control engineering fundamentals chapter 5 part 2 signal
TRANSCRIPT
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Control Engineering Fundamentals
Chapter 5 Part 2
Signal Flow Graph and Masonβs Gain Formula
Mason's Gain Formula for Signal Flow Graphs:
In many practical cases, we wish to determine the relationship between an
input variable and an output variable of the signal flow graph. The transmittance
between an input node and an output node is the overall gain, or overall
transmittance, between these two nodes. Mason's gain formula, which is applicable
to the overall gain, is given by
π. πΉ. =πΆ(π )
π (π )=
β ππΎβπΎππ=1
β
Where
PK = path gain or transmittance of Kth forward path.
Ξ = determinant of graph.
= 1 - (sum of all individual loop gains) + (sum of gain products of all
possible combinations of two non-touching loops) - (sum of gain
products
of all possible combinations of three non-touching loops) + . . .
β = 1 β β πΏπ
π
+ β πΏππΏπ
π,π
β β πΏππΏππΏπ
π,π,π
β πΏπ
π
= π π’π ππ πππ πππππ£πππ’ππ ππππ πππππ
β πΏππΏπ
π,π
= π π’π ππ ππππ πππππ’ππ‘π ππ πππ πππ π ππππ ππππππππ‘ππππ ππ π‘π€π
Nontouching loops.
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β πΏππΏππΏπ
π,π,π
= π π’π ππ ππππ πππππ’ππ‘π ππ πππ πππ π ππππ ππππππππ‘ππππ ππ
π‘βπππ Nontouching loops.
ΞK = Same as Ξ but formed by loops not touching the (PK) kth forwatd path.
(cofactor of the kth forward path determinant of the graph with the
loops
touching the kth forward path removed, that is, the cofactor ΞK, is
obtained from Ξ by removing the loops that touch path Pk).
(Note that the summations are taken over all possible paths from input to
output.)
In the following, we shall illustrate the use of Mason's gain formula by means
of these examples.
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Example (5.1): Consider the system shown in Figure (5-3). Obtain the closed-loop
transfer function C(s) /R(s) by use of Mason's gain formula.
Figure (5-3)
Solution:
In this system, there are three forward paths between the input R(s) and the
output C(s).
The forward path gains are
π1 = πΊ1πΊ2πΊ3πΊ4πΊ5
π2 = πΊ1πΊ6πΊ4πΊ5
π3 = πΊ1πΊ2πΊ7
There are four individual loops, the gains of these loops are
πΏ1 = βπΊ4π»1
πΏ2 = βπΊ2πΊ7π»2
πΏ3 = βπΊ6πΊ4πΊ5π»2
πΏ4 = βπΊ2πΊ3πΊ4πΊ5π»2
Loop L1 does not touch loop L2. Hence, the determinant A is given by
β= 1 β ( πΏ1 + πΏ2 + πΏ3 + πΏ4) + πΏ1πΏ2
β= 1 β (βπΊ4π»1 β πΊ2πΊ7π»2 β πΊ6πΊ4πΊ5π»2 β πΊ2πΊ3πΊ4πΊ5π»2) + πΊ4π»1πΊ2πΊ7π»2
The cofactor Ξ1 is obtained from Ξ by removing the loops that touch path P1.
Therefore, by removing L1, L2, L3, L4, and L1, L2 from Equation of Ξ, we obtain
β1= 1
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Similarly, the cofactor Ξ2 is
β2= 1
The cofactor Ξ3 is obtained by removing L2, L3, L4, and L1, L2 from Equation
of Ξ, giving
β3= 1 β πΏ1
The closed-loop transfer function C(s) /R(s) is then
π. πΉ. =πΆ(π )
π (π )=
π1β1 + π2β2 + π3β3
β
π. πΉ. =πΆ(π )
π (π )=
π1(1) + π2(1) + π3(1 β πΏ1)
1 β ( πΏ1 + πΏ2 + πΏ3 + πΏ4) + πΏ1πΏ2
π. πΉ. =πΆ(π )
π (π )=
πΊ1πΊ2πΊ3πΊ4πΊ5 + πΊ1πΊ6πΊ4πΊ5 + πΊ1πΊ2πΊ7(1 + πΊ4π»1)
1 β (βπΊ4π»1 β πΊ2πΊ7π»2 β πΊ6πΊ4πΊ5π»2 β πΊ2πΊ3πΊ4πΊ5π»2) + πΊ4π»1πΊ2πΊ7π»2
π. πΉ. =πΆ(π )
π (π )=
πΊ1πΊ2πΊ3πΊ4πΊ5 + πΊ1πΊ6πΊ4πΊ5 + πΊ1πΊ2πΊ7 + πΊ1πΊ2πΊ7πΊ4π»1)
1 + πΊ4π»1 + πΊ2πΊ7π»2 + πΊ6πΊ4πΊ5π»2 + πΊ2πΊ3πΊ4πΊ5π»2 + πΊ4π»1πΊ2πΊ7π»2
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Example (5-2): Consider the system shown in Figure (5-4). Obtain the closed-loop
transfer function C(s) /R(s) by use of Mason's gain formula.
Figure (5-4)
Solution:
First we draw a signal flow graph for this system.
In this system, there are three forward paths between the input R(s) and the
output C(s).
