InterJournal Complex Systems, 1690 Status: Accepted Manuscript Number: [1690] Submission Date: 2006 Feedback Linearization Control Of Systems With Singularities
Author(s): Fu Zhang ,Benito Fernndez-Rodriguez

Subject(s): CX.0

Category:

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.