O'Reilly logo

Signals and Systems by Smarajit Ghosh

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

9.8.4 Solution of Non-homogeneous State Equation

In this case u (t) is taken into account.

Taking Laplace transform of Eq.(9.49), we get,

 

s X(s) – x(0) = AX(s) + BU(s)

 

∴      (sIA) X(s) = x(0) + BU(s)

 

∴    X(s) = [sIA] −1 x(0) + (sIA) −1 BU(s)       (9.64)

 

Taking inverse Laplace transform of Eq. (9.64), we get,

 

x(t) = LT−1[(sIA)−1 x(0) + (sIA) −1BU(s)]
= LT−1[(sIA) −1 x(0) + LT−1[sIA) −1BU(s)]

 

Now  LT−1[(sIA)−1x(0)] = ϕ(t) x(0) = eAtx(0)

image

 

Eq. (9.67) represents the solution of non-homogeneous equation. It consists of (i) the term eAtx(0) called homogeneous or free response and (ii) the term called forced ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required