Question

The state model of a system is given as,

$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]\left[\begin{array}{c}{x}_2\\ x_1\end{array}\right]$

$x^T\left(0\right)=\left[10\right]$
The solution of homogenous equation is given by,

• A. $x\left(t\right)=\left[\begin{array}{c}{e}^t\\ 1\end{array}\right]$
• B. $x\left(t\right)=\left[\begin{array}{c}{e}^t\\ t{e}^t\end{array}\right]$
• C. $x\left(t\right)=\left[\begin{array}{c}0\\ e^t\end{array}\right]$
• D. $x\left(t\right)=\left[\begin{array}{c}{e}^t\\ e^t\end{array}\right]$

Answer 1

The correct option is B $x\left(t\right)=\left[\begin{array}{c}{e}^t\\ t{e}^t\end{array}\right]$

$x\left(t\right)=\varphi \left(t\right)x\left(0\right)$
Where, $\varphi \left(t\right)=$ state transition matrix
Given, $A=\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$

$[sI-A]=\left[\begin{array}{cc}s& 0\\ 0& s\end{array}\right]-\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]=\left[\begin{array}{cc}s-1& 0\\ -1& s-1\end{array}\right]$

$[sI-A{]}^{-1}=\frac{1}{(s-1)(s-1)}\left[\begin{array}{cc}s-1& 0\\ 1& s-1\end{array}\right]$

$\left[\begin{array}{cc}\frac{1}{s-1}& 0\\ \frac{1}{(s-1{)}^2}& \frac{1}{(s-1)}\end{array}\right]$

$L^{-1}[sI-A{]}^{-1}=\varphi (t)=\left[\begin{array}{cc}{e}^t& 0\\ t{e}^t& e^t\end{array}\right]$

$\therefore$ $x\left(t\right)=\left[\begin{array}{cc}{e}^t& 0\\ t{e}^t& e^t\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]$

$x\left(t\right)=\left[\begin{array}{c}{e}^t\\ t{e}^t\end{array}\right]$