4.1 Time Response Analysis.ppt

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Time Response AnalysisTime Response Analysis



ContentsContents

• Introduction

• Influence of Poles on Time Response

• Transient Response of First-OrderSystem

• Transient Response of Second-Order

System



IntroductionIntroduction• The concept of poles and zeros, fundamental to the

analysis of and design of control system, simplifiesthe ealuation of system response!

• The poles of a transfer function are"

i! #alues of the $aplace Transform aria%les s, thatcause the transfer function to %ecome infinite!

ii! &ny roots of the denominator of the transfer functionthat are common to roots of the numerator!

• The zeros of a transfer function are"

i! The alues of the $aplace Transform aria%le s, thatcause the transfer function to %ecome zero!

ii! &ny roots of the numerator of the transfer functionthat are common to roots of the denominator!



Influence of Poles on TimeInfluence of Poles on TimeResponseResponse

• The output response of a system is a sum of i! Forced response

ii! 'atural response

a) System showing an input and anoutput

b) Pole-zero plot of the system




c) Evolution of a system response. Followthe blue arrows to see the evolution ofsystem component generated by the

pole or zero




a) First-order system

b) Pole plot of thes stem

Eect of a real-ais pole upon transient

response



First-Order SystemFirst-Order System• (eneral form"

• Pro%lem" )erie the transfer function for the

follo*ing circuit

1)(

)()(

+==

s

K

s R

sC sG

τ

1

1)(

+= RCs

sG



First-Order SystemFirst-Order System• Transient Response" (radual change of output from

initial to the desired condition!

• +loc diagram representation"

• +y definition itself, the input to the system should %ea step function *hich is gien %y the follo*ing"

C(s)R(s) 1+ s

K

τ

s s R

1)( =

!here"

# $ %ainτ $ &ime constant



First-Order SystemFirst-Order System• (eneral form"

• Output response"

1)(

)()(

+==

s

K

s R

sC sG

τ

1

1

1)(

++=

+

=

s

B

s

A

s

K

s sC

τ

τ

τ

τ

t e B

At c −+=)(

)()()( s R sG sC =



First-Order SystemFirst-Order System• Pro%lem" Find the forced and natural responses for

the follo*ing systems



First Order SystemFirst Order System

First-order system response to a unit step



Transient ResponseTransient ResponseSpecificationsSpecifications

• Time constant, τ The time for e-at to decay ./0 of its

initial alue!

• Rise time, tr

The time for the *aeform to go

from 1!2 to 1!3 of its final alue!

• Settling time, ts

The time for the response to reach,

and stay *ithin 40 of its final alue!

a

1=τ

at r

2.2=

at s

4=




• Pro%lem" For a system *ith the transfer functionsho*n %elo*, find the releant responsespecifications

i! Time constant, τ

ii! Settling time, ts

iii! Rise time, tr

50

50)(

+= s

sG



Second-Order SystemSecond-Order System• (eneral form"

•Roots of denominator"

( )22

2

2nn

n

s s

K sG

ω ςω

ω

++=

!here"# $ %ain' $ (amping ratioωn $ ndamped natural

fre*uency

02 22

=++nn

s s ω ςω

12

2,1 −±−= ς ω ςω nn

s



Second-Order SystemSecond-Order System• 'atural fre5uency, ω

n

Fre5uency of oscillation of the system *ithoutdamping!

• )amping ratio, 6 7uantity that compares the e8ponential decay

fre5uency of the enelope to the natural fre5uency!

(rad/s)frequency Natural

frequencydecaylExponentia=ς



Second-Order SystemSecond-Order System• Pro%lem" Find the step response for the follo*ing

transfer function

• &ns*er"

( )22530

2252 ++

= s s

sG

( ) t t teet c

1515 151 −−−−=



Second-Order SystemSecond-Order System

Pro%lem" For each of the transfer function, find thealues of 6 and ωn, as *ell as characterize the nature

of the response!

a9

%9

c9

d9

( )40012

4002 ++

= s s

sG

( )000

002 ++

= s s

sG

( )

22530

2252

++

=

s s

sG

( )!25

!252 +

= s

sG



Second-Order SystemSecond-Order System• Step responses for second-order system damping

cases



Second-Order SystemSecond-Order System• Pole plot for the underdamped second-order system



Second-Order SystemSecond-Order System• Second-order response as a function of dampingratio



Second-Order SystemSecond-Order System• :hen 1 ; 6 ; 2, the transfer function is gien %y thefollo*ing!

• Pole position"

( )( ) ( )d nd n

n

j s j s

K sG

ω ςω ω ςω

ω

−+++=

2 !here"2

1 ς ω ω −=nd



Second-Order SystemSecond-Order System• Second-order response components generated %ycomple8 poles



Second-Order SystemSecond-Order System• Second-order underdamped responses for dampingratio alue




• Second-order underdamped response specifications




• Rise time, Tr

The time for the *aeform to go from 1!2 to 1!3 of itsfinal alue!

• Pea time, Tp

The time re5uired to reach the first

or ma8imum pea!

• Settling time, Ts

The time re5uired for the transient<s

damped oscillation to reach and stay

*ithin =40 of the steady-state alue!

2

1 ζ ω

π

−=n

pT

n

sT ζω

4=




• Percent oershoot, 0OS The amount that the *aeform oershoots the steady-

state, or final alue at pea time, e8pressed as apercentage of the steady-state alue!

"100" )1/( 2

×= −− ζ ζπ eOS

)100/("ln

)100/ln("

22 OS

OS

+

−=

π ζ



System PerformanceSystem Performance• Percent oershoot ersus damping ratio



System PerformanceSystem Performance• $ines of constant pea time Tp, settling time Ts andpercent oershoot 0OS

&s+ , &s

&p+ , &p

/S , /S+



System PerformanceSystem Performance• Step responses of second-order underdampedsystems as poles moe

a) !ith constantreal part

b) !ith constantimaginary

part



System PerformanceSystem Performance• Step responses of second-order underdampedsystems as poles moe

c) !ith constant dampingratio

4.1 Time Response Analysis.ppt

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