feedback linearization presented by : shubham bhat (eces-817)
TRANSCRIPT
Feedback Linearization
Presented by : Shubham Bhat
(ECES-817)
Feedback Linearization- Single Input case
3...)(
2...)()(
.0)0(
,
0)()(,
.0)0(,0
1...)()("
xTz
uxsxqv
defineWe
TwithRonT
hismdiffeomorplocalaandorigintheofodneighborhosomein
xallforxswithXSsqfunctionssmoothexistsThere
fandcontaining
RXsetopensomeonfieldsvectorsmootharegandfwhere
xugxfx
bydescribedsystemaConsider
n
Feedback Linearization- Single Input case
.,
,5
,
.)(
1
)(
)(
5...,)(
1
)(
)(
.,
..),(
4...
var
systemthetoappliedfilterpre
depenedentstatenonlinearaandfeedbacknonlineartherepresentsthen
systemthetoappliedinputexternaltheasvofthinkweifHence
functionssmoothalsoarexs
andxs
xqwhere
vxsxs
xqu
lelinearizabfeedbackcalledissystemthecasethisIn
lecontrollabisbApairthewhere
bvAzz
formtheof
equationaldifferentilinearasatisfyvandziablesresultingThe
1
0
1
1210
1
11
1
||
'
,
1
:
0
0
,
..
......
0.100
0010
,
n
ijj
n
i
n
sasAsI
polynomialsticcharacteritheoftscoefficienthearesatheand
bM
aaaa
AMM
where
bvMzAMMz
formcanonicallecontrollabinissystemresultingthethatsuchzMz
tiontransformastateaapplyingBy
Feedback Linearization- Contd.
]....[
)()()(),(
1
:
0
0
,
0000
.......
0..100
0010
,
]...[
110'
1'1
110
n
n
aaaa
where
uxsxTMaxqzavvxTMz
bA
where
vbzAz
systemloopclosedtheinresults
zaaavv
formfeedbackstatefurtherA
Feedback Linearization- Contd.
Problem Statement
vzzzzzzzthen
definedarezandviablesnewif
conditionsfollowingthesatisfying
TthatsuchRRThismdiffeomorplocalaiii
ofodneighborhothein
xallforxsthatsuchXSsfunctionsmoothaii
XSqfunctionsmoothai
existstheredoesinassystemtheGiven
nnn
nn
,,....,
,var
:
,0)0(:)(
.0
0)()()(
)()(
??,1
13221
Example- Controlling a fluid level in a tank
.
,
.
sectansec)(
2)(])([
tanmod
.
,tanint.
tan
0
0
0
problemregulationnonlinearainvolveshofcontrol
thehleveldesiredthefromdifferentquiteishlevelinitialtheIf
pipeoutlettheof
tioncrosstheisaandktheoftioncrosstheishAwhere
ghatudhhAdt
d
isktheofeldynamicThe
hislevel
initialtheandktheouflowtheisinputcontrolThehlevel
specifiedatokainfluidofhleveltheofcontroltheConsider
d
h
d
.0)(~
0~
,tan,)(~
~sin
,""
)(2)(
)(
2)(
tasththatimpliesThis
hh
isdynamicsloopclosedresultingthe
tconspositiveabeinganderrorlevelthebeinghthhwith
hvasvgChoo
vh
linearisdynamicsresultingthespecifiedbetoinputequivalentanbeingvwith
vhAghatu
aschosenistuIf
ghauhhA
aswrittenbecandynamicsThe
d
Example – Contd.
.0)(~
~)(
),(var
.sec
2
~)(2)(
mindet
tasthyieldstilltoasso
hthv
aschosenbecanvinputequivalentthe
thfunctionyingtimeknownaisleveldesiredtheIf
levelfluidtheraisetousedispartondthewhile
ghaflowoutputtheprovidetousedisRHStheonpartfirstThe
hhAghatu
lawcontrolnonlinearthebyederisflowinputactualThe
d
d
Example – Contd.
