======================
Control System Signals
======================
One or more sequences of user-defined **signals** are used to define a control-system
block diagram. The four types of signals are `measured <#measured-signals>`_, `demand
<#demand-signals>`_, `block <#block-signals>`_, and `control <#control-signals>`_. Each is
described in the sections that follow. Detailed information on the input requirements is
provided in the :ref:`Appendix<section-A6.1>`.
Measured Signals
================
Measured signals represent the value of plant-state variables, such as pressure,
temperature, or flow rate. Measured signals are implemented in SAS as a virtual data
acquisition system (DAS). Unless otherwise noted, measured signal values are obtained at
the beginning of a PRIMAR time step. These values are used to update the control system
state in effect during the time step. That state is then reported along with other simulation
output at the end of the time step.
Demand Signals
==============
Demand signals represent the value of an external, time-dependent load. Demand signals are
interpolated from a table of user-defined (time,value) pairs, where the values are assumed
to be normalized to an initial-condition value.
Block Signals
=============
Block signals represent mathematical or logical operations that define the control system's behavior. Input to block signals comes from measured, demand, or other block signals.
Variable Lag Compensator
------------------------
.. note:: New in Version 5.3.
.. rubric:: Theory
A *variable lag compensator* is similar to the lag compensator except
that the delay time parameter, :math:`\tau`, can vary during the Control System
simulation. An equation that expresses this behavior is
.. math::
y\left( t \right) + \tau\left( t \right)\frac{dy}{dt} = u\left( t \right)
where
:math:`u\left( t \right)` = input driving function
:math:`\tau\left( t \right)` = delay (lag)
:math:`y\left( t \right)` = output function result
Assuming that the lag time is always positive, a solution to the above
equation is
.. math::
y\left( t \right) = y_{0}e^{- \int_{0}^{t}\frac{dt'}{\tau\left( t \right)}} + e^{- \int_{t_{0}}^{t}\frac{dt'}{\tau\left( t \right)}}\int_{0}^{t}{\frac{u\left( t' \right)}{\tau\left( t' \right)}e^{\int_{t_{0}}^{t'}\frac{dt''}{\tau\left( t'' \right)}}dt'}
where :math:`y_{0} = y\left( 0 \right)` and :math:`t_{0}` is an
arbitrary integration constant.
The complexity of the above solution can be simplified if we assume the
input driving function and delay function are linear functions of time:
:math:`u\left( t \right) = u_{0} + u't` and
:math:`\tau\left( t \right) = \tau_{0} + \tau't`, respectively. With
this assumption, the solution becomes
.. math::
y\left( t \right) = y_{0}\left( 1 + \frac{\tau'}{\tau_{0}}t \right)^{- \frac{1}{\tau'}} + u_{0}\left\lbrack 1 - \left( 1 + \frac{\tau'}{\tau_{0}}t \right)^{- \frac{1}{\tau'}} \right\rbrack + \frac{u'}{1 + \tau'}\left\lbrack t - \tau_{0}\left\lbrack 1 - \left( 1 + \frac{\tau'}{\tau_{0}}t \right)^{- \frac{1}{\tau'}} \right\rbrack \right\rbrack
If :math:`\tau` is slowly varying with time and we take the limit that :math:`\tau' \rightarrow 0`, we arrive at the
solution for a constant :math:`\tau` during the time interval:
.. math::
y\left( t \right) = y_{0}e^{- \frac{t}{\tau}_{0}} + u_{0}\left( 1 - e^{- \frac{t}{\tau_{0}}} \right) + u'\left\lbrack t - \tau_{0}\left( 1 - e^{- \frac{t}{\tau_{0}}} \right) \right\rbrack
.. rubric:: Implementation
The analytic solution for the variable lag compensator can be used to
solve for the value of :math:`y_{1} = y\left( t_{0} + \Delta t \right)`
at the end of a Control System sub-interval (time step) given the
initial value :math:`y_{0} = y\left( t_{0} \right)`. Knowing the lag
time at the beginning of the time step, :math:`\tau_{0}`, and the input
driving function at the beginning and end of the time step,
:math:`u_{0} = u\left( t_{0} \right)` and
:math:`u_{1} = u\left( t_{0} + \Delta t \right)`, respectively, then the
solution for :math:`y_{1}` at the end of the time step is
.. math::
y_{1} = y_{0}e^{- \delta} + u_{0}\left( 1 - e^{- \delta} \right) + \left( u_{1} - u_{0} \right)\left\lbrack 1 - \frac{1 - e^{- \delta}}{\delta} \right\rbrack
where :math:`\delta = \frac{\Delta t}{\tau_{0}}`.
Because this solution is prone to numerical round-off errors, expansions
for the exponential function are used:
.. math::
e^{- \delta} = 1 - \delta + \frac{\delta^{2}}{2} - \frac{\delta^{3}}{6} + O\left\lbrack \delta^{4} \right\rbrack
and
.. math::
\frac{1 - e^{- \delta}}{\delta} = 1 - \frac{\delta}{2} + \frac{\delta^{2}}{6} - \frac{\delta^{3}}{24} + O\left\lbrack \delta^{4} \right\rbrack
Using these expansions, the solution is written as
.. math::
y_{1} = y_{0}\left\lbrack 1 - \delta + \frac{\delta^{2}}{2} \right\rbrack + u_{0}\left\lbrack \frac{\delta}{2} - \frac{\delta^{2}}{3} \right\rbrack + u_{1}\left\lbrack \frac{\delta}{2} - \frac{\delta^{2}}{6} \right\rbrack + O\left\lbrack \delta^{3} \right\rbrack
To minimize numerical round-off errors in the code, the following form
is used:
.. math::
y_{1} \approx y_{0}\left\lbrack 1 - \delta\left( 1 - \delta\left( \frac{1}{2} \right) \right) \right\rbrack + u_{0}\left\lbrack \delta\left( \frac{1}{2} - \delta\left( \frac{1}{3} \right) \right) \right\rbrack + u_{1}\left\lbrack \delta\left( \frac{1}{2} - \delta\left( \frac{1}{6} \right) \right) \right\rbrack
which is second-order accurate in :math:`\delta`. This form also avoids
the need to make expensive exponential function evaluations.
