Rezolvate 2 - Problema 1 Fie u(t un semnal continuu cu...

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Problema 1 Fie u ( t ) un semnal continuu cu derivata ˙ u ( t ) continu˘ a. Care dintre urm˘ atoarele poate fi o norm˘a pentru u ? 1. sup t | ˙ u ( t ) | 2. sup t | u ( t ) | + sup t | ˙ u ( t ) | Solut¸ie 1. Se constat˘ a, prin verificare direct˘ a, cu definit¸ia, c˘ a nu avem de a face cu o norm˘ a, ˆ ın schimb, aceasta este o semi-norm˘ a. 2. Verificˆ and cele patru ipoteze din definit¸ie, obt¸inem c˘ a aceasta este, ˆ ıntr-adev˘ ar, o norma. ± Problema 2 a consider˘ am sistemul cu funct¸ia de transfer s + 2 4 s + 1 , intrarea u ¸ si ie¸ sirea y . Calculat¸i sup k u k =1 k y k ¸ si g˘asit¸i o intrare pentru care se atinge acest supremum. Solut¸ie Conform tabelului 2 (curs, capitolul Norme pentru semnale ¸ si sisteme ), pozit¸ia (2,2), avem c˘ a, pentru k u k 1 sup k y k = k g ( t ) k 1 . Trebuie s˘ a determin˘ am k g ( t ) k 1 . Prin calcul direct obt¸inem g ( t ) = L - 1 { G ( s ) } ( s ) = 1 4 δ ( t ) + 7 16 e - 1 4 t · 1 ( t ) Deci, prin calcul direct, k g ( t ) k 1 = 2. S˘a consider˘ am acum, pentru t fixat, intrarea u ( t - τ ) := sgn ( g ( τ )) τ . Evident, k u k = 1 ¸ si y ( t ) = Z -∞ g ( τ ) u ( t - τ ) = Z -∞ | g ( τ ) | = k g k 1 ± Problema 3 Pentru un sistem liniar, cu intrarea u ( t ) ¸ si ie¸ sirea y ( t ), ar˘ atat¸i c˘ a sup k u k≤ 1 k y k = sup k u k =1 k y k unde, pentru norm˘ a se poate alege, de exemplu, norma 2. Solut¸ie
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This note was uploaded on 11/05/2011 for the course SYSTEM THE 14 taught by Professor Oaracristian during the Spring '11 term at UPB Colombia.

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Rezolvate 2 - Problema 1 Fie u(t un semnal continuu cu...

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