📄 rootlocusexamples .mht
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alt=3DRLFG=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Exam=
ple1/RLFindGain.png"></P>
<P>Characteristic Equation is 1+KG(s)H(s)=3D0, or 1+KN(s)/D(s)=3D0, =
or<BR>K =3D=20
-D(s)/N(s) =3D -( s<SUP>2</SUP> + 3 s ) / ( 1 )<BR>We can pick a value =
of s on the=20
locus, and find K=3D-D(s)/N(s).<BR><BR>For example if we choose s=3D =
-1.6 + 1.6j=20
(marked by asterisk),<BR>then D(s)=3D-4.87 + -0.243j, N(s)=3D 1 + =
0j,<BR>and=20
K=3D-D(s)/N(s)=3D 4.87 + 0.243j.<BR>This s value is not exactly on the =
locus, so K=20
is complex, (see note below), pick real part of K ( 4.87)<BR><BR>For =
this K=20
there exist 2 closed loop poles at s =3D -1.5=B1 1.6j. These poles =
are=20
highlighted on the diagram with dots, the value of "s" that was =
originally=20
specified is shown by an asterisk.<BR><BR>Note: it is often difficult to =
choose=20
a value of s that is precisely on the locus, but we can pick a point =
that is=20
close. If the value is not exactly on the locus, then the =
calculated value=20
of K will be complex instead of real. Just ignore the the imaginary part =
of K=20
(which will be small). </P>
<P>Note also that only one pole location was chosen and this determines =
the=20
value of K. If the system has more than one closed loop pole, the =
location of=20
the other poles are determine solely by K, and may be in undesirable=20
locations.<BR></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<HR>
<DIV style=3D"FONT-SIZE: smaller; TEXT-ALIGN: center">
<P>=A9 Copyright 2005-2007 <A=20
href=3D"mailto:erik_cheever@swarthmore.edu?subject=3DLinear Physical =
System Analysis (Copyright notice)">Erik=20
Cheever</A> This page may be freely used for =
educational=20
purposes.<BR></P>
<P>
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</P><A href=3D"http://www.swarthmore.edu/NatSci/echeeve1">Erik=20
Cheever</A> <A=20
href=3D"http://www.engin.swarthmore.edu/">Department of=20
Engineering</A> <A=20
href=3D"http://www.swarthmore.edu/">Swarthmore College</A> =
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<HR class=3DfullWidth>
<A name=3Dex2></A><!--webbot bot=3D"Include" =
u-include=3D"Example2/Example2.html" tag=3D"BODY" startspan -->
<H1> </H1>
<H1>Root Locus: Example 2</H1>
<H2>Transfer function</H2>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><IMG=20
alt=3DEx2=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Exam=
ple2/Example2TF.gif"> </P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DBckGrnd></A>Xfer Function Info</H2>
<P>For the open loop transfer function, G(s)H(s):<BR>We have n=3D3 poles =
at s =3D 0,=20
-3, -2. We have m=3D0 finite zeros. So there exists q=3D3 =
zeros as s=20
goes to infinity (q =3D n-m =3D 3-0 =3D 3).<BR><BR>We can rewrite the =
open loop=20
transfer function as G(s)H(s)=3DN(s)/D(s) where N(s) is the numerator =
polynomial,=20
and D(s) is the denominator polynomial. <BR>N(s)=3D 1, and D(s)=3D =
s<SUP>3</SUP> + 5=20
s<SUP>2</SUP> + 6 s.<BR><BR>Characteristic Equation is 1+KG(s)H(s)=3D0, =
or=20
1+KN(s)/D(s)=3D0,<BR>or D(s)+KN(s) =3D s<SUP>3</SUP> + 5 s<SUP>2</SUP> + =
6 s+ K( 1 )=20
=3D 0<BR></P>
<HR>
<P=20
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BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2>Completed Root Locus</H2>
<P></P>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><IMG=20
alt=3DRLTot=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Exam=
ple2/RLTotal.png"></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DRuleSym></A>Root Locus Symmetry</H2>
<P>As you can see, the locus is symmetric about the real axis.<BR></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DRuleNum></A>Number of Branches</H2>
<P>The open loop transfer function, G(s)H(s), has 3 poles, therefore the =
locus=20
has 3 branches. Each branch is displayed in a different color.<BR></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DRuleStart></A>Start/End Points</H2>
<P>Root locus starts (K=3D0) at poles of open loop transfer function,=20
G(s)H(s). These are shown by an "x" on the diagram above<BR><BR>As =
K→∞ the=20
location of closed loop poles move to the zeros of the open loop =
transfer=20
function, G(s)H(s). Don&apos;t forget we have we also =
have=20
q=3Dn-m=3D3 zeros at infinity. (We have n=3D3 finite poles, =
and m=3D0 finite=20
zeros).<BR></P>
<HR>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><A=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
href=3D"javascript:history.go(-1)"><IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif">=20
Back <IMG=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/images/LeftArro=
w.gif"></A></P>
<HR>
<H2><A name=3DRuleReal></A>Locus on Real Axis</H2>
<P=20
style=3D"MARGIN: 0px; COLOR: red; BORDER-TOP-STYLE: none; =
BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
TEXT-DECORATION: none; BORDER-BOTTOM-STYLE: none"><IMG=20
alt=3DRLAx=20
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Exam=
ple2/RLRealAxis.png"></P>
<P>The root locus exists on real axis to left of an odd number of poles =
and=20
zeros of open loop transfer function, G(s)H(s), that are on the real=20
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