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UniversityCollegeofSoutheastNorway

http://home.hit.no/~hansha

ControlTheorywithMathScriptExamples

Hans-PetterHalvorsen,2016.10.26

ii

TableofContentsTableofContents....................................................................................................................................ii

1 Introduction....................................................................................................................................1

1.1 MathScript...............................................................................................................................1

2 MathScriptBasics...........................................................................................................................2

2.1 Introduction.............................................................................................................................2

2.2 HowdoyoustartusingMathScript?.......................................................................................2

2.3 BasicOperations......................................................................................................................3

2.4 VectorsandMatrices..............................................................................................................4

2.5 LinearAlgebra.........................................................................................................................7

2.6 Plotting....................................................................................................................................8

2.6.1 Subplots...........................................................................................................................9

2.7 User-DefinedFunctionsinMathScript..................................................................................12

2.8 Scripts....................................................................................................................................12

2.9 FlowControl..........................................................................................................................15

2.10 ControlDesigninMathScript................................................................................................15

3 TransferFunctions........................................................................................................................17

3.1 Introduction...........................................................................................................................17

3.1.1 1.ordersystem...............................................................................................................17

3.1.2 1.ordersystemwithtime-delay....................................................................................18

3.1.3 2.ordersystem...............................................................................................................18

3.2 MathScript.............................................................................................................................19

3.3 FirstorderTransferFunction.................................................................................................21

3.3.1 1.ordersystemwithtime-delay....................................................................................23

iii TableofContents

Control Theory with MathScript Examples

3.4 SecondorderTransferFunction............................................................................................25

3.5 Simulation.............................................................................................................................26

3.6 BlockDiagrams......................................................................................................................26

3.7 AnalysisofStandardFunctions..............................................................................................27

3.7.1 Integrator......................................................................................................................28

3.7.2 1.ordersystem..............................................................................................................30

3.7.3 2.ordersystem..............................................................................................................33

4 State-spaceModels......................................................................................................................35

4.1 Introduction...........................................................................................................................35

4.2 MathScript.............................................................................................................................36

5 Time-delayandPade’-approximations.........................................................................................39

5.1 Introduction...........................................................................................................................39

5.2 MathScript.............................................................................................................................40

6 StabilityAnalysis...........................................................................................................................45

6.1 Introduction...........................................................................................................................45

6.2 Poles......................................................................................................................................46

6.3 FeedbackSystems.................................................................................................................49

6.3.1 LoopTransferfunction..................................................................................................49

6.3.2 Trackingtransferfunction.............................................................................................50

6.3.3 Sensitivitytransferfunction..........................................................................................50

6.3.4 CharacteristicPolynomial..............................................................................................50

7 FrequencyResponse.....................................................................................................................52

7.1 Introduction...........................................................................................................................52

7.2 MathScript.............................................................................................................................53

7.3 Examples...............................................................................................................................54

7.4 StandardTransferfunctions..................................................................................................61

7.4.1 Amplifier(Norwegian:“Forsterker”):............................................................................61

iv TableofContents


7.4.2 Integrator......................................................................................................................62

7.4.3 Derivator........................................................................................................................63

7.4.4 1.Ordersystem.............................................................................................................64

7.4.5 2.Ordersystem.............................................................................................................65

7.4.6 Zeropart(Norwegian:“Nullpunktsledd”).....................................................................66

7.4.7 Timedelay(Norwegian:“Tidsforsinkelse”)...................................................................66

8 FrequencyresponseAnalysis........................................................................................................68

8.1 Introduction...........................................................................................................................68

8.2 MathScript.............................................................................................................................70

9 StabilityAnalysisintheFrequencyDomain..................................................................................74

9.1 Introduction...........................................................................................................................74

9.2 MathScript.............................................................................................................................75

AppendixA–MathScriptFunctions......................................................................................................80

BasicFunctions.................................................................................................................................80

BasicPlottingFunctions....................................................................................................................80

FunctionsusedforControlandSimulation......................................................................................81

1

1 IntroductionThisdocumentpresentssomecontroltheoryandlotsofexamplesofhowyoumayimplementitinMathScript.

Thisdocumentgivesanintroductiontothefollowingtopics:

• TransferFunctionsandBlockDiagrams• State-SpaceModels• Time-delayandPade’approximations• FrequencyResponse• FrequencyResponseAnalysis• StabilityAnalysis

MathScripthaslotsofbuilt-infunctionalityfortheseapplications.Ineachchapterwewillgiveashortoverviewtothetheorybehind,beforewedigintotheMathScriptExamples.SinceMathScriptisalmostidenticaltoMATLAB,youcanuseMATLABinsteadinmostoftheexamplesshown.

1.1 MathScriptMathScriptisahigh-level,text-basedprogramminglanguage.MathScriptincludesmorethan800built-infunctionsandthesyntaxissimilartoMATLAB.Youmayalsocreatecustom-madem-filelikeyoudoinMATLAB.

MathScriptiswellsuitedforpracticalimplementationsofcontroltheory.

MathScript(LabVIEWMathScriptRTModule)isanadd-onmoduletoLabVIEWbutyoudon’tneedtoknowLabVIEWprogramminginordertouseMathScript,becauseMathScriptisatext-basedlanguagesimilartoMATLAB.

FormoreinformationaboutMathScript,pleasereadtheTutorial“LabVIEWMathScript”(http://home.hit.no/~hansha/?tutorial=mathscript).

Additionalexercisesaregiveninthecourse“SoYouThinkYouCanMathScript”(http://home.hit.no/~hansha/?lab=mathscript).

2

2 MathScriptBasics

2.1 IntroductionMathScriptisahigh-level,text-basedprogramminglanguage.MathScriptincludesmorethan800built-infunctionsandthesyntaxissimilartoMATLAB.Youmayalsocreatecustom-madem-filelikeyoudoinMATLAB.

MathScriptisanadd-onmoduletoLabVIEWbutyoudon’tneedtoknowLabVIEWprogramminginordertouseMathScript.

FormoreinformationaboutMathScript,pleasereadtheTutorial“LabVIEWMathScript”.

2.2 HowdoyoustartusingMathScript? YouneedtoinstallLabVIEWandtheLabVIEWMathScriptRTModule.Whennecessarysoftwareisinstalled,startMathScriptbyopenLabVIEW:

3 MathScriptBasics


IntheGettingStartedwindow,selectTools->MathScriptWindow...:

2.3 BasicOperationsVariables:

Variablesaredefinedwiththeassignmentoperator,“=”.MathScriptisdynamicallytyped,meaningthatvariablescanbeassignedwithoutdeclaringtheirtype,andthattheirtypecanchange.Valuescancomefromconstants,fromcomputationinvolvingvaluesofothervariables,orfromtheoutputofafunction.

Example:

4 MathScriptBasics


>> x = 17 x = 17 >> x = 'hat' x = hat >> x = [3*4, pi/2] x = 12.0000 1.5708 >> y = 3*sin(x) y = -1.6097 3.0000

[EndofExample]

Note!MathScriptiscasesensitive!ThevariablesxandXarenotthesame.

Note!Unlikemanyotherlanguages,wherethesemicolonisusedtoterminatecommands,inMathScriptthesemicolonservestosuppresstheoutputofthelinethatitconcludes.

Trythefollowing:

>> a=5 a = 5 >> a=6; >>

Asyousee,whenyoutypeasemicolon(;)afterthecommand,MathScriptwillnotrespond.

Itisnormalitenteronecommandineachline,likethis:

x = [0:0.1:1]; y = sin(x)

Butwecanalsoentermorethanonecommandononeline:

x = [0:0.1:1]; y = sin(x)

or:

x = [0:0.1:1], y = sin(x)

2.4 VectorsandMatricesVectors:

5 MathScriptBasics


Giventhefollowingvector:

𝑥 =123

ThiscanbeimplementedinMathScriptlikethis:

x =[1 2 3]

The“colonnotation”isveryusefulforcreatingvectors:

Example:

Thisexampleshowshowtousethecolonnotationcreatingavectoranddosomecalculations.

Matrices:

Giventhefollowingmatrix:

6 MathScriptBasics


𝐴 = 0 1−2 −3

MathScriptCode:

A=[0 1; -2 -3]

Giventhefollowingmatrix:

𝐶 =−1 2 04 10 −21 0 6

MathScriptCode:

C=[-1 2 0; 4 10 -2; 1 0 6]

Howtogetasubsetofamatrix:

→FindthevalueinthesecondrowandthethirdcolumnofmatrixC:

C(2,3)

Thisgives:

ans = -2

→FindthesecondrowofmatrixC:

C(2,:)

Thisgives:

ans =

4 10 -2

→FindthethirdcolumnofmatrixC:

C(:,3)

Thisgives:

ans = 0

-2

6

7 MathScriptBasics


DeletingRowsandColumns:

Youcandeleterowsandcolumnsfromamatrixusingjustapairofsquarebrackets[].

Example:

Given

𝐴 = 0 1−2 −3

WedefinethematrixA:

>>A=[0 1; -2 -3];

TodeletethesecondcolumnofamatrixA,weuse:

>>A(:,2) = [] A = 0 -2

[EndofExample]

2.5 LinearAlgebraLinearalgebraisabranchofmathematicsconcernedwiththestudyofmatrices,vectors,vectorspaces(alsocalledlinearspaces),linearmaps(alsocalledlineartransformations),andsystemsoflinearequations.

