Fakultät Forst-, Geo- und Hydrowissenschaften, Fachrichtung Wasserwesen, Institut für Abfallwirtschaft und Altlasten, Professur Systemanalyse Application of Fuzzy Logic and Uncertainties Measurement in Environmental Information Systems 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 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 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 We always want the most precise and accurate experimental data. The precision and accuracy are limited by the instrumentation and data gathering techniques.
Dealing with 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 news No matter how good you are there will always be errors. The question is How to deal with them? STATISTICS FUZZY THEORY
Uncertainty Uncertainty is defined as a gradual assessment of the truth content of a proposition in relation to the occurrence of an event. Uncertainty Stochastic Informal Lexical Type of uncertainty Randomness Fuzzy randomness Fuzziness Characteristic of uncertainty
Theories to Deal with 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 - Experimental analysis Black Box Neural Networks Knowledge-Based Analysis Rules Various Datasets - Numerical - Interval - Knowledge-based data Facts Integrated Model Numerically, based on knowledge and fuzzy logic
Fuzzy logic vs. Boolean logic Fuzzy logic is based on the idea that all things admit of degrees. Temperature, height, speed, distance, beauty all come on a sliding scale. Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1 (completely true). Instead of just black and white, it employs the spectrum of colours, accepting that things can be partly true and partly false at the same time. 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. Example: Tom is tall because his height is 181 cm. If we drew a line at 180 cm, we would find that David, who is 179 cm, is short. Is David really a short man or we have just drawn an arbitrary line in the sand?
Crisp and fuzzy sets of tall men D e g ree of Membership 1.0 Crisp Sets 0.8 0.6 Ta ll M e n 0.4 0.2 0.0 150 160 170 180 190 200 210 D e g ree of Membership 1.0 0.8 0.6 0.4 0.2 Fuzzy Sets H eig ht, cm 0.0 150 160 170 180 190 200 210 H eight, cm Boolean logic uses sharp distinctions. It forces us to draw lines between members of a class and non-members.
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 Systems Input Fuzzifier Inference Engine Defuzzifier Output Fuzzy Knowledge base
Fuzzy Control Systems Input Fuzzifier Inference Engine Defuzzifier Plant Output Fuzzy Knowledge base
Input Fuzzifier Inference Engine Defuzzifier Output Fuzzifier Fuzzy 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.
Input Fuzzifier Inference Engine Defuzzifier Output Inference Engine Fuzzy Knowledge base linguistic variables are used in fuzzy rules. Using If-Then type fuzzy rules converts the fuzzy input to the fuzzy output.
Mamdani Fuzzy models Original Goal: Control a steam engine & boiler combination by a set of linguistic control rules obtained from experienced human operators.
Input Fuzzifier Inference Engine Defuzzifier Output Defuzzifier Fuzzy Knowledge base Converts the fuzzy output of the inference engine to crisp using membership functions analogous to the ones used by the fuzzifier.
Nonlinearity In the case of crisp inputs & outputs, a fuzzy inference system implements a nonlinear mapping from its input space to output space.
Scheme Interface (Data Exchange) Environmental Information System Simulator e.g.: MODFLOW, SIWAPRO DSS Assessment Tool: Analyzing uncertainties in parameters and boundary conditions in the simulation results
Mathematical Background Flow and transport in the vadose zone: SiWaPro DSS Richards equation -> flow and water balance Parameterization of soil properties based on van Genuchten-Luckner r s r 1 h n m = volumetric water content t = time x i (i=1,2) = spatial coordinates K = hydraulic conductivity h = pressure head S = sink term
Mathematical Background Unsaturated hydraulic conductivity degree of mobility 0.00 0.25 0.50 0.75 1.00 K r k k 0 S S 0 1 1 S 1 1 S Parameter m - Transformations parameter (m= 1-1/n) - Scaling factor (=0,5) k 0, S 0 - Calibration point 1 m 1 m 0 m m relative permeability 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0.00 0.06 0.12 0.18 0.24 0.30 0.36 water content
Mathematical Background Convection-dispersion equation -> Solute Transportation r D s r fl,m u s r fl,m s m t m s m m q m dispersion convection change of mass storage degradation terms sinks/sources r D sfl,m, ss,m spatial coordinate dispersion coefficient specific mass in the liquid and/or solid phase m, m u 0 and 1. order degradation coefficient mean flux
Program 25
Representation of imprecision numbers as input of simulation programs Example: Triangular membership function for the saturated hydraulic conductivity
Example: Trapezoidal membership function for the saturated hydraulic conductivity
Minimal and/or Maximum Scenarios of Water Flow Model 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 Minimal and/or Maximum Scenarios
Plot for Pressure head with different membership functions Example for the use of fuzzy interval arithmetic for the Darcy Buckingham equation
Test Case: Using fuzzy modelling with optimization procedure NLPQLP for transient infiltration flow of water across an earth dam Structure of the dam and type of the boundary conditions
Transient infiltration flow of water across the dam after 18 minutes
Representation of the course of the minimum and maximum pressure head within the drainage range
Fuzzy modelling with the Fuzzy analysis LIBRARY 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