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Advanced Motion Control
Maarten Steinbuch
Control Systems Technology
Department of Mechanical Engineering
Eindhoven University of Technology
APC workshop, Vancouver, may 2005
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Contents
• background, motion systems
• control for dummies
• advanced motion control challenges
• embedded dynamical systems
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Eindhoven University of Technology
• 9 scientific departments, 10 academic Bachelor
programmes, 19 Master programmes, 3000 employees,
220 professors, 6800 students, 200 postgraduate students,
450 PhD students
• located in the Eindhoven-Aachen-Leuven triangle
• ‘mechatronics’ high tech industry:
• Philips, ASML, FEI, Assembleon
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Mechanical Engineering Department
• 9 full prof., 60 senior research staff,
• 18 Post Docs, 105 PhD students,
• 700 BSc and MSc students
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Structure of Mechanical Engineering
Thermo Fluids Engineering
Computational and Experimental Mechanics
Dynamical Systems Design
2 ‘theme Mastertracks’:
• Automotive Engineering Science
• Micro and Nano Technology
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Mechanical Engineering Department
Automotive Engineering Science (2001)
(Sub)-Micron Technology (2003)
Dynamical
Systems
Design
(DSD)
Thermo Fluids
Engineering
(TFE)
Computational &
Experimental
Mechanics
(CEM)
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TU/e - W : Full Chairs in DSD Division
• Dynamics and Control:
Prof.Dr. Henk Nijmeijer
• Control Systems Technology:
Prof.Dr.Ir. Maarten Steinbuch
• Systems Engineering :
Prof.Dr.Ir. Koos Rooda
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from Industry
…
…
to Academia
(1999)
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TU Eindhoven
Philips
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Isles of Academia
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…
in theory
there is no difference between theory and practice
in practice there is
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…2
simulation is like masturbation:
the more you do it
the more you think it is the real thing!
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bridging the gap
• education: merge classic & modern
• bring in real industrial systems
• confront PhDs with other disciplines
• learn from experimental experience
how to proceed with theory
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Control Systems Technology
• 1 full prof
• 2 part-time prof
• 7 associate and assistant prof
• 4 technical staff members
• 20 PhD students
• 40 MSc students/year
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• Motion Systems
– industrial applications (pick-and-place,
(bio)-robots)
– consumer applications (storage systems)
– hydraulic servo systems
• Automotive
– power trains (in particular CVT)
– (passive) car safety systems
– vehicle electrical power management
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DaVinci surgery robot © Intuitive Surgery Inc.
TU/e
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Automotive Safety Restraint Systems
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Zero-Inertia
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Continously Variable Transmission
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1/1,000 mm
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Source: intel, ICE
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Design
Reticle
Stepper Exposure Method
Lens
Reticle
Wafer
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Source: ASML
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Aberrations
Scatter
Illumination
Dose
optimization
Mask optimization software
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Source
optimization
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Mechanical
Electronic / Electrical
Thermal
Pneumatic
Optical
Hydraulic
Acoustical
Chemical
Software
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Motion Control for
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Motion Systems
m
F
x
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x
Disturbance F
Servo force ?
F
s
d
Mass
M
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Mechanical solution:
x
k
F
d
Mass
M
Force spring damper
F = −k ⋅ x − d ⋅ x
Eigenfrequency
1 k
f=
2π M
Disturbance Fd
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Servo analogon:
x
F
servo
Servo force
Fs = −k p ⋅ x − kv ⋅ x
Eigenfrequency
1
f =
2π
kp
M
Mass
M
k p : servo stiffness
kv : servo damping
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x
Disturbance F
Example:
d
F
s
Mass
M
Slide: mass = 5 kg
Required accuracy 10 µm at all times
Disturbance (f.e. friction) = 3 N
1. Required servo stiffness?
2. Eigenfrequency?
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How to move to / follow a setpoint:
x
h or x
s
Spring- damper:
F = k(h − x) + d (h − x)
Controller:
Fs = k p (xs − x) + kv (xs − x)
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x
xs
Fd
Fdisturbance
controller
xs
+
e
-
d
k p + kv
dt
process
+
F
Mass
x
+
Fservo
Kp/kv-controller or PD-controller
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xs
+
controller
e
-
kp
kv
kp = ?
process
kv = ?
