Double integrator explained

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

bf{u}

Differential equations

The differential equations which represent a double integrator are:

\ddot{q}=u(t)

y=q(t)

where both

q(t),u(t)\inR

Let us now represent this in state space form with the vector

bf{x(t)}=\begin{bmatrix} q\\

•

\\ \end{bmatrix}

•

bf{x

}(t)= \frac = \begin \dot\\ \ddot\\ \end

In this representation, it is clear that the control input

bf{u}

is the second derivative of the output

bf{x}

. 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

•

bf{x

}(t) = \begin 0& 1\\ 0& 0\\ \end\textbf(t) + \begin 0\\ 1\end\textbf(t)

bf{y}(t)=\begin{bmatrix}1&0\end{bmatrix}bf{x}(t).

According to this model, the input

bf{u}

is the second derivative of the output

bf{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

	Y(s)
	U(s)

	1
	s²

Using the differential equations dependent on

q(t),y(t),u(t)

and

bf{x(t)}

, and the state space representation:

Notes and References

Venkatesh G. Rao and Dennis S. Bernstein . Naive control of the double integrator . IEEE Control Systems Magazine . 2001 . 2012-03-04.