.form
companionlecontrollabthebydescribedsystemsnonlinearof
classsatoappliedbecanionlinearizatfeedbackofideaThe
.)()(
,
)()()(
statestheoffunctionnonlineararexbandxf
andoutputscalartheisxinputcontrolscalartheisuwhere
uxbxfx
aredynamicsitsifformcompanioninbetosaidissystemAn
vx
formegratormultiple
fvb
u
nonzerobetobgassuinputcontrolthegU
uxbxf
x
x
x
x
x
dt
d
n
n
n
n
int
][1
)min(sin
)()(
......2
1
1
Example – Contd.
0)(
0...
,
....
)1(....
,
0)1(
1)(
01
1
11
txthatimplieswhich
xkxkx
toleadingplanecomplexhalflefttheinstrictlyrootsitsallhas
kpkppolynomialthethatsochosenkthewith
nxkxkxkv
lawcontroltheThus
nn
n
nn
ni
no
Example – Contd.
)2cos(sincoscos2
2
,
sin
var,
)2cos(cos
sin2
111112
211
122
11
1122
1211
zauzzzzz
zzz
areequationsstatenewthethen
xaxz
xz
iablesofsetnewtheconsiderweifHowever
xuxxx
xaxxx
Input State Linearization
.sin
int
sin
,,
2
,
)cos2sincos()2cos(
1
:
2
211
11111
vinputnewthegudynamicsnewthe
gstabilizinofproblemtheodtransformebeenhasuinputcontrol
originalthegudynamicsoriginalthegstabilizinofproblemthe
tiontransformainputandtiontransformastatethethroughThus
vz
zzz
relationstateinputlinearato
leadingdesignedbetoinputequivalentanisvwhere
zzzzvza
u
lawcontrolthebycanceledbecantiesnonlineariThe
Input State Linearization-Contd.
111112
1
21
22
211
2
2211
cos2sincossin22()2cos(
1
,
.2
2
2
2,
.
xxxxxaxx
u
xandxstateoriginaltheoftermsIn
atplacedarepoleswhose
zz
zzz
dynamicsloopclosedstabletheinresulting
zvchoosemayweexampleFor
gainsfeedback
ofchoicesproperwithanywherepolestheplacecan
zkzkv
lawcontrolfeedbackstatelinearThe
Input State Linearization-Contd.
a
zzx
zx
byzfromgivenisxstateoriginalThe
)sin( 122
11
pole-placement loop
zKv T ),( vxuu ),( uxfx
)(xzz z
Linearization Loop
-
x0
Input State Linearization-Contd.
)(
),(
xhy
uxfx
systemtheConsider
Our objective is to make the output y(t) track a desired trajectory yd(t) while keeping the whole state bounded, where yd(t) and its time derivatives up to a sufficiently high order are assumed to be known and bounded.
1
213
35
12
3221 )1(sin
.
xy
uxx
xxx
xxxx
systemorderthirdtheConsiderge
Input Output Linearization
To generate a direct relationship between the output y and the input u,differentiate the output y
.mindet
)(1
1
.exp
)1()cos)(()(
)(
)()1(
.,
)1(sin
12
212233
511
1
12
3221
ederbetoinputnewaisvwhere
fvx
u
formtheininputcontrolachooseweIf
andybetweeniprelationshlicitanrepresentsThis
xxxxxxxf
bydefinedstatetheoffunctionaisxfwhere
xfuxy
againatedifferentiweuinputthetorelateddirectlynotisySince
xxxxy
Input Output Linearization-Contd.
.exp
0
.tan
)()(
.
int
,
int
12
21
21
dynamicserrorstableonentiallyanrepresentswhich
ekeke
bygivenissystemloopclosedtheoferrortrackingThe
tsconspositivebeingkandkwhere
tytyewhereekekyv
simpleis
egratordoublethisforcontrollertrackingaofdesignThe
vy
vinputnewtheandoutputthebetween
iprelationshegratordoublelinearsimpleaobtainWe
dd
Note : The control law is defined everywhere, except at the singularity points such that x2= -1.
