Approximation factor of the piecewise linear functions in Mamdani fuzzy system and its realization process1

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.

Neuro-fuzzy relational systems for nonlinear approximation and prediction

An improvement is presented for the tunnel algorithm introduced in [17] for determining the minimum number of rules required by a fuzzy system for modeling and identification in the SISO case. The original tunnel algorithm can deal with only Mamdani type fuzzy systems, while the Testing Point algorithm proposed in this paper, can be used to construct Takagi-Sugeno type fuzzy systems with consequent parts of any order and has the potential to be further extended to the MISO case. Numerical examples are given to illustrate the idea and improvement.

We consider iterated function systems on the real line that consist of continuous, piecewise linear functions. Under a mild separation condition, we show that the Hausdorff and box dimensions of the attractor are equal to the minimum of 1 and the exponent which comes from the most natural system of covers of the attractor.

<p style='text-indent:20px;'>We consider variational discretization [<xref ref-type="bibr" rid="b18">18</xref>] of a parabolic optimal control problem governed by space-time measure controls. For the state discretization we use a Petrov-Galerkin method employing piecewise constant states and piecewise linear and continuous test functions in time. For the space discretization we use piecewise linear and continuous functions. As a result the controls are composed of Dirac measures in space-time, centered at points on the discrete space-time grid. We prove that the optimal discrete states and controls converge strongly in <inline-formula><tex-math id="M1">\begin{document}$ L^q $\end{document}</tex-math></inline-formula> and weakly-<inline-formula><tex-math id="M2">\begin{document}$ * $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{M} $\end{document}</tex-math></inline-formula>, respectively, to their smooth counterparts, where <inline-formula><tex-math id="M4">\begin{document}$ q \in (1,\min\{2,1+2/d\}] $\end{document}</tex-math></inline-formula> is the spatial dimension. Furthermore, we compare our approach to [<xref ref-type="bibr" rid="b8">8</xref>], where the corresponding control problem is discretized employing a discontinuous Galerkin method for the state discretization and where the discrete controls are piecewise constant in time and Dirac measures in space. Numerical experiments highlight the features of our discrete approach.

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.

