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Derivative-Based Fuzzy System Optimization

Derivative-Based Fuzzy System Optimization. Dan Simon Cleveland State University. Suppose we have a fuzzy controller that operates for N time steps. The controller error can be measured as:. Input Modal Points c ij. Note that y q is constant. Therefore,.

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Derivative-Based Fuzzy System Optimization

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

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

  3. Input Modal Points cij Note that yq is constant. Therefore,

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

  5. Definition: ri1k = 1 if x1 fuzzy set i is a premise of the k-th rule and wk = fi1(x1), and ri1k = 0 otherwise. In other words, ri1k = 1 if x1 determines the activation level of the kth rule because of its membership in the i-th fuzzy set. Similarly, ri2k = 1 if x2 fuzzy set i is a premise of the k-th rule and wk = fi2(x2), and ri2k = 0 otherwise.

  6. Example: Error x1 x1 NS (0.8) and x1 Z (0.2) x2 Z (0.3) and x2 PS (0.7) 12-th rule: x1 NS and x2 PS  y  NS Therefore, r2,1,12 = 1. Change in Error x2 Throttle Position Change y x1 x2 ri,1,12 = 0 for i  {1, 3, 4, 5}, and ri,2,12 = 0 for i  [1, 5].

  7. Recall wk = firing strength of k-th rule, which is equal to the minimum of the two input membership functions. Therefore: Recall the membership functions fi1 (x1) are given by the following triangular functions:

  8. 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.

  9. 74% error decrease Example: Fuzzy Cruise Control – VehicleGrad.m

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

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

  12. 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

  13. 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?

  14. f11(x1) 1 f21(x1) 1 x1 b11– c11 b11+ x1 b21– c21

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

  16. Similar equality constraints for the input 2 MFs and the output MFs L1: 2( 1)  3 L2: 2( 1)  3 L3: 2( 1)  3

  17. In our case,  = 0

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

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

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

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

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