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\title{Modeling and Output Feedback Distributed Control for an Absorption Packed Column}
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             \author[1]{M.  Selatnia}

             \author[2]{illoul rachid  }

             \author[3]{Et M.S.  Boucherit}

             \affil[1]{  cole nationale polytechnique dalger}

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\date{\small \em Received: 10 December 2013 Accepted: 1 January 2014 Published: 15 January 2014}

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\begin{abstract}
        


This work consists of modeling, simulation and multiple models control of an industrial absorption packed column designated to remove CO2, from natural gas. The multiple models approach is an elegant way of turning nonlinear problems into linear ones. In this paper, we used the output feedback distributed control (ODC) coupled with local linearization of the model of the absorption packed column. We compared the results with those obtained with the traditional PID control and the results were satisfactory.

\end{abstract}


\keywords{multiple models, PDC control, methyldiethanolamine (MDEA) absorption packed column, LMI, lyapunov function.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
he absorption packed column is a physicochemical separation unit largely used in the chemistry industry. It consists of a tube where we send gas mixtures in order to separate one or more compounds from the principal mixture. It is largely used for the separation of acid gases (CO 2 , H 2 S) from natural gas.\par
The model presented in this paper is a dynamic model of the absorption packed column and consists of a set of non linear partial differential equations; it is elaborated starting from considerations on CO 2 and MDEA mass balance in gas and liquid phases and considers also the energy balance \hyperref[b2]{[2]}. We finally obtain a non linear distributed parameters system.\par
Few studies were carried out on modelling and controlling the absorption column. Crosby and Durbin \hyperref[b3]{[3]} studied the performance of a state controller. Roffel \hyperref[b4]{[4]} developed a sub-optimal output controller with state inequality constraint. Darwish and Fantin \hyperref[b5]{[5]} used a decentralized control with pole placement. Petrovsky \hyperref[b6]{[6]} developed a multivariable PI regulator. Najim \hyperref[b7]{[7]} developed a self-adjusting regulator in the case of CO 2 absorption by a diethanolamine solution and also multilevel learning control \hyperref[b8]{[8]}. It took again the problem later on with predictive control \hyperref[b9]{[9]}.\par
Few studies have also been published concerning the modelling and simulation of CO 2 absorption by aqueous solutions of MEA or MDEA on pilot and industrial columns \hyperref[b10]{[10]}\hyperref[b11]{[11]}\hyperref[b12]{[12]}.\par
For the model developed in our study, it seemed interesting for us to use the multiple models approach for the control of the absorption packed column because it enables us to obtain good performances for complex dynamics processes. We develop in first stage the PID regulation to compare the performances of the classical techniques with the performances of the Takagi-Sugeno multiple model approach. 
\section[{II.}]{II.}\par
Modeling and Open Loop Simulation of the Industrial Absorption Packed Column\par
The absorption packed column presented here is located at Khrechba and is part of the In Salah Gaz project \hyperref[b17]{[17]}, it removes CO 2 from natural gas by using an aqueous solution of methyldiethanolamine (MDEA) as a washing liquid, It is a packed type column measuring 8 meter height and 4 meter in diameter with Pall rings to improve the surface of contact between phases. For a better elimination of CO 2 from the natural gas, the liquid flow (water+ MDEA) is counter-current with gas flow. The working pressure and temperature are respectively 71.5 bar at 55°C \hyperref[b2]{[2]}.\par
At contact between liquid and gas phase occurs on the surface of the Pall rings, CO 2 passes from the gas phase to the liquid phase; this diffusion is accelerated by chemical reaction of CO 2 with the MDEA in the liquid phase. The liquid flow (water+ MDEA) and the CO 2 concentration in the gas mixture are respectively selected as control variable and output variable. In order to simplify the model, the following assumptions are done \hyperref[b2]{[2,}\hyperref[b13]{13]}:\par
? There is no resistance in gas phase\par
? The reaction between CO 2 and MDEA is fast (Ha>5)\par
? Axial dispersion is negligible in the gas phase and the liquid phase\par
? The MDEA does not pass in gas phase The mass balance on an elementary section dz of the column for CO 2 in the gas phase is written \hyperref[b2]{[2,}\hyperref[b13]{13,}\hyperref[b14]{14]}:\par
Quantity of aqueous solution at input Z = quantity of aqueous solution at the output (z+dz) + quantity of aqueous solution transferred from the liquid phase to the gas phase + accumulation. 
