##### Document Text Contents

Page 45

45

By substituting our values for ๐1, ๐2and ๐ into Equation 3.20 results in:

๐๐ =

0 + 0.935

2(0.7)

= 0.67 ๐๐๐/๐

Rearranging Equation 3.5 for ๐พ๐ results in:

๐พ๐ =

๐๐

2 โ ๐1๐2

๐พ โ ๐ป(๐ )

By substituting our values for ๐๐, ๐1, ๐2, ๐พ and ๐ป(๐ ) into Equation 3.21 results in:

๐พ๐ =

(0.67)2 โ (0)(0.935)

19.9(318.3)

= 7.04๐ฅ10โ5

๐

๐๐๐๐๐๐๐ ๐๐๐ข๐๐ก๐

In order to test this result, Iโve 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. Letโs 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:

๐บ๐๐๐ ๐๐๐๐๐๐(๐๐ต) = 20๐๐๐10 (

1

0.2

) = 20๐๐๐10(5) = 15๐๐ต

Figure 5.8 โ Sample Nyquist Plot

Weโll 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, โModern Control Engineeringโ, page 159, Prentice Hall, New Jersey, 5

th

Edition,

2010.

[2] โAvago Download Pageโ, Accessed June 1

st

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

[3] โAvago 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,โFeedback Control of Dynamic

Systemsโ, page 48, Prentice Hall, New Jersey, 6

th

Edition, 2010.

[5] โChirpโ, Wikipedia: The Free Encyclopedia, Wikimedia Foundation Inc., May 5, 2015, Accessed June

12

th

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

[6] Ogata, Katsuhiko, โModern Control Engineeringโ, page 161, Prentice Hall, New Jersey, 5

th

Edition,

2010.

[7] โExtras: System Identificationโ, Accessed June 13

th

,

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

[8] โRoot 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, โFeedback 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, โFeedback 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, โFeedback 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, โFeedback 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, โAdvanced 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

45

By substituting our values for ๐1, ๐2and ๐ into Equation 3.20 results in:

๐๐ =

0 + 0.935

2(0.7)

= 0.67 ๐๐๐/๐

Rearranging Equation 3.5 for ๐พ๐ results in:

๐พ๐ =

๐๐

2 โ ๐1๐2

๐พ โ ๐ป(๐ )

By substituting our values for ๐๐, ๐1, ๐2, ๐พ and ๐ป(๐ ) into Equation 3.21 results in:

๐พ๐ =

(0.67)2 โ (0)(0.935)

19.9(318.3)

= 7.04๐ฅ10โ5

๐

๐๐๐๐๐๐๐ ๐๐๐ข๐๐ก๐

In order to test this result, Iโve 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. Letโs 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:

๐บ๐๐๐ ๐๐๐๐๐๐(๐๐ต) = 20๐๐๐10 (

1

0.2

) = 20๐๐๐10(5) = 15๐๐ต

Figure 5.8 โ Sample Nyquist Plot

Weโll 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, โModern Control Engineeringโ, page 159, Prentice Hall, New Jersey, 5

th

Edition,

2010.

[2] โAvago Download Pageโ, Accessed June 1

st

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

[3] โAvago 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,โFeedback Control of Dynamic

Systemsโ, page 48, Prentice Hall, New Jersey, 6

th

Edition, 2010.

[5] โChirpโ, Wikipedia: The Free Encyclopedia, Wikimedia Foundation Inc., May 5, 2015, Accessed June

12

th

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

[6] Ogata, Katsuhiko, โModern Control Engineeringโ, page 161, Prentice Hall, New Jersey, 5

th

Edition,

2010.

[7] โExtras: System Identificationโ, Accessed June 13

th

,

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

[8] โRoot 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, โFeedback 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, โFeedback 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, โFeedback 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, โFeedback 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, โAdvanced 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