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In [[control theory|systems and control theory]], the '''double integrator''' is a canonical example of a [[second-order differential equation|second-order]] control system.<ref name=Maxwell1867>{{cite journal  | author = Venkatesh G. Rao and Dennis S. Bernstein | title = Naive control of the double integrator  | journal = IEEE Control Systems Magazine | version =  | publisher =  | year = 2001  | url = http://www-personal.umich.edu/~dsbaero/others/25-DoubleIntegrator.pdf  | accessdate = 2012-03-04
}}</ref> It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input <math>\textbf{u}</math>.
 
== State space representation ==
The normalized state space model of a double integrator takes the form
:<math>\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> \textbf{y}(t) = \begin{bmatrix} 1& 0\end{bmatrix}\textbf{x}(t).</math>
According to this model, the output <math>\textbf{y}</math> is the second derivative of the input <math>\textbf{u}</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>\frac{Y(s)}{U(s)} = \frac{1}{s^2}.</math>
 
== References ==
{{Reflist}}
 
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Latest revision as of 12:13, 5 January 2014


In systems and control theory, the double integrator is a canonical example of a second-order control system.[1] It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input u.

State space representation

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

x˙(t)=[0100]x(t)+[01]u(t)
y(t)=[10]x(t).

According to this model, the output y is the second derivative of the input u, 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

Y(s)U(s)=1s2.

References

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