Fuzzy Arbitrary Order System

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Fuzzy Arbitrary Order System

Author : Snehashish Chakraverty
Publisher : John Wiley & Sons
ISBN 13 : 1119004179
Total Pages : 324 pages
Book Rating : 4.1/5 (19 download)

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Book Synopsis Fuzzy Arbitrary Order System by : Snehashish Chakraverty

Download or read book Fuzzy Arbitrary Order System written by Snehashish Chakraverty and published by John Wiley & Sons. This book was released on 2016-08-09 with total page 324 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents a systematic treatment of fuzzy fractional differential equations as well as newly developed computational methods to model uncertain physical problems Complete with comprehensive results and solutions, Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications details newly developed methods of fuzzy computational techniquesneeded to model solve uncertainty. Fuzzy differential equations are solved via various analytical andnumerical methodologies, and this book presents their importance for problem solving, prototypeengineering design, and systems testing in uncertain environments. In recent years, modeling of differential equations for arbitrary and fractional order systems has been increasing in its applicability, and as such, the authors feature examples from a variety of disciplines to illustrate the practicality and importance of the methods within physics, applied mathematics, engineering, and chemistry, to name a few. The fundamentals of fractional differential equations and the basic preliminaries of fuzzy fractional differential equations are first introduced, followed by numerical solutions, comparisons of various methods, and simulated results. In addition, fuzzy ordinary, partial, linear, and nonlinear fractional differential equations are addressed to solve uncertainty in physical systems. In addition, this book features: Basic preliminaries of fuzzy set theory, an introduction of fuzzy arbitrary order differential equations, and various analytical and numerical procedures for solving associated problems Coverage on a variety of fuzzy fractional differential equations including structural, diffusion, and chemical problems as well as heat equations and biomathematical applications Discussions on how to model physical problems in terms of nonprobabilistic methods and provides systematic coverage of fuzzy fractional differential equations and its applications Uncertainties in systems and processes with a fuzzy concept Fuzzy Arbitrary Order System: Fuzzy Fractional Differential Equations and Applications is an ideal resource for practitioners, researchers, and academicians in applied mathematics, physics, biology, engineering, computer science, and chemistry who need to model uncertain physical phenomena and problems. The book is appropriate for graduate-level courses on fractional differential equations for students majoring in applied mathematics, engineering, physics, and computer science.

Computation and Modeling for Fractional Order Systems

Author : Snehashish Chakraverty
Publisher : Elsevier
ISBN 13 : 0443154058
Total Pages : 288 pages
Book Rating : 4.4/5 (431 download)

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Book Synopsis Computation and Modeling for Fractional Order Systems by : Snehashish Chakraverty

Download or read book Computation and Modeling for Fractional Order Systems written by Snehashish Chakraverty and published by Elsevier. This book was released on 2024-02-20 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computation and Modeling for Fractional Order Systems provides readers with problem-solving techniques for obtaining exact and/or approximate solutions of governing equations arising in fractional dynamical systems presented using various analytical, semi-analytical, and numerical methods. In this regard, this book brings together contemporary and computationally efficient methods for investigating real-world fractional order systems in one volume. Fractional calculus has gained increasing popularity and relevance over the last few decades, due to its well-established applications in various fields of science and engineering. It deals with the differential and integral operators with non-integral powers. Fractional differential equations are the pillar of various systems occurring in a wide range of science and engineering disciplines, namely physics, chemical engineering, mathematical biology, financial mathematics, structural mechanics, control theory, circuit analysis, and biomechanics, among others. The fractional derivative has also been used in various other physical problems, such as frequency-dependent damping behavior of structures, motion of a plate in a Newtonian fluid, PID controller for the control of dynamical systems, and many others. The mathematical models in electromagnetics, rheology, viscoelasticity, electrochemistry, control theory, Brownian motion, signal and image processing, fluid dynamics, financial mathematics, and material science are well defined by fractional-order differential equations. Generally, these physical models are demonstrated either by ordinary or partial differential equations. However, modeling these problems by fractional differential equations, on the other hand, can make the physics of the systems more feasible and practical in some cases. In order to know the behavior of these systems, we need to study the solutions of the governing fractional models. The exact solution of fractional differential equations may not always be possible using known classical methods. Generally, the physical models occurring in nature comprise complex phenomena, and it is sometimes challenging to obtain the solution (both analytical and numerical) of nonlinear differential equations of fractional order. Various aspects of mathematical modeling that may include deterministic or uncertain (viz. fuzzy or interval or stochastic) scenarios along with fractional order (singular/non-singular kernels) are important to understand the dynamical systems. Computation and Modeling for Fractional Order Systems covers various types of fractional order models in deterministic and non-deterministic scenarios. Various analytical/semi-analytical/numerical methods are applied for solving real-life fractional order problems. The comprehensive descriptions of different recently developed fractional singular, non-singular, fractal-fractional, and discrete fractional operators, along with computationally efficient methods, are included for the reader to understand how these may be applied to real-world systems, and a wide variety of dynamical systems such as deterministic, stochastic, continuous, and discrete are addressed by the authors of the book.

