Research Article Design and Experimental Evaluation of a ...position control system, a number of...

Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 590708 16 pageshttpdxdoiorg1011552013590708

Research ArticleDesign and Experimental Evaluation of a RobustPosition Controller for an Electrohydrostatic ActuatorUsing Adaptive Antiwindup Sliding Mode Scheme

Ji Min Lee Sung Hwan Park and Jong Shik Kim

School of Mechanical Engineering Pusan National University Jangjeon-dong Geumjeong-guBusan 609-732 Republic of Korea

Correspondence should be addressed to Jong Shik Kim jskimpnuedu

Received 13 May 2013 Accepted 5 June 2013

Academic Editors U Lee and S Liu

Copyright copy 2013 Ji Min Lee et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A robust control scheme is proposed for the position control of the electrohydrostatic actuator (EHA) when considering hardwaresaturation load disturbance and lumped systemuncertainties and nonlinearities To reduce overshoot due to a saturation of electricmotor and to realize robustness against load disturbance and lumped systemuncertainties such as varying parameters andmodelingerror this paper proposes an adaptive antiwindup PID sliding mode scheme as a robust position controller for the EHA systemAn optimal PID controller and an optimal anti-windup PID controller are also designed to compare control performance An EHAprototype is developed carrying out systemmodeling and parameter identification in designing the position controllerThe simplyidentified linear model serves as the basis for the design of the position controllers while the robustness of the control systems iscompared by experiments The adaptive anti-windup PID sliding mode controller has been found to have the desired performanceand become robust against hardware saturation load disturbance and lumped system uncertainties and nonlinearities

1 Introduction

Recently lots of hydraulic servo systems have been replacedby electric motor-driven systems to overcome the drawbacksof traditional hydraulic servo systems such as oil leakagemaintenance and laying complex pipe In spite of these draw-backs hydraulic systems are still used in fields that requirelarge force and high power In order to solve these issues elec-trohydrostatic actuator (EHA) systems are in the process ofbeing developed for industrial useThe EHA systems can alsobe referred to as a direct drive volume controlled (DDVC)actuator [1ndash3] or valveless hydraulic servo (VHS) actuator [4]

The EHA systems generally consist of an electric motorhydraulic pump hydraulic cylinder accumulator checkvalves and relief valves for safety An electric motor thatis directly tied to the hydraulic pump controls the cylinderwhose directional change depends upon the electric motorrsquosrotation direction Also an electric motor installed in theEHA system controls the flow rate and pressure of the work-ing fluid by regulating the velocity and torque of the motor

The EHA systems are kind of compact high-poweractuator modules that integrate the electric motor hydraulicpump hydraulic cylinder accumulator check valves andrelief valves into one component by using themanifold blockCompared to conventional hydraulic systems EHAs are verysmall and do not need complex pipe lines The EHA systemsare seen as environment-friendly hydraulic servo systembecause it does not leak oil

The main advantage of using EHA systems on the wholecomes from its increased energy efficiency [5ndash10] The EHAsystems designed in this research do not use any controlvalves such as a servo valve and can also supply theminimumenergy needed for the hydraulics system by controlling theservo motor meaning that the EHA systems are extremelyenergy efficient Also by using an electric servo motor asa control device EHA systems can quickly achieve fastresponse and high accuracy However since the EHA systemsare a kind of hydraulic servo system it has nonlinearityrelated to the friction force mainly located in the hydrauliccylinderTheEHAsystemalso has the closed hydraulic circuit

2 The Scientific World Journal

which can use relatively little working fluidThe performanceof the EHA system can also be sensitive to variation withinthe system parameters of the working fluid such as varietyof effective bulk modulus and the leakage coefficient due tochanges in the working environment [5 7 9 11ndash13] Thosenonlinearity and uncertainty may cause the position controlsystem of the EHA to be unstable or have a much degradedperformance

In order to obtain the desirable performance of an EHAposition control system a number of control strategies havebeen applied and investigated by many researchers Some ofthe work has used linear control theory for the position con-trol of EHA [14ndash16] The nonlinear robust control schemessuch as sliding mode control back-stepping control andFuzzy logic control have been proposed to achieve robustnessof position control system for EHA by compensating systemnonlinearity and uncertainties and to compensate for thenonlinear dynamics of EHA system such as friction [3 5 6 9ndash11 14 17 18] Adaptive control approaches have been also pro-posed to make up for parameter disturbances such as varyingthe viscous friction coefficient and the effective bulkmodulusdue to change of working condition [1ndash3 9 11 19 20]

However a fast response with small overshoot should bethe objective for the practical use of the EHA position controlsystems in industrial machine such as plastic injection mold-ing machine aircraft actuation system mobile machineryCNC pipe bending machines and hydraulic servo controlof steering system for ship and hydraulic press [1 2] eventhough there are lots of uncertainties and nonlinearity inposition control system of EHA The problem concerninglarge overshoot from the saturation in the electric motor hasbeen considered in recent studies [2] In [2] the saturationof voltage and current of the PWM driver and DC poweris taken into account for position control system of EHAThe researchers of [2] also consider dead zone torque due tothe static friction of motor pump and variable load torqueThe PD-PI hybrid control scheme is proposed to controlthe BLDC motor in order to achieve small overshoot andto compensate for the load disturbances However only acomputer simulation can verify the proposed linear controlscheme although there are lots of uncertainty and nonlinear-ity besides the dead zone and motor saturation In [2 19]they proposed a self-turning Fuzzy PID control scheme tocompensate for the dead zone saturation of motor angularvelocity parameter uncertainties and load disturbance withexperimental verification However improved accuracy andfast response appear necessary to apply the position controlresults to servo press Also the design of fuzzy rules dependslargely on the experience of experts or input-output data

In this research an adaptive antiwindup PID slidingmode control scheme is proposed to solve problems ofovershoot in position control caused by saturation of angu-lar velocity to the electric motor The previous researchesconsidered an antiwindup scheme based on the conventionalsliding mode controller [21 22] However the proposedcontroller consists of an adaptive control and a conventionalsliding mode control with antiwindup scheme Thereforethe proposed control scheme looks to implement robustnessto parameter variations in a variety of working condition

nonlinearity and load disturbance The validation of theproposed control scheme is verified by experiment

This paper will proceed as follows Section 2 reveals thecomposition and details of the position control test rig of anEHA prototype followed by systemmodeling and parameteridentification of the EHA systems in Section 3 In Section 4position controllers for the EHA system using traditionalPID antiwindup PID and adaptive antiwindup PID slidingmode control schemes are presentedThe comparative resultsof the experimental tests executed on the position controltest rig of EHA applying PID antiwindup PID and adaptiveantiwindup PID sliding mode control schemes are revealedin Section 5 followed by conclusions in Section 6

2 Experimental Setup of the ElectrohydraulicActuator Prototype

In this study the EHA system consists of a bidirectionalbrushless direct current (BLDC) motor axial displacementhydraulic piston pump acting doubly as a double rodhydraulic cylinder accumulator check valves and reliefvalves The accumulator serves as a hydraulic reservoir pres-surizing to obtain improved response speed by increasingthe suction ability of the hydraulic pump and to preventcavitations of the EHA system The electric motor drivingthe hydraulic pump directly controls the hydraulic cylinderThe direction change of the hydraulic cylinder depends onthe change of rotation direction of the electric motor Alsoan electric motor installed in the EHA system controls theflow rate and pressure of the working fluid by controllingthe velocity and torque of the electric motor The DC powersupply which has a maximum DC power supply capacityof 27 kW supplies electric power to the electric motorThe detailed specifications of the components are shown inTable 1

In this paper we developed an EHA prototype which canbe used for various purposes in industries such as an aircraftactuation system mobile construction machinery servopress steering gear and injection molding machine with theoverall specifications shown in Table 2 Figure 1 is a photo ofthe position control test rig of the EHA system followed bythe schematic diagram of hydraulic position control test rigfor the developed EHA prototype in Figure 2

The test rig for the developed EHA prototype consists oftwo hydraulic cylinders amain cylinder which is a part of theEHA and a load cylinder Two cylinders are directly coupledby a shaft and the load mass is installed between the twocylinders The electric motor controls the main cylinder ofEHA while the load cylinder is controlled by the servo valve

The hydraulic flow to the main cylinder is controlled bythe electric BLDC servo motor operated through a driverdirectly connected to the proposed controller The positionof the main cylinder gets measured by a linear variabledisplacement transducer (LVDT) installed on the bracketattached to the load mass and another one is also installedin the hollow of the main cylinder rod The signal from theLVDT is used for the closed-loop control of themain cylindervia proposed controllers The output force gets measured bythe load cell installed at the coupling between two cylinders

The Scientific World Journal 3

Table 1 Specification of components for the EHA prototype

Specification ValuePump

Type Fixed displacement axial pistonpump (bidirectional)

Number of pistons 7Displacement 5 times 10

minus6m3

Maximum pressure 206 times 106 Pa

Motor

Type Brushless DC motorDC link voltage 270VRated current 524 ArmsRated torque 134NsdotmMax speed 10000 rpmPower 9 kW

Hydraulic cylinder

Type Double rod double actingPiston diameter 0108mDiameter of rod 0044mLength of stroke 00650mMaximum pressure 206 times 10

6 PaAccumulator

Acc Type BladderGas precharge pressure 049 times 10

6 PaAccumulator volume 07 L

Load massWeight 50 kg

DS1104 dSPACE board is used as a main control device toimplement proposed position controllers optimal PID opti-mal antiwindup PID controller and an adaptive antiwindupPID sliding mode controller The real-time control system isprogrammed by MATLABSimulink and gets transferred tothe dSPACE board through the Real-Time Workshop Theparameters of the controller implemented on the board can bechanged and monitored through the Control-Desk softwarein real time

3 Modeling of Position ControlSystem of EHA

In this section we derive a mathematical model of EHAsystems identifying system parameters by the signal com-pression method (SCM) in order to design the positioncontroller of EHA

31 Mathematical Model of EHA System Figure 3 shows theschematic diagram of EHA system As shown in Figure 3the pump inoutflows 119876

119886

119876119887

can be represented with the

Table 2 Specifications of the EHA prototype

Specifications ValuesStall force Max 1541NStroke Max plusmn337mmVelocity 1230sim1425mmsPower Max 90 kW

Servo valve and fiter

EHALoad cell

Load cylinder

Motor amplifier

Load mass

LVDT

Figure 1 Photo of the position control test rig of the EHA

pump inletoutlet chamber pressures 119901119886

119901119887

and the cylinderdisplacement 119909 as follows

119876119886

= 119863119901

120596119901

minus 120585 (119901119886

minus 119901119887

) minus119881119886

120573119890

119889119901119886

119889119905minus 119862119890119901

(119901119886

minus 119901119903

)

119876119887

= 119863119901

120596119901

minus 120585 (119901119886

minus 119901119887

) +119881119887

120573119890

119889119901119887

119889119905+ 119862119890119901

(119901119887

minus 119901119903

)

(1)

where 119863119901

is displacement of the pump 120585 is the pump crossport leakage coefficient 119862

119890119901

is the pump leakage coefficient119901119903

is the accumulator pressure 120573119890

stands for the effectivebulk modulus of the working fluid and119881

119886

119881119887

are inletoutletvolumes of pipe and cylinder chamber respectively

In addition the cylinder inletoutlet flows 1198761

1198762

can berepresented by

1198761

= 119860 +119881119886

120573119890

1198891199011

119889119905+ 119871 (119901

1

minus 1199012

)

1198762

= 119860 minus119881119887

120573119890

1198891199012

119889119905minus 119871 (119901

2

minus 1199011

)

(2)

where 119860 is pressurized area of hydraulic piston 119871 is internalleakage coefficient of the cylinder and 119901

1

1199012

are the cylinderinletoutlet chamber pressures respectively In this study theexternal leakage of the cylinder is neglected

In this study 119881119886

and 119881119887

are assumed to be equal dueto the double rod and double acting hydraulic cylinder isused in EHA system Also 119881

119886

and 119881119887

can be expressed with


Force

Reservoir

Position

M

Computer system

MATLAB Real-TimeWorkshop Simulink

Optimal PID controller

DS1104 dSPACE

Load cylinder

Main cylinder

Servo valve

Mass

Servo motordirver

Loadcontroller

LoadLVDT

AMP

cell

Figure 2 Schematic diagram for the test rig of the EHA prototype

Bidirectional variable displacement

axial piston pump

MassCylinder rod

Electricmotor

Piston

Cylinder

p1

Q1

p2

Q2

pa

Qa

pb

Qb

120596p

J

x

Figure 3 Schematic diagram of an EHA system

the nominal volume of each EHA chamber 1198810

which equalsthe mean volume of pipe and cylinder chamber as follows

119881119886

= 1198810

+ 119860119909

119881119887

= 1198810

minus 119860119909

(3)

