Kaotiske systemer - analyse, simulering og eksperimenter på effekten av å legge til en integral regulator

Abstract

Model chaotic systems into Matlab/Simulink, or other programs that allow simulations. The systems should then be simulated for different disturbances to observer both the step response and how the amplitude, average value of amplitude and frequency, for one or more output variables, changes for different disturbance values. The results can be presented both as a time response or in bifurcation diagrams. Then the systems should be expanded to allow one or more I-controller and perform the same analysis again, to see the effect the I-controlled has on the system. Then if given the time create one of the systems as an electrical circuit. The models are created in Matlab using the ode45 solver on a Matlab script, made from the differential equations, for each of the chaotic models used. To create the Simulink model we simply started implementing the differential equations into Simulink. The next part is the expanded Simulink model with an I-controller, which was made possible by adding the integral response into the differential equation for x. Then adding an closed-feedback loop from x to calculate the error. To perform the simulation of the expanded chaotic systems we needed to find the integral gain, Ki, which was done by finding the transfer function from u, integral response, to x, the output. After finding the transfer function we made some assumptions, and then by comparing the closed-loop function, M(s), with a desired closed-loop function, Md(s), we can calculate the integral gain, Ki. From the simulations done on simple chaotic systems we can say that there is not a big difference to the results we get from a simulation of the Matlab script and on a model made in Simulink. Taking a look at the results from the calculations for Ki and the simulations using this value gave us some satisfying results for most of the chaotic systems.

Model chaotic systems into Matlab/Simulink, or other programs that allow simulations. The systems should then be simulated for different disturbances to observer both the step response and how the amplitude, average value of amplitude and frequency, for one or more output variables, changes for different disturbance values. The results can be presented both as a time response or in bifurcation diagrams. Then the systems should be expanded to allow one or more I-controller and perform the same analysis again, to see the effect the I-controlled has on the system. Then if given the time create one of the systems as an electrical circuit. The models are created in Matlab using the ode45 solver on a Matlab script, made from the differential equations, for each of the chaotic models used. To create the Simulink model we simply started implementing the differential equations into Simulink. The next part is the expanded Simulink model with an I-controller, which was made possible by adding the integral response into the differential equation for x. Then adding an closed-feedback loop from x to calculate the error. To perform the simulation of the expanded chaotic systems we needed to find the integral gain, Ki, which was done by finding the transfer function from u, integral response, to x, the output. After finding the transfer function we made some assumptions, and then by comparing the closed-loop function, M(s), with a desired closed-loop function, Md(s), we can calculate the integral gain, Ki. From the simulations done on simple chaotic systems we can say that there is not a big difference to the results we get from a simulation of the Matlab script and on a model made in Simulink. Taking a look at the results from the calculations for Ki and the simulations using this value gave us some satisfying results for most of the chaotic systems.