s,u)$. 
In other words, it allows that the occurring fault is not necessarily required to exactly match any of the pre-defined/pre-trained faults, which however is typically required by existing methods of \cite{CheW.IJACSP14, CheWH.TNNLS13, ZhaY.ACCESS19}. 
This property renders our FI scheme a better robust capability of preventing false/missed FI alarm in the presence of slight fault difference during the FI process.  

\end{remark}

Based on the above setup, to design  the FI adaptive threshold, we consider the $l$-th residual system in (\ref{FI_residual}), its time-domain solution can be derived as:
\begin{equation} \label{FI_match-residual0}
\begin{aligned}
	\tilde{x}_i^l (t) = \tilde{x}_i^l(t_d) e^{-b_i (t-t_d)} + \int_{t_d}^t e^{-b_i(t-\tau)} ( \rho_i^{l,l'}(x_s, u) - \varepsilon_{i}^l ) d \tau.

\end{aligned}
\end{equation} 
Note that $ \tilde{x}_{i}^l(t_d) =0$ and $\left| \varepsilon_{i}^l \right| < \xi_i^* $ from (\ref{Identify_funcapp-bar}), under Assumption \ref{ASS_fault-bound}, we have: 
\begin{equation} \label{FI_match-residual}
\begin{aligned}
	\left| \tilde{x}_i^l (t) \right|
	\leq &\, \int_{t_d}^t e^{-b_i(t-\tau)} ( \left| \varepsilon_{i}^l  \right| + \left| \rho_i^{l,l'}(x_s, u) \right| ) d \tau \\

	< &\, \frac{\xi_i^*}{b_i} + \int_{t_d}^t e^{-b_i(t-\tau)} \bar{\rho}_i^{l}(x_s, u) d \tau.
\end{aligned}
\end{equation} 
It guarantees that the FI residual signal $\left\| \tilde{x}_{i}^l(t) \right\|_1$ satisfies: 
\begin{equation} \label{FI_error-dynamics1}
	\left\| \tilde{x}_{i}^l(t) \right\|_1 < \frac{\xi_i^*}{b_i} + \left\| \int_{t_d}^t e^{-b_i(t-\tau)} \bar{\rho}_i^l (x_s, u) d \tau \right\|_1.
\end{equation}
Thus, the FI adaptive threshold $\bar{e}_i^l(t)$ can be designed as:
\begin{equation} \label{FI_threshold}
	\bar{e}_i^l(t) := \frac{\xi_i^*}{b_i} + \left\| \int_{t_d}^t e^{-b_i(t-\tau)} \bar{\rho}_i^{l}(x_s, u) d \tau \right\|_1,
\end{equation} 
for all $i=1,\cdots,m$, where $\xi_i^*$ is a small constant given in (\ref{Identify_funcapp-bar}), $b_i$ is a design constant from (\ref{FI_estimator}), and $\bar{\rho}_i^{l}(x_s, u)$ is a known function defined in Assumption \ref{ASS_fault-bound}. 

\begin{remark} \label{rem_FIthreshold-simplify}

The FI thresholds (\ref{FI_threshold}) can be implemented in a simplified form if the function $\bar{\rho}_i^l(x_s,u)$ is a constant $\bar{\rho}_i^l$. 
Specifically, with $\bar{\rho}_i^l(x_s,u)=\bar{\rho}_i^l$, the threshold (\ref{FI_threshold}) is given by $\bar{e}_i^l(t) = \frac{\xi_i^*}{b_i} + \left\| \frac{\bar{\rho}_i^l }{b_i} (1-e^{-b_i (t-t_d)}) \right\|_1 $, which can be further simplified as a constant threshold $\bar{e}_i^l = \frac{1}{b_i}(\xi_i^*+ \bar{\rho}_i^l) $. 

\end{remark}

Consequently, for the monitored system (\ref{FI_ODE-sys}), the proposed FI scheme consists of the FI estimators (\ref{FI_estimator}) and the adaptive threshold (\ref{FI_threshold}). 
Real-time FI decision making is based on the following principle.  
If there exists a unique residual system in (\ref{FI_residual}), say the $l$-th one, such that for all $ i=1,\cdots,m$ the residual signals $\tilde{x}_i^l$ satisfy $\left\| \tilde{x}_i^l(t) \right\|_1 \leq \bar{e}_i^l (t) $ for all time $t>t_d$, then, it can be deduced that the occurring fault $l'$ in (\ref{FI_ODE-sys}) is similar to the trained fault $l$. 
Using this idea, the FI decision making scheme can be devised as follows.


\textbf{Fault isolation decision making}: Compare the FI residual signals $\left\|\tilde{x}_i^k (t) \right\|_1$ with the FI adaptive thresholds $\bar{e}_i^k (t)$ for time $t \geq t_d$ and all $ i=1,\cdots,m$, $k=1,\cdots,N$. 
If there exists a unique $l \in \{1,\cdots,N\}$ such that: (i) $\forall i=1,\cdots,m$, $\left\|\tilde{x}_i^l(t)\right\|_1 \leq \bar{e}_i^l(t)$ holds for all time $t \geq t_d$; and (ii) $\forall k \in \{1,\cdots,N\} / \{l\}$, $\exists i\in \{1,\cdots,m\}$, $\left\|\tilde{x}_i^k(t^k)\right\|_1 > \bar{e}_i^k(t^k)$ holds at some time instant $t^k > t_d$. Then, the occurring fault $l'$ can be identified similar to the fault $l$, and the isolation time can be obtained as: $ t_{iso} = \max \left\{ t^k, k \in \{1,\cdots,N\} / \{l\} \right\}$. 



\begin{remark} \label{rem_FIestimator}


Similar to the FD scheme of Section \ref{sec_FD}, the proposed FI scheme can effectively deal with the effect of system uncertainty $f_s(x_s,u)$ in (\ref{FI_ODE-sys}) for accurate isolation, and the associated FI process can be achieved in a rapid manner. 
This is owing to the utilization of constant NN models $\bar{W}_i^{k\top}S(x_s,u)$ (obtained through the training process of Section \ref{sec_identification}) in the design of FI estimators (\ref{FI_estimator}). 

\end{remark}

\begin{remark} \label{rem_FIthreshold}

Existing FI schemes in \cite{BanK.ACC12, CaiJ.ACC16, GhaE.AUT09} rely on  ``constant'' thresholds for FI, which would limit the ability to separate the temporal dynamics of different type of faults for accurate isolation. These schemes can be applicable only to the faults that have sufficiently distinct differences.
For example, the FI schemes in \cite{BanK.ACC12, GhaE.AUT09} can distinguish the actuator faults occurring at different locations, but cannot recognize the actuator faults that occur at the same location but have different magnitudes. 
Advanced over these schemes, our approach design an ``adaptive'' threshold of (\ref{FI_threshold}) by using the nonlinear function $\bar{\rho}_i^{l}(x_s,u)$ that can accurately specify the similarity of each $l$-th type of faults, as defined in Assumption \ref{ASS_fault-bound}. 
With such a threshold, our FI scheme can achieve accurate isolation even for the faults that have relatively small differences. Improved FI accuracy and fault isolatability with our scheme compared to the ones in \cite{BanK.ACC12, CaiJ.ACC16, GhaE.AUT09} will be demonstrated later. 

\end{remark}

\begin{remark} \label{rem_FIapplicable}

Our FI scheme provides a unified framework to address the isolation problem of general faults occurring in the PDE system