Signal Processing and Linear Systems-B.P.Lathi copy

89b c i n general en a nie n 1 a le aoykj

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Unformatted text preview: e-Space A nalysis P roblems R epeat P rob. 13.3-5 i f A= [~: 835 Define a new s tate v ector w s uch t hat _ :] f (t) B = [ :] = u (t) x(O) C =[1 1J D =1 F ind t he s tate e quations o f t he s ystem w ith w a s t he s tate v ector. D etermine t he c haracteristic r oots ( eigenvalues) o f t he m atrix A i n t he o riginal a nd t he t ransformed s tate e quations. = [ :] 1 3.3-7 T he t ransfer f unction H (s) i n P rob. 13.2-8 is realized a s a c ascade o f H I(S) followed b y H2(S), w here and H 2( ) S 3s 1 3.4-2 T he s tate e quations o f a c ertain s ystem a re + 10 =~ L et t he o utputs o f t hese s ubsystems b e s tate v ariables X l a nd X 2, r espectively. Write t he s tate e quations a nd t he o utput e quation for t his s ystem a nd verify t hat H (s) = ( a) D etermine a n ew s tate v ector w ( in t erms o f v ector x ) s uch t hat t he r esulting s tate e quations a re i n d iagonalized form. ( b) I f t he o utput y is given b y C~(s)B+D. 1 3.3-8 F ind t he t ransfer-function m atrix H (s) for t he s ystem i n P rob. 13.3-5. 1 3.3-!l F ind t he t ransfer-function m atrix H (s) for t he s ystem i n P rob. 13.3-6. 1 3.3-10 y = C x+Df F ind t he t ransfer-function m atrix H (s) for t he s ystem w here x = A x+Bf C- [ 1 -1 y = Cx+Df 1] 2 a nd D=0 d etermine t he o utput y i n t erms o f t he n ew s tate v ector w . w here 1 3.4-3 G iven a s ystem 1 3.4-4 T he s tate e quations o f a c ertain s ystem a re g iven i n d iagonalized form a s d etermine a n ew s tate v ector w s uch t hat t he s tate e quations a re d iagonalized. 1 3.3-11 R epeat P rob. 13.3-1, u sing t he t ime-domain m ethod. 1 3.3-12 R epeat P rob. 13.3-2, u sing t he t ime-domain m ethod. 1 3.3-13 R epeat P rob. 13.3-3, u sing t he t ime-domain m ethod. 1 3.3-14 R epeat P rob. 13.3-4, using t he t ime-domain m ethod. 1 3.3-15 R epeat P rob. 13.3-5, using t he t ime-domain m ethod. 1 3.3-16 R epeat P rob. 13.3-6, u sing t he t ime-domain m ethod. 1 3.3-17 F ind t he u nit i mpulse r esponse m atrix h (t) for t he s ystem i n P rob. 13.3-7, using Eq. (13.65). 1 3.3-18 F ind t he u nit i mpulse r esponse m atrix h (t) for t he s ystem i n P rob. 13.3-6. 1 3.3-19 F ind t he u nit i mpulse r esponse m atrix h (t) for t he s ystem i n P rob. 13.3-10. 1 3.4-1 T he o utput e quation is given b y y =[1 3 1Jx D etermine t he o utput y i f T he s tate e quations o f a c ertain s ystem are given a s :i:t = X2 + 2 f a nd f (t) = u (t) 13 State-Space Analysis 8 36 S upplementary R eading y f F ig. P 13.5-1 1 3.5-1 W rite t he s tate e quations for t he s ystems d epicted in Fig. P I3.5-1. D etermine a n ew s tate v ector w s uch t hat t he r esulting s tate e quations a re i n diagonalized form. W rite t he o utput y in t erms o f w . D etermine i n each case w hether t he s ystem is controllable a nd observable. 1 3.6-1 A n LTI discrete-time s ystem is specified by A= [: :] B = [ :] Ambardar, Ashok, Analog and Digital Signal Processing, P WS Publishing, B oston, Mass, 1995. C =[O 1] a nd D =[I] a nd x(O) = [ :] a nd f [k] = u[k] ( a) F ind t he o utput y[k]...
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