Fuzzy two-term controllers with multi-fuzzy sets: mathematical models and analysis

This paper presents the mathematical model of Mamdani simplest fuzzy PI or PD Two-Input Two- Output (TITO) controller via Larsen Product (LP) inference method. There are two fuzzy sets on each of the four input variables and five fuzzy sets on each of the two output variables. L - type and Γ - type membership functions are chosen on each input variable, and trapezoidal and triangular membership functions are chosen on each output variable. The other components chosen are algebraic product t - norm, bounded sum s - norm, linear fuzzy control rules and Center of Sums (CoS) defuzzification. Mathematical model of the simplest fuzzy PI or PD TITO controller via Larsen product inference was not reported in the literature and hence a model is presented for the first time in this paper.

This paper reveals analytical structure for the simplest fuzzy PID controller which employs two fuzzy sets for each of the three input variables and four fuzzy sets for the output variable. Analytical structures are derived via left and right trapezoidal membership functions for each input, triangular membership functions for output, algebraic product triangular norm, bounded sum triangular co-norm, Mamdani minimum inference method, and center of area (COA) defuzzification method. Bounded-input bounded-output (BIBO) stability analysis is presented. Finally, a numerical example along with its simulation results is included to demonstrate the effectiveness of the simplest fuzzy PID controller.

This paper presents two new mathematical models of the simplest fuzzy PID controller which employ two fuzzy sets (negative and positive) on each of the three input variables (error, change in error and double change in error) and four fuzzy sets (-2, -1, +1, +2) on the output variable (incremental control). L-type, Γ-type and II-type membership functions are considered in fuzzification process of input and output variables. Controller modelling is done via algebraic product AND operation, maximum/bounded sum OR operation, Mamdani minimum inference method, and centre of sums (CoS) defuzzification. The new models obtained in this manner turn out to be nonlinear, and their properties are studied. Since digital controllers are implemented on the digital processors, the computational and memory requirements of the fuzzy controllers and conventional (non-fuzzy) controller are compared. Stability analysis of closed loop systems containing the fuzzy controller models is done using the small gain theorem.

Defuzzification is a process of getting a crisp value from the fuzzy data. Since the fuzzy data cannot be used directly for the real time applications, therefore it has to be converted into a crisp value. In designing the defuzzification unit in this work, the center of sums has been used, because of its computationally efficient nature. The Center of Sums (COS) is faster than many defuzzification method and the method is not restricted to symmetric membership function is simple and is being generally used in comparison to more complex Center of gravity (also called Center of area or Centroid method) defuzzification method. The proposed architecture has been modeled in VHDL and implemented in XILINX and Spartan field programmable gate arrays (FPGA). The proposed architecture is more efficient in area (the area of implementation of VLSI Architecture) and the speed of operation in comparison to a more complex architecture used for the Center of gravity. The functional analysis has revealed that the proposed architecture is implementing COS based defuzzifier efficiently and accurately.

In this paper mathematical models of Mamdani type fuzzy PI or PD controllers via Center of Gravity (CoG) defuzzification are obtained. These controllers are called the simplest fuzzy PI or PD controllers as minimal number of fuzzy sets (two fuzzy sets on each of the two-input variables and three fuzzy sets on the output variable) are employed. The basic components of these models are L-type and Y-type membership functions on each input variable, trapezoidal or triangular membership functions on output variable, minimum / algebraic product t-norm, maximum OR operator, three linear fuzzy control rules, minimum / algebraic product inference and CoG defuzzification. Mathematical models of fuzzy controller via CoG defuzzification were not reported in the literature and are therefore presented for the first time in this paper.

Multisets alias bags are similar to fuzzy sets but essentially different in basic concepts and operations. We overview multisets together with basics of fuzzy sets in order to observe differences between the two. We then introduce fuzzy multisets and the combination of the both concepts. There is another concept of real-valued multisets as a generalization of multisets. Rough approximations of multisets and fuzzy multisets are discussed which uses a natural projection of the universal set onto the set of equivalence classes.

