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עמוד:*22
) produce exact derivative estimations z i = f ( i t ) , i = 0 , 1 , . . . , n d , of the signal f 0 ( t ) , pro - ( 0 vided | f ( n d + 1 ) Lholds for some known L > 0 . So - called hybrid ( bi - homogeneous ) ≤ | 0 differentiators, are also effective for the variable bound L ( t ) . The accuracy of these differentiators is also analyzed in the presence of discrete noisy sampling . The asymptotics of errors z i − f ( i ) in the presence of bounded sampling noises 0 are proved to be the best possible . Filtering differentiators preserve the mentioned op - timality, whereas at the same time suppress potentially - unbounded noises having small local k th - order iterated integrals, k ≤ n f . The number n f is called the filtering order of the differentiator, whereas k is the filtering order of the noise signal . Homogeneous Differentiation The control approach to the n th - order differentiation of a noisy sampled function f 0 ( t ) t ) withsuggests constructing an observer for the disturbed integrator chain y ( n + 1 ) ( ξ = the output y and the unknown disturbance / input ξ = f ( n + 1 ) t ) . Its outputs z i , i = 0 , . . . , n , ( 0 t ) in spite of y = f 0 ( t ) being sampled with some noise andare to approximate the y ( i ) ( t ) being unknown . ( ξ Let the input f ( t ) take the form f ( t ) = f 0 ( t ) + η ( t ) ∈ R, where η ( t ) is a Lebesgue - measurable bounded noise, | η ( t ) | < ε 0 , and f 0 ( t ) is an n - times differentiable unknown function to be restored together with its n derivatives in spite of the unknown measure - ment noise intensity ε 0 . The last derivative f ( n ) is assumed to have a known Lipschitz 0 constant L > 0 , which means that f ( n + 1 ) t ) ∈ [ − L, L ] holds for almost all t . It is further ( 0 n L . denoted as f 0 ∈ Lip * 22
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אדמוני, אריאל
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