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Correct typos and clarify fix_controller.rst

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Fixes some typos and adds math expression c_0 for the initial control variable value; removes the "E-hat" term from equations (which doesn't appear elsewhere in the support doc or the source code)
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Shern Tee
2023-02-06 15:34:58 +10:00
committed by GitHub
parent a0a7e76cc3
commit fb5c0a4c87
1 changed files with 6 additions and 6 deletions
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12
doc/src/fix_controller.rst
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@ -96,7 +96,7 @@ the following dynamic equation:
.. math::
\frac{dc}{dt} = \hat{E} -\alpha (K_p e + K_i \int_0^t e \, dt + K_d \frac{de}{dt} )
\frac{dc}{dt} = -\alpha (K_p e + K_i \int_0^t e \, dt + K_d \frac{de}{dt} )
where *c* is the continuous time analog of the control variable,
*e* =\ *pvar*\ -\ *setpoint* is the error in the process variable, and
@ -106,7 +106,7 @@ keywords described above. The discretized version of this equation is:
.. math::
c_n = \hat{E} c_{n-1} -\alpha \left( K_p \tau e_n + K_i \tau^2 \sum_{i=1}^n e_i + K_d (e_n - e_{n-1}) \right)
c_n = c_{n-1} -\alpha \left( K_p \tau e_n + K_i \tau^2 \sum_{i=1}^n e_i + K_d (e_n - e_{n-1}) \right)
where :math:`\tau = \mathtt{Nevery} \cdot \mathtt{timestep}` is the time
interval between updates,
@ -129,7 +129,7 @@ magnitudes and signs of *pvar* and *cvar*\ . The magnitude of :math:`K_p`
should then be tested over a large positive range keeping :math:`K_i = K_d =0`.
A good value for :math:`K_p` will produce a fast response in *pvar*,
without overshooting the *setpoint*\ . For many applications, proportional
feedback is sufficient, and so :math:`K_i` = K_d =0` can be used. In cases
feedback is sufficient, and so :math:`K_i = K_d =0` can be used. In cases
where there is a substantial lag time in the response of *pvar* to a change
in *cvar*, this can be counteracted by increasing :math:`K_d`. In situations
where *pvar* plateaus without reaching *setpoint*, this can be
@ -138,12 +138,12 @@ counteracted by increasing :math:`K_i`. In the language of Charles Dickens,
the past, and :math:`K_d` the error yet to come.
Because this fix updates *cvar*, but does not initialize its value,
the initial value is that assigned by the user in the input script via
the initial value :math:`c_0` is that assigned by the user in the input script via
the :doc:`internal-style variable <variable>` command. This value is
used (by the other LAMMPS command that used the variable) until this
used (by every other LAMMPS command that uses the variable) until this
fix performs its first update of *cvar* after *Nevery* timesteps. On
the first update, the value of the derivative term is set to zero,
because the value of :math:`e_n-1` is not yet defined.
because the value of :math:`e_{n-1}` is not yet defined.
----------