A triangulation approach to asymptotically exact conditions for fuzzy summations

  • Authors:

  • Alexandre Kruszewski;Antonio Sala;Thierry M. Guerra;Carlos Ariño

  • Affiliations:

  • Laboratoire d'Automatique, Génie Informatique et Signal Unité Mixte de Recherche, Ecole Centrale de Lille, Villeneuve d'ascq, France;Department of Systems Engineering and Control, Instituto de Automática e Informática Industrial, Universidad Politécnica de Valencia, Valencia, Spain;Laboratoire d'Automatique, de Mécanique, et d'Informatique industrielles et Humaines, Unité Mixte de Recherche, Université de Valenciennes, Valenciennes Cedex 9, France;Department of Industrial Systems Engineering and Design, Universitat Jaume I, Spain

  • Venue:
  • IEEE Transactions on Fuzzy Systems
  • Year:
  • 2009

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Abstract

Many Takagi-Sugeno (T-S) fuzzy control-synthesis problems in the literature are expressed as the problem of finding decision variables in a double convex sum (fuzzy summation) of positive definite matrices. Matrices' coefficients in the summation take values in the standard simplex. This paper presents a triangulation approach to the problem of generating simplicial partitions of the standard simplex in order to set up a family of sufficient conditions and, in parallel, another family of necessary ones for fuzzy summations. The conditions proposed in this paper are asymptotically exact as the size of the involved simplices decreases; its conservativeness vanishes for a sufficiently fine partition (sufficiently dense mesh of vertex points). The set of conditions is in the form of linear matrix inequalities (LMIs), for which efficient software is available.