An improved fuzzy logic controller using genetic algorithmic techniques for building management systems

The Role of Vagueness in the Numerical Translation of Verbal Probabilities: A Fuzzy Approach Franziska Bocklisch 1 (franziska.bocklisch@psychologie.tu-chemnitz.de) Steffen F. Bocklisch 1 (steffen.bocklisch@etit.tu-chemnitz.de) Martin R. K. Baumann 2 (martin.baumann@dlr.de) Agnes Scholz 1 (agnes.scholz@psychologie.tu-chemnitz.de) Josef F. Krems 1 (josef.krems@psychologie.tu-chemnitz.de) Wilhelm-Raabe-Str. 43, Chemnitz University of Technology, Germany Lilienthalplatz 7, German Aerospace Center Braunschweig, Germany Abstract The paper describes a general two-step procedure for the numerical translation of linguistic terms using parametric fuzzy potential membership functions. In an empirical study 121 participants estimated numerical values that correspond to 13 verbal probability expressions. Among the estimates are the most typical numerical equivalent and the minimal and maximal values that just correspond to the given linguistic terms. These values serve as foundation for the proposed fuzzy approach. Positions and shapes of the resulting membership functions suggest that the verbal probability expressions are not distributed equidistantly along the probability scale and vary considerably in symmetry, vagueness and overlap. The role of vagueness for further investigations in reasoning and decision making is discussed and relations to knowledge representation and working memory are highlighted. Keywords: verbal probability expressions; vagueness; fuzzy potential membership functions; knowledge representation; diagnostic reasoning; working memory Introduction Since the 1960s up to the present time researchers of different scientific areas have sustained an interest in studying the relationship between verbal and numerical probability expressions (Lichtenstein & Newman, 1967; Teigen & Brun, 2003; Smits & Hoorens, 2005). Among these are cognitive psychologists that inquire about the influence of uncertainty expressions on basic cognitive processes such as reasoning and decision making (Windschitl & Wells, 1996) as well as engineers, computer scientists and others that focus on the characterization (Zadeh, 1978, 2002) or on the treatment of uncertainty in applications such as medical decision support systems (Boegl, Adlassnig, Hayashi, Rothenfluh & Leitich, 2004). This broad interdisciplinary interest may be motivated by the essential role language plays in our daily life. Verbal probability terms, such as probably or thinkable are very widely used to express uncertainty about the occurrence of future events or about the degree of belief in hypotheses. For example, a typical statement that illustrates the use of linguistic terms in the conversation of stock market traders could be: “It is very unlikely that there will be a significant increase in the price of oil in the next month vice future.”. Several studies consistently show that people prefer words over numbers to express uncertainty (e.g. Wallsten, Budescu, Zwick & Kemp, 1993). This preference may be explained by the possibility of saying something about two different kinds of subjective uncertainty by using only one word. First, the stochastic uncertainty about the occurrence of an event (e.g. the probability of an increase of the oil price) and second, the vagueness of the event (e.g. what is meant by “a significant increase”). The understanding of these two kinds of uncertainty, their relations to each other and the way in which they influence human reasoning and decision making is crucial for any application that aims to support decision makers for example in medicine, business, risk management, marketing or politics. In our view, in order to contribute to the understanding of uncertainty, it is essential to first uncover the underlying relationship between word meaning and mathematical concepts such as subjective probability or fuzzy membership. Therefore, we propose a general two- step procedure for the numerical translation of verbal probability expressions based on (1) empirical estimates modelled by (2) fuzzy membership functions (Zadeh, 1965, Bocklisch & Bitterlich, 1994). The paper is structured as follows: first, we compare verbal and numerical probability expressions and discuss existing translation approaches. Second, we present our proposal that goes beyond other methodical issues and the results of an empirical investigation. Thereafter, the results are discussed and conclusions (e.g. for the construction of verbal probability scales for questionnaires) are highlighted. Further, potentialities of the fuzzy pattern classification method for reasoning and decision processes are pointed out. Verbal and Numerical Probabilities There is broad agreement concerning the different features of verbal and numerical expressions (see Teigen & Brun, 2003 for an overview). Numerical probabilities are commonly described as precise, unambiguous and especially useful for calculations. Additionally, the quality of numerical expressions can be evaluated and compared to predictions of normative models such as Bayes nets. Currently many researchers in the area of cognitive

By regarding the set of fuzzy membership functions as a subset of the set of functions which depend on time, we show compactness in weak topology. We prove also that the function which composes the control of the function given by the center of gravity calculation is continuous for the topology of pointwise convergence. As an application, we discuss the existence of an optimal solution in fuzzy control for an open-loop system.

