\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 315, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2015/315\hfil Membrane bioreactor process]
{Mathematical model for a membrane \\ bioreactor process}

\author[M. El Hajji,  N. Chorfi, M. Jleli \hfil EJDE-2015/315\hfilneg]
{Miled El Hajji,  Nejmeddine Chorfi, Mohamed Jleli}

\address{Miled El Hajji \newline
General studies department, College of Telecom and Electronics,
Technical and Vocational Training Corporation, Jeddah 2146, Saudi Arabia}
\email{miled.elhajji@enit.rnu.tn}

\address{Nejmeddine Chorfi \newline
Department of Mathematics, College of Science, King Saud University,
Riyadh 11451, Saudi Arabia}
\email{nchorfi@ksu.edu.sa}

\address{Mohamed Jleli \newline
Department of Mathematics, College of Science, King Saud University,
Riyadh 11451, Saudi Arabia}
\email{jleli@ksu.edu.sa}

\thanks{Submitted April 20, 2015. Published December 24, 2015.}
\subjclass[2010]{35J15, 78M22}
\keywords{Membrane bioreactor; modeling; stability; observer}

\begin{abstract}
 In this article, we consider a simple mathematical model involving
 biomass growth on organic materials in a membrane bioreactor for
 a waste water treatment. Details of qualitative analysis are provided.
 We proposed a high gain observer that permits the reconstruction of
 the biomass concentration and the endegenous decay based on on-line
 measurements of the chemical oxygen demand (CDO).
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

Membrane bioreactors (MBRs) is a combination of a membrane process as
 microfiltration or ultrafiltration with a suspended growth bioreactor
to treat waste water where the main focus is to reduce the chemical oxygen
demand (COD) in the effluent discharged to natural waters.
Membrane bioreactor process for the sludge retention and separation
from the liquid has been one of the alternatives to the conventional
activated sludge process (Figure \ref{reactor}).
Two MBR configuration are possible, the submerged MBR, where the membrane
is placed in the reactionnal medium, and the side stream MBR, where the membrane
is out of the  reactionnal medium. The amount of organic pollutants found
in surface water, determined by COD measurements, gives us an idea on the
water quality.
The excess bacteria grown in the system are removed as sludge and this causes
high costs.

In this study, we used a submerged BRM configuration (Figure \ref{reactor}, center).
Assume that soluble COD in mixed liquor is equal to the effluent COD because
the submerged membranes used in MBR don't remove dissolved materials
(used membranes are mostly micro- or ultrafilters). Additionally, we assume
that all organic material in feed solution are soluble.  We neglect fouling
 phenomenon during membrane separation process. 

The features of this article are the following:

$\bullet$ A simple mathematical model involving biomass growth on organic materials
in a membrane bioreactor for a waste water treatment is proposed. Details of
qualitative analysis are provided.

$\bullet$ A high gain observer is proposed that permits the reconstruction of
the biomass concentration and the endegenous decay based on on-line measurements
of the chemical oxygen demand (CDO).

$\bullet$ Simulations are used to validate theoretical results provided
above, and finally, concluding remarks are given.


\begin{figure}[htb]
\begin{center}
\setlength{\unitlength}{1mm} 
\begin{picture}(105,41)(0,-3)
\put(0,0){\includegraphics[width=35mm,height=35mm]{fig1a}} % conventional.eps
\put(1,37){$S_{in}$}
\put(18,37){$S,X$}
\put(29,37){$S,X$}
\put(20,12){$X$}
\put(9,-4){$(a)$}
\put(36,0){\includegraphics[width=35mm,height=35mm]{fig1b}} % submerged.eps
\put(41,37){$S_{in}$}
\put(62,37){$S$}
\put(50,-4){$(b)$}
\put(72,0){\includegraphics[width=35mm,height=35mm]{fig1c}} % sidestream.eps
\put(73,37){$S,X$}
\put(92,19){$X$}
\put(104,19){$S$}
\put(88,-4){(c)}
\end{picture}
\end{center}
\caption{Membrane Bio-Reactor :
(a) Conventional activated suldge process,
(b) Submerged MBR configuration with integrated membrane unit,
(c) Side stream MBR configuration with external membrane unit.}
\label{reactor}
\end{figure}

