1. Model ReferenceModel Reference
Adaptive ControlAdaptive Control
Survey of Control Systems (MEM 800)Survey of Control Systems (MEM 800)
Presented byPresented by
Keith SevcikKeith Sevcik
2. ConceptConcept
Design controller to drive plant response to mimic idealDesign controller to drive plant response to mimic ideal
response (error = yresponse (error = yplantplant-y-ymodelmodel => 0)=> 0)
Designer chooses: reference model, controller structure,Designer chooses: reference model, controller structure,
and tuning gains for adjustment mechanismand tuning gains for adjustment mechanism
Controller
Model
Adjustment
Mechanism
Plant
Controller
Parameters
ymodel
u yplant
uc
3. MIT RuleMIT Rule
Tracking error:Tracking error:
Form cost function:Form cost function:
Update rule:Update rule:
– Change in is proportional to negative gradient ofChange in is proportional to negative gradient of
modelplant yye −=
)(
2
1
)( 2
θθ eJ =
δθ
δ
γ
δθ
δ
γ
θ e
e
J
dt
d
−=−=
θ J
sensitivity
derivative
4. MIT RuleMIT Rule
Can chose different cost functionsCan chose different cost functions
EX:EX:
From cost function and MIT rule, control law can beFrom cost function and MIT rule, control law can be
formedformed
<−
=
>
=
−=
=
0,1
0,0
0,1
)(where
)(
)()(
e
e
e
esign
esign
e
dt
d
eJ
δθ
δ
γ
θ
θθ
5. MIT RuleMIT Rule
EX: Adaptation of feedforward gainEX: Adaptation of feedforward gain
Adjustment Mechanism
ymodel
u yplantuc
Π
Π
θ
Reference
Model
Plant
s
γ−
)()( sGksG om =
)()( sGksGp =
-
+
6. MIT RuleMIT Rule
For system where is unknownFor system where is unknown
Goal: Make it look likeGoal: Make it look like
using plant (note, plant model isusing plant (note, plant model is
scalar multiplied by plant)scalar multiplied by plant)
)(
)(
)(
skG
sU
sY
= k
)(
)(
)(
sGk
sU
sY
o
c
=
)()( sGksG om =
7. 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 −=−=−= θ
δθ
δ
γ
θ
θθ
e
e
dt
d
eJ −=→= )(
2
1
)( 2
m
o
c y
k
k
kGU
e
==
δθ
δ
eyey
k
k
dt
d
mm
o
γγ
θ
−=−= '
8. 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
γ−
)()( sGksG om =
)()( sGksGp =
-
+
γ
9. MIT RuleMIT Rule
NOTE: MIT rule does not guarantee errorNOTE: MIT rule does not guarantee error
convergence or stabilityconvergence or stability
usually kept smallusually kept small
Tuning crucial to adaptation rate andTuning crucial to adaptation rate and
stability.stability.
γ
γ
10. SystemSystem
MRAC of PendulumMRAC of Pendulum
( ) TdmgdcJ c 1sin =++ θθθ
cmgdcsJs
d
sT
s
++
= 2
1
)(
)(θd2
d1dc
T
77.100389.0
89.1
)(
)(
2
++
=
sssT
sθ
11. MRAC of PendulumMRAC of Pendulum
Controller will take form:Controller will take form:
Controller
Model
Adjustment
Mechanism
Controller
Parameters
ymodel
u yplant
uc
77.100389.0
89.1
2
++ ss
12. MRAC of PendulumMRAC of Pendulum
Following process as before, writeFollowing process as before, write
equation for error, cost function, andequation for error, cost function, and
update rule:update rule:
modelplant yye −=
)(
2
1
)( 2
θθ eJ =
δθ
δ
γ
δθ
δ
γ
θ e
e
J
dt
d
−=−=
sensitivity
derivative
13. MRAC of PendulumMRAC of Pendulum
Assuming controller takes the form:Assuming controller takes the form:
( )
cplant
plantcpplant
cmpmodelplant
plantc
u
ss
y
yu
ss
uGy
uGuGyye
yuu
2
2
1
212
21
89.177.100389.0
89.1
77.100389.0
89.1
θ
θ
θθ
θθ
+++
=
−
++
==
−=−=
−=
14. MRAC of PendulumMRAC of Pendulum
( )
plant
c
c
cmc
y
ss
u
ss
e
u
ss
e
uGu
ss
e
2
2
1
2
2
2
1
2
2
2
2
1
2
2
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
θ
θ
θ
θ
θ
θθ
θ
θ
+++
−=
+++
−=
∂
∂
+++
=
∂
∂
−
+++
=
15. MRAC of PendulumMRAC of Pendulum
If reference model is close to plant, canIf reference model is close to plant, can
approximate:approximate:
plant
mm
mm
c
mm
mm
mm
y
asas
asae
u
asas
asae
asasss
01
2
01
2
01
2
01
1
01
2
2
2
89.177.100389.0
++
+
−=
∂
∂
++
+
=
∂
∂
++≈+++
θ
θ
θ
16. MRAC of PendulumMRAC of Pendulum
From MIT rule, update rules are then:From MIT rule, update rules are then:
ey
asas
asa
e
e
dt
d
eu
asas
asa
e
e
dt
d
plant
mm
mm
c
mm
mm
++
+
=
∂
∂
−=
++
+
−=
∂
∂
−=
01
2
01
2
2
01
2
01
1
1
γ
θ
γ
θ
γ
θ
γ
θ
17. MRAC of PendulumMRAC of Pendulum
Block DiagramBlock Diagram
ymodel
e
yplantuc
Π
Π
θ1
Reference
Model
Plant
s
γ−
77.100389.0
89.1
2
++ ss
Π
+
-
mm
mm
asas
asa
01
2
01
++
+
mm
mm
asas
asa
01
2
01
++
+
mm
m
asas
b
01
2
++
s
γ
Π
-
+
θ2
18. MRAC of PendulumMRAC of Pendulum
Simulation block diagram (NOTE:Simulation block diagram (NOTE:
Modeled to reflect control of DC motor)Modeled to reflect control of DC motor)
am
s+am
am
s+am
-gamma
s
gamma
s
Step
Saturation
omega^2
s+am
Reference Model
180/pi
Radians
to Degrees
4.41
s +.039s+10.772
Plant
2/26
Degrees
to Volts
35
Degrees
y m
Error
Theta2
Theta1
y
19. MRAC of PendulumMRAC of Pendulum
Simulation with small gamma = UNSTABLE!Simulation with small gamma = UNSTABLE!
0 200 400 600 800 1000 1200
-100
-50
0
50
100
150
ym
g=.0001
20. MRAC of PendulumMRAC of Pendulum
Solution: Add PD feedbackSolution: Add PD feedback
am
s+am
am
s+am
-gamma
s
gamma
s
Step
Saturation
omega^2
s+am
Reference Model
180/pi
Radians
to Degrees
4.41
s +.039s+10.772
Plant
1
P
du/dt
2/26
Degrees
to Volts
35
Degrees
1.5
D
y m
Error
Theta2
Theta1
y
21. MRAC of PendulumMRAC of Pendulum
Simulation results with varying gammasSimulation results with varying gammas
0 500 1000 1500 2000 2500
0
5
10
15
20
25
30
35
40
45
ym
g=.01
g=.001
g=.0001
707.
sec3
:such thatDesigned
56.367.2
56.3
2
=
=
++
=
ζ
s
m
T
ss
y
25. Experimental ResultsExperimental Results
PD feedback necessary to stabilizePD feedback necessary to stabilize
systemsystem
Deadzone necessary to prevent updatingDeadzone necessary to prevent updating
when plant approached modelwhen plant approached model
Often went unstable (attributed to inherentOften went unstable (attributed to inherent
instability in system i.e. little damping)instability in system i.e. little damping)
Much tuning to get acceptable responseMuch tuning to get acceptable response
26. ConclusionsConclusions
Given controller does not perform well enoughGiven controller does not perform well enough
for practical usefor practical use
More advanced controllers could be formed fromMore advanced controllers could be formed from
other methodsother 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)