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(1)

## System Control

6-2. Motor Control

P f K Yi

Professor Kyongsu Yi

Vehicle Dynamics and Control Laboratory Vehicle Dynamics and Control Laboratory

Seoul National University

(2)

Motor Position Control

c( )

G s 1

A

s

1 s e

R

u

1. Proportional

( ) G s  K

 A 1 

D  s  s 

c

( )

p

G s  K 1

1 ( )

e R

G s A

 

1 ( )

( 1) 1

1

c

p

G s s s K A R

 

 

: Type 1

p

1 

DG K A

s  s

 

1  s s (   1) s   s  1 

2

(3)

Motor Position Control

Unit step response : R=1 [rad]

0

1 1 1

lim 1 1 (0)

ss s

e s

DG s DG

 

1 0

 1 

 

: no steady state position error

Ramp input response : R=t [rad]p p p [ ]

0 2 0

1 1 1 1

lim lim

1 1

ss s s p

e s

DG s K A s

 

 

 

0

1 1

1 1

lim

s p p

s s K A K A

s

 

 

: finite error

1

p p

s   s 

3

(4)

## Motor Position Control

2 PI control

( )

P

K

I

D G s K

   s

2. PI control

: Type 2

  s

 

2

1

P I

K s K A DG s  s

 

Ramp input response : R=t [rad]

 

2

1 1 1

e 1 R

 DG 

 

 

2

2

1 1

1 1

p I

DG K s K A s

s  s

 

 

2

 

1 1

p I

K s K A

s  s

 

 

   

2

1

p I

s  s K s K A

   

lim ( ) 0

e

ss

 se s 

: no steady state position error

0

( )

ss s

4

(5)

Ch 4. Basic properties of Feedback

Feedback control system

### 1. Stability

2. Performance

Disturbance Rejection

transient (dynamic properties)

3. Robustness : sensitivity of the system to change in model parameter

5

(6)

## Stability of Feedback Systems

=

) (

) ( )

( ) ( 1

) ( )

( ) (

s Q

s P s

H s G

s G s

R s

C 

 

Poles : s

i

Re( )s  0

; stable

Re( )si  0

; stable Characteristic equation

Q s( )0

1

### 0

b b

b

b s

n

 b

1

s

n1

     b

1

s  b

0

 0 b

n n n n

6

(7)

Routh’s Stability Criterion

Ch t i ti ti

=

Characteristic equation

0

### 0

1 1

1

      

 b

s

b s b s

b

n n n n

Number of the characteristic equation with positive real parts

• Rooth’s Criterion

7

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

(8)

Routh’s Stability Criterion

E l 1)

=

## 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

 unstable

8

(9)

Routh’s Stability Criterion

E l 2)

=

Example 2)

 prevents completion of the array method 1)

s 11

x

4 2 3

4 2 2 1

) 5

( x x x x

s

Q    

method 2)

set Q s s ( )(   1) 0

x4

set

9

two sign change unstable

(10)

E l 3)

Routh’s Stability Criterion

=

## Example 3)

2

K

2

2 4 3 2

2

( ) ( 1)( 2)

( ) ( 1)( 2) 3 3 2

1 ( 1)( 2)

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

29K

K 149 K 0

10

0 14

K 9

  

(11)

Analog/Digital Controller

11

(12)

DC motor

J 



 b 

 K i  T

m m m t a l

a

a a a a e m

J b K i T

L di R i v K dt

 

  

  

 L

a

 0  T

l

J 



 b 

 K i  T

m m m t a l

e m a a a

J b K i T

K R i v

 

  

 

12

(13)

## DC motor

, ( ) ( )

m m m

s s s

    

Speed control

1 ( ) ( ) ( )

m a t a

m a l

J R K R

s s V s T s

bR K K bR K K bR K K

 

 

      

  

 

 bR

a

K K

t e

 bR

a

K K

t e

bR

a

K K

t e

  

 

 



  s 1 

m

( ) s AV s

a

( ) BT s

l

( )

    

: A

Time constant

:

: A B

13

(14)

## DC motor

( ) ( ) ( )

( ) ( ) ( )

m m m t a l

e m a a a

J s s b s K I T s K s R I s V s

    

  

Speed control

 

( ) ( ) ( ) /

a m m l t

I s J s b s T s K

R R

        

