diagram manipulation and reduction control systems
DESCRIPTION
Diagram Manipulation and ReductionControl SystemsTRANSCRIPT
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BLOCK DIAGRAM MANIPULATION AND
REDUCTIONBy Schck Tchamna Rodrigue, PhD
October 2014
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General Block Diagram of a Control System
( )G s( )R s ( ) ( )Y G s R s=Transfer Function
Reduced Block Diagram of a Control System
( )C s ( )pG s
( )H s
Summing point
E R F=
( )F s
Branchpoint
Forward Path
Feedback Path
Plant
Controller
+-
( )D s
Reference Input
( )R s
Main Feedback Signal
Disturbance
Manipulated Variable
Actuating Signal Systems Error ( )M s
Rodrigue Tchamna
( )Y s
-
( )C s ( )pG s
( )H s
E R F=
( )F s
+-
( )D s( )R s ( )M s ( )Y s
( )C s ( )pG s
( )H s
E R F=
( )F s
+-( )R s ( )M s ( )Y s
( ) 0if D s =
( ) ( )pG s C s
( )H s
E R F=
( )F s
+-( )R s ( )Y s
-
( ) ( )pG s C s
( )H s
E R F=
( )F s
+-( )R s ( )Y s
( )G s
( )H s
( )E s
( )F s
+-( )R s ( )Y s
( )G s Is called Direct or Forward transfer function( )H s Is called Feedback transfer function
( ) ( )G s H s Is called Open-loop transfer function
( )( )
Y sR s
Is called Closed-loop transfer function or Control ratio
( )( )
Y sE s
Is called Feedforward transfer function
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Closed Loop Transfer FunctionY GE=
E R F R HY= = ( )G s
( )H s
( )E s
( )F s
+-( )R s ( )Y s
( )Y G R HY GR GHY= = Y GHY GR+ =
( )I GH Y GR+ =( ) 1Y I GH G
R= +
For SISO 1Y GR GH=
+
-
( )G s
( )H s
( )E s
( )F s
++( )R s ( )Y s
For SISO 1Y GR GH=
Positive Feedback
-
The Characteristic equation of the system is defined as an equation obtained by setting thedenominator polynomial of the transfer function to zero.
For SISO 1Y GR GH=
+
1 0GH+ =The Characteristic equation of the above system is
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Block Diagram Reduction RulesFor many real systems, the block diagram of a Single Input-Single Output (SISO),feedback control system may involve several feedback loops and summing points.In principle, the block diagram of (SISO) closed loop system, no matter howcomplicated it is, it can be reduced to the standard single loop form.
( )G s( )R s ( ) ( )Y G s R s=Transfer Function
Block Diagram Reduction Rules
1 Combine all cascade blocks
2 Combine all parallel Blocks. Eliminate all minor (interior) feedback loops
3 Moving a pickoff point behind a block
4 Moving a pickoff point ahead a block
5 Moving a summing point behind a block
6 Moving a summing point ahead a block
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Manipulation Original Block Diagram Equivalent Block Diagram Equation
1Combine all cascade blocks
2 Combine all parallel Blocks. Eliminate all minor (interior) feedback loops
3 Moving a pickoff pointbehind a block
4 Moving a pickoff pointahead a block
5 Moving a summing pointbehind a block
6Moving a summing pointahead a block
1G +R Y
2G
1GR Y
2G
GR Y
R 1/ GG
R YR
GR Y GR Y
G
1R YG+2R
YG +
G1R2R
YG
+1
R
2R+1
R
1/ G2R
YG
1 2G GR Y
1 2G GR Y
1 2Y G G R=
( )1 2Y G G R=
YY GR RG
= =
Y GR=
( )1 2Y G G R=
1 2
21
Y GR RRG RG
=
=
-
( )G s
( )H s
( )E s
( )F s
+-( )R s ( )Y s
( )G s
( )H s
( )E s
( )F s
++( )R s ( )Y s
++ +- ++1R
2R 3R
-Y Y3R
2R
1R1 2 3Y R R R= +
1Y GHR GH=
+
1Y GHR GH=
-
( )1G s
( )D s
( )2G s( )E s
( )H s( )F s
( )R s ( )Y s-+ ++
Using the block diagrams rules, find the correct block Diagram among the block diagram a) b) or c)
Example 1
( )1G s ( )2G s( )E s
( )H s( )F s
( )R s ( )Y s-+
( )D s
++
( )21
G s
a)
( )1G s ( )2G s( )E s
( )H s( )F s
( )R s ( )Y s-+
( )D s
++ b)
( )1G s ( )2G s( )E s
( )H s( )F s
( )R s ( )Y s-+
( )D s
++
( )2G s
c)
c)
-
( )1G s( )R s
-+ ( )2G s
( )H s
-+( )Y s
( )1G s( )R s
-+( )Y s2
21GG H+
( )R s1 2
2
1 2
2
1
11
G GG HG G
G H
+
++
( )Y s ( )R s 1 22 1 21G G
G H G G+ +
( )Y s
Example 2
-
( )R s1 2 3
2 3 2
1 2 3 1
