SSIE 617 Fuzzy Sets Fuzzy Logic and Fuzzy Systems All Slides

8/2/2019 SSIE 617 Fuzzy Sets Fuzzy Logic and Fuzzy Systems All Slides

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SSIE 617: Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems

FUZZY MODELING

CASE: FUZZY MODELING OF UNSTEADY AERODYNAMICS

Prof Hal Lewis

Hesham Al-momani


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Table Of Contents

Abstract

Introduction

Fuzzy logic modeling

Goodness of Model

Fuzzy Modeling Steps/Procedures (Basic Concept)

Fuzzy Inference System (FIS)

Sugeno fuzzy system

Mamdani Fuzzy Inference

Tsukamoto Fuzzy models

Adaptive Neuro-fuzzy inference system (ANFIS)

Literature Survey on Unsteady Aerodynamics & Fuzzy Modeling

Research Example


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Abstract

In this paper fuzzy modeling concepts has been surveyed

and introduced concentration has been done on

unsteady nonlinear aerodynamic molding using the Fuzzy

logic. Presenting a case study FUZZY MODELING OF

UNSTEADY AERODYNAMICS where they used

experimental data taken from NASA wind tunnel .

A conclusion has been reached that fuzzy modeling is a

very handy tool which can be easily adapted to predict

and simulate any linear as well as nonlinear system and

in particular unsteady as well as steady aerodynamic

coefficients.


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INTRODUCTION

One of the particular application area of fuzzy sets, logic, and systems is the fuzzymodeling.

Unsteady complex aerodynamics are essential in the design of airfoil section. Thestudy of these types of movement is vital in the development of developing the wingaircraft and can not be ignored due to its important impact on todays aircraft specially

tactical fighters that needs high manuvarlibilty and air supremacy

The need for accurate aerodynamic model that can aid in control system design and beused for vehicle dynamic- nonlinear aerodynamic simulation is of great importance.

Presence methods & conventional approach for modeling unsteady aerodynamics areof limited value and perform poorly in handling complex nonlinear systems

Fuzzy Logic pitches in to solve this dilemma.. Since the aerodynamics represented bythe fuzzy-logic models is realistic, they can be coupled with the numerical integration of

flight dynamic equations to study possible improvement in controllability.

This gap filling is achieved through gradual smooth transition representedmathematically by membership functions Therefore, a more robust model

identification technique that satisfies these goals appeared ,

The Fuzzy Logic Modeling (FLM) technique is adopted to model the establishedaerodynamic models that are directly used in flight simulation in the presentapplication.


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Fuzzy Logic Modeling

The general idea of the FLM technique is to set up the relations between system

input and output variables.

FLM algorithm represents a multi-dimensional, nonlinear interpolation scheme

without requiring explicit functional forms between the input and output

variables.

Static and dynamic systems which make use of fuzzy sets is called a Fuzzy system

and they defined by if-then rules & called rule-based systems, or fuzzy models.

An if-then rule generally takes the form of If antecedent proposition then

consequent proposition. .The antecedent proposition is always a fuzzy proposition

of the type x is A, where x is a linguistic variable and A is a linguistic constant. In a

linguistic fuzzy model, both the antecedent and the consequent are fuzzypropositions

In the fuzzy-logic model, the model structure is indicated by the number of

membership functions for each variable.

For a fuzzy-logic model with multiple variables, the structure is the combination of

the numbers and forms of the membership functions assigned to all input

variables.


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FLM Technique Approaches &Tasks

There are two approaches in the FLM technique.1. The fuzzy set approach, involving fuzzy sets, membership functions,weighting factors, and the if-then fuzzy rules (Zadeh 1973). Theprocess involves three stages:

a-Fuzzification

b-Fuzzy rule inference

c-Defuzzification.

2-The internal function approach, involving the internal functions,membership functions, and the output cells (Takagi & Sugeno 1985).

There are Two main tasks are involved in the present FLM process.

a-Identification of the coefficients of the internal functions.

b- Structure identification to identify the optimal structure offuzzy cells of the model, in other words, the optimal number ofmembership functions for each variable.


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Model Construction with Fuzzy Reasoning

Below figure shows the model construction with fuzzy reasoning .


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Goodness of Model

The goodness of model can be checked by

1- Graphical presentations.

