Steady-State Response Analysis

System Control System Control

4. Transient and

Steady-State Response Analysis

Professor Kyongsu Yi

©2010 VDCL

©2010 VDCL

Vehicle Dynamics and Control Laboratory Seoul National University

Seoul National University

Systems

• Linear Time Invariant System

x  AxBu

Where A,B,C and D are constant matrix

• Linear Time Varying System

y Cx Du Where A,B,C and D are constant matrix

Linear Time Varying System

( ) ( ) ( ) ( ) x A t x B t u y C t x D t u

 

 



• Nonlinear System

( ( ), ( ), ( ), ) x  f x t y t u t t

( ( ), ( ), ( ), ) y  h x t u t y t t

Time Invariant System

• The Laplace Transform of Linear Time Invariant System

) ( )

( )

0 ( )

(s x AX s BU s

sX   

) ( )

0 ( ) ( ) (

) ( )

0 ( ) ( )

(

s BU x

s X A sI

s BU x

s AX s

sX













) ( )

( ) 0 ( ) (

) 0 ( ) (

)

(s sI A ¹x sI A ¹x sI A ¹BU s

X   ^   ^   ^

X(s) (sI A) x(0)(sI A) x(0)(sI A) BU(s)



Transfer function is derived from zero-initial condition

1 ] ( )

) (

[ )

(s C sI A ¹B D U s

Y   ^ 

, thus the Transfer function G(s) is

D B A sI C s

G   

G(s)  C(sI  A)^1B D

 ( ) ( )

In general the Transfer function is expressed as follows )

( )

) (

( _{1 1}

n b m

s s b

s G

Y ^m   _m 

) (

) ) (

( ) (

1 1

1

1 m n

s a s s

s G

U ^{n n}

m 







 

 _ ^





 



 

 

 

b as s

c s c p

s k p

s k

2

2 1

2 2

1 1

System Response

,,,

• Transient response

- Response goes from the initial state to the final state

• Steady state response

-The manner in which the system output behaves as t approaches infinityThe manner in which the system output behaves as t approaches infinity let then

th h t i ti ti

) (

) ) (

( P s

s s Q

G 

0 ) (s

P : the characteristic equation

: such that is characteristic roots or poles : such that are called zeros

0 ) (s  P

s

0 ) (s  Q

0 ) (s  P

s

) ( ) ( )

(s G s U s

Y 

 Partial Fraction ={ G(s) terms }+ U(s)

 Poles s₁,s₂,s₃(real),₁  j₁,₂  j₂

 Poles s₁,s₂,s₃(real),₁  j₁,₂  j₂

 Then the transient response becomes















C e D e t

e

C₁ ^s1t ₂ ^s2t ₁ ^1t sin₁



Stability

,,,

• Stable

If with no zero initial condition^{lim ( )  0}



 x t

t

A linear time invariant system is “stable” if the output eventually comes back to equilibrium state when the system is subject to an initial condition

• Equilibrium :

With no disturbance and input, the output stays in the same state, which is called equilibrium.

• Stable condition

for all , where is poles 0

)

Re(s_i 

s

s

• Critically stable

U t bl

Oscillations of the output continue forever some Re(s_i)  0

• Unstable

The output diverges without bound from its equilibrium state (when the system subjected to an initial conditions)

Stability

,,,

• Absolute Stability

Whether the system is stable or unstable

• Relative Stability

- Transient response

• Steady-state Error

p

- Damped Oscillation

• Steady state Error

The output does not exactly agree with the input ( Concerned with the Accuracy of the system)

First Order System

,,,

1 )

( C

1 1 )

( ) ) (

(   

Ts s

R s s C

G

When ^{R(s) 1} ; step input ( r(t)=u(t) )

• First Order System

When ; step input ( r(t)=u(t) )

s s R )(

s Ts

T s

s Ts

c 1

1 1

1 ) 1

( 



 

 



1 ) exp(

1 )

( t

t T

c    c t( ) ¹ e ^T1t T

 









 ) 1

( b b

G





 



 

 



1 1 ) 1

(

a s a b a s s b G

for First Order system T : time constant of First order system

For large T : 응답이 느리다 For small T : 응답이 빠르다

for First Order system, the time constant is _a¹

First Order System

,,,

When ; unit ramp input, that is, r(t)=t2

( ) 1 R s  s

T s T

c  1  1₂  ²  1₂  )

( ) Ts1 s² Ts1 s² s (

T t Te

t

c  ^tT  

 ( ) ^

Second Order System

,,,

b as s

s b s G

R s C



 

 ( ) ₂ )

( ) (

• Second Order System

b as s

s

R( )  

where a  2n b _n²

2 2

2

2 _{n n}

n

s

s  





 

n

 n

Note that poles : _n _{n 1} ² _j U it t

Unit step response

for ; step input

s s

R 1

) ( 

2 1 1 s 2

 

2 2

2

2 2

2 1

1 ) 2

(

n n

n n

n n

s s

s s

s s

s s

C  













 



 

 

1 ) tan sin(

1 1 ) (

2 1

2 

 



 







 e^ t ^

t

c _d

nt

1 ² 

where _{d n 1} ² : Damped natural frequency N t l f

n : Natural frequency

Second Order System

,,,

1. Rise time : 10%  90%

5%  95%

t

2. Max. Overshoot, M_p

3. Settling time, t_{g , ss} : 2% criterion t_s  4 / _n 5% criterion

4. Delay time, t_d = 50%

s n

s 3 / n

t   

Second Order System

,,,

• Step Response of Second Order System

2

2 2

( )