2 3 2 3
1
11
G G GG G H
G G G HG G H G
+
++
( )Y s
( )R s-+
( )Y s
1
3
HG
1 2 3
2 3 21G G GG G H+
( )R s1 2 3
2 3 2 1 2 11G G G
G G H G G H+ +( )Y s
( )R s-+ ( )1G s ( )2G s-+
( )Y s( )3G s
2H
1H
( )R s-+ ( )1G s ( )2G s-+
( )Y s( )3G s
2H
1
3
HG
( )R s-+ ( )1G s
( )Y s
1
3
HG
2 3
2 3 21G GG G H+
Example 3: Simplify the following block diagram First Method
Move pickoff point
Behind block
Negative feedback loop
Negative feedback loop
-
( )R s 1 2 32 3 2 1 2 11
G G GG G H G G H+ +
( )Y s
( )R s-+ ( )1G s ( )2G s-+
( )Y s( )3G s
2H
1H
( )R s-+
( )Y s
1
3
HG
1 2 3
3 2 21G G GG G H+
( )R s-+ ( )1G s ( )2G s -+
( )Y s( )3G s
2 2G H1H
( )R s-+ ( )1G s ( )2G s -+
( )Y s( )3G s
2 2G H1
3
HG
( )R s-+ 1 2G G
( )Y s
1
3
HG
3
3 2 21G
G G H+
Second Method (more complicated)
Move summing point
Behind block
Move pickoff point
Behind block
Negative feedback loop
Negative feedback loop
Example 3: Simplify the following block diagram
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Example 4 : Simplify the following block diagram
R-+ ++ 1G 4G
1H
3G
2G ++
2H
Y R-+ ++ 1 4G G 2 3G G+
2H
Y
1H
R-+ 2 3G G+
2H
Y1 41 4 11
G GG G H
Parallel and Cascade blocks
reduction
Positivefeedback loop
Cascade blocks reduction
R-+
2H
Y( )1 4 2 31 4 11
G G G GG G H
+
Negativefeedback loop R
Y( )
( )
1 4 2 3
1 4 1
1 4 2 32
1 4 1
1
11
G G G GG G H
G G G GH
G G H
+
++
R ( )( )
1 4 2 3
1 4 1 1 4 2 3 21G G G G
G G H G G G G H+
+ +
Y
-
R-+ -+ 1G 4G
1H
3G
2G ++
2H
Y
Example 5 : assume the following model is for a given plant
1 2sG
s=
+2
23
Gs
=+
332
Gs
=+ 4
35
sGs+
=+
1 1H = 226
sHs+
=+
R ( )( )
1 4 2 3
1 4 1 1 4 2 3 21G G G G
G G H G G G G H+
+ +
Y
R ( )( )
1 4 2 3
1 4 1 1 4 2 3 21G G G G
G G H G G G G H+
+ + +
Y
Using the original and the simplified block diagram. Simulate the system using Simulink :
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Multi inputs Single Output (MISO)
( )1G s ( )2G s( )E s
( )H s( )F s
( )R s ( )Y s-+ ++
1) Set one input to zero, 2) find the equivalent block Diagram, 3) then set the otherinput to zero, 4) find its equivalent block Diagram, 5) then sub up the two results
( )1G s
( )D s
( )2G s( )E s
( )H s( )F s
( )R s ( )Y s-+ ++
1 2G GE
HF
R Y-+
Step 1
Step 2
1 2
1 21G GY
R G G H=
+
-
( )1G s
( )D s
( )2G s( )E s
( )H s( )F s
( )R s ( )Y s-+ ++
2
1 21GY
D G G H=
+
Step3
1) Set one input to zero, 2) find the equivalent block Diagram, 3) then set the otherinput to zero, 4) find its equivalent block Diagram, 5) then sub up the two results
( )1G s
( )D s
( )2G s( )E s
( )H s
( )Y s-+ ++
Step 4
( )2G s
1G H
( )Y s( )D s
++ ( )2G s
1G H
( )Y s( )D s ++ ( )2G s
1G H
( )Y s( )D s +-
-
1 2
1 21G GY
R G G H=
+2
1 21GY
D G G H=
+
( )
1 2 2
1 2 1 2
21
1 2
1 1
1
G G GY R DG G H G G HG G R D
G G H
= ++ +
= ++
( )1G s
( )D s
( )2G s( )E s
( )H s
( )R s ( )Y s-+ ++
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Homework 1
( )R s-+ ( )1G s ( )2G s-+
( )Y s( )3G s
-+
1H
2H
( )4G s
3H
( )R s-+
( )Y s
1H
1 2 3 4
2 3 2 3 4 31G G G G
G G H G G H+ +
1) Prove that Block Diagram 1 = Block Diagram 2
Block Diagram 1
Block Diagram 2
2) Determine the final Transfer function of the system3) Simulate using Simulink
1 2sG
s=
+2
23
Gs
=+
332
Gs
=+ 4
35
sGs+
=+
1 1H = 226
sHs+
=+
3 2
16
Hs
=+
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Homework 2
Using the superposition principle, Compute the transfer function Y in terms of R and D
Simulate the system for a step input disturbance, for R = 0;
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LECTURE 6
Systems Time Domain Analysis
BLOCK DIAGRAM MANIPULATION AND REDUCTIONSlide Number 2Slide Number 3Slide Number 4Closed Loop Transfer FunctionSlide Number 6Slide Number 7Block Diagram Reduction RulesSlide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Multi inputs Single Output (MISO)Slide Number 20Slide Number 21Homework 1Homework 2LECTURE 6Systems Time Domain Analysis