IEEE/CAA Journal of Automatica Sinica  2018, Vol. 5 Issue(4): 869-877 PDF
Comparative Study of Different Decoupling Schemes for TITO Binary Distillation Column via PI Controller
1. Industrial Electronics and Control Engineering Department, Faculty of Electronic Engineering, Menoufia University, Menouf-23952, Egypt;
2. Badr Petroleum Company(BAPETCO), Cairo, Egypt
Abstract: This paper presents a comparative study of different decoupling control schemes for a two-input, two-output (TITO) binary distillation column via proportional-integral (PI) controller. The key idea behind this paper is designing two novel fuzzy decoupling schemes that depend on human knowledge, instead of the system mathematical model used in conventional decoupling schemes. Based on conventional and inverted decoupling schemes, fuzzy and inverted fuzzy decoupling schemes are developed. The control effect is compared using simulation results for the proposed two schemes with conventional decoupling and inverted decoupling. The proposed fuzzy decoupling schemes are easy to realize and simple to design, besides they have a good decoupling capability. Two methods are used to prove asymptotic stability of each loop and the entire closed-loop system by applying the proposed fuzzy decoupling-based PI controller. The Wood and Berry model of a binary distillation column is used to illustrate the applicability of the proposed schemes.
Key words: Binary distillation column     conventional decoupling     fuzzy decoupling     proportional-integral (PI) controller     stability analysis
Ⅰ. INTRODUCTION

Many researchers have focused on control methods for multiple-input and multiple-output (MIMO) systems. MIMO with $N$ input/output processes are characterized by significant interactions between their inputs and outputs. The control of MIMO processes is usually implemented using sets of single-input single-output (SISO) control loops. Interaction is a phenomenon that the loop gain in one loop depends on the loop gain in another loop, this interaction between controlled loops leads to deterioration in the control performance of each loop [1]-[3]. The control of MIMO processes requires proper input-output pairing and development of decoupling compensator unit. Decoupling control has emerged as one of the most popular techniques in the industrial process. The basic idea is to weaken, or even eliminate, the interactions between different input and output signals by decoupling methods. Then, the modern and advanced control methods could be applied to the separate SISO subsystems. And it is easy for industry engineers to use various advanced control strategies into practical works.

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 $n\times n$ processes by obtaining very simple inclusive expressions for the controller elements with a complexity independent from the system size. The last two methods are easy but require a complete model of the system. Usually operator's knowledge plays an important role in complex industrial processes. This knowledge may be integrated in the control system by the fuzzy inference system (FIS), independent of model-based analysis. FIS allows getting the control actions from input variables by using linguistic formulated rules, so their conception is more simplified. Fuzzy control has long been applied to industry with several important theoretical results and successful ones [12]-[17]. In recent years, many schemes of fuzzy decoupling have been developed in [18]-[22], such as the proposed fuzzy compensator of interactions in [19] acts as a feedforward controller. The outputs of the monovariable controllers are considered as the inputs of the fuzzy compensator, which produces compensation signals added to the single loop control signals for minimizing the effect of interactions. In [20], based on the conventional decoupling, fuzzy decoupling elements are developed to get the purpose of decoupling, the inputs of each fuzzy decoupling element are the output of PI controller and its differential, each element produces a compensation signal which is added to the single loop control signal for minimizing the effect of interactions.

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 model-based 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 MODEL

There are many types of distillation columns based on different classifications such as: batch, continuous, binary, multi-product, tray and packed [25]. Consider an MIMO binary distillation column with a given feed shown in Fig. 1, which has five manipulated inputs $U=[L\ V\ D\ B\ V_T]^T$. These are reflux flow rate, boil up vapour flow rate, top product flow rate, bottom product flow rate and overhead vapour flow rate, respectively, and five controlled outputs $Y$ $=$ $[X_D\ X_B\ M_D\ M_B\ P]^T$. These are top and bottom compositions, condenser and re-boiler levels and column pressure.

