Lecture 13
AO Control Theory
Claire Max
with many thanks to Don Gavel and Don Wiberg
UC Santa Cruz
February 19, 2013
Page 1
What are control systems?
• Control is the process of making a system variable
adhere to a particular value, called the reference
value.
• A system designed to follow a changing reference is
called tracking control or servo.
Page 2
Outline of topics
• What is control?
- The concept of closed loop feedback control
• A basic tool: the Laplace transform
- Using the Laplace transform to characterize the time
and frequency domain behavior of a system
- Manipulating Transfer functions to analyze systems
• How to predict performance of the controller
Page 3
Adaptive Optics Control
Aberrated wavefront
phase surfaces
Telescope Aperture
Wavefront
sensor
Corrected Wavefront
Computer
Wavefront corrector
(Deformable Mirror)
Imaging
dector
3
Page 4
Differences between open-loop and
closed-loop control systems
• Open-loop: control system
uses no knowledge of the
output
OPEN
• Closed-loop: the control
action is dependent on the
output in some way
• “Feedback” is what
distinguishes open from
closed loop
CLOSED
• What other examples can
you think of?
Page 5
More about open-loop systems
• Need to be carefully
calibrated ahead of time:
OPEN
• Example: for a
deformable mirror, need
to know exactly what
shape the mirror will have
if the n actuators are each
driven with a voltage Vn
• Question: how might you
go about this calibration?
Page 6
Some Characteristics of Feedback
• Increased accuracy (gets to the desired final position
more accurately because small errors will get corrected
on subsequent measurement cycles)
• Less sensitivity to nonlinearities (e.g. hysteresis in the
deformable mirror) because the system is always
making small corrections to get to the right place
• Reduced sensitivity to noise in the input signal
• BUT: can be unstable under some circumstances (e.g. if
gain is too high)
Page 7
Historical control systems: float valve
Credit: Franklin, Powell, Emami-Naeini
• As liquid level falls, so does float, allowing more liquid to flow
into tank
• As liquid level rises, flow is reduced and, if needed, cut off
entirely
• Sensor and actuator are both “contained” in the combination of
the float and supply tube
Page 8
Block Diagrams: Show Cause and Effect
Credit: DiStefano et al. 1990
• Pictorial representation of cause and effect
• Interior of block shows how the input and output are
related.
• Example b: output is the time derivative of the input
Page 9
“Summing” Block Diagrams are circles
X
Credit: DiStefano et al. 1990
• Block becomes a circle or “summing point”
• Plus and minus signs indicate addition or subtraction
(note that “sum” can include subtraction)
• Arrows show inputs and outputs as before
• Sometimes there is a cross in the circle
Page 10
A home thermostat from a control
theory point of view
Credit: Franklin, Powell, Emami-Naeini
Page 11
Block diagram for an automobile cruise
control
Credit: Franklin, Powell, Emami-Naeini
Page 12
Example 1
• Draw a block diagram for the equation
Page 13
Example 1
• Draw a block diagram for the equation
Credit: DiStefano et al. 1990
Page 14
Example 2
• Draw a block diagram for how your eyes and brain help
regulate the direction in which you are walking
Page 15
Example 2
• Draw a block diagram for how your eyes and brain help
regulate the direction in which you are walking
Credit: DiStefano et al. 1990
