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Signals and Systems by Smarajit Ghosh

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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

 

image

 

Taking Laplace transform of Eq. (9.55), we get,

 

sX(s) − x(0) = AX(s)

 

∴             (sIA) X(s) = x(0)

 

∴          X (s) = (sIA)−1x(0)       (9.56)

 

∴          X(s) = ϕ(s) x (0)       (9.57)

 

where             ϕ(s) = (sIA) −1       (9.58)

 

called the resolvent matrix.

 

image

 

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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