apart from the discretization time step size of ∆ and thereby make the implementation of the limit approach for the modification of an algorithm easier to follow, the time step size associated with step is identified as ∆ . It should be noted the load value is the value of the load at the lower end of the discontinuity at step . From the original formulation of a given integration algorithm, the information for step (which may include the displacement ( , velocity ( ) and/ or acceleration ( ), or an intermediate quantity defined by the particular integration algorithm to march forward) can be obtained using the information from step (see Fig. 1-c) and considering and ∆ . In order to obtain the information at the lower end of the discontinuity ∗ and thereby account for discontinuity effects properly, next step involves taking the limit of the expressions that define step information where ∆ goes to zero. On Fig. 1-d the information (i.e., the expressions for the load, displacement etc.) associated with ∗are the results of this limit process. In programming the resulting modified algorithm, a flag needs to be set in order to identify the time step when discontinuity (i.e., step ) takes place. When that happens, the numerical values of for ∗ , ∗etc. need to be evaluated from the expressions obtained by the limit approach, and using these, the integration algorithm in its original form can march forward to compute information for step 1.

111

Implementation of the Limit Approach

Central Difference Method

The discretized form of the equation of motion for a single-degree-of-freedom (SDOF) system at time step is:

(1) Central difference method uses a finite difference approximation for velocity and acceleration (Chopra, 2007). With a constant time step size of ∆ , the velocity and acceleration at step are expressed as:

2∆ (2)

1∆ ∆ ∆

2∆

(3)

Substituting the velocity and acceleration from Eqs. (2) and (3) in Eq. (1), and solving the expression for :

∆ 2∆2∆

/∆ 2 ∆

(4)

Implementation of Limit Approach to Obtain ( ∗) Information

Considering Fig. 1-c and rewriting Eqs. (2) and (3) for step , velocity and acceleration at that step are:

∆ ∆ ′ ,1

12 ∆ ∆ ′ ∆ ′ ∆

(5), (6)

Also, using the same equations, velocity and acceleration at step can be expressed as:

Fig. 1. (a) Load history with discontinuity, (b) Discretized load history, (c) Load history with step , (d) Load history after performing the limit

∆ ∆ ∆

∆ ∆ ∆

, , ,

, 1, 1, 1

1

, , ,

, ∗

∆ ∆ ∆∆ ∆ ′ ∆

load

time

load

time

load

time

load

time

, , ,

11 1

, 1, 1, 1

∗, ∗, ∗, ∗

112

∆ ∆ ′ ,

112 ∆ ∆ ′ ∆ ∆ ′

(7), (8)

Eqs. (9) and (10) express the equation of motion at time steps and , respectively:

(9)

(10) Once Eq. (9) is solved for (after substituting for and from Eqs. (5) and (6)), the result can be used to replace in Eq.(10) which then can be solved for (after substituting for and from Eqs. (7) and (8)). Performing a limit where ∆ ′ → 0 gives an expression for in terms of and :

2∗

∆ 2 ∆2∆

∗

/∆ 2 ∆

(11)

Comparing Eq. (4) with Eq.(11), application of limit approach to Central difference

method reveals that to account for the load discontinuity, only the load value for ( ∗) needs to be redefined as the average value of the loads at each end of the discontinuity.

Explicit Newmark method

Newmark family integration methods (Newmark, 1959) can be customized by selecting

two parameters ( and ) which specify the variation of acceleration over a time step. These parameters also determine the accuracy and stability characteristics of the method. For 0.5

0 Newmark's method becomes explicit and conditionally stable:

Δ 0.5Δ (12)

0.5 Δ0.5 Δ

(13)

Δ 0.5 0.5 (14) Implementation of Limit Approach to Obtain ( ∗) Information Defining the information for step using Eqs. (12)-(14) as required by the limit approach:

′ 0.5 ′ (15)

113

0.5 ′0.5 ′

(16)

0.5Δ (17) Performing a limit on the above expressions where∆ ′ → 0, will yield

∗ , ∗ (18), (19)

∗∗

(20)

where ∗

From above equations, it turns out that, while the values for displacement and velocity remain the same, the acceleration value at the lower end of the discontinuity ∗ needs to be updated. By stepping forward to step 1 using the updated information from Eqs. (18)- (20) derived consistently with Newmark explicit method formulation, the load discontinuity will be taken care of properly.

Rosenbrock

Considering an SDOF system the general implementation of Rosenbrock integration method proposed by Lamarche (2009) for real-time pseudo-dynamic testing is described in Fig. 2.

Implementation of Limit Approach to Obtain ( ∗) Information

Introducing step and taking the limit for the Rosenbrock algorithm, it can be easily shown that and approach to and , respectively (or equivalently ∗ and ∗ are the same as and , respectively). As a result, for Rosenbrock algorithm to handle the load

Fig. 2. Rosenbrock integration algorithm

1

Initial restoring force: Effective mass: Δ Δ

Δ12 

12

,12

Δ

Δ , 

Δ

Mid-step between and 1

Where

Impose and to the structure and measure the restoring force

,  

Δ2

Δ

Δ

Where

Impose and to the structure and measure the restoring force

2

3

4

5

1

114

discontinuity properly, the information at step 1 right after the discontinuity needs to be calculated using ∗, ∗ and ∗ which are the same as , , and , respectively.

