Bab 5 slide 1

7/23/2019 Bab 5 slide 1

http://slidepdf.com/reader/full/bab-5-slide-1 1/25

1st Revision 2007

Taufik

Cal Poly State University, 1

Test Input Signals

For the purposes of Analysis and Design, it is customary to consider the effect

of certain standard inputs, which subject the system to sudden changes

The Most Commonly sed Test Input Signals

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


Concept of !oles and "eros

!oles of Transfer Function

Those #alues of $aplace %ariable, s, that cause the transfer function to

become infinite

"eros of Transfer Function

Those #alues of $aplace %ariable, s, that cause the transfer function to

become &ero

'(ample) Determine the poles and &eros of the following transfer function2

2

5 6( )

( 1)( 13 42)

s sT s

s s s

+ +=

+ + +

2

25 6 ( 2)( 3)( )

( 1)( 13 42) ( 1)( 6)( 7) s s s sT s

s s s s s s+ + + += =

+ + + + + +

Answer)

"eros) s * +, +-

!oles) s * +., +/, +0

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


1oles of !oles and "eros

'(ample) 2i#en the following bloc3 diagram with a step input)

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik

Cal Poly State University,

! The Input !ole generates the form of Forced 1esponse

! A pole of the transfer function generates the form of 4atural 1esponse,

i5e5 the e(ponential response5 The further to the left a pole is on thenegati#e real a(is, the faster the e(ponential will decay to &ero

! A &ero helps generate the amplitude for both the steady state andtransient response

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik

Cal Poly State University, "

First 6rder System 1esponse and Specification

[ ]1 0b b

ya a

∞ = − =

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik

Cal Poly State University, #

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik

Cal Poly State University, $

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik

Cal Poly State University, %

Change the numerator to 78, such that T9s: * 78;9s<=8:

! It will not change the settling time, time constant, rise time

! It will only affect the steady state #alue, and hence steady state error

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


A Closed $oop First 6rder System) An '(ample

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


'rror can be controlled

through gain >

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


'(ample

Could we use the same e?uations used for the 6pen $oop First 6rder System@

'S5 6nce you simplify the T9s: of the Closed $oop to a form T9s: * b;9s<a:, then

you may use the same e?uations in#ol#ing a and b

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


Second 6rder System

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik

Cal Poly State University, 1"

Based on the system poles which determine the 4atural 1esponse of the

system)

! 6#erdamped 1esponse

! nderdamped 1esponse

! 6scillatory 1esponse! Critically 1esponse

6#erdamped 1esponse

!oles) Two 1'A$ at s * . and

Transient 1esponse) Two '(ponentials with time constants e?ual to the

reciprocal of the pole locations

'(ample)

Eith a nit step input 9.;s: the response is)

!oles * s * +.5.7/ and +05=7

1 2

1 2( ) t t y t K e K eσ σ − −= +

2

9 9( ) ( ) ( )

9 9 ( 1.146)( 7.854)Y s R s R s

s s s s= =

+ + + +

7.854 1.146( ) 1 0.171 1.171t t y t e e− −

= + −

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik

Cal Poly State University, 1#

nderdamped 1esponse

!oles) Two C6M!$'G at s * d H j d

Transient 1esponse) Damped Sinusoid with an e(ponential en#elope whose

time constant is e?ual to the reciprocal of the polesJ

real part5

( ) ( )1 2 3

1 3

2

( ) cos cos sin

: tan

d d t t

d d d y t K e t e K t K t

K where

K

σ σ ω φ ω ω

φ

− −

−

= − = +

=

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


'(ample)


!oles * s * +.5= H j 5=K

2

9 9( ) ( ) ( )

3 9 ( 1.5 2.59)( 1.5 2.59)Y s R s R s

s s s j s j= =

+ + + + + −

( )1.5( ) 1 1.55 cos 2.558 30t o y t e t −

= − −

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik

Cal Poly State University, 1$

6scillatory 1esponse

!oles) Two IMA2I4A1 at s * H j.

Transient 1esponse) ndamped Sinusoid with 1adian fre?uency e?ual to

the imaginary part of the poles

'(ample)


!oles * s * H j-

2

9 9( ) ( ) ( )

9 ( 3)( 3)Y s R s R s

s s j s j= =

+ + −

( )( ) 1 cos 3 y t t = −

( )1 1( ) cos y t K t ω φ = −

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik

Cal Poly State University, 1%

Critically Damped 1esponse

!oles) '?ual and 1'A$ at s * .

Transient 1esponse) 6ne term is an '(ponential whose time constant is

e?ual to the reciprocal of the !ole $ocation5 Another

is the product of time t and the same e(ponential term5

'(ample)


!oles * s * -

2 2

9 9( ) ( ) ( )

6 9 ( 3)

Y s R s R s

s s s

= =

+ + +

3 3( ) 1 3 t t y t te e− −

= − −

1 1

1 2( ) t t

y t K e K teσ σ − −

= +

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


nd 6rder System) nderdamped 1esponse Specifications

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


5 !ercent 6#ershoot) The Amount that the wa#eform o#ershoots the steady

state, or final #alue at the !ea3 Time, e(pressed as

percentage of the steady state #alue

-5 Settling Time) The Amount of time re?uired for the transientJs Damped

6scillations to stay within HL of the steady state

75 1ise Time) The time re?uired for the wa#eform to go from .8L to K8L of

its Final %alue

'(ample)

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


Ehat if the damping ratio is not within the specified range to find the 1ise Time@

From the Te(t Boo3, page -

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


The effect of #arying 4atural Fre?uency gi#en a damping ratio

The response is faster as the natural fre?uency increases5 owe#er, the

!ercent 6#ershoot remains the same, since)

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik


The effect of #arying Damping 1atio gi#en a 4atural Fre?uency

As Damping 1atio increases)

! 1esponse becomes less oscillatory

! The !ea3 Time Increases

! !ercent 6#ershoot Decreases

! 1ise Time Increases

7/23/2019 Bab 5 slide 1


1st Revision 2007

Taufik

Cal Poly State University, 2"

'STIMATI64 6F DAM!I42 1ATI6 2I%'4 y9t: with ST'! I4!T

If we 3now y9t: of a system, say from a step input, after applying a step input then

the damping ratio may be estimated as follows)

.5 Find a !oint on y9t: where the #alue settles within L of the final #alue

5 Count how many cycles from y9t: * 8 to the point obtained in . * N cycles

-5 Damping 1atio * 85== ; 9N cycles:

Bab 5 slide 1

Documents

Transcript of Bab 5 slide 1