Double integrator


In systems and control theory, the double integrator is a canonical example of a second-order control system. 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
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
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: