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
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  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.
2014. , 31 p.