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Exponential Observer for a Class of Exothermal Axial Dispersion Reactors

N. Barje*, M. E. Achhab, V. Wertz

Department of Mathematics, Faculty of Sciences, Université Chouaib Doukkali, El Jadida, Morocco

Abstract: In this work, an exponential observer is performed for an exothermal axial dispersion tubular reactor that involves one nonlinear sequential reaction. More precisely, the given state estimator is performed by using bounded observations and properties of the nonlinear set of partial differential equations. It is shown that the proposed observer admits a global unique solution and ensures asymptotic state estimator with exponentially decay error, when the temperature is available for measurement at the reactor outlet only. This result is confirmed by a numerical simulation.

Keywords: distributed parameter systems, dissipativity, exponential observer, invariance, nonlinear systems, perturbed systems, tubular reactors, C₀-semigroup

Received August 28, 2013; Revised September 21, 2013; Accepted September 25, 2013

Citation: N. Barje, M. E. Achhab, V. Wertz, Exponential Observer for a Class of Exothermal Axial Dispersion Reactors. Journal of Automation and Control, 1 (1). 14-20. January 2013. doi:10.12691/automation-1-1-3

1. Introduction

Over the last few years, extensive studies of the (bio)chemical processes monitoring and control have been elaborated and received considerable attentions, with a view to provide "good" controller and stabilizer designs for the (bio)process in consideration [1,2,3,4]. However, an important difficulty in the operation of a control (bio)process is related to the (poor) knowledge of the system’s states, due to the fact that, the variables of the models used to describe their dynamics depend on spatial variable. This dependence, along with additional aspects such as the boundary conditions and presence of nonlinear terms in the model, increase the complexity to have access to the necessary measurements. Installing all the requisite sensors may not be physically possible. In such cases, the states can be estimated using state estimators (observers) [5,6,7]. For surveys on observer design methods for this class of systems, see, [8] and the references quoted therein. In [3], the (bio) chemical non-isothermal tubular reactors are regionally stabilized around a desired profile using asymptotic observers, if the temperature in the reactor is available for measurement. In this direction, this paper investigates the question of the conception of an exponential observer for a class of (bio) chemical non-isothermal tubular reactors, in the case where the measurements of the temperature may occur at the reactor outlet only. The kinetics of the process is characterized by first-order kinetics with respect to the reactant concentration (mol/l) and by Arrhenius-type dependence with respect to the temperature (K). The dynamics of the process are described by the following two energy and mass balance partial differential equations (PDEs) (see [4,9]):

[picture] | (1)

[picture] | (2)

with the boundary conditions are given, for , by

[picture] | (3)

[picture] | (4)

[picture] | (5)

[picture] | (6)

and the initial conditions are given, for , by

[picture] | (7)

In the equations above, the following parameters and are the axial dispersion coefficient, the superficial fluid velocity, the heat of reaction, the density, the specific heat, the kinetic constant, the activation energy, the ideal gas constant, the wall heat transfer coefficient, the reactor diameter and the coolant temperature. and are respectively the inlet temperature and the inlet reactant concentration. and denote the time and space independent variables, and the length of the reactor, respectively. Finally and denote the initial temperature and reactant concentration profiles.

From a physical point of view it is expected that for all ,

and

where could possibly be equal to

Let consider the following dimensionless state transformation:

Let us consider also dimensionless time t and space z variables:

We shall assume in the rest of the paper that the coolant temperature is (i.e., ), since will be eliminated in the equation of the reconstruction error between the plan state and the observer state.

Then we obtain the following equivalent representation of the uncontrolled model (1)-(7):

[picture] | (8)

[picture] | (9)

with the boundary conditions

[picture] | (10)

[picture] | (11)

and the initial conditions

[picture] | (12)

The parameters and are related to the original parameters as follows:

The real constants and are strictly positive, whereas the constant is strictly positive () for the exothermic reactions and strictly negative ( ) for the endothermic reactions. In this paper, we investigate the case of exothermic reaction (i.e., ), or equivalently the case when .