The forward path gains are
π1 = πΊ1πΊ2πΊ3
From Figure for A signal flow graph, we see that there are three individual
loops. The gains of these loops are
πΏ1 = πΊ1πΊ2π»1
πΏ2 = βπΊ2πΊ3π»2
πΏ3 = βπΊ1πΊ2πΊ3
Note that since all three loops have a common branch, there are no
nontouching loops. Hence, the determinant A is given by
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β = 1 β ( πΏ1 + πΏ2 + πΏ3)
β = 1 β πΊ1πΊ2π»1 + πΊ2πΊ3π»2 + πΊ1πΊ2πΊ3
The cofactor Ξ1 of the determinant along the forward path connecting the
input node and output node is obtained from Ξ by removing the loops that touch
this path. Since path Pl touches all three loops, we obtain
β1= 1
Therefore, the overall gain between the input R(s) and the output C(s), or the
closed-loop transfer function, is given by
π. πΉ. =πΆ(π )
π (π )=
π1β1
β
π. πΉ. =πΆ(π )
π (π )=
π1(1)
1 β ( πΏ1 + πΏ2 + πΏ3)
π. πΉ. =πΆ(π )
π (π )=
πΊ1πΊ2πΊ3
1 β πΊ1πΊ2π»1 + πΊ2πΊ3π»2 + πΊ1πΊ2πΊ3
This is the same as the closed-loop transfer function obtained by block
diagram reduction. Mason's gain formula thus gives the overall gain C(s)/R(s)
without a reduction of the graph.
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Example (5.3): Find the transfer function for the system given by signal flow
graph of Figure below by using of Mason's gain formula.
Solution:
In this system, there are three forward paths between the input R(s) and the output C(s).
The forward path gains are
π1 =1
π3π3 =
π3
π3
π2 =1
π2π2 =
π2
π2
π3 =1
ππ1 =
π1
π
There are three individual loops, the gains of these loops are
πΏ1 = βπ1
π
πΏ2 = βπ2
π2
πΏ3 = βπ3
π3
The determinant β is given by
β= 1 β ( πΏ1 + πΏ2 + πΏ3)
β= 1 β (βπ1
πβ
π2
π2β
π3
π3)
β= 1 +π1
π+
π2
π2+
π3
π3
The cofactor Ξ1, Ξ2, Ξ3 is obtained
β1= 1
β2= 1
β3= 1
The closed-loop transfer function C(s) /R(s) is
π. πΉ. =πΆ(π )
π (π )=
π1β1 + π2β2 + π3β3
β
π. πΉ. =πΆ(π )
π (π )=
π3π3+
π2π2+
π1π
1+π1π
+π2π2+
π3π3
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Example (5.4): Find the transfer function for the system given by signal flow
graph of Figure (5.5) by using of Mason's gain formula.
Figure (5.5)
The forward path gains are
π1 = πΊ1πΊ2πΊ3πΊ4πΊ5
π2 = πΊ6
There are four individual loops, the gains of these loops are
πΏ1 = βπΊ2π»1
πΏ2 = βπΊ4π»2
πΏ3 = βπΊ6π»3
πΏ4 = βπΊ1πΊ2πΊ3πΊ4πΊ5π»3
β= 1 β ( πΏ1 + πΏ2 + πΏ3 + πΏ4) + (πΏ1πΏ2 + πΏ1πΏ3 + πΏ2πΏ3) β (πΏ1πΏ2πΏ3)
β= 1 β ( βπΊ2π»1 β πΊ4π»2 β πΊ6π»3βπΊ1πΊ2πΊ3πΊ4πΊ5π»3)
+ ( πΊ2π»1πΊ4π»2 + πΊ2π»1πΊ6π»3 + πΊ4π»2πΊ6π»3 ) β (βπΊ2π»1πΊ4π»2πΊ6π»3 )
β= 1 + πΊ2π»1 + πΊ4π»2 + πΊ6π»3+πΊ1πΊ2πΊ3πΊ4πΊ5π»3 + πΊ2π»1πΊ4π»2 + πΊ2π»1πΊ6π»3
+ πΊ4π»2πΊ6π»3 + πΊ2π»1πΊ4π»2πΊ6π»3
The cofactor Ξ1 is obtained from Ξ by removing the loops that touch path P1.
Therefore, by removing L1, L2, L3, L4, we obtain
β1= 1
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Similarly, the cofactor Ξ2 is obtained from Ξ by removing the loops that touch
path P2. Therefore, by removing L3, L4, we obtain
β2= 1 β (πΏ1 + πΏ2) + (πΏ1 πΏ2)
β2= 1 β ( βπΊ2π»1 β πΊ4π»2) + ( πΊ2π»1πΊ4π»2)
β2= 1 + πΊ2π»1 + πΊ4π»2 + πΊ2π»1πΊ4π»2
The closed-loop transfer function C(s) /R(s) is then
π. πΉ. =πΆ(π )
π (π )=
π1β1 + π2β2
β
π. πΉ. =πΆ(π )
π (π )=
π1(1) + π2[1 β (πΏ1 + πΏ2) + (πΏ1 πΏ2)]
1 β ( πΏ1 + πΏ2 + πΏ3 + πΏ4) + (πΏ1πΏ2 + πΏ1πΏ3 + πΏ2πΏ3) + (πΏ1πΏ2πΏ3)
π. πΉ. =πΆ(π )
π (π )=
πΊ1πΊ2πΊ3πΊ4πΊ5 + πΊ6(1 + πΊ2π»1 + πΊ4π»2 + πΊ2π»1πΊ4π»2)1 + πΊ2π»1 + πΊ4π»2 + πΊ6π»3+πΊ1πΊ2πΊ3πΊ4πΊ5π»3 + πΊ2πΊ4π»1π»2 + πΊ2πΊ6π»1π»3 + πΊ4πΊ6π»2π»3 + πΊ2πΊ4πΊ6π»1π»2π»3