Neuro-fuzzy relational systems for nonlinear approximation and prediction

The central point in designing a fuzzy system for modeling and control is the generation of a set of efficient and effective fuzzy rules. Generally, fuzzy rules can be generated either from heuristics or from experimental data. Heuristic fuzzy rule generation, as discussed in Chapter 1, consists of the following main steps: Determination of the structure of the fuzzy systems. This includes mainly the selection of the inputs and outputs. Usually, a multi-input multi-output fuzzy systems can be decomposed into a number of multi-input single-output fuzzy systems. Definition of linguistic terms and their fuzzy membership function for each fuzzy variables. If a Mamdani fuzzy system is designed, each input and output can be seen as a linguistic variable. Therefore, it is necessary to define the linguistic terms (e.g., Small, Medium, Large etc.) and their fuzzy membership function. Details about the definition of a linguistic variable is presented in Section 1.2.1. Determination of fuzzy inference method. For example, for fuzzy mapping rules, the fuzzy Cartesian product can be used to define the fuzzy relation. Refer to Section 1.2.2. Defuzzification. If the Mamdani type fuzzy rules are used, the fuzzy output derived from the fuzzy rule base should be defuzzified. Refer also to Section 1.2.2 for different defuzzification methods. If the Takagi-Sugeno fuzzy rules are used, the defuzzification is included in the fuzzy inference and a crisp output is obtained directly.

We have constructively proved a general class of multi-input single-output Takagi-Sugeno (TS) fuzzy systems to be universal approximators. The systems use any types of continuous fuzzy sets, fuzzy logic AND, fuzzy rules with linear rule consequent and the generalized defuzzifier. We first prove that the TS fuzzy systems can uniformly approximate any multivariate polynomial arbitrarily well, and then prove they can also uniformly approximate any multivariate continuous function arbitrarily well. We have derived a formula for computing the minimal upper bounds on the number of fuzzy sets and fuzzy rules necessary to achieve the prespecified approximation accuracy for any given bivariate function. A numerical example is furnished. Our results provide a solid-theoretical basis for fuzzy system applications, particularly as fuzzy controllers and models.

China started Sponge City Programme (SCP) to manage urban floods and to improve the quality of water environment in 2015. So far, the first two batches of pilot cities of SCP have been finished construction and two more batches of demonstration cities are ongoing. How to transfer the knowledge which learnt by the pilot cities to other cities is a key problem to enhance both the efficiency and the performance of SCP. This paper introduces the first step to develop a fuzzy system to represent knowledge which is acquired from stakeholders and decision makers, then to transform into if-then rules. This fuzzy logic inference system aims to simulate scenarios of real-life decision making. How to rate the performance of SCP in one city is chosen as an example in this paper. A Mamdani fuzzy inference system is applied, the fuzzy variables and fuzzy rules are hypothetical based on surveys of Qian’an, one of the pilot cities of SCP. Data from Qian’an is used to test and results show that the performance of SCP in Qian’an is good (74.1). The Mamdani fuzzy system is proved to be a powerful tool to represent knowledge and in the future other aspects of knowledge from SCP will be applied.

We examine the performance of compact fuzzy rule-based classification systems that consist of a small number of simple fuzzy rules with high comprehensibility. Those fuzzy systems are designed in a heuristic manner using rule selection criteria. We first describe fuzzy rule-based classification. Next we describe heuristic rule selection criteria using the terminology in data mining: confidence and support. A small number of fuzzy rules are extracted from numerical data based on each rule selection criterion. Then we examine the classification performance of extracted fuzzy rules through computational experiments on a number of benchmark data sets from the UCI ML Repository. Our results should be viewed as the lowest benchmark performance of fuzzy rule-based classification systems because fuzzy rules are extracted using a simple heuristic method with no optimization or tuning procedures. Nevertheless our results on some data sets are comparable to reported results by the C4.5 algorithm in the literature.

"The curse of dimensionality" has become a significant bottleneck for fuzzy system identification and approximation. In this paper, we cast the Takagi–Sugeno (T–S) fuzzy system identification into a hierarchical sparse representation problem, where our goal is to establish a T–S fuzzy system with a minimal number of fuzzy rules, which simultaneously have a minimal number of nonzero consequent parameters. The proposed method, which is called hierarchical sparse fuzzy inference systems ( H-sparseFIS), explicitly takes into account the block-structured information that exists in the T–S fuzzy model and works in an intuitive way: First, initial fuzzy rule antecedent part is extracted automatically by an iterative vector quantization clustering method; then, with block-structured sparse representation, the main important fuzzy rules are selected, and the redundant ones are eliminated for better model accuracy and generalization performance; moreover, we simplify the selected fuzzy rules consequent with sparse regularization such that more consequent parameters can approximate to zero. This algorithm is very efficient and shows good performance in well-known benchmark datasets and real-world problems.

