Lecture13 controls

Lecture 13
AO Control Theory
Claire Max
with many thanks to Don Gavel and Don Wiberg
UC Santa Cruz
February 19, 2013

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.

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

3
Aberrated wavefront
phase surfaces
Telescope Aperture
Imaging
dectorWavefront corrector
(Deformable Mirror)
Adaptive Optics Control
Wavefront
sensor
Corrected Wavefront
Computer

Differences between open-loop and
closed-loop control systems
• Open-loop: control system
uses no knowledge of the
output
• Closed-loop: the control
action is dependent on the
output in some way
• “Feedback” is what
distinguishes open from
closed loop
• What other examples can
you think of?
OPEN
CLOSED

More about open-loop systems
• Need to be carefully
calibrated ahead of time:
• 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?
OPEN

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)

Historical control systems: float valve
• 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
Credit: Franklin, Powell, Emami-Naeini

Block Diagrams: Show Cause and Effect
• 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
Credit: DiStefano et al. 1990

“Summing” Block Diagrams are circles
• 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
X

A home thermostat from a control
theory point of view

Block diagram for an automobile cruise
control

Example 1
• Draw a block diagram for the equation

Example 2
• Draw a block diagram for how your eyes and brain help
regulate the direction in which you are walking

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

18
The Laplace Transform Pair
( ) ( )
( ) ( )ò
ò
¥
¥-
¥
-
=
=
i
i
st
st
dsesH
i
th
dtethsH
p2
1
0
• Example 1: decaying exponential
( )
( ) ( )
( )
( ) s
s
s
s
s
s
->
+
=
+
-
=
=
=
¥
+-
¥
+-
-
ò
s
s
e
s
dtesH
eth
ts
ts
t
Re;
1
1
0
0
Re(s)
Im(s)
x
-
t0
Transform:

19
Inverse Transform:
( )
t
t
t
i
i
st
e
di
i
e
die
i
ds
s
e
i
th
s
p
p
s
p
p
s
q
p
qrr
p
sp
-
-
-
-
--
¥
¥-
=
=
=
+
=
ò
ò
ò
2
1
2
1
1
2
1
1
Re(s)
Im(s)
x
-
qr
r
s
rs
q
q
q
deids
e
s
es
i
i
i
=
=
+
+-=
--11


The above integration makes use of the Cauchy Principal Value Theorem:
If F(s) is analytic then ( ) ( )aiFds
as
sF p2
1
=
-ò
Example 1 (continued), decaying exponential

20
Example 2: Damped sinusoid
Re(s)
Im(s)
x
-
x -

t0
( ) ( )
( )
( ) ( )
( )
( ) ( ) s
wsws
w
wsws
wws
s
->÷
ø
ö
ç
è
æ
++
+
-+
=
+=
+=
=
+---
--
-
s
isis
sH
ee
eee
teth
titi
titit
t
Re;
11
2
1
2
1
2
1
cos

21
Laplace Transform Pairs
h(t) H(s)
unit step
0 t
s
1
(like lim s0 e-st )
t
e s-
s+s
1
( )te t
ws
cos- ÷
ø
ö
ç
è
æ
++
+
-+ wsws isis
11
2
1
( )te t
ws
sin-
÷
ø
ö
ç
è
æ
++
-
-+ wsws isisi
11
2
1
delayed step
0 tT s
e sT-
unit pulse
0 tT s
e sT-
-1
1

22
Laplace Transform Properties (1)
L ah t( )+ bg t( ){ }=aH s( )+ bG s( )
L h t + T( ){ }= esT
H s( )
L d t( ){ }=1
L h ¢t( )g t - ¢t( )d ¢t
0
t
ò
ì
í
î
ü
ý
þ
= H s( ) G s( )
L h ¢t( )d t - ¢t( )d ¢t
0
t
ò
ì
í
î
ü
ý
þ
= H s( )
h t - ¢t( ) eiw ¢t
d ¢t
0
t
ò = H iw( ) eiwt
Linearity
Time-shift
Dirac delta function transform
(“sifting” property)
Convolution
Impulse response
Frequency response
( )0£T