MathScriptarewellsuitedforLinearAlgebra.HerearesomeusefulfunctionsforLinearAlgebrainMathScript:

Function Description Examplerank Findtherankofamatrix.Providesanestimateofthenumberof

linearlyindependentrowsorcolumnsofamatrixA.>>A=[1 2; 3 4] >>rank(A)

det Findthedeterminantofasquarematrix >>A=[1 2; 3 4] >>det(A)

inv Findtheinverseofasquarematrix >>A=[1 2; 3 4] >>inv(A)

eig Findtheeigenvaluesofasquarematrix >>A=[1 2; 3 4] >>eig(A)

ones Createsanarrayormatrixwithonlyones >>ones(2) >>ones(2,1)

eye Createsanidentitymatrix >>eye(2)

diag Findthediagonalelementsinamatrix >>A=[1 2; 3 4] >>diag(A)

Type“helpmatfun”(Matrixfunctions-numericallinearalgebra)intheCommandWindowformoreinformation,ortype“helpelmat”(Elementarymatricesandmatrixmanipulation).

Youmayalsotype“help<functionname>”forhelpaboutaspecificfunction.

8 MathScriptBasics


2.6 PlottingMathScripthaslotsoffunctionalityforPlotting.Thesimplestandmostusedistheplotfunction.

Example:

>>t=[0:0.1:10]; >>y=cos(t); >>plot(t,y)

Thisgivesthefollowingplot:

[EndofExample]

MathScripthaslotsofbuilt-infunctionsforplotting:

Function Description Exampleplot Generatesaplot.plot(y)plotsthecolumnsofyagainstthe

indexesofthecolumns.>X = [0:0.01:1]; >Y = X.*X; >plot(X, Y)

figure Createanewfigurewindow >>figure >>figure(1)

subplot CreatesubplotsinaFigure.subplot(m,n,p)orsubplot(mnp),breakstheFigurewindowintoanm-by-nmatrixofsmallaxes,selectsthep-thaxesforthecurrentplot.TheaxesarecountedalongthetoprowoftheFigurewindow,thenthesecondrow,etc.

>>subplot(2,2,1)

grid Createsgridlinesinaplot.“gridon”addsmajorgridlinestothecurrentplot.“gridoff”removesmajorandminorgridlinesfromthecurrentplot.

>>grid >>grid on >>grid off

axis Controlaxisscalingandappearance.“axis([xminxmaxyminymax])”setsthelimitsforthex-andy-axisofthecurrentaxes.

>>axis([xmin xmax ymin ymax]) >>axis off >>axis on

title Addtitletocurrentplottitle('string')

>>title('this is a title')

xlabel Addxlabeltocurrentplot >> xlabel('time')

9 MathScriptBasics


xlabel('string')ylabel Addylabeltocurrentplot

ylabel('string')>> ylabel('temperature')

legend Createsalegendinthecorner(orataspecifiedposition)oftheplot

>> legend('temperature')

hold Freezesthecurrentplot,sothatadditionalplotscanbeoverlaid >>hold on >>hold off

Example:

Hereweseesomeexamplesofhowtousethedifferentplotfunctions:

[Figure:R.C.DorfandR.H.Bishop,ModernControlSystems,EleventhEdition:PearsonPrenticeHall]

[EndofExample]

2.6.1 Subplots

Thesubplotcommandenablesyoutodisplaymultipleplotsinthesamewindoworprintthemonthesamepieceofpaper.Typing“subplot(m,n,p)”partitionsthefigurewindowintoanm-by-nmatrixofsmallsubplotsandselectsthepthsubplotforthecurrentplot.Theplotsarenumberedalongthefirstrowofthefigurewindow,thenthesecondrow,andsoon.

10 MathScriptBasics


Thesyntaxisasfollows:

subplot(m,n,p)

Example:

x=0:0.1:2*pi; subplot(2,1,1) y=sin(x); plot(x,y) subplot(2,1,2) z=cos(x); plot(x,z)


11 MathScriptBasics


[EndofExample]

12 MathScriptBasics


2.7 User-DefinedFunctionsinMathScriptMathScriptincludesmorethan800built-infunctionsthatyoucanusebutsometimesyouneedtocreateyourownfunctions.

TodefineyourownfunctioninMathScript,usethefollowingsyntax:

function outputs = function_name(inputs) % documentation …

ThefigurebelowillustrateshowtocreateandusefunctionsinMathScript:

2.8 ScriptsAscriptisasequenceofMathScriptcommandsthatyouwanttoperformtoaccomplishatask.Whenyouhavecreatedthescriptyoumaysaveitasam-fileforlateruse.

13 MathScriptBasics


YoumayalsohavemultipleScriptWindowsopenatthesametimebyselecting“NewScriptEditor”intheFilemenu:

Thisgives:

14 MathScriptBasics


15 MathScriptBasics


2.9 FlowControlYoumayusedifferentloopsinMathScript

• Forloop• Whileloop

Ifyouwanttocontroltheflowinyourprogram,youmaywanttouseoneofthefollowing:

• If-elsestatement• Switchandcasestatement

Example:

function av = calc_average2(x) %This function calculates the average of a vector x N=length(x); tot=0; for i=1:N tot=tot+x(i); end av=tot;

[EndofExample]

2.10 ControlDesigninMathScriptType“helpcdt”intheCommandWindowintheMathScriptenvironment.TheLabVIEWHelpwindowappears:

16 MathScriptBasics


UsetheHelpwindowandreadaboutsomeofthefunctionsavailableforcontroldesignandsimulation.

SeeAppendixAforalistofsomeofthemostusedfunctionswithdescriptionandexamples.

17

3 TransferFunctions3.1 IntroductionTransferfunctionsareamodelformbasedontheLaplacetransform.Transferfunctionsareveryusefulinanalysisanddesignoflineardynamicsystems.

AgeneralTransferfunctionisontheform:

𝐻 𝑆 =𝑦(𝑠)𝑢(𝑠)

Whereyistheoutputanduistheinput.

Ageneraltransferfunctioncanbewrittenonthefollowinggeneralform:

𝐻 𝑠 =𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟(𝑠)𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟(𝑠)

=𝑏=𝑠= + 𝑏=?@𝑠=?@ + ⋯+𝑏@𝑠 + 𝑏C𝑎D𝑠D + 𝑎D?@𝑠D?@ + ⋯+𝑎@𝑠 + 𝑎C

TheNumeratorsoftransferfunctionmodelsdescribethelocationsofthezerosofthesystem,whiletheDenominatorsoftransferfunctionmodelsdescribethelocationsofthepolesofthesystem.

Belowwewilllearnmoreabout2importantspecialcasesofthisgeneralform,namelythe1.ordertransferfunctionandthe2.ordertransferfunction.

3.1.1 1.ordersystem

A1.ordertransferfunction:

𝐻 𝑠 =𝐾

𝑇𝑠 + 1

Where 𝐾 istheGainand 𝑇 istheTimeconstant.

Astepresponseofsuchatransferfunctionhasthefollowingcharacteristics:

18 TransferFunctions


3.1.2 1.ordersystemwithtime-delay

A1.ordertransferfunctionwithtime-delaymaybewrittenas:

𝐻 𝑠 =𝐾

𝑇𝑠 + 1𝑒?GH

Astepresponseofsuchatransferfunctionhasthefollowingcharacteristics:

[Figure:F.Haugen,AdvancedDynamicsandControl:TechTeach,2010]

Fromthestepresponseofsuchasystemwecaneasilyfind 𝐾, 𝑇 and 𝜏.

Moreabouttime-delaysinalaterchapter.

3.1.3 2.ordersystem



A2.ordertransferfunction:

𝐻 𝑠 =𝐾

𝑠𝜔C

K+ 2𝜁 𝑠

𝜔C+ 1

Moreabout1.orderand2.ordertransferfunctionslaterinthischapter.

3.2 MathScriptMathScripthasseveralfunctionsforcreatingtransferfunctions:

Function Description Exampletf Createssystemmodelintransferfunctionform.Youalsocan

usethisfunctiontostate-spacemodelstotransferfunctionform.

>num=[1]; >den=[1, 1, 1]; >H = tf(num, den)

Sys_order1 Constructsthecomponentsofafirst-ordersystemmodelbasedonagain,timeconstant,anddelaythatyouspecify.Youcanusethisfunctiontocreateeitherastate-spacemodeloratransferfunctionmodel,dependingontheoutputparametersyouspecify.

>K = 1; >tau = 1; >H = sys_order1(K, tau)

Sys_order2 Constructsthecomponentsofasecond-ordersystemmodelbasedonadampingratioandnaturalfrequencyyouspecify.Youcanusethisfunctiontocreateeitherastate-spacemodeloratransferfunctionmodel,dependingontheoutputparametersyouspecify.

>dr = 0.5 >wn = 20 >[num, den] = sys_order2(wn, dr) >SysTF = tf(num, den)

pid Constructsaproportional-integral-derivative(PID)controllermodelinparallel,series,oracademicform.RefertotheLabVIEWControlDesignUserManualforinformationaboutthesethreeforms.