Since a solid tubing is used in the prototype the pressureimpact of pipe expansion and flexibility on the effectivebulk modulus is assumed to be negligible Therefore therelationship pump port flows (119876

119886

119876119887

) and cylinder chamberflows (119876

1

1198762

) can be expressed as

1198761

= 119876119886

1198762

= 119876119887

(4)

Then the load 119876119871

can be defined as follows [21]

119876119871

=1198761

+ 1198762

2=119876119886

+ 119876119887

2 (5)

The relationship between the pump port pressures (119901119886

119901119887

) and cylinder chamber pressures (1199011

1199012

) can then beexpressed as

119901119886

= 1199011

+ 119901pipe

119901119887

= 1199012

minus 119901pipe(6)

where 119901pipe is pressure dropDue to the short length of tubing pump port pressures

are assumed to be equal that is the pressure drop can beneglected to the actuator inlet and outlet pressures such that[7 8 11 17 22]

119901119886

= 1199011

119901119887

= 1199012

(7)

Since the symmetrical double-rod hydraulic cylinder isused in this EHA system the following equation for thecylinder chamber pressures can be established

1198891199011

119889119905= minus

1198891199012

119889119905 (8)

By combining (1) (2) (5) (7) and (8) a simplifiedpumpcylinder model equation can be expressed as

119863119901

120596119901

= 119860 +1198810

120573119890

119871

+ 119862119879

119901119871

+ 1198911

(9)

where 119901119871

= 1199011

minus 1199012

119862119879

= 120585 + 119871 + (119862119901

2) is theequivalent leakage coefficient and 119891

1

is unmodeled dynamicsof hydraulic part of the EHA system

The actuator force and the displacement of the load canbe represented as

119865 = 119901119871

119860 = 119872 + 119861 + 1198912

(10)


where 119860 is pressurized area of hydraulic cylinder119872 is massof load 119861 is viscous friction coefficient and 119891

2

is lumpeduncertain nonlinearities due to external disturbance theunmodeled friction forces and other hard-to-model terms ofmechanical part of the EHA system

By combining (9) and (10) and neglecting 1198911

and 1198912

the linear transfer function between the angular velocity ofthe pump and the displacement of the main cylinder can beexpressed as

119866ℎ

(119904) =119909 (119904)

120596119901

(119904)

= (

119860119863119901

120573119890

1198721198810

)(1199043

+ (119861

119872+

119862119879

119863119901

120573119890

1198810

) 1199042

+

120573119890

(119861119862119879

119863119901

+ 1198602

)

1198721198810

119904)

minus1

(11)

where 119904 is the Laplace operatorBased on (11) the variation of the effective bulk modulus

of the working fluid can increase or decrease the compress-ibility of the working fluid and then the natural frequency ofthe EHA system can be adjusted meaning that the variationof the system parameters can affect the performance of theEHA position control system

On the other hand the dynamics of the BLDC motorwith controller that is electric part of the EHA system isconsidered as [14 23]

119866119898

(119904) =

120596119901

(119904)

V119903

(119904)=

119870119898

(1198791

119904 + 1)

119879119898

1

1199042

+ 119879119898

2

119904 + 1 (12)

where 119870119898

1198791

119879119898

1

and 119879119898

2

are parameters of the BLDCmotor with controller

32 Parameter Identification In general the unknown sys-tem parameters of the linear elements in a nonlinear systemcan be identified by the signal compression method (SCM)[24] In this study SCM is applied to identify the hydraulicpart of the EHA prototype

However hydraulic servo systems engage in significantnonlinear behavior due to system uncertainties such as pipelosses leakages and parameter variations of theworking fluid[9ndash11 16 17 19] Some of those uncertainties are sometimesneglected because the SCM can only identify the linear partof the system in the identification process This study thencompensates for the neglected nonlinear behavior of the EHAsystem by using a robust controller that is adaptive PIDsliding mode control scheme

As shown in Figure 4 the test signal that has the sameamplitude up to 4Hz in the frequency domain is applied tothe position control system of EHA by constructing a close-loop proportional control system to obtain a more accurateequivalent impulse response for the precise identification ofunknown parameters Figure 5 shows the comparison of thefrequency response of the position control system between

0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

Figure 4 Test signal

40

50

60

70

ExperimentModel

Mag

nitu

de (d

B)

Frequency (Hz)10minus1 100 101

Figure 5 Bode plots of the closed loop nominal model and theclosed loop actual system with a proportional controller

the closed-loop nominal model and the closed-loop actualsystem with a proportional controller whose gain is 1000

The identified transfer function of a closed-loop actualsystem with a proportional controller for EHA positioncontrol system is as follows

119870119866 (119904)

1 + 119870119866 (119904)=

118 times 107

1199043

+ 22311199042

+ 718 times 105

119904 + 118 times 107

(13)

where119870 is proportional gain of closed-loop actual systemFrom the identified nominal model for the closed-loop

position control system for EHA with the proportionalcontroller the parameters of the position control system canbe acquired by eliminating the effect of the proportionalcontroller mathematically


0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

ExperimentModel

(a) Response for the test signal

0 05 1 15 2 25 3 35 4 45

0

2

4

6

8

10

12

14

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

minus2

ExperimentModel

(b) Step response

Figure 6 Time responses of the position control system obtained by the experiment and nominal model

After eliminating the effect of the proportional controllerthe identified transfer function of the hydraulic part of theEHA prototype is as follows

119866 (119904) =118 times 10

4

1199043

+ 22311199042

+ 718 times 105

119904 (14)

The identified system parameters were verified on thetime domain Figures 6(a) and 6(b) show the responseagainst test signal and the step responses of the nominalmodel and actual system respectively The validations ofthe identified parameters are estimated quantitatively by thecross-correlation as shown in [24]

Corr =sum119873

119896=1

(119884119901

(119896) minus 119884119901

) (119884119898

(119896) minus 119884119898

)

radicsum119873

119896=1

(119884119901

(119896) minus 119884119901

)radicsum119873

119896=1

(119884119898

(119896) minus 119884119898

)2

(15)

where 119884119901

(119896) and 119884119898

(119896) are the responses of the real andnominal model in time train 119896 respectively 119884

119901

and 119884119898

arethe mean values of 119884

119901

(119896) and 119884119898

(119896) 119873 is the acquired datanumberThe value of the cross-correlation is 0904 which canbe calculated from the identification results The identifiedsystem parameters are then available to controller designThe identified transfer function in (14) is used to obtain theoptimal control gains for designed position controllers byusing MATLABSimulink

In addition the dynamics characteristic of the BLDCmotor with controller is also investigated by using SCMThetransfer function of the BLDC motor with controller can besimplified to a second-order linear system as described in(12) Figure 7 shows bode plot of the actual BLDC motorwith controllerThe bandwidth of the actual open loop BLDCmotor with controller is about 400 rads The bandwidth ofthe BLDC motor with controller is 20 times larger than thatof hydraulic part of the EHA systemTherefore in this study

110

115

120

125

Experiment

Mag

nitu

de (d

B)

Frequency (Hz)100 101 102

Figure 7 Bode plots of the open-loop actual BLDC motor withcontroller system

we consider that the dynamics of BLDCmotorwith controlleris negligible because the motor dynamic is sufficiently fasterthan hydraulic part of the EHA Then the electric part ofthe EHA system is taking account of the motor gain 119870V andcurrent saturation is reflected in the controller design and thecomputer simulations stage

4 Position Controller Design fora Prototype EHA

In this study an EHA prototype is developed for appli-cation in the industrial fields As mentioned in Section 2performance characteristics such as no overshoot and rapidresponse time in position control are the typical demandsof the EHA for industrial use Therefore appropriate control


Positioncontroller

BLDC motor with controller

Hydraulic part of EHA system

Motorcontroller

r e

120596

e xi

Ks

Km

Tm 120596p

plusmn100A

minus+ minus+

rpmplusmn10000Reference position (xd) 1

Jms + Bm

x(s)

120596p(s)=

ADp120573e

e

MV0

s3 + ((BM) + (CTDp120573eV0))s2 + (120573 (BCTDp + A2)MV0)s

Figure 8 Block diagram for the position control system of the EHA

strategy should be applied to EHA position control system inorder to achieve these outcomes

Figure 8 shows the block diagram of position controlsystem of the EHA As mentioned in the previous section inFigure 8 the dynamics of the BLDC motor with controlleris considered as just the motor gain 119870

119898

and saturation ofcurrent of the BLDC motor is considered as an actuatorconstraint

In order to compare and show the desirable performancecharacteristics for the proposed EHA control system threekinds of position controllers are designed based on identifiedsimple transfer function First an optimal PID controller isdesigned and applied to the system in order to understandthe basic characteristics of EHA position control systemThe optimal antiwindup PID controller is then designed toreduce overshoot due to saturation of current in the electricmotor Finally an adaptive antiwindup PID sliding modecontroller is designed to obtain the desirable performanceand robustness against the effect of hardware saturation loaddisturbance and parameter variation

41 Optimal PID and Optimal Antiwindup PID ControlSystems An optimal PID position control system is designedto understand basic control performance of EHA prototypeAlso an optimal antiwindup PID control system for the EHAis designed to obtain desirable performance characteristicconsidering saturation in electric motor

The cost function is considered during the design ofthe optimal PID and optimal antiwindup PID controllers asfollows [25]

119869 (119870119901

119870119894

119870119889

) =

infin

sum

119905=0

(119910step (119905) minus 119910119889

step (119905)) (16)

where 119910119889step(119905) is the desired step response of the optimalPID and optimal antiwindup PID control systems and 119910step(119905)is the step response by the identified transfer function in(14)The optimal PID and optimal antiwindup PID controllerdesigns can be stated as

min119870

119901119870

119894119870

119889

119869 (119870119901

119870119894

119870119889

) (17)

where 119870119901

119870119894

and 119870119889

are the proportional integral anddifferential control gains respectively

There are many optimization algorithms in the opti-mization toolbox of MATLABSimulink The cost functionis given by (119870

119901

119870119894

119870119889

) And the gains of the optimal PIDand optimal antiwindup PID controllers 119870

119901

119870119894

and 119870119889

can be found by using the optimization toolbox of MAT-LABSimulink To find optimal control gains the referencestep input is applied in optimization process and the responseperformance is set as follows The amplitude of reference is20mm the rising time is 04 seconds settling time is 08seconds percent overshoot is under 5 and steady-stateerror is under 01mm

In the optimization process of optimal PID and optimalantiwindup PID controllers system parameter uncertaintiesdue to the modeling error are considered Specifically plusmn10ofmodeling error is considered in terms of identified transferfunction associate with variable system parameters such aseffective bulk modulus of the working fluid and total leakagecoefficient

With respect to optimal antiwindup PID and antiwindupPID sliding mode controllers the antiwindup algorithmshown in Figure 9 is used to consider saturation in an electricmotor [2 26ndash28]

Figure 10 shows the results of a computer simulationwith optimal antiwindup PID control gains using MAT-LABSimulink optimization toolbox based on identifiedtransfer function considering model uncertainty The dottedline there shows upper bound and lower bound consideringthe plusmn10 of modeling error and straight line shows nominalsimulation result with optimal antiwindup PID control gainsAs shown in Figure 10 the simulation result with optimalgains satisfies set response performance

42 Adaptive Antiwindup SlidingModeController Consideringthe PID Sliding Surface In this study the adaptive anti-windup PID sliding mode control scheme is also designedto achieve desirable performance and robustness of EHAposition control systems Throughout the controller designprocess the effective bulk modulus and leakage coefficientsare considered as variable systemparameters In order to con-sider saturation in electric motor the antiwindup algorithmpresented in Figure 8 also applies to the adaptive antiwindupPID sliding mode control scheme


