Question

A state variable system $x\left(t\right)=\left[\begin{array}{cc}0& 1\\ 0& -3\end{array}\right]X\left(t\right)+\left[\begin{array}{c}1\\ 0\end{array}\right]U\left(t\right),$
with the initial condition $X\left(0\right)=[-13{]}^T$ and the unit step input $u\left(t\right)$ has

The state transition equation:

• A. $X\left(t\right)=\left[\begin{array}{c}t-e^{-t}\\ e^{-t}\end{array}\right]$
• B. $X\left(t\right)=\left[\begin{array}{c}t-e^{-t}\\ 3e^{-3t}\end{array}\right]$
• C. $X\left(t\right)=\left[\begin{array}{c}t-e^{-3t}\\ 3e^{-3t}\end{array}\right]$
• D. $X\left(t\right)=\left[\begin{array}{c}t-e^{-2t}\\ e^{-t}\end{array}\right]$

Answer 1

The correct option is C $X\left(t\right)=\left[\begin{array}{c}t-e^{-3t}\\ 3e^{-3t}\end{array}\right]$

ZIR (Zero Input Response)

$=\varphi \left(t\right)\times X\left(0\right)$

$=\left[\begin{array}{cc}1& \frac{1}{3}(1-e^{-3t})\\ 0& e^{-3t}\end{array}\right]\left[\begin{array}{c}-1\\ 3\end{array}\right]$

$=\left[\begin{array}{c}-1+1-e^{-3t}\\ 3e^{-3t}\end{array}\right]=\left[\begin{array}{c}-e^{-3t}\\ 3e^{-3t}\end{array}\right]$

ZSR (Zero State Response)

$=L^{-1}\left[\right(sI-A{)}^{-1}BU\left(s\right)]$

$=L^{-1}\left\{\left[\begin{array}{cc}\frac{1}{s}& \frac{1}{s(s+3)}\\ 0& \frac{1}{s+3}\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]\frac{1}{s}\right\}$

$=L^{-1}\left[\begin{array}{c}1/s^2\\ 0\end{array}\right]=\left[\begin{array}{c}t\\ 0\end{array}\right]$

$\therefore$ State transition equation

$=ZIR+ZSR=\left[\begin{array}{c}-e^{-3t}\\ 3e^{-3t}\end{array}\right]+\left[\begin{array}{c}t\\ 0\end{array}\right]$

$X\left(t\right)=\left[\begin{array}{c}t-e^{-3t}\\ 3e^{-3t}\end{array}\right]$