Model-Reference Adaptive
Systems
1,The idea
2,The MIT Rule
3,Determination of the adaptive gain
4,Lyapunov theory
5,Design of MRAS using Lyapunov
theory
6,Bounded-input,bounded-output stability
7,Applications to adaptive control
8,Output feedback
9,Relations between MRAS and STR
10,Nonlinear systems
11,Conclusions
Introduction
Adjustment
mechanism
u
Model
Controller parameters
Plant
y
Controller
y
m
u
c
Flight control in the 1950s
Two ideas Phil Whitaker MIT
– The Reference Model
– Parameter adjustment rule
The stability problem
– Lyapunov
– Input-output stability
Modified adjustment rules
Proliferation of algorithms
Unification
The MIT Rule
The idea
Examples
Error and parameter convergence
The MIT Rule
Tracking error
e y? y
m
Introduce
JHθI
1
2
e
2
Change parameters such that
dθ
dt
γ
partialJ
partialθ
γe
partiale
partialθ
where partiale/partialθ is the sensitivity derivative
dJ
dt
e
partiale
partialt
γ e
2
partiale
partialθ
2
Many alternatives
JHeI tet
gives
dθ
dt
γ
partialJ
partialθ
γ
partiale
partialθ
signHeI
ccirclebig K,J,?str?m and B,Wittenmark 1
Adjustment of Feedforward Gain
Process
y kGHsI
Desired response
y
m
k
0
GHsIu
c
Controller
u θu
c
e y? y
m
kGHpIθu
c
k
0
GHpIu
c
Sensitivity derivative
partiale
partialθ
kGHpIu
c
k
k
0
y
m
MIT rule
dθ
dt
γ
T
k
k
0
y
m
e?γy
m
e
Real examples
Robotics
CD player
Block Diagram
Model
e
y
Process
u
S
+
–
u
c
q
y
m
-
g
s
kG(s)
k
0
G(s)
P
P
dθ
dt
γ
T
k
k
0
y
m
e?γy
m
e
Example
Model
e
y
Process
u
S
+
–
u
c
q
y
m
-
g
s
kG(s)
k
0
G(s)
P
P
Simulation
0 5 10 15 20
2
0
0 5 10 15 20
0
2
Time
Time
θ
y
m
y
γ 2
γ 1
γ 0.5
A Remark
Notice that
GHpIHθuIrelationnegate θ GHpIu
Consequences for block diagram manipula-
tion
ccirclebig K,J,?str?m and B,Wittenmark 2
A First Order System
Process
dy
dt
ay bu
Model
dy
m
dt
a
m
y
m
b
m
u
c
Controller
uHtI θ
1
u
c
HtI?θ
2
yHtI
ideal parameters
θ
1
θ
0
1
b
m
b
θ
2
θ
0
2
a
m
a
b
Derivation of Adaptive Law
The error
e y? y
m
y
bθ
1
p a bθ
2
u
c
partiale
partialθ
1
b
p a bθ
2
u
c
partiale
partialθ
2
b
2
θ
1
Hp a bθ
2
I
2
u
c
b
p a bθ
2
y
Approximate
p a bθ
2
p a
m
Hence
dθ
1
dt
γ
a
m
p a
m
u
c
e
dθ
2
dt
γ
a
m
p a
m
y
e
Block Diagram
-
S
P
+
e
u y
S
P
P
P
-
+
u
c
G
m
(s)
G(s)
q
1
q
2
g
s
-
g
s
a
m
s + a
m
a
m
s + a
m
dθ
1
dt
γ
a
m
p a
m
u
c
e
dθ
2
dt
γ
a
m
p a
m
y
e
Example a 1,b 0.5,a
m
b
m
2.
Simulation
Input and output
0 2040608010
1
1
0 2040608010
5
0
5
Time
Time
y
m
y
u
Parameters
0 2040608010
2
4
0 2040608010
0
2
Time
Time
θ
1
θ
2
γ 5
γ 1
γ 0.2
γ 5
γ 1
γ 0.2
ccirclebig K,J,?str?m and B,Wittenmark 3
CONTINUOUS SYSTEM mras
"MRAS for first-order system with
" Gm=bm/(s+am)
INPUT y uc
OUTPUT u
STATE ym th1 th2 x1 x2
DER dym dth1 dth2 dx1 dx2
u=th1*uc-th2*y
dym=-am*ym+bm*uc
dx1=-am*x1+am*uc
dx2=-am*x2-am*y
e=y-ym
dth1=-gamma*e*x1
dth2=-gamma*e*x2
am:2 "model parameter
bm:2 "model parameter
gamma:2 "adaptation gain
END
Bad Parameters Good Control?
