INC 341PT & BP INC341 Root Locus (Continue) Lecture 8

INC 341 PT & BP

INC341Root Locus (Continue)

Lecture 8

INC 341 PT & BPINC 341 PT & BP

Sketching Root Locus(review)

1. Number of branches

2. Symmetry

3. Real-axis segment

4. Starting and ending points

5. Behavior at infinity


Refining the sketch

1.Real-axis breakaway and break-in points

2.Calculation of jω-axis crossing

3.Angels of departure and arrival

4.Locating specific points


Break-inpoint

Breakawaypoint

1. Real-axis breakaway andbreak-in points

point where the locus leaves the real axis

point where the locus returns to the real axis


1)()( sHsKG)()(

1

sHsGK

set s = σ (on the real axis))()(

1

HGK

Breakawaypoint

Break-inpoint


Example

)2)(1(

)5)(3()()(

ss

ssKsHsKG Find breakaway, break-in points

0158

612611

)158(

)23(

)23(

)158(1)()(

2

2

2

2

2

2

ss

ss

ds

dK

ss

ssK

ss

ssKsHsKG Condition of poles

then solve for s

s = -1.45, 3.82 is breakaway and break-in points


Another approach without derivative

n

i

m

i pszs 11

11

82.3,45.1

0612611

2

1

1

1

5

1

3

1

2

s

ss

ssss


2. Calculation of jω-axis crossing

Imaginary axis is a boundary of stabilityuse Routh-Hurwitz criterion!!!

Imaginary axis crossing


Review of Routh-Hurwitz

“the number of roots of the polynomial that are in the right half plane is equal to the number of sign changes in the first column”


Example

KsKsss

sKsT

3)8(147

)3()(

234

From the closed-loop transfer function, find an imaginary axis crossing


65.9

0720652

K

KK

Substitute K=9.65 in s2 to find the value of s

59.135.80

65.202

065.20235.8021)90( 22

js

sKsK

A complete row of zerosyields imag. axis roots


3. Angles of departure and Arrival

Fact: root locus starts at open loop poles and ends at open loop zeros

180)12()()( ksHsKG

Assume a point on the root locus close to a complex Pole, the sum of angles to this point is an odd multiple of 180.


180)12(654321 k

180)12(654312 k

Angel of arrival (zero)

Angel of departure (pole)


Example

sketch root locus and find angel of departureof complex poles

x

x

x

o-3 -2 -1

1


43.10857.251

180)2

1(tan)

1

1(tan90

1

111

4321


4. Calibrating root locus

Search a given line for the point yielding a summation of angles equal to an odd multiple of 180.

180)12()()( ksHsKG

Gain at this point = pole length/zero length


At r=0.747 71.1B

EDCAK

Intersection with damping ratio line

Coordinate on Damping line = (rcosθ, rsinθ)

Try r = 0.5, 1, 0.8, 0.7, 0.75, 0.725, …..

ζ=cosθ

θY=-mxM = tan(acos(damping ratio))


Examplesketching root locus

What is the exact point and gain where the locus crosses the imag. Axis?

Where is the breakaway point?

What range of K that keep the system stable?


Transient Response Designvia Gain Adjustment

Find K that gives a desired peak time, settling time, %OS (find K at the intersection)

Use 2 order approx. and consider only dominant pole


The third pole can be ignored (a gives a better approx. than b cause the third pole is further to the left)

Zero closed to the dominant poles can be cancelled by the third pole (c gives a better approx. than d)


Example

Find K that yields 1.52% overshoot.Also estimate settling time, peak time, steady-state error corresponding to the K

Step I: 1.52% overshoot ζ=0.8Step II: draw a root locus


Step III: draw a straight line of 0.8 damping ratio Step IV: find intersection points where the net angleis added up to 180*n, n=1,2,3,…


Step V: find the corresponding K at each pointStep VI: find peak time, settling time corresponding to the pole locations (assume 2nd order approx.) Step VII: calculate Kv and ss error

Note: case 1 and 2 cannot use 2nd order approx.cause the third pole and closed loop zero are far away.In case 3, the approx. is valid.


Generalized Root Locus

K is fixed, vary open loop pole instead!!!

Creating an equivalent system where p1 appearsas the forward path gain.


1022

10

)(1

)()(

112

psspssG

sGsT

Try to get a general TF )()(1

)()(

sHsKG

sKGsT

102

)2()()(

102)2(

1

10210

)(

)2(102

10)(

21

21

2

12

ss

spsHsKG

ssspsssT

spsssT



Using MATLAB with Root Locus

•tf

•pzmap

•rlocus

•sgrid

•sisotool

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INC 341PT & BP INC341 Root Locus (Continue) Lecture 8