5 - Block Diagrams (Read-Only)bw-drew/CSE_Lecture_5-UWE.pdfIndustrial Control UFMF6W-20-2 Control...
Transcript of 5 - Block Diagrams (Read-Only)bw-drew/CSE_Lecture_5-UWE.pdfIndustrial Control UFMF6W-20-2 Control...
06/03/2017
© 2017 University of the West of England, Bristol. 1
UWE Bristol
Industrial ControlUFMF6W-20-2
Control Systems EngineeringUFMEUY-20-3
Lecture 5: Block Diagrams and Steady State Errors
Today’s Lecture
• Block diagrams to represent control systems
• Block diagram manipulation• Example• Steady State Errors
Block Diagrams
• Block Diagrams provide a pictorial representation of a system
• Unidirectional operational block representing individual transfer functions
• Three basic elements:– Rectangles - operators– Lines - signals– Circles - additional or subtraction
Block Diagrams: Examples
• y = Ax
• e = r - c
Ax y
r e
c–
+
Block Diagrams: Examples
• y = Ax-Bz
Ax y–
+
zB
• Simple Closed Loop Control System
Closed Loop System
Process
Sensor
+
–Input Error
Feedback
Output
G(s)
H(s)
+
–R(s) E(s)
B(s)
C(s)
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• Simple Closed Loop Control System
– Transfer function from R(s) to C(s)
Closed Loop System
G(s)
H(s)
+
–R(s) E(s)
B(s)
C(s)
( ) ( ) ( )sBsRsE -=( ) ( ) ( )sCsHsB =
( ) ( ) ( ) ( ) ( )( )sGsCsEsEsGsC =®=
( )( ) ( ) ( ) ( )sCsHsRsGsC
-=®
ïïþ
ïïý
ü
• Simple Closed Loop Control System
– Transfer function from R(s) to C(s)
Closed Loop System
G(s)
H(s)
+
–R(s) E(s)
B(s)
C(s)
( )( ) ( ) ( ) ( )sCsHsRsGsC
-= ( ) ( ) ( ) ( )sRsHsG
sC =÷÷ø
öççè
æ+®
1
( )( )
( )( ) ( )sHsGsG
sRsC
+=\1
• Simple Closed Loop Control System
– Transfer function from R(s) to C(s)
Closed Loop System
G(s)
H(s)
+
–R(s) E(s)
B(s)
C(s)
( )( )
( )( ) ( )sHsGsG
sRsC
+=1
C(s)R(s)( )( ) ( )sHsGsG
+1
• Simple Closed Loop Control System
– With unity feedback, H(s) = 1
Closed Loop System
( )( ) ( )
( )( )sGsG
sHsGsG
+®
+ 11
G(s)
1
+
–R(s) E(s)
B(s)
C(s)
• Remove the feedback link from summing junction
Open Loop Transfer Function
G(s)
H(s)
+R(s) E(s)
B(s)
C(s)
( ) ( )sRsE =
( )( ) ( ) ( )sHsGsEsB=
Open Loop Transfer Function given by:
Block Diagram Manipulation
• Diagrams can be manipulated using the following transformations
• Combining Blocks in Series:
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Block Diagram Manipulation
• Moving a summing junction
13
Block Diagram Manipulation
• Moving a pickoff point ahead of a block
14
Block Diagram Manipulation
• Moving a pickoff point behind a block
15
Block Diagram Manipulation
• Moving a summing point ahead of a block
16
Block Diagram Manipulation
• Eliminating a feedback loop
17
Example
G1 G2 G3
H1
H2
1
+ + +
– – –R CE A B J D
HKC
Consider subgroup containing G3 and H1:
413
3
1G
HGG
JC
=+
=
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Example
G1 G2 G4
H2
1
+ +
– –R CE A B J
KC
Consider subgroup containing G3 and H1:
413
3
1G
HGG
JC
=+
=
Example
G1 G2 G4
H2
1
+ +
– –R CE A B J
KC
Consider subgroup containing G2 and G4:
542 GGGBC
==
Example
G1 G5
H2
1
+ +
– –R C
E A B
KC
Consider subgroup containing G2 and G4:
542 GGGBC
==
Example
G1 G5
H2
1
+ +
– –R C
E A B
KC
Consider subgroup containing G5 and H2:
625
5
1G
HGG
AC
=+
=
Example
G1 G6
1
+
–R C
E A
C
Consider subgroup containing G5 and H2:
625
5
1G
HGG
AC
=+
=
Example
G1 G6
1
+
–R C
E
C
Final Closed Loop Transfer Function
( ) 3212213
321
61
61
11 GGGHGHGGGG
GGGG
RC
+++=
+=
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Steady State Errors• Feedback control used to reduce steady-state
errors• Steady-state error is error after the transient
response has decayed• If error is unacceptable, the control system will
need modification• Errors are evaluated using standardised inputs
– Step inputs– Ramp inputs– Sinusoidal inputs
Example – No SS Error
Time
Res
pons
e
Transient
ts
Steady State
Example – No SS Error
Time
Res
pons
e
Transient
ts
Steady State
Error
Causes of Steady State Error
• Errors can be caused by factors including1. Instrumentation of measurement errors2. System non-linearities – deadbands,
hysteresis, saturation etc.3. Form of input signal4. Form of system transfer function5. External disturbances acting on the system,
for example: forces or torques
Error FunctionG(s)
H(s)
+
–R(s) E(s)
B(s)
