The document discusses fuzzy reasoning and fuzzy inferencing. It describes the three main steps in fuzzy inferencing: 1) fuzzification, which transforms crisp inputs into fuzzy inputs using membership functions, 2) rule evaluation using IF-THEN rules with fuzzy logic operators, and 3) defuzzification to produce crisp outputs from fuzzy outputs using methods like center of gravity. It provides examples of applications that have benefited from fuzzy systems like cement kiln control and expert systems. Finally, it presents some questions related to fuzzy logic concepts.
FUZZY CONTROL OF A SERVOMECHANISM: PRACTICAL APPROACH USING MAMDANI AND TAKAG...ijfls
The main objective of this work is to propose two fuzzy controllers: one based on the Mamdani inference
method and another controller based on the Takagi- Sugeno inference method, both will be designed for
application in a position control system of a servomechanism. Some comparations between the methods
mentioned above will be made with regard to the performance of the system in order to identify the
advantages of the Takagi- Sugeno method in relation to the Mamdani method in the presence of
disturbances and nonlinearities of the system. Some results of simulation and practical application are
presented and results obtained showed that controllers based on Takagi- Sugeno method is more efficient
than controllers based on Mamdani method for this specific application.
FUZZY CONTROL OF A SERVOMECHANISM: PRACTICAL APPROACH USING MAMDANI AND TAKAG...ijfls
The main objective of this work is to propose two fuzzy controllers: one based on the Mamdani inference
method and another controller based on the Takagi- Sugeno inference method, both will be designed for
application in a position control system of a servomechanism. Some comparations between the methods
mentioned above will be made with regard to the performance of the system in order to identify the
advantages of the Takagi- Sugeno method in relation to the Mamdani method in the presence of
disturbances and nonlinearities of the system. Some results of simulation and practical application are
presented and results obtained showed that controllers based on Takagi- Sugeno method is more efficient
than controllers based on Mamdani method for this specific application.
The Fuzzy Logic is discussed with three simple example problems all solved in MATLAB
1. Restaurant Problem
2. Temperature Controller
3. Washing Machine Problem
A step-by-step complete guide for Logistic Regression Classifier especially mentioning its Decision/Activation Function, Objective Function and Objective Function Optimization procedures.
Bayesian analysis of shape parameter of Lomax distribution using different lo...Premier Publishers
The Lomax distribution also known as Pareto distribution of the second kind or Pearson Type VI distribution has been used in the analysis of income data, and business failure data. It may describe the lifetime of a decreasing failure rate component as a heavy tailed alternative to the exponential distribution. In this paper we consider the estimation of the parameter of Lomax distribution. Baye’s estimator is obtained by using Jeffery’s and extension of Jeffery’s prior by using squared error loss function, Al-Bayyati’s loss function and Precautionary loss function. Maximum likelihood estimation is also discussed. These methods are compared by using mean square error through simulation study with varying sample sizes. The study aims to find out a suitable estimator of the parameter of the distribution. Finally, we analyze one data set for illustration.
ANALYTICAL FORMULATIONS FOR THE LEVEL BASED WEIGHTED AVERAGE VALUE OF DISCRET...ijsc
In fuzzy decision-making processes based on linguistic information, operations on discrete fuzzy numbers
are commonly performed. Aggregation and defuzzification operations are some of these often used
operations. Many aggregation and defuzzification operators produce results independent to the decisionmaker’s
strategy. On the other hand, the Weighted Average Based on Levels (WABL) approach can take
into account the level weights and the decision maker's "optimism" strategy. This gives flexibility to the
WABL operator and, through machine learning, can be trained in the direction of the decision maker's
strategy, producing more satisfactory results for the decision maker. However, in order to determine the
WABL value, it is necessary to calculate some integrals. In this study, the concept of WABL for discrete
trapezoidal fuzzy numbers is investigated, and analytical formulas have been proven to facilitate the
calculation of WABL value for these fuzzy numbers. Trapezoidal and their special form, triangular fuzzy
numbers, are the most commonly used fuzzy number types in fuzzy modeling, so in this study, such numbers
have been studied. Computational examples explaining the theoretical results have been performed.
