Jump to content

Double integrator

From Wikipedia, the free encyclopedia
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.[1] It models the dynamics of a simple mass in one-dimensional space under the effect of a time-varying force input .

Differential equations

[edit]

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

[edit]

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

[edit]

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:

References

[edit]
  1. ^ Venkatesh G. Rao and Dennis S. Bernstein (2001). "Naive control of the double integrator" (PDF). IEEE Control Systems Magazine. Retrieved 2012-03-04.