research article research on adaptive dual-mode switch control...

Research ArticleResearch on Adaptive Dual-Mode Switch Control Strategy forVehicle Maglev Flywheel Battery

Hui Gao Ying-Jun Wu and Jing-Jin Shen

Nanjing University of Posts and Telecommunications Nanjing 210023 China

Correspondence should be addressed to Hui Gao gaohui2005163com

Received 3 November 2014 Accepted 15 January 2015

Academic Editor Honglei Xu

Copyright copy 2015 Hui Gao et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Because of the jamming signal is real-time changeable and control algorithm cannot timely tracking control flywheel rotor thispaper takes vehicle maglev flywheel battery as the research object One kind of dual-model control strategy is developed based onthe analysis of the vibration response impact of the flywheel battery control system In view of the complex foundation vibrationproblems of electric vehicles the nonlinear dynamic simulation model of vehicle maglev flywheel battery is solved Throughanalyzing the nonlinear vibration response characteristics one kind of dual-mode adaptive hybrid control strategy based on 119867infin

control and unbalance displacement feed-forward compensation control is presented and a real-time switch controller is designedThe reliable hybrid control is implemented and the stability in the process of real-time switch is solved The results of this projectcan provide important basic theory support for the research of vehicle maglev flywheel battery control system

1 Introduction

As the future main traffic tools electric vehicle (EV) isrequired in the performance of starting acceleration andclimbing however this performance depends largely on thepower battery performance [1] But the faults of costly shortrange and short service life become the bottleneck to restrictEV development scale Specifically in the process of frequentstart-stop or climbing the chemical battery life is moreshorten because of fast and deep discharging [2 3] How toeliminate these shortcomings becomes the key EV to bequickly developed

Maglev flywheel can be applied in EV electric powersystem aerospace and other fields because of having highspecific energy high power fast charge and discharge longservice life no waste gas pollution environment-friendlyadvantages and so on [4ndash6] In the field of EV the maglevflywheel either can be as an independent power driving EV[3] or can be used as auxiliary power assisting the motivepower batteries work [7 8] However the maglev flywheelcontrol stability will be affected because of existing start-stopacceleration and deceleration steering and road randomvibration in the process of the EV driving and even causeinstability So the efficiency of magnetic suspension flywheelmust be reduced

For the scientific research and practical application ofmaglev flywheel a dual-mode adaptive hybrid control strat-egy is studied based on 119867infin control and AILC algorithmsand the state space equation ofmaglev flywheel was analyzedTo improve the robust stability of flywheel control systemand reduce the real-time interference one 119867infin controllerbased on the state space equation was solved To reducemaglev flywheel radial run-out one adaptive iterative learn-ing control theory was deduced and unbalance displacementcompensation was implemented The experimental resultshows the dual-model control strategy has better interfer-ence capability in maglev flywheel start-up process and hasstronger active control ability relative to only PID controlThemaglev flywheel based on the dual-model control can help theEVprimary battery improve its discharge characteristics andhelp to prolong its service life

2 Maglev Flywheel Nonlinear Dynamic Model

The below force and movement differential equation of theflywheel are discussed in order to solve the maglev flywheelnonlinear dynamic model Figure 1 shows the below force ofthe flywheel

In Figure 1 119874-119883119884119885 is the space coordinate 1 and 2 arerespectively the left and right position of the radial magnetic

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 327347 9 pageshttpdxdoiorg1011552015327347

2 Mathematical Problems in Engineering

ΔFx2

ΔFy2

l2

l

fzfy fx

Og

ΔFz

l1

ΔFx1

ΔFy1

X

Y

Z

O120579 120593

1

2

Figure 1 Flywheel below force analysis

bearing 119874119892 (119909119892 119910119892 119911119892) is the mass center of the flywheel 119897

is the distance between 1 and 2 1198971 is the distance from themass center to 1 1198972 is the distance from themass center to 2 120593and 120579 are respectively the angular displacements around the119909-axis and 119910-axis Δ1198651199091 Δ1198651199092 Δ1198651199101 Δ1198651199102 and Δ119865119911 are theelectromagnetic force of the flywheel in the three axes and119891119909119891119910 and119891119911 are the disturbing force and the unbalance forceof the flywheel in the three axes The flywheel movementdifferential equations are deduced according to the belowforce analysis in Figure 1 and the motion laws of particles [9]

119898119892 = Δ1198651199091 + Δ1198651199092 minus 119891119909

119898 119910119892 = Δ1198651199101 + Δ1198651199102 minus 119891119910

119868119903 minus 120596119868119886120579 = Δ1198651199101 sdot 1198971 minus Δ1198651199102 sdot 1198972

119868119903120579 + 120596119868119886 = minusΔ1198651199091 sdot 1198971 + Δ1198651199092 sdot 1198972

119898119892 = Δ119865119911 minus 119891119911

(1)

where 119898 is the flywheel quality 119868119903 and 119868119886 are respectivelythe radial inertia moment and axial inertia moment and 120596 isthe angular velocity of the flywheel The motion differentialequation of matrix form can be written by [10]

(((((((

(

1198972

119897119898

1198971

119897119898 0 0 0

0 01198972

119897119898

1198971

119897119898 0

0 0 minus119868119903

119897

119868119903

1198970

minus119868119903

119897

119868119903

1198970 0 0

0 0 0 0 119898

)))))))

)

(

1

2

1199101

1199102

)

+

(((((((

(

0 0 0 0 0

0 0 0 0 0

120596119868119886

119897minus

120596119868119886

1198970 0 0

0 0 minus120596119868119886

119897

120596119868119886

1198970

0 0 0 0 0

)))))))

)

(

1

2

1199101

1199102

)

=

(((((

(

1198961199091199091 1198961199091199092 0 0 0

0 0 1198961199091199101 1198961199091199102 0

0 0 11989711198961199091199101 minus11989721198961199091199102 0

minus11989711198961199091199091 11989721198961199091199092 0 0 0

0 0 0 0 119896119909119911

)))))

)

(

1199091

1199092

1199101

1199102

119911

)

+

(((((

(

1198961198941199091 1198961198941199092 0 0 0

0 0 1198961198941199101 1198961198941199102 0

0 0 11989711198961198941199101 minus11989721198961198941199102 0

minus11989711198961198941199091 11989721198961198941199092 0 0 0

0 0 0 0 119896119894119911

)))))

)

(

1198941198881199091

1198941198881199092

1198941198881199101

1198941198881199102

119894119888119911

)

+ (

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

) (

(

119891119909

119891119910

0

0

119891119911

)

)

(2)

where 1199091 1199092 1199101 1199102 and 119911 are the displacements of theflywheel in five freedomdegrees 1198941198881199091 1198941198881199092 1198941198881199101 1198941198881199101 and 119894119888119911 arerespectively the control current corresponding to each free-dom degree 1198961199091199091 1198961199091199092 1198961199091199101 1198961199091199102 and 119896119909119911 are respectivelythe displacement stiffness of each freedom degree and 11989611989411990911198961198941199092 1198961198941199101 1198961198941199102 and 119896119894119911 are respectively the current stiffness ofeach freedom degree

For the convenience of analysis type (2) can be expressedby type (3)