In the following sections, a conception of an exponential observer for the nonlinear system (8)-(12) is firstly given; then we state the existence of the global solution of the state estimator and prove that the estimation error converges exponentially to zero; at last, some conclusions and simulation results on the efficiency of the given conception can be obtained. The background of our approach can be found in [10,11,12].

2. Notations and Preliminaries

Let () be a real Banach space, is the infinitesimal generator of , is a continuous nonlinear function from a closed subset of into and is the identity operator on . Recall that

For the following uncontrolled abstract Cauchy problem

[picture] | (13)

we sate this important theorem that ensures the existence of the global unique solution .

Theorem 2.1 ([10], p. 355) If the following conditions are satisfied:

i) is -invariant, i.e., for all ;

ii) For all,

iii) is continuous in and there exists such that the operator is dissipative on (i.e., ).

Then, (13) has a unique mild solution on , for all . Furthermore, if is defined on by, for all and , it is a nonlinear semigroup on with as its generator.

We state also the following Theorem that will be needed to prove the exponential convergence of the estimation error.

Theorem 2.2 ([12], p. 109) Let be the infinitesimal generator of a -semigroup and is linear bounded operator on . The operator is the infinitesimal generator of a -semigroup which is the unique solution of the equation

For all . If in addition, , then

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 .

for all given for all given and is a positive cone. Let be a linear operator on , then is said to be positive linear operator if , for all , or equivalently.

As a useful criterion for the invariance condition given by (i) of Theorem 2.1, we have the following proposition.

Proposition 2.3 [9] Let T(t) be a strongly continuous semigroup of bounded linear operators on a real Banach lattice, generated by , such that for all for some and , then: is positive if and only if the resolvent operator is a positive linear operator for all

Where,

3. Semi-linear State Estimator

The (PDEs) (8)-(12) describing the exothermic reactor

[picture] | (14)

where is the linear operator defined by:

[picture] | (15)

It is shown in [13] that the linear operator A is the infinitesimal generator of a -semigroup of bounded linear operators on H, given by

Where and are the -semigroups generated, respectively, by and .

Remark 3.1 i) It is shown in [9] that for all , (which is equivalent, by Proposition 2.3, to ), and , for all .

ii) The -semigroup is exponentially stable (see [13]), i.e. there exist constants such that

In particular, , for all , with . Moreover, there exists a time, such that

The nonlinear operator N is defined on

for all in ,

[picture] | (16)

3.1. State Estimator Conception

Hereafter we consider measurements of the state vector are available at the reactor output only. In this case, the output function is defined as follows: we consider a (very small) finite interval at the reactor output such that:

[picture] | (17)

Where, if and elsewhere, with is a small number. The observer operator is linear bounded. For all ,

The adjoin operator of is defined for all by:

For all

Then,

An exponential observer for the system (8)-(12), when only the temperature is available for measurement, at the given reactor outlet, is given by

[picture] | (18)

[picture] | (19)

with the boundary conditions

[picture] | (20)

[picture] | (21)

and the initial conditions

[picture] | (22)

The system (18)-(22) can be written on its compact form as

[picture] | (23)

where, is the state variable of (13) and . The linear operator is the observer gain, satisfying

with is the identity operator of the Hilbert . The initial state of (12) is unknown while the initial stateof the observer can be assigned arbitrarily. Thus, the estimation error is still an unknown quantity even if we know.

3.2. Existence of the Global Solution

Let consider the following coupled system, given for all , by

[picture] | (24)

In aim to investigate the asymptotic behavior of the estimation error , we need to prove the existence of the solution of the augmented system (24), which remains in , on the whole interval , by applying Theorem 2.1. For this end we state the following lemmas concerning the nonlinearity involved in the dynamics (24). The proofs are similar to that given in [9].

Lemma 3.2.1 Consider the nonlinear operator (16). Then the operator is dissipative on , where .

Let define on , the distance

Lemma 3.2.2 For all ,

From the Theorem 2.2, the operator is the infinitesimal generator of a-semigroup

where is the -semigroup generated by the operator .

In the aim to study the invariance condition (i) of Theorem 2.1, it is useful to have the following Lemma:

Lemma 3.2.3 The -semigroup , is invariant.

