Transcript Chapter 4: Feedback Control System Characteristics Objectives In this chapter we extend the ideas of modeling to include control system characteristics, such as.

Chapter 4: Feedback Control System Characteristics
Objectives
In this chapter we extend the ideas of modeling to include control system
characteristics, such as sensitivity to model uncertainties, steady-state errors,
transient response characteristics to input test signals, and disturbance
rejection. We investigate the important role of the system error signal which we
generally try to minimize.
We will also develop the concept of the sensitivity of a system to a parameter
change, since it is desirable to minimize the effects of unwanted parameter
variation. We then describe the transient performance of a feedback system
and show how this performance can be readily improved. We will also
investigate a design that reduces the impact of disturbance signals.
Illustrations
Open-And Closed-Loop Control Systems
An open-loop (direct) system
operates without feedback and
directly generates the output in
response to an input signal.
A closed-loop system uses a
measurement of the output
signal and a comparison with
the desired output to generate
an error signal that is applied to
the actuator.
Illustrations
Open-And Closed-Loop Control Systems
H( s )
Y( s )
E( s )
1
G( s )
 R( s )
1  G( s )
1
1  G( s )
 R( s )
Error Signal
Thus, to reduce the error, the magnitu de of
1  G( s )

 1
H( s )  1
Y( s )
E( s )
G( s )
1  H( s ) G( s )
 R( s )
1
1  H[ ( s )  G( s ) ]
 R( s )
Thus, to reduce the error, the magnitu de of
Illustrations
1  G( s ) H( s )

 1
Sensitivity of Control Systems To Parameter Variations
For the closed-loop case if
Y( s )
1
H( s )
 R( s )
GH( s ) > 1
Output affected only by H(s)
G( s )  DG( s )
DY( s )
Open Loop
DG( s )  R( s )
Closed Loop
Y( s )  DY( s )
DY( s )
(G( s )  DG( s ) )
 R( s )
(
)


D

1
G( s )
G( s ) H( s )
DG( s )
(1  GH( s )  DGH( s ) )( 1  GH(s ))
 R( s )
GH( s ) > DGH( s )
DY( s )
DG( s )
( 1  GH( s ) )
Illustrations
2
 R( s )
The change in the output of the closed system
is reduced by a factor of 1+GH(s)
Sensitivity of Control Systems To Parameter Variations
Y (s )
R (s )
T (s )
d
D T ( s)
d
T ( s)
S
S
D G ( s)
G ( s)
T (s )
S G
S G
S H
Illustrations
 d T


d T 
 d 
G


d
G



1
T
(1 
T
 d T


d T 
 d G


d G 
T
 d G


d G


G



G
T
1
H [(s )G (s )]
1 
T
T
T
G
T
 d T


d T 
 d 

G
d
G



G
T
1
(1 
GH
)
2

G
G
( 1  GH
)
Sensitivity of the closed-loop to G variations reduced
GH
- GH
( 1  GH
)
)
Sensitivity of the closed-loop to H variations
When GH is large sensitivity approaches 1
Changes in H directly affects the output response
Example 4.1
Open loop
-Ka v i n
vo
T
-ka
T
SK a
C los ed loop

T
1
R2
Rp
R1
-Ka
1  Ka 
R1  R2
T
SK a
1
1  Ka 
I f Ka is large, the sens it iv ity is low.
4
Ka   1 0
Illustrations
   0 .1
T
SK a

-4
1
3
1  10
 9 .99 1 0
Control of the Transient Response of Control Systems
( s )
G( s )
Va( s )
K1
1 s  1
where,
K1
Illustrations
Km
Ra b  Kb Km
1
Ra J
Ra b  Kb Km
Control of the Transient Response of Control Systems
Illustrations
( s )
K a G( s )
K a K 1
R( s )
1  K a K t G( s )
 1 s  1  K a K t K 1
K a
s 
K1
1
 ( 1  K a K t K 1) 



1


Control of the Transient Response of Control Systems
Illustrations
Disturbance Signals In a Feedback Control Systems
R(s)
Illustrations
Disturbance Signals In a Feedback Control Systems
Illustrations
Disturbance Signals In a Feedback Control Systems
Km
G 1( s )
K a
E( s )
-( s )
G 2( s )
Ra
1
( J s  b )
G( s )
1  G 1( s )  G 2( s )  H( s )
H( s )
Kt 
Kb
Ka
 Td ( s )
G 1G 2H
(s ) > 1
E( s )
1
G 1( s )  H( s )
G 1( s ) H( s )
 Td ( s )
K a K m
Ra
Kt 

If G1(s)H (s ) v ery large the ef f ect of the disturbanc e
c an be minimized
Kb

Ka 
approxim ately
Striv e to maintain Ka large and Ra < 2 ohms
Illustrations
K a K m K t
Ra
s inc e Ka >> Kb
Steady-State Error
Eo ( s )
Ec( s )
R ( s ) - Y( s )
1
1  G( s )
( 1 - G( s ) )  R( s )
 R( s )
H( s )
1
St eady Stat e Error
l im e( t )
t 0
l im s  E( s )
s 0
For a st ep unit input
eo ( i nfi ni te)
ec( i nfi ni te)
Illustrations
l im s  ( 1 - G( s ) ) 
s 0
l im s  

s 0
1
s
 1

 1  G( s )  s
1
l im ( 1 - G( 0) )
s 0
l im
s
1




0  1  G( 0) 
The Cost of Feedback
Increased Number of components and Complexity
Loss of Gain
Instability
Illustrations
Design Example: English Channel Boring Machines
Y( s )
Y( s )
T( s )  R( s )  Td ( s )  D( s )
K  1 1 s
2
s  1 2 s  K
Illustrations
 R( s ) 
1
2
s  1 2 s  K
 D( s )
Design Example: English Channel Boring Machines
Study system for different
Values of gain K
St eady st ate error f or R (s )=1/s and D (s)=0
l im
e( t )
t  infinite
l im s  
  1
 1  K  1 1 s  s
2

s  s 

s 0
1
0
G
St eady st ate error f or R (s )=0 D (s )=1/ s
l im
t  infinite
Illustrations
y ( t)
l im s  
s 0
 1
 2
 s
 s  1 2 s  K 
1
Td
1
K
Study Examples of 4.9 - Control Systems Using MATLAB
And
Apply concepts performing Lab 3
Illustrations