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Subject: Root Locus Examples - Erik Cheever
Date: Wed, 16 Jan 2008 22:53:29 +0800
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<H1>Root Locus Examples</H1>
<UL id=3DNavHeader>
<LI><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Roo=
tLocus.html">Overview</A>=20
<LI><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Roo=
tLocusWhy.html">Why?</A>=20
<LI><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Der=
iveRootLocusRules.html">Derivations</A>=20
<LI id=3Dcurrent><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/RLo=
cusExamples.html">Examples</A>=20
<LI><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Roo=
tLocusCharacteristics.html">Characteristics</A>=20
<LI><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Roo=
tLocusReviewRules.html">Rules=20
Table</A>=20
<LI><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/RLo=
cusGui.html">RLocusGui</A>=20
<LI><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/RLo=
cusAll.html">Printable</A>=20
</LI></UL>
<HR class=3DfullWidth>
<DIV class=3DskinnyTable>
<TABLE class=3DskinnyTable>
<TBODY>
<TR>
<TH class=3DskinnyTable style=3D"HEIGHT: 21px" =
colSpan=3D5>Examples (Click=20
on Transfer Function)</TH></TR>
<TR>
<TD class=3DskinnyTable><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/RLo=
cusExamples.html#ex1"><STRONG>1<BR></STRONG><IMG=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Deri=
veRules/Example1TF.gif"></A></TD>
<TD class=3DskinnyTable><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/RLo=
cusExamples.html#ex2"><STRONG>2<BR></STRONG><IMG=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Deri=
veRules/Example2TF.gif"></A></TD>
<TD class=3DskinnyTable><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/RLo=
cusExamples.html#ex3"><STRONG>3<BR></STRONG><IMG=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Deri=
veRules/Example3TF.gif"></A></TD>
<TD class=3DskinnyTable><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/RLo=
cusExamples.html#ex4"><STRONG>4<BR></STRONG><IMG=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Deri=
veRules/Example4TF.gif"></A></TD>
<TD class=3DskinnyTable><A=20
=
href=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/RLo=
cusExamples.html#ex5"><STRONG>5<BR></STRONG><IMG=20
=
src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Deri=
veRules/Example5TF.gif"></A></TD></TR></TBODY></TABLE></DIV>
<HR class=3DfullWidth>
<A name=3Dex1></A><!--webbot bot=3D"Include" =
u-include=3D"Example1/Example1.html" tag=3D"BODY" startspan -->
<H1>Root Locus: Example 1</H1>
<H2>Transfer function</H2>
<P=20
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BORDER-RIGHT-STYLE: none; BORDER-LEFT-STYLE: none; TEXT-ALIGN: center; =
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src=3D"http://www.swarthmore.edu/NatSci/echeeve1/Ref/LPSA/Root_Locus/Exam=
ple1/imgAC.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=3D2 poles =
at s =3D 0,=20
-3. We have m=3D0 finite zeros. So there exists q=3D2 zeros =
as s goes to=20
infinity (q =3D n-m =3D 2-0 =3D 2).<BR><BR>We can rewrite the open loop =
transfer=20
function as G(s)H(s)=3DN(s)/D(s) where N(s) is the numerator polynomial, =
and D(s)=20
is the denominator polynomial. <BR>N(s)=3D 1, and D(s)=3D =
s<SUP>2</SUP> + 3=20
s.<BR><BR>Characteristic Equation is 1+KG(s)H(s)=3D0, or =
1+KN(s)/D(s)=3D0, <BR>or=20
D(s)+KN(s) =3D s<SUP>2</SUP> + 3 s+ K( 1 ) =3D 0<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>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=
ple1/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 2 poles, therefore the =
locus=20
has 2 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 =
q=3Dn-m=3D2=20
zeros at infinity. (We have n=3D2 finite poles, and m=3D0 finite=20
zeros).<BR></P>
<HR>
<P=20
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