d2y

dt2+ 5

dy

dt= 10x(t)

x(t) = �Ky(t) + r(t)

R(s)

Y (s)R(s) =

K

s

2+5s+10K

Y (s)R(s) =

10s

2+5s+10+K

Y (s)R(s) =

10s

2+5s+10K

Y (s)R(s) =

10Ks

2+5s+10K

Y (s)R(s) =

K

s

2+(5+K)s+10K

Y(s)(s2+5s) = 10X(s)

X(s) = �KY (s) +R(s)

Y (s)(s2 + 5s) = 10(�KY (s) +R(s))

Y (s)(s2 + 5s+ 10K) = 10R(s)

Y (s)

R(s)=

10

s2 + 5s+ 10K

2

d2u(t)

dt2+ 14

du(t)

dt+ 20u(t)

=

d2y(t)

dt2+ 4

dy(t)

dt+ 3y(t)

u(t) y(t)

20s2+14s+2s

2+4s+3

s

2+4s+3s

2+14s+20

2s2+14s+20s

2+4s+3

3s2+4s+12s2+14s+20

20s2+14s+23s2+4s+1

2s2U + 14sU + 20U = s2Y + 4sY + 3Y

(2s2 + 14s+ 20)U = (s2 + 4s+ 3)Y

Y

U=

2s2 + 14s+ 20

s2 + 4s+ 3

y(t)

1

K

1/K

10

1/10

P (s)

R Y

P (s) =G(s)

1 +G(s)

G(s) = fracKs(s+ 10)

P (s) =

K

s(s+10)

(1 +

K

s(s+10))=

K

s(s+ 10) +K

P (s) =K

s2 + 10s+K=

Y (s)

R(s)

y1 = lim

s 0sY (s)

R(s) = 1/s

Y (s) = P (s)R(s) =K

(s2 + 10s+K)

.1

s

y1 = lim

s 0

sK

(s2 + 10s+K)

.1

s

y1 =

K

(0 + 10.0 +K)

=

K

K= 1