\section{Mathematical model and results}

Let $S$  denote the soluble CDO and let $X$ denote the total microorganisms
present in the bioreactor at time $t$.
 The following ordinary system of differential equations describe the
growth of total microorganisms on soluble CDO:
\begin{equation}
\begin{gathered}
 \dot S=D (S_{in}-S) - \frac{\mu(S)}{Y} X\,,\\
 \dot X=  (\mu(S) -m ) X\,.
\end{gathered} \label{model}
\end{equation}
$S_{in}$ denotes the effluent CDO in the feed, $Y$ is the conversion factor
of COD converted to biomass, $D$ denotes the dilution rate through the
membrane and $m$ is the endogenous decay constant.

Note that equation which describes the biomass growth is similar to batch
culture \cite{elhajji2} however the substrate equation is similar to
classic continuous reactor \cite{smithbook}.
It appears reasonable to assume that the endogenous decay constant $m$ is smaller
that the dilution rate $D$ and that a priory bounds on parameter $m$ are known.
We use the following assumptions:
\begin{itemize}
\item[(A1)]  There exists constants $m^-$ and $m^+$ such that
$0<m^- \leq m \leq m^+ < D$.

\item[(A2)] The growth function $\mu(\cdot)$
is a smooth increasing function such that $\mu(0)=0$.

\end{itemize}
Let $ x=\frac{X}{Y}$, $s=S$ and $s_{in}=S_{in}$. Then one obtains
\begin{equation}
\begin{gathered}
 \dot s=D (s_{in}-s) - \mu(s) x,\\
 \dot x=  \mu(s) x-m x,
\end{gathered}\label{m0}
\end{equation}
with positive initial condition
$\big(s(0),x(0)\big)\in \mathbb{R}_+\times \mathbb{R}_+$.

\subsection*{General properties}

\begin{proposition} \label{prop1}
(1)  For any initial condition in $\mathbb{R}_+\times \mathbb{R}_+$,
the solution of\eqref{m0} is bounded and has positive components  and
thus is defined for all $t>0$.

(2) System \eqref{m0} admits a positive invariant  attractor set of
solution given by $ \Omega=\{(s,x)\in \mathbb{R}_+\times \mathbb{R}_+:
  s+x \leq \frac{D}{m}s_{in}\}$.
\end{proposition}

\begin{proof}
(1) The positivity of the solution is guaranteed by the fact that
If $ s=0$ then $\dot s= D s_{in}> 0$ and if $x=0$ then $\dot x=0$.
Next we have to prove the boundedness of solutions of \eqref{m0}.
 By adding the two equations of system \eqref{m0}, one obtains,
for $ z=s+x-\frac{D}{m}s_{in}$, a single equation
$$
\dot z\leq - m\big(s+x-\frac{D}{m}s_{in}\big)=- m z
$$
then
$$
0\leq s+x \leq \frac{D}{m}s_{in}+K e^{- m t}\quad \mbox{where}\quad
K=z(0)=s(0)+x(0)-\frac{D}{m}\;s_{in}.
$$
Since all terms of the sum are positive, then the solution is bounded

Part (2) is a direct consequence of the previous inequality.
\end{proof}


\subsection*{Stability}
Let $s^*$ be a solution of $\mu(s)=m$ and $ x^*=\frac{D}{m}(s_{in}-s^*)$.
The equilibrium points of system \eqref{m0} are
$$
F_0=(s_{in},0)\quad\mbox{and}\quad F^*=(s^*, x^*).
$$
Note that the trivial equilibrium point $F_0$ always exists and
that $F^*$ exists if and only if $ \mu(s_{in})>m$.

