Transfer Function of First Order Control Systems

Question one
Equation 2 is the transfer function of first order control systems. A transfer function describes the
relationship between the input and the output. In equation 2, X(s) is the input to the system and
Y(s) is the output. A transfer function helps in determining the type and the order of a control
system. Furthermore, frequency response of the control system, Nyquist, Root locus, Nichols and
bode plots can be drawn using the transfer function. The plots provide information that is useful
in study and analysis of control systems, for example information on stability of the system. The
denominator of the transfer function gives the characteristic equation, it is from the characteristic
equation that the roots of the system is derived.
Question two
First order transfer function of the system, T(s) is given by
T(s) = Y ( s)
X (s) = c
ds+1 .................................2
Y (s) = T (s) * X (s), but T(s) = c
ds+1 thus
= c
ds+1 * X (s).................................3
Since the input is the unit step function, X(s) = 1/s. substituting for X (s) in equation 3 we get
Y(s) =
1
s ∗c
(ds +1)
.....................................4
Rearranging equation 4 we get
Y(s) =
1
s ∗c
d∗(s +1/d )
Y(s) =
1
s ∗c /d
( s+1/d )
......................................5
Decomposing equation 5 using partial fraction gives
Y (s) = c / d
s (s +1/d ) ¿ A
s + B
s+1/d ........................6
A = c /d
s (s +1/d )∗s=¿ { c /d
( s+1 /d ) } s=0 = c

B = c /d
s (s +1/d )∗( s+1/d )={ c /d
s }s=-1/d= -c
Y (s) = c
s − c
s+1/ d = c { 1
s − 1
s+ 1
d
}......................7
Taking inverse Laplace transform of equation 7 we get
y(t) = c – c * exp{-(1/d)t},
y (t) = c{1-e-(1/d)t} ................................8
when t = 0, y (t) = c{1-e-0} = 0
when t = 1d, y(t) = c{1-e-1} = 0.632c
when t = 2d, y (t) = c{1-e-2} = 0.864c
when t = 3d, y (t) = c{1-e-3} = 0.950c
when t = 4d, y (t) = c{1-e-4} = 0.982c
when t = 5d, y (t) = c{1-e-5} = 0.993c
when t = ∞, y (t) = c{1-e-∞} = c
y (t) is explicitly zero when t is less than 1
Tabulating the information above:
t y(t)
0 0
1d 0.632c
2d 0.864c
3d 0.950c
4d 0.982c
5d 0.993c
∞ c
Table 1
u (t) = 1 when t is equal or greater than 1
u (t) = 0 when t is less than 1