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MARWAN AFFANDI SCHOOL OF MECHATRONICS

UNIMAP Email: [email protected]

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Adapted from Keith Sevcik and other sources

Model Adaptive Reference System Model Adaptive Reference System or

MRAS for short is an important adaptive model.

MRAS has been being used widely to solve various control, parameter identification and state estimation problems.

The original concept of MRAS is found in the article of the MIT group published in 1958 by Whitaker, Yarmon, and Kazer.

Dr. Marwan Affandi 2012

Model Adaptive Reference System They discussed about the problem of self

adjusting parameters of a controller in order to stabilize the dynamic characteristics of a feedback control when there were drift variations in the plant parameter.

But their method did not always guarantee the stability of adaptation because the gradient method was used in their approach.


Model Adaptive Reference System To solve the problem, Shackcloth and

Butchart proposed the stable adaptive control system with the application of Liapunov's stability theory.

In another approach, Landau used Popov's hyperstability theory.

However, some problems were still unsolved in these approaches.


Model Adaptive Reference System One of the biggest was that the parameter

adjustment laws involved state variables of the plant.

Parks indicated that the parameter adjustment was possible by using only input and output data of the plant.

However, his method had a shortcoming because the differentiation of input for the plant was used.


Model Adaptive Reference System Monopoli developed the input signal

synthesis adaptive control system, which could synthesize adaptive control by using only input and output data but without the differentiation of input for the plant.

Landau gave a good survey on various model reference adaptive techniques.


Model Adaptive Reference System Applications of MRAS include among

others: flight control of aircrafts, control of steering of ships, control of a steam cooled fast reactor, control of a nuclear rocket engine, parameter tracking of an air craft control system, optical tracking telescope, adaptive hydraulic servo mechanism, adaptive control for an internal combustion engine and dynamic input-output model in economics theory.


Model Adaptive Reference System A block diagram of the MRAS is shown in

Fig. 1. The system consists of two models, called

the reference model and the adaptive model.

In the reference model, it is assumed that the inputs determine outputs, and that the mutual interdependence structure is known.



Controller

Model

Adjustment Mechanism

Plant

Controller Parameters

ymodel

u yplant uc

Figure 1 Block diagram of an MRAS

Model Adaptive Reference System Design controller to drive plant response to

mimic ideal response (error = yplant-ymodel => 0) Designer chooses: reference model,

controller structure, and tuning gains for adjustment mechanism


Model Adaptive Reference System On the adaptive model, the structure is

perhaps not known and the output is to be controlled or whose parameters are to be identified.

In addition, the adaptive model is assumed to have the same dimension as the reference model but with unknown parameters (except for the initial conditions).


Model Adaptive Reference System When the same data is used as input for

both models, reference and adaptive outputs may differ.

The purpose of the MRAS method is to make the adaptive output asymptotically coincide with the output of the reference model by adjusting the parameters of the adaptive model.

This adaptive process has a close relationship with the stability theory of a parameter adjustment.


Model Adaptive Reference System In Fig. 1, there are an ordinary common

feedback loop composed of the process and the controller (called inner loop) and another feedback loop that changes the controller parameters (called outer loop).

The parameters are changed on the basis of feedback from the error.

Here, the error is the difference between the output of the system and the output of the reference model


Model Adaptive Reference System In the MRAS the desired behavior of the

system is specified by a model, and the parameters of the controller are adjusted based on the error.

Originally, MRAS were derived for deterministic continuous-time systems and then were later extended to discrete-time systems and systems with stochastic disturbance.


Model Adaptive Reference System The MIT rule was an original approach to

the MRAS developed at MIT. The rule has one parameter (the

adaptation gain) that must be chosen by the user.

Consider a closed-loop system where the controller has one adjustable parameter θ.

The desired closed-loop response is specified by a model whose output is ym.


Model Adaptive Reference System e is the error between the output y and the

output ym. Adjust parameters in such a way that the

loss function is minimized. To make J small, change the parameters in

the direction of the negative gradient of J, that is

This called the MIT rule.


Model Adaptive Reference System The partial derivative ∂e/∂θ (called the

sensitivity derivative of the system) describes how the error is affected by the adjustable parameters.

∂e/∂θ can be evaluated by assuming θ to be constant.

The following slides summary the MIT rule.


Model Adaptive Reference System MIT Rule Tracking error:

Form cost function:

Update rule:

Change in is proportional to negative

gradient of J. Dr. Marwan Affandi 2012

modelplant yye −=

)(21)( 2 θθ eJ =

δθδγ

δθδγθ eeJ

dtd

−=−=

sensitivity

derivative

Model Adaptive Reference System MIT Rule Can chose different cost functions Example

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


<−=>

=

−=

=

0 ,10 ,00 ,1

)( where

)(

)()(

eee

esign

esignedtd

eJ

δθδγθ

θθ

Model Adaptive Reference System MIT Rule Example: Adaptation of feedforward gain



ymodel

u yplant uc

Π

Π

θ

Reference Model

Plant sγ−

)()( sGksG om =

)()( sGksGp =

-

+

Model Adaptive Reference System MIT Rule For system where k is unknown

Goal: Make it look like

using plant (note, plant model

is scalar multiplied by plant)


