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DiscreteSystems
Hans-PetterHalvorsen
ControlEngineering
Transfer-funksjoner
Differensial-likninger
1.orden
2.orden
Laplace
Blokk-diagrammer
Tilstandsrom-modeller
Stabilitets-analyse
MathScript
LabVIEW
Dataverktøy
1.Systemetspoler Bodediagram
Analyse/Design
Realisering/Implementering
2.Frekvensrespons
DiskretiseringReguleringssystem
Detkomplekseplan
Detkomplekseplan
AsymptotiskstabiltsystemMarginaltstabiltsystem
Ustabiltsystem
Asymptotiskstabiltsystem
Marginaltstabiltsystem
Ustabiltsystem
TidsplanetAirHeater
1.ordenmedtidsforsinkelse
Tidsplanet S-planet
Reguleringsteknikk
K=ForsterkningT=Tidskonstant
Sprang-respons
Serie,Parallel,Feedback
Diskretisering
TableofContents1. WhatisDiscreteSignalsandDiscretization?2. WhyDiscretization?3. DiscretizationMethods
– Euler,Tustin,ZOH,etc.
4. DiscretizationExamples– DifferentialEquations->DifferenceEquations– Continous State-spacemodels->DiscreteState-spacemodels
5. DiscretizationinMathScript withExamples6. DiscretizationinLabVIEW withExamples7. DiscreteLowpass Filter
– DiscreteAlgorithm– LabVIEWImplementation(Example)
8. DiscretePI Controller– DiscreteAlgorith– LabVIEWImplementation(Example)
9. PC-basedControlSystem– LabVIEWExample
DigitalSignals&Values
1 0 1 00
E.g.8bit:00000000 ->0………11111111 ->255
Resolution:
E.g.8bitresolution
Resolution:5V/256=0.0195V
8bitshave256differentcombinationsorlevels
BinaryValues:“0”and“1”
ADConverter
DAConverter
MeasurementSignal
ControlSignal
AD&DAConverters
AD – AnalogtoDigitalDA – DigitaltoAnalog
ContinuousSignal
DiscretSignal Acomputercanonlydealwithdiscretesignals
DifferentDiscreteSymbolsandmeanings
Present Value:
Previous Value:
Next (Future)Value:
Note!DifferentNotationisusedindifferentlitterature!
SamplingandAliasingOriginalSignal
Aliasing(“Nedfolding”)->TheSamplingRateistolow!
SamplingTimeSamplingFrequency
DiscreteApproximationofthetimederivativeEulerbackwardmethod:
Eulerforwardmethod:
DiscretizationMethods
Eulerforwardmethod:
Eulerbackwardmethod:
OthermethodsareZeroOrderHold(ZOH),Tustin’smethod,etc.
Simplertouse!
MoreAccurate!
DiscretizationExampleGiventhefollowingcontinuoussystem:
WewillusetheEulerforwardmethod:
Students:Penandpaper:Findthedifferenceequation(discretedifferenitialequation)forthissystem.
DiscretizationExample- SolutionsGiventhefollowingcontinuoussystem:
WewillusetheEulerforwardmethod:
DiscretizationinMathScript- Alt.1
% Simulation of discrete modelclear, clc
% Model Parametersa = 0.25;b = 2;
% Simulation ParametersTs = 0.1; %sTstop = 20; %suk = 1; % Step in ux(1) = 0; % Initial value
% Simulationfor k=1:(Tstop/Ts)
x(k+1) = (1-a*Ts).*x(k) + Ts*b*uk;end
% Plot the Simulation Resultsk=0:Ts:Tstop;plot (k, x)grid on
Students:ImplementandSimulatethissysteminMathScript
DiscretizationinMathScript- Alt.2
% Find Discrete modelclearclc
% Model Parametersa = 0.25;b = 2;Ts = 0.1; %s
% State-space modelA = [-a];B = [b];C = [1];D = [0];
model = ss(A,B,C,D)model_discrete = c2d(model, Ts, 'forward')step(model_discrete)
Students:ImplementandSimulatethissysteminMathScript
Wegetthesameanswerasinpreviousexample!
