MIMO sliding mode control of a distributed parameter denitrifying biofilter

Applied Mathematical Modelling 25 (2001) 671±682

www.elsevier.nl/locate/apm

MIMO sliding mode control of a distributed parameter denitrifying bio®lter O. Boubaker a, J.P. Babary

b,*

, M. Ksouri

a

a

b

LACS, ENIT, B.P. 37-Tunis Belv ed ere 1002, Tunisia Laboratoire d'Analyse et d'Architecture des Systemes/CNRS, 7 avenue du Colonel Roche, F-31077 Toulouse Cedex 4, France

Received 18 January 2000; received in revised form 3 November 2000; accepted 4 December 2000

Abstract In this paper, the control problem of a distributed parameter system is developed. The process considered is a ®xed bed reactor in which drinkable water is treated. Micro-organisms ®xed in the reactor absorb the harmful components in such a way that their concentrations decrease in the out¯owing water. The addition of a carbon source is needed in this operation. The complexity of the control problem is due to the distributed and nonlinear nature of the model having time varying biological parameters. In drinkable water the harmful components are supposed to be nitrates, nitrites and ethanol supplied in the reactor inlet. The main contribution of this paper consists of designing a multi-input/multioutput (MIMO) sliding mode control of the distributed parameter bio®lter. Two approaches are developed: early lumping and late lumping sliding control. In both cases, not only are the harmful component concentrations controlled but moreover the addition of the carbon source is determined in such a way that the quality of water ful®ls international standards. The performances of the two approaches are compared and illustrated by means of simulations. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Bioprocess; Fixed bed reactor; Multivariable sliding mode control; Variable structure control

1. Introduction The quality of drinkable water is more and more damaged by human activities, e.g. the intensive use of nitrogen in agriculture and for domestic use. The high concentration of nitrogen contents a€ects human health; their regulation in drinkable water becomes then more stringent. Actually, biotechnological denitri®cation of drinkable water in ®xed bed bioreactors is a process which has gained popularity for compactness and low energy consumption. The ®xed bed reactor is a tubular reactor in which the water to be treated ¯ows. The reactor is packed with puzzolane, a very porous volcanic material which retains a large amount of micro-organisms. The micro-organisms ®xed in the reactor absorb the nutrients in such a way that the substrate concentrations (in this case nitrate and nitrite) decrease in the out¯owing water. The addition of a carbon source is needed in the denitri®cation operation. *

Corresponding author. Tel.: +33-05-61-33-63-15; fax: +33-05-61-33-69-69. E-mail address: [email protected] (J.P. Babary).

0307-904X/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 3 0 7 - 9 0 4 X ( 0 1 ) 0 0 0 0 5 - 1

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Nomenclature S1 …z; t† nitrate concentration S2 …z; t† nitrite concentration S3 …z; t† ethanol concentration X …z; t† biomass concentration maximum biomass concentration Xmax e bio®lter porosity death coecient of micro-organisms kd speci®c growth rate (denitratation) l1 speci®c growth rate (denitritation) l2 maximum speci®c growth rate (denitratation) l1 max maximum speci®c growth rate (denitritation) l2 max F ¯ow rate L bio®lter length a bio®lter section yield coecient (denitratation) Yh1 yield coecient (denitritation) Yh2 correction factor of anaerobic growth gg nitrate saturation coecient K1 nitrite saturation coecient K2 ethanol saturation coecient K3 The dynamics of bio®lters are generally described by partial di€erential equations, with nonlinearities and time varying parameters which make the control problem very complex. A sliding mode control can be useful for such a problem but the sliding control theory for distributed parameter systems is not enough developed. First, there is no general theory on partial di€erential equations with a discontinuous right-hand side. Secondly, application of variable structure techniques to distributed parameter systems must take into account boundary conditions in order to derive the control law. Because in many cases boundary conditions vary, the designed sliding mode may not exist. The main contribution of this paper consists of designing a multi-input/multi-output (MIMO) sliding control law for the distributed parameter system in order to control harmful substrates (nitrate and nitrite), and to optimise the addition of the carbon source (ethanol concentration at the input of the reactor), in order to improve the quality of water. Two approaches are developed. The control law is ®rst based on the lumped model. Then, it is shown that it is possible to derive the control law from the original distributed parameter system. The paper is organised as follows. Section 2 introduces the mathematical model of the distributed parameter bio®lter. In Section 3, sliding control laws are developed in order to regulate the harmful components at the outlet of the reactor at desired values. Simulation results are presented in Section 4. It is shown that it is possible to govern the behaviour of the bioreactor by controlling the addition of the carbon source and improving the quality of water. 2. Mathematical model of the bio®lter The process considered is a denitrifying ®xed bed reactor. The water to be treated ¯ows freely in the bio®lter. The bio®lter is packed with a very porous volcanic material which retains a large