The closed loop transfer function is
G
cl
HsI
θ
1
GHsI
1 θ
2
GHsI
θ
1
b
s a θ
2
b
01234
1
0
1
2
θ
2
θ
1
Error and Parameter Convergence
Consider adaptation of feedforward gain
e Hkθ?k
0
Iu
c
kHθ?θ
0
Iu
c
with θ
0
k
0
/k
dθ
dt
γk
2
u
2
c
Hθ?θ
0
I
Solution
θHtI θ
0
HθH0I?θ
0
Ie
γk
2
I
t
where
I
t
t
Z
0
u
2
c
HτI dτ
Exponential convergence with persistant
excitation
Determination of Adaptation Gain
A difficult problem
Approximations give insight
Leads to modified algorithms
ccirclebig K,J,?str?m and B,Wittenmark 4
An Example
daptation of feedforward gain
y kGHpIu
y
m
k
0
GHpIu
c
u θu
c
e y?y
m
dθ
dt
γy
m
e
Parameter equation
dθ
dt
γ y
m
HkGHpIθu
c
I γy
2
m
Why approximate? A thought experiment
dθ
dt
γ y
o
m
u
o
c
HkGHpIθI γHy
o
m
I
2
Characteristic equation
s γ y
o
m
u
o
c
kGHsI 0
Key parameter
μ γ y
o
m
u
o
c
k
Example GHsI
1
s 1
Example
Process
GHsI
1
s
2
a
1
s a
2
Approximate CE
s
3
a
1
s
2
a
2
s μ 0
μ γ y
o
m
u
o
c
k,Stability condition
γ y
o
m
u
o
c
k a
1
a
2
0 2040608010
0.1
0.1
0 2040608010
1
1
0 2040608010
10
10
Time
Time
Time
(a)
y
m
y
(b)
y
m
y
(c)
y
y
m
Modified Algorithm
MIT rule
dθ
dt
γ?e
Normalized adaptation law
dθ
dt
γ?e
α?
T
0 2040608010
0.1
0.1
0 2040608010
1
1
0 2040608010
10
10
Time
Time
Time
(a)
y
m
y
(b)
y
m
y
(c)
y
m
y
Summary
The idea
– Model following
– The MIT rule
The error equation
eHtI HGHp,θI?G
m
HpIIu
c
HtI
A gradient procedure
dθ
dt
γ?e
partialGHp,θI
partialθ
u
c
Approximations
Normalization
dθ
dt
γ
e
α?
T
ccirclebig K,J,?str?m and B,Wittenmark 5
Systems
1,The idea
2,The MIT Rule
3,Determination of the adaptive gain
4,Lyapunov theory
5,Design of MRAS using Lyapunov
theory
6,Bounded-input,bounded-output stability
7,Applications to adaptive control
8,Output feedback
9,Relations between MRAS and STR
10,Nonlinear systems
11,Conclusions
Introduction
Adjustment
mechanism
u
Model
Controller parameters
Plant
y
Controller
y
m
u
c
Flight control in the 1950s
Two ideas Phil Whitaker MIT
– The Reference Model
– Parameter adjustment rule
The stability problem
– Lyapunov
– Input-output stability
Modified adjustment rules
Proliferation of algorithms
Unification
The MIT Rule
The idea
Examples
Error and parameter convergence
The MIT Rule
Tracking error
e y? y
m
Introduce
JHθI
1
2
e
2
Change parameters such that
dθ
dt
γ
partialJ
partialθ
γe
partiale
partialθ
where partiale/partialθ is the sensitivity derivative
dJ
dt
e
partiale
partialt
γ e
2
partiale
partialθ
2
Many alternatives
JHeI tet
gives
dθ
dt
γ
partialJ
partialθ
γ
partiale
partialθ
signHeI
ccirclebig K,J,?str?m and B,Wittenmark 1
Adjustment of Feedforward Gain
Process
y kGHsI
Desired response
y
m
k
0
GHsIu
c
Controller
u θu
c
e y? y
m
kGHpIθu
c
k
0
GHpIu
c
Sensitivity derivative
partiale
partialθ
kGHpIu
c
k
k
0
y
m
MIT rule
dθ
dt
γ
T
k
k
0
y
m
e?γy
m
e
Real examples
Robotics
CD player
Block Diagram
Model
e
y
Process
u
S
+
–
u
c
q
y
m
-
g
s
kG(s)
k
0
G(s)
P
P
dθ
dt
γ
T
k
k
0
y
m
e?γy
m
e
Example
Model
e
y
Process
u
S
+
–
u
c
q
y
m
-
g
s
kG(s)
k
0
G(s)
P
P
Simulation
0 5 10 15 20
2
0
0 5 10 15 20
0
2
Time
Time
θ
y
m
y
γ 2
γ 1
γ 0.5
A Remark
Notice that
GHpIHθuIrelationnegate θ GHpIu
Consequences for block diagram manipula-
tion
ccirclebig K,J,?str?m and B,Wittenmark 2
A First Order System
Process
dy
dt
ay bu
Model
dy
m
dt
a
m
y
m
b
m
u
c
Controller
uHtI θ
1
u
c
HtI?θ
2
yHtI
ideal parameters
θ
1
θ
0
1
b
m
b
θ
2
θ
0
2
a
m
a
b
Derivation of Adaptive Law
The error
e y? y
m
y
bθ
1
p a bθ
2
u
c
partiale
partialθ
1
b
p a bθ
2
u
c
partiale
partialθ
2
b
2
θ
1
Hp a bθ
2
I
2
u
c
b
p a bθ
2
y
Approximate
p a bθ
2
p a
m
Hence
dθ
1
dt
γ
a
m
p a
m
u
c
e
dθ
2
dt
γ
a
m
p a
m
y
e
Block Diagram
-
S
P
+
e
u y
S
P
P
P
-
+
u
c
G
m
(s)
G(s)
q
1
q
2
g
s
-
g
s
a
m
s + a
m
a
m
s + a
m
dθ
1
dt
γ
a
m
p a
m
u
c
e
dθ
2
dt
γ
a
m
p a
m
y
e
Example a 1,b 0.5,a
m
b
m
2.
