Model-ReferenceAdaptive 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
Lecture 8 #7B The Input-OutputView
1,Introduction
#0F White Boxes and Black Boxes
#0F Input-output descriptions
#0F How to generalize from linearto
nonlinear?
2,The Small Gain Theorem #28SGT#29
#0F The notion of gain
#0F Examples
#0F The main result
3,The Passivity Theorem #28PT#29
#0F Passivity and phase
#0F Examples
#0F The Passivity Theorem
#0F Relations between SGT and PT
4,Applications to Adaptive Control
#0F The augmented error
#0F MRAS and STR
5,Conclusions
Introduction
#0F Black Boxes and White Boxes
#0F Input-Output Descriptions
#7B The Table
#7B Linear Systems
#7B Dualitybetween Signals and Systems
#0F Formalization of the input-output view
#7B Signals and signal spaces
#7B The notions of gain and phase
#7B The notion of passivity
#0F Stability
#7B Stability Concept BIBO
#7B Stability of a system!
#7B Stability criteria
#7B Extensions of Nyquist's theorem
#0F The Mathematical Framework
#7B Functional analysis
The Notion of Gain
Signal spaces
L
2
,kuk =
#10
R
1
,1
u
2
#28t#29dt
#111
2
L
1
,kuk = sup
0#14t#3C1
ju#28t#29j
Extended spaces
x
T
#28t#29=
#1A
x#28t#29 0#14t#14T
0 t#3ET
u2X
e
if x
T
2 X
The notion of gain #28=operatornorm#29
#0D#28S#29 = sup
u2X
e
kSuk
kuk
Gain smallest value #0D such that
kSuk#14#0D#28S#29kuk for all u 2 X
e
c#0D K,J,#C5str#F6m and B,Wittenmark 1
Examples
Linear systems with signals in L
2e
kyk#14max
!
jG#28i!#29j#01kuk
u
0
= sin!t
Linear Systems with signals in L
1
#0D#28G#29=
Z
1
0
jh#28#1C#29jd#1C
u
0
#28s#29=u
0
sign#28h#28t,s#29#29
Static nonlinear system
x
f(x)
f = g x
f =- g x
The SmallGain Theorem
Definition 1
A system is called bounded-input,bounded-
output #28BIBO#29 stable if the system has
bounded gain.
Theorem 1
Consider the system.
S
u ey
H
1
- H
2
Let #0D
1
and #0D
2
be the gains of the systems H
1
and H
2
,The closed-loop system is BIBO stable
if
#0D
1
#0D
2
#3C 1
and its gain is less than
#0D =
#0D
1
1,#0D
1
#0D
2
Passivity
#0F The idea
#7B Energy dissipation
#7B Capacitors,induktors,resistances
#7B Mass,spring,dashpot
#7B Circuit Theory
#7B Mechatronics
#0F Mathematical Formalization
#0F The Notion of Phase
#0F Examples
#0F Postive real linear system
#0F The passivity theorem
#0F Using passivity in system design
AFormal Statement
Definition 2
A system with input u and output y is passive
if
hyjui#150
The system is input strictly passive #28ISP#29 if
there exists "#3E0such that
hyjui#15"kuk
2
and output strictly passive #28OSP#29 if there exists
"#3E0such that
hyjui#15"kyk
2
Intuitively
#0F Think about u and v as voltage and
current orforce and velocity
#0F Causality?
c#0D K,J,#C5str#F6m and B,Wittenmark 2
The Notion of Phase
Let the signal space have an inner product
The phase for a given input u can then be
de#0Cned as
cos' =
hy jui
kukkyk
=
hHujui
kukkHuk
Passivity implies that the phase is in the range
,
#19
2
#14 ' #14
#19
2
Linear Time-invariantSystems
hyjui =
1
Z
0
y#28t#29u#28t#29 dt =
1
2#19
1
Z
,1
Y #28i!#29U#28,i!#29 d!
