4.1 time response analysis

7/24/2019 4.1 Time Response Analysis

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Time Response Analysis


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Introduction

Infuence o Poles on Time Response

Transient Response o First-OrderSystem

Transient Response o Second-OrderSystem


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The concept o poles andzeros, undamental to theanalysis o and design o control system, simpliesthe e!aluation o system response"

The poleso a transer unction are#

i" $alues o the %aplace Transorm !aria&les s, thatcause the transer unction to &ecome innite"

ii" Any roots o the denominator o the transer unctionthat are common to roots o the numerator"

The zeroso a transer unction are#

i" The !alues o the %aplace Transorm !aria&le s, thatcause the transer unction to &ecome zero"

ii" Any roots o the numerator o the transer unctionthat are common to roots o the denominator"


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The output response o a system is a sum oi" Forcedresponse

ii" 'aturalresponse

a( System sho)ing an input and anoutput

&( Pole-zero plot o the system


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c( *!olution o a system response" Follo)the &lue arro)s to see the e!olution osystem component generated &y the

pole or zero


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a( First-order system

&( Pole plot o thes stem

*+ect o a real-ais pole upon transientresponse


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eneral orm#

Pro&lem# .eri!e the transer unction or the

ollo)ing circuit

1)(

)()(

+==

s

K

sR

sCsG

1

1)(

+=RCs

sG


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Transient Response# radual change o output rominitial to the desired condition"

/loc0 diagram representation#

/y denition itsel, the input to the system should &ea step unction )hich is gi!en &y the ollo)ing#

C(s)R(s) 1+s

K

ssR

1)( =

1here,

2 # ain # Time constant


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eneral orm#

Output response#

1)(

)()(

+==

s

K

sR

sCsG

1

1

1)(

++=

+

=

s

B

s

A

s

K

ssC

teB

Atc +=)(

)()()( sRsGsC =


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Pro&lem# Find the orced and natural responses orthe ollo)ing systems


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First-order system response to a unit step


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Time constant, The time or e-atto decay 345 o its

initial !alue"

Rise time, tr The time or the )a!eorm to go

rom 6"7 to 6"8 o its nal !alue"

Settling time, ts The time or the response to reach,

and stay )ithin 95 o its nal !alue"

a

1=

atr

2.2=

ats

4=


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Pro&lem# For a system )ith the transer unctionsho)n &elo), nd the rele!ant responsespecications

i" Time constant,

ii" Settling time, ts

iii" Rise time, tr

50

50)(

+=s

sG


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eneral orm#

Roots o denominator#

( )22

2

2nn

n

ss

KsG

++=

1here,2 # ain: # .amping ration # ;ndamped natural

re


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'atural re


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Pro&lem# Find the step response or the ollo)ingtranser unction

Ans)er#

( )22530

2252 ++

=ss

sG

( ) tt teetc 1515 151 =


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Pro&lem# For each o the transer unction, nd the!alues o : and n, as )ell as characterize the nature

o the response"

a(

&(

c(

d(

( )40012

4002 ++

=ss

sG

( )000

002 ++

=ss

sG

( )

22530

2252

++

=

ss

sG

( )!25

!252 +

=s

sG


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Step responses or second-order system dampingcases


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Pole plot or the underdamped second-order system


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Second-order response as a unction o dampingratio


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Second-order response as a unction o dampingratio


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1hen 6 > : > 7, the transer unction is gi!en &y theollo)ing"

Pole position#

( )( ) ( )dndn

n

jsjs

KsG

+++=

2 1here,2

1 =nd


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Second-order response components generated &ycomple poles


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Second-order underdamped responses or dampingratio !alue


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Second-order underdamped response specications


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Rise time, Tr The time or the )a!eorm to go rom 6"7 to 6"8 o its

nal !alue"

Pea0 time, Tp The time re


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Percent o!ershoot, 5OS The amount that the )a!eorm o!ershoots the steady-

state, or nal !alue at pea0 time, epressed as apercentage o the steady-state !alue"

"100" )1/(2

= eOS

)100/("ln

)100/ln("

22 OS

OS

+

=


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Percent o!ershoot !ersus damping ratio


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%ines o constant pea0 time Tp, settling time Tsandpercent o!ershoot 5OS

Ts9> Ts7Tp9> Tp7

5OS7> 5OS9


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Step responses o second-order underdampedsystems as poles mo!e

a( 1ith constantreal part

&( 1ith constantimaginary

part


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Step responses o second-order underdampedsystems as poles mo!e

c( 1ith constant dampingratio

4.1 time response analysis

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