Question

Valid state transition matrix $\varphi \left(t\right)$ is,

• A. $\varphi \left(t\right)=\left(\begin{array}{cc}3t{e}^{-t}& t{e}^{-t}\\ -e^{-t}& e^{-t}\end{array}\right]$
• B. $\varphi \left(t\right)=\left(\begin{array}{cc}{e}^{-t}+t{e}^{-t}& t{e}^{-t}\\ -t{e}^{-t}& e^{-t}-t{e}^{-t}\end{array}\right]$
• C. $\varphi \left(t\right)=\left(\begin{array}{cc}t{e}^{-t}& e^{-t}\\ -e^{-t}+t{e}^{-1}& 2t{e}^{-t}\end{array}\right]$
• D. $\varphi \left(t\right)=\left(\begin{array}{cc}{e}^{-t}+t{e}^{-t}& e^{-t}\\ -t{e}^{-t}& e^{-t}-3e^{-t}\end{array}\right]$

Answer 1

The correct option is B $\varphi \left(t\right)=\left(\begin{array}{cc}{e}^{-t}+t{e}^{-t}& t{e}^{-t}\\ -t{e}^{-t}& e^{-t}-t{e}^{-t}\end{array}\right]$

State transition matrix, $\varphi \left(t\right)=e^{At}$

$\varphi \left(0\right)=I$

From property of state transmission matrix at t = 0,
$\varphi \left(0\right)=I$

$=\left(\begin{array}{cc}1+0& 0\\ -0& 1\end{array}\right]=\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]=I$