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KE31303KE31303 CONTROL SYSTEMSCONTROL SYSTEMS

Assoc. Prof Dr Yang Soo SiangBEng(Hons) MSc PhD

Room 28 Level 3School of Engineering and Information Technology

Universiti Malaysia Sabah

Assoc Prof Dr Yang Soo Siang 2

LECTURE 5: STABILITY ANALYSIS

KE31303 CONTROL SYSTEMS

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OVERVIEW General.

Routh-Hurwitz criterion.

Stability design.

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GENERAL

Transient requirements: time constant, risetime, settling time, peak overshoot,damping ratio etc

Steady state requirements: errors

Stability: actually MOST IMPORTANTSYSTEM SPECIFICATION!

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GENERAL IfIf systemsystem isis unstableunstable needneed notnot considerconsider otherother

specificationsspecifications nono basisbasis forfor controllercontroller designdesign..

FormalFormal definitionsdefinitions:: AnAn LTILTI systemsystem isis stablestable ifif thethe naturalnatural responseresponse approachesapproaches zerozero asas timetime

approachesapproaches infinityinfinity..

AnAn LTILTI systemsystem isis unstableunstable ifif thethe naturalnatural responseresponse growsgrows withoutwithout boundboundasas timetime approachesapproaches infinityinfinity..

AnAn LTILTI systemsystem isis marginallymarginally stablestable ifif thethe naturalnatural responseresponse neitherneitherdecaysdecays nornor growsgrows butbut remainsremains constantconstant oror oscillatesoscillates asas timetime approachesapproachesinfinityinfinity..

Assoc Prof Dr Yang Soo Siang 6

GENERAL

OR can be described as:OR can be described as:

A system is stable if every bounded inputA system is stable if every bounded inputyields a bounded outputyields a bounded output

A system is unstable if any bounded inputA system is unstable if any bounded inputyields an unbounded outputyields an unbounded output

thisthis definitiondefinition isis moremore relevantrelevant inin termsterms ofof controlcontrol systemsystemstability!stability!

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GENERAL Remember which part of the transfer function effectsRemember which part of the transfer function effects

system stability?system stability?

Hint:Hint:

if poles in left half plane (sif poles in left half plane (s--plane).plane).

If poles in right half plane (sIf poles in right half plane (s--plane).plane).

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GENERAL

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GENERAL UnstableUnstable systemssystems-- physicallyphysically resultsresults inin

damagedamage ofof equipment,equipment, adjacentadjacent propertiespropertiesandand mostmost importantlyimportantly humanhuman liveslives..

SystemsSystems areare designeddesigned withwith limitlimit stopsstops toto

preventprevent totaltotal runawayrunaway..

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GENERAL

NotNot allall mathematicalmathematical modelsmodels oror transfertransferfunctionsfunctions areare easilyeasily factorisedfactorised forfor youyou totoobserveobserve theirtheir polespoles conveniently!conveniently!

ForFor exampleexample

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ROUTH-HURWITZ CRITERION

ThisThis methodmethod-- yieldsyields stabilitystability informationinformation withoutwithoutthethe needneed toto solvesolve forfor thethe closedclosed looploop polespoles..

ResultsResults-- thethe numbernumber ofof closedclosed looploop systemsystem polespolesinin thethe leftleft halfhalf plane,plane, inin thethe rightright halfhalf planeplane andandonon thethe jj axisaxis..

HowHow manymany butbut notnot where!where! SoSo backback toto thethe previousprevious questionsquestions whywhyisis thisthis methodmethod stillstill relevant?relevant? HintHint:: forfor controllercontroller designdesign ableable toto yieldyieldaa rangerange ofof parametersparameters toto ensureensure stabilitystability ofof systemsystem..

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ROUTH-HURWITZ CRITERION

Requires two steps:Requires two steps:

GenerateGenerate aa datadata tabletable knownknown asas aa RouthRouth tabletable

InterpretInterpret thethe RouthRouth tabletable toto knowknow howhow manymanyclosedclosed looploop polespoles areare inin thethe leftleft halfhalf planeplaneetcetc......

