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ME322Module 6, Slide 1
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Outline
Build dynamic models
Study the dynamic responses of a system
Basic properties of feedback control
How to design control systems
Root-locus method
Frequency-response method
State-space method
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ME322Module 6, Slide 2
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Input, ROutput, Y
+_
KGG(s)KA
For a basic feedback control system, the
closed loop transfer function is:
)(1
)(
)(
)(
sGKK
sGKK
sR
sY
GA
GA
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ME322Module 6, Slide 3
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
The characteristic equation is:
Definition: The graph of all possible roots ofthe characteristic equation relative to someparticular variable is called the root locus.
Purpose: to show graphically the generaltrend of the roots of the closed-loop systemwhen some parameters are varied.
0)(1 sGKK GA
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ME322Module 6, Slide 4
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Assumptions:
The plant transfer function can be expressed
in general terms as the ratio of twopolynomials (m
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ME322Module 6, Slide 5
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Assumptions:
The plant gainKGis positive. n > m.
The root-locus parameter is defined as
Then, the root-locus form of the
characteristic equation is:
KsG
orsKG
1)(
0)(1
GAKKK
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ME322Module 6, Slide 6
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Root-locus for a DC-motor (Example)
DC Motor transfer function:
Characteristic equation of the closed loopsystem is [1+KG(s)]
)1)(1()(
21
ss
AsG
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ME322Module 6, Slide 7
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Root-locus for a DC-motor (Example)
Roots of the closed loop system
characteristic equation is
When K=0, open loop roots:
21
21
2
21212,1
21
2)1(4)()(
0)1)(1(
KAs
KAss
2
2
1
1
1;
1
ss
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ME322Module 6, Slide 8
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Example
j
0-1/t1-1/t2= pole
= zero
K=0 K=0
21
21
2)(
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ME322Module 6, Slide 9
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Root-locus for a DC-motor (Example)
Observations:
- One root locus branch per root
- Branches begin at open loop roots for K=0
since the system is open loop for K=0.
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ME322Module 6, Slide 10
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Root-locus for a DC-motor (Example)
Observations: when K increases,
- poles moving towards each other and intoimaginary conjugate poles at breakaway point(where roots move away from the real-axis)
- Decreasing time constant (faster response)
- Decreasing rise time- Decreasing damping ratio, more overshoot, less
stable
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ME322Module 6, Slide 11
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Phase condition
From the root-locus form of the characteristic
equation (assuming K is real and positive):
we can define that the root locus of G(s) is theset of points in the s-plane where thephase of G(s) is 180.
KsG
1)(
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ME322Module 6, Slide 12
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 1: Draw the pole and zero plot in the s-
plane.
]16)4[(
1)(
2
ss
sG
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ME322Module 6, Slide 13
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Pole and Zero Plot
j
0= pole
= zero
4
4
4
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ME322Module 6, Slide 14
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 2: Find the real axis portion of the locus.
The angles of complex poles cancel each other (theangles are all measured in a counterclockwisedirection from a horizontal line)
The angle of G(s0) for the test point s0 on the real
axis is given by the angles from poles and zeroson the real axis only.
The angle is 180 when the test point is to the left ofan odd number of poles plus zero.
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ME322Module 6, Slide 15
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Pole and Zero Plot
j
0= pole
= zero
4
4
4
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ME322Module 6, Slide 16
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 3: For large K, the branches of the root
locus go to either zeros or infinity. Drawthe asymptotes for large values of K.
At point
draw radial lines at the (n-m) distinct anglesmn
zps
ii
0
mnlmn
ll
,....,2,1,
)1(360180
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ME322Module 6, Slide 17
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 3: Draw the asymptotes for large values
of K.
At point
draw radial lines at the (n-m) distinct angles
67.203
)044(0
mn
zps
ii
;300;180;60
,....,2,1,)1(360180
321
mnlmn
ll
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ME322Module 6, Slide 18
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Pole and Zero Plot
j
0= pole
= zero
4
4
4
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ME322Module 6, Slide 19
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 4: Compute the departure angles from the poles
(q is the order of the multiple poles):
where i is the sum of the angles to the remainingpoles and i is the sum of the angles to all thezeros. lare positive or negative integers.
lq iidep 360180
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ME322Module 6, Slide 20
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 4: Compute the arrival angles at zeros (q is the
order of the multiple zeros):
where i is the sum of the angles to all the poles
and i is the sum of the angles to all theremaining zeros. lare positive or negativeintegers.
lq iiarr 360180
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ME322Module 6, Slide 21
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design MethodGuidelines for sketching a Root Locus:
Step 4: take the pole #2 at4 +4j, the angles to theremaining poles are 90 and 135, respectively.
