\begin{table*}[t] 
\centering 
\scriptsize
\tabcolsep=0.11cm 
\begin{tabular}{|p{1.5cm}|c|c|c|c|c|c|c|c|c|c|c|c|c|c|} 
\hline 
\multirow{2}{*}{\textbf{Benchmark}} &\multirow{2}{*}{\textbf{k}} & \multicolumn{2}{c|}{\textbf{MATISSE~\cite{girard2007approximate}}} & \multicolumn{3}{c|}{\textbf{Zhi Han~\cite{han2004reachability}}} & \multicolumn{4}{c|}{\textbf{Mixed bound}} & \multicolumn{4}{c|}{\textbf{Theoretical bound}} \\ 
\cline{3-15} 
&  & {$\delta$} & $t$ & $\delta$ & $t$ & $N$ & $ e_1 $ & $ e_2 $ & $\delta$ &$t$ & $e_1 $ & $e_2$ & $\delta$ &$t$ \\ 
\hline 
\multirow{2}{*}{\parbox{1.5cm}{Motor control system}} 
& $5$ 
%Prof Girard : bisimulation method 
& $2.1$ & $1$ 
%Zhi Han : simulation-based method 
&$\begin{pmatrix}0.0021  \\ 0.047\end{pmatrix}$ & $0.54 $ & {$5$} 
%Our method : mixed bound 
&$\begin{pmatrix}0.00015  \\ 0.00034 \end{pmatrix}$ &$\begin{pmatrix}0.002  \\ 0.047 \end{pmatrix}$ &$\begin{pmatrix}0.0022  \\ 0.047 \end{pmatrix}$ & $1.3$ 
%Our method : theoretical bound 
&$\begin{pmatrix}0.0098  \\ 0.0093 \end{pmatrix}$ &$\begin{pmatrix}0.53  \\ 0.53 \end{pmatrix}$ &$\begin{pmatrix}0.54  \\ 0.54 \end{pmatrix}$ & $0.25$  \\ 
\cline{2-15} 
& $4$ 
%Prof Girard : bisimulation method 
& $1.5$ & $0.88$ 
%Zhi Han : simulation-based method 
&$\begin{pmatrix}0.036  \\ 0.047\end{pmatrix}$ & $0.19 $ & {$5$} 
%Our method : mixed bound 
&$\begin{pmatrix}0.00062  \\ 0.00034 \end{pmatrix}$ &$\begin{pmatrix}0.035  \\ 0.047 \end{pmatrix}$ &$\begin{pmatrix}0.036  \\ 0.047 \end{pmatrix}$ & $1$ 
%Our method : theoretical bound 
&$\begin{pmatrix}0.009  \\ 0.009 \end{pmatrix}$ &$\begin{pmatrix}0.91  \\ 0.91 \end{pmatrix}$ &$\begin{pmatrix}0.92  \\ 0.92 \end{pmatrix}$ & $0.14$  \\ 
\hline 

