2. Badr Petroleum Company(BAPETCO), Cairo, Egypt
Many researchers have focused on control methods for multipleinput and multipleoutput (MIMO) systems. MIMO with
A survey on theory and developments of decoupling control has been discussed by [4]. Some of the decoupling control methods found in literatures could be classified under the following topics, showing the interest that rose in the last years: [5][8] based their design on decomposing the problem into two parts: 1) decoupling the system using conventional decoupling to decrease interaction, and 2) designing the controllers using some decentralized method. [9][11] included work that discusses a methodology of multivariable centralized control based on the structure of inverted decoupling. This method has presented for general
Distillation column is one of the most important operation units in terms of control in chemical engineering. It is used in chemical industry to separate liquid mixtures into their pure components by the application and removal of heat [23]. It consumes a huge amount of energy in both heating and cooling operations. A robust control of distillation column's operation is important, due to reduction of energy consumption during operation, but the robust control implementation is difficult due to the phenomenon of interaction or coupling, which exists between various control loops of the distillation column [24], [25]. In addition, distillation column is usually nonlinear, multivariable, and it is subject to constraints and disturbances.
In this paper, a control system for a binary distillation column which, dynamically, behaves as a TITO system has been implemented. The emphasis is to determine a decoupling control, where specific inputs and outputs are paired. This decoupling control is achieved using fuzzy logic, to reduce the reliance on modelbased analysis. Also, a comparative study is done using different decoupling schemes with two PI controllers to control the two composition control loops of the binary distillation column. Based on conventional and inverting decoupling schemes, fuzzy and inverting fuzzy decoupling schemes are developed. To the authors' best knowledge, there are no results considering inverted decoupling using fuzzy logic. The proposed schemes are simple and easy in driving fuzzy rules.
This paper is organized as follows. The distillation column model description is given in Section Ⅱ. Section Ⅲ presents the design of decoupling control schemes. Section Ⅳ presents the two PI controller designs with stability analysis. The simulation results are given in Section V. Finally, the conclusion is given in Section Ⅵ.
Ⅱ. DISTILLATION COLUMN MODELThere are many types of distillation columns based on different classifications such as: batch, continuous, binary, multiproduct, tray and packed [25]. Consider an MIMO binary distillation column with a given feed shown in Fig. 1, which has five manipulated inputs
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Fig. 1 Binary distillation column with LV configuration. 
Different control configurations result from different pairing between controlled and manipulated variables [26]. The distillation column configuration to be studied is
$ \left[\begin{array}{c} X_D(s)\\ X_B(s) \end{array}\right] = G(s) \left[\begin{array}{c} L(s)\\ V(s) \end{array}\right] $  (1) 
where
$ \begin{align*} &G_{11}(s)=\frac{12.8e^{s}}{16.7s+1}, \; G_{12}(s)=\frac{18.9e^{3s}}{20s+1}\\ &G_{21}(s)=\frac{6.6e^{7s}}{10.9s+1}, \; G_{22}(s)=\frac{19.4e^{3s}}{14.4s+1}. \end{align*} $ 
Consider a TITO system described in transfer function matrix
$ Q(s)=G(s)D(s) $  (2) 
$ D(s)=G^{1}(s)Q(s) $  (3) 
and
$ \begin{align*} &Q(s)=\left[\begin{array}{cc} G_{11}(s)&0\\0& G_{22}(s) \end{array}\right]\\ &D(s)=\left[\begin{array}{cc} R_{11}(s)&R_{12}(s)\\R_{21}(s)&R_{22}(s) \end{array}\right]. \end{align*} $ 
To eliminate the effect of the coupling between loops in the distillation column, a conventional decoupling is added before the distillation column as shown in Fig. 2. The transfer function between inputs and outputs is given by
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Fig. 2 Distillation column with conventional decoupling scheme. 
