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adding english and math description for actuator travel #646

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adding english and math description for actuator travel #646

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18 changes: 18 additions & 0 deletions docs/design/source/core_kpi.rst
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Expand Up @@ -227,6 +227,24 @@ Maximum allowed capital cost
relies on the BOPTEST user, who can use the objective quantification
of this KPI to take the decision.

Actuator travel
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The actuator travel measures how frequent and how strong an actuator
is turned on and off. This measure is important to assess the wear and
tear of the equipment, because frequent and erratic switching of control
signals can lead to equipment damages and shorter equipment lifespan.
Actuator travel also gives a measure of phenomena such as cycling and hunting.

The actuator travel :math:`AT(u)` of a trajectory :math:`u(t)` is calculated as
the ratio of the arc of :math:`u(t)` from :math:`t_0` to :math:`t_f`
to the time passed :math:`t_f-t_0`:

.. math:: AT(u) = \frac{\int_{t_0}^{t_f} \left \sqrt{(1+(u'(t))^2)} \right dt}{t_f-t_0},

where :math:`\int_{t_0}^{t_f} \left \sqrt{(1+(u'(t))^2)` is the formula for
calculating the arc of function :math:`u(t)` by integration.

Calculation Module
---------------------

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