INC 341 PT & BP

INC341Root Locus

Lecture 7

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s = 4 + j3

jω

σ

3

4

Rectangular form: 4 + j3

Polar form magnitude=5, angle = 37

Rectangular vs. polar

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Rectangular form

Add, Subtraction

Polar form

45)1()34( jjj

Multiplication

2510)1237(25

122375

Division

492

5

122375

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jba

a

b

r

θ

r

sin

cos

arctan

22

rb

ra

a

b

bar

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Vector representation of a transfer function

)(

)()(

1

1

i

n

i

i

m

i

ps

zssF

i

n

i

i

m

i

ps

zs

lengthspole

lengthszeroM

1

1

n

ii

m

ii pszs

anglespoleangleszero

11

)()(

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Vector s(s+a)

(s+a)(s+7)s = 5+j2

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Example

)2(

)1()(

ss

ssF

34.114217.0

04.1041787.1265

57.11620

)41)(43(

42

jj

j

Find F(s) at s = -3+j4

)2(

)1()(

ss

ssF

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What is root locus and why is it needed?

• Fact I: poles of closed-loop system are an important key to describe a performance of the system (transient response, i.e. peak time, %overshoot, rise time), and stability of the system.

• Fact II: closed-loop poles are changed when varying gain.

• Implication: Root locus = paths of closed-loop poles as gain is varied.

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Cameraman

Object Trackingusing infrared

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Varying gain (K)

Varying K, closed-loop poles are moving!!!

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Transient:• K<25 overdamped• K=25 critically damped• K>25 underdamped

• Settling time remains the same under underdamped responses.

Stability:• Root locus never crosses

over into the RHP, system is always stable.

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Concept of Root Locus

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)()(1

)()(

sHsKG

sKGsT

Characteristic equation

18011)()( sHsKG

Closed-loop transfer function

magnitude 1)()( sHsKG

,...3,2,1

180)12()()(

k

ksHsKG

phase

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magnitude 1)()( sHsKG

,...3,2,1

180)12()()(

k

ksHsKG phase

If there is any point on the root locus, its magnitude and phase will be consistant with the follows:

Note that: phase is an odd multiple of 180

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Example Is the point -2+3j a closed-loop pole for some value of gain? Or is the point on the root locus?

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55.7043.1089057.7131.564321

)2)(1(

)4)(3()()(

ss

ssKsHsKG

-2+3j is not on the root locus!!! What about ?)2/2(2 j

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)()(

1

1)()(

sHsGK

sHsKG

33.022.112.2

22.1707.021

43

LL

LLK

43

21)(LL

LLsG

The angles do add up to 180!!! is a point on the root locus )2/2(2 j

What is the corresponding K?

INC 341 PT & BPINC 341 PT & BP

Sketching Root Locus

1. Number of branches

2. Symmetry

3. Real-axis segment

4. Starting and ending points

5. Behavior at infinity

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1. Number of branchesNumber of branches = number of closed-loop poles

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2. SymmetryRoot locus is symmetrical about the real axis

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3. Real-axis segment

On the real axis, the root locus exists to the left of an odd number of real-axis

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180)12()()( ksHsKG

• Sum of angles on the real axis is either 0 or 180 (complex poles and zeroes give a zero contribution).

• Left hand side of even number of poles/zeros on the real axis give 180 (path of root locus)

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Example

root locus on the real axis

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4. Starting and ending points

Root locus starts at finite/infinite poles of G(s)H(s) And ends at finite/infinite zeros of G(s)H(s)

)()(1

)()(

sHsKG

sKGsT

closed-loop transfer function

)(

)()(

sD

sNsG

G

G )(

)()(

sD

sNsH

H

H

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)()()()(

)()()(

sNsKNsDsD

sDsKNsT

HGHG

HG

K=0 (beginning) poles of T(s) are

K=∞ (ending) poles of T(s) are

)()( sDsD HG

)()( sNsKN HG

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Example

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5. Behavior at infinity

Root locus approaches asymptote as the Locus approaches ∞, the asymptotes is given by

zeros finite#poles finite#

zeros finitepoles finite

a

...,2,1,0

zeros finite#poles finite#

)12(

k

ka

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)2)(1()()(

sss

KsHsKG

Rule of thumb

# of poles = # of zeroes

has 3 finite poles at 0 -1 -2, and 3 infinite zeroes at infinity

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Example

Sketch root locus

3

4

14

)3()421(0

2kfor , 3/5

1kfor ,

0kfor , 3/

zeros finite#poles finite#

)12(

k

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