double integrator

{{Short description|Second-order control system}}

File:PDcontrol3.png

In systems and control theory, the double integrator is a canonical example of a second-order control system.{{cite journal | author = Venkatesh G. Rao and Dennis S. Bernstein | title = Naive control of the double integrator | journal = IEEE Control Systems Magazine | year = 2001 | url = http://www-personal.umich.edu/~dsbaero/others/25-DoubleIntegrator.pdf | access-date = 2012-03-04

}} 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:

References