Double integrator

In systems and control theory, the double integrator is a canonical example of a second-order control system. It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input ${\textbf {u}}$.

State space representation

The normalized state space model of a double integrator takes the form

${\dot {\textbf {x}}}(t)={\begin{bmatrix}0&1\\0&0\\\end{bmatrix}}{\textbf {x}}(t)+{\begin{bmatrix}0\\1\end{bmatrix}}{\textbf {u}}(t)$

${\textbf {y}}(t)={\begin{bmatrix}1&0\end{bmatrix}}{\textbf {x}}(t).$

According to this model, the input ${\textbf {u}}$ is the second derivative of the output ${\textbf {y}}$, hence the name double integrator.

Transfer function representation

Taking the Laplace transform of the state space input-output equation, we see that the transfer function of the double integrator is given by

${\frac {Y(s)}{U(s)}}={\frac {1}{s^{2}}}.$