=
1
2#19
1
Z
,1
G#28i!#29U#28i!#29U#28,i!#29 d!
=
1
#19
1
Z
0
RefG#28i!#29gU#28i!#29U#28,i!#29 d!
Definition 3
A rational transfer function G with real
coe#0Ecients is positive real #28PR#29 if
ReG#28s#29 #15 0 for Res #15 0
A transfer function G is strictly positive real
#28SPR#29 if G#28s,"#29 is positive real for some real
"#3E0.
Characterizing Positive Real
Transfer Functions
Theorem 2
A rational transfer function G#28s#29 with real
coe#0Ecients is PR if and only if the following
conditions hold.
#0F #28i#29 The function has no poles in the right
half-plane.
#0F #28ii#29 If the function has poles on the
imaginary axis or at in#0Cnity,they are
simple poles with positive residues.
#0F #28iii#29 The real part of G is nonnegative
along the i! axis,that is,
Re#28G#28i!#29#29 #15 0
A transfer function is SPR if conditions #28i#29 and
#28iii#29 hold and if condition #28ii#29 is replaced by the
condition that G#28s#29 has no poles or zeros on
the imaginary axis.
Examples
Recall
hy jui =
1
#19
1
Z
0
RefG#28i!#29gU#28i!#29U#28,i!#29 d!
#0F Positive real PR
ReG#28i!#29 #15 0
#0F Input strictly passive ISP
ReG#28i!#29 #15 "#3E0
#0FOutput stricly passive OSP
ReG#28i!#29 #15 "jG#28i!#29j
2
G#28s#29=s+1SPR and ISP not OSP
G#28s#29=
1
s+1
SPR and OSP not ISP
G#28s#29=
s
2
+1
#28s+1#29
2
OSP and ISP not OSP
G#28s#29=
1
s
PR not SPR,OPS or ISP
c#0D K,J,#C5str#F6m and B,Wittenmark 3
Nonlinear Static Systems y = f#28u#29
hyjui =
Z
1
0
f#28u#28t#29#29u#28t#29 dt
#0F Passive if xf#28x#29 #15 0
#0F Input strictly passive #28ISP#29 if xf#28x#29 #15
#0Ejxj
2
#0F Output strictly passive if
xf#28x#29 #15 #0Ef
2
#28x#29
Geometric Interpretation
Example
#0F f#28x#29=x+x
3
input strictly passive
#0F f#28x#29=x=#281 + jxj#29 output strictly passive.
The Passivity Theorem
Theorem 3
Consider a system obtained by connecting two
systems H
1
and H
2
in a feedback loop
S
u ey
H
1
- H
2
Let H
1
be strictly output passive and H
2
be
passive,The closed-loop system is then BIBO
stable.
Passivity is an invariant under feedback.
Use of passivity in system design.
#0F Force control in robotics
#0F Remote manipulator
#0F How to think about the problem
Relations Between SmallGain and
Passivity Theorems
S
S
b)a)
I
d)
2
S
I S
c)
H
1
- I
- H
2
- I
I - H
2
S
1
- S
2
H
1
- H
2
I + H
1
()
-1
H
1
1
2
a ! b,H
1
! #28I + H
1
#29
,1
H
1;H
2
!I,H
2
b!c:S
i
=#28H
i
+I#29
,1
#28H
i
,I#29
Applicationsto adaptive control
#0F Structure of Adaptive Systems
#0F Apply passivity results
#0F Insight
#0F Modi#0Ced Algorithms
#0F PI adjustments
c#0D K,J,#C5str#F6m and B,Wittenmark 4
Adaptation of Feedforward Gain
q
S
–
Model
Process
+
S
–
Model
Process
+
y
y
e
e q
u
c
u
c
kG(s)
kG(s)
k
0
G(s)
k
0
G(s)
y
m
y
m
-
g
s
-
g
s
P
P
P
P
(a)
(b)
Redrawb#29as
0
G
e
S
q
S
-
H
g
s
u
c
q
0
P P
q - q
0
()u
c
-
Analysis
Lemma 1
Let r be a bounded square integrable function,
and let G#28s#29 be a transfer function that is
positive real,The system whose input-output
relation is given by
y = r#28G#28p#29ru#29
is then passive.
