\subsubsection{Differentiation according to Experimentees} \label{differentiation_according_to_experimentees}
Many records achieved a high $F_{1}score_{\mu}$s under the dominating simulation. However, there are also records that
did not worked that well under the best performing configuration. Figure \ref{fig:experimentee_result} shows the
distribution of the different records from different experimentees.

\begin{figure}
    \begin{center}
        \resizebox {\textwidth} {!} {
            \begin{tabular}{cc}
                \resizebox {!} {\height} {
                    \begin{tikzpicture}
                        \begin{axis}[
                            xmin=0.2,
                            xmax=1,
                            ymin=0.2,
                            ymax=1,
                            width=\axisdefaultwidth,
                            height=\axisdefaultwidth,
                            xlabel=$Precision_{\mu}$,
                            ylabel=$Recall_{\mu}$,
                            samples=100]
                            \addplot+[
                                blue,
                                only marks,
                                nodes near coords,
                                every node near coord/.style={at={(0.12,0.17)}, color=black},
                                point meta=explicit symbolic] table[x=x, y=y, meta=label] {../data/fig/experimentee_result/experimentee.dat};
                            \addplot[gray, domain=0.16:1] {(0.3 * x) / (2 * x - 0.3)};
                            \addplot[gray, domain=0.21:1] {(0.4 * x) / (2 * x - 0.4)};
                            \addplot[gray, domain=0.26:1] {(0.5 * x) / (2 * x - 0.5)};
                            \addplot[gray, domain=0.31:1] {(0.6 * x) / (2 * x - 0.6)};
                            \addplot[gray, domain=0.36:1] {(0.7 * x) / (2 * x - 0.7)};
                            \addplot[gray, domain=0.41:1] {(0.8 * x) / (2 * x - 0.8)};
                            \addplot[gray, domain=0.46:1] {(0.9 * x) / (2 * x - 0.9)};
                        \end{axis}
                    \end{tikzpicture}
                } &
                \resizebox {!} {\height} {
                    \begin{tikzpicture}
                        \begin{axis}[
                            xmin=0,
                            xmax=1,
                            ymin=0,
                            ymax=1,
                            width=\axisdefaultwidth,
                            height=\axisdefaultwidth,
                            xlabel=$Precision_{\mu}$,
                            ylabel=$Recall_{\mu}$,
                            samples=100]
                            \addplot+[
                                blue,
                                only marks,
                                nodes near coords,
                                every node near coord/.style={at={(-0.05,0)}, color=black},
                                point meta=explicit symbolic] table[x=x, y=y, meta=label] {../data/fig/experimentee_result/experimentee.dat};
                            \addplot[gray, domain=0.051:1] {(0.1 * x) / (2 * x - 0.1)};
                            \addplot[gray, domain=0.11:1] {(0.2 * x) / (2 * x - 0.2)};
                            \addplot[gray, domain=0.16:1] {(0.3 * x) / (2 * x - 0.3)};
                            \addplot[gray, domain=0.21:1] {(0.4 * x) / (2 * x - 0.4)};
                            \addplot[gray, domain=0.26:1] {(0.5 * x) / (2 * x - 0.5)};
                            \addplot[gray, domain=0.31:1] {(0.6 * x) / (2 * x - 0.6)};
                            \addplot[gray, domain=0.36:1] {(0.7 * x) / (2 * x - 0.7)};
                            \addplot[gray, domain=0.41:1] {(0.8 * x) / (2 * x - 0.8)};
                            \addplot[gray, domain=0.46:1] {(0.9 * x) / (2 * x - 0.9)};
                        \end{axis}
                    \end{tikzpicture}
                }
            \end{tabular}
        }
    \end{center}
    \caption{$Precision_{\mu}$ and $Recall_{\mu}$ of all 14 records from 14 different experimentees for the simulation
    that uses $\eta$DTW with a Sakoe-Chiba band of size 36\% depending on the input time series length, \textit{HAveD}
    as threshold determination and \textit{Mid} as window size determination without any filter. The left plot is just a
    zoomed version of the right plot. Gray lines are illustrating the distribution of $F_{1}score_{\mu}$ in
    $\frac{1}{10}$ steps.}
    \label{fig:experimentee_result}
\end{figure}

