Question

A linear system is described by the following state equation
$\dot{X}\left(t\right)=AX\left(t\right)+BU\left(t\right),A=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]$
The state- transition matrix of the system is

• A. $\left[\begin{array}{cc}\cos t& \sin t\\ -\sin t& \cos t\end{array}\right]$
• B. $\left[\begin{array}{cc}-\cos t& \sin t\\ -\sin t& -\cos t\end{array}\right]$
• C. $\left[\begin{array}{cc}-\cos t& -\sin t\\ -\sin t& \cos t\end{array}\right]$
• D. $\left[\begin{array}{cc}\cos t& -\sin t\\ \cos t& \sin t\end{array}\right]$

Answer 1

The correct option is A $\left[\begin{array}{cc}\cos t& \sin t\\ -\sin t& \cos t\end{array}\right]$

$\varphi \left(t\right)=L^{-1}[sI-A{]}^{-1}=L^{-1}{[\left[\begin{array}{cc}s& 0\\ 0& s\end{array}\right]-\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right]]}^{-1}=L^{-1}{\left[\begin{array}{cc}s& -1\\ 1& s\end{array}\right]}^{-1}=L^{-1}\left[\begin{array}{cc}\frac{s}{s^2+1}& \frac{1}{s^2+1}\\ \frac{-1}{s^2+1}& \frac{s}{s^2+1}\end{array}\right][D=s^2+1]=\left[\begin{array}{cc}\cos t& \sin t\\ -\sin t& \cos t\end{array}\right]$