lecture13.2013 controls v4

8/9/2019 Lecture13.2013 Controls v4

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What are control systems?What are control systems?

% Control is the process of making a system variableadhere to a particular value, called the referencevalue.

% A system designed to follow a changing reference iscalled tracking control or servo.


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Outline of topicsOutline of topics

%What is &ontrol'( )he &on&e*t o+ &losed loo* +eedba&k &ontrol

%, basi& tool- the .a*la&e trans+orm ( Using the .a*la&e trans+orm to &hara&terize the time

and +re/uen&y domain behavior o+ a system( Mani*ulating Transfer functions to analyze systems

%0ow to *redi&t *er+orman&e o+ the &ontroller


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Differences between open-loop andDifferences between open-loop andclosed-loop control systemsclosed-loop control systems

% Open-loop: control systemuses no knowledge of theoutput

% Closed-loop: the controlaction is dependent on theoutput in some way

% eedback! is whatdistinguishes open fromclosed loop

% "hat other e#amples canyou think of$

O%&'

C(O)&*


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!ome Characteristics of "eedbac#!ome Characteristics of "eedbac#

% ncreased accuracy gets to the desired final positionmore accurately because small errors will get correctedon subse/uent measurement cycles0

% (ess sensitivity to nonlinearities e.g. hysteresis in thedeformable mirror0 because the system is alwaysmaking small corrections to get to the right place

% 1educed sensitivity to noise in the input signal

% 234: can be unstable under some circumstances e.g. ifgain is too high0


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$istorical control systems% float &al&e$istorical control systems% float &al&e

% As li/uid level falls, so does float, allowing more li/uid to flowinto tank

% As li/uid level rises, flow is reduced and, if needed, cut offentirely

% )ensor and actuator are both contained! in the combinationof the float and supply tube

Credit: ranklin, %owell, &mami-'aeini


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'loc# Dia(rams% !how Cause and )ffect'loc# Dia(rams% !how Cause and )ffect

% %ictorial representation of cause and effect

% nterior of block shows how the input and output arerelated.

% ample b: output is the time derivative of the input

Credit- DiSte+ano et al 1 #


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** !ummin(+ 'loc# Dia(rams are circles!ummin(+ 'loc# Dia(rams are circles

% 2lock becomes a circle or summing point!

% %lus and minus signs indicate addition or subtraction

note that sum! can include subtraction0

% Arrows show inputs and outputs as before

% )ometimes there is a cross in the circle

X



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A home thermostat from a control theory A home thermostat from a control theory point of &iew point of &iew

Credit: ranklin, %owell, &mami-'aeini


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),ample 1),ample 1

% *raw a block diagram for the e/uation


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),ample 1),ample 1

% *raw a block diagram for the e/uation

Credit: *i)tefano et al. 5667


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),ample ),ample

% *raw a block diagram for how your eyes and brain helpregulate the direction in which you are walking


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),ample ),ample

% *raw a block diagram for how your eyes and brain helpregulate the direction in which you are walking



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!ummary so far !ummary so far

% *istinction between open loop and closed loop5 Advantages and disadvantages of each

% 2lock diagrams for control systems5 nputs, outputs, operations5 Closed loop vs. open loop block diagrams


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The Laplace Transform Pair

( ) ( )( ) ( )

=

=i

i

st

st

dsesHi

t h

dt et hsH

21

0

Example 1: decaying exponential

( )

( ) ( )

( )

( )

>+=

+=

=

=

+

+

ss

es

dt esH

et h

t s

t s

t

Re;1

1

0

0

Re(s)

Im(s)

x-

t0

Transform:


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Inverse Transform:

( )

t

t

t

i

i

st

e

dii

e

diei

dss

ei

t h

=

=

=

+=

21

21

121

1

Re(s)

Im(s)

x-

d eids

es

es

i

i

i

=

=+

+=11

The above integration makes u se of the Cauchy P rincipal Value Theorem:

If F(s) is analytic then ( ) ( )aiF dsas

sF 21 =

Example 1 (continued), decaying exponential


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Example 2: Damped sinusoid

Re(s)

Im(s)

x

-x -

t0( ) ( )

( )( ) ( )

( )( ) ( )

>

++++=+=

+=

=

+

sisis

sH

ee

eee

t et h

t it i

t it it

t

Re;11

212

121

cos


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Laplace Transform Pairs

h(t) H(s)

unit step0 t

s1(like lim 0 e

- t

)

t e +s1

( )t e t cos

++++ isis11

21

( )t e t sin

+++ isisi11

21

delayed step0 tT s

e sT

unit pulse 0 tT s

e sT 1

1


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Laplace Transform Properties ( 1)

L h t ( )+ g t ( ){ }= H s( )+ G s( )

L h t +T ( ){ }=esT H s( )

L t ( ){ }=1

L h t( )g t t( )d t0

t

=H s( ) G s( )

L h t( ) t t( )d t0

t

=H s( )h t t( ) e i t d t

0

t

=H i ( ) e i t

Linearity

Time-shift

Dirac delta function transform(sifting p roperty)

Convolution

Impulse response

Frequency response

( )0T


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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 h igh-gain feedback so that Y(s)~W(s)

W(s) +

C(s)

-

E(s)

Y(s)

residual

correction

disturbance

where = loop gain

What is a good choice for C(s)?...

Note: for consistency around t he l oop,the units of the gain must be theinverse of the units of C(s).


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The integrator , one choice for C(s)

( ) ( )=

lecture13.2013 controls v4

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