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TitlePractical Controller Design for Dummies
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Page 45

45


By substituting our values for ๐œŽ1, ๐œŽ2and ๐œ‰ into Equation 3.20 results in:

๐œ”๐‘› =
0 + 0.935

2(0.7)
= 0.67 ๐‘Ÿ๐‘Ž๐‘‘/๐‘ 

Rearranging Equation 3.5 for ๐พ๐‘ƒ results in:

๐พ๐‘ƒ =
๐œ”๐‘›

2 โˆ’ ๐œŽ1๐œŽ2
๐พ โˆ™ ๐ป(๐‘ )



By substituting our values for ๐œ”๐‘›, ๐œŽ1, ๐œŽ2, ๐พ and ๐ป(๐‘ ) into Equation 3.21 results in:

๐พ๐‘ƒ =
(0.67)2 โˆ’ (0)(0.935)

19.9(318.3)
= 7.04๐‘ฅ10โˆ’5

๐‘‰

๐‘’๐‘›๐‘๐‘œ๐‘‘๐‘’๐‘Ÿ ๐‘๐‘œ๐‘ข๐‘›๐‘ก๐‘ 




In order to test this result, Iโ€™ve created a Simulink model called base_closed_loop_model and added the

model elements particular to this example. I then setup a unit step input to inject into our model to see

the result. Shown in Figure 3.5 is a very smooth but slow response to a step input. Though it meets our

overshoot specification of 5%, it in no way comes close to our target rise time of 0.1s. Letโ€™s apply the 2
nd



Order Transient Response Equations for rise time and compare.





Figure 3.5 โ€“ Unit Step Response for KP of 7.04x10
-5



(3.21)

Page 90

90


If the Nyquist Plot intersects the real axis at the origin or on the positive real axis, the Gain Margin is

considered infinite.

Similarly, the Phase Margin is defined as the angle that the Nyquist Plot would have to change to move

to intersect the (-1, 0) point. On a Nyquist Plot, the angle is measured from the negative real axis about

the origin towards the intersection between the Nyquist Plot and the unit circle. If the angle is counter-

clockwise direction, it is considered positive. If the Nyquist Plot never intersects the unit circle, the

Phase Margin is considered infinite.

As shown below in Figure 5.8, the Sample Nyquist Plot has a Phase Margin of 112ยฐ. It also has a Distance

of 0.2, and using Equation 5.1, the Gain Margin is:

๐บ๐‘Ž๐‘–๐‘› ๐‘€๐‘Ž๐‘Ÿ๐‘”๐‘–๐‘›(๐‘‘๐ต) = 20๐‘™๐‘œ๐‘”10 (
1

0.2
) = 20๐‘™๐‘œ๐‘”10(5) = 15๐‘‘๐ต



Figure 5.8 โ€“ Sample Nyquist Plot



Weโ€™ll use the same conservative rule of thumb for Gain and Phase margins that we used stability

analysis using Bode Diagrams.

Unit Circle

Phase Margin = 112ยฐ

Distance = 0.2

Page 91

91


6.0 References

[1] Ogata, Katsuhiko, โ€œModern Control Engineeringโ€, page 159, Prentice Hall, New Jersey, 5
th

Edition,

2010.

[2] โ€œAvago Download Pageโ€, Accessed June 1
st

, 2015, http://www.avagotech.com/docs/AV02-1046EN.

[3] โ€œAvago Download Pageโ€, Accessed June 1
st

2015, http://www.avagotech.com/docs/AV02-0096EN.

[4] Franklin, Gene and Powell, J. David and Emami-Naeini, Abbas,โ€Feedback Control of Dynamic

Systemsโ€, page 48, Prentice Hall, New Jersey, 6
th

Edition, 2010.

[5] โ€œChirpโ€, Wikipedia: The Free Encyclopedia, Wikimedia Foundation Inc., May 5, 2015, Accessed June

12
th

, 2015, http://en.wikipedia.org/wiki/Chirp.

[6] Ogata, Katsuhiko, โ€œModern Control Engineeringโ€, page 161, Prentice Hall, New Jersey, 5
th

Edition,

2010.

[7] โ€œExtras: System Identificationโ€, Accessed June 13
th

,

http://ctms.engin.umich.edu/CTMS/index.php?aux=Extras_Identification.

[8] โ€œRoot Locusโ€, Wikipedia: The Free Encyclopedia, Wikimedia Foundation Inc., January 24, 2015,

Accessed June 20
th

, 2015, http://en.wikipedia.org/wiki/Root_locus.

[9] Franklin, Gene and Powell, J. David and Emami-Naeini, Abbas, โ€Feedback Control of Dynamic

Systemsโ€, page 120, Prentice Hall, New Jersey, 6
th

Edition, 2010.

[10] Franklin, Gene and Powell, J. David and Emami-Naeini, Abbas, โ€Feedback Control of Dynamic

Systemsโ€, page 119, Prentice Hall, New Jersey, 6
th

Edition, 2010.

[11] Franklin, Gene and Powell, J. David and Emami-Naeini, Abbas, โ€Feedback Control of Dynamic

Systemsโ€, pages 116-119, Prentice Hall, New Jersey, 6
th

Edition, 2010.

[12] Franklin, Gene and Powell, J. David and Emami-Naeini, Abbas, โ€Feedback Control of Dynamic

Systemsโ€, page 129, Prentice Hall, New Jersey, 6
th

Edition, 2010.

[13] SYDE 352, Introduction to Control Systems, Course Notes, Package 2 of 2, Winter 2006, Prof. Dan

Davison, Dept. of Electrical and Computer Engineering, University of Waterloo.

[14] Balemi, Silvano, โ€œAdvanced Controlโ€, June 3, 2011, Accessed August 13, 2015,

http://www.dti.supsi.ch/~smt/courses/DigImpl.pdf.

http://www.avagotech.com/docs/AV02-1046EN
http://www.avagotech.com/docs/AV02-0096EN
http://en.wikipedia.org/wiki/Chirp
http://ctms.engin.umich.edu/CTMS/index.php?aux=Extras_Identification
http://en.wikipedia.org/wiki/Root_locus
http://www.dti.supsi.ch/~smt/courses/DigImpl.pdf

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