Double integrator

Short description: Second-order control system

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 [math]\displaystyle{ \textbf{u} }[/math].

Differential equations

The differential equations which represent a double integrator are:

[math]\displaystyle{ \ddot{q} = u(t) }[/math]

[math]\displaystyle{ y = q(t) }[/math]

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

[math]\displaystyle{ \dot{\textbf{x}}(t)= \frac{d\textbf{x}}{dt} = \begin{bmatrix} \dot{q}\\ \ddot{q}\\ \end{bmatrix} }[/math]

In this representation, it is clear that the control input [math]\displaystyle{ \textbf{u} }[/math] is the second derivative of the output [math]\displaystyle{ \textbf{x} }[/math]. In the scalar form, the control input is the second derivative of the output [math]\displaystyle{ q }[/math].

State space representation

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

[math]\displaystyle{ \dot{\textbf{x}}(t) = \begin{bmatrix} 0& 1\\ 0& 0\\ \end{bmatrix}\textbf{x}(t) + \begin{bmatrix} 0\\ 1\end{bmatrix}\textbf{u}(t) }[/math]

[math]\displaystyle{ \textbf{y}(t) = \begin{bmatrix} 1& 0\end{bmatrix}\textbf{x}(t). }[/math]

According to this model, the input [math]\displaystyle{ \textbf{u} }[/math] is the second derivative of the output [math]\displaystyle{ \textbf{y} }[/math], 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

[math]\displaystyle{ \frac{Y(s)}{U(s)} = \frac{1}{s^2}. }[/math]

Using the differential equations dependent on [math]\displaystyle{ q(t), y(t), u(t) }[/math] and [math]\displaystyle{ \textbf{x(t)} }[/math], and the state space representation: