Page 114 - Modern Control Systems

88 Chapter 2 Mathematical Models of Systems

There are four self-loops:
= G 2H 2, = H 3G 3, = G 6H 6, and L 4 = G 7H 7.
L x L 2 L 3
Loops L x and L 2 do not touch L 3 and L 4. Therefore, the determinant is
A = 1 - {L x + L 2 + L 3 + L 4 ) + (L XL 3 + L ^ + L 2 L 3 + L 2L 4). (2.99)

The cofactor of the determinant along path 1 is evaluated by removing the loops
that touch path 1 from A. Hence, we have
= 0 and A! = 1 - (L 3 + L 4 ).
L x = L 2
Similarly, the cofactor for path 2 is
- 1 - + L 2 ).
A 2 (L t
Therefore, the transfer function of the system is
Y(s) AAj + /> 2 A 2
W) =T(S) = A (2.100)
G!G 2G 3G 4(1 - L 3 - L 4) + G 5G 6G 7G 8(1 - L t - L 2)
1 — Li\ — L> 2 — *->$ — Li\ + L,\L, 3 "t" LI\LI\ + L, 2L, 3 + L, 2L^

A similar analysis can be accomplished using block diagram reduction techniques.
The block diagram shown in Figure 2.31(b) has four inner feedback loops within the
overall block diagram. The block diagram reduction is simplified by first reducing
the four inner feedback loops and then placing the resulting systems in series. Along
the top path, the transfer function is

(hi*) G 3(s)
Y x{s) = G x{s) G 4(s)R(s)
1 - G 2(s)H 2(s) 1 - G 3(s)H 3(s)

G!(5)G 2 (5)G 3 (5)G 4 (5)
R(s).
(1 - G 2(s)H 2(s))(l - G 3(s)H 3(s))_

Similarly across the bottom path, the transfer function is
G 6(s) G 7(5)
Y 2(s) = G 5(s) Gs(s)R(s)
1 - G 6(s)H 6(s) 1 - GJ(S)HJ(S)

G 5( S)G 6(s)G 7(s)G*(s)
R(s).
(1 - G 6(s)H 6(s))(l - G 7(s)H 7(s))
The total transfer function is then given by
G 1( S)G 2(s)G 3( S)G 4(s)
Y(s) = y,(5) + Y 2(s) =
(1 - G 2(s)H 2(s))(l - G 3(s)H 3(s))

G 5(s)G 6(s)G 7(s)G 8(s)
+ R(s).
(1 - G 6(s)H 6(s))(l - G 7(s)H 7(s))