State Estimators for a Class of Isothermal Tubular Reactors

State Estimators for a Class of Isothermal Tubular Reactors

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

LINMA, Department of Mathematics, Université Chouaib Doukkali, El Jadida, Morocco.

LINMA, Department of Mathematics, Université Chouaib Doukkali, El Jadida, Morocco.

CESAME, Université Cathoulique de Louvain, Louvain-La-Neuve, Belgium.

Abstract

This work presents two state estimators for a class of isothermal tubular reactors involving sequential reactions for which the kinetics depends on the reactants concentrations only. These conceptions are performed by describing the model as infinite- dimensional state-space system, with bounded observation. It is shown that the given observers ensure asymptotic state estimator with exponentially decay error, when measurements (a) of both the reactant and product concentrations and (b) of only the product concentration, are available at the reactor output. Simulation results are also presented showing the effectiveness of the

reactions for which the kinetics only depends on the reactants concentrations involved in the following chemical reaction:

C1 bC2 (1)

where C1 is the reactant, C2 the product, and b 0 is the stoichiometric coefficient of the reaction. The dynamics of the process in a tubular reactor with axial dispersion are given, for all time

x x x

x x x

t 0 and for all z [0, L] where L is the reactor length, by mass balance equations (see [4]):

2

1 Da 1 1 r(x1 , x2 ),

proposed observers.

t z 2 z

x 2 x x

(2)

Keywords: Distributed parameter systems, state estimators, perturbed systems, tubular reactor, C0 – semigroup.

2 Da 2 2 br(x1 , x2 )

t z 2 z

with the boundary conditions:

1. Introduction

   D x1 (z 0,t) x (z 0,t) x

   (t),

   Although tubular reactors have been largely

   a z 1 in

   used in (bio)process industry for several decades,

   D x2 (z 0,t) x (z 0,t) 0,

   (3)

   system analysis and design of state observers has taken an increasing importance over the past decades (see [1], [2], [3], [4], [5], [6], [7], [8] and

   the references within).

   For the system control, the exact and full knowledge of system's states is important.

   a z 2

   D xi1,2 (z L,t) 0,

   a z

   and the initial conditions:

   However, in the mathematical model of the tubular

   x (z,t 0) x0 , x (z,t 0) x0

   (4)

   1 1 2 2

   reactors, the states depend on spatial variable and

   that makes it not possible to have full information of the system's states due to the fact that installing necessary sensors for measurements may not be physically possible or the costs may become excessive. In such a case, the states can be estimated using state estimators (observers).

   The motivation of this paper is to investigate

   where x1 (z,t), x2 (z,t), xin (z,t),, Da and r are the concentrations of C1 and C2 (mol l) , the influent reactant concentration l) and the fluid superficial velocity s) , the axial dispersion coefficient (m2 s), and the reactant rate

   this issue and provide an observer, that ensures (mol l s) . We assume that the kinetics depend only

   asymptotic state estimator with exponentially decay error, for the basic dynamical model of isothermal

   on the reactant concentration x1 and we consider a

   axial dispersion reactors involving sequential

   reaction rate model of the form

   r k0 x1 , where

   k is the kinetic constant (s1). x0 , x0

   are the

   on its domain

   0 1 2

   initial states. The purpose of this work is to reconstruct the state variables initially unknown,

   x (x , x )T H : x, dx

   H are

   absolutely

   when measurements may occur at the reactor output only, in the case (a) both reactant

   concentration and product concentration are

   1 2

   c

   c

   D( A) ontinuous ,

   dz

   d 2 x dx

   2 a i

   2 a i

   H , D i (0) x (0) 0,

   dz dz

   measured and (b) only the product concentration is available for measurements.

   dxi (L) 0, for

   dz

   dz

   i 1,2

2. State-space system framework

   The operator

   A1

   0 is the infinitesimal

   Let consider the Hilbert space

   H L2[0, L] L2[0, L], endowed with the usual inner

   A :

   0

   0

   A2

   product defined by

   generator of a

   C0 -semigroup

   (TA (t))t 0

   on H ,

   z1 , z2 x1 , x2 L2 y1 , y2 L2 ,

   exponentially stable (see [4]), i.e., there exists

   for all

   z (x , y )T and z (x , y )T in H , and

   constant M 0 such that,

   1 1 1 2 2 2

   the induced norm defined by 2

   (x , x )T x 2 x 2 ,

   TA M exp(

   t),

   t 0

   1 2 1 L2

   2 L2

   4Da

   for all (x , x )T H , where

   1 2

   L

   L

   L 2 L

   L 2 L

   f , g 2 f (z)g(z)dz and f f , f 2 .

