Nguyễn Công Phương

CONTROL SYSTEM DESIGN

Feedback Control System Characteristics

Contents

Introduction

I. II. Mathematical Models of Systems III. State Variable Models IV. Feedback Control System Characteristics V. The Performance of Feedback Control Systems VI. The Stability of Linear Feedback Systems VII. The Root Locus Method VIII.Frequency Response Methods IX. Stability in the Frequency Domain X. The Design of Feedback Control Systems XI. The Design of State Variable Feedback Systems XII. Robust Control Systems XIII.Digital Control Systems

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2

Feedback Control System Characteristics

1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter

Variations

4. Disturbance Signals in a Feedback Control

System

5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control

Design Software

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3

Introduction (1)

• An open – loop system operates without

feedback & directly generates the output in response to an input signal.

• It is highly sensitive to disturbances & to changes in parameters of the process.

Disturbance

Output

Controller

Actuator

Process

Desired output response

Open – loop control system (without feedback)

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4

Introduction (2)

• A closed – loop system uses a measurement of the input signal & a

comparison with the desired output to generate an error signal that is used by the controller to adjust the actuator.

• Advantages:

– Decreased sensitivity of the system to variations in the parameters of the

process.

– Improved rejection of the disturbances. – Improved measurement noise attenuation – Improved reduction of the steady – state error of the system. – Easy control & adjustment of the transient response of the system.

Disturbance

Error

Desired output response

Controller

Actuator

Process

Actual output

(–)

Measurement noise

Sensor

Measurement output

Feedback

Closed – loop control system with external disturbances & measurement noise

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5

Feedback Control System Characteristics

1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter

Variations

4. Disturbance Signals in a Feedback Control

System

5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control

Design Software

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6

Error Signal Analysis (1)

dT s ( )

aE s ( )

( )R s

( )Y s

cG s ( ) Controller

( )G s Process

( )

( )N s

( )H s Sensor

E s ( )

R s ( )

Y s ( )

c

c

c

Y s ( ) R s ( ) N s ( )    T s ( ) d 1 G s ( ) G s G s ( ) ( ) 1 1 G s G s ( ) ( ) c G s G s ( ) ( )   G s G s ( ) ( ) c G s G s ( ) ( ) 

c

c

c

( )

( )

L s G s G s ( ) c

E s ( ) R s ( ) N s ( )     T s ( ) d 1 G s G s ( ) ( ) 1 G s ( ) G s G s ( ) ( ) 1 1   G s G s ( ) ( ) c G s G s ( ) ( ) 

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E s ( ) R s ( ) N s ( )     T s ( ) d 1 L s ( ) 1 1 1  G s ( ) L s ( )  L s ( ) L s ( ) 

Error Signal Analysis (2)

dT s ( )

aE s ( )

( )R s

( )Y s

cG s ( ) Controller

( )G s Process

( )

( )N s

( )H s Sensor

( )

( )

L s G s G s ( ) c

F s

( ) 1

L s ( )

 

S s ( )

1 F s ( )

1 L s ( )

1

C s ( )

1

L s ( ) L s ( ) 

E s ( )

S s R s ( ) ( )

S s G s T s C s N s ( ) ( )

( )

( )

( )

d

( )

S s C s 

( ) 1 

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8

Feedback Control System Characteristics

1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter

Variations

4. Disturbance Signals in a Feedback Control

System

5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control

Design Software

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9

`

Sensitivity of Control Systems to Parameter Variations (1)

dT s ( )

aE s ( )

( )R s

( )Y s

cG s ( ) Controller

( )G s Process

( )

( )N s

( )H s Sensor

Y s ( )

R s ( )

N s ( )

T s ( ) d

1

G s ( ) G s G s ( ) ( )

1

1

G s G s ( ) ( ) c G s G s ( ) ( ) 

G s G s ( ) ( ) c G s G s ( ) ( ) 

c

c

G s G s ( ) ( )

1,

N s

( ) 0, 

( ) 0 

c

c T s d

Y s ( )

R s ( )

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R s ( )

E s ( )

N s ( )

T s ( ) d

1

1

1

Sensitivity of Control Systems to Parameter Variations (2) G s G s ( ) ( ) G s ( ) 1 c G s G s ( ) G s G s ( ) G s G s ( ) ( ) ( ) ( ) 

c

c

c

G s ( )

G s ( )

G s T s ( )

( ),

0,

N s ( )

0

 

d

E s ( )

E s ( )

R s ( )

 

1 G s G s ( ) ( )[

1

G s

( )]

 

c

E s ( )

R s ( )

 

( )

( )

[1

( )

( )]

G s G s ( ) ( )  c G s G s G s G s ( ) ( )][1  

G s G s 

c

c

c

G s G s ( ) ( )

( )

G s G s ( )

c

E s ( )

R s ( )

 

c cG s G s ( ) ( )  2 ( )] L s [1 

1

L s ( )

L s ( )

E s ( )

R s ( )

 

 

G s ( )  L s G s ( )

1 ( )

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Sensitivity of Control Systems to Parameter Variations (3)

dT s ( )

aE s ( )

( )R s

( )Y s

cG s ( ) Controller

( )G s Process

( )

( )N s

( )H s Sensor

T s ( )

( ) Y s R s ( )

S

T s T s ( ) / ( ) G s G s ( ) / ( )

ln ln

T G

 

T s T s ( ) / ( )  G s G s ( ) / ( ) 

 

System sensitivity is the ratio of the change in the system transfer function to the change of a process transfer function (or parameter) for a small incremental change.

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Sensitivity of Control Systems to Parameter Variations (4)

dT s ( )

aE s ( )

( )R s

( )Y s

cG s ( ) Controller

( )G s Process

( )

( )N s

( )H s Sensor

T s ( )

S

G s G s ( ) ( ) c G s G s ( ) ( ) 1  c T ln  G ln 

G

2

)

(1

.

T S   G

1

1 G s G s ( ) ( )

G c G G c G 

T G  . G T 

c

1

G G G c G G  c

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13

Ex.

K v

Sensitivity of Control Systems to Parameter Variations (5)  

v out

a in

T

K 

a

inv

outv

aK

K

a

S

.

1

T K

a

( 1)   K 

K a K 

K T  a . K T  a

a

a

a

inv

outv

T

aK

K K

1

G

1

 

G 

a

a

1

 

G  

inv

outv

1

T

T 

a

1

G

 a K K

1

K K  

1

1)

a

K  a K  ( a

1

 K

1)

K a ( 

a

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Ex.

a

inv

outv

T

G

G

1

1

Sensitivity of Control Systems to Parameter Variations (6) K  K 

G 

a

1

1)

K  a K  ( a

S

T K

T G S S G K

a

a

G

2

1

1)

(1

.

G S T

K (   a 1 K 

 

1

G

) G

1 

T G  . G T 

1 G  

a

1

1

1  K

1)

K a ( 

1

G

G G 

a

K

a

[1

K

2 1)]

 (

a

S

.

G K

a

1

K

1)

1 (

1   K 

K G  a . K G  a

a

a

1

1)

K a K  a K ( 

a

1

1)

.

T S   K

a

1 (

1

K

1)

1

1 K

K (   a 1 K 

 

a

a

a

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15

Sensitivity of Control Systems to Parameter Variations (7)

Ex.

inv

outv

1

T KS

aK

a

System sensitivity is the ratio of the change in the system transfer function to the change of a process transfer function (or parameter) for a small incremental change.

inv

outv

aK

S

T K

a

1

inv

outv

1 K  a

16

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G  1 1)   K  a K  ( a

Feedback Control System Characteristics

1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter

Variations

4. Disturbance Signals in a Feedback Control

System

5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control

Design Software

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17

Disturbance Signals in a Feedbak Control System (1)

E s ( )

R s ( )

N s ( )

T s ( ) d

1 L s ( )

1

1

1

G s ( ) L s ( ) 

L s ( ) L s ( ) 

R s ( )

N s

S s ( )

( ) 0, 

1 L s ( )

1

E s ( )

S s G s T s ( ) ( )

( )

 

 

T s ( ) d

d

1

G s ( ) L s ( ) 

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18

Ex.

s ( )

Disturbance Signals in a Feedbak Control System (2) dT s ( ) ( )

( )E s

( )mT s

aI

( )s1

aV s ( )

mK

Js b

1 aR

LT s ( )

( )

Motor back electromotive force

bK

http://ftpmirror.your.org/pub/wikimedia/images/wikipedia/fr/d/d9/

E s ( )

T s ( ) d

T s ( ) d

/

 

1 Js b K K R a

b m

1

K

K

b

m

1 Js b 

1 Js b  1 R a

( )

dT s D s / 

1

s

.

0

E t lim ( ) t 

sE s lim ( ) s 

lim s 0 

D Js b K K R s

/

 

b m

a

)

 

0 (

/

D b K K R a b m

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Ex.

s ( )

Disturbance Signals in a Feedbak Control System (3) dT s ( ) ( )

( )E s

( )mT s

aI

( )s1

aV s ( )

mK

Js b

1 aR

LT s ( )

( )

lim ( ) E t t 

/

D b K K R b m a

Motor back electromotive force

bK

dT s ( ) ( )

( )mT s

aE s ( )

( )R s

( )s1

Js b

( )

( )

K m R a

LT s ( )

aK Amplifier

bK

tV s ( )

D

tK Tachometer 

E s ( )

T s ( ) d

lim ( ) E t t 

R  a K K K a m t

b

K

1

.

t

K K

a

K K a m R a

  

1 Js b   1  Js b   sites.google.com/site/ncpdhbkhn

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Disturbance Signals in a Feedbak Control System (4)

dT s ( )

aE s ( )

