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Regional Stabilization of Exothermal Plug-Flow Tubular (Bio) Reactors

Nadia BARJE*, Farid BARJE, Mohamed EL AALLAOUI, Asmae KAMAL

Department of Mathematics, Faculté des Sciences, Université Chouaib Doukkali, Morocco

Abstract: This paper presents a regional stabilization of an exothermal (bio)chemical process around a specified steady-state temperature and concentration profiles. These desired profiles provide a constant temperature equilibrium that has lead to a closed-loop steady-state behavior which is close to that of an isothermal process. To achieve the regional stability a nonlinear state estimator based on the component temperature measurements is included into a state feedback system so that there is no need for measuring the process component concentration. Performance issues are illustrated in a simulation study.

Keywords: (bio)chemical process, compensator design, feedback stabilization, distributed parameter systems, nonlinear systems, tubular (bio)reactors

Received September 07, 2013; Revised October 28, 2013; Accepted November 03, 2013

Citation: Nadia BARJE, Farid BARJE, Mohamed EL AALLAOUI, Asmae KAMAL, Regional Stabilization of Exothermal Plug-Flow Tubular (Bio) Reactors. Automatic Control and Information Sciences, 1 (1). 6-10. January 2013. doi:10.12691/acis-1-1-2

1. Introduction

System theory properties for distributed parameter systems have been the object of active research over the last decades. In this direction, a large research activity has been dedicated, mainly to the analysis and to the control design of distributed parameter tubular (bio)reactors (see [1,2,3] and the references quoted therein). However, a number of important questions remained unsolved so far, in particular, the stability of unstable equilibrium points of non-isothermal tubular (bio)processes and, papers in this topic are scattered in the literature [4,5,6]. In [4], the authors provide a discontinuous feedback stabilization which globally and regionally asymptotically stabilizes non-isothermal chemical tubular reactors around desired steady-states.

In this direction, this paper proposes an exponentially feedback stabilization with a region of attraction around an unstable profile in steady-state of an exothermal plug-flow (bio)chemical reactor, corresponding to an optimal coolant temperature, this desired profile provides a constant temperature equilibrium, that forces the (bio)process to have a closed-loop steady-state behavior which is close to that of an isothermal reactor. To achieve the regional stability around the desired region of attraction, a component concentration estimator is constructed and included into a closed-loop system based on the component temperature measurements without need of component concentration measurement which is normally unavailable in practice.

The dynamics of the (bio)process are given, for all time and for all , by mass balance equations (see [4]):

[picture] | (1)

[picture] | (2)

with the boundary and the initial conditions:

[picture] | (3)

[picture] | (4)

where and are the temperature and the reactant concentration, the influent reactant concentration , the fluid superficial velocity , and the reactant rate . We assume that the kinetics depend only on the temperature and we consider a reaction rate model of the form , where is the kinetic constant . are the initial states.

The real constants are strictly positive, whereas the constant is strictly positive for of exothermic reaction and strictly negative for the endothermic reaction. In this paper, we investigate the case of exothermic reaction.

2. Notations and Preliminaries

Throughout the sequel, we assume , the Hilbert space with the usual inner product , for all and in , and the induced norm defined by for all .

Clearly the Hilbert space H is a real Banach Lattice (for more details, see [7]) where,

for almost all

for all given

In the (bio)chemical process (1)-(4), is considered as the control law. In order to facilitate our study we write the dynamic model (1)-(4) in terms of a nonlinear differential equation on , viz., for all positive t and all initial condition in , on its abstract form as

where, stands for the time derivative of the state , and the linear operator is defined by (see [1] for more details),

on its domain

Remark 2.1 The linear operator A is non-positive definite on its domain , for all

and

The control operator is a bounded linear operator from to , which is defined by . The control law .

The nonlinear operator N is defined on

for all in D,

[picture] | (5)

From physical point of view, it is expected that in the case of exothermic reaction, for all

where could possibly be equal to (see e.g. [4,8] and the references within)

Lemma 2.2 The nonlinear operator given by (5) is - Lipschitz on , where

Proof: Letand be in then,

Observe that , therefore it is sufficient to prove that is a Lipschitz operator on . Now for all

Thus,

Whence is a Lipschitz operator on D. Thus we can take as a Lipschitz constant of .

3. Feedback Stabilization

The problem that arises is how to stabilize the nonlinear system (1)-(4) around a desired profile solution of the following steady-state equation:

[picture] | (6)

under a prescribed control that ensures the (optimal) profile in the steady state.

In [5] temperature equilibrium profiles are studied that minimize different kinds of performance criterion. The following deals with the case where the performance criterion represents the energy consumption along the reactor. In [8] the author gives an unstable equilibrium profile that minimizes the energy consumption by

Where,

with are respectively the activation energy, the ideal gas constant, and the inlet temperature.

We aim in this paper to achieve a temperature feedback stabilization around this optimal profile.

3.1. State Observer Design

Hereafter we consider, as in [4], that the temperature is the only available measurement on the system. In this case, as in [4], a simple component state observer for the dynamical system (1)-(4) is constructed as:

[picture] | (7)

[picture] | (8)

with the boundary and the initial conditions:

[picture] | (9)

[picture] | (10)

The reconstruction error is shown to converge to zero.

