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Organization of the course:
• Meetings in A206 10:15-12:00March: 26, 30, 31;
April: 13, 14, 16, 20, 21, 23, 27, 28;May: 11, 12, 14, 25, 26, 28;
June: 1, 2, 4
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 1/20
Organization of the course:
• Meetings in A206 10:15-12:00March: 26, 30, 31;
April: 13, 14, 16, 20, 21, 23, 27, 28;May: 11, 12, 14, 25, 26, 28;
June: 1, 2, 4• Goals: Solving problems, familiarizing with software,
discussing homework from time to time
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 1/20
Organization of the course:
• Meetings in A206 10:15-12:00March: 26, 30, 31;
April: 13, 14, 16, 20, 21, 23, 27, 28;May: 11, 12, 14, 25, 26, 28;
June: 1, 2, 4• Goals: Solving problems, familiarizing with software,
discussing homework from time to time
• Homeworks to be assigned regularly
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 1/20
Organization of the course:
• Meetings in A206 10:15-12:00March: 26, 30, 31;
April: 13, 14, 16, 20, 21, 23, 27, 28;May: 11, 12, 14, 25, 26, 28;
June: 1, 2, 4• Goals: Solving problems, familiarizing with software,
discussing homework from time to time
• Homeworks to be assigned regularly
• Project at the end of the course.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 1/20
Organization of the course:
• Meetings in A206 10:15-12:00March: 26, 30, 31;
April: 13, 14, 16, 20, 21, 23, 27, 28;May: 11, 12, 14, 25, 26, 28;
June: 1, 2, 4• Goals: Solving problems, familiarizing with software,
discussing homework from time to time
• Homeworks to be assigned regularly
• Project at the end of the course.
• The textbook:
K.J. Åström and B. WittenmarkAdaptive Control
2nd Edition, Addison-Wesley Publishing Company, 1995
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 1/20
Lecture 1 topics:
• Discussion on difficulties and approaches for designingcontrollers for uncertain systems
◦ What are control design techniques you know?
◦ Are you ready to apply these techniques?
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 2/20
Lecture 1 topics:
• Discussion on difficulties and approaches for designingcontrollers for uncertain systems
◦ What are control design techniques you know?
◦ Are you ready to apply these techniques?
• Discussion on modeling of controlled mechanical systems
◦ How to build a model and formulate a control designproblem?
◦ What is a good model?
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 2/20
Lecture 1 topics:
• Discussion on difficulties and approaches for designingcontrollers for uncertain systems
◦ What are control design techniques you know?
◦ Are you ready to apply these techniques?
• Discussion on modeling of controlled mechanical systems
◦ How to build a model and formulate a control designproblem?
◦ What is a good model?
• Motivation for the problem formulation of adaptive controldesign
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 2/20
How to design a controller for this system?
The arm of the Furuta Pendulum with the bob at its end
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 3/20
Build a Model for the Dynamics
According to Newton’s law
α · φ = τ
where• φ is the angle of the arm,• α is the moment of inertia of the ram with the bob,• τ is the external torque
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 4/20
Build a Model for the Dynamics
According to Newton’s law
α · φ = τ
where• φ is the angle of the arm,• α is the moment of inertia of the ram with the bob,• τ is the external torque
The external torque τ consists of various components
τ = u − Ffriction(·) − Fdist.(·)
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 4/20
Build a Model for the Dynamics
According to Newton’s law
α · φ = τ
where• φ is the angle of the arm,• α is the moment of inertia of the ram with the bob,• τ is the external torque
Neglecting dynamics of voltage v and current i, we obtain
τ = KDC · v − Ffriction(·) − Fdist.(·)
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 4/20
Preliminary Model for Friction:
Models for friction torques (forces) can be classified by• materials and geometry in contact
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 5/20
Preliminary Model for Friction:
Models for friction torques (forces) can be classified by• materials and geometry in contact• complexity for computation
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 5/20
Preliminary Model for Friction:
Models for friction torques (forces) can be classified by• materials and geometry in contact• complexity for computation• purpose to use, for instance,
◦ to understand tire-road interaction as a function ofpattern, rubber properties etc of tire
◦ to use friction approximation for control◦ . . .
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 5/20
Preliminary Model for Friction:
Models for friction torques (forces) can be classified by• materials and geometry in contact• complexity for computation• purpose to use, for instance,
◦ to understand tire-road interaction as a function ofpattern, rubber properties etc of tire
◦ to use friction approximation for control◦ . . .
