Elsevier

Journal of Process Control

Volume 22, Issue 9, October 2012, Pages 1655-1669
Journal of Process Control

Infinite-dimensional LQ optimal control of a dimethyl ether (DME) catalytic distillation column

https://doi.org/10.1016/j.jprocont.2012.06.018Get rights and content

Abstract

This contribution addresses the development of a linear quadratic (LQ) regulator in order to control the concentration profiles along a catalytic distillation column, which is modelled by a set of coupled hyperbolic partial differential and algebraic equations (PDAEs). The proposed method is based on an infinite-dimensional state-space representation of the PDAE system which is generated by a transport operator. The presence of the algebraic equations, makes the velocity matrix in the transport operator, spatially varying, non-diagonal, and not necessarily negative through of the domain. The optimal control problem is treated using operator Riccati equation (ORE) approach. The existence and uniqueness of the non-negative solution to the ORE are shown and the ORE is converted into a matrix Riccati differential equation which allows the use of a numerical scheme to solve the control problem. The result is then extended to design an optimal proportional plus integral controller which can reject the effect of load losses. The performance of the designed control policy is assessed through a numerical study.

Highlights

► Infinite-dimensional system representation of a catalytic distillation model is treated for LQ optimal control design using an operator Riccati equation (ORE) approach. ► The existence and uniqueness of the non-negative solution to the ORE is explored. ► The solution to the ORE is found by converting it to an equivalent matrix Riccati differential equation. ► The result is then extended to design an optimal proportional plus integral controller which can reject the effect of load losses. ► The controller is implemented on the column model through a simulation study.

Introduction

Reactive distillation (RD) has received much attention in the literature over recent years due to its advantages and inherent complexities. This process is a combination of chemical reaction and multi-component distillation in a counter-current column. When solid catalyst is used to accelerate the reaction, the process is called catalytic distillation (CD). The most important advantages in use of RD include reduced downstream processing, using the heat of reaction for distillation, overcoming chemical equilibria by removing products from the reaction zone and etc.; however, the interaction between the simultaneous reaction and distillation introduces challenging problems in controlling the column operating conditions, which include the existence of steady-state multiplicity, strong interactions between process variables and process gain sign change [1].

The reactive distillation process can take place either in a trayed or a packed column [2]. In the case of packed column, the process belongs to the class of distributed parameter systems, meaning that the process variables are functions of both time and spatial coordinate. Such a system is modelled by a set of coupled PDAEs in which partial differential equations (PDEs) describe the transport-reaction phenomena, while algebraic equations represent the equilibrium condition. In the absence of the axial dispersion, the main transport mechanism is convection and the resulting PDEs are of first-order hyperbolic type [3]. The conventional approach to deal with PDE systems is early lumping. In this method the controller is designed using an approximate ODE model, which results from discretizing the PDE system at a finite number of points. This allows the use of standard control methods applicable to ODE systems, which has been widely addressed in the literature (e.g., [4], [5]); however, this approximation results in some mismatch in the dynamical properties of the original distributed parameter and the lumped parameter models [6], which affects the performance of the designed model-based controller. A more rigorous way to cope with such a distributed parameter process is to exploit the infinite-dimensional characteristic of the system [6], [7].

Despite the complexities in control of RD process, a relatively small amount of research work has been reported in this area and most of the publications deal with modelling, simulation, process design and the analysis of steady-state multiplicity (e.g., [2], [3], [8]). A significant portion of the literature in the area of control of RD concerns the effectiveness of different control structures including conventional proportional integral (PI) controllers (e.g., [9], [10], [11]). Linear model predictive control (MPC) technique has also been applied to control this process based on simplified dynamic models (e.g., [12], [1]). In addition, a limited number of papers in the literature have dealt with the advanced nonlinear control of the RD process (e.g., [13], [14], [15]). All of the existing research work deal with a system of ODEs either by considering a trayed column model or, equivalently, by discretizing a packed (distributed) column model using a finite number of discretization points. Considering the infinite-dimensional characteristic of the packed reactive distillation process, helps to capture all the dynamic modes of the system without using a large number of discretization points. In this approach the controller is designed based on the original PDE model rather than its discretized version; however, it introduces more challenging problem regarding the theoretical developments.