Time-Fractional Order Biological Systems with Uncertain Parameters

Author : Snehashish Chakraverty
Publisher : Springer Nature
ISBN 13 : 3031024230
Total Pages : 144 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Time-Fractional Order Biological Systems with Uncertain Parameters by : Snehashish Chakraverty

Download or read book Time-Fractional Order Biological Systems with Uncertain Parameters written by Snehashish Chakraverty and published by Springer Nature. This book was released on 2022-06-01 with total page 144 pages. Available in PDF, EPUB and Kindle. Book excerpt: The subject of fractional calculus has gained considerable popularity and importance during the past three decades, mainly due to its validated applications in various fields of science and engineering. It is a generalization of ordinary differentiation and integration to arbitrary (non-integer) order. The fractional derivative has been used in various physical problems, such as frequency-dependent damping behavior of structures, biological systems, motion of a plate in a Newtonian fluid, λμ controller for the control of dynamical systems, and so on. It is challenging to obtain the solution (both analytical and numerical) of related nonlinear partial differential equations of fractional order. So for the last few decades, a great deal of attention has been directed towards the solution for these kind of problems. Different methods have been developed by other researchers to analyze the above problems with respect to crisp (exact) parameters. However, in real-life applications such as for biological problems, it is not always possible to get exact values of the associated parameters due to errors in measurements/experiments, observations, and many other errors. Therefore, the associated parameters and variables may be considered uncertain. Here, the uncertainties are considered interval/fuzzy. Therefore, the development of appropriate efficient methods and their use in solving the mentioned uncertain problems are the recent challenge. In view of the above, this book is a new attempt to rigorously present a variety of fuzzy (and interval) time-fractional dynamical models with respect to different biological systems using computationally efficient method. The authors believe this book will be helpful to undergraduates, graduates, researchers, industry, faculties, and others throughout the globe.

Fractional-Order Modeling of Dynamic Systems with Applications in Optimization, Signal Processing, and Control

Author : Ahmed G. Radwan
Publisher : Academic Press
ISBN 13 : 0323902030
Total Pages : 530 pages
Book Rating : 4.3/5 (239 download)

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Book Synopsis Fractional-Order Modeling of Dynamic Systems with Applications in Optimization, Signal Processing, and Control by : Ahmed G. Radwan

Download or read book Fractional-Order Modeling of Dynamic Systems with Applications in Optimization, Signal Processing, and Control written by Ahmed G. Radwan and published by Academic Press. This book was released on 2021-10-22 with total page 530 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractional-order Modelling of Dynamic Systems with Applications in Optimization, Signal Processing and Control introduces applications from a design perspective, helping readers plan and design their own applications. The book includes the different techniques employed to design fractional-order systems/devices comprehensively and straightforwardly. Furthermore, mathematics is available in the literature on how to solve fractional-order calculus for system applications. This book introduces the mathematics that has been employed explicitly for fractional-order systems. It will prove an excellent material for students and scholars who want to quickly understand the field of fractional-order systems and contribute to its different domains and applications. Fractional-order systems are believed to play an essential role in our day-to-day activities. Therefore, several researchers around the globe endeavor to work in the different domains of fractional-order systems. The efforts include developing the mathematics to solve fractional-order calculus/systems and to achieve the feasible designs for various applications of fractional-order systems. - Presents a simple and comprehensive understanding of the field of fractional-order systems - Offers practical knowledge on the design of fractional-order systems for different applications - Exposes users to possible new applications for fractional-order systems

Affine Arithmetic Based Solution of Uncertain Static and Dynamic Problems

Author : Snehashish Chakraverty
Publisher : Springer Nature
ISBN 13 : 3031024249
Total Pages : 152 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Affine Arithmetic Based Solution of Uncertain Static and Dynamic Problems by : Snehashish Chakraverty

Download or read book Affine Arithmetic Based Solution of Uncertain Static and Dynamic Problems written by Snehashish Chakraverty and published by Springer Nature. This book was released on 2022-05-31 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: Uncertainty is an inseparable component of almost every measurement and occurrence when dealing with real-world problems. Finding solutions to real-life problems in an uncertain environment is a difficult and challenging task. As such, this book addresses the solution of uncertain static and dynamic problems based on affine arithmetic approaches. Affine arithmetic is one of the recent developments designed to handle such uncertainties in a different manner which may be useful for overcoming the dependency problem and may compute better enclosures of the solutions. Further, uncertain static and dynamic problems turn into interval and/or fuzzy linear/nonlinear systems of equations and eigenvalue problems, respectively. Accordingly, this book includes newly developed efficient methods to handle the said problems based on the affine and interval/fuzzy approach. Various illustrative examples concerning static and dynamic problems of structures have been investigated in order to show the reliability and efficacy of the developed approaches.

Justified Modeling Frameworks and Novel Interpretations of Ecological and Epidemiological Systems

Author : Bapan Ghosh
Publisher : Frontiers Media SA
ISBN 13 : 2832540147
Total Pages : 151 pages
Book Rating : 4.8/5 (325 download)

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Book Synopsis Justified Modeling Frameworks and Novel Interpretations of Ecological and Epidemiological Systems by : Bapan Ghosh

Download or read book Justified Modeling Frameworks and Novel Interpretations of Ecological and Epidemiological Systems written by Bapan Ghosh and published by Frontiers Media SA. This book was released on 2024-01-12 with total page 151 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Lotka-Volterra and the Kermack-McKendrick models are well celebrated and widely recognized in the field of ecology and epidemiology. Several modified ordinary differential equation models have been proposed over the last many decades to rationalize complex biological phenomena. In the current century, researchers have paid much attention to developing new modeling frameworks with delay differential equations, difference equations, fractional order systems, stochastic differential equations, etc. No doubt, these models have emerged many new bifurcations theory and methods which have equally contributed to the advances of Mathematics and interdisciplinary research. It is argued that these new modeling frameworks perform more effectively in analyzing and interpreting results compared to the conventional modeling frameworks with ordinary differential equations. However, implications of emerged bifurcations from new modeling approaches are often less interpreted from a biological viewpoint. Even, there is also a lack of understanding of how a fractional order model, for instance, displays a more realistic scenario to analyze a biological process. Therefore, a more serious justification is essential while modeling any biological event.

Advances in Differential and Difference Equations with Applications 2020

Author : Dumitru Baleanu
Publisher : MDPI
ISBN 13 : 3039368702
Total Pages : 348 pages
Book Rating : 4.0/5 (393 download)

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Book Synopsis Advances in Differential and Difference Equations with Applications 2020 by : Dumitru Baleanu

Download or read book Advances in Differential and Difference Equations with Applications 2020 written by Dumitru Baleanu and published by MDPI. This book was released on 2021-01-20 with total page 348 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is very well known that differential equations are related with the rise of physical science in the last several decades and they are used successfully for models of real-world problems in a variety of fields from several disciplines. Additionally, difference equations represent the discrete analogues of differential equations. These types of equations started to be used intensively during the last several years for their multiple applications, particularly in complex chaotic behavior. A certain class of differential and related difference equations is represented by their respective fractional forms, which have been utilized to better describe non-local phenomena appearing in all branches of science and engineering. The purpose of this book is to present some common results given by mathematicians together with physicists, engineers, as well as other scientists, for whom differential and difference equations are valuable research tools. The reported results can be used by researchers and academics working in both pure and applied differential equations.

Stability and Control of Nonlinear Time-varying Systems

Author : Shuli Guo
Publisher : Springer
ISBN 13 : 9811089086
Total Pages : 268 pages
Book Rating : 4.8/5 (11 download)

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Book Synopsis Stability and Control of Nonlinear Time-varying Systems by : Shuli Guo

Download or read book Stability and Control of Nonlinear Time-varying Systems written by Shuli Guo and published by Springer. This book was released on 2018-04-12 with total page 268 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents special systems derived from industrial models, including the complex saturation nonlinear functions and the delay nonlinear functions. It also presents typical methods, such as the classical Liapunov and Integral Inequalities methods. Providing constructive qualitative and stability conditions for linear systems with saturated inputs in both global and local contexts, it offers practitioners more concise model systems for modern saturation nonlinear techniques, which have the potential for future applications. This book is a valuable guide for researchers and graduate students in the fields of mathematics, control, and engineering.

Intelligent Fractional Order Systems and Control

Author : Indranil Pan
Publisher : Springer
ISBN 13 : 3642315496
Total Pages : 305 pages
Book Rating : 4.6/5 (423 download)

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Book Synopsis Intelligent Fractional Order Systems and Control by : Indranil Pan

Download or read book Intelligent Fractional Order Systems and Control written by Indranil Pan and published by Springer. This book was released on 2012-08-23 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fractional order calculus is finding increasing interest in the control system community. Hardware realizations of fractional order controllers have sparked off a renewed zeal into the investigations of control system design in the light of fractional calculus. As such many notions of integer order LTI systems are being modified and extended to incorporate these new concepts. Computational Intelligence (CI) techniques have been applied to engineering problems to find solutions to many hitherto intractable conundrums and is a useful tool for dealing with problems of higher computational complexity. This book borders on the interface between CI techniques and fractional calculus, and looks at ways in which fractional order control systems may be designed or enhanced using CI based paradigms. To the best of the author’s knowledge this is the first book of its kind exclusively dedicated to the application of computational intelligence techniques in fractional order systems and control. The book tries to assimilate various existing concepts in this nascent field of fractional order intelligent control and is aimed at researchers and post graduate students working in this field.

Multipoint Methods for Solving Nonlinear Equations

Author : Miodrag Petkovic
Publisher : Academic Press
ISBN 13 : 0123972981
Total Pages : 317 pages
Book Rating : 4.1/5 (239 download)

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Book Synopsis Multipoint Methods for Solving Nonlinear Equations by : Miodrag Petkovic

Download or read book Multipoint Methods for Solving Nonlinear Equations written by Miodrag Petkovic and published by Academic Press. This book was released on 2012-12-31 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the first on the topic and explains the most cutting-edge methods needed for precise calculations and explores the development of powerful algorithms to solve research problems. Multipoint methods have an extensive range of practical applications significant in research areas such as signal processing, analysis of convergence rate, fluid mechanics, solid state physics, and many others. The book takes an introductory approach in making qualitative comparisons of different multipoint methods from various viewpoints to help the reader understand applications of more complex methods. Evaluations are made to determine and predict efficiency and accuracy of presented models useful to wide a range of research areas along with many numerical examples for a deep understanding of the usefulness of each method. This book will make it possible for the researchers to tackle difficult problems and deepen their understanding of problem solving using numerical methods. Multipoint methods are of great practical importance, as they determine sequences of successive approximations for evaluative purposes. This is especially helpful in achieving the highest computational efficiency. The rapid development of digital computers and advanced computer arithmetic have provided a need for new methods useful to solving practical problems in a multitude of disciplines such as applied mathematics, computer science, engineering, physics, financial mathematics, and biology. - Provides a succinct way of implementing a wide range of useful and important numerical algorithms for solving research problems - Illustrates how numerical methods can be used to study problems which have applications in engineering and sciences, including signal processing, and control theory, and financial computation - Facilitates a deeper insight into the development of methods, numerical analysis of convergence rate, and very detailed analysis of computational efficiency - Provides a powerful means of learning by systematic experimentation with some of the many fascinating problems in science - Includes highly efficient algorithms convenient for the implementation into the most common computer algebra systems such as Mathematica, MatLab, and Maple

