Add control loop performance indicator function

[ufechner7]

Add a function that calculates a performance indicator to test_forcespeedcontroller2.jl. It should be zero without any controller and one with perfect control.

First idea: Use the sum of the mean square error of v_err and f_err. But both errors must be relative to some value.

See also: https://arrow.tudublin.ie/cgi/viewcontent.cgi?article=1034&context=engscheleart

[ufechner7]

Proposal:

The force error, the speed error and the control effort should be taken into account.

Relative force error

The relative force error is the maximum of the norm of the force errors, based on all samples where one of the force controllers was active, divided by the nominal winch force $F_\mathrm{max}$, as formula:

$F_\mathrm{err} = \frac{1}{F_\mathrm{max}}~{ \max |F-F_\mathrm{set}|}$

The reason to use the maximum here is that it must be avoided that the force goes below zero and also that it exceeds the maximum tether force.

Relative velocity error

The relative velocity error is the root mean square error (RMSE) of the velocity, based on all samples where one of the force controllers was active, divided by the average winch speed $\bar{v}$, as formula:

$v_\mathrm{err} = \frac{1}{\bar{v}}~\sqrt{\frac{1}{n} \sum_{i=1}^{n} (v-v_\mathrm{set})^2}$
with
$\bar{v} = \frac{1}{n} \sum_{i=1}^{n} \lvert v_\mathrm{set} \rvert$

For now, it is assumed that the maximal reel-in and reel-out speed of the winch are identical:

$v_\mathrm{max} = v_\mathrm{ro,max} = v_\mathrm{ri,max}$

The reason to use the mean square error for the velocity is that in this mode of operation the error only reduces the power output. Short peak errors are usually not a problem.

Combined performance

The combined performance indicator $\gamma$ in the range of 0..1 is defined as

$\gamma = 1 - \frac{1}{2}(F_\mathrm{err} + v_\mathrm{err})~$.

TODO

Take the control effort into account. This could be the average power needed to accelerate and to slow down the winch with respect to either the nominal winch power, or with respect to the average mechanical power at lower end of the tether.

[1-Bart-1]

I don't really get why you want a value between 0 and 1. The RMSE will be between 0 and $\infty$. It makes more sense to calculate the performance as the total or average error:
$\epsilon_{avg} = \frac{1}{2}(F_{err} + v_{err})$
Where $\epsilon = 0$ is a perfect controller.

edit: or does dividing the RMSE by the max force normalize it? It might distort the RMSE to do this, as a narrow and very high peak in the force could actually reduce this normalized RMSE due to the max force being in the denominator.

another edit: ok, so we have to normalize the RMSE so all errors are weighted correctly. But I think we should normalize by the mean force and mean velocity instead.

[ufechner7]

I updated the proposal.

[1-Bart-1]

$F_\mathrm{err} = \frac{1}{F_\mathrm{max}}~{ \max |F-F_\mathrm{set}|}$

Isn't that the same as:
$F_\mathrm{err} = \frac{1}{F_\mathrm{max}}~{ |F_\mathrm{max}-F_\mathrm{set}|}$

[ufechner7]

No, not at all. F_set is for example 300 N at (too) low wind speeds and 3700 N at (too) high wind speeds.

In between the speed controller that keeps the speed proportional to the square root of the force is active for optimal energy harvesting.

[ufechner7]

The proposal is implemented in main.

function gamma(set, f_err_, v_err_, v_set)
    1 - 0.5(f_err(set, f_err_) + v_err(v_err_, v_set))
end

Docstring is still missing.

TODO:

• move code into the package
• add control effort

[ufechner7]

Done (slightly differently than described here) and documented in the package documentation.