Section 5 Root Locus Analysis lecture 55

presentation

1. ESE 430 –Feedback Control Systems
SECTION 5:
ROOT-LOCUS ANALYSIS

2. K. Webb ESE 430
Introduction
2

3. K. Webb ESE 430
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Introduction
Consider a general feedback system:
Closed-loop transfer function is
????????????????????????=
????????????????????????????????????
1+????????????????????????????????????????????????????????????
????????????????????????is the forward- path transfer function
May include controller and plant
????????????????????????is the feedback- path transfer function
Each are, in general, rational polynomials in ????????????
????????????????????????=
????????????
????????????????????????
????????????
????????????
????????????
and ????????????????????????=
????????????
????????????????????????
????????????
????????????
????????????

4. K. Webb ESE 430
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Introduction
So, the closed- loop transfer function is
????????????????????????=
????????????
????????????
????????????????????????
????????????
????????????
????????????
1+????????????
????????????
????????????????????????
????????????
????????????
????????????
????????????
????????????????????????
????????????
????????????
????????????
=
????????????????????????
????????????????????????????????????
????????????????????????
????????????
????????????
????????????????????????
????????????????????????+????????????????????????
????????????????????????????????????
????????????????????????
Closed-loop zeros:
Zeros of ????????????????????????
Poles of ????????????????????????
Closed-loop poles:
A function of gain, ????????????
Consistent with what we’ve already seen –feedback moves
poles

5. K. Webb ESE 430
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Closed-Loop Poles vs. Gain
How do closed- loop poles vary as
a function of ???????????? ?
Plot for ???????????? =0,0.5,1,2,5,10,20
Trajectory of closed-loop poles vs.
gain (or some other parameter):
root locus
Graphical tool to help determine
the controller gain that will put
poles where we want them
We’ll learn techniques for
sketching this locus by hand

6. K. Webb ESE 430
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Root Locus
An example of the type of root locus we’ll learn to
sketch by hand, as well as plot in MATLAB:

7. K. Webb ESE 430
Evaluation of Complex Functions
7

8. K. Webb ESE 430
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Vector Interpretation of Complex Functions
Consider a function of a complex variable ????????????
????????????????????????=
????????????−????????????
1????????????−????????????
2⋯
????????????−????????????
1????????????−????????????
2⋯
where ????????????
????????????are the zerosof the function, and ????????????
????????????are
the polesof the function
We can write the function as
????????????????????????=
∏
????????????=1
????????????
????????????−????????????
????????????
∏
????????????=1
????????????
????????????−????????????
????????????
where ????????????is the # of zeros, and ????????????is the # of poles

9. K. Webb ESE 430
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Vector Interpretation of Complex Functions
At any value of ???????????? , i.e. any point in the complex plane, ????????????????????????
evaluates to a complex number
Another point in the complex plane with magnitude and phase
????????????????????????=??????????????????????????????
where
????????????=????????????????????????=
∏
????????????=1
????????????
????????????−????????????
????????????
∏
????????????=1
????????????
????????????−????????????
????????????
and
????????????=∠�
????????????=1
????????????
????????????−????????????
????????????−∠�
????????????=1
????????????
????????????−????????????
????????????
????????????=�
????????????=1
????????????
∠
????????????−????????????
????????????−�
????????????=1
????????????
∠
????????????−????????????
????????????

10. K. Webb ESE 430
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Vector Interpretation of Complex Functions
Each term ????????????−????????????
????????????represents a vectorfrom ????????????
????????????to the
point, ????????????, at which we’re evaluating ????????????????????????
Each ????????????−????????????
????????????represents a vectorfrom ????????????
????????????to ????????????
For example:
????????????????????????=
????????????+3
????????????+4????????????
2
+2????????????+5
Zero at: ????????????=−3
Poles at: ????????????
1,2=−1±??????????????????and ????????????
3=−4
Evaluate ????????????????????????at ????????????=−2+????????????
????????????????????????�
????????????=−2+????????????

11. K. Webb ESE 430
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Vector Interpretation of Complex Functions
First, evaluate the magnitude
????????????????????????=
????????????−????????????
1
????????????−????????????
1????????????−????????????
2????????????−????????????
3
????????????−????????????
1=1+????????????=2
????????????−????????????
1=−1−????????????=2
????????????−????????????
2=−1+????????????3=10
????????????−????????????
3=2+????????????=5
The resulting magnitude:
????????????????????????=
2
2105
=
2
10
????????????????????????=0.1414
????????????????????????=????????????????????????∠????????????????????????