Proof From Remark 3.1, we have , for all with , it follows by Theorem 2.2 that,

Let now prove that the semigroup is invariant, it is sufficient to demonstrate that the operator is positive (i.e., for all , according to the proposition 2.3).

Consider that and such that:

Let prove that , we have

(ie., ).

Whence,

Then, for almost all ,

It follows that,

From Remark 3.1, for all

Then,

i) For almost all [0,1] \ [1-,1], .

ii) For almost all , there are three cases:

a) If , thus .

b) If , thus .

c) If , suppose that there exist a subset non negligible , such that for almost all . We can suppose \ V negligible in the size, since is very small number, it follows,

what is contradictory.

Thus,

, for almost all .

It follows from (i) and (ii), that for almost all.

Consequently,

It follows that,

Besides, we have from Remark 3.1, .

Therefore,

The proof is thus complete.

The following proposition demonstrates the existence of the unique mild solution on of the coupled nonlinear system (24):

Proposition 3.2.4 For all , the dynamic system (24) has a unique mild solution in , for all .

Proof The linear bounded operator is the generator of a -semigroup defined for all by

(see [12], p., 30). In particular the -semigroup satisfies,

Indeed, it is proved in [9] that

Besides, from Lemma 3.2.3, for all in , and for all we have

Since

Then,

Hence,

It follows that,

Therefore, for all

The condition (i) of Theorem 2.1 is thus satisfied. Conditions (ii) and (iii) of Theorem 2.1 follows respectively by Lemma 3.2.1 and Lemma 3.2.2 Finally by applying Theorem 2.1, the augmented system (24) admits a unique mild solution on the whole interval.

3.3. Convergence of the Estimation Error

The following main result deals with the proof of the exponential convergence of the estimation error.

Theorem 3.3.1: Given the exothermal axial-dispersion reactor model (8)-(12). Suppose that there exists a bounded linear operator with g is a positive number, such that, then the dynamic system (18)-(22) is an exponential observer for the system (8)-(12).

Proof From Lemmas 3.2.1, 3.2.2 and 3.2.3, we prove by applying Theorem 2.1 that the evolution of the estimation error, given by

admits for all an unique mild solution on the whole interval , satisfying

such that, , for all .

It follows, for all , that

On the other hand, from Remark 3.1, there exists a time such that,

Consider. It follows from Theorem 2.2, that,

Thus, for all

By applying Gronwall's Lemma ([11], p 639), for all ,

Therefore, the estimation error converges exponentially to zero if

4. Simulation Result

In order to test the performance of the proposed observers, the equations have been integrated by using a backward finite difference approximation for the first-order space derivative, with the following set of parameter values (see [3,4]):

Table 1. Process parameters for numerical simulations

The measurements are taken on the length interval i.e., , and the process model has been arbitrary initialized with the constant profiles , and . In order to response to the assumptions of the Theorem 3.3.1 we set for the observer design parameter.

Figure 1 and Figure 2 shows respectively the time evolution of the temperature and concentration errors and related to the exponential observer (18)-(22).

Figure 1. Error evolution of the temperature

Figure 2. Error evolution of the product concentration

5. Conclusions and Prospects

In this work, we present an exponential observer to estimate the state variables initially unknown of a class of tubular reactor nonlinear models, namely exothermal axial dispersion tubular reactors involving sequential reactions for which the kinetics depends on temperature and reactant concentration. The given state estimator is based on measurements of the temperature at the reactor output only, and performed by a simulation study in which the parameters can be tuned by the user to satisfy specific needs in terms of convergence rate. It is shown in the simulations that the proposed observer is effective and satisfactory since it answers to difficulties of the reactant concentration measurements for a wide range of (bio)-chemical reactors.

An interesting topic for future research is to obtain a practicable feedback control (which is composed of original system and the observer) that stabilizes this class of non-isothermal tubular reactors around an optimal product concentration.

Acknowledgements

This paper presents research results of the Moroccan “Programme Thématique d’Appui à la Recherche Scientifique” PROTARS III, initiated by the Moroccan “Centre National de la Recherche Scientifique et Technique” (CNRST). The work is also supported by the Belgian Program on Interuniversity Poles of Attraction (PAI).

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