O. Boubaker et al. / Appl. Math. Modelling 25 (2001) 671±682

673

amount of micro-organisms. In the bio®lter the nitrates are reduced by micro-organisms into gaseous nitrogen by using an organic carbon and producing intermediate compounds, the nitrites. The bio®lter dynamics can be deduced from mass balance considerations of the di€erent component concentrations: nitrate, nitrite, carbon and biomass. More details about this system are given in [1]. 2.1. The distributed mathematical model Through an in®nitesimal volume the dynamical model of the process can be written as follows [2]: oS1 …z; t† ˆ ot

F oS1 …z; t† ae oz

1 Yh1 l …S1 ; S3 †X …z; t†; 1:14Yh1 e 1

oS2 …z; t† ˆ ot

F oS2 …z; t† 1 Yh1 ‡ l …S1 ; S3 †X …z; t† ae oz 1:14Yh1 e 1

oS3 …z; t† ˆ ot

F oS3 …z; t† ae oz

1 Yh1 e

l1 …S1 ; S3 †X …z; t†

oX …z; t† ˆ …l1 …S1 ; S3 † ‡ l2 …S2 ; S3 † ot X ; kd ˆ …l1 ‡ l2 † Xmax

1 Yh2 l …S2 ; S3 †X …z; t†; 1:71Yh2 e 2 1

Yh2 e

l2 …S2 ; S3 †X …z; t†;

…1a†

kd …S1 ; S2 ; S3 ; X ††X …z; t†;

for 0 < z 6 L and with the following initial and boundary conditions: S1 …z ˆ 0; t† ˆ S1;in …t†;

S1 …z; t ˆ 0† ˆ S1;0 …z†;

S2 …z ˆ 0; t† ˆ S2;in …t†;

S2 …z; t ˆ 0† ˆ S2;0 …z†;

S3 …z ˆ 0; t† ˆ S3;in …t†;

S3 …z; t ˆ 0† ˆ S3;0 …z†;

X …z ˆ 0; t† ˆ Xin …t†;

…1b†

X …z; t ˆ 0† ˆ X0 …z†:

The speci®c growth rate is expressed by the most classical MONOD-type expression: l1 …S1 ; S3 † ˆ gg l1 max

S1 S3 ; S1 ‡ K1 S3 ‡ K3

l2 …S2 ; S3 † ˆ gg l2 max

S2 S3 : S2 ‡ K2 S3 ‡ K3

…1c†

We suppose that all parameters of the model (1a)±(1c) are known or can be determined by using an estimation technique [3]. 2.2. The lumped parameter dynamical model The implementation of the control laws requires the computation of space derivatives. This is possible by using one of the weighted residual or ®nite di€erence methods [4]. In this paper simulation results have been carried out by using the orthogonal collocation method which is one of the weighted residual methods. The advantages of this method are detailed in [1,4]. The collocation method requires the selection of three parameters, namely the number of collocation points chosen on the space interval ‰0; LŠ such as z0 ˆ 0 < z1 <


