double integrator

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

Category:Control theory