Solution 9.10.3.4
S
G
a
=
a
G
@G
@a
=
a
Ka
s(s +a)
@
@a
Ka
s(s +a)
=
s(s +a)
K
;Kas
s
2
(s+ a)
2
+
K
s(s+ a)
= s(s +a)
"
;as+ s
2
+as
s
2
(s+ a)
2
#
=
s
(s+ a)
S
T
c
a
=
a
T
c
@T
c
@a
=
a
Ka
s
2
+ as+ Ka
@
@a
Ka
s
2
+ as+ Ka
=
s
2
+ as+ Ka
K
;Ka(s+K)
(s
2
+as +Ka)
2
+
K
s
2
+ as+ Ka
=
s
2
+ as+ Ka
1
;a(s+ K)
(s
2
+as +Ka)
2
+
1
s
2
+ as+ Ka
= (s
2
+ as+Ka))
"
;as;Ka+s
2
+ as+ Ka
(s
2
+as+ Ka)
2
#
=
s
2
s
2
+ as+ Ka
S
G
K
=
K
G
@G
@K
=
K
Ka
s(s +a)
@
@K
Ka
s(s +a)
=
s(s +a)
a
a
s(s +a)
= 1
S
T
c
K
=
K
T
c
@T
c
@K
=
K
Ka
s
2
+ as+ Ka
@
@K
Ka
s
2
+ as+ Ka
=
s
2
+ as+ Ka
a
;Ka(a)
(s
2
+as +Ka)
2
+
a
s
2
+ as+ Ka
=
s
2
+ as+ Ka
1
;Ka
(s
2
+as +Ka)
2
+
1
s
2
+ as+ Ka
= (s
2
+ as+Ka))
"
;Ka+s
2
+ as+ Ka
(s
2
+ as+Ka)
2
#
=
s(s +a)
s
2
+ as+ Ka