>Kc = 0.5; >Ti = 0.25; >SysOutTF = pid(Kc, Ti, 'academic');

Giventhegeneraltransferfunction:

𝐻 𝑠 =𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟(𝑠)𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟(𝑠)

=𝑏=𝑠= + 𝑏=?@𝑠=?@ + ⋯+𝑏@𝑠 + 𝑏C𝑎D𝑠D + 𝑎D?@𝑠D?@ + ⋯+𝑎@𝑠 + 𝑎C

InMathScriptwecandefinesuchatransferfunctionusingthebuilt-intffunctionasfollows:

num=[bm, bm_1, bm_2, … , b1, b0]; den=[an, an_1, an_2, … , a1, a0]; H = tf(num, den)

Example:

1.Giventhefollowingtransferfunction:

𝐻(𝑠) =2𝑠K + 3𝑠 + 4

5𝑠 + 9

MathScriptCode:

num=[2, 3, 4]; den=[5, 9]; H = tf(num, den)




𝐻(𝑠) =4𝑠O + 3𝑠 + 45𝑠K + 9

MathScriptCode:

num=[4, 0, 0, 3, 4]; den=[5, 0, 9]; H = tf(num, den)

Note!Ifsomeoftheordersaremissing,wejustputinzeros.Thetransferfunctionabovecanberewrittenas:

𝐻(𝑠) =4𝑠O + 0 ∙ 𝑠Q + 0 ∙ 𝑠K + 3𝑠 + 4

5𝑠K + 0 ∙ 𝑠 + 9


𝐻(𝑠) =7 + 3𝑠 + 2𝑠K

5𝑠 + 6𝑠K

Weneedtorewritethetransferfunctiontogetitincorrectorders:

𝐻(𝑠) =2𝑠K + 3𝑠 + 76𝑠K + 5𝑠

MathScriptCode:

num=[2, 3, 7]; den=[6, 5, 0]; H = tf(num, den)

[EndofExample]

Forcreatingmorecomplextransferfunctions,someofthefollowingfunctionsareuseful:

Function Description Exampleconv Computestheconvolutionoftwovectorsormatrices.Example:

𝐻 𝑠 =𝐾

2𝑠 + 1 (3𝑠 + 1)

>den1 = [2, 1]; >den2 = [3, 1]; >C = conv(den1, den2)

series ConnectstwosystemmodelsinseriestoproduceamodelSysSerwithinputandoutputconnectionsyouspecify.Example:

𝐻 𝑠 = 𝐻@ 𝑠 𝐻K(𝑠)

>Hseries = series(H1,H2)

feedback Connectstwosystemmodelstogethertoproduceaclosed-loopmodelusingnegativeorpositivefeedbackconnections

>SysClosed = feedback(SysIn_1, SysIn_2)



3.3 FirstorderTransferFunctionAfirstordertransferfunctionisgivenontheform:

𝐻 𝑠 =𝐾

𝑇𝑠 + 1

Where

𝐾 istheGain

𝑇 istheTimeconstant

Inthetimedomainwegetthefollowingdifferentialequation(usingInverseLaplace):

𝑥 =1𝑇(−𝑥 + 𝐾𝑢)

Wecandrawthefollowingblockdiagramofthesystem:

Example:

WewillusethetffunctioninMathScripttodefinethetransferfunction:

𝐻 𝑠 =𝐾

𝑇𝑠 + 1

Weset 𝐾 = 1 and 𝑇 = 1.

MathScriptCodeusingthetffunction:

K=1; T=1; num=[K]; den=[T, 1]; H = tf(num, den)

WeenterthecodeshownaboveintheScriptwindowasshownbelow:



MathScriptCodeusingthesys_order1function:

K = 1; T = 1; H = sys_order1(K, T)

[EndofExample]

StepResponse:

Thestepresponsefora1.ordertransferfunctionhasthefollowingcharacteristics(astep 𝑈 at 𝑡 =0):

ThetimeconstantTisdefinedasthetimewheretheresponsereaches63%ofthesteadystatevalue.

Example:

Giventhefollowing1.ordertransferfunction:

𝐻 𝑠 =1

𝑠 + 1



(𝐾 = 1, 𝑇 = 1)

Wecreatethefollowingcodeinordertoplotthestepresponseforthissystem:

K=1; T=1; num=[K]; den=[T, 1]; H = tf(num, den); Step(H)

Thisgivesthefollowingstepresponse:

[EndofExample]

3.3.1 1.ordersystemwithtime-delay

A1.ordersystemwithtime-delayhasthefollowingtransferfunction:

𝐻 𝑠 =𝐾

𝑇𝑠 + 1𝑒?GH

Inthetimedomainwegetthefollowingdifferentialequation(usingInverseLaplace):

𝑥 =1𝑇(−𝑥 + 𝐾𝑢(𝑡 − 𝜏))

Wecandrawthefollowingblockdiagramofthesystem:



StepResponse:


Example:

𝐻 𝑠 =1

2𝑠 + 1𝑒?QH

(𝐾 = 1, 𝑇 = 2, 𝜏 = 3)

TheMathScriptcodebecomes(usingthebuilt-insys_order1function):

K = 1; T = 2; delay=3; H = sys_order1(K, T, delay) step(H)

Theplotofthestepresponsebecomes:



[EndofExample]

3.4 SecondorderTransferFunctionAsecondordertransferfunctionisgivenontheform:

𝐻 𝑠 =𝐾

𝑠𝜔C

K+ 2𝜁 𝑠

𝜔C+ 1

Where

𝐾 isthegain

𝜁 zetaistherelativedampingfactor

𝜔C[rad/s]istheundampedresonancefrequency.

Example:

DefinethetransferfunctioninMathScript.Set 𝐾 = 1, 𝜁 = 1, 𝜔C = 1

Usethetffunctionorthesys_order2functioninMathScript

MathScriptCode:

num=[1]; den=[1, 2, 1]; H = tf(num, den)



or:

dr = 1 wn = 1 [num, den] = sys_order2(wn, dr) H = tf(num, den)

[EndofExample]

2.ordersystem-specialcase:When 𝜻 > 0 andthepolesarerealanddistinctwehave:

𝐻 𝑠 =𝐾

(𝑇@𝑠 + 1)(𝑇K𝑠 + 1)

Weseethatthissystemcanbeconsideredastwo1.ordersystemsinseries.

𝐻 𝑠 = 𝐻@ 𝑠 𝐻@ 𝑠 =𝐾

(𝑇@𝑠 + 1)∙

1(𝑇K𝑠 + 1)

=𝐾

(𝑇@𝑠 + 1)(𝑇K𝑠 + 1)

3.5 SimulationMathScripthasseveralfunctionsusedforsimulationpurposes:

Function Description Exampleplot Generatesaplot.plot(y)plotsthecolumnsofyagainstthe


step Createsastepresponseplotofthesystemmodel.Youalsocanusethisfunctiontoreturnthestepresponseofthemodeloutputs.Ifthemodelisinstate-spaceform,youalsocanusethisfunctiontoreturnthestepresponseofthemodelstates.Thisfunctionassumestheinitialmodelstatesarezero.Ifyoudonotspecifyanoutput,thisfunctioncreatesaplot.

>num=[1,1]; >den=[1,-1,3]; >H=tf(num,den); >t=[0:0.01:10]; >step(H,t);

lsim Createsthelinearsimulationplotofasystemmodel.Thisfunctioncalculatestheoutputofasystemmodelwhenasetofinputsexcitethemodel,usingdiscretesimulation.Ifyoudonotspecifyanoutput,thisfunctioncreatesaplot.

>t = [0:0.1:10] >u = sin(0.1*pi*t)' >lsim(SysIn, u, t)

Plotsfunctions:Herearesomeusefulfunctionsforcreatingplots:plot,figure,subplot,grid,axis,title,xlabel,ylabel,semilogx–formoreinformationabouttheplotsfunction,type“helpplots”.Ortype“help<functionname>”.

3.6 BlockDiagramsMathScripthavebuilt-infunctionsformanipulatingblockdiagramsandtransferfunctions.

Serial:



MathScript:

… H = series(h1,h2)

Parallel:

MathScript:

… H = parallel(h1,h2)

Feedback:

MathScript:

… H = feedback(h1,h2)

3.7 AnalysisofStandardFunctionsHerewewilltakeacloserlookatthefollowingstandardfunctions:



• Integrator• 1.Ordersystem• 2.Ordersystem

3.7.1 Integrator

ThetransferfunctionforanIntegratorisasfollows:

𝐻 𝑠 =𝐾𝑠

Pole(s):

TheIntegratorhasapoleinorigo: 𝑝 = 0

InMathScriptyoumayusethepolesfunctioninordertofindthepoles.

Example:

K=1; T=1; num=[K]; den=[T 1]; H=tf(num,den); p=poles(H)

[EndofExample]

Stepresponse:

Note!InMathScriptwecanusethestepfunctionforthispurpose.

Herewewillfindthemathematicalexpressionforthestepresponse(𝒚(𝒕)):

TheLaplaceTransformationpairforastepisasfollows:

1𝑠⇔ 1

Thestepresponseofanintegratorthenbecomes:

Im(s)

Re(s)



𝑦 𝑠 = 𝐻 𝑠 𝑢 𝑠 =𝐾𝑠∙𝑈𝑠= 𝐾𝑈

1𝑠K

WeusethefollowingLaplaceTransformationpairinordertofind 𝑦(𝑡):

1𝑠K⇔ 𝑡

Thenweget:

𝑦 𝑡 = 𝐾𝑈𝑡

→WeseethatthestepresponseoftheintegratorisaRamp.

Conclusion:AbiggerKwillgiveabiggerslope(InNorwegian:“stigningstall”)andtheintegrationwillgofaster.ThesimulationinMathScriptbelowwillalsoshowthis.

BelowwewillshowthisbyusingthestepfunctioninMathScript:

Example:

InMathScriptweusethestepfunctionforsimulationofastepresponse.WesetK=0.2,1,5.

MathScriptCode:

t=[0:0.5:5]; K=0.2 num=[K]; den=[1 0]; H1=tf(num,den); K=1 num=[K]; den=[1 0]; H2=tf(num,den); K=5 num=[K]; den=[1 0]; H3=tf(num,den); step(H1,H2,H3,t) axis([0, 5, 0, 5])

Note!UsingaForLoopinthiscasewouldbeabetterapproach.

Plot:



[EndofExample]

3.7.2 1.ordersystem

Thetransferfunctionfora1.ordersystemisasfollows:

𝐻 𝑠 =𝐾

𝑇𝑠 + 1

Pole(s):

A1.ordersystemhasapole: 𝑝 = − @[

InMathScriptyoumayusethepolesfunctioninordertofindthepoles.Thefunctionpzgraphplotsthepolesandzeros

Stepresponse:

Note!InMathScriptwecanusethestepfunctionforthispurpose.

