Feedback Linearization Control Of Systems With Singularities

Fu Zhang
The Mathworks, Inc

Benito Fernndez-Rodriguez
Department of Mechanical Engineering, UT Austin

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Abstract
When a nonlinear system approaches its singularities, the system is
hard to control. However, its behavior shows abundant information
about the system. This paper presents an approach for feedback
linearization control of a nonlinear system with singularities by
using high order derivatives to explore the detail of the dynamics
of the system near the singularities.

Around the singularity points, a system doesn't have well-defined
relative degree, and conventional feedback linearization techniques
fail. This paper presented, differentiates the output $r+1$ times
until $\dot {u}$ appears and a differential equation of the input
$u$ is acquired. It shows that at the singularity point, the $\dot {u}$ term disappears and the differential equation degenerates to a
quadratic equation that governs the dynamics of the system near the
singularities. The solutions to the quadratic equations are
discussed and shows that if the quadratic equation has only real
roots, the system has a well defined relative degree at the
singularity equal to $r+1$. It shows that the neighborhood of the
singularity can be divided into two sub-regions: in one region, it
is guaranteed that the quadratic equation will have only real
solutions and the other region it may have complex roots. By divided
the neighborhood of the singularity into the above regions, more
precise control of the system near singularity can be realized.
Switching controllers can be designed to switch from a r$^{th}$
controller when system is far away from the singularity to two
(r+1)$^{th}$ controllers when system is in the neighborhood of the
singularity. The ball and beam system is used as a motivation
example to show how the approach works. General formulation of
feedback linearization by using the presented approach is presented.
Numerical simulation results are also given.