1 mrac for inverted pendulum

Model ReferenceModel ReferenceAdaptive ControlAdaptive Control

Survey of Control Systems (MEM Survey of Control Systems (MEM 800)800)

Presented byPresented byKeith SevcikKeith Sevcik

ConceptConcept

Design controller to drive plant response to mimic Design controller to drive plant response to mimic ideal response (error = yideal response (error = yplantplant-y-ymodel model => 0)=> 0)

Designer chooses: reference model, controller Designer chooses: reference model, controller structure, and tuning gains for adjustment structure, and tuning gains for adjustment mechanismmechanism

Controller

Model

AdjustmentMechanism

Plant

Controller Parameters

ymodel

u yplant

uc

MIT RuleMIT Rule

Tracking error:Tracking error:

Form cost function:Form cost function:

Update rule:Update rule:

– Change in is proportional to negative Change in is proportional to negative gradient ofgradient of

modelplant yye

)(2

1)( 2 eJ

e

eJ

dt

d

J

sensitivity

derivative

MIT RuleMIT Rule Can chose different cost functionsCan chose different cost functions EX:EX:

From cost function and MIT rule, control law can From cost function and MIT rule, control law can be formedbe formed

0 ,1

0 ,0

0 ,1

)( where

)(

)()(

e

e

e

esign

esigne

dt

d

eJ

MIT RuleMIT Rule

EX: Adaptation of feedforward gainEX: Adaptation of feedforward gain

Adjustment Mechanism

ymodel

u yplantuc

Π

Π

θ

Reference Model

Plant

s

)()( sGksG om

)()( sGksGp

-

+

MIT RuleMIT Rule

For system where is For system where is unknownunknown

Goal: Make it look like Goal: Make it look like

using plant (note, plant using plant (note, plant model is scalar multiplied by plant)model is scalar multiplied by plant)

)()(

)(skG

sU

sY k

)()(

)(sGk

sU

sYo

c

)()( sGksG om

MIT RuleMIT Rule

Choose cost function:Choose cost function:

Write equation for error:Write equation for error:

Calculate sensitivity derivative:Calculate sensitivity derivative:

Apply MIT rule:Apply MIT rule:

coccmm UGkUkGUGkGUyye

ee

dt

deJ )(

2

1)( 2

mo

c yk

kkGU

e

eyeyk

k

dt

dmm

o

'

MIT RuleMIT Rule

Gives block diagram:Gives block diagram:

considered tuning parameterconsidered tuning parameter

Adjustment Mechanism

ymodel

u yplantuc

Π

Π

θ

Reference Model

Plant

s

)()( sGksG om

)()( sGksGp

-

+

MIT RuleMIT Rule

NOTE: MIT rule does not guarantee NOTE: MIT rule does not guarantee error convergence or stabilityerror convergence or stability

usually kept smallusually kept small

Tuning crucial to adaptation rate Tuning crucial to adaptation rate and stability.and stability.

SystemSystem

MRAC of PendulumMRAC of Pendulum

TdmgdcJ c 1sin

cmgdcsJs

d

sT

s

21

)(

)(d2d1dc

T

77.100389.0

89.1

)(

)(2

sssT

s


Controller will take form:Controller will take form:

Controller

Model

AdjustmentMechanism

Controller Parameters

ymodel

u yplant

uc

77.100389.0

89.12 ss


Following process as before, write Following process as before, write equation for error, cost function, and equation for error, cost function, and update rule:update rule:

modelplant yye

)(2

1)( 2 eJ

e

eJ

dt

d

sensitivity

derivative


Assuming controller takes the form:Assuming controller takes the form:

cplant

plantcpplant

cmpmodelplant

plantc

uss

y

yuss

uGy

uGuGyye

yuu

22

1

212

21

89.177.100389.0

89.1

77.100389.0

89.1


plant

c

c

cmc

yss

uss

e

uss

e

uGuss

e

22

1

2

22

12

2

22

1

22

1

89.177.100389.0

89.1

89.177.100389.0

89.1

89.177.100389.0

89.1

89.177.100389.0

89.1


If reference model is close to plant, If reference model is close to plant, can approximate:can approximate:

plantmm

mm

cmm

mm

mm

yasas

asae

uasas

asae

asasss

012

01

2

012

01

1

012

22 89.177.100389.0


From MIT rule, update rules are then:From MIT rule, update rules are then:

eyasas

asae

e

dt

d

euasas

asae

e

dt

d

plantmm

mm

cmm

mm

012

01

2

2

012

01

1

1

MRAC of PendulumMRAC of Pendulum Block DiagramBlock Diagram

ymodel

e

yplantuc

Π

Π

θ1

Reference Model

Plant

s

77.100389.0

89.12 ss

Π

+

-

mm

mm

asas

asa

012

01

mm

mm

asas

asa

012

01

mm

m

asas

b

012

s

Π

-

+

θ2


Simulation block diagram (NOTE: Simulation block diagram (NOTE: Modeled to reflect control of DC Modeled to reflect control of DC motor)motor)

am

s+amam

s+am

-gamma

s

gamma

s

Step

Saturation

omega^2

s+am

Reference Model

180/pi

Radiansto Degrees

4.41

s +.039s+10.772

Plant

2/26

Degreesto Vol ts

35

Degrees

y m

Error

Theta2

Theta1

y


Simulation with small gamma = Simulation with small gamma = UNSTABLE!UNSTABLE!

0 200 400 600 800 1000 1200-100

-50

0

50

100

150

ym

g=.0001


Solution: Add PD feedbackSolution: Add PD feedback

am

s+amam

s+am

-gamma

s

gamma

s

Step

Saturation

omega^2

s+am

Reference Model

180/pi

Radiansto Degrees

4.41

s +.039s+10.772

Plant

1

P

du/dt

2/26

Degreesto Volts

35

Degrees

1.5

D

y m

Error

Theta2

Theta1

y


Simulation results with varying Simulation results with varying gammasgammas

0 500 1000 1500 2000 25000

5

10

15

20

25

30

35

40

45

ym

g=.01

g=.001

g=.0001

707.

sec3

:such that Designed

56.367.2

56.32

s

m

T

ssy

LabVIEW VI Front PanelLabVIEW VI Front Panel

LabVIEW VI Back PanelLabVIEW VI Back Panel

Experimental ResultsExperimental Results

Experimental ResultsExperimental Results

PD feedback necessary to stabilize PD feedback necessary to stabilize systemsystem

Deadzone necessary to prevent Deadzone necessary to prevent updating when plant approached modelupdating when plant approached model

Often went unstable (attributed to Often went unstable (attributed to inherent instability in system i.e. little inherent instability in system i.e. little damping)damping)

Much tuning to get acceptable responseMuch tuning to get acceptable response

ConclusionsConclusions

Given controller does not perform well Given controller does not perform well enough for practical useenough for practical use

More advanced controllers could be More advanced controllers could be formed from other methodsformed from other methods– Modified (normalized) MITModified (normalized) MIT– Lyapunov direct and indirectLyapunov direct and indirect– Discrete modeling using Euler operatorDiscrete modeling using Euler operator

Modified MRAC methodsModified MRAC methods– Fuzzy-MRACFuzzy-MRAC– Variable Structure MRAC (VS-MRAC)Variable Structure MRAC (VS-MRAC)

1 mrac for inverted pendulum

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