Mx + Cx + Kx = Bic + If (3)

where M is the mass matrix C is the damping coefficientmatrixK is the displacement stiffnessmatrixB is the currentstiffness matrix I is the unit matrix x is the displacementvector ic is the control current vector and f is the unbalancedinertial force vector Type (3) can be rewritten as

x = minusCM

x minusKM

x +BM

ic +IM

f (4)

Mathematical Problems in Engineering 3

controller

controllerAILC

AMB

Flywheelrotor

Power amplifier

Displacementsensor

Dual-mode switch control strategy

Real-time switching control according to the flywheel speed

DA

AD

Tachometric survey

Reference signal

Hinfin

+minus

Real

-tim

e sw

itchi

ngco

ntro

ller

Reco

gniti

on ju

dgm

ent

Gen

eral

ized

cont

rol e

leph

ant

Figure 2 The dual-mode switch control strategy diagram

The state space of the flywheel system transfer functionmatrix 119866(119904) is

x = Ax + B1f + B2u

y = Cx

(5)

where x = [x x]T is the state vector u = ic is the vector

control y = x is the measured quantity namely the sensoroutput signal vector of the five freedom degrees Moreovertype (5) can also be expressed as

G (119904) = [11986611 (119904) 11986612 (119904)

11986621 (119904) 11986622 (119904)] = [

A B1 B2C 119874 119874

] (6)

where 119874 is the five-order zero matrix it can be solved outthrough types (4) and (5)

A = [

[

119874 IminusKM

minusCM

]

]

B1 = [

[

119874

IM

]

]

B2 = [

[

119874

minusIM

]

]

C = [I 119874]

(7)

So far the nonlinear dynamic model 119866(119904) of the maglevflywheel is solved and the precise model is provided for thedesign of the following dual-mode switch controller

3 Dual-Mode Switch Control Strategy

To improve the control stability and the energy storagedensity of vehicle maglev flywheel a dual-mode switchcontrol strategy is study based on 119867infin control algorithmand adaptive iterative learning control algorithm The con-trol strategy is shown in Figure 2 The control strategy

includes generalized control elephant and dual-mode switchcontroller Among them the generalized control elephantis composed of power amplifiers active magnetic bearing(AMB) flywheel rotor and displacement and the dual-mode switch controller is composed of 119867infin controller andunbalance compensation controller

The basic response is a random signal when EV is drivingon the bumpy road or start-stop acceleration-decelerationand steering To reduce the random impact and improve thecontrol robustness a 119867infin controller is solved Besides tolimit the flywheel radial run-out of the maglev flywheel bat-tery in the charging process and improve the energy storagedensity of the maglev flywheel an unbalance displacementfeed-forward compensation controller based on adaptiveiterative learning control (AILC) algorithm is adopted [10]The switching work between 119867infin control and AILC isimplemented by the real-time switching controller by judgingthe status of the EV and the maglev flywheel

31 119867infin Controller Design A kind of 119867infin control strategywithmixed sensitivity is designed according to the character-istic of the maglev flywheel state space equation it is shownin Figure 3

In Figure 3 KH is the transfer function matrix of the119867infin controller u is the input signal matrix 1198821 1198822 and 1198823

are the weighted functions G is the state space equation ofthe maglev flywheel (it contains the power amplifier transferfunction 119866119860 the sensor transfer function 119866119878 and the controlcurrent ic and so forth) y is the sensor output signal vector ris the system reference signal vector and e is the error signalmatrix

Tc is the closed transfer function in Figure 3 can bewritten as [11]

Tc = 119865119897 (GK119867) = G11 + G12K119867 (I minus G22K119867)minus1G21

(8)


Generalized controlled object G

Displacement sensor

Flywheel

W1 W2 W3

KH

KH

GAGS

ic

ic

I0

I0

+

+

+

+

minus

minus

uy

r e

Figure 3 The diagram of 119867infin control strategy

where type (8) is the linear fractional transformation of the119867infin controller K119867 The standard of 119867infin control problem isto find a real rational K119867 to make the controlled object Gstable work and to make the minimal 119867infin norm of Tc in thewhole frequency range [12] should be satisfied by

1003817100381710038171003817Tc1003817100381710038171003817infin

= sup120596

120590 (Tc (119895120596)) lt 120574 (9)

where 120590(Tc(119895120596)) is the biggest singular value of Tc ldquosuprdquo isthe supremum of 120590(Tc(119895120596)) in the whole frequency rangeand 120574 is a given positive number

The corresponding sensitivity and complementary sensi-tivity matrix functions S and T combined with Figure 2 canbe deduced

S =er

=1

I + K119867 (119904)G (119904)

T =yr

=K119867 (119904)G (119904)

I + K119867 (119904)G (119904)= I minus S

(10)

1198821(119904) is the weighted function of S its main purpose is tolimit the amplitude of S in a specified frequency range 1198823(119904)is the weighted function of T whose purpose is to limit theamplitude of T In addition the transfer function matrix ofthe 119867infin controller output is

R =ur

=K119867 (119904)

I + K119867 (119904)G (119904) (11)

R also has a weighted function 1198822(119904) its main purpose is tolimit the K119867 controller output 1198822(119904) should be chosen as arelatively small value in order to reduce the compensators andshorten operation cycle

The structural parameters of the maglev flywheel arecalculated and got as 119898 = 108 kg 1198961198941199091 = 1198961198941199092 = 1198961198941199101 = 1198961198941199102 =

9546NA 119896119894119911 = 15438NA 1198961199091199091 = 1198961199091199092 = 1198961199091199101 = 1198961199091199102 =

1517 times 106Nm and 119870119883119885 = 3216 times 10

6Nm Based on 119867infin

controller problem solving limit andMATLAB robust controlinstructions the solved parameters are involved in the G(119904)

solution procedure and the weighting functions 1198821(119904) and1198823(119904) are determined respectively by

1198821 (119904) =98

119904 + 2

1198823 (119904) =15119904

119904 + 3200

(12)

In addition 1198822(119904) is selected as 20 times 107 through the

simulation analysis Then the robust performance index(04232) is calculated based on the controlled object transferfunction matrix G and the three weighted functions if theindex greater than 1 the three weighted functions need to beselected Finally the four radial and one axial discrete transferfunctions of the 119867infin controller are concluded

1198701198671199091 = 1198701198671199092 = 1198701198671199101 = 1198701198671199102

=minus218119911

3+ 153119911

2minus 256119911 + 2132

1199113 minus 10611199112 + 869119911 minus 173

119870119867119911 =minus1013119911

3+ 805119911

2+ 468

1199113 minus 5871199112 + 232119911 minus 021

(13)

Because the maglev flywheel is axially symmetric distri-bution in four radial freedom degrees the four radial discretetransfer functions are the same In order to make the controlcycle consistent with the actual flywheel control system thesimulation sampling frequency is selected as 20000Hz

32 Feed-Forward Compensation Control Analysis To reducethe radial run-out of the flywheel and solve the problemof variable speed the AILC algorithm is adopted as thefeed-forward controller to implement vibratory displacementcompensation [10] In Figure 4 the control scheme consists ofPID feedback control system AILC feed-forward compensa-tion controller and generalized plantThe PID controller cansteady the whole system and improve the anti-interferenceability and the action of AILC is to make the learning gainaccurately track the expectant orbit The generalized plantincludes power amplifier electromagnetic coils and rotorsystem