\multirow{3}{*}{\parbox{1.5cm}{Helicopter}} 
& $20$ 
%Prof Girard : bisimulation method 
& $0.84$ & $5.2$ 
%Zhi Han : simulation-based method 
&$\begin{pmatrix}7.1e-05  \\ 4.3e-05\end{pmatrix}$ & $0.62 $ & {$5$} 
%Our method : mixed bound 
&$\begin{pmatrix}2.3e-05  \\ 0.00013 \end{pmatrix}$ &$\begin{pmatrix}5.7e-05  \\ 3.2e-05 \end{pmatrix}$ &$\begin{pmatrix}8e-05  \\ 0.00017 \end{pmatrix}$ & $9.8$ 
%Our method : theoretical bound 
&$\begin{pmatrix}0.28  \\ 0.95 \end{pmatrix}$ &$\begin{pmatrix}0.0017  \\ 0.0017 \end{pmatrix}$ &$\begin{pmatrix}0.28  \\ 0.95 \end{pmatrix}$ & $0.48$  \\ 
\cline{2-15} 
& $16$ 
%Prof Girard : bisimulation method 
& $29$ & $4.2$ 
%Zhi Han : simulation-based method 
&$\begin{pmatrix}0.00075  \\ 0.0013\end{pmatrix}$ & $0.6 $ & {$5$} 
%Our method : mixed bound 
&$\begin{pmatrix}0.0001  \\ 0.00054 \end{pmatrix}$ &$\begin{pmatrix}0.0007  \\ 0.00087 \end{pmatrix}$ &$\begin{pmatrix}0.0008  \\ 0.0014 \end{pmatrix}$ & $7.8$ 
%Our method : theoretical bound 
&$\begin{pmatrix}0.28  \\ 0.95 \end{pmatrix}$ &$\begin{pmatrix}0.029  \\ 0.029 \end{pmatrix}$ &$\begin{pmatrix}0.3  \\ 0.97 \end{pmatrix}$ & $0.46$  \\ 
\cline{2-15} 
& $10$ 
%Prof Girard : bisimulation method 
& $1.6e+02$ & $2.9$ 
%Zhi Han : simulation-based method 
&$\begin{pmatrix}0.024  \\ 0.038\end{pmatrix}$ & $0.59 $ & {$5$} 
%Our method : mixed bound 
&$\begin{pmatrix}0.0026  \\ 0.0072 \end{pmatrix}$ &$\begin{pmatrix}0.021  \\ 0.031 \end{pmatrix}$ &$\begin{pmatrix}0.024  \\ 0.039 \end{pmatrix}$ & $4.5$ 
%Our method : theoretical bound 
&$\begin{pmatrix}0.27  \\ 0.93 \end{pmatrix}$ &$\begin{pmatrix}1  \\ 1 \end{pmatrix}$ &$\begin{pmatrix}1.3  \\ 1.9 \end{pmatrix}$ & $0.47$  \\ 
\hline 

\multirow{3}{*}{\parbox{1.5cm}{Building model}} 
& $25$ 
%Prof Girard : bisimulation method 
& $0.0096$ & $38$ 
%Zhi Han : simulation-based method 
&$0.0051$& $20$ &{$5$} 
%Our method : mixed bound 
&$0.013$ &$6.2e-05$ &$0.013$ &$25$ 
%Our method : theoretical bound 
&$0.083$ &$0.0072$ &$0.09$ &$1$ \\ 
\cline{2-15} 
& $15$ 
%Prof Girard : bisimulation method 
& $0.069$ & $22$ 
%Zhi Han : simulation-based method 
&$0.005$& $16$ &{$5$} 
%Our method : mixed bound 
&$0.012$ &$0.00044$ &$0.013$ &$16$ 
%Our method : theoretical bound 
&$0.078$ &$0.084$ &$0.16$ &$1.2$ \\ 
\cline{2-15} 
& $6$ 
%Prof Girard : bisimulation method 
& $4.2$ & $11$ 
%Zhi Han : simulation-based method 
&$0.0058$& $14$ &{$5$} 
%Our method : mixed bound 
&$0.011$ &$0.00025$ &$0.011$ &$7.7$ 
%Our method : theoretical bound 
&$0.073$ &$0.21$ &$0.28$ &$1$ \\ 
\hline 

\multirow{3}{*}{\parbox{1.5cm}{Partial differential equation}} 
& $30$ 
%Prof Girard : bisimulation method 
& $0.009$ & $63$ 
%Zhi Han : simulation-based method 
&$5.6e-14$& $1.7$ &{$5$} 
%Our method : mixed bound 
&$1.8e-05$ &$5.6e-14$ &$1.8e-05$ &$3.3e+02$ 
%Our method : theoretical bound 
&$1$ &$5e-12$ &$1$ &$1.7$ \\ 
\cline{2-15} 
& $20$ 
%Prof Girard : bisimulation method 
& $0.038$ & $34$ 
%Zhi Han : simulation-based method 
&$3.5e-14$& $1.6$ &{$5$} 
%Our method : mixed bound 
&$6.8e-06$ &$3.5e-14$ &$6.8e-06$ &$2.4e+02$ 
%Our method : theoretical bound 
&$1$ &$5.4e-12$ &$1$ &$1.6$ \\ 
\cline{2-15} 
& $10$ 
%Prof Girard : bisimulation method 
& $0.086$ & $15$ 
%Zhi Han : simulation-based method 
&$9.8e-13$& $2$ &{$5$} 
%Our method : mixed bound 
&$1.1e-05$ &$9.8e-13$ &$1.1e-05$ &$1.4e+02$ 
%Our method : theoretical bound 
&$0.92$ &$2.7e-11$ &$0.92$ &$2$ \\ 
\cline{2-15} 
& $6$ 
%Prof Girard : bisimulation method 
& $4.2$ & $11$ 
%Zhi Han : simulation-based method 
&$3.5e-07$& $1.7$ &{$5$} 
%Our method : mixed bound 
&$1.8e-05$ &$3.5e-07$ &$1.8e-05$ &$1.1e+02$ 
%Our method : theoretical bound 
&$0.89$ &$5.5e-06$ &$0.89$ &$1.6$ \\ 
\hline 

