Double integrator

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

The differential equations which represent a double integrator are: where both

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

In this representation, it is clear that the control input

In the scalar form, the control input is the second derivative of the output

The normalized state space model of a double integrator takes the form According to this model, the input

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