The state variable formulation of a system is given as
$$\left[ {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr } } \right] = \left[ {\matrix{ { - 2} & 0 \cr 0 & { - 1} \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right] + \left[ {\matrix{ 1 \cr 1 \cr } } \right]u,\,\,{x_1}\left( 0 \right) = 0,$$
$${x_2}\left( 0 \right) = 0$$ and $$y = \left[ {\matrix{ 1 & 0 \cr } } \right]\left[ {\matrix{ {{x_1}} \cr {{x_2}} \cr } } \right]$$

The response $$y(t)$$ to a unit step input is

The state variable description of an $$LTI$$ system is given by $$$\left( {\matrix{ {\mathop {{x_1}}\limits^ \bullet } \cr {\mathop {{x_2}}\limits^ \bullet } \cr {\mathop {{x_3}}\limits^ \bullet } \cr } } \right) = \left( {\matrix{ 0 & {{a_1}} & 0 \cr 0 & 0 & {{a_2}} \cr {{a_3}} & 0 & 0 \cr } } \right)\left( {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr } } \right) + \left( {\matrix{ 0 \cr 0 \cr 1 \cr } } \right)u,$$$ $$$y = \left( {\matrix{ 1 & 0 & 0 \cr } } \right)\left( {\matrix{ {{x_1}} \cr {{x_2}} \cr {{x_3}} \cr } } \right)$$$

where $$y$$ is the output and $$u$$ is the input. The system is controllable for

The system $$\mathop X\limits^ \bullet = AX + BU$$ with $$A = \left[ {\matrix{ { - 1} & 2 \cr 0 & 2 \cr } } \right],$$ $$B = \left[ {\matrix{ 0 \cr 1 \cr } } \right]$$ is

A system is described by the following state and output equations $$${{d{x_1}\left( t \right)} \over {dt}} = - 3{x_1}\left( t \right) + {x_2}\left( t \right) + 2u\left( t \right)$$$ $$${{d{x_2}\left( t \right)} \over {dt}} = - 2{x_2}\left( t \right) + u\left( t \right)$$$

$$y\left( t \right) = {x_1}\left( t \right)$$ when $$u(t)$$ is the input and $$y(t)$$ is the output

The state $$-$$ transition matrix of the above system is