Different control configurations result from different pairing between controlled and manipulated variables [26]. The distillation column configuration to be studied is $LV$ control configuration as shown in Fig. 1. In this configuration, the manipulated variables are $\{L$ and $V\}$ and the controlled output variables are $\{X_D$ and $X_B\}$. The Wood and Berry distillation column is a TITO system that has been studied extensively. It is described by the following model [1]:

 $\left[\begin{array}{c} X_D(s)\\ X_B(s) \end{array}\right] = G(s) \left[\begin{array}{c} L(s)\\ V(s) \end{array}\right]$ (1)

where $G(s)=\left[\begin{array}{cc} G_{11}(s) & G_{12}(s)\\ G_{21}(s) & G_{22}(s) \end{array}\right]$ is the system matrix, and

 \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*}
Ⅲ. DECOUPLING CONTROL SCHEME DESIGN

Consider a TITO system described in transfer function matrix $G(s)$, the purpose of the decoupling matrix $D(s)$ is to transform the process as seen by the controller from $G(s)$, a full block matrix, to $Q(s)$ which is a diagonal matrix, and then each diagonal controller deals with each loop as SISO loop where

 $Q(s)=G(s)D(s)$ (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*}
A. Scheme 1 (Conventional Decoupling)

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

 \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 $C_1$ and $C_2$ are flow rate control output signals from each loop PI controller, and $D(s)=\left[\begin{array}{cc} 1 & R_{12}(s)\\R_{21}(s) & 1 \end{array}\right]$ is a decoupling matrix, where $R_{12}(s)$, $R_{22}(s)$ are the elements of the decoupling matrix.

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 $n\times n$ models.

B. Scheme 2 (Fuzzy Decoupling)

A fuzzy decoupling scheme consists of two elements: Fuzzy$_1$ and Fuzzy$_2$ as shown in Fig. 3, each one has two inputs and one output. Fuzzy$_1$ has inputs $(C_1, dX_B)$ and output signal $F_1$, Fuzzy$_2$ has inputs $(C_2, dX_D)$ and output signal $F_2$ where $C_1$ and $C_2$ are the outputs of two PI controller loops, also $dX_D$ and $dX_B$ are the rates of change of the top and bottom product composition, respectively. The output signals $F_1$ and $F_2$ are used as a flow rate compensation signal for the other loop, where $L=(C_1+F_2)$, and $V=(C_2+F_1)$. From the concept of fuzzy logic, each fuzzy decoupling element consists of the following four major components: Fuzzification, inference engine, rule base and defuzzification. Fuzzification comprises the process of transforming crisp values into grades of membership for linguistic terms of fuzzy sets. The membership function is used to associate a grade to each linguistic term. To the best of our understanding, the triangular membership function is simple to implement and fast for computation. So, three symmetrical triangular membership functions for Fuzzy$_1$ and Fuzzy$_2$ variables are named as Small$(S)$, Medium$(M)$, and High$(H)$ as shown in Fig. 4. The inference engine is considered as the decision-making-logic that determines how the fuzzy logic operations are performed based on the knowledge base (rule base), the inference engine of Fuzzy$_1$ and Fuzzy$_2$ is based on Mamdani inference system. The general form of the rule base is based on IF-THEN rule, for example

input$_1$ is $S$ AND input$_2$ is $S$ THEN output is $S$

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 $L$ will increase $X_D$ and $X_B$, and increasing $V$ will decrease $X_D$ and $X_B$ and vice versa, so changes in $L$ and $V$ will counteract each other. So:

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 $C_1$ is $M$ and $dX_B$ is ($S$ or $M$ or $H$) then $F_1$ is $M$.

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 $C_1$ is $S$ and $dX_B$ is $S$ then $F_1$ is $S$.

In this rule, in which $dX_B$ is small, this will be an indication that $C_2$ is high, i.e., opposite to the required compensation signal $F_1$. So, $F_1$ must be small.

If $C_1$ is $S$ and $dX_B$ is $H$ then $F_1$ is $M$.

In this rule, in which $dX_B$ is high, this will be an indication that $C_2$ is small, i.e., similar to the required compensation signal $F_1$. So, $F_1$ must be medium.