Example,PI adjustments
#12#28t#29=,#0D
1
u
c
#28t#29e#28t#29,#0D
2
t
Z
u
c
#28#1C#29e#28#1C#29d#1C
Explore the advantages of PI adjustments
analytically and by simulation!
A modi#0Cedalgorithm
Change
0
G
e
S
q
S
-
H
g
s
u
c
q
0
P P
q - q
0
()u
c
-
To
G
G
+
–
S
y
q
u
c
y
m
-
g
s
G
c
q
0
P
P
Make G
c
G SPR,Still a problem with pole
excess #3E 1.
The AugmentedError
Consider the error
e = G#28#12,#12
0
#29u
c
= G#28#12,#12
0
#29u
c
+#28#12,#12
0
#29Gu
c
,#28#12,#12
0
#29Gu
c
Introduce the augmented error
#0F = e + #11
where
#11 = G#28#12,#12
0
#29u
c
,#28#12,#12
0
#29Gu
c
= G#12u
c
,#12Gu
c
Notice that #11 is zero under stationary condi-
tions
Use the adaptation law
d#12
dt
=,#0D#0FG
2
u
c
Stabilitynow follows from the passivity
theorem
The idea can be extended to the general case,
details are messy.
c#0D K,J,#C5str#F6m and B,Wittenmark 5
A Minor Extension
Factor
G = G
1
G
2
where the transfer function G
1
is SPR,The
error e = y,y
m
can then be written as
e = G#28#12,#12
0
#29u
c
=#28G
1
G
2
#29#28#12,#12
0
#29u
c
= G
1
,
G
2
#28#12,#12
0
#29u
c
+#28#12,#12
0
#29G
2
u
c
,#28#12,#12
0
#29G
2
u
c
#01
Introduce
" = e+ #11
where #11 is the error augmentation de#0Cned by
#11 = G
1
#28#12,#12
0
#29G
2
u
c
,G#28#12,#12
0
#29u
c
= G
1
#28#12G
2
u
c
#29,G#12u
c
Use adaptation law
d#12
dt
=,#0D#0FG
2
u
c
MRAS with AugmentedError
–
+
Model
Process
y
–
e
S S
S
q
q
+
h
e
u
c
y
m
k
0
G
kG
-
g
s
G
1
G
2
u
c
P
P
P
Compare STR and MRAS
MRAS
d#12
dt
= #0D'
f
"
'
T
f
=,G
f
#28p#29grad
#12
"#28t#29
" = G
SPR
#28y,y
m
#29+#11=G
SPR
e+#11
Direct STR
y#28t#29='
T
f
#28t,d
0
#29#12
"#28t#29=y#28t#29,^=y#28t#29,'
T
f
#28t,d
0
#29
^
#12#28t,1#29
^
#12#28t#29=
^
#12#28t,1#29 + P#28t#29'
T
f
#28t,d
0
#29"#28t#29
Residual
"#28t#29=y#28t#29,^y#28t#29
=y#28t#29,y
m
#28t#29+y
m
#28t#29,^y#28t#29
=e#28t#29+#11#28t#29
What You ShouldKnow!
#0F The ideas
#7B How to make abstractions
#0F The concepts
#7B Notions of gain phase and passivity
#7B PR and SPR
#0F The key results
#7B The small gain theorem
#7B The passivity theorem
#7B The circle criterion
#0F Abilities
#7B Compute gain
#7B Determine passivity
#7B Apply to adaptive control
#0F Similarities between MRAS STR
c#0D K,J,#C5str#F6m and B,Wittenmark 6
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
Lecture 8 #7B The Input-OutputView
1,Introduction
#0F White Boxes and Black Boxes
#0F Input-output descriptions
#0F How to generalize from linearto
nonlinear?
