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

36b ut d fs t he s ubstitution of eq 1333b into

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Unformatted text preview: he presence of t he t erm u (t) in every element of"e At . This is t he case because t he limits of t he convolution integral r un from 0 t o t [Eq. (13.56)J. Thus (_~e-4t + ~e-9t)u(t) * ~u(t) eAt * B f(t) = [ ¥( _ e- 4t + e -9t )u(t) * ~u(t) --lse- 4t u(t) =[ i s(e- 4t 9t - e- )u(t) * u (t) (~e-4t - ~e-9' ) U(t) * u (t) * u (t) + ~e-9tu(t) * U(t)] _~e-4tu(t) * u (t) + ~e-9tu(t) * u (t) 1 = C [eAtx(O) + e AtB * f(t)J + D f(t) (13.62b) Now recall t hat t he c onvolution o f f (t) w ith t he u nit i mpulse 8(t) y ields f (t). L et u s define a j x j d iagonal m atrix 6(t) s uch t hat a ll i ts d iagonal t erms a re u nit i mpulse functions. I t is t hen o bvious t hat 6(t) * f (t) = f (t) a nd E q. ( 13.62b) c an b e e xpressed a s + e AtB * f(t)J + D 6(t) * f (t) = C eAtx(O) + [ CeAtB + D6(t)J * f (t) y (t) = C [eAtx(O) W ith t he n otation ~(t) for eAt, E q. ( 13.63b) m ay b e e xpressed a s ( 13.63a) (13.63b) ; 810 13 S tate-Space Analysis + [Ct/>(t)B + m (t)] * f(t) Ct/>(t)x(O) y et) = ~ .. (13.63c) 811 T he r eader c an verify t hat t he t ransfer-function m atrix H (s) i n Eq. (13.42) is t he L aplace t ransform o f t he u nit-impulse response m atrix h (t) in Eq. (13.66) . • J v z ero-state r esponse z ero-input r esponse T he z ero-state response; t hat is, t he response when x(O) = 0 , is y et) = [Ct/>(t)B = h (t) 13.4 Linear Transformation O f S tate Vector 1 3.4 + DS(t)] * ret) (13.64a) * ret) (13.64b) w here h (t) = Ct/>(t)B + DS(t) (13.65) T he m atrix h (t) is a k x j m atrix known as t he i mpulse r esponse m atrix. T he r eason for t his d esignation is obvious. T he i jth e lement of h (t) is h ;j(t), which r epresents t he z ero-state response Yi w hen t he i nput f j(t) = I i(t) a nd w hen all o ther i nputs ( and all t he i nitial conditions) are zero. I t c an also b e seen from Eq. (13.39) a nd (13.64b) t hat l inear Transformation o f S tate Vectors I n Sec. 13.1 we saw t hat t he s tate o f a system c an b e specified in several ways. T he s ets o f all possible s tate variables m ust b e r elated-in o ther words, if we are given one s et o f s tate variables, we should be able t o r elate i t t o a ny o ther s et. We are p articularly i nterested in a linear t ype o f relationship. L et X l, X 2, . .. , X n a nd W I, W 2, . .. , W n b e two different sets of s tate v ariables specifying t he s ame s ystem. L et t hese s ets b e r elated by linear equations as W I = P llXI + P l2 X 2 + . .. + P lnXn (13.67a) L:[h(t)] = H (s) • W n = P nlXI E xample 1 3.8 F or t he s ystem d escribed by Eqs. (13.40a) a nd ( 13.40b), d etermine eAt using Eq. (13.59b): t/>(t) = eAt = C 1+(s) :} T his p roblem was solved earlier w ith f requency-domain techniques. F rom E q. (13.41), we have t/>(t) = £ -1 [ ~ ( 8+1)(8+2) -2 ( 8+1)(8+2) 2 = £ -1 1 8 +1 [ 8 11 8+2 + 8 !2 -2t 2e - t - e = [ _ 2e- t + 2 e- 2t (8+1~(8+2) ] Wn ( 8+1)(8+2) '-v-' w 1 1] 8 +1 - -1 0 +1 8+2 o -2t] a nd -et e _ e- t + 2 e- 2t P l2 P In Xl P 21 P 22 P 2n X2 (13.67b) . .................. P nl P n2 P nn v p Xn ' '--v--' x w...
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This note was uploaded on 04/14/2013 for the course ENG 350 taught by Professor Bayliss during the Spring '13 term at Northwestern.

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