< zN‡1 ˆ L; the position of collocation points, and the so-called interpolation base functions li which must be evaluated at the collocation points. In this paper, the internal collocation points are considered as

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O. Boubaker et al. / Appl. Math. Modelling 25 (2001) 671±682

the zeros of the Nth order Jacobi polynomial. The Lagrangian polynomial interpolation is used to join the values together. A similar approximation has been presented in [5±7]. Each state variable of the system (1a) will be expressed as a ®nite sum of products of time functions and space functions as follows. Sm …z; t† ˆ

N ‡1 X

li …z†Sm …z ˆ zi ; t† for m ˆ 1; 2; 3;

X …z; t† ˆ

iˆ0

N‡1 X

li …z†X …z ˆ zi ; t†:

…2†

iˆ0

For the distributed parameter system (1a)±(1c), let k1 ˆ

1 Yh1 ; 1:14Yh1 e

k2 ˆ

1 Yh2 ; 1:71Yh2 e

k3 ˆ

1 Yh1 e

;

k4 ˆ

1 Yh2 e

;

and using (2) the following lumped system can be derived: 8i ˆ 1; . . . ; N ‡ 1; ds1i ˆ dt

N ‡1 F X li;j s1j ae jˆ1

ds2i ˆ dt

N ‡1 F X li;j s2j ‡ k1 l1i xi ae jˆ1

ds3i ˆ dt

N ‡1 F X li;j s3j ae jˆ1

dxi ˆ …l1i ‡ l2i dt

kdi †xi

F li;0 s1;in ae

k 1 l 1i x i ; k2 l2i xi ;

F s3;in li;0 ae

…3a† k 3 l 1i x i

k4 l2i xi ;

with kdi ˆ …l1i ‡ l2i †

sij ˆ Si …z ˆ zj ; t†; i ˆ 1; 2; 3; j ˆ 1; . . . ; N ‡ 1;

xi ; Xmax

dlj …z† li;j ˆ ; dz zˆzi

with the following initial conditions: s1i …t ˆ 0† ˆ s1i ;0 ;

s2i …t ˆ 0† ˆ s2i ;0 ;

s3i …t ˆ 0† ˆ s3i ;0 ;

xi …t ˆ 0† ˆ xi;0 :

…3b†

3. Control problem The objective of the process control is to regulate the total concentrations of the residuals' nitrogen contents y1 (nitrate and nitrite) at the reactor output at desired values imposed by international standards of drinkable water and also to optimise the addition of the carbon source (ethanol concentration in this case) S3;in by controlling the ethanol concentration y2 at the outlet reactor. The output variables are then expressed as follows: y1 …t† ˆ S1 …z ˆ L; t† ‡ S2 …z ˆ L; t†;

…4†

y2 …t† ˆ S3 …z ˆ L; t†:

…5†

The control inputs are the in¯uent ¯ow rate F …t† and the inlet ethanol concentration S3;in or equivalently: u1 ˆ

F …t† ; ae

…6†

O. Boubaker et al. / Appl. Math. Modelling 25 (2001) 671±682

u2 ˆ S3;in …t†:

675

…7†

In order to decouple the control variables in (3a), (3b), consider the following intermediate control variables: U1 ˆ

F …t† ae

…8†

and F …t† S3;in …t†: …9† ae The bioreactor to be controlled is a nonlinear and time varying system, which therefore needs a robust state feedback. The sliding control theory can be useful for this situation when model uncertainties, parameter variations and disturbances occur. The sliding mode control consists in bringing the system on a so-called sliding surface in the state space and maintaining it on this surface by using a switching algorithm toward an equilibrium state [8,9]. The ®rst step consists then in de®ning a so-called sliding surface ri . The second step is the choice of the control law that satis®es the following sliding condition: U2 ˆ

ri r_ i < 0;