The performance measures to rank different sliding window simulations $Precision_{\mu}$, $Recall_{\mu}$ and
$F_{1}score_{\mu}$ are explained in section \ref{performance_measure}. However, those measures do not show how often a
gesture was detected by a simulation. Therefore figure \ref{fig:experimentee_result2} illustrates all detected
gestures for all 14 test stream records under the best performing simulation. The green rectangles in the figure are true positives,
the red rectangles are false positives and the blue rectangles are false negatives. True negatives are invisble. The
blue and red rectangles can overlap. User marked gestures are visible by combining the green with the blue rectangles.
The pretended gestures detected by the simulation are visible by combining the green with the red rectangles. All green
rectangles of an experimentee summed up and divided by the sum of the green and red rectangles are resulting in the
$Precision_{\mu}$ value. The green rectangles summed up and divided by the sum of the green and blue rectangles are
resulting in the $Recall_{\mu}$ value. Those calculations are made over all test stream records for all 46920 different
simulations in section \ref{evaluation}.

A gesture is considered as detected if a green rectangle in figure \ref{fig:experimentee_result2} is surround by small
red or blue rectangles. 81 out of 112 gestures were detected right. A gesture is considered as wrongly
detected if a blue and a red rectangle are overlapping. 4 out of 112 gestures were detected wrongly. A
stand alone blue rectangle is considered as missed gesture. 27 out of 112 gestures were missed by the
simulation. A stand alone red rectangle does not exist in this simulation surprisingly. Experimentee $exp_4$ and
$exp_{13}$ are reaching a $F_{1}score_{\mu}$ greater than 0.9 and in both cases all eight gestures have been detected
right.
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                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(1953, -16) (1962, -16) (1962, 16) (1953, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(1962, -16) (2027, -16) (2027, 16) (1962, 16)} --cycle;
            \end{axis}
        \end{tikzpicture}
    } \\
    \resizebox {0.684\textwidth} {!} {
        \begin{tikzpicture}
            \begin{axis}[
                xmin=0,
                xmax=2216,
                ymin=-16,
                ymax=16,
                width=6.84*\axisdefaultwidth,
                height=\axisdefaultheight,
                xticklabels={,,},
                yticklabels={,,}]
                \addplot[blue, mark=none, opacity=0.4] table[x=t, y=x] {../data/fig/experimentee_result2/exp13.dat};
                \addplot[red, mark=none, opacity=0.4] table[x=t, y=y] {../data/fig/experimentee_result2/exp13.dat};
                \addplot[green, mark=none, opacity=0.4] table[x=t, y=z] {../data/fig/experimentee_result2/exp13.dat};
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(198, -16) (205, -16) (205, 16) (198, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(205, -16) (262, -16) (262, 16) (205, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(262, -16) (263, -16) (263, 16) (262, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(400, -16) (409, -16) (409, 16) (400, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(409, -16) (464, -16) (464, 16) (409, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(464, -16) (466, -16) (466, 16) (464, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(595, -16) (596, -16) (596, 16) (595, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(596, -16) (654, -16) (654, 16) (596, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(654, -16) (660, -16) (660, 16) (654, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(803, -16) (810, -16) (810, 16) (803, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(810, -16) (874, -16) (874, 16) (810, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(874, -16) (884, -16) (884, 16) (874, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(1294, -16) (1302, -16) (1302, 16) (1294, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(1302, -16) (1356, -16) (1356, 16) (1302, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(1356, -16) (1358, -16) (1358, 16) (1356, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(1643, -16) (1646, -16) (1646, 16) (1643, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(1646, -16) (1700, -16) (1700, 16) (1646, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(1700, -16) (1710, -16) (1710, 16) (1700, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(1893, -16) (1902, -16) (1902, 16) (1893, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(1902, -16) (1949, -16) (1949, 16) (1902, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(1949, -16) (1966, -16) (1966, 16) (1949, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(2094, -16) (2098, -16) (2098, 16) (2094, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(2098, -16) (2160, -16) (2160, 16) (2098, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(2160, -16) (2162, -16) (2162, 16) (2160, 16)} --cycle;
