Fuzzy System Identification Based Upon a Novel Approach to Nonlinear Optimization

Fuzzy systems are often used to model the behavior of nonlinear dynamical systems in process control industries because the model is linguistic in nature, uses a natural-language rule set, and because they can be included in control laws that meet the design goals. However, because the rigorous study of fuzzy logic is relatively recent, there is a shortage of well-defined and understood mechanisms for the design of a fuzzy system. One of the greatest challenges in fuzzy modeling is to determine a suitable structure, parameters, and rules that minimize an appropriately chosen error between the fuzzy system, a mathematical model, and the target system. Numerous methods for establishing a suitable fuzzy system have been proposed, however, none are able to demonstrate the existence of a structure, parameters, or rule base that will minimize the error between the fuzzy and the target system. The piecewise linear approximator (PLA) is a mathematical construct that can be used to approximate an input-output data set with a series of connected line segments. The number of segments in the PLA is generally selected by the designer to meet a given error criteria. Increasing the number of segments will generally improve the approximation. If the location of the breakpoints between segments is known, it is a straightforward process to select the PLA parameters to minimize the error. However, if the location of the breakpoints is not known, a mechanism is required to determine their locations. While algorithms exist that will determine the location of the breakpoints, they do not minimize the error between data and the model. This work will develop theory that shows that an optimal solution to this nonlinear optimization problem exists and demonstrates how it can be applied to fuzzy modeling. This work also demonstrates that a fuzzy system restricted to a particular class of input membership functions, output membership functions, conjunction operator, and defuzzification technique is equivalent to a piecewise linear approximator (PLA). Furthermore, this work develops a new nonlinear optimization technique that minimizes the error between a PLA and an arbitrary one-dimensional set of input-output data and solves the optimal breakpoint problem. This nonlinear optimization technique minimizes the approximation error of several classes of nonlinear functions leading up to the generalized PLA. While direct application of this technique is computationally intensive, several paths are available for investigation that may ease this limitation. An algorithm is developed based on this optimization theory that is significantly more computationally tractable. Several potential applications of this work are discussed including the ability to model the nonlinear portions of Hammerstein and Wiener systems

A Hopfield network approach to direct adaptive control of nonlinear systems

An automatic control system capable of controlling an unknown nonlinear system is designed using a direct adaptive control scheme, implemented with a Hopfield network. The application of this method to an arbitrary system is discussed in detail and three specific simulation studies are included. These studies include the implementation of the Hopfield network based direct adaptive control system to a linear system, an inverted pendulum, and a nonlinear model of the NPS Autonomous Underwater Vehicle (AUV) with six degrees of freedom. The AUV simulation includes a three dimensional trajectory following algorithm and shows the ability of the Hopfield network to adapt to simultaneous changes in the AUV's depth, speed and course. Additionally, an analog circuit design is proposed which implements the automatic control scheme without the support of a microprocessor. The circuit was set up in SPICE and the simulation results as well as some limitations of the analog circuit implementation of the Hopfield network are presented.http://archive.org/details/hopfieldnetworka00starLieutenant, United States NavyApproved for public release; distribution is unlimited