User:Egm6321.f10.team1.allen/HW3

Find
Generalize Eq 5.1 to the case of SC_L1_ODE_VC.

Solution
For a L1_ODE_CC we have

$$ x(t)=e^{a(t-t_0)}x(t_0)+\int_{t_0}^{t}[e^{a(t-\tau)}]bu(\tau)d\tau $$.

and for a L1_ODE_VC we have

$$ x(t)=e^{\int_{t_0}^{t}a(\tau)d\tau}x(t_0)+\int_{t_0}^{t}[e^{\int_{\tau}^{t}a(s)ds}]b(\tau)u(\tau)d\tau $$.

Similarly, with

$$\begin{align} \underline{x} (t) = [e^{ \underline{A} (t-t_0)} ] \underline{x} (t_0) + \int_{t_0}^{t} [e^ {\underline{A} (t- \tau\,)} ] \underline{B} \underline{u}(\tau\,) d\tau\, \end{align}$$

Author Egm6321.f10.team1.allen 04:30, 6 October 2010 (UTC)