Karadeniz Technical UniversityDepartment of Electrical and Electronics Engineering
61080 Trabzon, Turkey
Chapter 4-1
, y
Block Diagram ReductionBlock Diagram Reduction
B d tl d b d i l öğ il i k ll k lBu ders notları sadece bu dersi alan öğrencilerin kullanımına açık olup,üçüncü sahıslara verilmesi, herhangi bir yöntemle çoğaltılıp başkayerlerde kullanılması, yayınlanması Prof. Dr. İsmail H. ALTAŞ’ın yazılıiznine tabidir Aksi durumlarda yasal işlem yapılacaktıriznine tabidir. Aksi durumlarda yasal işlem yapılacaktır.
Chapter 4-2
Chapter 4-3
QUIZ 2- 6 Mayıs 2008 Salı Günü saat 15:30 da HalisDuman Amfisinde yapılacaktır.
6 Mayıs 2008 Salı günü saat 13:00 – 15:00 de arası Ave B gruplarına birlikte toplu ders yapılacaktır.
Ödev 2’nin bugün teslim edilmesi gerektiğiniunutmayınız.
HATIRLATMALAR
Chapter 4-4
BLOCK MANIPULATION RULES
Gain Block
Summing Junction
Pick-off Point
G(s)G(s)EEaa(s)(s) θθm m (s)(s)
+ _+ _
R(s)R(s) E(s)E(s)
Y(s)
Chapter 4-5
BLOCK MANIPULATION RULES
G1 G2X Y
G1G2X Y
COMBINING SERIAL BLOCKS
COMBINING PARALLEL BLOCKS
+_G1 G2X YG1
G2
+X Y
m
Chapter 4-6
BLOCK MANIPULATION RULES
CLOSING A FEEDBACK LOOP
Y=GE
X=HY
E=U X±
G
H±
+ EU Y
X
YE=
GE=U HY±
Y=U±HY
G
( )Y=G U±HY
Y=GU±GHY
Y GHY=GUm
( )1 GH Y=GUm
GY= U
1 GHm
G+_
1 GH
U Y
Chapter 4-7
BLOCK MANIPULATION RULES
CLOSING A FEEDBACK LOOP
G(s)
H(s)−
+ E(s)U(s) Y(s)
X
E(s)=U(s)-H(s)Y(s)
G(s)Y(s)= U(s)
1 G(s)H(s)+
Y(s)=G(s)E(s)
[ ]Y(s)=G(s) U(s)-H(s)Y(s)
Y(s)=G(s)U(s)-G(s)H(s)Y(s)
[ ]Y(s) 1+G(s)H(s) =G(s)U(s)G(s)
1 G(s)H(s)+
U(s) Y(s)
This is a rule that is used extensively.
Chapter 4-8
BLOCK MANIPULATION RULES
G1
H±
+ EU YG2
U Y1 2
1 21
G G
G G H±
Chapter 4-9
BLOCK MANIPULATION RULES
MOVING A SUMMING JUNCTION AHEAD OF A BLOCK
G
1/G
X
Y
Z
+-
+GX Z
+
+
_Y
MOVING A SUMMING JUNCION PAST A BLOCK
GX
Y
Z+
+
_GX Z
+
+
_
Y G
Chapter 4-10
MOVING A PICKOFF POINT AHEAD OF A BLOCK
GX
Y
Y
G
GXY
Y
BLOCK MANIPULATION RULES
MOVING A PICKOFF POINT PAST A BLOCK
GXY
X1/G
XY
X
G
Chapter 4-11
BLOCK DIAGRAM REDUCTION
TTdd(s)(s)
EEaa(s)(s) θθmm(s)(s)++++
++--
EEaa(s)(s) ++++
++--
aa LsR +1
aa LsR +1
GG11(s)(s)
KKiiKKii
GG22(s)(s)
mm JsB +1
mm JsB +1
GG33(s)(s)
s
1
s1
GG44(s)(s)
KKbb
HH11(s)(s)
With With TTdd(s)=0(s)=0 , first combine the inner forward path. , first combine the inner forward path.
GG11 GG22 GG33(s)(s)
HH11(s)(s)
+ _+ _ GG44(s)(s)EEaa(s)(s) θθmm(s)(s)
Chapter 4-12
Next combine the feedback loop.Next combine the feedback loop.
