December 2005
Intermediate to advanced
475 pages
12h 6m
English
Content preview from Signals and Systems
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9.8.1 Solution of Homogeneous State Equation
In homogeneous case, the term u(t) is not present.
The Eq. (9.49) becomes
Taking Laplace transform of Eq. (9.55), we get,
sX(s) − x(0) = AX(s)
∴ (sI – A) X(s) = x(0)
∴ X (s) = (sI – A)−1x(0) (9.56)
∴ X(s) = ϕ(s) x (0) (9.57)
where ϕ(s) = (sI – A) −1 (9.58)
called the resolvent matrix.
Taking inverse Laplace transform of Eq. (9.59), it can be written as
x(t) = ϕ(t) x(0) (9.60)
where ϕ(t) is given in Eq. (9.60) ...
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