The model of a harmonic oscillator corresponds to a second order system with  as the input and  as the output. The system is defined by the angular frequency , the attenuation , and the gain .s

The system is defined with the following differential equation

The transfer function that results from this differential equation can be obtained using the DynamicSystems[TransferFunction] command.

The step response for the corresponding system can be observed by changing the slider values for θ and ω in the following application.

In this example,  controls the damping, such that a system with  results in a system that is under damped and results in an overshoot.  For the cases with the system is over damped and the response has no overshoot.  If  the system is critically damped, resulting in the fastest rise time of the system without overshooting the final value. The parameter  is the natural frequency of the system.

The amplitude  is set to 1 for this example.

The response to the system to chaging proportional gain controller can be seen below. It is important to note that the P controller is not able to get the system to actually reach the desired final value. The controller has an offset error.

The effects of adjusting the proportional controller gain  and integral controller gain  values is displayed below. In most cases, the offset error can be eliminated by adding an integral component to the proportional controller.

It is important to note, that a system may become unstable and begin to oscillate out of control if the value of  is too small and the value of  to large.

The system response to changes in the proportional, integral and derivative gains are shown below.