\begin{proposition} \label{prop2}
\begin{enumerate}
\item There are no periodic orbits nor polycycles inside $\Omega$.
\item If $ \mu(s_{in})>m$, $F_0$ is a saddle point and $F^*$
is globally asymptotically stable.
\item If $ \mu(s_{in})<m$, $F_0$ is globally asymptotically stable.
\end{enumerate}
\end{proposition}

\begin{proof}
(1) Consider a trajectory of system \eqref{m0} belonging to $\Omega$.
Let us transform the system \eqref{m0} through the change of variables
$\xi_1 = s$ and $\xi_2 = \ln(x)$.
Then one obtains the  system
\begin{equation}
\begin{gathered}
\dot \xi_1= h_1(\xi_1,\xi_2):=D (s_{in}-\xi_1) - \mu(\xi_1) e^{\xi_2},\\
\dot \xi_2= h_2(\xi_1,\xi_2):=  \mu(\xi_1) -m\,.
\end{gathered} \label{orbit}
\end{equation}
We have
$$
 \frac{\partial h_1}{\partial \xi_1}
+ \frac{\partial h_2}{\partial \xi_2}
=  - D-\mu'(\xi_1)\; e^{\xi_2} < 0.
$$
From Dulac criterion \cite{smithbook}, we deduce that system \eqref{orbit}
has no periodic trajectory.
Hence system \eqref{m0} has no periodic orbit inside $\Omega$.

(2) Assume that $\mu(s_{in})>m$.
The Jacobian matrix $J^*$ of system \eqref{m0} at $(s^*,x^*)$ is
\[
J^*=\begin{bmatrix}
 -D - \mu'(s^*) x^*&  - m \\
 \mu'(s^*) x^*& 0
\end{bmatrix}.
\]
One can easily verify that
\[
\operatorname{tr}(J^*)= -D - \mu'(s^*)  x^*<0, \quad
 \det (J^*)= m \mu'(s^*) x^*>0,
\]
from where $F^*$ is a stable node.
The Jacobian matrix $J_0$ of system \eqref{m0} at $(s_{in},0)$ is
\[
J_0=\begin{bmatrix}
 -D&0\\
0& \mu(s_{in})-m
\end{bmatrix}
\]
One can easily verify that $F_0$ is a saddle point since $-D<0$
and $\mu(s_{in})-m>0$.
In this case $\Gamma_0=]0,+\infty[\times \{0\}$ is the stable manifold of the
saddle point $F_0$.

Let $s(0)\geq 0$ and $x(0)>0$. System \eqref{m0} has no periodic orbit
inside $\Omega$. Using the Poincar\'e-Bendixon Theorem
\cite{smithbook}, $F^*$ is a globally asymptotically stable
equilibrium point \cite{rad2}.

(3) If $\mu(s_{in})<m$, then  \eqref{m0} admits $F_0$ as the only equilibrium
point which is locally stable. As the omega limit set of any trajectory
have to be in the 2D compact and positively invariant set $\Omega$,
and since $F_0$ lies on the boundary of $\Omega$, $F_0$ must be
globally asymptotically stable by the Poincar\'e-Bendixson
Theorem \cite{rad1}.
\end{proof}

\begin{remark} \label{rmk1} \rm
$s^*$ is independent on the dilution rate $D$, contrarily to the
classical continuous reactor (chemostat) \cite{smithbook}.
Strict regulations regarding the maximum chemical oxygen demand allowed
in wastewater before they can be returned to the environment are
imposed by many governments. As COD at steady state
$(s^*)$ depends on the endegenous decay $(m)$ then it follows that
the used bacteria must have minimal decay constant.
\end{remark}

In the following we assume that  $ \mu(s_{in})>m$.

\subsection*{Observability}%\label{obs}
For the rest of this article, we shall use assumptions on the growth function
$\mu(\cdot)$ and the yield coefficient $Y$ of the classical Monod's system.
\begin{itemize}
\item[(A3)] $ \mu(\cdot)$ and $Y$ are known.
\end{itemize}

Our aim is to estimate on-line both parameter $m$ and unmeasured variable
$x$, based on the measurements of CDO $(s)$. Our system is not observable
if $s=0$ and/or $x=0$ that is why we proposed a set on which we are
in the ideal situation where the system is observable.
We considering the set
$$
\Omega=\big\{(s,x)\in\mathbb{R}_+^2: s>0,\; x>0, \;s+x<\frac{D}{m} s_{in} \big\}
$$
and deduced the following result.

\begin{proposition} \label{prop3}
Dynamics \eqref{m0} leaves the domain $ \Omega$ positively invariant.
\end{proposition}

