1. Derivative-Based Fuzzy System Optimization Dan Simon Cleveland State University 1

2. Suppose we have a fuzzy controller that operates for N time steps. The controller error can be measured as: 2

3. Input Modal Points c ij 3 Note that y q is constant. Therefore,

4. at each time step q, where w j is the firing strength,  j is the centroid, and J j is the area of the j-th output fuzzy membership function. 4

5. 5

6. Definition: r i1k = 1 if x 1  fuzzy set i is a premise of the k-th rule and w k = f i1 (x 1 ), and r i1k = 0 otherwise. In other words, r i1k = 1 if x 1 determines the activation level of the kth rule because of its membership in the i-th fuzzy set. Similarly, r i2k = 1 if x 2  fuzzy set i is a premise of the k-th rule and w k = f i2 (x 2 ), and r i2k = 0 otherwise. 6

7. x 1  NS (0.8) and x 1  Z (0.2) x 2  Z (0.3) and x 2  PS (0.7) 12-th rule: x 1  NS and x 2  PS  y  NS Therefore, r 2,1,12 = 1. 7 NLNSZPSPL NL NS NLNSZZZ ZNL NS ZPSPL PSZZZ PL PS PL Change in Error x 2 Error x 1 Throttle Position Change y x1x1 x2x2 Example: r i,1,12 = 0 for i  {1, 3, 4, 5}, and r i,2,12 = 0 for i  [1, 5].

8. Recall w k = firing strength of k-th rule, which is equal to the minimum of the two input membership functions. Therefore: 8 Recall the membership functions f i1 (x 1 ) are given by the following triangular functions:

9. 9

10. Summary: The expressions on pages 3, 5, 8, and 9, give us the partial derivative of the error with respect to the modal points of the input MFs. Similar methods are used to find the derivatives of the error with respect to input MF half-widths, output MF modal points, and output MF half- widths. Now we can use gradient descent (or another gradient-based method) to optimize the MFs. Reference: D. Simon, "Sum normal optimization of fuzzy membership functions," International Journal of Uncertainty, Fuzziness and Knowledge- Based Systems, Aug. 2002. 10

11. Example: Fuzzy Cruise Control – VehicleGrad.m 11 74% error decrease

12. Performance of cruise control after gradient descent optimization of MFs 12

13. 13 Default Output MFs Optimized Output MFs PlotMem( 'paramgu.txt', 2, [5 5], 1, 5) Throttle change (rad) The input MFs do not change as much as the output MFs

14. Some issues to think about: Why use 5 membership functions for the output and for each input? How can we make the response less oscillatory? How about something like: where  is a weighting parameter 14

15. Try different initial conditions (initial fuzzy membership functions) Try different gradient descent options Try different MF shapes How can we optimize while constraining the MFs to be sum normal? 15

16. f 11 (x 1 ) 16 c 11 b 11 – 1 x1x1 b 11 + f 21 (x 1 ) c 21 b 21 – 1 x1x1

17.  = number of input 1 fuzzy sets = number of input 2 fuzzy sets  = number of output fuzzy sets 17 Similar equality constraints can be written for the input 2 MFs, and for the output MFs.

18. Similar equality constraints for the input 2 MFs and the output MFs 18 L 1 : 2(   1)  3  L 2 : 2(  1)  3 L 3 : 2(   1)  3 

19. In our case,  = 0 19

20. 20 Example: Fuzzy Cruise Control – VehicleGrad(1); 70% error decrease (Recall that it was 74% for unconstrained optimization)

21. 21 Performance of cruise control after unconstrained and constrained gradient descent optimization of MFs

22. 22 Default Output MFs Optimized Constrained Output MFs PlotMem( 'paramgc.txt', 2, [5 5], 1, 5) Throttle change (rad) The input MFs do not change as much as the output MFs

23. Other gradient descent optimization algorithms: Chapter 3, “A Course in Fuzzy Systems and Control,” by Li-Xin Wang 23