From bfaae6860eea31802cebabb3fed866d8217c7fa8 Mon Sep 17 00:00:00 2001
From: Axel Kohlmeyer <akohlmey@gmail.com>
Date: Tue, 11 Feb 2020 16:28:45 +0100
Subject: [PATCH] fix controller

---
 doc/src/Eqs/fix_controller1.jpg | Bin 4832 -> 0 bytes
 doc/src/Eqs/fix_controller1.tex |  12 --------
 doc/src/Eqs/fix_controller2.jpg | Bin 5571 -> 0 bytes
 doc/src/Eqs/fix_controller2.tex |  12 --------
 doc/src/fix_controller.rst      |  48 ++++++++++++++++++--------------
 5 files changed, 27 insertions(+), 45 deletions(-)
 delete mode 100644 doc/src/Eqs/fix_controller1.jpg
 delete mode 100644 doc/src/Eqs/fix_controller1.tex
 delete mode 100644 doc/src/Eqs/fix_controller2.jpg
 delete mode 100644 doc/src/Eqs/fix_controller2.tex

diff --git a/doc/src/Eqs/fix_controller1.jpg b/doc/src/Eqs/fix_controller1.jpg
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diff --git a/doc/src/Eqs/fix_controller1.tex b/doc/src/Eqs/fix_controller1.tex
deleted file mode 100644
index 14f98fc303..0000000000
--- a/doc/src/Eqs/fix_controller1.tex
+++ /dev/null
@@ -1,12 +0,0 @@
-\documentclass[24pt]{article}
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-\pagestyle{empty}
-\Huge
-
-\begin{document}
-
-\begin{eqnarray*}
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-
-\end{document}
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diff --git a/doc/src/Eqs/fix_controller2.tex b/doc/src/Eqs/fix_controller2.tex
deleted file mode 100644
index 7bcaa333ad..0000000000
--- a/doc/src/Eqs/fix_controller2.tex
+++ /dev/null
@@ -1,12 +0,0 @@
-\documentclass[24pt]{article}
-
-\pagestyle{empty}
-\Huge
-
-\begin{document}
-
-\begin{eqnarray*}
-c_n  &=&Ê c_{n-1} -\alpha (K_p \tau e_n + K_i \tau^2 \sum_{i=1}^n e_i + K_d (e_n - e_{n-1}) )
-\end{eqnarray*}
-
-\end{document}
diff --git a/doc/src/fix_controller.rst b/doc/src/fix_controller.rst
index 7561df6856..839c01ef5f 100644
--- a/doc/src/fix_controller.rst
+++ b/doc/src/fix_controller.rst
@@ -97,43 +97,49 @@ The PID controller is invoked once each *Nevery* timesteps.
 The PID controller is implemented as a discretized version of
 the following dynamic equation:
 
-.. image:: Eqs/fix_controller1.jpg
-   :align: center
+.. math::
 
+   \frac{dc}{dt}  = \hat{E} -\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
-*alpha*\ , *Kp*\ , *Ki*\ , and *Kd* are constants set by the corresponding
+:math:`\alpha`, :math:`K_p`, :math:`K_i` , and :math:`K_d` are constants
+set by the corresponding
 keywords described above. The discretized version of this equation is:
 
-.. image:: Eqs/fix_controller2.jpg
-   :align: center
+.. math::
 
-where *tau* = *Nevery* \* *timestep* is the time interval between updates,
+   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)
+
+
+where :math:`\tau = \mathtt{Nevery} \cdot \mathtt{timestep}` is the time
+interval between updates,
 and the subscripted variables indicate the values of *c* and *e* at
 successive updates.
 
 From the first equation, it is clear that if the three gain values
-*Kp*\ , *Ki*\ , *Kd* are dimensionless constants, then *alpha* must have
+:math:`K_p`, :math:`K_i`, :math:`K_d` are dimensionless constants,
+then :math:`\alpha` must have
 units of [unit *cvar*\ ]/[unit *pvar*\ ]/[unit time] e.g. [ eV/K/ps
 ]. The advantage of this unit scheme is that the value of the
 constants should be invariant under a change of either the MD timestep
 size or the value of *Nevery*\ . Similarly, if the LAMMPS :doc:`unit style <units>` is changed, it should only be necessary to change
-the value of *alpha* to reflect this, while leaving *Kp*\ , *Ki*\ , and
-*Kd* unaltered.
+the value of :math:`\alpha` to reflect this, while leaving :math:`K_p`,
+:math:`K_i`, and :math:`K_d` unaltered.
 
 When choosing the values of the four constants, it is best to first
-pick a value and sign for *alpha* that is consistent with the
-magnitudes and signs of *pvar* and *cvar*\ .  The magnitude of *Kp*
-should then be tested over a large positive range keeping *Ki* = *Kd* =0.
-A good value for *Kp* will produce a fast response in *pvar*\ , without
-overshooting the *setpoint*\ .  For many applications, proportional
-feedback is sufficient, and so *Ki* = *Kd* =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 *Kd*\ . In situations
+pick a value and sign for :math:`\alpha` that is consistent with the
+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
+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
-counteracted by increasing *Ki*\ .  In the language of Charles Dickens,
-*Kp* represents the error of the present, *Ki* the error of the past,
-and *Kd* the error yet to come.
+counteracted by increasing :math:`K_i`.  In the language of Charles Dickens,
+:math:`K_p` represents the error of the present, :math:`K_i` the error of
+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
@@ -141,7 +147,7 @@ the :doc:`internal-style variable <variable>` command.  This value is
 used (by the other LAMMPS command that used 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 *e\_n-1* is not yet defined.
+because the value of :math:`e_n-1` is not yet defined.
 
 
 ----------