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ROUTH-HURWITZ CRITERION

Generating theGenerating the RouthRouth table:table:

For example:For example:

Begin by labeling the rows with powers ofBegin by labeling the rows with powers of ssfrom the highest of thefrom the highest of thedenominator of the closed loop transfer function todenominator of the closed loop transfer function to ss00

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ROUTH-HURWITZ CRITERION

StartStart withwith coeffcoeff ofof thethe highesthighest powerpower ofofss inin thethe denominatordenominator andand listlisthorizontallyhorizontally inin thethe firstfirst row,row, everyevery otherothercoeffcoeff..

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ROUTH-HURWITZ CRITERION

InIn thethe secondsecond row,row, listlist horizontallyhorizontallystartingstarting withwith thethe nextnext highesthighest powerpower ofof

ss,, everyevery coeffcoeff thatthat waswas skippedskipped inin thethefirstfirst rowrow..

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ROUTH-HURWITZ CRITERION

Generating theGenerating the RouthRouth table: the remainingtable: the remainingentryentry EachEach entryentry isis aa negativenegative determinantdeterminant ofof entriesentries inin thethe previousprevious

twotwo rowsrows divideddivided byby thethe entryentry inin thethe firstfirst columnscolumns directlydirectly aboveabove

thethe calculatedcalculated rowrow..

TheThe leftleft handhand columncolumn ofof thethe determinantdeterminant isis alwaysalways thethe firstfirstcolumncolumn ofof thethe previousprevious twotwo rows,rows, andand thethe rightright handhand columncolumn isisthethe elementselements ofof thethe columncolumn aboveabove andand toto thethe rightright..

TheThe tabletable isis completecomplete whenwhen allall thethe rowsrows areare completedcompleted downdown totoss00..

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ROUTH-HURWITZ CRITERION

Generating theGenerating the RouthRouth table:table:

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ROUTH-HURWITZ CRITERION

For example:For example:

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ROUTH-HURWITZ CRITERION

for convenience any row can be multiplied by a positive constantwithout changing the value of the rows below. Not to be multiplied bynegative constants!

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ROUTH-HURWITZ CRITERION

Interpretation:Interpretation: TheThe RouthRouth--Hurwitz criterion declares that the numberHurwitz criterion declares that the number

of roots of the polynomial that are in the right halfof roots of the polynomial that are in the right halfplane is equal to the number of sign changes in theplane is equal to the number of sign changes in the

first column.first column.

IfIf closedclosed looploop tftf hashas allall polespoles inin LHPLHP thenthen systemsystem isisstablestable;; nono signsign changechange inin thethe firstfirst column!column!

FromFrom thethe exampleexample shownshown twotwo polespoles inin rightright RHPRHPWhy?Why? (based(based onon RouthRouth--HurwitzHurwitz criterioncriterion))

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ROUTH-HURWITZ CRITERION

Positive number

Positive number

Negative number

2 sign change = 2 RHP poles exist! Hence system is unstable2 sign change = 2 RHP poles exist! Hence system is unstable!

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ROUTH-HURWITZ CRITERION

Special cases:Special cases:

Zero in first column of a rowZero in first column of a row

Entire row consisting of zeroEntire row consisting of zero

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ROUTH-HURWITZ CRITERION

For example (zero in first column):For example (zero in first column):

T(s)= 10/sT(s)= 10/s55+2s+2s44+3s+3s33+6s+6s22+5s+3+5s+3

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ROUTH-HURWITZ CRITERION

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ROUTH-HURWITZ CRITERION

Replace zero by a small number,Replace zero by a small number, ..

Assume a sign, positive or negative for theAssume a sign, positive or negative for thequantityquantity ..

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ROUTH-HURWITZ CRITERION

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ROUTH-HURWITZ CRITERION

WhetherWhether positivepositive oror negative,negative, resultsresults ofofinterpretationinterpretation willwill bebe thethe samesame..

ForFor thethe example,example, systemsystem isis unstableunstable withwith 22polespoles inin RHPRHP..

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ROUTH-HURWITZ CRITERION

AlternativelyAlternatively

WriteWrite aa polynomialpolynomial thatthat hashas reciprocalreciprocalrootsroots ofof thethe denominatordenominator writewrite thethedenominatordenominator inin reversereverse order,order,

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ROUTH-HURWITZ CRITERION

ViaVia thethe samesame exampleexample::

T(s)=T(s)= 1010/s/s55++22ss44++33ss33++66ss22++55s+s+33

TheThe denominatordenominator inin reversereverse orderorder

D(s)=D(s)= 33ss55++55ss44++66ss33++33ss22++22s+s+11

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ROUTH-HURWITZ CRITERION

Two sign changes hence systemis unstable and has two RHPpoles!

