Root Locus (2A) Young Won Lim 10/15/14
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1 Root Locus (2A
2 Copyright (c 2014 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions to youngwlim@hotmail.com. This document was produced by using OpenOffice and Octave.
3 The Original Transfer Function G(s = N (s D(s R(s G(s Y (s = sm + b 1 s m b m s n + a 1 s n a n = (s (s z 2 (s z m (s p 1 (s p 2 (s p n Root Locus (2A 3
4 The New Transfer Function KG(s 1+ KG(s = K N (s/ D(s 1+ K N (s/ D(s R(s + G(s Y (s = K N (s D(s+K N (s K D(s + K N (s = 0 Root Locus (2A 4
5 Conditions for new roots (1 Find roots s D(s + K N (s = 0 Conditions for a root s 0 (K = 0 ( = N (s = 0 (K 0 ( 0 ( D( S 0 ( 0 (K = N (s 0 = real 0 N (s 0 = real + ( N (s N (s 0 = 0 0 Root Locus (2A 5
6 Conditions for new roots (2 Find roots s D(s + K N (s = 0 N (s 0 = 0 0 ( no common root + K N (s 0 = 0 cannot be a root N (s 0 0 for a root s 0 K = N (s 0 = 0 = 0 : original poles K 0 + K N (s 0 = 0 0 = K N (s 0 0 K = N (s 0 0 Root Locus (2A 6
7 Check for new roots Find roots s D(s + K N (s = 0 = 0 K = 0 + K N (s 0 = 0 s 0 is on the root locus of the pair (N, D 0 N (s 0 = real K = N (s 0 s 0 is on the root locus of the pair (N, D + K N (s 0 = 0 K > G(s 0 N (s 0 > 0 positive root locus of the pair (N, D K < G(s 0 N (s 0 < 0 negative root locus of the pair (N, D Root Locus (2A 7
8 Rule: Uniqueness of K either s 0 is not on the root locus for any s 0 or s 0 is on the root locus then there is a unique K + K N (s 0 = 0 Root Locus (2A 8
9 Rule: Real number condition of K 0 N (s 0 = real K = N (s 0 s 0 + K N (s 0 = 0 is on the root locus of the pair (N, D { arg N (s } 0 D(s = q π 0 N (s 0 = 1 K K > 0 K < 0 Arg { N ( s 0 D( s 0 } = π Arg { N ( s 0 D( s 0 } = 0 arg { N (s 0 } = arg { (s 0 (s 0 z 2 (s 0 z m (s 0 p 1 (s 0 p 2 (s 0 p n } = arg(s 0 + arg(s 0 z arg(s 0 z m [arg(s 0 p 1 + arg(s 0 p arg(s 0 p n ] arg(s 0 + arg( s 0 z arg( s 0 z m [arg(s 0 p 1 + arg(s 0 p arg(s 0 p n ] = (2 q+1 π K > 0 positive root locus (2 qπ K < 0 negative root locus Root Locus (2A 9
10 Number of Roots D(s + K N (s = 0 n roots G(s = N (s D(s = sm + b 1 s m b m s n + a 1 s n a n = (s (s z 2 (s z m (s p 1 (s p 2 (s p n K = 1 n-1 roots 1 root at infinity n-2 roots 2 root at infinity Root Locus (2A 10
11 Roots when K=0 K = N (s 0 = 0 = 0 The roots of D(s : The original poles G(s = N (s D(s = sm + b 1 s m b m s n + a 1 s n a n = (s (s z 2 (s z m (s p 1 (s p 2 (s p n Root Locus (2A 11
12 Roots on real axis (1 positive root locus s 0 p 0 z 0 p 1 z 1 p 2 { arg N (s } 0 D(s = (2q+1π 0 (s 0 (s 0 p 2 (s 0 p 1 (s 0 z 0 (s 0 p 0 Arg(s 0 p 1 = 0 Arg(s 0 z 0 = 0 Arg(s 0 p 0 = 0 Arg(s 0 = π Arg(s 0 p 2 = π Arg { N (s 0 } = Arg (s 0 z 0 + Arg(s 0 [ Arg(s 0 p 0 + Arg(s 0 p 1 + Arg (s 0 p 2 ] = 0 (2 q+1π Not on the positive root locus Root Locus (2A 12
13 Roots on real axis (2 positive root locus s 0 p 0 z 0 p 1 z 1 p 2 { arg N (s } 0 D(s = (2q+1π 0 (s 0 p 1 (s 0 (s 0 p 2 (s 0 z 0 (s 0 p 0 Arg(s 0 z 0 = 0 Arg(s 0 p 0 = 0 Arg(s 0 p 1 = π Arg(s 0 = π Arg(s 0 = π Arg { N (s 0 } = Arg (s 0 z 0 + Arg(s 0 [ Arg(s 0 p 0 + Arg(s 0 p 1 + Arg (s 0 p 2 ] = π = (2 q+1π on the positive root locus Root Locus (2A 13
14 Roots when K (1 K D(s + K N (s K N (s N (s = 0 The roots of N(s : The original zeros G(s = N (s D(s = sm + b 1 s m b m s n + a 1 s n a n K = D(s N (s = sn + a 1 s n a n s m + b 1 s m b m s + (a 1 b 1 s 1 + = s [ 1 + a 1 b 1 s + ] s m + b 1 s m b m s n + a 1 s n a n s n + b 1 s n 1 + (a 1 b 1 s n 1 + Root Locus (2A 14