\multirow{3}{*}{\parbox{1.5cm}{International space station}} 
& $25$ 
%Prof Girard : bisimulation method 
& $0$ & $0$ 
%Zhi Han : simulation-based method 
&$\begin{pmatrix}0.001  \\ 4.6e-05\end{pmatrix}$ & $15 $ & {$5$} 
%Our method : mixed bound 
&$\begin{pmatrix}0  \\ 0 \end{pmatrix}$ &$\begin{pmatrix}0.001  \\ 4.6e-05 \end{pmatrix}$ &$\begin{pmatrix}0.001  \\ 4.6e-05 \end{pmatrix}$ & $12$ 
%Our method : theoretical bound 
&$\begin{pmatrix}0.00026  \\ 0.00026 \end{pmatrix}$ &$\begin{pmatrix}0.47  \\ 0.47 \end{pmatrix}$ &$\begin{pmatrix}0.47  \\ 0.47 \end{pmatrix}$ & $12$  \\ 
\cline{2-15} 
& $10$ 
%Prof Girard : bisimulation method 
& $0$ & $0$ 
%Zhi Han : simulation-based method 
&$\begin{pmatrix}5.6e-05  \\ 9e-05\end{pmatrix}$ & $13 $ & {$5$} 
%Our method : mixed bound 
&$\begin{pmatrix}0  \\ 0 \end{pmatrix}$ &$\begin{pmatrix}5.6e-05  \\ 9e-05 \end{pmatrix}$ &$\begin{pmatrix}5.6e-05  \\ 9e-05 \end{pmatrix}$ & $11$ 
%Our method : theoretical bound 
&$\begin{pmatrix}0.00022  \\ 0.00021 \end{pmatrix}$ &$\begin{pmatrix}1.7  \\ 1.7 \end{pmatrix}$ &$\begin{pmatrix}1.7  \\ 1.7 \end{pmatrix}$ & $11$  \\ 
\hline 

\multirow{3}{*}{\parbox{1.5cm}{FOM model}} 
& $20$ 
%Prof Girard : bisimulation method 
& $0$ & $0$ 
%Zhi Han : simulation-based method 
&$2.7e-07$& $80$ &{$5$} 
%Our method : mixed bound 
&$0$ &$2.7e-07$ &$2.7e-07$ &$80$ 
%Our method : theoretical bound 
&$1.3$ &$1.1e-05$ &$1.3$ &$50$ \\ 
\cline{2-15} 
& $15$ 
%Prof Girard : bisimulation method 
& $0$ & $0$ 
%Zhi Han : simulation-based method 
&$0.00021$& $81$ &{$5$} 
%Our method : mixed bound 
&$0$ &$0.00021$ &$0.00021$ &$81$ 
%Our method : theoretical bound 
&$1.3$ &$0.0065$ &$1.3$ &$51$ \\ 
\cline{2-15} 
& $10$ 
%Prof Girard : bisimulation method 
& $0$ & $0$ 
%Zhi Han : simulation-based method 
&$0.1$& $80$ &{$5$} 
%Our method : mixed bound 
&$0$ &$0.1$ &$0.1$ &$80$ 
%Our method : theoretical bound 
&$1.3$ &$2.2$ &$3.5$ &$50$ \\ 
\hline 

\end{tabular} 
 \caption{The error bounds and computing times obtained from different methods on different benchmarks in which: $k$ is the dimension of the output abstraction, $\delta$ is total error bound, $e_1$ is the zero input response error, $e_2$ is the zero state response error, $t$ is the error computing time (in second) and $N$ is the number of simulations.} 
\tablabel{experiment_test} 
\end{table*} 