)(1
)(
1321
321
sHGGG
sGGG
+ GG44(s)(s)θθmm(s)(s)EEaa(s)(s)
The final series combination isThe final series combination is
)(1
)(
1321
4321
sHGGG
sGGGG
+EEaa(s)(s) θθmm(s)(s)
Note:Note: The defined values of the components may be The defined values of the components may be substituted insubstituted in to get the final transfer function in terms to get the final transfer function in terms of system parameters.of system parameters.
BLOCK DIAGRAM REDUCTION
Chapter 4-13
BLOCK DIAGRAM REDUCTION
Given a control system represented in the block diagram shown. Determine the relationship Y(s)/R(s).
R(s) +G1 G2 G3
H2
H1
Y(s)
_
++
_
+
(a)
Chapter 4-14
BLOCK DIAGRAM REDUCTION
R(s) +G1 G2 G3
H2
H1
Y(s)
_
++
_
+
(a)
Y(s)R(s)
H2
G1
G1 G2
H1
+ _
+
+G3
+_
Chapter 4-15
BLOCK DIAGRAM REDUCTION
Y(s)R(s)
H2
G1
G1 G2
H1
+ _
+
+G3
+_
Y(s)R(s)
H2
G1
G3+
_+
_
G1G2
1- G1G2H1
+
Chapter 4-16
BLOCK DIAGRAM REDUCTION
+_R(s) Y(s)G1G2G3
1- G1G2H1+G2G3H2-
R(s) Y(s)G1G2G3
1- G1G2H1+G2G3H2+G1G2G3
Chapter 4-17
Signal Flow Graphs
The block diagram reduction method works well for relatively simple block diagrams, but it gets very confusing for more complicated models.
A signal flow graph represents the same information as the block diagram, however it leads to a set of rules that allow a systematic approachto finding the overall input/output transfer function.
Basic definitions
Chapter 4-18
Signal Flow Graphs
- It is a graphical tool for control systems analysis and design
- It consists of nodes and branches
- The relationship between the inputs(s) and output(s) are determined by Mason’s gain formula
DEFINITION:
Chapter 4-19
Signal Flow Graphs
• Each branch is unilateral (one direction)
• Each node transmits the sum of all entering signals along each “output” branch
• A forward path is the path travelled by the signal in a forward direction
• A loop is formed when the signal travels and returns to its original source
• Special nodes: Source node - no inputsSink node - no outputs
PROPERTIES OF FLOW GRAPHS:
Chapter 4-20
Signal Flow Graphs
a) construct the signal flow graph either from ablock diagram or from the basic physicalconnection of system components (the transferfunctions of the components must be known).
b) Identify and calculate the various paths andloops in the signal flow graph.
c) With the results from b), apply a formula,Mason’s formula, to determine the overalltransfer function.
The main steps are as follows:
Chapter 4-21
Signal Flow Graphs
node
x1 x2t12
nodebranch
x2 = t12 x1
t12
x2x1
t12 ≡ G12(s)
Construction
Nodes, branches and transmission elements
Chapter 4-22
Signal Flow Graphs Construction
x2
x1
t12
t13
t14
x3
x4
x2 = t12 x1
x3 = t13 x1
x4 = t14 x1
Distribution node
t14
t24
t34
x2
x3
x4
x1
x4 = t14 x1 + t24 x2 + t34 x3
Summation node
Chapter 4-23
Signal Flow Graphs Construction
V1 V2T12
1. A SINGLE BRANCH
V1,V2 are called nodes and T12 is called a branch
This single branch represents the equation
V2 = T12 V1
Note: V1 = V2/T12 (each branch is unilateral)
Chapter 4-24
Signal Flow Graphs Construction
V3 = T13V1 + T23V2
V1 V3
V2
T13
T23
2. SUM OF TWO BRANCHES
Chapter 4-25
Signal Flow Graphs Construction
3. PARALLEL BRANCHES
V2 = (T12a + T12b) V1
V2 = T12 V1
V1 = T21 V2
V1 V2
T12b
T12a
T12
V1
T21
V2
Chapter 4-26
Signal Flow Graphs Construction
V3 = T12 T23 V1
4. CASCADED BRANCHESV1 T12 V2 V3T23
5. NODE ELIMINATION
V3 = T13V1 + T23V2 and V4 = T34V3, then
V4 = (T34T13)V1 + (T34T23)V2
T13 T34V1 V3
T23
V2
V4 V1T34 T13
V4
T34 T23
Chapter 4-27
Signal Flow Graphs Construction
•• Write down and Write down and label the nodeslabel the nodes from input to from input to output, representing all the important signals.output, representing all the important signals.
•• Draw in all the Draw in all the branchesbranches connecting the nodes connecting the nodes and write down their transmission functions.and write down their transmission functions.
•• Check for any additional nodes and branches Check for any additional nodes and branches required in the required in the feedback pathsfeedback paths..
Chapter 4-28
Signal Flow Graphs Example 1
Ea(s) ++
+-
Ki
Kb
aa LsR +1
mm JsB +1
s
1
Td(s)Td(s)
Ea(s)
+Ki
aa LsR +1
mm JsB +1
s
1
G1(s) G2(s) G3(s) G4(s)
H1(s)
x1 x2 x3 x4
x5
θm(s)
xx11 GG11xx22 GG22
xx33 xx44GG33 GG4411EEaa(s)(s) θθmm(s)(s)
--HH11
TTdd(s)(s)11
Servomotor System
Chapter 4-29
Signal Flow Graphs
Mason’s FormulaSource node:Source node: only has outgoing branches.only has outgoing branches.Sink node:Sink node: only has incoming branches.only has incoming branches.Path:Path: continuous unidirectional succession of branchescontinuous unidirectional succession of branches
(passes through no node more than once).(passes through no node more than once).Forward path:Forward path: a path from input to output.a path from input to output.Feedback path or loop:Feedback path or loop: originates and terminates at theoriginates and terminates at the
same node.same node.NonNon--touching paths:touching paths: paths with paths with nono common nodes.common nodes.PathPath gain or loop gain:gain or loop gain: product of branch gains orproduct of branch gains or
transmission functions along the path.transmission functions along the path.
Chapter 4-30
paths.forwardofnumber the;11
ppkMTk
kk L=ΔΔ
= ∑wherewhere 1=Δ
fourofgroupsngnon touchi
ofproductsgain loop⎟⎟⎠
⎞⎜⎜⎝
⎛+∑
threeofgroupsngnon touchi
ofproductsgain loop⎟⎟⎠
⎞⎜⎜⎝
⎛− ∑
pairsngnon touchiof
productsgain loop⎟⎟⎠
⎞⎜⎜⎝
⎛+ ∑
( )gainsloopall−∑
∑T =
FP (1 -∑ ∑ Loops not touching FP)
1 - Loops
∑Δ
Δ= kkM
Signal Flow Graphs Mason’s Formula
Chapter 4-31
Signal Flow Graphs Mason’s Formula
paths.forwardofnumber the;11
ppkMTk
kk L=ΔΔ
= ∑
wherewhere
path.forwardthengnot touchiloopsusingonly defined thk kΔ=Δ
gainpath forwardthk kM =
... continued
Chapter 4-32
Signal Flow Graphs Mason’s Formula
1. Identify all forward paths and write the path gains Mk.
2. Identify all loops and write the loop gains.
3. Identify all non touching loop pairs and write down the loop gain products.
4. Do the same for groups of 3, 4, … non touching loops.
5. Calculate Δ as defined.
6. Identify all loops not touching forward path k, and repeat steps 2 → 5 to calculate Δk.
7. Apply Mason’s formula to calculate the overall transfer function.
Chapter 4-33
Signal Flow Graphs Mason’s Formula
Example
Forward paths:Forward paths:
MM11 ⇒⇒ EEaa xx1 1 xx2 2 xx3 3 xx4 4 θθm m Gain = Gain = GG11GG22GG33GG44
Feedback loops:Feedback loops:
LL11 ⇒⇒ xx1 1 xx2 2 xx3 3 xx4 4 xx11 Loop gainLoop gain = = -- GG11GG22GG33HH11
Non touching loop pairs: noneNon touching loop pairs: none
xx11 GG11xx22 GG22
xx33 xx44GG33 GG4411EEaa(s)(s) θθmm(s)(s)
--HH11
TTdd(s)(s))(
)(
sE
s
a
mθServomotor System
Chapter 4-34
Signal Flow Graphs Mason’s Formula
Servomotor System ...continued
then, then, ΔΔ = 1 + = 1 + GG11GG22GG33HH11
Loops not touching forward path 1: Loops not touching forward path 1: nonenone
then,then, ΔΔ11 = 1= 1
Apply Mason’s formula.Apply Mason’s formula.
