A second order LTI system is described by the following state equation. $$$\eqalign{ & {d \over {dt}}{x_1}\left( t \right) - {x_2}\left( t \right) = 0 \cr & {d \over {dt}}{x_2}\left( t \right) + 2{x_1}\left( t \right) + 3{x_2}\left( t \right) = r\left( t \right) \cr} $$$

When x₁(t) and x₂(t) are the two state variables and r(t) denotes the input. The output c(t)=X₁(t). The systyem is

A second-order linear time-invariant system is described by the following state equations $$$\eqalign{& {d \over {dt}}{x_1}\left( t \right) + 2{x_1}\left( t \right) = 3u\left( t \right) \cr & {d \over {dt}}{x_2}\left( t \right) + {x_2}\left( t \right) = u\left( t \right) \cr} $$$

Where x₁(t), then the system is

A network is described by the state model as $$$\eqalign{ & {\mathop x\limits^ \bullet _1} = 2{x_1} - {x_2} + 3u, \cr & \mathop {{x_2}}\limits^ \bullet = - 4{x_2} - u, \cr & y = 3{x_1} - 2{x_2} \cr} $$$

the transfer function H(s)$$\left[ { = {{Y\left( s \right)} \over {U\left( s \right)}}} \right]is$$

The state variable representation of a system is given as $$$\eqalign{ & \mathop x\limits^ \bullet = \left[ {\matrix{ 0 & 1 \cr 0 & { - 1} \cr } } \right]x;x\left( 0 \right) = \left[ {\matrix{ 1 \cr 0 \cr } } \right] \cr & y = \left[ {\matrix{ 0 & 1 \cr } } \right]x \cr} $$$

The response y(t) is