Advances and Applications in Chaotic Systems

Author : Sundarapandian Vaidyanathan
Publisher : Springer
ISBN 13 : 3319302795
Total Pages : 599 pages
Book Rating : 4.3/5 (193 download)

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Book Synopsis Advances and Applications in Chaotic Systems by : Sundarapandian Vaidyanathan

Download or read book Advances and Applications in Chaotic Systems written by Sundarapandian Vaidyanathan and published by Springer. This book was released on 2016-03-22 with total page 599 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book reports on the latest advances and applications of chaotic systems. It consists of 25 contributed chapters by experts who are specialized in the various topics addressed in this book. The chapters cover a broad range of topics of chaotic systems such as chaos, hyperchaos, jerk systems, hyperjerk systems, conservative and dissipative systems, circulant chaotic systems, multi-scroll chaotic systems, finance chaotic system, highly chaotic systems, chaos control, chaos synchronization, circuit realization and applications of chaos theory in secure communications, mobile robot, memristors, cellular neural networks, etc. Special importance was given to chapters offering practical solutions, modeling and novel control methods for the recent research problems in chaos theory. This book will serve as a reference book for graduate students and researchers with a basic knowledge of chaos theory and control systems. The resulting design procedures on the chaotic systems are emphasized using MATLAB software.

Mathematical Methods in Interdisciplinary Sciences

Author : Snehashish Chakraverty
Publisher : John Wiley & Sons
ISBN 13 : 1119585503
Total Pages : 464 pages
Book Rating : 4.1/5 (195 download)

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Book Synopsis Mathematical Methods in Interdisciplinary Sciences by : Snehashish Chakraverty

Download or read book Mathematical Methods in Interdisciplinary Sciences written by Snehashish Chakraverty and published by John Wiley & Sons. This book was released on 2020-07-15 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: Brings mathematics to bear on your real-world, scientific problems Mathematical Methods in Interdisciplinary Sciences provides a practical and usable framework for bringing a mathematical approach to modelling real-life scientific and technological problems. The collection of chapters Dr. Snehashish Chakraverty has provided describe in detail how to bring mathematics, statistics, and computational methods to the fore to solve even the most stubborn problems involving the intersection of multiple fields of study. Graduate students, postgraduate students, researchers, and professors will all benefit significantly from the author's clear approach to applied mathematics. The book covers a wide range of interdisciplinary topics in which mathematics can be brought to bear on challenging problems requiring creative solutions. Subjects include: Structural static and vibration problems Heat conduction and diffusion problems Fluid dynamics problems The book also covers topics as diverse as soft computing and machine intelligence. It concludes with examinations of various fields of application, like infectious diseases, autonomous car and monotone inclusion problems.

Fuzzy Systems and Data Mining VII

Author : C. Shen
Publisher : IOS Press
ISBN 13 : 1643682156
Total Pages : 494 pages
Book Rating : 4.6/5 (436 download)

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Book Synopsis Fuzzy Systems and Data Mining VII by : C. Shen

Download or read book Fuzzy Systems and Data Mining VII written by C. Shen and published by IOS Press. This book was released on 2021-11-04 with total page 494 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fuzzy systems and data mining are indispensible aspects of the computer systems and algorithms on which the world has come to depend. This book presents papers from FSDM 2021, the 7th International Conference on Fuzzy Systems and Data Mining. The conference, originally due to take place in Seoul, South Korea, was held online on 26-29 October 2021, due to ongoing restrictions connected with the COVID-19 pandemic. The annual FSDM conference provides a platform for knowledge exchange between international experts, researchers, academics and delegates from industry. This year, the committee received 266 submissions, and this book contains 52 papers, including keynotes and invited presentations, oral and poster contributions. The papers cover four main areas: 1) fuzzy theory, algorithms and systems – including topics like stability; 2) fuzzy applications – which are widely used and cover various types of processing as well as hardware and architecture for big data and time series; 3) the interdisciplinary field of fuzzy logic and data mining; and 4) data mining itself. The topic most frequently addressed this year is fuzzy systems. The book offers an overview of research and developments in fuzzy logic and data mining, and will be of interest to all those working in the field of data science.