 

 

( )

a

( )

a

( ) ( )

e m m m l a

t t

R R

K s J s b s T s V s

K K

          

  ( ) ( ) ( )

1 ( ) ( )

e t a a m m t a a l

m a t

m a

a t e a t e

K K bR R J s s K V s R T s

J R K

s s V s

bR K K bR K K

     

 

         

a l

( )

a t e

R T s

bR K K

 

a t e a t e

 

a t e

( ) ( ) ( )

1 1

m a l

A B

s V s T s

s s

 

  

 

v

a

 a T ,

l

 b w

ss

 Aa  Bb

14

(15)

## DC motor

Open loop control

B Tl

ref

Hr Dol

1 A

s

m

A Va

1 sec

r

H volt

 

( open-loop transfer function)

ol ol

K volt D open loop controller

volt

   

m

A

  H K

( open loop transfer function)

D  K 

 

r ol

1

ref

H K  s

  

1

a ol ref ol ref

ss ol ref

v D K

w A K K

 

    

   

 

 

 K

ol

  A



T

l

 0

15

(16)

## DC motor

Closed-loop(transfer function) control

B A Tl

ref

Hr Dcl

1 A

 s

m

A Va

H

1 volt sec

H 

cl cl

K volt D closed loop controller

volt

   

Hy

y

1

H sensor

1 sec

r

H Input shaping volt

   

K  A B

( ) ( ) ( )

1 1

cl

m ref l

cl cl

K A B

s s T s

s K A s K A

 

    

     

( ) ( ) 1

m cl r cl

D A

s T s H   s H D A

  ( )     

( ) 1 1

1

cl r

ref cl y

cl y

T s H

s D H A s D H A

s

 

  

      

cl

f

  A K  

, then

,

1

m ss ref

A K

cl

 

 

cl

1

A K 

m ss,

 

ref

16

(17)

DC motor

Disturbance Rejection (non zero )

ss

A K

ol ref

B T

l

      

T

l

ss ol ref l

ref

B T

l

  

Closed-loop

1 1

cl

ss ref l

cl cl

A K B

A K A K T

     

   

17

(18)

## DC motor

Numerical Examples

1 sec, 10, 50, 100 / sec, 0.1

60 A B

ref

l

      

### Open-loop

  

100 50 0.1 95 / sec 5%

ss

   

### Closed-loop

A K B

0, 99 100 99 / sec

1 100

cl

l ss ref

cl

 A K  

   

 

0.5

99 0.05 98.95

1 1

cl

ss ref l

cl cl

A K B

A K A K T

        

 



 

로 만들자

Reject disturbances by a factor of at least 100 compared with the open loop system, 1 A Kclcl 100

cl

9.9

K 

0.051% variation from the speed w/o disturbance

18

(19)

DC motor

Sensitivity of System Gain to Parameter Changes

Open-loop

( ) ( ) ( )

1 1

m ol ref l

A B

s K s T s

s s

 

     

 

Closed-loop

( ) ( ) ( )

1 1

cl

m ref l

cl cl

K A B

s s T s

s K A s K A

 

     

     

 

cl( ) T s



19

(20)

## DC motor

Sensitivity of System Gain to Parameter Changes

### Motor Gain Variations

A   A  A

Open-loop

: the nominal gain ol ol ol

 

ol ol

T T K A A

K A K A

 

  

 

ol ol

T K  A

  ( 0)

ol ol

T  T s 

ol ol

T K A

ol

ol

 

ol ol

T A

T A

 

; 10% error in A would yield a 10% error inTol

ol

1

ol

T A

T A

  

; sensitivity, s, of the gain from to with respect to the parameter A

ref

m

20

(21)

## DC motor

Sensitivity of System Gain to Parameter Changes

### Motor Gain Variations

A   A  A

AA K

A K

 

 

1

cl

cl cl

cl

A A K

T T

A A K

 

 

 

  

1

cl cl

cl

T A K

A K

 

 

dTcl

T A

  Tcl A dTcl A

Closed-loop

cl

Tcl A

  dA 

cl cl

T T dA A

cl cl

cl cl

dT A

T A

T A T dA

  ; sensitivity

Tcl

A cl

cl cl

S sensitivity of T with respect to A A dT

T dA

 

   

 