Page 16
Summary so far
• Distinction between open loop and closed loop
– Advantages and disadvantages of each
• Block diagrams for control systems
– Inputs, outputs, operations
– Closed loop vs. open loop block diagrams
Page 17
The Laplace Transform Pair
¥
H (s ) = ò h(t )e - st dt
0
i¥
1
st
(
)
h (t ) =
H
s
e
ds
ò
2pi -i¥
• Example 1: decaying exponential
Transform:
h (t ) = e -st
¥
H (s ) = ò e
- (s +s )t
0
dt
t
Im(s)
0
¥
=
=
- 1 -(s +s )t
e
s +s
0
1
;
s +s
Re(s ) > -s
-
x
Re(s)
18
The Laplace Transform Pair
Example 1 (continued), decaying exponential
Inverse Transform:
i¥
1
1
st
h (t ) =
e
ds
ò
2pi -i¥ s + s
p
1
-1 -st
=
r
e ir dq
ò
2pi -p
= e -st
p
s = -s + reiq
Im(s)
1
= r -1e -iq
s +s
ds = ireiq dq
-

x

Re(s)
1
i dq
ò
2pi -p
= e -st
The above integration makes use of the Cauchy Principal Value Theorem:
If F(s) is analytic then
1
ò F (s )s - a ds = 2piF (a )
19
The Laplace Transform Pair
Example 2: Damped sinusoid
h (t ) = e -st cos(wt )
t
0
1
= e -st (eiwt + e -iwt )
2
1 -(s -iw )t
= (e
+ e -(s +iw )t )
2
1æ
1
1
ö
H (s ) = ç
+
÷; Re(s ) > -s
2 è s + s - iw s + s + iw ø
Im(s)
x

-
x
-
Re(s)
20
Laplace Transform Pairs
H(s)
h(t)
(like lim s0 e-st )
1
unit step
0
t
1
s +s
e -st
e
-st
1æ
1
1
ö
+
ç
÷
2 è s + s - iw s + s + iw ø
cos(wt )
1æ
1
1
ö
ç
÷
2i è s + s - iw s + s + iw ø
e-st sin (wt )
t
e - sT
s
t
1 - e - sT
s
delayed step
0 T
unit pulse
0 T
1
s
21
Laplace Transform Properties (1)
L{ ah(t ) + bg(t ) } = aH ( s) + bG( s)
L{ h( t + T ) } = e sT H ( s)
L{ d( t ) } =1
ìt
ü
Lí ò h( t ¢) g( t - t ¢) dt ¢ý = H ( s) G( s)
î0
þ
ìt
ü
Lí ò h( t ¢) d ( t - t ¢) dt ¢ý = H ( s)
î0
þ
Linearity
Time-shift
(T £ 0)
Dirac delta function transform
(“sifting” property)
Convolution
Impulse response
t
iwt ¢
iwt
¢
¢
h
t
t
e
d
t
=
H
i
w
e
( )
ò ( )
0
Frequency response
22
Laplace Transform Properties (2)
System Block Diagrams
y(t)
x(t)
Y(s)
X(s)
H(s)
h(t)
product of input spectrum X(s) with
frequency response H(s)
(H(s) in this role is called the transfer
function)
convolution of input x(t)
with impulse response h(t)
ìt
ü
Lí ò h( t ¢) x ( t - t ¢) dt ¢ý = H ( s) X ( s) = Y ( s)
î0
þ
Power or Energy
¥
ò h(t ) dt =
2
0
i¥
1
2
(
)
H
s
ds
ò
2pi -i¥
1
=
2p
¥
ò H (iw ) dw
-¥
2
“Parseval’s Theorem”
23
Closed loop control (simple example, H(s)=1)
W(s)
disturbance
+
-
E(s)
residual
Y(s)
correction
gC(s)
where g = loop gain
Our goal will be to suppress X(s) (residual) by high-gain feedback so that Y(s)~W(s)
E ( s) = W ( s) - gC ( s) E ( s)
solving for E(s),
W ( s)
E ( s) =
1+ gC ( s)
What is a good choice for C(s)?...
Note: for consistency “around the
loop,” the units of the gain g must be
the inverse of the units of C(s).
24
The integrator, one choice for C(s)
A system whose impulse response is the unit step
0
t
ì0 t < 0
1
c(t ) = í
« C (s ) =
s
î1 t ³ 0
acts as an integrator to the input signal:
y(t)
x(t)
c(t)
t
t
0
0
y (t ) = ò c(t - t ¢)x (t ¢)dt ¢ = ò x (t ¢)dt ¢
c(t-t’)
x(t’)
0
t
that is, C(s) integrates the past history of inputs, x(t)
t’
25
Control Loop Arithmetic
W(s)
+
input
A(s)
-
output
Y(s)
B(s)
Y ( s) = A(s)W ( s ) - A( s) B ( s) Y ( s)
Solving for Y(s),
Y ( s) =
A ( s )W ( s )
1+ A( s ) B ( s )
Instability if any of the roots of the polynomial 1+A(s)B(s)
are located in the right-half of the s-plane
26
Page 26
The Integrator (2)
In Laplace terminology:
X (s )
Y (s ) =
s
Y(s)
X(s)
C(s)
An integrator has high gain at low frequencies, low gain at high frequencies.