Position controller BLDC motor with controller

Motorcontroller

r i re

120596

i

Ks

Ks Km

Tm

plusmn100A

minus+

+++

+

rpmplusmn10000

minus

ka

ei

s

k2s

orKi

s

k3 or Kd

k1 or Kpe 120596p1

Jms + Bm

Figure 9 Block diagram for adopted antiwindup algorithm in controller design

06 07 08 09 1 1118

19

20

21

22Upper boundary

Lower boundary

Nominal response

0 02 04 06 08 1 12 14 16 18 20

5

10

15

20

25

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

Position referencePosition

Figure 10 Step response of the optimal antiwindup PID controlsystem in computer simulation based on identified transfer functionof EHA system

The PID sliding surface 120577 for the design of the adaptivesliding mode control system is defined as [29ndash32]

120577 = 1198961

119890 + 1198962

int 119890 119889119905 + 1198963

119890 (18)

where the tracking error of the position 119890 = 119909119889

minus 119909 119909 isthe displacement of hydraulic cylinder and 119896

1

1198962

and 1198963

arepositive design parameters

The sliding mode control law consists of equivalent androbust control term that is

120596119901

= 120596119901119890119902

+ 120596119901robust (19)

By combining (11) and (18) with consideration of the noiseterm in the jerk =

119898

+ 119899

the derivative of the slidingsurface 120577 can be written as

120577 = minus 1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

120596119901

+ 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119898

+ 119899

)

+ 1198963

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+ 1198963

119889

+ 1198961

119890 + 1198962

119890

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891

(20)

where 119891 = 1198911

+ (119862119879

119860)1198912

+ (1198810

119860119863119901

120573119890

) 1198912

is lumped uncer-tain nonlinearities of the EHA system and the subscripts 119898and 119899 denote the nominal and noisy values respectively

To determine the equivalent control term 120596119901119890119902

the noiseterm in the jerk is neglected and it is assumed that the slidingsurface 120577 is at steady-state that is 120577 = 0 and then theequivalent control law can be determined as

120596119901119890119902

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119889

(21)

Thus 120577 can be rewritten as

120577 = 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119899

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891 (22)

In the standard sliding mode control to satisfy the reach-ability condition that directs system trajectories toward asliding surface where they remain and to attenuate chatteringin the control input the derivative of the sliding surface isselected as [33 34]

120577 = minus119863120577 minus 119870 sgn (120577) (23)

where119863 and119870 are positive design parameters


To determine a robust control term120596119901robust that achieves

robustness to uncertainties such as noise terms in the jerk andunmodeled system dynamics as well as the disturbance forceit is assumed that

1198630

10038161003816100381610038161205771003816100381610038161003816 + 1198700 gt

100381610038161003816100381610038161003816100381610038161003816

1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119899

+1

1198721198810

119891)

100381610038161003816100381610038161003816100381610038161003816

+ 120578

(24)

where 1198700

= (119860119863119901

120573119890

(1198611198810

+ 119862119879

119863119901

119872120573119890

))119870 and 1198630

=

(119860119863119901

120573119890

(1198611198810

+119862119879

119863119901

119872120573119890

))119863 are positive design parametersand 120578 is sufficiently small positive parameter On the assump-tion of (24) the robust control law is determined as

120596119901robust = 119863120577 + 119870 sgn (120577) (25)

Finally the sliding mode control law is selected as

120596119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 119889

+1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119863120577 + 119870 sgn (120577)

(26)

If the total leakage coefficient 119862119879

and the effective bulkmodulus of EHA systems 120573

119890

are considered as variableparameters that is 119862

119879

= 119862119879119899

+ 119862119879119901

and 120573119890

= 120573119890119899

+ 120573119890119901

then the uncertainty 120595 is defined as

120595 = 120579119879

120593 = (1198721198810

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

) 119898

+ (

(1198602

+ 119861119862119879119901

119863119901

) 120573119890119901

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)

(27)

where 120579119879

= [(1198721198810

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)) ((1198602

+

119861119862119879119901

119863119901

)120573119890119901

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

))] and 120593 = [119909

119898

119909

] Theparameter vector 120579 is considered as an unknown parametervector and it can be estimated by using the update lawFrom (26) and estimated unknown parameter vector 120579 theestimated sliding mode control law can be selected as

119901

=

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119860119863119901

120573119890119899

times 119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

+ 119863120577 + 119870 sgn (120577)

(28)

In order to obtain the update law for the unknownparameters the Lyapunov candidate is defined as

119881 =1

21205772

+1

2120574120579119879

120579 (29)

where 120579 = 120579 minus 120579 120574 is a positive parameter Also 120579 and 120579 arethe nominal and estimated parameter vectors respectivelyThe derivative of the Lyapunov candidate including slidingdynamics is expressed as

= 120577 [119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

minus1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

119901

] minus1

120574120579119879120579

(30)

Substituting the estimated angular velocity of the pump 119901

given by (28) into (30) yields

= minus1198631205772

minus 119870120577 sgn (120577) + 120579119879 (120577120593 minus 1120574

120579) (31)

From the derivative of the Lyapunov included sliding dynam-ics the update law for the unknown parameters is selected as

120579 = 120574120577120593 (32)

Therefore the derivative of the Lyapunov is given as

= minus1198631205772

minus 119870120577 sgn (120577) le 0 (33)

By applying invariant set theorem for (33) the asymptoticalstability of the EHAposition control system is guaranteed [3334]

Finally the sliding mode control law can be selected as

119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times (120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890)

+ 119863120577 + 119870 sgn (120577) (34)

where 120573119890

(119905 + 119879119904

) = (119889120573119890

(119905)119889119905)119879119904

+ 120573119890

(119905) 119862119879

(119905 + 119879119904

) =

(119889119862119879

(119905)119889119905)119879119904

+ 119862119879

(119905) and 119879119904

is the sampling time


Table 3 Optimal gains of optimal PID antiwindup PID and adaptive anti-windup PID sliding mode control systems in experiments

Gain Optimal PID Optimal antiwindup PID Adaptive antiwindup PID sliding modeProportional (119870

119901

or 1198961

) 10343 16947 950Integral (119870

119894

or 1198962

) 184 10877 0001Derivative (119870

119889

or 1198963

) 47 4160 145Antiwindup (119870

119886

) mdash 0001 0001

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)

Reference 10 mmReference 20 mm


(b) Position errors

Figure 11 Performances of optimal PID position control system for EHA without load disturbance (only mass load is applied)

5 Experimental Results and Discussion

This section looks to the experimental results in order tocompare the performance of a position control systemsThreekinds of controllers are compared in order to show improvedperformance of overshoot reduction and robustness to theparameter change and load disturbance of the proposedcontroller for EHA position control system

Firstly optimal PID controller is designed and imple-mented by experiments Figure 11 shows experimental resultsof the PID position controller with step references of fourdifferent amplitudes which are 10 20 30 and 40mmwithoutload disturbance that is only mass load is applied as load tothe position control system

The gains of the three-position controllers are optimallyselected by using MATLABSimulink optimization toolboxbased on identified system dynamic model of (14) In theprocess of optimal gain selection the amplitude of positionreference is set to 20mm and target performance is 04seconds of the rising time 08 seconds of settling timeunder 5 of percent overshoot and under 01mm of steady-state error In experiments the optimal gains derived fromMATLABSimulink optimization toolbox are tuned to obtainbest responses The finally tuned optimal gains of the threecontrollers are represented in Table 3

As shown in Figure 11 large overshoot is observed inposition control result for using the optimal PID positioncontroller The performance of optimal PID position control

system is represented in Table 4The performance of optimalPID controller in experiment is not satisfactory the aimedvalue In detail large overshoot is observed in experimentsThe current and angular velocity saturations of BLDC motorin position experiment result of optimal PID controller areshown in Figure 12The large overshoot in position controllercan be caused due to saturation of BLDC motor as shown inFigure 12

To reduce such a large overshoot an antiwindup algo-rithmwhich is described in Figure 9 is applied to the positioncontrol system of the EHA The performance characteristicsof an antiwindup PID position controller are depicted inFigure 13 and also the values of performance indices areshown in Table 4 As shown in Figure 13 the overshoot ofthe optimal antiwindup PID controller is more diminishedthan that of the optimal PID controller However as shown inTable 4 the antiwindupPID controller still has a performanceof relatively long settling times which do not meet the aimedperformance Moreover as the amplitude of the referenceinput increases the steady-state error is increasing as shownin Figure 13

Figure 14 shows the experimental results of optimal adap-tive antiwindup sliding mode position control system for theEHA system As shown in Figure 14 the optimal antiwindupPID sliding mode position control system has a better per-formance than the system with optimal PID and antiwindupPID controllers such that relatively small overshoot and shortsettling time are obtained by the adaptive antiwindup PID


Table 4 Comparison of the performance of optimal PID optimal antiwindup PID and adaptive antiwindup PID sliding mode controlsystems without load disturbance

Performance Reference Optimal PID Optimal antiwindup PID Adaptive antiwindup PIDsliding mode

Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time

10mm 178 sec 22 sec 06 sec20mm 056 sec 22 sec 06 sec30mm 058 sec 063 sec 07 sec40mm 069 sec 26 sec 07 sec

0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)



(a) Saturation of impressed current

0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)



(b) Saturation of angular velocity

Figure 12 Saturation of BLDCmotor for optimal PID position control system for EHAwithout load disturbance (only mass load is applied)

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors

Figure 13 Performances of optimal antiwindup PID position control system for EHA without load disturbance (only mass load is applied)


0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors

Figure 14 Performances of optimal adaptive antiwindup PID sliding mode position control system for EHA without load disturbance (onlymass load is applied)

0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)

Pressure at port number 1Pressure at port number 2

Figure 15 Pressure variations of optimal PID position control system for EHA without load disturbance (only mass load is applied)

sliding mode control system As shown in Figure 14 andTable 4 the consistent performance is realized by the optimaladaptive antiwindup PID sliding mode control system Thetime-domain quantitative performance indicators are alsosummarized in Table 4 for optimal adaptive antiwindup PIDsliding mode control system without load

On the other hand Figure 15 depicts pressure variationsof the optimal PID EHAposition control systemwithout loaddisturbance As shown in Figure 15 the range of operatingpressure of the EHA system is under 3MPa inwhich the effec-tive bulk modulus can be significantly changed according tocontained air ratio of working fluid [5 7 11ndash13] as shownin Figure 16 The range of operating pressure for antiwindup

PID and antiwindup PID sliding mode control system is alsounder 3MPa

Figure 17 depicts pressures of the EHA system when theposition control systems are operating with load disturbanceAs shown in Figure 17 the maximum operating pressures ofthe EHA system is about 10MPa when the position controlsystems are working under the condition of adding distur-bance loadThemaximumoperating pressure for antiwindupPID and antiwindup PID sliding mode control system isalso up to 10MPa Thus the effective bulk modulus can bechanged greatly according to variation of operating pressureof the EHA systems In addition total leakage coefficient alsocan be changed in response to the level of working pressure


0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)

Contained air ratio ()001

110

Figure 16 Example of the relationship between pressure and effective bulk modulus [12 13]

0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)


Figure 17 Pressure variations of optimal PID position control system for EHA with load disturbance

In order to investigate robustness to the lumped systemuncertainties such as varying parameters and modeling errorand load disturbance the performances of the three-positioncontrollers of EHA system are compared in experiments byapplying load disturbance The experiments are conductedwith 20mm position reference and the load disturbance isapplied by controlling the opening area of servo valve whichis installed in a meter outline of load cylinder

As shown in Figure 18 the optimal PID and optimalantiwindup PID control systems do not achieve robustnessto the parameter change and load disturbance In spite of theload disturbance and saturation of the BLDC motor in theposition control system for EHA the adaptive antiwindupPID sliding mode control system has a better performanceand robustness than the optimal PID and optimal antiwindupPID control systems such that the adaptive antiwindup PID

sliding mode control system recovers the parameter changeand external disturbance in about 09 sec with a small over-shootThe time-domain quantitative performance indicatorsare summarized in Table 5 for optimal PID antiwindup PIDand adaptive antiwindup PID sliding mode control systemswith load disturbance The list of terms are represented inTable 6

6 Conclusion

This paper discussed the application of the optimal PIDcontroller with optimal control gains using the optimizationtoolbox ofMATLABSimulink to the position control systemof EHA in order to understand the basic performancecharacteristics of the system In experiment with optimal PIDcontroller large overshoot due to saturation of electric motoris observed


0 05 1 15 2 25 30

5

10

15

20

25

30

35

Position referenceOptimal PIDOptimal antiwindup PIDAdaptive antiwindup PID sliding mode

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)

