#### 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*)

∴ (*s***I** – **A**) **X**(*s*) = **x**(0) + **BU**(*s*)

∴ *X*(*s*) = [*s***I** – **A**] ^{−1} **x**(0) + (*s***I** – **A**)^{ −1} **BU**(*s*) (9.64)

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

**x**(*t*) = *LT*^{−1}[(*s***I** – **A**)^{−1} **x**(0) + (*s***I** – **A**) ^{−1}**BU**(*s*)]

= *LT*^{−1}[(*s***I** – **A**) −1 **x**(0) + *LT*^{−1}[*s***I** – **A**) ^{−1}**BU**(*s*)]

Now *LT*^{−1}[(*s***I** – **A**)^{−1}**x**(0)] = *ϕ*(*t*) **x**(0) = *e*^{Atx}(0)

Eq. (9.67) represents the solution of non-homogeneous equation. It consists of (i) the term *e*^{At}*x*(0) called homogeneous or free response and (ii) the term called forced ...