Controlled Invariant Hypersurfaces of Polynomial Control Systems

• Eva Zerz ^[1] ; Sebastian Walcher ^[1]
  1. [1] Rheinisch-Westfälische Technische Hochschule Aachen University

     Rheinisch-Westfälische Technische Hochschule Aachen University

     Städteregion Aachen, Alemania

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 11, Nº 1, 2012 (Ejemplar dedicado a: Algebraic and Analytic Techniques for Polynomial Vector Fields), págs. 145-158
• Idioma: inglés
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• Resumen

  • We study input-affine control systems with polynomial nonlinearity. A variety V is said to be controlled invariant if there exists a feedback law of polynomial type that causes the closed loop system to have V as an invariant variety. Using the theory of Gröbner bases, we show how to constructively decide whether a given variety is controlled invariant for a given system, and if so, how to determine all feedback laws achieving the task. We also describe a set of “trivial” vector fields for which V is invariant. If V is a smooth hypersurface, then V is only invariant for its trivial vector fields. We discuss conditions under which the converse is also true.