Optimal PIDOptimal antiwindup PIDAdaptive antiwindup PID sliding mode

(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force

Figure 18 Performances of optimal PID antiwindup PID and adaptive antiwindup PID sliding mode control systems for EHA with loaddisturbance

Table 5 Comparison of the performance of optimal PID optimalantiwindup PID and adaptive antiwindup PID slidingmode controlsystems with load disturbance

Performance OptimalPID

Optimalantiwindup PID

Adaptive antiwindupPID sliding mode

Overshoot 45 39 12Settling time 38 sec 18 sec 08 sec

In order to achieve the desirable performance of EHAposition control system optimal antiwindup PID controller

and adaptive antiwindup PID sliding mode controller arealso applied to control the position of EHA systems Inthe process of robust control system design mathematicalmodeling was carried out based on theoretical analysisand the simple third-order nominal model was derivedThe system parameters of the simple second-order nominalmodel were identified by the signal compression method

Based on the experimental results and the time-domainquantitative performance indicators the performance of theEHA position control system can be improved and robustto the load disturbance parameter variation and system


Table 6 List of terms

Notation Description UnitA Pressurized area of hydraulic cylinder of EHA m2

119861 Viscous friction coefficient Nmsec119862119879

Equivalent leakage coefficient m3secPa119863119901

Pump displacement m3rad119890 Position error mm119894 Current Arms119870pipe Pipe coefficient relating pressure drop to flow Pam6sec2

119870119904

Angular velocity to voltage gain mdash119870119901

Proportional control gain mdash119870119894

Integral control gain mdash119870119889

Differential control gain mdash119896119886

Antiwindup gain mdash1198961

1198962

1198963

119863 119870 120574 Positive design parameters of adaptive PID sliding mode controller mdash119870119898

1198791

1198791198981 1198791198982

Parameters of dynamic model for the BLDC motor with controller mdashL Internal leakage coefficient of the cylinder m3PasecM Mass load of position control test rig of EHA N1199011

1199012

Cylinder inletoutlet chamber pressure Pa119901pipe Pressure drop of pipe between the pump and the cylinder Pa1198761

1198762

Cylinder inletoutlet flow m3sec119904 Laplace operator mdash119879119898

Torque of BLDC motor Nm(radsec)119879119904

Sampling time sec119881119886

119881119887

Volumes of inletoutlet pipe and cylinder chamber m3

1198810

Mean volume of pipe and cylinder chamber m3

V119890

Voltage signal of error for angular velocity of BLDC motor VV119903

Voltage signal of reference angular velocity of BLDC motor VV120596

Voltage signal of angular velocity V119909 Cylinder displacement m119910119889

step(119905) Desired step response of the optimal PID and antiwindup PID control system mdash119910step(119905) Step response by the identified transfer function mdash120573119890

Effective bulk modulus Pa120596119901

Pump angular velocity radsec120577 Sliding surface mdash120585 Pump cross port leakage coefficient m3Pasec

uncertainty such as saturation of electric motor by using theadaptive antiwindup PID sliding mode controller

References

[1] J-MZheng S-D Zhao and S-GWei ldquoFuzzy iterative learningcontrol of electro-hydraulic servo system for SRM direct-drive volume control hydraulic pressrdquo Journal of Central SouthUniversity of Technology vol 17 no 2 pp 316ndash322 2010

[2] J-M Zheng S-D Zhao and S-G Wei ldquoApplication of self-tuning fuzzy PID controller for a SRM direct drive volumecontrol hydraulic pressrdquoControl Engineering Practice vol 17 no12 pp 1398ndash1404 2009

[3] D K Pitipongsa Guansak and W Po-Ngaen ldquoTwo layer fuzzygeneralized Maxwell-slip compensator in direct drive servohydraulic systemrdquo inAppliedMechanics andMaterials pp 420ndash427 2012

[4] Y Cao X Y Tian GQWuQM Li andYHWu ldquoSimulationexperiment of roll bending machine in valveless servo systemrdquoAdvanced Materials Research vol 619 pp 410ndash414 2012

[5] Y Chinniah R Burton and S Habibi ldquoFailure monitoringin a high performance hydrostatic actuation system using theextended Kalman filterrdquo Mechatronics vol 16 no 10 pp 643ndash653 2006

[6] Y Chinniah R Burton S Habibi and E Sampson ldquoIdentifi-cation of the nonlinear friction characteristics in a hydraulicactuator using the extended Kalman filterrdquo Transactions of theCanadian Society for Mechanical Engineering vol 32 no 2 pp121ndash136 2008

[7] Y Chinniah S Habibi R Burton and E Sampson ldquoParameteridentification in a high performance hydrostatic actuationsystem using the Unscented Kalman filterrdquo Transactions of theCanadian Society for Mechanical Engineering vol 30 no 3 pp375ndash389 2006


[8] K Rongjie J Zongxia W Shaoping and C Lisha ldquoDesign andsimulation of electro-hydrostatic actuator with a built-in powerregulatorrdquoChinese Journal of Aeronautics vol 22 no 6 pp 700ndash706 2009

[9] HMKim SH Park JH Song and J S Kim ldquoRobust positioncontrol of electro-hydraulic actuator systems using the adaptiveback-stepping control schemerdquo Journal of Systems and ControlEngineering vol 224 no 6 pp 737ndash746 2010

[10] Y Lin Y Shi and R Burton ldquoModeling and robust discrete-time sliding-mode control design for a fluid power electrohy-draulic actuator (EHA) systemrdquo IEEEASME Transactions onMechatronics vol 18 no 1 pp 1ndash10 2013

[11] HM Kim S H Park J M Lee and J S Kim ldquoA robust controlof electro hydrostatic actuator using the adaptive back-steppingscheme and fuzzy neural networksrdquo International Journal ofPrecision Engineering andManufacturing vol 11 no 2 pp 227ndash236 2010

[12] J Kajaste H Kauranne A Ellman and M Pietola ldquoExperi-mental validation of differentmodels for effective bulkmodulusof hydraulic fluidrdquo in Proceedings of the 9th ScandinavianInternational Conference on Fluid Power (SICFP rsquo05) p 16Linkoping Sweden 2005

[13] K Sunghun and M Hubertus Measurement of Effective BulkModulus for Hydraulic Oil at Low Pressure vol 134 ETATS-UNIS American Society of Mechanical Engineers New YorkNY USA 2012

[14] M A El Sayed and S Habibi ldquoInner-loop control for electro-hydraulic actuation systemsrdquo Journal of Dynamic SystemsMeasurement and Control vol 134 no 1 2012

[15] S Habibi and A Goldenberg ldquoDesign of a new high perfor-mance electrohydraulic actuatorrdquo IEEEASME Transactions onMechatronics vol 5 no 2 pp 158ndash164 2000

[16] M Pachter C H Houpis and K Kang ldquoModelling and controlof an electro-hydrostatic actuatorrdquo International Journal ofRobust and Nonlinear Control vol 7 no 6 pp 591ndash608 1997

[17] SWang R Burton and S Habibi ldquoSlidingmode controller andfilter applied to an electrohydraulic actuator systemrdquo Journal ofDynamic SystemsMeasurement andControl vol 133 no 2 2011

[18] S Wang S Habibi and R Burton ldquoSliding mode control for anelectrohydraulic actuator systemwith discontinuous non-linearfrictionrdquo Journal of Systems and Control Engineering vol 222no 8 pp 799ndash815 2008

[19] K K Ahn and Q T Dinh ldquoSelf-tuning of quantitative feedbacktheory for force control of an electro-hydraulic test machinerdquoControl Engineering Practice vol 17 no 11 pp 1291ndash1306 2009

[20] S H Cho and R Burton ldquoPosition control of high performancehydrostatic actuation system using a simple adaptive control(SAC) methodrdquoMechatronics vol 21 no 1 pp 109ndash115 2011

[21] H E Merritt Hydrostatic Control Systems Wiley InterscienceNew York NY USA 1967

[22] S R Habibi and R Burton ldquoParameter identification fora high-performance hydrostatic actuation system using thevariable structure filter conceptrdquo Journal of Dynamic SystemsMeasurement and Control vol 129 no 2 pp 229ndash235 2007

[23] J Y Oh G-H Jung G-H Lee Y-J Park and C-S SongldquoModeling and characteristics analysis of single-rod hydraulicsystemusing electro-hydrostatic actuatorrdquo International Journalof Precision Engineering and Manufacturing vol 13 no 8 pp1445ndash1451 2012

[24] N Aoshima ldquoMeasurement of nonlinear vibration by signalcompression methodrdquo Journal of the Acoustical Society ofAmerica vol 76 no 3 pp 794ndash801 1984

[25] G P Liu and S Daley ldquoOptimal-tuning nonlinear PID controlof hydraulic systemsrdquo Control Engineering Practice vol 8 no 9pp 1045ndash1053 2000

[26] S-U Lee and P H Chang ldquoThe development of anti-windupscheme for time delay control with switching action using inte-gral sliding surfacerdquo Journal of Dynamic Systems Measurementand Control vol 125 no 4 pp 630ndash638 2003

[27] M Yokoyama G-N Kim and M Tsuchiya ldquoIntegral slidingmode control with anti-windup compensation and its applica-tion to a power assist systemrdquo Journal of Vibration and Controlvol 16 no 4 pp 503ndash512 2010

[28] M Yokoyama Y Tohta and K Saitoh ldquoA design method ofsliding mode controller for servo-systems subject to actuatorsaturationrdquo JSME International Journal Series C vol 46 no 3pp 960ndash966 2003

[29] I Eker ldquoSecond-order sliding mode control with PI slidingsurface and experimental application to an electromechanicalplantrdquo Arabian Journal for Science and Engineering vol 37 no7 pp 1969ndash1986 2012

[30] I Eker ldquoSecond-order sliding mode control with experimentalapplicationrdquo ISA Transactions vol 49 no 3 pp 394ndash405 2010

[31] I Eker ldquoSliding mode control with PID sliding surface andexperimental application to an electromechanical plantrdquo ISATransactions vol 45 no 1 pp 109ndash118 2006

[32] I Eker and S A Akınal ldquoSliding mode control with integralaugmented sliding surface design and experimental applicationto an electromechanical systemrdquo Electrical Engineering vol 90no 3 pp 189ndash197 2008

[33] J J E Slotine and W Li Applied Nonlinear Control PearsonEducation Upper Saddle River NJ USA 1991

[34] M Krstic I Kanellakopoulos and P Kokotovic Nonlinear andAdaptive Control Design Wiley Interscience New York NYUSA 1995

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DistributedSensor Networks



which can use relatively little working fluidThe performanceof the EHA system can also be sensitive to variation withinthe system parameters of the working fluid such as varietyof effective bulk modulus and the leakage coefficient due tochanges in the working environment [5 7 9 11ndash13] Thosenonlinearity and uncertainty may cause the position controlsystem of the EHA to be unstable or have a much degradedperformance

In order to obtain the desirable performance of an EHAposition control system a number of control strategies havebeen applied and investigated by many researchers Some ofthe work has used linear control theory for the position con-trol of EHA [14ndash16] The nonlinear robust control schemessuch as sliding mode control back-stepping control andFuzzy logic control have been proposed to achieve robustnessof position control system for EHA by compensating systemnonlinearity and uncertainties and to compensate for thenonlinear dynamics of EHA system such as friction [3 5 6 9ndash11 14 17 18] Adaptive control approaches have been also pro-posed to make up for parameter disturbances such as varyingthe viscous friction coefficient and the effective bulkmodulusdue to change of working condition [1ndash3 9 11 19 20]

However a fast response with small overshoot should bethe objective for the practical use of the EHA position controlsystems in industrial machine such as plastic injection mold-ing machine aircraft actuation system mobile machineryCNC pipe bending machines and hydraulic servo controlof steering system for ship and hydraulic press [1 2] eventhough there are lots of uncertainties and nonlinearity inposition control system of EHA The problem concerninglarge overshoot from the saturation in the electric motor hasbeen considered in recent studies [2] In [2] the saturationof voltage and current of the PWM driver and DC poweris taken into account for position control system of EHAThe researchers of [2] also consider dead zone torque due tothe static friction of motor pump and variable load torqueThe PD-PI hybrid control scheme is proposed to controlthe BLDC motor in order to achieve small overshoot andto compensate for the load disturbances However only acomputer simulation can verify the proposed linear controlscheme although there are lots of uncertainty and nonlinear-ity besides the dead zone and motor saturation In [2 19]they proposed a self-turning Fuzzy PID control scheme tocompensate for the dead zone saturation of motor angularvelocity parameter uncertainties and load disturbance withexperimental verification However improved accuracy andfast response appear necessary to apply the position controlresults to servo press Also the design of fuzzy rules dependslargely on the experience of experts or input-output data