To improve control performance and enhance the con-vergence rate of the learning law there are two modificationsin AILC The first one is enhancing the error informationaction of previous control period and the second one isproposing a novel impacting factor 120573 as the coefficient of thelearning gain V119896 120573 can reduce the effects of learning gain tothe control system when rotor speed does not coincide withthe learning cycle of AILC AILC can implement vibratorydisplacement compensation without any information of thegeneralized plant and it will not increase the interference ofthe feedback controller Only the expectant signal 119910119889 and theoutput of the sensor 119910119896 are needed in AILC here 119910119889 is 25 Vwitch is defined as the balance position of rotor during staticsuspension The error signal between 119910119889 and 119910119896 is iterativelylearned then the perfect and unknown control signal 119906119896 isobtained as the input signal of the power amplifier


Sensor

120573

yd

yk

= 25V+

++

+ +

+

+

minus

Q1

ek ekminus1

c uk

P

k+1

k

QMemory II Memory I

PID controller Power amplifier AMB

Flywheel

Feedback control system

AILC feed-forward controller

Generalized plant

Figure 4 AILC compensation principle of AMB system

The functions of AILC can be introduced by the iterativeformulas in discrete domain The error formula is given

The update learning law of AILC is summarized as

V119896+1 (119899) = 120573119875V119896 (119899) + 119876119890119896 (119899) + 1198761119890119896 (119899 minus 1) (14)

To understand the action ofAILCbetter the control inputsignal of generalized plant is written as

119906119896 (119899) = 119888 (119899) + 120573V119896 (119899) (15)

where 119906119896(119899) is the controller input 119888(119899) is PID controlleroutput V119896(119899) is the learning gain of AILC and 120573 is theimpacting factor of V119896(119899) The last objective is to obtainperfect controller signal 119906119889 when having infinitely iterativelearning and then make the error signal become zero Itshould be satisfied as [13]

lim119896rarrinfin

119906119896 (119899) = 119906119889 lim119896rarrinfin

119890119896 (119899) = 0 (16)

The discrete transfer function of the learning law of (14)can be calculated by

119881119896+1 (119911) = 120573119875119881119896 (119911) + 119876119864119896 (119911) + 1198761119864119896 (119911) 119911minus1

= 120573119875119881119896 (119911) + (119876 + 1198761119911minus1

) 119864119896 (119911)

(17)

where 119911minus1 represents the lag operator in time domain and

it could make the sampled signal lag one period This is thereason why the former error information is added into themodified learning law of AILC

The PID controller with incomplete differential part isgiven in time domain

119888 (119905) = 119866119875 (119905)

= 119870119901 [119890 (119905) +1

119879119894

int

119905

0

119890 (119905) 119889119905 +119879119889

1 + 119879119891

119889 (119890 (119905))

119889119905]

(18)

The discrete function of (18) is deduced as

119862 (119911)

= 119870119901 [1 +1198790

119879119894

1

1 minus 119911minus1+

119879119889

(1 + 119879119891) 1198790

(1 minus 119911minus1

)] 119864 (119911)

(19)

The 119911-transform of the control signal function of (15) isdeduced by

119880119896 (119911) = 119862 (119911) + 120573119881119896 (119911)

= 119896119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)] 119864 (119911)

+ 120573119881119896 (119911)

(20)

where defining

119867 (119911)

= 1 (119896119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)])

minus1

(21)

The discrete function of error signal can be obtained as

119864 (119911) = (119880119896 (119911) minus 120573119881119896 (119911)) 119867 (119911) (22)

Putting (22) into (17) it has

119881119896+1 (119911) = 120573119875119881119896 (119911) + (119876 + 1198761119911minus1

) (119880119896 (119911) minus 120573119881119896 (119911)) 119867 (119911)

= 120573 [119875 minus (119876 + 1198761119911minus1

) 119867 (119911)] 119881119896 (119911)

+ (119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

(23)


Equation (24) gives the transformation in limit calcula-tion at the two sides of (23)

119881infin (119911) = lim119896rarrinfin

119881119896 (119911)

= lim119896rarrinfin

(119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

1 minus 120573 [119875 + (119876 + 1198761119911minus1) 119867 (119911)]

(24)

If 119875 and 120573 all are 1 at the same time the followinginequality is satisfied by [14]

10038171003817100381710038171003817120573119875 minus 120573[119876 + 1198761119911

minus1]119867(119911)

10038171003817100381710038171003817infinlt 1 (25)

Equation (24) can be simplified by


119880119896 (119911) = 119880119889 (119911) (26)

That is the controller signal 119906119896(119911) is replaced by 120573V119896(119911)

when infinite iteration is operated and the error signal willbecome zero According to (22) and (26) the error signal canbe shown as

lim119896rarrinfin

119864 (119911) = 119867 (119911) [ lim119896rarrinfin

119880119896 (119911) minus 120573 lim119896rarrinfin

119881119896 (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119881infin (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119880119889 (119911)] = 0

(27)

The convergence of AILC has been demonstrated accord-ing to (26) and (27) and a perfect controller signal 119906119889 hasbeen obtained as power amplifier input Therefore the AILCalgorithm as the feed-forward compensation controller canbe adopted in the application of maglev flywheel unbalancevibratory compensation

According to the start-up time and the variability of therotor frequency from static suspension to one fixed speed theequation of 120573 is given as

120573 = (119891

119891119889

)

119899

(28)

where 119891 is the flywheel frequency 119891119889 is a given frequencyand the action of 119899 is to reduce the value of 120573 when 119891 is faraway from 119891119889 (usually 119899 is greater than 2) In the start-upprocess due to 119891 ≪ 119891119889 the value of 120573 should be very smallit can weaken the influence of 119881119896 on the generalized plantHowever 120573 will be close to 1 if 119891 asymp 119891119889 it can enhance theeffect of repetitive learning and can make the error signal toconverge toward zero

4 Simulation and Experimental

41 Simulation Analysis First the stability of 119867infin controlleris attested through analyzing the singular values relationshipbetween 119878(119904) and 1198821(119904) as well as between 119879(119904) and 1198823(119904)The sensitivity and complementary sensitivity function andthe corresponding weighted function singular value relations

0

20

40

60

80

100

10minus1 100 101 102 103 104 105

Angular frequency 120596 (rads)

minus20

minus40

minus60

S(s)1W1(s)

T(s)

1W3(s)

Sing

ular

val

ue (d

B)Figure 5 Sensitivity complementary sensitivity and the corre-sponding weighted function singular value relations

are shown in Figure 5 The curves in Figure 5 are created inthe ldquoMrdquo file of MATLAB through some command functionsthe transfer functions of Tc K119867 and G(119904) and so on

