1. Neuro-Fuzzy Control Adriano Joaquim de Oliveira Cruz NCE/UFRJ adriano@nce.ufrj.br

2. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 2 Neuro-Fuzzy Systems = Usual neural networks that simulate fuzzy systems = Introducing fuzziness into neurons

3. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 3 ANFIS architecture = Adaptive Neuro Fuzzy Inference System = Neural system that implements a Sugeno Fuzzy model.

4. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 4 Sugeno Fuzzy Model = A typical fuzzy rule in a Sugeno fuzzy model has the form If x is A and y is B then z = f(x,y) = A and B are fuzzy sets in the antecedent. = z=f(x,y) is a crisp function in the consequent. = Usually z is a polynomial in the input variables x and y. = When z is a first-order polynomial the system is called a first-order Sugeno fuzzy model.

5. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 5 Sugeno Fuzzy Model  x y  x y w1w1 w2w2 A2A2 A1A1 B1B1 B2B2 z 1 =p 1 x+q 1 y+r 1 z 2 =p 2 x+q 2 y+r 2

6. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 6 Sugeno First Order Example = If x is small then y = 0.1x + 6.4 = If x is median then y = -0.5x + 4 = If x is large then y = x – 2 Reference: J.-S. R. Jang, C.-T. Sun and E. Mizutani, Neuro-Fuzzy and Soft Computing

7. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 7 Comparing Fuzzy and Crisp

8. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 8 Sugeno Second Order Example = If x is small and y is small then z = -x + y +1 = If x is small and y is large then z = -y + 3 = If x is large and y is small then z = -x + 3 = If x is large and y is large then z = x + y + 2 Reference: J.-S. R. Jang, C.-T. Sun and E. Mizutani, Neuro-Fuzzy and Soft Computing

9. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 9 Membership Functions

10. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 10 Output Surface

11. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 11 ANFIS Architecture = Output of the ith node in the l layer is denoted as O l,i A1A1 A2A2 B1B1 B2B2 x y      Layer 1Layer 3Layer 2Layer 4Layer 5 xy xy w1w1 w2w2 f O 1,2

12. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 12 ANFIS Layer 1 = Layer 1: Node function is = x and y are inputs. = A i and B i are labels (e.g. small, large).   (x) can be any parameterised membership function. = These nodes are adaptive and the parameters are called premise parameters.

13. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 13 ANFIS Layer 2 = Every node output in this layer is defined as: = T is T-norm operator. = In general, any T-norm that perform fuzzy AND can be used, for instance minimum and product. = These are fixed nodes.

14. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 14 ANFIS Layer 3 = The ith node calculates the ratio of the ith rule’s firing strength to the sum of all rules’ firing strength = Outputs of this layer are called normalized firing strengths. = These are fixed nodes.

15. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 15 ANFIS Layer 4 = Every ith node in this layer is an adaptive node with the function = Outputs of this layer are called normalized firing strengths. = p i, q i and r i are the parameter set of this node and they are called consequent parameters.

16. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 16 ANFIS Layer 5 = The single node in this layer calculates the overall output as a summation of all incoming signals.

17. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 17 ANFIS Layer 5 = Every ith node in this layer is an adaptive node with the function = Outputs of this layer are called normalized firing strengths.

18. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 18 Alternative Structures = The structure is not unique. = For instance layers 3 and 4 can be combined or weight normalisation can be performed at the last layer.

19. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 19 Alternative Structure cont. A1A1 A2A2 B1B1 B2B2 x y    w1w1 w2w2 f O 1,2  Layer 1Layer 2Layer 5Layer 3 xy xy Layer 4

20. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 20 Training Algorithm = The function f can be written as = There is a hybrid learning algorithm based on the least-squares method and gradient descent.

21. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 21Example = Modeling the function = Input range [-10,+10]x[-10,+10] = 121 training data pairs = 16 rules, with four membership functions assigned to each input. = Fitting parameters = 72; 24 premise and 48 consequent parameters.

22. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 22 Initial and Final MFs

23. *@2001 Adriano Cruz *NCE e IM - UFRJ Neuro-Fuzzy 23 Training Data