In this research an adaptive antiwindup PID slidingmode control scheme is proposed to solve problems ofovershoot in position control caused by saturation of angu-lar velocity to the electric motor The previous researchesconsidered an antiwindup scheme based on the conventionalsliding mode controller [21 22] However the proposedcontroller consists of an adaptive control and a conventionalsliding mode control with antiwindup scheme Thereforethe proposed control scheme looks to implement robustnessto parameter variations in a variety of working condition

nonlinearity and load disturbance The validation of theproposed control scheme is verified by experiment

This paper will proceed as follows Section 2 reveals thecomposition and details of the position control test rig of anEHA prototype followed by systemmodeling and parameteridentification of the EHA systems in Section 3 In Section 4position controllers for the EHA system using traditionalPID antiwindup PID and adaptive antiwindup PID slidingmode control schemes are presentedThe comparative resultsof the experimental tests executed on the position controltest rig of EHA applying PID antiwindup PID and adaptiveantiwindup PID sliding mode control schemes are revealedin Section 5 followed by conclusions in Section 6

2 Experimental Setup of the ElectrohydraulicActuator Prototype

In this study the EHA system consists of a bidirectionalbrushless direct current (BLDC) motor axial displacementhydraulic piston pump acting doubly as a double rodhydraulic cylinder accumulator check valves and reliefvalves The accumulator serves as a hydraulic reservoir pres-surizing to obtain improved response speed by increasingthe suction ability of the hydraulic pump and to preventcavitations of the EHA system The electric motor drivingthe hydraulic pump directly controls the hydraulic cylinderThe direction change of the hydraulic cylinder depends onthe change of rotation direction of the electric motor Alsoan electric motor installed in the EHA system controls theflow rate and pressure of the working fluid by controllingthe velocity and torque of the electric motor The DC powersupply which has a maximum DC power supply capacityof 27 kW supplies electric power to the electric motorThe detailed specifications of the components are shown inTable 1

In this paper we developed an EHA prototype which canbe used for various purposes in industries such as an aircraftactuation system mobile construction machinery servopress steering gear and injection molding machine with theoverall specifications shown in Table 2 Figure 1 is a photo ofthe position control test rig of the EHA system followed bythe schematic diagram of hydraulic position control test rigfor the developed EHA prototype in Figure 2

The test rig for the developed EHA prototype consists oftwo hydraulic cylinders amain cylinder which is a part of theEHA and a load cylinder Two cylinders are directly coupledby a shaft and the load mass is installed between the twocylinders The electric motor controls the main cylinder ofEHA while the load cylinder is controlled by the servo valve

The hydraulic flow to the main cylinder is controlled bythe electric BLDC servo motor operated through a driverdirectly connected to the proposed controller The positionof the main cylinder gets measured by a linear variabledisplacement transducer (LVDT) installed on the bracketattached to the load mass and another one is also installedin the hollow of the main cylinder rod The signal from theLVDT is used for the closed-loop control of themain cylindervia proposed controllers The output force gets measured bythe load cell installed at the coupling between two cylinders






minus6m3


Motor


Hydraulic cylinder


6 PaAccumulator








119886

119876119887





EHALoad cell

Load cylinder

Motor amplifier

Load mass

LVDT



119901119887


119876119886

= 119863119901

120596119901

minus 120585 (119901119886

minus 119901119887

) minus119881119886

120573119890

119889119901119886

119889119905minus 119862119890119901

(119901119886

minus 119901119903

)

119876119887

= 119863119901

120596119901

minus 120585 (119901119886

minus 119901119887

) +119881119887

120573119890

119889119901119887

119889119905+ 119862119890119901

(119901119887

minus 119901119903

)

(1)

where 119863119901


119890119901




119886

119881119887



1198762


1198761

= 119860 +119881119886

120573119890

1198891199011

119889119905+ 119871 (119901

1

minus 1199012

)

1198762

= 119860 minus119881119887

120573119890

1198891199012

119889119905minus 119871 (119901

2

minus 1199011

)

(2)


1

1199012



and 119881119887


119886

and 119881119887



Force

Reservoir

Position

M

Computer system



DS1104 dSPACE

Load cylinder

Main cylinder

Servo valve

Mass

Servo motordirver

Loadcontroller

LoadLVDT

AMP

cell



axial piston pump

MassCylinder rod

Electricmotor

Piston

Cylinder

p1

Q1

p2

Q2

pa

Qa

pb

Qb

120596p

J

x




119881119886

= 1198810

+ 119860119909

119881119887

= 1198810

minus 119860119909

(3)


119886

119876119887


1

1198762


1198761

= 119876119886

1198762

= 119876119887

(4)



119876119871

=1198761

+ 1198762

2=119876119886

+ 119876119887

2 (5)


119901119887


1199012


119901119886

= 1199011

+ 119901pipe

119901119887

= 1199012

minus 119901pipe(6)



119901119886

= 1199011

119901119887

= 1199012

(7)


1198891199011

119889119905= minus

1198891199012

119889119905 (8)


119863119901

120596119901

= 119860 +1198810

120573119890

119871

+ 119862119879

119901119871

+ 1198911

(9)

where 119901119871

= 1199011

minus 1199012

119862119879

= 120585 + 119871 + (119862119901


1



119865 = 119901119871

119860 = 119872 + 119861 + 1198912

(10)



2



and 1198912


119866ℎ

(119904) =119909 (119904)

120596119901

(119904)

= (

119860119863119901

120573119890

1198721198810

)(1199043

+ (119861

119872+

119862119879

119863119901

120573119890

1198810

) 1199042

+

120573119890

(119861119862119879

119863119901

+ 1198602

)

1198721198810

119904)

minus1

(11)




119866119898

(119904) =

120596119901

(119904)

V119903

(119904)=

119870119898

(1198791

119904 + 1)

119879119898

1

1199042

+ 119879119898

2

119904 + 1 (12)

where 119870119898

1198791

119879119898

1

and 119879119898

2





0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


40

50

60

70

ExperimentModel

Mag

nitu

de (d

B)





119870119866 (119904)

1 + 119870119866 (119904)=

118 times 107

1199043

+ 22311199042

+ 718 times 105

119904 + 118 times 107

(13)




0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

ExperimentModel


0 05 1 15 2 25 3 35 4 45

0

2

4

6

8

10

12

14

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

minus2

ExperimentModel

(b) Step response



119866 (119904) =118 times 10

4

1199043

+ 22311199042

+ 718 times 105

119904 (14)


Corr =sum119873

119896=1

(119884119901

(119896) minus 119884119901

) (119884119898

(119896) minus 119884119898

)

radicsum119873

119896=1

(119884119901

(119896) minus 119884119901

)radicsum119873

119896=1

(119884119898

(119896) minus 119884119898

)2

(15)

where 119884119901

(119896) and 119884119898


119901

and 119884119898


119901

(119896) and 119884119898



110

115

120

125

Experiment

Mag

nitu

de (d

B)







Positioncontroller



Motorcontroller

r e

120596

e xi

Ks

Km

Tm 120596p

plusmn100A

minus+ minus+


Jms + Bm

x(s)

120596p(s)=

ADp120573e

e

MV0





119898





119869 (119870119901

119870119894

119870119889

) =

infin

sum

119905=0

(119910step (119905) minus 119910119889

step (119905)) (16)


min119870

119901119870

119894119870

119889

119869 (119870119901

119870119894

119870119889

) (17)

where 119870119901

119870119894

and 119870119889



119901

119870119894

119870119889


119901

119870119894

and 119870119889








Motorcontroller

r i re

120596

i

Ks

Ks Km

Tm

plusmn100A

minus+

+++

+

rpmplusmn10000

minus

ka

ei

s

k2s

orKi

s

k3 or Kd

k1 or Kpe 120596p1

Jms + Bm


06 07 08 09 1 1118

19

20

21

22Upper boundary

Lower boundary

Nominal response

0 02 04 06 08 1 12 14 16 18 20

5

10

15

20

25

Pisto

n di

spla

cem

ent (

mm

)

Time (s)




120577 = 1198961

119890 + 1198962

int 119890 119889119905 + 1198963

119890 (18)



1

1198962

and 1198963



120596119901

= 120596119901119890119902

+ 120596119901robust (19)


119898

+ 119899


120577 = minus 1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

120596119901

+ 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119898

+ 119899

)

+ 1198963

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+ 1198963

119889

+ 1198961

119890 + 1198962

119890

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891

(20)

where 119891 = 1198911

+ (119862119879

119860)1198912

+ (1198810

119860119863119901

120573119890

) 1198912




120596119901119890119902

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119889

(21)


120577 = 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119899

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891 (22)


120577 = minus119863120577 minus 119870 sgn (120577) (23)





1198630

10038161003816100381610038161205771003816100381610038161003816 + 1198700 gt

100381610038161003816100381610038161003816100381610038161003816

1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119899

+1

1198721198810

119891)

100381610038161003816100381610038161003816100381610038161003816

+ 120578

(24)

where 1198700

= (119860119863119901

120573119890

(1198611198810

+ 119862119879

119863119901

119872120573119890

))119870 and 1198630

=

(119860119863119901

120573119890

(1198611198810

+119862119879

119863119901

119872120573119890


120596119901robust = 119863120577 + 119870 sgn (120577) (25)


120596119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 119889

+1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119863120577 + 119870 sgn (120577)

(26)



119890


119879

= 119862119879119899

+ 119862119879119901

and 120573119890

= 120573119890119899

+ 120573119890119901


120595 = 120579119879

120593 = (1198721198810

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

) 119898

+ (

(1198602

+ 119861119862119879119901

119863119901

) 120573119890119901

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)

(27)

where 120579119879

= [(1198721198810

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)) ((1198602

+

119861119862119879119901

119863119901

)120573119890119901

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

))] and 120593 = [119909

119898

119909


119901

=

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119860119863119901

120573119890119899

times 119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

+ 119863120577 + 119870 sgn (120577)

(28)


119881 =1

21205772

+1

2120574120579119879

120579 (29)


= 120577 [119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

minus1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

119901

] minus1

120574120579119879120579

(30)



= minus1198631205772

minus 119870120577 sgn (120577) + 120579119879 (120577120593 minus 1120574

120579) (31)


120579 = 120574120577120593 (32)


= minus1198631205772

minus 119870120577 sgn (120577) le 0 (33)



119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times (120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890)

+ 119863120577 + 119870 sgn (120577) (34)

where 120573119890

(119905 + 119879119904

) = (119889120573119890

(119905)119889119905)119879119904

+ 120573119890

(119905) 119862119879

(119905 + 119879119904

) =

(119889119862119879

(119905)119889119905)119879119904

+ 119862119879

(119905) and 119879119904





119901

or 1198961

) 10343 16947 950Integral (119870

119894

or 1198962

) 184 10877 0001Derivative (119870

119889

or 1198963

) 47 4160 145Antiwindup (119870

119886

) mdash 0001 0001

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors













Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time


0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)




0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)





0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors



0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














minus6m3


Motor


Hydraulic cylinder


6 PaAccumulator








119886

119876119887





EHALoad cell

Load cylinder

Motor amplifier

Load mass

LVDT



119901119887


119876119886

= 119863119901

120596119901

minus 120585 (119901119886

minus 119901119887

) minus119881119886

120573119890

119889119901119886

119889119905minus 119862119890119901

(119901119886

minus 119901119903

)

119876119887

= 119863119901

120596119901

minus 120585 (119901119886

minus 119901119887

) +119881119887

120573119890

119889119901119887

119889119905+ 119862119890119901

(119901119887

minus 119901119903

)

(1)

where 119863119901


119890119901




119886

119881119887



1198762


1198761

= 119860 +119881119886

120573119890

1198891199011

119889119905+ 119871 (119901

1

minus 1199012

)

1198762

= 119860 minus119881119887

120573119890

1198891199012

119889119905minus 119871 (119901

2

minus 1199011

)

(2)


