Phase portrait modeling of a nonlinear system with a dynamic fuzzy network
Fuzzy logic and neural networks are two important technologies for modeling and control of dynamical systems and have been constrained by the non-dynamical nature of their some popular architectures. There exist problems such as large rule bases (i.e., curse of dimensionality), long training times, the need to determine buffer lengths. This article proposes to overcome these major problems in phase portrait modeling of a nonlinear system with a dynamic fuzzy network (DFN) with unconstrained connectivity and with dynamic fuzzy processing units called "feurons". Nonlinear physical system properties can be encapsulated by DFN. As an example, DFN has been used as the modeler for some nonlinear physical system such as chaotic, limit cycle, oscillator. The minimization of an integral quadratic performance index subject to dynamic equality constraints is considered for a phase portrait modeling application. For gradient computation adjoint sensitivity method has been used. Its computational complexity is significantly less than direct sensitivity method, but it requires a backward integration capability. We used first and approximate second order gradient-based methods including Broyden-Fletcher-Golfarb-Shanno algorithm to update the parameters of the dynamic fuzzy networks yielding faster rate of convergence.
phase portrait neural networks limit cycle dynamic fuzzy logic chaos mathematical foundations