Lecture 12 AO Control Theory - Lick Observatorymax/289/Lectures 2016/Lecture 12 Control...
-
Upload
vuongquynh -
Category
Documents
-
view
217 -
download
2
Transcript of Lecture 12 AO Control Theory - Lick Observatorymax/289/Lectures 2016/Lecture 12 Control...
Page 1
Lecture 12
AO Control Theory
Claire Max with many thanks to Don Gavel and Don Wiberg
UC Santa Cruz February 18, 2016
Page 2
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 3
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 4 3!
Aberrated wavefront phase surfaces!
Telescope Aperture!
Imaging dector!Wavefront corrector
(Deformable Mirror)!
Adaptive Optics Control
Wavefront sensor!
Corrected Wavefront!
Computer!
Page 5
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
Page 6
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
Page 7
Some Characteristics of Closed- Loop Feedback Control
• 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 8
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
Page 9
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
Page 10
“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
Credit: DiStefano et al. 1990
Page 11
A home thermostat from a control theory point of view
Credit: Franklin, Powell, Emami-Naeini
Page 12
Block diagram for an automobile cruise control
Credit: Franklin, Powell, Emami-Naeini
Page 13
Example 1
• Draw a block diagram for the equation
Page 14
Example 1
• Draw a block diagram for the equation
Credit: DiStefano et al. 1990
Page 15
Example 2
• Draw a block diagram for how your eyes and brain help regulate the direction in which you are walking
Page 16
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 17
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"
( ) ( )
( ) ( )∫
∫∞
∞−
∞−
=
=
i
i
st
st
dsesHi
th
dtethsH
π210
• Example: decaying exponential"
( )
( ) ( )
( )
( ) σσ
σσ
σ
σ
−>+
=
+−=
=
=
∞+−
∞+−
−
∫
ss
es
dtesH
eth
ts
ts
t
Re;1
1
0
0
Re(s)"
Im(s)"
x!-σ
t"0"Transform:"
19
The Laplace Transform Pair"
Inverse Transform:"
( )
t
t
t
i
i
st
e
dii
e
diei
dss
ei
th
σ
π
π
σ
π
π
σ
θπ
θρρπ
σπ
−−
−
−
−−
∞
∞−
=
=
=
+=
∫
∫
∫
21
21
121
1
Re(s)"
Im(s)"
x!-σ
θρ
ρσ
ρσ
θ
θ
θ
deids
es
es
i
i
i
=
=+
+−=
−−11
ρθ
The above integration makes use of the Cauchy Principal Value Theorem:"
If F(s) is analytic then" ( ) ( )aiFdsas
sF π21 =−∫
Example (continued), decaying exponential"
20
Laplace Transform Pairs"
h(t)" H(s)"
unit step"0" t"
s1(like lim sè0 e-st )!
te σ− σ+s1
( )te t ωσ cos− ⎟⎠⎞⎜
⎝⎛
+++
−+ ωσωσ isis11
21
( )te t ωσ sin− ⎟⎠⎞⎜
⎝⎛
++−
−+ ωσωσ isisi11
21
delayed step"0" t"T" s
e sT−
unit pulse"0" t"T" s
e sT−−1
1"
21
Laplace Transform Properties (1)"
�
L αh t( ) + βg t( ){ } = αH s( ) + βG s( )
�
L h t + T( ){ } = esT H s( )
�
L δ t( ){ } = 1
�
L h ′ t ( )g t − ′ t ( )d ′ t 0
t
∫⎧ ⎨ ⎩
⎫ ⎬ ⎭
= H s( ) G s( )
�
L h ′ t ( ) δ t − ′ t ( )d ′ t 0
t
∫⎧ ⎨ ⎩
⎫ ⎬ ⎭
= H s( )
�
h t − ′ t ( ) eiω ′ t d ′ t 0
t
∫ = H iω( ) eiωt
Linearity"
Time-shift"
Dirac delta function transform"(“sifting” property)"
Convolution"
Impulse response"
Frequency response"
( )0≤T
22
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"
Conservation of Power or Energy"
[h t( )]2 dt0
∞
∫ = 12πi
H s( ) 2 ds−i∞
i∞
∫
= 12π
H iω( ) 2 dω−∞
∞
∫“Parseval’s Theorem”"
�
L h ′ t ( )x t − ′ t ( )d ′ t 0
t
∫⎧ ⎨ ⎩
⎫ ⎬ ⎭
= H s( )X s( ) = Y s( )
23
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 gain"
Note: for consistency “around the loop,” the units of the gain g must be the inverse of the units of C(s)."
g ≡
X(s)"
Page 24
Back Up: Control Loop Arithmetic
Y s( ) = A(s)W s( )− A s( )B s( )Y s( )
Y s( ) = A s( )W s( )
1+ A s( )B s( )
Unstable if any roots of 1+A(s)B(s) = 0 are in right-half of the s-plane: exponential growth
Y(s)W(s) +
B(s)-
input A(s) output
exp(st)
Page 25
Stable and unstable behavior
Stable Stable
Unstable
Unstable
Page 26
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 !"
where ! = loop gain!
