The closed-loop transfer function of a control system is

The closed-loop transfer function of a control system is given by
C(s) = 2(s - 1)
R(s) (s + 2)(s + 1)

For a unit step input the output is—

Given r (t) = 1 then R (s) = 1/s

C(s)	=	2 (s - 1)
R(s)	=	(s + 2)(s + 1)

C(s) = R(s).	2 (s - 1)
	s(s + 2)(s + 1)

=	0.2 (s - 1)
	s(s + 2)(s + 1)

C(s) =	A	+	B	+	C
	s		s + 1		s + 2

A =	2 (s + 1)	\|_s=0 = -1
	(s + 1)(s + 2)

B =	2 (s - 1)	\|_s=1 = 4
	s(s + 2)

C =	2 (s - 1)	\|_s=2 = -3
	s(s + 1)

on putting the value of A, B, C we get

C(s) =	-1	+	4	-	3
	s		s + 1		s + 2

Taking inverse Laplace transform, we get
C (t) = – 1 + 4e^{– t} – 3e^{– 2t}