12. K. Webb ESE 430
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Vector Interpretation of Complex Functions
Next, evaluate the angle
∠????????????????????????=∠????????????−????????????
1−∠????????????−????????????
1
−∠????????????−????????????
2−∠????????????−????????????
3
∠????????????−????????????
1=∠1+????????????=45°
∠????????????−????????????
1=∠−1−????????????=−135°
∠????????????−????????????
2=∠−1+????????????3=108.4°
∠????????????−????????????
3=∠2+????????????=26.6°
The result:
????????????????????????�
????????????=−2+????????????
=0.1414∠ 45°
????????????????????????=????????????????????????∠????????????????????????

13. K. Webb ESE 430
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Finite vs. Infinite Poles and Zeros
Consider the following transfer function
????????????????????????=
????????????+8
????????????????????????+3????????????+10
One finite zero: ????????????=−8
Three finite poles: ????????????=0, ????????????=−3, and ????????????=−10
But, as ????????????→∞
lim
????????????→∞
????????????????????????=
∞
∞
3
=0
This implies there must be a zero at ???????????? =∞
All functions have an equal number of poles and zeros
If ????????????????????????has ????????????poles and ????????????zeros, where ????????????≥????????????, then ????????????????????????has
????????????−????????????zeros at ???????????? =????????????
∞
????????????
∞
is an infinite complex number –infinite magnitude and someangle

14. K. Webb ESE 430
The Root Locus
14

15. K. Webb ESE 430
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Root Locus – Definition
Consider a general feedback system:
Closed-loop transfer function is
????????????????????????=
????????????????????????????????????
1+????????????????????????????????????????????????????????????
Closed-loop poles are roots of
1+????????????????????????????????????????????????????????????
That is, the solutions to
1+????????????????????????????????????????????????????????????=0
Or, the values of ????????????for which
????????????????????????????????????????????????????????????=−1 (1)

16. K. Webb ESE 430
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Root Locus – Definition
Because ????????????????????????and ????????????????????????are complex functions, (1) is really two
equations:
∠????????????????????????????????????????????????=2????????????+1180°
that is, the angle is an odd multiple of 180° , and
????????????????????????????????????????????????????????????=1
So, if a certain value of ????????????satisfies the angle criterion
∠????????????????????????????????????????????????=2????????????+1180°
then that value of ????????????is a closed-loop pole for some value of ????????????
And, that value of ????????????is given by the magnitude criterion
????????????=
1
????????????????????????????????????????????????

17. K. Webb ESE 430
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Root Locus – Definition
The rootlocusis the set of all points in the s-plane
that satisfy the anglecriterion
∠????????????????????????????????????????????????=2????????????+1180°
The set of all closed-looppolesfor 0≤????????????≤∞
We’ll use the angle criterion to sketch the root locus
We will derive rules for sketching the root locus
Not necessary to test all possible s- plane points

18. K. Webb ESE 430
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Angle Criterion –Example
Determine if ????????????
1=−3+??????????????????is on this system’s root locus
????????????
1is on the root locus if it satisfies the angle criterion
∠????????????????????????
1=2????????????+1180°
From the pole/zero diagram
∠????????????????????????
1=−135°+90°
∠????????????????????????
1=−225°≠2????????????+1180°
????????????
1does not satisfy the angle
criterion
It is not on the root locus

19. K. Webb ESE 430
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Angle Criterion –Example
Is ????????????
2=−2+????????????on the root locus?
Now we have
∠????????????????????????
2=−135°+45°=−180°
????????????
2is on the root locus
What gain results in a closed- loop pole at ????????????
2?
Use the magnitude criterion to determine ????????????
????????????=
1
????????????????????????
2
=????????????
2+1????????????
2+3=2⋅2=2
????????????=2yields a closed-loop pole at ????????????
2=−2+????????????
And at its complex conjugate, ̅????????????
2=−2−????????????

20. K. Webb ESE 430
Root Locus –Real-axis segments
20

21. K. Webb ESE 430
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Real-Axis Root-Locus Segments
We’ll first consider points on the real axis, and whether or
not they are on the root locus
Consider a system with the following open-loop poles
Is ????????????
1on the root locus? I.e., does it satisfy the angle criterion?
Angle contributions from
complex poles cancel
Pole to the rightof ????????????
1:
−∠????????????
1−????????????
1=−180°
All poles/zeros to the left
of ????????????
1:
−∠????????????
1−????????????
2=−∠????????????
1−????????????
3=∠????????????
1−????????????
1=0°
????????????
1satisfies the angle criterion, ∠???????????? ????????????
1=−180°, so it is
on the root locus

