Applications - A Guide to MATLAB for Beginners and Experienced Users - page 199

Here the small solid circle corresponds to λ T = 1 / 2, and we are just at the
limit of stability, since this circle does not cross the spiral produced by
z exp( z ) for z a complex number on the critical line Re z = 0, though it “hugs”
the spiral closely. The dotted and dashed circles, corresponding to λ T = 1or
2, do cross the spiral, so they correspond to unstable traffic flow.
We can check these theoretical predictions with a simulation using
SIMULINK. We'll give a picture of the SIMULINK model and then
explain it.

5 5

.8

In1 Out1

5

5

sensitivity
parameter

5

Reaction-time
Delay

Subsystem:
computes velocity
differences

car speeds

1
s

5

5

5

x o
Integrate
u" to get u'

-C- 1

s

5

5

5

x o
Integrate
u' to get u

5

initial velocities 5

relative
car positions

5

5

Ramp

carpositions

5

5

-C-

To Workspace

initial car positions

Here the subsystem, which corresponds to multiplication by S − I , looks like
this:

5

1
Out1

5

4

5

5 5

1
In1

em

5

4

Here are some words of explanation. First, we are showing the model using
the options Wide nonscalar lines and Signal dimensions in the Format