FACTORS AFFECTING SEASONING OF TIMBER USING SAWDUST OPERATED KILN Hiran Amarasekera
P L A M C Wijewarnasuriya and H S Amarasekera
University of Sri Jayewardenepura, Sri Lanka
International Forestry and Environment Symposium 2010 Annual Symposium organized by Department of Forestry and Environmental Science, University of Sri Jayewardenepura, Nugegoda, Sri Lanka http://fesympo.sjp.ac.lk/
Full Paper
http://staff.sjp.ac.lk/hiran/publications/factors-affecting-seasoning-timber-using-sawdust-operated-kiln
The Fuzzy Logic is discussed with three simple example problems all solved in MATLAB
1. Restaurant Problem
2. Temperature Controller
3. Washing Machine Problem
A step-by-step complete guide for Logistic Regression Classifier especially mentioning its Decision/Activation Function, Objective Function and Objective Function Optimization procedures.
Bayesian analysis of shape parameter of Lomax distribution using different lo...Premier Publishers
The Lomax distribution also known as Pareto distribution of the second kind or Pearson Type VI distribution has been used in the analysis of income data, and business failure data. It may describe the lifetime of a decreasing failure rate component as a heavy tailed alternative to the exponential distribution. In this paper we consider the estimation of the parameter of Lomax distribution. Baye’s estimator is obtained by using Jeffery’s and extension of Jeffery’s prior by using squared error loss function, Al-Bayyati’s loss function and Precautionary loss function. Maximum likelihood estimation is also discussed. These methods are compared by using mean square error through simulation study with varying sample sizes. The study aims to find out a suitable estimator of the parameter of the distribution. Finally, we analyze one data set for illustration.
ANALYTICAL FORMULATIONS FOR THE LEVEL BASED WEIGHTED AVERAGE VALUE OF DISCRET...ijsc
In fuzzy decision-making processes based on linguistic information, operations on discrete fuzzy numbers
are commonly performed. Aggregation and defuzzification operations are some of these often used
operations. Many aggregation and defuzzification operators produce results independent to the decisionmaker’s
strategy. On the other hand, the Weighted Average Based on Levels (WABL) approach can take
into account the level weights and the decision maker's "optimism" strategy. This gives flexibility to the
WABL operator and, through machine learning, can be trained in the direction of the decision maker's
strategy, producing more satisfactory results for the decision maker. However, in order to determine the
WABL value, it is necessary to calculate some integrals. In this study, the concept of WABL for discrete
trapezoidal fuzzy numbers is investigated, and analytical formulas have been proven to facilitate the
calculation of WABL value for these fuzzy numbers. Trapezoidal and their special form, triangular fuzzy
numbers, are the most commonly used fuzzy number types in fuzzy modeling, so in this study, such numbers
have been studied. Computational examples explaining the theoretical results have been performed.
FACTORS AFFECTING SEASONING OF TIMBER USING SAWDUST OPERATED KILN Hiran Amarasekera
P L A M C Wijewarnasuriya and H S Amarasekera
University of Sri Jayewardenepura, Sri Lanka
International Forestry and Environment Symposium 2010 Annual Symposium organized by Department of Forestry and Environmental Science, University of Sri Jayewardenepura, Nugegoda, Sri Lanka http://fesympo.sjp.ac.lk/
Full Paper
http://staff.sjp.ac.lk/hiran/publications/factors-affecting-seasoning-timber-using-sawdust-operated-kiln
Although fuzzy systems demonstrate their ability to
solve different kinds of problems in various applications, there is an increasing interest on developing solid mathematical implementations suitable for control applications such as that used in fuzzy logic controllers (FLC). It is well known that, wide range of parameters is needed to be specified before the construction of a fuzzy system. To simplify in a systematic way the design and construction of a general fuzzy system, and without loss for generality a full parameterization process for a singleton type FLC is proposed in this paper. The resented methodology is very helpful in developing a universal computing algorithm for a standard fuzzy like PID controllers. An illustrative example shows the simplicity of applying the new paradigm.
Fuzzy Control of a Servomechanism: Practical Approach using Mamdani and Takag...ijfls
The main objective of this work is to propose two fuzzy controllers: one based on the Mamdani inference method and another controller based on the Takagi- Sugeno inference method, both will be designed for application in a position control system of a servomechanism. Some comparations between the methods mentioned above will be made with regard to the performance of the system in order to identify the advantages of the Takagi- Sugeno method in relation to the Mamdani method in the presence of disturbances and nonlinearities of the system. Some results of simulation and practical application are presented and results obtained showed that controllers based on Takagi- Sugeno method is more efficient than controllers based on Mamdani method for this specific application.