Alpha method

Alpha method (Shing et al 2002) is an implicit method used in real-time pseudo-dynamic

testing which is based on Hilber -method (Hilber et al 1977). As can be seen from Fig. 3, Alpha method starts by computing a predictor displacement in stage 1 , which is followed by a fixed number of iteration substeps in stages 2 and 3 ; during the last iteration substep an equilibrium correction is performed. Through the equilibrium error correction (stages 4 and 5 ) the displacement and restoring force values are made available for the computation of the next step predictor displacement and, as a result, the actuator moves without interruption.

Implementation of Limit Approach to Obtain ( ∗) Information

Upon the application of the limit approach starting from stage 1 in Fig. 3, the predicted displacement at step approaches to after setting∆ ′ → 0 (or equivalently ∗ ) and the displacement term in stage 2 becomes constant (i.e., ∗ ). Considering that both parameters ∗ and approach to application of the limit to stages 4 and 5 gives:

Δ Δ 0.5Δ

1 1 1 Δ

Predicted displacement

Δ 1 

 

Impose , measure the displacement and measure the restoring force

1Δ

Δ Δ 0.5 β  

Δ 1

Δ 1

1

1

1

2

4

3

∗

∗

5

6

∗ Δ 1 1 Δ

Note that: 0 1,

Fig. 3. Alpha method algorithm

115

(21)

lim∆ ′→

∗ (22)

lim∆ ′→

∗ (23)

Eq. (23) is true for linear elastic systems. In stage 6 the expression for acceleration can be revised for step as

1

Δ ′Δ ′ Δ ′ 0.5 β

(24)

Upon substituting the expression of from stage 2 which has the expression for embedded from stage 1 , and as a result of the cancellations that take place, the zero over zero indeterminacy as ∆ ′ approaches to zero is eliminated; and the acceleration for ∗ is obtained:

∗ 1 ∗

(25)

where ∗ And the expression for velocity in stage 6 yields

∗ (26)

The above application of the limit approach in modifying Alpha method reveals that, to

compute the information at step 1 right after the discontinuity, information from ∗ needs to be used where the updated acceleration from Eq.(26) is used together with the value of the load at the lower end of the discontinuity.

Numerical Simulations

In order to verify the success of the proposed modifications in handling the load discontinuity, numerical simulation results for each integration algorithm are presented here. An undamped linear SDOF system with 0.2533 . / , and 10 / (i.e.,undamped natural period 1 .) subjected to the two loading cases shown in Fig.4 is considered. Fig. 4 (a) is a step pulse with an amplitude of 10 kips and duration of 0.1 sec.; whereas in Fig. 4 (b) the load value changes from +10 to -10 kips at the discontinuity and then increases to zero linearly over a duration of 0.1 sec.

W

the equatturn the amodifiedsimulatiothat of Chduration

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-0.5

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izing the loaon, selectionistortion intrin handling cted as 0.1 s), where it womentum eq

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Fig. 5. Sim

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0.4 0.6time (sec)

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Rosenbrock-W

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ading historyn of a very smoduced in thlarge extra i

sec.; which awas shown thquation is use

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modifiedtheoreticalunmodified

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modifiedtheoreticalunmodified

(sec) 0.2

ad cases with

y for subsequmall time stehe response. impulses, thealso made theat time step ed. It needs

∆

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0

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)

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0.4 0.6time (sec)

Newmark Explicit

0.4 0.6time (sec)

Alpha Method

(sec) 0.2

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numerical omparable w

as the impuor Central

beyond whic

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116

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117

discontinuity, the proposed modifications provide considerable improvements in the accuracy of the numerical results. The unmodified responses from each algorithm between Fig. 5 and 6 are the same as expected, since with a time step size of 0.1 sec., the discretized versions of the load cases in Fig. 4 (a) and (b) are the same (see Fig. 1 (b)). Depending on the accuracy characteristics of the particular integration algorithm, the agreement between the theoretical response and the modified numerical solution can be improved by using a smaller time step.

Fig. 6. Simulation results for the loading case in Fig. 4(b).

Conclusion

Pseudodynamic testing method has been implemented successfully both in slow and real-

time for seismic loading of structures. However, when a sharp discontinuity exists in the loading history as in the case of pulse loading, the discretized version of the load will have an extra distortion which manifests itself as an amplitude distortion in the numerical response and may render the pseudodynamic test results inaccurate. Other than using very small time steps in discretizing the load, previous studies proposed the use of numerical solution of the momentum equation of motion that replaces the force equation of motion. Although the success of the momentum approach has been presented using Newmark explicit integration algorithm, replacing the force equilibrium with the momentum equation may not be a simple task if one wishes to use an integration algorithm customized for particular testing needs. The study presented here introduces a limit approach that considers the force equation of motion and modifies the integration algorithms in their final form to account for the load discontinuity. The general modification approach and its implementation to four integration algorithms are provided together with numerical simulation results that show the improved accuracy of the modified algorithms.

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0 0.2 0.4 0.6 0.8 1-1

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men

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