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Computational Intelligence in Automotive Applications by Danil Prokhorov_7

Tham khảo tài liệu 'computational intelligence in automotive applications by danil prokhorov_7', kỹ thuật - công nghệ, điện - điện tử phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 108 D. Prokhorov the second trajectory starts at t 0 in x 0 Xo 2 etc. The coverage of the domain X should be as broad as practically possible for a reasonably accurate approximation of I. Training the NN controller may impose computational constraints on our ability to compute 4 many times during our iterative training process. It may be necessary to contend with this approximation of R 1 A A W i s Yu i t . 5 X0 s ex s 1 2 . 5 t 0 The advantage of A over R is in faster computations of derivatives of A with respect to W i because the number of training trajectories per iteration is s N and the trajectory length is H T. However A must still be an adequate replacement of R and possibly I in order to improve the NN controller performance during its weight training. And of course A must also remain bounded over the iterations otherwise the training process is not going to proceed successfully. We assume that the NN weights are updated as follows W i 1 W i d i 6 where d i is an update vector. Employing the Taylor expansion of I around W i and neglecting terms higher than the first order yields I W i 1 I W i dI i W i 1 - W i . 7 I W i 1 I W i dW i W i 1 VV i . 7 Substituting for W i 1 W i from 6 yields I W i 1 I W i -AA d i . 8 ỠW i The growth of I with iterations i is guaranteed if dW i T d i 0. 9 Alternatively the decrease of I is assured if the inequality above is strictly negative this is suitable for cost minimization problems e.g. when u t yr t yp t 2 which is popular in tracking problems. It is popular to use gradients as the weight update d i n i dW 10 where n i 0 is a learning rate. However it is often much more effective to rely on updates computed with the help of second-order information see Sect. 4 for details. The condition 9 actually clarifies what it means for A to be an adequate substitute for R. The plant model is often required to train the NN controller. The model needs to provide accurate enough d such that 9 is satisfied. Interestingly from the .