Properties of Adaptive Systems
1,Introduction
2,Nonlinear Dynamics
3,Adaptation of Feedforward Gain
4,Stability of DSTR
5,Averaging
6,Applications of Averaging
7,Robustness
8,Conclusions
Lecture 10 - Averaging
Lecture 11 - Robustness and
ConvergenceRate
1,The idea
2,Averaging theorems
3,Howtodoit?
4,Applications to adaptive control
#0F Performance
#0F Convergence rates
#0F Sensitivity to assumptions
5,A new look at MRAS
The Idea
#0F Separate Fast and Slow Motions
#0F The Origin of the Idea
#7B Classical Mechanics
#7B Numerical Analysis
#7B Nonlinear Oscillations
#7B Krylov and Boboliubov 1937
#7B Minorsky 1962
#0F Two Time Scales
#7B Slowparameters
#7B Fast states
#7B What did we really assume,Think!
TypicalResult
dx
dt
= f#28x;y;t#29
dy
dt
= #0Fg#28x;y;t#29
Assume y is constant and solve the fast
equation
dx
dt
= f#28x;y;t#29
Let the solution be #18#28t#29,The averaged equa-
tion is then
d#16y
dt
= #0Favgg#28#18#28t#29; #16y;t#29=#0FG#28#16y#29
Consider solutions such that
y#280#29 = #16y#280#29 = y
0
Then
jy#28t#29,#16y#28t#29j #3CK#0F
for 0 #14 t#3CT=#0F,If the averaged equation is
stable then y is also stable and the inequality
holds for 0 #14 t#3C1
c#0D K,J,#C5str#F6m and B,Wittenmark 1
Recipe
#0F Consider parameters constant when
analysing system
#0F Average fast motion when analysing the
estimate
#0F Many ways to take averages
Many Ways to Compute Averages
avg
n
f
#10
^
#12;#18#28
^
#12;t#29;t
#11o
=
1
T
Z
T
0
f
#10
^
#12;#18#28
^
#12;t#29;t
#11
dt
avg
n
f
#10
^
#12;#18#28
^
#12;t#29;t
#11o
= lim
T!1
Z
T
0
f
#10
^
#12;#18#28
^
#12;t#29;t
#11
dt
avg
n
f
#10
^
#12;#18#28
^
#12;t#29;t
#11o
= Ef
#10
^
#12;#18#28
^
#12;t#29;t
#11
Sinusoidal signals
V#28s#29=G
v
#28s#29U#28s#29
W#28s#29=G
w
#28s#29U#28s#29
If u = sin!t we have
avg#28vw#29
=
u
2
0
2
jG
v
#28i!#29jjG
w
#28i!#29jcos#28argG
v
#28i!#29,argG
w
#28i!#29#29
=
u
2
0
2
Re#28G
v
#28i!#29G
w
#28,i!#29#29
ATypicalSelf-tuningRegulator
Process parameters
Controller
design
Estimation
Controller
Process
Controller
parameters
Reference
Input Output
Specification
Self-tuning regulator
#0F Nonlinear system
#0F What can wesay apart from stability?
#0F Notice two loops
#7B Fast feedback loop
#7B Slower parameter adjustment loop
#0F Use di#0Berence in time scales in analysis
Structure of Adaptive Systems
Continuous time MRAS
d#18
dt
= A#28#23#29#18 + B#28#23#29#17
#11 =
#12
e
'
#13
= C#28#23#29#18 + D#28#23#29#17
d
^
#12
dt
= #0D
'#28#23;#18#29e#28#23;#18#29
#0B+'#28#23;#18#29
T
'#28#23;#18#29
Alternative representation
d
^
#12
dt
= #0D
#28G
'#17
#17#29#28G
e#17
#17#29
#0B +#28G
'#17
#17#29
T
G
'#17
#17
Discrete time STR
#18#28t+1#29=A#28#23#29#18#28t#29+B#28#23#29#17#28t#29
#11#28t#29=
#12
e#28t#29
'#28t#29
#13
=C#28#23#29#18#28t#29+D#28#23#29#17#28t#29
^
#12#28t+1#29=
^
#12#28t#29+P#28t+1#29'#28t#29e#28t#29
P#28t+1#29=P#28t#29,
P#28t#29'#28t#29'
T
#28t#29P#28t#29
#15+'
T
#28t#29P#28t#29'#28t#29
c#0D K,J,#C5str#F6m and B,Wittenmark 2
The Robustness Issue
#0F What is Robustness?
