IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 7, NO. 2, APRIL 1999 151 Heuristic Constraints Enforcement for Training of and Rule Extraction from a Fuzzy/Neural Architecture— Part II: Implementation and Application Sinan Altug, Student Member, IEEE, Mo-Yuen Chow, Senior Member, IEEE, and H. Joel Trussell, Fellow, IEEE Abstract—This paper is the second of two companion papers. The foundations of the proposed method of heuristic constraint enforcement on membership functions for knowledge extraction from a fuzzy/neural architecture was given in Part I. Part II develops methods for forming constraint sets using the constraints and techniques for finding acceptable solutions that conform to all available a priori information. Moreover, methods of integration of enforcement methods into the training of the fuzzy-neural architecture are discussed. The proposed technique is illustrated on a fuzzy–AND classification problem and a motor fault de- tection problem. The results indicate that heuristic constraint enforcement on membership functions leads to extraction of heuristically acceptable membership functions in the input and output spaces. Although the method is described on a specific fuzzy/neural architecture, it is applicable to any realization of a fuzzy inference system, including adaptive and/or static fuzzy inference systems. Index Terms— Constraint enforcement, knowledge extraction, neural-fuzzy architectures, set theory. I. IMPLEMENTATION TECHNIQUES FOR CONSTRAINT ENFORCEMENT METHODS T HE CONSTRAINTS discussed in Part I of this paper [1] can be enforced by different methods, depending on the space in which they are imposed. In this paper the constraints are enforced in to demonstrate the proposed method. A. Constraint Enforcement by Projections onto Constraint Sets As discussed in Part I, enforcing a constraint on an el- ement is equivalent to replacing it with an element of the corresponding constraint set, which by definition conforms to the constraint. For this purpose, we will use the projection of an element onto the constraint set [2]. The projection of an element onto a constraint set is the element of that is closest to with respect to a metric. Mathematically, this is given by solving the minimization problem (1) subject to (2) Manuscript received November 11, 1997; revised September 14, 1998. This work was supported in part by the National Science Foundation under Grant ECS-9 521 609. The authors are with the Department of Electrical and Computer Engineer- ing, North Carolina State University, Raleigh, NC 27695 USA. Publisher Item Identifier S 1063-6706(99)02794-0. where is the projection of onto and is a metric defined on same space as . In order for (1) to have a unique solution, it is required that the set is closed and convex in a Hilbert space [2], [3]. In Part I it was noted that the candidate solution set is the intersection of the constraint sets. The method of projections onto convex sets (POCS) converges to a point in the intersection of these sets provided that the intersection is not empty [2]. The details regarding the method of POCS are given in [2]–[5]. Thus, if the constraint sets formed by using the a priori information are closed convex sets, the point of convergence of successive projections onto these sets is an element in the intersection of the constraint sets. By definition, this point conforms to the information given in each set, and is a feasible solution [4]. Continuing from Part I, our aim is to construct closed and convex constraint sets that result from the constraints given in (12)–(17), so that POCS can be utilized to find a common point off all constraint sets that conforms to all constraints. B. Issues Regarding Constraint Enforcement in Constructing the constraints set in the space, , the prob- lem in (1) is interpreted as finding a , where is a constraint set constructed in the parameter space (3) where is the element of to be projected onto a constraint set is its projection onto and is the Euclidian 2-norm in . The constraint sets are constructed on the parameter space so that enforcement is applied on the parameters of the membership functions. Sets of interest are those defined by the constraints discussed in Section IV of part I. This paper uses constraint enforcement in to illustrate the method. C. Constraint Enforcement Methods in Let be the projection operator that projects a vector onto a closed convex constraint set . The iteration given by (4) 1063–6706/99$10.00  1999 IEEE