C(s)
( ) ( ) ( )sBsRsE -=
( ) ( ) ( ) ( ) ( ) ( )sEsGsHsCsHsB ==
( ) ( ) ( ) ( ) ( )sEsHsGsRsE -=
( ) ( ) ( )[ ] ( )sRsHsGsE =+1
( )( ) ( ) ( )sHsGsRsE
+=1
1
Calculating Value
• Use the final value theorem:
• Inputs can be– Step
– Ramp
( )( ) ( ) ( ) ( ) ( ) ( ) ( )sR
sHsGsE
sHsGsRsE
´+
=®+
=1
11
1
System dependent
Input dependent
( ) ( )ssEtEEstss 0limlim®¥®
==
( ) 2sAsR =
( )sAsR =
A is step amplitude
A is step velocity
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Example
• System and Feedback Transfer functions:
• Error
( ) ( ) ( ) 1 and 1
1=
+= sH
sssG
t
( )( ) ( ) ( )
( )
( )( ) 111
111
11
1++
+=
++
=+
=ssss
sssHsGsR
sEtt
t
Example
• Step input
• Steady State Error:
( )( )
( )( ) ( ) ( )
( ) 111
111
+++
=®++
+=
ssss
sAsE
ssss
sRsE
tt
tt
( ) ( )( )
( )( ) 0
1010010
111lim
0=
+++
=++
+==
® tt
tt Assss
sAsssEE
sss
• as Ess = 0 and H(s) = 1, steady state output will be same as input
Rh=1
C
Example
• Ramp input
• Steady State Error:
( )( )
( )( ) ( ) ( )
( ) 111
111
2 +++
=®++
+=
ssss
sAsE
ssss
sRsE
tt
tt
( ) ( )( )
( )( ) AA
ssss
sAsssEE
sss =++
+=
+++
==® 1010
0111
1lim 20 tt
tt
• in this case Ess is not zero and during application of ramp input C will lag R by A
Time - s
Response
A
R
C
Example
• System and Feedback Transfer functions:
• Error
( ) ( ) 2 and 1
1=
+= sH
ssG
t
( )( ) ( ) ( ) 3
1
121
11
1++
=
++
=+
=ss
ssHsGsR
sEtt
t
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Example
• Step input
• Steady State Error:
( )( ) ( )
31
31
++
=®++
=ss
sAsE
ss
sRsE
tt
tt
( ) ( )330
1031lim
0
AAss
sAsssEE
sss =++
=++
==® t
ttt
• in this case, error is not zero and the output will not be equal to the input
• steady state output Css can also be found using the Final Value Theorem as follows :
( ) ( ) ( ) ( ) ( ) ( ) ( )( )
( )23
11 sRss
sHsEsRsCsCsHsRsE ÷
øö
çèæ
++
-=-
=®-=tt
32311
2311lim)(lim
00
AAsA
sssssCC
ssss =÷øö
çèæ -=÷
øö
çèæ
++
-==®® t
t
• resulting response for A = 1:
R
C
(b) ramp input:)(lim
0ssEE
sss ®= ÷÷
ø
öççè
æ++
=® 20 )3(
)1(limssAss
s tt
¥=´+´+
=0)03()01( A
tt
error continues to increase which is not acceptable
Time - s
Response R
C
Today’s lecture• Block Diagrams pictorial representation of
control system• Rectangles represent operations• Lines are signals• Summing junctions enable addition/subtraction• Manipulation Techniques to reduce block
diagrams to transfer function• Steady State Errors help to determine what
happens to signal in steady state
An electrical motor is used in a closed loop system to control the angular position of an inertial load. The position of the load, which is directly connected to the motor, is measured by a simple rotary potentiometer. The output signal from the transducer is compared with the input demand and the resulting error signal is passed to a voltage/current amplifier. The input demand is converted from angular displacement to voltage before being connected to the summing junction.
Example Block Diagram
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• Closed Loop
Example
DCAmplifier
Battery
Anglesetting
DC motor
Angle
Turntable
+–
Tachometer
Amplifier DCmotor Turntable
ControlDevice Actuator Process
Desired angle(voltage)
Actual angle+–
Potentiometer
Sensor
Error
Measured angle(voltage)
Gain
Gain
system block diagram:
An electrical motor is used in a closed loop system to control the angular position of an inertial load.
system block diagram:
The position of the load, which is directly connected to the motor, is measured by a simple rotary potentiometer.
system block diagram:
The output signal from the transducer is compared with the input demand and the resulting error signal is passed to a voltage/current amplifier.
The input demand is converted from angular displacement to voltage before being connected to the summing junction.
system block diagram: • system equations:
IkT mm =-torqueMotor(a)
eAVI =-Amplifier(b)
ooL csJsT qq +=- 2Load(c)
ipi kV q=-demandInput(d)
oTo kV q=-Feedback(e)
oie VVV -=-Error(f)
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• system block diagram
reduced block diagram
csJs +21Tm oq
mkI
APk
Tk
VeVi
Vo
iq +
-
+G(s)
H(s)-
R C csJskAksG Pm
+= 2)(
P
T
kksH =)(
• Closed Loop Transfer function for system:
Tm
Pm
i
o
kAkcsJskAks++
= 2)(qq