Adaptive fuzzy modeling and control of chaotic dynamical systems

This research investigates the modeling and adaptive fuzzy control of chaotic dynamical systems using fuzzy rules for the description of the underlying plant. The fuzzy rule description becomes the basis for building an indirect adaptive fuzzy controller. The bisection and homogeneity algorithm is introduced as a modification and an extension of a recursive partitioning 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 nonautnomous chaotic periodically perturbed 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 nonautnomous 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. To answer the question, “How does one know that numerically generated computer maps of chaos are real?”, the fuzzy shadowing property is introduced. Sufficiency conditions, in the form of two corollaries and a constructive theorem, are proven. These state conditions for an orbit, generated by an additive fuzzy logic system, to be ε-shadowed by a true orbit of the dynamical system. 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. Fuzzy functional decomposition is introduced as a method for decomposing hazy rules based upon conditional expectations for modeling unknown functions of a second-order system of relative degree two. A gradient projection method is introduced into the algorithm for adapting system parameters for control. 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.