            \end{axis}
        \end{tikzpicture}
    } \\
    \resizebox {0.743\textwidth} {!} {
        \begin{tikzpicture}
            \begin{axis}[
                xmin=0,
                xmax=2405,
                ymin=-16,
                ymax=16,
                width=7.43*\axisdefaultwidth,
                height=\axisdefaultheight,
                xticklabels={,,},
                yticklabels={,,}]
                \addplot[blue, mark=none, opacity=0.4] table[x=t, y=x] {../data/fig/experimentee_result2/exp14.dat};
                \addplot[red, mark=none, opacity=0.4] table[x=t, y=y] {../data/fig/experimentee_result2/exp14.dat};
                \addplot[green, mark=none, opacity=0.4] table[x=t, y=z] {../data/fig/experimentee_result2/exp14.dat};
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(264, -16) (272, -16) (272, 16) (264, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(272, -16) (329, -16) (329, 16) (272, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(329, -16) (336, -16) (336, 16) (329, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(473, -16) (476, -16) (476, 16) (473, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(476, -16) (538, -16) (538, 16) (476, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(538, -16) (543, -16) (543, 16) (538, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(686, -16) (758, -16) (758, 16) (686, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(982, -16) (993, -16) (993, 16) (982, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(993, -16) (1058, -16) (1058, 16) (993, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(1058, -16) (1064, -16) (1064, 16) (1058, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(676, -16) (741, -16) (741, 16) (676, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(1370, -16) (1380, -16) (1380, 16) (1370, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(1380, -16) (1435, -16) (1435, 16) (1380, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(1435, -16) (1436, -16) (1436, 16) (1435, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(1639, -16) (1647, -16) (1647, 16) (1639, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(1647, -16) (1704, -16) (1704, 16) (1647, 16)} --cycle;
                \addplot+[fill, opacity=0.5, red, mark=none] coordinates {(1998, -16) (2007, -16) (2007, 16) (1998, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(2007, -16) (2063, -16) (2063, 16) (2007, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(2063, -16) (2064, -16) (2064, 16) (2063, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(2264, -16) (2267, -16) (2267, 16) (2264, 16)} --cycle;
                \addplot+[fill, opacity=0.5, green, mark=none] coordinates {(2267, -16) (2332, -16) (2332, 16) (2267, 16)} --cycle;
                \addplot+[fill, opacity=0.5, blue, mark=none] coordinates {(2332, -16) (2337, -16) (2337, 16) (2332, 16)} --cycle;
            \end{axis}
        \end{tikzpicture}
    }
    \caption{An illustration of all 14 test streams from $exp_1$ at the top to $exp_14$ at the bottom under the best
    performing gesture detection simulation. The green rectangles are true positive marks of a gesture class. The red
    rectangles are marking false positives of a gesture class the blue rectangles are marking false negatives. True
    negatives are invisible.}
    \label{fig:experimentee_result2}
\end{figure}


% Copyright 2021 UChicago Argonne, LLC
% Author:
% - Haoyu Wang and Roberto Ponciroli, Argonne National Laboratory
% - Andrea Alfonsi, Idaho National Laboratory

% Licensed under the Apache License, Version 2.0 (the "License");
% you may not use this file except in compliance with the License.
% You may obtain a copy of the License at

%   https://www.apache.org/licenses/LICENSE-2.0.txt

% Unless required by applicable law or agreed to in writing, software
% distributed under the License is distributed on an "AS IS" BASIS,
% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
% See the License for the specific language governing permissions and
% limitations under the License.

