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

1. – 3e^{– 2t^{+ 4e^{– t} – 1}}
2. – 3e^{– 2t} – 4e^{– t} + 1
3. zero
4. infinity

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

∴ C(s) =	2(s - 1)
	(s + 1)(s + 2)

Expanding in partial fractions

=	k₁	+	k₂	+	k₃
	s		s + 1		s + 3

∴ k₁ =	- 2	= - 1
	2

k₂ =	2(s - 1)	\|_s=-1 =	2(- 2)	= 4
	s(s + 2)		- 1(1)

k₃ =	2(s - 1)	\|_s=-2 =	2(- 3)	= - 3
	s(s + 2)		- 2( - 2)

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

and output, c(t) = [– 1 + 4e^{– t} – 3e^{– 2t}] u(t).