2

1 1 1

cl

cl cl cl cl

cl cl

cl

A K K K A K

A

A K A K

A K

    

    

  1

1 A Kcl

   ; less sensitive compared to open loop

STol 1 ;

1 1

1 ;

  

1 10 9.9 100

ol

SA   

21

(22)

## Open/Closed loop control systems

controller

=

R Ts 1

k kc

controller open loop y

controller closed loop R

1 Ts

k

kc y

Open loop Transfer function 에서

k k k Ts

R y

c c

1 1

1

이 되어야 error=0 이 된다. Closed loop Transfer function 에서

k k Ts

k k k

Ts k k R

y p p

1 1

k k Ts

Ts k k

R p

p

1

1 1

( ) R s

s

Open loop Transfer function 에서 e ss 0

k k k

k k e k

p p

p

ss

1

1 1 1

Closed loop Transfer function 에서

22

p p

(23)

## Open/Closed loop control systems

=

예외의 경우에 대한 예>

1 ,

10  

k

k model error 가 있을 경우

Open loop Transfer function 에서

1 , ( ) 1.1

y k k k k

  y t  

 , ( )  

1

c

R k Ts  y k

( y )

Closed loop Transfer function 에서

k e k

p

ss

 

1 1

101 1 1 100

1

100 

k k e

kp ss

let

1 k

k

1 1 1

, 0.009

100 ( )

k   k , e

ss

   

100 1 10(11) 111

1 ( )

ss

k k k

   

23

(24)

The Classical Three-term Controller (PID controller)

## Motor Speed Control (unity feedback)

Tl

 

0

( )

( ) ( )

t

a P I t D

a I

v K e K e d K de dt

V s K

U s D s K K s

 

   

    

Dc

e

ref

va

1 A

s /

B A

m

( ) ( ) D sc( ) KP K sD E sE s    s

s1

P- control

( ) ( ) ( )

1 1

cl

m ref l

cl cl

A K B

s s T s

s A K s A K

 

    

     

0 0

Kcl KP, KI 0, KD 0 KK KK

PI- control

 

ref m

a P ref m I

v K K

s

  

    

 

   

2 2

( ) ( ) ( )

1 1

P I

m ref l

A K s K Bs

s s T s

A K A K A K A K

 

    

( ) I

c P

s D s K K

  s

24

   

2 2

( ) ( ) ( )

1 1

m ref l

P I P I

s A K s A K s A K s A K

           

Steady statem ref   0 Tlref

(25)

The Classical Three-term Controller (PID controller)

Motor Speed Control (unity feedback)

PID- control ( )

( ) ( )

I

c P D

K

D s U s K K s

E s s

   

( )

1 1

P D

I

E s s

K T s

T s

 

    

 

0

Non-negligible armature inductance LLa  0

    

0 2 1 2

( )

 

a a m t e m t a a a l

m t a a a l

L s R J s b K K K v L s R T a s a s a s K v L s R T

        

 

 

 

    

PID control, Va KP KI K sD

ref m

s

 

       

K K

0 2 1 2 tP KI D  m( ) tP KI Dref( )

a a

l

a s a s a K K K s s K K K s s L s R T

s s

    

               

   

2

3 2

( ) t D t P t I

m

K K s K K s K K

s K K K K K K

 

  

   

 

   

3 2

0 1 2

2

3 2

0 1 2

m( )

t D t P t I

a a

l

t D t P t I

a s a K K s a K K s K K L s R s

a s a K K s a K K s K K T

    

 

     

25

Ex 4.6) Ch.eq.

3 parameters for 3 ch. Roots ; arbitrary

   

3 2

0 1 t D 2 t P t I 0

a s a K K s a K K sK K

(26)

## Ziegler-Nicholas Tuning Rules for PID controllers

+ e

_

r

1

c

c

1

d

i

K T s

T s

 

 

 

 

u

plant

Method 1) Transient response method

R(max slope)

Method 1) Transient response method

PLANT

L

Stepinput

1 0.9, 3.3

c

c i

K for P control RL

K T L for PI control

RL

input

1.2, 2 , 0.5

c i d

K T L T L for PID control

RL



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

26

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

## shaped

(27)

Ziegler-Nicholas Tuning Rules for PID controllers

Method 2) Ultimate sensitivity method Method 2) Ultimate sensitivity method

PLANT

+

_

K

c

1

K

c

 1, 

n

 increase

K increase

decrease

 