Letting  $(s(0),x(0))\in \bar\Omega$ and considering the state vector
\[
\xi=\begin{bmatrix} s & \dot s &\ddot s \end{bmatrix}^{T}
=\begin{bmatrix} \xi_1 & \xi_2 &\xi_3 \end{bmatrix}^{T} \, ,
\]
one obtains the dynamics
\begin{gather*}
\dot \xi = A\xi + \begin{pmatrix} 0\\0\\ \varphi(y,\xi)
\end{pmatrix} \\
y=C\xi
\end{gather*}
with
\begin{align*}
  \varphi(y,\xi)
&=\Big(\xi_2-D(s_{in}-y)\Big)
\Big[\Big(\mu''(y)\mu(y)-(\mu'(y))^2\Big)
 \frac{\xi_2^2}{\mu^2(y)}+\frac{\mu'(y)}{\mu(y)}\xi_3+\mu'(y)\xi_2\Big]\\
&\quad - D \xi_3 + \frac{(\xi_3 +D \xi_2)^2}{\xi_2-D(s_{in}-y)}\,.
\end{align*}
and the pair $(A,C)$ in the Brunovsky's canonical form
\[
A=\begin{pmatrix} 0&1&0\\0&0&1\\0&0&0 \end{pmatrix}\quad
\text{and}\quad  C=\begin{pmatrix} 1 & 0 & 0
\end{pmatrix}.
\]
The unknown parameter $m$ and the unknown state variable $x$ are then made explicit
as functions of the state vector $\xi$:
\begin{gather*}
 m= l_{m}(y,\xi)=\mu(y)+\frac{\mu'(y)}{\mu(y)}\xi_2
 +\frac{D\,\xi_2+\xi_3}{-\xi_2+D(s_{in}-y)}\,, \\
 x= l_{x}(y,\xi)=\frac{-\xi_2+D(s_{in}-y)}{\mu(y)} .
\end{gather*}
One can notice that functions $\varphi(y,\cdot)$ and
$l_{m}(y,\cdot)$ are not well defined on $\mathbb{R}^{3}$, but
using the fact that $m^-\leq m \leq m^+$, we can consider
(globally) Lipschitz extension of function $l_m(y,\cdot)$ (and
then $\varphi(y,\cdot)$) away from the trajectories of the system,
as follows:
\begin{gather*}
\tilde l_{m}(y,\xi)
=   \max\Big(m^-,\min\Big(m^+, \mu(y)+\frac{\mu'(y)}{\mu(y)}\xi_2
+\frac{D\xi_2+\xi_3}{-\xi_2+D(s_{in}-y)}\Big)\Big)\,,\\
\begin{aligned}
 \tilde \varphi(y,\xi)
&= \Big(\xi_2-D(s_{in}-y)\Big)\Big[\Big(\mu''(y)\mu(y)
-(\mu'(y))^2\Big)\frac{\xi_2^2}{\mu^2(y)}+\frac{\mu'(y)}{\mu(y)}\xi_3
+\mu'(y)\xi_2\Big]\\
&\quad  - D \xi_3 + \Big(\xi_3 +D \xi_2\Big)
\Big(\tilde l_{m}(y,\xi)-\mu(y)-\frac{\mu'(y)}{\mu(y)}\xi_2\Big)\,.
\end{aligned}
\end{gather*}
Then one obtains a construction of a high gain observer.

\begin{proposition} \label{propobs1}
There exist numbers $a>0$ and $b>0$ such that the observer
\begin{equation}\label{obs1}
\begin{gathered}
 \dot{\hat\xi} = A\hat\xi +\begin{bmatrix}
0\\0\\ \tilde \varphi(y,\hat \xi)\end{bmatrix}
-\begin{bmatrix} 3\theta\\ 3\theta^{2}\\\theta^{3}
\end{bmatrix}(\hat\xi_{1}-y)\,,\\
 (\hat m,\hat x)=\big(\tilde l_{m}(y,\hat \xi)l_{x}(y,\hat \xi)\big)
\end{gathered}
\end{equation}
guarantees the convergence
\begin{equation}\label{convobs1}
\max\Big(|\hat m(t)-m|,|\hat x(t)-x(t)|\Big)
\leq a e^{-b\theta t}\|\hat\xi(0)-\xi(0)\|
\end{equation}
for any $\theta$ large enough and $t \geq 0$.
\end{proposition}

