Solution 4.6.5.2
(a)
~
G =
G
1;G+GH
=
K
s(s +1)
1;
K
s(s+1)
;
10K
s(s+1)(s+10)
=
K(s+10)
s(s +1)(s+10);K(s+10)+10K
=
K(s+10)
s
3
+11s
2
+10s;Ks
=
K(s+10)
s[s
2
+11s+(10;K)]
(b)
~
G =
G
1;G+GH
=
K(s+1)
s
2
1;
K(s+1)
s
2
;
K(s+1)
s
2
(s+5)
=
K(s+1)(s+5)
s
2
(s+5);K(s+1)(s+5)+K(s+1)
=
K(s+ 1)(s+5)
s
3
+5s
2
;K(s
2
+6s+5)+Ks+ K
=
K(s+1)(s+5)
s
3
+(5;K)s
2
;5Ks;4K
:
1
(c)
~
G =
G
1;G+ GH
=
1
(s + 1)(s+5)
1;
1
(s+1)(s+5)
;
K
s(s+1)(s+5)
=
s
s(s+1)(s+5);s+ K
=
s
s
3
+6s
2
+5s;s +K
=
s
s
3
+6s
2
+4s+K
:
(d)
~
G =
G
1;G+GH
=
1
s(s+10)
1;
1
s(s+10)
;
K
s(s +10)
=
1
s(s+10);1+K
=
1
s
2
+10s+(K ;1)
:
2