• Increase until you hit the stability limit K

c 0.5 U

KK for P control

 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

   

    

27

(28)

PID controllers – reviews, gain tuning

Gd

d

c( ) e G s

R

u y

GP

1) ( )

P

( )

I t

( )

D

de u t K e t K e t dt K

  

0

 d

( ) ( ) ( )

( )

P I t D

I

c P D

dt

G s K K K s

  s 

 

1 1

P D

i

K T s

T s

 

    

 

## 2) Ziegler-Nichols Tuning Rules

3) I - control : no steady-state error, disturbance rejection

28

3) I control : no steady state error, disturbance rejection

D - control : additional damping

(29)

Integrator Anti-windup

PI- control

saturation

u I

P

K K

s e

R

uc y

Plant u

uc

u um in

um ax

1. Large R Large saturation integration of error, Large increases until the sign of the error changescvery large overshoot

u u u

c

u

increases until the sign of the error changes very large overshoot (wind up) poor transient response

u

c

### 2. Solution to this problem : integrator anti-windup

- “ turn off” the integral action when the actuator saturates Digital control :

(i l t ti t max) Fi 4 22

" if u  u , K

I

 0"

(implementation-next page) Fig 4.22

29

(30)

## Integrator Anti-windup

PI- controller with anti-windup (Fig 4.22)

i KP

KI

s e

R

uc u

saturation

uc

u um in

um ax

uI

K u

y

m a x

u

Ka uc

KP

(a)

KI

s e

R

uc u

saturation

uc

u um in

um ax

KP

uI

u  u

y

Ka

saturation :

u   u

max

30

(31)

## Integrator Anti-windup

First order lag equivalent (during saturation)

e K sP KI uc

a I

sK K

(d)

Should be large enough

K

a

The anti-windup : provide local feedback to make the controller stable alone when the main loop is opened by signal saturation (and any circuit which does this will perform as anti-windup)

e

R

w

e

R

r cl

H D G y

Hy

H

v

Hr

31

(32)

## Integrator Anti-windup

If , then

y r

H  H

e

R

D G y

w

D G y

1

v

32

(33)

Integrator Anti-windup

Ex) The Effect of anti-windup

### Consider a plant with the following transfer function for small signals Consider a plant with the following transfer function for small signals

( ) 1 G s  s

and a PI controller,

( )

I

2 4

c P

D s k k

s s

   

in the unity feedback configuration. The input to the plant is limited to .

 1.0

Study the effect of anti-windup on the response of the system

33

(34)

## Integrator Anti-windup

) i h i i d

2 Kp

Control

### 1) Without anti-windup

Step Saturation

1/s

Plant Output

4 Ki

1/s Integrator

2 K 1

Control1

2) With anti-windup

Step1 Saturation1

1/s

Plant1 Output1

Kp1

4 Ki1

1/s Integrator1

Step1 Ki1 Saturation1 Plant1 Output1

10 Ka

Integrator1

34

(35)

## Integrator Anti-windup

0 4 0.6 0.8

1 without anti-windup

with anti-windup

-0.4 -0.2 0 0.2 0.4

Control

0 1 2 3 4 5 6 7 8 9 10

-1 -0.8 -0.6

ti [ ]

time [sec]

1 2 1.4 1.6

without anti-windup with anti-windup

0 6 0.8 1 1.2

Output

0 0.2 0.4 0.6

35

0 1 2 3 4 5 6 7 8 9 10

0

time [sec]

(36)

Fig 4 26 w

Fig 4.26

R

w

Hr Dcl G

y

Hy

v

Fig 4.26, Fig 4.27 Feedback control block diagram Fig 4.28 unity feedback systemg y y w

R

D u G y

v

DG G DG

Y R W V

1 1 1

1 1 1

DG DG DG

D DG D

U R W V

DG DG DG

1 G DG

Actual error

(not error signal in feedback loop) 1

1 1 1

G DG

E R Y R W V

DG DG DG

  

36

(37)

## Steady-state Tracking and System Type

• Definition

The sensitivity : 1 S 1

DG

The complementary sensitivity : DG

1 1

T S DG

   DG

ESRSGW TV

A major goal of control

S T

p

~ frequency band

S T

37

(38)

## 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 Steady state Error 를 생각해보자

) 1 (

)

(s R s

E  ( )

) ( ) 1

( R s

s s G

E  

1) 일 경우 (unit step input), s s

R 1

) (

1 1 1

s

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

k Type system kp

0

;

lim ( ) (0) ; 1

; 2

p s

yp y

k G s G Type system

Type system

   

 1

38

; 0

1 1

; 1, 2,..