2- Various measures, measuring or by Root Mean

Square Of Errors ((RMSE): (di-pi)2/n .

3- Or by Statistical analysis of errors methods!!.


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Fuzzy Modeling Steps/Procedures (Basic Concept)

1-Identification and Carefully selection of the most significant parameters that

affect the desired output. This requires aa-Detailed study of the target system

b-Ifexperimental or field data of the target system is not available, then ahighly monitored experiment is to be performed& data obtained should cover allthe necessary ranges of concerned parameters also the deviations from actualdata due should be accounted for.

c-Two data groups are to be prepared: one for checking the model and theother for testing model generality Both groups are to contain minimum andmaximum as well as intermediate values for all concerned inputs

d-It's vital to use standard definitions of parameters and mathematicalrelations otherwise they should be clearly defined.

e-The final selection of the model inputs and outputs depends on the relativeimportance of every one and on the required degree of model accuracy.

2-Selection of the proper inference systems based on existing fuzzy model orhuman experience .If similar experiments exist then they should be fully studiedand understood so that the positive aspects are stressed and the negative onesare avoided and ddetermine suitable universe of discourse and a term set for thevariables.


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3-Determination of the order of Sugeno consequent equation and Mamdani fuzzy

inference system4-Partitioning of input space.

5-Selection of the proper inference system based on existing fuzzy model

6-Determine model choices and parameters (including inference operators by selectingthe best number type of MF and this done by

a-By plotting the output versus the concerned input.

b- The number of membership function is chosen equal or close to the numberof distinct areas. MF that has a shape closest to the shape of the plot (output versusinput) is selected.

c-Selecting ranges for both and calculating the root mean square error (RMSE)for each case. lowest RMSE MF are chosen.

d-Determine the values of linguistic terms and MFs are then defined byoptimization and regression techniques.

7-Use of Defuzzification techniques to extract a crisp overall output value.

8-check model generality by prediction the output of an experimentallypredetermined data

9-Check accuracy and simplicity of the model

Fuzzy Modeling Steps/Procedures )


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Definition:A Fuzzy Inference System (FIS) is a way of mapping an input space to an output

space using fuzzy logic. A FIS tries to formalize the reasoning process of human language bymeans of fuzzy logic (that is, by building fuzzy IF-THEN rules). If the service is good, even ifthe food is not excellent, the tip will be generous

It also have Different names; fuzzy rule-based system, fuzzy model, fuzzy associativememory, fuzzy logic controller & fuzzy system Fuzzy inference system

(FIS) has found many applications, such as data classification, pattern recognition, robotics,

automatic control, and many others. A design of a fuzzy inference system is based on the past known behavior of a target system,

a developed FIS should reproduce the behavior of the target system .

FIS can be constructed for a specific application by :

Incorporate human expertise about the target system: it is called the domainknowledge (linguistic data!)

Use conventional system identification techniques for fuzzy modeling when input-

output data of a target system are available (numerical data)


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Why should we use Fuzzy Inference Systems

?Why should we use Fuzzy Inference Systems

a-Fuzzy logic does not solve new problems. It uses new methods to solveeveryday problems.

b-Mathematical concepts within fuzzy reasoning are very simple.

c-Fuzzy logic is flexible: it is easy to modify a FIS just by adding or deleting rules.There is no need to create a new FIS from scratch.

d-Fuzzy logic allows imprecise data (it does NOT work with uncertainty): ithandles elements in a fuzzy set, i.e. membership values. For instance, fuzzy logic workswith 'He is tall to the degree 0.8' instead of 'He is 180cm tall'.

e-Fuzzy logic is built on top of the knowledge of experts: it relies on the know-how of the ones who understand the system.

f-Fuzzy logic can be blended with other classic control techniques.

When shouldn't we use fuzzy logic?

Fuzzy logic is based on natural language. It is the codification of common sense.Thus, we shall not use it when our common sense tells us not to do so.


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Basically FIS has three major components as follows :

a-Fuzzy rule selection ,Rule base selects the set of fuzzy rules)

b-Membership function identification (Database (or dictionary) defines themembership functions used in the fuzzy rules)

c- Reasoning or interference procedure mechanism.

