feedback linearization presented by : shubham bhat (eces-817)

Feedback Linearization

Presented by : Shubham Bhat

(ECES-817)

Feedback Linearization- Single Input case

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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

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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

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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.

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Checking controllability and involuvity conditions.

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Example system- Contd.

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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.

)(00

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asymptotictoextended

becancontrollerplacementpolesimpleThe

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sin

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:

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onentiallyzerotoconverges

errortrackingtheandboundedremainsstatewholethe

kkyLLL

u

lawcontrolthegubyThen

stableallyasymptoticuniformlyisandboundedisexists

equationtheofsolutionthethatand

boundedandsmoothisthaterestofregiontheover

tconsanddefinedrreerelativehassystemtheAssume

Theorem

rrr

dr

frfg

dddd

d

d

Tracking Control

Inverse Dynamics

)(),(),()(

)(,

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

,

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.0),()(.)(

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.)(

)0(

,sec

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tubaty

satisfymusttuinputcontroltheThus

ttytytytt

scoordinatenormaloftermsIn

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referencethetoidenticalistyoutputsystemthethatassumeusLet

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outputreferenceatracktooutputplantthefororderinbeshould

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tionpreviousbydescribedsystemsFor

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r

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kr

k

rr

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.,

).(

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).(

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systemtheofdynamicsinversecalledaretheyTherefore

tyhistoryoutputreferencetoingcorrespond

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yidenticalltyoutputforinputcontrolrequired

theobtaincanwetytrajectoryreferenceaGiven

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)](),([)(

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equationaldifferentitheofsolutionistwhere

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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


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


).(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

feedback linearization presented by : shubham bhat (eces-817)

Documents

system output

inputstate linearizationhow

inputstate linearizable

example system contd

input output linearizationto

control input u

state equation

state bounded