⇒
x
Fservo
error
⇔
stability
Trade off
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Setpoints:
xs
x
What should xs look like as a function of time, when moving the mass?
(first order, second order, third order,….?)
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x
Apply a force F (step profile):
F (t ) = Mx(t )
F
M
⇓
x(t) is second order, when F constant
Second order profile requires following information:
- maximum acceleration
- maximum velocity
- travel distance
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Example
Pos
=
2π ≈ 6.3rad
Velmax
Acc max
=
=
20π ≈ 63rad / sec
500π ≈ 1.6e3rad / sec 2
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Concluding remarks time domain tuning
A control system, consisting of only a single mass m and a
kp/kv controller (as depicted below), is always stable.
kp will act as a spring; kv will act as a damper
As a result of this: when a control system is unstable, it
cannot be a pure single mass + kp/kv controller
(With positive parameters m, kp and kv)
x
xs
kp
M
kv
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Frequency domain
Time domain:
Monday and Thursday at 22:10
Frequency domain:
twice a week
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x
F
M
weak spring
(f = 2.5 Hz)
excitation force in N and displacement in m
M = 5 kg
0.2
excitation force (offset 0.1 &scaling 1e-4)
0.15
F (t ) = 400 sin(2π 7t )
0.1
0.05
H (7 Hz ) ≈ 0.045 / 400 = 1e − 4 m / N
∠H ( 7 Hz ) ≈ −180
0
0
-0.05
response
-0.1
-0.15
0
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0.5
1
1.5
time in sec
2
2.5
42
3
measurement mechanics stage
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amplitude in dB
-20
-40
-60
-80
2
10
frequency in Hz
3
10
phase in deg
200
0
-200 2
10
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3
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Transfer function:
x( s )
1
H ( s) =
=
F ( s ) Ms 2 + ds + k
F
1
Ms 2 + ds + k
x
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Bode plot of the PD-controller:
kp = 1500 N/m; kv = 20 Ns/m
amplitude in dB
100
80
+1
+0
60 0
10
phase in deg
100
1
10
frequency in Hz
2
10
1
10
2
10
50
0 0
10
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Fd
xs +
e
-
C(s)
+
Fs+
x
H(s)
Four important transfer functions
1. Open loop
2. Closed loop
3. Sensitivity
L(s) = C (s) H (s)
x
C (s) H (s)
T (s) = (s) =
xs
1 + C (s) H (s)
1
e
S(s) = (s) =
1+C(s)H(s)
xs
H ( s)
x
4. Proces Sensitivity H ps ( s) =
( s) =
Fd
1 + C ( s) H ( s)
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Filters
Mimo
PeeDee
PeeEye
•Integral action
•Differential action
•Low-pass
•High-pass
•Band-pass
•Notch filter
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Integral action :
X(t)
1
τis
Y(t)
τI integral time constant τI =1/ki
-1
0°
ω=2πf
-90°
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Differential action
+1
u
u
ε
ε
H = ks = ; s = jω;
ε
ks
“tamme” differentiator :
= kω
ω
+90°
0°
u
ks
=
ε τ d s +1
1
u
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τd
= ωd = 2πfd
ω
+90°
0°
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H=
u
ε
=
“lead” filter
1 + τ 1s 1 + τ d s
=
1+ τ 2s 1+ τ d s
γ
τ1
≈ 2−5
τ2
γ
γ>1
ω c = ω1ω 2 =
1
1
τ 1τ 2
ω
+90°
0°
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ω1 =
1
τ1
ω2 =
1
τ2
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⎛
⎞
⎜
⎟
⎞
⎛
u
1 ⎜ 1+τ d s ⎟
⎟⎟
⎜
H = = k ⎜1 +
ε
⎝ τ i s ⎠⎜ 1 + τ d s ⎟
⎜
⎟
γ
⎝
⎠
P+I+D
-1
ωi =
+1
1
τi
ωd =
1
τd
ω
+90°
0°
-90°
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ε(t)
-
2nd order filter
ω1
ω1
s
s
H=
u
ε
=
k
s2
ω
u(t)
1
2β
2β
2
1
+ 2β
Top:
s
ω1
+1
ωo = ω1 1 − β 2 .
