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 ${\textbf {u}}$.

Differential equations

The differential equations which represent a double integrator are:

${\ddot {q}}=u(t)$

$y=q(t)$

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

${\dot {\textbf {x}}}(t)={\frac {d{\textbf {x}}}{dt}}={\begin{bmatrix}{\dot {q}}\\{\ddot {q}}\\\end{bmatrix}}$

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

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

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

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

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

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