   0

   The control operator B is a bounded linear operator from IR2 to H , which is defined by

   In the dynamic model (2)-(4), xin is considered

   , where

   (z)

   is the Dirac delta

   as the control at z 0 . In order to facilitate our

   B (.)

   0

   study we have to carry out some transformations. If

   0 1 2

   0 1 2

   we extract the boundary controlled part, the basic model given by (2)-(3) and the unknown initial condition (4), when expended with an output equation, is given a description in terms of a linear differential equation on H , viz., for all positive t and all initial conditions x : (x0 , x0 )T in H , (see

   [4] and the references within),

   distribution. The control u(t) xin (t) , and the output trajectory C is a bounded linear operator.

   The following Theorem will be needed in the

   sequel for the state observer conceptions.

   Theorem 2.1 ([9], p., 109) Let A be the infinitesimal generator of a C0 -semigroup

   (TA (t))t 0 and D is linear bounded operator on

   x(t) Ax(t) Bu(t)

   y(t) Cx(t), x(0) x

   (5)

   H . The operator A D is the infinitesimal

   0 generator of a

   C0 -semigroup

   (TAD (t))t0 which

   where, x stands for the time derivative of the state

   is the unique solution of the equation

   T

   T

   T

   T

   x(t) (x1(.,t), x2 (.,t)) , and the linear operator A is

   T (t)x T (t)x T (t s)DT

   (s)x ds,

   defined by

   AD 0

   0 0

   AD 0

   Ax

   D1

   d 2 .

   dz 2

   d.

   dz

   • k0

     0

     d 2 .

     d.

     x1 (t)

     x (t)

     For all x0 H . If in addition, then

     TA

     Met ,

     bk0

     D2

     2

     2

     dz

     2

     dz

     TAD

     (t)

     Me(M D )t .

     2.1 State observer

     Hereafter we consider measurements of the state

     vector

     x(t) are available at the reactor output only.

     In this case, the output function

     y(.)

     is defined as

     follows: we consider a (very small) finite interval at the reactor output [1 ,1] such that:

     y(t)

     (Cx)(t)

     1

     • wih

       C C

       C T defined by (6) and g is a

       : 0 [1,1] (a)x(a,t)da,

       t IR

       (6)

       1 2

       positive number.

       Where,

       [1,1](a) 1 if

       a [1 ,1]

       and

       The system (7)-(9) can be written on its compact form

       [1,1](a) 0

       elsewhere, with

       0 1

       is a

       x(t) Ax(t) Bu(t) GC*C(x(t) x(t))

       small number. The observer operator C : H IR2

       (10)

       is linear bounded. For all x, y H IR2 ,

       1

       y(t) Cx(t), x(0) x0

       Cx, y 2 [1,1](a)x(a,.)da, y 2

       where, x(t) (x (., t), x (., t))T

       is the state

       IR 0

       1

       1

       x(a,.),

       IR

       (a) y da

       1 2

       T

       T

       variable of (5) and x(t) (x1 (.,t), x2 (.,t)) . The

       0 [1,1]

       IR 2

       linear operator G is the observer gain, satisfying

       The adjoint operator

       C of C is defined for all

       G gI with I is the identity operator of the

       (z,t) [0,1] IR by:

       Hilbert H .

       The initial state (x (0), x (0))T

       of (5) is unknown

       1 2

       while the initial state (x (0), x (0))T of the observer

       (C y)(z)

       (z) y 1 2

       For all x H ,

       [1,1]

       can be assigned arbitrarily. Thus, the estimation error is still an unknown quantity even if we know

       (x (0), x (0))T .

       CCx 2 1 (

       (z)(

       1 2

       (a)x(a,.)da))2 dz

       1

       1

       0 [1 ]

       [1,1]

       Then,

       CC

       0 [1,1]

       2 2 2

       2 2 2

       x 2 x

       1. Full-order observer

          In this section a full-order observer, when both the reactant concentration and the product concentration are available for measurements at the reactor output, is provided as an asymptotic state estimator with exponentially decay error.

          A candidate observer for the system (2)-(4), is obtained as the output of the following dynamic system

          Proposition 2.1: Given the isothermal axial- dispersion reactor basic dynamical model (2)-(4). Suppose that there exists a bounded linear operator

          x

          2 x

          x

          G gI , where G is a positif number, such that

          1 Da 1 1 k0 x1 xin (t) 2

          t z 2 z

          g , the dynamic system (7)-(9) is an

          • gC* (C x C x )

          8Da

          1 1 1 1 1

          (7)

          exponential observer for the system (2)-(4).

          x

          2 x

          x

          2 Da 2 2 bk0 x1

          t z 2 z

          • gC* (C x C x )

          Proof 2.1 Let consider the linear operator G gI , where g is a positive number. The operator

          2 2 2 2 2

          GCC is a bounded linear operator on H , such

          with the boundary conditions:

          that

          GCC

          g .

          D x1 (z 0,t) x (z 0,t) x

          (t),

          On the other hand, the C0 -semigroup (TA (t))t0

          a z 1

          2 x

          in

          (8)

          on H , is exponentially stable, such that

          2

          D 2 (z 0,t) x (z 0,t) 0,

          a z 2 2

          x

          TA M exp(

          4Da

          t),

          t 0

          D i1,2 (z L,t) 0,

          a z

          Thus,

          log TA

          (t)

          log M

          2

          ,

          t 0

          and the initial conditions:

          t t 4Da

          x (z,t 0) x0 , x (z,t 0) x0

          (9)

          1 1 2 2

          There exists a time tM such that,

          log M

          t

          2

          ,

          4Da

          for

       2. Reduced-order observer

          In this section, we will present an observer in

          all t tM

          (since log M converges to zero).

          t

          the case where only the product concentration is available for measurements. Let consider the

          It follows,

          TA

          2

          exp( ),

          8D

          t tM

          dynamic

          2

          2

          x1 D

          x1 x1 k x

          a t

          a z 2

          z 0 1

          Now, by the Theorem 2.1, the linear operator

          A GCC is the infinitesimal generator of a C –

          x

          xin (t),

          2 x

          x

          (11)

          0 2 Da 2 2 bk0 x1

          AGC C

          AGC C

          semigroup (T

          (t))

          t0

          satisfying, for all

          t z 2 z

          • gC* (C x C x )

            t tM :

            2 2 2 2 2

            TAGC*C

            (t)

            exp((

            2

            8Da

            • GC*C )t)

              with the boundary and initial conditions (8)-(9).

              Proposition 2.2 Given the isothermal axial-

              2

              exp((

              8D

            • g

            )t)

            dispersion reactor basic dynamical model (2)-(4). Suppose that there exists a bounded linear operator

            a G gI

            with g is a positif number, such that

            Let consider the dynamics (5) and (10), the evolution of the estimation error e(t) x(t) x(t) ,

            2

            g

            8Da

            , then the dynamic system (11) and (8)-(9)

            given by

            is an exponential observer for the system (2)-(4).

            Proof 2.2 It is proved in the previous section that

            e(t) ( A GC*C)e(t)

            there exists a time tM

            such that the C0 -semigroup

            e

            e

            0

            0

            (0) x

   • x0

   (TA

   (t))

   t0

   satisfies for all t tM ,

   2

   admits a unique mild solution on the interval

   TA (t) M exp( 8D

   t).

   [0,[

   given by:

   e(t) T

   AGC*C

   (t)e(0), for all

   a

   Thus, for all t tM

   e(0) H

   Hence,

   and t 0 (see [10]).

   TA1

   (t)

   esp(

   2

   t)

   8D

   e(t)

   TAGC*C

   (t)

   e(0) ,

   t 0

   T (t)

   a

   2

   exp( t),

   It follows, if

   2

   g

   8Da

   A2

   , the estimation error

   8Da

   converges exponentially to zero.

   where

   (TA (t))t0

   and

   (TA (t))t0

   are the

   C0 –

   1

   1

   2

   2

   That means that the dynamic system (7)-(9) is

   an exponential observer for the system (2)-(4).

   semigroup generated respectively by Let consider the linear operator

   A1 and

   A2 .

   More precisely the reconstruction error

   satisfies, for all t tM

   2

   x(t) x(t)

   G : G1

   0

   0

   2

   2

   ,

   ,

   gI G2

   x(t) x(t)

   x(0) x(0) exp((

   8Da

   g)t)

   where g is a positive number, and

   C 0

   C T .

