Week 3: Harmonic Oscillator#
What you see#
The example show the Harmonic Oscillator (see (Herlau [Her25], Subsection 11.1.2)) example. There is no control signal, \(u(t) = 0\), so the system just oscillates forever.
How it works#
The bead has a position and a velocity which form the two coordinates of the state
\[\begin{split}\mathbf{x}(t) = \begin{bmatrix}\text{position} \\ \text{velocity} \end{bmatrix} = \begin{bmatrix} x_1(t) \\ x_2(t) \end{bmatrix}\end{split}\]
The dynamics of the environment is then governed by the differential equation:
\[\begin{split}\dot{\mathbf x} = \begin{bmatrix}0 & 1 \\ -\frac{k}{m} & 0 \end{bmatrix}\mathbf{x}\end{split}\]