2,The Small Gain Theorem #28SGT#29
#0F The notion of gain
#0F Examples
#0F The main result
3,The Passivity Theorem #28PT#29
#0F Passivity and phase
#0F Examples
#0F The Passivity Theorem
#0F Relations between SGT and PT
4,Applications to Adaptive Control
#0F The augmented error
#0F MRAS and STR
5,Conclusions
Introduction
#0F Black Boxes and White Boxes
#0F Input-Output Descriptions
#7B The Table
#7B Linear Systems
#7B Dualitybetween Signals and Systems
#0F Formalization of the input-output view
#7B Signals and signal spaces
#7B The notions of gain and phase
#7B The notion of passivity
#0F Stability
#7B Stability Concept BIBO
#7B Stability of a system!
#7B Stability criteria
#7B Extensions of Nyquist's theorem
#0F The Mathematical Framework
#7B Functional analysis
The Notion of Gain
Signal spaces
L
2
,kuk =
#10
R
1
,1
u
2
#28t#29dt
#111
2
L
1
,kuk = sup
0#14t#3C1
ju#28t#29j
Extended spaces
x
T
#28t#29=
#1A
x#28t#29 0#14t#14T
0 t#3ET
u2X
e
if x
T
2 X
The notion of gain #28=operatornorm#29
#0D#28S#29 = sup
u2X
e
kSuk
kuk
Gain smallest value #0D such that
kSuk#14#0D#28S#29kuk for all u 2 X
e
c#0D K,J,#C5str#F6m and B,Wittenmark 1
Examples
Linear systems with signals in L
2e
kyk#14max
!
jG#28i!#29j#01kuk
u
0
= sin!t
Linear Systems with signals in L
1
#0D#28G#29=
Z
1
0
jh#28#1C#29jd#1C
u
0
#28s#29=u
0
sign#28h#28t,s#29#29
Static nonlinear system
x
f(x)
f = g x
f =- g x
The SmallGain Theorem
Definition 1
A system is called bounded-input,bounded-
output #28BIBO#29 stable if the system has
bounded gain.
Theorem 1
Consider the system.
S
u ey
H
1
- H
2
Let #0D
1
and #0D
2
be the gains of the systems H
1
and H
2
,The closed-loop system is BIBO stable
if
#0D
1
#0D
2
#3C 1
and its gain is less than
#0D =
#0D
1
1,#0D
1
#0D
2
Passivity
#0F The idea
#7B Energy dissipation
#7B Capacitors,induktors,resistances
#7B Mass,spring,dashpot
#7B Circuit Theory
#7B Mechatronics
#0F Mathematical Formalization
#0F The Notion of Phase
#0F Examples
#0F Postive real linear system
#0F The passivity theorem
#0F Using passivity in system design
AFormal Statement
Definition 2
A system with input u and output y is passive
if
hyjui#150
The system is input strictly passive #28ISP#29 if
there exists "#3E0such that
hyjui#15"kuk
2
and output strictly passive #28OSP#29 if there exists
"#3E0such that
hyjui#15"kyk
2
Intuitively
#0F Think about u and v as voltage and
current orforce and velocity
#0F Causality?
c#0D K,J,#C5str#F6m and B,Wittenmark 2
The Notion of Phase
Let the signal space have an inner product
The phase for a given input u can then be
de#0Cned as
cos' =
hy jui
kukkyk
=
hHujui
kukkHuk
Passivity implies that the phase is in the range
,
#19
2
#14 ' #14
#19
2
Linear Time-invariantSystems
hyjui =
1
Z
0
y#28t#29u#28t#29 dt =
1
2#19
1
Z
,1
Y #28i!#29U#28,i!#29 d!