   The above proposition presents a "full-order"

   The operator

   GCC is a bounded linear operator

   observer when both reactant concentration and product concentration are measured at the reactor output. In most cases it is not possible to have

   on H , such that

   G C C

   g .

   access to measure the reactant concentration, in 2 2 2

   such a case the states can be estimated using a

   "reduced-order" observer based on measurements

   By the Theorem 2.1, the linear operator

   at the reactor output of the product concentration

   A G C C

   is the infinitesimal generator of a

   2 2 2 2

   (T *

   (T *

   only.

   C0 -semigroup

   A2 G2C2C2

   (t))

   t0

   satisfying, for all t tM

   the following set of parameter values (see [7,6]):

   2 Da

   0.167 m2 s,

   0.025 m s ,

   L 1m,

   2 2

   2 2

   0

   0

   TA2 G2C*C

   (t)

   exp((

   8D

   g)t)

   k 106 s 1 ,

   xin

   0.02,

   b 2 mol l.

   a

   Let now consider the dynamics (5) and (10), the evolution of the estimation error e(t) x(t) x(t) ,

   The measurements are taken on the length interval [3 4, L] i.e., 3 4 , and the process model has been arbitrary initialized with the constnt

   given by

   profiles

   x1 (0, z) 1, x2 (0, z) 0, x1 (0, z) 0 , and

   e1 (t) A1e1 (t),

   e1 (0) x1 (0) x1 (0)

   x2 (0, z) 1. In order to response to the assumptions of the Propositions 2.1 and 2.2, we set

   * 2

   for the observer design parameter.

   e2 (t) ( A2 gC2C2 )e2 (t),

   e2 (0) x2 (0) x2 (0)

   g

   16Da

   Admits a unique mild solution on the interval

   Figure 1. shows the time evolution of the

   T estimation error e (e ,e )T

   related to the

   [0,[ given for all (e (0),e (0)) H by: 1 2

   1 2 exponential observer (7)-(9).

   (e (t), e (t))T (T (t)e (0),T (t)e (0))T ,

   1 2 A 1

   AGCC 2

   For all t 0 . That implies,

   e1

   TA

   e1

   e2

   A G C C

   (t)

   e2

   T *

   T *

   2 2 2 2

   Hence, for all t tM

   2

   a

   a

   x1 (t) x1 (t) x1 (0) x1 (0) exp( 4D t)

   x2 (t) x2 (t)

   2

   a

   a

   x2 (0) x(0) exp(( 8D

   g)t)

   T

   T

   Figure1: Evolution in time and space of the

   It follows that, if

   2

   g

   8Da

   , the estimation errors

   estimation error e (e1 ,e2 ) .

   converge exponentially to zero, and that mains that the dynamic system (11) and (8)-(9) is an exponential observer for the system (2)-(4).

   Commentaire 2.1 In this section, we have described two different exponential observers for the isothermal Axial-Dispersion reactor basic dynamical model. The first one (eq. (7)-(9)) improves the convergence rate of the concentration error by reintroducing a measurement of both the reactant and product concentrations. The second one (eq. (11) and (8)-(9)) shows that an exponential observer can be constructed even if the reactant concentration is not measured. The Proposition 2.2 provides a simple conception of observer but less effective than that given by Proposition 2.1, since the dynamic of the state error on the reactant concentration remains dependent on the system's dynamic.

   1. Simulation result

   In order to test the performance of the proposed observers, numerical simulations will be given with

   Figures (a), (b) and (c) show respectively the time evolution of the estimation error at the positions 3*L/4, 2*L/4 and L/4, for the case where only the product concentration is measured (the plot '- -') i.e the exponential observer (11) and (8)- (9), and for the case where both the reactant and the product concentrations are measured with the exponential observer (7)-(9).

   1. Estimation error at z=0.9L

   2. Estimation error at z=0.5L

   3. Estimation error at 0.1L

   It is seen as expected that the product concentration error related to the exponential observer (7)-(9) is faster than the one related to the exponential observer (11) and (8)-(9).

3. Conclusions and prospects

In this paper we present two observers to estimate the state variables initially unknown of isothermal tubular reactor models, namely axial dispersion reactors involving sequential reactions for which the kinetics only depends on the reactants concentrations involved in the reaction. The proposed observers are based on measurements available 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 theoretical setting and in the simulations that the "Full-order" observer is effective relatively to the convergence time. However, the "Reduced-order" observer is more satisfactory since it answers to difficulties of the reactant concentration measurements for a wide range of (bio)-chemical reactor.

1. Knowlegments

   This paper presents research results of the Moroccan Programme Thématique dAppui 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 Programme on Interuniversity Poles of Attraction (PAI).

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