1 0

ss p

p

Type system e k

Type system

k

 

(39)

Steady state Error in Unity Feedback control systems

1

=

2) 12 일 경우 (unit step input), ( )

R s 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

lim ( ) (0) ; 1

v

## Type system

k SG s G k Type system

 

   

; 0

1 1

; 1

ss

Type system

e Type system

k k



  

; 2, 3..

s p

Type system



 

v 0 ; 2, 3,..

k k

Type system



3) 1 일 경우 ( i i )

3) 13 일 경우 (unit step input), ( )

R s 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 ka

0 ; 0

0 ; 1

Type system Type system



 ; 0,1

1 1 Type system



2 0

0 ; 1

lim ( ) (0)

; 2

; 3

a s

Type system

k S G s G

k Type system Type system



   



1 1

; 2

; 3, 4,..

0

ss a

e Type system

k k

Type system

  



39

(40)

## Open/Closed loop control systems

controller

=

R Ts 1

k kc

controller open loop y

controller closed loop R

1 Ts

k

kc y

Open loop Transfer function 에서

k k k Ts

R y

c c

1 1

1

이 되어야 error=0 이 된다. Closed loop Transfer function 에서

k k Ts

k k k

Ts k k R

y p p

1 1

k k Ts

Ts k k

R p

p

1

1 1

Steady/ state error for 1 ( ) R ss

Open loop Transfer function 에서 ess 0

k k k

k k e k

p p

p

ss

1

1 1 1

Closed loop Transfer function 에서

40

p p

(41)

## Open/Closed loop control systems

=

예외의 경우에 대한 예>

1 ,

10  

k

k model error 가 있을 경우

Open loop Transfer function 에서

1 , ( ) 1.1

y k k k k

  y t  

 , ( )  

1

c

R k Ts  y k

( y )

Closed loop Transfer function 에서

k e k

p

ss

 

1 1

101 1 1 100

1

100 

k k e

kp ss

let

1 k

k

1 1 1

, 0.009

100 ( )

k   k , e

ss

   

100 1 10(11) 111

1 ( )

ss

k k k

   

41

(42)

System Type for Reference Tracking

## • System type : the degree of the polynomial for which the steady-state system error is a nonzero finite constant

• type 0 : finite error to be a step input which is an input polynomial of zero degree

 W   V 0 

1

E 1 R SR

 DG 

1

( )

k

1( ) 1

k

r t t t R

s

 

0

## lim ( ) lim ( ) 1

t

e t e

ss s

sE s



 

0

0 1

lim 1 ( )

1

1 1

lim 1

s

s k

s R s

DG s DG s

 

 

42

1  DG s

(43)

System Type for Reference Tracking

### Step input

if DG has no pole at the origin, i.e.

DG (0)  0

0

1 1 1

lim 0

1 1 (0)

ss s

e s

DG s DG

 

 

Type 0

K

DG (0)

: “position error constant Type 0 , : position error constant Type 1 ,

P

(0) K

DG

0

1 ( ) DG G s

 s

Type 2 ,

2 0

1 ( )

DG G s

 s

43

(44)

## System Type for Reference Tracking

### System

0

0

( ) 1

n n ( )

DG G s G s

s s

: no pole at the origin

G s

0

( )

0

( )

n

G s  K

0 1

0

1 1

lim 1

1 ( )

ss s k

n

e s

G s s s

0

( )

n

0 0

lim 1

( )

n

n k

s

s

s G s s

If , then

n  k e  0

, then

n  k e  

0 1

n   k

, then

1

n

## e Type n

 K 0

n   k

, then

1 1 0

e Type

 K

1  K

0

Type 0 ; 0, 0

1 1

ss

n k

e

K : position error const

K

0 0

1 ( ) 1

ss G s K K0: position error const

Kp

Type 1 ; 1, 1

1 1 1

ss ( )

n k

e G K K

Kv

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