Therefore, the fuzzy decoupling control rules can be easily obtained as shown in Tables Ⅰ and .

Table Ⅰ
Fuzzy$_1$ Rules for Fuzzy Decoupling Control
Table Ⅱ
FUZZY2 RULES FOR FUZZY DECOUPLING CONTROL
C. Scheme 3 (Inverted Decoupling)

With inverted decoupling approach the decoupling transfer function complexity is independent of the system size [9]. As shown in Fig. 5, the decoupling matrix $D(s)$ is split into two matrices: a matrix $D_d(s)$ in the direct path (between controller outputs $C$ and process inputs $U$) and a matrix $D_o(s)$ in a feedback loop (between process inputs $U$ and controller outputs $C$). The matrix $D_d(s)$ must have only non-zero diagonal elements, since there must be only a direct connection for each process input. Note that these relationships are not required in the matrix $D_o(s)$. Additionally, since the signal flow direction in $D_o(s)$ is opposite to that of $D_d(s)$, the corresponding elements of $D_o(s)$ are opposite to that of $D_d(s)$, the corresponding elements of $D_o(s)$ must equal zero and are the same as transposing non-zero elements of $D_d(s)$. For instance, in a $2\times2$ process, if element $D_d(2, 1)$ is specified as a direct path between $U_2$ and $C_1$, there will be no feedback from $U_2$ toward $C_1$, and thus, the element $D_o(1, 2)$ must be zero. Following the decoupling representation shown in Fig. 6, the expression of the decoupling matrix $D(s)$ is obtained as follows:

 Download: larger image Fig. 5 Matrix representation of inverted decoupling for SISO.
 $D(s)=\frac{U}{C}=\frac{D_d}{I-D_oD_d}$ (10)
 $D^{-1}(s)=D_d^{-1}(I-D_oD_d).$ (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 (1-2) [9] is used as follows:

Configuration (1-2): In this configuration elements $D_d(1, 1)$, $D_d(2, 2)$, diagonal elements in $D_d$, and $D_o(1, 2)$, $D_o(2, 1)$, off diagonal elements of $D_o$, are selected, other elements equal zero.

 $D_d=\left[\begin{array}{cc} d_{11}&0\\ 0&d_{22} \end{array}\right], \; D_o=\left[\begin{array}{cc} 0&do_{12}\\ do_{21}&0 \end{array}\right].$

From (12), one can get

 $$$D^{-1}(s)= \left[\begin{array}{cc} \frac{1}{d_{11}}&-do_{12}\\-do_{21}&\frac{1}{d_{22}} \end{array}\right] = \left[\begin{array}{cc} \frac{g_{11}}{q_1}&\frac{g_{12}}{q_1}\\ \frac{g_{21}}{q_2}&\frac{g_{22}}{q_2} \end{array}\right].$$$ (13)

Thus $d_{11}=q_1/g_{11}$, $do_{12}=-g_{12}/q_1$, $do_{21}=-g_{21}/q_2$, $d_{22}$ $=$ $q_2/g_{22}$.

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 $D_d$. However, this case is only one of the four possible cases according to the two elements chosen to be equal to unity. Fig. 6 shows case 1 scheme which is applied to binary distillation column.

D. Scheme 4 (Inverted Fuzzy Decoupling)

From Case 1, configuration (1-2) 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$_1$ and Fuzzy$_2$, each one has two inputs and one output. Fuzzy$_1$ has inputs $(U_1, dX_B)$ and output signal $F_1$, Fuzzy$_2$ has inputs $(U_2, dX_D)$, and output signal $F_2$, where $U_1=L= (C_1+F_2)$ and $U_2=V$ $=$ $(C_2+F_1)$. Each output signal ($F_1$ or $F_2$) is used as a compensation signal for the other loop. The membership functions of input and output variables for Fuzzy$_1$ and Fuzzy$_2$ are shown in Fig. 4, also according to the dynamic behavior of distillation column mentioned in previous section. The inverted fuzzy decoupling control rules are shown in Tables Ⅲ and . Inverted fuzzy decoupling has the advantage of simplicity in design for $n\times n$ processes and using fuzzy logic in designing decoupling elements.