Fuzzy Logic and Mathematics

Author : Radim Bělohlávek
Publisher : Oxford University Press
ISBN 13 : 0190200014
Total Pages : 545 pages
Book Rating : 4.1/5 (92 download)

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Book Synopsis Fuzzy Logic and Mathematics by : Radim Bělohlávek

Download or read book Fuzzy Logic and Mathematics written by Radim Bělohlávek and published by Oxford University Press. This book was released on 2017 with total page 545 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main part of the book is a comprehensive overview of the development of fuzzy logic and its applications in various areas of human affair since its genesis in the mid 1960s. This overview is then employed for assessing the significance of fuzzy logic and mathematics based on fuzzy logic.

Fuzzy Arbitrary Order System

Author : Snehashish Chakraverty
Publisher :
ISBN 13 :
Total Pages : pages
Book Rating : 4.:/5 (96 download)

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Book Synopsis Fuzzy Arbitrary Order System by : Snehashish Chakraverty

Download or read book Fuzzy Arbitrary Order System written by Snehashish Chakraverty and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Fuzzy Differential Equations and Applications for Engineers and Scientists

Author : S. Chakraverty
Publisher : CRC Press
ISBN 13 : 1315355531
Total Pages : 138 pages
Book Rating : 4.3/5 (153 download)

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Book Synopsis Fuzzy Differential Equations and Applications for Engineers and Scientists by : S. Chakraverty

Download or read book Fuzzy Differential Equations and Applications for Engineers and Scientists written by S. Chakraverty and published by CRC Press. This book was released on 2016-11-25 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: Differential equations play a vital role in the modeling of physical and engineering problems, such as those in solid and fluid mechanics, viscoelasticity, biology, physics, and many other areas. In general, the parameters, variables and initial conditions within a model are considered as being defined exactly. In reality there may be only vague, imprecise or incomplete information about the variables and parameters available. This can result from errors in measurement, observation, or experimental data; application of different operating conditions; or maintenance induced errors. To overcome uncertainties or lack of precision, one can use a fuzzy environment in parameters, variables and initial conditions in place of exact (fixed) ones, by turning general differential equations into Fuzzy Differential Equations ("FDEs"). In real applications it can be complicated to obtain exact solution of fuzzy differential equations due to complexities in fuzzy arithmetic, creating the need for use of reliable and efficient numerical techniques in the solution of fuzzy differential equations. These include fuzzy ordinary and partial, fuzzy linear and nonlinear, and fuzzy arbitrary order differential equations. This unique work?provides a new direction for the reader in the use of basic concepts of fuzzy differential equations, solutions and its applications. It can serve as an essential reference work for students, scholars, practitioners, researchers and academicians in engineering and science who need to model uncertain physical problems.

Fuzzy Control and Identification

Author : John H. Lilly
Publisher : John Wiley & Sons
ISBN 13 : 1118097815
Total Pages : 199 pages
Book Rating : 4.1/5 (18 download)

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Book Synopsis Fuzzy Control and Identification by : John H. Lilly

Download or read book Fuzzy Control and Identification written by John H. Lilly and published by John Wiley & Sons. This book was released on 2011-03-10 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives an introduction to basic fuzzy logic and Mamdani and Takagi-Sugeno fuzzy systems. The text shows how these can be used to control complex nonlinear engineering systems, while also also suggesting several approaches to modeling of complex engineering systems with unknown models. Finally, fuzzy modeling and control methods are combined in the book, to create adaptive fuzzy controllers, ending with an example of an obstacle-avoidance controller for an autonomous vehicle using modus ponendo tollens logic.