Lecture Control system design: Feedback control system characteristics - Nguyễn Công Phương

Nguyễn Công Phương

CONTROL SYSTEM DESIGN
Feedback Control System
Characteristics


Contents
I. Introduction
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
Desired output
response

Controller

Actuator

Process

Output

Open – loop control system (without feedback)

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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.
Desired
output
response

Disturbance
Error
(–)

Controller

Actuator

Sensor

Process

Actual
output
Measurement
noise

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)
Td ( s )

R( s)

Ea ( s )

( )

Gc ( s )

G( s)

Controller

Process

H ( s)

Y ( s)
N ( s)

Sensor

E ( s)  R( s)  Y ( s)
Y ( s) 

Gc ( s )G ( s )
Gc ( s )G ( s )
G( s)
R( s) 
Td ( s ) 
N ( s)
1  Gc ( s )G ( s )
1  Gc ( s )G ( s )
1  Gc ( s )G ( s )

 E ( s) 

Gc ( s )G ( s )
1
G( s)
R( s) 
Td ( s ) 
N ( s)
1  Gc ( s )G ( s )
1  Gc ( s )G ( s )
1  Gc ( s )G ( s )
L( s )  Gc ( s )G ( s )

 E ( s) 

1
G( s)

L( s )
R( s) 
Td ( s ) 
N (s)
1  L( s )
1  L( s )
1  L( s )
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Error Signal Analysis (2)
Td ( s )

R( s)

Ea ( s )

( )

Gc ( s )

G( s)

Controller

Process

H ( s)

Y ( s)
N ( s)

Sensor

L( s )  Gc ( s )G ( s )
F ( s )  1  L( s )
S ( s) 

1
1

F ( s ) 1  L( s )

C ( s) 

L( s )
1  L( s )

E ( s )  S ( s ) R ( s )  S ( s )G ( s )Td ( s )  C ( s ) N ( s )
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)
Td ( s )

R( s)

Ea ( s )

( )

Gc ( s )

G( s)

Controller

Process

H ( s)

Y ( s)
N ( s)

Sensor

Gc ( s )G ( s )
Gc ( s )G ( s )
G( s)
Y ( s) 
R( s) 
Td ( s ) 
N ( s)
1  Gc ( s )G ( s )
1  Gc ( s )G ( s )
1  Gc ( s )G ( s )

Gc ( s )G ( s )  1, Td ( s )  0, N ( s )  0
 Y ( s)  R( s)
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Sensitivity of Control Systems to
Parameter Variations (2)
E ( s) 

Gc ( s )G ( s )
1
G( s)
R( s) 
Td ( s ) 
N ( s)
1  Gc ( s )G ( s )
1  Gc ( s )G ( s )
1  Gc ( s )G ( s )
G ( s )  G ( s )  G ( s ), Td ( s )  0, N ( s )  0

 E ( s )  E ( s ) 
 E ( s ) 

1
R( s)
1  Gc ( s )[G ( s )  G ( s )]

Gc ( s ) G ( s )
R( s)
[1  Gc ( s )G ( s )  Gc ( s ) G ( s )][1  Gc ( s ) G ( s )]

Gc ( s )G ( s )  Gc ( s ) G ( s )

 E ( s ) 

Gc ( s ) G ( s )
R( s)
2
[1  L( s )]
1  L( s )  L( s )

 E ( s )  

1 G ( s )
R( s)
L( s ) G ( s )

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Sensitivity of Control Systems to
Parameter Variations (3)
Td ( s )

R( s)

Ea ( s )

( )

Gc ( s )

G( s)

Controller

Process

H ( s)

Y ( s)
N ( s)

Sensor

Y ( s)
T ( s) 
R( s)
T ( s ) / T ( s )
T ( s ) / T ( s )  ln T
S


G ( s ) / G ( s ) G ( s ) / G ( s )  ln G
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)
Td ( s )

R( s)

Ea ( s )

( )

Gc ( s )

G( s)

Controller

Process

H ( s)

Y ( s)
N ( s)

Sensor

Gc ( s )G ( s )
T ( s) 
1  Gc ( s )G ( s )
 ln T
S
 ln G

Gc
G
2
T G (1  GcG )
G
1
T
 SG 
. 
.

GcG
G T
G
1  Gc ( s )G ( s )
1  GcG
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Ex.

Sensitivity of Control Systems to
Parameter Variations (5)
vout   K a vin
T   Ka

S KT a 

vin

T K a ( 1)K a K a
.

.
1
K a T
K a
 Ka
 Ka
G
T

1  Ka  1  G

 Ka
1  Ka 
T
G 

1  T 1   Ka
1  Ka 


vin

 Ka

 Ka

vout

vout



vin

 Ka
G
1  K a (   1)

vout

 Ka
1  K a (   1)
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Sensitivity of Control Systems to
Parameter Variations (6)

Ex.

