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SVM Viability Kernel Approximation (3/3)

Example: population system

Description of the system

We consider a simple dynamical system of population growth on a limited space. The state (x(t), y(t)) of the system represents the size of a population x(t), which grows or diminishes with the evolution rate y(t). The size of the population must remain in an interval K=[a,b], with a > 0. The inertia bound c limits the derivative of the evolution rate at each time step. The system in discrete time defined by a time interval dt can be written as follows:

$x(t+dt) = x(t)+x(t)y(t)dt$ \\
  $y(t+dt) = y(t) + u(t)$


$-c \leq u(t) \leq c$ .

The viability constraint set is the set K=[a,b] x [d,e]. For the following example, we used the following parameters: a=0.2, b=3, c=0.5, d=-2, e=2.

Progressive approximation of the viability kernel

The following pictures show the progressive approximation of the viability kernel. The points in blue are viable states, those in red non-viable states and in blue, the approximated viability kernel. The black lines represents the theoretical curves of the viability kernel.

SVM heavy controller

The following two pictures present a trajectory obtained by using the SVM heavy controller. On the left, the number of steps of anticipation is 2 while on the right, it is 10. The light blue zone inside the approximation of the viability kernel correponds to the security distance.

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