DiscretizationinLabVIEW usingtheFormulaNode
Example (bacteriapopulation):
ImplementingdicreteequationsinLabVIEWusingCsyntax
Students:ImplementandSimulatethissysteminLabVIEWusingaFormulaNodeandaForLoop.ASubVIcouldbeusedtoimplementthemodel.
Wewillsimulatethenumberofbacteriainthejarafter1hour(simulationperiod),assumingthatinitiallythereare100bacteriapresent(x(0)).Setb=1,p=0.5.SetTs=0.01s.
SubVI
DiscretizationinLabVIEWusingtheFormulaNode- Solutions
Note!ASubVIisusedtoimplementthemodel.Recommended!!
SimulationResults:SubVI:
Note!Right-ClickandselectProperties.SetMultiplier=0.01(=Ts)intheScalesTab.
DiscreteState-spaceModels
Orusinganothernotation:
DiscreteState-spaceModels- Example
Giventhefollowingcontinuoussystem:
WewillusetheEulerforwardmethod:
Students:Students:Findthediscretestate-spacemodel(usingpenandpaper)
DiscreteState-spaceModels- Example(Solutions)
Giventhefollowingcontinuoussystem:
WewillusetheEulerforwardmethod:
DiscreteState-spaceModelsExampleinMathScript
Giventhefollowingcontinuoussystem:
% PropertiesTs = 0.1;
% Continuous System:A = [0, -Ts; 0, 0];B = [2*Ts, 0]';C = [1 0];D = [0];ssmodel = ss(A, B, C, D);figure(1)step(ssmodel)
% Discrete System:ssmodel_discete = c2d(ssmodel, Ts, 'forward’)figure(2)step(ssmodel_discete)
MathScript
Students:ImplementthesystemaboveandfindthestepresponseusingMathScript.
TrywithdifferentvaluesforTs!
ImplementingaControlSystemWhiletherealprocessiscontinuous,normallytheControllerandtheFilterisimplementedinacomputer.
WearegoingtofinddiscretealgorithmsfortheLowpassFilterandthePIControllerandimplementtheminLabVIEW
PC-basedControlSystem–LabVIEWExample(HMI/GUI)
LowpassFilterLowpassFilterTransferfunction:
Students:PenandPaper:Findthediscretealgorithmforthisfilter,whichcanbeimplmentedinaprogramminglanguagelikeCorLabVIEWFormulaNode
UsetheEulerBackwardmethod:
DiscreteLowpassFilter
WeusetheEulerBackwardmethod:
InverseLaplacegivesthedifferentialEquation:
Thisgives:
LowpassFilterTransferfunction:
Wedefine:
Thisgives:
ThisalgorithmcanbeeaslyimplementedinaProgramminglanguage
Filteroutput Noisyinputsignal
DiscreteLowpassFilter- LabVIEW
Students:ImplementaDiscreteLowpassFilterandmakesuretotestitproperly
ExampleofImplementation
TestingtheDiscreteLowpassFilter
PIControllerContinuousPIController:
Students:PenandPaper:FindthediscretealgorithmforthisPIcontroller,whichcanbeimplmentedinaprogramminglanguagelikeCorLabVIEWFormulaNode
UsetheEulerBackwardmethod:
DiscretePIControllerContinuousPIController:
WeusetheEulerBackwardmethod:
Wemayset:
ThisgivesthefollowingdiscretePIalgorithm:
ThisalgorithmcanbeeaslyimplementedinaProgramminglanguage
DiscretePIController- LabVIEW
Students:ImplementaDiscretePIControllerandmakesuretotestitproperly
ExampleofImplementation
MathematicalModeloftheProcess
FeedbackNode
TestofDiscretePIControllerinLabVIEWNote!MakesuretocompareyourPIControllerwiththebuilt-inPIDController!
PC-basedControlSystem–LabVIEWExample(Realprocess)
SubVIthathandlestheI/ObetweenLabVIEWandtheRealProcess
SubVI:
Scaling:1-5V->20-50degreesCelsius
Ts
Tf
PC-basedControlSystem–LabVIEWExample(HMI/GUI)
Hans-PetterHalvorsen,M.Sc.
UniversityCollegeofSoutheastNorwaywww.usn.no
E-mail:[email protected]:http://home.hit.no/~hansha/