1

1199012



and 119881119887


119886

and 119881119887



Force

Reservoir

Position

M

Computer system



DS1104 dSPACE

Load cylinder

Main cylinder

Servo valve

Mass

Servo motordirver

Loadcontroller

LoadLVDT

AMP

cell



axial piston pump

MassCylinder rod

Electricmotor

Piston

Cylinder

p1

Q1

p2

Q2

pa

Qa

pb

Qb

120596p

J

x




119881119886

= 1198810

+ 119860119909

119881119887

= 1198810

minus 119860119909

(3)


119886

119876119887


1

1198762


1198761

= 119876119886

1198762

= 119876119887

(4)



119876119871

=1198761

+ 1198762

2=119876119886

+ 119876119887

2 (5)


119901119887


1199012


119901119886

= 1199011

+ 119901pipe

119901119887

= 1199012

minus 119901pipe(6)



119901119886

= 1199011

119901119887

= 1199012

(7)


1198891199011

119889119905= minus

1198891199012

119889119905 (8)


119863119901

120596119901

= 119860 +1198810

120573119890

119871

+ 119862119879

119901119871

+ 1198911

(9)

where 119901119871

= 1199011

minus 1199012

119862119879

= 120585 + 119871 + (119862119901


1



119865 = 119901119871

119860 = 119872 + 119861 + 1198912

(10)



2



and 1198912


119866ℎ

(119904) =119909 (119904)

120596119901

(119904)

= (

119860119863119901

120573119890

1198721198810

)(1199043

+ (119861

119872+

119862119879

119863119901

120573119890

1198810

) 1199042

+

120573119890

(119861119862119879

119863119901

+ 1198602

)

1198721198810

119904)

minus1

(11)




119866119898

(119904) =

120596119901

(119904)

V119903

(119904)=

119870119898

(1198791

119904 + 1)

119879119898

1

1199042

+ 119879119898

2

119904 + 1 (12)

where 119870119898

1198791

119879119898

1

and 119879119898

2





0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


40

50

60

70

ExperimentModel

Mag

nitu

de (d

B)





119870119866 (119904)

1 + 119870119866 (119904)=

118 times 107

1199043

+ 22311199042

+ 718 times 105

119904 + 118 times 107

(13)




0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

ExperimentModel


0 05 1 15 2 25 3 35 4 45

0

2

4

6

8

10

12

14

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

minus2

ExperimentModel

(b) Step response



119866 (119904) =118 times 10

4

1199043

+ 22311199042

+ 718 times 105

119904 (14)


Corr =sum119873

119896=1

(119884119901

(119896) minus 119884119901

) (119884119898

(119896) minus 119884119898

)

radicsum119873

119896=1

(119884119901

(119896) minus 119884119901

)radicsum119873

119896=1

(119884119898

(119896) minus 119884119898

)2

(15)

where 119884119901

(119896) and 119884119898


119901

and 119884119898


119901

(119896) and 119884119898



110

115

120

125

Experiment

Mag

nitu

de (d

B)







Positioncontroller



Motorcontroller

r e

120596

e xi

Ks

Km

Tm 120596p

plusmn100A

minus+ minus+


Jms + Bm

x(s)

120596p(s)=

ADp120573e

e

MV0





119898





119869 (119870119901

119870119894

119870119889

) =

infin

sum

119905=0

(119910step (119905) minus 119910119889

step (119905)) (16)


min119870

119901119870

119894119870

119889

119869 (119870119901

119870119894

119870119889

) (17)

where 119870119901

119870119894

and 119870119889



119901

119870119894

119870119889


119901

119870119894

and 119870119889








Motorcontroller

r i re

120596

i

Ks

Ks Km

Tm

plusmn100A

minus+

+++

+

rpmplusmn10000

minus

ka

ei

s

k2s

orKi

s

k3 or Kd

k1 or Kpe 120596p1

Jms + Bm


06 07 08 09 1 1118

19

20

21

22Upper boundary

Lower boundary

Nominal response

0 02 04 06 08 1 12 14 16 18 20

5

10

15

20

25

Pisto

n di

spla

cem

ent (

mm

)

Time (s)




120577 = 1198961

119890 + 1198962

int 119890 119889119905 + 1198963

119890 (18)



1

1198962

and 1198963



120596119901

= 120596119901119890119902

+ 120596119901robust (19)


119898

+ 119899


120577 = minus 1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

120596119901

+ 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119898

+ 119899

)

+ 1198963

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+ 1198963

119889

+ 1198961

119890 + 1198962

119890

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891

(20)

where 119891 = 1198911

+ (119862119879

119860)1198912

+ (1198810

119860119863119901

120573119890

) 1198912




120596119901119890119902

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119889

(21)


120577 = 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119899

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891 (22)


120577 = minus119863120577 minus 119870 sgn (120577) (23)





1198630

10038161003816100381610038161205771003816100381610038161003816 + 1198700 gt

100381610038161003816100381610038161003816100381610038161003816

1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119899

+1

1198721198810

119891)

100381610038161003816100381610038161003816100381610038161003816

+ 120578

(24)

where 1198700

= (119860119863119901

120573119890

(1198611198810

+ 119862119879

119863119901

119872120573119890

))119870 and 1198630

=

(119860119863119901

120573119890

(1198611198810

+119862119879

119863119901

119872120573119890


120596119901robust = 119863120577 + 119870 sgn (120577) (25)


120596119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 119889

+1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119863120577 + 119870 sgn (120577)

(26)



119890


119879

= 119862119879119899

+ 119862119879119901

and 120573119890

= 120573119890119899

+ 120573119890119901


120595 = 120579119879

120593 = (1198721198810

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

) 119898

+ (

(1198602

+ 119861119862119879119901

119863119901

) 120573119890119901

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)

(27)

where 120579119879

= [(1198721198810

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)) ((1198602

+

119861119862119879119901

119863119901

)120573119890119901

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

))] and 120593 = [119909

119898

119909


119901

=

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119860119863119901

120573119890119899

times 119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

+ 119863120577 + 119870 sgn (120577)

(28)


119881 =1

21205772

+1

2120574120579119879

120579 (29)


= 120577 [119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

minus1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

119901

] minus1

120574120579119879120579

(30)



= minus1198631205772

minus 119870120577 sgn (120577) + 120579119879 (120577120593 minus 1120574

120579) (31)


120579 = 120574120577120593 (32)


= minus1198631205772

minus 119870120577 sgn (120577) le 0 (33)



119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times (120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890)

+ 119863120577 + 119870 sgn (120577) (34)

where 120573119890

(119905 + 119879119904

) = (119889120573119890

(119905)119889119905)119879119904

+ 120573119890

(119905) 119862119879

(119905 + 119879119904

) =

(119889119862119879

(119905)119889119905)119879119904

+ 119862119879

(119905) and 119879119904





119901

or 1198961

) 10343 16947 950Integral (119870

119894

or 1198962

) 184 10877 0001Derivative (119870

119889

or 1198963

) 47 4160 145Antiwindup (119870

119886

) mdash 0001 0001

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors













Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time


0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)




0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)





0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors



0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Force

Reservoir

Position

M

Computer system



DS1104 dSPACE

Load cylinder

Main cylinder

Servo valve

Mass

Servo motordirver

Loadcontroller

LoadLVDT

AMP

cell



axial piston pump

MassCylinder rod

Electricmotor

Piston

Cylinder

p1

Q1

p2

Q2

pa

Qa

pb

Qb

120596p

J

x




119881119886

= 1198810

+ 119860119909

119881119887

= 1198810

minus 119860119909

(3)


119886

119876119887


1

1198762


1198761

= 119876119886

1198762

= 119876119887

(4)



119876119871

=1198761

+ 1198762

2=119876119886

+ 119876119887

2 (5)


119901119887


1199012


119901119886

= 1199011

+ 119901pipe

119901119887

= 1199012

minus 119901pipe(6)



119901119886

= 1199011

119901119887

= 1199012

(7)


1198891199011

119889119905= minus

1198891199012

119889119905 (8)


119863119901

120596119901

= 119860 +1198810

120573119890

119871

+ 119862119879

119901119871

+ 1198911

(9)

where 119901119871

= 1199011

minus 1199012

119862119879

= 120585 + 119871 + (119862119901


1



119865 = 119901119871

119860 = 119872 + 119861 + 1198912

(10)



2



and 1198912


119866ℎ

(119904) =119909 (119904)

120596119901

(119904)

= (

119860119863119901

120573119890

1198721198810

)(1199043

+ (119861

119872+

119862119879

119863119901

120573119890

1198810

) 1199042

+

120573119890

(119861119862119879

119863119901

+ 1198602

)

1198721198810

119904)

minus1

(11)




119866119898

(119904) =

120596119901

(119904)

V119903

(119904)=

119870119898

(1198791

119904 + 1)

119879119898

1

1199042

+ 119879119898

2

119904 + 1 (12)

where 119870119898

1198791

119879119898

1

and 119879119898

2





0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


40

50

60

70

ExperimentModel

Mag

nitu

de (d

B)





119870119866 (119904)

1 + 119870119866 (119904)=

118 times 107

1199043

+ 22311199042

+ 718 times 105

119904 + 118 times 107

(13)




0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

ExperimentModel


0 05 1 15 2 25 3 35 4 45

0

2

4

6

8

10

12

14

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

minus2

ExperimentModel

(b) Step response



119866 (119904) =118 times 10

4

1199043

+ 22311199042

+ 718 times 105

119904 (14)


Corr =sum119873

119896=1

(119884119901

(119896) minus 119884119901

) (119884119898

(119896) minus 119884119898

)

radicsum119873

119896=1

(119884119901

(119896) minus 119884119901

)radicsum119873

119896=1

(119884119898

(119896) minus 119884119898

)2

(15)

where 119884119901

(119896) and 119884119898


119901

and 119884119898


119901

(119896) and 119884119898



110

115

120

125

Experiment

Mag

nitu

de (d

B)







Positioncontroller



Motorcontroller

r e

120596

e xi

Ks

Km

Tm 120596p

plusmn100A

minus+ minus+


Jms + Bm

x(s)

120596p(s)=

ADp120573e

e

MV0





119898





119869 (119870119901

119870119894

119870119889

) =

infin

sum

119905=0

(119910step (119905) minus 119910119889

step (119905)) (16)


min119870

119901119870

119894119870

119889

119869 (119870119901

119870119894

119870119889

) (17)

where 119870119901

119870119894

and 119870119889



119901

119870119894

119870119889


119901

119870119894

and 119870119889








Motorcontroller

r i re

120596

i

Ks

Ks Km

Tm

plusmn100A

minus+

+++

+

rpmplusmn10000

minus

ka

ei

s

k2s

orKi

s

k3 or Kd

k1 or Kpe 120596p1

Jms + Bm


06 07 08 09 1 1118

19

20

21

22Upper boundary

Lower boundary

Nominal response

0 02 04 06 08 1 12 14 16 18 20

5

10

15

20

25

Pisto

n di

spla

cem

ent (

mm

)

Time (s)




120577 = 1198961

119890 + 1198962

int 119890 119889119905 + 1198963

119890 (18)



1

1198962

and 1198963



120596119901

= 120596119901119890119902

+ 120596119901robust (19)


119898

+ 119899


120577 = minus 1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

120596119901

+ 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119898

+ 119899

)

+ 1198963

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+ 1198963

119889

+ 1198961

119890 + 1198962

119890

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891

(20)

where 119891 = 1198911

+ (119862119879

119860)1198912

+ (1198810

119860119863119901

120573119890

) 1198912




120596119901119890119902

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119889

(21)


120577 = 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119899

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891 (22)


120577 = minus119863120577 minus 119870 sgn (120577) (23)





1198630

10038161003816100381610038161205771003816100381610038161003816 + 1198700 gt

100381610038161003816100381610038161003816100381610038161003816

1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119899

+1

1198721198810

119891)

100381610038161003816100381610038161003816100381610038161003816

+ 120578

(24)

where 1198700

= (119860119863119901

120573119890

(1198611198810

+ 119862119879

119863119901

119872120573119890

))119870 and 1198630

=

(119860119863119901

120573119890

(1198611198810

+119862119879

119863119901

119872120573119890


120596119901robust = 119863120577 + 119870 sgn (120577) (25)


120596119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 119889

+1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119863120577 + 119870 sgn (120577)

(26)



119890


119879

= 119862119879119899

+ 119862119879119901

and 120573119890

= 120573119890119899

+ 120573119890119901


120595 = 120579119879

120593 = (1198721198810

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

) 119898

+ (

(1198602

+ 119861119862119879119901

119863119901

) 120573119890119901

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)