Herewewillfindthemathematicalexpressionforthestepresponse(𝒚(𝒕)):

𝑦 𝑠 = 𝐻 𝑠 𝑢(𝑠)

Im(s)

Re(s)-1/T



Where

𝑢 𝑠 =𝑈𝑠

WeuseinverseLaplaceandfindthecorrespondingtransformationpairinordertofind 𝑦(𝑡)).

𝑦 𝑠 =𝐾

𝑇𝑠 + 1∙𝑈𝑠

WeusethefollowingLaplacetransformpair:

𝑘𝑇𝑠 + 1 𝑠

⇔ 𝑘(1 − 𝑒?]/[)

Thisgives:

𝑦 𝑡 = 𝐾𝑈(1 − 𝑒?]/[)

Thestepresponseisasfollows:

BelowwewillshowthisbyusingthestepfunctioninMathScript:

Example:

FordifferentvaluesforK,eg.,K=0.5,1,2andT=1.WeusethestepfunctioninMathScript.

t=[0:0.5:10]; den=[1 1]; K=0.5; num=[K]; H1=tf(num,den);



K=1; num=[K]; H2=tf(num,den); K=2; num=[K]; H3=tf(num,den); step(H1,H2,H3,t); axis([0, 10, 0, 2]);


Belowweseetheplotforthis:

[EndofExample]

Example:

FordifferentvaluesforT:T=0.2,0.5,1,2,4andK=1




WeseefromFigureabovethatsmallerT(Timeconstant)givesfasterresponse.

[EndofExample]

3.7.3 2.ordersystem


𝐻 𝑠 =𝐾𝜔CK

𝑠K + 2𝜁𝜔C𝑠 + 𝜔CK=

𝐾𝑠𝜔C

K+ 2𝜁 𝑠

𝜔C+ 1

Where

• 𝐾 isthegain• 𝜁 zetaistherelativedampingfactor• 𝜔C[rad/s]istheundampedresonancefrequency.

Wehavethat:



Figure:F.Haugen,AdvancedDynamicsandControl:TechTeach,2010.

→ShowthisinMathScript

Specialcase:When 𝜻 > 0 andthepolesarerealanddistinctwehave:

𝐻 𝑠 =𝐾

(𝑇@𝑠 + 1)(𝑇K𝑠 + 1)

Weseethatthissystemcanbeconsideredastwo1.ordersystemsinseries:

𝐻 𝑠 = 𝐻@ 𝑠 𝐻@ 𝑠 =𝐾

(𝑇@𝑠 + 1)∙

1(𝑇K𝑠 + 1)

=𝐾

(𝑇@𝑠 + 1)(𝑇K𝑠 + 1)

35

4 State-spaceModels

4.1 IntroductionAstate-spacemodelisastructuredformorrepresentationofasetofdifferentialequations.State-spacemodelsareveryusefulinControltheoryanddesign.Thedifferentialequationsareconvertedinmatricesandvectors,whichisthebasicelementsinMathScript.

Wehavethefollowingequations:

𝑥@ = 𝑎@@𝑥@ + 𝑎K@𝑥K + ⋯+ 𝑎D@𝑥D + 𝑏@@𝑢@ + 𝑏K@𝑢K + ⋯+ 𝑏D@𝑢D

⋮

𝑥D = 𝑎@=𝑥@ + 𝑎K=𝑥K + ⋯+ 𝑎D=𝑥D +𝑏@=𝑢@ + 𝑏K=𝑢K + ⋯+ 𝑏D@𝑢D

⋮

Thisgivesonvectorform:

𝑥@𝑥K⋮𝑥D`

=𝑎@@ ⋯ 𝑎D@⋮ ⋱ ⋮

𝑎@= ⋯ 𝑎D=b

𝑥@𝑥K⋮𝑥D`

+𝑏@@ ⋯ 𝑏D@⋮ ⋱ ⋮

𝑏@= ⋯ 𝑏D=c

𝑢@𝑢K⋮𝑢Dd

𝑦@𝑦K⋮𝑦De

=𝑐@@ ⋯ 𝑐D@⋮ ⋱ ⋮𝑐@= ⋯ 𝑐D=

g

𝑥@𝑥K⋮𝑥D`

+𝑑@@ ⋯ 𝑑D@⋮ ⋱ ⋮

𝑑@= ⋯ 𝑑D=h

𝑢@𝑢K⋮𝑢Dd

ThisgivesthefollowingcompactformofagenerallinearState-spacemodel:

𝑥 = 𝐴𝑥 + 𝐵𝑢

𝑦 = 𝐶𝑥 + 𝐷𝑢

Example:

Wehavethefollowingsystem:

𝑥@ = 𝑥K

36 State-spaceModels


2𝑥K = −2𝑥@−6𝑥K+4𝑢@+8𝑢K

𝑦 = 5𝑥@+6𝑥K+7𝑢@

Converttothefollowingstate-spaceform:

𝑥 = 𝐴𝑥 + 𝐵𝑢

𝑦 = 𝐶𝑥 + 𝐷𝑢

Firstwedo:

𝑥@ = 𝑥K

𝑥K = −𝑥@−3𝑥K+2𝑢@+4𝑢K

𝑦 = 5𝑥@+6𝑥K+7𝑢@

Thisgives:

𝑥@𝑥K

= 0 1−1 −3

b

𝑥@𝑥K + 0 0

2 4c

𝑢@𝑢K

𝑦 = 5 6g

𝑥@𝑥K + 7 0

h

𝑢@𝑢K

→TrytodefinethisState-SpacemodelinMathScript.

[EndofExample]

4.2 MathScriptMathScripthasseveralfunctionsforcreatingstate-spacemodels:

Function Description Exampless Constructsamodelinstate-spaceform.Youalsocanusethis

functiontoconverttransferfunctionmodelstostate-spaceform.

>A = [1 2; 3 4] >B = [0; 1] >C = B' >SysOutSS = ss(A, B, C)


>K = 1; >tau = 1; > [A, B, C, D] = sys_order1(K, tau)


>dr = 0.5 >wn = 20 >[A, B, C, D] = sys_order2(wn, dr) >SysSS = ss(A, B, C, D)

Example:



Givenamass-spring-dampersystem:

Wherec=dampingconstant,m=mass,k=springconstant,F=u=force

Thestate-spacemodelforthesystemis:

𝑥@𝑥K

=0 1

−𝑘𝑚

−𝑐𝑚

𝑥@𝑥K +

01𝑚

𝑢

𝑦 = 1 0𝑥@𝑥K

Definethestate-spacemodelaboveusingthessfunctioninMathScript.Setsomearbitraryvaluesforc=dampingconstant,m=mass,k=springconstant.

WewilluseMathScripttodefinethestatespacemodel:

MathScriptCode:

c=1; m=1; k=1; A = [0 1; -k/m -c/m]; B = [0; 1/m]; C = [1 0]; SysOutSS = ss(A, B, C)

WeusethestepfunctioninMathScriptinordertosimulatethestepresponse:



Trywithdifferentvaluesofc,mandkandwatchtheresults.

[EndofExample]

39

5 Time-delayandPade’-approximations

5.1 IntroductionTime-delaysareverycommonincontrolsystems.TheTransferfunctionofatime-delayis:

𝐻 𝑠 = 𝑒?GH

A1.ordertransferfunctionwithtime-delaymaybewrittenas:

𝐻 𝑠 =𝐾

𝑇𝑠 + 1𝑒?GH

Insomesituationsitisnecessarytosubstitute 𝑒?GH withanapproximation,e.g.,thePadé-approximation:

𝑒?GH ≈1 − 𝑘@𝑠 + 𝑘K𝑠K + ⋯± 𝑘D𝑠D

1 + 𝑘@𝑠 + 𝑘K𝑠K + ⋯+ 𝑘D𝑠D

Belowweseea1.orderanda2.orderPadé-approximation:



5.2 MathScriptMathScripthasabuilt-infunctioncalledpadeforcreatingtransferfunctionsfortime-delays:

Function Description Examplepade IncorporatestimedelaysintoasystemmodelusingthePade

approximationmethod,whichconvertsallresiduals.Youmustspecifythedelayusingthesetfunction.Youalsocanusethisfunctiontocalculatecoefficientsofnumeratoranddenominatorpolynomialfunctionswithaspecifieddelay.

>[num, den] = pade(delay, order) >[A, B, C, D] = pade(delay, order)

Sys_order1 >K=4; T=3; delay=5; >H = sys_order1(K, T, delay)

set >H = set(H1, 'inputdelay', delay);

series >H = series(H1,H2);

Example:

Herearesomeexamplesofhowtousethepadefunction:

SysCon = zpk(1, 3.2, 6) SysCon = set(SysCon, 'inputdelay', 6, 'outputdelay', 1.1) SysDel = pade(SysCon, 2) delay = 1.2 order = 3 [num, den] = pade(delay, order)

Orherisanalternativewithoutusingthepadefunction:

s=tf('s'); %Defines s to be the Laplace variable used in transfer functions K=1; T=1; %Gain and time-constant



H1=tf(K/(T*s+1)); %Creates H as a transfer function delay=1; %Time-delay H2=set(H1,'inputdelay',delay);%Defines H2 as H1 but with time-delay figure(1) %Plot of simulated responses will shown in Figure 1 step(H1,H2) %Simulates with unit step as input, and plots responses.

[EndofExample]

Example:

ThisexampleshowsPade’approximationswithdifferentorders:

[EndofExample]

Ifwehavea1.ordersystem,wecanalsousethefunctionsys_order1().

