Application of Fuzzy Logic and Uncertainties Measurem ent in - - PowerPoint PPT Presentation

Application of Fuzzy Logic and Uncertainties Measurem ent in Environm ental I nform ation System s

Fakultät Forst-, Geo- und Hydrowissenschaften , Fachrichtung Wasserwesen, Institut für Abfallwirtschaft und Altlasten , Professur Systemanalyse

Dresden, 21 July 2011

Goal

Installing Fuzzy Control System in Environmental Information System

Developing a Tool for Identification of Parameters and Boundary Conditions Uncertainties in Water Balance and Solute Transport Simulation

Contribution Contribution

• So far decision making took place based on objective

information, not subjective information

• So Measurements were always somewhat different

from the “true value” .

• These deviations from the true value are called

errors.

• Consideration of Uncertainties in the input data of

simulation programs and generating more prcise and accurate outputs

Dartboard analogy Dartboard analogy

Precision: How reproducible are measurements? Accuracy: How close are the measurements to the true value? Imagine a person throwing darts, trying to hit the bulls-eye.

Not accurate Not precise Accurate Not precise Not accurate Precise Accurate Precise

Data Data

The precision and accuracy are limited by the instrumentation and data gathering techniques. We always want the most precise and accurate experimental data.

Dealing w ith Errors Dealing w ith Errors

• Identify the errors and their magnitude.
• Try to reduce the magnitude of the error.

HOW ?

• Better instruments
• Better experimental design
• Collect a lot of data

Bad new s Bad new s… …

• No matter how good you

are… there will always be errors.

• The question is… How to

deal with them?

STATI STI CS STATI STI CS FUZZY THEORY FUZZY THEORY

Uncertainty Uncertainty

Uncertainty is defined as a gradual assessm ent of the truth content of a proposition in relation to the

• ccurrence of an event.

Uncertainty Uncertainty

Lexical Lexical Informal Informal Stochastic Stochastic Type of uncertainty Fuzziness Fuzziness Fuzzy randomness Fuzzy randomness Randomness Randomness Characteristic

• f uncertainty

Theories to Deal w ith Uncertainty Theories to Deal w ith Uncertainty

• Bayesian Probability
• Hartley Theory
• Chaos Theory
• Dempster-Shafer Theory
• Robust Optimization
• Markov Models
• Neural Networks
• Zadeh’s Fuzzy Theory

• Diffrent Modeling Methods
• Theoretical analysis



PDE

• Experim ental analysis

 Black Box  Neural Netw orks  Know ledge-Based Analysis Rules

• Various Datasets
• Num erical
• I nterval
• Know ledge-based data  Facts

I ntegrated Model

Num erically, based on know ledge and fuzzy logic

Fuzzy logic vs. Fuzzy logic vs. Boolean logic Boolean logic

Fuzzy logic is based on the idea that all things adm it of degrees. Tem perature, height, speed, distance, beauty  all com e on a sliding scale.

• Fuzzy logic uses the continuum of logical values betw een 0 ( com pletely false) and 1

( com pletely true) . I nstead of just black and w hite, it em ploys the spectrum of colours, accepting that things can be partly true and partly false at the sam e tim e.

(a) Boolean Logic. (b) Multi-valued Logic.

0 1 1 0.2 0.4 0.6 0.8 1 1 1

Exam ple: Tom is tall because his height is 1 8 1 cm . I f w e drew a line at 1 8 0 cm , w e w ould find that David, w ho is 1 7 9 cm , is short. I s David really a short m an or w e have just draw n an arbitrary line in the sand?

Crisp and fuzzy sets of Crisp and fuzzy sets of “ “tall m en tall m en” ”

1 5 0 2 1 0 1 7 0 1 8 0 1 9 0 2 0 0 1 6 0 H e ig h t, c m D e g re e o f M e m b e rsh ip T a ll M e n 1 5 0 2 1 0 1 8 0 1 9 0 2 0 0 1 .0 0 .0 0 .2 0 .4 0 .6 0 .8 1 6 0 D e g re e o f M e m b e rsh ip 1 7 0 1 .0 0 .0 0 .2 0 .4 0 .6 0 .8 H e ig h t, c m F u z z y S e ts C risp S e ts

Boolean logic uses sharp distinctions. I t forces us to draw lines betw een m em bers of a class and non-m em bers.

Fuzzy Logic Fuzzy Logic

Fuzzy logic reflects how people think. It attempts to model our sense of words, our decision making and our common sense. As a result, it is leading to new, more human, intelligent systems. The basic idea of the fuzzy set theory is that an element belongs to a fuzzy set with a certain degree of membership. Thus, a proposition is not either true or false, but may be partly true (or partly false) to a degree. This degree is usually taken as a real number in the interval [0,1].

In the fuzzy theory, fuzzy set A of universe X is defined by function A(x) called the membership function of set A A(x): X  [0, 1], where A(x) = 1 if x is totally in A; A(x) = 0 if x is not in A; 0 < A(x) < 1 if x is partly in A.