The Architecture of Fuzzy Inference Systems consists of

Fuzzy Models:

Mamdani Fuzzy models

Sugeno Fuzzy Models

Tsukamoto Fuzzy modelsBelow figure shows the structure of fuzzy systems

Fuzzy Systems

FuzzyKnowledge base

Input FuzzifierInferenceEngine

Defuzzifier Output

Fuzzy Inference &Components & Architecture


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Fuzzy Control Systems

FuzzyKnowledge base

FuzzifierInference

EngineDefuzzifier Plant Output

Input

The Fuzzifier converts the crisp input to a linguistic variable using the membership

functions stored in the fuzzy knowledge base.

Inference Engine Use If-Then type fuzzy rules converts the fuzzy input to the fuzzy

output,

Fuzzification stage here many internal functions are defined to cover the ranges of

the influencing variables (i.e. input variables). The ranges of the input variables are

all transformed into the domain of [0,1]. The membership grading also ranges from

0 to 1.0, with "0" meaning no effect from the corresponding internal function, and

"1" meaning a full effect whilein Defuzzification: extraction of a crisp value that best represents a fuzzy set, i.e. In

each fuzzy cell, the contribution to the outcome (i.e. the cell output) is based on the

internal function, The final prediction of the outcome is the weighted average of all

cell outputs after the process of reasoning algorithm. Because of this weighting

among many factors over large ranges of possibilities

it is necessary to have a crisp output in some situations where an inference system

is used as a controller there are Five commonly used defuzzifying methods:

Centroid of area (COA)

Bisector of area (BOA)

Mean of maximum (MOM)

Smallest of maximum (SOM)

Largest of maximum (LOM)



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Sugeno Fuzzy System

(TSK Fuzzy Model)

The Sugeno fuzzy system process , Also known as TSK fuzzy model ,Takagi, Sugeno & Kang,1985, was established to develop a system to generate a fuzzy rules from a given input-

output data.

,Sugeno role TSK fuzzy rule is of the form:

If x is A & y is B then z = f(x, y) :Where A & B are fuzzy sets in the antecedent, while z =

f(x, y) is a crisp function

Sugeno fuzzy model rules state that (Each rule has a crisp output, Overall output is obtained

via weighted average and No defuzzyfication required).

The Case of two rules with a first-order Sugeno fuzzy model state that (Each rule has a crisp

output, Overall output is obtained via weighted average and No defuzzyfication required).


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Sugeno Fuzzy System

(TSK Fuzzy Model)

Sugeno fuzzy system example are shown below

R1: if X is small and Y is small then z = x +y +1

R2: if X is small and Y is large then z = y +3

R3: if X is large and Y is small then z =

x +3R4: if X is large and Y is large then z = x + y + 2


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Mamdani Fuzzy Inference

.

Mamdani's method is the most commonly used in applications, due to its simple structure of'min-max' operations.

In Mamdani Fuzzy models the Original Goal was to control a steam engine & boiler combination

by a set of linguistic control rules obtained from experienced human operators.

Illustrations of how a two-rule Mamdani fuzzy inference system derives the overall output z

when subjected to two crisp input x & y as shown in the fig

It is worth mentioning that Mamdani's method is useful when there is a small number of

variables.

Mamdani steps :

Step 1: Evaluate the antecedent for each rule.

Step 2: Obtain each rule's conclusion.

Step 3: Aggregate conclusions.Step 4: Defuzzification
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Mamdani Fuzzy Inference System Operations

operators function:

AND operator (usually T-norm) for the rule firing strength computation with

ANDed antecedents

OR operator (usually T-conorm) for calculating the firing strength of a rulewith ORed antecedents

Implication operator (usually T-norm) for calculating qualified consequent

MFs based on given firing strength

Aggregate operator (usually T-conorm) for aggregating qualified consequent

MFs to generate an overall output MF composition of facts & rules to derive

a consequent

Defuzzification operator for transforming an output MF to a crisp single

output value


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Tsukamoto Fuzzy Models

It is characterized by the following, the consequent of each fuzzy if-then-

rule is represented by a fuzzy set with a monotonical MF The inferred

output of each rule is a crisp value induced by the rules firing strengths

shown fig below

Tsukamoto Fuzzy models


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Partition Styles for Fuzzy Models

There are 3 partitioning Styles for Fuzzy Models

a. Grid partition

b. Tree partition

c. Scatter partition

Partition Styles for Input Space

Grid

Partition

Tree

Partition

Scatter

Partition

Figure (12)


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Literature Survey on Unsteady Aerodynamics & Fuzzy Modeling

(Unsteady aerodynamics modeling for flight dynamics application) Qing,

He & Wu

The research results show that:

1) It seems that the unsteady aerodynamic modeling is still more or less of academic

significanceonly, not yet applied to aircraft design.