1
-2
0°
-90 °
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-180°
ω = ω1
52
ε
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General 2nd order filters
General: ω1≠ω2
s2
H=
u
ε
=
ω
s
ω
2
1
2
2
2
+ 2 β1
+ 2β 2
s
ω1
s
ω2
⎛ ω2 ⎞
⎜⎜ ⎟⎟
⎝ ω1 ⎠
+1
2
+2
1
+1
+180°
0°
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ω1 ≥ ω 2
+2
1
⎛ ω22 ⎞
⎜ 2⎟
⎜ω ⎟
⎝ 1 ⎠
-2
ω
0°
-180°
ω2
ω1
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“Notch”-filter :ω1= ω2
ampl.
fase
β1
β2
0°
-180°
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Loop shaping procedure for motion systems
1.
stabilize the plant:
add lead/lag with zero = bandwidth/3 and pole = bandwidth*3,
adjust gain to get stability;
or add a pure PD with break point at the bandwidth
2.
add low-pass filter:
choose poles = bandwidth*6
3.
add notch if necessary,
or apply any other kind of first or second order filter and shape the loop
4.
add integral action:
choose zero = bandwidth/5
5.
increase bandwidth:
increase gain and zero/poles of integral action, lead/lag and
other filters
during steps 2-5: check all relevant transfer functions,
and relate to disturbance spectrum
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Implementation issues
1. sampling = delay: linear phase lag
for example: sampling at 4 kHz gives phase lag
due to Zero-Order-Hold of:
180º @ 4 kHz
18º
@ 400 Hz
9º
@ 200 Hz
2. Delay due to calculations
3. Quantization (sensors, digital representation)
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Feedforward design
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Feedforward based on inverse model
ms 2
xs
K p + Kv s
1
ms 2
x
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Example: m=5 [kg], b=1 [Ns/m],
2nd degree setpoint
xs [m]
1
0.5
vs [ms-1]
0
1.5
1
0.5
as [ms-2]
0
4
2
0
-2
-4
0
0.2
0.4
0.6
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0.8
1
1.2
1.4
1.6
t [s]
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Example: tracking error, no feedforward
-3
2
x 10
1.5
viscous damping effect
error [m]
1
0.5
0
-0.5
-1
-1.5
-2
0
0.2
0.4
0.6
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0.8
1
1.2
1.4
1.6
1.8
t [s]
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Example: tracking error, with feedforward
-3
2
x 10
K fv= 0.9, K fa= 0
1.5
error [m]
1
K fv= 0.9, K fa= 4.5
0.5
0
-0.5
-1
-1.5
-2
0
0.2
0.4
0.6
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0.8
1
1.2
1.4
1.6
1.8
t [s]
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sign( x s )
xs
xs
xs
feedforward structure
K fc
K fa
K fv
C(s)
H(s)
x
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3rd degree setpoint trajectory
xs [m]
1.5
1
0.5
vs [ms-1]
0
1
0.5
as [ms-2]
0
4
2
0
-2
-4
0
0.2
0.4
0.6
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0.8
1
1.2
1.4
1.6
1.8
t [s]
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Parasitic Dynamics
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Three Types of Dynamic Effects
- Actuator flexibility
- Guidance flexibility
- Limited mass and stiffness of frame
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1. Actuator flexibility
x
k
F
s
Sensor
Motor
d
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x1
c
F
amplitude in dB
-50
x1
F
-100
-150 0
10
d
-2
-2
1
10
frequency in Hz
2
10
phase in deg
200
0
-200 0
10
M2
M1
1
10
2
10
x2
c
F
amplitude in dB
-50
x
H= 2
F
-100
M1
d
x2
F
x1
F
-150 0
10
phase in deg
200
1
10
frequency in Hz
2
10
0
-200 0
10
M2
1
10
2
10
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2. Guidance flexibility
x
F
s
M,
J
k
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3. Limited mass and stiffness of frame
x
F
s
Motor
Frame
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Motion Control Properties
• experimentation is ‘cheap’
Disturbance Design Cycle: 7 min FRF measurement,
model, loopshape, implementation
•
•
•
•
plant decoupling, i.e. SISO
feedforward: low-order model-based
feedback: loopshaping
key limitation: bode gain/phase - sensitivity integral
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Gunter Stein’s Bode Lecture, CDC 1989
IEEE Control Systems Magazine, 23 (2003), pp 12- 25
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Motion Control Challenge:
how to cope with Bode sensitivity limitation?