In accordance with the provisions of the Energy Strategy of Ukraine for the period until 2035, despite the significant development of green energy, natural gas remains the main energy source in the country. In connection with the accession to the common European space of regulation of natural gas tradein the country, all payments for fuel consumption should be made in units of energy. Thus, the main question is the quality of natural gas. One of the peculiarities of the gas supply system is the considerable degree of uncertainty in the change of a large number of disturbing factors and constantly changing parameters of its functioning. A mathematical model based on fuzzy logic theory isexpedient for gas quality assessment. According to the results of consideration of the factors characterizing the physico-chemical properties of the natural gas extracted, the quality of its preparation for transportation and the technical operation conditions of the gas distribution system ina settlement, a fuzzy set was obtained to determine the quality of the fuel. According to the "center of mass" method for defuzzification, the fuzzy set corresponds to a fuzzy set that is in the range from one to five points. As an example, for natural gas with limit values in accordance with the Gas Transmission System Code, the quality is estimated at 3.1 points. The fuzzy inference model, together with the dephasification procedure, provides the ability to observe changes in the baseline – the quality of natural gas – depending on changes in quantitative and qualitative factors on the path from the gas field to the consumer. Using the proposed mathematical model in a complex for all levels and sublevels, it is possible to obtain a predicted estimate of the impact of the above factors on fuel quality. In this case, the value obtained is taken from the results of a virtual experiment, which is based on an expert knowledge base.

Type 2 fuzzy multiset is a new concept formed by integrating type 2 fuzzy set and multiset which are very useful for modelling uncertainties. The purpose of this paper is to introduce the concept of type 2 fuzzy α multicut set and to show how α- cut in secondary level helps to relate type 2 fuzzy α- multicuts and type 2 fuzzy multisets. The unique representation of type 2 fuzzy multisets using either the family of all type 2 fuzzy α multicuts or the family of all type 2 fuzzy α strong multicuts are also showed. The construction of membership grades in different levels are explained. Furthermore, we investigate the multicriteria group decision making based on type 2 fuzzy multisets. Finally, we give an example to show the validity of type 2 fuzzy multiset in group decision making problem.

Several extensions and generalizations of fuzzy sets have been introduced in the literature, for example, Atanassov's intuitionistic fuzzy sets, type 2 fuzzy sets, and fuzzy multisets. In this paper, we propose hesitant fuzzy sets. Although from a formal point of view, they can be seen as fuzzy multisets, we will show that their interpretation differs from the two existing approaches for fuzzy multisets. Because of this, together with their definition, we also introduce some basic operations. In addition, we also study their relationship with intuitionistic fuzzy sets. We prove that the envelope of the hesitant fuzzy sets is an intuitionistic fuzzy set. We prove also that the operations we propose are consistent with the ones of intuitionistic fuzzy sets when applied to the envelope of the hesitant fuzzy sets. © 2010 Wiley Periodicals, Inc.

This study aimed to compare fuzzy systems with different configurations to predict the surface temperature (ts) of broiler chickens subjected to different intensities and durations of thermal challenges in the second week of life. Data on the ts of broiler chickens aged 8 to 11 days were acquired by infrared thermography and subjected to combinations of four dry-bulb temperatures (tdb) (24, 27, 30, and 33 °C) and four durations of thermal challenges (DTC) (1, 2, 3, or 4 days). The input variables of the fuzzy systems were tdb and DTC, and the output variable was ts. The Mamdani inference method involving five defuzzification methods [center of gravity (centroid), bisector of the area (bisector), largest of maximum (lom), middle of maximum (mom), and smallest of maximum (som)], and Sugeno inference with two defuzzification methods [weighted average (wtaver) and weighted sum (wtsum)] were evaluated. For both inference methods, triangular and Gaussian pertinence functions were tested for input and output variables, except for Sugeno inference, which used singletons functions as output variables. While developing fuzzy systems, different configurations must be compared, and the system with smaller simulation errors should be selected.

