Lecture 9 – Analysis of Adaptive
Systems
Theme,Adaptive systems are nonlinear.
What can we say about their behavior?
1,Introduction
2,Nonlinear dynamics
3,Time variability
4,Stability of STR
5,Summary
Properties of Adaptive Systems
1,Introduction
2,Nonlinear Dynamics
3,Adaptation of Feedforward Gain
4,Stability of DSTR
5,Averaging
6,Applicatins of Averaging
7,Robustness
8,Conclusions
1,Introduction
Investigate a given system
– Stability
– Convergence
– Convergence rate
Improved algorithms
General principles
– Understand behavior
– Can difficulties occur?
– Unification
– Structures
– Identifiability
– Excitation
– Achievable performance
2,Nonlinear Dynamics
Structure of equations
What can be done?
– Equilibria
– Local properties
– Global properties
A simple example
Structural stability
ccirclebig K,J,?str?m and B,Wittenmark 1
A Typical Self-tuning Regulator
Process parameters
Controller
design
Estimation
Controller
Process
Controller
parameters
Reference
Input Output
Specification
Self-tuning regulator
Nonlinear system
What can we say apart from stability?
Notice two loops
– Fast feedback loop
– Slower parameter adjustment loop
Use difference in time scales in analy-
sis
Structure of Equations
Continuous time MRAS
dξ
dt
AH?Iξ BH?Iν
η
e
CH?Iξ DH?Iν
d
θdt γ
H?,ξIeH?,ξI
α?H?,ξI
T
H?,ξI
Alternative representation
d
θ dt γ
HG
ν
νIHG
eν
νI
α HG
ν
νI
T
G
ν
ν
Discrete time STR
ξHt 1I AH?IξHtI BH?IνHtI
ηHtI
eHtI
HtI
CH?IξHtI DH?IνHtI
θHt 1I
θHtI PHt 1I?HtIeHtI
PHt 1I PHtI?
PHtI?HtI?
T
HtIPHtI
λ?
T
HtIPHtI?HtI
Example
Process model
yHt 1I θyHtI uHtI
Controller
uHtI?
θHtIyHtI y
0
θHt 1I
θHtI γ
yHtI
yHt 1I?
θHtIyHtI?uHtI
α y
2
HtI
True system
yHt 1I θ
0
yHtI a uHtI
Example
Simplification
yHt 1I
θ
0
θHtI
yHtI a y
0
θHt 1I
θHtI γ
yHtI
θ
0
θHtI
yHtI a
α y
2
HtI
Equilibrium solutions
y y
0
θ θ
0
a
y
0
Local behavior
A
a
y
0
y
0
γ
a
α y
2
0
1?γ
y
2
0
α y
2
0
!
Characteristic equation
z
2
a
1
z a
2
where
a
1
ay
0
1 γ
y
2
0
α y
2
0
a
2
a
y
0
ccirclebig K,J,?str?m and B,Wittenmark 2
Local Analysis
Characteristic equation
z
2
a
1
z a
2
0
Stability conditions (Schur-Cohn)
(i) a
2
1
(ii) a
2
a
1
1 0
(iii) a
2
a
1
1 0
gives
(i)
a
y
0
1
(ii) γ 2
H1?a/y
0
IHα y
2
0
I
y
2
0
(iii) γ 0
g
y
0
a
- y
0
2(a + y
0
2
)
y
0
2
Global Properties - Stable
Equilibrium
Explore the properties at the boundaries
(home work)!
Simulation α 0.1,γ 0.1,θ
0
1.5,
y
0
1,and a 0.9,Unique stable
equilibrium
10 0 10 20
0
1
2
θ
y
Consider
θ and y equations separately
yHt 1I Hθ
0
θIyHtI a y
0
θHt 1I
1?γ
y
2
HtI
α y
2
HtI
θHtI γ
ayHtI
α y
2
HtI
Intuitive Discussion
yHt 1I Hθ
0
θIyHtI a y
0
θHt 1I
1?γ
y
2
HtI
α y
2
HtI
θHtI γ
ayHtI
α y
2
HtI
Thefrozenmotionfix
θ.
yHt 1I Hθ
0
θIyHtI a y
0
Equilibrium
y
a y
0
1
θ?θ
0
What is the character of the motion of y
for different
θ?
When is is stable?
When does it converge fast?
When is is oscillatory?
When is it monotone?