\section{Introduction}
\label{sec:Introduction}
Feasible Actuator Range Modifier (FARM) is a RAVEN (\cite{RAVEN} and \cite{RAVENtheoryMan})plugin designed to solve the
supervisory control problem in Integrated Energy System (IES) project. FARM utilizes the linear state-space representation
(A,B,C matrices) of a model to predict the system state and output in the future time steps, and adjust the actuation variable
to avoid the violation of implicit thermal mechanical constraints.

\subsection{FARM Plugin: Reference Governor SIMO}
\label{RG_SIMO}
  The first external model in FARM plugin is a scalar Reference Governor(\cite{kolmanovsky2014reference} and
  \cite{garone2017reference}) for single input multiple output (SIMO) system. Assume
  the system has the following linear state space representation(\cite{rowell2002state}) in \textbf{discrete} time domain:
  \begin{itemize}
  \item \begin{math} \overrightarrow{x}[k+1]=\textbf{A}*\overrightarrow{x}[k]+\textbf{B}*u[k] \end{math}
  \item \begin{math} \overrightarrow{y}[k+1]=\textbf{C}*\overrightarrow{x}[k+1] \end{math}
  \end{itemize}
  where:
  \begin{itemize}
  \item \begin{math} u\in\mathbb{R} \end{math}, is system actuation variable, scalar;
  \item \begin{math} \overrightarrow{x}\in\mathbb{R}^n \end{math}, is system state vector, n elements;
  \item \begin{math} \overrightarrow{y}\in\mathbb{R}^p \end{math}, is system output vector, p elements;
  \item \begin{math} \textbf{A}\in\mathbb{C}^{n \times n} \end{math} is the state matrix;
  \item \begin{math} \textbf{B}\in\mathbb{C}^{n \times 1} \end{math} is the input matrix;
  \item \begin{math} \textbf{C}\in\mathbb{C}^{p \times n} \end{math} is the output matrix;

  \end{itemize}

  The Reference Governor SIMO use the linear matrices \begin{math} \textbf{A},\textbf{ B}, \textbf{ C} \end{math} and the
  current system state vector \begin{math} \overrightarrow{x}[k] \end{math} to predict the system state vectors
  \begin{math} \overrightarrow{x}[k+1],\overrightarrow{x}[k+2], ... ,\overrightarrow{x}[k+g] \end{math} and output vectors
  \begin{math} \overrightarrow{y}[k+1],\overrightarrow{y}[k+2], ... ,\overrightarrow{y}[k+g] \end{math} for the next
  \begin{math} \textbf{g} \end{math} time steps assuming a constant actuation variable \begin{math} u \end{math} is
  applied to the system. In addition, the steady state vector
  \begin{math} \overrightarrow{x}[\infty]\end{math} and output vector \begin{math} \overrightarrow{y}[\infty]\end{math}
  under the same system actuation variable \begin{math} u \end{math} is predicted.

  With the knowledge of the constraints
  (\begin{math} \overrightarrow{y_{min}} \end{math} and \begin{math} \overrightarrow{y_{max}} \end{math})
  on system output vector \begin{math} \overrightarrow{y} \end{math}, the predicted system output vectors
  (\begin{math} \overrightarrow{y}[k+1],\overrightarrow{y}[k+2], ... ,\overrightarrow{y}[k+g] \end{math},
  and \begin{math} \overrightarrow{y}[\infty]\end{math})
  will be compared to the constraints element-wisely. Once a violation was found, the actuation variable
  \begin{math} u \end{math} will be scaled down to avoid the violation.

  In addition to the scaled down system actuation variable \begin{math} u \end{math}, the admissible range of actuation
  variable (\begin{math} u_{min} \end{math} and \begin{math} u_{max} \end{math}) is also calculated. The actuation
  variable with a value within this admissible range will not cause any violation of the constraints in both the next consecutive
  \begin{math} \textbf{g} \end{math} time steps and the steady state.