5. Basic Control Actions and Response of Control Systems

System Control

Professor Kyongsu Yi

©2014 VDCL

Vehicle Dynamics and Control Laboratory Seoul National University

5. Basic Control Actions and Response of Control Systems

(1) PID Control (2) Routh’s stability tests

(3) Systems Type

Open loop and closed loop : model sensitivity

2

•Routh stability criterion

•Effects of Integral and derivative control

•Steady state error in unity feedback control systems

•Open loop and closed loop systems

Control Systems

u=f(r,c)

u

4

1. ON-OFF control

Classifications of controllers

h(t)

u

e

u

t

2. Proportional Control

Classifications of controllers

e k

U =

_p

k

_p

: The proportional Gain

J

MOTOR

+

-

Disturbance

Antenna Position

Control System

6

2. Proportional Control

Classifications of controllers

e k

U =

_p

k

_p

: The proportional Gain

k

^p _Js²¹_+Bs ^c

r e

2

2 2

c(s) Js Bs

r(s) Js Bs

1 Js Bs

p

p

p p

K

K

K K

= + =

+ +

+ +

=

+ -

2. Proportional Control

Classifications of controllers

=

• Step input response 1 ( ) r s = s

• steady state response

0

0 2

( ) lim ( )

lim 1 1

s

p

s p

c t sC s s K

Js Bs K s

→

→

=

= =

+ +

θ

8

2. Proportional Control

Classifications of controllers

=

• Response to torque disturbance

+ _

+ D

_ ₂1

Js +Bs

K

P

r=0 c

2 2

( )

^p

( ) 1 ( )

p p

C s K r s D s

Js Bs K Js Bs K

= +

+ + + +

Assume that 1 ( )

D s = s steady state response

p

s

s Js Bs K

p

s K

t

c 1 1 1

)

( = lim ⋅

₂

+ + ⋅ =

∞

>

−

: steady-state error

Large  small steady-state error  large motor power is needed  oscillations

 large

 small damping ratio K

p

w

n w_n ^Kp J

 

=

 

 

 

JKp

B

= 2 ζ

3. Proportional-Integral Control (PI control)

Classifications of controllers

=

• Response to torque disturbance

( ) 1

( ) ^P 1 _i

U s K

E s T s

 

=  + 

 

+ e _

D

r

1

u c

P

1

i

K T s

 

 + 

 

²

1 Js + Bs

0

( ) ( ) ( )

t P p

i

u t K e t K e t dt

= + T

∫

3 2

1 ( )

( )

P P

i

P P

i

K s K

T C s

R s Js Bs K s K T +

=

+ + +

No steady-state error for reference input

10

3. Proportional-Integral Control (PI control)

Classifications of controllers

=

+ e _

D

r

1

u c

P

1

i

K T s

 

 + 

 

²

1 Js + Bs

2

2

3 2

1 ( )

1 1

( ) 1

_P

(1 )

i

P P

i

C s Js Bs

D s K

T s Js Bs s

Js Bs K s K T

= +

+ +

+

=

+ + +

( ) 1 D s = s

0 3 2

( ) lim 1 0

s P

P

i

C t s s

K s Js Bs K s

T

=

→

=

+ + +

No steady-state error for step disturbance

4. Proportional-Derivative Control (PD control)

Classifications of controllers

=

+ e

r _ c

P d

K + K s

₂

1

Js + Bs

( ) _P ( ) _d d ( ) u t K e t K e t

= + dt

2

( )

( ) ( )

P d

d P

K K s C s

R s Js B K s K

= +

+ + +

P d

JK K B 2

= +

ζ : increased effective damping

u

12

5. Proportional-Integral-Derivative Control (PID control)

Classifications of controllers

=

2

3 2

2

1 1 1

( )

1 1 1

( ) 1

i i

i i

T s Js Bs T

C s

R s Js Bs

T s Js Bs T

= + =

+ + +

+

+ e

r _

1

c

c

1

d

i

K T s

T s

 

+ +

 

 

²

1 Js + Bs

6. I control

+ e

r _

1

c

T s

i ²

1 Js + Bs

u

u

Ziegler-Nicholas Tuning Rules for PID controllers

=

+ e

r _

1

c

c

1

d

i

K T s

T s

 

+ +

 

  plant

u

Method 1) Transient response method

PLANT

R(max slope)

L

Step input

1 0.9, 3.3

1.2, 2 , 0.5

c

c i

c i d

K for P control RL

K T L for PI control

RL

K T L T L for PID control

RL

 = −



 = = −



 = = = −



• This method works good if the unit step response is (sigmod) shaped

∫

14

Ziegler-Nicholas Tuning Rules for PID controllers

=

Method 2) Ultimate sensitivity method PLANT

+

_

K

^c

c

1,

n

K increase

decrease ω

ζ

〉〉 →

→

• Increase until you hit the stability limit K

c

0.5

0.45 , 0.83

0.6 , 0.5 0.125

c U

c U i U

c U i U i U

K K for P control

K K T P for PI control

K K T P T P for PID control

= −

  = = −

  = = = −



Routh’s stability criterion

16

Stability of Feedback Systems

=

) (

) ( )