The problem of mean square bounded-input bounded-output(BIBO) stability is investigated for a class of discrete-time stochastic control systems with time delays and non-linear perturbations. In this paper, a special point δ in the time delay's variation interval is introduced, and the variation interval is divided into two subintervals. Then, by defining a special Lyapunov-Krasovskii functional and checking its variation in the two subintervals, respectively, some novel delay-dependent stability criteria for the discrete-time stochastic control systems are derived. These conditions are expressed in the forms of linear matrix inequalities(LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Meanwhile, this paper provides a new method for studying discrete-time stochastic mean square BIBO stability. Finally, a numerical example is given to illustrate the validity of the main results.

Delay-dependent BIBO stability analysis of switched uncertain neutral systems

In many contexts, type-2 fuzzy sets (T2 FS) are obtained from a type-1 fuzzy set to which we wish to add uncertainty. However, in the current type-2 representation, there is no restriction on the shape of the footprint of uncertainty and the embedded sets (ESs) that can be considered acceptable. This leads, usually, to the loss of the semantic relationship between the T2 FS and the concept it models. As a consequence, the interpretability of some of the ESs and the explainability of the uncertainty measures obtained from them can decrease. To overcome these issues, constrained type-2 (CT2) fuzzy sets have been proposed. However, no formal definitions for some of their key components [e.g., acceptable ESs (AESs)] and constrained operations have been given. In this article, we provide some theoretical underpinning for the definition of CT2 sets, their inferencing and defuzzification method. To conclude, the constrained inference framework is presented, applied to two real-world cases and briefly compared to the standard interval type-2 inference and defuzzification method.

In previous publications, the authors presented novel techniques for the design and realization of Bode-type variable-amplitude digital equalizers. The frequency response of the resulting variable-amplitude digital equalizers can be varied from that of an arbitrary shaping transfer function to its inverse by varying the value of a single variable digital multiplier. This paper is concerned with the development of an exact bounded-input bounded-output (BIBO) stability condition for general-order Bode-type variable-amplitude digital equalizers. The salient feature of the resulting BIBO stability condition is that it guarantees the stability of the Bode-type variable-amplitude digital equalizers over an entire range of values for the constituent variable digital multiplier. Moreover, this BIBO stability condition can simply be evaluated by examining the shaping transfer function associated with the variable-amplitude digital equalizer proper.

Many fuzzy controller structures based on various inference methods have been presented [1–14]. Among them the most widely used methods in practice are: the Mamdani method proposed by Mamdani and his associates [2] who adopted the min-max compositional rule of inference based on an interpretation of a control rule as a conjunction of the antecedent and consequent; and the product-sum method proposed by Mizumoto [10–12] who suggested introducing the product and arithmetic mean aggregation operators to replace the logical AND (minimum) and OR (maximum) calculations in the min-max compositional rule of inference. The product-sum method greatly simplifies the algorithm of the fuzzy controller. In the algorithm of a fuzzy controller, the defuzzification calculation is also a complicated and time-consuming task. Using different defuzzification methods would not give a distinct improvement of performance. Hans and Christoph [22] have given a review on the defuzzyfication. Takagi and Sugeno proposed a crisp-type model in which the consequent parts of the fuzzy control rules are crisp functional representation or crisp real numbers in the simplified case instead of fuzzy sets [13, 14]. With this model of crisp real number output, the fuzzy set of the inference consequence will be a discrete fuzzy set with a finite number of points, this can greatly simplify the calculation of the defuzzification algorithm. The product-sum inference method and the crisp output model are often applied in a mixed manner. The mixed product-sum crisp model has a fine performance and the simplest algorithm that is very easy to be implemented in hardware system and converted into a fuzzy neural network model.