H(f) = Camera Exposure x DM Response x Computer Delay!C(f) = Controller Transfer Function!
!(t) +!
H(f)
-!
e(t)
"DM(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!
(!)! (! )!f!H!f!gC!f!e!
)!(!1!1!
+!=! !( f )!
g
27
The integrator, one choice for C(s)"
( ) ( )⎩⎨⎧
=↔≥<
=s
sCtt
tc 10100
A system whose impulse response is the unit step"
acts as an integrator to the input signal:"
( ) ( ) ( ) ( ) tdtxtdtxttctytt
′′=′′′−= ∫∫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)"
28
An integrator has high gain at low frequencies, low gain at high frequencies."
C(s)!X(s)! Y(s)! ( ) ( )
ssXsY =
In Laplace terminology:"The Integrator, continued"
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)"
≡
The closed loop transfer function with the integrator in the feedback loop is:"
input disturbance (e.g. atmospheric wavefront)"
closed loop transfer function"output (e.g. residual wavefront to science camera)"
C s( ) = 1s
⇒ X s( ) = W s( )1+ g s
= ss + g
⎛⎝⎜
⎞⎠⎟W s( ) = HCL s( )W s( )
29
HCL s( ) = ss + g
HCL(s),viewedasasinusoidalresponsefilter:
DCresponse=0(“Type-0”behavior)HCL s( )→ 0 as s → 0
HCL s( )→1 as s →∞ High-passbehavior
andthe“break”frequency(transiFonfromlowfreqtohighfreqbehavior)isarounds~g
Theintegratorinclosedloop(1)
30
Theintegratorinclosedloop(2)
ThebreakfrequencyisoKencalledthe“half-power”frequency
s
( )2γiHCL
1
1/2
g
(log-logscale)
• Notethatthegain,g,isthebandwidthofthecontroller:• Frequenciesbelowgarerejected,frequenciesaboveg arepassed.• ByconvenFon,gisknownasthegain-bandwidthproduct.
HCL (s) 2
HCL s( ) = ss + g
Page 31
Disturbance Rejection Curve for Feedback Control With Compensation
19!
Increasing Gain (g)
Much better rejection!
Starting to resonate!
(! )!f!H!f!gC! )!(!1!1!
+!=!
!( f )!e( f )!
C(f) = Integrator = e-sT / (1 – e-sT)
Page 32
Assume that residual wavefront error is introduced by only 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 33
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"
Page 34
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"
(")" (" )"f"H"f"gC"f"e"
)"("1"gC( f )H( f )!+"
=" n( f )
Noise feeds through"
Noise averaged out"
Page 35
Residual from atmosphere + noise ! Conditions
! RMS uncorrected turbulence: 5400 nm ! RMS measurement noise: 126 nm ! gain = 0.4
! Total Closed Loop Residual = 118 nm RMS 24"
Residual Turbulence Dominates"
Noise Dominates"
Page 36
Increased Measurement Noise ! Conditions
! RMS uncorrected turbulence: 5400 nm ! RMS measurement noise: 397 nm ! gain = 0.4
! Total Closed Loop Residual = 290 nm RMS 25"
Residual Turbulence Dominates"
Noise Dominates"
Page 37
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
! Total Closed Loop Residual = 186 nm RMS 26"
Residual Turbulence Dominates"
Noise Dominates"
Page 38
What we have learned
• Pros and cons of 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 39
!"#"$"%&"'(")*+,")-"'&$./"-)#""-/+&0)&1%2$13)1%34)#1$)56)'4'2"7'8
91$)+%).%2$1-:&2.1%)21)&1%2$13)1#);"%"$+3)'4'2"7'<)'17");11-)
2"=2')+$">
!"#$"#!%%&'()*#+"#,"#!-./0/*#.)"#2"#+.34.&%*#5$%)6-%3
+786/9#:/8(4);*#<-/)6(=/#>.33*#?@@A
?8)98)9$+%03.%<)@8)A8)B1C"33<)+%-)58)D7+7.EB./()(*#5,//&0.=C
$%)6-%3#%D#:7).9(=#+786/98;*#E6F#/&"*#<-/)6(=/#>.33##?@@?"
91$)#:$2*"$).%#1$7+2.1%)1%)&1%2$13)'4'2"7')$"'"+$&*).%)56<
'"")2*")F#56)C"/'.2")G:/3.&+2.1%')+%-)2*".$)$"#"$"%&"'8