CHAPTER 1. AN INTRODUCTION TO DATA AND FUNCTIONS. 1.1 Describing Single-Variable Data. 1.2 Describing Relationships between Two Variables. 1.3 An Introduction to Functions. 1.4 The Language of Functions. 1.5 Visualizing Functions. CHAPTER 2. RATES OF CHANGE AND LINEAR FUNCTIONS. 2.1 Average Rates of Change. 2.2 Change in the Average Rate of Change. 2.3 The Average Rate of Change Is a Slope. 2.4 Putting a Slant on Data. 2.5 Linear Functions: When Rates of Change Are Constant. 2.6 Visualizing Linear Functions. 2.7 Constructing Graphs and Equations of Linear Functions. 2.8 Special Cases. 2.9 Breaking the Line: Piecewise Linear Functions. 2.10 Constructing Linear Models of Data. 2.11 Looking for Links between Education and Earnings: A Case Study on Using Regression Lines. CHAPTER 3. WHEN LINES MEET: LINEAR SYSTEMS. 3.1 Interpreting Intersection Points: Linear and Nonlinear Systems. 3.2 Visualizing and Solving Linear Systems. 3.3 Reading between the Lines: Linear Inequalities. 3.4 Systems with Piecewise Linear Functions: Tax Plans. CHAPTER 4. THE LAWS OF EXPONENTS AND LOGARITHMS: MEASURING THE UNIVERSE. 4.1 The Numbers of Science: Measuring Time and Space. 4.2 Positive Integer Exponents. 4.3 Zero, Negative, and Fractional Exponents. 4.4 Converting Units. 4.5 Orders of Magnitude. 4.6 Logarithms as Numbers. CHAPTER 5. GROWTH AND DECAY: AN INTRODUCTION TO EXPONENTIAL FUNCTIONS. 5.1 Exponential Growth. 5.2 Exponential Decay. 5.3 Comparing Linear and Exponential Functions. 5.4 Visualizing Exponential Functions. 5.5 Exponential Functions: A Constant Percent Change. 5.6 More Examples of Exponential Growth and Decay. 5.7 Compound Interest and the Number e. 5.8 Semi-Log Plots of Exponential Functions. CHAPTER 6. LOGARITHMIC LINKS: LOGARITHMIC AND EXPONENTIAL FUNCTIONS. 6.1 Using Logarithms to Solve Exponential Equations. 6.2 Using Natural Logarithms to Solve Exponential Equations Base e. 6.3 Visualizing and Applying Logarithmic Functions. 6.4 Using Semi-Log Plots to Construct Exponential Models for Data. C H A P T E R 7. POWER FUNCTIONS. 7.1 The Tension between Surface Area and Volume. 7.2 Direct Proportionality: Power Functions with Positive Powers. 7.3 Visualizing Positive Integer Power Functions. 7.4 Comparing Power and Exponential Functions. 7.5 Inverse Proportionality: Power Functions with Negative Powers. 7.6 Visualizing Negative Integer Power Functions. 7.7 Using Logarithmic Scales to Find the Best Functional Model. CHAPTER 8. QUADRATICS AND THE MATHEMATICS OF MOTION. 8.1 An Introduction to Quadratic Functions: The Standard Form. 8.2 Visualizing Quadratics: The Vertex Form. 8.3 The Standard Form vs. the Vertex Form. 8.4 Finding the Horizontal Intercepts: The Factored Form. 8.5 The Average Rate of Change of a Quadratic Function. 8.6 The Mathematics of Motion. CHAPTER 9. NEW FUNCTIONS FROM OLD. 9.1 Transformations. 9.2 The Algebra of Functions. 9.3 Polynomials: The Sum of Power Functions. 9.4 Rational Functions: The Quotient of Polynomials. 9.5 Composition and Inverse Functions. 9.6 Exploring, Extending & Expanding. APPENDIX Student Data Tables for Exploration 2.1. Data Dictionary for FAM1000 Data. SOLUTIONS For all Algebra Aerobics and Check Your Understanding problems for odd-numbered problems in the Exercises and Chapter Reviews. All solutions are grouped by chapter ANS-1. INDEX.

We focus on the analysis and design of two different sliding mode observers for dynamic Takagi-Sugeno (TS) fuzzy systems. A nonlinear system of this class is composed of multiple affine local linear models that are smoothly interpolated by weighting functions resulting from a fuzzy partitioning of the state space of a given nonlinear system subject to observation. The Takagi-Sugeno fuzzy system is then an accurate approximation of the original nonlinear system. Our approach to the analysis and design of observers for Takagi-Sugeno fuzzy systems is based on extending sliding mode observer schemes to the case of interpolated multiple local affine linear models. Thus, our main contribution is nonlinear observer analysis and design methods that can effectively deal with model/plant mismatches. Furthermore, we consider the difficult case when the weighting functions in the Takagi-Sugeno fuzzy system depend on the estimated state.

This paper presents a novel fuzzy controller that controls speed on induction machines using direct torque control. DTC is used to facilitate the control of the induction machine. Speed control is based in non-interactive PID control theory and Mamdani fuzzy systems. Genetic algorithm techniques are used for offline optimizations of the normalization factors (they dictate what portion of the decision table is used) and limits of membership functions. Simulations are presented, results and conclusions are discussed.