Example:

Giventhefollowingtransferfunction:

𝐻 𝑠 =3

2𝑠 + 1𝑒?OH

Wedefinethetransferfunctionusingsys_order1withthefollowingcode:

K = 3; T = 2;



delay = 4; H = sys_order1(K, T, delay)

Inadditionwefindthestepresponse:

step(H)


Weknowthatastepresponseofsuchatransferfunctionhasthefollowingcharacteristics:




Sowecaneasilyfind 𝐾, 𝑇 and 𝜏 fromthegraph:

Anotherwaytodefine 𝐻 𝑠 = QKHn@

𝑒?OH isusingthetf()andset()functions:

s = tf('s') H1 = tf(K/(T*s+1)); H = set(H1, 'inputdelay', delay); step(H)

Wecanalsocombinetf()andthepade()functionlikethis:

num = [K]; den = [T, 1]; H1 = tf(num, den); order = 5; H2 = pade(delay, order) H = series(H1, H2) step(H)

Inthelastexamplewefirstdefinedthetransferfunction QKHn@

usingthetf()function,thenwe

definedthetimedelay 𝑒?OH usingthepade()function.Finallywehavecombinedthese2functionsusingtheseries()function.

Inthelastexamplewegetthefollowingplot:



[EndofExample]

45

6 StabilityAnalysis6.1 IntroductionAdynamicsystemhasoneofthefollowingstabilityproperties:

• Asymptoticallystablesystem• Marginallystablesystem• Unstablesystem

Belowweseethebehaviorofthese3differentsystemsafteranimpulse[F.Haugen,AdvancedDynamicsandControl:TechTeach,2010]:

Asymptoticallystablesystem:

lim]→s

ℎ 𝑡 = 0

Marginallystablesystem:

0< lim]→s

ℎ 𝑡 < ∞

Unstablesystem:

lim]→s

ℎ 𝑡 = ∞

46 StabilityAnalysis


6.2 PolesThepolesareimportantwhenanalysisthestabilityofasystem.Thefigurebelowgivesanoverviewofthepolesimpactonthestabilityofasystem:


Thus,wehavethefollowing:

Asymptoticallystablesystem:

Eachofthepolesofthetransferfunctionliesstrictlyinthelefthalfplane(hasstrictlynegativerealpart).

Marginallystablesystem:

Oneormorepolesliesontheimaginaryaxis(haverealpartequaltozero),andallthesepolesaredistinct.Besides,nopoleslieintherighthalfplane.



Unstablesystem:

Atleastonepoleliesintherighthalfplane(hasrealpartgreaterthanzero).

Or:Therearemultipleandcoincidentpolesontheimaginaryaxis.

Example:doubleintegrator 𝐻(𝑠) = @Hw

Example:

Giventhefollowingsystem:

𝐻 𝑠 =𝑠 + 1

𝑠K − 𝑠 + 3

Wewillanalyzethestabilityofthissystem.Inordertodothatwewillplotthestepresponseandfindthepolesforthissystem.

Westartbydefiningthetransferfunctionandplottingtheimpulseresponse:

clear clc % Define Transfer Function num=[1,1]; den=[1,-1,3]; H=tf(num,den); % Step Response t=[0:0.01:10]; impulse(H,t);

Theimpulsebecomes:



→Fromtheplotweseethatthesystemisunstable.

Nextwefindthepolesforthesystem:

poles(H) pzgraph(H)

Thepolesareasfollows(foundfromthebuilt-inpolesfunction):

0.5 + 1.6583i

0.5 - 1.6583i

Wehavealsousedthebuilt-inpzgraphfunctioninordertoplotthepoles(andzeros):



→Weseefromtheplotthatthepoles(redcross)liesintherighthalfplane(hasrealpartgreaterthanzero)andthattherearemultiplepolesontheimaginaryaxis,whichindicatethatthesystemisunstable.

[EndofExample]

6.3 FeedbackSystemsHerearesomeimportanttransferfunctionstodeterminethestabilityofafeedbacksystem.Belowweseeatypicalfeedbacksystem.

6.3.1 LoopTransferfunction

TheLooptransferfunction 𝑳(𝒔) (Norwegian:“Sløyfetransferfunksjonen”)isdefinedasfollows:

𝐿 𝑠 = 𝐻{ 𝑠 𝐻|(𝑠)𝐻=(𝑠)



Where

𝐻{(𝑠) istheControllertransferfunction

𝐻|(𝑠) istheProcesstransferfunction

𝐻=(𝑠) istheMeasurement(sensor)transferfunction

Note!Anothernotationfor 𝐿 is 𝐻C

6.3.2 Trackingtransferfunction

TheTrackingtransferfunction 𝑻(𝒔) (Norwegian:“Følgeforholdet”)isdefinedasfollows:

𝑇 𝑠 =𝑦(𝑠)𝑟(𝑠)

=𝐻{𝐻|𝐻=

1 + 𝐻{𝐻|𝐻==

𝐿(𝑠)1 + 𝐿(𝑠)

= 1 − 𝑆(𝑠)

TheTrackingProperty(Norwegian:“følgeegenskaper”)isgoodifthetrackingfunctionThasvalueequaltoorcloseto1:

𝑇 ≈ 1

6.3.3 Sensitivitytransferfunction

TheSensitivitytransferfunction 𝑺(𝒔) (Norwegian:“Sensitivitetsfunksjonen/avviksforholdet”)isdefinedasfollows:

𝑆 𝑠 =𝑒(𝑠)𝑟(𝑠)

=1

1 + 𝐿(𝑠)= 1 − 𝑇(𝑠)

TheCompensationPropertyisgoodifthesensitivityfunctionShasasmallvalueclosetozero:

𝑆 ≈ 0𝑜𝑟 𝑆 ≪ 1

Note!

𝑇 𝑠 + 𝑆 𝑠 =𝐿(𝑠)

1 + 𝐿(𝑠)+

11 + 𝐿(𝑠)

≡ 1

6.3.4 CharacteristicPolynomial

Wehavethat:

𝐿 𝑠 =𝑛�(𝑠)𝑑�(𝑠)

And:




=𝐿(𝑠)

1 + 𝐿(𝑠)=

𝑛�(𝑠)𝑑�(𝑠)

1 + 𝑛�(𝑠)𝑑�(𝑠)

=𝑛�(𝑠)

𝑑� 𝑠 + 𝑛�(𝑠)

Where 𝑛�(𝑠) and 𝑑� 𝑠 numeratorandthedenominatoroftheLooptransferfunction 𝐿(𝑠).

TheCharacteristicPolynomialforthecontrolsystemthenbecomes:

𝑎(𝑠) = 𝑑� 𝑠 + 𝑛�(𝑠)

52

7 FrequencyResponse7.1 IntroductionThefrequencyresponseofasystemisafrequencydependentfunctionwhichexpresseshowasinusoidalsignalofagivenfrequencyonthesysteminputistransferredthroughthesystem.Eachfrequencycomponentisasinusoidalsignalhavingacertainamplitudeandacertainfrequency.

Thefrequencyresponseisanimportanttoolforanalysisanddesignofsignalfiltersandforanalysisanddesignofcontrolsystems.Thefrequencyresponsecanbefoundexperimentallyorfromatransferfunctionmodel.

WecanfindthefrequencyresponseofasystembyexcitingthesystemwithasinusoidalsignalofamplitudeAandfrequencyω[rad/s](Note: 𝜔 = 2𝜋𝑓)andobservingtheresponseintheoutputvariableofthesystem.

Thefrequencyresponseofasystemisdefinedasthesteady-stateresponseofthesystemtoasinusoidalinputsignal.Whenthesystemisinsteady-stateitdiffersfromtheinputsignalonlyinamplitude/gain(A)andphaselag(𝜙).

Ifwehavetheinputsignal:

𝑢 𝑡 = 𝑈𝑠𝑖𝑛𝜔𝑡

Thesteady-stateoutputsignalwillbe:

𝑦 𝑡 = 𝑈𝐴�sin(𝜔𝑡 + 𝜙)

Where 𝐴 = ��

is theratiobetweentheamplitudesoftheoutputsignalandtheinputsignal(in

steady-state).

Aand 𝜙 isafunctionofthefrequencyωsowemaywrite 𝐴 = 𝐴 𝜔 , 𝜙 = 𝜙(𝜔)

Foratransferfunction

𝐻 𝑆 =𝑦(𝑠)𝑢(𝑠)

Wehavethat:

𝐻 𝑗𝜔 = 𝐻(𝑗𝜔) 𝑒�∠�(��)

53 FrequencyResponse


Where 𝐻(𝑗𝜔) isthefrequencyresponseofthesystem,i.e.,wemayfindthefrequencyresponsebysetting 𝑠 = 𝑗𝜔 inthetransferfunction.Bodediagramsareusefulinfrequencyresponseanalysis.TheBodediagramconsistsof2diagrams,theBodemagnitudediagram, 𝐴(𝜔) andtheBodephasediagram, 𝜙(𝜔).

TheGainfunction:

𝐴 𝜔 = 𝐻(𝑗𝜔)

ThePhasefunction:

𝜙 𝜔 = ∠𝐻(𝑗𝜔)

The 𝐴(𝜔)-axisisindecibel(dB),wherethedecibelvalueofxiscalculatedas: 𝒙 𝒅𝑩 = 𝟐𝟎𝒍𝒐𝒈𝟏𝟎𝒙

The 𝜙(𝜔)-axisisindegrees(notradians!)

HereyouwilllearntoplotthefrequencyresponseinaBodediagram.

