Lecture 10-2 Linear Systems Analysis in the Time Domain II

Lecture 10-2

Linear Systems Analysis

in the Time Domain II

- Transient Response -

System Analysis Spring 2015

2

2

2 2

2

2 2

( ) 2

1 1

( ) ( ), ( )

2

n

n n

n

n n

G s s s

R s step input C s

s s s s



 



 

  

  

 

G(s) ( )

R s C s ( )

Second Order Systems

2 2

1

2

( ) 1 cos sin 1 cos( )

1 1

tan

1

n n

t t

d d d

c t e t t e t



  



 

 

 



 



 

 

           

 

2 3

1

2 2

2

_{n n}

K s K K

s s  s 

  

 



²



²

2 2 2

( ) / 1 1

1

( ) (1 )

n n

n n

s

s s

    

  

   

    

Damping ratio and Pole placement

i) ζ.>1 : poles are real and distinct (over damped)

ii) ζ.=1: poles are real and coincident (critically damped)

iii) 0 < ζ < 1 : pole are complex conjugates (under damped)

iv) ζ. = 0 : The pole are purely imaginary (undamped)

System Analysis Spring 2015

4

Step Response of Second-Order Systems

Step Response of Second-Order Systems

1. Over damped Case

1 2

1 2

( ) 1

^{p t p t}

y

step

t   C e

^

 C e

^

2. Critically damped Case

3. Under damped Case

4. Undamped Case

1 2

( ) 1

^{pt pt}

y

step

t   C e

^

 C te

^

( ) 1

2

cos( )

1

nt

step d

y t e t



 



 







1

tan

2

1

 









( ) 1 cos( )

step n

y t    t

2

1

,

2 _{n n}

1

p p       

1

,

2 _n

p p   

2

1

,

2 _{n n}

1

p p     j   

1

,

2 _n

p p   j 

System Analysis Spring 2015

6

Under-damped Second-Order System

2

1

,

2 _{n n}

1

p p     j   

1

2

d n

    

Damped Natural Frequency:

Influence of ω n and ζ on the pole locations

System Analysis Spring 2015

8

2 2

2 1

( ) 1 cos sin 1 sin( )

1 1

tan 1

n

n n

t

t t

step d d d

y t e t e t e t



 







 

 

 



  



     

 

 

Influence of ζ

T

r 0.1

0.9 1.0

t

T

p

Step Response Based Second -Order System Specifications

T

s

1) Peak Time: The time required to reach the first or maximum peak

2) Settling Time: The time required for the transeints’ damped oscillations to reach and stay within of the steady-state value.

3) Rise time : The time required to go from 0.1 to 0.9 of the final value

4) Percent Overshoot: % OS The amount that the waveform overshoots the steady-state at the peak time, expressed as a percentage of the steady-state value

( ) y t

 2%

T

s

T

p

T

r

%OS

System Analysis Spring 2015

10

2 2

2 2 2 2 2

( ) 2 ( ) (1 )

n n

n n n n

sY s s s s

 

    

 

    

1) Peak Time T

p

Step Response Based Second -Order System Specifications

, 2 ,

d

t

    

2 2

2 2 2

1 1

( ) (1 )

n

n

n n

s

  



  

 

   



²



( )

2

sin 1

1

nt n

y t  e

^



n

 t









 

Set , y t  ( )  0

1

2 p

d n

T  

  

 



1 2 2

2 2

( ) 1 1 sin 1

1 1

n n

p p n

n

M y T e

 

 

   

  

 



 

 

           

2) % OS M

_o

Step Response Based Second -Order System Specifications

% OS = ( )

p steady state

100

steady state

y T y

y





 

1 exp

2

1





 

 

       

100 exp

2

100

1

p s

o

s

percent overshoot

M y

M y





 

  

         

System Analysis Spring 2015

12 3) Settling time :

2

sin( )

1

nt

d

e y r e t



 





   



2% , 4 1

5% , 3 1

s

n

s

n

case T

case T





 

 

e

Step Response Based Second -Order System Specifications

2

0.2

1

nt

e

^









ln(0.2 1

2

)

s

n

T 



 



4) Rise time (01. to 0.9) :

Step Response Based Second -Order System Specifications

System Analysis Spring 2015

14

 

 

2

2 2

2 2

1

2 2

0, (0) 0

2 0

1

2 2

( ) (0) (0) 2 ( ) (0) ( ) 0

2 (0)

( ) 2

( ) (0) sin (0) cos (0) cos tan

1 1

n n

n n

n

n n

n

n n

t t

d d d

mx bx kx x

x x x

b b

m mk

s X s sx x sX s x X s

s x

X s s s

x t e

^

x t x t x e

^

t

 

 

 



 

    

 

  

   

  

 

       

 

 

 

 

 

     

 

 

 

  

 



1 

2

 

 

  

 

Experimental Determination of Damping Ratio

m

b k

1

1

( 1) 1

( ( 1) )

n

n n

t

n T

t n T

n

x e

x e e

 



 

  

 

Experimental Determination of Damping Ratio

1 1

2 2

1

1

2

2 1

2 2 1

ln ln

1 1

ln ( 1)

1 ln 1

4 1 ln

1

n n

d n

n n

n

n

x x

x T n x

x n T

x

x

n x

x

n x

 

 

 







 

          

 

 

 

  

 

   

 

           

Logarithmic decrement

System Analysis Spring 2015

16

1

2 2

2 2

1

2

( ) (0) cos tan

1 1

1 1

1 , ,

tan 1 2

w tn

d

d n d

n

x t x e t

t t



 

 

    

 

  



 



 

 

       

    

  





n

1 2

d n

  

Estimate of Response Time

End of Lecture 10-2