Simulation
Input and output
0 2040608010
1
1
0 2040608010
5
0
5
Time
Time
y
m
y
u
Parameters
0 2040608010
2
4
0 2040608010
0
2
Time
Time
θ
1
θ
2
γ 5
γ 1
γ 0.2
γ 5
γ 1
γ 0.2
ccirclebig K,J,?str?m and B,Wittenmark 3
CONTINUOUS SYSTEM mras
"MRAS for first-order system with
" Gm=bm/(s+am)
INPUT y uc
OUTPUT u
STATE ym th1 th2 x1 x2
DER dym dth1 dth2 dx1 dx2
u=th1*uc-th2*y
dym=-am*ym+bm*uc
dx1=-am*x1+am*uc
dx2=-am*x2-am*y
e=y-ym
dth1=-gamma*e*x1
dth2=-gamma*e*x2
am:2 "model parameter
bm:2 "model parameter
gamma:2 "adaptation gain
END
Bad Parameters Good Control?
The closed loop transfer function is
G
cl
HsI
θ
1
GHsI
1 θ
2
GHsI
θ
1
b
s a θ
2
b
01234
1
0
1
2
θ
2
θ
1
Error and Parameter Convergence
Consider adaptation of feedforward gain
e Hkθ?k
0
Iu
c
kHθ?θ
0
Iu
c
with θ
0
k
0
/k
dθ
dt
γk
2
u
2
c
Hθ?θ
0
I
Solution
θHtI θ
0
HθH0I?θ
0
Ie
γk
2
I
t
where
I
t
t
Z
0
u
2
c
HτI dτ
Exponential convergence with persistant
excitation
Determination of Adaptation Gain
A difficult problem
Approximations give insight
Leads to modified algorithms
ccirclebig K,J,?str?m and B,Wittenmark 4
An Example
daptation of feedforward gain
y kGHpIu
y
m
k
0
GHpIu
c
u θu
c
e y?y
m
dθ
dt
γy
m
e
Parameter equation
dθ
dt
γ y
m
HkGHpIθu
c
I γy
2
m
Why approximate? A thought experiment
dθ
dt
γ y
o
m
u
o
c
HkGHpIθI γHy
o
m
I
2
Characteristic equation
s γ y
o
m
u
o
c
kGHsI 0
Key parameter
μ γ y
o
m
u
o
c
k
Example GHsI
1
s 1
Example
Process
GHsI
1
s
2
a
1
s a
2
Approximate CE
s
3
a
1
s
2
a
2
s μ 0
μ γ y
o
m
u
o
c
k,Stability condition
γ y
o
m
u
o
c
k a
1
a
2
0 2040608010
0.1
0.1
0 2040608010
1
1
0 2040608010
10
10
Time
Time
Time
(a)
y
m
y
(b)
y
m
y
(c)
y
y
m
Modified Algorithm
MIT rule
dθ
dt
γ?e
Normalized adaptation law
dθ
dt
γ?e
α?
T
0 2040608010
0.1
0.1
0 2040608010
1
1
0 2040608010
10
10
Time
Time
Time
(a)
y
m
y
(b)
y
m
y
(c)
y
m
y
Summary
The idea
– Model following
– The MIT rule
The error equation
eHtI HGHp,θI?G
m
HpIIu
c
HtI
A gradient procedure
dθ
dt
γ?e
partialGHp,θI
partialθ
u
c
Approximations
Normalization
dθ
dt
γ
e
α?
T
ccirclebig K,J,?str?m and B,Wittenmark 5