The Coulomb friction model is often used for feedforwardcompensation of friction
FC
(
dφdt
)
=
F+, dφ
dt> 0
F−, dφdt
< 0
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 5/20
Preliminary Model for Friction (Cont’d):
To identify parameters of the linear part of the model (α, KDC),we need to compensate nonlinearities.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 6/20
Preliminary Model for Friction (Cont’d):
To identify parameters of the linear part of the model (α, KDC),we need to compensate nonlinearities.
⇓Compensating friction based on the Coulomb friction model can
help us to reduce nonlinearity due to friction.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 6/20
Preliminary Model for Friction (Cont’d):
To identify parameters of the linear part of the model (α, KDC),we need to compensate nonlinearities.
⇓Compensating friction based on the Coulomb friction model can
help us to reduce nonlinearity due to friction.
Important Observation: we can estimate values v+, v−
KDC · v+ = F+, KDC · v− = F−
without knowing KDC . Apply a ramp voltage, and record theconstant values v+ and v−, when the arm starts moving.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 6/20
Preliminary Model for Friction (Cont’d):
To identify parameters of the linear part of the model (α, KDC),we need to compensate nonlinearities.
⇓Compensating friction based on the Coulomb friction model can
help us to reduce nonlinearity due to friction.
Important Observation: we can estimate values v+, v−
KDC · v+ = F+, KDC · v− = F−
without knowing KDC . Apply a ramp voltage, and record theconstant values v+ and v−, when the arm starts moving.
Important Observation: Only these values v+, v− are neededfor feedforward compensation of the Coulomb friction.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 6/20
Preliminary Model for Friction (Cont’d):
The experiments show that these values are
vCoulomb ≈
0.032, if ddtφ > 0
−0.033, if ddtφ < 0
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 7/20
Preliminary Model for Friction (Cont’d):
The experiments show that these values are
vCoulomb ≈
0.032, if ddtφ > 0
−0.033, if ddtφ < 0
Adding to control a feedforward signal vCoulomb
v = vnominal + vCoulomb,
removes strong nonlinearity, making the dynamics close to linear
α · φ(t) = KDC · v − Ffriction(·) − Fdist.(·)
= KDC · vnominal(t) − β · φ(t) + e(t, ·)
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 7/20
Still Some Problems!
We have to choose the input signal vnominal for the system
αφ(t) = KDC vnominal(t)−βφ(t)+e(t), y(t) = φ(t)+em(t)
Suggestion?
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 8/20
Still Some Problems!
We have to choose a good input signal vnominal for the system
αφ(t) = KDC vnominal(t)−βφ(t)+e(t), y(t) = φ(t)+em(t)
Suggestion? The system is unstable!
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 8/20
Still Some Problems!
We have to choose a good input signal vnominal for the system
αφ(t) = KDC vnominal(t)−βφ(t)+e(t), y(t) = φ(t)+em(t)
Suggestion? The system is unstable!
A possible scheme is• introducing a feedback action to stabilize the system,• using a reference signal with a particular pass-band
characteristic.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 8/20
Still Some Problems!
We have to choose a good input signal vnominal for the system
αφ(t) = KDC vnominal(t)−βφ(t)+e(t), y(t) = φ(t)+em(t)
Suggestion? The system is unstable!
A possible scheme is• introducing a feedback action to stabilize the system,• using a reference signal with a particular pass-band
characteristic.
The simplest choice is the proportional feedback
vnominal(t) = Kp
(
r(t) − y(t))
= Kp
(
r(t) − {φ(t) + em(t)})
where r(t) is the reference signal.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 8/20
Model Used in Experiments:
Combining
αφ(t) = KDC vnominal(t)−βφ(t)+e(t), y(t) = φ(t)+em(t)
with
vnominal(t) = Kp
(
r(t) − y(t))
= Kp
(
r(t) − {φ(t) + em(t)})
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 9/20
Model Used in Experiments:
Combining
αφ(t) = KDC vnominal(t)−βφ(t)+e(t), y(t) = φ(t)+em(t)
with
vnominal(t) = Kp
(
r(t) − y(t))
= Kp
(
r(t) − {φ(t) + em(t)})
we get the input-output description (r → y) with stable model
α·φ(t)+β·φ(t)+KDC ·Kp·φ(t) =
= KDC · Kp · r(t) +{
e(t) + Kp · em(t)}
y(t) = φ(t) + em(t)
It will be used in experiment!