Input Output Linearization-Contd.
Internal Dynamics
•If we need to differentiate the output r times to generate an explicit relationship between output y and input u, the system is said to have a relative degree r.
•The system order is n. If r<= n, there is an part of the system dynamics which has been rendered “unobservable”. This part is called the internal dynamics, because it cannot be seen from the external input-output relationship.
•If the internal dynamics is stable, our tracking control design has been solved. Otherwise the tracking controller is meaningless.
•Therefore, the effectiveness of this control design, based on reduced-order model, hinges upon the stability of the internal dynamics.
1
32
2
1
xy
u
uxx
x
systemnonlineartheConsider
Assume that the control objective is to make y track yd(t).Differentiating y leads to the first state equation.Choosing control law
dynamicsernalthetoleadingequationdynamic
ondthetoappliedalsoisinputcontrolsameThe
ee
zerotoeofeconvergenconentialyieldswhich
tytexu d
int,
sec
0
exp
)()(32
Internal Dynamics
Internal Dynamics- Contd
.)()(
.0
,0sin
,
.tan
)(
,
3/122
3/122
3/12
boundedistyderivativewhosetytrajectoryanygiven
controlrysatisfactoprovidedoescontrollerabovetheTherefore
Dxwhenxand
Dxwhenxce
Dx
thatconcludeweThus
tconspositiveaisDwhere
Dety
getweboundedbetoassumedisyandboundediseIf
dd
d
d
.,
322
nonlinearandautonomousnonticallycharactersiswhich
eyxx d
.,
),()(()()(
)(int)(
0
,.exp
,
).()(
2
22
2
2
1
2
2
1
udoessoboundedremainsx
tytotendstytytotendstywhilethatseeWe
teyxxdynamicsernaltheandyyewhere
eeequationtrackingtheyieldseyxu
lawcontroltheThusucontainslicitlywhich
uxy
getweoutputtheofationdifferentioneWith
tyoutputdesiredatracktorequiredistywhere
xy
u
ux
x
x
systemlinearobservableandlecontrollabsimpletheConsider
dd
dd
d
d
Internal Dynamics in Linear Systems
.
.inf
,,
)(
int,
:
2
22
1
2
2
1
systemtheforcontrollersuitableanotisthisTherefore
tasinity
togobothuyaccordinglandxthatimpliesThis
ytexx
isdynamicsernalthebutdynamicserrortracking
sametheyieldsaboveaslawcontrolsameThe
xy
u
ux
x
x
systemdifferentslightlyaConsider
d
Internal Dynamics in Linear Systems
.inf
,min
.min
int,
.1,sec
.1,,
.
1)(
1)(
,
tan
22
21
effortinite
requirestrackingperfectsystemsphaseimumnonFor
systemphaseimum
aisitbecausestableissystemfirstofdynamicsernalThus
atzeroplanehalfrightaistherecaseondtheforwhile
atzeroplanehalfleftaistherecasefirsttheforySpecficall
zerosdifferentbutpolessamethehavesystemstheBoth
p
ppW
p
ppW
functionstransfertheconsiderwe
systemstwothebetweendifferencelfundamentathisdundersTo
Internal Dynamics in Linear Systems
•Extending the notion of zeros to nonlinear systems is not a trivial proposition.
•For nonlinear systems, the stability of the internal dynamicsmay depend on the specific control input.
•The zero-dynamics is defined to be the internal dynamicsof the system when the system output is kept at zero by the input.
•A nonlinear system whose zero dynamics is asymptotically stable is an asymptotically minimum phase system.
•Zero-Dynamics is an intrinsic feature of a nonlinear system, which does not depend on the choice of control law or the desired trajectories.
Extension of Internal Dynamics to Zero Dynamics
•Lie derivative and Lie bracket
•Diffeomorphism
•Frobenius Theorem
•Input-State Linearization
•Examples
•The zero dynamics with examples
•Input-Output Linearization with examples
•Opto-Mechanical System Example
Mathematical Tools
Lie Derivatives
.
,,)(
ftorespectwithhofderivativeLiethecalled
hLfieldvectoraandxhfunctionscalaraGiven f
fhhLbydefinedfunctionscalaraisftorespectwithhof
derivativeLiethethenRonfieldvectorsmoothabeRRf
andfunctionscalarsmoothabeRRhLetDefinition
f
nn
n
,:
,::
hLxx
hLyhLx
x
hy
areoutputtheofsderivativeThe
xhy
xfx
Example
ff
f2][
;
)(
)(
:
Lie Brackets
,....2,1],[
Re
int)tan
(],[
,
.
1
iforgadfgad
ggad
byyrecursiveldefinedbecanBracketsLiepeated
adjofordss
adwheregadaswrittencommonlyisgfBracketLieThe
gffggf
bydefinedvectorthirdaisgandf
ofBracketLieTheRonfieldsvectortwobegandfLet
if
if
of
f
n
)sin2)(2sin(2)2cos(cos
)2cos(
)2cos(
0
cossin
cos2
cos
)sin(2
0)sin(2
00],[
)2cos(
0)(
cos
sin2
)()(
121111
1
1112
1
12
121
1
112
121
xaxxxxx
xa
xxxx
ax
xx
xaxx
xgf
ascomputedbecanbracketLieThe
xxg
xx
xaxxf
bydefinedgandffieldsvectortwothewith
uxgxfxLet
Example
Example - Lie Brackets
Properties of Lie Brackets
.)(
:)(
],[],[
:)(
.tan
,,,,,,
],[],[],[
],[],[],[
:)(
21
2121
22112211
22112211
xoffunctionsmoothaisxhwhere
hLLhLLhL
identityJacobiiii
fggf
itycommutativskewii
scalarstconsareand
andfieldsvectorsmootharegggfffwhere
gfgfggf
gfgfgff
ybilineariti
propertiesfollowinghaveBracketsLie
fggfgad f
Diffeomorphisms and State transformations
.)(
,intsin
.)(
:
.
,
,,:
:
0
1
ofsubregionainhismdiffeomorplocaladefinesxthen
ofxxpoaatgularnonismatrixJacobianthe
IfRinregionaindefinedfunctionsmoothabexLet
Lemma
smoothisand
existsinverseitsifandsmoothisitifhismdiffeomorp
acalledisregionaindefinedRRfunctionA
Definition
n
nn
Example
zxwhere
zhy
uzgzfz
tionrepresentastateNew
uxgxfx
xx
z
yieldszofationDifferenti
xz
bydefinedbestatesnewtheletand
xhy
uxgxfx
bydescribedsystemdynamictheConsider
1
*
**
)(
)()(
:
))()((
)(
)(
)()(
Frobenius Theorem- Completely Integrable
.
,1,1
0
)(),...(),(
,,int
],...,[
:int
21
21
tindependen
linearlyarehgradientstheandmjmniwhere
fh
equationsaldifferentipartialofsystem
thesatisfyingxhxhxhfunctionsscalarmn
existsthereifonlyandifegrablecompletelybetosaidis
RonffffieldsvectorofsettindependenlinearlyA
egrablecompletelyofDefinition
i
ji
mn
nm
jixfxxff
thatsuchRR
functionsscalararethereifonlyandifinvolutivebeto
saidisffffieldsvectorofsettindependenlinearlyA
conditiontyinvolutiviofDefinition
m
kkijkji
nijk
n
,)()()](,[
:
,,
],...,[
:
1
21
Frobenius Theorem- Involutivity
Frobenius theorem
.
,,int
.
