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Unformatted text preview: X1“) .2. cm :1;
02mm (5)]: (A) G30)Glawmsmu) Comroller Plant Feedback Output
transducer (a) Plan: and
Controllsr Acmaxing
signal
(error) Output Mason’s Rule
Earlier in this chapter, we showed how to reduce block diagrams to single transfer
functions. Now we are ready to show a technique for reducing signal~ﬂow graphs
to single transfer functions that relate the output of a system to its input. The block diagram reduction technique we studied in Section 5.2 requires suc—
cessive application of fundamental relationships in order to arrive at the system
transfer function. On the other hand, Mason’s rule for reducing a signal—ﬂow graph
to a single transfer function requires the application of one formula. The formula
was derived by S. J. Mason when he related the signal—ﬂow graph to the simulta—
neous equations that can be written from the graph (Mason, 1953). In general, it can be complicated to implement the formula without making
mistakes. Speciﬁcally, the existence of what we will later call nontouching loops
increases the complexity of the formula. However, many systems do not have non—
touching loops. For these systems, you may ﬁnd Mason’s rule easier to use than
block diagram reduction. Mason’s formula has several components that must be evaluated. First, we
must be sure that the deﬁnitions of the components are well understood. Then we
must exert care in evaluating the components. To that end, we discuss some basic deﬁnitions applicable to signalﬂow graphs; then we state Mason’s rule and do an
example. Deﬁnitions
Loop gain: The product of branch gains found by traversing a path that starts at a node and ends at the same node without passing through any other node more than
once and following the direction of the signal ﬂow. For examples of loop gains, see
Figure 5.20. There are four loop gains: 1. G2(s)H1(s) (5.25a)
2. G4(s)H2(s) (5.25b)
3. G4(s)Gs(s)H3(s) (5.25c)
4. G4(S)Gé(S)H3(S) (5.25d) F orward—path gain: The product of gains found by traversing a path from the
input node to the output node of the signal—ﬂow graph in the direction of signal Figure 5.20 SignaIﬂow graph
for demonstrating Gl(s)
Mason’s rule R“) . ﬂow. Examples of forward—path gains are also shown in Figure 5.20. There are
forward—path gains: ‘ 1. Gi(S)G2(S)G3(S)G4(S)Gs(S)G7(S) (5.2;
2. G1(S)G2(S)G3(S)G4(S)G6(S)G7(S) (52 Nontouching loops: Loops that do not have any nodes in common. In '3
ure 5.20, loop Gz(s)H1(s) does not touch loops G4(s)H2(s), G4(s)G5(s)H3(s), G4<s)06<s)H3<s). Nontouchingloop gain: The product of loop gains from nontouching is:
taken two, three, four, etc., at a time. In Figure 5.20, the product of 1' 1. [Gz(s)Hi (s)] [G4(S)H2(S)] (52 :
2 [G2(S)H1(s)][G4(s)Gs(s)H3(s)] (52
3. [Gz(s)Hi(s)][G4(s)G6(S)H3(S)] (5.27
The product of loop gains [G4 (s)Gs (s)H3 (s)] [G4 (s)G6 (s)H3 (s)] is not a nontouchi loop gain since these two loops have nodes in common. In our example there .: no nontouchingloop gains taken three at a time since three nontouching loops Ii
not exist in the example. ‘ We are now ready to state Mason’s rule. Mason's Rule i
The transfer function, C(s)/R(s), of a system represented by a signalﬂow graph I (52* where k = number of forward paths