Global Properties - Unstable
Equilibrium
Choose θ
0
1,α 0.1,y
0
1,and
a 0.9,Equilibrium is y 1 and θ 1.9.
Stable if γ 0.22,Choose γ 0.5.
0 50 100 150 200
1
0
1
2
3
0 50 100 150 200
1
2
3
Time
Time
y
θ
Bursts
Explain intuitively!
ccirclebig K,J,?str?m and B,Wittenmark 3
Intuitive Discussion
yHt 1I Hθ
0
θIyHtI a y
0
θHt 1I
1?γ
y
2
HtI
α y
2
HtI
θHtI γ
ayHtI
α y
2
HtI
θ
θ?θ
0
Equilibrium
θ
1
y
Phase plane γ 0.5 a 0.9
1 0 1 2
1
2
3
θ
y
Chaos!!
Show the double pendulum!
Phase plane γ 0.5 a?1.1
2?1 0 1 2
0
1
2
3
4
3,Adaptation of Feedforward Gain
dy
dt
k
θHtIu
c
HtI?yHtI
Parameter adjustment MIT rule
d
θ
dt
γy
m
HtIeHtI?γy
m
HtIHyHtI?y
m
HtII
Complete system
d
dt
θ
y
0?γ y
m
HtI
ku
c
HtI?1
θ
y
γy
2
m
HtI
0
0 5 10 15 20
0
1
0 5 10 15 20
0
1
0 5 10 15 20
0
1
Time
Time
Time
(a)
θ
(b)
θ
(c)
θ
ccirclebig K,J,?str?m and B,Wittenmark 4
Linear Periodic Systems
Consider the system
dΦ
dt
AHtIΦ
with
AHt τI AHtI
Solution has the form
ΦHtI DHtIe
Ct
where DHtI DHt τI.
Stability condition,All eigenvalues of C less
than one in magnitude.
How to compute C?
Typical Stability Boundary
Adjustment of feedforward gain MIT rule
Process model
GHsI
1
s 1
Command signal
u
c
HtI sin ω t
100
0
01 2w
g
Stable
Summary
Adaptive systems are nonlinear!
Analysis
Equilibria
Local analysis - Linearization
Global behavior
– Difficult
– Behavior can be very complicated
Simulation
Approximations
Use special structure
More of this to follow
4,Stability of DSTR
Review of Algorithm
Properties of Estimator
Main Result
Discuss Assumptions
Disturbances
ccirclebig K,J,?str?m and B,Wittenmark 5
The Algorithm
Process model
A
Hq
1
IyHtI B
Hq
1
IuHt?dI
Desired response
A
m
Hq
1
IyHtI t
0
u
c
Ht?dI
Notice all process zeros cancelled Estimate
parameters of the model
A
o
A
m
yHt dI R
uHtI S
yHtI?
T
HtIθ
θHtI
θHt?1I
γ?Ht?dI
α?
T
Ht?dI?Ht?dI
eHtI
eHtI yHtI
T
Ht?dI
θHt?1I
Control law
R
uHtI
S
yHtI t
0
A
o
u
c
HtI
Main Result
Theorem 1
Assume that
A1,The time delay d is known.
A2,Upper bounds on the degrees of the
polynomials A
and B
are known.
A3,The polynomial B has all its zeros
inside the unit disc.
A4,The sign of b
0
r
0
is known.
Then
(i) The sequences QuHtIR and QyHtIR are
bounded
(ii) lim
t3‘
A
m
Hq
1
IyHtI?t
0
u
c
Ht?dI
0
Idea of Proof Disturbances
Process model
A
Hq
1
IyHtI B
Hq
1
IuHt?dI vHtI
Assume
Upper bounds on degrees
Minimum phase
Sign of b
0
known
Stability results (Egardt)
Conditional Updating
sup
t
t
R
1
A
o
A
m
vt c
Update only when
tet
2c
2?maxHb
0
/
b
0
,1I
Projection
Parameters bounded apriori,Modify
estimator to give estimates inside prior
bounds.
ccirclebig K,J,?str?m and B,Wittenmark 6
Summary
1,Nonlinear Nature of Adaptive Systems
Equilibria
Local properties
Global properties
Complicated behavior
2,Adaptive Feedforward
Much simpler
Linear but time-varying
Periodic case tractable
Also complicated,not as bad as 1
3,Direct STR
Use special properties
Assumptions required
4,Next we will exploit difference in time
scales
ccirclebig K,J,?str?m and B,Wittenmark 7