,...,
:
21
involutiveisit
ifonlyandifegrablecompletelyissetThe
fieldsvector
tindependenlinearlyofsetabefffLet
Theorem
m
02)3(
04
33
22
23
11
213
x
hx
x
hxx
x
hx
x
h
x
hx
equationsaldifferentipartialofsettheConsider
.,
2121
321
21
322
31231
21
.,03],[
]0312[],[
].,[
,mindet
]2)3([]014[
],[
solvableareequationsaldifferentipartialtwotheTherefore
T
TT
involutiveisfieldsvectorofsetthisffffSince
xff
fffieldsvectorofsettheoftyinvolutivithe
checkusletsolvableissetthiswhethereertoorderIn
xxxxfxf
withffarefieldsassociatedThe
Frobenius theorem- example
Input-State Linearization
.
1
.
.
0
0000
.....
...100
0..010
var
)(var
)()(
,:
,
,)()(
)()(sin
:
lawglinearizinthecalledis
lawcontroltheandstateglinearizinthecallediszstatenewThe
bA
where
bvAzz
relationiantintimelinearasatisfy
vinputnewtheandxziablesstatenewthethatsuch
vxxu
lawcontrolfeedbacknonlinearaRpismdiffeomorha
Rinregionaexiststhereiflelinearizabstateinputbe
tosaidisRonfieldsvectorsmoothbeingxgandxfwith
uxgxfxformtheinsystemnonlinearinputgleA
Definition
n
n
n
Conditions for Input-State Linearization
.,
.int
.
int
:Re
},...,{)(
},...,{)(
:
,,
,)()(
:
1
1
casenonlineartheinsatisfiedgenerallynotbutsystems
linearforsatisfiedtriviallyisItuitivelessisconditiontyinvolutiviThe
systemnonlinear
theforconditionilitycontrollabaserpretedbecanconditionfirstThe
marksFew
ininvolutiveisgadgadgsettheii
intindependenlinearlyaregadgadgfieldsvectorthei
holdconditionsfollowingthethat
suchregionaexiststhereifonlyandiflinearizedstateinputis
fieldsvectorsmoothbeingxgandxfwithsystemnonlinearThe
Theorem
nff
nff
How to perform input-state Linearization
11
111
11
11
11
11
1
1
1)(
)(
,
]....[)()(
0
2,...00
)deg
(,)(
.)(
.,...,)(
:
zLLx
zLL
zLx
withtiontransforma
inputtheandzLzLzxztiontransformastatetheComputeiv
gadz
nigadz
nreerelativetheofionlinearizatoutputinputto
leadingfunctionoutputthezstagefirstthefindsatisfiedarebothIfiii
satisfiedareconditionstyinvolutiviandilitycontrollabthewhetherCheckii
systemgiventheforgadgadgfieldsvectortheConstructi
stepsfollowingthethrough
performedbecansystemnonlinearaofionlinearizatstateinputThe
nfg
nfg
nf
Tnff
nf
nf
nff
uqqkqJ
qqkqMgLqI
)(
0)(sin
212
211
Consider a mechanism given by the dynamics which represents a single link flexible joint robot.Its equations of motion is derived as
Because nonlinearities ( due to gravitational torques) appear in the first equation, While the control input u enters only in the second equation, there is no easy wayto design a large range controller.
T
T
Jg
xxj
kxxx
l
kx
l
MgLxf
aswrittenbecangandffieldsvectoringcorrespondand
qqqqx
]1
000[
)()(sin[
][
3143112
2211
Example system
Checking controllability and involuvity conditions.
001
01
0
000
000
][
2
2
32
J
k
J
J
k
J
IJ
kIJ
k
gadgadgadg fff
It has rank 4 for k>0 and IJ> infinity. Furthermore, since the above vector fields are constant, they form an involutive set.Therefore the system is input-state linearizable.
Example system- Contd.
Let us find out the state-transformation z = z(x) and the inputtransformation so that input-state linearization is achieved.vxxu )()(
)(cos
)(sin
.,
0000
421234
31123
212
1
11
11
1
1
4
1
3
1
2
1
xxI
kxx
I
MgLfzz
xxI
kx
I
MgLfzz
xfzz
zfromobtainedbecanstatesotherThe
xz
isequationabovethetosolutionsimplestThe
onlyxoffunctionabemustzThus
x
z
x
z
x
z
x
z
Example system - Contd.