22. K. Webb ESE 430
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Real-Axis Root-Locus Segments
Now, determine if point ????????????
2is on the root locus
Again angles from complex poles
cancel
Always true for real- axis points
Pole and zero to the left of ????????????
2
contribute 0°
Always true for real- axis points
Two poles to the right of ????????????
1:
−∠????????????
2−????????????
1−∠????????????
2−????????????
2=−360°
Angle criterion is not satisfied
∠????????????????????????
2=−360°≠2????????????+1180°
????????????
2is noton the root locus

23. K. Webb ESE 430
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Real-Axis Root-Locus Segments
From the preceding development, we can conclude the
following concerning real- axis segments of the root
locus:
Allpointsontherealaxistotheleftofanodd
numberofopen-looppolesand/orzerosare
ontherootlocus

24. K. Webb ESE 430
Root Locus – Non-Real-Axis Segments24

25. K. Webb ESE 430
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Non-Real-Axis Root-Locus Segments
Transfer functions of physically-realizable systems
are rational polynomials with real-valued
coefficients
Complex poles/zeros come in complex-conjugate pairs
Root locus is a plot of closed loop poles as ????????????varies
from 0→∞
Where does the locus start? Where does it end?
Root locus is symmetric about the real axis

26. K. Webb ESE 430
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Non-Real-Axis Root-Locus Segments
????????????????????????=
????????????????????????????????????
1+????????????????????????????????????????????????????????????
We’ve seen that we can represent this closed-loop
transfer function as
????????????????????????=
????????????????????????
????????????????????????????????????
????????????????????????
????????????
????????????
????????????????????????
????????????????????????+????????????????????????
????????????????????????????????????
????????????????????????
The closed-loop poles are the roots of the closed-loop
characteristic polynomial
Δ????????????=????????????
????????????????????????????????????
????????????????????????+????????????????????????
????????????????????????????????????
????????????????????????
As ????????????→0
Δ????????????→????????????
????????????????????????????????????
????????????????????????
Closed-loop poles approach the open- loop poles
Root locus starts at the open- loop poles for ????????????=0

27. K. Webb ESE 430
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Non-Real-Axis Root-Locus Segments
As ????????????→∞
Δ????????????→????????????????????????
????????????????????????????????????
????????????????????????
So, as ????????????→∞, the closed-loop poles approach the open-
loop zeros
Root locus ends at the open-loop zeros for ????????????=∞
Including the ????????????−????????????zeros at ????????????
∞
Previous example:
????????????=5poles, ????????????=1zero
One pole goes to the finite zero
Remaining poles go to the ????????????−????????????=4zeros at ????????????
∞
Where are those zeros? (what angles?)
How do the poles get there as ????????????→∞?

28. K. Webb ESE 430
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Non-Real-Axis Root-Locus Segments
As ????????????→∞, ????????????of the ????????????poles approach the ????????????finite zeros
The remaining ????????????−????????????poles are at ????????????
∞
Looking back from ????????????
∞
, it appears that these ????????????−????????????
poles all came from the same point on the real axis, ????????????
????????????Considering only these ????????????−????????????poles, the
corresponding root locus equation is
????????????
????????????=1+????????????
1
????????????−????????????
????????????
????????????−????????????
=0
These poles travel from ????????????
????????????(approximately) to ????????????
∞
along
????????????−????????????asymptotesat angles of ????????????
????????????,????????????

29. K. Webb ESE 430
Root Locus –Asymptote Angles
29

30. K. Webb ESE 430
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Asymptote Angles –????????????
????????????,????????????
To determine the angles of the ????????????−????????????asymptotes,
consider a point, ????????????
1, very far from ????????????
????????????If ????????????
1is on the root locus, then
∠????????????
????????????????????????
1=2????????????+1180°
That is, the ????????????−????????????angles from ????????????
????????????to ????????????
1sum to an odd
multiple of 180°
????????????−????????????????????????
????????????,????????????=2????????????+1180°
Therefore, the angles of the asymptotes are
????????????
????????????,????????????=
2????????????+1180°
????????????−????????????

31. K. Webb ESE 430
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Asymptote Angles –????????????
????????????,????????????
For example
????????????=5poles and ???????????? =3zeros
????????????−????????????=2poles go to ????????????
∞
as ????????????→∞
Poles approach ????????????
∞
along asymptotes at angles of
????????????
????????????,0=
2⋅0+1180°
5−3
=
180°
2
=90°
????????????
????????????,1=
540°
2
=270°
If ????????????−????????????=3
????????????
????????????,0=
180°
3
=60°, ????????????
????????????,1=
540°
3
=180°, ????????????
????????????,2=
900°
3
=300°

