1. 4.3. Time Response Specification in Design
Settling time, Ts ,is the time required for the output to
settle within a certain percent of its final value.
Common values are 5% and 2%. Settling time is
proportional to the , k
Ts = k  = ———
n
If 2% is used to specify the settling value then k=4.
Tr, Ts,and css, are equally meaningful for, over,
critically, or under-damped cases, while Tp,and Mp,are
meaningful only for under-damped cases.
Under-damped cases.
To find Tp,we differentiate c(t) and equalizing the
result to zero and solve the equation for t. We will
find that
The typical of unit step response of a system is as
in figure below
nt
c(t)
Mpt
1.0
0.9
0.1 Tr Tp
1+ d
1 d
css Ts
Rise time Tr is the time required for the response to
rise from 10% to 90% of the final value css.
Mpt is the peak value, and Tp is the time required to
reach Mpt.
Mpt  css
percent overshoot = ————— 100%
css

2. 4.3. Time Response Specification in Design
Example:
Servomotor is used to control the position of plotter
pen as in the following figure.
Recall that
0.5/s(s+2)
motor
R(s) Ka
Amplifier  +
and
Both %over-shoot and nTp are functions only
of  and can be plotted as in the following figure
%over-shoot
nTp
100 80 60 40 20
5.0
4.6
4.2
3.8
3.4
3.0 
Here we have
Suppose that we want to have =1, the fastest
response with no overshoot then
n= 1 and Ka=2.
Note that the settling time is
Tp= 4/  n= 4s
It is not fast enough. If we want the faster response
First we have to choose a different motor, then
Redesign the compensator.

3. 4. 3. Time Response Specification in Design
4.3. Time Response Specification in Design. Step response parameters relation in terms of pole location.
The poles of the TF is
s = n (12)
And is plotted in the following figure
If we draw a line from origin to s1 then the line and
the real axis make angle . Percent over-shoot can
be expressed in term of this angle. s   s1 s2
If we want that the %over-shoot to be less then
specified values then  must be less then a value
that can be calculated. The poles is then restricted
in the region shown in the following figure.. s
The settling time is Ts= k/n and is inversely related
to the real part of the poles. If it is specified that the
settling time to be less then Tsm then n>k/Tsm
k/Tsm s

4. 4.4. Frequency response of Systems.
If input of a system is sinusoidal
r(t)=A cos (t), then the output is still
sinusoidal, c(t)=A|G(j)| cos(t+) with
amplitude = A|G(j)|
phase shift =  =  G(j).
We define the frequency response to be
the complex function G(j).
First Order System
We are interested primarily in the magnitude. We plot it
|G(j)| K B
0.707K 
System bandwidth B is  at which the gain 1/2K, B can
be found to be B = 1/. It is convenient to have a plot in
normalized frequency v = , vB = 1 an the normalized time
is tv = t/. Consequently the normalized plot of freq
and time response is
0.9K
0.1K
t/=tv
c(t) K
Tr/
normalized step response
|Gn(jv)| K
vB=1
0.707K
v= 
normalized frequency response
where
|G(j)| is also called the gain

5. 4.4. Frequency response of Systems.
= 0.1
Second Order System
The transfer function of second order system is
|Gn(j)|
0.25
The frequency response can be written as
(1)
0.5
0.707 1
The magnitude is
B / n
0.707
v =/ n
Define the normalized frequency as v= /n
we have the normalized magnitude |G(j)|
The bandwidth varies with  and n . Recall that
then for constant  increasing n will decrease Tp
with the same factor
Plots of this frequency response for various 
is given in the following figure

6. 4.4. Frequency response of Systems.
|Gn(j)v|
B/n
0.707
v =/n
0.5
0.25
= 0.1 1
4.4. Frequency response of Systems.
The maximum magnitude denoted by Mp 
The peak value of the magnitude of frequency
response is function only of , as is the peak
value of the step response.
The peak value of |Gn(j)v| can be determine by
differentiating |Gn(j)v| and set this to zero, this yields
The peaking frequency response indicates the condition
called resonance, and r is called the resonant frequency

7. 4.5. Time and Frequency Scaling
Time is often scaled to set unity .
Then the T.F is said to be time scaled.
Scaling the time will scales the frequency also.
Let t denote the time before scaling,
tv denotes time after scaling, and
 denotes the scaling parameter thus
t = tv and tv = t/
Let c(t) the function to be scaled and
cs(tv) be the scaled function, thus
cs(tv) = c(t)
Let  = 10 then typical c(t) and cs(tv) are
For >1 the signal will be sped up, hence will
increase the frequency and vice versa.
Consequently the frequency will be scaled as
 = v and v = /
Time derivative will be affected by time scaling
as follow
Time scaling will make:
The amplitude form is unaffected
The time constant  will be changed to /
The damping ratio  is unaffected.
c(t) t
cs(tv) tv

8. 4.6. Response of Higher-Order System
First consider non-unity second order system
Where z and p denotes the zero and pole.
Since (2) is real T.F then if there are
complex zeros or poles they must be present
in conjugate pairs.
Using partial fractional expansion The T.F
(2) can be expressed as the sum of non-
unity gain first order terms and second
order terms.
(1)
The time and freq responses of this system will
be K times of those of unity second order system.
Now consider the response of
Gs(s) = sG(s)
From L.T properties we see that
gs(t) = g’(t)
where g’(t) represent the derivative of g(t)
The general T.F of Higher order system is
4.6. Reduced order models
Reduced order model is obtained by ignoring
insignificant poled
Dominant poles and zeros
Insignificant poles and zeros
unstable region
(2)
Eq. (2) can be expressed as
(3)