The choice principle of 1198821(119904) is guaranteeing the anti-interference and tracking ability of the flywheel system andthe smaller the singular value of 119878(119904) is the better the systemtracking ability is The curve of 119878(119904) should be under thecurve of the 11198821(119904) if it does not conform to the demandthe weighted function 1198821(119904) must be chosen again Thechoice principle of 1198823(119904) is guaranteeing the flywheel systemoutput to recurrence the input and the smaller the singularvalue of 119879(119904) is the smaller the system impact by compounddisturbance because of model uncertainty is The curve of119879(119904) should also be under the curve of the 11198823(119904) if it doesnot conform to the demand the weighted function 1198823(119904)

must be chosen again Figure 5 shows that the curve of 119878(119904)

is under the curve of 11198821(119904) and the curve of 119879(119904) is underthe curve of 11198823(119904) which demonstrates that the selection ofweighted function in types (14) and (15) can meet the designrequirements and the solved 119867infin controller is appropriate

Second the simulation parameters of AILC are selected as119891119889 = 600Hz 119875 = 0995 119876 = 075 1198761 = 025 119899 = 0 1 99and the definition of ldquo119899rdquo indicates that the AILC algorithmhas 100 memory points in one control period Figure 6 showsthe rotor simulation orbits and radial run-out displacementswith AILC compensation

The flywheel has a regular circular trajectory in Fig-ure 6(a) and a sine radial run-out displacement in Fig-ure 6(b) without adding AILC compensation When theAILC algorithm starts to work at 005 s the circular trajectoryis gradual convergence to a point and the amplitude ofthe radial run-out displacement is obvious attenuation Thecurves change simulation result in Figure 6 testifies the AILCalgorithm having good displacement compensation effect


0 10 20

0

5

10

15D

istan

cey1

(120583m

)

Distance x1 (120583m)

minus5

minus10

minus15minus20 minus10

(a) Rotor position orbit

0 002 004 006 008 01

0

5

10

15

Dist

ance

x1

(120583m

)

minus5

minus10

minus15

Time t (s)

(b) Rotor run-out in axis 1199091

Figure 6 Flywheel trajectory with AILC compensation

00

4

3

2

5

100 600500400300200 700

4

2

6

Flywheel radial displacement curve

Control current curve

Time t (ms)

0 100 600500400300200 700Time t (ms)

Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)

Control current curveeee


Figure 7 Flywheel displacement and control current in start-upprocess by 119867infin control

and ensuring the flywheel to rotate around its collectioncenter

42 Experimental Results In Figure 7 the radial displace-ment voltage curve and the control current curve of theflywheel are shown in start-up process by the solved 119867infin

controller control the displacement voltage quickly drops tothe equilibrium position (25 V) from the sensor calibrationposition (4V) It illustrates that the 119867infin controller has goodrobust stability and can be used in the control system ofvehicle maglev flywheel

Figure 8 includes the radial displacement curve thecontrol current curve and the speed measuring pulse whenflywheel normal is rotating by 119867infin control and flywheel

0 10 50403020

2

4

3

2

1

06

Time t (ms)

C1 flywheel radial displacement (1Vdiv)

C1

C2

C4 C4 control current (2Adiv)

C2 speed measuring signal (0sim33V)

Con

trol c

urre

nti

(A) Ra

dial

disp

lace

men

tx(V

)

2

4

6


C1

C2



Figure 8 Flywheel displacement and control current in rotatingprocess by 119867infin control and the speed measuring pulse at 600Hz

rotational frequency is 600Hz The displacement curve is asine wave and it can show the flywheel has mass unbalanceTo restrict flywheel radial run-out the control current alsois sine having basic consistent phase with the displacementvoltage The speed measuring pulse has a voltage range from0V to 33 V the purpose is to guarantee the pulse be capturedby DSP acquisition circuit

The compensated effect by AILC algorithm at 600Hzis shown in Figure 9 including the radial displacementcurve and the control current curve after compensationCompared with the radial sine displacement in Figure 8 theradial displacement is balance in the position of 25 V whichindicates the flywheel is limited rotating around its geometriccenter To limit the radial run-out and increase the activecontrol effect the control current amplitude is significantlylarger than without compensation


00

2

1

0

3

10 50403020

4

2

6



Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)

Time t (ms)

0 10 50403020Time t (ms)


Figure 9 Flywheel displacement and control current in AILC com-pensation process at 600Hz

0 50 100 150 2000

5

10

15

20

25

30

35

Start Speed upConstant speed

Flywheel not workingOnly PID controlDual-model switch control

Constantspeed

Time t (s)

Out

put p

ower

P(k

W)

Figure 10 EV primary battery output power in different modes

Figure 10 gives the output power curves of EV primarybattery in three different modes If flywheel is not workingthe output power is about 35 kW when EV starts up if theflywheel battery is working and only PID controlling theoutput power of the primary battery is about 25 kW in EVstart-up process if the dual-model switch control strategy isacting on the flywheel the output power reduces to 23 kW

The power change shows the flywheel has better auxiliaryeffect to EV primary battery when it is controlled by the dual-model switch control strategy Similarly the output powerof EV primary battery is the smallest in the accelerationprocess when the dual-model switch control strategy controlsthe flywheel Moreover because the flywheel rotate speedis higher when the dual-model switch control strategy isworking EV needs lower motive power when restartingTherefore when maglev flywheel participates in dischargeand is controlled by dual-model switch control strategyEV primary battery output power is obviously decreasedin the whole driving process the influence on chemicalcharacteristics of primary battery is reduced because theinstantaneous discharge depth is small and the service lifeof EV primary battery can be improved

5 Conclusion

Through analyzing the nonlinear dynamic characteristicsand the application on EV one kind of dual-model controlstrategy based on 119867infin control and AILC algorithms hasbeen studied in this paper and the state space equationof maglev flywheel was analyzed To improve the robuststability of flywheel control system and reduce the real-time interference one 119867infin controller based on the statespace equation was solved To reduce maglev flywheel radialrun-out one adaptive iterative learning control theory wasdeduced and unbalance displacement compensation wasimplementedThe experimental result shows the dual-modelcontrol strategy has better interference capability in maglevflywheel start-up process and has stronger active controlability relative to only PID control The maglev flywheelbased on the dual-model control can help the EV primarybattery improve its discharge characteristics help to prolongits service life and accelerate the development scale of electricvehicles

Conflict of Interests

The authors declare that they have no conflict of interestsregarding this paper

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (51405244) the China PostdoctoralScience Foundation (2014M551634) and the Jiangsu ProvinceNatural Science Foundation of China (BK20140880) Theauthors would like to thank the editor and the reviewers fortheir constructive comments and suggestions to improve thequality of the paper

References

[1] E Sortomme Optimal aggregator bidding strategies for vehicle-to-grid [PhD thesis of Philosophy] University of Washington2011

[2] H-F Dai Z-C Sun and X-Z Wei ldquoTechnologies to relief un-uniformity of power batteries used in electrical vehiclesrdquo Jour-nal of Automotive Safety and Energy vol 2 no 1 pp 62ndash67 2011


[3] B Bolund H Bernhoff and M Leijon ldquoFlywheel energyand power storage systemsrdquo Renewable and Sustainable EnergyReviews vol 11 no 2 pp 235ndash258 2007

[4] M Jiang J Salmon and A M Knight ldquoDesign of a permanentmagnet synchronous machine for a flywheel energy storagesystem within a hybrid electric vehiclerdquo in Proceedings of theIEEE International Electric Machines and Drives Conference(IEMDC rsquo09) pp 1736ndash1742 May 2009