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 9/20
10 20 30 40 50 60 70−6
−4
−2
0
2
4
6
φ(t)
, (ou
tput
)
RECORDED DATA (EXPERIMENT 1)
10 20 30 40 50 60 70
−0.05
0
0.05
Time
r(t)
, (re
fere
nce)
Experiment 1: Kp = 0.05 and r(t) is the sum of sinusoids
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 10/20
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2x 10
5
FF
T o
f φ(t
), (
outp
ut)
0 2 4 6 8 10 12 14 16 18 200
200
400
600
800
1000
FF
T o
f r(t
), (
refe
renc
e)
frequency [rad/sec]
Experiment 1: Kp = 0.05 and r(t) is the sum of sinusoids
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 10/20
10 20 30 40 50 60 70−6
−4
−2
0
2
4
6
φ(t)
, (ou
tput
)
RECORDED DATA (EXPERIMENT 2)
10 20 30 40 50 60 70
−0.05
0
0.05
Time
r(t)
, (re
fere
nce)
Experiment 2: Kp = 0.1 and r(t) is the sum of sinusoids
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 10/20
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2x 10
5
FF
T o
f φ(t
), (
outp
ut)
0 2 4 6 8 10 12 14 16 18 200
200
400
600
800
1000
FF
T o
f r(t
), (
refe
renc
e)
frequency [rad/sec]
Experiment 2: Kp = 0.1 and r(t) is the sum of sinusoids
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 10/20
10 20 30 40 50 60 70−6
−4
−2
0
2
4
6
φ(t)
, (ou
tput
)
RECORDED DATA (EXPERIMENT 3)
10 20 30 40 50 60 70
−0.05
0
0.05
Time
r(t)
, (re
fere
nce)
Experiment 3: Kp = 0.15 and r(t) is the sum of sinusoids
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 10/20
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2x 10
5
FF
T o
f φ(t
), (
outp
ut)
0 2 4 6 8 10 12 14 16 18 200
200
400
600
800
1000
FF
T o
f r(t
), (
refe
renc
e)
frequency [rad/sec]
Experiment 3: Kp = 0.15 and r(t) is the sum of sinusoids
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 10/20
Impulse Response for Non-parametric Identification:
If the system is linear and the input signal is impulse
r(t) ={
r(0), r(ts), r(2ts), . . . , r(nts), . . .}
={
ρ, 0, 0, 0, . . .}
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 11/20
Impulse Response for Non-parametric Identification:
If the system is linear and the input signal is impulse
r(t) ={
r(0), r(ts), r(2ts), . . . , r(nts), . . .}
={
ρ, 0, 0, 0, . . .}
then the output is
y (t = k · ts) =
+∞∑
i=1
g(i·ts)r(t−i·ts)+e(t)
= g(ts)r(t − ts)+g(2ts)r(t − 2ts)+. . .+g(kts)r(t − kts)+. . .
= g(ts) · 0 + g(2ts) · 0 + · · · + g(kts) · ρ + · · · + e(t)
= g(kts) · ρ + e(t)
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 11/20
Impulse Response for Non-parametric Identification:
If the system is linear and the input signal is impulse
r(t) ={
r(0), r(ts), r(2ts), . . . , r(nts), . . .}
={
ρ, 0, 0, 0, . . .}
then the output is
y (t = k · ts) =
+∞∑
i=1
g(i·ts)r(t−i·ts)+e(t)
= g(ts)r(t − ts)+g(2ts)r(t − 2ts)+. . .+g(kts)r(t − kts)+. . .
= g(ts) · 0 + g(2ts) · 0 + · · · + g(kts) · ρ + · · · + e(t)
= g(kts) · ρ + e(t)
Estimates{
g(kts)}
used for guessing delay, gain, stability. . .
