Double integrator

Feedback system with a PD controller and a double integrator plant

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 $\mathbf{\text{𝐮}}$.

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 $\mathbf{\text{𝐱(𝐭)}}=\left[\begin{array}{c}q\\ \dot{q}\end{array}\right]$

$\dot{\mathbf{\text{𝐱}}}(t)=\frac{d\mathbf{\text{𝐱}}}{dt}=\left[\begin{array}{c}\dot{q}\\ \ddot{q}\end{array}\right]$

In this representation, it is clear that the control input $\mathbf{\text{𝐮}}$ is the second derivative of the output $\mathbf{\text{𝐱}}$. 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

$\dot{\mathbf{\text{𝐱}}}(t)=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]\mathbf{\text{𝐱}}(t)+\left[\begin{array}{c}0\\ 1\end{array}\right]\mathbf{\text{𝐮}}(t)$

$\mathbf{\text{𝐲}}(t)=\left[\begin{array}{cc}1& 0\end{array}\right]\mathbf{\text{𝐱}}(t).$

According to this model, the input $\mathbf{\text{𝐮}}$ is the second derivative of the output $\mathbf{\text{𝐲}}$, 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 $\mathbf{\text{𝐱(𝐭)}}$, and the state space representation: