Lecture 15 Guidelines for Root Locus Summary Process Control - - PowerPoint PPT Presentation

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Lecture 15 Guidelines for Root Locus Summary Process Control Prof. Kannan M. Moudgalya IIT Bombay Monday, 26 August 2013 1/31 Process Control Root locus guidelines Outline: Guidelines for Root Locus 1. Summary Real axis portion

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SLIDE 1

Lecture 15 Guidelines for Root Locus Summary

Process Control

  • Prof. Kannan M. Moudgalya

IIT Bombay Monday, 26 August 2013

1/31 Process Control Root locus guidelines

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SLIDE 2

Outline: Guidelines for Root Locus

  • 1. Summary

◮ Real axis portion ◮ Symmetry ◮ Number of branches, starting and ending points ◮ Asymptotes ◮ Multiple roots ◮ Angles of arrival and departure

  • 2. Imaginary axis intercept - implications
  • 3. Example

2/31 Process Control Root locus guidelines

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SLIDE 3

Examples

Will use G1(s) = s + 0.5 s(s + 1)(s + 2)(s + 10) G2(s) = (s + 2) s(s + 1) G3(s) = 1 (s + 1)(s + 2)(s + 3)

3/31 Process Control Root locus guidelines

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SLIDE 4

Recall: Root locus, a definition

◮ Root locus is the locus of roots of

1 + KG(s) = 0, as K goes from 0 to ∞. 1 + KG(s) = 0 or KG(s) = −1

◮ Recall that we are checking this for an arbitrary

point s

◮ We get the following magnitude and phase

relations: |KG(s)| = 1 ∠KG(s) = (2l + 1)180◦, l = 0, ±1, ±2, . . .

4/31 Process Control Root locus guidelines

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SLIDE 5
  • 1. Summary

5/31 Process Control Root locus guidelines

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SLIDE 6

Real axis portion

To the left of odd number of poles + zeros

6/31 Process Control Root locus guidelines

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SLIDE 7

midsem-RL-4.pdf

asymptotic directions

  • pen loop poles
  • pen loop zeroes

r: - 0.005216 - 5.075 i K: 15.73

  • 6
  • 4
  • 2

2 4 6

  • 14
  • 12
  • 10
  • 8
  • 6
  • 4
  • 2

2 Evans root locus Real axis Imaginary axis

7/31 Process Control Root locus guidelines

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SLIDE 8

Root locus of real systems are Symmetric

Im(s) Re(s) Im(s) Re(s) Im(s) Re(s)

8/31 Process Control Root locus guidelines

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SLIDE 9

midsem-RL-4.pdf

asymptotic directions

  • pen loop poles
  • pen loop zeroes

r: - 0.005216 - 5.075 i K: 15.73

  • 6
  • 4
  • 2

2 4 6

  • 14
  • 12
  • 10
  • 8
  • 6
  • 4
  • 2

2 Evans root locus Real axis Imaginary axis

9/31 Process Control Root locus guidelines

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SLIDE 10
  • 3. Number of branches, starting and

ending points

◮ Number of branches = degree of the

denominator polynomial (n > m)

◮ Branches of root locus start at open loop poles ◮ Branches of root locus terminate at open loop

zeros

10/31 Process Control Root locus guidelines

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SLIDE 11

midsem-RL-4.pdf

asymptotic directions

  • pen loop poles
  • pen loop zeroes

r: - 0.005216 - 5.075 i K: 15.73

  • 6
  • 4
  • 2

2 4 6

  • 14
  • 12
  • 10
  • 8
  • 6
  • 4
  • 2

2 Evans root locus Real axis Imaginary axis

11/31 Process Control Root locus guidelines

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SLIDE 12

Angles and real axis intercepts of asymptotes

Let G(s) = b(s) a(s) = sm + b1sm−1 + · · · + bm−1s + bm sn + a1sn−1 + · · · + an−1s + an Angles: φl = 2l + 1 n − m180◦, l = 0, 1, . . . , n − m − 1 Asymptote intersects real axis at α, given by α = pi − zi n − m = −a1 + b1 n − m

12/31 Process Control Root locus guidelines

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SLIDE 13

midsem-RL-4.pdf

asymptotic directions

  • pen loop poles
  • pen loop zeroes

r: - 0.005216 - 5.075 i K: 15.73

  • 6
  • 4
  • 2

2 4 6

  • 14
  • 12
  • 10
  • 8
  • 6
  • 4
  • 2

2 Evans root locus Real axis Imaginary axis

13/31 Process Control Root locus guidelines

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SLIDE 14

Multiple Roots

◮ G(s) = b(s)/a(s) ◮ Condition for multiple roots:

adb(s) ds − bda(s) ds = 0

◮ Applies only when s belongs to the root locus

14/31 Process Control Root locus guidelines

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SLIDE 15

Example

◮ G(s) =

s + 2 s(s + 1)