.. rubric:: Use
A *variable lag compensator* is identified in the Control System input
as signal type ITYPE = 23. It requires two input signals to define the
time-dependent functions :math:`u\left( t \right)` and
:math:`\tau\left( t \right)`. These are identified by the input
parameters J1 and J2, respectively. Input for the variable lag
compensator is summarized in the table below.
========== =========================================================
Input Description
========== =========================================================
ISIG User-defined signal number
ITYPE = 23 Variable Lag Compensator type
J1 Signal # of driving function, :math:`u(t)`
J2 Signal # of delay time, :math:`\tau\left( t \right)`
F1 Multiplier for driving function
[default: 1]
F2 (unused)
F3 (unused)
F4 Steady-state initial condition, :math:`y\left( 0 \right)`
[default: :math:`y\left( 0 \right) = u\left( 0 \right)`]
F5 Zero-crossing parameter (F5 > 0)
========== =========================================================
The input includes two optional parameters, F1 and F4, and one
required parameter, F5. If the optional parameters are blank or zero,
defaults will be applied as indicated in the table. For example, if F1
is non-zero, then the actual driving function will include a multiplier:
.. math::
u\left( t \right) = \mathrm{F1} \times \mathrm{\text{sig}}\left( \mathrm{J1} \right)
Likewise, if F4 is non-zero, it is used to define the initial
steady-state condition as :math:`y\left( 0 \right) = \mathrm{F4}`.
Otherwise, the steady-state solution to the equation is
:math:`y\left( 0 \right) = u\left( 0 \right)`.
The zero-crossing parameter, F5, is used in the same manner as with
other Control System signals to stabilize convergence testing when the
solution for :math:`y\left( t \right)` is near zero. The normalized
error is approximated as follows to avoid divide-by-zero errors:
.. math::
\mathrm{\text{error}} = \left| \frac{\Delta y}{y} \right| \cong \frac{\left| \Delta y \right|}{\left| y \right| + \mathrm{F5}}
Therefore, users can select a value for F5 that is small enough to
ensure convergence when :math:`y\left( t \right)` is small, but large
enough to avoid excessive iterations when small values for
:math:`y\left( t \right)` do not have a significant impact on the
overall solution.
Like the lag compensator, the variable lag compensator shifts both the
phase and amplitude of the input signal :math:`u\left( t \right)`.
However, the phase and amplitude vary as a function of time through the
additional input for :math:`\tau\left( t \right)`. To demonstrate this,
consider the following conditions:
.. math::
u\left( t \right) &= \sin\left( \frac{2\pi}{5}t \right) \\
\tau\left( t \right) &= 1.9\sin\left( \frac{2\pi}{50}t \right) + 2 \\
y\left( 0 \right) &= 2 \\
These can be represented in the Control System input as follows:
::
INCONT 5
# Time: t
1 -55 0 0 0.0 0.0 0.0 0.0 0.0
# Driving Signal: Sin(2Ï€t/5)
11 22 1 0 1.0 1.2566370 0.0 0.0
# Lag Time Variation: Sin(2Ï€t/50)
12 22 1 0 1.0 .12566370 0.0 0.0
# Constant: 2
13 21 0 0 2.0 0.0 0.0 0.0 0.0
# Lag Time: 1.9Sin(2Ï€t/50) + 2
17 1 12 13 1.9 1.0 1.0 0.0 0.0
# Variable Lag Compensator(11,17)
# ITYPE J1 J2 F1 F4 F5
18 23 11 17 1.0 0.0 0.0 2.0 0.001
999 1
8001 2 0.00000001 0.00000001
END
Using the above Listing in a SAS4A/SASSYS-1 input file would produce the
following Control System output:
.. image:: media/VarLag.png
:width: 5.5in
:align: center
Note that because F4 = 2, the variable lag compensator starts at the
non-equilibrium value :math:`y\left( 0 \right) = 2`. As the lag time
:math:`\tau\left( t \right)` rises and falls, the delay in the input
signal becomes longer and shorter, respectively. Similarly, the
amplitude of the output decreases and increases.
Control Signals
===============
Control signals are used to set the value of a SAS variable during the simulation.
Currently, a limited number of simulation variables can be controlled. The primary purpose
of control signals is to affect reactivity (e.g. scram), pump torque, and valve
coefficients.
.. _simulation_trip:
Simulation Trip
---------------
The simulation trip control signal (JTYPE=-10) is used to stop the SAS simulation based on the value
of an input signal. When the input signal changes from false (a value of zero) to true (a non-zero value), code execution will end at the end of the main time step.
Since the input signal is treated as a logical, a large zero crossing parameter (e\ :sub:`z`\, entered in F6SIG) is recommended to avoid control system cutbacks.
A value of 1E+09 is suggested.
The control system trips the simulation by setting the maximum number of time steps to the current main time step number
when the input signal triggers the trip. To restart from a case that was ended by the simulation trip control signal, the user should reset :sasinp:`MAXSTP`
in the restart case input file to the desired value so that the simulation can continue.
A simulation trip control signal can be configured as unlatched (J2SIG = 0) or latched (J2SIG = 1).
An unlatched simulation trip control signal is capable of tripping the simulation again after a restart, while a
latched simulation trip control signal cannot.