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| This research investigates the adaptive modeling and control of chaotic dynamical systems using fuzzy rules for structural and parameter identification. The fuzzy rule description becomes the basis for building an indirect adaptive fuzzy controller for the parametrically perturbed pendulum. The bisection and homogeneity algorithm is introduced as a modification and an extension of a recursive partitioning clustering algorithm that generates rules directly from the data. Experimental results in the fuzzy modeling of chaos are presented for the three-dimensional autonomous Lorenz attractor and for the non-autonomous chaotic pendulum. A variable step size Euler's method performs trajectory reconstruction over multiple Standard Additive Model fuzzy systems. Domain decomposition splits regions prior to running the algorithm. Domain decomposition enforces a bound on the training time, the time necessary for fuzzy rule generation, and, for non-autonomous system modeling, it imposes a temporal ordering on fuzzy rules. Research results indicate that domain decomposition, Standard Additive Model fuzzy inference, and a one-step Euler's method produce smooth trajectory approximations on the order of Runge-Kutta numerical simulations. The final emphasis of this research is the building of an indirect adaptive fuzzy controller to train the chaotic pendulum to follow a periodic reference trajectory. Functional decomposition is introduced as a method for splitting fuzzy rules, based upon the linearity property of the conditional expectations of the rules’ consequents. In the controller equation, the decomposed fuzzy rule bases represent the unknown functions of a second-order system of relative degree two. Experimental results demonstrate that the fuzzy adaptive controller has a mean square tracking error that converges asymptotically to zero with a convergence rate on the order of one thousand times faster than conventional state feedback linearizing controllers. |
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