\section[{Which gives:}]{Which gives:}dz dt dC S dz S C G C G Ag dz z Ag z Ag + + = + ? ) ( ) (\textbf{(1)}\par
Where G (m3/s) is the volumetric gas flow, ? the CO 2 flow transferred from the gas phase to the liquid phase, S the section of the column and C Ag (mol/m 3 ) the CO 2 concentration in the gas phase. Given U G =G/S (m/s) the gas flow velocity, we obtain then:dt dC dz dC U Ag Ag g ? = + ?\textbf{(2)}\par
The chemical reaction between CO2 and the MDEA is \hyperref[b10]{[10]}\hyperref[b11]{[11]}\hyperref[b12]{[12]}:? + + ? + + 3 4 2 1 2 3 2 1 2 HCO NCH R R O H NCH R R CO (3)\par
The reaction rate rA has the following form \hyperref[b14]{[14,}\hyperref[b15]{15]}:BL AL A C C k r = (\textbf{4})\par
Where k is the constant for reaction rate \hyperref[b14]{[14,}\hyperref[b15]{15]}:?? = 2,9610 5 exp( ? 5332 .8 ?? )\textbf{(5)}\par
CAL is the CO 2 concentration in the liquid phase and CBL the MDEA concentration in the liquid phase. The mass balance for CO 2 in the liquid phase gives finally:] [ BL AL C kC = ?\textbf{(6)}\par
Which means that the totality of CO 2 transferred to the liquid phase reacts with the MDEA.\par
The mass balance for the MDEA in the liquid phase gives:dz dt dC S dz S C C k C L C L Bl Bl Al dz z Bl z Bl ? ? = + ] [ ) ( ) (\textbf{(7)}\par
Where L is the volumetric liquid flow. By taking account of (5) and noting by UL = L/S (m/s) the mean liquid flow velocity, we obtain:dt dC dz dC U Bl Bl l = ? ? (8)\par
Our absorption packed column is finally described by the following set of partial derivative equations:? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? = + ? ? t C z C U t C z C U BL BL L Ag Ag g ? ?\textbf{(9)}\par
The procedure to compute flow ? is given in \hyperref[b2]{[2]} according to \hyperref[b14]{[14]}\hyperref[b15]{[15]}\hyperref[b16]{[16]}.\par
We have finally to consider the boundary conditions which for gas phase are the CO 2 concentration at the column bottom or input concentration CAGe and for liquid phase the MDEA concentration at the column top or input concentration CBLe.? ? ? ? ? = = = = BLe h z BL Age z Ag C C C C 0 (10)\par
Chemical reactions within the industrial column induces a strong heat emission and the appearance of a temperature gradient throughout the column; the temperature variation is approximately 5°C between the input and the output of the column, which leads us to establish an energy balance in order to describe the temperature changes which affects the various concentrations along the column \hyperref[b16]{[16]}:( ) ( ) [ ] ? ? ? ? ? ? ? ? ? ? ? = ? ? + ? ? ? ? ? = ? ? ? ? ? ? ? + ? ? ? ? t T T T a.h r Î?"H C cp z T U t T C cp T T a.h z T U l g l l g A r i i l i l l L g i i g i g g l l g g g 1\textbf{(11)}\par
With: C i g : concentration in gas phase at the interface (mol/m 3 ) C i l : concentration in liquid phase at the interface (mol/m 3 )cp i g :\par
Specific heat in the gas phase at the interface (J/mol.K) h g/l : coefficient of heat transfer (convection) (J/m2.K.s) T l : Liquid temperature (K) Tg: Gas temperature (K) Î?"Hr: enthalpy of the reaction (J/mol) cp i l : specific heat in the liquid phase at the interface (J/mol.K)\par
We finally take into account the boundary conditions for the temperature which are the temperatures for gas and the liquid at the column input.? ? ? ? ? ? ? = ? ? = = ? ? = = = = = 0 0 0 0 h z g le h z l z l ge z g z T , T T z T , T T (12) b) Model validation\par
A test was carried out on our industrial absorption column to compare the output CO 2 concentration given by the model with the real one and this for a step input variation of 10 t/h. the data were collected on a horizon of 6800 seconds. The results are grouped in figure \hyperref[fig_1]{2} where we represent respectively, the flows of MDEA and gas at the output and then the concentrations of CO 2 at the column output either experimental or given by the model \hyperref[b2]{[2]}.  Simulations show that the system is stable. It presents a dead time in response to a step input disturbance on the CO 2 concentration due to the gas propagation along the absorption column. 