(27)

where 120579119879

= [(1198721198810

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)) ((1198602

+

119861119862119879119901

119863119901

)120573119890119901

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

))] and 120593 = [119909

119898

119909


119901

=

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119860119863119901

120573119890119899

times 119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

+ 119863120577 + 119870 sgn (120577)

(28)


119881 =1

21205772

+1

2120574120579119879

120579 (29)


= 120577 [119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

minus1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

119901

] minus1

120574120579119879120579

(30)



= minus1198631205772

minus 119870120577 sgn (120577) + 120579119879 (120577120593 minus 1120574

120579) (31)


120579 = 120574120577120593 (32)


= minus1198631205772

minus 119870120577 sgn (120577) le 0 (33)



119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times (120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890)

+ 119863120577 + 119870 sgn (120577) (34)

where 120573119890

(119905 + 119879119904

) = (119889120573119890

(119905)119889119905)119879119904

+ 120573119890

(119905) 119862119879

(119905 + 119879119904

) =

(119889119862119879

(119905)119889119905)119879119904

+ 119862119879

(119905) and 119879119904





119901

or 1198961

) 10343 16947 950Integral (119870

119894

or 1198962

) 184 10877 0001Derivative (119870

119889

or 1198963

) 47 4160 145Antiwindup (119870

119886

) mdash 0001 0001

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors













Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time


0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)




0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)





0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors



0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











2



and 1198912


119866ℎ

(119904) =119909 (119904)

120596119901

(119904)

= (

119860119863119901

120573119890

1198721198810

)(1199043

+ (119861

119872+

119862119879

119863119901

120573119890

1198810

) 1199042

+

120573119890

(119861119862119879

119863119901

+ 1198602

)

1198721198810

119904)

minus1

(11)




119866119898

(119904) =

120596119901

(119904)

V119903

(119904)=

119870119898

(1198791

119904 + 1)

119879119898

1

1199042

+ 119879119898

2

119904 + 1 (12)

where 119870119898

1198791

119879119898

1

and 119879119898

2





0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


40

50

60

70

ExperimentModel

Mag

nitu

de (d

B)





119870119866 (119904)

1 + 119870119866 (119904)=

118 times 107

1199043

+ 22311199042

+ 718 times 105

119904 + 118 times 107

(13)




0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

ExperimentModel


0 05 1 15 2 25 3 35 4 45

0

2

4

6

8

10

12

14

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

minus2

ExperimentModel

(b) Step response



119866 (119904) =118 times 10

4

1199043

+ 22311199042

+ 718 times 105

119904 (14)


Corr =sum119873

119896=1

(119884119901

(119896) minus 119884119901

) (119884119898

(119896) minus 119884119898

)

radicsum119873

119896=1

(119884119901

(119896) minus 119884119901

)radicsum119873

119896=1

(119884119898

(119896) minus 119884119898

)2

(15)

where 119884119901

(119896) and 119884119898


119901

and 119884119898


119901

(119896) and 119884119898



110

115

120

125

Experiment

Mag

nitu

de (d

B)







Positioncontroller



Motorcontroller

r e

120596

e xi

Ks

Km

Tm 120596p

plusmn100A

minus+ minus+


Jms + Bm

x(s)

120596p(s)=

ADp120573e

e

MV0





119898





119869 (119870119901

119870119894

119870119889

) =

infin

sum

119905=0

(119910step (119905) minus 119910119889

step (119905)) (16)


min119870

119901119870

119894119870

119889

119869 (119870119901

119870119894

119870119889

) (17)

where 119870119901

119870119894

and 119870119889



119901

119870119894

119870119889


119901

119870119894

and 119870119889








Motorcontroller

r i re

120596

i

Ks

Ks Km

Tm

plusmn100A

minus+

+++

+

rpmplusmn10000

minus

ka

ei

s

k2s

orKi

s

k3 or Kd

k1 or Kpe 120596p1

Jms + Bm


06 07 08 09 1 1118

19

20

21

22Upper boundary

Lower boundary

Nominal response

0 02 04 06 08 1 12 14 16 18 20

5

10

15

20

25

Pisto

n di

spla

cem

ent (

mm

)

Time (s)




120577 = 1198961

119890 + 1198962

int 119890 119889119905 + 1198963

119890 (18)



1

1198962

and 1198963



120596119901

= 120596119901119890119902

+ 120596119901robust (19)


119898

+ 119899


120577 = minus 1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

120596119901

+ 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119898

+ 119899

)

+ 1198963

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+ 1198963

119889

+ 1198961

119890 + 1198962

119890

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891

(20)

where 119891 = 1198911

+ (119862119879

119860)1198912

+ (1198810

119860119863119901

120573119890

) 1198912




120596119901119890119902

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119889

(21)


120577 = 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119899

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891 (22)


120577 = minus119863120577 minus 119870 sgn (120577) (23)





1198630

10038161003816100381610038161205771003816100381610038161003816 + 1198700 gt

100381610038161003816100381610038161003816100381610038161003816

1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119899

+1

1198721198810

119891)

100381610038161003816100381610038161003816100381610038161003816

+ 120578

(24)

where 1198700

= (119860119863119901

120573119890

(1198611198810

+ 119862119879

119863119901

119872120573119890

))119870 and 1198630

=

(119860119863119901

120573119890

(1198611198810

+119862119879

119863119901

119872120573119890


120596119901robust = 119863120577 + 119870 sgn (120577) (25)


120596119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 119889

+1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119863120577 + 119870 sgn (120577)

(26)



119890


119879

= 119862119879119899

+ 119862119879119901

and 120573119890

= 120573119890119899

+ 120573119890119901


120595 = 120579119879

120593 = (1198721198810

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

) 119898

+ (

(1198602

+ 119861119862119879119901

119863119901

) 120573119890119901

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)

(27)

where 120579119879

= [(1198721198810

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)) ((1198602

+

119861119862119879119901

119863119901

)120573119890119901

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

))] and 120593 = [119909

119898

119909


119901

=

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119860119863119901

120573119890119899

times 119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

+ 119863120577 + 119870 sgn (120577)

(28)


119881 =1

21205772

+1

2120574120579119879

120579 (29)


= 120577 [119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

minus1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

119901

] minus1

120574120579119879120579

(30)



= minus1198631205772

minus 119870120577 sgn (120577) + 120579119879 (120577120593 minus 1120574

120579) (31)


120579 = 120574120577120593 (32)


= minus1198631205772

minus 119870120577 sgn (120577) le 0 (33)



119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times (120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890)

+ 119863120577 + 119870 sgn (120577) (34)

where 120573119890

(119905 + 119879119904

) = (119889120573119890

(119905)119889119905)119879119904

+ 120573119890

(119905) 119862119879

(119905 + 119879119904

) =

(119889119862119879

(119905)119889119905)119879119904

+ 119862119879

(119905) and 119879119904





119901

or 1198961

) 10343 16947 950Integral (119870

119894

or 1198962

) 184 10877 0001Derivative (119870

119889

or 1198963

) 47 4160 145Antiwindup (119870

119886

) mdash 0001 0001

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors













Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time


0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)




0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)





0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors



0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 1 2 3 4 5 6 7 8minus3

minus25minus2

minus15minus1

minus050

051

152

253

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

ExperimentModel


0 05 1 15 2 25 3 35 4 45

0

2

4

6

8

10

12

14

Pisto

n di

spla

cem

ent (

mm

)

Time (s)

minus2

ExperimentModel

(b) Step response



119866 (119904) =118 times 10

4

1199043

+ 22311199042

+ 718 times 105

119904 (14)


Corr =sum119873

119896=1

(119884119901

(119896) minus 119884119901

) (119884119898

(119896) minus 119884119898

)

radicsum119873

119896=1

(119884119901

(119896) minus 119884119901

)radicsum119873

119896=1

(119884119898

(119896) minus 119884119898

)2

(15)

where 119884119901

(119896) and 119884119898


119901

and 119884119898


119901

(119896) and 119884119898



110

115

120

125

Experiment

Mag

nitu

de (d

B)







Positioncontroller



Motorcontroller

r e

120596

e xi

Ks

Km

Tm 120596p

plusmn100A

minus+ minus+


Jms + Bm

x(s)

120596p(s)=

ADp120573e

e

MV0





119898





119869 (119870119901

119870119894

119870119889

) =

infin

sum

119905=0

(119910step (119905) minus 119910119889

step (119905)) (16)


min119870

119901119870

119894119870

119889

119869 (119870119901

119870119894

119870119889

) (17)

where 119870119901

119870119894

and 119870119889



119901

119870119894

119870119889


119901

119870119894

and 119870119889








Motorcontroller

r i re

120596

i

Ks

Ks Km

Tm

plusmn100A

minus+

+++

+

rpmplusmn10000

minus

ka

ei

s

k2s

orKi

s

k3 or Kd

k1 or Kpe 120596p1

Jms + Bm


06 07 08 09 1 1118

19

20

21

22Upper boundary

Lower boundary

Nominal response

0 02 04 06 08 1 12 14 16 18 20

5

10

15

20

25

Pisto

n di

spla

cem

ent (

mm

)

Time (s)




120577 = 1198961

119890 + 1198962

int 119890 119889119905 + 1198963

119890 (18)



1

1198962

and 1198963



120596119901

= 120596119901119890119902

+ 120596119901robust (19)


119898

+ 119899


120577 = minus 1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

120596119901

+ 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119898

+ 119899

)

+ 1198963

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+ 1198963

119889

+ 1198961

119890 + 1198962

119890

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891

(20)

where 119891 = 1198911

+ (119862119879

119860)1198912

+ (1198810

119860119863119901

120573119890

) 1198912




120596119901119890119902

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119889

(21)


120577 = 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119899

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891 (22)


120577 = minus119863120577 minus 119870 sgn (120577) (23)





1198630

10038161003816100381610038161205771003816100381610038161003816 + 1198700 gt

100381610038161003816100381610038161003816100381610038161003816

1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119899

+1

1198721198810

119891)

100381610038161003816100381610038161003816100381610038161003816

+ 120578

(24)

where 1198700

= (119860119863119901

120573119890

(1198611198810

+ 119862119879

119863119901

119872120573119890

))119870 and 1198630

=

(119860119863119901

120573119890

(1198611198810

+119862119879

119863119901

119872120573119890


120596119901robust = 119863120577 + 119870 sgn (120577) (25)


120596119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 119889

+1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119863120577 + 119870 sgn (120577)

(26)



119890


119879

= 119862119879119899

+ 119862119879119901

and 120573119890

= 120573119890119899

+ 120573119890119901


120595 = 120579119879

120593 = (1198721198810

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

) 119898

+ (

(1198602

+ 119861119862119879119901

119863119901

) 120573119890119901

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)

(27)

where 120579119879

= [(1198721198810

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)) ((1198602

+

119861119862119879119901

119863119901

)120573119890119901

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

))] and 120593 = [119909

119898

119909


119901

=

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119860119863119901

120573119890119899

times 119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

+ 119863120577 + 119870 sgn (120577)

(28)


119881 =1

21205772

+1

2120574120579119879

120579 (29)


= 120577 [119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

minus1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

119901

] minus1

120574120579119879120579

(30)



= minus1198631205772

minus 119870120577 sgn (120577) + 120579119879 (120577120593 minus 1120574

120579) (31)


120579 = 120574120577120593 (32)


= minus1198631205772

minus 119870120577 sgn (120577) le 0 (33)



119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times (120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890)

+ 119863120577 + 119870 sgn (120577) (34)

where 120573119890

(119905 + 119879119904

) = (119889120573119890

(119905)119889119905)119879119904

+ 120573119890

(119905) 119862119879

(119905 + 119879119904

) =

(119889119862119879

(119905)119889119905)119879119904

+ 119862119879

(119905) and 119879119904





119901

or 1198961

) 10343 16947 950Integral (119870

119894

or 1198962

) 184 10877 0001Derivative (119870

119889

or 1198963

) 47 4160 145Antiwindup (119870

119886

) mdash 0001 0001

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors













Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time


0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)




0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)





0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors



0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Positioncontroller



Motorcontroller

r e

120596

e xi

Ks

Km

Tm 120596p

plusmn100A

minus+ minus+


Jms + Bm

x(s)

