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Computacion Inteligente

Learn how to tune fuzzy systems using least-squares estimation. Explore the singleton and linear Takagi-Sugeno fuzzy models with practical examples. MATLAB programming exercises included.

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Computacion Inteligente

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  1. Computacion Inteligente Tuning fuzzy systems: Least-Squares Estimation of Consequents

  2. Contents • The singleton fuzzy model • Linear Takagi-Sugeno fuzzy systems • Example: Sinedata

  3. The singleton fuzzy model

  4. The singleton fuzzy model • Let’s consider the Singleton fuzzy model: • where the normalized degree of fulfillment is,

  5. The singleton fuzzy model • Expressing output y in vector form, • where we define

  6. Linear Least Squares tuning • The form of the model to be tuned is in only a slightly different form from the standard linearmodel. • This means that Linear Least Squares can be used to train certain types of fuzzy systems Ones that can be parameterized so that they are“linear in the parameters”

  7. Linear Takagi-Sugeno fuzzy systems

  8. Linear Takagi-Sugeno fuzzy systems • Let’s consider the linear Takagi-Sugeno system: • Denote,

  9. Linear Takagi-Sugeno fuzzy model • Then, we can write where we define This means that Linear Least Squares can be used

  10. Least-squares tuning • Let’s find the parameters using Least Squares • we can write

  11. Least-squares tuning • Assume a set of N input-output data pairs • Denote

  12. Least-squares tuning • Denote • Create the extended matrix • Further, denote

  13. Least-squares tuning • Now vector y can be written in a matrix form, • Then the optimal batch least-squares solution which gives the minimal prediction error is

  14. Least-squares tuning • Then the optimal batch least-squares solution which gives the minimal prediction error is • This is an optimal batch least-squares solution which gives the minimal prediction error, and as such is suitable for prediction models.

  15. Least-squares tuning • At the same time, however, it may bias the estimates of the consequent parameters as parameters of local models. • If an accurate estimate of local model parameters is desired, a least-squares approach applied per rule may be used: Prove it!

  16. Example: Sinedata

  17. Example: Sinedata • Ejemplo: Modelo TS lineal y sinedata • Dados el juego de pares de datos (x,y) en sinedata.mat, construir un sistema fuzzy Takagi-Sugeno y ajustar los parametros del consecuente por el metodo de los minimos cuadrados. • Solucion: VER sinedata_FIS.m.

  18. Exercise • Ejercicio: Minimos cuadrados aplicado por regla • Dados el juego de pares de datos (x,y) en sinedata.mat, desarrollar un programa en MATLAB para construir un sistema fuzzy Takagi-Sugeno y ajustar los parametros del consecuente usando la expresion:

  19. Sources • Kevin M. Passino, Stephen Yurkovich, Fuzzy Control. Addison Wesley Longman, Inc. 1998 • Robert Babuska. Course Fuzzy and Neural Control, 2001/2002

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