Control System ZN Method, PID tuner detailed notes with MATLAB

Chapter 2

2.1




Figure 1: Response of a Second Order System (i.e., Amplitude vs. Time)

i) After observing the step response we can observe that the first peak overshoot of
1.26−1
magnitude Mp= =0.26 or 26%.
1
Damping ratio=ς
−πς
√ 1−ς2
Mp=e

−πς
√1−ς 2
0.26=e

−πς
ln ⁡(0.26)=
√ 1−ς 2

, −πς
−1.34707=
√1−ς2
Squaring both sides, we will get the value of the damping ratio;



( πς )2
1.8146= 2
1−ς


Damping ratio=¿ ς =0.39408

The damping ratio is 0< ς <1 so the system shown in the figure is underdamped.

Natural frequency
The natural frequency can be calculated by using 2% settling band.
T s=4 T

4
T s=
ς wn

T s=2

4
2=
ς wn

rad
w n=5.075
s

The natural frequency calculated from the above calculations is 5.075 rad/s.

ii) Transfer function of the system
The transfer function can be calculated by using the general response of second order
system which can be derived by the equations below:-
Here we denote transfer function of the system by G(s)

2
wn
G ( s )= 2 2
s +2 ς wn + wn

, Now we can substitute the values of natural frequency and damping ratio,

25.756
G ( s )= 2
s +4 s+ 25.756
This is the final transfer function that is developed using the equation and denoted by G(s).

iii)




iv)