On the Local Controllability of a Class of Quadratic Systems

TL;DR

This paper investigates the local controllability of a broad class of affine nonlinear quadratic systems, providing necessary and sufficient conditions, and establishing controllability results for notable examples like the Lorenz and Sprott systems.

Contribution

It derives a generalized Kalman-like condition for strong accessibility and establishes local controllability under mild assumptions for quadratic systems including Lorenz and Sprott.

Findings

A necessary and sufficient condition for strong accessibility is established.

Local controllability is proven for certain systems under mild assumptions.

Sharp controllability conditions are obtained for Lorenz and Sprott systems in the single-input case.

Abstract

The local controllability of a rich class of affine nonlinear control systems with nonhomogeneous quadratic drift and constant control vector fields is analyzed. The interest in this particular class of systems stems from the ubiquity in science and engineering of some of its notable representatives, namely the Sprott system, the Lorenz system and the rigid body among others. A necessary and sufficient condition for strong accessibility reminiscent of the Kalman rank condition is derived, and it generalizes Crouch's condition for the rigid body. This condition is in general not sufficient to infer small-time local controllability. However, under some additional mild assumptions local controllability is established. In particular for the Sprott and Lorenz systems, sharp conditions for small-time local controllability are obtained in the single-input case.