MATLABExamples

Hans-PetterHalvorsen

DiscreteSystems

DiscreteSystems• MATLABhasbuilt-inpowerfulfeaturesforsimulationofcontinuousdifferentialequationsanddynamicsystems.

• SometimeswewanttoorneedtodiscretizeacontinuoussystemandthensimulateitinMATLAB.

• Thismeansweneedtomakeadiscreteversionofourcontinuousdifferentialequations.• Actually,thebuilt-inODEsolversinMATLABusedifferentdiscretizationmethods

DiscretizationMethods

• Euler;–Eulerforwardmethod,–Eulerbackwardmethod

• ZeroOrderHold(ZOH)• Tustin• ...

DiscreteApproximationofthetimederivativeEulerbackwardmethod:

Eulerforwardmethod:

DiscreteSystems

DiscretizationMethods

Eulerforwardmethod:

Eulerbackwardmethod:

OthermethodsareZeroOrderHold(ZOH),Tustin’smethod,etc.

Simplertouse!

MoreAccurate!

DiscreteSystems

DifferentDiscreteSymbolsandmeanings

Present Value:

Previous Value:

Next (Future)Value:

Note!DifferentNotationisusedindifferentlitterature!

DiscreteSystems

DiscreteSimulationGiventhefollowingdifferentialequation:

�̇� = 𝑎𝑥

where𝑎 = − &',where𝑇 isthetimeconstant

Note!�̇� = )*)+

FindthediscretedifferentialequationandplotthesolutionforthissystemusingMATLAB.Set𝑇 = 5andtheinitialcondition𝑥(0) = 1.CreateascriptinMATLAB(.mfile)whereweplotthesolution𝑥 𝑡 inthetimeinterval0 ≤ 𝑡 ≤ 30

DiscreteSimulationWecanusee.g.,theEulerApproximation:

�̇� ≈𝑥67& − 𝑥6

𝑇8Thenweget:

𝑥67& − 𝑥6𝑇8

= 𝑎𝑥6

Whichgives:𝑥67& = (1 + 𝑎𝑇8)𝑥6

clear, clc

% Model ParametersT = 5;a = -1/T;

% Simulation ParametersTs = 0.1; %sTstop = 30; %sx(1) = 1;

% Simulationfor k=1:(Tstop/Ts)

x(k+1) = (1+a*Ts)*x(k);end

% Plot the Simulation Resultst=0:Ts:Tstop;plot(t,x)grid on

BacteriaSimulationInthistaskwewillsimulateasimplemodelofabacteriapopulationinajar.Themodelisasfollows:

birthrate=𝑏𝑥deathrate=𝑝𝑥2

Thenthetotalrateofchangeofbacteriapopulationis:�̇� = 𝑏𝑥 − 𝑝𝑥=

Setb=1/hourandp=0.5 bacteria-hourWewillsimulatethenumberofbacteriainthejarafter1hour,assumingthatinitiallythereare100bacteriapresent.→FindthediscretemodelusingtheEulerForwardmethodbyhandandimplementandsimulatethesysteminMATLABusingaForLoop.

BacteriaSimulationWecreateadiscretemodel.anduseEulerForwarddifferentiationmethod:

�̇� ≈𝑥67& − 𝑥6

𝑇8Where𝑇8 istheSamplingTime.Weget:

𝑥67& − 𝑥6𝑇8

= 𝑏𝑥6 − 𝑝𝑥6=

Thisgives:

𝑥67& = 𝑥6 + 𝑇8(𝑏𝑥6 − 𝑝𝑥6=)

clear, clc% Model Parametersb = 1;p = 0.5;

% Simulation ParametersTs=0.01;Tstop = 1; %sx(1)=100;

% Simulation Loopfor k=1:(Tstop/Ts)

% Discrete modelx(k+1) = x(k) + Ts*(b*x(k) - p*x(k)^2);

end

% Plot the simulation resultst=0:Ts:Tstop;plot(t,x)grid on

Simulationwith2variables

Giventhefollowingsystem:

𝑑𝑥&𝑑𝑡 = −𝑥=

𝑑𝑥=𝑑𝑡 = 𝑥&

FindthediscretesystemandsimulatethediscretesysteminMATLAB.

Simulationwith2variablesUsingEuler:

�̇� ≈𝑥67& − 𝑥6

𝑇8Thenweget:

𝑥& 𝑘 + 1 = 𝑥& 𝑘 − 𝑇8𝑥= 𝑘

𝑥= 𝑘 + 1 = 𝑥= 𝑘 + 𝑇8𝑥& 𝑘

WhichwecanimplementinMATLAB.Theequationswillbesolvedinthetimespan[−11] withinitialvalues[1, 1].

clear, clc

Ts = 0.1;

Tstart = -1;Tstop = 1;

x1(1) = 1;x2(1) = 1;

for k=1:((Tstop-Tstart)/Ts)x1(k+1) = x1(k) - Ts.*x2(k);x2(k+1) = x2(k) + Ts.*x1(k);

end

t = Tstart : Ts : Tstop;plot(t,x1,t,x2)grid on

Hans-PetterHalvorsen,M.Sc.

UniversityCollegeofSoutheastNorwaywww.usn.no

E-mail:hans.p.halvorsen@hit.noBlog:http://home.hit.no/~hansha/