控制系统设计_根轨迹法

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14
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
R (s )
+
Roots to find out the solution of the characteristic equation.
E (s )
q( s ) = 1 + G ( s ) H ( s ) = 0
q( s ) = 1 + KG1 ( s ) H1 ( s ) = 0 G1 ( s ) H1 ( s ) = −1 • 1 K
21
1 + G1 ( s ) H1 ( s ) = 1 + K
Root Locus 10
s +1 =0 s (s + a)
2
Root Locus 8
Root Locus 8
8
6
6
6
4
4
4
2 Imaginary Axis
2 Imaginary Axis
2 Imaginary Axis
-8 -7 -6 -5 -4 -3 -2 -1 0
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.
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)
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
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)
0
0
0
-2
-2
-4
-2
-6
-4
-4
-8
-6
-10 -10 -9 -8 -7 -6 -5 Real Axis -4 -3 -2 -1 0
-6
a = 10
-8 -9
-8 -8
-7
-6
-5
-4 Real Axis
-3
-2
-1
0
Real Axis
a=9
Root Locus 5
0.4
a =8
Root Locus
∏ ( 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.
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
-1.5
-2 -3
-6 -4
-3
-2
-1 Real A xis
0
1
2
Effects adding poles to G1(s)H1(s)
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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)
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
n−m n−m ( 2 k + 1)π n − m , ∀K ≥ 0 ; k = 0,1,2,⋯ , n − m − 1 θk = 2kπ , ∀K < 0 n−m
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9. Breakaway points (saddle points) on the Roots of d G ( s) H ( s ) = 0 or d K = 0 1 1 root loci. ds ds 10. Intersection of root loci with imaginary Routh-Hurwitz criterion. axis. 11. Angles of departure and angles of arrival ( 2k + 1)π ∀K ≥ 0 ∠G1 ( s ) H 1 ( s ) = , k = 0,1,2,⋯ of the root loci. ∀K < 0 2 kπ at s = pi or s = z j . n 12. Calculation of K at a specific root si . ∏ (s + p )
≥ 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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| 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
Root Locus 8
6
4
0.3
4
3
0.2
2
2
0.1
Imaginary Axis
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.
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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
A
nБайду номын сангаасm
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
The Root Locus Method
The Root Locus Concept The Root Locus Procedure The Root contour The Root Locus Using Matlab
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T ( s) =
Y ( s ) p( s ) = R ( s ) q( s )
180 ± 2kπ , for K > 0 ∠G1 ( s ) H1 ( s ) = 00 ± 2kπ , for K < 0 where k = 0,1,2...
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q( s ) = 1 + G ( s ) H ( s ) = 1 + K
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
−
G (s )
C(s)
H (s )
| G1 ( s ) H1 ( s ) |=
1 |K |
q( s ) = s 3 + 2 s 2 + Ks + 4 K = 0 = s 3 + 2 s 2 + K ( s + 4) = 0 s+4 =0 s 2 ( s + 2) s+4 G1 ( s ) H1 ( s ) = 2 s ( s + 2) = 1+ K