23
Laplace Transform Properties (2)
h(t)
x(t) y(t)
H(s)
X(s) Y(s)
convolution of input x(t)
with impulse response h(t)
product of input spectrum X(s) with
frequency response H(s)
(H(s) in this role is called the transfer
function)
System Block Diagrams
Power or Energy
( ) ( )
( )ò
òò
¥
¥-
¥
¥-
¥
=
=
ww
p
p
diH
dssH
i
dtth
i
i
2
2
0
2
2
1
2
1
“Parseval’s Theorem”
L h ¢t( )x t - ¢t( )d ¢t
0
t
ò
ì
í
î
ü
ý
þ
= H s( )X s( ) = Y s( )

24
Closed loop control (simple example, H(s)=1)
E s( )= W s( )- gC s( )E s( )
solving for E(s),
E s( ) =
W s( )
1+ gC s( )
Our goal will be to suppress X(s) (residual) by high-gain feedback so that Y(s)~W(s)
W(s) +
gC(s)
-
E(s)
Y(s)
residual
correction
disturbance
where g = loop gain
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).

25
The integrator, one choice for C(s)
( ) ( )
î
í
ì
=«
³
<
=
s
sC
t
t
tc
1
01
00
A system whose impulse response is the unit step
acts as an integrator to the input signal:
( ) ( ) ( ) ( ) tdtxtdtxttcty
tt
¢¢=¢¢¢-= òò 00
t’
0
c(t-t’)
x(t’)
t
0 t
c(t)
x(t) y(t)
that is, C(s) integrates the past history of inputs, x(t)

26
Control Loop Arithmetic
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
Y(s)W(s) +
B(s)
-
input
A(s) output

27
An integrator has high gain at low frequencies, low gain at high frequencies.
C(s)
X(s) Y(s)
( ) ( )
s
sX
sY =
In Laplace terminology:
The Integrator (2)
Write the input/output transfer function for an integrator in closed loop:
+
gC(s)
-
X(s)
Y(s)
W(s)
HCL(s)
W(s) X(s)
º
The closed loop transfer function with the integrator in the feedback loop is:
C s( ) =
1
s
Þ X s( ) =
W s( )
1+ g s
=
sW s( )
s + g
= HCL s( )W s( )
input disturbance (e.g. atmospheric wavefront)
closed loop transfer function
output (e.g. residual wavefront to science camera)

Block Diagram for Closed Loop Control
9
Our goal will be to ﬁnd a C(f) that suppress e(t) (residual) so that ! DM tracks ! ²
where ! = loop gain
H(f) = Camera Exposure x DM Response x Computer Delay
C(f) = Controller Transfer Function
! (t)
+
H(f)
-
e(t)
²DM(t)
residual
correction
disturbance
gC(f)
We can design a ﬁlter, C(f), into the feedback loop to:
a) Stabilize the feedback (i.e. keep it from oscillating)
b) Optimize performance
( )
( )fHfgC
fe
)(1
1
+
= ! ( f )
g

Disturbance Rejection Curve for Feedback
Control Without Compensation
17
Increasing
Gain (g)
Not so
great
rejection
Starting to
resonate
( )fHfgC )(1
1
+
=
! ( f )
e( f )
C(f) =1

30
The integrator, one choice for C(s)
( ) ( )
î
í
ì
=«
³
<
=
s
sC
t
t
tc
1
01
00
A system who’s impulse response is the unit step
acts as an integrator to the input signal:
( ) ( ) ( ) ( ) tdtxtdtxttcty
tt
¢¢=¢¢¢-= òò 00
t’
0
c(t-t’)
x(t’)
t
0 t
c(t)
x(t) y(t)
that is, C(s) integrates the past history of inputs, x(t)

31
An integrator has high gain at low frequencies, low gain at high frequencies.
C(s)
X(s) Y(s)
( ) ( )
s
sX
sY =
In Laplace terminology:
The Integrator
Write the input/output transfer function for an integrator in closed loop:
+
gC(s)
-
X(s)
Y(s)
W(s)
HCL(s)
W(s) X(s)
º
The closed loop transfer function with the integrator in the feedback loop is:
C s( ) =
1
s
Þ X s( ) =
W s( )
1+ g s
=
sW s( )
s + g
= HCL s( )W s( )
input disturbance (e.g. atmospheric wavefront)
closed loop transfer function
output (e.g. residual wavefront to science camera)

