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Lista1_SCII2010 - 1 SEM 538 SISTEMAS DE CONTROLE II 1a...

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1 SEM 538 - SISTEMAS DE CONTROLE II 1 a Lista de Exerc´ ıcios 1. Encontre o termo gen´ erico y ( k ) de uma progress˜ao geom´ etrica de raz˜ao r ∈ ℜ . 2. Encontre o termo gen´ erico s ( k ) da soma dos k primeiros termos da progress˜ao do exerc´ ıcio 1, considerando r = 3 e r = 1. 3. Encontre a solu¸ c˜ao para a equa¸ c˜ao a diferen¸ ca de segunda ordem y ( k + 2) y ( k + 1) 2 y ( k ) = 5 . 2 k com y (0) = 3 e y (1) = 0, utilizando o M´ etodo dos Coeficientes a Determinar. 4. Calcule a Transformada Z das seguintes fun¸ c˜oes discretas (a) x ( kT 0 ) = kT 0 (b) x ( kT 0 ) = sen ( θkT 0 ) (c) x ( kT 0 ) = e akT 0 cos ( θkT 0 ) (d) x ( kT 0 ) = 0 , 1 , 2 , 3 , 0 , 0 , 0 , 0 , · · · 5. Obter a transformada Z inversa para os seguintes casos, sendo T 0 = 1: (a) X ( z ) = 1 , 6 z 2 0 , 8 z 3 2 , 6 z 2 2 , 2 z 0 , 6 . (b) X ( z ) = 0 , 6 z z 2 1 , 7 z +0 , 7 . (c) X ( z ) = 0 , 1( z +1) z ( z 1) 2 ( z 0 , 6) . Itens a) e b) usar o M´ etodo da Fatora¸ c˜ao. Item c) usar M´ etodo da Divis˜ao at´ e o quinto termo. Em todos os casos, quando aplic´avel, calcular: o valor final ( x ( k ) | k →∞ ), o valor inicial ( x ( k ) | k 0 ).
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