\documentclass[preview]{standalone}
\usepackage[utf8]{inputenc}
\usepackage{wangyan}
\begin{document}
\newsavebox\fa
\begin{lrbox}{\fa}
$\begin{aligned}
\overline{f}^2\leq \frac{\overline{p}_0-\overline{q}}{\overline{p}_0}
\end{aligned} $
\end{lrbox}
\newsavebox\pa
\begin{lrbox}{\pa}
$\begin{aligned}
\overline{p}_0\overline{f}^{2} +\overline{q}
\end{aligned} $
\end{lrbox}
%-------------------------------
\newsavebox\fb
\begin{lrbox}{\fb}
$\begin{aligned}
\frac{\overline{p}_0-\overline{q}}{\overline{p}_0} < \overline{f}^2< 1
\end{aligned} $
\end{lrbox}
\newsavebox\pb
\begin{lrbox}{\pb}
$\begin{aligned}
\overline{p}_0\overline{f}^{2} +\frac{\overline{q}}{1-\overline{f}^2}
\end{aligned} $
\end{lrbox}
%----------------------
\newsavebox\fc
\begin{lrbox}{\fc}
$ \begin{aligned}
\overline{f}^2\geq 1
\end{aligned} $
\end{lrbox}
%-------------
\newsavebox\flessone
\begin{lrbox}{\flessone}
$ \begin{aligned}
\overline{f}^2< 1
\end{aligned} $
\end{lrbox}
%-------------------------------
\newsavebox\pc
\begin{lrbox}{\pc}
$ \begin{aligned}
&\max \Big\{ \overline{p}_0\overline{f}^{2k} \\ & +\overline{q}\sum_{n=0}^{k-1}\overline{f}^{2n}, \\ &\quad \frac{1}{ \alpha^m \underline{\gamma} },\\ &\quad k=1, 2, \cdots, m-1 \Big\}
\end{aligned} $
\end{lrbox}
%---------------------------------
\newsavebox\pgreaterq
\begin{lrbox}{\pgreaterq}
$ \begin{aligned}
\overline{p}_0> \overline{q}
\end{aligned} $
\end{lrbox}
%---------------------
\newsavebox\plesseqq
\begin{lrbox}{\plesseqq}
$ \begin{aligned}
\overline{p}_0\leq \overline{q}
\end{aligned} $
\end{lrbox}
%----------------------------------
\begin{table}
\caption{The uniform upper bounds $\overline{p}$ of error covariance with regards to the values of $\overline{f}^2$ }\label{ta:upper_bound}
\centering
\begin{threeparttable}
\begin{tabular}{c|l|l|l}
\midrule\midrule
\multicolumn{3}{c|}{$\overline{f}^2$} & \multicolumn{1}{c}{$\overline{p}$} \\ \midrule\midrule
\multirow{3}{*}{ \usebox{\flessone}}&
\multirow{2}{*}{\usebox{\pgreaterq} } & \usebox{\fa}& \usebox{\pa} \\\cline{3-4}
& & \usebox{\fb} & \usebox{\pb} \\\cline{2-4}
& \multicolumn{2}{c|}{\usebox{\plesseqq}} & \usebox{\pb} \\ \cmidrule(l r){1-4}
\multicolumn{3}{c|}{ \usebox{\fc}\tnote{*}} & Eq. (37) \\ \midrule\midrule
\end{tabular}
\begin{tablenotes}
\item[*] The condition of \textit{uniform observability} needs to be satisfied simultaneously.
\end{tablenotes}
\end{threeparttable}
\end{table}
\end{document}