T

 Ka

G

1  Ka  1  G

vin

 Ka
G
1  K a (   1)

vout

S KT a  SGT S KGa
1
G
2
1  K a (   1)
1
T G (1  G )
G
1


STG 
. 
.

 Ka
G
1  Ka 

G T
G
1 G 1
1  K a (   1)
1 G

S KGa

1
K a
2
G K a [1  K a (   1)]

.

.
K a G
K a

 S KT a 

Ka
1

 Ka
1  K a (   1)
1  K a (   1)

1  K a (   1)
1

1
.

1  Ka 
1  K a (   1) 1  K a 
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15


Ex.

Sensitivity of Control Systems to
Parameter Variations (7)
S KT a

vin

1

 Ka

vout

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.

vin

 Ka

vout



S KT a 

1
1  Ka 

vin

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 Ka
G
1  K a (   1)

vout

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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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17


Disturbance Signals
in a Feedbak Control System (1)

E ( s) 

1
G( s)
L( s )
R( s) 
Td ( s ) 
N ( s)
1  L( s )
1  L( s )
1  L( s )
R ( s )  N ( s )  0, S ( s ) 

1
1  L( s )

G( s)
 E ( s)  
Td ( s )   S ( s )G ( s )Td ( s )
1  L( s )

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Ex.

Disturbance Signals
in a Feedbak Control System (2)
Va ( s )

E ( s)

( )

1 I a ( s)
Km
Ra

Td ( s )
Tm ( s ) (  )

Motor back electromotive force
/>

1 ( s)
TL ( s ) Js  b

Kb

1
1
Js  b
E ( s) 
Td ( s ) 
Td ( s )
1
1
Js

b

K
K
/
R
b m
a
1  Kb
Km
Ra
Js  b

Td ( s )  D / s
 lim E (t )  lim sE ( s )  lim s

t 

s 0

s0


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1
D
.
Js  b  K b K m / Ra s

D
 0 (  )
b  K b K m / Ra
19


Ex.

Disturbance Signals
in a Feedbak Control System (3)
Va ( s )

lim E (t ) 

t 

E ( s)

( )

D
b  K b K m / Ra

1 I a ( s)
Km
Ra

Td ( s )
Tm ( s ) (  )

1 ( s)
TL ( s ) Js  b

Motor back electromotive force

R( s)

Ea ( s )

Ka

(  ) Amplifier

Km
(  ) Ra

Vt ( s )

Kb

Td ( s )
Tm ( s ) (  )

1 ( s)
TL ( s ) Js  b

Kb

Kt

1
Tachometer
 Ra
Js  b
 lim E (t ) 
D
E ( s) 
Td ( s )
t

Ka Km Kt
K K
Kb 
1 
1 a m .
K


 t

Ra Js  b 
Ka 
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20


Disturbance Signals
in a Feedbak Control System (4)
Td ( s )

R( s)

Ea ( s )

( )

Gc ( s )

G( s)

Controller

Process

H ( s)

Y ( s)
N ( s)

Sensor

E ( s) 

Gc ( s )G ( s )
1
G( s)
R( s) 
Td ( s ) 
N ( s)
1  Gc ( s )G ( s )
1  Gc ( s )G ( s )
1  Gc ( s )G ( s )

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21


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)
T ( s)
d

Tm ( s ) (  )
Km
1 ( s) 1  ( s)
Ra
s
TL ( s ) Js  b
( )
Load
Armature
K
Back electromotive force b

Va ( s )

Km

 ( s)

Va ( s ) s[( Js  b) Ra  K b K m ]

 G( s) 

Km
( s)

Va ( s ) [( Js  b) Ra  K b K m ]

Km
Ra b  K b K m
K1


Ra J
s  1  1s  1
Ra b  K b K m
Va ( s ) 

 ( s) 

k2 E
s

K1 k2 E
.
  (t )  K1k2 E (1  e  t / 1 )
 1s  1 s

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23


Control of the Transient
Response (2)
T ( s)
d

 (t )  K1k2 E (1  e  t / 1 )

1 

Tm ( s ) (  )
Km
1 ( s) 1  ( s)
Ra
s
TL ( s ) Js  b
( )
Load
Armature
K
Back electromotive force b

Va ( s )

Ra J
Km

, K1 
Ra b  K b K m
Ra b  K b K m

k E
R( s)  2
s Ea ( s )

KaG( s)
( s)

R( s ) 1  K a KtG ( s )


  (t ) 

K a K1
 1s  1  K a K t K1

Ka

(  ) Amplifier

Km
(  ) Ra

Vt ( s )

Td ( s )
Tm ( s ) (  )

1 ( s)
TL ( s ) Js  b

Kb

Kt
Tachometer

 K a K t K1   k2 E 
 K a K t K1  
K a K1k2 E 
1

exp

t

1

exp
t 





1  K a K t K1 
1
Kt 

1




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Control of the Transient
Response (3)
1

0.8

Speed

With feedback
Without feedback
0.6

0.4

0.2

0

0

2

4

6

8

10
Time (s)

12

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