Tk = the kth forwardpath gain A = 1 — 2 loop gains + Z nontouching—loop gains taken two at atime — I nontouchingloop gains taken three at a time + Z nontouchingloo
gains taken four at a time — ~ Ak = A — 2 loop gain terms in A that touch the kth forward path. In other
words, Ak is formed by eliminating from A those loop gains that touch
the kth forward path. Notice the alternating signs for the components of A. The following example will
help clarify Mason’s rule. Transfer function via Mason’s rule Problem Find the transfer function, C(s)/R(s), for the signal—ﬂow graph in Figure
5.21 . Gl(S) 02(5) G3(S) 04(5) G5(3) MS) 0 O C(s) H 4(3) Solution First, identify the forwardpath gains. In this example there is only one: Gi(S)Gz(S)G3(s)G4(S)Gs(s) (529)
Second, identify the loop gains. There are four, as follows:
1. Gz(s)H1(s) (5.30a)
2. G4(S)H2(s) (5.30b)
3. G7(s)H4(s) (5.30c)
4 G2(S)G3(S)G4(S)Gs(S)G6(S)G7(S)Gs(s) (530d) Third, identify the nontouching loops taken two at a time. From Eqs. (5.30) and
Figure 5.21, we can see that loop 1 does not touch loop 2, loop 1 does not touch
loop 3, and loop 2 does not touch loop 3. Notice that loops 1, 2, and 3 all touch loop
4. Thus, the combinations of nontouching loops taken two at a time are as follows: Loop 1 and loop 2: G2(s)H1(s)G4(s)H2(s) (5.313)
Loop 1 and loop 3: Gz(s)H1(s)G7(s)H4(s) (5.31b)
Loop 2 and loop 3: G4(s)H2(s)G7(s)H4(s) (5.31c) Finally, the nontouching loops taken three at a time are as follows:
Loops 1, 2, and 3: G2(S)H1(s)G4(s)H2(s)G7(s)H4(s) (5.32) i ._,. 74W” 7, 7 Now, from Eq. (5.28) and its deﬁnitions, we form A and Ak. Hence A = 1 — [02(S)H1(S) + G4(S)H2(S) + G7(S)H4(S) , + 020003 (S)G4(S)Gs (S)G6(S)G7 (9% 6;. 't [(?2(S)111(S)(34(S)I¥é(s) 4' (72(5)111(S)(37(S)114(S) i “t (34(3)172(S)(;7(S)114(S)] _ "— [(32(S)1¥i(S)(?4(S)I¥2(S)(77(S)IY4(S)] l We form Ak by eliminating from A the loop gains that touch the kth forward
A1 = 1 — 67(s)H4(s) <5; Expressions (5.29), (5.33), and (5.34), are now substituted into Eq. yielding the transfer function: _ T1A1 _ [G1(S)Gz(S)G3(S)G4(S)Gs(S)l[1 — G7(S)H4(S)] Since there is only one forward path, G(s) consists of only one term rather thy
sum of terms, each coming from a forward path. Figure 5.19
Signalﬂow graph
development: a. signal nodes; b. signalﬂow graph; 0. simpliﬁed signal
flow graph Converting a block diagram to a signalﬂowgraph
Problem Convert the block diagram of Figure 5.11 to a signal—ﬂow graph. R(S) O O
V1(S) O
V6(S) O
V2(S) H1(S) O
V3 (5) (a) (b) O
V7(S) O
V4(S) O
V5(S) Problem Reduce the system shown in Figure 5.11 to a‘single transfer function. Vg(S) O C(s) F orwardpath gains are Gle G3 and G103. Loop gains are —G1G2H], —G2H2, and —G3H3. Nontouching loops are [—GIGZHJ ][—G3H,] = GszG3HlH3 and [—GzH2][—G3H3] = G2G3H2H3. Also, A = l + GleHl + 0sz + G3H3 + (;1G2G3I1f115I3 + G2G3H2H3.
Finally, Al :1 and A2 = 1. C (S) 2 TkAk
Substituting these values into T (s) 2 ~— = " yields
R(s) A ‘
T(S)_ G1(s)Gg(s)[1+Gz<s)] _ [1+ G2 (S)Hz(S) + G (SW2 (S)Hl (3)][1 + G3 (5)113 (5)] ...
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This note was uploaded on 01/14/2012 for the course EEL 3657 taught by Professor Staff during the Spring '08 term at University of Central Florida.
 Spring '08
 Staff

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