Double integrator

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

thumb | right | alt=Feedback system with a PD controller and a double integrator plant | Feedback system with a PD controller and a double integrator plant In systems and control theory, the '''double integrator''' is a canonical example of a 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 | year = 2001 | url = http://www-personal.umich.edu/~dsbaero/others/25-DoubleIntegrator.pdf | access-date = 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>.

== Differential equations == The differential equations which represent a double integrator are: :<math>\ddot{q} = u(t)</math> :<math>y = q(t)</math>

where both <math>q(t), u(t) \in \mathbb{R} </math> Let us now represent this in state space form with the vector <math>\textbf{x(t)} = \begin{bmatrix} q\\ \dot{q}\\ \end{bmatrix}</math>

:<math> \dot{\textbf{x}}(t)= \frac{d\textbf{x}}{dt} = \begin{bmatrix} \dot{q}\\ \ddot{q}\\ \end{bmatrix} </math>

In this representation, it is clear that the control input <math>\textbf{u}</math> is the second derivative of the output <math>\textbf{x}</math>. In the scalar form, the control input is the second derivative of the output <math>q</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 input <math>\textbf{u}</math> is the second derivative of the output <math>\textbf{y}</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>

Using the differential equations dependent on <math> q(t), y(t), u(t)</math> and <math>\textbf{x(t)}</math>, and the state space representation:

== References == {{Reflist}}

Category:Control theory