04/19/23 6.5 Root-locus: Other Rules & Examples

1

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Total angle contributedby “other” singularities

’

’

’

Rule 11:Departure (arrival) angle

s0

- + + =(2k+1)

s0 is on the RL if

-’+’+’

p1

Angle of departure from p1

04/19/23 6.5 Root-locus: Other Rules & Examples

2

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Rule 11 (Cont’d)

Angle of departure from a pole=(2k+1)π+total angle contributed by “other” singularities

Angle of arrival at a zero=(2k+1)π-total angle contributed by “other” singularities

For the RL: For the CRL:

04/19/23 6.5 Root-locus: Other Rules & Examples

3

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Example

))()(( 321

1

6575

12234

sss

s

ssss

sGH

n=4; m=1

# of branches=max(4,1)=4

Starting at 4 poles

1 end at the zero

3 end at infinity

04/19/23 6.5 Root-locus: Other Rules & Examples

4

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Example (cont’d)

3 asymptotes at angles:(2k+1)π/3=180o,60o

intersecting at:{-3-2-0-0+(-1)}/(4-1)=-2

Real axis loci:

Between -2 & 1

& to the left of -3

04/19/23 6.5 Root-locus: Other Rules & Examples

5

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Example (cont’d)

jω-axis crossing: R-H test yields

Single-pole crossing for K=6

Two-pole crossing for K=0 (start)& K=50 at s=j11= j 3.31

04/19/23 6.5 Root-locus: Other Rules & Examples

6

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Departure angle from +j1:(2k+1)180o

+135

-90

-tan-1(1/2)

-tan-1(1/3)

=180o

04/19/23 6.5 Root-locus: Other Rules & Examples

7

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

BA points?

0GHds

d yields

3s4+6s3-6s2-14s-11=0

Roots are:1.6677-2.2888-0.6894j0.6966

None on the RL

04/19/23 6.5 Root-locus: Other Rules & Examples

8

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

COMBINE!

04/19/23 6.5 Root-locus: Other Rules & Examples

9

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Check by MATLAB

rlocus([1 -1],[1 5 7 5 6]),grid,figure(1)

04/19/23 6.5 Root-locus: Other Rules & Examples

10

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Rule 12:Calibration

))H(sG(s

1K

00

)p(s

)z(sGH

i

i

i0

i00 ps

zsGH(s )

i0

i0

zs

psK

Gain for a root to be at s0

04/19/23 6.5 Root-locus: Other Rules & Examples

11

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Note:

|z|z 00-z-z 11

||z0

z1

zerosto distances of product

polesto distances of productK

04/19/23 6.5 Root-locus: Other Rules & Examples

12

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Apply to example by MATLAB

rlocfind([1 -1],[1 5 7 5 6]), grid,figure(1)

04/19/23 6.5 Root-locus: Other Rules & Examples

13

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Examples

2s

1GH

04/19/23 6.5 Root-locus: Other Rules & Examples

14

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Introduce a zero in LHP

2s

1sGH

rlocus([1 1],[1 0 0])

04/19/23 6.5 Root-locus: Other Rules & Examples

15

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Introduce a 2nd pole in LHP

2s

1sGH

2

04/19/23 6.5 Root-locus: Other Rules & Examples

16

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Introduce a zero to 2nd order system

1)s(s

1GH

2s

04/19/23 6.5 Root-locus: Other Rules & Examples

17

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

Introduce a pole

1)s(s

2sGH

3)(s

04/19/23 6.5 Root-locus: Other Rules & Examples

18

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

COMPARE

1)s(s

1GH

3s

2s

1)s(s

1GH

04/19/23 6.5 Root-locus: Other Rules & Examples

19

Introduction to Feedback Systems / © 1999 Önder YÜKSEL

End of Chapter

General index

Restart section

Next chapter Restart chapter

End show

The Endi