4
3
2
2
1 Imag A xis
1 0 Imag Axis -5 -4 -3 -2 Real A xis -1 0 1 2 0 -1 -2 -2 -3 -3 -4 -5 -5
-1
-4 -6
-4
-3
-2 -1 Real A xis
0
1
2
Effects adding poles to G1(s)H1(s)
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q( s ) = s 3 + s 2 + β s + α = 0
s3 + s 2 + α = 0 1+
1+
βs
s + s +α
3 2
=0
α
s ( s + 1)
2
=0
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R(s )
k1 s(s + 2)
Y (s )
k2s
Specifications: 1. 2. 3. Steady-state error for a ramp input ≤ 35% Damping ratio of dominant roots
0
0
-1 -2 -2 -4
-3
-6 -6
-5
-4
-3
-2 Real A xis
-1
0
1
2
-4 -6
-5
-4
-3
-2 -1 Real Axis
0
1
2
3
Effects adding poles to G1(s)H1(s)
-∞<K<∞
Lab #3 Written a M-file to plot the root loci step by step. (rlocfind)
Odd segments
Step 5: Determine the number of separate loci,SL.The number of separate loci is equal to the number of poles
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Step 6: The root loci must be symmetrical with respect to the horizontal real axis. Step 7: The linear asymptotes are centered at a point on the real axis given by σ = ∑ poles − ∑ zeros The angle of the asymptotes with respect to the
| G1 ( s ) H1 ( s ) |s = s1 = ∠G1 ( s ) H1 ( s ) |s = s1 1 = s( s + 2) s = s1
1 =0 s ( s + 2)
s1
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The Root Locus Procedure
Step Related equation or Rule F ( s ) = 1 + KG1 ( s ) H 1 ( s ) = 0 1. Write the characteristic equation so that the parameter of interest K appears as a multiplier. m 2. Factor G1 ( s ) H 1 ( s ) in terms of n poles ∏(s + z j ) j =1 and m zeros. G1 ( s ) H 1 ( s ) = n
≥ 0.707
sec.
≤3
Settling time to within 2 % of the final value
sec.
1 + GH ( s ) = s 2 + 2 s + βs + α = 0 β = k2 k1 , α = k1
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∏ ( s + pi )
i =1
3. Locate the open-loop poles and zeros of 5 : poles, : zeros, F (s ) in the s-plane with selected ∆ or : roots of characteristic equation symbols. 4. Locate the segments of the real axis that a). Locus begins at a pole and ends at zero. are root locus. b). Locus lies to left of an odd number of poles and zeros ( K ≥ 0 ). ρ = n , when n ≥ m; 5. The number of branch on the root loci, ρ. n: number of finite poles, m: number of finite zeros 6. The root loci are symmetrical with respect to the horizontal real axis. 7. Intersect of the asymptotes (Centroid) ∑ pi − ∑ z j or σ = ∑ Re( pi ) − ∑ Re( z j ) σ = 8. Angles of asymptotes of the root loci.
dK =0 ds
the tangents to the loci at the breakaway point are equally over 3600
1 + KG1 ( s ) = 1 + K
1 =0 ( s + 2)( s + 4)
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K = −( s + 2)( s + 4) dK = −( 2 s + 6) = 0, s = −3 ds
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Examples
1 F ( s) = 1 + K s ( s + 2)
2
F ( s) = 1 + K
1 s ( s + 2)( s + 3)
6
1.5
4
1 2
0.5 Imag A xis
0
Imag A xis
-2 -1 Real Axis 0 1 2
0
-0.5 -2 -1 -4
A
n−m
real axis is
φA =
(2 q + 1) × 1800 n−m
n = 4, m = 1, φ1 =
(2 q + 1) × 1800 , q = 0,1,2, n − m − 1 4 −1
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Step 8:The actual point at which the root locus crosses the imaginary axis is readily evaluated by utilizing the Routh-Hurwitz criterion. Step 9: Determine the breakaway point
Effects adding zeros to G1(s)H1(s)
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F ( s) = 1 + K
6
1 s 4 + 12 s 3 + 64 s 2 + 32 s
F ( s) = 1 + K
4
1 s ( s + 3)( s 2 + 2 s + 2)
3
4
2
2 Imag Axis
1 Imag Axis
K=
i =1 m i
∏ (s + z j )
j =1
s = si
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Step 4: The root locus on the real axis always lies in a section of the
real axis to the left of an odd number of poles and zeros.

-2
-1 Real A xis
0
1
2
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F ( s) = 1 + K
1 s ( s + 2)( s + 3)( s + 4)
5 4 3
F ( s) = 1 + K
s+4 s ( s + 2)( s + 3)
dK ds
2 s -2
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Step 10:Determine the angle of departure of the locus from a pole and the angle of arrival of the locus at a zero,using the phase angle criterion.
−
G (s )
C(s)
H (s )
| G1 ( s ) H1 ( s ) |=