Page 388 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 71 STATE SPACE ANALYSIS

C. Solution in the Time Domain:
Following

we define

where k! = k(k - 1) . . -2.1. If t = 0, then Eq. (7.53) reduces to
e0 = I (7.54)

where 0 is an N x N zero matrix whose entries are all zeros. As in ea('-') = ea'e-a' =
e-ar at
e , we can show that
eA(t-r) = eA~e-Ar = e-AreA~ (7.55)

Setting T = t in Eq. (7.55), we have
eAte-Ar = e-AteAt = eO=I (7.56)

Thus,

which indicates that e-A' is the inverse of eA'.
The differentiation of Eq. (7.53) with respect to t yields

which implies

d
-eAt = AeA' = eAt~
dt
Now using the relationship [App. A, Eq. (A.70)]

and Eq. (7.581, we have