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עמוד:*23
The general differentiator is usually of the form z i = ϕ i ( z 0 − f ( t ) ) + z i + 1 , i = 0 , . . . , n − 1 , ˙ z n = ϕ n ( z 0 − f ( t ) ) , ( 1 ) ˙ where ϕ i is a scalar function, z i ∈ R . The system is understood in the Filippov sense to allow discontinuities of ϕ i . The equivalent recursive form of ( 1 ) is z 0 = ϕ i ( z 0 − f ( t ) ) + z 1 , ˙ z i = ϕ i ( z i − ˙ z i − 1 ) + z i + 1 , i = 1 , . . . , n − 1 , ˙ z n = ϕ n ( z 0 − f ( t ) ) . ( 2 ) ˙ assuming the noise is absent ( i . e . ε = 0 ) , subtracting f ( i + 1 ) from both sides of ( 1 ) , and 0 denoting σ i = z i − f ( i ) derive , 0 i = ϕ i ( σ 0 ) + σ i + 1 , i = 0 , . . . , n − 1 σ ˙ n ∈ ϕ n ( σ 0 ) + [ − L, L ] , ( 3 ) σ ˙ which is a DIin the error space →− σ n . DI ( 1 ) becomes homogeneous and FTstable for properly chosen functions ϕ i . The ”standard” n th - order homogeneous SM - based diffrerentiator has the form z 0 = − ˜ λ n L 1 ˙ n + 1 z 0 − f � n � n + 1 + z 1 , z 1 = − ˜ λ n − 1 L 2 ˙ n + 1 z 0 − f � n − 1 � n + 1 + z 2 , . . . z n − 1 = − ˜ λ 1 L n ˙ n + 1 z 0 − f � 1 � n + 1 + z n , z n = − ˜ λ 0 Lsign ( z 0 − f ) , ˙ ) 4 ( where the parameters ˜ λ i > 0 of the differentiator ( 5 ) are to be chosen in advance for each n, i = 0 , 1 , 2 , . . . , n . * 23
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אדמוני, אריאל
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