A Mixed Binary-Real NSGA II Algorithm Ensuring Both Accuracy and Interpretabi...IJECEIAES
In this work, a Neuro-Fuzzy Controller network, called NFC that implements a Mamdani fuzzy inference system is proposed. This network includes neurons able to perform fundamental fuzzy operations. Connections between neurons are weighted through binary and real weights. Then a mixed binaryreal Non dominated Sorting Genetic Algorithm II (NSGA II) is used to perform both accuracy and interpretability of the NFC by minimizing two objective functions; one objective relates to the number of rules, for compactness, while the second is the mean square error, for accuracy. In order to preserve interpretability of fuzzy rules during the optimization process, some constraints are imposed. The approach is tested on two control examples: a single input single output (SISO) system and a multivariable (MIMO) system.
VALIDATION METHOD OF FUZZY ASSOCIATION RULES BASED ON FUZZY FORMAL CONCEPT AN...cscpconf
In order to treat and analyze real datasets, fuzzy association rules have been proposed. Several
algorithms have been introduced to extract these rules. However, these algorithms suffer from
the problems of utility, redundancy and large number of extracted fuzzy association rules. The
expert will then be confronted with this huge amount of fuzzy association rules. The task of
validation becomes fastidious. In order to solve these problems, we propose a new validation
method. Our method is based on three steps. (i) We extract a generic base of non redundant
fuzzy association rules by applying EFAR-PN algorithm based on fuzzy formal concept analysis.
(ii) we categorize extracted rules into groups and (iii) we evaluate the relevance of these rules
using structural equation model.
User_42751212015Module1and2pagestocompetework.pdf
User_42751212015Module1and2pagestocompetework_1.pdf
User_42751212015Module2Homework(CIS330).docx
[INSERT TITLE HERE] 1
Running head: [INSERT TITLE HERE]
[INSERT TITLE HERE]
Student Name
Allied American University
Author Note
This paper was prepared for [INSERT COURSE NAME], [INSERT COURSE ASSIGNMENT] taught by [INSERT INSTRUCTOR’S NAME].
Directions: Please complete each of the following exercises. Please read the instructions carefully.
For all “short programming assignments,” include source code files in your submission.
1. Short programming assignment. Combine the malloc2D function of program 3.16 with the adjacency matrix code of program 3.18 to write a program that allows the user to first enter the count of vertices, and then enter the graph edges. The program should then output the graph with lines of the form:
There is an edge between 0 and 3.
2. Short programming assignment. Modify your program for question 2.1 so that after the adjacency matrix is created, it is then converted to an adjacency list, and the output is generated from the list.
3. Short programming assignment. Modify program 4.7 from the text, overloading the == operator to work for this ADT using a friend function.
4. Is the ADT given in program 4.7 a first-class ADT? Explain your answer.
5. Suppose you are given the source code for a C++ class, and asked if the class shown is an ADT. On what factors would your decision be based?
6. How does using strings instead of simple types like integers alter the O-notation of operations?
User_42751212015Module1Homework(CIS330)Corrected (1).docx
[INSERT TITLE HERE] 1
Running head: [INSERT TITLE HERE]
[INSERT TITLE HERE]
Student Name
Allied American University
Author Note
This paper was prepared for [INSERT COURSE NAME], [INSERT COURSE ASSIGNMENT] taught by [INSERT INSTRUCTOR’S NAME].
Directions: Please refer to your textbook to complete the following exercises.1. Refer to page 12 of your text to respond to the following:Show the contents of the id array after each union operation when you use the quick find algorithm (Program I.I) to solve the connectivity problem for the sequence 0-2, 1-4, 2-5, 3-6, 0-4, 6-0, and 1-3. Also give the number of times the program accesses the id array for each input pair.2. Refer to page 12 of your text to respond to the following:Show the contents of the id array after each union operation when you use the quick union algorithm (Program I.I) to solve the connectivity problem for the sequence 0-2, 1-4, 2-5, 3-6, 0-4, 6-0, and 1-3. Also give the number of times the program accesses the id array for each input pair.3. Refer to figures 1.7 and 1.8 on pages 16 and 17 of the text. Give the contents of the id array after each union operation for the weighted quick union algorithm running on the examples corresponding to figures 1.7 and 1.84. For what value is N is 10N lg N>2N2 ...