Table Ⅲ
FUZZY1 RULES FOR INVERTED FUZZY DECOUPLING CONTROL
Table Ⅳ
FUZZY2 RULES FOR INVERTED FUZZY DECOUPLING CONTROL
Ⅳ. CONTROLLER DESIGN

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 Elements

The conventional and inverted decoupling consists of two terms $D_{12}(s)$ and $D_{21}(s)$. From (1), (8), (9) and (14), these terms can be written as

 $D_{12}(s)=R_{12}(s)=do_{12}(s)=\frac{18.9(16.7s+1)}{12.8(21s+1)}e^{-2s}$ (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$_1$ and Fuzzy$_2$ are shown in Tables Ⅰ-, three symmetrical triangular membership functions for the Fuzzy$_1$ and Fuzzy$_2$ inputs, and output variables are chosen as shown in Fig. 4. The low and high limits of the universe of discourses for these variables are as follows.

For the fuzzy decoupling elements: $C_1= [0.18\ 0.33]^T$, $C_2$ $=$ $[-0.042 \ 0]^T$, $dX_B= [-0.2 \ 0.1]^T$, and $dX_D= [-0.025$ $0.025]^T$, $F_1= [0.01 \ 0.172]^T$, and $F_2= [-0.05 \ 0]^T$.

For the inverted fuzzy decoupling elements: $U_1= [0.1$ $0.22]^T$, $U_2= [0 \ 0.09]^T$, $dX_B= [-0.012 \ 0.006]^T$, and $dX_D$ $=$ $[-0.01 \ 0.08]^T$, $F_1= [0 \ 0.01]^T$, and $F_2= [0 \ 0.12]^T$.

B. PI Parameters Design

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.

Table Ⅴ
PI CONTROLLER PARAMETERS
B.1. Stability Analysis

Two different methods are used to perform stability analysis. In the first method, multivariable systems are decomposed into $n$ SISO subsystems using system structural decomposition, and then stability analysis is performed for each loop independently. In the second one, stability analysis is performed using TITO tool.

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 $L$ to the free output $X_D$ is

 $g_1=G_{11}-(G_{12}K_2(I+G_{22}K_2)^{-1} G_{21}).$ (16)

Also, the steady state transfer function of the proposed system "seen" from the free input $V$ to the free output $X_B$ is

 $g_2=G_{22}-(G_{21}K_1(I+G_{11}K_1)^{-1}G_{12})$ (17)

where $K_1$, $K_2$ are the transfer functions of PI controllers for loop$_1$ and loop$_2$, respectively, and formulated as

 $K_i=KP_i+\frac{KI_i}{s}$ (18)

where $i=1, 2$.

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 $G(s)$ and its SISO independent subsystems do not have poles in the right hand plane, so the system with the decentralized PI controller is stable as the Nyquist contour of each equivalent open loop transfer function does not encircle point $(-1, 0)$, i.e.,

 $N(-1, g_1 K_1 )=0, \; N (-1, g_2 K_2) =0.$

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 $(-1, 0)$.

 Download: larger image Fig. 10 DNA diagram using TITO tool (left) loop1 and (right) loop2.
Ⅴ. SIMULATION RESULTS

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 ($r_1=1$, $r_2=0$) at simulation time (0 to 500). From the results, the performance of the top composition obtained with inverted fuzzy decoupling is better than those obtained with other schemes as shown in Fig. 11. Moreover, bottom composition has smaller change with inverted fuzzy decoupling than others as shown in Fig. 12. The control signals of two PI controllers are shown in Figs. 13 and 14. The final control signals $(U_1, U_2)$ are shown in Figs. 15 and 16.

 Download: larger image Fig. 11 Response of top composition with different decoupling.
 Download: larger image Fig. 12 Response of bottom composition with different decoupling.
 Download: larger image Fig. 15 Response of $U_1$.
 Download: larger image Fig. 16 Response of $U_2$.