$ \begin{align} &\left[\begin{array}{c} X_D(s)\\X_B(s) \end{array}\right]\\ &\;\ = \left[\begin{array}{cc} G_{11}(s)&G_{12}(s)\\ G_{21}(s)&G_{22}(s) \end{array}\right] \left[\begin{array}{cc} 1&R_{12}(s)\\R_{21}(s)&1 \end{array}\right] \left[\begin{array}{c} C_1(s)\\C_2(s) \end{array}\right] \end{align} $  (4) 
where
Equation (4) can be written as
$ \begin{align} &\left[\begin{array}{c} X_D(s)\\X_B(s) \end{array}\right]\\ &\; = \left[\begin{array}{cc} G_{11}(s)+G_{12}(s)R_{21}(s)&G_{12}(s)+G_{11}(s)R_{12}(s)\\ G_{12}(s)+G_{22}(s)R_{21}(s)&G_{22}(s)+G_{21}(s)R_{12}(s) \end{array}\right]\\ &\qquad\ \times \left[\begin{array}{c} C_1\\C_2 \end{array}\right]. \end{align} $  (5) 
To remove the interaction, the following terms in (5) should be equal zero
$ G_{12}(s)+G_{11}(s)R_{12}(s)=0 $  (6) 
$ G_{21}(s)+G_{22}(s)R_{21}(s)=0. $  (7) 
From (6) and (7)
$ R_{12}(s)=\frac{G_{12}(s)}{G_{11}(s)} $  (8) 
$ R_{21}(s)=\frac{G_{21}(s)}{G_{22}(s)}. $  (9) 
The disadvantage of this approach is that the decoupling matrix will be complex with
A fuzzy decoupling scheme consists of two elements: Fuzzy
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Fig. 3 Block diagram of fuzzy decoupling scheme. 
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Fig. 4 Triangular membership function. 
input
The transformation from a fuzzy set to a crisp number is done through defuzzification; center of gravity method is used for defuzzification.
The derivation of the fuzzy decoupling control rules can be obtained directly from the dynamic behavior of binary distillation column [27], as follows: increasing
1) When the input of a certain loop is medium (steady state region) then its compensation signal to the other loop will be medium. For example
If
2) The direction of the change in the compensation signal should follow the direction of the change in the input signal, taking into account the change in the output direction of the other loop. For example
If
In this rule, in which
If
In this rule, in which
Therefore, the fuzzy decoupling control rules can be easily obtained as shown in Tables Ⅰ and Ⅱ.
With inverted decoupling approach the decoupling transfer function complexity is independent of the system size [9]. As shown in Fig. 5, the decoupling matrix
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Fig. 5 Matrix representation of inverted decoupling for SISO. 
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Fig. 6 Distillation column with inverted decoupling. 
$ D(s)=\frac{U}{C}=\frac{D_d}{ID_oD_d} $  (10) 
$ D^{1}(s)=D_d^{1}(ID_oD_d). $  (11) 
From (3) and (11), one can get
$ D^{1}(s)=(D_d^{1}D_o)=Q^{1}(s)G(s). $  (12) 
Depending on the system size, there are different configurations with different cases for the choice of inverted decoupling elements; the proper configuration with proper case depends on the realizability of the decoupling elements. In this work, case 1 from configuration (12) [9] is used as follows:
Configuration (12): In this configuration elements
$ D_d=\left[\begin{array}{cc} d_{11}&0\\ 0&d_{22} \end{array}\right], \; D_o=\left[\begin{array}{cc} 0&do_{12}\\ do_{21}&0 \end{array}\right]. $ 
From (12), one can get
$ \begin{equation} D^{1}(s)= \left[\begin{array}{cc} \frac{1}{d_{11}}&do_{12}\\do_{21}&\frac{1}{d_{22}} \end{array}\right] = \left[\begin{array}{cc} \frac{g_{11}}{q_1}&\frac{g_{12}}{q_1}\\ \frac{g_{21}}{q_2}&\frac{g_{22}}{q_2} \end{array}\right]. \end{equation} $  (13) 
Thus
Two of these elements are set to unity, and so only two elements need to be implemented. Usually, the two elements chosen to be equal to one are the elements in the direct way, that is, the elements of the matrix
From Case 1, configuration (12) of inverted decoupling and the concept of fuzzy decoupling, the fuzzy inverted decoupling is developed as shown in Fig. 7. Inverted fuzzy decoupling consists of two elements: Fuzzy
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Fig. 7 Block diagram of inverted fuzzy decoupling. 
The controller design consists of two steps, 1) the decoupling elements based on Wood and Berry distillation column mathematical model are calculated, 2) the PI controller parameters for each loop are designed individually according to stability analysis.