…10†

which ensures the attractivity of the sliding surface in the state space. It has been shown in [8] that the controlled system is very robust with respect to model uncertainties and parameter variations. 3.1. Early lumping MIMO sliding control Consider the lumped model of the denitrifying process (3a), (3b). It can be written in the following nonlinear form: g_ ˆ f …g; t† ‡ g…g; t†u;

…11†

where g…t† 2 Rn is the state vector, u…t† 2 Rm is the control vector, and f 2 Rn is a nonlinear vector. g is an n  m-dimensional matrix. The sliding control law is obtained by forcing each control variable ui of the control vector to satisfy the following law:  ‡ ui …g; t† if ri …g† > 0; i ˆ 1; . . . ; m; …12† ui ˆ ui …g; t† if ri …g† < 0; on m sliding surfaces of dimension …n 1† designed by ri ˆ fg=ri …g; t† ˆ 0g; i ˆ 1; . . . ; m. It is shown in [8] that the control law u can be the sum of two components: ui ˆ ueqi ‡ Dui ;

i ˆ 1; . . . ; m:

…13†

ueqi is the so-called equivalent control law which is obtained for an ideal sliding mode for which the system state is maintained on the sliding surface: ri …g† ˆ 0;

i ˆ 1; . . . ; m:

…14†

An ideal sliding mode is ensured only if r_ i …g† ˆ 0;

i ˆ 1; . . . ; m;

…15†

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then ori ‰f …g; t† ‡ g…g; t†ueq Š ˆ 0; og

i ˆ 1; . . . ; m:

…16†

The equivalent control is then derived:   1  ueqi ˆ …ori =og†T g…t; g† …ori =og†T f …t; g† ; i ˆ 1; . . . ; m:   The matrix …ori =og†T g…t; g† is supposed to be singular. Dui is the high-frequency component de®ned by Dui ˆ

Mi sgn…ri …g††;

…17†

i ˆ 1; . . . ; m:

…18†

The gains Mi are determined by considering the sliding condition (10). The robustness property is preserved with respect to uncertainties and parameter variations. The sgn function is de®ned by 8 if ri …g† > 0; 1;

679

i ˆ 1; . . . ; m;

/i being as small as possible. Simulation runs were carried out with the following initial conditions: S1 …z; t ˆ 0† ˆ 16:93 g‰NŠ=m3 ;

S2 …z; t ˆ 0† ˆ 0 g‰NŠ=m3 ;

S3 …z; t ˆ 0† ˆ 101:5 g‰DCOŠ=m3 ;

Xa …z; t ˆ 0† ˆ 625 g‰DCOŠ=m3 :

The parameters values are: e ˆ 0:52;

gg ˆ 0:8;

l2 max ˆ 0:32 h 1 ; yh1 ˆ 0:56;

Xa max ˆ 675 g‰DCOŠ=m3 ;

K1 ˆ 1:5 g‰NŠ=m3 ;

l1 max ˆ 0:36 h 1 ;

K2 ˆ 1 g‰NŠ=m3 ;

K3 ˆ 40 g‰NŠ=m3 ;

yh2 ˆ 0:54:

The control variables are supposed to be bounded: u1max ˆ 12 m=h;

u1min ˆ 2 m=h;

u2max ˆ 185 g‰DCOŠ=m3 ;

u2min ˆ 0 g‰DCOŠ=m3 :

During the ®rst 20 h, the system runs in open loop with a speed ¯ow rate of 9 m/h, then the MIMO control law is applied. Some set point changes occur at di€erent times, as described in Table 1 and are shown in Fig. 1 (early lumping sliding control law) and in Fig. 2 (late lumping sliding control law).

Fig. 2. Late lumping sliding control laws: robustness to set point changes (see Table 1).

