Control-Root Locus

8/21/2019 Control-Root Locus

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Aims

Learn a specific techniquewhich shows how

changes in one of a systems parameter(usually the controller gain, K)

will modify thelocation of the closed-loop poles

in the s-domain.


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Definition

01 sHsKG

The closed-loop poles of the negative feedback control:

are the roots of the characteristic equation:

01 sHsKG

The root locus is the locus of the closed-loop poles

when a specific parameter (usually gain, K)

is varied from 0 to infinity.


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Root Locus Method The value of sin the s-plane that make the loop gain

KG(s)H(s) equal to -1 are the closed-loop poles

(i.e. )

KG(s)H(s) = -1 can be split into two equations byequating the magnitudes and anglesof both sides

of the equation.

101 sHsKGsHsKG


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Angle and Magnitude Conditions

Independent of K

,,,l 210

,,,l 210

12180

1

12180

1

1

0

lsHsG

KsHsG

lsHsKGsHsKG

sHsKG

o


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Learning Steps1) Sketch the root locus of the following system:

2) Determine the value of Ksuch that the dampingratio of a pair of dominant complex conjugateclosed-loop is 0.5.


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Rule #1Assuming npoles and mzeros for G(s)H(s):

The nbranchesof the root locus start at the n

poles. mof these nbranches end on the mzeros

The n-mother branches terminate at infinityalong asymptotes.

First step: Draw the npoles and mzeros of G(s)H(s)using x and o respectively


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Step #1Draw the npoles and m

zeros of G(s)H(s) using xand o respectively.

3 poles:

p1 = 0; p2 = -1; p3 = -2

No zeros

21

1

ssssHsG


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Step #1Draw the npoles and m

zeros of G(s)H(s) using xand o respectively.

3 poles:

p1 = 0; p2 = -1; p3 = -2

No zeros

21

1

ssssHsG


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Rule #2

The loci on the real axisare to the leftof an ODDnumberof REAL poles and REAL zerosof

G(s)H(s)

Second step: Determine the loci on the real axis.Choose a arbitrary test point. If the TOTAL number

of both real poles and zeros is to the RIGHT of thispoint is ODD, then this point is on the root locus


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Step #2Determine the loci on the

real axis:

Choose a arbitrary testpoint.

If the TOTAL number ofboth real poles and zeros

is to the RIGHT of thispoint is ODD, then thispoint is on the root locus


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Step #2Determine the loci on the

real axis:

Choose a arbitrary testpoint.

If the TOTAL number ofboth real poles and zeros

is to the RIGHT of thispoint is ODD, then thispoint is on the root locus


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Rule #3

Assuming npoles and mzeros for G(s)H(s): The root loci for very large values of s must be

asymptotic to straight lines originate on the real axisat point:

radiating out from this point at angles:

Third step: Determine the n - masymptotes of the root loci.Locate s = on the real axis. Compute and draw angles.Draw the asymptotes using dash lines.

mn

lo

l

12180

mn

zp

s mi

n

i


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Step #3

Determine the n - masymptotes: Locate s = on the real axis:

Compute and draw angles:

Draw the asymptotes usingdash lines.

13

210

03

321

ppps

mn

ll

12180

0

0

1

0

0

0

18003

112180

6003

102180

,,,l 210


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Step #3

Determine the n - masymptotes: Locate s = on the real axis:

Compute and draw angles:

Draw the asymptotes usingdash lines.

13

210

03

321

ppps

mn

ll

12180

0

0

1

0

0

0

18003

112180

6003

102180

,,,l 210


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Breakpoint Definition The breakpoints are the points in the s-domain

where multiplesroots of the characteristic

equation of the feedback control occur.

These points correspond to intersection points onthe root locus.


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Rule #4Given the characteristic equation is KG(s)H(s) = -1

The breakpoints are the closed-loop poles thatsatisfy:

Fourth step: Find the breakpoints. Express Ksuch as:

Set dK/ds = 0 and solve for the poles.

0dsdK

.

sHsGK

1


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Step #4Find the breakpoints.

Express Ksuch as:

Set dK/ds = 0 and solve for the

poles.

4226057741

0263

21

2

.s,.s

ss

sssK

sss

)s(H)s(G

K

23

211

23


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Step #4Find the breakpoints.

Express Ksuch as:

Set dK/ds = 0 and solve for the

poles.

4226057741

0263

21

2

.s,.s

ss

sssK

sss

)s(H)s(G

K

23

211

23


19/23

Rule #1 AgainAssuming npoles and mzeros for G(s)H(s):

The nbranchesof the root locus start at the n

poles. mof these nbranches end on the mzeros

The n-mother branches terminate at infinityalong asymptotes.

Last step: Draw the n-mbranches that terminate atinfinity along asymptotes


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Last Step

Draw the n-mbranches thatterminate at infinity along

asymptotes


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Points on both root locus & imaginary axis?

Points on imaginary axissatisfy:

Points on root locus satisfy:

Substitute s=jinto thecharacteristic equation andsolve for.

jsj?

- j

01 sHsKG

20 or


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Learning Steps1) Sketch the root locus of the following system:

2) Determine the value of Ksuch that the dampingratio of a pair of dominant complex conjugateclosed-loop is 0.5.

See class notes


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Control-Root Locus

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