∞
∫ log S ( jω ) dω = 0
0
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directions of motion control research
•
•
•
•
MIMO loopshaping
nonlinear control of linear systems (reset…)
disturbance-based modelling and control
data-driven control
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directions of motion control research
•
•
•
•
MIMO loopshaping
nonlinear control of linear systems (reset…)
disturbance-based modelling and control
data-driven control
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MIMO integral constraints…
‘Loop’ 1
‘Loop’ 2
… ‘Loop’ n
Sensitivity dirt
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directions of motion control research
•
•
•
•
MIMO loopshaping
nonlinear control of linear systems (reset…)
disturbance-based modelling and control
data-driven control
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Problem formulation
• Do there exist nonlinear feedback controllers
that give better ‘performance’ for linear motion
systems than linear solutions?
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Approach
•
•
•
•
•
Performance measures?
Plant is linear, but
disturbances and specifications ‘change’
Use LPV for synthesis?
How about non-smooth (reset) filters
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Bode Gain/Phase relation
Slope = n
means
phase = n*90 degrees
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SPAN- filter
rectifier
low-pass
X
sign
lead
be creative with control!
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directions of motion control research
•
•
•
•
MIMO loopshaping
nonlinear control of linear systems (reset…)
disturbance-based modelling and control
data-driven control
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disturbance-based modelling and control
• disc errors vs shocks optical storage
• stochastic vs deterministic disturbances
• repetitive vs a-periodic setpoints or disturbances
Internal model principle….
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Iterative Learning Control (ILC)
ek +1 < ek
Q(1 − L PS ) < 1
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directions of motion control research
•
•
•
•
MIMO loopshaping
nonlinear control of linear systems (reset…)
disturbance-based modelling and control
data-driven control
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Machine-in-the-loop design
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Data-driven control
• Examples:
– data-based LQG control
– iterative feedback tuning
– virtual reference feedback tuning
– unfalsified control
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Problem statement
• Design a SISO LTI controller C for LTI plant P
r +
-
e
C
u
P
y
• Control objective: realize the desired So and To
• Ideal controller Co:
PCo
1
.
, To =
So =
1+ PCo
1+ PCo
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Data-based controller design
• The controller class: {C( z, θ)} = {Cp ( z)βT ( z)θ}.
•
Cp(z) is directly prescribed by the designer: notches,
integrators, etc.
• Basis functions:
β( z) = [ β0 ( z) β1( z) … βn ( z)]T .
• Tuning parameters: θ = [θ0 θ1 … θn ]T .
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• Constraint on Co:
To ( z ) = Co ( z ) So ( z ) P( z )
• Model-based cost function:
J MB(θ) = (To ( z) − C( z, θ)So ( z)P( z))W ( z) 2
2
• Processing the measurements:
To ( z)u(t ) = C( z, θ)So ( z)P( z)u(t ) ⇒ To ( z)u(t ) = C( z, θ)So ( z) y(t )
• Data-based cost function:
1 N
N
J DB (θ) = ∑[ L( z)(To ( z)u(t ) − C( z, θ)So ( z) y(t ))]2
N t =1
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N
ˆ
C1 ( z , θ DB )
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Unfalsified control
Given a set of controllers, implement one, use
the I/O data, and check which part of the set
is not feasible, then change the set and
iterate
Safanov, Tsao 1997
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Explore Motion Control Properties
• experimentation is ‘cheap’
Disturbance Design Cycle: 7 min FRF measurement,
model, loopshape, implementation Data based
• plant decoupling, i.e. SISO MIMO disturbances
• feedforward: low-order model-based Learning control
• feedback: loopshaping nonlinear control
• key limitation: bode gain/phase - sensitivity integral
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The End
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