A. Calculating Decoupling ElementsThe conventional and inverted decoupling consists of two terms
$ D_{12}(s)=R_{12}(s)=do_{12}(s)=\frac{18.9(16.7s+1)}{12.8(21s+1)}e^{2s} $  (14) 
$ D_{21}(s)=R_{21}(s)=do_{21}(s)=\frac{6.6(14.4s+1)}{19.4(10.9s+1)}e^{4s}. $  (15) 
Fuzzy and inverted fuzzy decoupling schemes are designed in the following. The suggested fuzzy rules of Fuzzy
For the fuzzy decoupling elements:
For the inverted fuzzy decoupling elements:
The design of PI parameters of each loop has been done using signal constraint block from MATLAB library (Simulink design optimization), by specifying the desired signal response and attaching the signal constraint block to the feedback signal to optimize the model response to known inputs. Simulink design optimization software tunes parameters in the model to meet specified constraints. Each loop has been designed individually with closing the other loop. Trial and error was used to reach the optimum performance by making little changes in the parameters. The parameters of the two PI controllers are shown in Table Ⅴ. The top composition PI controller is chosen as the direct action controller which means that the controller output rises if the measurement increases, but the bottom composition PI controller is chosen as the reverse action controller which means that the controller output drops if the measurement increases. PI controllers are chosen since the flow control is considered as a fast process, and using derivative action with a fast process usually leads to instability.
Two different methods are used to perform stability analysis. In the first method, multivariable systems are decomposed into
For system structural decomposition in [8] stability analysis is done without using decoupling element as follows:
The steady state transfer function of the proposed system "seen" from the free input
$ g_1=G_{11}(G_{12}K_2(I+G_{22}K_2)^{1} G_{21}). $  (16) 
Also, the steady state transfer function of the proposed system "seen" from the free input
$ g_2=G_{22}(G_{21}K_1(I+G_{11}K_1)^{1}G_{12}) $  (17) 
where
$ K_i=KP_i+\frac{KI_i}{s} $  (18) 
where
Applying Nyquist theorem to each individual loop transfer function in (16) and (17) which is multiplied by the transfer function of the controller (18) of the same loop is shown in Figs. 8 and 9. According to Theorem 1 in [8], each individual element of
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Fig. 8 Nyquist diagram for loop1. 
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Fig. 9 Nyquist diagram for loop2. 
$ N(1, g_1 K_1 )=0, \; N (1, g_2 K_2) =0. $ 
TITO tool is a tool developed by [8] to design decentralized PID controllers for multivariable systems with conventional decoupling, and specifications analysis and time response simulations can also be obtained using this tool. In this work TITO tool is used to obtain the structural direct Nyquist arrays (SDNA) for the proposed distillation column model with the designed conventional decoupling elements as shown in Fig. 10. The system with two designed decentralized PI controllers is stable as the Nyquist contour of each loop does not encircle point
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Fig. 10 DNA diagram using TITO tool (left) loop1 and (right) loop2. 
The decoupling schemes proposed in previous section have been tested on Wood and Berry binary distillation column mathematical model. Simulation work has been done in Simulink and Fuzzy toolbox in the environment of MATLAB.
The simulation results are obtained for the closed loop response of each loop, by changing the decoupling scheme with the same PI controller parameters. Figs. 11 and 12 show the response of top and bottom compositions for changing the top composition set point and keeping bottom composition without change (
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Fig. 11 Response of top composition with different decoupling. 
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Fig. 12 Response of bottom composition with different decoupling. 
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Fig. 13 Loop1 PI control signal. 
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Fig. 14 Loop2 PI control signal. 
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Fig. 15 Response of 
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Fig. 16 Response of 
In this paper, applying decoupling control using different decoupling schemes for a binary distillation column is presented, besides the PI controller is designed for each loop after decoupling. The key idea of the design procedure is to introduce decoupling elements in cascade with the distillation column to reduce the interactions that occur between their strategic variables. The main advantage of using decoupling control is the simplicity in determining the controller parameters since each loop is treated independently. The problem of mathematical models is solved by using fuzzy methods in implementing decoupling elements. Fuzzy and inverted fuzzy decoupling schemes are developed based on conventional and inverted decoupling ones. A comparison between fuzzy and inverted fuzzy decoupling schemes and other decoupling schemes is presented. The obtained simulation results show that the performances obtained with fuzzy and inverted fuzzy decoupling schemes are better than those obtained by using other decoupling schemes. Future research will extend the results of this paper to develop adaptive fuzzy and inverted adaptive fuzzy decoupling schemes.
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