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# fix controller command

## Syntax

    fix ID group-ID controller Nevery alpha Kp Ki Kd pvar setpoint cvar

- ID, group-ID are documented in fix command

- controller = style name of this fix command

- Nevery = invoke controller every this many timesteps

- alpha = coupling constant for PID equation (see units discussion below)

- Kp = proportional gain in PID equation (unitless)

- Ki = integral gain in PID equation (unitless)

- Kd = derivative gain in PID equation (unitless)

- pvar = process variable of form c_ID, c_ID[I], f_ID, f_ID[I], or v_name

      c_ID = global scalar calculated by a compute with ID
      c_ID[I] = Ith component of global vector calculated by a compute with ID
      f_ID = global scalar calculated by a fix with ID
      f_ID[I] = Ith component of global vector calculated by a fix with ID
      v_name = value calculated by an equal-style variable with name

- setpoint = desired value of process variable (same units as process variable)

- cvar = name of control variable

## Examples

    fix 1 all controller 100 1.0 0.5 0.0 0.0 c_thermo_temp 1.5 tcontrol
    fix 1 all controller 100 0.2 0.5 0 100.0 v_pxxwall 1.01325 xwall
    fix 1 all controller 10000 0.2 0.5 0 2000 v_avpe -3.785 tcontrol

## Description

This fix enables control of a LAMMPS simulation using a control loop feedback mechanism known as a proportional-integral-derivative (PID) controller. The basic idea is to define a “process variable” which is a quantity that can be monitored during a running simulation. A desired target value is chosen for the process variable. A “control variable” is also defined which is an adjustable attribute of the running simulation, which the process variable will respond to. The PID controller continuously adjusts the control variable based on the difference between the process variable and the target.

Here are examples of ways in which this fix can be used. The examples/pid directory contains a script that implements the simple thermostat.

Goal | process variable | control variable
Simple thermostat | instantaneous T | thermostat target T
Find melting temperature | average PE per atom | thermostat target T
Control pressure in non-periodic system | force on wall | position of wall

Note

For this fix to work, the control variable must actually induce a change in a running LAMMPS simulation. Typically this will only occur if there is some other command (e.g. a thermostat fix) which uses the control variable as an input parameter. This could be done directly or indirectly, e.g. the other command uses a variable as input whose formula uses the control variable. The other command should alter its behavior dynamically as the variable changes.

Note

If there is a command you think could be used in this fashion, but does not currently allow a variable as an input parameter, please notify the LAMMPS developers. It is often not difficult to enable a command to use a variable as an input parameter.

The group specified with this command is ignored. However, note that the process variable may be defined by calculations performed by computes and fixes which store their own “group” definitions.

The PID controller is invoked once each Nevery timesteps.

The PID controller is implemented as a discretized version of the following dynamic equation:

\[\frac{dc}{dt} = -\alpha (K_p e + K_i \int_0^t e \, dt + K_d \frac{de}{dt} )\]

where c is the continuous time analog of the control variable, e =pvar-setpoint is the error in the process variable, and \(\alpha\), \(K_p\), \(K_i\) , and \(K_d\) are constants set by the corresponding keywords described above. The discretized version of this equation is:

\[c_n = c_{n-1} -\alpha \left( K_p \tau e_n + K_i \tau^2 \sum_{i=1}^n e_i + K_d (e_n - e_{n-1}) \right)\]

where \(\tau = \mathtt{Nevery} \cdot \mathtt{timestep}\) 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 \(K_p\), \(K_i\), \(K_d\) are dimensionless constants, then \(\alpha\) 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 unit style is changed, it should only be necessary to change the value of \(\alpha\) to reflect this, while leaving \(K_p\), \(K_i\), and \(K_d\) 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 \(K_p\) should then be tested over a large positive range keeping \(K_i = K_d =0\). A good value for \(K_p\) will produce a fast response in pvar, without overshooting the setpoint. For many applications, proportional feedback is sufficient, and so \(K_i = K_d =0\) 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 \(K_d\). In situations where pvar plateaus without reaching setpoint, this can be counteracted by increasing \(K_i\). In the language of Charles Dickens, \(K_p\) represents the error of the present, \(K_i\) the error of the past, and \(K_d\) the error yet to come.

Because this fix updates cvar, but does not initialize its value, the initial value \(c_0\) is that assigned by the user in the input script via the internal-style variable command. This value is 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 \(e_{n-1}\) is not yet defined.

The process variable pvar can be specified as the output of a compute or fix or the evaluation of a variable. In each case, the compute, fix, or variable must produce a global quantity, not a per-atom or local quantity.

If pvar begins with “c_”, a compute ID must follow which has been previously defined in the input script and which generates a global scalar or vector. See the individual compute doc page for details. If no bracketed integer is appended, the scalar calculated by the compute is used. If a bracketed integer is appended, the Ith value of the vector calculated by the compute is used. Users can also write code for their own compute styles and add them to LAMMPS.

If pvar begins with “f_”, a fix ID must follow which has been previously defined in the input script and which generates a global scalar or vector. See the individual fix page for details. Note that some fixes only produce their values on certain timesteps, which must be compatible with when fix controller references the values, or else an error results. If no bracketed integer is appended, the scalar calculated by the fix is used. If a bracketed integer is appended, the Ith value of the vector calculated by the fix is used. Users can also write code for their own fix style and add them to LAMMPS.

If pvar begins with “v_”, a variable name must follow which has been previously defined in the input script. Only equal-style variables can be referenced. See the variable command for details. Note that variables of style equal define a formula which can reference individual atom properties or thermodynamic keywords, or they can invoke other computes, fixes, or variables when they are evaluated, so this is a very general means of specifying the process variable.

The target value setpoint for the process variable must be a numeric value, in whatever units pvar is defined for.

The control variable cvar must be the name of an internal-style variable previously defined in the input script. Note that it is not specified with a “v_” prefix, just the name of the variable. It must be an internal-style variable, because this fix updates its value directly. Note that other commands can use an equal-style versus internal-style variable interchangeably.

## Restart, fix_modify, output, run start/stop, minimize info

Currently, no information about this fix is written to binary restart files. None of the fix_modify options are relevant to this fix.

This fix produces a global vector with 3 values which can be accessed by various output commands. The values can be accessed on any timestep, though they are only updated on timesteps that are a multiple of Nevery.

The three values are the most recent updates made to the control variable by each of the 3 terms in the PID equation above. The first value is the proportional term, the second is the integral term, the third is the derivative term.

The units of the vector values will be whatever units the control variable is in. The vector values calculated by this fix are “extensive”.

No parameter of this fix can be used with the start/stop keywords of the run command. This fix is not invoked during energy minimization.

## Restrictions

This fix is part of the EXTRA-FIX package. It is only enabled if LAMMPS was built with that package. See the Build package page for more info.

## Related commands

fix adapt

## Default

none