  We study dynamics of the inverted pendulum on the wheel on a soft surface and
under a proportional-integral-derivative controller. The behaviour of such
pendulum is modelled by a system with a differential inclusion. If the the
system has a sensor for the rotational velocity of the pendulum, the tilt
sensor and the encoder for the wheel then this system is observable. The using
of the observed data for the controller brings stochastic perturbations into
the system. The properties of the differential inclusion under stochastic
control is studied for upper position of the pendulum. The formula for the
time, which the pendulum spends near the upper position, is derived.
