Double integrator

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 ==