Root Locus

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Chapter 5

• Root Locus

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Introduction

• Root Locus illustrates how the poles of the
  closed-loop system vary with the closed-loop
  gain.

• Graphically, the locus is the set of paths in
  the complex plane traced by the closed-loop poles
  as the root locus gain is varied from zero to
  infinity.

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Example of Root Locus
Locus for a system with three poles and no zeros
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Locus for a system with three poles and two
zeros. Note that the part of the locus off the
real axis is close to joining with the real axis,
in which case break points would occur.
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Locus for a system with 5 poles and 2 zeros.
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5
-36.87o
4
-3
0
-2
-4
36.87o
-5
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block diagram of the closed loop system
given a forward-loop transfer function
KGc(s)H(s)
where K is the root locus gain, and the
corresponding closed-loop transfer function
the root locus is the set of paths traced by the
roots of
1 KGc(s)H(s) 0

• as K varies from zero to infinity. As K changes,
  the solution to this equation changes.

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So basically, the root locus is sketch based on
the characteristic equation of a given transfer
function.
Let say
Thus, the characteristic equation
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Root locus starts from the characteristic
equation.
Split into 2 equations
Magnitude condition
Angle condition
where
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Remarks
If Si is a root of the characteristic equation,
then 1G(s)H(s) 0 OR Both the magnitude and
Angle conditions must be satisfied
If not satisfied? Not part of root locus
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Let say my first search point, S1
S1
0
-1
-2
(A) Satisfy the angle condition FIRST
(B) Magnitude condition to find K
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Let say the next search point
0
-1
-2
Check whether satisfy angle condition
Since angle condition was not satisfied ? not
part of root locus
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CONSTRUCTION RULES OF ROOT LOCUS

• 8 RULES TO FOLLOW

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Construction Rules of Root Locus
Then
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Thus
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Rule 1 When K 0
? Root at open loop poles

• The R-L starts from open loop poles
• The number of segments is equal to the number of
  open loop poles

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Rule 2 When K 8

• The R-L terminates (end) at the open loop zeros

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Rule 3 Real-axis segments
On the real axis, for K gt 0 the root locus exists
to the left of an odd number of real-axis, finite
open-loop poles and/or finite open-loop zeros
Example 2
R-L doesnt exist here
R-L exist here
R-L exist here
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Rule 4 Angle of asymptote
NP number of poles NZ number of zeros
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Example 3
NP3
NZ0
Centroid
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Rule 5 Centroid
From example 3
Recall
Thus,
-1
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Rule 6 Break away break in points (if exist)
0
0
-1
-1
-2
-2
Break away point
Break in point
How to find these points ?
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Differentiate K with respect to S equate to zero
How ?
Solve for S ? this value will either be the break
away or break in point
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Let say
Thus,
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Example 4
Solution
0
-1
-2
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Example Root Locus
0
-1
-2
NP 3 NZ 0
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0
-1
-2
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Break away point
Invalid, why ???
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How to determine these values ?
0
-1
-2
-0.42
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Rule 7 R-L crosses j?-axis (if exist)
If there is a breakaway/ break in point
Use Routh Hurwitz
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From characteristic equation
Construct the Routh array
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Since
Force S1 row to zero or K6
Replace K6 into S2 row
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0
-1
-2
-0.42
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Rule 8 Angle of departure (arrival) (if exist)
No complex conjugate poles/zeros
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