2) In the Unsteady aerodynamics the building-up of aerodynamics is regarded as

a blackbox,and the dependence of aerodynamics on flight states is described

by newly developed technologies, such as fuzzy logic and neural network with the

complex flow mechanism not involved.

3) Special cases the post-stall maneuvering .one of their conclusions that the

unsteady aerodynamics affects observably the post-stall maneuver performance and

unsteady aerodynamic models have been adopted.


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The application of FNN in unsteady aerodynamics modeling based on fuzzy

clustering SHI Zhi-wei ChinaSHI Zhi-weiMING Xiao

In this paper a Fuzzy Neural Network(FNN)model based on fuzzy clustering is

developed

The fuzzy space and the number of fuzzy roles of this model are defined by the

fuzzy clustering method and weight coefficients of the model are adjusted by the

BP algorithm

Using the model the unsteady aerodynamics of one aircraft in pitching-wiling

motion is identified

It is suggested that the fuzzy clustering method Can be used to design fuzzy

neural network structures and the developed model earl be used to identify the

nonlinear unsteady aerodynamics of many complicated maneuvers

In Fuzzy-Logic Analysis of the FDR Data of a Transport Aircraft in Atmospheric

Paper objective was to illustrate the nonlinear unsteady aerodynamic models

based on the FLM technique having the capability to evaluate the variations in

stability of commercial aircraft with adverse weather effects.


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Fuzzy logic modeling of nonlinear unsteady aerodynamics. Wang, Z.J., Lan, C.E., Brandon,J.M.:

Their study focused on: the wind shear for takeoff and landingprocess on a Boeing 737-800

passenger plane, fuzzy logic modeling techniques can handle the multi-variable parameters of the

nonlinear behavior, and provide a reasonable dynamic and unsteady aerodynamic modelfrom analog shock .

The results showed that the crosswind will affect the flight stability and control capabilities.

Application of artificial neural networks in nonlinear aerodynamics and aircraft design.Rokhsaz, K., Steck, J.E.

They used neural networks in analysis of three aerodynamics problems and two in flightdynamics.

The aerodynamics cases are those of a harmonically oscillating airfoil, a pitching delta wing,and airfoil design.

The flight dynamic examples involve control of a super maneuver and a decoupled controlcase.

It is demonstrated that highly nonlinear aerodynamic cases can be generalized with

sufficient accuracy for design purposes. It is shown that although neural networks generalize well on the aerodynamic problems,

they appear lacking comparable robustness in modeling dynamic systems & shown thatgeneralization appears to become weak outside of the training domain


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Successive identification of a fuzzy model and its application toprediction of a

complex system. Fuzzy Sets and Systems 42, 315334(5) Sugeno, M. and K.

Tanaka (1991). Successive identification of a fuzzy model.

The process consisted of two levels. The first level (called supervisor level) &other level, named adjustment level,

It was explained that the fuzzy model could be successively improved andmore accurate modeling would be achieved.

A Fuzzy Logic Based Approach To Qualitative Modeling. IEEE TransactionsOn Fuzzy Systems, Vol.1 no.1 February 1993M.Sugeno and T Yasukawa,

Presented an approach to qualitative modeling based on fuzzy logic

The model was derived from a fuzzy model using the linguistic approximationmethod.

The structure of a fuzzy model was obtained through fuzzy c-means clusteringmethod.

The applicability of the method was justified by presenting a model ofdynamical process and a model of a human operators control action.


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Adaptive Based Fuzzy Inference System IEEE Transactions On Systems Man &

Cybernetics Vol 23 No 3 PP665-685 May 1993J R Jang ANFIS:

Discussed the architectural procedure of ANFIS which could construct an input-output mapping based mostly on human expertise and stipulated data pairs.

