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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 et  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.65 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 et _ 2 e 2t oCt) + 2 e 2t
[  6e t + 8 e2t (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 tateSpace 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 APIw + P Bf [2 0] (13.68c) = Aw+:Bf 3 (13.68d) w here [! !] 1 a nd A = P API (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 (pI 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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 Spring '13
 Bayliss
 Signal Processing, The Land

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