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Root Locus MethodRoot Locus Method446446--2020

Prof. Neil A.Prof. Neil A. DuffieDuffie

University of WisconsinUniversity of Wisconsin--MadisonMadison

Neil A. Neil A. DuffieDuffie, 1996, 1996

All rights reservedAll rights reserved

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Importance of Pole LocationImportance of Pole Location

Performance is a function of pole locationPerformance is a function of pole location

-- transient responsetransient response

-- absolute stability (stable or not?)absolute stability (stable or not?)

-- relative stability (how stable?)relative stability (how stable?)

Poles migrate as control parameters varyPoles migrate as control parameters vary

-- function of controller gains, zeros, polesfunction of controller gains, zeros, poles-- what values produce good locations?what values produce good locations?

-- design (place poles) using root locusdesign (place poles) using root locus

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Transient ResponseTransient Response

Re

Im

00

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Absolute StabilityAbsolute Stability

00

Stable regionStable region(negative real parts)(negative real parts)

Unstable regionUnstable region(non(non--negativenegative

real parts)real parts)

ImIm

ReRe

left 1/2left 1/2--planeplane

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RouthRouth--Hurwitz Stability CriterionHurwitz Stability Criterion

RouthsRouths criterion is a method forcriterion is a method forassessing stability without finding roots.assessing stability without finding roots.

The method is tabular, finds the numberThe method is tabular, finds the numberof roots with positive real parts, and isof roots with positive real parts, and is

described in most controls textbooks.described in most controls textbooks.

The method was developed in the lateThe method was developed in the late

1800s when finding roots was difficult.1800s when finding roots was difficult.

Powerful calculation tools on the desktopPowerful calculation tools on the desktop

have made the method less useful.have made the method less useful.

Review it at a high level at this point.Review it at a high level at this point.

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Characteristic EquationCharacteristic Equation

ProcessProcessControlControl

++--

R(s)R(s) C(s)C(s)E(s)E(s) M(s)M(s)GG

cc

(s)(s)

C(s)C(s)

GGpp

(s)(s)

System transfer function:System transfer function:

Characteristic equation:Characteristic equation:

D (z) = 1 + Gc(s)Gp(s) = 0

C(s)

R(s) =Gc (s)Gp (s)

1+ Gc (s)Gp (s) =N(s)

D(s)

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Signs in Characteristic EquationSigns in Characteristic Equation

All coefficients of characteristic equation:All coefficients of characteristic equation:

-- must have the same signmust have the same sign

-- must be nonmust be non--zerozero

Necessary (but not sufficient) condition forNecessary (but not sufficient) condition for

absolute stability (from Rouths Criterion)absolute stability (from Rouths Criterion)

Examples:Examples:

ss33 + 2s+ 2s22 + s + 5 = 0+ s + 5 = 0 (may be stable)(may be stable)

ss33 + 2s+ 2s22 -- s + 5 = 0s + 5 = 0 (unstable)(unstable)

ss33 + 2s+ 2s22 + 5 = 0+ 5 = 0 (unstable)(unstable)

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Relative StabilityRelative Stability

How stable is a system?How stable is a system?

-- compared to another systemcompared to another system

-- distance to the border of instabilitydistance to the border of instability

Measures of relative stabilityMeasures of relative stability

-- damping associated with each rootdamping associated with each root

-- real parts of rootsreal parts of roots-- gain and phase marginsgain and phase margins

(frequency response concept:(frequency response concept:

study later)study later)

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Relative StabilityRelative Stability

ReRe

ImIm

00ReRe

ImIm

00

11 22dd22

SystemSystem

#2#2

dd11

SystemSystem

#1#1

11 > 22))dd11 < d< d22

System #2 is relatively moreSystem #2 is relatively morestable than System #1!stable than System #1!

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Step Response of Systems #1 and #2Step Response of Systems #1 and #2

System #2System #2

System #1System #1

System #1:System #1:

-- is relatively less stable than System #2is relatively less stable than System #2-- has more oscillatory step responsehas more oscillatory step response

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Root LocusRoot Locus

Definition: The root locus is the path of theDefinition: The root locus is the path of theroots of the characteristic equationroots of the characteristic equation

plotted in the splotted in the s--plane as a systemplane as a system

parameter is changed.parameter is changed.

Design: Choose a parameter value forDesign: Choose a parameter value for

which the locus lies in a good area ofwhich the locus lies in a good area ofthe sthe s--plane (where dynamics meet specs).plane (where dynamics meet specs).

Iteration: If no part of the root locus lies inIteration: If no part of the root locus lies ina good area of the sa good area of the s--plane, then changeplane, then change

the structure of the controller to modifythe structure of the controller to modify

the locus. Then choose parameter value.the locus. Then choose parameter value.

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Root Locus of 3Root Locus of 3rdrd--Order SystemOrder System

k < 2:k < 2: > 1> 1k > 2:k > 2: < 1< 1k < 30: stablek < 30: stablek > 30: unstablek > 30: unstable

--66 --44 --22 00 44 66--66

--44

--22

22

44

66

ReRe

ImIm

00

22

G(s)= ks s+ 3 s+ 2 C(s)R(s) = ks s+ 3 s+ 2+ k

00 00 00

3030

3030

3030 2222

300300

300300

3003003.53.5

3.53.5

3.53.5

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Choice of k forChoice of k for = 0.707= 0.707

--66 --44 --22 00 44 66--66

--44

--22

22

44

66

ReRe

ImIm

00

22

00 00 00

3030

3030

3030 2222ForFor = 0.707:= 0.707:k 3.5k 3.5

300300

300300

3003003.53.5

3.53.5

3.53.5

4545

G(s)= ks s+ 3 s+ 2 C(s)R(s) = ks s+ 3 s+ 2+ k

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Unit Step Response for k = 3.5Unit Step Response for k = 3.5

00 11 22 33 44 55 66 77 88 99 101000

0.20.2

0.40.4

0.60.6

0.80.8

11

1.21.2

tt

0.7070.707c(t)c(t)