◮ K = 3.414 × 2.414

1.414 = 5.828, double root

15/31 Process Control Root locus guidelines

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SLIDE 16

Double Roots

asymptotic directions

  • pen loop poles
  • pen loop zeroes
  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

  • 4.5
  • 4.0
  • 3.5
  • 3.0
  • 2.5
  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 Evans root locus Imaginary axis 16/31 Process Control Root locus guidelines

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SLIDE 17
  • 3. Angles of arrival and departure

17/31 Process Control Root locus guidelines

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SLIDE 18

Arrival, departure angles at double roots

asymptotic directions

  • pen loop poles
  • pen loop zeroes

r: - 3.414 + 3.918e- 08 i K: 5.828

  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 1.0 1.5 2.0

  • 4.5
  • 4.0
  • 3.5
  • 3.0
  • 2.5
  • 2.0
  • 1.5
  • 1.0
  • 0.5

0.0 0.5 Evans root locus Real axis Imaginary axis

18/31 Process Control Root locus guidelines

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SLIDE 19

Arrival and departure angles

Explained using G(s) = s + 2 s(s + 1)

◮ Arrive at ±90◦ ◮ Depart at ±90◦

19/31 Process Control Root locus guidelines

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SLIDE 20

midsem-RL-4.pdf

asymptotic directions

  • pen loop poles
  • pen loop zeroes

r: - 0.005216 - 5.075 i K: 15.73

  • 6
  • 4
  • 2

2 4 6

  • 14
  • 12
  • 10
  • 8
  • 6
  • 4
  • 2

2 Evans root locus Real axis Imaginary axis

20/31 Process Control Root locus guidelines

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SLIDE 21
  • 2. Imaginary axis intercepts

21/31 Process Control Root locus guidelines

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SLIDE 22

Imaginary axis intercepts

Gcl = KG 1 + KG

◮ Root locus is the locus of roots of

1 + KG(s) = 0, as K goes from 0 to ∞.

◮ Assume that the system is open loop stable ◮ For some value of K, suppose that the root

locus crosses the imaginary axis

◮ Is there any relation with stability? ◮ How do we calculate the intercepts?

22/31 Process Control Root locus guidelines

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SLIDE 23

Scilab code for root locus - 13-RL-2.sce

1

s = %s;

2 G = s y s l i n ( ’ c ’ , . . . 3

1/( s +1) /( s +2) /( s +3) ) ;

4

evans (G, 2 0 0 ) ;

5

d a t a t i p T o g g l e

23/31 Process Control Root locus guidelines

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SLIDE 24

Imaginary axis intercepts

◮ Substitute s = jω

◮ Obtain two equations: real and imaginary ◮ Solve for ω and the corresponding K

◮ Routh-Hurwitz criterion

24/31 Process Control Root locus guidelines

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SLIDE 25

Routh array

Closed loop characteristic equation is given by ansn + an−1sn−1 + · · · + a1s + a0 = 0 Form Routh Array: an an−2 an−4 . . . a1 an−1 an−3 an−5 . . . a0 b1 b2 b3 . . . b1 = an−1an−2 − anan−3 an−1 b2 = an−1an−4 − anan−5 an−1

25/31 Process Control Root locus guidelines

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Routh-Hurwitz Criterion

an an−2 an−4 . . . a0 an−1 an−3 an−5 . . . a1 b1 b2 b3 . . . c1 c2 . . . c1 = b1an−3 − an−1b2 b1 c2 = b1an−5 − an−1b3 b1 Necessary and suff. conditions for stability:

◮ i.e. for all poles to have negative real part ◮ entries in first column of Routh Array > 0

26/31 Process Control Root locus guidelines

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SLIDE 27

Guidelines

Guidelines for plotting root locus plots are available

27/31 Process Control Root locus guidelines

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SLIDE 28

Plot the root locus of

G3(s) = s + 0.5 s(s + 1)(s + 2)(s + 10)

28/31 Process Control Root locus guidelines

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SLIDE 29

What we learnt today

  • 1. Summary

◮ Real axis portion ◮ Symmetry ◮ Number of branches, starting and ending points ◮ Asymptotes ◮ Multiple roots ◮ Angles of arrival and departure

  • 2. Imaginary axis intercept - implications
  • 3. Example

29/31 Process Control Root locus guidelines

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SLIDE 30

Thank you

30/31 Process Control Root locus guidelines