#7B How sensitive is the result to the
assumptions?
#7B What are the critical assumptions?
#7B What about the assumptions in the
stabilityproof
#7B Other #0Celds
#0F Adaptation of Feedforward Gain
#7B Lyapunov versus MIT once more
#0F A #0Crst order system
MRAS with MIT and Lyapunov Rule
#0F What do we know?
#0F What happens when G 6= G
0
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)
d
^
#12
dt
=,#0Dy
m
e #28MIT#29
d
^
#12
dt
=,#0Du
c
e #28SPR#29
Analysis
d
^
#12
dt
=,#0Dy
m
e #28MIT#29
d
^
#12
dt
=,#0Du
c
e #28SPR#29
e = y,y
m
= kG#28p#29
#10
^
#12#28t#29u
c
#28t#29
#11
,k
0
G
m
#28p#29u
c
#28t#29
Hence
d
^
#12
dt
+#0D#28k
0
G
m
u
c
#29kG#28
^
#12u
c
#29=#0Dk
0
G
m
u
c
d
^
#12
dt
+ #0Du
c
kk
0
G#28
^
#12u
c
#29=#0Dk
0
G
m
u
c
Averaged equations
d
#16
#12
dt
+#0D
#16
#12kk
0
avgf#28G
m
u
c
#29#28Gu
c
#29g = #0Dk
2
0
avgf#28G
m
u
c
#29
2
g
d
#16
#12
dt
+ #0D
#16
#12kavgfu
c
#28Gu
c
#29g = #0Dk
0
avgfu
c
#28G
m
u
c
#29g
The equilibrium parameters are
#16
#12
MIT
=
k
0
k
avgf#28G
m
u
c
#29
2
g
avgf#28G
m
u
c
#29#28Gu
c
#29g
#16
#12
SPR
=
k
0
k
avgfu
c
#28G
m
u
c
#29g
avgfu
c
#28Gu
c
#29g
Analysis
d
#16
#12
dt
+#0D
#16
#12kk
0
avgf#28G
m
u
c
#29#28Gu
c
#29g = #0Dk
2
0
avgf#28G
m
u
c
#29
2
g
d
#16
#12
dt
+ #0D
#16
#12kavgfu
c
#28Gu
c
#29g = #0Dk
0
avgfu
c
#28G
m
u
c
#29g
The equilibrium parameters are
#16
#12
MIT
=
k
0
k
avgf#28G
m
u
c
#29
2
g
avgf#28G
m
u
c
#29#28Gu
c
#29g
#16
#12
SPR
=
k
0
k
avgfu
c
#28G
m
u
c
#29g
avgfu
c
#28Gu
c
#29g
#0Davgf#28G
m
u
c
#29#28Gu
c
#29g #3E 0 #28MIT#29
#0Davgfu
c
#28Gu
c
#29g #3E 0 #28SPR#29
c#0D K,J,#C5str#F6m and B,Wittenmark 3
An Example
G
m
#28s#29=
a
s+a
G#28s#29=
ab
#28s + a#29#28s + b#29
Stability conditions
#16
#12
MIT
=
k
0
k
b
2
+!
2
b
2
#16
#12
SPR
=
k
0
k
a#28b
2
+ !
2
#29
b#28ab,!