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ROUTH-HURWITZ CRITERION

For example (row of zeros*):For example (row of zeros*):

T(s)= 10/sT(s)= 10/s55+7s+7s44+6s+6s33+42s+42s22+8s+56+8s+56

* zero in magnitude NOT zero of transfer function!* zero in magnitude NOT zero of transfer function!

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ROUTH-HURWITZ CRITERION

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ROUTH-HURWITZ CRITERION

Second row divided by 7 for convenience.Second row divided by 7 for convenience.

Third row all zeros henceThird row all zeros hence

ObserveObserve thethe rowrow immediatelyimmediately aboveabove thethe rowrow ofof zeros,zeros,useuse entriesentries inin thatthat rowrow forfor coeffcoeff toto formform polynomialpolynomial totoreplacereplace allall zeroszeros inin thethe 33rdrd rowrow..

P(s)=sP(s)=s44++66ss22++88

Differentiating,Differentiating, dPdP(s)/(s)/dsds==44ss33++1212s+s+00

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ROUTH-HURWITZ CRITERION

UUsese thethe coeffcoeff fromfrom thethe differentiateddifferentiated polynomialpolynomial toto replacereplace thethezeros,zeros,

44ss33++1212s+s+00

ForFor convenience,convenience, multipliedmultiplied dividedivide byby 44 afterafter replacingreplacing thethe zeroszeros..

RemainderRemainder ofof rowrow isis formedformed inin aa straightforwardstraightforward mannermanner byby followingfollowingthethe standardstandard formform..

Obviously,Obviously, therethere areare nono RHPRHP poles!poles!

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STABILITY DESIGN

StabilityStability designdesign exampleexample:: findfind thethe rangerange ofof gaingain KK forforsystemsystem toto bebe stable,stable, unstableunstable andand marginallymarginally stablestable..AssumeAssume K>K>00..

FindFind thethe closedclosed looploop transfertransfer functionfunction..

FormForm thethe RouthRouth tabletable

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STABILITY DESIGN

StabilityStability designdesign exampleexample::

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STABILITY DESIGN

StabilityStability designdesign exampleexample::

SinceSince KK isis assumedassumed positive,positive, wewe seesee allall elementselements inin thethe firstfirstcolumncolumn areare alwaysalways positivepositive exceptexcept forfor thethe ss11 rowrow..

ThisThis entryentry cancan bebe positive,positive, negativenegative oror zerozero..

IfIf KK > 13861386 thethe ss11 termterm willwill bebe negative,negative, hencehence 22 signsign changechange--22 polespoles onon thethe RHPRHP andand 11 polepole inin LHP,LHP, systemsystem unstableunstable..

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STABILITY DESIGN

StabilityStability designdesign exampleexample::

IfIf KK == 13861386 wewe havehave entireentire rowrow ofof zerozero

ReturningReturning toto thethe ss22 rowrow andand replacingreplacing KK withwith 13861386,, formform thethepolynomial,polynomial,

P(s)=18sP(s)=18s22+1386+1386

DifferentiatingDifferentiating withwith respectrespect toto s,s,

dPdP(s)/(s)/dsds = 36s+0= 36s+0

hence replace hence replace coeffcoeff in the row of zerosin the row of zeros

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STABILITY DESIGN

StabilityStability designdesign exampleexample::

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STABILITY DESIGN

StabilityStability designdesign exampleexample::

NoNo changechange ofof signsign hencehence thethe eveneven polynomialpolynomial downdowntoto bottombottom ofof tabletable..

EvenEven polynomialpolynomial hashas twotwo rootsroots onon thethe jwjw axisaxis..

NoNo signsign changechange aboveabove eveneven polynomial,polynomial, hencehenceremainingremaining rootroot isis inin LHPLHP-- systemsystem isis marginallymarginallystablestable..

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NEXT LECTURE Root Locus:

Basic stuff- significance etc.

Plotting and sketching

What you need to do! Review this lecture and try out examples in Chp 5, Nise till pg

305. They are all relevant for your understanding and for you to

be familiar with forming the Routh table and stability design viaRouth Hurwitz.

In addition, read about root locus of course!