Compensator Design

Tutorial Set - 6

Problem - 1

A unity feedback system is operating with 15% overshoot .

Using frequency response technique , design a compensator to yield Kv = 50 with the gain crossover frequency and phase margin remaining approximately same as that of uncompensated system

)7()(

+=

ssKsG

Solution

Computation of Gain:

Hence, the forward path transfer function is 350

507

1)7(

lim)(lim 20

12

0

=

==+

==

K

Ksss

KssGsKsssv

)7(350)(+

=ss

sG

Solution - 1

Given that the system is operating with 15% overshoot.

The relation between the phase margin and the damping ration is

517.0

15.01

exp2

=

=

17.53

412

2tan4

1

=

++=

M

M

Solution - 2 Because of the influence of the compensator in the

compensated system, we consider additional 10. Hence, the phase margin is considered as 63.17.

From the Bode plot of the forward path transfer function we get the frequency, ,3.54 rad/sec and magnitude 22 dB at a phase (180+63.17) or, 116.83.

This gain is to be attenuated by a Lag compensator The high frequency asymptote of the compensator should

contribute -22 dB at such that the compensated system magnitude is 0 dB at this frequency.

We chose the gain crossover frequency 1 decade below 3.54 rad/sec

M

M

Bode Plot of Gain Adjusted OLTF

It is observed that gain is 22 dB at -116.83 degree. This gain is to be attenuated by Lag Compensator

Solution - 3

If the structure of the compensator is

1/T=0.345 rad/sec Then,

The corner frequency are chosen as

The compensator TF:

1;1)(1

1

>++

= T

TC s

ssG

58.12221log20 ==

028.058.12

354.01 ==T

028.0354.00794.0)(

++

=sssGC

Bode Plot

Bode Plot

It is verified from Bode plot that GM &PM remains unchanged for the compensated system

Compensator DesignProblem - 1SolutionSolution - 1Solution - 2Bode Plot of Gain Adjusted OLTFSolution - 3Bode Plot