Question

A second order system starts with an initial condition of $\left[\begin{array}{c}2\\ 3\end{array}\right]$ without any external input. The state transition matrix for the system is given by $\left[\begin{array}{cc}{e}^{-2t}& 0\\ 0& e^{-t}\end{array}\right]$. The state of the system at the end of $1$ second is given by

• A. $\left[\begin{array}{c}0.271\\ 1.100\end{array}\right]$
  
• B. $\left[\begin{array}{c}0.135\\ 0.368\end{array}\right]$
• C. $\left[\begin{array}{c}0.271\\ 0.736\end{array}\right]$
• D. $\left[\begin{array}{c}0.135\\ 1.100\end{array}\right]$

Answer 1

The correct option is A $\left[\begin{array}{c}0.271\\ 1.100\end{array}\right]$


State transition matrix,
$\varphi \left(t\right)=\left[\begin{array}{cc}{e}^{-2t}& 0\\ 0& e^{-t}\end{array}\right]$
Initial conditions,
$x\left(0\right)=\left[\begin{array}{c}2\\ 3\end{array}\right]$
Zero input response is given by
$x\left(t\right)=\varphi \left(t\right)x\left(0\right)$
$=\left[\begin{array}{cc}{e}^{-2t}& 0\\ 0& e^{-t}\end{array}\right]\left[\begin{array}{c}2\\ 3\end{array}\right]=\left[\begin{array}{c}2e^{-3t}\\ 3e^{-1}\end{array}\right]$
State of the system at $t=1s$
$x\left(t\right){|}_{t=1}\left[\begin{array}{c}2e^{-2}\\ 3e^{-1}\end{array}\right]=\left[\begin{array}{c}0.271\\ 1.100\end{array}\right]$