The theory of feedback control for distributed parameter systems in the context of infinite-dimensional system representation has received a lot of attention in the control community. A body of this research work deals with parabolic PDE systems by using modal analysis approach [6], [7]. The basic idea in this approach is to derive finite-dimensional systems that capture the dominant dynamics of the parabolic PDE, which are subsequently used for controller design. This method is not applicable to first-order hyperbolic PDE systems where all the eigenmodes of the spatial differential operator contain the same amount of energy. In [16] geometric control approach is used for a class of first-order quasi-linear hyperbolic systems. This work assumes a symmetric velocity matrix with partially negative eigenvalues to preserve the C0-semigroup generation property; however, this assumption is not valid for counter-current two-phase contactors such as multi-component (reactive) distillation in which the chemical equilibrium is established. These systems are modelled by coupled first-order hyperbolic PDEs and algebraic equations where the algebraic equations can be solved for the algebraic variables to yield a set of pure PDEs in which the velocity matrix is spatially varying, non-symmetric, and its eigenvalues are not necessarily negative through of the domain. A combination of method of characteristics and model predictive control is used in [17] to convert hyperbolic PDEs to equivalent ODE system; however, applicability of the method of characteristics is restricted to systems having one or two characteristic curves. In [18] a predictive control algorithm is proposed for nonlinear parabolic and first-order hyperbolic PDEs with state and control constraints; however, in case of hyperbolic PDEs, this work assumes that the system under consideration has a very fast dynamics and therefore, the control objectives are satisfied only at the steady-state.

Recently, linear quadratic (LQ) regulator for first-order hyperbolic PDEs has been studied by solving an operator Riccati equation (ORE) for a given infinite-dimensional state-space model. The solution of the ORE is strongly based on the form of the hyperbolic operator, in particular, the velocity matrix. This method is used in [19] for a class of hyperbolic PDEs in which the velocity matrix is a negative identity matrix. The method is then extended to a more general class of hyperbolic systems where the velocity matrix is diagonal with not necessarily identical entries [20].

The purpose of the present work is to develop an ORE-based infinite-dimensional LQ control for controlling the concentration profiles in a DME packed catalytic distillation column modelled by a set of first-order hyperbolic PDAEs. The PDAE system is converted to a system of pure first-order hyperbolic PDEs by solving the algebraic equations, and a linear system is then calculated through linearizing the nonlinear PDEs around the desired steady-state profiles. In contrast to the previous work in the area of LQ control of hyperbolic systems [19], [20], the resulting linear infinite-dimensional model involves a hyperbolic operator in which the velocity matrix is spatially varying, non-diagonal, and not necessarily negative through of the domain. Therefore, the approaches developed in previous contributions, are not applicable to this kind of operator. In order to solve the LQ control problem, a state transformation is used to make the velocity matrix diagonal. The ORE for the resulting system is then formulated and the existence of its unique and non-negative solution is explored. The solution of the ORE is found through converting it to an equivalent matrix Riccati differential equation which can be solved numerically to calculate the optimal feedback operator. Subsequently, the result is extended to obtain an optimal proportional plus integral controller which can reject the effect of load losses such as changes in the feed flowrate. Finally, a set of numerical simulations is performed to assess the performance of the designed controller.

Section snippets

Process model

The process under consideration is shown in Fig. 1. This catalytic distillation column is used for producing DME through the liquid phase dehydration of methanol. The column has an effective packing height of 4 m, which consists of four sections of 1 m each in height. The rectifying zone at the column top and the stripping zone at the column bottom are filled with packing in which separation is taking place. The two middle sections are catalytically packed reaction zones in which the following

Infinite-dimensional state-space representation

In this section we derive an infinite-dimensional state-space setting for PDAE model (3), (4), (5), (6), (7), (8), (9), (10), (11) which is used for formulating the LQ control problem in the next section. Let us use notation x = [x1, x2], u1 = L, u2 = G and u = [u1, u2]. By taking the derivatives of y1 and y2 in (6), (7) with respect to the spatial coordinate, and by substituting them into (3), (4), we get:xt=F(x,u)xz+Q(x)x(0,t)=xin,x(z,0)=xe(z)wherext=x1t,x2tT,xz=x1z,x2zT,Q(x)=rDϕUL,2r

LQ control synthesis

In this section, we are interested in developing an LQ control policy in order to regulate the mole fraction profiles along the catalytic distillation column modelled by PDAE system (3), (4), (5), (6), (7), (8), (9), (10), (11) and represented in infinite-dimensional state-space setting (19). The multivariate nature of the catalytic distillation process and its high energy requirements justify the use of an optimal control policy. Model predictive control (MPC) and LQ control are the most

Simulation results

In this section, a set of numerical simulations is performed to assess the performance of the LQ controller developed in Section 4. The control objective is to regulate the mole fraction profiles along the catalytic distillation column described by model equations (3), (4), (5), (6), (7), (8), (9), (10), (11). In the first part of the simulation study, we deal with a regulation problem in which the controller designed in Section 4.1 is tested against rejecting the effect of a non-zero initial

Conclusions

In this work an ORE-based infinite-dimensional LQ control policy is developed to control the mole fraction profiles along a packed catalytic distillation column. The column is modelled by a set of coupled hyperbolic PDAEs. By solving the algebraic equations for the algebraic variables and substituting them into the PDEs, a model consisting of a set of pure hyperbolic PDEs is obtained. The resulting infinite-dimensional system, involves a hyperbolic operator in which the velocity matrix is

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