15 Roots when K (2 K = D(s N ( s = sn + a 1 s n a n s m + b 1 s m b m [ = s 1 + a 1 b 1 s + ] = s [ 1 + a 1 b 1 s + ] 1 [ ( K s 1 + a b ] 1 1 s 1 1 [ ( K s 1 + a 1 b 1 ] ( s (1+x k 1+k x (x 1 K s 0 1 e j ( π+2 q π 1 K e j ( π+2qπ λ = a 1 b 1 s + a 1 b 1 a 1 b 1 centroid (K e j π 1 s + a 1 b 1 2 π Root Locus (2A 15
16 Centroid (1 G(s = N (s D(s = sm + b 1 s m b m b 1 = s n + a 1 s n 1 i + + a n = (s (s z 2 (s z m (s p 1 (s p 2 (s p n a 1 = j z i p j = i = j R{z i } R{p j } s 0 1 K e j ( π+2qπ a 1 b 1 λ = a 1 b 1 = p j z i = R {p j } R{z i } Centroid θ = (2q+1π q = 0, 1,, +1 Angle of asymptotes Root Locus (2A 16
17 Centroid (2 G(s = N (s D(s = (s (s z 2 (s z m (s p 1 (s p 2 (s p n λ = a 1 b 1 = p j z i = R {p j } R{z i } Centroid G(s = N (s D(s = (s+z 1 (s+z 2 (s+ z m (s+ p 1 (s+ p 2 (s+ p n λ = + a 1 b 1 = p j + z i = R {p j }+ R{z i } Centroid Root Locus (2A 17
18 Angle of departure from a pole arg(s 0 + arg( s 0 z arg( s 0 z m [arg(s 0 p 1 + arg(s 0 p arg(s 0 p n ] = (2 q+1 π K > 0 positive root locus s 0 = p 1 + ϵe jθ d arg( p 1 + ϵe j θ d z1 + arg( p 1 + ϵ e jθ d z2 + + arg( p 1 + ϵe jθ d zm [arg( p 1 + ϵe j θ d p 1 + arg( p 1 + ϵe j θ d p arg( p 1 + ϵ e j θ d p n ] p 1 arg( p 1 + arg( p 1 z arg( p 1 z m [arg(ϵ e jθ d + arg( p1 p arg( p 1 p n ] = (2q+1 π arg( p 1 + arg( p 1 z arg( p 1 z m [θ d + arg( p 1 p arg( p 1 p n ] = (2q+1π π + arg( p 1 + arg( p 1 z arg( p 1 z m [arg(p 1 p arg( p 1 p n ] = θ d Root Locus (2A 18
19 Angle of arrival from a zero arg(s 0 + arg( s 0 z arg( s 0 z m [arg(s 0 p 1 + arg(s 0 p arg(s 0 p n ] = (2 q+1 π K > 0 positive root locus s 0 = z 1 + ϵe jθ a arg(z 1 + ϵe j θ a z1 + arg(z 1 + ϵe j θ a z2 + + arg( z 1 + ϵ e jθ a zm [arg( z 1 + ϵe jθ a p 1 + arg( z 1 + ϵ e jθ a p arg(z 1 + ϵe j θ a p n ] z 1 arg(ϵe j θ a + arg( z1 z arg(z 1 z m [arg(z 1 p 1 + arg(z 1 p arg(z 1 p n ] = (2q+1π θ a + arg(z 1 z arg( z 1 z m [arg(z 1 p 1 + arg(z 1 p arg(z 1 p n ] = (2q+1π π [arg(z 1 z arg( z 1 z m ] + [arg( z 1 p 1 + arg(z 1 p arg( z 1 p n ] = θ a Root Locus (2A 19
20 Breakaway Points (Repeated Roots D(s + K N ( s = ( s s 1 r (s s 2 [ D (s + K N (s] s=s0 = 0 [ d d s ( D(s + K N (s ]s=s 0 = 0 [ d d s D (s + K d d s N (s ]s=s 0 = 0 d d s D(s 0 d N (s 0 d s N (s 0 = 0 N (s 0 d d s D (s 0 D (s 0 d d s N (s 0 = 0 Check this for repeated roots N (s d d D (s D (s d s d s N (s = 0 d d s K = d [ d s D (s ] N (s = 0 Root Locus (2A 20
21 A p 4 ϵ p 4 + ϵ e jθ d s 0 = p 4 + ϵ e jθ d p 3 p 1 p 4 p 2 arg(s 0 + arg(s 0 z arg(s 0 z m [arg(s 0 s p 1 + arg(s 0 p arg(s 0 p n ] = (2 q+1 π K > 0 positive root locus = [arg(s 0 s p 1 + arg(s 0 p arg(s 0 p n ] = Root Locus (2A 21
22 A Root Locus (2A 22
23 References [1] [2]
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andpass Sapling (2) Copyright (c) 2009-2012 Young W. Li. Perission is granted to copy, distribute and/or odify this docuent under the ters of the GNU Free Docuentation License, Version 1.2 or any later
More informationBackground LTI Systems (4A) Young Won Lim 4/20/15
Background LTI Systems (4A) Copyright (c) 2014-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationCT Correlation (2A) Young Won Lim 9/9/14
CT Correlation (2A) Copyright (c) 200-204 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later
More informationProbability (10A) Young Won Lim 6/12/17
Probability (10A) Copyright (c) 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later