Write the input/output transfer function for an integrator in closed loop:
The closed loop transfer function with the integrator in the feedback loop is:
W(s)
+
X(s)
º
-
Y(s)
gC(s)
X(s)
W(s)
HCL(s)
W ( s) sW ( s)
1
C ( s) =
Þ X ( s) =
=
= HCL ( s) W ( s)
s
1+ g s s + g
closed loop transfer function
output (e.g. residual wavefront to science camera)
27
input disturbance (e.g. atmospheric wavefront)
Block Diagram for Closed Loop Control
! (t)
disturbance
+
residual
e(t)
² DM(t)
gC(f)
H(f)
correction
where g
! = loop gain
H(f) = Camera Exposure x DM Response x Computer Delay
C(f) = Controller Transfer Function
Our goal will be to find a C(f) that suppress e(t) (residual) so that ! DM tracks ! ²
e(f )=
1
!( f )
1 + gC ( f ) H ( f )
We can design a filter, C(f), into the feedback loop to:
a)
b)
Stabilize the feedback (i.e. keep it from oscillating)
Optimize performance
9
Page 28
Disturbance Rejection Curve for Feedback
Control Without Compensation
e( f )
1
=
! ( f ) 1 + gC ( f ) H ( f
)
C(f) =1
Increasing
Gain (g)
Not so
great
rejection
Starting to
resonate
17
Page 29
The integrator, one choice for C(s)
A system who’s impulse response is the unit step
0
t
ì0 t < 0
1
c(t ) = í
« C (s ) =
s
î1 t ³ 0
acts as an integrator to the input signal:
y(t)
x(t)
c(t)
t
t
0
0
y (t ) = ò c(t - t ¢)x (t ¢)dt ¢ = ò x (t ¢)dt ¢
c(t-t’)
x(t’)
0
t
that is, C(s) integrates the past history of inputs, x(t)
t’
30
The Integrator
In Laplace terminology:
X (s )
Y (s ) =
s
Y(s)
X(s)
C(s)
An integrator has high gain at low frequencies, low gain at high frequencies.
Write the input/output transfer function for an integrator in closed loop:
The closed loop transfer function with the integrator in the feedback loop is:
W(s)
+
-
X(s)
Y(s)
gC(s)
º
X(s)
W(s)
HCL(s)
W ( s) sW ( s)
1
C ( s) =
Þ X ( s) =
=
= HCL ( s) W ( s)
s
1+ g s s + g
closed loop transfer function
output (e.g. residual wavefront to science camera)
31
input disturbance (e.g. atmospheric wavefront)
The integrator in closed loop (2)
where
s
H CL (s ) =
g +s
HCL(s), viewed as a sinusoidal response filter:
HCL (iw )® 0 as w ® 0
DC response = 0
(“Type-0” behavior)
HCL (iw )® 1 as w ® ¥
High-pass behavior
and the “break” frequency (transition from low freq to high freq
behavior) is around w ~ g
32
The integrator in closed loop (3)
At the break frequency, w = g, the power rejection is:
H CL (ig)
2
ig
-ig
g2
1
=
= 2 =
( g + ig) ( g - ig) 2g 2
Hence the break frequency is often called the “half-power” frequency
(ig )
HHCLCL(ig)
22
1
1/2
~w2
g
w
(log-log scale)
Note also that the loop gain, g, is the bandwidth of the controller.
Frequencies below g are rejected, frequencies above g are passed. By
convention, g is thus known as the gain-bandwidth product.