32. K. Webb ESE 430
Root Locus –Asymptote Origin
32

33. K. Webb ESE 430
33
Asymptote Origin
The ????????????−????????????asymptotes come from a point, ????????????
????????????, on the real axis -
where is ????????????
????????????located?
The root locus equation can be written
1+????????????
????????????????????????
????????????????????????
=0
where
????????????????????????=????????????
????????????
+????????????
1????????????
????????????−1
+⋯+????????????
????????????
????????????????????????=????????????
????????????
+????????????
1????????????
????????????−1
+⋯+????????????
????????????
According to a property of monicpolynomials:
????????????
1=−Σ????????????
????????????
????????????
1=−Σ????????????
????????????
where ????????????
????????????are the open-loop poles, and ????????????
????????????are the open-loop zeros

34. K. Webb ESE 430
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Asymptote Origin
The closed-loop characteristic polynomial is
????????????
????????????
+????????????
1????????????
????????????−1
+⋯+????????????
????????????+????????????????????????
????????????
+????????????
1????????????
????????????−1
+⋯+????????????
????????????
If ????????????<????????????−1, i.e. at least two more poles than zeros, then
????????????
1=−Σ????????????
????????????
where ????????????
????????????are the closed-loop poles
The sum of the closed-loop poles is:
Independent of ????????????
Equal to the sum of the open-loop poles
−Σ????????????
????????????=−Σ????????????
????????????=????????????
1The equivalent open-loop location for the ????????????−????????????poles going to
infinity is ????????????
???????????? These poles, similarly, have a constant sum:
????????????−????????????????????????
????????????

35. K. Webb ESE 430
35
Asymptote Origin
As ????????????→∞, ????????????of the closed-loop poles go to the open
loop zeros
Their sum is the sum of the open- loop zeros
The remainder of the poles go to ????????????
∞
Their sum is ????????????−????????????????????????
????????????
The sum of all closed-loop poles is equal to the sum of
the open- loop poles
Σ????????????
????????????=Σ????????????
????????????+????????????−????????????????????????
????????????=Σ????????????
????????????
The origin of the asymptotesis
????????????
????????????=
Σ????????????
????????????−Σ????????????
????????????
????????????−????????????

36. K. Webb ESE 430
36
Root Locus Asymptotes – Example
Consider the following system
????????????=1open-loop zero and ????????????=5open-loop poles
As ????????????→∞:
One pole approaches the open- loop zero
Four poles go to ????????????
∞
along asymptotes at angles of:
????????????
????????????,0=
180°
4
=45°, ????????????
????????????,1=
540°
4
=135°
????????????
????????????,2=
900°
4
=225°, ????????????
????????????,3=
1260°
4
=315°

37. K. Webb ESE 430
37
Root Locus Asymptotes – Example
The origin of the asymptotes is
????????????
????????????=
Σ????????????
????????????−Σ????????????
????????????
????????????−????????????
????????????
????????????=
−1+−4+−5+−2+????????????+−2−????????????−−3
5−1
????????????
????????????=
−14+3
4
=−2.75
As ????????????→∞, four poles approach ????????????
∞
along four
asymptotes emanating from ????????????=−2.75at angles of
45°, 135°, 225°, and 315°

38. K. Webb ESE 430
38
Root Locus Asymptotes – Example

39. K. Webb ESE 430
Refining the Root Locus
39

40. K. Webb ESE 430
40
Refining the Root Locus
So far we’ve learned how to accurately sketch:
Real-axis root locus segments
Root locus segments heading toward ????????????
∞
, but only far from ????????????
????????????
Root locus from previous
example illustrates
additional characteristics
we must address:
Real-axis
breakaway/break-in
points
Angles of
departure/arrival at
complex poles/zeros
????????????????????????-axis crossing locations

41. K. Webb ESE 430
Real-Axis Breakaway/Break-In Points41

42. K. Webb ESE 430
42
Real-Axis Breakaway/Break-In Points
Consider the following system and its root locus
Two finite poles approach
two finite zeros as ???????????? →∞
Where do they leave the
real axis?
Breakaway point
Where do they re-join the
real axis?
Break- in point

43. K. Webb ESE 430
43
Real-Axis Breakaway Points
Breakaway point occurs somewhere
between ????????????=−1and ????????????=−2
Breakaway angle:
????????????
????????????????????????????????????????????????????????????????????????????????????????????????????????????=
180°
????????????
where ????????????is the number of poles that
come together – here, ±90°
As gain increases, poles come together
then leave the real axis
Along the real-axis segment, maximum gain occurs at the breakaway
point
To calculate the breakaway point:
Determine an expression for gain, ????????????, as a function of ????????????
Differentiate w.r.t. ????????????
Find ????????????for ????????????????????????/????????????????????????=0to locate the maximum gain point

44. K. Webb ESE 430
44
Real-Axis Breakaway Points
All points on the root locus satisfy
????????????=−
1
????????????????????????????????????????????????
On the segment containing the breakaway point, ????????????=????????????,
so
????????????=−
1
????????????????????????????????????????????????
The breakaway point is a maximum gain point, so
????????????????????????
????????????????????????
=
????????????
????????????????????????
−
1
????????????????????????????????????????????????
=0
Solving for ????????????yields the breakaway point

45. K. Webb ESE 430
45
Real-Axis Breakaway Points
For our example, along the real axis
????????????=−
1
????????????????????????
=−
????????????+1????????????+2
????????????+3????????????+4
=−
????????????
2
+3????????????+2
????????????
2
+7????????????+12
Differentiating w.r.t. ????????????
????????????????????????
????????????????????????
=−
????????????
2
+7????????????+122????????????+3−????????????
2
+3????????????+22????????????+7
????????????
2
+7????????????+12
2
=0
Setting the derivative to zero
????????????
2
+7????????????+122????????????+3−????????????
2
+3????????????+22????????????+7=0
4????????????
2
+20????????????+22=0
????????????=−1.63,−3.37
The breakaway point occurs at ????????????=−1.63

46. K. Webb ESE 430
46
Real-Axis Break-In Points
The poles re- join the real axis at a break -in point
A minimum gain point
As gain increases, poles move apart
Break-in angles are the same as breakaway angles
????????????
????????????????????????????????????????????????????????????−???????????????????????? =
180°
????????????
As for the breakaway point, the break-in point satisfies
????????????????????????
????????????????????????
=
????????????
????????????????????????
−
1
????????????????????????????????????????????????
=0
In fact, this yields both breakaway and break-in points
For our example, we had ????????????=−1.63,−3.37
Breakaway point: ????????????=−1.63
Break-in point: ????????????=−3.37

47. K. Webb ESE 430
47
Real-Axis Breakaway/Break-In Points
????????????=−1.63????????????=−3.37

48. K. Webb ESE 430
Angles of Departure/Arrival
48

49. K. Webb ESE 430
49
Angles of Departure/Arrival
Consider the following two systems
????????????
1????????????=
????????????
2
+0.2????????????+2.26
????????????
2
+0.2????????????+4.01
????????????
2
????????????=
????????????
2
+0.2????????????+4.01
????????????
2
+0.2????????????+2.26
Similar systems, with very different stability behavior
Understanding how to determine angles of departure from complex
poles and angles of arrival at complex zeros will allow us to predict this

50. K. Webb ESE 430
50
Angle of Departure
To find the angle of departure from a pole, ????????????
1:
Consider a test point, ????????????
0, very close to ????????????
1
The angle from ????????????
1to ????????????
0is ????????????
1
The angle from all other poles/zeros, ????????????
????????????/????????????
????????????, to ????????????
0are approximated as the angle
from ????????????
????????????or ????????????
????????????to ????????????
1
Apply the angle criterion to find ????????????
1
�
????????????=1
????????????
????????????
????????????−????????????
1−�
????????????=2
????????????
????????????
????????????=
2????????????+1180°
Solving for the departure angle, ????????????
1:
????????????
1=�
????????????=1
????????????
????????????
????????????−�
????????????=2
????????????
????????????
????????????−180°
In words:
????????????
????????????????????????????????????????????????????????????????????????=Σ∠????????????????????????????????????????????????????????????−Σ∠??????????????????????????????????????????????????????????????????????????????????????????????????????????????????−180°

51. K. Webb ESE 430
51
Angle of Departure
If we have complex-conjugate open-loop poles with
multiplicity ????????????, then
�
????????????=1
????????????
????????????
????????????−????????????????????????
1−�
????????????=????????????+1
????????????
????????????
????????????=
2????????????+1180°
The ????????????different angles of departure from the
multiple poles are
????????????
1,????????????=
∑
????????????=1
????????????????????????
????????????−∑
????????????=????????????+1
????????????????????????
????????????−
2????????????+1180°
????????????
where ????????????=1,2,…????????????

52. K. Webb ESE 430
52
Angle of Arrival
Following the same procedure, we can derive an
expression for the angle of arrivalat a complex zero
of multiplicity ????????????
????????????
1,????????????=
∑
????????????=1
????????????????????????
????????????−∑
????????????=????????????+1
????????????????????????
????????????+
2????????????+1180°
????????????
In summary
????????????
????????????????????????????????????????????????????????????????????????,????????????=
Σ∠????????????????????????????????????????????????????????????−Σ∠??????????????????????????????????????????????????????????????????????????????????????????????????????????????????−2????????????+1180°
????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????
????????????
????????????????????????????????????????????????????????????????????????,????????????=
Σ∠????????????????????????????????????????????????????????????−Σ∠??????????????????????????????????????????????????????????????????????????????????????????????????????????????????+2????????????+1180°
????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????