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Several parameter perturbations are applied as shown in Table 2. Simulation results are shown in Fig. 3 (early lumping sliding control law) and Fig. 4 (late lumping sliding control law). The simulation results show good tracking and excellent performance of the sliding mode controller for set point or parameter changes. The set point has been reached and tracked in spite of the distributed nature, the nonlinearities, the time varying behaviour of the bio®lter and the external disturbances. The late lumping control law gives better results than the early lumping one because in the ®rst case the e€ect of coupling control inputs is less observed. Moreover the quality of the treated water is better because there is less ethanol concentration. Compared to results obtained when a single-input/single-output sliding mode control is used [1,10], the multi-input/ multi-output sliding mode control is more interesting. Not only are the residuals' nitrogen contents regulated but also the ethanol concentration in drinkable water. Then the quality of water is improved. Table 2

List of parameter changes Time (h)

Parameter changes

S1;in g‰NŠ=m3

y1d

y2d

0 20 30 40 50 60 100

± ± l1 max ˆ 0:4 h 1 ; l2 max ˆ 0:35 h K2 ˆ 1:3 g‰NŠ=m3 K3 ˆ 45 g‰NŠ=m3 ± ±

16.93 ± ± ± ± 18.5 ±

Open loop 5 ± ± ± ± ±

20 ± ± ± ± ±

1

Fig. 3. Early lumping sliding control laws: robustness to parameter changes (see Table 2).

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Fig. 4. Late lumping sliding control laws: robustness to parameter changes (see Table 2).

5. Conclusion This paper introduces the multivariable sliding mode control of a ®xed bed biological reactor in which drinkable water is treated. The bio®lter must control at the reactor outlet the total concentration of harmful substrates at desired values, by acting on the ¯ow rate and the addition of the carbon source. Compared to previous studies, the main contribution of this paper is the design of the multi-input/multi-output sliding mode control law of a distributed parameter when the control variables are coupled. The robustness of the control laws is presented and compared in simulation results. References [1] J.P. Babary, S. Bourrel, Sliding mode control of a denitrifying bio®lter, Applied Mathematical Modelling 23 (1999) 609±620. [2] J. Jacob, H. Pingaud, J.M. Le Lann, S. Bourrel, J.P. Babary, B. Capdeville, Dynamic simulation of bio®lters, Simulation ± Practice and Theory 4, 335±348. [3] G. Bastin, D. Dochain, On Line Estimation and Adaptative Control of Bioreactors, Elsevier, Amsterdam, 1990. [4] J.V. Villadsen, M.L. Michelsen, Solution of di€erential equation models by polynomial approximation, in: Internal Series in the Physical and Chemical Engineering Sciences, Prentice-Hall, Englewood Cli€s, NJ, 1978. [5] D. Dochain, J.P. Babary, N. Tali Maamar, Modelling and adaptive control of nonlinear distributed parameter bioreactors via orthogonal collocation, Automatica 28 (5) (1992) 873±883. [6] O. Boubaker, R. M'hiri, M. Ksouri, J.P. Babary, Sliding control of linear input delay systems, in: 15th IMACS World Congress on Scienti®c Computation Modelling and Applied Mathematics (IMACS'97), Berlin, Germany, 24±29 August 1997, vol. 5, pp. 91±96.

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[7] O. Boubaker, R. M'hiri, M. Ksouri, J.P. Babary, MIMO variable structure control for non linear distributed parameter systems: application to ®xed bed bioreactors, in: Second IMACS Multiconference IEEE CESA'98 Computational Engineering in Systems Applications, Symposium on Modelling, Analysis and Control, NabeulHammamet, Tunisia, 1±4 April 1998, vol. 1, pp. 79±84. [8] V.I. Utkin, Variable structure systems with sliding modes: survey paper, IEEE Transactions On Automatic Control 22 (2) (1977) 212±222. [9] V.I. Utkin, Sliding modes in control optimization, Springer, Berlin, 1992. [10] O. Boubaker, J.P. Babary, On SISO and MIMO sliding control of a parameter distributed biological process, in: 1999 IEEE International Conference on Systems, Man and Cybernetics (SMC'99), Tokyo, Japan, 1999.