Set l= - 1, we obtain:
Based on the complex conjugate symmetry, thedeparture angle at pole4-4j is 45. Thedeparture angle for the pole in the origin is 180.
(No zero here, hence no arrival angle calculation)
45180225
)1(*360180)13590(0
360180_2
liidep
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ME322Module 6, Slide 22
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Pole and Zero Plot
Test point S0 is close to
the pole.
j
0= pole
s0
4
4
4
1
2
3
= zero
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ME322Module 6, Slide 23
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Pole and Zero Plot
j
0= pole
= zero
4
4
4
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ME322Module 6, Slide 24
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 5: estimate the cross-over points on the
imaginary axis.The characteristic equation is:
0328
0]16)4[(
1
23
2
Ksss
ss
K
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ME322Module 6, Slide 25
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design MethodGuidelines for sketching a Root Locus:
Step 5: estimate the cross-over points on theimaginary axis.
Build Routh array:
Cross-over at K=2560:
08
256:
8:
321:
0
1
2
3
Ks
Ks
Ks
s
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ME322Module 6, Slide 26
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 5: estimate the cross-over points on theimaginary axis.
Asymptote cross-over points based on
trigonometric relationships:
(2.67*tan60)j=4.62j. Conjugate: -4.62j
66.5
0256)(32)(8)(
0
0
2
0
3
0
jjj
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ME322Module 6, Slide 27
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Pole and Zero Plot
j
0= pole
= zero
4
4
4
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ME322Module 6, Slide 28
Northern Illinois UniversitySummer 2005
Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 6: estimate the locations of multiple
closed-loop roots, and find the arrival anddeparture angles at these locations.
We can find K at breakaway point by taking a
derivative:
0)(1
)(
;0)1
(
2
0
ds
dba
ds
dab
bb
a
ds
d
Gds
dss
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ME322Module 6, Slide 29 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 6:
Rule of thumb: Two locus segments comingtoward each other will break away with a+/- 90 change of direction. 3 locussegments will approach at 120 wrt each
other and depart with a 60 changerelative to the arrival angles.
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ME322Module 6, Slide 30 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Guidelines for sketching a Root Locus:
Step 7: complete the sketch. The root locus
branches start at poles and end at zeros orinfinity.
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ME322Module 6, Slide 31 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Pole and Zero Plot
j
0
= pole
= zero
4
4
4
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ME322Module 6, Slide 32 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Applications of the Guidelines
Consider the root locus of the characteristic
equation 1+KG(s)=o, where
Construct the root locus for various values ofP.
)(
1)(
2 pss
ssG
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ME322Module 6, Slide 33 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Applications of the Guidelines
Case 1: Large values of P.
2
1)(
s
ssG
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ME322Module 6, Slide 34 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Step 1: mark poles & zeros
j
0
= pole
= zero
1
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ME322Module 6, Slide 35 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Step 2: draw locus on real axis to the left of an
odd number of poles plus zeros:j
0
= pole
= zero
1
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ME322Module 6, Slide 36 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Step 3: Find the number of asymptotes and draw
them
number of asymptotes = n-m=2 - 1=1
does not matter for there is only one asymptote.
(its the root locus on the real axis)
180
12
)11(360180
,....,2,1,)1(360180
1
mnlmn
ll
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ME322Module 6, Slide 37 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Step 4: Calculate departure and arrival angles
j
0
= pole
= zero
1
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ME322Module 6, Slide 38 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Step 4: Calculate departure and arrival angles
Draw a small circle around the
double poles.
180360180)180180(
360180
;90;90;360180)00(2
360180
_2_1
l
lq
l
lq
arr
iiarr
depdepdep
iidep
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ME322Module 6, Slide 39 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Step 4: Calculate departure and arrival angles
j
0
= pole
= zero
1
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ME322Module 6, Slide 40 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design MethodStep 5: Cross-over on the imaginary axis
The characteristic equation is
Build Routh array:
For K>0, all roots are in LHP and do not crossimaginary axis.
02 KKss
0:
0:
1:
0
1
2
Ks
Ks
Ks
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ME322Module 6, Slide 41 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design MethodStep 6: estimate the locations of multiple roots, and find the
arrival and departure angles at these locations.
2;0
;0)2(;01*2*)1(
1;1)(
2;)(
0)(1
2
2
2
ss
sssssds
dbssb
sds
dassa
ds
dba
ds
dab
b
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ME322Module 6, Slide 42 Northern Illinois UniversitySummer 2005 Peter A. LinCopyright 2001-2005
Root-locus Design Method
Step 7: Sketch the locus
j
0
= pole
= zero
1
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Basic Properties of Feedback
Homework Assignment
Draw the root locus for the following:
1).
2).
)5)(1(
4)(
ss
sG
2
)3(
)( s
s
sG