Fuller problem

Here is a very classical example of a chattering phenomenon [1]:

Latex formula
The solution is, for large enough T, bang-bang (i.e., with values alternately Latex formula), the switching times geometrically converging to a value Latex formula, and then the (trivial) singular arc Latex formula and Latex formula. These switches are not easy to reproduce numerically. We display in figure 1 the control, with a zoom on the entry point of the singular arc.

Numerical simulations: problem fuller
Discretization: Gauss II with 1000 steps.
We take there
Latex formula, Latex formula, Latex formula and Latex formula.


Figure 1: Fuller problem: chattering control (with zoom); x and v.
A.T. Fuller. Study of an optimum non-linear control system. J. of Electronics and Control, 15:63-71, 1963

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