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

Fakultät Forst-, Geo- und Hydrowissenschaften , Fachrichtung Wasserwesen, Institut für Abfallwirtschaft und Altlasten , Professur Systemanalyse Application of Fuzzy Logic and Uncertainties Measurem ent in Environm ental I nform ation System s Dresden, 21 July 2011

Goal Developing a Tool for Installing Fuzzy Control Identification of Parameters and System in Environmental Boundary Conditions Uncertainties Information System 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 Accurate Not Accurate Not precise Not accurate Precise precise Precise

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

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 occurrence of an event. Uncertainty Uncertainty Type of Stochastic Informal Lexical Stochastic Informal Lexical uncertainty Characteristic Fuzzy Fuzzy Randomness Fuzziness Randomness Fuzziness of uncertainty randomness randomness

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. Boolean logic Boolean logic Fuzzy logic vs. 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. 0 0 0 1 1 1 0 0 0.2 0.4 0.6 0.8 1 1 ( a ) Boolean Logic. ( b ) Multi-valued Logic. 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 “ “tall m en tall m en” ” Crisp and fuzzy sets of D e g re e o f C risp S e ts M e m b e rsh ip 1 .0 0 .8 T a ll M e n 0 .6 0 .4 0 .2 0 .0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 2 1 0 H e ig h t, c m D e g re e o f F u z z y S e ts M e m b e rsh ip 1 .0 0 .8 0 .6 0 .4 0 .2 0 .0 1 5 0 1 6 0 1 7 0 1 8 0 1 9 0 2 0 0 2 1 0 H e ig h t, c m 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 Inference Input Output Fuzzifier Defuzzifier Engine Fuzzy Knowledge base

Fuzzy Control System s Fuzzy Control System s Input Inference Output Fuzzifier Defuzzifier Plant Engine Fuzzy Knowledge base

Inference Inference Fuzzifier Fuzzifier Defuzzifier Defuzzifier Input Input Output Output Engine Engine Fuzzifier Fuzzifier Fuzzy Fuzzy Knowledge base Knowledge base 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.

Inference Inference Fuzzifier Fuzzifier Defuzzifier Defuzzifier Input Input Output Output Engine Engine I nference Engine I nference Engine Fuzzy Fuzzy Knowledge base Knowledge base linguistic variables are used in fuzzy rules. Using If-Then type fuzzy rules converts the fuzzy input to the fuzzy output.

Mam dani Fuzzy m odels Fuzzy m odels Mam dani  Original Goal: Control a steam engine & boiler combination by a set of linguistic control rules obtained from experienced human operators.

Inference Inference Fuzzifier Fuzzifier Defuzzifier Defuzzifier Input Input Output Output Engine Engine Defuzzifier Defuzzifier Fuzzy Fuzzy Knowledge base Knowledge base 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 output space.

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

Mathem atical Background Background Mathem atical Flow and transport in the vadose zone: SiWaPro DSS = volumetric water  Richards equation -> flow and water balance content t = time �� �� � � � �� x i (i=1,2) = spatial � �� � � �� �� �� � � �� �� � �� � coordinates K = hydraulic conductivity h = pressure head S = sink term Parameterization of soil properties based on van Genuchten-Luckner     m   n   r    1 | | h        s r

Mathem atical Background Background Mathem atical Unsaturated hydraulic conductivity degree of mobility 0.00 0.25 0.50 0.75 1.00 1.0   m    1       1 1 0.9 m   S        k S         0.8 K     r m    k   S 1 r elative permeability 0.7 0  0    1 1 m  S 0     0.6 0.5 0.4 Parameter m - Transformations parameter 0.3 (m= 1-1/n) 0.2 - Scaling factor (  =0,5)  0.1 k 0 , S 0 - Calibration point 0.0 0.00 0.06 0.12 0.18 0.24 0.30 0.36 water content

Mathem atical Background Background Mathem atical Convection-dispersion equation -> Solute Transportation Convection-dispersion equation -> Solute Transportation         s u s  s fl , m fl , m   m            D s q   m m m m     r r r   t dispersion convection change of mass degradation terms sinks/sources storage r spatial coordinate  m ,  0 and 1. order degradation D dispersion coefficient m coefficient s fl,m , s s,m specific mass in the liquid u mean flux 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 Flow Flow Model Model W ater Richards equation -> flow and water balance  = volumetric water content t = time x i (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