( ) 2

C s n

R s s s



 

  

0  1 (1) Under damped :

( ) 2 _{n n}

R s s   s

1 2

d n

  

  

( ) 2

( )

n

n d n d

C s

R s s j s j



   

    

St ¹

Step response

s s

R 1

) ( 

 

1

( ) ( ) 1 cos 2 sin

1

nt

d d

c t L C s e ^  t   t





  

 

   _   _

2 1

2

1 cos tan 1

1

nt

d

e t

  

 

     

  

 

 

   

Second Order System

,,,

• Step Response of Second Order System

 1 (2) Critically damped :

 

 

2

( ) 2

( ) 1 ⁿ 1

n n t

n

C s

s s

c t e ^ t









 

 





 1 (3) Critically damped :

  

2

2 2

( ) ⁿ

C s 



  ^{ }   ^{ }

2 1 2 1

2 2 2 2

1 1

( ) 1

2 1 1 2 1 1

nt nt

c t e ^{  } e ^{  }

     

     

  







   

2 2 2 2

2  1    1 2  1    1

1 2

2

1 1

2 1

s t s t

e e

s s



 

 

    

 

 

 

2 1

2

1 1

s   n

 

  

2

1 2

2   1 s s  ^{s2 }





1 2

s  s

Effect of Pole Locations

,,,

(1) First Order System

( ) ; 1

1 Y G s

R s Ts



   

 1

R s Ts

Step response :

s s

R 1

) ( 

1 1

( ) 1

Y s  Ts s



1 T

1

1 T s Ts

  

 

1 1

1 /

s s T

 ^{s s}^





1

( ) 1 ^Tt

y t  e^ for t  0 Pole : s     1 /T

,,,

Effect of Pole Locations

2

Y b

R  s as b

  where a  2n b _n²

(2) Second Order System

R s as b

Poles :   1  ² j

2

2 2

2

n

n n

s s



 

  

2

n

Step response :

s s

R 1

) ( 

2 1

Poles :  _n  _n 1 ² j









   

1 2

d n

  

2

2 2

( ) 1

2

n

n n

Y s s s s



 

 

 

2 2

2 1

2 s n

 

  

 

n



2 1

2

( ) 1 sin tan 1

1

nt

d

y t e t

  

 

    

 

     

2 2

2 _{n n}

s s   s

n







 



: damping ratio



: natural freq.

n ^_ⁿ_^cos_cos^{ }_^{ n}



Pole Locations and Transient Response (Impulse)

,,,

Effects of Zeros

2 2 2

1. The effect of zero near poles (cancel the pole response)

  

 

  

1

2

2 2 2

( ) 1 2 1 2

2 1.1 2 0.1 0.9 0.18 1.64

( ) 1 1 1 2 1 1 1 2 1 2

H s s s s s

H s s

s s s s s s

  

   

  

           

• If we put the zero exactly at s=-1, this term will vanish completely

  

1.1 s1 s2 1.1s1 s2 s1 s2

If we put the zero exactly at s 1, this term will vanish completely

• The coefficient of the term (s+1) has been modified from 2 in H₁(s) to 0.18 in H₂(s)

I l l d th t f th t t i th t t l

In general, a zero near a pole reduces the amount of that term in the total response

coefficient ^{c s}₁^{( ) }





zero near the pole P₁, F(s) will be small

 

1 1 s p 1

,,,

Effects of Zeros

Two complex poles and one zero

2. Effect of zeros on the transient response

 

   

2

2 2 2

/ 1

( ) / 2 / 1 2

n

n n

n n

n n

s s

H s s s s s

 

 

 

  

 

 

 

 

poles : zero :

1 2

n n

s     j

s  n

 1 : the value of the zero will be close to that of the real part of the poles

 3 : very little effect on Mp

3₃ : increasing effect as  decreases below 3

  : increasing effect as decreases below 3

,,,

Effects of Zeros

Replacing with

2. Effect of zeros (L.T. Analysis) s

/

s 

2

/ 1

( ) 2 1

H s s

s s





 

 

2 2

1 1

2 1 2 1

s

s s  s s

 

   

 

0 0

0

( ) ( )

( ) 1 _d( )

h t d

dth t

H s H s

 

: produce overshoot

,,,

Effects of Zeros

: the zero is in the RHP where s>0

3. Nonminimum-phase zero

  0

RHP

; RHP zero

nonminimum-phase zero

,,,

The Effect of an extra pole

   

 

( ) 1

/ 1 / 2 / 1

H s

s  s   s 

     

• Effect on the Standard Second-order step response



 







DC i f t

• DC gain of a system

: the ratio of the output of a system to its input (presumed constant) after all transients have decayed

DC gain 1

0 0

lim ( )1 lim ( )

s s G s s G s

 s 

  

Effect of Poles-Zeros on Dynamic System

1. 2

order system with no finite zeros

Rise time : Overshoot :tr  ^1.8

5%,

p 16%, M



 

0.7 0.5







 Settling time :

r

n ^6%,

35%,

p 

 0.3



  4.6

ts

    _n

2 A Zero in the LHP 2. A Zero in the LHP

Increase the overshoot

(if the zero is within a factor of 4 of the real part of the complex poles)

3. A Zero in the RHP (nonminimum-phase zero)

- Depress the overshootDepress the overshoot

- May cause the step response to start out in the wrong direction

4. An additional pole in the LHP

- Increase the rise time significantly if the extra pole is within a factor of 4 of the real part of the complex poles

of 4 of the real part of the complex poles