Nonlinear effects of a power amplifier (PA) in communications systems include spectral spreading, which causes adjacent channel interference. For predistorter design, it is crucial to employ a model that can sufficiently model the intermodulation distortion (IMD) caused by the PA. The order of IMD generated by a polynomial is determined by its highest order basis. For a piecewise linear function, the order of nonlinearity cannot be directly deduced from its basis function. This paper investigates how IMD is introduced by the simplicial canonical piecewise linear (SCPWL) function. We examine the operation imposed by the SCPWL basis function on sinusoid and multi-tone signal in the time domain and frequency domain. Our analysis results in expressions that can be used to predict the output spectrum of the SCPWL basis. The SCPWL basis operation involves spreading of the input signal spectrum. The output spectrum of each SCPWL basis includes infinite IMD components. The IMD order of the SCPWL model is not determined by the highest order basis. Instead, the number of SCPWL parameters, i.e., the number of linear affine segments that defines the function, determines how accurately it can model the IMD components.

Topology design of remote patient monitoring system concerning qualitative and quantitative issues

In this paper, we propose a new variant for incremental, evolving fuzzy systems extraction from data data streams, termed as GEN-FLEXFIS (short for Generalized Flexible Fuzzy Inference Systems). It builds upon the FLEXFIS methodology (published by the authors before) and extends it for generalized Takagi-Sugeno (TS) fuzzy systems, which implement generalized rotated rules in arbitrary position, employing a high-dimensional kernel rather than a connection of one-dimensional components (fuzzy sets) with t-norms. The extension includes the development of the evolving clustering learning engine, termed as eVQ-A, to extract ellipsoidal clusters in arbitrary position. Furthermore, a new merging concept based on a combined adjacency-homogenuity relation between two clusters (rules) is proposed in order to prune unnecessary rules and to keep the complexity of the generalized TS fuzzy systems low. Equipped with a new projection concept for high-dimensional kernels onto one-dimensional fuzzy sets, the new approach also provides equivalent conventional TS fuzzy systems, thus maintaining interpretability when inferring new query samples. GEN-FLEXFIS will be evaluated based on high-dimensional real-world data (streaming) sets in terms of accuracy versus final model complexity, compared with conventional FLEXFIS and other well-known (evolving) fuzzy systems approaches.

&amp;lt;p&amp;gt;Fuzzy logic is often used for calculation and simulation of real environmental situations. Wind erosion can often be complex, and from various erosion situations it is one of the hardest to be calculated and exactly described. In our research, we based the structure of the fuzzy system on the soil loss of six soils with different mechanical compositions measured in wind channels. Measurement of soil loss in four wind speed ranges during soil channel testing of soils. During the wind tunnel analysis of the soils, the topsoil loss was measured in four wind speed ranges (I. 11,2-11,6 m/s; II. 12.5-13.3 m/s, III. 14.4-14.7 m/s, IV. 15.5-15.7 m/s) on six soils with different mechanical compositions (four sand and two clayey sand soil). The mathematical model programmed and built up in MATLAB, this mamdani type fuzzy evaluation system uses two input parameters wind speed and ErosionFactor. The mathematical model requests the soils mechanical composition and identifies it based on the USDA triangle diagram. Many mathematical methods applicable to fine tune a fuzzy system. We have chosen the method of exhaustive design to cover the whole parameter space. The mathematical model calculated the soil loss. Model runs were also performed with the SWEEP model according to the soils examined in the wind tunnel. Based on our results, we found that using our fuzzy mathematical model, we obtained estimated soil loss values similar to the SWEEP model compared to the soil loss measured in the wind tunnel. However, it should be noted that the USDA SWEEP model requires a much larger amount of data to estimate the extent of soil loss caused by a wind erosion damage event.&amp;lt;/p&amp;gt;

This paper focuses on the stability analysis of T-S fuzzy control systems. The artificial T-S model method is utilized with piecewise linear interpolations into the normalized fuzzy membership functions. The stability conditions are derived to a series of LMIs. Using piecewise linear interpolation functions, we obtain finite LMIs and only solve them at the interpolation points. Furthermore, because of the dramatically improved approximation accuracy of piecewise linear interpolations, the method presented in this paper can provide a wider stable region for T-S fuzzy control systems, compared with the approach with staircase (zero-order) interpolations. A simulation example is adopted to illustrate the advantage of the proposed method.