Fuzzy concepts are represented by fuzzy sets, so a fuzzy knowledge is defined to be a collection of fuzzy sets (concepts) which includes two constants, 1 (true) and 0 (false). Abstractly, a logical derivation is a finite series of logical operations acting on a given knowledge. In fuzzy world, logical operations are defined by mathematical operations. So fuzzy logical derivations are mathematical derivations, and vice versa In this paper, the closure of fuzzy reasoning is characterized by fuzzy rough sets, and by topologies (neighborhood systems). A new fuzzy concept F N is derivable from the old knowledge K iff the membership function of F N is continuous in K-topology, or equivalently, iff F N is a rough fuzzy set (concept) of I N D(K),or plainly, iff the membership function of F N is a “step” function in the sense that it takes a constant value in each equivalence class of I N D(K).A fuzzy set L partitions the universe into equivalence classes; each equivalence class consists of those elements which have the same degree of membership. The indiscernibility relation I N D(K)over K is the “intersection” of all the equivalence relations induced by the fuzzy sets in K. The membership function of a fuzzy set is a real valued function, so U can be given the minimal topology such that the membership function of each fuzzy set in K is a continuous function. Such topology is called K-topology.

Automatically determine the membership function based on the maximum entropy principle

This paper presents a fuzzy goal programming (FGP) approach for modeling and solving multiobjective fractional programming problems (MOFPPs) with fuzzy numbers parameter sets. In the proposed approach, first the notion of α-cut in fuzzy sets (FSs) is used to transform a problem into conventional MOFPP by using the tolerance membership functions in FSs. In model formulation, membership functions are converted into fuzzy goals for measuring the degree of satisfaction of decision maker (DM) with the solution for achievement of fuzzily described objectives of the problem. In the solution process, an iterative parametric method is addressed within the framework of minsum FGP model to reach the highest membership value (unity) to the extent possible in the decision making environment. The efficiency of the proposed approach is illustrated by a numerical example. The model solution is also compared with the solution obtained by using other approaches studied previously.

This paper describes a fuzzy logic controller for car-like mobile robots. It introduces a simple heuristic that helps a designer in specifying the fuzzy input and output sets. The design of fuzzy rules follows intuitively from the design of the fuzzy input sets. In practical tests, this fuzzy logic controller resulted in greatly reduced errors and also in a control law with 75% less control work than a traditional sliding mode controller.

The Ball and Beam system (BBS) is a nonlinear and unstable system which resembles with many real-time complicated systems. Providing an appropriate beam angle to give the stability to the ball on the beam in a specific position, is a challenging task for the control system researchers. In this paper a robust interval type-2 fuzzy logic controller (IT2FLC) is designed. The dimension of the rule base is reduced by using signed distance method. This signed distance method makes the IT2FLC to a Single input Interval Type-2 Fuzzy logic Controller (SIIT2FLC). The type-2 fuzzy sets resolve the problem of determining membership functions in type-1 fuzzy systems. The membership function of a type-2 fuzzy set is three dimensional, where third dimension is the value of membership function at each point on its two dimensional domain that is called its footprint of uncertainty (FOU). The ability of FOU to represent more uncertainties enables one to cover the input and output domains with less number of fuzzy sets. This SIIT2FLC gives the smooth 2-D control surface and robustness to the system. The simulation work is carried out in simulink environment of MATLAB (7.8.0) software. The simulation results are also validated in the real-time implementation of the BBS, designed by Googol Technology. Experimental results show that the performance of the proposed controller is better than the single input type-1 fuzzy logic controller in terms of transient and steady state response. The ability of handling uncertainty (robustness) of proposed controller has been checked by applying a disturbance signal to the position sensor’s output and also by parameter variation of the BBS.

