Flatness and defect of non-linear systems: introductory theory and examples

TLDR

Flat systems generalize Kalman controllability by subsuming physical properties within a linearizing output. The paper introduces flat systems as feedback‑equivalent to linear ones and proposes a high‑frequency control strategy to render averaged systems flat. Flatness is quantified by a non‑negative defect integer defined via differential algebra, and illustrated on cranes, car‑with‑trailer systems, and pendulums, with a high‑frequency control scheme to achieve flatness. Many realistic system classes, such as cranes and car‑with‑trailer configurations, are shown to be flat.

Abstract

Abstract We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.