[5] J Abrahamsson J de Santiago and J G Oliveira ldquoPrototypeof electric driveline with magnetically levitated double woundmotorrdquo in Proceedings of the 19th International Conference onElectrical Machines (ICEM rsquo10) pp 1ndash5 September 2010

[6] M S Raymond M E F Kasarda and P E Allaire ldquoWindagepower lossmodeling of a smooth rotor supported by homopolaractive magnetic bearingsrdquo Journal of Tribology vol 130 no 2pp 1ndash8 2008

[7] Y-L Xu and J-C Fang ldquoDevelopment of low power loss energystorage flywheelrdquo Transactions of China Electrotechnical Societyvol 23 no 12 pp 11ndash16 2008

[8] T A Aanstoos J P Kajs W G Brinkman et al ldquoHigh voltagestator for a flywheel energy storage systemrdquo IEEE Transactionson Magnetics vol 37 no 1 pp 242ndash247 2001

[9] H Qing-kai Nonlinear Vibration Analysis and DiagnosisMethod for Fault Rotor System Science Press Beijing China2010

[10] H Gao L Xu and Y Zhu ldquoUnbalance vibratory displacementcompensation for active magnetic bearingsrdquo Chinese Journal ofMechanical Engineering vol 26 no 1 pp 95ndash103 2013

[11] Z Gosiewski and A Mystkowski ldquoRobust control of active mag-netic suspension analytical and experimental resultsrdquoMechan-ical Systems and Signal Processing vol 22 no 6 pp 1297ndash13032008

[12] J-M Kim C-K Kim M-K Park and K-H Kim ldquoA robustcontrol of a magnetic suspension system with a flexible railrdquo inProceedings of the IEEE International Symposium on IndustrialElectronics (ISIE rsquo01) vol 2 pp 1309ndash1312 Pusan Republic ofKorea June 2001

[13] H G Chiacchiarini and P S Mandolesi ldquoUnbalanced compen-sation for active magnetic bearings using ILCrdquo in Proceedings ofthe IEEE International Conference on Control Applications pp58ndash63 Mexico City Mexico September 2001

[14] C Bi D Wu Q Jiang and Z Liu ldquoAutomatic learning controlfor unbalance compensation in active magnetic bearingsrdquo IEEETransactions on Magnetics vol 41 no 7 pp 2270ndash2280 2005

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


ΔFx2

ΔFy2

l2

l

fzfy fx

Og

ΔFz

l1

ΔFx1

ΔFy1

X

Y

Z

O120579 120593

1

2

Figure 1 Flywheel below force analysis

bearing 119874119892 (119909119892 119910119892 119911119892) is the mass center of the flywheel 119897

is the distance between 1 and 2 1198971 is the distance from themass center to 1 1198972 is the distance from themass center to 2 120593and 120579 are respectively the angular displacements around the119909-axis and 119910-axis Δ1198651199091 Δ1198651199092 Δ1198651199101 Δ1198651199102 and Δ119865119911 are theelectromagnetic force of the flywheel in the three axes and119891119909119891119910 and119891119911 are the disturbing force and the unbalance forceof the flywheel in the three axes The flywheel movementdifferential equations are deduced according to the belowforce analysis in Figure 1 and the motion laws of particles [9]

119898119892 = Δ1198651199091 + Δ1198651199092 minus 119891119909

119898 119910119892 = Δ1198651199101 + Δ1198651199102 minus 119891119910

119868119903 minus 120596119868119886120579 = Δ1198651199101 sdot 1198971 minus Δ1198651199102 sdot 1198972

119868119903120579 + 120596119868119886 = minusΔ1198651199091 sdot 1198971 + Δ1198651199092 sdot 1198972

119898119892 = Δ119865119911 minus 119891119911

(1)

where 119898 is the flywheel quality 119868119903 and 119868119886 are respectivelythe radial inertia moment and axial inertia moment and 120596 isthe angular velocity of the flywheel The motion differentialequation of matrix form can be written by [10]

(((((((

(

1198972

119897119898

1198971

119897119898 0 0 0

0 01198972

119897119898

1198971

119897119898 0

0 0 minus119868119903

119897

119868119903

1198970

minus119868119903

119897

119868119903

1198970 0 0

0 0 0 0 119898

)))))))

)

(

1

2

1199101

1199102

)

+

(((((((

(

0 0 0 0 0

0 0 0 0 0

120596119868119886

119897minus

120596119868119886

1198970 0 0

0 0 minus120596119868119886

119897

120596119868119886

1198970

0 0 0 0 0

)))))))

)

(

1

2

1199101

1199102

)

=

(((((

(

1198961199091199091 1198961199091199092 0 0 0

0 0 1198961199091199101 1198961199091199102 0

0 0 11989711198961199091199101 minus11989721198961199091199102 0

minus11989711198961199091199091 11989721198961199091199092 0 0 0

0 0 0 0 119896119909119911

)))))

)

(

1199091

1199092

1199101

1199102

119911

)

+

(((((

(

1198961198941199091 1198961198941199092 0 0 0

0 0 1198961198941199101 1198961198941199102 0

0 0 11989711198961198941199101 minus11989721198961198941199102 0

minus11989711198961198941199091 11989721198961198941199092 0 0 0

0 0 0 0 119896119894119911

)))))

)

(

1198941198881199091

1198941198881199092

1198941198881199101

1198941198881199102

119894119888119911

)

+ (

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

) (

(

119891119909

119891119910

0

0

119891119911

)

)

(2)

where 1199091 1199092 1199101 1199102 and 119911 are the displacements of theflywheel in five freedomdegrees 1198941198881199091 1198941198881199092 1198941198881199101 1198941198881199101 and 119894119888119911 arerespectively the control current corresponding to each free-dom degree 1198961199091199091 1198961199091199092 1198961199091199101 1198961199091199102 and 119896119909119911 are respectivelythe displacement stiffness of each freedom degree and 11989611989411990911198961198941199092 1198961198941199101 1198961198941199102 and 119896119894119911 are respectively the current stiffness ofeach freedom degree

For the convenience of analysis type (2) can be expressedby type (3)

Mx + Cx + Kx = Bic + If (3)

where M is the mass matrix C is the damping coefficientmatrixK is the displacement stiffnessmatrixB is the currentstiffness matrix I is the unit matrix x is the displacementvector ic is the control current vector and f is the unbalancedinertial force vector Type (3) can be rewritten as

x = minusCM

x minusKM

x +BM

ic +IM

f (4)


controller

controllerAILC

AMB

Flywheelrotor

Power amplifier

Displacementsensor



DA

AD

Tachometric survey

Reference signal

Hinfin

+minus

Real

-tim

e sw

itchi

ngco

ntro

ller

Reco

gniti

on ju

dgm

ent

Gen

eral

ized

cont

rol e

leph

ant



x = Ax + B1f + B2u

y = Cx

(5)



G (119904) = [11986611 (119904) 11986612 (119904)

11986621 (119904) 11986622 (119904)] = [

A B1 B2C 119874 119874

] (6)


A = [

[

119874 IminusKM

minusCM

]

]

B1 = [

[

119874

IM

]

]