Theorem 3.1: Consider the observer dynamics (7)-(10) for the controlled system (1)-(4). Let for all , then the reconstruction error has the property that for all initial error satisfying the condition with a positive constant .

Proof: Representing the observer dynamics (7)-(8) in terms of the reconstruction error and differentiating the functional

along the trajectories of the resulting error system, we obtain for all

by applying Remark 2.1, we have

Thus,

and that ensures the exponential stability of the error system, thereby yielding that

for all initial error satisfying the condition . The proof of the Theorem 3.1 is thus completed.

3.2. Temperature Feedback Stabilization

The aim of this section is to involve the system observer into a closed-loop system to achieve feedback stabilization of the temperature of (1)-(4) , with the region of attraction containing a prescribed set of the form

In the sequel, such a stabilization is referred to as a regional stabilization.

The following control law is shown to regionally stabilize system (1)-(4) around the desired steady state:

[picture] | (11)

where is the output of the concentration observer (7)-(10) and is a positive number .

Theorem 3.2: Let consider the dynamic temperature feedback controller (7)-(10) and (11) such that Then, the (bio)chemical process (1)-(4) is regionally exponentially stable around the optimal steady-state profile (6).

Proof: Let us represent the feedback controller (7)-(10) and (11) in term of the reconstruction error

by

[picture] | (12)

[picture] | (13)

[picture] | (14)

[picture] | (15)

Let now differentiating the functionals

along the trajectory (12)- (15), yields

From Remark 2.1 and Lemma 2.2, we obtain for all

Thus

Suppose , that ensures the exponential stability of the temperature error, yielding that

for all initial error satisfying the condition , with a positive constant .

Then, there exists a time such that for all the reconstruction error is maintained within the manifold

[picture] | (16)

It follows that for all the system motion along the manifold (16) is governed by

[picture] | (17)

with the conditions (13)-(14).

The derivative of the functional yields from Remark 2.1, for all

Thus,

More precisely, for all

Now, if , then the reconstruction errors and are exponential stability. The proof of the Theorem 3.2 is thus completed.

4. Simulation Results

In order to test the performance of the proposed observers, numerical simulations will be given with the following set of parameter values (see [4,5,8]):

The process model has been arbitrary initialized with the constant profiles, and . In order to response to the assumption of the Theorem 3.2 we set for the compensator design parameter.

Figure 1. Evolution of the component temperature error

Figure 2. Evolution of the component concentration error

Figure 1 and Figure 2 show respectively the time evolution in time and space of the temperature and concentration errors and , it can be observed that the optimal equilibrium is exponentially stable and, more precisely, the temperature converges exponentially to the constant temperature equilibrium.

It is proved as expected in the theoretical study that the error between the component state (1)-(4) and the optimal steady-state (6) converges exponentially to zero.

5. Conclusions

The exothermic reactors represent an interesting class of systems that may exhibit multiple steady states, either stable or unstable. In this paper, we present a conception of a regional exponentially feedback stabilization around an optimal unstable profile in steady-state when the temperature is the only available measurement on the system. This desired profile provides a constant temperature equilibrium, that has lead to a closed-loop steady-state behavior which is close to that of an isothermal reactor model. It is shown in the simulations that the given regional compensator is effective and satisfactory since it answers to difficulties of the reactant concentration measurements for a wide range of (bio)-chemical reactors.

References

[1]   | J. Winkin, D. Dochain, P. Ligarius, “Dynamical Analysis of Distributed Parameter Tubular Reactors”, Automatica, 2000, Vol. 36, 349-361.
[2]   | C. Antoniades, P. D. Christofides, “Studies on nonlinear dynamics and control of tubular reactor with recycle,” Nonlinear Anal, 2001, Vol.47, pp.5933-5944.
[3]   | N. Barje, F. Barje, M. El Aallaoui, A. Kamal, “State Estimators for Isothermal Plug-Flow (Bio)Reactors”, International Journal of Applied Mathematics, 2013, Vol.28, Issue.2.
[4]   | Y. Orlov, D. Dochain, “Discontinuous Feedback Stabilization of Minimum-Phase Semilinear Infinite-Dimensional Systems With Application to Chemical Tubular Reactor Models”, IEEE Trans. Aut. Contr, 2002, Vol. 47, 1293-1304.
[5]   | I.Y. Smets, D. Dochain, J.F. Van Impe, “Optimal Temperature Control of a Steady-State Exothermic Plug-Flow Reactor”, AIChE Journal, 2002, Vol. 48, No. 2, 279-286.
[6]   | I. Aksikas, J.F. Forbes, “On asymptotic stability of semi-linear distributed parameter dissipative systems”, Automatica, 2010, Vol. 46:6, pp. 1042-1046.
[7]   | R., NAGEL, “One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics”, Vol. 1184, Springer, New York, 1986.
[8]   | Aksikas, “Analysis and LQ-Optimal Control of Infinite Dimensional Semi-linear Systems,” PhD Thesis, University Catholic, Louvain, Belgium, 2005.