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,
∴ (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
where ϕ(t) is given in Eq. (9.60) ...
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