Fuzzy Expert System s Fuzzy Expert System s

Fuzzy Knowledge base

Input

Fuzzifier Inference Engine Defuzzifier

Output

Fuzzy Control System s Fuzzy Control System s

Fuzzy Knowledge base Fuzzifier Inference Engine Defuzzifier Plant

Output

Input

Fuzzifier Fuzzifier

Fuzzy Knowledge base Fuzzy Knowledge base

Input

Fuzzifier Inference Engine Defuzzifier

Output Input

Fuzzifier Inference Engine Defuzzifier

Output

Converts the crisp input to a linguistic variable using the membership functions stored in the fuzzy knowledge base. A linguistic variable is a fuzzy variable. For example, the statement “John is tall” implies that the linguistic variable John takes the linguistic value tall.

I nference Engine I nference Engine

Fuzzy Knowledge base Fuzzy Knowledge base

Input

Fuzzifier Inference Engine Defuzzifier

Output Input

Fuzzifier Inference Engine Defuzzifier

Output

linguistic variables are used in fuzzy rules. Using If-Then type fuzzy rules converts the fuzzy input to the fuzzy output.

Mam dani Mam dani Fuzzy m odels Fuzzy m odels

 Original Goal: Control a steam engine &

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

Defuzzifier Defuzzifier

Fuzzy Knowledge base Fuzzy Knowledge base

Input

Fuzzifier Inference Engine Defuzzifier

Output Input

Fuzzifier Inference Engine Defuzzifier

Output

Converts the fuzzy output of the inference engine to crisp using membership functions analogous to the ones used by the fuzzifier.

Nonlinearity Nonlinearity

In the case of crisp inputs & outputs, a fuzzy inference system implements a nonlinear mapping from its input space to

• utput space.

Schem e Schem e

Environmental Information System Simulator e.g.: MODFLOW, SIWAPRO DSS Assessment Tool: Analyzing uncertainties in parameters and boundary conditions in the simulation results

Interface (Data Exchange)

Mathem atical Mathem atical Background Background

• 



m n h r s r     

     

| | 1     

Flow and transport in the vadose zone: SiWaPro DSS Richards equation -> flow and water balance Parameterization of soil properties based on van Genuchten-Luckner  = volumetric water content t = time xi (i=1,2) = spatial coordinates K = hydraulic conductivity h = pressure head S = sink term

Mathem atical Mathem atical Background Background

Unsaturated hydraulic conductivity

   

                                      

m m r

m m

S S S S k k K

1 1

1 1 1 1   

0.00 0.06 0.12 0.18 0.24 0.30 0.36

water content

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

relative permeability

0.00 0.25 0.50 0.75 1.00

degree of mobility

Parameter m

• Transformations parameter

(m= 1-1/n) 

• Scaling factor (=0,5)

k0, S0

• Calibration point

Mathem atical Mathem atical Background Background

Convection-dispersion equation -> Solute Transportation Convection-dispersion equation -> Solute Transportation

change of mass storage sinks/sources degradation terms

 

m m m m m m , fl m , fl

q s t s r s u r s D r                            

dispersion convection

0 and 1. order degradation coefficient u mean flux

m m,



r spatial coordinate D dispersion coefficient sfl,m, ss,m specific mass in the liquid and/or solid phase

Program Program

25

Representation of im precision Representation of im precision num bers as input of sim ulation num bers as input of sim ulation program s program s

Example: Triangular membership function for the saturated hydraulic conductivity

Example: Trapezoidal membership function for the saturated hydraulic conductivity

Minim al and/ or Maxim um Scenarios of Minim al and/ or Maxim um Scenarios of W ater W ater Flow Flow Model Model

Richards equation -> flow and water balance  = volumetric water content t = time xi (i=1,2) = spatial coordinates K = hydraulic conductivity h = pressure head S = sink term

Plot for Minim al and/ or Maxim um Plot for Minim al and/ or Maxim um Scenarios Scenarios

Plot for Pressure head w ith Plot for Pressure head w ith different m em bership functions different m em bership functions

Example for the use of fuzzy interval arithmetic for the Darcy Buckingham equation

Test Test Case Case: : Using Using fuzzy fuzzy m odelling m odelling w ith w ith optim ization

• ptim ization procedure

procedure NLPQLP NLPQLP for for transient infiltration transient infiltration flow of w ater across an earth dam flow of w ater across an earth dam

Structure of the dam and type of the boundary conditions

Transient infiltration flow of w ater Transient infiltration flow of w ater across the dam after 1 8 m inutes across the dam after 1 8 m inutes

Representation of the course of the minimum and maximum pressure head within the drainage range

Fuzzy m odelling w ith the Fuzzy Fuzzy m odelling w ith the Fuzzy analysis LI BRARY of Fortran analysis LI BRARY of Fortran

• Developed by Institute for statics and

dynamics of civil engineering faculty , TU Dresden)

• A tool for modelling uncertainties by Fuzzy

Randomness

• Comparison of the simulation results for both

procedures

• References to the application of the two

procedures (advantages, advantages)

Simulation result of dam flow with FALIB