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• W632454714 abstract "The ODE- and DDE models are based on the same model with the only exceptionthat the DDE model features reaction times. They are defined by considerationforces and sub-consideration forces. The values(h; k) determines the number ofcars that each driver considers and therefore adds to the system as additionalterms that are of the same form as the consideration forces, hence the sub prefix.The basic case where there are no sub-considerations involved is called the basecase of the system and equals to (h; k) = (1; 1). The (h; k) of the system is determiningthe matrix B in equation (11) by the number of sub diagonals h and super diagonalsk that are filled by weights of the forces.The time integrations can result in three base cases, unstable, stable oscillatingand exponentially stable. These cases refer to the behavior of all system velocities.The unstable case can for limited time frames predict collisions between cars butotherwise diverge and cannot generally be used. Oscillating stable systems reacha constant velocity after a settling time and fits well into a realistic scenario. Theexponential case reaches a constant velocity the fastest and is therefore the soughtafter solution. Both models are similar in this regard apart from the fact that theDDE model generally have a lot more system energy. Figures 1 and 4 are empiricalproof that the models works as defined and can predict some traffic behavior.An interesting observation during testing was that the ODE exponential case wouldalways remain exponential no matter the multiplication(; ; ) = C(; ; ), theonly difference would be the system energy since larger acting forces are coupledwith larger energies. The DDE model however is dependent on the system energyfor stability since the delay sets a system energy limit for stability since too largeforces coupled with delay will not achieve the optimum distance d.The system stability analysis can be reduced in both models to analyzing the homogeneousand particular parts separate. The expansions confirms in both caseswhat the time integrations shows and can give an idea of how the stability changeswith one parameter changing. However, this is where the DDE model behavescompletely different from the ODE model. For the ODE case it is possible toplot a complete eigenvalue chart whereas the DDE case has an infinite number ofeigenvalues and is therefore impossible to completely chart. A conclusion that isin common between the models is that Fd(t) inherently is dominant and shouldas such be at lower priority compared to the other consideration forces in orderto help system stability. Comparisons to the spring equation revealed that systemsthat prioritize Fd(t) too high converges to a system of particles in a chainconnected by springs with no friction giving the observed behavior. PrioritizingF fr(t) help stability in both cases with the exception that DDE case will be stablefor a sub interval (since the top limit comes from the system total energy) withinthe expansion whereas the ODE case remains stable through the whole intervall.The problems that come with larger systems are stability- and computation complexityrelated. All through the project has the models focused on the base casewith no sub-considerations. The thesis is that adding sub-considerations will againstabilize an unstable system with the addition that each consecutive weight shoulddeflate its value exponentially. The results proves that an unstable system can bestabilized by simply increasing the (h; k) of the system. This can have applicationswhen the optimal weights are not enough to stabilize a large system.When computing the eigenvalues for large systems it puts strain on the algorithm’eig’. According to [1] is the computation time complexity proportional to n3.What the resulting fit shows is that the relation is more quadratic than cubic andthe reason is described to be the appearance of the system matrix for the basecase. The matrix structure is similar to the one of the upper Hessenbergs whichas a result saves time when transforming the input matrix which is the reason whythe complexity is weakly cubic." @default.
• W632454714 created "2016-06-24" @default.
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• W632454714 date "2014-01-01" @default.
• W632454714 modified "2023-09-26" @default.
• W632454714 title "TRAFFIC SIMULATIONS THROUGH ODE-, DDE SYSTEM MODELING AND NUMERICAL COMPUTATIONS" @default.
• W632454714 hasPublicationYear "2014" @default.
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