Signals and Systems

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) ...