The new approach to the construction of parametric membership functions for fuzzy sets with unequal supports

This paper presents a novel implementation of a Takagi-Sugeno-type fuzzy logic controller (FLC) on a two-wheeled mobile robot (2WMR), which consists of two wheels in parallel and an inverse pendulum. The control objective of the 2WMR is to achieve position control of the wheels while keeping the pendulum around the upright position that is an unstable equilibrium. The novelties of this work lie in three aspects. First, the FLC is a synthesized design which utilizes both heuristic knowledge and model information of the 2WMR system. The FLC structure, including the fuzzy labels, membership functions, and inference, is chosen based on heuristic knowledge about the 2WMR. The output parameters of the FLC are determined by comparing the output of the FLC with that of a linear controller at certain operating points, which avoids the difficulty and tediousness in manual tuning. The linear controller is designed based on a linearized model of the 2WMR system. Second, the proposed FLC is a simple and realizable design for real implementation. Only two fuzzy labels are adopted for each fuzzy variable. Sixteen fuzzy rules are used with eight output parameters and four range parameters for the membership functions to be determined. Third, the proposed FLC is successfully implemented on a real-time 2WMR for regulation and setpoint control tasks. Satisfactory responses are achieved when the 2WMR travels not only on a flat surface but also on an inclined surface. Through comprehensive experiment-based investigations, the effectiveness of the proposed FLC is validated, and the FLC shows superior performance than the existing methods.

Approximations of fuzzy numbers by step type fuzzy numbers

Most existing QoS-aware Service Discovery and Selection (QSDS) schemes use linguistic terms to describe the QoS satisfaction degree when ranking competing alternatives. However, such schemes determine the degree of QoS satisfaction using fuzzy set membership functions, and cannot therefore take negative evidence (i.e., consumer dissatisfaction) into account. By contrast, vague sets provide the ability to represent both positive and negative evidence when modeling uncertain objects. Accordingly, the present study proposes a new satisfaction-based Web service ranking method for QSDS problems involving a group of online service consumers with imprecise and inconsistent expectations and degrees of satisfaction regarding multiple QoS criteria. Importantly, the proposed approach overcomes the uncertainty regarding the difference between the users' expectations and their degree of satisfaction inherent in Dempster---Shafer (D-S) evidence theory. Overall, the results show that the proposed scheme provides an effective means of solving the problem of vague and ill-defined information in the QSDS decision-making process.

Music-likeness is an indicator for evaluating temporal structure properties of environmental soundscape, of which fuzzy set membership functions are constructed on the basis of 1/fγ behavior of selected music fragments. The temporal dynamics in intensity and pitch vibrational sequences of Western Classical Music and Chinese National Music were analyzed by fitting their fluctuation spectra. Data comparison and statistical results show that temporal structures of these music fragments are geared towards 1/f behavior on the whole, but the probabilistic distributions of several parameters have significant difference between these two cultures. Finally, two suggestions on expansion and modification of music-likeness were presented on the basis of analytical results.

Reconstructability analysis is viewed as a process of investigating the possibilities of reconstructing desirable properties or overall systems from the knowledge of the corresponding properties of their various subsystems. Overall systems are represented by n-dimensional relations, each of which is viewed as a finite set of aggregate states (n-tuples) formed by individual stales of n variables. Depending on the context or investigation, the relations may include various measures defined on them, such as probability distributions or fuzzy set membership functions. Each subsystem is defined as a projection of the overall relation into k or its n dimensions (k < n). Every set of subsystems or an overall system is referred to as a structure system and is viewed as a possible basis for reconstructing specified properties of the overall system This paper focuses on the problem of generating, in an orderly fashion, all meaningful structure systems of a given overall system which ore desirable in each particular...

Bee Colony Optimization metaheuristic for fuzzy membership functions tuning

It is well known that the process of tuning a fuzzy logic controller is almost always a very complex task, which is time consuming, very laborious and often requires expert knowledge of the controlled system. Mapping of fuzzy logic controller's parameters (rule base and membership functions of input(s) and output(s)) into a performance measure in a closed analytical form is near impossible to get, and thus the use of any classical optimization method is automatically ruled out. Knowing this, genetic algorithms with a fitness function in a form of cumulative response error represent a good choice of the optimization method. This approach enables the use of offline optimization of membership functions' parameters (which are being coded into chromosomes). Sugeno-Takagi fuzzy logic controllers with a proportional and a derivative component, and also with a fixed rule base are used in this approach. Experimental results of both simulations and validations on real systems are given in this paper and they show the good performance of this approach.