Accordingly, the input transformation is
)cos)(()cos(sin)(
))((
exp
)/()(
13112
21
44
xI
MgL
J
k
I
kxx
I
k
I
kx
I
MgLxx
I
MgLxa
where
xavk
IJu
aslicitlywrittenbecanwhich
gzfzvu
.4
43
32
21
ionlinearizatstateinputthecompletingthus
vz
zz
zz
zz
equationslinearofsetfollowingthewithupendWe
Example system- Contd.
Finally, note thatThe above input-state linearization is actually global, because the diffeomorphism z(x) and the input transformation are well defined everywhere.Specifically, the inverse of the state transformation is
.
)cos(
)sin(
12424
1313
22
11
everywhereabledifferentianddefinedwelliswhich
zzI
MgLz
k
Izx
zI
MgLz
k
Izx
zx
zx
Example system- Contd.
Input-Output Linearization of SISO systems
?)(
?
int)(
?)(
:
.
)(
)()(
sin
ionslinearizatoutputinputonbasedscontrollerstabledesigntoHowiii
ionlinearizatoutputinputthe
withassociateddynamicszeroanddynamicsernaltheareWhatii
systemnonlinearaforrelationoutputinputlinearageneratetoHowi
Issues
outputsystemtheisywhere
xhy
uxgxfx
tionrepresenta
spacestatethebydescribedsysteminputglenonlinearaGiven
Generating a linear input-output relation
0)(
100)(
,
deg:
.deg,,
.,
1
xhLL
rixhLL
xifregion
ainrreerelativehavetosaidissystemSISOTheDefinition
undefinedisreerelativesystemsthecasessomeinHowever
tynonlinearithecanceltoudesignthenandappears
uinputtheuntilrepeatedlyyfunctionoutputtheatedifferenti
tosimplyisionlinearizatoutputinputofapproachbasicThe
rfg
ifg
Normal Forms
)(
),(
),(),(
..
,int
]...[]....[
0
1
1
0
)1(21
xatorin
statesnormalorscoordinatenormalastoreferredareandThe
y
asdefinedoutputthewith
w
uba
aswritten
becansystemtheofformnormalthexpoaofodneighborhoaIn
yyy
Let
ii
r
TrTr
Zero Dynamics
.int
.int
int'
int,
.,
.),(
)(
int
dynamicsernalthe
ofstabilitytheaboutsconclusionsomemaketousallow
willdynamicszeroStudyingzeroatainedmaisy
outputthethatsuchisinputcontrolthewhendynamics
ernalssystemthegconsiderinbysystemnonlinear
theofpropertyrinsicandefinecanweHowever
statesoutputtheondependsdynamicsthisGenerally
formnormaltheofw
equationsrnlastthetoscorrespondsimplyionlinearizat
outputinputthewithassociateddynamicsernalThe
)()()(
,,
)(
)()(
,
.,
dim)(
,.
int
*
1*
xuxgxfx
toaccording
evolvesxstatessystemthedynamicszerothetoingcorrespondTherefore
xhLL
xhLtu
feedbackstatethebygivenbemustuinputoriginalThe
dynamicszeroinoperatetosystemthefororderIn
zeroatstaysythatsuchbemustinputFurther
Rinsurfacesmoothensionalrnthetorestrictedismotionits
whendynamicsitsissystemaofdynamicszerotheThuszeroaresderivative
timeitsofallthatimplieszeroyidenticallisyoutputthethatconstraThe
rfg
rf
n
Zero Dynamics
),0(
),0(
int
,
),0(
0
,0)0(..,
'min
0
0
b
au
statesernaltheof
onlyfunctionaaswrittenbecanuinputcontrolThe
systemnonlineartheof
dynamicszerotheisequationabovethedefinitionBy
w
asformnormalinwrittenbecandynamicssystemthe
eisurface
theonisstateinitialssystemthethatgAssu
Zero Dynamics- Contd.
