Feature request: Analytical solutions for simple cases

[cmichelenstrofer]

We often compare to the analytical linear solutions. We should write a function maybe in utils that does this genearlly instead of writting the same code over and over for each specific case.

• This could also be used as initial guess for the state x for non-linear cases.
• We could also include the solution for saturated PI, see this comment.

[cmichelenstrofer]

Comment by @rebeccamccabe:

For a better initial guess, instead of using the unsaturated amplitude Fp, the fundamental of the saturated waveform will have an amplitude of alpha * Fmax / Fp, where alpha is a value ranging from 1 (no saturation) to 4/pi (full saturation, the fundamental of a square wave):

(from the Franklin-Powell-Emami Control textbook section on describing functions)

To go even further (unclear this is actually necessary) I've also derived the math for the higher harmonics too, which I think should be exact for a linear problem.

which is derived here https://github.com/symbiotic-engineering/MDOcean/blob/rgm-force-saturation/dev/describing_fcn_error.mlx

Originally posted by @rebeccamccabe in #269 (comment)