Convoy Car example

Convoy Car example

The example

In this example, there is leading car followed by one or two cars behind it, trying to maintain their position at 50m from each other.

The equations are:

for i in [1,N-1], with x"_i, x'_i and x_i the acceleration, speed and position of vehicle i. i=0 corresponds to the leading car, N being the toal number of cars.

To ease the analysis of the results, we reformulate the problems by considering the deltas of positions and speeds, dv_i=x'_i-x'_i+1 and dp_i = x_i-x_i+1. After discretization using the time step delta, we have the following system:

The purpose of the analysis is to show that there is no collision, and to compute bounds on the relative positions and speeds of the cars.

In experiments, we will use a constant acceleration for the leading car inside an interval, and we will constrain other initial values by intervals as well.

Structure of the real Jordan form

This example is interesting as the real Jordan form of the loop body has blocks associated to complex eigenvalues of size N-1, where N is the number of cars.

With a constant acceleraton for the leading car, for N=2 we have the real Jordan form

(the 6th dimension corresponds to the constant dimension added to model the affine transformation with a linear one).

For N=3 we have the real Jordan form

In practice, this corresponds to the common experience that with such a control law, the farther is a car from the leading car, the larger will be its oscillations (in relative position and speed).

Analysis

Some comments:

• We succeed to compute lower and upper bounds on the state variables of system, for an initial condition which much more general than a single point.
• Here the number of dimensions begins is higher, and we have bounded rotating behavior. This generates "ellipsoidal" polyhedra with many constraints and generators, hence running times are significanlty higher than in other examples.
• The good news is that even with the paramater "l" set to zero (that is, we just compute lower and upper bounds on the coefficients of integer powers of the Jordan normal form), we still obtain interesting bound.
• As expected, the variations of relative position and speed of car 0 are higher than those of car 1 (and car 2 for the 3-cars examples).

Convoy Car example