\section[{III.}]{III.} 
\section[{Pid Regulation of the Industrial Absorption Packed Column}]{Pid Regulation of the Industrial Absorption Packed Column}\par
We apply a PID regulation to the dynamic model of our absorption packed column. We choose a sampled control with a sampling period of 10 seconds. The reference for the input CO 2 concentration is 0.25mole \%, which corresponds to a concentration of  The simulation results are satisfactory; the PID regulator cancels the permanent error and ensures a quick response due to the derivative action. The regulation shows a net asymmetrical behaviour between responses to positive and negative step input disturbances due to the strong non linearity of the relationship between the input and the output IV. 
\section[{Multiple Models Control of the Industrial Absorption Packed Column a) Introduction}]{Multiple Models Control of the Industrial Absorption Packed Column a) Introduction}\par
The multiple models is a non linear modeling technique which allows to achieve a good compromise between precision and model complexity. In the light of the numerous work related to it in recent years \hyperref[b18]{[18]}, \hyperref[b19]{[19]}, \hyperref[b20]{[20]}, it arises a great interest, especially in applications dealing with simulation and control. It can also be seen as a particular fuzzy modeling technique \hyperref[b21]{[21]}, \hyperref[b22]{[22]}, corresponding to a Takagi Sugeno (TS) approach \hyperref[b23]{[23]}. A TS model is a composed of a finite number of linear models connected with non linear functions called membership functions and verifying the convex mapping property (i.e. they are non negative and their sum is equal to 1). It allows us to solve various problems of control, observation and diagnosis for non linear systems with linear techniques.\par
The approach associated with multiple models in control is known as the Parallel Distributed Compensation (PDC) \hyperref[b24]{[24]}. This method is based on a set of linear controllers designed for each linear model, and stability of the overall closed loop is guaranteed via a Lyapunov function common to all the linear models.\par
In this paper, we identify the absorption packed column as a multiple model of the TS type and proposes a control based on Output feedback Distributed Control (ODC). 
\section[{b) Problem formulation i. The Multiple model approach}]{b) Problem formulation i. The Multiple model approach}\par
The multiple models have three basic structures: Coupled states (TS), uncoupled states \hyperref[b25]{[25]} and hierarchical structure. The coupled states Structure (TS) is the most popular in the analysis and synthesis of the multiple models. It is written in the following form:? ??(??) = ? ?? ?? (??(??)) ?? ??=1\par
(?? ?? ??(??) + ?? ?? ??(??)) ??(??) = ? ?? ?? (??(??))?? ??=1 ?? ?? ??(??)\textbf{(13)}\par
??(??) ? ?? ?? being the state vector, ??(??) ? ?? ?? the input vector, ??(??) ? ?? ?? the output vector z(t)? R is the decision variable or premises and the matrices ?? ?? ? ?? ?? ×?? , ?? ?? ? ?? ??×?? , ???? ?? ?? ? ?? ??×?? ? ?? = 1, ? . . , ?? are constant and supposed to be known.\par
The activation function or membership function ?? ?? (??(??) determines the degree of activation of the i th local model. It allows a progressive passage from this model to the other close local models. These functions can depend of the measurable variables of the system (the input and output signals) or of the non measurable variables of the system (the states). They can be of triangular or Gaussian form and satisfy the properties of convex mapping:? ??(??) = ? ?? ?? (??(??)) ?? ??=1 (?? ?? ??(??) + ?? ?? ??(??)) ??(??) = ? ?? ?? (??(??)) ?? ??=1 ?? ?? ??(??)\textbf{(14)}\par
