problem 6 7 8 frequency response

7/27/2019 Problem 6 7 8 Frequency Response

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Bode Diagram

Gm=6.02 dB (at 13.1 rad/s) , Pm =5.23 deg (at 9.23 rad/s)

Frequency (rad/s)

-150

-100

-50

0

50

100

System: u

Frequency (rad/s): 13.1

Magnitude (dB): -6.02

Magnitude(dB)

10-1

100

101

102

103

104

-270

-225

-180

-135

-90

System: u

Frequency (rad/s): 9.23

Phase (deg): -175Phase(deg)

Problem 6: Bode Diagram

In this section, we will plot the Bode diagram of the open-loop system with a proportional controller

with gain equal to

2 . With the diagram, we will determine the gain and phase margin.

The open-loop system which include the gain obtained from root-locus and the potentiometer give usthe transfer function:

ssssss

sss

K

GCU

c

ss

s

s

17171.101

86961

17171.101

83.20

2

2623

1

17171.101

83.20

2

1

2323

23

)()()(

)(

++=

++

=

++

=

=

The open-loop transfer function then input to Matlab to tabulate the bode diagram, which we get:

From the bode diagram, the gain is -6.02dB and the phase margin is -175 deg.


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-140 -120 -100 -80 -60 -40 -20 0-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500Nyquist Diagram

Real Axis

ImaginaryAxis

-1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

Nyquist Diagram

Real Axis

ImaginaryAxis

Problem 7 & 8: Nyquist Diagram & Nyquist Stability Criterion

From the open-loop transfer function of problem 6, we plot the Nyquist diagram using Mathlab and we

get:

We zoom in to the area of1 +0 point can see that there are one clockwise encirclements from the

gain cK

2 around that point:

2

cK

cK

cK2

Critical Pt


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In stability criterion it is express as

= +

where is the number of zeros in the right hand side of the system;

is the number of clockwise encirclement of the 1 +0 point;

is the number of right hand pole of the system;

For a system does not have a right hand pole, then = . This implies that there should not be any

encirclement around 1 +0 point.

But for the Nyquist diagram above, there is one clockwise encirclements from the gainc

K2 . The system

also does not have any zeros in right hand pole which mean , this conclude that the open-loop

system with bigger gain is not stable.

In this system the critical point lie between -0.98 and and the root locus encircles the critical point onetime clockwise. Therefore for the system to be stable , we require.

0.98 = 1

The range of the gain to be stable is,

1.02 <

Comparison of the result with Root Locus

For the result of we get from section 4.6, which is 2623 is fall within the range of 1.02 < . Hencethe open-loop response is concluded stable.

problem 6 7 8 frequency response

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