120596p(s)=

ADp120573e

e

MV0





119898





119869 (119870119901

119870119894

119870119889

) =

infin

sum

119905=0

(119910step (119905) minus 119910119889

step (119905)) (16)


min119870

119901119870

119894119870

119889

119869 (119870119901

119870119894

119870119889

) (17)

where 119870119901

119870119894

and 119870119889



119901

119870119894

119870119889


119901

119870119894

and 119870119889








Motorcontroller

r i re

120596

i

Ks

Ks Km

Tm

plusmn100A

minus+

+++

+

rpmplusmn10000

minus

ka

ei

s

k2s

orKi

s

k3 or Kd

k1 or Kpe 120596p1

Jms + Bm


06 07 08 09 1 1118

19

20

21

22Upper boundary

Lower boundary

Nominal response

0 02 04 06 08 1 12 14 16 18 20

5

10

15

20

25

Pisto

n di

spla

cem

ent (

mm

)

Time (s)




120577 = 1198961

119890 + 1198962

int 119890 119889119905 + 1198963

119890 (18)



1

1198962

and 1198963



120596119901

= 120596119901119890119902

+ 120596119901robust (19)


119898

+ 119899


120577 = minus 1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

120596119901

+ 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119898

+ 119899

)

+ 1198963

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+ 1198963

119889

+ 1198961

119890 + 1198962

119890

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891

(20)

where 119891 = 1198911

+ (119862119879

119860)1198912

+ (1198810

119860119863119901

120573119890

) 1198912




120596119901119890119902

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119889

(21)


120577 = 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119899

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891 (22)


120577 = minus119863120577 minus 119870 sgn (120577) (23)





1198630

10038161003816100381610038161205771003816100381610038161003816 + 1198700 gt

100381610038161003816100381610038161003816100381610038161003816

1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119899

+1

1198721198810

119891)

100381610038161003816100381610038161003816100381610038161003816

+ 120578

(24)

where 1198700

= (119860119863119901

120573119890

(1198611198810

+ 119862119879

119863119901

119872120573119890

))119870 and 1198630

=

(119860119863119901

120573119890

(1198611198810

+119862119879

119863119901

119872120573119890


120596119901robust = 119863120577 + 119870 sgn (120577) (25)


120596119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 119889

+1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119863120577 + 119870 sgn (120577)

(26)



119890


119879

= 119862119879119899

+ 119862119879119901

and 120573119890

= 120573119890119899

+ 120573119890119901


120595 = 120579119879

120593 = (1198721198810

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

) 119898

+ (

(1198602

+ 119861119862119879119901

119863119901

) 120573119890119901

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)

(27)

where 120579119879

= [(1198721198810

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)) ((1198602

+

119861119862119879119901

119863119901

)120573119890119901

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

))] and 120593 = [119909

119898

119909


119901

=

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119860119863119901

120573119890119899

times 119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

+ 119863120577 + 119870 sgn (120577)

(28)


119881 =1

21205772

+1

2120574120579119879

120579 (29)


= 120577 [119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

minus1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

119901

] minus1

120574120579119879120579

(30)



= minus1198631205772

minus 119870120577 sgn (120577) + 120579119879 (120577120593 minus 1120574

120579) (31)


120579 = 120574120577120593 (32)


= minus1198631205772

minus 119870120577 sgn (120577) le 0 (33)



119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times (120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890)

+ 119863120577 + 119870 sgn (120577) (34)

where 120573119890

(119905 + 119879119904

) = (119889120573119890

(119905)119889119905)119879119904

+ 120573119890

(119905) 119862119879

(119905 + 119879119904

) =

(119889119862119879

(119905)119889119905)119879119904

+ 119862119879

(119905) and 119879119904





119901

or 1198961

) 10343 16947 950Integral (119870

119894

or 1198962

) 184 10877 0001Derivative (119870

119889

or 1198963

) 47 4160 145Antiwindup (119870

119886

) mdash 0001 0001

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors













Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time


0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)




0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)





0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors



0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Motorcontroller

r i re

120596

i

Ks

Ks Km

Tm

plusmn100A

minus+

+++

+

rpmplusmn10000

minus

ka

ei

s

k2s

orKi

s

k3 or Kd

k1 or Kpe 120596p1

Jms + Bm


06 07 08 09 1 1118

19

20

21

22Upper boundary

Lower boundary

Nominal response

0 02 04 06 08 1 12 14 16 18 20

5

10

15

20

25

Pisto

n di

spla

cem

ent (

mm

)

Time (s)




120577 = 1198961

119890 + 1198962

int 119890 119889119905 + 1198963

119890 (18)



1

1198962

and 1198963



120596119901

= 120596119901119890119902

+ 120596119901robust (19)


119898

+ 119899


120577 = minus 1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

120596119901

+ 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119898

+ 119899

)

+ 1198963

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+ 1198963

119889

+ 1198961

119890 + 1198962

119890

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891

(20)

where 119891 = 1198911

+ (119862119879

119860)1198912

+ (1198810

119860119863119901

120573119890

) 1198912




120596119901119890119902

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119889

(21)


120577 = 1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119899

+1198963

1198611198810

+ 119862119879

119863119901

119872120573119890

119891 (22)


120577 = minus119863120577 minus 119870 sgn (120577) (23)





1198630

10038161003816100381610038161205771003816100381610038161003816 + 1198700 gt

100381610038161003816100381610038161003816100381610038161003816

1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119899

+1

1198721198810

119891)

100381610038161003816100381610038161003816100381610038161003816

+ 120578

(24)

where 1198700

= (119860119863119901

120573119890

(1198611198810

+ 119862119879

119863119901

119872120573119890

))119870 and 1198630

=

(119860119863119901

120573119890

(1198611198810

+119862119879

119863119901

119872120573119890


120596119901robust = 119863120577 + 119870 sgn (120577) (25)


120596119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 119889

+1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119863120577 + 119870 sgn (120577)

(26)



119890


119879

= 119862119879119899

+ 119862119879119901

and 120573119890

= 120573119890119899

+ 120573119890119901


120595 = 120579119879

120593 = (1198721198810

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

) 119898

+ (

(1198602

+ 119861119862119879119901

119863119901

) 120573119890119901

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)

(27)

where 120579119879

= [(1198721198810

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)) ((1198602

+

119861119862119879119901

119863119901

)120573119890119901

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

))] and 120593 = [119909

119898

119909


119901

=

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119860119863119901

120573119890119899

times 119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

+ 119863120577 + 119870 sgn (120577)

(28)


119881 =1

21205772

+1

2120574120579119879

120579 (29)


= 120577 [119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

minus1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

119901

] minus1

120574120579119879120579

(30)



= minus1198631205772

minus 119870120577 sgn (120577) + 120579119879 (120577120593 minus 1120574

120579) (31)


120579 = 120574120577120593 (32)


= minus1198631205772

minus 119870120577 sgn (120577) le 0 (33)



119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times (120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890)

+ 119863120577 + 119870 sgn (120577) (34)

where 120573119890

(119905 + 119879119904

) = (119889120573119890

(119905)119889119905)119879119904

+ 120573119890

(119905) 119862119879

(119905 + 119879119904

) =

(119889119862119879

(119905)119889119905)119879119904

+ 119862119879

(119905) and 119879119904





119901

or 1198961

) 10343 16947 950Integral (119870

119894

or 1198962

) 184 10877 0001Derivative (119870

119889

or 1198963

) 47 4160 145Antiwindup (119870

119886

) mdash 0001 0001

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors













Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time


0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)




0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)





0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors



0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












1198630

10038161003816100381610038161205771003816100381610038161003816 + 1198700 gt

100381610038161003816100381610038161003816100381610038161003816

1198963

1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

(119899

+1

1198721198810

119891)

100381610038161003816100381610038161003816100381610038161003816

+ 120578

(24)

where 1198700

= (119860119863119901

120573119890

(1198611198810

+ 119862119879

119863119901

119872120573119890

))119870 and 1198630

=

(119860119863119901

120573119890

(1198611198810

+119862119879

119863119901

119872120573119890


120596119901robust = 119863120577 + 119870 sgn (120577) (25)


120596119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times 119889

+1198721198810

1198611198810

+ 119862119879

119863119901

119872120573119890

119898

+

(1198602

+ 119861119862119879

119863119901

) 120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

+1198961

1198963

119890 +1198962

1198963

119890 + 119863120577 + 119870 sgn (120577)

(26)



119890


119879

= 119862119879119899

+ 119862119879119901

and 120573119890

= 120573119890119899

+ 120573119890119901


120595 = 120579119879

120593 = (1198721198810

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

) 119898

+ (

(1198602

+ 119861119862119879119901

119863119901

) 120573119890119901

1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)

(27)

where 120579119879

= [(1198721198810

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

)) ((1198602

+

119861119862119879119901

119863119901

)120573119890119901

(1198611198810

+ 119862119879119901

119863119901

119872120573119890119901

))] and 120593 = [119909

119898

119909


119901

=

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119860119863119901

120573119890119899

times 119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

+ 119863120577 + 119870 sgn (120577)

(28)


119881 =1

21205772

+1

2120574120579119879

120579 (29)


= 120577 [119889

+ 120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890

minus1198963

119860119863119901

120573119890

1198611198810

+ 119862119879

119863119901

119872120573119890

119901

] minus1

120574120579119879120579

(30)



= minus1198631205772

minus 119870120577 sgn (120577) + 120579119879 (120577120593 minus 1120574

120579) (31)


120579 = 120574120577120593 (32)


= minus1198631205772

minus 119870120577 sgn (120577) le 0 (33)



119901

=

1198611198810

+ 119862119879

119863119901

119872120573119890

119860119863119901

120573119890

times (120579119879

120593 +1198721198810

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

119898

+

(1198602

+ 119861119862119879119899

119863119901

) 120573119890119899

1198611198810

+ 119862119879119899

119863119901

119872120573119890119899

+1198961

1198963

119890 +1198962

1198963

119890)

+ 119863120577 + 119870 sgn (120577) (34)

where 120573119890

(119905 + 119879119904

) = (119889120573119890

(119905)119889119905)119879119904

+ 120573119890

(119905) 119862119879

(119905 + 119879119904

) =

(119889119862119879

(119905)119889119905)119879119904

+ 119862119879

(119905) and 119879119904





119901

or 1198961

) 10343 16947 950Integral (119870

119894

or 1198962

) 184 10877 0001Derivative (119870

119889

or 1198963

) 47 4160 145Antiwindup (119870

119886

) mdash 0001 0001

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors













Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time


0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)




0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)





0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors



0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119901

or 1198961

) 10343 16947 950Integral (119870

119894

or 1198962

) 184 10877 0001Derivative (119870

119889

or 1198963

) 47 4160 145Antiwindup (119870

119886

) mdash 0001 0001

0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors













Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time


0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)




0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)





0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors



0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Overshoot

10mm 389 208 1020mm 161 133 0530mm 147 119 0240mm 122 82 01

Settling time


0 05 1 15 2 25 3minus120

minus90

minus60

minus30

0

30

60

90

120

150

Impr

esse

d cu

rren

t (A

)

Time (s)




0 05 1 15 2 25 3minus6000

minus4000

minus2000

0

2000

4000

6000

8000

10000

12000

14000

Mot

or an

gula

r spe

ed (r

pm)

Time (s)





0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors



0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1 15 2 25 305

10152025303540455055

Pisto

n di

spla

cem

ent (

mm

)

Time (s)


(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)



(b) Position errors


0 05 1 15 2 25 30

1

2

3

4

5

Pres

sure

(MPa

)

Time (s)








0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 5 10 15 20 25 300

400

800

1200

1600

2000

Bulk

mod

ulus

(MPa

)

Pressure (MPa)


110


0 05 1 15 2 25 30123456789

101112

Pres

sure

(MPa

)

Time (s)






6 Conclusion



0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 05 1 15 2 25 30

5

10

15

20

25

30

35


Pisto

n di

spla

cem

ent (

mm

)

Time (s)

(a) Step responses

0 05 1 15 2 25 3minus1

minus08

minus06

minus04

minus02

0

02

04

06

08

1

Posit

ion

erro

r (m

m)

Time (s)


(b) Position errors

0 05 1 15 2 25 30

2

4

6

8

10

Load

forc

e (kN

)

Time (s)


(c) Load force
















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation















119870119904






1198962

1198963


1198791

1198791198981 1198791198982


1199012


1198762




119881119887


1198810


V119890








References






































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation







































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Design and Experimental Evaluation of a ...position control system, a number of...

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Transcript of Research Article Design and Experimental Evaluation of a ...position control system, a number of...