( ) ( 1

) ( )

( ) (

s Q

s P s

H s G

s G s

R s

C =

= +

Poles : s

i

Re( )s_i <0

; stable Characteristic equation

Q s( ) =0

0

0

1 1

1

+ ⋅ ⋅ ⋅ + + =

+ b

₋

s

⁻

b s b s

b

_n ⁿ _n ⁿ

Routh’s Stability Criterion

=

Characteristic equation

0

0

1 1

1

+ ⋅ ⋅ ⋅ + + =

+ b

₋

s

⁻

b s b s

b

_n ⁿ _n ⁿ

Number of the characteristic roots with positive real parts

=Number of sign changes of the first column have the same sign

• Rooth’s Criterion

18

=

Routh’s Stability Criterion

Example 1)

240 152

72 10

) (

2 3

4

5 + + + + +

= s s s s s

s P R

C

Two sign change

 Q(s)=0 has two roots with positive real parts

 unstable

=

Routh’s Stability Criterion

Example 2)

 prevents completion of the array method 1) 1

s = x

4 2 3

4

2 2 1

) 5

( x

x x x

s x

Q = + + + +

two sign change  unstable

method 2)

set Q s s ( )( + = 1) 0

20

=

2

2 4 3 2

2

( ) ( 1)( 2)

( ) ( 1)( 2) 3 3 2

1 ( 1)( 2)

K

C s s s s s K K

R s K s s s s K s s s s K

s s s s

+ + +

= = =

+ + + + + + + +

+ + + +

7 0

2− 9K >

⇒

^{K <14}₉ ^{K >0}

0 14 K 9

∴ < <

Example 3)

Routh’s Stability Criterion

Analog/Digital Controller

22

Steady-state error and system types

Steady state Error in Unity Feedback control systems

=

R

₊

E (s ) G ( s ) C ( s ) -

) 1 (

) 1 )(

1 (

) 1 (

) 1 )(

1 ) (

(

2

1

+ + ⋅ ⋅ ⋅ +

+

⋅

⋅

⋅ +

= +

s T s

T s

T S

s T s

T s

T s K

G

p N

n b

a

N=0 ; Type 0, N=1 ; Type 1, N=2 ; Type 2

) ) (

( 1

) 1

( R s

s s G

E = +

Steady state Error

24

Steady state Error in Unity Feedback control systems

= 1) 일 경우 (unit step input), s s

R 1

)

( =

0 0

1 1 1

lim ( ) lim

1 ( ) 1 (0)

ss s s

e sE s s

G s s G

−> −>

= = =

+ +

Static position error constant

0

; 0

lim ( ) (0) ; 1

; 2

p s

k Type system

k G s G Type system

Type system

−>



= = = ∞

∞

1 ; 0

1 1

1 0 ; 1, 2,..

ss p

p

Type system e k

k Type system

  +

= + =  



kp

Steady state Error in Unity Feedback control systems

=

2) ^{R s( ) 1}₂ 일 경우 (unit-ramp input),

= s

0 2 0 0

1 1 1 1

lim lim lim

1 ( ) ( ( ) 1) ( )

ss s s s

e s

G s s s G s sG s

−> −> −>

= = =

+ +

Static velocity error constant _kv

0

0 ; 0

lim ( ) ; 1

; 2, 3..

v s

Type system

k SG s k Type system

Type system

−>

 

= = 

∞ 

; 0

1 1

; 1

; 2, 3,..

0

ss v

Type system

e Type system

k k

Type system

 ∞

= =  

 

26

Steady state Error in Unity Feedback control systems

3) ^{R s( ) 1}₃ 일 경우 (unit-acceleration input),

= s

3 2

0 0

1 1 1

lim lim

1 ( ) ( )

ss s s

e s

G s s s G s

−> −>

= =

+

Static acceleration error constant

k

_a

2 0

0 ; 0

0 ; 1

lim ( )

; 2

; 3

a s

Type system Type system

k S G s

k Type system Type system

−>



= = 

∞



; 0,1

1 1

; 2

0 ; 3, 4,..

ss a

Type system

e Type system

k k

Type system

 ∞

= =  

 

Open/Closed loop control systems

=

R Ts +1

k kc

controller open loop y

closed loop R

+1 Ts

k kc

controller

y

Open loop Transfer function 에서

1

c

1

c

y k

k k

R = Ts → = k

+

^{이되어야 error=0 이된다.}

Closed loop Transfer function 에서

k k Ts

k k Ts

k k Ts k k R

y

p p

p p

+

= + + +

= +

1 1 1

1

Steady/ state error for 1 ( ) R s = s

Open loop Transfer function 에서 _{= 0} ess

k k k

k k e k

p p

p

ss = +

− +

= 1

1 1 1

Closed loop Transfer function 에서 +

-

28

Open/Closed loop control systems

=

Model error sensitivity 1 ,

10 ∆ =

= k

k model error

Open loop Systems

1 , ( ) 1.1

1

y k k k k

R k Ts y t k

+ ∆ + ∆

= = =

+

(10% steady state error)

