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

I n o ther w ords t he p oles o f all t ransfer f

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Unformatted text preview: =Px (13.67c) (13.67d) 0] Ii(t) S ubstituting t he m atrices q,(t), Set), C , D , a nd B [Eq. (13.40c)] i nto Eq. (13.65), we have -2t] 1 0] + [~ + e -2t [1 1 e-t - e - e - +2 P ll Defining t he v ector w a nd m atrix P , as shown above, we c an w rite E q. (13.67b) as 2 + 0+2 T he s ame r esult is o btained i n Sec. B.6-5 by using Eq. (13.59a) [see Eq. (B.84)]. Also, Set) is a diagonal j x j o r 2 x 2 m atrix: Set) = [OCt) + P n2X2 + . .. + P nnXn or o :] [ Ii(t) 0 0] Thus, t he s tate v ector x is t ransformed i nto a nother s tate v ector w t hrough t he l inear t ransformation i n E q. (13.67c). I f we know w , we c an d etermine x from Eq. (13.67d), provided t hat p - l e xists. T his is equivalent t o s aying t hat P is a n onsingular m atrixt ( IPI =f 0 ). T hus, i f P is a nonsingular m atrix, t he v ector w defined by Eq. (13.67c) is also a s tate vector. Consider t he s tate e quation o f a s ystem X = A x+Bf Ii(t) 1 (13.68a) w =Px (13.68b) If t hen 3 e-t _ 2 e- 2t oCt) + 2 e- 2t [ - 6e- t + 8 e-2t (13.66) t This condition is equivalent t o saying t hat all n equations in Eq. (13.67a) are linearly independent; t hat is, none of the n equations can be expressed as a linear combination of the remaining equations. 13 S tate-Space A nalysis 8 12 813 13.4 Linear Transformation O f S tate Vector a nd A =PAP- l =[11 1 ][0 -1 -2 1 ][11]-1 -3 1 -1 = [: H ence t he s tate e quation (13.68a) now becomes -:] [-: -:] or w = P AP-Iw + P Bf [-2 0] (13.68c) = Aw+:Bf 3 (13.68d) w here [! -!] -1 a nd A = P AP-I (13.69a) a nd (13.69b) T herefore E quation (13.68d) is a s tate e quation for t he s ame system, b ut now i t is expressed i n t erms of t he s tate v ector w. T he o utput e quation is also modified. Let t he o riginal o utput e quation b e T his is t he d esired s tate e quation for t he s tate v ector w . T he s olution o f t his e quation requires a knowledge o f t he i nitial s tate w(O). T his c an b e o btained from t he given initial s tate x(O) b y using Eq. (13.70b) . • C x+Df y= I n t erms of t he new s tate variable w, t his e quation becomes y = C (p-I W ) o + Df = Cw+Df where C= • C p- I (13.69c) E xample 1 3.9 T he s tate e quations o f a c ertain s ystem a re given by WI WI a nd a re W2 = X l + X2 or (13.70b) According t o E q. (13.70b), t he s tate e quation for t he s tate v ariable w is given by Vol = where [see E q. (13.69)] A w+Bf A =[O 1 ;-2 - 3]; B =[I; 2 ]; P =[1 1 ;1 - 1]; A hat=P* A * inv(P) B hat=P*B 0 Invariance o f Eigenvalues (13.70a) F ind t he s tate e quations for this system when t he new s tate v ariables C omputer E xample C 13.4 Solve E xample 13.9 using MATLAB. We have seen (Sec. 13.3) t hat t he poles o f all possible transfer functions of a system are t he eigenvalues of t he m atrix A . I f we t ransform a s tate v ector from x t o w , t he v ariables W I, W 2, . .. , W n a re linear combinations of X I, X 2, . .. , X n a nd t herefore may be considered as o utputs. Hence, t he p oles o f t he t ransfer functions relating W I, W 2, . .. ,...
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