Linear predictor function: Difference between revisions
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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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[[Category:Control theory]] | |||
Latest revision as of 13:13, 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 .
State space representation
The normalized state space model of a double integrator takes the form
According to this model, the output is the second derivative of the input , 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
References
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