Closed loop Systems

k e k

p

ss

= +

1

1 100 1 1

0.0099 100 101

1

p ss

k e

k k

k

= ⇒ = = =

+

let

,

1 1 1

0.00901

100 1 10(11) 111

1 ( )

ss

k k

e

k k

k + ∆

= = = =

+ + ∆ +

End of section 5

30

Higher Order Systems

= n n

n n

m m

m m

a s a s

a s a

b s b s

b s b s H s G

s G s

R s C

+ +

⋅

⋅

⋅ + +

+ +

⋅

⋅

⋅ +

= +

= +

−

− −

−

1 1

1 0

1 1

1 0

) ( ) ( 1

) ( )

( ) (

• unit step response

R s( ) ¹

= s

s a s

a

b s

s b

C _n ^m

m 1

) (

0 0

0 ⋅

+

⋅

⋅

⋅ +

+

⋅

⋅

⋅

= + ,

characteristic equation

1 0

1 1

0sⁿ +asⁿ⁻ +⋅⋅⋅+a_n− s+a_n = a

pi

s= i=1,⋅⋅⋅q

j

s=−ζ_kω_k ± 1−ζ_k²ω_k k =1,⋅⋅⋅r

zero ;

s=Zi i=1,⋅⋅⋅m

2 1

2 2

2 2 1 1

1 1

( ) ( ) 1

( ) 2

( ) ( 2 )

m

q r

i j k k k k k k

i

q r

j i r k k k

i k k k

j k

k s z a a b s C

C s s s p s s

s s p s

ζ ω ω ζ

ζ ω ω ζ ω ω

=

= =

= =

Π − + + −

= = + +

− + +

Π − Π + +

∑ ∑

∑

∑

∑

=

−

=

−

=

− +

− +

+

= ^r

k

k k

t k r

k

k k

t k q

j

t p

je b e t C e t

a a

t

C ^{i k k k k}

1

2 1

2 1

1 sin 1

cos )

( ^{ζ ω} ω ζ ^ζω ω ζ

Effect of sensors on system performance

=

• Fast sensor dynamics ; H(s)=constant - First order sensor

- Overdamped second order - underdamped second order

32

=

Phase Lead and Phase Lag in Sinusoidal Response

) (

) (

s X

t x

) (

) (

s Y

t ) y

(s G

em Linearsyst

ω ω

ω ω ω

ω

j s

a j

s a s

s b s

s b s

s b

s s X G s X s G s Y

s s X X

n n

+ − + +

+ +

⋅

⋅

⋅ + +

+ +

=

= +

=

⇒

= +

2 2 1

1

2 2 2

2

) ( ) ( ) ( ) ( ) (

위 system 에 x t( )= X sinωt 를 넣어주면 y t( )는 ?

1) Transient response  Stable system인경우 = 0 2) Stable system인 경우 Steady state response 는

) ( ,

) ( ),

sin(

) sin(

) 2 (

) ( )

(

, ) ( )

(

) ( )

(

2 ) ) (

( )

(

2 ) ) (

( )

( ) (

) ( )

( 2 2

2 2

ω φ

ω φ

ω

φ ω ω

ω

φ ω

ω

ω ω

ω ω ω

ω

ω ω ω

ω

φ ω φ

ω φ φ

ω ω ω

ω

j G j

G X Y t

Y

t j

G j X

e j e

G X t y

phase e

j G j

G

e j G j

G

j j j XG

s s s X G a

j j j XG

s s s G a

e a ae

t y

t j t

j j j

j s

j s t

j t

j

∠

=

= +

=

+

− =

=

∴

=

=

−

=

= + −

=

− −

= +

+ −

=

+

=

+

− +

−

=

−

=

−

=

Phase Lead and Phase Lag in Sinusoidal Response

2

1 2 2

1 2 2

( ) 1

( ) , ( )

1 1

( ) , ( ) , ( ) tan

1 1

( ) sin( ), tan

ss 1 G s K

Ts

K K

G j G j

Tj T

K K

G j G j G j T

Tj T

y t X K t T

T

ω ω

ω

ω ω ω ω

ω ω

ω φ φ ω

ω

−

−

= +

= =

+ +

= = ∠ = −

+ +

∴ = + = −

+

< 0

φ : phase lag, φ > 0 : phase lead

Example) First order system

0 2 4 6 8 10 12 14 16 18 20

-3 -2 -1 0 1 2 3

x(t) y(t)

0 2 4 6 8 10 12 14 16 18 20

-3 -2 -1 0 1 2 3

End of section 5

5-6 routh stability criterion

5-7 Effects of Integral and derivative control 5-8 steady state error in unity feedback

control systems

Next Root locus analysis