AN IMPROVED DECISION SUPPORT SYSTEM BASED ON THE BDM (BIT DECISION MAKING) ME...ijmpict
Based on the BDM (Bit Decision Making) method, the present work presents two contributions: first, the
illustration of the use of the technique known as SOP (Sum Of Products) in order to systematize the
process to obtain the correlation function for sub-system’s mathematical modelling, and second,the provision of capacity to manage a greater than binary but a finite - discrete set of possible subjective qualifications of suppliers at any criterion.
The critical routines within signal processing algorithms are typically data intensive and iterate many times, carrying out the same functions on multi-dimensional arrays and input streams. These routines use the vast majority of an implementation’s resources, and for the longest time over the course of the algorithms execution. Such routines are classed as Nested-Loop Programs. Whether the implementation is software running on a processor or a higher performance customized hardware implementation, there is always a tradeoff between the throughput performance (execution-time) and the resources used such as the amount of processor memory and registers in a software implementation or the gate-count in a hardware implementation. This article shows the reader techniques and methods for manipulating a signal processing algorithm, in particular those conforming to a Nested-Loop Program, and the different generic implementation architectures along this tradeoff spectrum, as well as the different methods for describing and analyzing an algorithm implementation. Each of these methods for describing an algorithm is shown to correspond to a different abstraction-level view and as such exposes different features and properties for ease of analysis and manipulation. Manipulation techniques, mainly algorithmic-transformations are described and it is shown how these transformations take an implementation and form a new one at a different point in the trade-off spectrum of resources used versus throughput by performing calculations in a different order and a different way to arrive at the same result.
AN OPTIMAL FUZZY LOGIC SYSTEM FOR A NONLINEAR DYNAMIC SYSTEM USING A FUZZY BA...IJCNCJournal
The impetus for this paper is the development of Fuzzy Basis Function “FBF” that assigns in an optimal fashion, a function approximation for a nonlinear dynamic system. A fuzzy basis function is applied to find the best location of the characteristic points by specifying the set of fuzzy rules. The advantage of this technique is that, it may produce a simple and well-performing system because it selects the most significant fuzzy basis functions to minimize an objective function in the output error for the fuzzy rules. The fuzzy basis function is a linguistic fuzzy IF_THEN rule. It provides a combination of the numerical information and the linguistic information in the form input-output pairs and in the form of fuzzy rules. The proposed control scheme is applied to a magnetic ball suspension system.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Connector Corner: Automate dynamic content and events by pushing a buttonDianaGray10
Here is something new! In our next Connector Corner webinar, we will demonstrate how you can use a single workflow to:
Create a campaign using Mailchimp with merge tags/fields
Send an interactive Slack channel message (using buttons)
Have the message received by managers and peers along with a test email for review
But there’s more:
In a second workflow supporting the same use case, you’ll see:
Your campaign sent to target colleagues for approval
If the “Approve” button is clicked, a Jira/Zendesk ticket is created for the marketing design team
But—if the “Reject” button is pushed, colleagues will be alerted via Slack message
Join us to learn more about this new, human-in-the-loop capability, brought to you by Integration Service connectors.
And...
Speakers:
Akshay Agnihotri, Product Manager
Charlie Greenberg, Host
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
Neuro-symbolic (NeSy) AI is on the rise. However, simply machine learning on just any symbolic structure is not sufficient to really harvest the gains of NeSy. These will only be gained when the symbolic structures have an actual semantics. I give an operational definition of semantics as “predictable inference”.
All of this illustrated with link prediction over knowledge graphs, but the argument is general.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024
Lesson 32
1. Module
11
Reasoning with
uncertainty-Fuzzy
Reasoning
Version 2 CSE IIT, Kharagpur
2. Lesson
32
Fuzzy Reasoning -
Continued
Version 2 CSE IIT, Kharagpur
3. 11.4 Fuzzy Inferencing
The process of fuzzy reasoning is incorporated into what is called a Fuzzy Inferencing
System. It is comprised of three steps that process the system inputs to the appropriate
system outputs. These steps are 1) Fuzzification, 2) Rule Evaluation, and
3) Defuzzification. The system is illustrated in the following figure.
Each step of fuzzy inferencing is described in the following sections.
11.4.1 Fuzzification
Fuzzification is the first step in the fuzzy inferencing process. This involves a domain
transformation where crisp inputs are transformed into fuzzy inputs. Crisp inputs are
exact inputs measured by sensors and passed into the control system for processing, such
as temperature, pressure, rpm's, etc.. Each crisp input that is to be processed by the FIU
has its own group of membership functions or sets to which they are transformed. This
group of membership functions exists within a universe of discourse that holds all
relevant values that the crisp input can possess. The following shows the structure of
membership functions within a universe of discourse for a crisp input.