)()()( skGsUsY

=

)()()( sGksUsY

oc

=

)()( sGksG om =

Model Adaptive Reference System MIT Rule Choose cost function:

Write equation for error:

Calculate sensitivity derivative:

Apply MIT rule:


δθδγθθθ ee

dtdeJ −=→= )(

21)( 2

coccmm UGkUkGUGkGUyye −=−=−= θ

mo

c ykkkGUe ==

δθδ

eyeykk

dtd

mmo

γγθ−=−= '

Model Adaptive Reference System MIT Rule Gives block diagram:

considered tuning parameter



ymodel

u yplant uc

Π

Π

θ

Reference Model

Plant sγ−

)()( sGksG om =

)()( sGksGp =

-

+

γ

Model Adaptive Reference System MIT Rule NOTE: MIT rule does not guarantee error

convergence or stability

usually kept small

Tuning crucial to adaptation rate and stability.


γ

γ

Model Adaptive Reference System MRAC of Pendulum System:


d2 d1 dc

T

( ) TdmgdcJ c 1sin =++ θθθ

cmgdcsJsd

sTs

++= 2

1

)()(θ

77.100389.089.1

)()(

2 ++=

sssTsθ

Model Adaptive Reference System MRAC of Pendulum Controller will take form:


Controller

Model


Controller Parameters

ymodel

u yplant uc

77.100389.089.1

2 ++ ss

Model Adaptive Reference System MRAC of Pendulum Following process as before, write

equation for error, cost function, and update rule:


modelplant yye −=

)(21)( 2 θθ eJ =

δθδγ

δθδγθ eeJ

dtd

−=−=

sensitivity

derivative

Model Adaptive Reference System MRAC of Pendulum Assuming controller takes the form:


( )

cplant

plantcpplant

cmpmodelplant

plantc

uss

y

yuss

uGy

uGuGyyeyuu

22

1

212

21

89.177.100389.089.1

77.100389.089.1

θθ

θθ

θθ

+++=

−

++==

−=−=

−=

Model Adaptive Reference System MRAC of Pendulum


( )plant

c

c

cmc

yss

uss

e

uss

e

uGuss

e

22

1

22

21

2

2

22

1

22

1

89.177.100389.089.1

89.177.100389.089.1

89.177.100389.089.1

89.177.100389.089.1

θθ

θθ

θ

θθ

θθ

+++−=

+++−=

∂∂

+++=

∂∂

−+++

=

Model Adaptive Reference System MRAC of Pendulum If reference model is close to plant, we can

approximate:


plantmm

mm

cmm

mm

mm

yasas

asae

uasas

asaeasasss

012

01

2

012

01

1

012

22 89.177.100389.0

+++

−=∂∂

+++

=∂∂

++≈+++

θ

θ

θ

Model Adaptive Reference System MRAC of Pendulum From MIT rule, update rules are then:


eyasas

asaeedt

d

euasas

asaeedt

d

plantmm

mm

cmm

mm

++

+=

∂∂

−=

++

+−=

∂∂

−=

012

01

2

2

012

01

1

1

γθ

γθ

γθ

γθ

Model Adaptive Reference System MRAC of Pendulum


ymodel

e

yplant uc

Π

Π

θ1

Reference Model

Plant

sγ−

77.100389.089.1

2 ++ ss

Π

+

-

mm

mm

asasasa

012

01

+++

mm

mm

asasasa

012

01

+++

mm

m

asasb

012 ++

sγ

Π

-

+

θ2

Model Adaptive Reference System MRAC of Pendulum Simulation block diagram (NOTE: Modeled

to reflect control of DC motor)


am

s+amam

s+am

-gamma

sgamma

s

Step

Saturation

omega^2

s+amReference Model

180/pi

Radiansto Degrees

4.41

s +.039s+10.772

Plant

2/26

Degreesto Volts

35

Degrees

y m

Error

Theta2

Theta1

y

Model Adaptive Reference System Simulation with small gamma =

UNSTABLE!


0 200 400 600 800 1000 1200-100

-50

0

50

100

150

ymg=.0001

MRAC of Pendulum Solution: Add PD feedback

am

s+amam

s+am

-gamma

sgamma

s

Step

Saturation

omega^2

s+amReference 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

Model Adaptive Reference System MRAC of Pendulum Simulation results with varying gammas


0 500 1000 1500 2000 25000

5

10

15

20

25

30

35

40

45

ymg=.01g=.001g=.0001

707.sec3

:such that Designed

56.367.256.3

2

==

++=

ζs

m

T

ssy

Model Adaptive Reference System LabVIEW VI Front Panel


Model Adaptive Reference System Experimental Results


Model Adaptive Reference System Experimental Results PD feedback necessary to stabilize system Deadzone necessary to prevent updating

when plant approached model Often went unstable (attributed to inherent

instability in system i.e. little damping) Much tuning to get acceptable response


Model Adaptive Reference System Conclusions Given controller does not perform well

enough for practical use More advanced controllers could be

formed from other methods Modified (normalized) MIT Lyapunov direct and indirect Discrete modeling using Euler operator

Modified MRAC methods Fuzzy-MRAC Variable Structure MRAC (VS-MRAC)


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