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 $𝐮$ .

Differential equations

The differential equations which represent a double integrator are:

\ddot{q} = u (t)

y = q (t)

where both $q (t), u (t) \in ℝ$ . Let us now represent this in state space form with the vector $𝐱(𝐭) = [\begin{matrix} q \\ \dot{q} \end{matrix}]$

\dot{𝐱} (t) = \frac{d 𝐱}{d t} = [\begin{matrix} \dot{q} \\ \ddot{q} \end{matrix}]

In this representation, it is clear that the control input $𝐮$ is the second derivative of the output $𝐱$ . 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{𝐱} (t) = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}] 𝐱 (t) + [\begin{matrix} 0 \\ 1 \end{matrix}] 𝐮 (t)

𝐲 (t) = [\begin{matrix} 1 & 0 \end{matrix}] 𝐱 (t) .

According to this model, the input $𝐮$ is the second derivative of the output $𝐲$ , 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}} .

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