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KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2014 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

This report is written as a project to conclude the three year bachelor part of a

five year degree in engineering and will as such target students finishing their three

year bachelor degrees. The main topic of this project is traffic simulation through

numerical analysis, with the accompanying subtopics time integration, eigenvalue

analysis and computation complexity. The tool used through the project is Matlab

computational software. This project features two general traffic models where the

first is based on a system of ordinary differential equations (ODE’s) and the second

one is based on a system of delay differential equations (DDE’s). The project will

highlight the implications of driver considerations in terms of stability, stability

related to system size, stability related to reaction times and the relation between

the large system stability and increasing the (h; k) values of the model.


(h; k) values of the model

Abstract [en]

The ODE- and DDE models are based on the same model with the only exception

that the DDE model features reaction times. They are defined by consideration

forces and sub-consideration forces. The values

(h; k) determines the number of

cars that each driver considers and therefore adds to the system as additional

terms 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 base

case of the system and equals to (h; k) = (1; 1). The (h; k) of the system is determining

the matrix B in equation (11) by the number of sub diagonals h and super diagonals

k that are filled by weights of the forces.

The time integrations can result in three base cases, unstable, stable oscillating

and exponentially stable. These cases refer to the behavior of all system velocities.

The unstable case can for limited time frames predict collisions between cars but

otherwise diverge and cannot generally be used. Oscillating stable systems reach

a constant velocity after a settling time and fits well into a realistic scenario. The

exponential case reaches a constant velocity the fastest and is therefore the sought

after solution. Both models are similar in this regard apart from the fact that the

DDE model generally have a lot more system energy. Figures 1 and 4 are empirical

proof that the models works as defined and can predict some traffic behavior.

An interesting observation during testing was that the ODE exponential case would

always remain exponential no matter the multiplication

(; ; ) = C(; ; ), the

only difference would be the system energy since larger acting forces are coupled

with larger energies. The DDE model however is dependent on the system energy

for stability since the delay sets a system energy limit for stability since too large

forces coupled with delay will not achieve the optimum distance d.

The system stability analysis can be reduced in both models to analyzing the homogeneous

and particular parts separate. The expansions confirms in both cases

what the time integrations shows and can give an idea of how the stability changes

with one parameter changing. However, this is where the DDE model behaves

completely different from the ODE model. For the ODE case it is possible to

plot a complete eigenvalue chart whereas the DDE case has an infinite number of

eigenvalues and is therefore impossible to completely chart. A conclusion that is

in common between the models is that Fd(t) inherently is dominant and should

as such be at lower priority compared to the other consideration forces in order

to help system stability. Comparisons to the spring equation revealed that systems

that prioritize Fd(t) too high converges to a system of particles in a chain

connected by springs with no friction giving the observed behavior. Prioritizing

F fr(t) help stability in both cases with the exception that DDE case will be stable

for a sub interval (since the top limit comes from the system total energy) within

the expansion whereas the ODE case remains stable through the whole intervall.

The problems that come with larger systems are stability- and computation complexity

related. All through the project has the models focused on the base case

with no sub-considerations. The thesis is that adding sub-considerations will again

stabilize an unstable system with the addition that each consecutive weight should

deflate its value exponentially. The results proves that an unstable system can be

stabilized by simply increasing the (h; k) of the system. This can have applications

when 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 and

the reason is described to be the appearance of the system matrix for the base

case. The matrix structure is similar to the one of the upper Hessenbergs which

as a result saves time when transforming the input matrix which is the reason why

the complexity is weakly cubic.

Place, publisher, year, edition, pages
2014. , 31 p.
National Category
Engineering and Technology
URN: urn:nbn:se:kth:diva-148377OAI: diva2:736211
Available from: 2014-08-05 Created: 2014-08-05 Last updated: 2014-08-05Bibliographically approved

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Niklas Carl Simon Kleyer kandidatexamensarbete Bachelor degree mathematics, KTH.(1346 kB)78 downloads
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