53. K. Webb ESE 430
????????????
1
????????????
1
????????????
2
????????????
2
????????????
3
53
Departure/Arrival Angles –Example
Angle of departure from ????????????
1
????????????
1=�
????????????=1
????????????
????????????
????????????−�
????????????=2
????????????
????????????
????????????−180°
????????????
1=90°+90°−90°+92.9°−180°
????????????
1=−182.9°
Due to symmetry:
????????????
2=−????????????
1=182.9°
Angle of arrival at ????????????
1
????????????
1=�
????????????=1
????????????
????????????
????????????−�
????????????=2
????????????
????????????
????????????+180°
????????????
1=−90°+90°+93.8°−90°+180°
????????????
1=183.8°,????????????
2=−183.8°

54. K. Webb ESE 430
54
Departure/Arrival Angles –Example

55. K. Webb ESE 430
????????????
1
????????????
1
????????????
2
????????????
2
????????????
3
55
Departure/Arrival Angles –Example
Next, consider the other system
Angle of departure from ????????????
1
????????????
1
=−90°+90°−90°+93.8°
−180°
????????????
1=−363.8°→−3.8°
????????????
2=3.8°
Angle of arrival at ????????????
1????????????
1
=90°+90°+92.9°−90°
+180°
????????????
1=362.9°→2.9°
????????????
2=−2.9°

56. K. Webb ESE 430
56
Departure/Arrival Angles –Example

57. K. Webb ESE 430
????????????????????????-Axis Crossing Points
57

58. K. Webb ESE 430
58
????????????????????????-Axis Crossing Points
To determine the location of a
????????????????????????-axis crossing
Apply Routh- Hurwitz
Find value of ????????????that results in a
row of zeros
Marginal stability
????????????????????????-axis poles
Roots of row preceding the
zero row are ????????????????????????-axis crossing
points
Or, plot in MATLAB
More on this later

59. K. Webb ESE 430
Sketching the Root Locus -Summary59

60. K. Webb ESE 430
60
Root Locus Sketching Procedure –Summary
1.Plotopen-loop polesand zerosin the s-plane
2.Plot locus segments on the real axis to the left of an odd
number of poles and/or zeros
3.For the ????????????−????????????poles going to ????????????
∞
, sketch asymptotesat
angles ????????????
????????????,????????????, centered at ????????????
????????????, where
????????????
????????????,????????????=
2????????????+1180°
????????????−????????????
????????????
????????????=
∑
????????????=1
????????????????????????
????????????−∑
????????????=1
????????????????????????
????????????
????????????−????????????

61. K. Webb ESE 430
61
Root Locus Sketching Procedure –Summary
4.Calculate departure anglesfrom complex poles of multiplicity ????????????≥1
????????????
????????????=
Σ∠????????????????????????????????????????????????????????????−Σ∠??????????????????????????????????????????????????????????????????????????????????????????????????????????????????−2????????????+1180°
????????????
and arrival anglesat complex zeros of multiplicity ????????????≥1
????????????
????????????=
Σ∠????????????????????????????????????????????????????????????−Σ∠??????????????????????????????????????????????????????????????????????????????????????????????????????????????????+2????????????+1180°
????????????
5.Determine real-axis breakaway/break-in points as the solutions to
????????????
????????????????????????
1
????????????????????????????????????????????????
=0
Breakaway/break-in angles are 180°/????????????to the real axis
6.If desired, apply Routh- Hurwitz to determine ????????????????????????-axis crossings

62. K. Webb ESE 430
62
Sketching the Root Locus – Example 1
Consider a satellite, controlled by a proportional-
derivative (PD) controller
A example of a double -integratorplant
We’ll learn about PD controllers in the next section
Closed-loop transfer function
????????????????????????=
????????????????????????+1
????????????
2
+????????????????????????+????????????
Sketch the root locus
Two open-loop poles at the origin
One open-loop zero at ????????????=−1

63. K. Webb ESE 430
63
Sketching the Root Locus – Example 1
1.Plotopen-loop poles and
zeros
Two poles, one zero
2.Plot real-axis segments
To the left of the zero
3.Asymptotes to ????????????
∞
One pole goes to the finite
zero
One pole goes to ∞ at 180°-
along the real axis