More informationCLTI Differential Equations (3A) Young Won Lim 6/4/15
CLTI Differential Equations (3A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
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Hilbert nner Product Space (2B) Copyright (c) 2009-2011 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
More informationSignal Processing. Young Won Lim 2/20/18
Copyright (c) 2016 2018 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published
More informationPropositional Logic Logical Implication (4A) Young W. Lim 4/21/17
Propositional Logic Logical Implication (4A) Young W. Lim Copyright (c) 2016-2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation
More informationUndersampling (2B) Young Won Lim 4/4/12
Undersapling (2) Copyright (c) 2009-2012 Young W. Li. Perission is granted to copy, distribute and/or odify this docuent under the ters of the GNU Free Docuentation License, Version 1.2 or any later version
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urface Integrals (6A) urface Integral tokes' Theorem Copright (c) 2012 Young W. Lim. Permission is granted to cop, distribute and/or modif this document under the terms of the GNU Free Documentation License,
More informationBilateral Laplace Transform (6A) Young Won Lim 2/16/15
Bilterl Lplce Trnsform (6A) 2/6/5 Copyright (c) 25 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version.2 or ny
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Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
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Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationBilateral Laplace Transform (6A) Young Won Lim 2/23/15
Bilterl Lplce Trnsform (6A) Copyright (c) 25 Young W. Lim. Permission is grnted to copy, distribute nd/or modify this document under the terms of the GNU Free Documenttion License, Version.2 or ny lter
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Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
More informationSchool of Mechanical Engineering Purdue University. DC Motor Position Control The block diagram for position control of the servo table is given by:
Root Locus Motivation Sketching Root Locus Examples ME375 Root Locus - 1 Servo Table Example DC Motor Position Control The block diagram for position control of the servo table is given by: θ D 0.09 See
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Spectra (A) /8/ Copyright (c) 0 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version. or any later version
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Second Order ODE's (2A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or
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Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
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Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
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Capacitor in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
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Finite State Machine (A) 6/9/8 Copyright (c) 23-28 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free ocumentation License, Version.2 or
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Capacitors in an AC circuit Copyright (c) 2011 2017 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
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