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.{{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 .
Differential equations
The differential equations which represent a double integrator are:
:
:
where both
Let us now represent this in state space form with the vector
q\\
\dot{q}\\
\end{bmatrix}
:
\dot{q}\\
\ddot{q}\\
\end{bmatrix}
In this representation, it is clear that the control input is the second derivative of the output . In the scalar form, the control input is the second derivative of the output .
State space representation
The normalized state space model of a double integrator takes the form
:
0& 1\\
0& 0\\
\end{bmatrix}\textbf{x}(t) +
\begin{bmatrix} 0\\ 1\end{bmatrix}\textbf{u}(t)
:
According to this model, the input is the second derivative of the output , 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
:
Using the differential equations dependent on and , and the state space representation:
References
{{Reflist}}