SYS635 Adaptive Control Systems
Ka C. Cheok
MRAS Gradient Method 0th 1st 2nd Order Systems
22 Oct ‘05
1
MODEL REFERENCE ADAPTIVE SYSTEM (MRAS)
MRAS Objective:
Based on information y
m
, y, u and u
c
, devise a controller that automatically adjusts
itself so that the behavior of the closedloop control plant output
(y) closely follows that (y
m
) of the
reference model.
In other words, make y mimic y
m
.
Illustration:
Can you think of an example for such a
system?
Dance choreographer? Coach?
5.2 The MIT Rule (read Chap 5.2)
Example of a
cost function
2
1
( )
( )
where
is a function of
2
J
θ
ε θ
ε
θ
=
MIT Rule
says that the time rate of change of
θ
is
proportional to negative gradient of
J
w.r.t.
θ
. That is
d
dJ
d
dt
d
d
θ
ε
γ
γε
θ
θ
= −
= −
Integrating yields the continuoustime equation for
updating
θ
:
0
0
0
0
( )
(
)
(
)
t
t
t
t
d
dJ
t
t
dt
t
dt
dt
d
θ
θ
θ
θ
γ
θ
=
+
=
−
∫
∫
Since we can
d
t
dt
θ
θ
∆
≈
∆
,
this leads to the Delta Rule
that says the increment (delta) in
θ
is approximately
(
)
dJ
t
d
θ
γ
θ
∆
≈ −
∆
and the discretetime equation
for updating
θ
is
new
old
θ
θ
θ
=
+ ∆
Water surface
He/she is looking for the deepest spot in the river, should he/she go
forward or backward? What logic did he used?
This may be a vector although drawn as a scalar
0
J
θ
∂
<
∂
0
J
θ
∂
>
∂
( )
J
θ
θ
y
u
u
c
y
m
Reference Model (in
computer/or a physical system)
Controller
Physical
Plant
Adjustment
Mechanism
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SYS635 Adaptive Control Systems
Ka C. Cheok
MRAS Gradient Method 0th 1st 2nd Order Systems
22 Oct ‘05
2
(
)
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)
(
)
(
)
(
)
(
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(
)
(
)
1
2
2
1
Sigmoid function
=
1
1
1
y
1
Derivative
( 1) 1
1
1
1
1
1
1
1
1
1
1
1
h
h
h
h
h
h
h
h
h
h
h
h
y
e
e
e
e
e
e
h
h
h
e
e
y y
e
e
e
e
−
−
−
−
−
−
−
−
−
−
−
−
−
−
+
∂
∂
+
∂
+
=
=
= −
+
−
=
∂
∂
∂
+
=
=
−
=
−
+
+
+
+
Application of MIT Rule to a single neuron model
Let’s suppose we are shown a inputoutput
pattern or phenomenon in the form of
y
m
versus
x
curve. In the following example,
it so happens that the pattern behaves as
shown in the figure.
We introduce a neuron model whose math
description tends to produce an I/O
relationship similar to the given pattern.
That single neuron math model is
described by a sigmoidal funtion
1
1
h
y
h
wx
b
e
−
=
=
+
+
where
w
is called a weight and
b
is a
bias.
The error between the pattern and the neuron model is given by
(
)
1
1
1
1
m
m
m
h
wx
b
y
y
y
y
e
e
ε
−
−
+
=
−
=
−
=
−
+
+
We would like to find
w
and
b
so that
y
follows
m
y
.
A way to do this is to find
w
and
b
so that they
minimize the error cost function
(
)
2
1
,
2
J
w b
ε
=
The gradients of J w.r.t. w and b are:
(
)
(
)
(
)( )(
)(
)
2
1
1
(
)
2
1
1
(1
)
(1
)
h
m
y
y
wx
b
J
J
y
h
e
y y
x
y yx
w
y
h
w
y
h
w
ε
ε
ε
ε
ε
ε
−
∂
∂
∂
−
∂
+
∂
∂
∂
∂
∂
+
=
=
=
−
=
−
∂
∂
∂
∂
∂
∂
∂
∂
∂
(
)
(
)
(
)( )(
)( )
2
1
1
(
)
2
1
1
(1
)
1
(1
)
h
m
y
y
wx
b
J
J
y
h
e
y y
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 Spring '11
 re
 Trigraph, adaptive control systems, Ka C. Cheok, MRAS Gradient Method

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