Stable controller design for T--S fuzzy systems based on Lie algebras

  • Authors:

  • Stephen P. Banks;Evren Gürkan;İsmet Erkmen

  • Affiliations:

  • Department of Automatic Control and Systems Engineering, University of Sheffield, Mappin Street, Sheffield S1 3JD, England;Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, USA;Department of Electrical and Electronics Engineering, Middle East Technical University, 06531 Ankara, Turkey

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2005

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Abstract

In this paper, we study the stability of fuzzy control systems of Takagi-Sugeno-(T-S) type based on the classical theory of Lie algebras. T-S fuzzy systems are used to model nonlinear systems as a set of rules with consequents of the type x@?(t)=A"lx(t)+B"lu(t). We conduct the stability analysis of such T-S fuzzy models using the Lie algebra L"A generated by the A"l matrices of these subsystems for each rule in the rule base. We first develop our approach of stability analysis for a commuting algebra L"A, where all the consequent state matrices A"l's commute. We then generalize our results to the noncommuting case. The basic idea here is to approximate the noncommuting Lie algebra with a commuting one, such that the approximation error is minimum. The results of this approximation are extended to the most general case using the Levi decomposition of Lie algebras. The theory is applied to the control of a flexible-joint robot arm, where we also present the decomposition procedure.