The model of a harmonic oscillator corresponds to a second order system with $u\left(t\right)$ as the input and $y\left(t\right)$ as the output. The system is defined by the angular frequency $\mathrm{\ω}$, the attenuation $\mathrm{\θ}$, and the gain $A$.

The system is defined with the following differential equation

$\mathrm{sysde}≔\frac{{\ⅆ}^2}{\ⅆt^2}y\left(t\right)\+2\mathrm{\omega }\mathrm{\theta } \left(\frac{\ⅆ}{\ⅆt}y\left(t\right)\right)\+{\mathrm{\omega }}^{2}y\left(t\right)\={\mathrm{\omega }}^{2}Au\left(t\right)$

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

$\mathrm{with}\left(\mathrm{DynamicSystems}\right)\:$

$$\begin{gathered} \mathrm{systf}≔\mathrm{TransferFunction}\left(\mathrm{sysde}\,\mathrm{inputvariable}\=\left[u\left(t\right)\right]\,\mathrm{outputvariable}\=\left[y\left(t\right)\right]\right)\: \\ \mathrm{PrintSystem}\left(\mathrm{systf}\right) \end{gathered}$$

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, $\mathrm{\theta }$ controls the damping, such that a system with $\mathrm{\&theta;}<1$ results in a system that is under damped and results in an overshoot.  For the cases with $\mathrm{\&theta;}\&gt;1$the system is over damped and the response has no overshoot.  If $\mathrm{\&theta;}\&equals;1$ the system is critically damped, resulting in the fastest rise time of the system without overshooting the final value. The parameter $\mathrm{\omega }$ is the natural frequency of the system.

The amplitude $A$ is set to 1 for this example.

$\mathrm{Pcontroller}≔\mathrm{Kp}\&colon;$

$\mathrm{OpenLoopEq}≔\mathrm{Pcontroller}\cdot \mathrm{systf}:-\mathrm{tf}\left[1\&comma;1\right]$

$\mathrm{ClosedLoopEq}≔\mathrm{simplify}\left(\frac{\mathrm{OpenLoopEq}}{1\&plus;\mathrm{OpenLoopEq}}\right)$

$\mathrm{ClosedLoopSys}≔\mathrm{TransferFunction}\left(\mathrm{ClosedLoopEq}\right)\&colon;$

$\mathrm{PrintSystem}\left(\mathrm{ClosedLoopSys}\right)$

The response to the system to changing 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.

$\mathrm{PIcontroller}≔\mathrm{Kp}\&plus;\frac{\mathrm{Ki}}{s}$

$\mathrm{OpenLoopEq1}≔\mathrm{PIcontroller}\cdot \mathrm{systf}:-\mathrm{tf}\left[1\&comma;1\right]$

$\mathrm{ClosedLoopEq1}≔\mathrm{simplify}\left(\frac{\mathrm{OpenLoopEq1}}{1\&plus;\mathrm{OpenLoopEq1}}\right)$

$\mathrm{ClosedLoopSys1}≔\mathrm{TransferFunction}\left(\mathrm{ClosedLoopEq1}\right)\&colon;$

$\mathrm{PrintSystem}\left(\mathrm{ClosedLoopSys1}\right)$

The effects of adjusting the proportional controller gain $\left(\mathrm{Kp}\right)$ and integral controller gain $\left(\mathrm{Ki}\right)$ 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 $\mathrm{Ki}$ is too large and the value of $\mathrm{Kp}$ too small (relative to $\mathrm{Ki}$).

$\mathrm{PIDcontroller}≔\mathrm{Kp} \&plus; \frac{\mathrm{Ki}}{s}\&plus;\mathrm{Kd}\cdot s$

$\mathrm{OpenLoopEq2}≔\mathrm{PIDcontroller}\cdot \mathrm{systf}:-\mathrm{tf}\left[1\&comma;1\right]$

$\mathrm{ClosedLoopEq2}≔\mathrm{simplify}\left(\frac{\mathrm{OpenLoopEq2}}{1\&plus;\mathrm{OpenLoopEq2}}\right)$

$\mathrm{ClosedLoopSys2}≔\mathrm{TransferFunction}\left(\mathrm{ClosedLoopEq2}\right)\&colon;$

$\mathrm{PrintSystem}\left(\mathrm{ClosedLoopSys2}\right)$

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