ADAPTIVE CONTROL

Presented byHARIKRISHNA SATISH.T

Roll no:000910802023

ADAPTIVE CONTROLLER An adaptive controller is a controller with

adjustable parameters and mechanism for adjusting the parameters.

BLOCK DIAGRAM OF ADAPTIVE CONTROL SYSTEM

THE ADAPTIVE CONTROL PROBLEM

The construction of the adaptive controller contains the following steps

• Characterize the desired behavior of the closed loop system.

• Determine the suitable control law with adjustable parameters.

• Find the mechanism for adjusting the parameters.

• Implement the control law.

ADAPTIVE SCHEMES

There are four different schemes we follow while constructing an adaptive control

system.

• Gain scheduling. • Model reference adaptive control.• Self tuning regulators.• Dual control.

MODEL REFERENCE ADAPTIVE CONTROL

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

• Designer chooses: reference model, controller structure, and Designer chooses: reference model, controller structure, and tuning gains for adjustment mechanism.tuning gains for adjustment mechanism.

• Basic methods used to design adjustment mechanism are Basic methods used to design adjustment mechanism are 1.MIT Rule 2.Lyapunov rule1.MIT Rule 2.Lyapunov rule

Controller

Model

AdjustmentMechanism

Plant

Controller Parameters

ymodel

u yplant

uc

Adaptation of feed forward gain Adaptation of feed forward gain

• Tracking error:Tracking error:

• Form cost function:Form cost function:

• Update rule:Update rule:

– Change in Change in θθ is proportional to negative gradient of is proportional to negative gradient of JJ

modelplant yye

e

eJ

dt

d

)(2

1)( 2 eJ

MIT Rule

• For system where k is unknown.For system where k is unknown.

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

using reference model . using reference model .

)()(

)(skG

sU

sY

)()(

)(sGk

sU

sYo

c

)()( sGksG om

MIT 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:

e

edt

deJ )(

2

1)( 2

mo

c yk

kkGU

e

eyeyk

k

dt

dmm

o

'

coccmm UGkUkGUGkGUyye

MIT Rule

• Adaptation feedforward gain

Adjustment Mechanism

ymodel

u yplantuc

Π

Π

θ

Reference Model

Plant

s

)()( sGksG om

)()( sGksGp

-

+

MRAC Example

)uk (kθ -ae dt

de

y-ye

θuu

u k -ay dt

dy

ku -ay dt

dy

m

m

c

mmm

c

:Error of Derivative

0e :mEquilibriu Desired

:Error ofEquation

:Equation Controller

: Model Reference

: Model Process

APPLYING MIT RULE

• Cost function :

• Error equation :

• Updating rule :

)(2

1)( 2 eJ

modelplant yye

e

eJ

dt

d

CONSTRUCTION OF BLOCK DIAGRAM

Adjustment Mechanism

ymodel

u yplantuc

Π

Π

θ

Reference Model

Plant

s

as

km

_____

as

k

_____

-

+

eydt

dm

MIT Rule to Lyapunov Transition1. Several Problems were encountered in the usage of the MIT rule.

2. Also, it was not possible in general to prove closed loop stability, or convergence of the output error to zero.

3. A new way of redesigning adaptive systems using Lyapunov theory was proposed by Parks.

4. This was based on Lyapunov stability theorems, so that stable and provably convergent model reference schemes were obtained.

5. The update laws are similar to that of the MIT Rule, with the sensitivity functions replaced by other functions.

6. The theme was to generate parameter adjustment rule which guarantee stability

LYAPUNOV STABILITY

• Lyapunov’s method states that a system has a uniform asymptotically stable equilibrium x=0.if a Lyapunov function V(x)exists that satisfies:

- V(x) > 0 for X≠0 (positive definite) - < 0 for x≠0 (negative definite) - V(∞) ∞ for x∞ - V(0) =0.

)(XV

Adaption of feed forward gain using Lyapunov function

)uk (kθ -ae dt

de :Error of Derivative

0e :mEquilibriu Desired

y-y e :Error of Equation

θuu :Equation Controller

u k -ay dt

dy :Model Reference

ku -ay dt

dy :Model Process

m

m

c

mmm

c

Cont……• Lyapunov Function::

• Derivative of Lyapunov Function:

• Choosing the Adjustment Rule:

• Adaptation Law according Lyapunov Methods:

22

22)

k

k(θ

ke

γV m

))((2

)())((

eudt

d

km

kkae

dt

d

km

kku

km

kkaee

dt

dV

2aedt

dVue

dt

d

uedt

d

Adaptive Feed Forward gain Block Diagram

Adjustment Mechanism

ymodel

u yplantuc

Π

Π

θ

Reference Model

Plant

s

)()( sGksG om

)()( sGksGp

-

+

MRAC for first order closed loop system

• Process Model:

• Reference Model:

• Controller Structure:

• Error Equation:

• Desired Equilibrium:

• Derivative of Error:

cbuaydt

dy

ubaydt

dymm

m

yθuθu 21c

myye

0e

)ub(bθy)aa(bθeadt

dem1m2m

conti……..

• Candidate for Lyapunov Function:

• Derivative of Lyapunov Function:

• Adaptation Law:

))b(bθbγ

1)aa(bθ

bγ

1(e

2

1)θ,θ, V(e 2

m12

m22

21

γue)dt

dθ)(b(bθ

γ

1γye)

dt

dθ)(aa(bθ

γ

1e-a

dt

dθ)b(bθ

γ

1

dt

dθ)aa(bθ

γ

1

dt

dee

dt

dV

1m1

2m2

2

m

1m1

2m2

γyedt

dθ γue

dt

dθ 21

BLOCK DIAGRAM

ADVANTAGES OF USING LYAPUNOV FUNCTION

• The Analysis of system equations is difficult in M.I.T rule. where as in Lyapunov method is easy.

• M.I.T rule does not guarantee error convergence or stability

• Lyapunov Adaptive laws gives guaranteed stability.(i.e. error (e)=0).

References

1. Karl j.Astrom,Bjorn Wittenmark;second edition ,1995.”adaptive control”.pearson education,london.

2. Hans Butler ;1992.”Model reference adaptive control” prentice hall international ltd.

3. Petros A. Ioannou,Jing Sun;1995” Robust adaptive control”,Prentice Hall ,englewood cliffs,NJ .