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 11/20
Spectral Analysis for Non-parametric Identification:Computing fast Fourier transforms
YN(ω) =1
√N
N∑
t=1
y(t)ejωt, RN(ω) =1
√N
N∑
t=1
r(t)ejωt
can be used for estimation the transfer function of the system
GN(ejω) = YN(ω)/RN(ω)[
≈ G(ejω)]
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 12/20
Spectral Analysis for Non-parametric Identification:Computing fast Fourier transforms
YN(ω) =1
√N
N∑
t=1
y(t)ejωt, RN(ω) =1
√N
N∑
t=1
r(t)ejωt
can be used for estimation the transfer function of the system
GN(ejω) = YN(ω)/RN(ω)[
≈ G(ejω)]
Such estimate can be computed from input and cross spectrum
ΦNyr(ω) =
∫ π
−π
W (ξ − ω)YN(ξ)RN(ξ)dξ
ΦNr (ω) =
∫ π
−π
W (ξ − ω)RN(ξ)RN(ξ)dξ
asGN(ejω) =
ΦNyr(ω)
ΦNr (ω)
[
≈ G(ejω)]
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 12/20
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2x 10
5
FF
T o
f φ(t
), (
outp
ut)
0 2 4 6 8 10 12 14 16 18 200
200
400
600
800
1000
FF
T o
f r(t
), (
refe
renc
e)
frequency [rad/sec]
Experiment 1: Kp = 0.05 and r(t) is the sum of sinusoids
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 13/20
100
101
100
101
102
103
Am
plitu
de
From u1 to y1
100
101
−300
−250
−200
−150
−100
−50
0
Pha
se (
degr
ees)
Frequency (rad/s)
g_spa15g_spa25g_spa50
g_spa15g_spa25g_spa50
Experiment 1 (Kp = 0.05), we expect a pick around 2.5-3 rad/s
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 13/20
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2x 10
5
FF
T o
f φ(t
), (
outp
ut)
0 2 4 6 8 10 12 14 16 18 200
200
400
600
800
1000
FF
T o
f r(t
), (
refe
renc
e)
frequency [rad/sec]
Experiment 2: Kp = 0.1 and r(t) is the sum of sinusoids
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 13/20
100
101
101
102
103
Am
plitu
de
From u1 to y1
100
101
−300
−200
−100
0
100
200
Pha
se (
degr
ees)
Frequency (rad/s)
g_spa15g_spa25g_spa50
g_spa15g_spa25g_spa50
Experiment 2 (Kp = 0.1), we expect a pick around 3.5-4 rad/s
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 13/20
Prediction Error Method:
Both the model of the system obtained from basic principles andthe spectral analysis of the data suggest• the order of the model equals to 2• delay in response is not substantial
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 14/20
Prediction Error Method:
Both the model of the system obtained from basic principles andthe spectral analysis of the data suggest• the order of the model equals to 2• delay in response is not substantial
PEM computes an estimate
y(t) = G(s)r(t), G(s) =b1s + b0
s2 + a1s + a0
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 14/20
Prediction Error Method:
Both the model of the system obtained from basic principles andthe spectral analysis of the data suggest• the order of the model equals to 2• delay in response is not substantial
PEM computes an estimate
y(t) = G(s)r(t), G(s) =b1s + b0
s2 + a1s + a0
for the 2nd order system
α·φ(t)+β·φ(t)+KDC ·Kp·φ(t) =
= KDC · Kp · r(t) +{
e(t) + Kp · em(t)}
y(t) = φ(t) + em(t)
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 14/20
Prediction Error Method:
Both the model of the system obtained from basic principles andthe spectral analysis of the data suggest• the order of the model equals to 2• delay in response is not substantial
PEM computes an estimate
y(t) = G(s)r(t), G(s) =b1s + b0
s2 + a1s + a0
for the 2nd order system y(t) = φ(t) + em(t),
α · φ(t)+β · φ(t)+KDC ·Kp ·φ(t) =
= KDC · Kp · r(t) +{
e(t) + Kp · em(t)}
Guesses:
b1 ≈ 0, b0 ≈ a0, a1 ≈ β
α, a0 ≈ KDCKp
α
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 14/20
Comments on PEM:
Prior to start searching for parameters, one should take intoaccount the following.• The spectra of data was within interval [2, 6] [rad/sec] ⇒
an estimate for b0 of our model
G(s) =b1s + b0
s2 + a1s + a0
will be unreliable and of a large variance,
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 15/20
Comments on PEM:
Prior to start searching for parameters, one should take intoaccount the following.• The spectra of data was within interval [2, 6] [rad/sec] ⇒
an estimate for b0 of our model
G(s) =b1s + b0
s2 + a1s + a0
will be unreliable and of a large variance,
• The result of estimation might be an unstable system. Theviscous friction in the system is small; hence, one shouldexpect to have two resonance poles close to imaginary axis,which can be identified wrongly.