Local Asymptotic Stabilization
systemloopclosedstableallyasymptoticlocallyatoleads
ykykykyLyLL
xu
lawcontroltheThen
planehalflefttheinstrictlyrootsitsallhas
kpkpkppK
polynomialthethatsuchktsconsChoose
kwhereykykykvthatassumeusLet
stableallyasymptoticlocallyis
dynamicszeroitsandrreerelativehassystemthethatAssume
Theorem
rr
rfr
fg
rr
r
i
ir
r
]...[1
)(
,
.
...)(
tan
....
.
,deg
:
01)1(
11
011
1
01)1(
1
Example System
.
242
,.
2
)0(
322
,1deg,
2
.intmod
3
0
(0'
3
1222
1
311
222
121
21
22
1
21
22
22
11
systemnonlinearthestablizeslocally
xxxxu
lawcontroltheThereforestableallyasymptoticisthusand
xx
simplyisysettingbyobtaineddynamicszeroassociatedThe
uxxxxxdt
dy
becauseissystemtheofreerelativetheoutputthistoingCorrespond
xxyfunctionoutputthedefineusLet
egratorpureatoingcorrespondeableuncontrollanhasthusand
uxx
x
isxxxwherexationlinearizatssystemThe
uxx
xxx
T
Global Asymptotic Stability
Zero Dynamics only guarantees local stability of a control system based on input-output linearization.
Most practically important problems are of global stabilization problems.
An approach to global asymptotic stabilization based on partial feedback linearization is to simply consider the control problem as a standard lyapunov controller problem, but simplified by the fact that putting the systems in normal form makes part of the dynamics linear.
The basic idea, after putting the system in normal form, is to view as the “input” to the internal dynamics, and as the “output”.
Steps for Global Asymptotic Stability
•The first step is to find a “ control law” which stabilizes the internal dynamics.
•An associated Lyapunov function demonstrating the stabilizing property.
•To get back to the original global control problem.
•Define a Lyapunov function candidate V appropriately as a modified version of
•Choose control input v so that V be a Lyapunov function for the whole closed-loop dynamics.
)(00
0V
0V
Local Tracking Control
)()()(~
]...[
.)1(
ttt
byvectorerrortrackingthedefineand
yyy
Lettaskscontroltracking
asymptotictoextended
becancontrollerplacementpolesimpleThe
d
Trdddd
.exp
~
]~....[1
sin
.,,
0)0(),(
,),int
tan(deg
:
101)(
11
1
onentiallyzerotoconverges
errortrackingtheandboundedremainsstatewholethe
kkyLLL
u
lawcontrolthegubyThen
stableallyasymptoticuniformlyisandboundedisexists
equationtheofsolutionthethatand
boundedandsmoothisthaterestofregiontheover
tconsanddefinedrreerelativehassystemtheAssume
Theorem
rrr
dr
frfg
dddd
d
d
Tracking Control
Inverse Dynamics
)(),(),()(
)(,
0)](....)()([)()(
,
01,....,.1,0)()(
.0),()(.)(
)(
.)(
)0(
,sec
)1(
tubaty
satisfymusttuinputcontroltheThus
ttytytytt
scoordinatenormaloftermsIn
trktyty
ttytyeityoutput
referencethetoidenticalistyoutputsystemthethatassumeusLet
perfectlyty
outputreferenceatracktooutputplantthefororderinbeshould
uinputcontrolandxconditionsinitialthewhatoutfinduslet
tionpreviousbydescribedsystemsFor
rrr
r
Trrrrr
kr
k
rr
r
.,
).(
)(Pr
).(
)(
,)(
systemtheofdynamicsinversecalledaretheyTherefore
tyhistoryoutputreferencetoingcorrespond
tuinputthecomputetousallowequationsevious
tytoequal
yidenticalltyoutputforinputcontrolrequired
theobtaincanwetytrajectoryreferenceaGiven
r
r
r
)](),([)(
)(
ttwt
equationaldifferentitheofsolutionistwhere
r
Inverse Dynamics- Contd.
