Jorma K. Mattila

Training RBF networks using a DE algorithm with adaptive control

This paper concerns the application of differential evolution to training radial basis function networks. The algorithm consists of initial tuning, local tuning, and global tuning. The last two tunings both use a cycle-increased searching scheme, and global tuning employs fuzzy adaptive control. The mean square error from desired to actual outputs is applied as the objective function. Four standard test functions is used for demonstration. A comparison of net performances with two approaches reported in the literature shows the resulting network performs better in terms of a lower mean square error with a smaller network

Modifiers based on some t-norms in fuzzy logic

Modifiers generated by n-placed functions are considered. The subject matter of modifiers are fuzzy sets, i.e. membership functions defined on the unit interval I = [0, 1]. Two sets of modifiers are considered as example cases. One of them is a set of modifiers generated by t-norms and t-conorms. Here different dual pairs of norms create modifiers of different grade of strength. These norms are examples of two-placed functions. Another case is to generate a series of modifiers using only one DeMorgan class of norms. Norms are generalized to be n-placed functions. The place number n takes effect to the strength of a modifier. Two different DeMorgan classes are taken into the consideration. The first steps to the direction of many-valued modifier logics are taken.

Solving leontief input-output model with fuzzy entries

A general fuzzy linear system is investigated using fuzzy numbers and Gauss-Seidel iteration formula. We have used our fuzzy linear system to solve Leontief input-output model with fuzzy entries. When solving Leontief input-output model we are usually making the assumption that we know entirely the consumption matrix from industrial entries and we are certain about the final demand. These assumptions however depend heavily on estimates and information received from the industry and hence in these estimates, uncertainty plays a crucial role. To address this type of uncertainty fuzzy methods are needed to model this and in this article we are giving a procedure to solve this problem. Numerical example is given to illustrate the procedure.

Leontief input-output model with trapezoidal fuzzy numbers and Gauss-Seidel algorithm

In this work a fuzzy linear system is used to solve Leontief input-output model with fuzzy entries. For solving this model, we assume that the consumption matrix from different sectors of the economy and final demand are known. These assumptions heavily depend on the information obtained from the industries. Hence it is clear that uncertainties are involved in this information. The aim of this work is to model these uncertainties and to address them by fuzzy entries such as fuzzy numbers and LR-type fuzzy numbers (triangular and trapezoidal). Fuzzy linear system has been developed and it is solved using Gauss-Seidel algorithm. Numerical examples show the efficiency of this algorithm. The famous example from Prof. Leontief, where he solved the production levels for US economy in 1958, is also further analysed to demonstrate the efficiency of our approach.

Zadeh algebra as the basis of Łukasiewicz logics

An algebra of the theory of standard fuzzy sets are considered and a definition for Zadeh algebra is constructed. Basic things of the semantics of Łukasiewicz many-valued logics are derived from Zadeh algebra.

On Modal Operators in Łukasiewicz’ n -Valued Logics

In this paper, we study alethic modal operators in Łukasiewicz’ many-valued logics where truth values are certain rational numbers from the closed unit interval [0,1]. The alethic modal operators necessary and possible are added to Ł3 using Tarski’s idea. The result is a modal logic denoted by Ł\(_{3}^{\mathrm{mod}}\). A formula-equivalency between Ł n and Ł\(_{n}^{\mathrm{mod}}\) is stated for n ≥ 3. Truth value assignments, or valuations, in Łukasiewicz’ n-valued modal logic are considered. The laws of excluded middle and contradiction are considered and found that they are possible in Ł\(_{n}^{\mathrm{mod}}\). Normal modal many-valued systems based on axiom schemes (K), (T), (S4), (B) and (S5) are considered.

Possibility Based Modal Semantics for Graded Modifiers

A brief introduction to basic modifiers is given. Any modifier with its dual and the corresponding negation form a DeMorgan triple similar to that of t-norms, t-conorms, and negation. The lattice structure of the unit interval with the usual partial order is similar to that of the set of all membership functions. This structure has a certain connection to implication, by means of the subsethood of fuzzy sets, and it is possible to create a similar expression for modifiers as the axiom of reflexivity is in modal logic. Also, other connections to modal logics can be found. This motivates to develop a formal semantics to modifier logic by means of that of modal logic. Actually, this kind of logic is so-called metalogicconcerning either true or false statements about properties of modifiers. This version is based on graded possibility operations. Hence, semantic tools for weakening modifiers are derived. The corresponding things for substantiating modifiers are constructed by means of duality. Finally, some outlines for modifier systems are considered.

Application of Fuzzy Gain-Scheduling in Position Control of a Servo Hydraulic System with a Flexible Load

Abstract The control of hydraulic servo-systems has been the focus of intense research over the past decades. Hydraulic position servos with an asymmetrical cylinder are commonly used in industry. These kinds of systems are nonlinear in nature and generally difficult to control. Changing system parameters using the same gains will cause overshoot or even loss of system stability. The highly non-linear behaviour of these devices makes them ideal subjects for applying different types of sophisticated controllers. The paper is concerned with a second order model reference to positioning control of a flexible load servo-hydraulic system using fuzzy gain-scheduling. In the present study, to compensate the lack of damping in a hydraulic system, an acceleration feedback was used. To compare the results, a p-controller with feedforward acceleration and different gains in extension and retraction is used. The design procedure for the controller and experimental results are discussed. The results suggest that using the fuzzy gain-scheduling controller decrease the error of position reference tracking.

On models in fuzzy propositional logic

A model theory of fuzzy propositional logic is considered. The basic frame for fuzzy propositional logics are Zadeh-algebras, i.e., special quasi-Boolean algebras, where valuation functions are universes of these algebras. There are two levels of truth-values, numerical (usually the unit interval[0,1], or in general, a lattice L) and linguistic. Linguistic truth-values are fuzzy subsets of the set of numerical truth-values. Fuzzy model is defined based on numerical truth-values, i.e. it is the set of designated truth-values. Its linguistic label is true. Truth conditions and the concepts validity, satisfiability, refutability, and invalidity are considered.

On Aristotelian tradition and fuzzy revolution

Aristotelian tradition of science and Aristotelian attitude to world are considered. Some considerations of the relationship between Aristotelian tradition and modal-based logics and between that and many-valued logics are done. Prof. Zadehs early fuzzy set theory is presented as a special quasi-Boolean algebra. It is shown by examples that Aristotelian tradition of science remains as a part of new tradition of science, and Aristotelian attitude to world is ruined.