33
Disturbance Rejection Curve for
Feedback Control With Compensation
e( f )
1
=
! ( f ) 1 + gC ( f ) H ( f
)
C(f) = Integrator = e-sT / (1 – e-sT)
Increasing
Gain (g)
Much
better
rejection
Starting to
resonate
19
Page 34
Residual wavefront error is introduced
by two sources
1. Failure to completely cancel atmospheric phase
distortion
2. Measurement noise in the wavefront sensor
Optimize the controller for best overall
performance by varying design parameters such
as gain and sample rate
Page 35
Atmospheric turbulence
! Temporal power spectrum of atmospheric phase:
S! (f) = 0.077 (v/r0)5/3 f -8/ 3
Increasing
wind
! Power spectrum of residual phase
Se(f) = | 1/(1 + g C(f) H(f)) |2 S! (f)
Uncontrolled
Closed Loop
Controlled
22
Page 36
Noise
! Measurement noise enters in at a different point in
the loop than atmospheric disturbance
! (t)
disturbance
+
residual
e(t)
! DM(t)
gC(f)
n(t)
H(f)
correction
noise
! Closed loop transfer function for noise:
e(f )=
gC( f )H( f )
n( f )
1 + gC ( f ) H ( f )
Noise feeds
Noise
through
averaged out
23
Page 37
Residual from atmosphere +
noise
! Conditions
! RMS uncorrected turbulence: 5400 nm
! RMS measurement noise: 126 nm
! gain = 0.4
Noise Dominates
Residual Turbulence
Dominates
! Total Closed Loop Residual = 118 nm RMS
24
Page 38
Increased Measurement
Noise
! Conditions
! RMS uncorrected turbulence: 5400 nm
! RMS measurement noise: 397 nm
! gain = 0.4
Noise Dominates
Residual Turbulence
Dominates
! Total Closed Loop Residual = 290 nm RMS
25
Page 39
Reducing the gain in the higher
noise case improves the residual
! Conditions
! RMS uncorrected turbulence: 5400 nm
! RMS measurement noise: 397 nm
! gain = 0.2
Noise Dominates
Residual Turbulence
Dominates
! Total Closed Loop Residual = 186 nm RMS
26
Page 40
Quick Review
System Block Diagrams
y(t)
x(t)
h(t)
convolution of input x(t)
with impulse response h(t)
Y(s)
X(s)
H(s)
product of input spectrum X(s) with
frequency response H(s)
(H(s) in this role is called the transfer
function)
ìt
ü
Lí ò h( t ¢) x ( t - t ¢) dt ¢ý = H ( s) X ( s) = Y ( s)
î0
þ
41
Stable input-output property: system response consists
of a series of decaying exponentials
Transform Domain
Im(s)
Time Domain
h (t ) = e-s t
0
-s
t
h (t ) = e-s t cos (wt )
0
t
x
1
H (s ) =
s +s
Re(s)
Im(s)
x
w
-s
x
-w
Re(s)
1æ
1
1
ö
H (s ) = ç
+
÷
2 è s + s - iw s + s + iw ø
42
What Instability Looks Like
Transform Domain
Time Domain
Im(s)
h (t ) = e+s t
0
H (s ) =
-s
t
1
s -s
x
Im(s)
h (t ) = e+s t cos (wt )
wx
-s
0
t
Re(s)
-w x
Re(s)
1æ
1
1
ö
H (s ) = ç
+
÷
2 è s - s - iw s - s + iw ø
43
Control Loop Arithmetic
W(s)
input
Y(s)
+
A(s)
output
B(s)
Y (s ) = A(s)W (s )- A (s ) B (s )Y (s )
solving for Y(s),
Y (s ) =
A (s )W (s )
1 + A (s ) B (s )
Instability if any of the roots of the polynomial 1+A(s)B(s)
are located in the right-half of the s-plane
44
Summary
• The architecture and problems associated with feedback
control systems
• The use of the Laplace transform to help characterize
closed loop behavior
• How to predict the performance of the adaptive optics
under various conditions of atmospheric seeing and
measurement signal-to-noise
• A bit about loop stability, compensators, and other good
stuff
Page 45
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