BelowweseeanexampleofaBodeplotcreatedinMathScript:

7.2 MathScriptMathScripthasseveralfunctionsforFrequencyresponses:

Function Description Examplebode CreatestheBodemagnitudeandBodephaseplotsofasystem

model.Youalsocanusethisfunctiontoreturnthemagnitudeandphasevaluesofamodelatfrequenciesyouspecify.Ifyou

>num=[4]; >den=[2, 1]; >H = tf(num, den) >bode(H)



donotspecifyanoutput,thisfunctioncreatesaplot.bodemag CreatestheBodemagnitudeplotofasystemmodel.Ifyoudo

notspecifyanoutput,thisfunctioncreatesaplot.>[mag, wout] = bodemag(SysIn) >[mag, wout] = bodemag(SysIn, [wmin wmax]) >[mag, wout] = bodemag(SysIn, wlist)

margin Calculatesand/orplotsthesmallestgainandphasemarginsofasingle-inputsingle-output(SISO)systemmodel.Thegainmarginindicateswherethefrequencyresponsecrossesat0decibels.Thephasemarginindicateswherethefrequencyresponsecrosses-180degrees.UsethemarginsfunctiontoreturnallgainandphasemarginsofaSISOmodel.

>num = [1] >den = [1, 5, 6] >H = tf(num, den) margin(H)

margins Calculatesallgainandphasemarginsofasingle-inputsingle-output(SISO)systemmodel.Thegainmarginsindicatewherethefrequencyresponsecrossesat0decibels.Thephasemarginsindicatewherethefrequencyresponsecrosses-180degrees.UsethemarginfunctiontoreturnonlythesmallestgainandphasemarginsofaSISOmodel.

>[gmf, gm, pmf, pm] = margins(H)

7.3 Examples

Example:

Wehavethefollowingtransferfunction

𝐻 𝑠 =𝑦(𝑠)𝑢(𝑠)

=1

𝑠 + 1

BelowweseethescriptforcreatingthefrequencyresponseofthesysteminabodeplotusingthebodefunctioninMathScript.Usethegridfunctiontoapplyagridtotheplot.

% Transfer function H=1/(s+1) num=[1]; den=[1, 1]; H = tf(num, den) bode (H);

[EndofExample]

Example:

Ithisexamplewewilluse3differentmethodstofind 𝐴 and 𝜙 foragivenfrequency 𝜔.


𝐻 =𝐾

𝑇𝑠 + 1

Set 𝐾 = 1, 𝑇 = 1

Theinputsignalisgivenby:

u t = Usinωt

Thesteady-stateoutputsignalwillthenbe:



y t = UA¢sin(ωt + ϕ)

Thegainisgivenby:

𝐴 =𝑌𝑈

Thephaselagisgivenby:

𝜙 = −𝜔Δ𝑡[𝑟𝑎𝑑/𝑠]

Method1:WecreateaMathScriptprogramwherewedefinethetransferfunctionanddefinetheinputsignalandplotit.

WewillthenusethelsimfunctionisMathScripttoplottheoutputsignalforagivenfrequency, 𝜔 =1,inthesameplot.Weset 𝑈 = 1 inthisexample.

Thefollowingcodewilldothis:

% Define Transfer function K = 1; T = 1; num = [K]; den = [T, 1]; H = tf(num, den); % Define input signal t = [1: 0.1 : 12]; w = 1; U = 1; u = U*sin(w*t); figure(1) plot(t, u) % Output signal hold on lsim(H, 'r', u, t) grid on hold off legend('input signal', 'output signal')

Thisgivesthefollowingplot



Fromtheplotabovewegetthefollowingvalues:

𝑌 = 0.68

Δ𝑡 = 0.8

WeusethefollowingScripttocalculate 𝐴©c and 𝜙©ª«¬ªªH:

% Values found from plot1 for w=1 Y = 0.68; A = Y/U; AdB = 20*log10(A) dt = 0.8; phi = -w*dt; %[rad] phi_degrees = phi*180/pi %[degrees]

Thisgives:

𝐴 = 0.68, 𝐴©c = −3.35[𝑑𝐵]

𝜙 = −0.8𝑟𝑎𝑑, 𝜙©ª« = −45.9𝑑𝑒𝑔𝑟𝑒𝑒𝑠

Method2:Nextwewillusethebodefunctiontoplotthefrequencyresponse/Bodeplottoseeifwegetthesameresults.

Thecodeforthisis:

% Define Transfer function K = 1; T = 1;



num = [K]; den = [T, 1]; H = tf(num, den); %Bode plot figure(2) bode(H) subplot(2,1,1) grid on subplot(2,1,2) grid on

Thisgives:

WeusetheBodeplottofind 𝐴©c and 𝜙©ª«¬ªªH for 𝜔 = 1

→Asyoucanseefromtheplotabovewegetthesameresults.

Method3:Herewewillusethebodefunctiontocalculatetheexactvaluesandcomparewiththeothermethods.

TheMathScriptbecomes:

% Define Transfer function K = 1; T = 1; num = [K]; den = [T, 1]; H = tf(num, den);



%Calculated magnitude and phase values for some given frequencies wlist = [0.001, 0.01, 0.1, 1, 3, 5, 10, 100]; [mag, phase, wout] = bode(H, wlist); magdB = 20*log10(mag) phase

Thisgives:

magdB = -4.3429e-006 -0.0004 -0.0432 -3.0103 -10 -14.1497 -20.0432 -40.0004 phase = -0.0573 -0.5729 -5.7106 -45 -71.5651 -78.6901 -84.2894 -89.4271

→wegetthesameresultsherealso(asexpected).

[EndofExample]

Example:

Wehavethefollowingtransferfunction:

𝐻 𝑆 =4

2𝑠 + 1

Breakfrequency:

𝜔 =1𝑇=12= 0.5

Themathematicalexpressionsfor 𝐴(𝜔) and 𝜙(𝜔):

𝐻(𝑗𝜔) ©c = 20𝑙𝑜𝑔4 − 20𝑙𝑜𝑔 (2𝜔)K + 1

∠𝐻(𝑗𝜔) = −arctan(2𝜔)



FrequencyresponseofthesysteminabodeplotusingthebodefunctioninMathScript:

MathScriptCode:

% Transfer function num=[4]; den=[2, 1]; H = tf(num, den) % Bode Plot bode(H) % Margins and Phases wlist=[0.1, 0.16, 0.25, 0.4, 0.625,2.5]; [mag, phase,w] = bode(H, wlist); magdB=20*log10(mag); %convert to dB mag_data = [w, magdB] phase_data = [w, phase]

Thisgives:

Fromthecodeaboveweget 𝐴(𝜔) and 𝜙(𝜔) forthefollowingfrequenciesusingMathScriptcode:



𝜔 𝐴(𝜔) 𝜙(𝜔)

0.1 11.9 -11.3

0.16 11.6 -17.7

0.25 11.1 -26.5

0.4 9.9 -38.7

0.625 7.8 -51.3

2.5 -2.1 -78.6

Wefind 𝐴(𝜔) and 𝜙(𝜔) forthesamefrequenciesaboveusingthemathematicalexpressionsfor𝐴(𝜔) and 𝜙(𝜔) andaForLoopinMathScript.Wedefineavectorw=[0.1,0.16,0.25,0.4,0.625,2.5].

→Weseetheresultsarethesameastheresultfoundusingthebodefunction.

[EndofExample]



7.4 StandardTransferfunctionsHerewewillfindthefrequencyresponseforthefollowingtransferfunctions:

• Amplifier• Integrator• Derivator• 1.ordersystem• 2.ordersystem• Zero-part• Timedelay

7.4.1 Amplifier(Norwegian:“Forsterker”):

ThetransferfunctionforanAmplifierisasfollows:

𝐻(𝑠) = 𝐾

Where

𝐾 isthegain

Themathematicalexpressionsfor 𝐴(𝜔) and 𝜙(𝜔) isasfollows:

Gain:

𝐴 𝜔 = 𝐻(𝑗𝜔) = 𝐾

orindB:

𝐻(𝑗𝜔) ©c = 20𝑙𝑜𝑔𝐾

Phase:

𝜙 𝜔 = ∠𝐻 𝑗𝜔 = 0

Example:

WeplottheBodeplotfortheAmplifierusingthebodefunctioninMathScript(K=1):



→Weseethatboth 𝐴(𝜔) and 𝜙(𝜔) areindependentofthefrequency𝜔.

[EndofExample]

7.4.2 Integrator

ThetransferfunctionforanIntegratorisasfollows:

𝐻 𝑠 =𝐾𝑠

Where

𝐾 isthegain


Gain:

𝐴 𝜔 = 𝐻(𝑗𝜔) =𝐾𝜔

orindB:

𝐻(𝑗𝜔) ©c = 20𝑙𝑜𝑔𝐾𝜔

Phase:

𝜙 𝜔 = ∠𝐻 𝑗𝜔 = −𝜋2𝑟𝑎𝑑 = −90°



Example:

WeplottheBodeplotfortheIntegratorusingthebodefunctioninMathScript:

[EndofExample]

7.4.3 Derivator

ThetransferfunctionforanDerivatorisasfollows:

𝐻 𝑠 = 𝐾𝑠

Where

𝐾 isthegain


Gain:

𝐴 𝜔 = 𝐻(𝑗𝜔) = 𝐾𝜔

orindB:

𝐻(𝑗𝜔) ©c = 20𝑙𝑜𝑔𝐾𝜔

Phase:

𝜙 𝜔 = ∠𝐻 𝑗𝜔 = +𝜋2𝑟𝑎𝑑 = +90°

Example:



WeplottheBodeplotfortheDerivatorusingthebodefunctioninMathScript:

[EndofExample]

7.4.4 1.Ordersystem


𝐻 𝑠 =𝐾

𝑇𝑠 + 1

Where

𝐾 isthegain

TistheTimeconstant


𝐴 𝜔 = 𝐻(𝑗𝜔) =𝐾

𝜔K𝑇K + 1

𝜙 𝜔 = ∠𝐻 𝑗𝜔 = −arctan(𝜔𝑇)

Example:

WeplottheBodeplotforthe1.ordersystemusingthebodefunctioninMathScript:



[EndofExample]

7.4.5 2.Ordersystem


𝐻 𝑠 =𝐾𝜔CK

𝑠K + 2𝜁𝜔C𝑠 + 𝜔CK=

𝐾𝑠𝜔C

K+ 2𝜁 𝑠

𝜔C+ 1

Where

𝐾 isthegain

𝜁 zetaistherelativedampingfactor

𝜔C[rad/s]istheundampedresonancefrequency.