Aluminium alloy (Al 63400) is burnished using different burnishing parameters. It deals with the modelling of nonlinear characteristics of ball burnishing using Sugeno fuzzy neural system. A fuzzy neural network or neuro-fuzzy system is a learning machine that finds the parameters of the fuzzy systems (i.e. fuzzy sets, fuzzy rules) by exploiting approximation techniques from neural networks. Input parameters are designed for experimental process using Box–Behnken method. Ball burnishing tool is used in CNC machining centre to surface finish the milling process. The tool and work-piece material are tungsten carbide and aluminium, respectively. The input parameters are force, table feed, step over, ball diameter and number of passes. The output parameters are surface roughness along feed direction, surface roughness across feed direction and surface micro-hardness. The minimum surface roughness obtained for ball burnishing process along the tool path (x-axis) is 0.032 μm and across the tool path (y-axis) is 0.232 μm and its micro-hardness is 91.63 HV. Sugeno fuzzy neuro system is used to model the nonlinear characteristics of the surface roughness (along the tool path and across the tool path) and micro-hardness. The Pearson product moment correlation is used to validate the fuzzy neuro model.

Induction of fuzzy rules and membership functions from training examples

On the basis of PID engineering tuning methods, a fuzzy PID control algorithm based on the fuzzy rules is put forward. To design this kind of algorithm, firstly, simplify PID formula according to the Z-N and critical proportion methods, and then, construct a piece of fuzzy control table according to fuzzy control rules and fuzzy control reasoning. In the paper, three-dimensional fuzzy control is used, which is more predominant than that with two-dimension. In real-time control, by looking up fuzzy control table, the PID parameter can be adjusted on line. The simulation results of controlling the angle of the helm and course of the ship show that the algorithm raised in this paper has good performances such as simple operation, excellent personality, etc.

We prove that a general class of multi-input single-output Takagi-Sugeno (TS) fuzzy systems are universal approximators. The fuzzy systems studied in this paper are general and typical: they use any types of continuous fuzzy sets, any types of fuzzy logic AND, fuzzy rules with linear TS rule consequent and the generalized defuzzifier containing the popular centroid defuzzifier as a special case. We first constructively prove that the general TS fuzzy systems can uniformly approximate any multivariate polynomial with arbitrarily small error and then prove that the general fuzzy systems can uniformly approximate any multivariate continuous function with arbitrarily high approximation accuracy.

The traditional PID control method has the disadvantages of slow response speed, low precision and poor anti-interference ability when controlling the brushless DC motor system. In view of these shortcomings, a fuzzy PID control based brushless DC motor control system was proposed. The principle and characteristics of fuzzy control were introduced; The control principle of PID control based on fuzzy rules was analyzed; The fuzzy subsets were established and fuzzy control rules were established; The simulation of brushless DC motor control system was simulated by Matlab/Simulink. The results showed that the PID control algorithm based on fuzzy logic control improved the dynamic performance and stronger adaptive ability of the DC brushless motor.

Fuzzy controller design of micro-unmanned helicopter relying on improved genetic optimization algorithm

A genetic algorithm for generating fuzzy classification rules

Several non-linear systems show complex behaviors. For example, some of those plants present a high degree of oscillations throughout the time. Adaptive algorithms used to approximate such difficult neural behaviors can show important definciencies. The differential network is not an exception. Indeed, when just one neural network is applied to get an adequate approximation, the identification error could be not so close to zero. One possible suggestion to solve this problem is to define a set of neuronal networks that works together. The members of such set will work each one on well defined trajectories subspaces of the uncertain system. In this paper, it is discussed how to combine the identification properties offered by the continuous neural network and the characteristic decision capabilities arised by fuzzy methods. The selection of which neural network is activated depends on decision achieved by a takagi-sugeno fuzzy system. The Chen circuit will be used to demonstrate the superior performance achieved by the suggested class of mixed neural network and fuzzy system, usually so-called neuro-fuzzy system.

Piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics.

The use of fuzzy rule-based systems in regression problems is widely extended due to the precision of the obtained models. Moreover, the use of Mamdani models is usually referred to as a good choice in many real problems, since it provides an interpretable and precise functional relationship between the output and input variables. In this paper we present a new leaning Mamdani fuzzy system FRLC-Rgress (Fuzzy Rule Learning through Clustering for Regression Problems). This provides an accurate fuzzy system and simple Mamdani fuzzy rule bases for regression problems. FRLC-Rgress based on linguistic modifiers and fuzzy clustering achieves a low complexity of the learned models while keeping a high accuracy, by following two stages: multi- granularity, fuzzy discretization of the variables, and perceptual learning of the fuzzy rules. FRLC-Rgress is experimented using six real-world datasets. It outperforms two of the most and simple fuzzy systems (genetic fuzzy systems) in state of the art.