32
( )
s
s
sHCL
+
=
g
HCL(s), viewed as a sinusoidal response filter:
DC response = 0
(“Type-0” behavior)( ) 0as0 ®® wwiHCL
( ) ¥®® ww as1iHCL
High-pass behavior
and the “break” frequency (transition from low freq to high freq
behavior) is around w ~ g
The integrator in closed loop (2)
where

33
The integrator in closed loop (3)
At the break frequency, w = g, the power rejection is:
HCL ig( )
2
=
ig
g +ig( )
-ig
g -ig( )
=
g2
2g2
=
1
2
Hence the break frequency is often called the “half-power” frequency
w
( )2
giHCL
1
1/2
g
~w2
(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.
HCL (ig)
2

Disturbance Rejection Curve for
Feedback Control With Compensation
19
Increasing
Gain (g)
Much
better
rejection
Starting to
resonate
( )fHfgC )(1
1
+
=
! ( f )
e( f )
C(f) = Integrator = e-sT / (1 – e-sT)

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

Atmospheric turbulence
! Temporal power spectrum of atmospheric phase:
S! (f) = 0.077 (v/r0)5/3 f -8/ 3
! Power spectrum of residual phase
Se(f) = | 1/(1 + g C(f) H(f)) |2 S! (f)
22
Increasing
wind
Uncontrolled
Closed Loop
Controlled

Noise
! Measurement noise enters in at a different point in
the loop than atmospheric disturbance
! Closed loop transfer function for noise:
23
! (t)
+
H(f)
-
e(t)
! DM(t)
residual
correction
disturbance
gC(f) n(t)
noise
( )
( )fHfgC
fe
)(1
gC( f )H( f )
+
= n( f )
Noise feeds
through
Noise
averaged out

Residual from atmosphere +
noise
! Conditions
! RMSuncorrected turbulence: 5400 nm
! RMSmeasurement noise: 126 nm
! gain = 0.4
! Total Closed Loop Residual = 118 nm RMS
24
Residual Turbulence
Dominates
Noise Dominates

Increased Measurement
Noise
! Conditions
! gain = 0.4
25
Residual Turbulence
Dominates
Noise Dominates

Reducing the gain in the higher
noise case improves the residual
! Conditions
! gain = 0.2
26
Residual Turbulence
Dominates
Noise Dominates

41
Quick Review
h(t)
x(t) y(t)
H(s)
X(s) Y(s)
convolution of input x(t)
with impulse response h(t)
product of input spectrum X(s) with
frequency response H(s)
(H(s) in this role is called the transfer
function)
System Block Diagrams
L h ¢t( )x t - ¢t( )d ¢t
0
t
ò
ì
í
î
ü
ý
þ
= H s( )X s( ) = Y s( )

42
( ) t
h t e s-
=
t0
Re(s)
Im(s)
x
-s
( )
1
H s
s s
=
+
Re(s)
Im(s)
x
-s
x -w
w
( )
1 1 1
2
H s
s i s is w s w
æ ö
= +ç ÷
+ - + +è ø
t0
( ) ( )cost
h t e ts
w-
=
Stable input-output property: system response consists
of a series of decaying exponentials
Time Domain Transform Domain

43
Re(s)
Im(s)
x
-s
( )
1
H s
s s
=
-
t0
( ) t
h t e s+
=
What Instability Looks Like
Re(s)
Im(s)
x
-s
x-w
w
( )
1 1 1
2
H s
s i s is w s w
æ ö
= +ç ÷
- - - +è ø
t0
( ) ( )cost
h t e ts
w+
=
Time Domain Transform Domain

44
Control Loop Arithmetic
( ) ( ) ( ) ( ) ( )( )Y s A s W s A s B s Y s= -
solving for Y(s), ( )
( ) ( )
( ) ( )1
A s W s
Y s
A s B s
=
+
W(s) +
B(s)
-
input
Instability if any of the roots of the polynomial 1+A(s)B(s)
are located in the right-half of the s-plane
A(s)
Y(s)
output

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

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