=
1
2#19
1
Z
,1
G#28i!#29U#28i!#29U#28,i!#29 d!
=
1
#19
1
Z
0
RefG#28i!#29gU#28i!#29U#28,i!#29 d!
Definition 3
A rational transfer function G with real
coe#0Ecients is positive real #28PR#29 if
ReG#28s#29 #15 0 for Res #15 0
A transfer function G is strictly positive real
#28SPR#29 if G#28s,"#29 is positive real for some real
"#3E0.
Characterizing Positive Real
Transfer Functions
Theorem 2
A rational transfer function G#28s#29 with real
coe#0Ecients is PR if and only if the following
conditions hold.
#0F #28i#29 The function has no poles in the right
half-plane.
#0F #28ii#29 If the function has poles on the
imaginary axis or at in#0Cnity,they are
simple poles with positive residues.
#0F #28iii#29 The real part of G is nonnegative
along the i! axis,that is,
Re#28G#28i!#29#29 #15 0
A transfer function is SPR if conditions #28i#29 and
#28iii#29 hold and if condition #28ii#29 is replaced by the
condition that G#28s#29 has no poles or zeros on
the imaginary axis.
Examples
Recall
hy jui =
1
#19
1
Z
0
RefG#28i!#29gU#28i!#29U#28,i!#29 d!
#0F Positive real PR
ReG#28i!#29 #15 0
#0F Input strictly passive ISP
ReG#28i!#29 #15 "#3E0
#0FOutput stricly passive OSP
ReG#28i!#29 #15 "jG#28i!#29j
2
G#28s#29=s+1SPR and ISP not OSP
G#28s#29=
1
s+1
SPR and OSP not ISP
G#28s#29=
s
2
+1
#28s+1#29
2
OSP and ISP not OSP
G#28s#29=
1
s
PR not SPR,OPS or ISP
c#0D K,J,#C5str#F6m and B,Wittenmark 3
Nonlinear Static Systems y = f#28u#29
hyjui =
Z
1
0
f#28u#28t#29#29u#28t#29 dt
#0F Passive if xf#28x#29 #15 0
#0F Input strictly passive #28ISP#29 if xf#28x#29 #15
#0Ejxj
2
#0F Output strictly passive if
xf#28x#29 #15 #0Ef
2
#28x#29
Geometric Interpretation
Example
#0F f#28x#29=x+x
3
input strictly passive
#0F f#28x#29=x=#281 + jxj#29 output strictly passive.
The Passivity Theorem
Theorem 3
Consider a system obtained by connecting two
systems H
1
and H
2
in a feedback loop
S
u ey
H
1
- H
2
Let H
1
be strictly output passive and H
2
be
passive,The closed-loop system is then BIBO
stable.
Passivity is an invariant under feedback.
Use of passivity in system design.
#0F Force control in robotics
#0F Remote manipulator
#0F How to think about the problem
Relations Between SmallGain and
Passivity Theorems
S
S
b)a)
I
d)
2
S
I S
c)
H
1
- I
- H
2
- I
I - H
2
S
1
- S
2
H
1
- H
2
I + H
1
()
-1
H
1
1
2
a ! b,H
1
! #28I + H
1
#29
,1
H
1;H
2
!I,H
2
b!c:S
i
=#28H
i
+I#29
,1
#28H
i
,I#29
Applicationsto adaptive control
#0F Structure of Adaptive Systems
#0F Apply passivity results
#0F Insight
#0F Modi#0Ced Algorithms
#0F PI adjustments
c#0D K,J,#C5str#F6m and B,Wittenmark 4
Adaptation of Feedforward Gain
q
S
–
Model
Process
+
S
–
Model
Process
+
y
y
e
e q
u
c
u
c
kG(s)
kG(s)
k
0
G(s)
k
0
G(s)
y
m
y
m
-
g
s
-
g
s
P
P
P
P
(a)
(b)
Redrawb#29as
0
G
e
S
q
S
-
H
g
s
u
c
q
0
P P
q - q
0
()u
c
-
Analysis
Lemma 1
Let r be a bounded square integrable function,
and let G#28s#29 be a transfer function that is
positive real,The system whose input-output
relation is given by
y = r#28G#28p#29ru#29
is then passive.
