International Journal of Approximate Reasoning
Given a fuzzy logic system, how can we determine the membership functions that will result in the best performance? If we constrain the membership functions to a specific shape (e.g., triangles or trapezoids) then each membership function can be parameterized by a few variables and the membership optimization problem can be reduced to a parameter optimization problem. The parameter optimization problem can then be formulated as a nonlinear filtering problem. In this paper we solve the nonlinear filtering problem using H∞ state estimation theory. However, the membership functions that result from this approach are not (in general) sum normal. That is, the membership function values do not add up to one at each point in the domain. We therefore modify the H∞ filter with the addition of state constraints so that the resulting membership functions are sum normal. Sum normality may be desirable not only for its intuitive appeal but also for computational reasons in the real time implementation of fuzzy logic systems. The methods proposed in this paper are illustrated on a fuzzy automotive cruise controller and compared to Kalman filtering based optimization.
Simon, Daniel J., "H-infinity Estimation for Fuzzy Membership Function Optimization" (2005). Electrical Engineering & Computer Science Faculty Publications. 60.
Simon, D. (2005). H-Infinity Estimation for Fuzzy Membership Function Optimization. International Journal of Approximate Reasoning, 40(3), 224-242. doi:10.1016/j.ijar.2005.04.002
NOTICE: this is the author’s version of a work that was accepted for publication in International Journal of Approximate Reasoning. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in International Journal of Approximate Reasoning, 40, 3, (11-01-2005); 10.1016/j.ijar.2005.04.002