Version 2 CSE IIT, Kharagpur
4. where:
degree of membership: degree to which a crisp value is compatible to a membership
function, value from 0 to 1, also known as truth value or fuzzy input.
membership function, MF: defines a fuzzy set by mapping crisp values from its domain
to the sets associated degree of membership.
crisp inputs: distinct or exact inputs to a certain system variable, usually measured
parameters external from the control system, e.g. 6 Volts.
label: descriptive name used to identify a membership function.
scope: or domain, the width of the membership function, the range of concepts, usually
numbers, over which a membership function is mapped.
universe of discourse: range of all possible values, or concepts, applicable to a system
variable.
When designing the number of membership functions for an input variable, labels must
initially be determined for the membership functions. The number of labels correspond to
the number of regions that the universe should be divided, such that each label describes
a region of behavior. A scope must be assigned to each membership function that
numerically identifies the range of input values that correspond to a label.
The shape of the membership function should be representative of the variable. However
this shape is also restricted by the computing resources available. Complicated shapes
require more complex descriptive equations or large lookup tables. The next figure
shows examples of possible shapes for membership functions.
Version 2 CSE IIT, Kharagpur
5. When considering the number of membership functions to exist within the universe of
discourse, one must consider that:
i) too few membership functions for a given application will cause the response of the
system to be too slow and fail to provide sufficient output control in time to recover from
a small input change. This may also cause oscillation in the system.
ii) too many membership functions may cause rapid firing of different rule consequents
for small changes in input, resulting in large output changes, which may cause instability
in the system.
These membership functions should also be overlapped. No overlap reduces a system
based on Boolean logic. Every input point on the universe of discourse should belong to
the scope of at least one but no more than two membership functions. No two
membership functions should have the same point of maximum truth, (1). When two
membership functions overlap, the sum of truths or grades for any point within the
overlap should be less than or equal to 1. Overlap should not cross the point of maximal
truth of either membership function. Marsh has proposed two indices to describe the
overlap of membership functions quantitatively. These are overlap ratio and overlap
robustness. The next figure illustrates their meaning.
Version 2 CSE IIT, Kharagpur
6. The fuzzification process maps each crisp input on the universe of discourse, and its
intersection with each membership function is transposed onto the µ axis as illustrated in
the previous figure. These µ values are the degrees of truth for each crisp input and are
associated with each label as fuzzy inputs. These fuzzy inputs are then passed on to the
next step, Rule Evaluation.
Fuzzy Rules
We briefly comment on so-called fuzzy IF-THEN rules introduced by Zadeh. They may
be understood as partial imprecise knowledge on some crisp function and have (in the
simplest case) the form IF x is Ai THEN y is Bi. They should not be immediately
understood as implications; think of a table relating values of a (dependent) variable y to
values of an (independent variable) x:
x A1 ... An
y B1 ... Bn
Ai, Bi may be crisp (concrete numbers) or fuzzy (small, medium, …) It may be
understood in two, in general non-equivalent ways: (1) as a listing of n possibilities,
called Mamdani's formula:
n
MAMD(x,y) ≡ (Ai(x) & Bi(y)).
B
i=1
Version 2 CSE IIT, Kharagpur
7. (where x is A1 and y is B1 or x is A2 and y is B2 or …). (2) as a conjunction of
implications:
n
RULES(x,y) ≡ (Ai(x) → Bi(y)).
B
i=1
(if x is A1 then y is B1 and …).
Both MAMD and RULES define a binary fuzzy relation (given the interpretation of Ai's,
Bi's and truth functions of connectives). Now given a fuzzy input A*(x) one can consider
B
the image B* of A*(x) under this relation, i.e.,
B*(y) ≡ x(A(x) & R(x,y)),
where R(x,y) is MAMD(x,y) (most frequent case) or RULES(x,y). Thus one gets an
operator assigning to each fuzzy input set A* a corresponding fuzzy output B*. Usually
this is combined with some fuzzifications converting a crisp input x0 to some fuzzy A*(x)
(saying something as "x is similar to x0") and a defuzzification converting the fuzzy image
B* to a crisp output y0. Thus one gets a crisp function; its relation to the set of rules may
be analyzed.