64. K. Webb ESE 430
64
Sketching the Root Locus – Example 1
4.Departure/arrival angles
No complex poles or zeros
5.Breakaway/break-in points
Breakaway occurs at multiple
roots –at ????????????=0
Break-in point:
????????????
????????????????????????
????????????
2
????????????+1
=0
????????????+12????????????−????????????
2
????????????+1
2
=0
????????????
2
+2????????????=0→????????????=−2,0
????????????=−2

65. K. Webb ESE 430
65
Sketching the Root Locus – Example 2
Now consider the same satellite with a different
controller
A lead compensator–more in the next section
Closed-loop transfer function
????????????????????????=
????????????????????????+1
????????????
3
+12????????????
2
+????????????????????????+????????????
Sketch the root locus

66. K. Webb ESE 430
????????????=−5.5
66
Sketching the Root Locus – Example 2
1.Plotopen-loop poles and zeros
Now three open-loop poles and
one zero
2.Plot real- axis segments
Between the zero and the pole at
????????????=−12
3.Asymptotes to ????????????
∞
????????????
????????????,1=
180°
2
=90°
????????????
????????????,2=
540°
2
=270°
????????????
????????????=
−12−−1
2
=−5.5

67. K. Webb ESE 430
67
Sketching the Root Locus – Example 2
4.Departure/arrival angles
No complex open-loop poles or zeros
5.Breakaway/break-in points
????????????
????????????????????????
????????????
2
(????????????+12)
????????????+1
=0
????????????+13????????????
2
+24????????????−????????????
3
+12????????????
2
????????????+1
2
=0
2????????????
3
+15????????????
2
+24????????????=0
????????????=0,−2.31,−5.19
Breakaway: ????????????=0, ????????????=−5.19
Break-in: ????????????=−2.31
????????????=−2.31
????????????=−5.19

68. K. Webb ESE 430
68
Sketching the Root Locus – Example 3
Now move the controller’s pole to ???????????? =−9
Closed-loop transfer function
????????????????????????=
????????????????????????+1
????????????
3
+9????????????
2
+????????????????????????+????????????
Sketch the root locus

69. K. Webb ESE 430
????????????=−4
69
Sketching the Root Locus – Example 3
1.Plotopen-loop poles and zeros
Again, three open-loop poles and one
zero
2.Plot real-axis segments
Between the zero and the pole at ???????????? =
−9
3.Asymptotes to ????????????
∞
????????????
????????????,1=90°
????????????
????????????,2=270°
????????????
????????????=
−9−−1
2
=−4
4.Departure/arrival angles
No complex open-loop poles or zeros

70. K. Webb ESE 430
70
Sketching the Root Locus – Example 3
4.Breakaway/break-in points
????????????
????????????????????????
????????????
2
(????????????+9)
????????????+1
=0
????????????+13????????????
2
+18????????????−????????????
3
+9????????????
2
????????????+1
2
=0
2????????????
3
+12????????????
2
+18????????????=0
????????????=0,−3,−3
Breakaway: ????????????=0, ????????????=−3
Break-in: ????????????=−3
Three poles converge/diverge at ????????????=
−3
Breakaway angles: 0°, 120°, 240°
Break-in angles: 60°, 180°, 300°
????????????=−3

71. K. Webb ESE 430
Root Locus in MATLAB
71

72. K. Webb ESE 430
72
feedback.m
sys=feedback(G,H,sign)
G: forward- path model –tf, ss, zpk, etc.
H: feedback-path model
sign: -1for neg. feedback, +1for pos. feedback –
optional–default is -1
sys: closed-loop system model object of the same
type as G and H
Generates a closed-loop system model from
forward-path and feedback-path models
For unity feedback, H=1

73. K. Webb ESE 430
73
feedback.m
For example:
T=feedback(G,H);
T=feedback(G,1);
T=feedback(G1*G2,H);

74. K. Webb ESE 430
74
rlocus.m
[r,K] =rlocus(G,K)
G: open-loop model –tf, ss, zpk, etc.
K: vector of gains at which to calculate the locus – optional –
MATLAB will choose gains by default
r: vector of closed-loop pole locations
K: gains corresponding to pole locations in r
If no outputs are specified a root locus is plotted in the
current (or new) figure window
This is the most common use model, e.g.:
rlocus(G,K)

75. K. Webb ESE 430
Generalized Root Locus
75

76. K. Webb ESE 430
76
Generalized Root Locus
We’ve seen that we can plot the root locus as a function of
controller gain, ????????????
Can also plot the locus as a function of other parameters
For example, open-loop pole locations
Consider the following system:
Plot the root locus as a function of pole location, ????????????
Closed-loop transfer function is
????????????????????????=
1
????????????????????????+????????????
1+
1
????????????????????????+????????????
=
1
????????????
2
+????????????????????????+1