( )1321
432111
a
m
HGGG1
GGGGM
)s(E
sT
+=
Δ
Δ=
θ=
Chapter 4-35
Signal Flow Graphs Mason’s Formula
Consider the transfer function from the disturbance input, Td(s) to the output, θm(s) , with (Ea = 0).
The forward path is now
M1 → Td x3 x4 θm Gain = G3G4
The loops are not changed, so Δ and Δ1 are unchanged.
Applying Mason’s formula;
Note: The denominator has not changed.Note: The denominator has not changed.
Servomotor System ...continued
( )1321
4311
1)( HGGG
GGM
sT
sT
d
m
+=
Δ
Δ==
θ
Chapter 4-36
Signal Flow Graphs Mason’s Formula
Example
Forward Paths:
M1 ⇒ R’R x3 x4 C Gain = G6G4G5
M2 ⇒ R’R x1 x2 x3 x4 C Gain = G1G2G3G4G5
C(s)C(s)GG11xx11 GG22
xx22 xx33GG33 GG4411RR’’(s)(s)
--HH11
xx44 GG55
R(s)R(s)
--HH22
GG66
Chapter 4-37
Signal Flow Graphs Mason’s Formula
Feedback loops:
L1 ⇒ x1 x2 x1 Loop gain = – G2H1
L2 ⇒ x3 x4 x3 Loop gain = – G4H2
Non touching loop pairs:
L1 L2 ⇒ Loop gain = G2G4H1H2
then Δ = 1 – (– G2H1 – G4H2) + (G2G4H1H2)
= 1 + G2H1 + G4H2 + G2G4H1H2
... continued
Chapter 4-38
Signal Flow Graphs Mason’s Formula
Loops not touching forward path 1 : Loops not touching forward path 1 : LL11
then, then, ΔΔ11 = 1= 1 –– ((–– GG22HH1 1 ) = 1 + ) = 1 + GG22HH11
Loops not touching forward path 2 : Loops not touching forward path 2 : nonenone
then, then, ΔΔ22 = 1= 1
Now applying Mason’s formulaNow applying Mason’s formula
... continued
21422412
54321125462211
1
)1(
)(
)(
HHGGHGHG
GGGGGHGGGGMM
sR
sCT
+++++
=Δ
Δ+Δ==
Chapter 4-39
Signal Flow Graphs Mason’s Formula
Example
CVin A V1 B
F E
V4
V2 V3 DVout
Vout / Vin =ABCD
1 - CEF
Vout / Vin =FP
1 - LP
Chapter 4-40
Signal Flow Graphs Mason’s Formula
Vin A V1 B
G
C
D
V2 E
F
Vout
FP1 = ACE , FP2 = ABDE
LOOP #1 = BG , LOOP #2 = EF
T = Vout/Vin =ACE [ 1 - 0 ] + ABDE [ 1 - 0 ]
1 - ( BG + EF ) + ( BGEF )
= ACE + ABDE1 - ( BG + EF ) + ( BGEF )
Example
Chapter 4-41
Signal Flow Graphs Mason’s Formula
Vin A B
C
D
I
E F
H
G Vout.
F.P.1 = ACFG , LOOP #1 = DI
F.P.2 = ABDEFG , LOOP #2 = FH
T = Vout/Vin = ACFG [ 1 - DI ] + ABDEFG [ 1 - 0 ]
1 - ( DI + FH ) + ( DIFH )
= ACFG - ACFGDI + ABDEFG1 - DI - FH + DIFH
Example
Chapter 4-42
Signal Flow Graphs Mason’s Formula
Vin +1
H
B
CA D E
G
F
+1 Vout.FP1 = (1) (B) (1) = B LOOP #1 = CHFP2 = (1) (A) (C) (D) (E) (F) (1) = ACDEF LOOP #2 = EG
T = Vout/Vin = B [ 1 - ( CH + EG ) + ( CHEG ) ] + ACDEF [ 1 - 0 ]
1 - ( CH + EG ) + ( CHEG )
T = Vout/Vin =B - BCH + BEG + BCHEG + ACDEF
1 - CH - EG + CHEG
Example
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