The paper included comparisons with artificial neural networks and earlier work onfuzzy modeling.

The ANFIS example used contained 16 rules and four membership functions assignedto each input variable.

The example proved that ANFIS can effectively model a highly nonlinear surface.

Fuzzy Neural Networks For Nonlinear Systems Modeling. IEE Proceedings-ControlTheory Applications, Vol. 142 no 6 PP 551-561 November 1995,J Zhang and A.JMorris,

Presented a method for modeling of non-linear systems based on fuzzy neuralnetwork. In this process the input space of a nonlinear system was partitioned intoseveral regions.

Then in each region a reduced order linear model was used to represent the system.

Defuzzificaton to produce the overall output was based on center of gravity method.The method was effective with systems having major non-linarites.


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Rule-Base Structure Identification in an. Adaptive-Network-Based Fuzzy

Inference System. Chuen-Tsai Sun

Presented a paper deals with rule-base structure identification.

The paper proposed a general modeling scheme for an adaptive-network-based fuzzy inference system which would be used in data compression,pattern recognition, and other fields.

Linguistic model identification for fuzzy system. Y.H. Joo, H.S. Hwang, K.B.Kim and K.B. Woo (10) (Fuzzy-Logic Analysis of the FDR Data of aTransport Aircraft, in Atmospheric Turbulence).

Proposed an approach for identifying a linguistic fuzzy model for a multi-input-singe output system.

They utilized a c-means clustering and genetic algorithm scheme.

The approach was tested through examples and the simulation results showedthat the number of the rules identified by the method was small without loss ofmodel accuracy.

For a system with more than three input variables, it would be better toconsider the linguistic terms of triangular form, in which the sum of all themembership values of each input to its linguistic terms become one.


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Fuzzy Modeling Of Unsteady Aerodynamics done by Almahadin,Jordan university ofscience and technology ,Jordan

The present FLM technique was explained in detail and verified with simple examples and

wind-tunnel data. It was shown that the FLM technique was capable of handling nonlinear and unsteady

aerodynamic environment exhibited for a twin-jet transport in severe atmospheric turbulencewith sudden plunging motion in transonic flight.

The predicted results showed that the models could produce reasonable aerodynamiccoefficients and several derivatives for the assessment of stability characteristics,

Research on Unsteady Aerodynamic Models and Flight Simulation for the Aircraft Oscillation atHigh Angle of Attack

The unsteady aerodynamics of the fighter at high angle of attack was tested at low speedwind tunnel. model was measured during in oscillating in pitching motion, yawing motion,rolling motion or yawing-rolling coupled-motion.

the applicability of 4 kinds of unsteady aerodynamic models was analyzed in this paper. S

ome of the unsteady aerodynamic characteristics were acquired by analyzing the wind tunnel

test results and investigate the the modeling method of fuzzy logic. It is indicated that the value of the convergence coefficient is a basic factor determining the

speed and precision of modeling.

It is indicated that the aerodynamic characteristics of coupled-motion are reflected well to thetest results by the unsteady aerodynamic model of fuzzy logic.


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Fuzzy Modeling of Unsteady Aerodynamics

Almahadin Master Thesis


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Fuzzy Modeling of Unsteady Aerodynamics

Almahadin Master Thesis

Table of contentsIntroduction

Background on fighters super maneuvers

Unsteady Aerodynamics Literature Review

Experimental Programs

Other Related Investigation

Fuzzy logic approach

Introduction To Fuzzy Logic

Fuzzy Inference System

Sugeno Fuzzy SystemFuzzy Modeling

Adaptive Neuro-fuzzy Inference System (Anfis)

Fuzzy Logic Literature survey


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Summary

One of the many examples of the fuzzy modeling is a master thesis of

Fuzzy Modeling Of Unsteady Aerodynamics done by Al-mahadin In this project he investigated and established fuzzy model and compared

it with already known experimental data taken from NASA experiment

about Semi Empirical Method used for PHD project.

The objective of his investigation is motivated by the need to have an

accurate aerodynamic model that can be used for vehicle preliminarydesign purposes.

Also, to aid in control system design which is a challenging problem in thepresence of highly nonlinear aerodynamic loading with large hysteresisloops.