( )R s

( )Y s

cG s ( ) Controller

( )G s Process

( )

( )N s

( )H s Sensor

c

c

c

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E s ( ) R s ( ) N s ( )    T s ( ) d 1 G s G s ( ) ( ) 1 G s ( ) G s G s ( ) ( ) 1 1   G s G s ( ) ( ) c G s G s ( ) ( ) 

Feedback Control System Characteristics

1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter

Variations

4. Disturbance Signals in a Feedback Control

System

5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control

Design Software

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22

Control of the Transient Response (1)

dT s ( ) ( )

( )mT s

( )s

( )s1

aV s ( )

1 s

LT s ( )

( )

Js b Load

K m R a Armature

K K

]

s ( )  V s ( ) a

K m s Js b R [( )  a

b m

bK

Back electromotive force

m

b m

m

b m

1

K 1 s 

1

s

1

K R b K K  a R J a R b K K 

b m

a

V s ( ) a

k E 2 s

t

1/ 

G s ( )    ) [( K K ]  s ( )  V s ( ) a K Js b R  a

t ( )

(1

e

)

K k E 1 2

1

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s ( ) .    1 k E 2 s K 1 s  

Control of the Transient Response (2)

dT s ( ) ( )

( )mT s

( )s

( )s1

aV s ( )

t

1/ 

t ( )

(1

e

)

K k E 1 2

1 s

LT s ( )

( )

Js b Load

K m R a Armature

m

bK

Back electromotive force

a

b m

a

b m

R s ( )

dT s ( ) ( )

( )mT s

aE s ( )

k E 2 s

( )s1

Js b

( )

( )

K m R a

LT s ( )

aK Amplifier

,    1 K 1 R J a R b K K  K R b K K 

a

bK

tV s ( )

 s ( )  R s ( ) 1  K G s ( ) a K K G s ( ) t

tK Tachometer

1

a

t

t

a

t

a

 s K K a 1 1    K K K 1

a

t

1

t

1

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t ( ) t t         1 k E 2 K K K K 1  K K K 1  K K k E a 1 2 K K K  1              1 exp       1 exp     

Control of the Transient Response (3)

1

0.8

With feedback Without feedback

0.6

d e e p S

0.4

0.2

0

0

2

4

6

8

12

14

16

18

20

10 Time (s)

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Feedback Control System Characteristics

1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter

Variations

4. Disturbance Signals in a Feedback Control

System

5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control

Design Software

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26

Steady – State Error (1)

E

s ( )

R s ( )

Y s ( )

G s R s ( )] ( )

[1  

open loop 

1

E s ( ) o

G s ( ) s

( )R s

1 s

1

(

s

G

(0)

)  

1  

e o

sE s ( ) o

e t lim ( ) o t 

lim s 0 

lim s 0 

G s ( ) s

E

s ( )

R s ( )

closed loop 

1 G s G s ( ) ( )

1

c

.

E s ( ) c

1 G s G s ( ) ( )

1 s

1

c

( )R s

1 s

(

s

.

)  

e c

sE s ( ) c

lim ( ) e t c t 

lim s 0 

lim s 0 

1

1 G s G s ( ) ( )

1 s

1 (0)

1

G

(0)

c

G c

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27

Steady – State Error (2)

Open – loop

Closed – loop

(

)  

(

G

(0)

) 1   

e c

oe

1 (0)

1

G

(0)

G c

(0)G

K

G

(0)

K

(0) 1; 

cG

) 1

(

K

   

oe

)

ce (   

1

K

1 

K

) 0

1

(

e     o

K

100

) 1/101

(

e     c

0.1

) 0.1

(

    

e o

0.1

(

)

    

e c

K  K

K  K

1 101

1 91

)

)

0.1

0.0011

oe (   | ( ) | r t

0.1 1

ce (   | ( ) | r t

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Feedback Control System Characteristics

1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter

Variations

4. Disturbance Signals in a Feedback Control

System

5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control

Design Software

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29

The Cost of Feedback

• Feedback control is very good, but ... • An increased number of components &

complexity: – Requires sensor(s) – The sensor is often the most expensive component in a

control system

– The sensor introduces noise & inaccuracies into the

system

G s G s ( ) ( )

c

• The loss of gain:

1

G s G s ( ) ( ) c G s G s ( ) ( ) 

c

• The possibility of instability.

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30

Feedback Control System Characteristics

1. Introduction 2. Error Signal Analysis 3. Sensitivity of Control Systems to Parameter

Variations

4. Disturbance Signals in a Feedback Control

System

5. Control of the Transient Response 6. Steady – State Error 7. The Cost of Feedback 8. Control System Characteristics Using Control

Design Software

sites.google.com/site/ncpdhbkhn

31

Control System Characteristics Using Control Design Software

Ex.

dT s ( ) ( )

( )R s

( )Y s

1)

( )

11s K Controller

1 s s  ( Load

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32