Stochastic properties of an inverted pendulum on a wheel on a soft surface

TL;DR

This paper analyzes the stochastic dynamics of an inverted pendulum on a wheel on a soft surface, incorporating control and sensor data, and derives formulas for the time spent near the unstable equilibrium.

Contribution

It models the system with differential inclusion, studies stochastic effects under control, and provides a formula for the duration near the upper position.

Findings

Derived formula for time near the upper position

Analyzed stochastic properties of the controlled system

Established observability with sensors

Abstract

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.