Application of Feedback Linearization to Opto-Mechanics
)))(sin(tan2
(cos)))((tansin2
(sin)( 1212
z
xka
z
xkbcAxI
For the double slit aperture, the irradiance at any point in space is given as:
= wavelength = 630 nmk = wave number associated with the wavelength a = center-to-center separation = 32 umb = width of the slit = 18 umz = distance of propagation =1000 um
Plant Model
-
+ )1(
1
sMotor Dynamics Plant Model
UX2 Y= X1
1
1
1
2
sXU
X)(sin 21 XcAXY
UXXt
X
122
UXXct
X
222 )(sin
)(1)(sin 2 assumeAXcY
)(1)(sin 2 assumeAXcY
Plant Model
Input-State Linearization
.
,
)(sin)()(
)(sin
)(sin
)(sin
sin
1111
11
1
11
111
111
21
dynamicsloop
closedstableagetcanwegainsfeedbackofchoiceproperBy
zzczfwherezfzku
zkvSelect
zv
zzcvu
zzczuTherefore
uzzczgetWe
xztiontransformaagU
UXXct
X
222 )(sin
Pole-Placement loop21 xz z
Plant Model
-
+ )1(
1
sMotor Dynamics Plant Model
U(x,v)X2 Y
11zKv -
0
Input-State Linearization- Block diagram
22
22222
2222
2222
22
2222222
22
22222
22
22222
2
22
)sin()sin()(sin)sin(
)cos()cos()(sin)cos(
])(sin)[sin(])(sin)[cos(
)sin())(cos(
)sin(])[cos(
)sin()(sin
x
xuxxxcx
uxxxxxcxx
x
uxxcxuxxcxxy
x
xxxxx
x
xxxxxy
x
xxcy
Input-Output Linearization
22
2222
22222
22
222
22
222222
2222
22
222
22
22222
2222
2222
)]sin()cos()}cos(){sin(([sin
])sin()cos(
[
)]sin()(sin)sin()cos()(sin)cos([
])sin()cos(
[
)sin()sin()(sin)sin(
)cos()cos()(sin)cos(
x
xxxxxxxxc
x
xxxu
x
xxxcxxxxcxx
x
uxuxx
x
xuxxxcx
uxxxxxcxx
y
Input-Output Linearization
0]
)sin()cos([int
sin
][
0,1,
])sin()cos(
[
)(
)(])sin()cos(
[
22
2
2
2
1
22
2
2
2
22
2
2
2
x
x
x
xwherespo
gularityatexcepteverywheredefinedislawControl
yyeekyvSelect
BAgetweBuAxxwithComparing
vy
x
x
x
xxfv
u
wherexfx
x
x
xuy
dd
Input-Output Linearization
).(deg
int
.
,
:
22
ordersystemnrreerelativebecause
systemthiswithassociateddynamicsernalnoareThere
system
nonlinearthestabilizeslocallycontrollerfeedbackthehence
stableallyasymptoticisdynamicszeroThis
uxx
bygivenisDynamicsZero
Zero Dynamics
Conclusion
Control design based on input-output linearization can be made in 3 steps:
•Differentiate the output y until the input u appears
•Choose u to cancel the nonlinearities and guarantee tracking convergence
•Study the stability of the internal dynamics
If the relative degree associated with the input-output linearization is the same as the order of the system, the nonlinear system is fully linearized.
If the relative degree is smaller than the system order, then the nonlinear system is partially linearized and stability of internal dynamics has to be checked.
Homework Problems
2
122
211
1
222
112
21
2
)1(
xy
uxxx
uxkxx
fordynamicszerotheofstabilityglobalCheck
xy
uxxaxxx
xx
forcontrolleroutputinputlinearaDesign