Post on 18-Nov-2014
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 .