Feedback Bhat

transcript

Feedback LinearizationPresented by : Shubham Bhat

(ECES-817)

Feedback Linearization- Single Input case

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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.

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Input Output Linearization

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Input Output Linearization-Contd.

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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.

systemnonlineartheConsider

Assume that the control objective is to make y track yd(t).Differentiating y leads to the first state equation.Choosing control law

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•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

fhhLbydefinedfunctionscalaraisftorespectwithhofderivativeLiethethenRonfieldvectorsmoothabeRRf

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areoutputtheofsderivativeThexhyxfx

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Lie Brackets

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Example

Example - Lie Brackets

Properties of Lie Brackets

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Frobenius theorem- example

Input-State Linearization

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Conditions for Input-State Linearization

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holdconditionsfollowingthethatsuchregionaexiststhereifonlyandiflinearizedstateinputis

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How to perform input-state Linearization

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systemgiventheforgadgadgfieldsvectortheConstructi

stepsfollowingthethroughperformedbecansystemnonlinearaofionlinearizatstateinputThe

uqqkqJqqkqMgLqI

)(0)(sin

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.

xxjkxxx

lMgLxf

aswrittenbecangandffieldsvectoringcorrespondandqqqqx

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Example system

Checking controllability and involuvity conditions.

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 )()(

421234

xxIkxx

IMgLfzz

xfzzzfromobtainedbecanstatesotherThe

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onlyxoffunctionabemustzThusxz

Example system - Contd.

Accordingly, the input transformation is

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exp)/()(

IMgLxx

IMgLxa

xavkIJu

aslicitlywrittenbecanwhichgzfzvu

ionlinearizatstateinputthecompletingthusvzzzzzzz

equationslinearofsetfollowingthewithupendWe

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

everywhereabledifferentianddefinedwelliswhich

MgLzkIzx

Input-Output Linearization of SISO systems

int)(?)(

)()()(

ionslinearizatoutputinputonbasedscontrollerstabledesigntoHowiiiionlinearizatoutputinputthe

withassociateddynamicszeroanddynamicsernaltheareWhatiisystemnonlinearaforrelationoutputinputlinearageneratetoHowi

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Normal Forms

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Zero Dynamics

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formnormaltheofwequationsrnlastthetoscorrespondsimplyionlinearizatoutputinputthewithassociateddynamicsernalThe

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zeroatstaysythatsuchbemustinputFurtherRinsurfacesmoothensionalrnthetorestrictedismotionits

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Zero Dynamics

),0(),0(

,0)0(..,'min

statesernaltheofonlyfunctionaaswrittenbecanuinputcontrolThe

systemnonlineartheofdynamicszerotheisequationabovethedefinitionBy

asformnormalinwrittenbecandynamicssystemtheeisurface

theonisstateinitialssystemthethatgAssu

Zero Dynamics- Contd.

Local Asymptotic Stabilization

systemloopclosedstableallyasymptoticlocallyatoleads

ykykykyLyLL

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stableallyasymptoticlocallyisdynamicszeroitsandrreerelativehassystemthethatAssume

Theorem

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Example System

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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.

Local Tracking Control

)()()(~

tttbyvectorerrortrackingthedefineand

Lettaskscontroltrackingasymptotictoextended

becancontrollerplacementpolesimpleThe

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equationtheofsolutionthethatandboundedandsmoothisthaterestofregiontheover

tconsanddefinedrreerelativehassystemtheAssumeTheorem

Tracking Control

Inverse Dynamics

)(),(),()(

)(,0)](....)()([)()(

,01,....,.1,0)()(

.0),()(.)()(

)0(,sec

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Inverse Dynamics- Contd.

Application of Feedback Linearization to Opto-Mechanics

)))(sin(tan2

(cos)))((tansin2

(sin)( 1212

zxkbcAxI

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(

Motor Dynamics Plant Model

UX2 Y= X1

)(sin 21 XcAXY

222 )(sin

)(1)(sin 2 assumeAXcY

Plant Model

Input-State Linearization

)(sin)()(

)(sin)(sin

)(sinsin

dynamicsloopclosedstableagetcanwegainsfeedbackofchoiceproperBy

zzczfwherezfzkuzkvSelect

zvzzcvu

zzczuThereforeuzzczgetWe

xztiontransformaagU

222 )(sin

Pole-Placement loop21 xz z

Plant Model

-+ )1(

Motor Dynamics Plant Model

U(x,v)X2 Y

11zKv -

Input-State Linearization- Block diagram

2222222

)sin()sin()(sin)sin(

)cos()cos()(sin)cos(

])(sin)[sin(])(sin)[cos(

)sin())(cos(

)sin(])[cos(

)sin()(sin

xxuxxxcx

uxxxxxcxx

xuxxcxuxxcxx

xxxxxx

xxxxxxy

Input-Output Linearization

222222

)]sin()cos()}cos(){sin(([sin

])sin()cos([

)]sin()(sin)sin()cos()(sin)cos([

])sin()cos([

)sin()sin()(sin)sin()cos()cos()(sin)cos(

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xxxxcxxxxcxx

xuxuxxx

xuxxxcxuxxxxxcxx

0])sin()cos([int

])sin()cos([

)(])sin()cos([

xxwherespo

gularityatexcepteverywheredefinedislawControlyyeekyvSelect

BAgetweBuAxxwithComparingvy

wherexfx

).(degint

ordersystemnrreerelativebecausesystemthiswithassociateddynamicsernalnoareThere

systemnonlinearthestabilizeslocallycontrollerfeedbackthehence

stableallyasymptoticisdynamicszeroThis

uxxbygivenisDynamicsZero

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

xyuxxx

uxkxxfordynamicszerotheofstabilityglobalCheck

xyuxxaxxx

xxforcontrolleroutputinputlinearaDesign

Feedback Bhat

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