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Local Stabilization of Time-Delay Nonlinear Discrete-Time Systems Using Takagi-Sugeno Models and Convex Optimization

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  • Luís F. P. Silva
  • Valter J. S. Leite
  • Eugênio B. Castelan
  • Michael Klug

Abstract

A convex condition in terms of linear matrix inequalities (LMIs) is developed for the synthesis of stabilizing fuzzy state feedback controllers for nonlinear discrete-time systems with time-varying delays. A Takagi-Sugeno (T-S) fuzzy model is used to represent exactly the nonlinear system in a restricted domain of the state space, called region of validity. The proposed stabilization condition is based on a Lyapunov-Krasovskii (L-K) function and it takes into account the region of validity to determine a set of initial conditions for which the actual closed-loop system trajectories are asymptotically stable and do not evolve outside the region of validity. This set of allowable initial conditions is determined from the level set associated to a fuzzy L-K function as a Cartesian product of two subsets: one characterizing the set of states at the initial instant and another for the delayed state sequence necessary to characterize the initial conditions. Finally, we propose a convex programming problem to design a fuzzy controller that maximizes the set of initial conditions taking into account the shape of the region of validity of the T-S fuzzy model. Numerical simulations are given to illustrate this proposal.

Suggested Citation

  • Luís F. P. Silva & Valter J. S. Leite & Eugênio B. Castelan & Michael Klug, 2014. "Local Stabilization of Time-Delay Nonlinear Discrete-Time Systems Using Takagi-Sugeno Models and Convex Optimization," Mathematical Problems in Engineering, Hindawi, vol. 2014, pages 1-10, August.
    • Handle: RePEc:hin:jnlmpe:587510
    DOI: 10.1155/2014/587510
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