An appropriate projection ensuring stability for an estimatedsystem should be implemented in such a case.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 15/20
Results of PEM:
Two guesses from the first principle modeling are wrong for allthe estimated models:• the estimate for b1 differs from zero;• the estimate for b0 is 10 times bigger that the estimate of a0.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 16/20
Results of PEM:
Two guesses from the first principle modeling are wrong for allthe estimated models:• the estimate for b1 differs from zero;• the estimate for b0 is 10 times bigger that the estimate of a0.
Estimates fora0
Kp
=KDC
αare close to each other
Experiment 1: a0/Kp = 121.37, σ = 1.15
Experiment 2: a0/Kp = 126.16, σ = 0.347
Experiment 3: a0/Kp = 124.69, σ = 0.236
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 16/20
Results of PEM:
Two guesses from the first principle modeling are wrong for allthe estimated models:• the estimate for b1 differs from zero;• the estimate for b0 is 10 times bigger that the estimate of a0.
Estimates fora0
Kp
=KDC
αare close to each other
Experiment 1: a0/Kp = 121.37, σ = 1.15
Experiment 2: a0/Kp = 126.16, σ = 0.347
Experiment 3: a0/Kp = 124.69, σ = 0.236
These mean values and their variances are used to obtainKDC
α≈ 124.86, σ = 0.125
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 16/20
What is Left?
• Have we found the DC-gain KDC and the ineartia α? NO!
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 17/20
What is Left?
• Have we found the DC-gain KDC and the ineartia α? NO!
• We have estimated that
KDC
α≈ 124.86
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 17/20
What is Left?
• Have we found the DC-gain KDC and the ineartia α? NO!
• We have estimated that
KDC
α≈ 124.86
• How to obtain estimates for KDC and α?
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 17/20
What is Left?
• Have we found the DC-gain KDC and the ineartia α? NO!
• We have estimated that
KDC
α≈ 124.86
• How to obtain estimates for KDC and α?
• Suggestion: Change the inertia α by adding known value!
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 17/20
What is Left?
• Have we found the DC-gain KDC and the ineartia α? NO!
• We have estimated that
KDC
α≈ 124.86
• How to obtain estimates for KDC and α?
• Suggestion: Change the inertia α by adding known value!
• Denote it by Ja, then the same procedure will result inestimation of
KDC
α + Ja
≈ A
Two equations will be enough for estimating KDC and α!
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 17/20
Summary
• To choose a reasonable linear mathematical model forcontrol design one have to reduce effect of nonlinearitiessuch as friction.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 18/20
Summary
• To choose a reasonable linear mathematical model forcontrol design one have to reduce effect of nonlinearitiessuch as friction.
• Finding reliable parameters for a model is a verychallenging task.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 18/20
Summary
• To choose a reasonable linear mathematical model forcontrol design one have to reduce effect of nonlinearitiessuch as friction.
• Finding reliable parameters for a model is a verychallenging task.
• It is not clear neither
◦ how accurately parameters must be estimated nor◦ how precise we need to know them to succeed with a
particular control design (see Examples in Chapter 1 ofthe book!).
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 18/20
Summary
• To choose a reasonable linear mathematical model forcontrol design one have to reduce effect of nonlinearitiessuch as friction.
• Finding reliable parameters for a model is a verychallenging task.
• It is not clear neither
◦ how accurately parameters must be estimated nor◦ how precise we need to know them to succeed with a
particular control design (see Examples in Chapter 1 ofthe book!).
• Perhaps, one can attempt to estimate parameters on-line,adapting the accuracy to what is enough to achieve aparticular control goal.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 18/20
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 19/20
Adaptive control has two components:• on-line estimating of some parameters• simultaneous redesigning of the control law.
Note: a parametric model is needed and a knowledge on how todesign a controller for each possible set of parameters.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 19/20
Next Lecture / Assignments:
• Next recitation: March 30, 10:00-12:00, in A206Tekn.Software for off-line identification to be discussed anddistributed.
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 20/20
Next Lecture / Assignments:
• Next recitation: March 30, 10:00-12:00, in A206Tekn.Software for off-line identification to be discussed anddistributed.
• Read Chapter 1 of the book
c©Leonid Freidovich. March 26, 2010. Elements of Iterative Learning and Adaptive Control: Lecture 1 – p. 20/20