1. Introduction

Distillation is the process to separate chemical substances most used in the industry, with the petrochemical (petroleum products) [1] and food (alcoholic beverages production) industries being the most important due to the current people lifestyle in which the daily use of petroleum fuels is essential for both transport and energy generation.

Fractional distillation is used to separate homogeneous liquid mixtures in which the difference between the boiling points of the components is less than 25°C. Each of the separate components is called fractions. In general, there are two operating modes: continuous and batch. In the continuous mode, the feeding of the liquid

mixture and the extraction of the distilled product are carried out continuously. In the batch distillation, the mixture is initially deposited in the boiler; at the end of the process, the distillate and bottom product are extracted.

present a discrete-time D-LPV observer to estimate sensors and actuators states and

Fuzzy Logic Modeling and Observers Applied to Estimate Compositions in Batch Distillation…

In [14], the authors present a pair of extended Luenberger observers (complete and reduced order) to estimate the compositions of a multicomponent mixture from temperature measurements of the distillation column plates. The observers' gains are calculated from the location of the closed-loop eigenvalues using a math-

In [15], a full-order nonlinear observer is presented to estimate the composition and temperatures of a distillation column. A nonlinear model obtained by the mass balance in each plate of the column is used, resulting in a set of high-order differential equations with nonlinear terms. The observer is validated in simulation to demonstrate his behavior and his robustness. The parametric representation or identification is another methodology used to estimate certain variables of the

The difficulty of designing and implementing the observers lies mainly in the nonlinear dynamics of the distillation column; thus, having a linear system would facilitate the design of observers and controllers to implement control strategies such as fault detection and diagnosis systems and automatic control and tolerant control in order to improve the performance and safety of the process, as well as the

The Takagi-Sugeno fuzzy modeling is a tool to model and control complex systems using a nonlinear system decomposition in a multi-model structure formed by linear and not necessarily independent and fuzzy logic models [18, 19], where the representation of the nonlinear system is achieved by a weighted summation of the whole subsystems. The Takagi-Sugeno representation provides a solution to solve the problems in the design and implementation of control strategies for

In [20], the authors propose a methodology to design control techniques for systems represented in the Takagi-Sugeno form. In [21] the identification of a model of a binary distillation column, based on fuzzy models, taking into account 6 system inputs and 2 outputs for 64 rules is presented. The model is simulated using

Authors in [22] present a controller of the molar composition of the distilled and bottom products for a binary distillation column using neural networks and fuzzy logic (ANFIS) based on a 2 2 MIMO system. In [23] an adaptive PID controller based on Takagi-Sugeno modeling to control the distilled and bottom products of a

Due to the close relationship between the fuzzy representation of nonlinear systems and the theory of linear matrix inequalities, different works based on both techniques have been developed, allowing to find solutions to the calculations corresponding to the observer and controller gains and the Lyapunov stability analysis. In [24], a methodology to design observers and controllers for a fuzzy

The main contribution of this work is the design of a fuzzy observer based on a Takagi-Sugeno model to estimate the molar compositions and temperatures of the light component in each plate of a binary distillation column. The observer performance is validated for applications such as system monitoring and fault detection.

The objective of a batch distillation process is to separate two or more elements

2. Takagi-Sugeno fuzzy model for a batch distillation column

from a mixture, where the most volatile element is obtained as the distilled

faults, using the H∞ approach applied to the estimation error.

distillation columns, as presented in [16, 17].

DOI: http://dx.doi.org/10.5772/intechopen.83479

quality of the distilled product.

real data to validate its performance.

binary distillation column is presented.

nonlinear systems.

system is proposed.

11

ematical software.

The batch operation is mainly used to separate small amounts of mixture, to obtain different qualities of the distilled product from the same mixture, or to separate multicomponent mixtures.

A batch distillation column is not operated using constant parameters; the control actions are continuously adjusted according to the state of the distillation; therefore, monitoring and controlling all the variables of the process are essential to improve the quality and quantity of the distilled product, as well as to guarantee the safety of the process and the operators. To fulfill or facilitate this objective, it is necessary to implement control techniques such as models, observers, and controllers.

In the literature, modeling and control techniques such as estimators, observers, fault detection systems, and control systems are applied to distillation columns in order to obtain a better analysis and understanding of the dynamics of the process, improving the quality of the distilled product and enhancing the user safety, among other tasks.

Distillation column simplified models present the basic principles of the process and its operation taking into account several considerations to describe the dynamics of the system in a simpler but understandable form. Authors in [2] present a simplified model of a binary distillation column, based on the liquid-vapor equilibrium of the binary mixture and the mass balance considering all the elements of the distillation column as plates.

Authors in [3] design a model based on the existence of the liquid and vapor molar fluids that vary in each column plate; the compositions of the bottom product and the distilled product are estimated using a dynamic model based on the mass and component balances. In [4] a low-order model of an ideal multicomponent distillation using the theory of nonlinear wave propagation is presented. Authors in [5] present a low-order model for a reactive multicomponent distillation column, in addition to designing a predictive control to obtain the best quality of the distilled product.

Rigorous models are more complete because they represent plate by plate the element balance of phases in each element of the distillation column (boiler, condenser, and plates). In these models, the mathematic expressions are determined by a series of differential equations given by mass, light component, or energy balances depending on the application, the control strategy, or the operation type. An important advantage of the rigorous modeling is the high resolution of the dynamics, having the disadvantage of combining a greater number of variables and expressions that make difficult the design, simulation, and implementation of controllers.

In [6], a model based on neural networks is presented in order to optimize the energy efficiency in a binary distillation column. Authors in [7] present a model of a binary distillation column based on neural networks. The neural network training and validation are performed using real data from a nine-plate pilot plant for a mixture of methanol and water. Authors in [8] present the simulation and optimization of a rigorous model for a batch reactive distillation column. Authors in [9] present the design and simulation of a discrete Kalman filter to estimate the molar compositions of the light component in a batch distillation column.

Generally, the light component composition measurement is performed offline using expensive instruments, so the implementation of state observers to estimate online this composition has become a frequent and important task. Authors in [10–12] present high-gain observers to estimate the light component composition in all the distillation column plates from the measurement of the temperature of some plates and the column actual inputs.

Due to the different distillation types and their mathematical representation, there are different types of observers for different applications. In [13] the authors