B2 = [

[

119874

minusIM

]

]

C = [I 119874]

(7)











(8)



Displacement sensor

Flywheel

W1 W2 W3

KH

KH

GAGS

ic

ic

I0

I0

+

+

+

+

minus

minus

uy

r e



1003817100381710038171003817Tc1003817100381710038171003817infin

= sup120596

120590 (Tc (119895120596)) lt 120574 (9)



S =er

=1

I + K119867 (119904)G (119904)

T =yr

=K119867 (119904)G (119904)

I + K119867 (119904)G (119904)= I minus S

(10)


R =ur

=K119867 (119904)

I + K119867 (119904)G (119904) (11)



9546NA 119896119894119911 = 15438NA 1198961199091199091 = 1198961199091199092 = 1198961199091199101 = 1198961199091199102 =





1198821 (119904) =98

119904 + 2

1198823 (119904) =15119904

119904 + 3200

(12)



1198701198671199091 = 1198701198671199092 = 1198701198671199101 = 1198701198671199102

=minus218119911

3+ 153119911

2minus 256119911 + 2132

1199113 minus 10611199112 + 869119911 minus 173

119870119867119911 =minus1013119911

3+ 805119911

2+ 468

1199113 minus 5871199112 + 232119911 minus 021

(13)





Sensor

120573

yd

yk

= 25V+

++

+ +

+

+

minus

Q1

ek ekminus1

c uk

P

k+1

k

QMemory II Memory I


Flywheel



Generalized plant




V119896+1 (119899) = 120573119875V119896 (119899) + 119876119890119896 (119899) + 1198761119890119896 (119899 minus 1) (14)


119906119896 (119899) = 119888 (119899) + 120573V119896 (119899) (15)


lim119896rarrinfin

119906119896 (119899) = 119906119889 lim119896rarrinfin

119890119896 (119899) = 0 (16)


119881119896+1 (119911) = 120573119875119881119896 (119911) + 119876119864119896 (119911) + 1198761119864119896 (119911) 119911minus1

= 120573119875119881119896 (119911) + (119876 + 1198761119911minus1

) 119864119896 (119911)

(17)




119888 (119905) = 119866119875 (119905)

= 119870119901 [119890 (119905) +1

119879119894

int

119905

0

119890 (119905) 119889119905 +119879119889

1 + 119879119891

119889 (119890 (119905))

119889119905]

(18)


119862 (119911)

= 119870119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)] 119864 (119911)

(19)


119880119896 (119911) = 119862 (119911) + 120573119881119896 (119911)

= 119896119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)] 119864 (119911)

+ 120573119881119896 (119911)

(20)

where defining

119867 (119911)

= 1 (119896119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)])

minus1

(21)


119864 (119911) = (119880119896 (119911) minus 120573119881119896 (119911)) 119867 (119911) (22)


119881119896+1 (119911) = 120573119875119881119896 (119911) + (119876 + 1198761119911minus1

) (119880119896 (119911) minus 120573119881119896 (119911)) 119867 (119911)

= 120573 [119875 minus (119876 + 1198761119911minus1

) 119867 (119911)] 119881119896 (119911)

+ (119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

(23)




119881119896 (119911)


(119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

1 minus 120573 [119875 + (119876 + 1198761119911minus1) 119867 (119911)]

(24)


10038171003817100381710038171003817120573119875 minus 120573[119876 + 1198761119911

minus1]119867(119911)

10038171003817100381710038171003817infinlt 1 (25)



119880119896 (119911) = 119880119889 (119911) (26)



lim119896rarrinfin

119864 (119911) = 119867 (119911) [ lim119896rarrinfin


119881119896 (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119881infin (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119880119889 (119911)] = 0

(27)



120573 = (119891

119891119889

)

119899

(28)




0

20

40

60

80

100

10minus1 100 101 102 103 104 105


minus20

minus40

minus60

S(s)1W1(s)

T(s)

1W3(s)

Sing

ular

val

ue (d









0 10 20

0

5

10

15D

istan

cey1

(120583m

)


minus5

minus10



0 002 004 006 008 01

0

5

10

15

Dist

ance

x1

(120583m

)

minus5

minus10

minus15

Time t (s)



00

4

3

2

5

100 600500400300200 700

4

2

6



Time t (ms)

0 100 600500400300200 700Time t (ms)

Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)








0 10 50403020

2

4

3

2

1

06

Time t (ms)


C1

C2



Con

trol c

urre

nti

(A) Ra

dial

disp

lace

men

tx(V

)

2

4

6


C1

C2







00

2

1

0

3

10 50403020

4

2

6



Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)

Time t (ms)

0 10 50403020Time t (ms)



0 50 100 150 2000

5

10

15

20

25

30

35



Constantspeed

Time t (s)

Out

put p

ower

P(k

W)




5 Conclusion




Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










controller

controllerAILC

AMB

Flywheelrotor

Power amplifier

Displacementsensor



DA

AD

Tachometric survey

Reference signal

Hinfin

+minus

Real

-tim

e sw

itchi

ngco

ntro

ller

Reco

gniti

on ju

dgm

ent

Gen

eral

ized

cont

rol e

leph

ant



x = Ax + B1f + B2u

y = Cx

(5)



G (119904) = [11986611 (119904) 11986612 (119904)

11986621 (119904) 11986622 (119904)] = [

A B1 B2C 119874 119874

] (6)


A = [

[

119874 IminusKM

minusCM

]

]

B1 = [

[

119874

IM

]

]

B2 = [

[

119874

minusIM

]

]

C = [I 119874]

(7)











(8)



Displacement sensor

Flywheel

W1 W2 W3

KH

KH

GAGS

ic

ic

I0

I0

+

+

+

+

minus

minus

uy

r e



1003817100381710038171003817Tc1003817100381710038171003817infin

= sup120596

120590 (Tc (119895120596)) lt 120574 (9)



S =er

=1

I + K119867 (119904)G (119904)

T =yr

=K119867 (119904)G (119904)

I + K119867 (119904)G (119904)= I minus S

(10)


R =ur

=K119867 (119904)

I + K119867 (119904)G (119904) (11)



9546NA 119896119894119911 = 15438NA 1198961199091199091 = 1198961199091199092 = 1198961199091199101 = 1198961199091199102 =





1198821 (119904) =98

119904 + 2

1198823 (119904) =15119904

119904 + 3200

(12)



1198701198671199091 = 1198701198671199092 = 1198701198671199101 = 1198701198671199102

=minus218119911

3+ 153119911

2minus 256119911 + 2132

1199113 minus 10611199112 + 869119911 minus 173

119870119867119911 =minus1013119911

3+ 805119911

2+ 468

1199113 minus 5871199112 + 232119911 minus 021

(13)





Sensor

120573

yd

yk

= 25V+

++

+ +

+

+

minus

Q1

ek ekminus1

c uk

P

k+1

k

QMemory II Memory I


Flywheel



Generalized plant




V119896+1 (119899) = 120573119875V119896 (119899) + 119876119890119896 (119899) + 1198761119890119896 (119899 minus 1) (14)


119906119896 (119899) = 119888 (119899) + 120573V119896 (119899) (15)


lim119896rarrinfin

119906119896 (119899) = 119906119889 lim119896rarrinfin

119890119896 (119899) = 0 (16)


