root locus2 2

8/6/2019 Root Locus2 2

http://slidepdf.com/reader/full/root-locus2-2 1/16

1

Example #2C(s)+

-

R(s)K

)20(

1

+ss

Closed-Loop Transfer Function is:

K ssK

ssK

ss

K

s RsC ++=

++

+= )20(

)20(

11

)20(

1

)()(

Use Matlab to plot Root Locus

-20 -15 -10 -5 0 5-15

-10

-5

0

5

10

15

Real Axis

I m a g A x i s



2

-20 -15 -10 -5 0 5-15

-10

-5

0

5

10

15

Real Axis

I m a g A x i s

Find K for roots with ζ=.707

-p2=-20 -p1=0

s=-10+j10

200

010101

=

++−=+ j ps

200

2010102

=

++−=+ j ps

Use Magnitude Criterion

1*

1

)(

)(

21

=++

= ps ps

K s Den

s NumK

Once a point, s, is known to be on the root locus, then the

magnitude criterion can be used to find the gain K for that

point (i.e., find K at desired root locations)

2001200*200

1=⇒= K K



3

-20 -15 -10 -5 0 5-15

-10

-5

0

5

10

15

Real Axis

I m a g A x i s

Find K for a C.E. root at s=-8

-p2=-20 -p1=0

s=-8

8081 =+−=+ ps

122082 =+−=+ ps

96

1

12*8

1

=⇒

=

K

K

Example #3

C(s)+

-

R(s)K

)2)(1(

)3(

+++ss

s


)3()2)(1(

)3(

)(

)(

++++

+

= sK ss

sK

s R

sC

Use Matlab to find the root locus



4

Root Locus for Example #3

-6 -5 -4 -3 -2 -1 0 1 2-1.5

-1

-0.5

0

0.5

1

1.5

Real Axis

I m a g A x i s

Modify Axes>> axis([-5 0 -1 3]) axis([Xmin Xmax Ymin Ymax])

Find K for

minimum

peak time

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-1

-0.5

0

0.5

1

1.5

2

2.5

3

Real Axis

I m a g A x i s



5

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0-1

-0.5

0

0.5

1

1.5

2

2.5

3

Real Axis

I m a g A x i s

K for minimum peak time

-p1=-1

4.24.12

14.13

22

1

=+=

++−=+ j ps

s=-3+j1.4

-p2=-2

7.14.11

24.13

22

2

=+=

++−=+ j ps

-z1=-34.14.10

34.13

221

=+=

++−=+ j zsXXO

Use magnitude criterion

1*)(

)(

21

1 =++

+=

ps ps

zsK

s Den

s NumK

9.217.1*4.2

4.1≈⇒= K K

0)3(9.2)2)(1( =++++ sssCheck roots of:



6

Example #4C(s)+

-

R(s)K

)3)(1(

)5(

+++ss

s


)5()3)(1(

)5(

)(

)(

++++

+=

sK ss

sK

s R

sC

Is the point s=-4+j2 on the root locus?

Plot the vectors

-5 -4 -3 -2 -1 0

+j1

+j2

-j1

-j2

s=-4+j2

)( 1 zs+∠ )( 1 ps +∠)( 2 ps +∠



7

Use Phase Criterion

oo

oo

360180)()()(

360180)(

)(

211 k ps ps zs

k s Den

s Numangle

±=+∠−+∠−+∠

±=

++

=++−∠=+∠ −1

2tan)524()( 1

1 j zs

o4.6312tan)( 1

1 =

++=+∠ − zs

Continue Phase Criterion

−+

=++−∠=+∠ −3

2tan)124()( 1

1 j ps

o3.1463

2tan)( 1

1 =

−+

=+∠ − ps

−+

=++−∠=+∠ −1

2tan)324()( 1

2 j ps

o6.11612tan)( 1

1 =

−+=+∠ − ps



8

Add Phase Angles

oooo 5.1996.1163.1464.63

)()()( 211

−=−−

=+∠−+∠−+∠ ps ps zs

s=-4+j2 is not on this

root locus

Rules #1-#3 for Root Locus

• Put polynomial in standard form,

• Plot open-loop poles (X) and zeros (O)on complex plane

• Root locus is on the real axis to the leftof an odd number of open-loop polesand zeros (gives angle of 180°)

0))...()((

))...()((1

)(

)(1)(1

21

21 =++++++

+=+=+n

m

ps ps ps

zs zs zsK

s Den

s NumK sKf



9

Rules #4-#5 for Root Locus• Number of linear asymptotes = n-m,

angle to 1st one (equally spaced) is

• If n-m>1, center of linear asymptotesfound by

mna −=

°180

φ

−−−

−= ∑∑

==

m

j j

n

iia z p

mn11

)()(1σ

Rules #6-#8 for Root Locus

• Determine the breakaway points on the real axis (if any).

• Use the Routh-Hurwitz criteria to evaluateimaginary axis crossings (if any)

– look for a row of zeros in the S0 or S1 rows

• Evaluate the angle of departure from complex poles (and angle of arrival at complex zeros) by the angle criteria



10

Rules #9-#10 for Root Locus• Sketch in the remainder of the root locus.

– Use the angle criteria to determine suitability of any

questionable points.

• The gain K at any root location can befound by the magnitude criteria,

1...

...

2121 =+++

+++

nm ps ps ps

zs zs zsK

Example #5

C(s)+

-

R(s)K

)4)(3)(1(

)2(

++++

ssss

s


)2()4)(3)(1(

)2(

)(

)(

+++++

+

= sK ssss

sK

s R

sC

Apply root locus steps 1, 2, 3, 4, 5, and 7 to this problem



11

Root Locus #5

-6 -5 -4 -3 -2 -1 0 1 2-8

-6

-4

-2

0

2

4

6

8

Real Axis

I m a g A x i s

Example #6

C(s)+

-

R(s)K

)25)(4(

1

++ sss


K sss

K

s R

sC

+++= )25)(4()(

)(




12

Root Locus #6

-30 -25 -20 -15 -10 -5 0 5-30

-20

-10

0

10

20

30

Real Axis

I m a g A x i s

Example #7

C(s)+

-

R(s)K

)5)(1(

)4(

++−ss

ss


)2()4)(3)(1(

)2(

)(

)(

++++−

+

= sK sss

sK

s R

sC




14

Root Locus #8

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1-4

-3

-2

-1

0

1

2

3

4

Real Axis

I m a g A x i s

Example #9

C(s)+

-

R(s)K

)84(

1

2 ++ sss


K sss

K

s R

sC

+++=

)84()(

)(

2




15

Root Locus #9

-8 -6 -4 -2 0 2 4-5

-4

-3

-2

-1

0

1

2

3

4

5

Real Axis

I m a g A x i s

Example #10

C(s)+

-

R(s)K

)172)(5(

)1012(2

2

+++

++

sss

ss


)1012()172)(5(

)1012(

)(

)(

22

2

++++++

++

= ssK sss

ssK

s R

sC




16

Root Locus #10

-7 -6 -5 -4 -3 -2 -1 0 1 2-10

-8

-6

-4

-2

0

2

4

6

8

10

Real Axis

I m a g A x i s

root locus2 2

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