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')+$">

!"#$"#!%%&'()*#+"#,"#!-./0/*#.)&#1"#2"#+.34.&%*#5$%)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