Page 386 - Applied Numerical Methods Using MATLAB

SIMILARITY TRANSFORMATION AND DIAGONALIZATION 375
we use the modal matrix obtained as (E8.2.2) in Example 8.2 to make a substi-
tution of variable
√
x 1 (t) 1 1/ 2 w 1 (t)
x(t) = V w(t), = √ (E8.3.2)
x 2 (t) 0 −1/ 2 w 2 (t)
which converts Eq. (E8.3.1) into

V w (t) = AV w(t) + Bu s (t) (E8.3.3)

We premultiply (E8.3.3) by V −1 to write it in a decoupled form as

w (t)=V −1 AV w(t) + V −1 Bu s (t)= w(t) + V −1 Bu s (t) with w(0) = V −1 x(0);

w 1 (t) 0 0 w 1 (t) 1 1 0 u s (t)
= + √ u s (t) = √

w 2 (t) 0 −1 w 2 (t) 0 − 2 1 −w 2 (t) − 2u s (t)
(E8.3.4)

w 1 (0) 1 1 1 0
with = √ = √
w 2 (0) 0 − 2 −1 2
where there is no correlation between the variables w 1 (t) and w 2 (t).Thenwe
can solve these two equations separately to have

w 1 (t) = u s (t) with w 1 (0) = 0;
1 1
sW 1 (s) − w 1 (0) = ; W 1 (s) = ; w 1 (t) = tu s (t) (E8.3.5a)
s s 2
√ √

w 2 (t) =−w 2 (t) − 2u s (t) with w 2 (0) = 2;
√
2
sW 2 (s) − w 2 (0) =−W 2 (s) − ;
s
√ √ √
w 2 (0) 2 2 2 2
W 2 (s) = − =− + ,
s + 1 s(s + 1) s s + 1
√
−t
w 2 (t) = 2(−1 + 2e )u s (t) (E8.3.5b)
and substitute this into Eq. (E8.3.2) to get

√ √
x 1 (t) 1 1/ 2 w 1 (t) 1 1/ 2 t
= √ = √ √ u s (t)
−t
x 2 (t) 0 −1/ 2 w 2 (t) 0 −1/ 2 2(−1 + 2e )
−t
t − 1 + 2e
= −t u s (t) (E8.3.6)
1 − 2e
This is the same result as Eq. (6.5.10) obtained in Section 6.5.1.