He wanted to produce a fuzzy model of the unsteady aerodynamic loadsoccurring during high angle of attack maneuvering. T

The fuzzy model should be able to predict the aerodynamic loads for agiven delta wing performing a prescribed maneuver.


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Summary

The model has three inputs : angle of attack, non-dimensional pitch rate, and aspectratio. Its output is the unsteady aerodynamic normal force coefficient (CN.

A highly reliable model was found which predicts the CN with a RMSE (Root meansquare Error) of less than 6 % (94%) for the 0 to 90 degrees angle of attack range andresults were BETTER than the semi-empirical method results.

Based on the experience gained throughout this research, a proposed fuzzy modeling

procedure was suggested & recommended. Conclusion that fuzzy modeling is an excellent method to simulate nonlinear systems

He started the introduction by viewing the historical background on tactical fighters

super maneuvers

He Defined the super maneuverability and superiority.

Investigated Prediction tools and models.

Introduced the principles of Fuzzy Inference System (FIS) , fuzzy modeling systems

like Sugeno fuzzy system, Mamdani fuzzy & Adaptive Neuro-fuzzy inference system

(ANFIS)fuzzy logic.

He also investigated and stated the Fuzzy modeling Steps


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Chapter Two

Literature Survey

The researcher presented a lot of Unsteady aerodynamic models and

fuzzy logic models


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Experimental Unsteady Aerodynamic Data Analysis

The experiment data taken from an experiment took place at NASA Ames

research center by semi empirical method( Low speed 7ft x 10 ft wind

tunnel) They include drag, lift, normal force, and pitching moment

coefficients.

Data analysis presented for the purpose of determining the most important

aerodynamic parameters which will be used as inputs of the fuzzy model.

A brief discussion of the data to be used to develop the fuzzy model along

with data analysis are given.

Chapter Three


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Chapter Three

Modeling Discussion

The research is launched by a thorough investigation of unsteady aerodynamic

data in all aspects: theoretical , experimental, and modeling In existing flight simulation few methods, beside wind-tunnel

experimentation, are commonly used to deal with high angle of attack

unsteady aerodynamics, namely: tabulated quasi steady data , local linearized model that forms apiecewise continues fit of the nonlinear response and some other methods also.

In this research he attempted to device a simple fuzzy model which simulatehigh angle of attack aerodynamics specifically normal force coefficient (CN)

The other aerodynamic coefficients are expected to be similarly modeled. ,

(CL, CN, CD, CM, CRM)

Versatility, generality, validity and accuracy of this model checked by other

extra data obtained also by Jarrah and other data obtained by Bragg andSultani


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Model generality

One of the most important criteria of any model is the extent of model generality.

This fuzzy model has no exception to achieve this

Model generality also checked by predicting other experimental data not used for training

were used and termed extra-Jarrah data. & Bragg & Sultani were tested

The initial model has predicted this data with an RMSE of 8.35% for all overall extra-Jarrah

data.

The accuracy and value of the final model depended greatly on the accuracy of the

experimental data themselves and on the degree of matching between the different data

groups.

Based on the analysis carried out the final fuzzy model which appears to be more

reasonable is the initial model itself. Since it best fits the training and checking data as well

as extra Jarrah data except the last two groups due to the difference in angle of attack ranges.

This leaded to the conclusion max & minwould have to be included in the model inputs if

different angle of attack ranges are to be predicted and RMSE to be below 10%.


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Modelaccuracy

He checked the accuracy analysis and mentioned that the modeling accuracy shows

a varied dependency on the three concerned parameters : MF number, MF type , and

epochs number.

The model is more sensitive to the selection of the number of MF for the rate of

change of angle-of attack itself & individual model inputs

He also stated that Fuzzy logic modeling accuracy depends greatly on the accuracy

of the experimental data used for training. However, errors (high RMSE) in this

model are probably the cause of the following:

a-Errors in obtaining experimental data and those used to compare the initial/final model predictions.

b-No standard experimental procedure.

c- Differences in data reduction and experiments.

d-Tare measurements differences

e-Non standard variable definitions.

f-Ignoring both min and max as model inputs.

h- Lack of through understanding of some of the aerodynamic phenomenon such as bursting.I-Lack of detailed information regarding some of the fuzzy logic aspects.

j- Errors in predicting Bragg and Sultani data are mostly the cause of differences between their experiment

and Jarrah experiment with respect to the following : tare measurements, data filtering, data acquisition, variable

definitions specially k, angle of attack ranges, experimental setting and others

He improved the accuracy analysis modeling by number of means used accurate experimental data as

possible. And found that he should included min and max as model inputs to improve the prediction of data

with different range

Chapter Four


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p

Models Comparisons

Fuzzy modeling comparison with semi-empirical method

The powerfulness of fuzzy modeling can proved by a comparison with a publishedmethod namely semi-empirical method

Fuzzy modeling gives better prediction than semi-empirical method for the 0 to 90degrees angle of attack range.

