FLATNESS AND DEFECT OF NONLINEAR-SYSTEMS - INTRODUCTORY THEORY AND EXAMPLES

We introduce flat systems, which are equivalent to linear ones via a s pecial type of feedback called endogenous. Their physical properties a re subsumed by a linearizing output and they might be regarded as prov iding another nonlinear extension of Kalman's controllability. The dis tance to flatness is measured by a non-negative integer, the defect. W e utilize differential algebra where flatness and defect are best defi ned without distinguishing between input, state, output and other vari ables. Many realistic classes of examples are flat. We treat two popul ar ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The thre e non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strate gy is proposed such that the averaged systems become flat.