Example:

WeplottheBodeplotforthe2.ordersystemusingthebodefunctioninMathScript:



[EndofExample]

7.4.6 Zeropart(Norwegian:“Nullpunktsledd”)

ThetransferfunctionforaZeropartsystemisasfollows:

𝐻 𝑠 = 𝐾(𝑇𝑠 + 1)

Where

𝐾 isthegain

TistheTimeconstant


Gain:

𝐴 𝜔 = 𝐻(𝑗𝜔) = 𝐾 𝜔𝑇 K + 1

orindB:

𝐻(𝑗𝜔) ©c = 20𝑙𝑜𝑔𝐾 𝜔𝑇 K + 1

Phase:

𝜙 𝜔 = ∠𝐻 𝑗𝜔 = +arctan(𝜔𝑇)

7.4.7 Timedelay(Norwegian:“Tidsforsinkelse”)

ThetransferfunctionforaTimeDelayisasfollows:



𝐻 𝑠 = 𝐾𝑒?GH

Where

𝐾 isthegain

𝜏 isthetime-delay


Gain:

𝐴 𝜔 = 𝐻(𝑗𝜔) = 𝐾

Phase:

𝜙 𝜔 = ∠𝐻 𝑗𝜔 = −𝜔𝜏𝑟𝑎𝑑 = −𝜔𝜏180𝜋

𝑑𝑒𝑔𝑟𝑒𝑒𝑠

Note!

2𝜋𝑟𝑎𝑑 = 360°

𝜋𝑟𝑎𝑑 = 180°

Example:

WeplottheBodeplotfortheTimedelayusingthebodefunctioninMathScript:

[EndofExample]

68

8 FrequencyresponseAnalysis

8.1 IntroductionHerearesomeimportanttransferfunctionstodeterminethestabilityofafeedbacksystem.Belowweseeatypicalfeedbacksystem.

TheLooptransferfunction 𝑳(𝒔) (Norwegian:“Sløyfetransferfunksjonen”)isdefinedasfollows:

𝐿 𝑠 = 𝐻{𝐻|𝐻=

Where

𝐻{ istheControllertransferfunction

𝐻| istheProcesstransferfunction

𝐻= istheMeasurement(sensor)transferfunction

Note!Anothernotationfor 𝐿 is 𝐻C

TheTrackingtransferfunction 𝑻(𝒔) (Norwegian:“Følgeforholdet”)isdefinedasfollows:


=𝐻{𝐻|𝐻=

1 + 𝐻{𝐻|𝐻==

𝐿(𝑠)1 + 𝐿(𝑠)

= 1 − 𝑆(𝑠)

TheTrackingProperty(Norwegian:“følgeegenskaper”)isgoodifthetrackingfunctionThasvalueequaltoorcloseto1:



𝑇 ≈ 1

TheSensitivitytransferfunction 𝑺(𝒔) (Norwegian:“Sensitivitetsfunksjonen/avviksforholdet”)isdefinedasfollows:

𝑆 𝑠 =𝑒(𝑠)𝑟(𝑠)

=1

1 + 𝐿(𝑠)= 1 − 𝑇(𝑠)

TheCompensationPropertyisgoodifthesensitivityfunctionShasasmallvalueclosetozero:

𝑆 ≈ 0𝑜𝑟 𝑆 ≪ 1

Note!

𝑇 𝑠 + 𝑆 𝑠 =𝐿(𝑠)

1 + 𝐿(𝑠)+

11 + 𝐿(𝑠)

≡ 1

FrequencyResponseAnalysisoftheTrackingProperty:

Fromtheequationsabovewefind:

TheTrackingProperty(Norwegian:“følgeegenskaper”)isgoodif:

𝐿(𝑗𝜔) ≫ 1

TheTrackingProperty(Norwegian:“følgeegenskaper”)ispoorif:

𝐿(𝑗𝜔) ≪ 1

Bandwidths 𝝎𝒕,𝝎𝒄, 𝝎𝒔 (seethesketchbelow)



𝝎𝒄 –crossover-frequency–thefrequencywherethegainoftheLooptransferfunction 𝐿(𝑗𝜔) hasthevalue:

1 = 0𝑑𝐵

𝝎𝒕 –thefrequencywherethegainoftheTrackingfunction 𝑇(𝑗𝜔) hasthevalue:

12≈ 0.71 = −3𝑑𝐵

𝝎𝒔 -thefrequencywherethegainoftheSensitivitytransferfunction 𝑆(𝑗𝜔) hasthevalue:

1 −12≈ 0.29 = −11𝑑𝐵

8.2 MathScriptMathScripthasseveralfunctionsforfrequencyresponseanalysis:

Function Description Exampletf Createssystemmodelintransferfunctionform.Youalsocan

usethisfunctiontostate-spacemodelstotransferfunctionform.


poles Returnsthelocationsoftheclosed-looppolesofasystemmodel.

>num=[1] >den=[1,1] >H=tf(num,den) >poles(H)

tfinfo Returnsinformationaboutatransferfunctionsystemmodel. >[num, den, delay, Ts] = tfinfo(SysInTF)

series ConnectstwosystemmodelsinseriestoproduceamodelSysSerwithinputandoutputconnectionsyouspecify






bode CreatestheBodemagnitudeandBodephaseplotsofasystemmodel.Youalsocanusethisfunctiontoreturnthemagnitudeandphasevaluesofamodelatfrequenciesyouspecify.Ifyoudonotspecifyanoutput,thisfunctioncreatesaplot.


bodemag CreatestheBodemagnitudeplotofasystemmodel.Ifyoudonotspecifyanoutput,thisfunctioncreatesaplot.

>[mag, wout] = bodemag(SysIn) >[mag, wout] = bodemag(SysIn, [wmin wmax]) >[mag, wout] = bodemag(SysIn, wlist)





Example:


Processtransferfunction:

𝐻| =𝐾𝑠𝑒?GH

Where 𝐾 = µ́¶b,where 𝐾H = 0,556, 𝐴 = 13,4, 𝜚 = 145 and 𝜏 = 250

Measurement(sensor)transferfunction:

𝐻= = 𝐾=

WhereKm=6,67%/m.

Controllertransferfunction(PIController):

𝐻{ = 𝐾| +𝐾|𝑇 𝑠

SetKp=1,5ogTi=1000sec.

WeshallfindtheLooptransferfunction (𝒔) ,Sensitivitytransferfunction 𝑺(𝒔),Trackingtransferfunction 𝑻(𝒔) usingtheseriesandfeedbackfunctionsinMathScript.

MathScriptCode:

%Calculating control system transfer functions: L=series(Hc,series(Hp,Hs)); %Calculating loop tranfer function T=feedback(L,1); %Calculating tracking transfer function S=1-T; %Calculating sensitivity transfer function



WeplottheBodeplotforL,TandSandfindtheBandwidths 𝜔], 𝜔{, 𝜔H:

MathScriptCode:

bodemag(L,T,S), grid %Plots maginitude of L, T, and S in Bode diagram

Bodeplot(Magnitudeonly)ofL,TandS:

Wefindthestabilitymargins(GM,PM)ofthesystem(𝐿(𝑠)):

margin(L), grid %Plotting L and stability margins and crossover frequencies in Bode diagram

Bodeplotwiththestabilitymargins(GM,PM)markedontheplot:



[EndofExample]

74

9 StabilityAnalysisintheFrequencyDomain

9.1 IntroductionGainMargin(GM)andPhaseMargin(PM)areimportantdesigncriteriaforanalysisoffeedbackcontrolsystems.

Adynamicsystemhasoneofthefollowingstabilityproperties:

• Asymptoticallystablesystem• Marginallystablesystem• Unstablesystem

TheGainMargin–GM(Δ𝐾)ishowmuchtheloopgaincanincreasebeforethesystembecomeunstable.

ThePhaseMargin-PM(𝜑)ishowmuchthephaselagfunctionoftheloopcanbereducedbeforetheloopbecomesunstable.

Where:



• 𝝎𝟏𝟖𝟎(gainmarginfrequency-gmf)isthegainmarginfrequency/frequencies,inradians/second.Againmarginfrequencyindicateswherethemodelphasecrosses-180degrees.

• GM(Δ𝐾)isthegainmargin(s)ofthesystem. • 𝝎𝒄 (phasemarginfrequency-pmf)returnsthephasemarginfrequency/frequencies,in

radians/second.Aphasemarginfrequencyindicateswherethemodelmagnitudecrosses0decibels.

• PM(𝜑)isthephasemargin(s)ofthesystem.

