The Dynamic Search for the Minimal Dynamic Extension
Rollen S. D'Souza
TL;DR
The paper tackles the problem of identifying the minimal dynamic extension needed to render a nonlinear control system feedback linearizable. It introduces a differential-geometric, category-theoretic framework where dynamic extensions are arrows in a category of regular nonlinear control systems and uses dynamic programming, possibly with a heuristic, to find minimal extensions efficiently. Central contributions include linking regular zero dynamics foliations to dynamic extensions, formulating a functorial loss and DP equations on categories, and proposing a Leading Integrability Defect Heuristic to guide search toward involution. Through one- and two-step dynamic precompensator examples, it demonstrates how minimal extensions can be discovered even when pure prolongations fail, offering a scalable alternative to solving general dynamic feedback linearization with prohibitive computation.
Abstract
Identifying the dynamic precompensator that renders a nonlinear control system feedback linearizable is a challenging problem. Researchers have explored the problem -- dynamic feedback linearization -- and produced existence conditions and constructive procedures for the dynamic precompensator. These remain, in general, either computationally expensive or restrictive. Treating the challenge as intrinsic, this article views the problem as a search problem over a category. Dynamic programming applies and, upon restriction to a finite category, classic search algorithms find the minimal dynamic extension. Alternatively, a heuristic aiming towards feedback linearizable systems can be employed to select amongst the infinitely-many extensions. This framing provides a distinctive, birds-eye view of the search for the dynamic precompensator.