Table comparison of semi-empirical and initial final fuzzy model outputs to semi-empirical method experimental output ,comparison is greatly in favor of fuzzymodeling


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ComparisonTableModel parameters

Reynolds number = 4.56 X 105

Percent toot mean

Square error ( % RMSE)

Aspect ratio Non-dimensional pitch

rate

Angle of attack range

(degrees)

Fuzzy final model Semi-empirical method

1 0.02 0 to 90 8.26 11.12

1 0.04 0 to 90 8.53 9.18

1 0.06 0 to 90 4.57 9.92

1.5 0.04 0 to 60 14.63 8.1

2 0.04 0 to 90 6.94 11.76

2 0.06 0 to 90 5.47 17.28

1 & 2 Range above 0 to 90 6.75 11.85

Range above Range above Range above 8.17 12.37

chapter five


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chapter five

Results Discussion

he find out that the Out of the five unsteady aerodynamic coefficient (CN, CL, CD, CM,

CRM), the normal force coefficient (CN) was picked to be the model output todemonstrate the present modeling procedure

Studying the effect of aerodynamic variables on these coefficients that resulted on

setting three model inputs: angle of attack ( ), rate of change of angle of attack( ),

and aspect ratio (AR). They were found to be the most significant parameters affecting

the aerodynamic loading.

The fuzzy modeling parameters, which were manipulated to obtain the optimum

model, (number &Type of membership functions for each input, and number of

epochs.

The measure of acceptable accuracy was the root mean square error (RMSE).

It was believed that number of MF could be approximated by plotting the

experimental data and dividing visually the figure into distinct segments.

R lt Di i


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Results Discussion

Seven types of MF were implemented. And it was found that MF type has negligible

effect on modeling of this system. Therefore the simplest one (triangular) was set asinitial model parameter. Moreover this MF type was found to equally satisfies the data

and finally found that Sometimes the resulted initial and final membership functions

are different.

He got the Initial model which refers to the simplest model that best fits and tracks the

training and checking data with the minimal root mean square error (RMSE).

The choice of the number of membership functions and their types for each input done

arbitrary since no general rule presents in this research a general guide lines is sought

based on the idea that the required MF is the one which best fits the input data from the

point of view of simplicity, convenience, speed and efficiency.

It is suspected that the choice of MF type and number can be guessed by plotting

experimental data and visually dividing the plot into distinct segments. The mostcommonly used membership functions, investigated and implemented in this research,

The initial model was obtained with the following parameters: membership functions

type is triangular, MF numbers are [4 3 3] for the three inputs , ,AR respectively .


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Conclusion

The research has resulted in highly reliable fuzzy model able to predict

normal force coefficient with an overall root mean square error of lessthan 6 % for angle of attack range of 0 to 90 degrees

The model accuracy depends greatly on the selection of membership

functions number while it shows much less sensitivity of MF type.

the model accuracy is affected most by the rate of change of angle of

attack then the aspect ration followed by the angle of attack.

Initial/final fuzzy model was compared favorably with semi-empirical

method for 0 to 90 degrees range.


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Recommendation

For better prediction of unsteady aerodynamic loads it is wise to

include the minimum and maximum Values!!( angle of attack) as model

inputs.

For the purpose of obtaining a complete unsteady aerodynamic model

that predict all five coefficients, its recommended that the modelingprocedure is repeated while using other coefficients as the model

outputs.

The fuzzy modeling in this research was based on Suggeno scheme. It

would be of great importance that the research is repeated using

Mamdani method.


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Thank you

SSIE 617 Fuzzy Sets Fuzzy Logic and Fuzzy Systems All Slides

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