2
#29
!#3C
p
ab
0 100 300 500
0.0
0.5
1.0
0 100 300 500
0.0
0.5
1.0
0 100 300 500
0
5
10
0 100 300 500
0
5
10
Time Time
Time Time
#28a#29 #28b#29
#28c#29 #28d#29
^
#12
^
#12
^
#12
^
#12
Analysis of MRAS for First Order
System
#0F Equilibrium conditions
#0F Local stability
#0F Convergence rate
#0F Robustness
The System
Design Model
G#28s#29=
b
s+a
Desired Response
G
m
#28s#29=
b
m
s+a
m
Controller
u#28t#29=#12
1
u
c
,#12
2
y
Lyapunov design
-
S
P
+
e
u y
S
P
P
P
-
+
u
c
G
m
(s)
G(s)
q
1
q
2
g
s
-
g
s
Analysis
d
^
#12
1
dt
=,#0Du
c
e
d
^
#12
2
dt
= #0Dye
e = y,y
m
y = G#28p#29u
y
m
= G
m
#28p#29u
c
u=
^
#12
1
u
c
,
^
#12
2
y
c#0D K,J,#C5str#F6m and B,Wittenmark 4
EquilibriumValues
Closed loop transfer function
G
c
=
^
#12
1
G
1+
^
#12
2
G
Control error
e#28t#29=y#28t#29,y
m
#28t#29=#28G
c
#28p#29,G
m
#28p#29#29u
c
#28t#29
Sinusoidal signals
^
#12
0
1
G#28i!#29=
^
#12
0
2
G
m
#28i!#29G#28i!#29+G
m
#28i!#29
Hence
^
#12
0
1
=
Imf1=G#28i!#29g
Imf1=G
m
#28i!#29g
^
#12
0
2
=,
ImfG
m
#28i!#29=G#28i!#29g
ImG
m
#28i!#29
Discuss high and low !,signs etc.
Averaging Analysis
G
e#17
=
^
#12
1
G
1+
^
#12
2
G
,G
m
G
T
'#17
=
#12
,1
^
#12
1
G
1+
^
#12
2
G
#13
d
#16
#12
1
dt
=,
#0Du
2
0
2
Re
#08
F#28!;
#16
#12
1;
#16
#12
2
#29
#09
d
#16
#12
2
dt
=
#0Du
2
0
2
Re
#1A
F#28!;
#16
#12
1;
#16
#12
2
#29
#16
#12
1
G#28,i!#29
1+
#16
#12
2
G#28,i!#29
#1B
F#28!;
#16
#12
1;
#16
#12
2
#29=
#16
#12
1
G#28i!#29
#281 +
#16
#12
2
G#28i!#29
,G
m
#28i!#29
Accuracy of Averaging
Consider the case a =1,b=2,a
m
=b
m
=3,
#0D=1,u
c
= sin!t
G#28s#29=
2
s+1
G
M
#28s#29=
2
s+2
0 2040608010
0
1
Time
Local Stability
Linearize the averaged equations
A =
#0Du
2
0
jG
m
j
2
#16
#12
0
1
#12
,cos#12
m
jG
m
jcos2#12
m
jG
m
j,jG
m
j
2
cos#12
m
#13
where #12
m
= arctan#28!=a
m
#29
#15
2
+#0B#15#281 + cos
2
#12
m
#29+#0B
2
sin
2
#12
m
=0
where
#0B =
#0Du
2
0
a
m
b
2#28a
2
m
+!
2
#29
Discuss convergence rate as a function of
frequency.
c#0D K,J,#C5str#F6m and B,Wittenmark 5
UnmodeledDynamics
Review basic assumptions,Robustness.
G#28s#29=
458
#28s + 1#29#28s
2
+30s+229#29
#0F What happens?
#7B Step commands
#7B Sinusoidal command signals
#0F Analysis
#0F Intuitive insight
Step Commands and Sinusoidal
Measurement Errors
0 10203040
0.2
0.0
0.2
0.4
0.6
Time
^
#12
1
^
#12
1
#28no noise#29
^
#12
2
^
#12
2
#28no noise#29
Hint,Number of parameters and excitation.
SinusoidalCommands
Command signal sinusoidal with frequencies
#28a#29 ! =1; #28b#29 ! =3; #28c#29 ! =6; #28d#29 ! =20.