Note! 𝝎𝟏𝟖𝟎 and 𝝎𝒄arecalledthecrossover-frequencies

Thedefinitionsareasfollows:

GainCrossover-frequency- 𝝎𝒄:

𝐿 𝑗𝜔{ = 1 = 0𝑑𝐵

PhaseCrossover-frequency- 𝝎𝟏𝟖𝟎:

∠𝐿 𝑗𝜔@»C = −180°

GainMargin-GM(𝚫𝑲):

𝐺𝑀 = @� ��ÀÁÂ

or:

𝐺𝑀 𝑑𝐵 = − 𝐿 𝑗𝜔@»C 𝑑𝐵

PhasemarginPM(𝝋):

𝑃𝑀 = 180° + ∠𝐿(𝑗𝜔{)

Wehavethat:

• Asymptoticallystablesystem: 𝝎𝒄 < 𝝎𝟏𝟖𝟎• Marginallystablesystem: 𝝎𝒄 = 𝝎𝟏𝟖𝟎• Unstablesystem: 𝝎𝒄 > 𝝎𝟏𝟖𝟎

9.2 MathScriptMathScripthasseveralfunctionsforstabilityanalysis:

Function Description Examplebode CreatestheBodemagnitudeandBodephaseplotsofasystem

model.Youalsocanusethisfunctiontoreturnthemagnitudeandphasevaluesofamodelatfrequenciesyouspecify.Ifyoudonotspecifyanoutput,thisfunctioncreatesaplot.


bodemag CreatestheBodemagnitudeplotofasystemmodel.Ifyoudo >[mag, wout] = bodemag(SysIn)



notspecifyanoutput,thisfunctioncreatesaplot. >[mag, wout] = bodemag(SysIn, [wmin wmax]) >[mag, wout] = bodemag(SysIn, wlist)





Example:


𝐻 𝑆 =1

𝑠 𝑠 + 1 K

Wewillfindthecrossover-frequenciesforthesystemusingMathScript.Wewillalsofindalsothegainmarginsandphasemarginsforthesystem.

Weget:



BelowweseetheBodediagramwiththecrossover-frequencyandthegainmarginandphasemarginforthesystemplottedin:

[EndofExample]

Example:


𝐻 𝑠 =𝑠 + 1

𝑠K − 𝑠 + 3

→ThesystemisunstableandFrequencyResponsegivesmeaningonlyforstablesystems.

Note!Thefrequencyresponseofasystemisdefinedasthesteady-stateresponseofthesystemtoasinusoidalinputsignal.

TheBodediagramforunstablesystemsdon’tshowwhathappenswiththesinusoidalsignalofagivenfrequencywhenthesysteminputistransferredthroughthesystembecauseitneverreachsteadystate.

Weseethatthesystemisunstablebecausesomeofthecoefficientsinthedenominatorpolynomial𝑠K − 𝑠 + 3 arenegative.



Weconfirmthisbysomesimulationsandfindingthepolesforthesystem:

poles(H) pzgraph(H)

Thisgives:

→Weseethepolesarecomplexconjugateandthattheyliesintherighthalf-plane.

Weplotthestepresponseforthetransferfunctionusingthestepfunction:

num=[1,1]; den=[1,-1,3]; H=tf(num,den); t=[0:0.01:10]; step(H,t);




→Weseethesystemisunstable

[EndofExample]

80

AppendixA–MathScriptFunctions

BasicFunctionsHerearesomedescriptionsforthemostusedbasicMathScriptfunctions.

Function Description Examplehelp MathScriptdisplaysthehelpinformationavailable >>help

help<function>

Displayhelpaboutaspecificfunction >>help plot

who,whos wholistsinalphabeticalorderallvariablesinthecurrentlyactiveworkspace.

>>who >>whos

clear Clearvariablesandfunctionsfrommemory. >>clear >>clear x

size Sizeofarrays,matrices >>x=[1 2 ; 3 4]; >>size(A)

length Lengthofavector >>x=[1:1:10]; >>length(x)

format Setoutputformat

disp Displaytextorarray >>A=[1 2;3 4]; >>disp(A)

plot Thisfunctionisusedtocreateaplot >>x=[1:1:10]; >>plot(x) >>y=sin(x); >>plot(x,y)

clc CleartheCommandwindow >>cls

rand Createsarandomnumber,vectorormatrix >>rand >>rand(2,1)

max Findthelargestnumberinavector >>x=[1:1:10] >>max(x)

min Findthesmallestnumberinavector >>x=[1:1:10] >>min(x)

mean Averageormeanvalue >>x=[1:1:10] >>mean(x)

std Standarddeviation >>x=[1:1:10] >>std(x)

BasicPlottingFunctionsFunction Description Exampleplot Generatesaplot.plot(y)plotsthecolumnsofyagainstthe


figure Createanewfigurewindow >>figure >>figure(1)

subplot CreatesubplotsinaFigure.subplot(m,n,p)orsubplot(mnp),breakstheFigurewindowintoanm-by-nmatrixofsmallaxes,selectsthep-thaxesforthecurrentplot.TheaxesarecountedalongthetoprowoftheFigurewindow,thenthesecondrow,etc.

>>subplot(2,2,1)

81 AppendixA–MathScriptFunctions


grid Createsgridlinesinaplot.“gridon”addsmajorgridlinestothecurrentplot.“gridoff”removesmajorandminorgridlinesfromthecurrentplot.

>>grid >>grid on >>grid off

axis Controlaxisscalingandappearance.“axis([xminxmaxyminymax])”setsthelimitsforthex-andy-axisofthecurrentaxes.

>>axis([xmin xmax ymin ymax]) >>axis off >>axis on

title Addtitletocurrentplottitle('string')

>>title('this is a title')

xlabel Addxlabeltocurrentplotxlabel('string')

>> xlabel('time')

ylabel Addylabeltocurrentplotylabel('string')

>> ylabel('temperature')

legend Createsalegendinthecorner(orataspecifiedposition)oftheplot

>> legend('temperature')

hold Freezesthecurrentplot,sothatadditionalplotscanbeoverlaid >>hold on >>hold off

Formoreinformationabouttheplotsfunction,type“helpplots”.

FunctionsusedforControlandSimulationFunction Description Example

plot Generatesaplot.plot(y)plotsthecolumnsofyagainsttheindexesofthecolumns.

>X = [0:0.01:1]; >Y = X.*X; >plot(X, Y)

tf Createssystemmodelintransferfunctionform.Youalsocanusethisfunctiontostate-spacemodelstotransferfunctionform.


poles Returnsthelocationsoftheclosed-looppolesofasystemmodel.

>num=[1] >den=[1,1] >H=tf(num,den) >poles(H)

tfinfo Returnsinformationaboutatransferfunctionsystemmodel. >[num, den, delay, Ts] = tfinfo(SysInTF)

step Createsastepresponseplotofthesystemmodel.Youalsocanusethisfunctiontoreturnthestepresponseofthemodeloutputs.Ifthemodelisinstate-spaceform,youalsocanusethisfunctiontoreturnthestepresponseofthemodelstates.Thisfunctionassumestheinitialmodelstatesarezero.Ifyoudonotspecifyanoutput,thisfunctioncreatesaplot.

>num=[1,1]; >den=[1,-1,3]; >H=tf(num,den); >t=[0:0.01:10]; >step(H,t);

lsim Createsthelinearsimulationplotofasystemmodel.Thisfunctioncalculatestheoutputofasystemmodelwhenasetofinputsexcitethemodel,usingdiscretesimulation.Ifyoudonotspecifyanoutput,thisfunctioncreatesaplot.

>t = [0:0.1:10] >u = sin(0.1*pi*t)' >lsim(SysIn, u, t)


>K = 1; >tau = 1; >H = sys_order1(K, tau)


>dr = 0.5 >wn = 20 >[num, den] = sys_order2(wn, dr) >SysTF = tf(num, den) >[A, B, C, D] = sys_order2(wn, dr) >SysSS = ss(A, B, C, D)

damp Returnsthedampingratiosandnaturalfrequenciesofthepolesofasystemmodel.

>[dr, wn, p] = damp(SysIn)

pid Constructsaproportional-integral-derivative(PID)controllermodelineitherparallel,series,oracademicform.RefertotheLabVIEWControlDesignUserManualforinformationaboutthesethreeforms.

>Kc = 0.5; >Ti = 0.25; >SysOutTF = pid(Kc, Ti, 'academic');

conv Computestheconvolutionoftwovectorsormatrices. >C1 = [1, 2, 3]; >C2 = [3, 4]; >C = conv(C1, C2)

82 AppendixA–MathScriptFunctions


series ConnectstwosystemmodelsinseriestoproduceamodelSysSerwithinputandoutputconnectionsyouspecify




ss Constructsamodelinstate-spaceform.Youalsocanusethisfunctiontoconverttransferfunctionmodelstostate-spaceform.

>A = eye(2) >B = [0; 1] >C = B' >SysOutSS = ss(A, B, C)

ssinfo Returnsinformationaboutastate-spacesystemmodel. >A = [1, 1; -1, 2] >B = [1, 2]' >C = [2, 1] >D = 0 >SysInSS = ss(A, B, C, D) >[A, B, C, D, Ts] = ssinfo(SysInSS)

pade IncorporatestimedelaysintoasystemmodelusingthePadeapproximationmethod,whichconvertsallresiduals.Youmustspecifythedelayusingthesetfunction.Youalsocanusethisfunctiontocalculatecoefficientsofnumeratoranddenominatorpolynomialfunctionswithaspecifieddelay.

>[num, den] = pade(delay, order) >[A, B, C, D] = pade(delay, order)

bode CreatestheBodemagnitudeandBodephaseplotsofasystemmodel.Youalsocanusethisfunctiontoreturnthemagnitudeandphasevaluesofamodelatfrequenciesyouspecify.Ifyoudonotspecifyanoutput,thisfunctioncreatesaplot.


bodemag CreatestheBodemagnitudeplotofasystemmodel.Ifyoudonotspecifyanoutput,thisfunctioncreatesaplot.

>[mag, wout] = bodemag(SysIn) >[mag, wout] = bodemag(SysIn, [wmin wmax]) >[mag, wout] = bodemag(SysIn, wlist)





Formoredetailsaboutthesefunctions,type“helpcdt”togetanoverviewofallthefunctionsusedforControlDesignandSimulation.Fordetailedhelpaboutonespecificfunction,type“help<function_name>”.

Hans-PetterHalvorsen,M.Sc.

E-mail:[email protected]

Blog:http://home.hit.no/~hansha/

UniversityCollegeofSoutheastNorway

www.usn.no

control theory with mathscript exampleshome.hit.no/~hansha/documents/control/documents/lecture...

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