\begin{proof}
Consider a trajectory of dynamics \eqref{m0}. 
Define $K_{\theta}=-\begin{bmatrix} 3\theta&3\theta^{2}&\theta^{3} 
\end{bmatrix}^{T}$.
One can check that $K_{\theta}=-P_{\theta}^{-1}C^{T}$, where 
$P_{\theta}$ is solution of the algebraic equation
\[
\theta P_{\theta}+A^T P_{\theta}+P_{\theta}A=C^{T}C.
\]
Let $e=\hat\xi-\xi$ be the error vector. One has
\[
\dot e = (A+K_{\theta}C)e+\begin{bmatrix}
0\\0\\ \tilde \varphi(y,\hat\xi)-\tilde \varphi(y,\xi)
\end{bmatrix}
\]
where $\tilde \varphi(y,\cdot)$ is (globally) Lipschitz on $\mathbb{R}^{3}$.
Using the result in \cite{gauthier}, there exists two constants $\alpha>0$ 
and $\beta>0$ such that
$\|e(t)\|\leq \alpha e^{-\beta\theta t}\|e(0)\|$ for $\theta$ large enough.
Finally, functions $\tilde l_{m}(y,\cdot)$,  
$l_{x}(y,\cdot)$ being also (globally) Lipschitz on $\mathbb{R}^{3}$, 
one obtains the inequality \eqref{convobs1}. 
\end{proof}

\section{Numerical examples}

Consider a Monod's growth function where $\mu_{\rm max}=5$ and $k_{s}=1$ 
then  system \eqref{m0} becomes 
\begin{gather*}
 \dot s=-  \frac{5s}{1+s}x +2 (s_{in}-s) \,,\\
 \dot x=  (\frac{5s}{1+s} -m)\; x\,.
\end{gather*}

 In a first step we suppose that $m=0.5$, $D=2$ and we validate the stability 
results presented in Proposition \ref{prop2} (see Figure \ref{stability}).

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.48\textwidth]{fig2a} % comport1.eps
\includegraphics[width=0.48\textwidth]{fig2b} % comport2.eps
\end{center}
\caption{The $s-x$ behaviour for $s_{in}=1$ (left) which satisfies 
$\mu(s_{in})>m$ then $E^*$ is GAS and for $s_{in}=0.1$ (right) which 
satisfies $\mu(s_{in})<m$ then $E_0$ is GAS.} 
\label{stability}
\end{figure}

 In a second step, we suppose that parameter $m$ is unknown and we used 
the observer proposed in Proposition \ref{obs1} to reconstruct parameter 
$m$ and state variable $x$. Considered initial conditions are $s(0)=1$
 and $x(0)=1$ where $s_{in}=5$ and $D=2$.
By assumption A1, parameter $m$ is chosen, along with a priory bounds 
$m^-=0.1\leq m=0.5\leq m^+=1$. We have chosen a gain parameter $\theta=4$ 
that provides a small error on the estimation of the parameter $m$ and 
the state variable $x$ (see Figure \ref{observer}).

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.32\textwidth]{fig3a} % y.eps
\includegraphics[width=0.32\textwidth]{fig3b} % mhat.eps
\includegraphics[width=0.32\textwidth]{fig3c} % xhat.eps
\end{center}
\caption{Graph of observation $y$, estimation of parameter $m$ and
state variable $x$ in the case of noised measurements.}\label{observer}
\end{figure}


\subsection*{Conclusion}
We considered a simple mathematical model involving biomass growth 
on organic materials in a waste water treatment plants
 (membrane bioreactor process). Details of qualitative analysis are given. 
A high gain observer is proposed that permits the reconstruction 
of the biomass concentration and the endegenous decay based on on-line
 measurements of the chemical oxygen demand (CDO).

\subsection*{Acknowledgments}
The authors extend their sincere appreciations to the Deanship
of Scientific Research at King Saud University for its
funding of this Prolific Research group (PRG-1436-20).

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\bibitem{smithbook} H. L. Smith, P. Waltman; 
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\end{thebibliography}

\end{document}