0 50 100 150 200
0
1
0 50 100 150 200
0
2
4
0 50 100 150 200
0
1
2
0 50 100 150 200
0
5
10
15
Time Time
Time Time
#28a#29
^
#12
1
^
#12
2
#28b#29
^
#12
2
^
#12
1
#28c#29
^
#12
2
^
#12
1
#28d#29
^
#12
1
^
#12
2
Analysis - Step Commands
Closed loop characteristic equation for #0Cxed
parameters
#28s + 1#29#28s
2
+30s+ 229#29 + 458#12
2
=0
or
s
3
+31s
2
+ 259s + 229 + 458#12
2
=0
Stable if,0:5 #3C
^
#12
2
#3C 17:03 = #12
stab
2
d
#16
#12
1
dt
=,
#0Du
2
0
2
#12
#16
#12
1
G#280#29
1+
#16
#12
2
G#280#29
,G
m
#280#29
#13
d
#16
#12
2
dt
=
#0Du
2
0
2
#16
#12
1
G#280#29
1+
#16
#12
2
G#280#29
#12
#16
#12
1
G#280#29
1+
#16
#12
2
G#280#29
,G
m
#280#29
#13
Equilibrium values
#16
#12
2
=
#16
#12
1
G
m
#280#29
,
1
G#280#29
c#0D K,J,#C5str#F6m and B,Wittenmark 6
Local and Global Behavior
Linearize around the equilibrium set
dx
dt
=
#0Du
2
0
2#12
0
1
#12
,1 1
1,1
#13
x
Stability boundary
q
2
q
1
Global behavior
20?10 0 10 20
0
10
20
20?10 0 10 20
0
10
20
#28a#29
^
#12
2
^
#12
1
#28b#29
^
#12
2
^
#12
1
Analysis of Global Behavior
d
#16
#12
1
dt
=,
#0Du
2
0
2
#12
#16
#12
1
G#280#29
1+
#16
#12
2
G#280#29
,G
m
#280#29
#13
d
#16
#12
2
dt
=
#0Du
2
0
2
#16
#12
1
G#280#29
1+
#16
#12
2
G#280#29
#12
#16
#12
1
G#280#29
1+
#16
#12
2
G#280#29
,G
m
#280#29
#13
d
#16
#12
2
d
#16
#12
1
=,
G#280#29
#16
#12
1
1+
#16
#12
2
G#280#29
This di#0Berential equation has the solution
#16
#12
2
2
+
2
G#280#29
#16
#12
2
+
#16
#12
2
1
= const
Measurement Noise
0 0.2 0.4 0.6 0.8 1
0.2
0.0
0.2
0.4
^
#12
2
^
#12
1
#28a#29
#28b#29
Noise makes the system drift along the
equilibrium line!
Summary
Input Exact Model Unmodeled
Dynamics
Step Equilibrium is Stability lost
half line for some IC
Step & Drift along Drift along EQ
noise equilibrium set until unstable
Sine Correct Equilibrium
function equilibrium input dependent
Unstable for
high frequencies
c#0D K,J,#C5str#F6m and B,Wittenmark 7
Robust Adaptive Control
#0F What do the di#0Eculties depend on?
#0F How can the di#0Eculties be avoided?
