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 $\textbf$.

Differential equations

The differential equations which represent a double integrator are:
:$\ddot = u(t)$
:$y = q(t)$

where both $q(t), u(t) \in \mathbb$
Let us now represent this in state space form with the vector $\textbf = \begin q\\ \dot\\ \end$

:$\dot(t)= \frac = \begin \dot\\ \ddot\\ \end$

In this representation, it is clear that the control input $\textbf$ is the second derivative of the output $\textbf$. In the scalar form, the control input is the second derivative of the output $q$.

State space representation

The normalized state space model of a double integrator takes the form
:$\dot(t) = \begin 0& 1\\ 0& 0\\ \end\textbf(t) + \begin 0\\ 1\end\textbf(t)$
:$\textbf(t) = \begin 1& 0\end\textbf(t).$
According to this model, the input $\textbf$ is the second derivative of the output $\textbf$, 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
:$\frac = \frac.$

Using the differential equations dependent on $q(t), y(t), u(t)$ and $\textbf$, and the state space representation:

References

{{Reflist

Control theory