11.4.2 Rule Evaluation
Rule evaluation consists of a series of IF-Zadeh Operator-THEN rules. A decision
structure to determine the rules require familiarity with the system and its desired
operation. This knowledge often requires the assistance of interviewing operators and
experts. For this thesis this involved getting information on tremor from medical
practitioners in the field of rehabilitation medicine.
There is a strict syntax to these rules. This syntax is structured as:
IF antecedent 1 ZADEH OPERATOR antecedent 2 ............ THEN consequent 1 ZADEH
OPERATOR consequent 2..............
The antecedent consists of: input variable IS label, and is equal to its associated fuzzy
input or truth value µ(x).
The consequent consists of: output variable IS label, its value depends on the Zadeh
Operator which determines the type of inferencing used. There are three Zadeh
Operators, AND, OR, and NOT. The label of the consequent is associated with its output
membership function. The Zadeh Operator is limited to operating on two membership
functions, as discussed in the fuzzification process. Zadeh Operators are similar to
Boolean Operators such that:
AND represents the intersection or minimum between the two sets, expressed as:
Version 2 CSE IIT, Kharagpur
8. OR represents the union or maximum between the two sets, expressed as:
NOT represents the opposite of the set, expressed as:
The process for determining the result or rule strength of the rule may be done by taking
the minimum fuzzy input of antecedent 1 AND antecedent 2, min. inferencing. This
minimum result is equal to the consequent rule strength. If there are any consequents that
are the same then the maximum rule strength between similar consequents is taken,
referred to as maximum or max. inferencing, hence min./max. inferencing. This infers
that the rule that is most true is taken. These rule strength values are referred to as fuzzy
outputs.
11.4.3 Defuzzification
Defuzzification involves the process of transposing the fuzzy outputs to crisp outputs.
There are a variety of methods to achieve this, however this discussion is limited to the
process used in this thesis design.
A method of averaging is utilized here, and is known as the Center of Gravity method or
COG, it is a method of calculating centroids of sets. The output membership functions to
which the fuzzy outputs are transposed are restricted to being singletons. This is so to
limit the degree of calculation intensity in the microcontroller. The fuzzy outputs are
transposed to their membership functions similarly as in fuzzification. With COG the
singleton values of outputs are calculated using a weighted average, illustrated in the next
figure. The crisp output is the result and is passed out of the fuzzy inferencing system for
processing elsewhere.
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9. 11.5 APPLICATIONS
Areas in which fuzzy logic has been successfully applied are often quite concrete.
The first major commercial application was in the area of cement kiln control, an
operation which requires that an operator monitor four internal states of the kiln,
control four sets of operations, and dynamically manage 40 or 50 "rules of thumb"
about their interrelationships, all with the goal of controlling a highly complex set
of chemical interactions. One such rule is "If the oxygen percentage is rather high
and the free-lime and kiln- drive torque rate is normal, decrease the flow of gas
and slightly reduce the fuel rate".
Other applications which have benefited through the use of fuzzy systems theory
have been information retrieval systems, a navigation system for automatic cars, a
predicative fuzzy-logic controller for automatic operation of trains, laboratory
water level controllers, controllers for robot arc-welders, feature-definition
controllers for robot vision, graphics controllers for automated police sketchers,
and more.
Expert systems have been the most obvious recipients of the benefits of fuzzy
logic, since their domain is often inherently fuzzy. Examples of expert systems
with fuzzy logic central to their control are decision-support systems, financial
planners, diagnostic systems for determining soybean pathology, and a
meteorological expert system in China for determining areas in which to establish
rubber tree orchards.
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10. Questions
1. In a class of 10 students (the universal set), 3 students speaks German to some degree,
namely Alice to degree 0.7, Bob to degree 1.0, Cathrine to degree 0.4. What is the size of
the subset A of German speaking students in the class?
2. In the above class, argue that the fuzzy subset B of students speaking a very good
German is a fuzzy subset of A.
3. Let A and B be fuzzy subsets of a universal set X. Show that
4. For arbitrary fuzzy subsets A and B, show that
5. Let X = {0, 1, 2, …., 6}, and let two fuzzy subsets, A and B, of X be defined by:
Find:
and
Solutions
1. |A| = 0.7 + 1.0 + 0.4 = 2.1
2. The addition of “very” strengthens the requirement, which consequently will be less
satisfied. Thus for all which is precisely what characterized the fuzzy
subset relation
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