119881119896+1 (119911) = 120573119875119881119896 (119911) + 119876119864119896 (119911) + 1198761119864119896 (119911) 119911minus1

= 120573119875119881119896 (119911) + (119876 + 1198761119911minus1

) 119864119896 (119911)

(17)




119888 (119905) = 119866119875 (119905)

= 119870119901 [119890 (119905) +1

119879119894

int

119905

0

119890 (119905) 119889119905 +119879119889

1 + 119879119891

119889 (119890 (119905))

119889119905]

(18)


119862 (119911)

= 119870119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)] 119864 (119911)

(19)


119880119896 (119911) = 119862 (119911) + 120573119881119896 (119911)

= 119896119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)] 119864 (119911)

+ 120573119881119896 (119911)

(20)

where defining

119867 (119911)

= 1 (119896119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)])

minus1

(21)


119864 (119911) = (119880119896 (119911) minus 120573119881119896 (119911)) 119867 (119911) (22)


119881119896+1 (119911) = 120573119875119881119896 (119911) + (119876 + 1198761119911minus1

) (119880119896 (119911) minus 120573119881119896 (119911)) 119867 (119911)

= 120573 [119875 minus (119876 + 1198761119911minus1

) 119867 (119911)] 119881119896 (119911)

+ (119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

(23)




119881119896 (119911)


(119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

1 minus 120573 [119875 + (119876 + 1198761119911minus1) 119867 (119911)]

(24)


10038171003817100381710038171003817120573119875 minus 120573[119876 + 1198761119911

minus1]119867(119911)

10038171003817100381710038171003817infinlt 1 (25)



119880119896 (119911) = 119880119889 (119911) (26)



lim119896rarrinfin

119864 (119911) = 119867 (119911) [ lim119896rarrinfin


119881119896 (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119881infin (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119880119889 (119911)] = 0

(27)



120573 = (119891

119891119889

)

119899

(28)




0

20

40

60

80

100

10minus1 100 101 102 103 104 105


minus20

minus40

minus60

S(s)1W1(s)

T(s)

1W3(s)

Sing

ular

val

ue (d









0 10 20

0

5

10

15D

istan

cey1

(120583m

)


minus5

minus10



0 002 004 006 008 01

0

5

10

15

Dist

ance

x1

(120583m

)

minus5

minus10

minus15

Time t (s)



00

4

3

2

5

100 600500400300200 700

4

2

6



Time t (ms)

0 100 600500400300200 700Time t (ms)

Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)








0 10 50403020

2

4

3

2

1

06

Time t (ms)


C1

C2



Con

trol c

urre

nti

(A) Ra

dial

disp

lace

men

tx(V

)

2

4

6


C1

C2







00

2

1

0

3

10 50403020

4

2

6



Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)

Time t (ms)

0 10 50403020Time t (ms)



0 50 100 150 2000

5

10

15

20

25

30

35



Constantspeed

Time t (s)

Out

put p

ower

P(k

W)




5 Conclusion




Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Displacement sensor

Flywheel

W1 W2 W3

KH

KH

GAGS

ic

ic

I0

I0

+

+

+

+

minus

minus

uy

r e



1003817100381710038171003817Tc1003817100381710038171003817infin

= sup120596

120590 (Tc (119895120596)) lt 120574 (9)



S =er

=1

I + K119867 (119904)G (119904)

T =yr

=K119867 (119904)G (119904)

I + K119867 (119904)G (119904)= I minus S

(10)


R =ur

=K119867 (119904)

I + K119867 (119904)G (119904) (11)



9546NA 119896119894119911 = 15438NA 1198961199091199091 = 1198961199091199092 = 1198961199091199101 = 1198961199091199102 =





1198821 (119904) =98

119904 + 2

1198823 (119904) =15119904

119904 + 3200

(12)



1198701198671199091 = 1198701198671199092 = 1198701198671199101 = 1198701198671199102

=minus218119911

3+ 153119911

2minus 256119911 + 2132

1199113 minus 10611199112 + 869119911 minus 173

119870119867119911 =minus1013119911

3+ 805119911

2+ 468

1199113 minus 5871199112 + 232119911 minus 021

(13)





Sensor

120573

yd

yk

= 25V+

++

+ +

+

+

minus

Q1

ek ekminus1

c uk

P

k+1

k

QMemory II Memory I


Flywheel



Generalized plant




V119896+1 (119899) = 120573119875V119896 (119899) + 119876119890119896 (119899) + 1198761119890119896 (119899 minus 1) (14)


119906119896 (119899) = 119888 (119899) + 120573V119896 (119899) (15)


lim119896rarrinfin

119906119896 (119899) = 119906119889 lim119896rarrinfin

119890119896 (119899) = 0 (16)


119881119896+1 (119911) = 120573119875119881119896 (119911) + 119876119864119896 (119911) + 1198761119864119896 (119911) 119911minus1

= 120573119875119881119896 (119911) + (119876 + 1198761119911minus1

) 119864119896 (119911)

(17)




119888 (119905) = 119866119875 (119905)

= 119870119901 [119890 (119905) +1

119879119894

int

119905

0

119890 (119905) 119889119905 +119879119889

1 + 119879119891

119889 (119890 (119905))

119889119905]

(18)


119862 (119911)

= 119870119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)] 119864 (119911)

(19)


119880119896 (119911) = 119862 (119911) + 120573119881119896 (119911)

= 119896119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)] 119864 (119911)

+ 120573119881119896 (119911)

(20)

where defining

119867 (119911)

= 1 (119896119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)])

minus1

(21)


119864 (119911) = (119880119896 (119911) minus 120573119881119896 (119911)) 119867 (119911) (22)


119881119896+1 (119911) = 120573119875119881119896 (119911) + (119876 + 1198761119911minus1

) (119880119896 (119911) minus 120573119881119896 (119911)) 119867 (119911)

= 120573 [119875 minus (119876 + 1198761119911minus1

) 119867 (119911)] 119881119896 (119911)

+ (119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

(23)




119881119896 (119911)


(119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

1 minus 120573 [119875 + (119876 + 1198761119911minus1) 119867 (119911)]

(24)


10038171003817100381710038171003817120573119875 minus 120573[119876 + 1198761119911

minus1]119867(119911)

10038171003817100381710038171003817infinlt 1 (25)



119880119896 (119911) = 119880119889 (119911) (26)



lim119896rarrinfin

119864 (119911) = 119867 (119911) [ lim119896rarrinfin


119881119896 (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119881infin (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119880119889 (119911)] = 0

(27)



120573 = (119891

119891119889

)

119899

(28)




0

20

40

60

80

100

10minus1 100 101 102 103 104 105


minus20

minus40

minus60

S(s)1W1(s)

T(s)

1W3(s)

Sing

ular

val

ue (d









0 10 20

0

5

10

15D

istan

cey1

(120583m

)


minus5

minus10



0 002 004 006 008 01

0

5

10

15

Dist

ance

x1

(120583m

)

minus5

minus10

minus15

Time t (s)



00

4

3

2

5

100 600500400300200 700

4

2

6



Time t (ms)

0 100 600500400300200 700Time t (ms)

Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)








0 10 50403020

2

4

3

2

1

06

Time t (ms)


C1

C2



Con

trol c

urre

nti

(A) Ra

dial

disp

lace

men

tx(V

)

2

4

6


C1

C2







00

2

1

0

3

10 50403020

4

2

6



Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)

Time t (ms)

0 10 50403020Time t (ms)



0 50 100 150 2000

5

10

15

20

25

30

35



Constantspeed

Time t (s)

Out

put p

ower

P(k

W)




5 Conclusion




Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Sensor

120573

yd

yk

= 25V+

++

+ +

+

+

minus

Q1

ek ekminus1

c uk

P

k+1

k

QMemory II Memory I


Flywheel



Generalized plant




V119896+1 (119899) = 120573119875V119896 (119899) + 119876119890119896 (119899) + 1198761119890119896 (119899 minus 1) (14)


119906119896 (119899) = 119888 (119899) + 120573V119896 (119899) (15)


lim119896rarrinfin

119906119896 (119899) = 119906119889 lim119896rarrinfin

119890119896 (119899) = 0 (16)


119881119896+1 (119911) = 120573119875119881119896 (119911) + 119876119864119896 (119911) + 1198761119864119896 (119911) 119911minus1

= 120573119875119881119896 (119911) + (119876 + 1198761119911minus1

) 119864119896 (119911)

(17)




119888 (119905) = 119866119875 (119905)

= 119870119901 [119890 (119905) +1

119879119894

int

119905

0

119890 (119905) 119889119905 +119879119889

1 + 119879119891

119889 (119890 (119905))

119889119905]

(18)


119862 (119911)

= 119870119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)] 119864 (119911)

(19)


119880119896 (119911) = 119862 (119911) + 120573119881119896 (119911)

= 119896119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)] 119864 (119911)

+ 120573119881119896 (119911)

(20)

where defining

119867 (119911)

= 1 (119896119901 [1 +1198790

119879119894

1


119879119889

(1 + 119879119891) 1198790


)])

minus1

(21)


119864 (119911) = (119880119896 (119911) minus 120573119881119896 (119911)) 119867 (119911) (22)


119881119896+1 (119911) = 120573119875119881119896 (119911) + (119876 + 1198761119911minus1

) (119880119896 (119911) minus 120573119881119896 (119911)) 119867 (119911)

= 120573 [119875 minus (119876 + 1198761119911minus1

) 119867 (119911)] 119881119896 (119911)

+ (119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

(23)




119881119896 (119911)


(119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

1 minus 120573 [119875 + (119876 + 1198761119911minus1) 119867 (119911)]

(24)


10038171003817100381710038171003817120573119875 minus 120573[119876 + 1198761119911

minus1]119867(119911)

10038171003817100381710038171003817infinlt 1 (25)



119880119896 (119911) = 119880119889 (119911) (26)



lim119896rarrinfin

119864 (119911) = 119867 (119911) [ lim119896rarrinfin


119881119896 (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119881infin (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119880119889 (119911)] = 0

(27)



120573 = (119891

119891119889

)

119899

(28)




0

20

40

60

80

100

10minus1 100 101 102 103 104 105


minus20

minus40

minus60

S(s)1W1(s)

T(s)

1W3(s)

Sing

ular

val

ue (d









0 10 20

0

5

10

15D

istan

cey1

(120583m

)


minus5

minus10



0 002 004 006 008 01

0

5

10

15

Dist

ance

x1

(120583m

)

minus5

minus10

minus15

Time t (s)



00

4

3

2

5

100 600500400300200 700

4

2

6



Time t (ms)

0 100 600500400300200 700Time t (ms)

Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)








0 10 50403020

2

4

3

2

1

06

Time t (ms)


C1

C2



Con

trol c

urre

nti

(A) Ra

dial

disp

lace

men

tx(V

)

2

4

6


C1

C2







00

2

1

0

3

10 50403020

4

2

6



Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)

Time t (ms)

0 10 50403020Time t (ms)



0 50 100 150 2000

5

10

15

20

25

30

35



Constantspeed

Time t (s)

Out

put p

ower

P(k

W)




5 Conclusion




Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119881119896 (119911)


(119876 + 1198761119911minus1

) 119867 (119911) 119880119896 (119911)

1 minus 120573 [119875 + (119876 + 1198761119911minus1) 119867 (119911)]

(24)


10038171003817100381710038171003817120573119875 minus 120573[119876 + 1198761119911

minus1]119867(119911)

10038171003817100381710038171003817infinlt 1 (25)



119880119896 (119911) = 119880119889 (119911) (26)



lim119896rarrinfin

119864 (119911) = 119867 (119911) [ lim119896rarrinfin


119881119896 (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119881infin (119911)]

= 119867 (119911) [119880119889 (119911) minus 120573119880119889 (119911)] = 0

(27)



120573 = (119891

119891119889

)

119899

(28)




0

20

40

60

80

100

10minus1 100 101 102 103 104 105


minus20

minus40

minus60

S(s)1W1(s)

T(s)

1W3(s)

Sing

ular

val

ue (d









0 10 20

0

5

10

15D

istan

cey1

(120583m

)


minus5

minus10



0 002 004 006 008 01

0

5

10

15

Dist

ance

x1

(120583m

)

minus5

minus10

minus15

Time t (s)



00

4

3

2

5

100 600500400300200 700

4

2

6



Time t (ms)

0 100 600500400300200 700Time t (ms)

Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)








0 10 50403020

2

4

3

2

1

06

Time t (ms)


C1

C2



Con

trol c

urre

nti

(A) Ra

dial

disp

lace

men

tx(V

)

2

4

6


C1

C2







00

2

1

0

3

10 50403020

4

2

6



Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)

Time t (ms)

0 10 50403020Time t (ms)



0 50 100 150 2000

5

10

15

20

25

30

35



Constantspeed

Time t (s)

Out

put p

ower

P(k

W)




5 Conclusion




Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 10 20

0

5

10

15D

istan

cey1

(120583m

)


minus5

minus10



0 002 004 006 008 01

0

5

10

15

Dist

ance

x1

(120583m

)

minus5

minus10

minus15

Time t (s)



00

4

3

2

5

100 600500400300200 700

4

2

6



Time t (ms)

0 100 600500400300200 700Time t (ms)

Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)








0 10 50403020

2

4

3

2

1

06

Time t (ms)


C1

C2



Con

trol c

urre

nti

(A) Ra

dial

disp

lace

men

tx(V

)

2

4

6


C1

C2







00

2

1

0

3

10 50403020

4

2

6



Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)

Time t (ms)

0 10 50403020Time t (ms)



0 50 100 150 2000

5

10

15

20

25

30

35



Constantspeed

Time t (s)

Out

put p

ower

P(k

W)




5 Conclusion




Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










00

2

1

0

3

10 50403020

4

2

6



Con

trol c

urre

nti

(A)

Radi

al d

ispla

cem

entx

(V)

Time t (ms)

0 10 50403020Time t (ms)



0 50 100 150 2000

5

10

15

20

25

30

35



Constantspeed

Time t (s)

Out

put p

ower

P(k

W)




5 Conclusion




Acknowledgments


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article research on adaptive dual-mode switch control...

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