77. K. Webb ESE 430
77
Generalized Root Locus
????????????????????????=
1
????????????
2
+????????????????????????+1Want the denominator to be in the root-locus form:
1+????????????????????????????????????????????????????????????
First, isolate ????????????in the denominator
????????????????????????=
1
????????????
2
+1+????????????????????????
Next, divide through by the remaining denominator terms
????????????????????????=
1
????????????
2
+1
1+????????????
????????????
????????????
2
+1

78. K. Webb ESE 430
78
Generalized Root Locus
????????????????????????=
1
????????????
2
+1
1+????????????
????????????
????????????
2
+1
The open- loop transfer function term in this form is
????????????????????????????????????????????????=
????????????
????????????
2
+1
Sketch the root locus:
1.Plot poles and zeros
A zero at the origin and poles at ???????????? =±????????????
2.Plot real-axis segments
Entire negative real axis is left of a single zero

79. K. Webb ESE 430
79
Generalized Root Locus
3.Asymptote to ????????????
∞
Single asymptote along negative
real axis
4.Departure angles
????????????
1=90°−90°−180°
????????????
1=−180°=−????????????
2
5.Break-in point
????????????
????????????????????????
1
????????????????????????????????????????????????
=
????????????
????????????????????????
????????????
2
+1
????????????
=0
????????????2????????????−????????????
2
+1
????????????
2
=0
????????????
2
−1=0→????????????=+1,−1
????????????=+1is not on the locus
Break-in point: ????????????=−1

80. K. Webb ESE 430
Design via Gain Adjustment
80

81. K. Webb ESE 430
81
Design via Gain Adjustment
Root locus provides a graphical representation of
closed-loop pole locations vs. gain
We have known relationships (some approx.) between
pole locations and transient response
These apply to 2
nd
-order systems with no zeros
Often, we don’t have a 2
nd
-order system with no zeros
Would still like a link between pole locations and transient
response
Can sometimes approximate higher-order systems as
2
nd
-order
Valid only under certain conditions
Always verify response through simulation

82. K. Webb ESE 430
82
Second-Order Approximation
A higher-order system with a pair of second-order poles
can reasonably be approximated as second- order if:
1)Any higher-order closed-loop poles are either:
a)at much higher frequency (>~5×) than the dominant
2
nd
-order pair of poles, or
b)nearly canceled by closed-loop zeros
2)Closed-loop zeros are either:
a)at much higherfrequency (>~5×) than the dominant
2
nd
-order pair of poles, or
b)nearly canceled by closed-loop poles

83. K. Webb ESE 430
83
Design via Gain Adjustment –Example
Determine ????????????for 10%overshoot
Assuming a 2
nd
-order approximation applies:
????????????=
−ln????????????????????????
????????????
2
+ln
2
????????????????????????
=0.59
Next, plot root locusin MATLAB
Find gaincorresponding to 2
nd
-order poles with ????????????=0.59
If possible–often it is not
Determine if a 2
nd
-order approximationis justified
Verify transient response through simulation

84. K. Webb ESE 430
84
Design via Gain Adjustment –Example
Root locus shows that
a pair of closed-loop
poles with ????????????=0.59
exist for ???????????? =5.23:
????????????
1,2=−1.25±??????????????????.71
Where is the third
closed-loop pole?

85. K. Webb ESE 430
85
Design via Gain Adjustment –Example
Third pole is at
????????????=−3.5
Not high enough in
frequency for its effect to
be negligible
But, it is in close
proximity to a closed-
loop zero
Is a 2
nd
-order
approximation justified?
Simulate

86. K. Webb ESE 430
86
Design via Gain Adjustment –Example
Step response
compared to a true 2
nd
-
order system
No third pole, no zero
Very similar response
11.14%overshoot
2
nd
-order
approximation is valid
Slight reduction in gain
would yield 10%overshoot

87. K. Webb ESE 430
87
Design via Gain Adjustment –Example
Step response
compared to systems
with:
No zero
No third pole
Quite different
responses
Partial pole/zero
cancellation makes 2
nd
-
order approximation
valid, in this example

88. K. Webb ESE 430
88
When Gain Adjustment Fails
Root loci do not go through every point in the s-plane
Can’t always satisfy a single performance specification, e.g.
overshoot orsettling time
Can satisfy two specifications, e.g. overshoot andsettling
time, even less often
Also, gain adjustment affects steady-state error
performance
In general, cannot simultaneously satisfy dynamic
requirements and error requirements
In those cases, we must add dynamics to the controller
A compensator