Kaotiske systemer - analyse, simulering og eksperimenter på effekten av å legge til en integral regulator
Master thesis
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https://hdl.handle.net/11250/3032547Utgivelsesdato
2022Metadata
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- Studentoppgaver (TN-IDE) [877]
Sammendrag
Model chaotic systems into Matlab/Simulink, or other programs that allowsimulations. The systems should then be simulated for different disturbancesto observer both the step response and how the amplitude, average valueof amplitude and frequency, for one or more output variables, changes fordifferent disturbance values. The results can be presented both as a timeresponse or in bifurcation diagrams. Then the systems should be expandedto allow one or more I-controller and perform the same analysis again, to seethe effect the I-controlled has on the system. Then if given the time createone of the systems as an electrical circuit.The models are created in Matlab using the ode45 solver on a Matlabscript, made from the differential equations, for each of the chaotic modelsused. To create the Simulink model we simply started implementing the differentialequations into Simulink. The next part is the expanded Simulinkmodel with an I-controller, which was made possible by adding the integralresponse into the differential equation for x. Then adding an closed-feedbackloop from x to calculate the error. To perform the simulation of the expandedchaotic systems we needed to find the integral gain, Ki, which wasdone by finding the transfer function from u, integral response, to x, theoutput. After finding the transfer function we made some assumptions, andthen by comparing the closed-loop function, M(s), with a desired closed-loopfunction, Md(s), we can calculate the integral gain, Ki.From the simulations done on simple chaotic systems we can say thatthere is not a big difference to the results we get from a simulation of theMatlab script and on a model made in Simulink. Taking a look at the resultsfrom the calculations for Ki and the simulations using this value gave ussome satisfying results for most of the chaotic systems. Model chaotic systems into Matlab/Simulink, or other programs that allowsimulations. The systems should then be simulated for different disturbancesto observer both the step response and how the amplitude, average valueof amplitude and frequency, for one or more output variables, changes fordifferent disturbance values. The results can be presented both as a timeresponse or in bifurcation diagrams. Then the systems should be expandedto allow one or more I-controller and perform the same analysis again, to seethe effect the I-controlled has on the system. Then if given the time createone of the systems as an electrical circuit.The models are created in Matlab using the ode45 solver on a Matlabscript, made from the differential equations, for each of the chaotic modelsused. To create the Simulink model we simply started implementing the differentialequations into Simulink. The next part is the expanded Simulinkmodel with an I-controller, which was made possible by adding the integralresponse into the differential equation for x. Then adding an closed-feedbackloop from x to calculate the error. To perform the simulation of the expandedchaotic systems we needed to find the integral gain, Ki, which wasdone by finding the transfer function from u, integral response, to x, theoutput. After finding the transfer function we made some assumptions, andthen by comparing the closed-loop function, M(s), with a desired closed-loopfunction, Md(s), we can calculate the integral gain, Ki.From the simulations done on simple chaotic systems we can say thatthere is not a big difference to the results we get from a simulation of theMatlab script and on a model made in Simulink. Taking a look at the resultsfrom the calculations for Ki and the simulations using this value gave ussome satisfying results for most of the chaotic systems.