#0F Excitation
#0F Dead zones
#0F Leakage
d
^
#12
dt
= #0D
'e
#0B + '
T
'
+ #0B
1
#28#12
0
,
^
#12#29
d
^
#12
dt
= #0D
'e
#0B + '
T
'
+ #0B
1
jej#28#12
0
,
^
#12#29
#0F Normalization
Cr#28t#29 = max #28ju#28t#29j;jy#28t#29j#29
Normalized signals
~y =
y
r; ~u = ur; ~v =
v
r
Stochastic Self-tuner
Process
A
#03
#28q
,1
#29y#28t#29=B
#03
#28q
,1
#29u#28t,d#29+C
#03
#28q
,1
#29e#28t#29
Model
y#28t#29=R
#03
#28q
,1
#29u#28t,d#29+S
#03
#28q
,1
#29y#28t,d#29
Least squares parameter estimation
^
#12#28t#29=
^
#12#28t,1#29+ #0D#28t#29R#28t#29
,1
'#28t,d#29e#28t#29
e#28t#29=
#10
y#28t#29,'
T
#28t,d#29
^
#12#28t,1#29
#11
R#28t#29=R#28t,1#29+#0D#28t#29
,
'#28t,d#29'
T
#28t,d#29,R#28t,1#29
#01
Control
u#28t#29=,
^
S
#03
#28q
,1
#29
^
R
#03
#28q
,1
#29
y#28t#29
This implies '#28t#29
T
^
#12#28t#29=0Notice
P#28t#29=#0D#28t#29R#28t#29
,1
where #0D#28t#29=1=t.
Stochastic Averaging
Approximate
y#28t#29='
T
#28t,d#29#12#19'
T
#28t,d;
#16
#12#29
#16
#12
The equations then becomes
#16
#12#28t#29=
#16
#12#28t,1#29 +#0D#28t#29
#16
R#28t#29
,1
f#28
#16
#12#29
#16
R#28t#29=
#16
R#28t,1#29+#0D#28t#29
,
G#28
#16
#12#29,
#16
R#28t,1#29
#01
f#28
#16
#12#29=E
#08
'#28t,d;
#16
#12#29
,
y#28t#29,'
T
#28t,d;
#16
#12#29
#16
#12
#01#09
G#28
#16
#12#29=E
#08
'#28t,d;
#16
#12#29'
T
#28t,d;
#16
#12#29
#09
De#0Cne #01#1C =
P
t
0
k=t
#0D#28k#29,then
#16
#12#28t
0
#29=
#16
#12#28t#29+#01#1C
#16
R#28t#29
,1
f
,
#16
#12#28t#29
#01
#16
R#28t
0
#29=
#16
R#28t#29+#01#1C
,
G
,
#16
#12#28t#29
#01
,
#16
R#28t#29
#01
Change time scale t = #1C and t
0
= t +#01#1C
d
#16
#12
d#1C
=
#16
R#28#1C#29
,1
f
,
#16
#12#28#1C#29
#01
d
#16
R
d#1C
= G
,
#16
#12#28#1C#29
#01
,
#16
R#28#1C#29
An example
Process
y#28t#29+ay#28t,1#29 = u#28t,1#29+ bu#28t,2#29+ e#28t#29+ce#28t,1#29
Parameters,a =,0:99,b =0:5,and c =,0:7
Estimation model
y#28t#29=u#28t,1#29 +r
1
u#28t,2#29+ s
0
y#28t,1#29
Controller
u#28t#29=,s
0
y#28t#29,r
1
u#28t,1#29
0123
1
0
1
0123
1
0
1
#28a#29 #16r
1
#16s
0
#28b#29 ^r
1
^s
0
c#0D K,J,#C5str#F6m and B,Wittenmark 8
Local Instabilities
Process
y#28t#29,1:6y#28t,1#29,0:75y#28t,2#29 =
u#28t,1#29+ u#28t,2#29+ 0:9u#28t,3#29
e#28t#29+1:5e#28t,1#29+0:75e#28t,2#29
The B-polynomial has zeros at
z
1;2
=,0:50#060:81i
and
C#28z
1;2
#29=,0:40#060:40i
0 500 1000 1500 2000
1
0
1
2
3
Time
^s
0
^r
1
^r
2
^s
1
Conclusions
#0F Averaging a useful tool
#0F Captures a key feature of adaptive systems
#7B Parameters change slowly
#0F Simpli#0Ccation
#0F Problem split in two
#7B Linear system
#12 = constant
#7B Nonliear equation of lower dimension
d#12
dt
= f#28#12;t#29
#0F Good insight into robustness issues
#0F Good complement to simulation
#0F But still di#0Ecult
c#0D K,J,#C5str#F6m and B,Wittenmark 9