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طراحي يك سيستم فازي Designing of a Fuzzy System Vali Derhami Yazd University, Computer Department [email protected] FIS is Universal Approximator Theorem 9.1 (Universal Approximation Theorem). Suppose that the input universe of discourse U is a compact set in Rn. Then, for any given real continuous function g(x) on U and arbitrary > 0, there exists a COG fuzzy system with Gaussian membership function such that: Also study 14.1 pp.278-287 from Tanaka’s book 2/14 Author: Vali Derhami Design of Fuzzy Systems Structure and Parameters (Ch. 5.1 Babuska’s book): structure selection involves: Input and output variables. Structure of the rules: This choice involves the model type (linguistic, singleton, relational, Takagi-Sugeno) and the antecedent form Number and type of membership functions for each variable. Type of the inference mechanism, connective operators, defuzzification method. 3/14 Author: Vali Derhami Design of Fuzzy Systems (Cont.) After the structure is fixed, the performance of a fuzzy model can be finetuned by adjusting its parameters. Tunable parameters: parameters of antecedent & consequent membership functions, (determine their shape and position) and the rules (determine the mapping between the antecedent and consequent fuzzy regions). 4/14 Author: Vali Derhami Design of Fuzzy Systems Design methods: Domain Knowledge ( as a Fuzzy Expert System) (Ch. 5.1 Babuska’s book) Numerical Data (Rule Extraction) Learning Algorithms Hybrid Mathematical Modeling 5/14 Author: Vali Derhami Domain Knowledge 1- Select the input and output variables, the structure of the rules, and the inference and defuzzification methods. 2. Decide on the number of linguistic terms for each variable and define the corresponding membership functions. 3. Formulate the available knowledge in terms of fuzzy ifthen rules. 4. Validate the model (typically using data). If the model does not meet the expected performance, iterate on the above design steps. 6/14 Author: Vali Derhami Design based on numerical data (Look-Up scheme) (Ch. 12, wang) The analytic formula of g(x) is unknown, a limited number of input-output pairs [xk, yk; zk] , k=1, 2, …, N Step1: Define fuzzy sets to cover the input and output spaces. X=[min(xk) max(xk)]; Y=[min(yk) max(yk)]; Z=[min(zk) max(zk)]; Define (nx,ny, nz) normal fuzzy sets in each space. Tx, Ty, Tz : completeness and consistency 7/14 Author: Vali Derhami Step 2: Generate one rule from one input-output pair. 8/14 Author: Vali Derhami Step 3. Assign a degree to each rule generated in Step 2 to resolve conflicts keep only one rule from a conflicting group that has the maximum degree. D( Rulek ) ai ( xk )b j ( yk )cl ( zk ) 9/14 Author: Vali Derhami Step 4. Create the fuzzy rule base. The rules generated in Step 2 that do not conflict with any other rules. The rule from a conflicting group that has the maximum degree, where a group of conflicting rules consists of rules with the same IF parts. Linguistic rules from human experts (due to conscious knowledge). 10/14 Author: Vali Derhami Step 5. Construct the fuzzy system based on the fuzzy rule base. For example, fuzzy systems with product inference engine, singleton fuzzifier, and center average defuzzifier. Study Ch. 3.4 Tanaka’s book Study “Truck upper-backer control” in page 157 and “Time series prediction” in page 162 from Wanag’s MATLAB command: FIS = GENFIS1(DATA, NUMMFS, INPUTMF, OUTPUTMF) The designed FIS can be improved by learning algorithms (See Ch. 5.3 Babuska’s book Author: Vali Derhami 11/14 Learning Algorithms الگوريتمهاي آموزشي مي توانندبراي تنظيم سيستم استفاده شوند Learning Structure Learning Parameters: yˆ f (u; ) Methods: Supervised Learning Unsupervised Learning Reinforcement Learning 12/14 Author: Vali Derhami مثال Example 13/14 Author: Vali Derhami Mathematical Modeling Read 2.2.1 from Tanaka’s book 14/14 Author: Vali Derhami