Example,PI adjustments
#12#28t#29=,#0D
1
u
c
#28t#29e#28t#29,#0D
2
t
Z
u
c
#28#1C#29e#28#1C#29d#1C
Explore the advantages of PI adjustments
analytically and by simulation!
A modi#0Cedalgorithm
Change
0
G
e
S
q
S
-
H
g
s
u
c
q
0
P P
q - q
0
()u
c
-
To
G
G
+
–
S
y
q
u
c
y
m
-
g
s
G
c
q
0
P
P
Make G
c
G SPR,Still a problem with pole
excess #3E 1.
The AugmentedError
Consider the error
e = G#28#12,#12
0
#29u
c
= G#28#12,#12
0
#29u
c
+#28#12,#12
0
#29Gu
c
,#28#12,#12
0
#29Gu
c
Introduce the augmented error
#0F = e + #11
where
#11 = G#28#12,#12
0
#29u
c
,#28#12,#12
0
#29Gu
c
= G#12u
c
,#12Gu
c
Notice that #11 is zero under stationary condi-
tions
Use the adaptation law
d#12
dt
=,#0D#0FG
2
u
c
Stabilitynow follows from the passivity
theorem
The idea can be extended to the general case,
details are messy.
c#0D K,J,#C5str#F6m and B,Wittenmark 5
A Minor Extension
Factor
G = G
1
G
2
where the transfer function G
1
is SPR,The
error e = y,y
m
can then be written as
e = G#28#12,#12
0
#29u
c
=#28G
1
G
2
#29#28#12,#12
0
#29u
c
= G
1
,
G
2
#28#12,#12
0
#29u
c
+#28#12,#12
0
#29G
2
u
c
,#28#12,#12
0
#29G
2
u
c
#01
Introduce
" = e+ #11
where #11 is the error augmentation de#0Cned by
#11 = G
1
#28#12,#12
0
#29G
2
u
c
,G#28#12,#12
0
#29u
c
= G
1
#28#12G
2
u
c
#29,G#12u
c
Use adaptation law
d#12
dt
=,#0D#0FG
2
u
c
MRAS with AugmentedError
–
+
Model
Process
y
–
e
S S
S
q
q
+
h
e
u
c
y
m
k
0
G
kG
-
g
s
G
1
G
2
u
c
P
P
P
Compare STR and MRAS
MRAS
d#12
dt
= #0D'
f
"
'
T
f
=,G
f
#28p#29grad
#12
"#28t#29
" = G
SPR
#28y,y
m
#29+#11=G
SPR
e+#11
Direct STR
y#28t#29='
T
f
#28t,d
0
#29#12
"#28t#29=y#28t#29,^=y#28t#29,'
T
f
#28t,d
0
#29
^
#12#28t,1#29
^
#12#28t#29=
^
#12#28t,1#29 + P#28t#29'
T
f
#28t,d
0
#29"#28t#29
Residual
"#28t#29=y#28t#29,^y#28t#29
=y#28t#29,y
m
#28t#29+y
m
#28t#29,^y#28t#29
=e#28t#29+#11#28t#29
What You ShouldKnow!
#0F The ideas
#7B How to make abstractions
#0F The concepts
#7B Notions of gain phase and passivity
#7B PR and SPR
#0F The key results
#7B The small gain theorem
#7B The passivity theorem
#7B The circle criterion
#0F Abilities
#7B Compute gain
#7B Determine passivity
#7B Apply to adaptive control
#0F Similarities between MRAS STR
c#0D K,J,#C5str#F6m and B,Wittenmark 6