The multiple models can be viewed as universal approximators since any nonlinear system can be approximated by a multiple models representation with sufficient accuracy and this simply by increasing the number of sub-models. In practice, a reduced number of sub-models can be sufficient to obtain a satisfactory approximation, and we can use the tools of linear systems analysis to achieve this goal.\par
There are three approaches largely used in the literature allowing us to obtain a TS model: Identification, transformation by nonlinear sectors \hyperref[b26]{[26]}, or linearization. This last one is used in this work (figure  {\ref 7}).\par
Figure  {\ref 7} : PDC control for TS multiple models Using convex analysis for regulator synthesis, the multiple models allows us to obtain control laws by the simultaneous resolution of a finite number of Linear Matrix Inequalities (LMI). In this case, the number of LMI inequalities is polynomial with respect to the number of local models. Thus, it is advisable to minimize the number of local models to limit the conservatism of the method.\par
In the case of TS multiple models, this technique of regulator synthesis corresponds to the PDC method, It supposes that all the linear sub-models are at least stabilizable. Subsequently, they will also be supposed controlled.\par
Given the TS model given by equation ( \hyperref[formula_15]{13}), a control law resulting from PDC synthesis will thus be the combination of linear control laws for each sub-model, given by: ??(??) = ? ? ?? ?? ???(??)??? ?? ??(??) ?? ??=1 \hyperref[b14]{(14)} Applying this control law to the TS multiple model, we obtain in closed loop:? ??(??) = (?? ?? ? ?? ?? ?? ?? )??(??) ??(??) = ?? ?? ??(??)\textbf{(15)}\par
Or, in a more explicit way:\{??(??) = ? ? ?? ?? ???(??)? ?? ?? =1 ?? ??=1\par
?? ?? ???(??)???? ?? ? ?? ?? ?? ?? ???(??) \hyperref[b16]{(16)} The stability conditions for the closed loop system amounts to find a control gain Fj such that the derivative of the candidate Lyapunov function associated with the system is negative. Stabilizing the system thus amounts to solve the following problem: Find a positive definite matrix P and Fi matrices, i=1. , M such that:\par
(?? ?? ? ?? ?? ?? ?? ) ?? ?? + ??(?? ?? ? ?? ?? ?? ?? ) < 0 \hyperref[b17]{(17)} We notice that this inequality is non linear with respect to P and Fi. By using the congruence of the symmetrical full row matrix:?? = ?? ?1 (\textbf{18})\par
We get:???? ?? ?? + ?? ?? ?? ? ???? ?? ?? ?? ?? ?? ? ?? ?? ?? ?? ?? < 0 (\textbf{19})\par
By using the bijective variable change?? ?? = ?? ?? ??, ?? = 1, ? , ??,\textbf{(20)}\par
The problem becomes LMI in variables X and Mi.?? ???? = ???? ?? ?? ? ?? ?? ?? ?? ?? ?? + ?? ?? ?? ? ?? ?? ?? ?? < 0 (\textbf{21})\par
We finally get:? ? ?? ?? ?? ?? =1 ?? ??=1 ???(??)??? ?? (??(??))?? ???? (\textbf{22})\par
And we can express the following result:\par
Theorem 4.1 \hyperref[b24]{[24]} Given a continuous TS model, the PDC control law ( \hyperref[formula_16]{14}) and the ?ij, if it exists a positive definite matrix X and Mi matrices, such that ( \hyperref[formula_24]{21}) is satisfied for all i,j=1,?, M, then the closed loop is overall asymptotically stable. Moreover, if the problem has a solution, the gains of the PDC control are given by:?? ?? = ?? ?? ?? ?1 (23)\par
And the PDC control is:??(??) = ? ? ?? ?? ?? ??=1 ???(??)??? ?? ??(??)\textbf{(24)}\par
If ?? ?? = ??, ??? = 1, ? , ??, then we define a linear control law. In practice, to determine the matrix P and the control gain F i , we have to solve \hyperref[b21]{(21)} for all i,j=1,?,M. In the particular case where the multiple models verify the positive co linearity of the input matrices, thais:?? ?? = ??, ??? ? ?? ?? (\textbf{25})\par
The closed loop multiple models system of (  {\ref 16}) is rewritten without the crossing terms B i F j :??(??) = ? ?? ?? ?? ??=1 ???(??)?(?? ?? ? ?? ?? ?? ?? )??(??)\textbf{(26)}\par
