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עמוד:*24
. Aproper choice of the parameters ˜ λ i renders the error dynamics finite - time stable Correspondingly, in the absence of noises the equalities z i = f ( i ) are established in finite - 0 time . It is not simple to properly choose the differentiator parameters for each n . The task is facilitated by employing its recursive form z 0 = − λ n L 1 ˙ n + 1 z 0 − f ( t ) � n � n + 1 + z 1 , z 1 = − λ n − 1 L 1 ˙ n z 1 − ˙ z 0 � n − 1 � n z 2 , + . . . z n − 1 = − λ 1 L 1 ˙ 2 z n − 1 − ˙ z n − 2 � 1 � z n , + 2 z n = − λ 0 Lsign ( z n − ˙ z n − 1 ) , ˙ ) 5 ( for some positive λ i > 0 , i = 0 , 1 , . . . , n . Excluding ˙ z i reduce ( 5 ) to the general struc - ture ( 1 ) and the standard form ( 4 ) . It is easily verified that ˜ λ 0 = λ 0 , ˜ λ n = λ n , and i = λ i ˜ λ i λ ˜ i + 1 i + 1 , i = n − 1 , n − 2 , . . . , 1 . In the case f ( t ) ≡ 0 systems ( 4 ) and ( 5 ) become homogeneous of the HD− 1 with deg t = 1 , deg z i = deg σ i = n − i + 1 , i = 0 , . . . , n . An infinite sequence of parameters →− λ = { λ 0 , λ 1 , . . . } can be built, providing coeffi - cients ˜ λ i of ( 4 ) for all natural n . For this end, one simply starts with any λ 0 > 1 and recursively adds a sufficiently large value λ n > 0 for each n = 1 , 2 , . . . . The parameters are surprisingly easily found by simulation . In particular, →− λ = are well checked for n ≤ 9 . } . . . , 16 , 14 , 12 , 10 , 7 , 5 , 3 , 2 , 5 . 1 , 1 . 1 { For the future usage introduce the number n d currently equal to the differentiation * 24
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אדמוני, אריאל
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