The stability conditions of theorem 4.1 reduce then to the stability of the dominant models: P>0 ,(?? ?? ? ?? ?? ?? ?? )?? + ??(?? ?? ? ?? ?? ?? ?? ) < 0, ??? = 1, ? , ??\textbf{(27)}\par
Substituting Bi by B, the control law leads to similar conditions. Year 2014 c) Multiple models identification\par
The structural identification of a multiple models representation consists in the determination of the local models structures and the operation zones (or validity zones) for each local model \hyperref[b25]{[25]}. The local models can be of various structures but in general we use simple structures, such as linear models.\par
The identification leads to a family of functions parameterized by the parameters vector i ? defining the structure of the i th local model, and the parameters vector i ? characterizing the zone of validity of this local model. The parametric estimate consists in determining for each local model i the parameters vector:T T T i i i ? ? ? ? ? = ? ?\par
The parametric estimation (also called training) is done on the basis of minimization of a functional binding the inputs and outputs system to the characteristics parameters of the model.?? ?? = 1 2 ? (?? ?? (??) ? ?? ?(??)) 2 ?? ??=1 = 1 2 ? ??? ?? (??) ? ? ð??"ð??" ?? ?? ??=1 ???(??), ?? ?? ? ð??"ð??" ?? ???(??), ?? ?? ?? 2 ?? ??=1\textbf{(28)}\par
In order to simplify the model, we choose linear sub-models of ARX type (auto regressive with exogenous inputs) with 3 inputs (system MISO):? Liquid Flow Ul [ton/h] ? Gaz flow Ug [ton/h] ? Input CO 2 concentration Cge [mol\%]\par
The choosen local models are second order ones, they are written in the following form: The decision variable z(t) is in our case the state vector x(t) output gas concentration of the industrial column C gs, ??(??) = ??? ð??"ð??"?? (??) ?? ð??"ð??"?? (?? ? 1)? ?? . A multiple models with four sub-models can easily be obtained in the form:?? ð??"ð??"???? (?? + 1) = ?? ??1 ?? ð??"ð??"?? (??) + ?? ??2 ?? ð??"ð??"?? (?? ? 1) + ?? ?? ?? ?? (??) + ?? ?? ?? ð??"ð??" (??) + ?? ?? ?? ð??"ð??"?? (??) + ?? ?? + ?? ?? (?? + 1)\textbf{(29)}??(??) = ? ?? ?? 4 ??=1 ???(??)?(?? ?? ? ?? ?? ?? ?? )??(??)\textbf{(31)}\par
With the memberships functions µ i (x(t)) either triangular or Gaussian. The parameters of the four models are:     ?? 1 = ? ?0. 
\section[{Discussions}]{Discussions}\par
The oscillations of the output CO 2 concentration for the industrial column are mainly due to fuzzy control, but the operating point of is quickly reached in less than 50 seconds . We note on all the curves  {\ref (5, 6 and 11-16)} that disturbance is always rejected with both PID and ODC regulation. But comparison of the peak values as well as the oscillations show us that ODC control acts more quickly than PID control. 
\section[{VI.}]{VI.} 
\section[{Conclusion}]{Conclusion}\par
In this paper, a TS multiple models obtained by linearization was used for the regulation of an industrial absorption packed column used for gas washing. We take a reduced number of submodels, four, to ease identification. The results obtained with output distributed feedback (ODC) are better than those by classical PID regulation, for a method which is not significantly complicated. Further investigations will be undertaken in the use of multiple models in control, multiple observers and multiple models in systems diagnosis.\begin{figure}[htbp]
\noindent\textbf{1}\includegraphics[]{image-2.png}
\caption{\label{fig_0}Figure 1 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{2}\includegraphics[]{image-3.png}
\caption{\label{fig_1}Figure 2 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{34}\includegraphics[]{image-4.png}
\caption{\label{fig_2}Figure 3 :Figure 4 :}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{56}\includegraphics[]{image-5.png}
\caption{\label{fig_3}Figure. 5 :Figure 6 :}\end{figure}
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